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Page 1: "Seismic Behaviour of Cable-Stayed Bridges"

Department of continuum mechanics and theory of structures

Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos

Universidad Politécnica de Madrid

SEISMIC BEHAVIOUR OF

CABLE-STAYED BRIDGES: DESIGN,

ANALYSIS AND SEISMIC DEVICES

Doctoral Thesis

Alfredo Cámara Casado

Ingeniero de Caminos, Canales y Puertos

Advisor; Prof. Dr. Miguel Ángel Astiz Suárez

Madrid, October 2011

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Seismic Behaviour of Cable-Stayed Bridges: Design, Analysis and Seismic Devices

Doctoral ThesisUniversidad Politécnica de Madrid

Madrid, October 2011

The composition of the text has been made using LATEXand GNU applications

Author; Alfredo Cámara CasadoIngeniero de Caminos, Canales y Puertos (6-year MEng in Civil Engineering)

Advisor; Miguel Ángel Astiz SuárezProf. Dr. Ingeniero de Caminos, Canales y Puertos

Escuela Técnica Superior de Ingenieros de Caminos, Canales y PuertosDepartment of continuum mechanics and theory of structures

Technical University of MadridProfesor Aranguren s/nMadrid 28040

Phone: (+34) 91 336 5358

Fax: (+34) 91 336 6702

e-mail: [email protected]

Home page: http://w3.mecanica.upm.es/∼alfredo

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Tribunal nombrado por el Mgfco. y Excmo. Sr. Rector de la UniversidadPolitécnica de Madrid, el día

Examination panel appointed by the Rector of the Technical University of Madrid

on

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Presidente / President D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Vocal / Vowel D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Vocal / Vowel D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Vocal / Vowel D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Secretario / Secretary D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Suplente / Substitute D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Suplente / Substitute D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Realizado el acto de defensa y lectura de la Tesis el día (Once defended the Doc-

toral Thesis on) . . . . . . de . . . . . . . . . . . . . . . de 2011 en la (at) E.T.S. de Ingenieros deCaminos, Canales y Puertos de la U.P.M.

Calicación / Mark: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

EL PRESIDENTE LOS VOCALESPresident Vowels

EL SECRETARIOSecretary

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Dedicado a mis padres y a mi hermana

The important thing is not to stop questioning

Albert Einstein

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Agradecimientos /

Acknowledgements

Muchas veces me he preguntado si merecía la pena tanto esfuerzo dedicado a latesis doctoral, y en más de una ocasión me ha resultado difícil justicar el sacricionecesario para alcanzar el objetivo. Sin embargo, la satisfacción que produce verterminado el trabajo compensa con creces todo lo demás. Sirvan estas líneas paraanimar a cualquiera que comience el desafío del doctorado, y como reconocimientoa las personas que ya lo terminaron.

Siempre he pensado que la parte más difícil de escribir serían los agradecimientos,y no me equivocaba, resulta imposible agradecer con unas simples palabras todo elapoyo que he recibido hasta aquí.

Miguel Á. Astiz, gracias por guiar mi investigación en estos primeros años dededicación, ha sido un auténtico honor tenerte como director de tesis.

José M. Goicolea, aprovecho esta ocasión para agradecerte todo el apoyo que mehas ofrecido, incluso desde antes de terminar la carrera.

Ana M. Ruiz Terán y Peter J. Staord, sois el mejor ejemplo de lo que quierollegar a ser, gracias por toda vuestra ayuda, tanto en Londres como aquí.

Ma Dolores G. Pulido, va a ser difícil que pueda devolverte todo lo que me hasdado, espero tener la ocasión de hacerlo. Gracias.

Es injusto que me limite a nombrar simplemente a todas las personas que me hanacompañado estos años en la 9a planta de nuestra escuela, espero que sepáis com-prenderlo y perdonadme los que se me olviden, gracias por vuestra ayuda, compren-sión y cariño; Inés Cano, Khanh Nguyen Gia, Cesar A. Polindara, Javier Oliva, PabloAntolín, Mario Bermejo, Sergio Blanco, Mustapha El Hamdaoui, Óscar González,Yolanda Cabrero, Eloína Fernández y tantos y tantas otras. Pasquale Dinoi, heaprendido mucho trabajando contigo, gracias por tu excelente esfuerzo.

Thank you very much Bradleys, you made me feel part of your great familyduring my time in London, sometimes dicult, but with your company alwaysmarvelous.

Gracias a todos mis amigos y amigas, los que me quedan después de tanto tiempodedicado en exclusiva al doctorado, por vuestra compresión y apoyo.

Agradezco profundamente la ayuda económica e institucional de la UniversidadPolitécnica de Madrid, de la Universitat Politècnica de Catalunya y de la Universityof East London, sin la cual hubiera sido muy complicado realizar la tesis doctoral.

Por último, y especialmente, gracias a mi familia. Donde estoy, y a donde quieraque llegue, os lo debo a vosotros.

Parece que ha llegado el momento de apagar el ordenador y afrontar nuevosretos, lo hago con la ilusión del primer día. Espero estar en contacto con todos losque he citado aquí, los que se me han olvidado y los que me quedan por conocer.

En Madrid, a 19 de Octubre de 2011

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Summary

The social and economical importance of long-span bridges is extremely large; cable-stayed bridges currently span distances ranging from 200 to even more than 1000 m,representing key points along infrastructure networks and requiring an outstandingknowledge of their seismic response. The objective of the study is three-fold; (i) todiscern how project decisions aect the seismic behaviour of cable-stayed bridges;(ii) to shed light on appropriate analysis strategies in order to address their linearand nonlinear dynamic response; and (iii) to compare dierent control schemes withpassive seismic devices disposed along the towers.

The organization of the content follows a natural progression, starting with themotivation of the work and presenting the state of art on this topic, followed by thedescription of the framework where the research is developed; the studied structuresand the simplifying assumptions employed throughout the document are clearlydened. Modal analysis of these bridges precedes the presentation of the seismicaction and the validation of the accelerograms which have been used. At this point,the discussion about the seismic results starts with a chapter completely devotedto the linear and nonlinear analysis procedures available to date, being followed bythe comparison of the elastic and inelastic response of cable-stayed bridges, focusedon the eect of dierent project decisions. Finally, several control strategies withseismic devices are addressed in order to maintain the towers in the elastic range.The main conclusions are drawn to close the thesis, and new lines of research aresuggested.

Satisfying the proposed objectives by means of rigorous nite element models,validated through the comparison with experimental tests conducted by other au-thors, the main contributions of the present work are highlighted:

• The localization of plastic strain deformation in nite element models repre-senting hollow-section reinforced concrete structures has been studied, observ-ing that the plastic hinge length obtained by classical expressions, is twice theoptimum dimension for the linear `beam'-type elements forming the towers.

• A specic procedure to generate synthetic accelerograms for nonlinear analysiswith Rayleigh damping has been introduced and validated, imposing actionscoherent with the design spectrum when damping varies with the frequency.

• The seismic consequences of key features like the tower shape, main spanlength, cable-system arrangement and type of foundation soil are analyzed.

• Dierent calculation methodologies are validated in the linear range, and new`pushover' procedures are proposed in order to study the nonlinear response ofthese exible and strongly coupled structures when three-dimensional seismicexcitations are imposed.

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vi SUMMARY

• Due to the large inuence of the transverse seismic reaction of the deck againstthe towers, analytical models are proposed and validated in order to predictthis eect prior to the denition of nite element models representing the fullbridge, providing valuable information for the designer in the early stages ofthe project.

• In light of the unacceptable damage recorded in several models of diamond-type pylons, specially in the lower part, the project of this element in bridgeslocated in seismic areas has been also addressed; from these analyses, newdesign recommendations are obtained to minimize the inelastic demand in thetowers and other features related to the economic cost of the foundations.Based on the energy balance, a scalar parameter that quanties in a simpleyet practical manner the damage caused by the earthquake has been proposed,facilitating the comparison between models.

• Large inelastic excursions of the reinforcement rebars and the concrete, besidesextensive cracking at key locations of the tower, have been observed in severalmodels, which may compromise their structural integrity. In order to preventsuch inadvisable behaviour, and trying to maintain the towers in the linearrange, the incorporation of dierent devices to control the seismic behaviouris explored through parameters based on the extreme seismic response and theenergy dissipation. Both yielding metallic dampers and viscous uid dampershave been considered with several designs and congurations, obtaining rele-vant conclusions for the designer.

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Resumen

La importancia social y económica de los grandes puentes atirantados es ex-tremadamente elevada; sus vanos principales varían típicamente entre 200 y 600metros, llegando incluso a sobrepasar los 1000 metros. Estas estructuras represen-tan puntos clave en las redes de transporte y requieren un estricto conocimiento desu respuesta sísmica. El objetivo del presente estudio consta de tres partes, que dannombre a la tesis doctoral: (i) obtener conclusiones sobre el efecto que tienen difer-entes decisiones de proyecto en el comportamiento sísmico de los puentes atirantados;(ii) explorar los diversos procedimientos de análisis para abordar con garantías elcomportamiento sísmico de estas estructuras, tanto en rango lineal como no lineal;y (iii) comparar diferentes estrategias de control sísmico con dispositivos pasivoscolocados en las torres.

La organización de los contenidos sigue una progresión natural, comenzandopor la motivación del trabajo y presentando el estado del conocimiento sobre estetema, seguido de la descripción de las estructuras consideradas en el estudio y de lashipótesis que simplican el problema. A continuación, se incluye el análisis modaly la acción sísmica al detalle, así como la validación de los acelerogramas sintéticosempleados. Llegados a este punto, comienza la presentación de los resultados delestudio sísmico con un capítulo dedicado en exclusiva a los procedimientos de análi-sis disponibles, tanto en rango lineal como no lineal, seguido por la comparacióntipológica de la respuesta sísmica elástica e inelástica de todos los puentes atiranta-dos analizados, con especial atención al efecto causado por diferentes decisiones deproyecto. Por último, se han abordado diversas estrategias de control con el objetivode mantener la torre en rango elástico. El trabajo concluye recogiendo las princi-pales conclusiones obtenidas y abriendo nuevas líneas de investigación que podríancontinuar el estudio.

Cumpliendo con los objetivos establecidos y empleando rigurosos modelos deelementos nitos, validados exhaustivamente con ensayos experimentales llevados acabo por otros autores, deben destacarse las siguientes contribuciones de la presentetesis doctoral:

• Se ha estudiado la localización de la deformación plástica en modelos de ele-mentos nitos que representan estructuras de hormigón con secciones huecas,como las empleadas en las torres de atirantamiento, observando que la di-mensión óptima en los elementos nitos lineales tipo `viga' es la mitad de lalongitud de la rótula plástica en piezas de hormigón armado.

• Un algoritmo de generación de acelerogramas sintéticos ha sido presentadoy validado con el objetivo de obtener señales adecuadas y coherentes con elespectro de diseño en cálculos no lineales con amortiguamiento en función dela frecuencia, como el de Rayleigh.

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viii SUMMARY

• El efecto en la respuesta sísmica que tienen la forma de la torre, la luz principaldel puente, el tipo de atirantamiento y la clase de terreno de cimentación, entreotros aspectos, ha sido analizado en detalle.

• Se han estudiado diferentes procedimientos de análisis sísmico, tanto en régi-men lineal como no lineal, y se han propuesto modicaciones de los métodos`pushover' para abordar el cálculo no lineal de unas estructuras tan exiblesy fuertemente acopladas como los puentes atirantados cuando se someten aexcitaciones sísmicas tridimensionales.

• Dada la importancia del empuje transversal del tablero en la respuesta sísmicade las torres, se ha propuesto y validado un modelo analítico para poder pre-decir dicha acción sin necesidad de establecer un modelo de elementos nitosque represente el puente completo, lo cual puede ser de gran utilidad para elproyectista en las primeras fases de diseño.

• Debido al inaceptable daño sísmico que ha sido registrado en varios modelosde torre con diamante inferior, se ha optimizado el diseño de este elemento y sehan obtenido recomendaciones de diseño que minimizan tanto la disipación deenergía por parte de la propia torre, como factores directamente relacionadoscon el coste de la cimentación. Un parámetro escalar que cuantica sim-plicadamente el daño estructural en las torres debido al terremoto ha sidopropuesto en función del balance energético, facilitando la comparación entredistintos modelos.

• Se han observado relevantes incursiones inelásticas, tanto de las armadurascomo del hormigón en varias estructuras, así como una importante suraciónen zonas clave para la seguridad de la torre y, por tanto, de todo el puente.Con el objetivo de evitar este inapropiado comportamiento, y de acercar larespuesta de las torres al rango puramente elástico, se ha estudiado la incor-poración de dispositivos sísmicos. Para ello, la respuesta sísmica extrema y laenergía disipada han sido contrastadas antes y después de incluir disipadoresbasados en la plasticación de metales y amortiguadores de uidos viscosos.Han sido considerados en cada caso varios diseños y diversas posiciones de estosdispositivos en la torre, obteniendo conclusiones relevantes para el proyectista.

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Contents

1 Motivation and scope 3

2 State of the art 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Seismic behaviour of cable-stayed bridges . . . . . . . . . . . . . . . 10

2.2.1 Vibration modes . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.3 Dynamic analysis procedures . . . . . . . . . . . . . . . . . . 202.2.4 Seismic response of the towers . . . . . . . . . . . . . . . . . 272.2.5 Spatial variability: multi-component seismic excitation . . . 272.2.6 Inuence of tower-deck connection on the seismic response . 302.2.7 Soil-structure interaction . . . . . . . . . . . . . . . . . . . . 312.2.8 Seismic behaviour of multiple-span cable-stayed bridges . . . 322.2.9 Near-eld earthquakes and vertical excitation . . . . . . . . . 33

2.3 Capacity versus mitigation design . . . . . . . . . . . . . . . . . . . 332.4 Mitigation design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.1 Energy-based design . . . . . . . . . . . . . . . . . . . . . . . 382.4.2 Passive dampers . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.3 Active and semi-active devices in cable-stayed bridges . . . . 532.4.4 Compendium of the seismic device typologies . . . . . . . . . 54

2.5 Cable-stayed bridges constructed in seismic areas . . . . . . . . . . . 542.5.1 Seismic failures reported in cable-stayed bridges . . . . . . . . 56

3 Modelling and basic assumptions 61

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2 Cable-stayed bridges description . . . . . . . . . . . . . . . . . . . . 62

3.2.1 Geometric aspects . . . . . . . . . . . . . . . . . . . . . . . . 623.2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 653.2.3 Deck-tower connection . . . . . . . . . . . . . . . . . . . . . 663.2.4 Prestress of the lower strut . . . . . . . . . . . . . . . . . . . 70

3.3 Materials and damping . . . . . . . . . . . . . . . . . . . . . . . . . . 733.4 Loading scheme and analysis . . . . . . . . . . . . . . . . . . . . . . 763.5 Finite element model description . . . . . . . . . . . . . . . . . . . . 78

3.5.1 Discretization of the towers: localization phenomena . . . . . 793.5.2 Discretization of the cable-system: cable-structure interaction 823.5.3 Discretization of the deck . . . . . . . . . . . . . . . . . . . . 843.5.4 Special-purpose elements . . . . . . . . . . . . . . . . . . . . . 85

3.6 Spatial variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.7 Symmetry of the seismic response . . . . . . . . . . . . . . . . . . . . 903.8 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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4 Modal analysis 95

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2 Fundamental vibration modes . . . . . . . . . . . . . . . . . . . . . . 964.3 Higher mode contribution . . . . . . . . . . . . . . . . . . . . . . . . 994.4 Modal coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5 Seismic action 107

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.2 Eurocode 8 spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.3 Natural records versus synthetic accelerograms . . . . . . . . . . . . 1105.4 Synthetic accelerograms description . . . . . . . . . . . . . . . . . . . 111

5.4.1 Signicant duration . . . . . . . . . . . . . . . . . . . . . . . 1115.4.2 Calculation scheme . . . . . . . . . . . . . . . . . . . . . . . . 1125.4.3 MRHA accelerograms: constant damping . . . . . . . . . . . 1155.4.4 NL-RHA accelerograms: Rayleigh damping . . . . . . . . . . 1155.4.5 Number of required records . . . . . . . . . . . . . . . . . . . 121

5.5 Synthetic accelerograms validation . . . . . . . . . . . . . . . . . . . 1225.6 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6 Seismic analysis 129

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.2 Mathematical approach . . . . . . . . . . . . . . . . . . . . . . . . . 1306.3 Elastic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.3.1 Direct Response History Analysis: DRHA . . . . . . . . . . . 1316.3.2 Modal Response History Analysis: MRHA . . . . . . . . . . 1336.3.3 Modal Response Spectrum Analysis: MRSA . . . . . . . . . 1366.3.4 Previous studies . . . . . . . . . . . . . . . . . . . . . . . . . 1406.3.5 Comparison of the results . . . . . . . . . . . . . . . . . . . . 141

6.4 Inelastic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.4.1 Modal Pushover Analysis: MPA . . . . . . . . . . . . . . . . 1516.4.2 Extended Modal Pushover Analysis: EMPA . . . . . . . . . 1586.4.3 Coupled Nonlinear Static Pushover: CNSP. A new method . 1616.4.4 Discussion of the results . . . . . . . . . . . . . . . . . . . . . 166

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7 Typological study of the elastic seismic behaviour 183

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1837.2 Transverse static response of the towers . . . . . . . . . . . . . . . . 1847.3 Extreme seismic response of the towers . . . . . . . . . . . . . . . . 185

7.3.1 Extreme seismic forces . . . . . . . . . . . . . . . . . . . . . 1857.3.2 Extreme total stresses . . . . . . . . . . . . . . . . . . . . . . 189

7.4 Eect of the accelerogram component . . . . . . . . . . . . . . . . . 1957.5 Eect of the transition between sections . . . . . . . . . . . . . . . . 1977.6 Extreme transverse deck reaction . . . . . . . . . . . . . . . . . . . . 199

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7.6.1 Full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2007.6.2 Simplied model . . . . . . . . . . . . . . . . . . . . . . . . . 2007.6.3 Analytical model . . . . . . . . . . . . . . . . . . . . . . . . . 208

7.7 Performance of the lower strut . . . . . . . . . . . . . . . . . . . . . 2187.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

8 Inelastic seismic behaviour 227

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2278.2 Extreme seismic forces . . . . . . . . . . . . . . . . . . . . . . . . . . 2288.3 Seismic demand of inelastic deformation . . . . . . . . . . . . . . . . 231

8.3.1 Damage in the lower diamond . . . . . . . . . . . . . . . . . 2338.3.2 Typological study . . . . . . . . . . . . . . . . . . . . . . . . 235

8.4 Specic studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2388.5 Dissipation factor Ω: an energetic approach . . . . . . . . . . . . . . 239

8.5.1 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . 2398.5.2 Dissipation factor denition . . . . . . . . . . . . . . . . . . 2418.5.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

8.6 Lower diamond optimization . . . . . . . . . . . . . . . . . . . . . . 2438.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

9 Seismic devices 251

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2519.2 Initial design considerations . . . . . . . . . . . . . . . . . . . . . . . 2539.3 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . 2559.4 Yielding Metallic Dampers: MD . . . . . . . . . . . . . . . . . . . . 257

9.4.1 Design basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2579.4.2 Triangular plates: TADAS . . . . . . . . . . . . . . . . . . . 2599.4.3 Shear Links: SL . . . . . . . . . . . . . . . . . . . . . . . . . 2709.4.4 Buckling-Restrained Braces: BRB . . . . . . . . . . . . . . . 280

9.5 Viscous uid Dampers: VD . . . . . . . . . . . . . . . . . . . . . . . 2829.5.1 Design and optimization basis . . . . . . . . . . . . . . . . . 2829.5.2 Optimization of the velocity exponent; αd . . . . . . . . . . . 287

9.6 Comparison between dierent strategies . . . . . . . . . . . . . . . . 2919.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

10 Conclusions and further studies 297

10.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29810.2 Further studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

A Constructed cable-stayed bridges 309

A.1 Rion-Antirion bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . 309A.2 Memorial Bill Emerson bridge . . . . . . . . . . . . . . . . . . . . . . 312A.3 Tsurumi Fairway bridge . . . . . . . . . . . . . . . . . . . . . . . . . 314A.4 Yokohama Bay bridge . . . . . . . . . . . . . . . . . . . . . . . . . . 315

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A.5 Ting Kau bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319A.6 Stonecutters bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

B Dimensions and characteristics of the models 323

B.1 Tower height and span distribution . . . . . . . . . . . . . . . . . . 323B.2 Cross-section of the deck . . . . . . . . . . . . . . . . . . . . . . . . 325B.3 Dimensions of the towers . . . . . . . . . . . . . . . . . . . . . . . . 327

B.3.1 Thickness of the tower sections . . . . . . . . . . . . . . . . . 336B.4 Characteristics of the foundations . . . . . . . . . . . . . . . . . . . 337B.5 Cable-system arrangement . . . . . . . . . . . . . . . . . . . . . . . 341B.6 Characteristics of each stay. Cable cross-section . . . . . . . . . . . 342

C Nonlinear nite element model 347

C.1 Materials: constitutive properties and assumptions . . . . . . . . . . 347C.1.1 Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347C.1.2 Reinforcement steel . . . . . . . . . . . . . . . . . . . . . . . 351

C.2 Mesh sensitivity: localization in hollow sections . . . . . . . . . . . . 353C.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353C.2.2 Plastic hinge length . . . . . . . . . . . . . . . . . . . . . . . 354C.2.3 Solid cantilever under monotonic loads . . . . . . . . . . . . . 355C.2.4 Hollow cantilever under monotonic loads . . . . . . . . . . . . 357C.2.5 Full cable-stayed bridge FEM . . . . . . . . . . . . . . . . . . 359C.2.6 Conclusions and application rules . . . . . . . . . . . . . . . . 361

C.3 Experimental verication employing cyclic tests on columns . . . . . 363C.3.1 Takahashi and Iemura's tests . . . . . . . . . . . . . . . . . . 363C.3.2 Sakai and Unjoh's tests . . . . . . . . . . . . . . . . . . . . . 365C.3.3 Orozco and Ashford's tests . . . . . . . . . . . . . . . . . . . 366

C.4 Model optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

D Spatial variability 371

D.1 Background theory and analysis framework . . . . . . . . . . . . . . 372D.2 Static procedures based on imposed displacements . . . . . . . . . . 373

D.2.1 Eurocode 8 (part 2) proposal . . . . . . . . . . . . . . . . . . 373D.2.2 Spanish code proposal: NCSP . . . . . . . . . . . . . . . . . . 375D.2.3 Priestley's proposal . . . . . . . . . . . . . . . . . . . . . . . . 376D.2.4 Results obtained with the static procedures . . . . . . . . . . 378

D.3 Dynamic procedure based on delayed accelerograms . . . . . . . . . . 378D.3.1 Results obtained with the dynamic procedure . . . . . . . . . 381

D.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

E Validation of synthetic accelerograms 385

E.1 Denition of earthquake scenarios . . . . . . . . . . . . . . . . . . . 385E.1.1 Empirical spectrum . . . . . . . . . . . . . . . . . . . . . . . . 386E.1.2 Specic horizontal earthquake scenarios . . . . . . . . . . . . 388

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Contents xiii

E.2 Seismological features . . . . . . . . . . . . . . . . . . . . . . . . . . 389E.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391E.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

F Pushover analysis in seismic codes and guidelines 397

F.1 Eurocode 8 - Part 2: Bridges . . . . . . . . . . . . . . . . . . . . . . 397F.2 ATC-40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399F.3 FEMA-356 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

G Step-by-step description of advanced pushover 401

G.1 Modal Pushover Analysis: MPA . . . . . . . . . . . . . . . . . . . . 401G.2 Extended Modal Pushover Analysis: EMPA . . . . . . . . . . . . . . 403G.3 Coupled Nonlinear Pushover Analysis: CNSP . . . . . . . . . . . . . 406

H Nonlinear SDOF integration 409

I Transverse reaction of the deck 413

J Tributary mass of the deck: tower model 417

List of Abbreviations 422

List of Symbols 425

Bibliography 435

Abstract 456

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Chapter 1

Motivation and scope

Why is it important to study the seismic behaviour of cable-stayed bridges?.Concern about the safety of infrastructure networks under the earthquake strike

has been historically a matter of great concern for the society, and specially for thecivil engineering community. Along such network, bridges represent key points sincetheir failure may be associated with the interruption of infrastructures, preventingthe access of health personnel and the evacuation of injured, besides immense eco-nomical costs. Unfortunately examples of partially or completely damaged bridgesare not an exception, as occurred in past Kobe (1995), Chi-Chi (1999) or recentlyin Tohoku (2011) earthquakes.

The present thesis is focused on the seismic response of several classical cable-stayed typologies, which may span natural barriers insurmountable in the past; themain span of these outstanding structures range competitively from 200 to 1000meters. Cable-stayed bridges represent iconic symbols for the region where theyare located and conform neuralgic points, their failure due to extreme events, likeearthquake ground motions, is therefore inadmissible. Dierent analysis strategies,besides the eect of several project decisions and the incorporation of seismic devicesto improve their behaviour are explored in the present research.

Contents organization

In order to address the study, chapter 2 starts describing the current state of knowl-edge on the seismic behaviour of cable-stayed bridges and the incorporation of pas-sive devices to control their response.

Once the framework established by current experience is set, the descriptionof the considered structures and their nonlinear nite element modelling followsin chapter 3, along with the basic assumptions employed thereafter. A detailedresearch on the section dimensions and proportions of several constructed cable-stayed bridges has been performed to establish the extensive parametrization ofproposed bridges; the main span ranges from 200 to 600 meters and is the keyparameter which denes completely the rest of the structure. Furthermore, vetypes of tower shapes have been considered, besides two cable-system arrangementsand two extremely dierent types of foundation soils; rocky or soft.

Modal analysis of proposed structures is accomplished in chapter 4, sheddingsome light on the dynamic properties of cable-stayed bridges. The rest of the work isrooted in several conclusions obtained in this chapter about the modal characteristicof these bridges.

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4 Chapter 1. Motivation and scope

Eurocode 8 compliant seismic action is dened in chapter 5, representing largeearthquakes that may occur in seismic prone areas worldwide. An ad hoc codeis presented in order to obtain synthetic accelerograms coherent with the designspectra when the damping is constant or variable with the frequency. The discussionabout the suitability of such articial signals compared with empirical predictionsof natural ground motions is also included.

The applicability of dierent analysis procedures in order to study the linear andnonlinear seismic behaviour of cable-stayed bridges is discussed in chapter 6. Newnonlinear static procedures which extend the classical pushover to three-axially ex-cited cable-stayed bridges and takes into account the characteristic modal couplingof these structures are proposed, and their results compared with the ones obtainedby means of rigorous nonlinear response history analysis and other pushover strate-gies.

Conclusions drawn from above comparison of analysis techniques serve as a start-ing point for linear elastic seismic analysis in chapter 7. First, the study of dierentstructural typologies under the proposed seismic excitation is accomplished in elasticrange, obtaining consequences of dierent project decisions on the seismic demandrecorded along the towers, which assume the key role in the global integrity of thestructure. Specic research about the transverse reaction in the towers due to thedeck and the eectiveness of the lower strut distributing such action between bothsides are included, besides other studies. An analytical procedure is proposed toobtain approximately the dynamic strike of the deck against the towers, before thedevelopment of a full-bridge nite element model.

A remarkable stress concentration was observed in several sections resulting fromthe elastic analysis, far exceeding cracking and yielding limit deformations. Chapter8 takes the seismic analysis to its more advanced and rigorous expression, includingnonlinear yielding and cracking eects, in order to obtain an accurate response whichis rst compared with the elastic solution presented before to gain some informationon the ductility demand along the studied types of towers. A damage factor isproposed in order to compare the dissipated hysteretic energy recorded in dierenttypologies, which helps to optimize the parameters dening the lower diamond inbridges located in seismic areas.

Even optimizing the tower design, the seismic excitation studied is strong enoughto cause relevant structural damage in several parts of this vital member, speciallyif central cable arrangement is employed and if the foundation soil stiness is re-duced. Traditional capacity design relies on the dissipation of energy by means ofsuch inelastic demand produced in the structure itself. Nowadays, this approach isdeemed unsafe in cable-supported bridges, since the towers are critical members forthe structural integrity and should remain practically elastic under strong groundshaking. Large cable-stayed bridges in seismic areas currently include seismic de-vices to centralize the earthquake demand, easier to repair (if required) than thegreat damaged towers sections resulting from betting on capacity design. Chapter9 addresses the study of these innovative techniques to reduce the tower inelasticresponse, obtaining useful design recommendations and applicability ranges.

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5

Chapter 10 sets the end of the present doctoral thesis about cable-stayed bridges,collecting the relevant conclusions and proposing further studies to continue theresearch.

Several appendices are included at the end of the document with additional ma-terial, which have been separated from the text to facilitate the reading; appendix Acollects technical information about some major cable-stayed bridges constructed inseismic zones; appendix B presents the research on real cable-stayed bridge dimen-sions, the proposed parametrization and complementary data; appendix C includesan elaborated review of the constitutive behaviour describing the employed materi-als, both in linear an nonlinear range, besides specic studies validating the niteelement discretization employed, furthermore, results about sensitivity studies deal-ing with the optimization of the parameters which dene the model and the analysisare gathered in this section; appendix D addresses the eects caused by the spatialvariability of the seismic action in the proposed cable-stayed bridges, considering dif-ferent wave-propagation velocities, incidence angles and analysis strategies proposedby seismic codes or research works; appendix E presents a thorough verication ofthe synthetic records considered in this thesis; a revision of pushover in seismiccodes and guidelines is collected in appendix F; appendix G summarizes the pro-posed advanced pushover methodologies in a step-by-step form; appendix H includesan integration algorithm for the dynamic response of a Single Degree Of Freedomsystem with combined isotropic/kinematic hardening; the analytical model and thecomplete expressions proposed to predict the extreme seismic reaction of the deckagainst the towers is included in appendix I; nally, appendix J studies the portionof the deck aecting the tower in transverse direction during the earthquake, inorder to obtain simple and accurate models representing exclusively the towers.

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Chapter 2

State of the art

Contents

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Seismic behaviour of cable-stayed bridges . . . . . . . . . . 10

2.2.1 Vibration modes . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.3 Dynamic analysis procedures . . . . . . . . . . . . . . . . . . 20

2.2.4 Seismic response of the towers . . . . . . . . . . . . . . . . . 27

2.2.5 Spatial variability: multi-component seismic excitation . . . 27

2.2.6 Inuence of tower-deck connection on the seismic response . 30

2.2.7 Soil-structure interaction . . . . . . . . . . . . . . . . . . . . 31

2.2.8 Seismic behaviour of multiple-span cable-stayed bridges . . . 32

2.2.9 Near-eld earthquakes and vertical excitation . . . . . . . . 33

2.3 Capacity versus mitigation design . . . . . . . . . . . . . . . 33

2.4 Mitigation design . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.1 Energy-based design . . . . . . . . . . . . . . . . . . . . . . . 38

2.4.2 Passive dampers . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4.3 Active and semi-active devices in cable-stayed bridges . . . . 53

2.4.4 Compendium of the seismic device typologies . . . . . . . . . 54

2.5 Cable-stayed bridges constructed in seismic areas . . . . . 54

2.5.1 Seismic failures reported in cable-stayed bridges . . . . . . . 56

2.1 Introduction

This chapter gives an overview about the state of knowledge on the seismic responseof cable-stayed bridges, and the solutions that have been carried out in real casesconstructed worldwide.

Cable-stayed structural solution has been known for centuries, but only becamepracticable with the advent of high-strength steel wire and the proposal to perma-nently prestress the stays by Eduardo Torroja (1927) and Franz Dischinger (1934),thereby preventing the excessive deections and aerodynamic instabilities sueredby the deck until that time. Since then, this kind of bridges have been employed

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10 Chapter 2. State of the art

successfully with spans ranging from 100 to more than 1000 meters, and it seemsthat in the near future the limits will be pushed.

Before looking at the seismic behaviour of these structures, their static responseshould be known since this base plays a key role in the design of their principalconstitutive elements; towers, cable-system and deck, which governs in turn theirdynamic behaviour. The static response of cable-stayed bridges is a widely doc-umented topic which is not included in the present thesis, the interested reader isdirectly referred to classical texts; [Gimsing 1998], [Walther 1988], [Manterola 2006],[Virlogeux 2001], [Fernandez-Troyano 2004], among others.

Both dynamic and static responses of cable-stayed bridges may present relevantnonlinearities1, whose origin has been observed by many authors; [Walther 1988],[Abdel-Ghaar 1991a], [Morgenthal 1999], [Ren 1999], [Ali 1995], [Ren 2005], amongothers. Two sources of nonlinearity have been clearly distinguished; material nonlin-earity, which depends on the materials employed and the relationship between eachother; and geometric nonlinearity, which is characteristic of cable-stayed bridges andis in turn composed of:

• Geometric nonlinearity due to the response of cable-stays: It is produced bythe modication in the catenary shape because of changes in the cable stress.

• Geometric nonlinearity due to second order moments (P −∆ eects) in com-pressed elements (the deck and the towers).

• Geometric nonlinearity due to large movements: Cable-stayed bridges are veryexible structures and may undergo important movements.

The geometric nonlinearities introduced by stay-cables lead to the increase ofthe structural stiness when demanding forces are enlarged, presenting cable-stayedbridges a slight geometric hardening in the elastic range which distinguishes thistypology from the rest [Fleming 1980] [Karoumi 1998]. During early loading stages,geometric nonlinearities domain the response and dierentiate cable-stayed bridgesfrom other structures, however, material nonlinearity governs advanced demandstages and the stiness is degraded as in conventional structures. Figure 2.1 illus-trates the dierence in the elastic response of cable-stayed bridges compared withother types of structures, and the importance of nonlinear calculations under strongloads.

2.2 Seismic behaviour of cable-stayed bridges

Due to the great exibility of cable-stayed bridges and consequent long fundamentalvibration periods with low spectral accelerations associated, these structures present

1In the present work, it has been observed that material nonlinearities could become veryimportant in the towers and seismic devices if included.

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2.2. Seismic behaviour of cable-stayed bridges 11

Figure 2.1. Schematic comparison between the elastic response of classical cable-stayed

bridges and structures without cable-system. An idea of the analysis accuracy with dierent

linearization approaches depending on the stage of loading is included. O-B: area governed

by geometric nonlinearity. Beyond B: area governed by material nonlinearity.

in principle a good seismic behaviour2. Furthermore, the number of supports isreduced (abutments, towers and intermediate piers), allowing the large expectedrelative movements without introducing important forces3.

However, this important exibility, added to their light weight and low associateddamping, causes large amplitude oscillations when they are excited by an earthquakeor another dynamic action, thus reducing such vibrations with added dampers eitherpassive, active, semi-active, or hybrid (combinations of the previous types) could bespecially recommendable. As long as the main span of the bridge is increased, itbecomes more susceptible to environmental dynamic actions like earthquakes orwind [He 2001]. Section 2.5.1 collects the most important failures reported in cable-stayed bridges due to ground motions.

The seismic response of cable-stayed bridges have focused the attention of the sci-entic community since early 80's, being remarkable the work of Abdel-Ghaar andhis co-authors [Abdel-Ghaar 1991a] [Abdel-Ghaar 1991b], with special attentionon the nonlinear behaviour, the sensitivity to support conditions, cable-vibrationphenomena and spatial variability. More recently, the concern about curved cable-stayed bridges and the incorporation of seismic devices to control the seismic re-sponse have raised new research trends.

2The problems caused by the seismic action in the deck are normally related to the horizontalcomponent of the ground motion; the vertical action aects more to the cable-system and thetowers because the deck is isolated from vibrations in this direction, behaving like a beam overelastic foundation [Walther 1988].

3The analysis of the spatial variability eect in the seismic action included in chapter 3, anddiscussed further in appendix D, suggests that dierential imposed movements at the foundationsdo not introduce severe loads generally, due to the large exibility of these structures.

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12 Chapter 2. State of the art

2.2.1 Vibration modes

The study of the dynamic response of a cable-stayed bridge, particularly if it issubjected to earthquake excitations, requires a thorough previous analysis of thevibration properties (frequencies, modal deformations, participation factors and ac-tivated modal mass) [Walther 1988], even if the employed methodology does notaccomplish the modal decomposition approach. Modal analysis is important toknow the way the structure vibrates and to nd out if there are any element withinappropriate dynamic behaviour prior to time-consuming seismic calculations.

Modal coupling is a characteristic feature of cable-stayed bridges, specially be-tween the transverse exure of the deck and its torsional response, which dieren-tiates their dynamic response from suspension bridges. This coupling is governedto a large extent by the mass distribution through the deck cross-section and thegeometry of the stay system. Due to this modal interaction, the distinction be-tween pure vertical, transverse and torsional movements is complicated and forcesthe designer to employ three-dimensional models in order to represent the structure[Abdel-Ghaar 1991a]. Modal coupling is stronger as long as the main span is longer(more generally speaking; as long as the exibility is higher).

First vibration modes have long periods and are generally associated with thedeck, followed by modes which excite the cable-system and may be coupled withthe deck. Higher frequencies appear deforming mainly the towers and their cou-pling with the deck depends on the connection conditions between both elements[Morgenthal 1999]. Due to the complexity and modal couplings inherent in cable-stayed bridges, a large number of modes are usually required in their dynamic anal-ysis.

Bruno and Leonardi [Bruno 1997] performed analytical and numerical studiesabout vibration modes presented in cable-stayed bridges, observing the small con-tribution of deck stiness and the negligible eect of the tower shape in the case oflateral cable arrangement4, with the exception of torsional modes, which are clearlyaected.

A review of the main body of knowledge in vibration modes of cable-stayedbridges is summarized below, more information may be found in other researchworks [Walker 2009] [Morgenthal 1999] [Valdebenito 2005].

Pure vertical deck modes

The vertical stiness of modern cable-stayed bridges with closely spaced stays andslender decks is dominated by the cable-system, being well characterized by geo-metric parameters like the main span length, the ratio between side and main spanlengths, or the tower height. Wyatt [Wyatt 1991] proposed analytical expressions toestimate the fundamental vertical vibration of cable-stayed bridges, neglecting thestiness of the deck.

Kawashima et al [Kawashima 1993] published the following expression for the

4This result has been veried in the present work and is presented in chapter 4.

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2.2. Seismic behaviour of cable-stayed bridges 13

rst frequency associated with the vertical bending exure of the deck, fv, correlatedfrom eld forced excitation tests conducted for thirteen constructed cable-stayedbridges in Japan5:

fv = 33.8L−0.763P (2.1)

LP being the main span in meters.Higher vertical deck modes present more zero-displacements nodes along this ele-

ment, reducing the inuence of axial deformation in the cable-system and increasingthe importance of deck stiness.

Pure torsional deck modes

Unlike vertical stiness, which is governed by the cable-system, torsional stinessmay arise in equal measure from the cable-system geometry or the deck cross-section,and it is thus more readily inuenced by the form of the tower and deck sections.

Kawashima et al [Kawashima 1993] also proposed an expression for the rsttorsional frequency of the deck, rooted in the correlation of experimental results:

fθ = 17.5L−0.453P (2.2)

The fundamental torsional frequency is dominated by the torsional rigidity of thedeck if this is a box-girder and, consequently, the torsional rigidity `GJ '6 has a sig-nicant magnitude; this is the case of bridges with central cable plane arrangement,where the cables only provide up to 10-20 % additional rigidity [Virlogeux 1999].Wyatt [Wyatt 1991] proposed the following expression to obtain the rst natural tor-sional frequency of the deck, neglecting the contribution of the stays (beam model),which is appropriate in bridges with moderate spans and central cable arrangement:

Torsional (beam model) fθ =1

2

√GJ

mr2L2tor

(2.3)

Ltor being the length of the deck between the points where torsion is prevented,m the mass of the deck per unit length and r the radius of gyration.

However, since the torsional stiness of the deck cross-section is GJ/Ltor, in caseof open cross-sections (associated with lateral cable arrangements) and/or cable-stayed bridges with long spans, the torsional stiness is governed by the cable-system. Gimsing [Gimsing 1998] proposed an idealized model with two springsrepresenting the cables in order to study (for lateral cable arrangements) the verticaland torsional fundamental frequencies of the deck neglecting its stiness;

Torsional (stay model) fv =1

√2CcpMd

; fθ =1

√CcpB

2

2Im,x(2.4)

5New proposals for the vertical and transverse fundamental frequencies of cable-stayed bridgeshave been proposed in this thesis (chapter 4).

6`GJ ' is the torsional rigidity; G shear modulus, J torsion constant.

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14 Chapter 2. State of the art

Where Ccp is the vertical stiness of each cable plane and Md, B, Im,x are re-spectively the mass of the deck, the width between both lateral cable planes andthe deck torsional mass moment of inertia. Gimsing proposed two extreme ideal-ized models in order to obtain Im,x, presented in gure 2.2; (a) if two concentratedmasses are considered in the cable planes, hence Im,x = MdB

2/4; (b) if uniformlydistributed mass of the deck is considered across the section width B, thereforeIm,x = MdB

2/12. In real structures, the mass distribution will be somewhere be-tween both extremes and hence, considering expression 2.4 and both values of Im,x,the ratio fθ/fv is seen to lie between 1 and

√3 ≈ 1.73 (which is further increased

if the stiness of deck cross-section is included). Observed ratios fθ/fv are between1.5 and 1.6 in practice [Wyatt 1991], which agree with the proposal of Gimsing.

(a) Concentrated masses(lateral cable arrangement)

(b) Distributed mass(central cable arrangement)

Figure 2.2. Ideal dynamic models for the estimation of deck vertical and torsional frequen-

cies. Proposed by [Gimsing 1998].

Therefore, cable-stayed bridges may present very closely spaced vertical andtorsional frequencies, specially if: (1) the deck has negligible torsional and verticalstiness related to the cable-system, and (2) the mass of the deck is concentratedout to the edges. This suggests that lateral cable plane arrangement, which includesopen deck sections with two lateral girders, could maximize couplings between thetorsional and vertical exure, aecting the accuracy of modal combination rules inseismic analysis [Walker 2009] and also the critical speed for utter.

The stiness of the cable-system, Ccp, may be selected by the designer to some ex-tent. In lateral cable arrangements, pure torsional mode requires anti-phase motionof the two cable planes and compatible motion of the towers; `H'-shaped towers al-low dierential movements between both lateral cable planes, however `A'-, inverted`Y'- or single mast-shaped towers with two cable planes prevent such movementsbecause these two planes meet at the same point and therefore pure torsional modesrequire axial extension of the stays, which is signicantly stier than the dierentiallongitudinal movement of the legs in `H'-shaped towers.

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2.2. Seismic behaviour of cable-stayed bridges 15

Transverse deck modes

The cable-system oers small transverse restraint to the deck, unless the cable planesare signicantly inclined (which is only appreciable in bridges with moderate spans).Neglecting the contribution of the stays and the exibility of the towers7, the trans-verse frequencies of the deck may be approximated from those of a continuous beamwith the same span arrangement, being dominated by the transverse exural sti-ness of the deck, EIH (respectively E is the Young's elasticity modulus and IHthe transverse moment of inertia of the deck). Wyatt [Wyatt 1991] proposed thefollowing expression for the rst transverse frequency related to the deformation ofthe deck:

Transverse (beam model) fy =1

2

√π2CyEIHmL4

P

(2.5)

Cy being a factor depending on the main to side span length. The transversefrequencies of the deck are not easily controlled by the designer since m and EIHare governed by the width of the deck (imposed by infrastructure requirements).

Kawashima et al [Kawashima 1993] again employed experimental results to cor-relate an expression for the rst transverse frequency of cable-stayed bridges in termsof the main span8:

fy = 482L−1.262P (2.6)

The coupling between the transverse exure and torsional response of the deck ischaracteristic in cable-stayed bridges, as it was already stated, and this interactionis important if fy and fθ are close to each other, observing no signicant reductionin the natural frequencies themselves, otherwise this coupling is weak [Wyatt 1991].

Tower modes

The cable-system strongly coerces the tower in longitudinal direction whereas itcauses a negligible eect along the transverse axis (unless the cable planes are sig-nicantly inclined). Therefore, pure transverse tower modes, which may be ap-proximated by cantilever models of the tower, appear before the longitudinal ones,estimated through encastred-pinned beam models; encastred at the foundationsand pinned at the top due to the constraint exerted by the cables, specially theones anchored to the points of the deck with prevented vertical movements, i.e. theabutments and intermediate piers.

7Section 7.6 proposes a methodology to estimate the transverse reaction of the deck against thetower, which includes an analytical expression to obtain the rst frequency associated with thetransverse exure of the deck, neglecting its torsion and the cable-system contribution, but takinginto account the exibility of the towers.

8An alternative expression for the rst transverse frequency related to the deck, fy, (also interms of the main span length exclusively) has been proposed in chapter 4.

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16 Chapter 2. State of the art

Cable-structure interaction

Another distinctive property of cable-stayed bridges is the transferred energy be-tween local cable and global modes, which is usually referred as cable-structureinteraction.

This eect was rst studied by Leonhardt and Zellner [Leonhardt 1980] anddeveloped before by many others. Normally, cable-structure interaction plays abenecial role in the seismic response of cable-stayed bridges, however, when sub-jected to earthquakes with specic dominant frequencies, the eect could lead to asignicant increase of the seismic forces.

The discretization employed to represent the cable-system could be therefore im-portant in terms of the accuracy of the dynamic analysis in cable-stayed bridges9.If only one nite element per stay-cable is employed (sometimes referred in the lit-erature as OECS; One Element Cable-System), their local vibration is neglectedand the dynamic interaction between the cable-system and the deck is also ig-nored, which may be important in the seismic analysis of cable-stayed bridges[Abdel-Ghaar 1991a]. The cable vibration could introduce a signicant amountof energy through higher modes, which are relevant in terms of seismic force contri-butions but in lesser extent in terms of displacements [Abdel-Ghaar 1991a][Abdel-Ghaar 1991b].

A large number of new vibration modes with low frequencies appears if multipleelements are employed in each cable (referred as MECS; Multiple Element Cable-System). Many of these modes are associated with pure local cable in-plane orout-of-plane lateral exure, and do not change the structural response of the bridgesubjected to a broadband seismic excitation. However, a smaller number of coupleddeck-cable modes arises, modifying pre-existing global modes and hence aectingthe dynamic response. Thuladar et al [Tuladhar 1995] conrmed this conclusion,observing important eects if the rst natural frequencies of the cables overlap withthe rst frequencies of the bridge.

Caetano et al [Caetano 2000] compared analytical OECS and MECS models withexperimental shaking-table results of Jindo bridge physical model (South Korea).No signicant dierences between models with single or multiple elements per cablewere observed and, hence, the physical model was modied to bring local and globalmodes together. Such `articial' model exhibited low seismic eects, precisely dueto cable-structure interaction, when it was subjected to broadband earthquakes,but modifying the signal and employing narrow-band records tuned to the rstglobal and local cable modes, signicant eects arise. Simplied models consisting ofmass-cable systems were later studied by Caetano [Caetano 2007] applying `external-excitation' perpendicular to the stay axis10, and verifying that, if the structure hasglobal modes with frequencies close to the fundamental cable ones (or twice thisvalue), cable-structure interaction may be important.

9A specic study about the eect of cable modelling has been conducted in the present thesisand is presented in chapter 3.

10On the other hand, the excitation parallel to the cable chord is termed `parametric-excitation'.

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2.2. Seismic behaviour of cable-stayed bridges 17

Cable-structure interaction is likely to be benecial if broadband excitationstrikes the bridge, reducing the seismic forces due to moderate cable vibrations.However, it could be unfavourable if the structure is subjected to narrow-bandearthquakes with important energy associated with coupled frequencies, presentinglarge cable oscillations which may act to increase the overall response.

The eect of local cable vibrations in the global behaviour of the bridge has beenwidely studied and is now a source of major research. Sometimes special devicesare installed in the cables in order to control the vibration caused by rain and windeects, subsequently improving their seismic eect. Such devices increase the energydissipation in the stays, which otherwise present reduced damping by themselves,about 0.05 % to 4 % [Abdel-Ghaar 1991b].

Several authors have veried the improvement in the accuracy of the recordedseismic demand if multiple elements per cable are employed [Abdel-Ghaar 1991b],[Tuladhar 1995], [Au 2003] [Ko 2001]. These authors recommend the discretizationof cables with multiple elements in the seismic analysis of cable-stayed bridges,however Ni et al [Ni 2000] studied the modal properties of Ting Kau bridge (China)and veried that the eect of multi-element discretization is only appreciable in thelongest cables, 465 m long, whilst the rest of the stays could be modeled with oneelement per cable without losing accuracy, which is also defended by Wilson andGravelle [Wilson 1991].

How many elements should be included per element to take into account re-alistically the cable-structure interaction?. Caetano [Caetano 2007] conducted asensitivity analysis on this topic studying Vasco da Gama bridge (Portugal), con-cluding that 9 elements per cable yields errors less than 5 %, even in the longestcables with 226 m long, being able to obtain the rst three local vibration modes ofthese stays accurately.

2.2.2 Damping

Cable-stayed bridges present low damping and assuming standard values for thefraction of critical damping (ξ) of 5 % falls on the unsafe side [Kawashima 1991].The estimation of damping is important, but also complicated since it depends onthe relative damping of each constitutive element (towers, cable-system and deck)and even on their conguration and interaction between each other.

Broadly speaking, the eect of damping has been implemented by means of threeprocedures in previous research works and also in the present doctoral thesis:

• Through the realistic representation of the nonlinearity sources that may bedeveloped during the earthquake (e.g. hysteresis loops in structural membersand seismic devices, radiation of energy by means of supports, etc.). This isthe most precise methodology in order to take into account the dissipation ofenergy under large seismic events, strong enough to develop such nonlineari-ties. It could be implemented in nonlinear dynamic calculations, or by meansof reduced spectra (which is less accurate).

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18 Chapter 2. State of the art

• Using Rayleigh or Caughey damping theory to decompose the structure damp-ing matrix c and perform modal dynamic analysis. Dierent damping is asso-ciated with each vibration mode, and previous modal analysis is required inorder to nd the range of frequencies with the most signicant contribution inthe response. Values of ξ ≈ 2− 4 % are imposed on the borders of such inter-val, obtaining higher values of damping associated with higher modes, whoseparticipation is assumed negligible and may cause numerical instabilities indynamic calculations. However, Yamaguchi and Furukawa [Yamaguchi 2004],in their work about Yokohama Bay cable-stayed bridge (Japan), concludedthat Rayleigh (or Caughey) damping is not appropriate in the seismic analy-sis of this structure due to the special connection which is established betweenthe deck and the towers (see appendix A). This approach is further developedin chapter 5.

• Considering a fraction of the critical damping constant for all vibration modes,ξ = constant ≈ 2 − 4 %. This is the simplest procedure and has beenadopted by many researchers (e.g. Ali and Ghaar [Ali 1995], Morgenthal[Morgenthal 1999]), seismic codes and guidelines. This approach can only bedirectly used in calculations based on modal decomposition (spectrum analy-sis or modal dynamics) and, therefore, is valid for bridges behaving in linearrange without seismic devices.

The research of Kawashima and Unjoh [Kawashima 1991] concluded that damp-ing is strongly inuenced by the considered vibration mode, since dierent consti-tutive elements are excited depending on the mode shape, which in turn presentdierent damping values. Furthermore, it depends on modal coupling, velocity ofwave propagation, dimensions of the foundations and the direction of the studiedresponse11, among other parameters. These authors proposed a methodology to es-timate the damping by dividing the structure in elements with the same dissipationmechanism, obtaining the contribution of each sub-structure and aggregating the re-sults to approximate the modal damping. The study demonstrated that damping islargely aected by the amplitude of ground excitation, and thus harp cable-systemarrangement presents higher damping values associated with longitudinal oscilla-tions than analogous bridges with fan cable-system. These cable arrangementsare illustrated in gure 2.3. Such dependency of the rst modes on the ampli-tude of the seismic excitation has been experimentally contrasted in constructedcable-stayed bridges, as it was published in the research of Siringoringo and Fujino[Siringoringo 2005] about Yokohama Bay cable-stayed bridge (Japan).

An improved damping denition is achieved in structures with passive seismicdevices (chapter 9) since dissipation in these elements is realistically dened throughtheir constitutive properties, and are responsible for most of the dissipated energy[Soong 1997].

11For example, the towers have lower damping values in transverse direction due to the negligiblebending stiness of the cable-system, however such cables in longitudinal direction involve theresponse of the tower and the deck, resulting higher damping values.

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2.2. Seismic behaviour of cable-stayed bridges 19

Figure 2.3. Cable layout solutions in elevation view for cable-stayed bridges.

2.2.2.1 Damping mechanisms

Walker [Walker 2009] presented a practical revision of damping sources in classicalcable-stayed bridges, here, the concluding remarks are included:

Structural damping

The cyclic demand of materials in the structure dissipates energy due to the hys-teresis loops associated with plastic deformation, which may arise even below thematerial elastic limit because of stress concentrations that occur at microscopic lo-calized plastic ow. However, this type of damping increases dramatically whenthe plastic limit is exceeded, governing the dissipation of the bridge. This source ofdamping depends on the vibration amplitude but not on its frequency [Chopra 2007].

Friction at bearings

When relative movements between the deck and the abutments (incorporating sup-port devices) are recorded, damping is produced due to coulomb friction, or bymeans of hysteresis loops if support devices include a lead core (LRB). Such damp-ing depends on the amplitude and the vibration mode to the extent this relativemovement is achieved.

Cable-slip in the cable-system

Energy may be dissipated through internal slip between wires forming the stays if athreshold amplitude is exceeded, overcoming its internal friction. Consequently, thiskind of damping depends on the amplitude of vibration and type of cables employed.

Foundation radiation damping

Vibrations in the foundations originated by the seismic excitation cause energydissipation due to the radiation to surrounding subsoil, which could lead to higherdamping levels than those associated with the superstructure, hence highly dampedsubstructures are often treated as a secondary dynamic system. This dampingclearly depends on the vibration mode.

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20 Chapter 2. State of the art

Aerodynamic damping

Superstructure vibration is coerced by the surrounding air, providing a resistanceproportional to the square of the relative velocity; air damping is viscous, i.e. rate-dependent. This kind of damping is typically modest in conventional cable-stayedbridges due to the low air density, the reduced superstructure surface area and thelarge associated inertia forces involved in its seismic movement.

System damping

It is due to the interaction between the deck, cable-system and towers. A signicantamount of energy may be dissipated by means of this source of damping in classicalcable-stayed bridges if vibration modes of the stay-cables are similar to the onesassociated with the deck and coupling eects arise [Caetano 2000], as it has beenalready discussed in section 2.2.1.

2.2.2.2 Practical simulation of damping sources

Energy dissipation, despite represents a key issue in any dynamic analysis, is farfrom being accurately simulated in the engineering practice due to the complex andmiscellaneous nature of damping processes.

Conventional bridges and buildings typically avoid specic considerations aboutany of the dissipation sources presented above, and simply consider viscous damp-ing through constant factors (ξ) provided by relevant codes, hence preventing theintroduction of additional uncertainties in the model and obtaining solutions on thesafe side, since specic dissipation mechanisms are ignored12.

2.2.3 Dynamic analysis procedures

Several seismic analysis strategies may be employed to obtain the response of astructure subjected to earthquake excitations, with inherent advantages and dis-advantages which may support or discourage their use depending on the type ofstructure, expected nonlinearities and type of response measure extracted. Wethya-vivorn and Fleming [Wethyavivorn 1987] and Abdel-Ghaar [Abdel-Ghaar 1991a]published reference works on the seismic analysis procedures and their viability inthe study of cable-stayed bridges.

In this section, the available seismic analysis techniques are briey discussed,introducing their key features in order to better understand the limits of applicabilityand sources of errors associated with each procedure. A detailed mathematicaldescription of each method is included in chapter 6. For a thorough treatment,the works of Chopra [Chopra 2007], Clough and Penziem [Clough 1993], Villaverde[Villaverde 2009] or García Reyes [García-Reyes 1998] are strongly recommended.

12In the present thesis, the structural damping is rigorously included in nonlinear analysisthrough realistic material properties (chapters 8 and 9), being the most important dissipationmechanism. The other sources of damping are deliberately ignored, which is a safe assumption.

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2.2. Seismic behaviour of cable-stayed bridges 21

The dynamic response of a N -degree of freedom structure subjected to the earth-quake action is expressed by means of the following system of N coupled dierentialequations (hereinafter referred as system of dynamics):

mu + cu + fS(u, u) = −mιug (2.7)

Where u(t) is the relative displacement vector, m and c are respectively themass and damping matrices of the structure, fS is the stiness component of theforce vector in the structure and denes the relationship between force and displace-ment vectors; while the structure behaves in the linear elastic range fS = ku, beingk the elastic stiness matrix of the structure. Finally, ι is the inuence matrixconnecting the degrees of freedom of the structure and the imposed accelerogramdirections ug(t), which generally, and neglecting imposed rotations at the founda-tions, is a vector with three components uTg (t) = (uXg , u

Yg , u

Zg ), each one related to

the accelerogram component along each principal direction.

Inelastic seismic analysis procedures

There are several ways to face the general nonlinear dynamic problem, next orderedfrom more to less time-demanding procedures.

• Non-Linear Response History Analysis (NL-RHA):

One rigorous way to address the dynamic response of the structure is to solvedirectly the complete coupled system of dynamics (2.7); Non-Linear ResponseHistory Analysis (NL-RHA) integrates this system step-by-step, consideringthe tangent stiness at each iteration in order to linearize the problem. Severalalgorithms may be employed to solve directly the coupled system, being themost commonly used the HHT scheme [Hilber 1977].

NL-RHA is the most accurate methodology to predict the inelastic seismic de-mand in a structure; the procedure may fully take into account the geometricand material nonlinearities (e.g. cyclic stiness degradation, hysteretic dissi-pation), and is able to analyze realistically the eect of seismic devices if theyare equipped. However, a set of three-axial representative accelerograms isrequired, besides mathematical models capable of representing adequately thecyclic load-deformation characteristics of all the important elements, and e-cient computing tools in order to deal with time-consuming calculations, whichare not justied in every structure and engineering oce [Krawinkler 1998],[Bommer 2003], [Chopra 2007].

Uncertainties arise when the nonlinear cyclic behaviour of concrete, steel andbonding interfaces have to be described. Priestley [Priestley 1996] recom-mended direct nonlinear dynamics only when specic aspects of the design ofthe bridge need to be veried. Furthermore, the integration scheme introducesphase errors, which are higher as long as the ratio of the step-time (∆t) overthe considered vibration period (T ) is increased [Hilber 1977], as it may be

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22 Chapter 2. State of the art

appreciated in gure 2.4. There are seismic regulations which preclude thisprocedure, like the National Annex of Eurocode 8 in Germany [EC8 2011].

NL-RHA is largely the most demanding method available but is gaining trac-tion nowadays, since earthquake engineering is moving towards performance-based design and away from force-based design, thus reliable methods arerequired in order to obtain the realistic nonlinear seismic demand. Moreover,nonlinear dynamic analysis benets from the improvement in computers ca-pabilities.

Incremental Dynamic Analysis (IDA, sometimes referred as `dynamic pushover')[Vamvatsikos 2002] performs sets of nonlinear dynamic calculations by scalingseveral accelerograms with dierent intensities, in order to obtain an accuratesight of the nonlinear response of the bridge. It is commonly considered themost precise way to explore the behaviour of structures under large groundmotion excitations. However, its computational cost is dicult to justify in allcases, being used as reference result in many research works in order to stressthe accuracy of simplied methods, like pushover analysis.

Per

iod

elon

gatio

n / T

a

a

Figure 2.4. Relative phase error in terms of the ratio of the step-time and the vibration

period (∆t/T ) in several direct integration methods; Hilber - Hughes - Taylor (HHT) with

dierent numerical damping values αa; Wilson; Newmark; and Houbolt. Taken from Hilber

et al [Hilber 1977].

• Non-linear Static Procedures (NSP):In recent years, Nonlinear Static Procedures (NSP), commonly named pushovermethods, have received a great deal of research, specially since seismic designguidelines ATC-40 [ATC 1996] and FEMA 273 [fem 1997] were published13.Their main goal is to estimate the nonlinear seismic response by means ofstatic calculations, pushing the structure up to certain target displacementusing load patterns which try to represent the distribution of inertia forces.

13Normative pushover strategies are presented in appendix F.

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2.2. Seismic behaviour of cable-stayed bridges 23

These methodologies expose design weaknesses that may remain hidden inan elastic analysis and yield good estimations of the nonlinear seismic per-formance under certain conditions (which will be discussed later), drasticallyreducing the computational cost [Krawinkler 1998]. For these reasons manycodes and guidelines recommend the use of pushover to evaluate the inelasticseismic behaviour [EC8 2005a], [ATC 1996], [fem 2000], [fem 2005]. Howeverthe mathematical basis of the procedure is far from accurate; it is assumed thatthe nonlinear response of a Multi Degree Of Freedom structure (MDOF) maybe related to the response of an equivalent Single Degree Of Freedom system(SDOF), which implies that the response is controlled by a single mode, whoseshape remains constant through the analysis [Krawinkler 1998]. Despite theseassumptions are clearly incorrect, the estimated results have been found to beaccurate many times, compared with the realistic nonlinear dynamic analy-sis14, when the structure is dominated by the rst mode of vibration [Lu 2004][Gosh 2008] [Krawinkler 1998].

Dierent proposals have been made to overcome the shortcomings of pushoveranalysis, forming what is called as `advanced pushover'; (1) in order to takeinto account the higher mode eect N2 [Fajfar 2000] and Modal PushoverAnalysis (MPA) [Chopra 2002] were developed, among others, but they com-bine the modal contributions with standard rules like SRSS and hence equi-librium may be not satised and the signs are lost, misleading the results[Ferracuti 2009], there are currently dierent attempts to solve this problem(e.g. [López-Menjibar 2004]); (2) several adaptive pushover analyses havebeen proposed in order to change the load pattern along the structure aslong as yielding mechanisms are developed, they can be based either on im-posed load [Gupta 2000] or displacements patterns [Antoniou 2004]. Despitepromising adaptive pushover methodologies normally improve the accuracy[Antoniou 2004] [Ferracuti 2009] [López-Menjibar 2004] [Sei 2009], its dif-culty is inevitably increased and is somewhat away from the initial ob-jective of a simplied yet accurate method. Moreover, Papanikolau et al

[Papanikolaou 2005] pointed out the misleading results that adaptive pushovercould yield, besides the numerical diculties involved in the extraction of vi-bration modes if large inelastic deformations arise.

Another pitfalls of pushover analysis are the diculty in the representation ofthree-dimensional and torsional eects, as well as the consideration of multi-directional simultaneous seismic excitations, which are found to be importantin structures with strong modal couplings like cable-stayed bridges15, see sec-tion 2.2.1. In this direction, Lin and Tsai [Lin 2008] proposed an extensionof MPA substituting the SDOF by a three Degree Of Freedom system which

14A thorough comparison between the direct nonlinear analysis and several pushover proceduresis conducted in chapter 6.

15Chapter 6 address the problem of three-axially excited structures under strong ground shaking,proposing new pushover procedures to obtain the response.

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24 Chapter 2. State of the art

takes into account the coupling between the two horizontal translations andthe rotation about the vertical axis (torsion), increasing the complexity of theprocedure. More practically, Huang and Gould [Huang 2007] performed a si-multaneous bi-directional pushover considering two load patterns along bothhorizontal directions.

So far, most of the research is currently focused on buildings and few worksaddress the problem of the applicability of advanced pushover to bridges (e.g.[Lu 2004], [Gosh 2008], [Shattarat 2008], [Kappos 2005]), whereas no specicstudies on this topic about cable-stayed bridges have been found by the author.On the other hand, bridges are usually more aected by higher modes and,therefore, developing a modal pushover procedure for such structures is evenmore of a challenge than in the case of buildings16.

The seismic action in nonlinear static calculations may be introduced bymeans of a set of accelerograms or through their equivalent spectrum, in fact,pushover analysis could be considered as the nonlinear extension of the modalresponse spectrum analysis (MRSA) [Chopra 2007], further information is in-cluded in chapter 6.

Elastic seismic analysis procedures

The seismic analysis is strongly simplied if the forces on the stiness componentof the structure are related to the deformation by means of a linear elastic stinessmatrix; fS = ku. The procedures based on this assumption are briey discussednext, ordered again from more to less computational cost.

• Modal Response History Analysis (MRHA):

Modal Response History Analysis (simply referred as modal dynamics) is basedon modal superposition to uncouple the system of dynamics (2.7), which isonly possible if the behaviour is elastic (fS = ku) and if the damping matrix c

responds to what is called `classical damping' and hence is able to be decom-posed. Such damping property is automatically achieved using Rayleigh orCaughey damping matrices, which is the reason to employ these hypotheses,based on unrealistic viscous damping (see section 2.2.2.2). These assumptionspreclude the analysis of structures beyond the elastic range and the use of seis-mic devices, with the exception of linear viscous uid dampers or viscoelasticdampers [Villaverde 2009], where complex eigenvectors may be extracted.

Once the modal decomposition of N -degree of freedom system of dynamics isdone, N independent dierential equations are obtained, representing each onethe response of the corresponding SDOF. Furthermore, taking into account the

16Two new methodologies are proposed in this thesis (chapter 6) to estimate the nonlinear three-dimensional response of cable-stayed bridges.

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2.2. Seismic behaviour of cable-stayed bridges 25

whole set of N equations17 is not necessary, instead only the most contributingvibration modes should be included.

The number of modes to be considered could be established as the requiredmodes to activate more than 90 % of the total model mass, according to theseismic codes (e.g. Eurocode 8 [EC8 2004], NCSP [NCSP 2007]). However,cable-stayed bridges include very exible elements and only high order modesexcite the mass of the towers close to the foundation level, which represents asignicant percentage of the total mass. In these structures, exceeding 70 80% of the modal participation without including a large number of vibrationmodes is complicated [Chen 2007] [Morgenthal 1999]18. The modes may becomputed using Ritz-vector or Subspace iteration procedures, among others.

Dealing with linear elastic seismic analysis, the only numerical source of errorsintroduced in modal dynamics is the possible omission of the eect of highermode contributions [Chopra 2007] [Clough 1993], being the most accurate pro-cedure for linear dynamics. Since it is based on time history integration ofSDOF equations, the seismic action needs to be introduced in time-historydomain (like in direct nonlinear dynamics or in several pushover strategies),i.e. through accelerogram records.

• Modal Response Spectrum Analysis (MRSA):

Unlike time-domain modal dynamics, spectrum analysis (Biot's PhD thesis,[Biot 1932]) obtains directly the peak response of each considered SDOF (rep-resenting the structure vibration modes) by means of the design spectrumemployed to describe the seismic excitation. Such modal maxima is obtainedwithout the introduction of any approximation, unfortunately, the informationregarding the instant along the earthquake when each modal extreme responseoccurs is lost, and combination rules (more or less inaccurate) are employedto estimate the global behaviour. Moreover, errors may be introduced in priorSDOF integration required to obtain the spectrum.

This procedure is also based on modal decomposition and thus it is intendedfor linear analysis. Nonetheless, there are approximations to include the dis-sipated energy by means of the structure itself (through hysteresis or sti-ness degradation) by reducing the spectra in several ways. The reduction ofthe design spectrum to consider the eect of seismic devices [Mayes 1993] isadopted in Eurocode 8 [EC8 2005a], more information may be found in chap-ter 9. Modifying elastic design spectra to involve complex nonlinear processesis undoubtedly far from rigorous, and is used mainly in the design stage of con-ventional structures which may be subjected to moderate to medium seismic

17Considering complex structures like cable-stayed bridges, many degrees of freedom should beincluded in the analysis to accurately represent their dynamic behaviour, and thus the model couldeasily exceed N = 5000 degrees of freedom.

18Chapter 4 deals with the number of required modes in cable-stayed bridges.

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26 Chapter 2. State of the art

events, where strong nonlinearities are not expected19.

A complete review on dierent existing combination rules and their applica-bility to the seismic analysis of cable-stayed bridges is presented in the workof Walker [Walker 2009], concluding that Complete Quadratic Combination(CQC) approach [Der Kiureghian 1981] is the most adequate modal combina-tion option due to the strong modal coupling which characterize the dynamicbehaviour of these bridges. Walker also contributed with a number of keyobservations, included in chapter 6 besides the theoretic treatment of thisprocedure.

The aforementioned modal combination is carried out along each directionwhere the earthquake is imposed20. Once the extreme response is obtained,assuming the seismic event acting separately in each direction, the resultsare combined by means of SRSS rule [Goodman 1955], thus accepting thecomplete independence between each other in this `directional combination',which is suggested by seismic codes [EC8 2004] [NCSP 2007]. Both modal anddirectional combination rules assume the superposition principle, and thereforeare strictly valid only if the response is maintained in the linear elastic range.

The use of spectrum analysis in large cable-stayed bridges has been criticizedby a number of authors [Ren 1999] [Valdebenito 2005] since it is only valid inlinear range. Ren and Obata [Ren 1999] studied the inelastic seismic behaviourof one 600 m main span cable-stayed bridge, concluding that nonlinearitieswere negligible even considering strong earthquakes. However, these authorsrecognized that in such studied case, performing a linear analysis was correct,but the superposition principle is not equally appropriate due to the complexdynamic coupling of cable-stayed bridges.

Considering these shortcomings and accepting its limitations, spectrum analy-sis is the preferred tool for practicing engineers because of the following decisiveadvantages; (i) the extreme envelope of the response is directly obtained (un-like procedures based on time-domain history of the response, like direct andmodal dynamics), which is a key point in the design or retrot of structures;(ii) seismic codes are committed to this procedure, describing the seismic ac-tion directly by means of acceleration spectra (e.g. Eurocode 8 [EC8 2005a]and NCSP [NCSP 2007]), despite they recommend the verication of the solu-tion by means of response history analysis dealing with important structures,specially if seismic devices are incorporated.

19In the present thesis, a high level of accuracy is pursued in order to gain knowledge on therealistic nonlinear response of complex cable-stayed bridges, hence spectrum analysis is strictlyapplied to elastic calculations.

20In the present thesis it is generally considered a three-directional excitation; X,Y, Z, unlessthe opposite is stated in some specic studies.

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2.2. Seismic behaviour of cable-stayed bridges 27

Recommended analysis scheme in cable-stayed bridges

Many authors recommend the next step-by-step approach to study the seismic be-haviour of structures; (1) The rst step in the analysis scheme is to obtain thedeformed conguration due to the self-weight of the structure, by means of a linearor nonlinear static analysis; (2) starting from such deformed conguration, vibra-tion modes are extracted (see gure 2.1); (3) a linear or nonlinear seismic analysisfollows. Regardless of the type of dynamic analysis selected, it should start fromthe deformed conguration, which, according to Ren and Obata [Ren 1999] and Renand Peng [Ren 2005], could be obtained by means of a linear elastic static analysiseven considering large cable-stayed bridges; extracting the initial vibration prop-erties from the deformed conguration exerts negligible inuence in modal resultsas other researchers have stated (and also the present work), however it is a keypremise to obtain an accurate dynamic response in the opinion of Ren and his co-authors. Information about the analysis scheme adopted in this thesis may be foundin chapter 3.

2.2.4 Seismic response of the towers

Hayashikawa et al [Hayashikawa 2000] studied the seismic behaviour of steel tow-ers in cable-stayed bridges with isolation devices and dierent structural typologies.Subsequent research by Abdel-Raheem and Hayashikawa [Abdel-Raheem 2003] ver-ied the inuence that the following factors exert on the seismic response of thetowers; (i) damping associated with the tower; (ii) transverse and longitudinal towershape, specially the size and distribution along the tower of transverse struts in `H'-shaped models; (iii) accelerogram characteristics, being recommended the study ofa large number of records; (iv) initial construction imperfections in metallic towers(geometric imperfections and residual stresses); (v) soil-structure interaction (sec-tion 2.2.7); (vi) cable arrangement; (vii) type of connection between the deck andthe tower; and (viii) the material of the tower (steel or concrete), steel towers arelight-weight and hence lower seismic forces are associated, however they presentreduced damping values.

Despite the major importance of the towers in the global resistance of cable-stayed bridges, there are only a few research works about the seismic behaviour ofthese elements [Valdebenito 2005]. Chapters 7 and 8 aim to shed some light onthis topic, considering the linear and nonlinear response of towers parameterized interms of the main span.

2.2.5 Spatial variability: multi-component seismic excitation

The same ground motions may be imposed in all the foundations along the bridgeif its length is reduced. However, considering long structures like cable-stayedbridges, the eect of the earthquake could be appreciably dierent between dif-ferent foundations due to the large distance between. Many studies concluded thatnon-synchronous excitations could exert an important eect in cable-stayed bridges;

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28 Chapter 2. State of the art

[Walther 1988], [Abdel-Ghaar 1991a], [Nazmy 1992], [Tuladhar 1999], [Soyluk 2004],among others.

Considering two supports of the structure, some distance apart, the long periodseismic movement received is nearly the same, but the short period component of theaction is dierent and uncoupled due to the spatial variability of the earthquake.Such eect leads to the asynchronous movement of the supports, adding pseudo-static forces which should be taken into account in bridges with signicant spans(where the distance between piers is important compared with the wavelength of theseismic excitation in the range of the most contributing frequencies of the bridge,see gure 2.5), or in bridges spanning soils with strong discontinuities which maychange the properties of the seismic waves [Nazmy 1992]. In light of these sources ofasynchronism in the signal, Eurocode 8 [EC8 2004] recommends the considerationof this eect dealing with bridges longer than 240 or 120 meters (total length of thedeck) for rocky and soft soils respectively, or with signicant changes in the foun-dation subsoil along their length. In the Spanish code [NCSP 2007], non-uniformground motions should be addressed if the total length of the deck is larger than600 m.

Studying in more detail the phenomena which cause the loss of correlation in theseismic action between separated supports, Priestley [Priestley 1996] distinguishedthe following eects;

1. Time shift and incidence angle of the wave-train: The wave-trainreaches dierent foundations with a time-delay (δt) with respect to the supportcloser to the epicenter, due to the nite velocity (vs) of seismic waves (whichis increased with the stiness of the surrounding subsoil). On the other hand,the axis of the bridge forms generally an incidence angle (θ) with the wave-train (see gure 3.20), hence the delay recorded between two piers separateda distance Li and assuming a constant propagation velocity of the waves vs istheoretically;

δt =Licosθvs

(2.8)

2. Loss of correlation due to complex refractions and reexions of

waves: This phenomena, strongly frequency dependant, may be neglected (ina rst approximation) dealing with exible structures where the fundamentalperiod is larger than 1 s [Priestley 1996], like cable-stayed bridges.

3. Filtering and local amplications: Changes in the properties and orogra-phy of the soil could lead to the loss of correlation between the excitation indierent supports, including modications of the frequency content. The prob-lem may be simplied by means of an average or envelope spectrum includingeach type of soil in which the bridge is supported [EC8 2004] [NCSP 2007].

The eect of the spatial variability of the seismic action is more pronouncedthe stier is the structure (e.g. concrete decks in cable-stayed bridges behave worst

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2.2. Seismic behaviour of cable-stayed bridges 29

Rotational - translational soil movement (surface R waves)

Horizontal movement(SH waves)

Longitudinal movement(P waves)

Vertical movement(SV waves)

wavelength

Main span length (Lp)

Figure 2.5. Dierent types of seismic waves exciting the supports of a long cable-stayed

bridge. Adapted from the work of Abdel-Ghaar [Abdel-Ghaar 1991a] and Nazmy and

Ghaar [Nazmy 1992].

than metallic ones in this sense), since pseudo-static forces due to dierential supportdisplacements are maximized. As the main span of the bridge gets larger, the time-delay is consequently higher but the increase in the exibility could be even moreimportant and the global eect in large cable-stayed bridges may be the reductionof the spatial variability inuence [Abdel-Ghaar 1991a].

Nazmy and Ghaar [Nazmy 1992] stated that neglecting the spatial variabilityof the earthquake strike may underestimate the response in cable-stayed bridges,but the level of error depends on each particular case of study, specically aspectslike the foundation soil, main span, stiness and level of hyperstatic response arefound to be important21.

There are several approaches to take into account the eect of multi-componentseismic excitations, depending on the sophistication of the employed analysis. Broadlyspeaking, simplest methods are based on imposed displacements at foundations(which in turn depend on the propagation velocity of the waves vs and distancebetween supports Li) obtaining this way the pseudo-static forces, which are com-bined with inertial results using SRSS22 rule in order to obtain the total response.More sophisticated methodologies take into account the loss of coherency of the

21The inuence of spatial variability in cable-stayed bridges is explored in chapter 3 and, specially,in appendix D.

22Due to the use of SRSS combination rule, simplied static procedures based on imposed dis-placements consider that pseudo-static and inertial responses are fully uncoupled.

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30 Chapter 2. State of the art

seismic excitation, and its delay, through the correction of the acceleration spec-trum by means of statistical theories [Der Kiureghian 1992] [Liang 2006]. Finally,there are one group of procedures which employs conveniently delayed records ormodies their frequency content to take into account the spatial variability of thesignal; a rst approach in this case is to consider the propagation velocity constantalong the soil where the bridge is located, hence assuming that seismic waves arenot dispersive [Abdel-Ghaar 1991a], leading to accelerograms imposed in dier-ent foundations which are simply delayed. Some of these procedures are furtherdiscussed in appendix D, where specic implementation examples in cable-stayedbridges have been also included.

2.2.6 Inuence of tower-deck connection on the seismic response

The typology of the connection between the deck and the towers represents akey factor in the static and dynamic behaviour of cable-stayed bridges [He 2001].Two extreme cases may be considered in the project of these structures: oatingor sti connection. Both design philosophies and their associated seismic eectshave been considered in several research works: [Ye 2002], [Ali 1995], [Nazmy 1992],[Morgenthal 1999], [Fujino 2006] about Yokohama Bay bridge (Japan) and [Liu 2006]about Hangzhou bridge (China), among others.

• Floating connection completely releases the relative movement between thetowers and the deck, which is supported exclusively by means of the cable-system (as well as abutments and intermediate piers in lateral spans). Theincreased structural exibility reduces seismic forces thanks to the lower asso-ciated spectral accelerations. However, the displacements recorded could beinadmissible and impacts between the deck and the tower may occur.

• Sti connection avoids the relative movement between the towers and the deck,reducing the displacements and maximizing seismic forces along the towers.

The project solution must be a compromise between both extreme options,adapted to the probability of earthquake occurrence and the importance of thestructure (which is high for cable-stayed bridges). Seismic energy may be dissipatedby incorporating dampers to the connection (chapter 9), or including other typesof seismic devices to limit the movement under slow actions (e.g. moderate earth-quakes) and to release it under large seismic events [Abdel-Ghaar 1991a]. Betweenboth extreme design possibilities, cable-stayed bridges constructed in seismic areastend to present connections closer to the oating solution, incorporating seismicdevices to control movements and to include additional damping sources, as it maybe observed in tables 2.5 and 2.6 (thoroughly detailed in appendix A).

In light of the seismic response of Rion-Antirion cable-stayed bridge (Greece)obtained by [Morgenthal 1999], sliding of the cars is much more likely if the deckis xed transversely to the towers than if it is released, due to higher accelerationsalong the girder, furthermore, stronger seismic demands are observed in the towers.

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2.2. Seismic behaviour of cable-stayed bridges 31

Abdel Ghaar [Abdel-Ghaar 1991a] veried that rst vibration modes arenotably elongated by releasing the movement of the deck from the towers and abut-ments, due to the increase of the structure exibility, however higher vibration modesremain almost unchanged.

Tuladhar and Dilger [Tuladhar 1999] veried the increase in the sensitivity of theresponse on the wave-train propagation velocity (vs) if the deck is xed to the towers(sti connection). Such variation does not aect the structure if slide or elasticbearings (neoprene supports) are disposed between the deck and the supports.

Khan [Khan 1994] studied a cable-stayed bridge in seismic zone, stating thatexible supports with damping abilities located in deck-tower connections improveits behaviour due to the energy dissipation, the increase of the frequency in criticaltorsional vibration modes and the improved modal participation factors.

2.2.7 Soil-structure interaction

The interaction between the soil and the structure modies the earthquake signalin terms of frequencies, amplitude and duration, and hence considering this eectmay be important, specially if the foundation soil is soft and with characteristicfrequencies close to governing modes of the bridge [Zheng 1995]. The foundationcould be considered rigidly joined to the ground if it is anchored to rocky soil, buthigher modes exciting the mass of the towers close to support levels are decisivelyaected by the foundation exibility, and hence these modes could lead to misleadingresults if a sti connection is supposed between the ground and the towers23.

If there is some inclination between the seismic waves and the foundation, arocking movement in this element is produced, resulting a positive eect in theisolation of seismic vibrations of the cable-stayed bridge [Betti 1993].

It is possible to discretize in the numerical model a large enough region of the sur-rounding soil close to the foundation in order to take into account the soil-structureinteraction, or to employ contour nite elements which prevents the rebound of thewaves in the borders of the model. Another possibility, more practical, is to sub-stitute the eect of the soil by means of springs and dashpots associated with theappropriate degrees of freedom, which presents the diculty of assigning the actualdynamic stiness24 (inuenced by the frequency content of the excitation). Zhengand Takeda [Zheng 1995] observed that springs represent a valid approach to simu-late the eect of surrounding soil if the governing frequency of its movement is low,but it is less accurate if the frequency is higher. Thus, mass-spring models are ade-quate for the denition of the subsoil stiness in exible structures like cable-stayedbridges, with low dominating frequencies, but might underestimate higher-modecontribution, which is representative in terms of seismic forces [Morgenthal 1999].

Abdel-Raheem and Hayashikawa [Abdel-Raheem 2003] veried that nonlinear

23This eect has been observed in the modal analysis included in chapter 4.24The exibility of the subsoil has been modelled only by means of linear elastic springs in the

present thesis, whose stiness responses exclusively to static criteria and is constant along theearthquake, hence neglecting soil-structure interaction (see chapter 3).

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32 Chapter 2. State of the art

subsoil seismic behaviour and its interaction with the super-structure may lead toreduced seismic forces along the towers of cable-stayed bridges. However, Fan et al

[Fan 1994] concluded that such interaction might increase forces and displacementsrecorded in the towers in comparison with the sti-foundation model, specially con-sidering bridges with oating deck-tower connections. Further studies are requiredto overcome the discrepancy among researchers about soil-structure interaction incable-stayed bridges.

Cable-stayed bridges are relatively light-weight exible structures and transmitreduced energy to the foundations during the earthquake [Clough 1993] [Walker 2009],hence it seems justied to neglect this eect in the present study.

2.2.8 Seismic behaviour of multiple-span cable-stayed bridges

Multiple-span cable-stayed bridges normally require the analysis of a signicant por-tion of the bridge (properly extracted from the entire structure) due to its importantlength. Tuladhar and Dilger [Tuladhar 1994] studied the required number of spansand the corresponding boundary conditions to represent the rest of the bridge, be-sides the consideration of the spatial variability of the seismic action, which wasobserved extremely important since unsafe results were obtained if ignored.

Ye et al [Ye 2002] studied the conceptual design of continuous cable-stayedbridges with three towers, considering several ways to improve the eectiveness ofthe cable-system, like diagonal stay-cables (gure 2.6(a)) or intersected cable pairs(gure 2.6(b)). Both solutions reduced the longitudinal displacement of the towers,specially the one in the center of the bridge, but the static response was optimizedin terms of economy and performance in the rst proposal.

(a) Diagonal cables

(b) Intersected cables

Figure 2.6. Types of cable-stiening proposals in multiple-span cable-stayed bridges stud-

ied by Ye et al [Ye 2002].

Despite there is a lack of research about the behaviour of multi-span cable-stayedbridges under earthquake ground motions [Valdebenito 2005], several works havebeen published about the seismic response of Ting Kau bridge (Hong Kong), withthree towers and diagonal stabilization cables (gure 2.6(a)); [Ni 2000], [Ko 2001].Moreover, the seismic behaviour of Rion-Antirion bridge (with four pylons) has beentreated in depth; [Infanti 2004], [Morgenthal 1999], among others.

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2.3. Capacity versus mitigation design 33

2.2.9 Near-eld earthquakes and vertical excitation

Broadly speaking, two major particularities may arise if the distance between thestructure and the epicenter is reduced; (i) pulse-like eects increase notably thespectral acceleration in a narrow range of frequencies (centered in the pulse fre-quency); (ii) the vertical component of the seismic action may be even higher thanthe horizontal one, and both sometimes triplicate the design values. If the bridge islocated close to an active fault, the general recommendation of Spanish code NCSP[NCSP 2007] may be criticized [Valdebenito 2005], since the vertical component issimply assumed to be 70 % of the horizontal action, whilst Eurocode 8 assumes 90%.

The vertical component of the seismic action hardly aects the deck since itis isolated by means of the large number of anchored cables behaving like elasticsupports [Walther 1988]. However, recent research pointed out that seismic axialloads along the towers due to the vertical component of the earthquake excitationmay be important [Valdebenito 2005] (also veried here). Furthermore, the inuenceof velocity pulse-like eects may reduce the eectiveness of supplemental dampingadded by seismic devices.

Eurocode 8 (part 2) [EC8 2005a] requires the use of near-eld spectra (largerthan conventional ones) if the foundation soil is soft, if the bridge is located lessthan 15 km away any active fault, or if fundamental periods are longer than 3seconds; cable-stayed bridges with main spans over 400 m generally present higherfundamental periods (chapter 4), and hence near-eld (or near-fault) eects shouldbe considered25.

2.3 Capacity versus mitigation design

There are basically three philosophies dealing with the seismic design of structures:

• To design the structure with large sections so that it behaves in linear elasticrange during strong earthquakes. This philosophy, not discussed further inthe present thesis, may lead to undesirable designs from the constructive andeconomic points of view if medium to large seismic events could strike thebridge, which currently discourages its application in modern codes.

• To endow certain elements of the structure with sucient ductility, so that thecontrolled damage in these sections (plastic hinges) dissipates seismic energythrough hysteresis cycles. The designer locates the damage caused by theearthquake in the most appropriate areas of the towers, moving the seismicdemand away from the most sensitive elements, like the deck. This option isreferred as capacity design.

• To limit (or bypass) the action of the earthquake in the structure (insteadof resisting its attack as in the previous design) through the use of properly

25No attempt has been made in order to include near-eld eects in this thesis.

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34 Chapter 2. State of the art

arranged seismic devices in the structure, so that these auxiliary elements con-centrate the seismic demand. This novel methodology is known asmitigation

design.

Figure 2.7, adapted from the research of Huber and Medeot [Huber 2005], sum-marizes both seismic design trends and corresponding seismic devices, developedfurther in subsequent sections.

SEISMIC DESIGN

Structural dissipation (capacity) Mitigation (passive or active)

Permanent Temporal

Structural design

(reinforcement)

Seismic

connectorsAux. Devices

Elastomeric

bearings

Isolation +

DampingDampers

Energy

dissipation

Seismic

isolation

Figure 2.7. Possible control strategies in the seismic behaviour of a structure and corre-

sponding seismic devices. Adapted from the research of Huber and Medeot [Huber 2005].

Dealing with capacity design, it is of vital importance to provide enough ductilityin the sections where plastic hinges are expected, in order to achieve the demandingrotation capacity. This is essentially accomplished through detailed design of thelongitudinal and transverse reinforcement in such areas, which normally requiresconnement reinforcement26.

Nowadays, seismic mitigation is the preferred solution in the design of cable-stayed bridges located in earthquake-prone areas27 because towers are intended toremain in elastic range, which is advisable since they assume the main part of theglobal structural resistance (appendix A collects information about major cable-stayed bridges in seismic areas). Seismic devices locate the earthquake demandand, after that, they are much easier to repair (if needed) than large tower sections.Furthermore, important displacements are assumed and expected in such exiblebridges (and the displacements could be not increased with the fundamental periodif it is very large28, see gure 2.8), therefore, the increase in the exibility of thesestructures due to the incorporation of seismic devices is not normally problematic.

Several important cable-stayed bridges with seismic devices constructed in seis-mic areas also allow some structural damage in the towers in order to reduce the

26The present thesis is focused on mitigation design, however conned concrete has been alsoconsidered (see chapter 3).

27Chapter 9 includes the design of cable-stayed bridges with seismic devices.28The spectral displacement may be even reduced when the governing period is enlarged for

specially exible bridges, since the spectral displacement converges to the maximum ground dis-placement dg.

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2.3. Capacity versus mitigation design 35

uncertainty under unexpectedly large earthquakes. This approach is sometimesreferred as `partial isolation' and has been employed in Rion-Antirion and Stonecut-ters bridges (see sections A.1 and A.6 respectively in appendix A.). On the otherhand, Abdel-Ghaar [Abdel-Ghaar 1991a] suggested that the seismic-control sys-tem should be composed of several sub-systems (e.g. limiters and initiators devices),including robust elements in order to ensure the structural integrity.

Hyperstatic structures improve the seismic behaviour compared with their iso-static counterparts because of the larger participation of dierent members in theglobal response. With this in mind, seismic connectors may be established in strate-gic points along the structure allowing the slow movements of dierent parts (as-sociated with creeping, shrinkage, slow movements of the supports, etc.), but con-necting them like a sti link under fast dynamic excitations, hence increasing thelevel of hyperstatism. Neither energy is dissipated by these devices nor is reected,hence they do not belong to mitigation devices, instead dierent elements of thestructure are temporally joined increasing the seismic resistance (capacity) of thestructure [Huber 2005], [Brown 1995], [Morgenthal 1999], [Dyke 2003]. A remark-able example of a cable-stayed structure with this kind of devices is the MemorialBill Emerson bridge (USA) (see appendix A), which incorporates Shock Transmis-sion Units (STU), belonging to the Rigid Connection Devices category in EN 15129[en1 2009], a key normative in the design with seismic devices.

Mitigation design leads to slenderer structures, which mainly remain in elasticrange under strong earthquakes, furthermore cable-stayed bridges present very lowdamping values and hence it is in principle a good idea to add auxiliary sourcesof energy dissipation, however, these advantages are marred by the increased costof the seismic devices and, specially, of their maintenance. Furthermore, there is alack of normative support, design guidelines and experimental practice about bridgeswith seismic devices; only a reduced number of bridges employ these avant-gardesolution (the number is growing rapidly). Despite added seismic devices have beenveried to eectively dissipate seismic energy since decades, not every damper maybe applied to every structure and their applicability should be thoroughly studiedtaking into account the following issues [Villaverde 2009]; (i) performance objective(dampers are less eective controlling loading stages near the collapse since, in thatcase, the structural damage is much more important in the energy balance than theequivalent viscous damping introduced by seismic devices); (ii) secondary eects(which depends on the type of dampers and structure, e.g. the eect of releasingsome degrees of freedom to introduce dampers); (iii) environmental eects (seismicdevices are sensitive to `age-eects' in greater or lower extent); (iv) cost.

Assuming the mentioned shortcomings, the dissipation of seismic energy bymeans of special devices will play a key role in the future of earthquake engineering;the present thesis is focused on this approach. Capacity design is not discussedfurther, but the interested reader may nd valuable information about the imple-mentation of this procedure in cable-stayed bridges elsewhere [Camara 2008].

Below, some key aspects of mitigation design are included, besides informationabout seismic devices and their historical introduction in cable-stayed bridges.

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36 Chapter 2. State of the art

2.4 Mitigation design

Two objectives are pursued when mitigation design is selected; (1) to elongate thefundamental vibration periods of the structure in order to reduce the associatedspectral accelerations; and/or (2) to increase the dissipation by means of addeddamping. Considering a design acceleration spectrum in ADRS29 coordinates, rep-resented in gure 2.8, it is useful to visualize graphically both eects; when theperiod is elongated from Tf,0 to Tf,1, the spectral acceleration is reduced eectivelyand the spectral displacement is maintained constant if the vibration period is largerthan TD = 2 s, which is the period marking in Eurocode 8 [EC8 2004] the start ofthe displacement-sensitive area of the spectrum, i.e. the constant displacement seg-ment (the transverse fundamental period of cable-stayed bridges is typically above2 s if the main span is larger than 400 m, chapter 4), however, if the vibration pe-riod is slightly below TD, its increment supposes large displacements and moderateacceleration reductions (which may be the undesirable case in important transversevibration modes of cable-stayed bridges). On the other hand, increasing the damp-ing always reduces the seismic response, regardless of the type of result consideredand the governing vibration period, but the eectiveness is lower with increments be-yond certain damping levels [Priestley 1996] [Villaverde 2009], as it is schematicallyillustrated in gure 2.8, varying from ξ = 4, 15 and 30 %.

Figure 2.8.Mitigation design objectives ((1) and (2)) presented by means of the Eurocode

8 [EC8 2004] seismic action in ADRS coordinates. (1) Fundamental period elongation. (2)

Damping increment.

In this direction, the incorporation of seismic devices tries to achieve one of theseeects, or both at the same time, resulting the following classication focused onthe way the seismic energy in the main structure is reduced:

29A plot in ADRS coordinates represents the acceleration spectrum in terms of the spectraldisplacements and accelerations.

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2.4. Mitigation design 37

• Base isolation devices increase the fundamental period of the structure:strictly speaking, only laminated Rubber Bearings (RB) meet purely this re-quirement since their energy dissipation capability is reduced, they just reectpart of the seismic energy.

• Dampers transform the energy in heat (e.g. yielding metallic dampers orviscous uid dampers) or in vibrational energy (e.g. tuned mass or liquiddampers).

• Base isolation plus damping. Usually both strategies are employed inthe same device, like High-Damping Rubber bearings (HDR), Lead RubberBearings (LRB, where the lead core introduces the dissipation), FrictionalPendulum Systems (FPS), etc. (see table 2.4). Nonetheless, the current trendin seismic mitigation design is to combine both isolation and damping in dif-ferent devices located in several positions along the bridge; one of the mostcommonly used scheme is the incorporation of classical rubber bearings (RB)and viscous uid dampers (VD), conferring VD the required energy dissipationand restoring forces.

Apart from aforementioned dampers or isolators, there are auxiliary devicessometimes referred as limiters and initiators; limiters are employed if limiting thedisplacement of the deck is required in case the main seismic device is too exibleand this element could strike the towers; initiators (also called fuse restrainers30)prevent the action of seismic devices if the structure is subjected to wind or moderateground motions. Rion Antirion cable-stayed bridge (Greece) includes fuse restrainers(appendix A).

Regarding the energy required by the seismic device in order to face the earth-quake, there are two major trends in mitigation design; passive and active control:

• Passive devices do not require any external added energy to help the struc-ture when the earthquake strikes, transforming the introduced seismic energy.They are the most commonly used seismic devices due to their reliability,robustness, simplicity and economy.

• Active devices, in contrast, require external energy to actuate. The earth-quake action is detected by means of several actuators disposed in criticalpoints along the structure, which include detection and classication algo-rithms involving real-time signal processing. External energy is sent to theseismic devices when large enough motions are detected, which produce anopposite movement, or any other benecial action, to control the seismic re-sponse of the bridge. In other words, active devices deliberate energy to thestructural system in the opposite sense to that deliberated by the distur-bances. Sometimes important amounts of energy are required, which couldbe not available during the earthquake due to a power outage for example.Furthermore, the introduced external energy could destabilize the structure.

30Fuse restrainers are employed in chapter 9.

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38 Chapter 2. State of the art

• In order to solve the shortcomings in active control, semi-active devices

are currently studied in many research works, specially magnetorheologicaldampers (see section 2.4.3). This promising strategy also requires actuatorsin order to detect the seismic event, however, semi-active devices do not addany external energy and thus structural stability is not compromised; theyjust need energy to change their own behaviour (e.g. modifying their force-displacement response). The energy requirements are less demanding and theirreliability is higher than the one associated with active devices [Jung 2003].Notwithstanding, the use of seismic devices is yet in an experimental stageand there are very few examples implemented in constructed bridges (some ofthese are included in section 2.4.3). Another possibility is to combine activeand passive seismic devices resulting the hybrid control, more robust thanactive control and more eective than passive one [Park 2003b].

2.4.1 Energy-based design

Considering the energy as a design criterion is conceptually very appealing, speciallydealing with structures that include seismic devices, since the designer is concernednot so much with resistance of lateral loads, but rather with energy dissipation;energy management is a key to the successful design in bridges with seismic devices,being the main objective to minimize the hysteretic energy dissipated by the struc-ture itself [Soong 1997]. As it has been already discussed, this may be accomplishedwith base isolation and/or adding dampers.

Whilst the energy concept does not currently provide the basis for aseismic designcodes, there is a considerable body of knowledge on the topic [Soong 1997]. Housner[Housner 1956] was the rst to propose an energy-based philosophy in the middleof the twentieth century, but this approach was largely ignored a number of yearsand, instead, ductility displacement philosophies proposed by Veletsos and Newmark[Veletsos 1960] were used to construct inelastic spectra in limit-state design method-ologies. More recently, there has been a resurgence of interest in energy-based design;Uang and Bertero [Uang 1988] introduced an alternative denition of the energy bal-ance with the input energy based on absolute displacements, however they veriedthat the classical expressions based on relative displacements (equations (8.3) and(8.4)) are more appropriate for long-period structures like cable-stayed bridges.

The dependence of the seismic damage on the load path was also veried byUang and Bertero; the damage is not only dened by the magnitude of dissipatedenergy, which is a serious shortcoming for the use of energy concepts for limit designof traditional structures. However, in structures with seismic devices, the energyapproach is much more appropriate since the emphasis is directly on seismic dissi-pation31 [Soong 1997].

With the strong advent of structural dampers in the last ten years, many re-searchers select performance criteria based on the energy balance. In this sense, the

31Chapters 8 and 9 employ energy-design concepts to optimize the tower shape and the passivecontrol system.

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2.4. Mitigation design 39

performance index proposed by Pinkaew et al [Pinkaew 2003] is remarkable, whichtakes into account both displacements and hysteresis energy dissipated by the mainstructure. Among others, Xu et al [Xu 2007] also considered energy principles intheir study about passive seismic devices.

2.4.2 Passive dampers

The purpose of this section is to describe the basic concepts behind the supplemen-tal passive energy dissipation technology in structures. Many devices have beensuggested over the past few years for this purpose, however most of the researchworks and implementation in constructed structures have been concentrated onfour of these passive devices [Villaverde 2009]: (a) friction dampers, (b) viscoelasticdampers, (c) viscous uid dampers and (d) yielding metallic dampers (hystereticdampers). This section is devoted to yielding metallic dampers and viscous uiddampers (which will be considered in chapter 9) due to their broad application, butrst, other devices are briey discussed for completeness and a short summary ofkey research works about passive dampers in cable-stayed bridges is included.

Friction dampers (FD) dissipate kinetic energy through the sliding of surfaceswith high friction coecients. More than 30 years of experience and the large num-ber of buildings equipped with these devices guarantee their incorporation. How-ever the diculty to maintain their properties over long time intervals is also wellknown due to the corrosion of metallic surfaces and relaxation of the normal load[Soong 1997] [Villaverde 2009].

Viscoelastic dampers (VE) use the phase-lag between the shear strain and thecorresponding stress in viscoelastic materials as the way to dissipate energy. Thesedampers have been successfully employed in the stabilization of buildings subjectedto wind actions and present good re-centering capabilities, but the required levelof dissipated energy is higher for seismic applications. Furthermore, the energycontent of the earthquakes is usually spread over a wider range of frequencies and theremarkable dependency of VE on the frequency content could be clearly a problem,without forgetting the inuence of ambient temperature [Soong 1997].

Both friction and viscoelastic dampers are strongly aected by `age-problems',specially installed in bridges, where environmental conditions are adverse (morethan in the case of buildings), and the lack of condence on the response of thesedevices whenever the earthquake strikes the structure has lead the author to ignoretheir use in the present doctoral thesis.

Tuned dampers, either with mass (TMD) or uids (TLD) are very appealingwhen controlling the dynamic response of a structure, and the number of bridgesequipped with these devices seems to be increased considerably in the near future.With this control strategy, some structural vibrational energy is transferred to tuneddampers, which simply oscillate in an elastic way (linear or nonlinear). Figure2.9 includes the basic idea of a vibration absorber (TMD) subjected to harmonicexcitations, introduced by Frahm [Frahm 1909] more than one century ago, anddeveloped further by Den Hartog [Den Hartog 1956] and many others subsequently.

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40 Chapter 2. State of the art

Figure 2.9.Model of SDOF structure subjected to harmonic excitations with an attached

Tuned Mass Damper (TMD), besides the main parameters controlling the response.

The Tuned Liquid Damper (TLD) operates in the same principle as the TMD.Basically there are two types of TLD; dampers based on sloshing liquids, and columnTLD (TLCD, currently the focus of numerous research works). Figure 2.10 includesan example of damper which combines both responses depending on the direction ofthe excitation, which has been recently developed by Lee et al [Sung-Kyung 2011].TLCD sometimes includes orices to provide added dissipation; these solutions withmultiple dampers have been recently proposed in order to control the seismic re-sponse of exible structures [Gosh 2011].

(a) Scheme of the absorber (b) Laboratory test

Figure 2.10. Scheme and laboratory model of a bidirectional TLCSD [Sung-Kyung 2011].

Nonetheless, the broad frequency content of the earthquake excitation (far dif-ferent than harmonic loads), besides the detuning due to inelastic demand andconsequent vibration period elongation, may be detrimental for tuned damper per-formance, being key drawbacks that question their use as passive systems for theseismic control of a structure under large earthquakes. However, several research

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2.4. Mitigation design 41

works have suggested that strong ground shaking and signicant modications ofnatural periods do not lead to inadmissible reductions of the eectiveness in TMD[Pinkaew 2003] [Sgobba 2010]. In principle, structures governed by one vibrationmode are good candidates to be controlled by tuned dampers properly adjusted tothis frequency, however, cable-stayed bridges involve complex modal couplings andseveral modes contribute signicantly to their response (see section 6.4.4.1).

Numerical considerations

Structures with added dampers present inherent nonlinear behaviour which may beevident even under moderate earthquakes due to several reasons [Villaverde 2009];(1) increases and redistributions of damping between several modes are produced;(2) uneven distribution of high damping sources which precludes the considerationof classical damping, and hence modal analysis with complex eigenmodes or nonlin-ear dynamic analysis (NL-RHA) should be selected (chapter 6); (3) most dampersexhibit nonlinear constitutive behaviour as it will be presented next, in such casesthe nonlinearities must be explicitly included in the nal design.

Research about passive dampers in cable-stayed bridges

First works about passive isolation in cable-stayed bridges correspond to Ali andAbdel-Ghaar [Ali 1991], which considered the installation of Lead Rubber Bearings(LRB) between the deck and the supports. Such devices increase vibration periodsas it has been explained previously, and despite these structures are very exibleby themselves, the eective reduction of the seismic demand due to the energydissipated by hysteresis loops was veried, besides an increment of displacements.The same research stated the reduction of passive devices eciency of as long as themain span is increased32.

Abdel-Raheem and Hayashikawa [Abdel-Raheem 2003] proposed an eective andeconomic seismic protection by means of viscoelastic isolating devices and alsothrough hysteresis loops in transverse struts linking both sides of `H'-shaped towers,verifying the elastic behaviour of the main structural parts of the towers.

Recently, the concern about the performance of passive energy dissipation sys-tems during near-eld ground motions with velocity pulse-like eects has promotedseveral research works; [Xu 2007], [Valdebenito 2009].

2.4.2.1 Yielding Metallic Dampers (MD)

Yielding Metallic Dampers (MD) take advantage of the hysteretic properties ofmetals deformed in the post-elastic range in order to dissipate energy. When thestructure is subjected to moderate earthquakes, these devices act as sti memberswhich help to reduce structural deformations, however, plastic excursions in thesedampers arise under strong ground motions if properly designed and located alongthe bridge.

32This result has been observed in the present work (chapter 9).

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42 Chapter 2. State of the art

The idea of using yielding metallic dampers began more than thirty years agowith the conceptual and experimental work by Kelly et al [Kelly 1972] and Skinneret al [Skinner 1975]. During ensuing years, considerable progress has been madein the development of these devices; as a result of this ongoing research program,several products have been developed and equipped in both new and retrot projectsworldwide [Soong 1997]. Usually, the metal employed is mild steel, although thereare devices with lead or other metal alloys. Tables 2.1 and 2.2 summarize some ofthe most relevant yielding dampers found by the author reviewing the bibliographyavailable to date (most of them rst designed for building applications).

Triangular and X-shaped dampers present constant curvature along their heightdue to the specic shape and bending moment distribution associated with theboundary conditions in each case (cantilever and double-cantilever respectively).Therefore, yielding of the plates is uniformly distributed along their length, prevent-ing the concentration of plastic deformation and hence maximizing their stabilityand energy dissipation. X-shaped devices employ bolts at both connections and thesensitivity of damper stiness to tightness of these bolts has been observed. On theother hand, triangular metallic dampers are welded only in one side of the platesand hence their behaviour seems to be more robust and recommendable for bridgeengineering applications33 (see gure 2.14). Moreover, the axial load in triangu-lar MD is negligible due to the slotted-pin connection at the apex, favouring thestability.

Despite not generally considered as yielding metallic damper, Buckling-RestrainedBraces (BRB) share the same energy dissipation principle, based on axial hystereticbehaviour. This damper34, studied and employed in Japan and other countries formore than 30 years, consists of a core steel plate with cruciform, rectangular oranother cross-section in a concrete lled steel tube (other possibilities like sand orno lling material has been used inside the tube). Figures 2.11(a) and 2.11(c) il-lustrate the main parts of this device and one application case respectively. Theaxial load transfer between metallic core and concrete is eliminated or reduced bymeans of special coatings or layers of rubber or silicone to de-bond the contact. Themission of the concrete-lled tube is to prevent the buckling of the steel core whenit is under compression, obtaining the same tensile and compressive yielding loadsand hence maximizing the energy dissipated by means of hysteresis loops (gure2.11(b)) [Uang 2004] [Villaverde 2009]. In addition to their large energy dissipationand cost-eectiveness, BRB are capable of furnishing stiness requirements to meetstructural drift limits, furthermore, excellent low-cycle fatigue life has been observedin these devices.

Another type of yielding damper are the Shear Links (SL), widely employed sinceearly 1980's in building structures forming eccentrically braced frames [Roeder 1978].The objective now is to concentrate the plastic deformation in short metallic links,properly stiened, in order to obtain a ductile yielding response dominated by

33Triangular MD (also referred as TADAS in the literature) have been considered in the numericalstudies of chapter 9.

34BRB has been considered in the proposed towers of cable-stayed bridges in chapter 9.

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2.4. Mitigation design 43

Name Scheme Main features References

Triangularsteel plates Cantilever triangular [Tsai 1993](TADAS) steel plates [Parulekar 2009]

X-shaped Double cantilever [Bergman 1987]steel plates `X'-shaped [Tena-Colunga 1997](ADAS) steel plates [Bakre 2006]

Honeycomb In-plane behaviour [Kobori 1992]damper Hexagonal holes

[Chan 2008]Steel Slit In-plane behaviour [Oh 2009]

Damper (SSD) Longitudinal slits [Ghabraie 2010]

Dual FunctionMetallic Damper Additional stiness [Li 2007]

(DFMD) In-plane behaviour

Shear panel Stiened panel [Chen 2005]damper In-plane behaviour [Mazzolani 2008]

Sliding Bearings (SB)C-shaped Bi-directional response [Priestley 1996]damper (crescent moon) [Battaini 2007]

Pipe Damper Concrete inll [Maleki 2010a](optional) [Maleki 2010b]

Torsional responseTorsional beam Rangitikei bridge [Skinner 1975]

damper (New Zealand)

Torsional tube Torsional response [Franco 2010]damper (without bending)

Table 2.1. Summary of the most relevant yielding Metallic Dampers (MD) in the literature.

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44 Chapter 2. State of the art

Name Scheme Main features References

Tube-in-Tube Bracing system [Benavent-Climent 2010]Damper (TTD)

Lead forcedLead Extrusion through [Robinson 1976]Damper (LED) an orice [Skinner 1993]

No fatigue

Table 2.2. Summary of the most relevant yielding Metallic Dampers (MD) in the literature

(continued).

shear forces. Recently, Berman and Bruneau [Berman 2008] published laboratorytest results about shear links with tubular sections, being recommended in bridgeapplications to prevent lateral torsional buckling. The seismic response of thenew San Francisco - Oakland Bay cable-supported bridge (USA) relies on multi-ple shear links between both vertical shafts conforming the pylon; [Dusicka 2002],[McDaniel 2003], [McDaniel 2005], [Uang 2005], among others. Figure 2.12 depictsthe solution adopted in this structure and the laboratory tests conducted. The eec-tiveness of energy dissipation by means of hysteresis loops in such links was evident,specially the ones located in the intermediate part of the tower height35. The newRichmond - San Rafael bridge (USA) also includes these links, and the future Ger-ald Desmond cable-stayed bridge (USA) will present designs for the tower and shearlinks similar to those adopted in Oakland bridge. These dampers are integrated inthe main structure and, therefore, they are not clearly identied as auxiliary.

Low-cycle fatigue in yielding metallic dampers

It is imperative to consider the durability of any damper design since the sur-vivability of the structure strongly depends on these elements and, therefore, theirfailure should be considered in the design by means of phenomenological theories.It is well known that metals subjected to a limited number of excursions (e.g. <1000) well into the inelastic range could experience severe problems, phenomenoncalled `low-cycle fatigue'. Such mechanism involve the growth and interconnectionof micro-cracks that eventually cause the failure [Soong 1997].

Priestley [Priestley 1996], in light of experimental low-fatigue results obtained byTyler [Tyler 1978] on typical yielding dampers, recommended the limitation of themaximum strain range during ground motions; normally several design earthquakesand one extreme seismic event should be resisted without problems, and thereforetypical values of maximum strain amplitudes for mild steel dampers fall in the rangeof 3 % for the design earthquake and 5 % for the extreme earthquake, verifying withthis limits a suciently large number of cycles to failure, Nf , in gure 2.13.

35Chapter 9 delves into the seismic control of cable-stayed bridges with shear links.

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2.4. Mitigation design 45

Encasing concrete

Yieldingsteel core

Steel tube

(a) 3D viewDuctile Design of Steel Structures

(b) Force-deformation behaviour [Uang 2004]

Encasing concrete

Yieldingsteel core

Steel tube

Restrained yielding segment

Encasing concrete

Restrained nonyielding segment (enlarged cross-section)

Unrestrained nonyieldingsegment (extension forbolted connection)

(c) Elevation

Encasing concrete

Yieldingsteel core

Steel tube

Restrained yielding segment

Encasing concrete

Restrained nonyieldingsegment (enlarged cross-section)

Unrestrained nonyieldingsegment (extension forbolted connection)

(d) Building with BRB frames

Figure 2.11. Conguration of Buckling-Restrained Braces (BRB).

A more thorough understanding of the phenomenon is required due to the im-portance of metallic dampers in the global structural response. Con-Manson tra-ditional approach to low-cycle fatigue is valid for constant amplitude strain cycling.However, the strain cycles due to seismic excitation are clearly not constant interms of amplitude; the Palmgren-Miner's cumulative damage rule is the simplestapproach to adapt the information provided by Con-Manson's rule to the fatiguelife under variable cycling36.

The inuence in the order of the cycles, neglected in the preceding approach, hasbeen experimentally observed [Soong 1997]. There are proposals in order to take

36Chapter 9 includes a practical implementation of the Palmgren-Miner's rule.

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46 Chapter 2. State of the art

(a) Schematic plan and elevation of the towershafts and shear links [Uang 2005]

(b) Laboratory test of the shear links[Dusicka 2004]

Figure 2.12. Positioning and testing of shear links in San Francisco - Oakland Bay cable-

stayed bridge (USA).

10 100 1000 10000

1

2

3

4

5

6

Figure 2.13. Fatigue life for typical yielding Metallic Dampers (MD). Adapted from the

work of Tyler [Tyler 1978].

into account the load-path dependency of low-cycle fatigue but they are beyondthe scope of this thesis, considering accurate enough the previous Palmgren-Miner'sprocedure.

Structural implementation of yielding metallic dampers

The incorporation of MD along the structure obviously depends on the specicdevice conguration presented in tables 2.1 and 2.2. Triangular and `X'-shapeddampers in building applications require sti support framing to ensure that thedisplacement is maximized, being approximately equal to the inter-story drift (see

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2.4. Mitigation design 47

gure 2.14), such framing is likely more expensive than the damper it supports37

[Whittaker 2004]. Figure 2.14 presents the typical conguration of triangular metal-lic dampers in building frames.

Free verticalmovement

Chevron braces

Figure 2.14. Typical implementation of triangular yielding metallic dampers in building

frames.

Buckling-Restrained Braces avoid the sti support framing required in otherhysteretic dampers and are usually disposed in chevron38 conguration (see gure2.11(d)), reducing the large out-of balance vertical forces typically associated withother types of concentrically braced frames [Bruneau 1998], and allowing doors inbuilding applications. The connection between BRB and the gusset plate of theframe is generally performed by means of bolts and splice plates (see gure 2.15(a)),however some proprietary solutions include pinned brace-to-gusset connection (seegure 2.15(b)) which has the following advantages; (i) it completely isolates thebrace from any moment or shear that could be transmitted as a result of the lateraldrift [Uang 2004]; (ii) the restrained yielding segment (gure 2.11(c)) is longer, hencereducing the axial strain; (iii) the collar assembly of the connection helps preventingthe out-of-plane buckling of the core; (iv) allows the use of ganging multiple bracestogether, increasing the capacity of the conguration.

Niihara et al [Niihara 1994] studied the consequences in the seismic response ofcable-stayed bridges when yielding metallic dampers are incorporated, and severaldeck-tower connections are designed; the reduction of seismic displacements selectingharp cable-system arrangement was veried.

Specic advantages and disadvantages of yielding metallic dampers

Yielding dampers are perhaps the most economic and robust seismic devices.Another clear advantage over other solutions is their virtually insensitive propertiesto environmental actions and age eects. A signicant portion of energy dissipatedby hysteresis loops in these dampers will be converted into heat, however, for rea-sonable devices, a signicant change in the mechanical properties due to the increaseof temperature is not expected (which instead may be important in lead dampers,LED) [Soong 1997].

37However, triangular MD are employed connecting the deck and the lower strut (or abutments)of cable-stayed bridges in chapter 9, therefore it is neither necessary nor possible to design a stiframe to support the damper.

38Chevron bracing scheme is sometimes referred as inverted `V' conguration.

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48 Chapter 2. State of the art

Bolts

Gusset plate

UnrestrainedNon-yielding segment

(a) Bolted connection

UnrestrainedNon-yielding segment

Gusset plate

Pin connection

(b) Pinned connection

Figure 2.15. Buckling-Restrained Braces (BRB): frame connection detail.

On the other hand, they have also associated disadvantages; (i) the possibility ofpremature fatigue failure; (ii) if not properly controlled, steels commonly employedto fabricate the MD may have a wide range of yield strength, which introducesuncertainties; (iii) they may leave the structure with signicant permanent oset af-ter an earthquake (they have no re-centering mechanism by themselves); (iv) thesedevices could generate high-frequency vibrations due to the sudden change in theglobal structure stiness after the damper yielding; (v) the response of the structuremight be worst with yielding metallic dampers than without them for specic con-gurations and earthquakes, being necessary a complete nonlinear dynamic studyof several design possibilities.

2.4.2.2 Viscous uid Dampers (VD)

Modern Viscous uid Dampers (VD) dissipate energy by pushing uid throughorices, transforming the external work into heat and consequently elevating thetemperature of the viscous uid and its mechanical parts (depicted in gure 2.16),nally, the heat is transferred to the environment by radiation heat transfer. Firstviscous dampers were used late XIX century for military purposes and from the1920s in the automobile industry, their incorporation to civil structures as the wayto control their response under earthquakes or wind actions began in the 1990s.Since then, viscous dampers have been employed for the seismic protection of alarge number of buildings and bridges throughout the world, being remarkable theinstallation of mega-brace 20 m long dampers (including braces) in a building inMexico (2003) [Villaverde 2009].

A thorough description of VD architecture is beyond the scope of this work,which is instead interested in their macroscopic behaviour and implementation is-sues in structures, furthermore, most of viscous dampers intended for structuralapplications are proprietary and generally the engineer does not interact directlyin the design of the device, being their mechanical properties and seal guaranteedduring a service life of at least 25 years, and eective operation of the device ensured

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2.4. Mitigation design 49

Figure 2.16.Mechanical parts of a viscous uid damper (VD). Courtesy of Alga S.p.A.

over a wide range of temperatures from -40oC to 70oC. Practical descriptions of VDare presented elsewhere [Soong 1997] [Villaverde 2009].

Viscous damper behaviour and macroscopic modelling

The reaction force of the viscous uid damper may be expressed as:

Fd = Cd |x|αd sign (x) (2.9)

Where Cd is the damping coecient, x the relative velocity and αd the velocityexponent (between 0.15 and 2).

Dierent values of the velocity exponent are obtained with specially shaped ori-ces to attain dierent ow characteristics; a value of αd = 2 results using cylindrical(Bernoullian) orices; αd = 1 corresponds to the classical linear damping39; damperswith αd = 0.5, and generally with low values of this exponent, are useful for applica-tions involving extremely high velocity shocks in order to obtain moderate damperreactions [Villaverde 2009] [Martinez-Rodrigo 2003] [Valdebenito 2009].

Low velocity exponents (αd) are recommended in cable-stayed bridges to obtaingreat energy dissipation and nearly constant reaction forces for a wide range of veloc-ities [Infanti 2004]. Three remarks strengthen this conclusion; (i) the increase of thedissipated energy by reducing the nonlinear exponent may be theoretically demon-strated (a SDOF system subjected to a sinusoidal excitation dissipates 31 % moreenergy with αd = 0.5 than considering αd = 2, see chapter 9); (ii) low exponents leadto reductions in the amplitude per cycle of motion and, consequently, the improve-ment of the overall performance [Villaverde 2009]; (iii) the temperature dependenceof modern seismic viscous dampers with special orices and low values of αd is lessimportant [Soong 1997]. In this direction, Valdebenito [Valdebenito 2009] observedthat reduced exponents always improve the seismic response of cable-stayed bridgesunder strong ground motions in comparison with the solution incorporating lineardampers, due to the large velocities recorded in such long-period structures, how-ever the nonlinearity of the damper has a reduced eect in the peak response of thestructures dominated by vibration periods in the velocity-sensitive spectrum region,similar results were obtained by Lin and Chopra [Lin 2002]. On the other hand,Martinez-Rodrigo and Romero [Martinez-Rodrigo 2003] observed that reduced val-ues of αd may lead to poor structural performances and considerable damage in

39If αd = 1, a modal dynamic analysis with complex eigenvalues may be conducted.

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50 Chapter 2. State of the art

buildings with chevron frames, recommending values of the velocity exponent closeto the unity. Nonetheless, Dicleli and Mehta [Dicleli 2007] observed in analogousbuilding bracing structures that VD are eective under earthquake excitation, beingthe sensitivity to the characteristics of the record especially reduced with low αd.

Large velocity exponents, i.e. αd = 2, may be desired in order to allow slowmovements (due to thermal eects, creeping, shrinkage, etc.) without the intro-duction of important reactions, but locking-up under fast dynamic actions (due toimportant earthquakes or winds). These specic viscous dampers are referred asShock Transmission Units (STU) or buers and are not designed to dissipate largeenergy quantities like dampers with low values of αd, instead these devices link tem-porary several parts of the structure, normally in longitudinal direction, increasingits resistance. As it has been previously mentioned, STU are considered part ofcapacity design and therefore they are not going to be discussed here.

On the other hand, Valdebenito [Valdebenito 2009] veried the important roleof the damping coecient (Cd) in the seismic behaviour of cable-stayed bridges;low values of this parameter lead to important forces transmitted to the structure.Yamaguchi and Furukawa [Yamaguchi 2004] considered the incorporation of viscousdampers in deck-tower and deck-pier connections of Yokohama Bay bridge (Japan),observing that there are one optimum value of the damper stiness which minimizesthe recorded shear at towers foundations. The same conclusion was published inthe work of Soneji and Jangid [Soneji 2007], where dierent passive control schemeswere proposed in order to improve the seismic response of Quincy Bay - View cable-stayed bridge (USA), employing viscous dampers in longitudinal direction to dissi-pate energy, along with the isolation obtained either by means of Friction PendulumSystems (FPS), High-Damping Rubber Bearings (HDR) or Lead Rubber Bearings,concluding that αd = 0.4 is in the studied case the optimum velocity exponent indamper constitutive behaviour.

Most of the viscous uid dampers installed in cable-stayed bridges include apressure control system to provide the device with a extreme value of its reactionFd,max, limiting this way the damage in the structural elements which receive thisforce.

In order to give a reference to the reader, the parameters of viscous dampersemployed in some major cable-stayed bridges are presented in table 2.3.

Structural implementation of viscous uid dampers

Small relative displacements may be not enough to create a uid ow whichdissipates an appreciable amount of energy, and consequently dampers have beendiscouraged by many authors and guidelines [fem 1994] in the control of sti frames,like cable-stayed bridge towers, under ground motions. The most reasonable positionfor viscous dampers is, in principle, connecting the deck and the abutments ortowers, where the extreme dierence in the exibilities of both members maximizesthe relative displacements.

The limitation of VD to control large concrete frames is valid for conventionaldamper congurations like Chevron or Diagonal solutions (gure 2.17 left and center

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2.4. Mitigation design 51

Bridge nameMain

αd Cd [kN/(m/s)αd ] Fd,max [kN]span [m]

Basarab (Romania, 2011) 150∗ 0.20 2800 4× 2500

Rion-Antirion (Greece, 2004) 560 0.15 3000 4× 3500

Sutong (China, 2008) 1088 0.40 3750 4× 10000

Tsurumi (Japan, 1994) 510 2∗∗ 10000 2× 2000

Stonecutters (China, 2009) 1018 2∗∗ Not found 4× 8000

Table 2.3. Parameters of Viscous uid Dampers (VD) equipped in several constructed

cable-stayed bridges at each deck-tower connection. (∗) denotes that the bridge has only

one tower and the main span should be doubled for comparison purposes. (∗∗) refers to

Shock Transmission Unit (STU, dampers not intended for large energy dissipation).

respectively), nonetheless, relatively recent research of the university at Bualo(USA) has sought to expand the utility of VD to sti frames or wind applications,magnifying the relative damper displacements by means of special congurations,thus reducing the cost of the devices due to the increase in the stroke and reductionof damper volume required [Constantinou 2000] [Sigaher 2003]. In this sense, Mainand Krenk [Main 2005] veried40 that the maximum modal damping is introducedby VD if modal relative displacements are maximized, hence Cd, αd and the positionof the devices along the structure should be selected in order to facilitate suchmovements.

The magnication factor fm is an useful parameter usually dened to study theeectiveness of the damper conguration; let x be the relative displacement of theupper point where the device is attached in comparison with the lower one (seegure 2.17), and xd the relative displacement of the damper along its axis; dierentdamper assemblies yield dierent magnication factors relating both displacements;

fm =xdx

(2.10)

So far, most of the seismic devices installed in constructed cable-stayed bridges(e.g. Rion-Antirion, Stonecutters, Sutong, among many others) have been disposedbetween the deck and the towers, intermediate piers or abutments in diagonal41

(fm ≈ 1) or perfectly horizontal (fm = 1) assemblies (see details in appendix A).Valdebenito [Valdebenito 2009] observed that the optimal layout of VD equippedin cable-stayed bridges is achieved with longitudinal dampers at the connections ofthe deck with the abutments and towers, specially under near-fault records; in theopinion of this author, transverse VD at deck-tower connection are not very ecientin controlling the response because the movement of the towers reduce the relative

40It was veried by means of an approximate solution for the complex eigenfrequencies of adiscrete system with several viscous dampers.

41If dampers are located below the deck, connecting this element with the legs of the towerin diagonal assembly, small values of the angle between the horizontal line and the damper (θ)are dened because the vertical space is typically reduced due to the transverse strut, obtainingmagnication factors close to the unity (see gure 2.17, center).

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52 Chapter 2. State of the art

displacements42, which contrasts with the transverse dampers layout in some majorconstructed cable-stayed bridges, like Rion Antirion (Greece).

In contrast with the familiar diagonal and chevron brace congurations, whichpresent moderate values of the magnication factor, special assemblies eectivelymaximize the relative displacements by increasing fm and, specially, the equivalentdamping (which depends on the square of fm if linear VD are employed). The mosteective conguration seems to be the upper toggle assembly, resulting a dampingfactor approximately ten times higher than the one obtained with chevron bracing[Sigaher 2003] [Whittaker 2004]. Figure 2.17 presents schematically several dampercongurations, besides their magnication factors.

Figure 2.17. Several congurations for the installation of dampers (MD or VD) in a frame.

Despite the special solutions for the installation of dampers in sti frames havebeen clearly developed for building structures, they may be applicable to the sticoncrete frame conformed transversely by the lateral legs and transverse struts ofthe towers in cable-stayed bridges. Vader and McDaniel [Vader 2004] studied theincorporation of several dampers with dierent congurations (diagonal, chevronand toggle brace), substituting the shear links between the shafts of Oakland cable-supported bridge, observing improved seismic responses in transverse directions withseveral dampers arranged in toggle congurations.

Specic advantages and disadvantages of viscous uid dampers

Viscous dampers are the unique seismic devices which simultaneously reduceboth displacements and stresses of the structure under ground motions; this is sobecause the velocity is inherently out-of-phase with the deformations that producethe stresses43. With viscous devices the peak stress of the structure and the extremereaction introduced by the damper never occurs at the same instant, which is indeedthe case in other seismic devices, that may reduce the displacement response at theexpense of increasing its stresses. Viscous dampers have been experimentally proved

42However, the eciency of VD between the deck and the tower in transverse direction has beenobserved in chapter 9, even in large models with main spans of 400 m.

43Considering the seismic response of a cantilever with VD attached to its top section, thefollowing is observed; when the deformation and stresses are maximum the velocity and hence thereaction force of the damper is zero, analogously, when the deformation is zero the velocity anddamper reaction is maximum.

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2.4. Mitigation design 53

to be an eective way to enhance the seismic response of the structure, in additionthey are easy to install due to their relatively compact size [Villaverde 2009].

However, they have several disadvantages; (i) designers must be concerned withthe long-term durability of their seals and the cost of maintenance; (ii) anotherconcern is the thermal resistance of VD under very high temperatures (they maydissipate more than 100 MJ during a strong seismic event), despite commercialbrands defend that the duration of the earthquake is not long enough to increasethe temperature of these devices up to dangerous limits; (iii) they require largerelative displacements to eectively dissipate energy, which discourages their usein sti structures like reinforced concrete frames, unless the aforementioned specialcongurations are employed, which complicates the installation of the dampers; (iv)they make the structure more expensive.

2.4.3 Active and semi-active devices in cable-stayed bridges

Early analytic and experimental studies in cable-stayed bridges with these inno-vative devices were conducted by Schemmann and Smith [Schemmann 1996]. Thesubstantial reduction of the extreme seismic forces was veried, observing also thatthe optimal position of the actuators (which detect the strike of the earthquake) isclose to the center of the main span. Furthermore, it was concluded that in order toreduce the displacements, it is only necessary to control the rst vibration modes,however higher modes need to be also controlled by the seismic device if it is desiredto reduce eectively the seismic forces, which again states the importance of suchhigh frequencies in the seismic response of cable-stayed bridges.

Research about hybrid control schemes including passive Lead Rubber Bearings(LRB), reducing the subsequent increase of displacements by means of active orsemi-active devices, is being developed. Example of such control systems are theproposals of Park et al [Park 2003a] [Park 2003b] about Memorial Bill Emersonbridge (USA), more information is included in appendix A (section A.2).

Li et al [Li 2001] analyzed the incorporation of Active Mass Dampers (AMD) incable-stayed bridges, observing their eectiveness in the reduction of seismic forces,specially since it was found enough to control the rst vibration mode in order toreduce lateral displacements signicantly. These devices modify the properties ofconventional Tuned Mass Dampers (TMD) taking into account the properties of theseismic excitation in real-time.

Currently, the use of Magnetorheological Dampers (MRD) in structures is veryattractive. When subjected to a magnetic eld, magnetorheological uid greatlyincreases its apparent viscosity, to the point of becoming a viscoelastic solid. Theyield stress of the uid, when it is activated, can be accurately controlled by mod-ifying a magnetic eld created with a relatively small amount of added externalenergy. Despite the important number of research works recently published (or un-der development) about these devices, already implemented in automobile industry,few constructed bridges incorporate them. Two remarkable examples which includeMRD in the cable-system are located in China; Dongting Lake and Binzhou Yellow

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54 Chapter 2. State of the art

bridges.Iemura and Pradono [Iemura 2003b] [Iemura 2003a] concluded that passive (vis-

cous dampers plus elastic bearings) and specially semi-active devices (variable oriceviscous dampers) could result very eective when controlling the seismic response ofcable-stayed bridges, in light of the numerical results obtained in Tempozan (Japan)and Memorial Bill Emerson bridges (USA). Variable orice damper, employed in thesemi-active control, presented the advantage of requiring actuators only in the deviceitself, which showed a pseudo-negative stiness suited to dissipate large amounts ofseismic energy.

2.4.4 Compendium of the seismic device typologies

Currently, there are a great number of seismic devices, some of them remain at thelevel of research and found no practical application yet. The following works estab-lish the framework in seismic devices for structural implementations; [Soong 1997],[Jara 2002], [Kunde 2003], [Valdebenito 2005], [Huber 2005], among others.

Table 2.4 collects most of the seismic devices employed to date in the structuralseismic control, ordered according to their energy requirements and eect within thestructure, besides the reference keywords employed in the literature.

2.5 Cable-stayed bridges constructed in seismic areas

Tables 2.5 and 2.6 summarize the main characteristics and specic seismic controlstrategies adopted in some of the major cable-stayed bridges constructed in seismicareas worldwide.

In order to improve the readability of this thesis, detailed information about thebridges briey presented in tables 2.5 and 2.6 is moved to appendix A, including theirprincipal dimensions, design objectives, technical characteristics of seismic devices,etc. Project details observed in these cable-stayed structures located in zones withhigh seismicity have inspired some of the decisions assumed in the bridges studiedalong the present work, which will be exposed in chapters 3 and 9.

The structures collected in these tables are not the only important cable-stayedbridges with seismic devices, other outstanding examples like Sutong or Higashi-Kobe bridges, among others, incorporate dampers to control their response duringlarge ground motions. However, they are not included since the information foundby the author about these structures is not complete. Nonetheless, it is worth giv-ing a brief sight of both bridges, due to their importance; (i) Sutong bridge (China)is the longest cable-stayed bridge in the world with a span arrangement of 500 +1088 + 500 m and two 300 m tall inverted `Y'-shaped pylons, longitudinal dampers(αd = 0.4) connect the girder and these towers in longitudinal direction, which donot restrain the displacement of the steel girder induced by temperature, moderatewinds, and vehicle trac, but instead transfer the loads induced by gusts, earth-quakes, and other fast actions; (ii) Higashi-Kobe bridge (Japan) is a cable-stayedstructure (spans: 200 + 485 + 200 m) with metallic `H'-shaped towers and sti

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2.5. Cable-stayed bridges constructed in seismic areas 55

Category

StructuralAction

Examples

PassiveControl

Baseisolation

Periodelongation

Rub

ber

Bearings(R

B)

Isolationplus

Dam

ping

High-Dam

ping

Rub

ber

bearings(H

DR)

Periodelongation

LeadRub

ber

Bearings(LRB)

plus

addedenergy

dissipation

ResilientFrictionBaseIsolator

(R-FBI)

FrictionPendu

lum

System

s(FPS)

SlidingBearings(SB)

Dam

pers

Viscous

uid

Dam

pers(V

D)

YieldingMetallic

Dam

pers(M

D)

Add

edenergy

dissipation

FrictionDam

pers(FD)

(dam

ping)

ViscoelasticDam

pers(V

E)

LeadExtrusion

Dam

pers(LED)

Shape-Mem

oryAlloys

(SMA)

ParticleDam

per

SeismicConnectors∗

Linkdierentelem

ents

ofSh

ockTransmission

Units

(STU)

thestructurewhenthe

Displacem

entControl

Devices

(DCD)

earthquake

strikes

Rigid

ConnectionDevices

(RCD)

ResonantDam

pers

Balance

ofinertiaforces

Tun

edMassDam

pers(T

MD)

Tun

edLiquidDam

pers(T

LD)

ActiveControl

Baseisolation

Activeperiodelongation

ActiveBaseIsolation(A

BI)

Resonantdampers

Activeinertiaforces

ActiveMassDam

pers(A

MD)

ActiveCables

Activeforces

throughcables

ActiveCables(A

C)

Jetdampers

Activeairjet

JetDam

pers

Semi-activeControl

Dam

pers

Variableorice

dampers

Add

moredamping

Variablefriction

dampers

whenearthquake

strikes

Semi-active

Tun

edMassDam

pers(STMD)

ElectrorheologicalDam

pers(ERD)

MagnetorheologicalDam

pers(M

RD)

Table2.4.Types

andmain

examplesofseismicdevices.(∗)Seism

icconnectors

belongto

capacity

designandtherest

tomitigationdesign.

Page 74: "Seismic Behaviour of Cable-Stayed Bridges"

56 Chapter 2. State of the art

girder, a special aseismic design was adopted by completely releasing the deck inlongitudinal direction, hence increasing the vibration period and reducing the spec-tral accelerations, but enlarging the displacements, this problem was eased by ar-ranging the cable-system in harp pattern and by installing vane-type oil dampers indeck-piers connections, intended as Shock Transmission Units (STU) [Ganev 1998][Naganuma 2000].

2.5.1 Seismic failures reported in cable-stayed bridges

An evident lack of reports about signicant failures in cable-stayed bridges causedby earthquakes have been found by this author and also by Valdebenito and Aparicio[Valdebenito 2005], which suggests their adequate seismic behaviour.

Nevertheless, Filiatrault et al [Filiatrault 1993] published the damage exertedby Saguenay earthquake (1988, moment magnitude MW = 6.0) in Shipshaw bridge(Canada, 183 m long), conrming the failure of one anchorage plate connecting thesteel box girders to one abutment, but recognizing an important stress concentrationin that point prior to the earthquake.

Perhaps, the most important damages resulting from ground motions in a cable-stayed structure were the ones recorded in Chi-Lu bridge (Taiwan, two 120 m spans)after the great Chi-Chi earthquake (1999, MW = 7.3). The structure was almostcomplete at the time of the seismic event. Among other problems, signicant damageoccurred in the deck and moderate cracking was observed in the pylon below decklevel, however, above the roadway, there was severe spalling of the cover and crackingextended in the tower upward nearly to the level of the lowest cables [Chang 2004].

Cable-stayed bridges with seismic devices located in seismic-prone areas appearto have performed well in recent large earthquakes; no information about any failurecaused by seismic events has been found by the author in the structures mentionedin tables 2.5 and 2.6, some of them (Tsurumi and Yokohama Bay bridges) subjectedto the extremely large Tohoku earthquake (2011, MW = 9.0). However, the vane-type dampers of the Higashi-Kobe bridge (Japan) were broken and taken o duringthe near-fault Kobe earthquake (1995, MW = 7.2) and buckling in one of the pierswas observed44, besides other damages in several supports [Naganuma 2000], despitethese failures, the structure performed outstandingly during the Kobe earthquakein the opinion of Ganev et al [Ganev 1998].

44The vane-type dampers of Higashi-Kobe bridge are intended as Shock Transmission Units(STU) and, therefore, mitigation design was not strictly adopted in this structure.

Page 75: "Seismic Behaviour of Cable-Stayed Bridges"

2.5. Cable-stayed bridges constructed in seismic areas 57

Name

Elevationscheme

Transverse

Structural

Deck-towerconnection

Other

(longitudinaldeck-tow

erconnection

features

Vertical

Longit.

Transverse

seismic

conn

ection)[m

eters]

(Z)

(X;tra

c)(Y

)measures

Es

qu

ema

de

l alz

ad

o [

Co

tas

en

met

ros]

Co

nex

ión

Otr

as m

edid

as

Co

nex

ión

To

rre

- T

ab

lero

lon

git

ud

inal

tran

sv

ers

alV

ert

ical

(Z

)L

on

git

ud

inal

(X

; tr

áfi

co

)T

ran

sv

ers

al (

Y)

de

pro

tec

ció

n s

ísm

ica

286

+ 3

x 5

60

+ 2

86

Mu

ltiva

noA

t. L

ate

ral e

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aT

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ctu

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tern

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cia

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F)

en lo

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l

1

43

+ 3

51 +

143

A

t. L

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ral e

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Em

ers

on

Tab

lero

mix

to(U

SA

)T

orr

es d

e h

orm

igón

Un

idad

es

de

tra

nsm

isió

ne

n fo

rma

de

HA

po

yos

PO

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e c

ho

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(ST

U)

Fijo

25

5 +

51

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25

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t. C

ent

ral s

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ap

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con

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ica

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+ 3

x 5

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tern

os

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cia

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orr

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en lo

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1

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143

A

t. L

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ral e

n s

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ónU

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smis

ión

en

form

a d

e H

Ap

oyo

s P

OT

de

ch

oqu

e (S

TU

)F

ijo

25

5 +

51

0 +

25

5A

t. C

ent

ral s

em

iarp

aT

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airw

ayT

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ero

me

tálic

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ap

ón

)T

orr

es d

e h

orm

igón

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ble

s d

e a

ncla

je +

con

fo

rma

de

dia

ma

nte

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ort

igu

ador

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ipo

vele

taF

ijo

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nex

ión

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rre

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ab

lero

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om

bre

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rac

terí

stic

as

Lateral

free

free

Dam

pers

Internal

semi-harp

(VD)

dampers

Rion-Antirion

Com

posite

+(incables)

(Greece,2004)

deck

Fuse

Special

`Y'Diamond

Restrainers

foun

dation

286

+3×

560

+28

6Mult.-span

(parallel)

Es

qu

ema

de

l alz

ad

o [

Co

tas

en

met

ros]

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nex

ión

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lon

git

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ical

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)L

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git

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inal

(X

; tr

áfi

co

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ran

sv

ers

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Y)

de

pro

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ció

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ica

286

+ 3

x 5

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1

43

+ 3

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143

A

t. L

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aT

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hor

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ónU

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en

form

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e H

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oyo

s P

OT

de

ch

oqu

e (S

TU

)F

ijo

25

5 +

51

0 +

25

5A

t. C

ent

ral s

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aT

su

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i F

airw

ayT

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ero

me

tálic

o(J

ap

ón

)T

orr

es d

e h

orm

igón

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ble

s d

e a

ncla

je +

con

fo

rma

de

dia

ma

nte

Fijo

Am

ort

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ador

es t

ipo

vele

taF

ijo

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nex

ión

To

rre

- T

ab

lero

N

om

bre

Ca

rac

terí

stic

as

Es

qu

ema

de

l alz

ad

o [

Co

tas

en

met

ros]

Co

nex

ión

Otr

as m

edid

as

Co

nex

ión

To

rre

- T

ab

lero

lon

git

ud

inal

tran

sv

ers

alV

ert

ical

(Z

)L

on

git

ud

inal

(X

; tr

áfi

co

)T

ran

sv

ers

al (

Y)

de

pro

tec

ció

n s

ísm

ica

286

+ 3

x 5

60

+ 2

86

Mu

ltiva

noA

t. L

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ral e

n s

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iarp

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ro m

ixto

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ctu

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tern

os

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rio

n (

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cia

)T

orr

es r

ígid

as d

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orm

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ortig

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ore

s (V

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en lo

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ante

sco

n f

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a d

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iam

an

teL

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re(e

n p

aral

elo)

Cim

enta

ción

esp

ecia

l

1

43

+ 3

51 +

143

A

t. L

ate

ral e

n s

em

iarp

aT

able

ro m

ixto

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rres

de

hor

mig

ónU

nid

ade

s d

e tr

an

smis

ión

en

form

a d

e H

Ap

oyo

s P

OT

de

ch

oqu

e (S

TU

)F

ijo

25

5 +

51

0 +

25

5A

t. C

ent

ral s

em

iarp

aT

su

rum

i F

airw

ayT

abl

ero

me

tálic

o(J

ap

ón

)T

orr

es d

e h

orm

igón

Ca

ble

s d

e a

ncla

je +

con

fo

rma

de

dia

ma

nte

Fijo

Am

ort

igu

ador

es t

ipo

vele

taF

ijo

Co

nex

ión

To

rre

- T

ab

lero

N

om

bre

Ca

rac

terí

stic

as

Lateral

Fixed

semi-harp

Shock

Bill

Emerson

Com

posite

POT

Transm.

(USA

,2003)

deck

supp

orts

Units

`H'towers

(STU)

143

+35

1+

143

Es

qu

ema

de

l alz

ad

o [

Co

tas

en

met

ros]

Co

nex

ión

Otr

as m

edid

as

Co

nex

ión

To

rre

- T

ab

lero

lon

git

ud

inal

tran

sv

ers

alV

ert

ical

(Z

)L

on

git

ud

inal

(X

; tr

áfi

co

)T

ran

sv

ers

al (

Y)

de

pro

tec

ció

n s

ísm

ica

286

+ 3

x 5

60

+ 2

86

Mu

ltiva

noA

t. L

ate

ral e

n s

em

iarp

aT

able

ro m

ixto

Fu

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es e

stru

ctu

rale

s +

Am

ort

igu

ador

es in

tern

os

Rio

n A

nti

rio

n (

Gre

cia

)T

orr

es r

ígid

as d

e h

orm

igón

Am

ortig

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ore

s (V

F)

en lo

s tir

ante

sco

n f

orm

a d

e d

iam

an

teL

ibre

Lib

re(e

n p

aral

elo)

Cim

enta

ción

esp

ecia

l

1

43

+ 3

51 +

143

A

t. L

ate

ral e

n s

em

iarp

aT

able

ro m

ixto

To

rres

de

hor

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ónU

nid

ade

s d

e tr

an

smis

ión

en

form

a d

e H

Ap

oyo

s P

OT

de

ch

oqu

e (S

TU

)F

ijo

25

5 +

51

0 +

25

5A

t. C

ent

ral s

em

iarp

aT

su

rum

i F

airw

ayT

abl

ero

me

tálic

o(J

ap

ón

)T

orr

es d

e h

orm

igón

Ca

ble

s d

e a

ncla

je +

con

fo

rma

de

dia

ma

nte

Fijo

Am

ort

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ador

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ipo

vele

taF

ijo

Co

nex

ión

To

rre

- T

ab

lero

N

om

bre

Ca

rac

terí

stic

as

Es

qu

ema

de

l alz

ad

o [

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tas

en

met

ros]

Co

nex

ión

Otr

as m

edid

as

Co

nex

ión

To

rre

- T

ab

lero

lon

git

ud

inal

tran

sv

ers

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ical

(Z

)L

on

git

ud

inal

(X

; tr

áfi

co

)T

ran

sv

ers

al (

Y)

de

pro

tec

ció

n s

ísm

ica

286

+ 3

x 5

60

+ 2

86

Mu

ltiva

noA

t. L

ate

ral e

n s

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able

ro m

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es e

stru

ctu

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s +

Am

ort

igu

ador

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tern

os

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n A

nti

rio

n (

Gre

cia

)T

orr

es r

ígid

as d

e h

orm

igón

Am

ortig

uad

ore

s (V

F)

en lo

s tir

ante

sco

n f

orm

a d

e d

iam

an

teL

ibre

Lib

re(e

n p

aral

elo)

Cim

enta

ción

esp

ecia

l

1

43

+ 3

51 +

143

A

t. L

ate

ral e

n s

em

iarp

aT

able

ro m

ixto

To

rres

de

hor

mig

ónU

nid

ade

s d

e tr

an

smis

ión

en

form

a d

e H

Ap

oyo

s P

OT

de

ch

oqu

e (S

TU

)F

ijo

25

5 +

51

0 +

25

5A

t. C

ent

ral s

em

iarp

aT

su

rum

i F

airw

ayT

abl

ero

me

tálic

o(J

ap

ón

)T

orr

es d

e h

orm

igón

Ca

ble

s d

e a

ncla

je +

con

fo

rma

de

dia

ma

nte

Fijo

Am

ort

igu

ador

es t

ipo

vele

taF

ijo

Co

nex

ión

To

rre

- T

ab

lero

N

om

bre

Ca

rac

terí

stic

as

Central

Fixed

Vane

Fixed

Tsurumi

semi-harp

dampers

Fairway

Metallic

+(Japan,1994)

deck

Anchor

`Y'diam

ond

cables

255

+51

0+

255

towers

Table2.5.Summary

ofsomeofthemajorcable-stayedbridges

inseismicareas.

Page 76: "Seismic Behaviour of Cable-Stayed Bridges"

58 Chapter 2. State of the art

Name

Elevatio

nsch

eme

Transverse

Stru

ctural

Deck-to

werconnectio

nOther

(longitudinaldeck-tow

erconnectio

nfeatures

Vertical

Longit.

Transverse

seism

ic

connection)[m

eters](Z)

(X;tra

c)(Y

)measures

Esq

ue

ma d

el alzad

o [C

otas en

metro

s]C

on

exión

Otras

med

idas

Co

nex

ión

To

rre - T

ablero

lon

gitu

din

al

tran

sve

rsalV

ertical (Z

)L

on

gitu

din

al (X; tráfico

)T

ransv

ers

al (Y)

de p

rote

cción

sísm

ica20

0 + 4

60 + 20

0A

t. Lateral en

sem

iarpa

LibreT

able

ro: celosía

metálica

Lim

itado m

vto. máxim

oY

oko

ha

ma B

ay

de gran rigidez

med

iante cone

xión LB

C(J

apó

n)

Torres m

etálica

s en H

Libre

(Link Bearing C

onne

ction)

Fijo

127

+ 4

48 + 475

+ 1

27 M

ultivanoA

t. Latera

l 4 pla

nosC

oaccion

adoT

ablero

mixto

en el mástil cen

tral C

oaccionad

o por

Am

ortiguado

res (VE

)T

ing

Kau

(Ch

ina

)T

orres; Mástile

spo

r apoyo

s PO

T

apoyo

s PO

Tintern

os en los cables

de ho

rmig

ónA

poyos PO

Tverticale

svertica

lesde esta

bilización

28

9 + 1

018 +

289

At. La

teral en se

mia

rpaA

mo

rtiguad

ores internos

Sto

nec

utters (C

hin

a)T

ablero; acero + horm

igón

en los tirante

s[E

n co

nstru

cció

n]

Torres; M

ástiles

Am

ortiguadores

Ap

oyos

Am

ortiguadores de

de ho

rmig

ónLib

rehidráulicos

deslizantes

ma

sa sinton

izada (T

MD

)en

las torres

No

mb

reC

aracterística

sC

on

exió

n T

orre

- Ta

blero

Esq

ue

ma d

el alzad

o [C

otas en

metro

s]C

on

exión

Otras

med

idas

Co

nex

ión

To

rre - T

ablero

lon

gitu

din

al

tran

sve

rsalV

ertical (Z

)L

on

gitu

din

al (X; tráfico

)T

ransv

ers

al (Y)

de p

rote

cción

sísm

ica20

0 + 4

60 + 20

0A

t. Lateral en

sem

iarpa

LibreT

able

ro: celosía

metálica

Lim

itado m

vto. máxim

oY

oko

ha

ma B

ay

de gran rigidez

med

iante cone

xión LB

C(J

apó

n)

Torres m

etálica

s en H

Libre

(Link Bearing C

onne

ction)

Fijo

127

+ 4

48 + 475

+ 1

27 M

ultivanoA

t. Latera

l 4 pla

nosC

oaccion

adoT

ablero

mixto

en el mástil cen

tral C

oaccionad

o por

Am

ortiguado

res (VE

)T

ing

Kau

(Ch

ina

)T

orres; Mástile

spo

r apoyo

s PO

T

apoyo

s PO

Tintern

os en los cables

de ho

rmig

ónA

poyos PO

Tverticale

svertica

lesde esta

bilización

28

9 + 1

018 +

289

At. La

teral en se

mia

rpaA

mo

rtiguad

ores internos

Sto

nec

utters (C

hin

a)T

ablero; acero + horm

igón

en los tirante

s[E

n co

nstru

cció

n]

Torres; M

ástiles

Am

ortiguadores

Ap

oyos

Am

ortiguadores de

de ho

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Page 77: "Seismic Behaviour of Cable-Stayed Bridges"
Page 78: "Seismic Behaviour of Cable-Stayed Bridges"
Page 79: "Seismic Behaviour of Cable-Stayed Bridges"

Chapter 3

Modelling and basic assumptions

Contents

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2 Cable-stayed bridges description . . . . . . . . . . . . . . . . 62

3.2.1 Geometric aspects . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 65

3.2.3 Deck-tower connection . . . . . . . . . . . . . . . . . . . . . 66

3.2.4 Prestress of the lower strut . . . . . . . . . . . . . . . . . . . 70

3.3 Materials and damping . . . . . . . . . . . . . . . . . . . . . . 73

3.4 Loading scheme and analysis . . . . . . . . . . . . . . . . . . . 76

3.5 Finite element model description . . . . . . . . . . . . . . . . 78

3.5.1 Discretization of the towers: localization phenomena . . . . . 79

3.5.2 Discretization of the cable-system: cable-structure interaction 82

3.5.3 Discretization of the deck . . . . . . . . . . . . . . . . . . . . 84

3.5.4 Special-purpose elements . . . . . . . . . . . . . . . . . . . . 85

3.6 Spatial variability . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.7 Symmetry of the seismic response . . . . . . . . . . . . . . . 90

3.8 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.1 Introduction

Earthquakes are certainly the most unpredictable and devastating environmental ac-tions among all that may aect a bridge; their frequency content, intensity, length,or strike direction among other seismological considerations, strongly inuence thestructural response. Furthermore, the bridge itself modies the seismic input, andits exibility and orientation with respect to the fault also represent key variables.As if that were not enough, strong ground motions may cause important nonlin-earities and subsequent degradation and changes in the dynamic properties of thestructure. The scientic treatment of the structural seismic response began in theearly twentieth century and, despite major advances have been made, the problemis far from being solved [Villaverde 2009].

Therefore, clearly establishing the modelling assumptions and simplicationsconducted prior to the analysis is a key point, in order to be able to address it,nding a compromise point between accuracy and analysis eciency.

Page 80: "Seismic Behaviour of Cable-Stayed Bridges"

62 Chapter 3. Modelling and basic assumptions

The task is accomplished in this chapter; rst the idealized cable-stayed bridgesconsidered are geometrically described in section 3.2, conguring the frameworkwhere the thesis is developed; this is followed by the material constitutive assump-tions in linear and nonlinear analysis (section 3.3); next, a detailed study on thelocalization phenomenon and local cable vibration is briey summarized, leading tothe description of the optimized nite element mesh employed in the towers and therest of the bridge (section 3.5); nally, sections 3.6 and 3.8 respectively gather re-sults about the spatial variability and symmetry of the seismic action, which yieldsto defend the simplications collected in the last paragraph of the chapter (section3.6).

Appendices B and C contain detailed information about the models, which havebeen extracted from the main body of the thesis to improve its legibility. Valuabledetails about the parametrization of the studied models are moved to appendix B,along with the dimensions of a large number of constructed cable-stayed bridgeswhich have been used to establish the proportions of all the considered structures;sections of the towers (gures B.3 to B.7), deck, foundations and cable-stays, in-cluding the span arrangement in elevation and tower geometry. In this chapter, themost outstanding aspects of the parameterized models collected in appendix B arehighlighted, whereas the support conditions are also discussed here. On the otherhand, appendix C includes a detailed presentation of nonlinear material constitutivemodels, besides the results of specic studies about several modelling factors, likethe appropriate nite element length in the towers taking into account the localiza-tion phenomenon. Furthermore, the comparison between experimental results onreinforced concrete cantilever beams and the numerical simulation with proposednite element models is presented in appendix C, validating their use in nonlinearanalysis.

Finally, appendix D addresses the eects of the spatial variability of the seis-mic action in one of the proposed models through dierent analysis strategies, andconsidering several wave-propagation velocities or incidence angles, which completesthe additional information.

3.2 Cable-stayed bridges description

3.2.1 Geometric aspects

One of the main objectives of the present thesis is to obtain general conclusions aboutthe inuence in the seismic response of dierent decisions in the project of cable-stayed bridges. There are many choices involved in the design of these structures; thecable arrangement, the tower shape and its connection with the deck, the boundaryconditions in the supports, among others. Therefore, an important variety in thetypology is available for the designer, who is also dealing with the main span andtype of foundation soil as key parameters in the project. This casuistry is attemptedto be framed with the following parameters:

Page 81: "Seismic Behaviour of Cable-Stayed Bridges"

3.2. Cable-stayed bridges description 63

• Main span of the bridge, LP , covering a range between 200 and 600 meters1

(every 100 meters). Figure 3.1 illustrates the schematic elevation of the stud-ied models, being the symbols included self-explanatory (they are detailed inappendix B).

Figure 3.1. Schematic bridge elevation. Measurements in meters.

• Shape of the tower; ve types of towers have been considered and they arecollected in gure 3.2 along with the keywords used hereinafter to present theresults, being the correlation with the labels2 and the shape of the towersrather obvious; the letter `D' indicates that the tower has lower diamondconguration. All the towers are made of reinforced concrete.

Figure 3.2. Types of towers considered in the study, and keywords used as a reference in

the results.

• Two cable-system arrangements have been considered; two Lateral Cable Planes(LCP), which was applied to all tower types, and one Central Cable Plane(CCP), which was applied to the inverted `Y'-shaped3 towers (with and with-

1In chapter 7, a wider range of main spans have been considered when the transverse reactionof the deck against the towers is studied, reaching up to 1000 m.

2Inverted `Y'-shaped towers are sometimes simply referred as `Y'-shaped towers in the body ofthe thesis.

3Central Cable Plane conguration is only possible in inverted Y-shaped towers.

Page 82: "Seismic Behaviour of Cable-Stayed Bridges"

64 Chapter 3. Modelling and basic assumptions

out lower diamond).

• Type of foundation soil; two extremely dierent types of foundation groundsproposed by Eurocode 8 [EC8 2004] have been considered; rocky soil (TA)and soft soil (TD), which denes not only the seismic excitation (presentedin chapter 5) but also the exibility of the foundation. No attempt has beenmade to consider the soil-structure interaction but its eect is not expected tobe relevant due to the light-weight and great exibility of cable-stayed bridges(see section 2.2.7).

Altogether, 35 nite element models have been analysed with two types of foun-dation soil. Figure 3.3 illustrates a schematic 3D view of one considered model(referred as YD-CCP), including the global reference axes employed in the work.

X (traffic)

Y

Z

Figure 3.3. 3D view of the model with inverted `Y'-shaped towers and one central cable

plane: YD-CCP. Global axis system (X,Y,Z) employed hereinafter.

Specic studies, like the analysis of the eect of the support conditions in theabutments (chapter 7) or the optimization of lower diamond (chapter 8), considerslightly dierent structural congurations in order to shed some light on particularaspects. Details about such modications are oered in the specic sections, beforeexamining the analysis results.

The deck holds four road trac lanes and has a total width (B) of 25 meters.The section of the deck is composite, which is the normal solution in the studiedspan range (200 to 600 m). The deck has an open section in bridges with lateralcable-system assemblies (LCP), and closed-box section in the case of central cablearrangements (CCP) in order to withstand the torsion that the cable-system is notable to resist. Figures 3.18 and 3.19 present the model of both deck cross-sections(more information is collected in appendix B).

The general proportions of the towers and their sections have a signicant impacton the seismic behaviour of cable-stayed bridges (chapter 7). To select carefully

Page 83: "Seismic Behaviour of Cable-Stayed Bridges"

3.2. Cable-stayed bridges description 65

these values in order to represent real constructed bridges is therefore paramount,which has been the objective of the study on the dimensions of the towers includedin appendix B. Due to the large number of cable-stayed bridges considered in thisthesis, the simplicity is strongly recommended in the model design; constant sectionsbetween dierent parts of the towers have been adopted in the parametric denition.In order to facilitate the comparison of the results between analogous models withdierent towers, the sections have been established as similar as possible, takinginto account an important number of constructed cable-stayed bridges.

The cable-system is disposed in semi-harp conguration, which is the normaloption in modern designs, representing a compromise solution between structuraleciency and ease of construction. Details about the specic semi-harp layoutemployed are included in appendix B.

The height of the deck over the foundation level of the towers (Hi) marks theheight of the gravity center and, therefore, it results a key variable for the seismicresponse of cable-stayed bridges [Fujino 2006]. Nevertheless, this value is imposedby constraints of the infrastructure and is hardly modiable by the designer; in orderto minimize the broad range of variables involved in the study (presented above), theheight of the deck is considered proportional to the main span length (LP ) throughthe expression:

Hi =H

2(3.1)

Where H = LP /4.8 is the height of the tower above the deck4, the interestedreader is referred to appendix B for more information about the dimensions ofthe proposed structures. Expression (3.1) yields reasonable values of the verticaldistance between the deck and the foundations of the towers in the considered mainspan range of 200 to 600 m; (i) if LP = 200 m, the height of the deck is Hi = 20.8 m,which is more or less the case of many cable-stayed bridges, like Sama de Langreobridge (Spain); (ii) if LP = 400 m, the height of the deck is Hi = 41.6 m, Helgeland(Germany) or Baytown (USA) bridges, among others, present similar congurations;(iii) if LP = 600 m, the height of the deck is Hi = 62.5 m, being the case of referencecable-stayed bridges like Sutong (China).

3.2.2 Boundary conditions

The exibility associated with the tower foundation, which depends on the sur-rounding subsoil category, has been taken into account by means of linear springswith stiness properties independent on the excitation (i.e. neglecting soil-structureinteraction). Including the foundation exibility in dynamic analysis was found im-portant, since higher vibration modes are required (see chapter 4) and they involve

4If the height of the deck over the foundation level of the towers is considered independent ofthe main span length, varying this value would change signicantly the proportional position ofthe gravity center, and consequently the dynamic response, which would mask the seismic eectsof other project decisions, and this is precisely one of the main objectives of the present doctoralthesis.

Page 84: "Seismic Behaviour of Cable-Stayed Bridges"

66 Chapter 3. Modelling and basic assumptions

certain deformation of tower foundations5. The denition of the foundation exibil-ity associated with the displacement along the three principal directions (X, Y andZ, gure 3.3) in terms of its dimensions is also included in appendix B (see gureB.10). The three rotations of the towers at foundation level are completely avoided.

The intermediate piers located in both lateral spans (see gure 3.1) only avoidthe vertical movement of the deck, being released all the other degrees of freedom,according to the classical6 design of cable-stayed bridges.

The connection between the deck and the abutments prevents the vertical (UZ),transverse (UY ) and longitudinal (UX) displacements of the deck, and also its tor-sional rotation (θXX), whereas the rotation along the transverse axis (θY Y ) is com-pletely released. The seismic eect of the stiness provided in the abutments to therotation of the deck along the vertical axis (θZZ), i.e. its transverse rotation, hasbeen explicitly studied in chapter 7, and it is dened in the relevant sections priorto the presentation of the results.

3.2.3 Deck-tower connection

As it was discussed previously in section 2.2.6, the connection between the deckand the towers plays an important role in the global static and seismic behaviourof the structure. Following the current trend in the design of cable-stayed bridgesin seismic areas, the deck-tower connection is close to the oating solution; only thetransverse displacement is constrained between both elements, releasing the rest ofdegrees of freedom7. If no seismic devices are involved, in real projects this transverseconnection is performed usually by means of two concrete protrusions in both laterallegs of the towers, separated from the deck a small distance (δc) which ensuresfree vertical and longitudinal movements (gure 3.4(a)) [Gimsing 1998]. Therefore,during the earthquake, the deck alternatively exerts a reaction against the towers,or in other words, the deck rst strikes one leg of the tower (say leg 1) whilst theopposite is free of this load (say leg 2), later the deck aects only the leg 2 directlyand so on.

Simulating this realistic connection requires the denition of the contact betweenthe deck and the tower (gure 3.4(a)), which inherently leads to nonlinear analysis(even if the materials remain elastic) and consequently prevents procedures basedon modal decomposition (see section 2.2.3). In order to overcome this shortcoming,a simplied connection is proposed here; the deck is rigidly connected to one leg8

of the tower (leg 1) and no reaction is directly exerted to the opposite leg by this

5This was also concluded by [Zheng 1995], being further discussed in chapter 4.6The transverse stiness of the towers is much larger than the one of the intermediate piers

(due to the dimensions required in the towers to transfer the vertical loads from the deck), henceconnecting the deck and the towers in transverse direction is preferable, whilst this degree offreedom is released in exible elements like the intermediate piers.

7The solution adopted for the deck-tower connection is the same as the one designed in Rion-Antirion cable-stayed bridge, see table 2.5 or appendix A.

8The rigid connection in both legs at the same time would lead to half reactions in the towersdue to the deck, compared with the real results.

Page 85: "Seismic Behaviour of Cable-Stayed Bridges"

3.2. Cable-stayed bridges description 67

element. Undoubtedly, the results of the seismic analysis are dierent in leg 2 whensuch simple connection is implemented, but it will be demonstrated in this paragraphthat both linear and nonlinear calculations yield similar results with the realistic orsimplied solutions along the leg which receive the connection (leg 1). The simpliedsolution is therefore strongly recommended, in such case the design should consideronly the forces and displacements recorded in leg 1, being the realistic response ofleg 2 quite similar due to the symmetry of the real connection.

(a) Realistic connection (alternate contact) (b) Simplied connection (sti link to one leg)

Figure 3.4. Dierent numerical deck-tower connection strategies.

Figure 3.5 presents average extreme seismic responses9 along the leg 1 with bothproposed connections10 in two models of cable-stayed bridges subjected to a setof twelve synthetic accelerograms corresponding to rocky subsoil class (TA), thematerials are completely elastic and direct response history analysis is performed(more information about the earthquake action and the analysis procedures areincluded in chapters 5 and 6 respectively). It is clear that the elastic results obtainedwith both connection strategies are nearly the same in the leg where the link isestablished (leg 1), in fact the extreme reaction of the deck (responsible for the peakin the transverse shear) is similar.

The nonlinear behaviour of the lower strut may be inuenced by the connectionmodel, since the realistic response of the deck always introduces tension in thistransverse element and, consequently, it may degrade the stiness of the concretesection through extensive cracking. However, in the response history of the simpliedmodel, both tension and compression are exerted alternatively. This theoreticalvariation in the cyclic response of the concrete along the strut is presented in gure3.6 to clarify the dierence in the energy dissipated by means of this element withboth connections. The simplied connection could lead to non-conservative modelsin the nonlinear range.

It was indented to conduct analysis like those presented above for elastic calcu-lations, but unfortunately, the convergence was not achieved in the models with gap

9The seismic response (rd) is obtained by extracting the eects due to the self-weight (rG) fromthe total response (r); rd = r − rG.

10In the realistic connection, zero-gap (δc = 0) has been considered in order to avoid numericalinstabilities due to the rattle of the deck. A very small gap is left in real cable-stayed bridges,normally obtained with a thin layer of some light material.

Page 86: "Seismic Behaviour of Cable-Stayed Bridges"

68 Chapter 3. Modelling and basic assumptions

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14 16 180.0

0.2

0.4

0.6

0.8

1.0

A) Torre en “H” B) Torre en “Y” invertida C) Torre en “Y” invertida con diamante

D) Torre en “A” E) Torre en “A” con diamante

Figure 3.5. Comparison of extreme elastic seismic forces considering both connections

proposed in gure 3.4. Left: Axial load along the leg 1 in Y-CCP model with LP = 300 m

and foundation on rocky soil (TA). Right: Transverse shear along the leg 1 in Y-LCP model

with LP = 400 m and foundation on rocky soil (TA). Direct elastic dynamics (DRHA).

(a) Realistic connection (alternatecontact)

(b) Simplied connection (stilink to one leg)

Figure 3.6. Theoretical performance dierences in a reinforced concrete element subjected

only to tension cycles (realistic connection) or to alternating tension-compression cycles

(simplied connection).

connection and material nonlinear behaviour, which is by itself another reason torecommend the simplied option depicted in gure 3.4(b). However, an alternativestudy was carried out in order to address the importance of the connection modelin analysis with material nonlinearities; a nite element model of the lower strutwas extracted from the global model, representing the exibility of both legs of thetower by means of springs11. A cyclic static analysis is proposed, imposing dier-

11Their stiness kpile represents the transverse cantilever response of the tower legs below thelower strut.

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3.2. Cable-stayed bridges description 69

ent histories of displacements12 at the ends of the struts in order to represent bothconnections, as it is shown in gure 3.7. The analysis has been performed in modelswith `H'-shaped towers and main spans of 200 or 600 m, the results are presentedin gure 3.8. It is observed that the ductility demand in the lower strut is reducedin the studied cases because of the exibility of the lower piers and, therefore, theresults are nearly the same considering xed or alternate connection also in thenonlinear range.

Bonvalot

Abaqus

= =

X

YZ

YF

Figure 3.7. Sketch of the cyclic static test of the model representing only the strut to

clarify the eect of deck-tower connection in nonlinear range. The eect of the rest of the

tower in the transverse response is included by means of appropriate springs.

(a) LP = 200 m

-400 -300 -200 -100 0 100 200 300 400Transverse displacement; uY,cp [mm]

-100

-50

0

50

100

Tra

nsve

rse

forc

e;FY

[MN

]

LP = 600 m

Fixed connectionAlternate connection

(b) LP = 600 m

Figure 3.8. Force-displacement results in the cyclic static analysis proposed to address

the eect of deck-tower connection, including material nonlinearities in the strut. H-LCP

model with dierent main spans.

However, it should be noted here that large ductility demands are recordedin nonlinear seismic analysis of towers with lower diamond and moderate spans

12The extreme displacement imposed is equal to the maximum transverse displacement of thelower strut in the corresponding linear time history analysis of the complete models.

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70 Chapter 3. Modelling and basic assumptions

(chapter 8) considering the simplied connection. In these cases, more signicanterrors introduced by the simplication of the connection are expected.

To sum up, the simplied connection depicted in gure 3.4(b), which consists of asti link between the deck and one leg of the tower (leg 1), is strongly recommendedin the elastic and inelastic analysis of cable-stayed bridges with the proposed deck-tower conguration (without seismic devices), assuming certain errors on the unsafeside if extensive cracking appears along the lower strut. Special congurations of theconnection may be dened when seismic devices are equipped, these congurationsare detailed in chapter 9.

3.2.4 Prestress of the lower strut

The diversion of the path followed by the compressive axial loads along the legs ofthe tower, introduces tension in the lower strut, which retain the separation of theselateral members. The larger the dierence between the slope of the lower piers andthe legs above the deck, the greater this tension, and consequently, bridges withlower diamond are more sensitive to this eect, which may dangerously crack theconcrete of the lower strut even before the action of the earthquake. `H'-shapedtowers also present this response, in a lower extent. On the other hand, bridgeswith moderate span enlarge the mentioned tensile stress of the strut, since thewidth of the deck (B = 25 m) is not inuenced by the main span, hence the lowerthis value, the smaller the height of the towers and, thus, the diversion in the pathof compressions is sharper. Particularized for the most sensitive models to thiseect, gure 3.9 schematically presents the tension of the lower strut due to thecompression introduced by the tower self-weight and, specially, by the cable-systemcarrying the loads from the deck.

Including the material nonlinearities detailed in the following section, the lowerstrut in models with lower diamond and 200 m main span13 has been observed topresent uniform cracking due to the application of the self-weight of the structure,prior to the ground motion. Despite the reinforcement remains elastic in these cases,the initial damage of the tower due to self-weight loads is clearly undesirable, andprestressing the strut is immediately presented as the solution of the problem.

The objective of this paragraph is to compare the seismic response of two mod-els, the most sensitive ones to the initial tension of the strut, with and withoutprestress in this member, including the nonlinear properties of the materials. Therequired prestress is obtained in order to avoid tension in the strut due to the appli-cation of two loading hypotheses, considering not only the self-weight (which is thedominant action), but also the live load and the wind14 combined according to Eu-rocode 0 [EC0 2002]; (i) the self-weight of the bridge, the maximum live load along

13The models referred here include smooth transitions between the lower piers below the deckand the vertical element of the lower diamond, see section 7.5 for more information.

14The transverse wind along the deck exerts a reaction against the towers through the stitransverse connections and consequently introduces tension in the lower strut. Furthermore, asmall bending moment is applied to the strut due to the distance between this transverse memberand the connection (see appendix B for more details about the geometry of the tower).

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3.2. Cable-stayed bridges description 71

Figure 3.9. Schematic explanation of the tension exerted in the lower strut due to the

compression introduced in the anchorage area.

the deck and the maximum concomitant transverse wind; and (ii) the self-weightof the bridge, the maximum transverse wind and the maximum concomitant liveload along the deck. The live load is obtained following IAP [IAP 1998], which isclose to Eurocode 1 (part 2) [EC1 2003] prescriptions, whereas the wind load meetsEurocode 1 (part 4) [EC1 2005] (more information about the applied wind action,only considered in specic calculations of this thesis, and ignored in seismic analysis,may be found in section 9.2). The most demanding load combination is consideredto obtain the prestress, which yields values around 9 MPa in the model with lowerdiamond and 4 MPa in the `H'-shaped tower (both with LP = 200 m); these valuesare feasible considering that the maximum reasonable prestress is 10 MPa.

Figure 3.10 presents the extreme transverse shear force along the tower withlower diamond, exclusively due to the earthquake, when the initial uniform pre-stress of 9 MPa is applied to the lower strut or when this load is not considered.The selected model has a main span of 200 m and foundation on soft soil (the mostdemanding one). The small inuence of the prestress in the lower strut to avoidinitial cracking may be observed; preloading this element leads to transverse seis-mic forces up to 12 % larger than neglecting such action, which is explained by theincrease of the nonlinear response in the model without prestress (with the con-sequent increment of energy dissipation and the slight reduction of the maximumforces, specially in the lower legs below the deck, which concentrate the demand,see chapter 8). Taking into account that the dispersion of the results with the set of12 records rounds 10 % (chapter 6), the inuence of the prestress might be ignored.

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72 Chapter 3. Modelling and basic assumptions

Figure 3.10. Extreme transverse seismic shear force VY along the tower height, with and

without (W/O) prestress in the lower strut. YD-CCP model, LP = 200 m, soft soil (TD),

θZZ free at deck-abutments connection. Nonlinear time-domain analysis; the average value

of the set of 12 records is presented.

The eect of the prestress in a pushover analysis up to the collapse (consideringthe rst transverse mode) is illustrated in gure 3.11(a) for one `H'-shaped tower15

with uniform 4 MPa prestress in the strut, observing that prestressing this elementleads to reductions in the global ductility of the structure. The extreme seismicforces along the lower strut in the corresponding full structure are depicted in gure3.11(b), resulting from nonlinear dynamic analysis; the small inuence of the preloadis again veried.

In light of these results, it is clear that prestressing the strut to control theserviceability state causes a negligible eect on the seismic response of the tower,which is the objective of this doctoral thesis. In fact, the uniform demand of tensiledeformation along the strut due to the self-weight, live load and wind is around 0.025% in the worst case, whereas the total averaged demand including the earthquakeis typically ten times larger (at least) in the same model subjected to earthquakesassociated with soft soil conditions. Although prestressing the strut is important forthe serviceability state of cable-stayed bridges with lower diamond, this load mayinvolve numerical problems related to convergence under severe ground motions (asit has been observed), furthermore, its inuence in the seismic response is marginaland complicates modelling. Therefore, no prestress has been applied to the strut ofany tower hereinafter.

15In this case the pushover is conducted in the model representing only the tower, conductingbi-dimensional analysis. More realistic three-dimensional pushover studies of the full cable-stayedbridges are carried out in chapter 6, including both towers, the deck and the cable-system in theanalysis.

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3.3. Materials and damping 73

0.00 0.05 0.10 0.15 0.20

Displacement; u [m]

0

2

4

6

8

10

For

ce;P

[MN

]Mode 1 (transverse)

W/O prestress

With prestress

(a) Nonlinear static analysis (pushover) of thetower (load path extracted from the rst trans-verse vibration mode). Control point at deck-tower connection.

0.0 0.2 0.4 0.6 0.8 1.00.0

2.5

5.0

7.5

10.0

(a) N

W/O prestress With prestress

3.6

4.0

4.4

4.8

(b) VZ

0.8

1.6

2.4

3.2

4.0

(c) VX

lower strut. Soil type D

Dimensionless strut distance; y = y / Y∗

Maxim

um

seis

mic

forc

e[M

N]

tot

yYtot

(b) Extreme seismic forces along the strut result-ing from a nonlinear dynamic analysis (averagedvalue with 12 records)

Figure 3.11. Eect of the prestress in the lower strut of the H-LCP model, LP = 200 m,

soft soil (TD), and θZZ xed at deck-abutment connections.

3.3 Materials and damping

The materials employed are described numerically through advanced constitutivemodels. Their mean properties have been considered, instead of the reduction withsafety factors leading to design values, since the purpose of this thesis is the studyof the realistic seismic response with the most accurate analysis procedures andmodelling techniques.

Only the key features of the material behaviour are summarized here, whichhave been selected using relevant Eurocodes (Eurocode 2 [EC2 2004] and Eurocode8 [EC8 2005a]). A detailed description of the conned concrete and yielding rein-forcement steel models is included in appendix C.

The elastic damping assumed in dynamic analysis does not discern the materialtype (concrete or steel) in order to avoid uncertainties with the dissipation properties(one of the most unpredictable features involved with dynamics [Chopra 2007]), thereader is referred to section 2.2.2 for a detailed discussion about several factorswhich may decisively inuence the damping of the structure. The idea in the presentwork is to minimize the uncertainties and focus the interest on the seismic eectof dierent project decisions or analysis strategies, therefore the elastic damping issimplied and assumed independent of the material type. Moreover, the structuraldamping introduced through the hysteretic response of the materials is accuratelyconsidered by means of nonlinear constitutive models, and it is the most importantsource of dissipation [Soong 1997]. Further information about how the dampingis included in this thesis for nonlinear dynamics (Rayleigh's damping theory) inChapter 5, but, regardless of the analysis procedure employed, the eective criticaldamping factor is ξ = 4 % for all the materials included in the models and presentednext.

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74 Chapter 3. Modelling and basic assumptions

Concrete

The concrete employed in the towers is conned due to the transverse reinforcementof the sections, represented in gure 3.13. The model proposed by Mander et al[Mander 1988] is adopted here, which is widely accepted and included in seismiccodes like Eurocode 8 [EC8 2005a] or NCSP [NCSP 2007]. The concrete categoryis HA-40. The tension-stiening behaviour of the concrete is a key property16 innonlinear analysis, being the resistance fc,t = 3 MPa. The main characteristics ofthe proposed concrete are summarized in table 3.1, more information may be foundin appendix C.

Young's modulus; Ec [GPa] 35Poisson's ratio; νc 0.2Density; ρc [kg/m3] 2500Max. elastic stress (compression softening); fcm [MPa] -48Elastic limit (compression); εcy -0.14 %Maximum stress (compression); fcm,c [MPa] -64.28Ultimate deformation (compression); εcu,c -2.6 %Maximum stress (tension); fc,t [MPa] 3Cracking initiation (tension); εc,crack 0.0086 %Max. deformation (tension-stiening); εmax

c,t 0.035 %

Table 3.1. Summary of the properties of conned HA-40 concrete employed in this work.

Negative sign denotes compression and positive sign denotes tension.

According to Priestley [Priestley 1996], typical values of the ultimate deforma-tion in conned concrete members under compression range from −1.2 to −5 %,which supposes a 4- to 16-fold increase over traditionally assumed values for un-conned concrete. The value adopted in the study for the ultimate deformation,εcu,c = −2.6 %, is discussed further in appendix C, besides the rest of materialparameters.

The nonlinear model adopted in nite element analysis to represent the connedconcrete of the tower is a `smeared crack' concrete model [Abaqus 2010], which con-siders the global damage of concrete elements by means of the tension-stiening orsoftening behaviour, instead of tracking with individual failure models like cracking,spalling, pinching or rebar-buckling, among others (more information is collected inappendix C). Unfortunately, the degradation in the elastic stiness of the concretedue to cyclic loading beyond the linear range is not captured in this model which,on the other hand, is the only one available using `beam' elements [Abaqus 2010](like the ones used to dene the towers).

The behaviour of the concrete in the upper slab of the deck, and also the con-crete of the towers in linear elastic analysis conducted in chapter 7, corresponds

16A sensitivity analysis about the eect of variations in concrete tension-stiening behaviourhave been conducted, and the results are presented in appendix C.

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3.3. Materials and damping 75

to the elastic branch of the nonlinear conned model mentioned above, extendedindenitely under tension and compression.

Reinforcement and structural steel

The steel employed in the longitudinal and transverse reinforcement along the toweris B-500 SD. In nonlinear analysis of chapters 8 and 9, the reinforcement steel in-cludes the yielding at fs,y = 552 MPa and the eects associated with cyclic plas-ticity by means of linear kinematic hardening (e.g. Bauschinger eect), but specicfeatures like the reduction of the constitutive properties due to buckling or low-cycle fatigue of the steel rebars are ignored17, which is reasonable since the extremerecorded plastic deformation in the reinforcement, during the proposed earthquakes,is typically far below the ultimate strain of the steel (see chapter 8). The main char-acteristics of this yielding steel are included in table 3.2. The interested reader isreferred to appendix C for more information.

Young's modulus; Es [GPa] 210Poisson's ratio; νs 0.3Density; ρs [kg/m3] 7850Yield stress; fs,y [MPa] 552Elastic limit; εsy 0.26 %Maximum stress; fs,t [MPa] 665Ultimate deformation; εsu 11.4 %

Table 3.2. Summary of the properties of yielding steel B-500 SD employed in this work.

Under monotonic loads, the steel behaviour is the same in tension or compression.

The structural steel employed in the metallic members of the deck (longitudinaland transverse beams in lateral cable arrangements or box metallic sheet if a centralcable plane is dened) is always elastic and is also B-500 SD.

Prestressing steel in the cable-stays

The cables employed are made of steel Y-1770, being its main characteristics in-cluded in table 3.3.

Young's modulus; Ep [GPa] 195Poisson's ratio; νs 0.3Density; ρs [kg/m3] 7850Maximum stress; σT,ult [MPa] 1770

Table 3.3. Summary of the properties of prestressing steel Y-1770 employed in the cables.

17The low-cycle fatigue has been explored in depth within the study of yielding metallic dampersin chapter 9.

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76 Chapter 3. Modelling and basic assumptions

The real response of the stays presents geometric nonlinearity, since the cablesag of the catenary is reduced by increasing the stress, and thus the stiness ofthe cable is increased, as it has been briey suggested in section 2.1. Usually, thisbehaviour is simplied with the tangent modulus given by Ernst (Ev,0) [ACHE 2007][Walther 1988] [Manterola 2006] [Morgenthal 1999].

Ev,0 =Ep

1 +w2pd

2pEpAT

12F 3p,0

(3.2)

Where Ep = 195 GPa is the elasticity modulus18 of the prestressing steel in thecables [ACHE 2007], wp is the weight of the cable-stay per unit length (wp = ρsAT g,ρs = 7800 kg/m3 and g = 9.81m/s2 is the earth gravitational acceleration), dp isthe horizontal projection of the stay, AT is the cross-section of the stay and Fp,0 isthe initial preload in this element19.

The correction of elasticity modulus in order to take into account the cable sagis higher the larger values of the horizontal projection (dp) and lower the prestress(Fp,0). Employing expression (3.2) in the proposed cable-stayed bridges, it wasveried that the strongest reduction of the elasticity modulus is around 6 to 7 %for the longest cables. Therefore, it is assumed valid to consider completely linearthe behaviour in all the cables, being their modulus the same as the steel whichconforms them; Ep = Ev,0 = constant = 195 GPa.

This simplication is deemed acceptable in all the proposed models and for allthe cables. Furthermore, the variation of the cable stress have been studied in severalmodels under the strongest earthquakes and it was observed that it remains low-to-moderate during the records, being unnecessary to update the Young's modulus ofthe stays along the response history analysis. The conclusions of Wilson and Gravelle[Wilson 1991] or Morgenthal [Morgenthal 1999] support this simplication.

3.4 Loading scheme and analysis

The dierent seismic analysis procedures are thoroughly discussed in chapter 6,however, it is important to clarify here the general loading scheme adopted, since ittacitly introduces several assumptions.

Despite Eurocode 8 [EC8 2005a] states that 20 % of the live load should beincluded in terms of added mass along the deck during the seismic analysis if thetrac is intense (ψ21 = 0.2), it is currently questioned and may be changed insubsequent revisions. The mass of the live load is neglected during the analysis(ψ21 = 0) in this thesis, assuming a normal trac. In principle, this assumption

18The elasticity modulus (Ep) of the stays is lower than the modulus of the steel that are madeup, in order to take into account the loss of stiness due to the helical winding of the wires[ACHE 2007].

19The calculation of the cross-section and the prestress of each stay is discussed in appendix B.

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3.4. Loading scheme and analysis 77

leads to conservative results since the mass and modal periods are slightly lower,but the associated spectral accelerations are higher for the fundamental modes.

The denition of the cross-section and prestress in each stay of the cable-systemgenerally involves an onerous iterative procedure utterly intractable in all the studiedmodels. However, the purpose of this thesis is the seismic response of the bridge,not the eects associated with the self-weight, the live load or the fatigue in theanchorages of the stays. Therefore, a simplied procedure to obtain the area ofthe cables is detailed in appendix B, considering the same prestress in each stayand hence facilitating the parametrization of their areas in terms of the main span,which is essential due to the large number of structures studied.

Nonetheless, despite the simplied procedure is rooted on physical basis, whenequal prestress is applied to all the cables, redistributing eects leads to a certainlevel of longitudinal unbalanced shear forces at the foundation level of the towers,which may yield misleading results in nonlinear analysis. No prestress has beenapplied to the cables in order to avoid this shortcoming, and the eect of the self-weight of the deck and the cable-system, which is important in nonlinear analysisdue to the benecial compression introduced in concrete sections (except in thestruts), is represented by means of concentrated constant loads in the anchorages ofthe cables along the tower.

To clarify the accuracy of this approximation, gure 3.12 includes the extremeseismic response (averaged for three records) obtained by considering material andgeometrical nonlinearities in one model with both rigorous and simplied procedures;(i) the self-weight distributed along the whole model, and the optimized cable sectionand prestress applied to each stay as a result of an iterative calculation; (ii) thesections of the cables obtained through the simplied procedure, the weight of thedeck and the cable-system imposed as simplied loads in cable-anchorages of bothtowers and no prestress imposed in the stays. Very similar seismic responses havebeen observed in the tower with this simplied procedure or considering the prestressof the cables and the real self-weight distribution, which supports the use of thesimplied method hereinafter in this thesis. On the other hand, gure 3.12 alsosuggests that the geometric nonlinearity introduced by the axial load of the cables(which is only fully considered if the cable prestress is included) is negligible formain spans below 400 m; slightly stronger inuence is expected with longer spans,but the material nonlinearity is reduced in turn (see chapter 8).

With this approach in mind, the general analysis scheme considered, regardlessof the procedure selected, is analogous to that recommended by [Fleming 1980],[Nazmy 1992], [Ren 1999], [Ren 2005], among others (section 2.2.3):

1. Application of the self-weight of the towers and the concentrated anchorageloads representing the self-weight of the deck and cable-system20.

2. Extraction of the vibration modes from the deformed conguration21.20Material nonlinearities are included here if the subsequent seismic analysis incorporates them,

geometric nonlinearities are always included in the static step.21The stiness matrix is linearized to obtain the modes. Damping is ignored in this step.

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78 Chapter 3. Modelling and basic assumptions

Figure 3.12. Extreme averaged seismic axial and shear forces recorded along the tower

height with and without prestress in the cables. Y-CCP model with LP = 400 m and θZZfree. Rocky soil type A. Inelastic dynamic analysis (NL-RHA).

3. Seismic analysis (MRSA, MRHA, NSP or NL-RHA, see chapter 6).

3.5 Finite element model description

Finite Element Models (FEM) have been developed in order to represent the staticand dynamic behaviour of the proposed cable-stayed bridges in a realistic manner.Several parameters are required to dene the FEM, specially if material nonlin-earities are expected; some of them have a physical meaning and are dened bythe general practice and/or seismic regulations (such as the concrete compressivestrength, reinforcement area, etc.), whilst others are simply numerical (e.g. the num-ber of integration points, the residual force in dynamic analysis, etc.) and should bevalidated through the comparison with experimental results or sensitivity analysis.

The commercial software Abaqus [Abaqus 2010] has been employed in all theanalysis carried out in this thesis. This software is a general nite element packagewidely used in civil engineering research, which has been validated by the authorfor the purpose of this study, starting from simple SDOF models (comparing thesolution with analytic results) and increasing step-by-step the complexity of themodel until one of the full cable-stayed bridges was achieved. The numerical resultsbeyond the linear range obtained with the proposed FEM have been also success-fully compared with experimental laboratory tests under monotonic and cyclic loads(appendix C).

Several types of nite elements have been employed in dierent parts of thebridges, and verications based on modal properties of models representing only

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3.5. Finite element model description 79

specic parts of the structure have been performed in order ensure their properdynamic representation. Furthermore, sensitivity analysis were carried out to maxi-mize both the accuracy and the calculation eectiveness, including their conclusionsin appendix C. These studies address the number of integration points, concreteYoung's modulus, connement eectiveness, concrete tension-stiening, reinforce-ment hardening and residual force in HHT integration algorithm.

3.5.1 Discretization of the towers: localization phenomena

The tower discretization is established by means of `beam'-type nite elementsthrough the gravity center of each section. A model based on stresses (`ber sec-tion') rather than resultants has been considered, which is the most accurate rep-resentation since the equilibrium is obtained in each section and step-time throughthe integration of the actual stress in each concrete ber and longitudinal rebar(which in turn depend on the history of the response). Many research works rec-ommend `ber section' models, instead of classical discretizations where moment-curvature section properties are directly introduced [Légeron 2005] [Bonvalot 2006][Sakai 2006], which are not able to consider accurately the variation of seismic axialload [Bai 2004], characteristically important in cable-stayed bridges. Furthermore,the moment-curvature properties are developed for independent 2D analysis in eachprincipal direction, and this thesis considers the multi-directional excitation of fully3D models in the inelastic range. Therefore, simply superimposing the responses inlongitudinal and transverse directions, without considering their nonlinear coupling,is deemed inadequate22.

The sections of the towers are strongly reinforced in order to achieve the men-tioned connement of the concrete23 and to prevent numerical instabilities associatedwith cracking [Abaqus 2010]. Figure 3.13 depicts the longitudinal and transversereinforcement dened in a generic tower section; the position and area of each lon-gitudinal rebar has been assigned to all the elements in the towers, whereas thetransverse reinforcement is indirectly represented by means of the conned con-crete model (see appendix C). The Spanish seismic code [NCSP 2007] has beenemployed to dene the volume of transverse reinforcement. Care should be takenwhen designing the reinforcement in hollow sections, because conning the con-crete is dicult and the resulting behaviour is rather complex [Papanikolaou 2009b][Papanikolaou 2009a], strong amounts of transverse and longitudinal rebars are re-quired to avoid the characteristic spalling of the concrete cover, prior to advancedloading stages [Takahashi 2000]; furthermore, this kind of sections present complex

22A thorough study about the coupled longitudinal and transverse response in cable-stayedbridges is discussed in chapter 6, where a new pushover procedure is presented in order to takeinto account this eect.

23The same reinforcement ratio along the whole tower is considered for the sake of simplicity,despite only the parts close to the foundations and connections with other members present suchstrong amounts of steel in real projects. Numerical results are not aected by this simplicationsince the areas where connement should not be provided behave in elastic range, which is thesame in both conned and unconned concrete.

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80 Chapter 3. Modelling and basic assumptions

interactions between bending and shear in advanced loading stages.

Figure 3.13. Description of the longitudinal and transverse reinforcement in the towers.

Figure 3.14 illustrates the reinforced concrete sections associated with each nodeof the `beam'-type FEM of the tower, represented by concrete bers and discreterebars (`ber-section' approach).

Figure 3.14. Schematic representation of the `ber-section' approach adopted in this thesis

to dene the towers.

On the other hand, no attempt has been made to dene bar slip eects in towerconnections.

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3.5. Finite element model description 81

Localization phenomenon: mesh sensitivity in nonlinear problems

The accuracy of FEM in linear elastic analysis is increased, to a greater or lesserextent, the smaller nite element length employed in the discretization. However,if the stiness of some element is notably reduced, it localizes the deformation andthe smaller the element size may lead to worst results. Therefore, the nite elementlength in models which might undergo plastic deformations must be carefully studiedprior to the nal analysis [Légeron 2005]. An extensive work has been performed inthis thesis to ensure the accuracy of FEM when capturing the nonlinear behaviourof concrete, steel and their interaction, by contrasting the results with experimentallaboratory tests published by other authors. Such information is transferred toappendix C in order to improve the readability of the document, here just theconcluding remarks are presented.

Coinciding with other recent research works [Légeron 2005] [Bonvalot 2006], itwas found that the optimum element length is related to the length of the plastichinge (lp) developed in the concrete member, which undoubtedly has a physicalsupport. There are several proposals to predict the size of the area where the damageis localized in concrete structures, but perhaps the most accepted expression is theone suggested by Priestley [Priestley 1996]:

lp = 0.08Lc + 0.022 fs,ye φbl ≥ 0.044 fs,ye φbl Stress in [MPa] (3.3)

Where lp is the plastic hinge length (in meters), Lc is the distance between thecritical section of the plastic hinge and the point of contra-exure (in meters), φblm is the diameter of longitudinal rebars (in this case φbl = 0.032 m) and fs,ye =

1.1fs,y = 607 MPa is the design yield stress of reinforcement steel.However, the plastic hinge length expressed in equation (3.3) is the optimum

element length only if `beam' elements with quadratic interpolation are employed,as it was concluded by comparing numerical and experimental results (appendix C).In this work, linear elements (with shear deformation) are used in the discretizationof the towers, and it was observed that in this case the optimum element length ishalf the value obtained with expression (3.3); Lelem = lp/2.

In light of the observed deformation of the towers during the seismic analysis,a reasonable value Lc = 0.2Htot has been considered for both legs of the towerregardless of the response direction (Htot being the total height of the tower, fromthe foundation level to the top section, see gure 3.1) and Lc = 0.5B for transversestruts (where B = 25 m is the width of the deck). Therefore, the discretizationof the tower legs depends on their dimensions and hence on the main span (LP ),however, the element length in the anchorage area of the tower is always 2 meters(the distance between anchorages), in order to facilitate modelling.

Priestley's proposal was developed for solid columns, and some concern arisesaround the plastic hinge length in hollow sections, like the ones employed in thetowers. A specic study was conducted by the author and is briey discussed inappendix C, concluding that the expression (3.3) also yields good results in hollowsections. Figure 3.15 presents the comparison between FEM results and the ex-

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82 Chapter 3. Modelling and basic assumptions

perimental tests conducted by Takahashi and Iemura [Takahashi 2000], consideringcantilever hollow sections with reduced connement eect.

(a) Elevation and cross-section of the columnunder test

Quasi-static test

FEA Results

(b) Results of the comparison

Figure 3.15. Correlation between nonlinear nite element model proposed here and the

experimental results reported by Takahashi and Iemura [Takahashi 2000].

3.5.2 Discretization of the cable-system: cable-structure interac-tion

The energy transfer between the cables and the tower, or specially the deck, isa characteristic dynamic eect in cable-stayed bridges, as it was discussed previ-ously in section 2.2.1, which may be taken into account by representing each cablewith multiple elements (MECS). However, due to the great number of cable-stayedbridges considered, and the required parametrization, a simplication of the cable-system model is strongly recommended, which suggests the description of each stayby means of only one element (OECS). Furthermore, MECS models require thedenition of the prestress in each cable, which would force us to perform iterativeanalysis in each model to obtain the appropriate preload in all the cables (see section3.4). Here, the dierences between both approaches are briey addressed in termsof modal and seismic analysis. In MECS model, 16 elements per stay-cable havebeen employed24 (regardless of the total cable length).

Figure 3.16 presents the fundamental transverse vibration mode considering sin-gle and multiple element cable discretizations in the model with a main span of 400m, inverted `Y'-shaped towers and central cable layout. The exure of the cables in

24Recall that, according to Caetano [Caetano 2007], 9 elements per cable yields errors less than5 % compared with more rened mesh, as it was mentioned in section 2.2.1; the larger elementsemployed by Caetano in the cables are approximately 25 m, whilst the larger elements used herein MECS round 20 m, ensuring an accurate representation of cable-structure interactions.

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3.5. Finite element model description 83

MECS model is clearly observed, being coupled with the global transverse exureof the deck, which in this case does not modify signicantly the vibration period.The strongest modal coupling eect in bridges with LP = 400 m was also veried incomparison with other main spans. A great number of local cable modes appearedbetween the rst global modes of the bridge, with reduced participation factors andhence negligible seismic eect in the global response of the structure.

(a) Single-cable element (OECS): T2 = 2.65 s (b) Multi-element cable (MECS). T2 = 2.65 s

Figure 3.16. First transverse vibration mode obtained with single and multiple element

cable discretizations. Model Y-CCP with LP = 400 m and θZZ free.

The extreme transverse seismic shear along one leg of the tower (leg 1) is pre-sented in gure 3.17, averaged from the results of twelve three-directional records(rocky soil, TA), comparing both single and multi-element models. The seismicdemand in the towers when single-cable elements (OECS) are employed is largerthan the corresponding values with multiple elements (MECS) in all the consideredmodels and types of response measures (agrees with [Caetano 2000] since the earth-quakes are broadband). The dierences could be appreciable (up to 40 %) if modalcoupling is important, like in the case of the bridge with LP = 400 m presentedin gure 3.16. The reduction of the response in MECS models, observed also inother research works (section 2.2.1), is not surprising because of two reasons; (1)the vibration of the cables may absorb part of the seismic energy, acting as a TunedMass Damper (TMD) (section 2.4.2); and (2) with multiple element per cable, thevibration periods of important modes may be elongated (not in the studied case)and therefore the associated spectral accelerations decreased.

Despite the seismic forces are inuenced by the discretization of the cables,considering the cable-structure interaction (more realistic) leads to reduced seismicdemands. In light of the results obtained, one element per cable (OECS) is employedin all the models in this thesis, which falls on the safe side, and the discussion about

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84 Chapter 3. Modelling and basic assumptions

0 3 6 9 120.0

0.2

0.4

0.6

0.8

1.0

6 12 18 24 8 16 24 32

22

17

OECSMECS

Maximum seismic transverse shear force; V [MN]Y

Soil A. Y-CCP model

Figure 3.17. Extreme averaged seismic transverse shear force VY recorded along tower

height with dierent main spans and cable discretizations. Y-CCP model. θZZ free. Rocky

soil type A. Elastic analysis.

cable-structure interaction will be nished here. Each cable is thus represented withone single `truss'25 element.

3.5.3 Discretization of the deck

The deck in cable-stayed bridges with lateral cable layout (LCP) has been modelledseparating the metallic structure (longitudinal and transverse girders) from the con-crete slab, linking both parts by means of innitely sti connectors. Both transverseand longitudinal girders are discretized with linear 3D `beam' elements, whereas theupper slab is represented by means of `shell' elements with constant thickness (0.25m). The distance between the level of steel beams and the concrete slab is equalto the separation of the corresponding gravity centers, in order to capture properlythe exural and torsional behaviour of the deck (as it is described in gure 3.18). Itwas decided to use this shell-beam hybrid representation, in detriment of the classi-cal discretization using simply `beam' elements disposed along the gravity center ofeach deck cross-section (which is less expensive computationally), in order to avoidthe uncertainties associated with the torsional dynamic response of composite opensections26.

The torsional response of box-shaped deck in bridges with central cable-system

25`Truss' elements only resist axial forces, not exure.26However, Wilson and Gravelle [Wilson 1991] support the employment of one longitudinal line

of beam elements to represent the dynamic behaviour of open-section decks in LCP cable-stayedbridges, considering two lumped masses at each section in order to take into account properly thetorsional inertia.

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3.5. Finite element model description 85

Figure 3.18.Modelling detail of the deck in bridges with lateral cable-system (LCP).

assemblies (CCP) is more clear and, therefore, a simple line of beam elements havebeen used to describe their response (gure 3.19). A ctitious density (ρ∗) havebeen dened in these cases, taking into account the contribution of the concreteslab in the beam in terms of mass, since it has been homogenized to the steel:

ρ∗ =ρcAc + ρsAs

Ahom(3.4)

Where Ac and As are respectively the area of the concrete and the steel of deckcross-section, Ahom is the total cross-section area homogenized to steel, whilst ρcand ρs are the concrete and steel densities.

Figure 3.19.Modelling detail of the deck in bridges with central cable-system (CCP).

Regardless of the cable-system arrangement selected, the length of the elementsin the deck is about 5 meters, disposing two elements between consecutive cableanchorages.

3.5.4 Special-purpose elements

Non-structural mass

The realistic distribution of the mass along the bridge is an important aspect of themodels employed in dynamic analysis. Point `mass' elements have been employedto describe the mass of the asphalt27 (density 2300 kg/m3), lateral and centralbarriers, diaphragms of the deck (in CCP models) and the anchorages, among other

27The asphalt in the case of CCP models has been considered by increasing the density of thesteel associated with the beam model. In LCP models, however, point masses uniformly distributedover the surface of the shell representing the slab dene the mass of the asphalt (it is recognizedthat the same accuracy would be achieved by increasing properly the density of the concrete).

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86 Chapter 3. Modelling and basic assumptions

non-structural members in specic types of towers28. These `mass' elements areproperly located in the model, respecting the position of the gravity center of eachnon-structural member, like the barriers showed in gure 3.18.

Anchorages of the deck and the towers

The real separation between the gravity center of the deck (or the tower) and thecable ends, which is generated by means of the anchorages, is represented throughrigid links. The distances between the cable ends and the gravity center of thetower are presented in appendix B, both for lateral and central cable-system models(gures B.3 to B.7).

Foundations of the tower

As it has been previously mentioned, the exibility of the surrounding subsoil issimulated by means of linear springs at the bottom of the towers (more informationis included in appendix B).

Seismic devices

The seismic devices introduced in chapter 9 consist of springs, dashpots and/orconnectors, which are described in that chapter.

3.6 Spatial variability of the seismic action

This phenomenon is characteristic in long bridges and it was already discussed insection 2.2.5. Eurocode 8 [EC8 2005a] suggests that asynchronous seismic excitationshould be considered29 if the length of the bridge is larger than 240 m on rocky soils(TA) or 120 m on soft soils (TD). The proposed cable-stayed bridges have mainspans ranging from 200 to 400 m, and the total length of the deck takes values from360 to 1080 m. Therefore, the spatial variability should be considered in all themodels proposed in this work.

The problem of the loss of synchronization in the seismic action has been studiedin some detail in one of the most sensitive cable-stayed bridges to this phenomenon,and the discussion of the results has been moved to appendix D. The attention isfocused on dierent numerical approaches suggested by several authors and seismiccodes, which could be grouped with generality in three main categories; (1) staticprocedures based on imposed dierential displacements at foundation levels to rep-resent the pseudostatic forces; (2) modal spectrum methods with modied spectra(this approach is not considered in the study); and (3) response history analysis

28`A'-shaped towers include the mass of the upper concrete slab, which links both legs in theanchorage area. Models with lower diamond include the mass of the massive element in theconnection between the inclined legs below the deck and the vertical pier.

29The Spanish code, NCSP [NCSP 2007], states that the spatial variability needs to be consideredif the total length of the deck is larger than 600 m.

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3.6. Spatial variability 87

with delayed accelerograms depending on the position of the foundation and thecharacteristics of the wave-train (as it is depicted in gure 3.20). In this paragraph,only the most relevant results and conclusions of the study are briey presented.

Plan of the bridge

Bridge axis

Seismic waves propagation

Elevation of the bridge

Delayed record

Figure 3.20. Schematic representation of the loss of synchronization in the seismic waves

due to the separation of bridge supports. The parameters involved are completely described

in section D.2.3.

The longitudinal and transverse pseudostatic bending moments obtained in thefurthest tower from the epicenter (the right one) by means of dierent static proce-dures are presented in gure 3.21 for the studied model; inverted `Y'-shaped towerswith 400 m main span, lateral cable-arrangement, xed transverse rotation of thedeck in the connections with the abutments (θZZ xed) and foundation on rocky soil(TA). The strongest inuence of the spatial variability in the longitudinal responseof the towers, specially at the bottom section of the anchorage area, is clear fromthis plot.

Figure 3.22 presents the evolution of the longitudinal bending moment at thebottom section of the anchorage area in the right tower, obtained with dynamic anal-ysis based on delayed accelerograms. Two realistic wave-propagation velocities forrocky soils have been considered, and the response compared with the synchronousexcitation. The incidence angle, formed between the axis of the deck and the propa-gation wave-train is θ = 0. A strong increase in the seismic response under certainpropagation velocities is veried; ignoring the spatial variability could yield resultsup to 20 % on the unsafe side for the discussed results if vs = 1330 m/s, which is acritical wave-propagation velocity since it causes delays of the signal proportional toimportant vibration modes with longitudinal movement of the towers in the modelwith 400 m main span.

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88 Chapter 3. Modelling and basic assumptions

0

0.2

0.4

0.6

0.8

1

0 5000 10000 15000 20000 25000 30000

0

0.2

0.4

0.6

0.8

1

0 5000 10000 15000 20000 25000 30000

NCSPPriestley

EC8 - set AEC8 - set B (+)EC8 - set B (-)

NCSPPriestley

EC8 - set AEC8 - set B (+)EC8 - set B (-)

Dim

en

sio

nle

ss t

ow

er

he

igh

t; z

= z

/ H

*tot

Pseudostatic longitudinal bending moment; Myy [kNm]

Spatial variability. Static procedure

Dim

en

sio

nle

ss t

ow

er

he

igh

t; z

= z

/ H

*tot

Spatial variability. Static procedure

Pseudostatic transverse bending moment; Mxx [kNm]

Figure 3.21. Longitudinal (upper box) and transverse (lower box) pseudostatic bending

moments obtained in the right tower through dierent static procedures, based on imposed

displacements to address the spatial variability of the seismic action. Y-LCP model, LP =

400 m, θZZ xed and rocky soil (TA).

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3.6. Spatial variability 89

−150000

−100000

−50000

0

50000

100000

150000

0 5 10 15 20−150000

−100000

−50000

0

50000

100000

150000

0 5 10 15 20

Figure 3.22. Longitudinal bending moment time-history at the lowest section of the an-

chorage area in the right tower, considering synchronous and asynchronous excitation with

dierent propagation velocities. θ = 0. The result of the static procedure proposed by

Eurocode 8 [EC8 2005a] is also included. Y-LCP model, LP = 400 m, θZZ xed and rocky

soil (TA).

Several conclusions were drawn from this study, highlighting the following30;(i) the loss of the accelerogram correlation may be important (in a benecial ordangerous sense) in cable-stayed bridges with main spans between 300 and 500meters, which are long enough to present signicant delays, whereas the deck issti enough to exert important pseudo-static seismic forces in the towers due todierential imposed displacements in the foundations; (ii) the maximum error on theunsafe side, which may be introduced by neglecting the spatial variability, is below 20% in the most critical model and wave-propagation velocity studied (for rocky soil),nevertheless, larger errors are expected on soft soils due to the increment in the delayof the records (caused by the reduced wave-propagation velocity); (iii) the eect ofthe spatial variability was found to be more signicant for the longitudinal responseof the towers, specially in the anchorage area, due to the constraint exerted by thecable-system, which relates the towers with the deck; (iv) code-compliant proceduresbased on imposed displacements at the foundations in static analysis and simpliedcombination rules may neglect the eect of the spatial variability, whereas morerigorous strategies with time-domain calculations and delayed accelerograms maycapture appreciable dierences between uniform and non-uniform excitations.

Despite, in light of the results, the eect of the earthquake asynchronism maybe signicant for some specic main spans and types of response, the considerationof these phenomena is not the scope of the present thesis and the discussion aboutmulti-component excitations is nished here. This is reasonably supported becausethe transverse seismic response is the most demanding one (as it is observed in subse-quent chapters), and the inuence of the spatial variability is lower in this direction

30Further details may be found in appendix D.

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90 Chapter 3. Modelling and basic assumptions

(because the response is more exible than in the longitudinal one), furthermore,the maximum error recorded by ignoring the non-uniform nature of the excitationis below 20 % on rocky soil. Therefore, the seismic input in all the supports isconsidered the same hereinafter.

3.7 Symmetry of the seismic response

The large number of studied cable-stayed bridges, and the overwhelming computa-tional cost associated with some of the performed analysis in this work, stronglyrecommend the minimization of the required output information in the numericalresults. In the previous paragraph, it has been decided to consider the same excita-tion in all the supports of the bridge, on the other hand, we are interested mainly inthe seismic response of the towers (the critical members of the bridge). Therefore,it is very appealing the idea of concentrate the study only in the response of onetower, assuming that the other one presents the same results.

This would be true if both the structure and the excitation were symmetrical;the rst point is denitely true in all the cases, but the seismic input, causes the lossof the symmetry in the longitudinal direction (X) despite the same accelerogramsstrike all the supports; if the same displacements are imposed along the longitudinalaxis in the foundations of the abutments and the towers31, the cable-system tries tofollow the deformation of the deck, introducing tension in one tower and compressionin the opposite, as it is illustrated in gure 3.23.

Figure 3.23. Schematic representation of the loss of symmetry in the response of the towers

due to the longitudinal seismic action.

However, the relative longitudinal movements of the deck are moderate sinceit is xed at both abutments and, consequently, the longitudinal reaction is largelyconcentrated in these extremes, so the axial load in the towers due to the longitudinalaction should be reduced in the proposed structures.

This section aims to demonstrate that the seismic response of both towers couldbe assumed identical. The averaged extreme seismic forces recorded along the heightof the left and right towers is presented in gure 3.24. Nonlinear material behaviourand the most demanding seismic actions have been considered in a cable-stayed

31The intermediate piers only constrain the vertical movement of the deck, and hence the seismicexcitation is not imposed in these points along the longitudinal direction.

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3.8. Basic assumptions 91

bridge model with high seismic demands (YD-CCP, see chapters 7 and 8) in orderto maximize the nonlinear seismic response and, therefore, the expected dierencesbetween the response of each tower. In light of the results, both transverse and longi-tudinal responses are nearly independent of the tower considered, even in nonlinearrange.

Figure 3.24. Extreme averaged seismic longitudinal (left) and transverse (right) shear

forces along the leg of both towers. YD-CCP model with LP = 400 m and θZZ free. Soil

type D. Inelastic analysis.

The seismic response of both towers is assumed the same in linear and nonlinearrange, and only the left tower is studied in this thesis (except in the study of thespatial variability in section 3.6 and appendix D).

3.8 Basic assumptions

Detailed studies about physical and modelling aspects of proposed cable-stayedbridges have been performed in order to obtain a compromise solution betweenaccuracy and calculation agility, without losing sight of the objectives pursued bythe thesis. A brief summary of these verication studies has been presented inthis chapter, which is complemented by appendices B, C, and D, justifying severalassumptions.

As a general conclusion of the chapter, it could be said that it is neither possiblenor interesting to address all the seismic eects that could be associated with cable-stayed bridges, and it is therefore paramount to clearly establish a set of basicassumptions prior to the analysis, which have been presented throughout the chapterand will be summarized below:

1. The properties of the bridge are fully parameterized in terms of the main spanlength (LP ), taking into account the dimensions of a number of constructedcable-stayed bridges (section 3.2.1).

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92 Chapter 3. Modelling and basic assumptions

2. No attempt has been made to consider the soil-structure interaction (section3.2).

3. The transverse deck-tower connection in bridges without seismic devices issimplied; a sti link between the deck and one leg of the tower is disposed(section 3.2.3). Consequently, the seismic response along the opposite leg isnot considered.

4. The prestress of the lower strut to correct the tension observed in severalmodels prior to the earthquake is ignored (section 3.2.4).

5. The elasticity modulus of the cable-stays is deemed constant during the earth-quake, neglecting the eect of the cable-sag (section 3.3).

6. The concrete and steel constitutive properties in nonlinear range have beenxed following relevant codes and widely accepted research works. Despitethe degradation in the elastic stiness of the concrete due to cyclic loadingbeyond the linear range is not captured with the adopted model, cracking andother important eects are taken into account, and the good correlation withexperimental investigations reported by other authors was veried, validatingboth the material models and the discretization employed (section 3.3).

7. No prestress has been applied to the cables, and their sections have beenobtained through a simplied proposed procedure which avoids the classicaliterations. The self-weight of the deck and the cable-system is introduced inthe models by means of concentrated loads in their anchorages (section 3.4).

8. One element per stay-cable (OECS) has been used, neglecting the cable-structure interaction, which is a safe assumption (section 3.5.2).

9. The spatial variability of the seismic action has been neglected. Therefore, thesame accelerograms or spectra are imposed in all the supports of the bridge(section 3.6).

10. The seismic response of both towers is assumed identical, and hereinafter onlythe results of the one located on the left side in gure 3.23 are presented(section 3.8).

11. Sign criteria in normal stresses and deformations; positive sign indicates ten-sion and negative sign represents compression. Forces and bending momentsare presented in local coordinates (according to the orientation of the localcoordinate system, tangent to each `beam'-type nite element of the tower).

Other sources of uncertainties are involved with the seismic excitation and theanalysis procedures, being discussed in chapters 5 and 6 respectively.

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Chapter 4

Modal analysis

Contents

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2 Fundamental vibration modes . . . . . . . . . . . . . . . . . . 96

4.3 Higher mode contribution . . . . . . . . . . . . . . . . . . . . 99

4.4 Modal coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.1 Introduction

The study about the vibration modes of a structure is an essential step prior to theseismic analysis, even if the procedure selected is not based on modal decomposition,since valuable information about the dynamic behaviour is obtained and severalimportant design conclusions may be extracted in this initial stage.

A complete modal analysis is presented here, taking into account the backgroundincluded in section 2.2.1. The chapter starts with the study of the fundamentalvibration modes in all the proposed models and suggesting expressions in order toapproximate the rst vibration periods in terms of the main span (LP ), followedby the consideration of higher mode contribution in the dynamic response and thenumber of modes required in seismic analysis procedures based on vibration modes(modal dynamics and spectrum analysis, chapter 6). The chapter concludes witha brief study on the modal coupling of cable-stayed bridges and the appropriatemodal combination rules.

As it has been discussed in section 3.4, the vibration modes are extracted fromthe deformed conguration, after imposing the self-weight and obtaining the equi-librium (in this chapter the static step is elastic), but before the seismic action.

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96 Chapter 4. Modal analysis

4.2 Fundamental vibration modes

The rst vibration modes concentrate most of the contribution in the global seismicresponse of a structure, therefore it is important to analyze both the vibration periodand the deformed shape associated with these modes.

The type of surrounding subsoil is negligible in the fundamental vibration modes,since the foundations of the towers are too sti to be excited in the rst modes, hencethe type of soil is not distinguished in this paragraph.

Figures 4.1 and 4.2 respectively present the vibration period of the rst twomodes obtained in all the studied cable-stayed bridges in terms of the main span(LP ), distinguishing the cable-system arrangement. The empirical proposal ofKawashima et al [Kawashima 1993] to estimate the rst vertical vibration modeis also included (expression (2.1)), which is based on measurements in real cable-stayed bridges (section 2.2.1); a good correlation with the FEM results is evidencedboth in lateral and central cable assemblies. However, the rst transverse periodproposed by these authors (expression (2.6)) is not clearly applicable here, since it isgoverned by the transverse inertia of the deck and hence its width is a fundamentalfactor, being the value considered in this work (B = 25 m wide) dierent than thewidth of the deck in the structures studied by Kawashima and his co-authors. Onthe other hand, the rst transverse mode in the models with a main span of 200 mis misleading due to the coupling with the torsional response of the deck, and thusit has been removed from the results.

The following conclusions may be drawn from gures 4.1 and 4.2:

• The shape of the tower is almost irrelevant in terms of the rst vibration modeassociated with the vertical exure of the deck. The moderate movement ofthe towers in this mode, which is dominated by the deformation of the deck,explains this result. This conclusion is also supported by the the negligibletransverse tower displacement in this mode; their movement is slightly longi-tudinal (see gures 4.3(a) and 4.4(b)), being the inertia of the towers in thisdirection nearly the same among all the proposed structures.

• The rst transverse vibration mode mainly involves the deformation of thedeck, but also excites the towers in transverse direction, specially if the mainspan is moderate (lower than 300 m), because the transverse stiness of thedeck and the towers is more comparable and some coupling arises (see gure4.3(c)). This reason, added to the fact that the tower shape strongly dierenti-ates its transverse stiness, explains why the inuence of the tower typology islarger in transverse modes, specially for moderate spans. Section 7.6 exploresthe modal couplings between the deck and the towers.

• The fundamental vibration mode presents vertical exure of the deck if themain span is lower or equal than 400 m, above this limit, the fundamentalmode turns to a transverse exure of the deck; by increasing the main span(LP ) the width of the deck (which governs the transverse stiness), remains

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4.2. Fundamental vibration modes 97

Figure 4.1. First vibration periods obtained in models with lateral cable-system (LCP) in

terms of the main span, besides the empirical approximation proposed by Kawashima et al

[Kawashima 1993] for the rst vertical period.

Figure 4.2. First vibration periods obtained in models with central cable-system (CCP)

in terms of the main span, besides the empirical approximation proposed by Kawashima et

al [Kawashima 1993] for the rst vertical period.

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98 Chapter 4. Modal analysis

constant and hence the transverse stiness falls rapidly, becoming smaller thanin the vertical direction, where it is strengthened by the cable-system1.

• By comparing the rst vibration periods in bridges with lateral and central ca-ble arrangements, a modest decrease of these values in the range of consideredmain spans is observed if central-cable system is implemented. Several reasonsmay aect this result2 and no concluding remarks could be established.

Several research works propose expressions to estimate the rst vibration modesin cable-stayed bridges, as it was discussed in section 2.2.1, currently some ofthem could be considered classical formulae with indubitable utility [Wyatt 1991][Kawashima 1993]. However, having developed rigorous models based on the pro-portions of a signicant number of cable-stayed bridges in this thesis (appendix B),it may be useful to suggest new expressions in order to estimate these fundamentalmodes, based on modern structures with oating deck-tower connection (character-istic of seismic-prone areas), improving the correlation with the rst transverse modeand distinguishing the cable-arrangement. Least squares adjustments to the FEMresults included in gures 4.1 and 4.2 have been performed to obtain the followingoptimized expressions for the fundamental periods in vertical (Tv) and transverse(Ty) directions:

TV,LCP = 0.088L0.592P → 1st vertical mode. LCP (4.1a)

TY,LCP = 4.689 · 10−4L1.463P → 1st transverse mode. LCP (4.1b)

TV,CCP = 0.08L0.583P → 1st vertical mode. CCP (4.1c)

TY,CCP = 3.275 · 10−4L1.5P → 1st transverse mode. CCP (4.1d)

Where the units of LP and T are meters and seconds respectively.The predictions obtained with these expressions are close to the proposal by

[Kawashima 1993] (eq. (2.1)), based on real cable-stayed bridges, which suggeststhe accuracy of the formulae if the considered structure has span distribution, deckcharacteristics (the width should be around 25 m) and deck-tower connections sim-ilar to the ones considered in this work (established in chapter 3).

At this point, the mode shape of the rst vibration modes is considered. Similarresults have been obtained regardless of the tower shape and cable arrangement,except the torsional response of the deck, which is more exible in bridges with`H'-shaped towers (as it was discussed previously in section 2.2.1). Figures 4.3 and4.4 collect the rst seven vibration modes in `Y'-shaped tower models with lateral

1The vertical stiness of the deck is also strongly inuenced by its depth, which is slightlyincreased with the main span (mainly due to aerodynamic reasons [Astiz 2001], see appendix B).

2For example, the modications in the mass and stiness along the deck considering one cablearrangement or another.

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4.3. Higher mode contribution 99

cable-system and 200 or 600 m main span respectively (xed transverse rotation indeck-abutments connection, θZZ xed).

The transverse exure is the most demanding response in the towers due to theirconnection with the deck (demonstrated in chapter 7), in this sense, the vibrationmodes which excite the towers transversely deserve special attention; the transverseexure of the deck is coupled with its torsion in the main span3 (gures 4.3(c)and 4.4(a)), specially if the length is moderate, presenting in this case a signicantcoupling also with the transverse exure of the towers (the eect of the transversemodes in the reaction exerted by the deck against the towers has been addressedin chapter 7). Bridges collecting the cable planes in one point above the deck (i.e.inverted `Y'- and `A'-shaped towers with and without lower diamond) present modesassociated with the transverse antisymmetric exure of the towers that are coupledwith the second order torsion of the deck (gures 4.3(g) and 4.4(e)), and may appearbefore the rst order pure torsion of the deck.

The longitudinal exure of both towers is associated with the vertical exure ofthe deck by means of the cable-system, and therefore several vibration modes withthese characteristics appear among the rst ones, introducing signicant longitudinalseismic forces in the towers, specially in the anchorage area (chapter 7).

Finally, the uncoupled transverse and longitudinal response of the towers in therst modes should be remarked, only slightly compromised by modes with decktorsion. This property, besides the previous comment, are both important featuresfurther discussed in chapter 6, dealing with advanced pushover methods.

4.3 Higher mode contribution

The number of modes with signicant contribution depends both on the type ofstructure and studied response (axial loads, bending moments, displacements, etc.).As an attempt to ensure the proper representation of the global dynamic response4,seismic codes [EC8 2005a] [NCSP 2007] simply establish that, at least, 90 % of thetotal mass of the model (MT ) should be activated by the considered vibration modesin the three principal directions (j = X, j = Y , j = Z). The cumulative eectivemass ratio (ηjmodal) in j-direction is dened as follows:

ηjmodal =

∑nM

jeff,n

MT> 0.9 (4.2)

Where M jeff,n is the eective modal mass of the n-mode in the direction j,

obtained as follows:

M jeff,n =

(Γjn)2 Mn︷ ︸︸ ︷φTnmφn (4.3)

3The coupling between the transverse and torsional response of the deck is characteristic incable-stayed bridges [Wyatt 1991].

4A sucient number of modes included in the analysis is required in procedures based on modaldecomposition; modal dynamics (MRHA) and modal spectrum analysis (MRSA) (see chapter 6).

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100 Chapter 4. Modal analysis

Particularized for the nth vibration mode, Γjn is the participation factor in j-direction, φn the modal displacement vector and Mn the generalized mass, whereasm is the mass matrix of the structure. The participation factor is dened as:

Γjn =φTnmι

j

φTnmφn=φTnmι

j

Mn(4.4)

Finally, ιj is an inuence matrix, obtained as the movement of each degree offreedom when the unit rigid body motion in direction j is imposed at the founda-tions.

The great exibility of cable-stayed bridges causes problems in the applicabilityof expression (4.2), since a large number of modes is usually required to activate90 % of the total mass, specially if the towers have lower diamond founded onrocky soil, because a large part of the total bridge mass (MT ) is concentrated closeto their foundation, only excited by means of very high-order modes which alsoinvolve the exibility of the surrounding subsoil5. Figure 4.5 presents the evolutionof the cumulative eective mass with the number of modes considered in one of theproposed models with both types of foundation soil; normally the vertical directiongoverns the number of required modes, due to the large stiness of the towers inthis direction. The inuence of the subsoil category on the governing higher verticalmodes is also remarked; more modes are required to meet code provisions if thesubsoil is sti (e.g. class TA), since the order of the vertical tower modes is increasedin comparison with bridges on soft soil.

Normally, the frequency of the last mode required to meet Eurocode 8 provisions(flim) is about 7.5 and 10 Hz, as depicted in the left box of gure 4.6. However,the signicant inuence of higher modes in the extreme seismic forces of the towersis a matter of some concern, specially in terms of the axial load; this response isin part governed by vertical modes of the towers6 (with frequencies around 12 Hzin some cases with lower diamond on sti soils). These modes may fall beyond thelimits proposed by the seismic codes. The number of required modes to reach alimit frequency flim = 35 Hz has been explored7 in all the models, and the resultsare presented in the right box of gure 4.6, particularized for soft soil (TD). Thenumber of modes to be included signicantly increases with this new limit, butit has been veried that the results of spectrum analysis with both requirementsare generally the same, due to the small value of the participation factors in suchhigh-order vibration modes, which suggests that the number of modes proposed byEurocode 8 is adequate in nearly all the cable-stayed bridges considered (see section6.3.5 for more details).

5A complete discussion about the applicability of the seismic regulations in cable-stayed bridgeswas performed by the author in previous research [Camara 2008].

6Both pure vertical and transverse responses of the towers contribute to their axial load, butthe eect of the transverse accelerogram component is more pronounced (chapter 7).

7The limit frequency flim = 33 Hz is thought to be the upper limit of interesting frequencies inearthquake engineering [Chopra 2007], which is a bit enlarged here for research purposes.

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4.3. Higher mode contribution 101

(a) Mode 1: 1st vertical exure of the deck. T1 = 2.05 s

(b) Mode 2: 2nd vertical exure of the deck. T2 = 1.56 s

(c) Mode 3: 1st transverse exure + torsion of the deck +transverse exure of the towers. T3 = 1.30 s

(d) Mode 4: 3rd vertical exure of the deck. T4 = 1.16 s

(e) Mode 5: 1st torsion of the deck. T5 = 0.99 s

(f) Mode 6: 4th vertical exure of the deck. T6 = 0.91 s

(g) Mode 7: 2nd transverse exure of the towers + torsionof the deck. T7 = 0.90 s

Figure 4.3. First seven vibration periods obtained in one of the studied cable-stayed

bridges. Y-LCP model with LP = 200 m and θZZ xed.

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102 Chapter 4. Modal analysis

(a) Mode 1: 1st transverse exure of the deck. T1 = 5.55 s

(b) Mode 2: 1st vertical exure of the deck. T2 = 3.90 s

(c) Mode 3: 2nd vertical exure of the deck. T3 = 3.17 s

(d) Mode 4: 3rd vertical exure of the deck. T4 = 2.34 s

(e) Mode 5: 1st transverse exure of the towers + torsionof the deck. T5 = 2.05 s

(f) Mode 6: 1st torsion of the deck. T6 = 2.01 s

(g) Mode 7: 4th vertical exure of the deck. T7 = 1.95 s

Figure 4.4. First seven vibration periods obtained in one of the studied cable-stayed

bridges. Y-LCP model with LP = 600 m and θZZ xed.

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4.3. Higher mode contribution 103

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150 200 250 300 350 400

Mod Nº 131; T131 = 0,14 s

Mod Nº 113; T113 = 0,16 s

Dirección longitudinal (X). Terreno A (roca sana)Dirección transversal (Y). Terreno A (roca sana)

Dirección vertical (Z). Terreno A (roca sana)Dirección longitudinal (X). Terreno D (suelo)Dirección transversal (Y). Terreno D (suelo)

Dirección vertical (Z). Terreno D (suelo)

e

e

X direction. Soil TAY direction. Soil TAZ direction. Soil TAX direction. Soil TDY direction. Soil TDZ direction. Soil TD

Figure 4.5. Evolution of the cumulative eective mass ratio (ηjmodal) with the number of

included modes in dierent foundation soils, particularized for the three principal directions

j = X,Y, Z. Important vertical modes are highlighted. A-LCP model with LP = 400 m

and θZZ xed.

200 250 300 350 400 450 500 550 6000

200

400

600

800

1000EC8 f = Hz

200 250 300 350 400 450 500 550 6000

5

10

15

20

25

30

35lim

Figure 4.6. Left: frequency of the last vibration mode to be included in the analysis (flim)

in order to meet EC8 [EC8 2005a] in terms of the main span (LP ). Right: number of

required modes considering EC8 rule or imposing flim = 35 Hz, in terms of LP . Soft soil

(TD). θZZ xed in lateral cable-system (LCP) and free in central cable (CCP) models.

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104 Chapter 4. Modal analysis

4.4 Modal coupling

The characteristic modal coupling in cable-stayed bridges is well known in the lit-erature [Abdel-Ghaar 1991a] [Morgenthal 1999] (section 2.2.1), presenting manyvibration periods very close to each other. The strong modal coupling is indi-rectly veried in this paragraph by means of the ratio between consecutive periods;ρij = Tj/Ti (being Tj < Ti), included in gure 4.7 considering one of the pro-posed models with dierent main spans. It is observed that modal coupling is moreimportant between high-order modes.

0.5

0.6

0.7

0.8

0.9

1

50 100 150 200 250 300

Aco

plam

ient

o m

odal

; ρij

= Tj/ Ti

Número del modo de vibración

ρij = 0.714

Luz principal LLuz principal LLuz principal LLuz principal LLuz principal L

P = 200 mP = 300 mP = 400 mP = 500 mP = 600 m

LLLLL

Figure 4.7.Modal coupling (ρij) in H-LCP models with dierent main spans. θZZ xed.

Rocky soil (TA).

Seismic regulations [EC8 2005a] [NCSP 2007] preclude modal combination rulesdierent than CQC (equation (6.24) in chapter 6) in Modal Response SpectrumAnalysis (MRSA) if the following expression is satised:

ρij ≥0.1

0.1 + ξ(4.5)

Considering the damping ratio ξ = 0.04 yields the limit coupling ρij = 0.714,which is included in gure 4.7, verifying that CQC rule should be considered inresponse spectrum procedures (chapter 6) to perform the combination of modalmaxima in all the cases.

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Chapter 5

Seismic action

Contents

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2 Eurocode 8 spectra . . . . . . . . . . . . . . . . . . . . . . . . 108

5.3 Natural records versus synthetic accelerograms . . . . . . . 110

5.4 Synthetic accelerograms description . . . . . . . . . . . . . . 111

5.4.1 Signicant duration . . . . . . . . . . . . . . . . . . . . . . . 111

5.4.2 Calculation scheme . . . . . . . . . . . . . . . . . . . . . . . . 112

5.4.3 MRHA accelerograms: constant damping . . . . . . . . . . . 115

5.4.4 NL-RHA accelerograms: Rayleigh damping . . . . . . . . . . 115

5.4.5 Number of required records . . . . . . . . . . . . . . . . . . . 121

5.5 Synthetic accelerograms validation . . . . . . . . . . . . . . . 122

5.6 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.1 Introduction

Uncertainties associated with the seismic action may be disheartening; both inten-sity, duration, frequency content, epicentral distance and orientation of the structurewith respect to the wave-train (among others) are fundamental variables very dif-cult to predict, even conducting detailed studies based on the geology of the areaand the history of past seismic events. Furthermore, the return period of importantearthquakes is normally larger than the lifetime of the structure and, therefore, anuncomfortable question needs to be faced by the engineer: to what extent the struc-ture should be seismically protected?. The answer should take into account boththe importance of the structure and the seismic risk of its emplacement. This lastpart of the equation is addressed in the present chapter, introducing unavoidableand justied simplications in order to represent such uncertain action.

There are many research works and doctoral theses devoted exclusively to theearthquake excitation, which is also closely related to the geological branch of knowl-edge. Consequently, a thorough review of this topic is beyond the scope of thepresent thesis, which relies on the simplied seismic input indicated in seismic reg-ulations. However, several specic contributions have been made in order to deneand justify the accelerograms proposed here, which may be considered innovative; (i)the explicit consideration of constant or variable Rayleigh damping in these records,

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108 Chapter 5. Seismic action

distinguishing the analysis procedure in which they will be employed; (ii) the gen-eration of synthetic records based on specic frequencies of the analyzed structure;and (iii) the validation of the resulting articial signals with natural earthquakepredictions, verifying better performance of the proposed algorithm compared withother software.

The design action established by Eurocode 8 [EC8 2004] through elastic spectrais adopted here as a reference, and presented in section 5.2. However, the analysisprocedures based on the history of the response performed in this thesis requirethe denition of accelerograms compliant with the design spectra; a brief discussionabout pros and cons of natural and synthetic records in section 5.3 precedes thedetails about the employed articial signals and the number of required recordsin section 5.4, which are validated in section 5.5 and appendix E, by comparingseveral seismic measurements with empirical predictions about natural earthquakes.Section 5.6 collects the basic assumptions introduced to simplify the problem of theseismic action, closing the chapter.

5.2 Design seismic action: Eurocode 8 spectra

The elastic design spectra proposed by Eurocode 8 (part 1) [EC8 2004] have beenadopted in the present work, considering two extremely dierent subsoil classes;rocky soil (A) and soft soil (D), both selecting type 1 spectrum1 and design groundacceleration (PGA): ag = 0.5 g (dened on type A ground), which is representa-tive of high-prone seismic areas worldwide. The prescriptions of this code aboutspectra for vibration periods longer than 4 s have been followed, since models withmain span of 600 meters have fundamental periods about 6 seconds (see chapter4, gures 4.1 and 4.2). The damping ratio is ξ = 4 %, being lower than typical 5

% in order to consider the reduced damping of cable-stayed bridges (section 2.2.2),furthermore, the Spanish code [NCSP 2007] suggests this damping ratio (4 %) incomposite bridges under extreme seismic events, which is the case considered here.On the other hand, the design ground acceleration in vertical direction is simply90 % of the horizontal one2: avg = 0.9ag. Figure 5.1 presents the horizontal andvertical3 design spectra employed in this research as the reference seismic action.

The eect of the spatial variability in the seismic action is neglected, and theearthquake equally aects all the bridge supports (section 3.6). Figure 5.2 illustratesschematically how this action is three-directionally imposed in the coerced degrees

1Previous research performed by the author concluded that the design spectra associated withtype 1 events are much more dangerous for a cable-stayed bridge than the type 2 [Camara 2008],which was expected taking into account the higher spectral accelerations in type 1 spectra for thedominant long period area characteristic of cable-stayed bridges.

2However, several authors criticize this simplistic selection of the vertical spectrum, which mayunderestimate the real records if near-fault eects are present (see section 2.2.9).

3Eurocode 8 [EC8 2004] only prescribes the vertical spectra for periods lower than 4 s, despitethe fundamental vertical deck modes in the studied structures are always below this limit (seegures 4.1 and 4.2), it is assumed expandable for longer values due to possible period elongationsduring inelastic analyses.

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5.2. Eurocode 8 spectra 109

0.0 0.5 1.0 1.5 2.0Per iodo; T [s]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Aceleracionespectral;Sa[g]

Espect ro hor izontal

Espect ro ver t ical

0 1 2 3 4 5 6 7Per iodo; T [s]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Aceleracionespectral;S a[g]

Ter reno A

Espect ro hor izontal

Espect ro ver t ical

0 1 2 3 4 5 6 70.0

0.5

1.0

1.5

2.0

Vertical spectrum

Horizontal. Type D

Horizontal. Type A

EC8 spectra

Period; T [s]

Spec

tral

acc

eler

atio

n; Sa

[g]

Figure 5.1. Elastic design spectra (proposed by Eurocode 8 [EC8 2004]) taken as the

reference seismic action. Ultimate Limit State (ULS).

of freedom of the foundations along the structure4, either by means of the spectrapresented in gure 5.1 or through the accelerograms described below. Both horizon-tal components (EH) are considered equal in terms of the spectrum, and completelyindependent in time domain5, whereas the ground acceleration associated with thevertical excitation (EV ) is generated from a dierent spectrum, ensuring its inde-pendence from the horizontal records.

Seismic analysis procedures based on the history of the response (modal dynamicanalysis, direct dynamics and advanced pushover proposed in chapter 6) requiresthe denition of the seismic action in time domain, hence appropriate accelerograms(natural or synthetic) are obtained from the spectra depicted in gure 5.1, and ap-plied three-dimensionally6 to the structure following the scheme presented in gure5.2.

Strictly speaking, in addition to the three orthogonal components of the groundmotion, the rotational components may also induce signicant eects. The inuenceof these imposed rotations is usually small and it has been traditionally neglected[Kubo 1979]. However, rotational eects may be of importance in extended struc-tures such as pipelines [Nigbor 1994] and it has been included in Eurocode 8 (part6) [EC8 2005b], dealing with towers and masts, which suggests that it could be rel-evant in long cable-stayed bridges. More research on this topic is ongoing but theimposed rotations due to the earthquake are not considered in this thesis.

4The vertical piers do not constrain the horizontal movement of the deck, and therefore onlythe vertical seismic action is imposed in these points.

5The accelerogram components in 3D analysis must be independent, even if they are generatedfrom the same spectrum, according to current seismic codes [EC8 2004].

6Several specic studies in chapters 7, 8 and 9 consider the seismic excitation imposed only inone principal direction. If the opposite is not stated, the seismic action is three-dimensional.

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110 Chapter 5. Seismic action

Figure 5.2. Orthogonal components of the seismic action along the supports of a cable-

stayed bridge, expressed by means of spectra or accelerograms.

5.3 Natural records versus synthetic accelerograms

It is well known that stochastic-based synthetic accelerograms, despite matchingquite well the target design spectrum, do not represent the real characteristics of anatural seismic event, specially in terms of the stationary phase content, the rate ofinput energy and the number of strong motion peaks, which is higher in comparisonwith recorded accelerograms and leads to an unreasonably high energy content7.Due to these features, the use of synthetic accelerograms has been demonstrated tobe not recommendable in nonlinear calculations [Naeim 1995]. Many seismic codesand guidelines do not allow the use of synthetic accelerograms, although Eurocode8 [EC8 2004] does it.

One of the main drawbacks of employing natural records is the selection of com-pliant accelerograms; although there are extensive earthquake databases currentlyavailable for the engineer (e.g. PEER-NGA, ESD or ITACA) and specic soft-ware to perform the selection (e.g. REXEL [Iervolino 2009]), large scaling factorswhich may introduce some bias in the response are normally required to match theimportant PGA employed in the design action (since there are very few real earth-quakes which achieve such intensity). Furthermore, the t of the three componentsof the average spectrum in the range of important periods of the structure within

7There are nowadays new proposals trying to avoid the traditional problems associated withsynthetic accelerograms, like the ones included in Buratti's PhD thesis [Buratti 2009]. However,such procedures require information about the specic earthquake scenario, which is ignored in thegeneral study conducted here, discouraging their application.

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5.4. Synthetic accelerograms description 111

the tolerance criteria specied by seismic codes is sometimes intractable, as it wasexperienced by the author, specially for soft soils due to the lack of records in thisclass of terrain. On the other hand, matching the average spectrum (as it is recom-mended in seismic regulations), and not the particular spectrum of each record, maylead to signals with very dierent seismic demand in the structure, and consideringthe average response could be misleading in nonlinear analyses. To complete thepicture, it is known that there is invariably a degree of noise in all natural records,particularly at long periods, requiring the application of lters which limit the usableperiod range [Boore 2005], a factor that may be important in exible structures likecable-stayed bridges8. Moreover, the accelerometers may also introduce errors whenrecording the ground motion history of the high-order pulses; frequencies above 25Hz are not captured with enough accuracy [Chopra 2007] (which is not a problemin cable-stayed bridges as it is discussed in section 5.4.2).

Despite the lack of research and normative support about synthetic accelero-grams, they have been widely used in the seismic analysis of major structures world-wide, because very good matching with the target spectrum can be easily achieved,reducing the number of time calculations needed to obtain a robust result. In thepresent research, synthetic accelerograms represent the optimal way to compare theresults between models in linear and nonlinear range, avoiding the introduction ofbias in the results caused by scaling and/or matching natural records to the targetspectrum, and minimizing the number of computationally expensive time-domaincalculations. Synthetic accelerograms are also the most straightforward seismic in-put for dynamic calculations if only the elastic design spectrum is known, and thisis the case in the general study of the seismic behavior of cable-stayed bridges ad-dressed here, where no specic earthquake scenario (basically dened by the momentMagnitude and the Rupture distance) is introduced trying to maintain generality.

The question about the use of natural or synthetic records seems far from beingresolved today; in this thesis, synthetic records seems the only reasonable optiondue to the vibration properties associated with the large number of models studied,the large PGA considered and the generality of the research.

5.4 Synthetic accelerograms description

5.4.1 Signicant duration

Nowadays, there is a strong debate in the literature about the importance of mag-nitude on the structural response, and consequently, about the relevance of theearthquake duration, which is directly related to its magnitude. Several authorssuggest that magnitude should be considered in the selection of natural records fornonlinear dynamic calculations [Bommer 2000] [Bommer 2004], others argue thatmagnitude and rupture distance are not important if the records match the target

8For example, the European Strong-motion Database (ESD) collects accelerograms using anelliptical lter with cut-o frequencies of 0.25 and 25 Hz, and the rst interesting vibration modein this work is 0.15 Hz (section 5.4.2), which would be ltered.

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112 Chapter 5. Seismic action

spectrum [Bazzurro 1994] [Bazzurro 1998]. The results of Hancock and Bommer[Hancock 2006] conrmed that if the objective of the seismic study is the maxi-mum response of the structure, then the duration has little inuence (even if theresponse is nonlinear), but if we are interested in cumulative parameters (like energydissipation) the duration of the earthquake is a key factor in the response.

There are several denitions available for the signicant duration of the earth-quake, like DSR,5−75 and DSR,5−95 [Bommer 2009], which normally employ the con-cept of Arias Intensity measure (Ia), expressed below:

Ia =π

2g

∫ ttotal

0x2g(t) dt (5.1)

Where xg(t) is the ground acceleration, g = 9.81 [m/s2] is the earth gravitationalacceleration, and ttotal is the total duration of the signal above threshold. TheArias Intensity determines the intensity of shaking by measuring the acceleration oftransient seismic waves.

The Signicant Relative Duration DSR,5−75 aims to consider the strong pulsephase of the record; the interval starts at the instant when 5 % of the total AriasIntensity is reached and concludes when 75 % is achieved (see gure 5.10), measuringthe energy of the body waves. On the other hand, the purpose of DSR,5−95 is totake into account the full wave-train, extending the duration until 95 % of the totalArias Intensity is obtained.

NCSP [NCSP 2007] recommends values of DSR,5−95 above 10 to 60 seconds,whereas Eurocode 8 [EC8 2005a] indicates that the stationary part of the signalshould be larger than 10 s, but there is a deep lack of recommendations in thisdirection, which is likely due to the strong support that natural records currentlyreceive from the seismic codes. The study about the eects of the earthquake dura-tion is not the purpose of this thesis, and a reasonable value for the total durationof the record (ttotal) is considered; always equal to 20 s (being DSR,5−95 ≈ 14 s, asit is demonstrated in appendix E).

Priestley [Priestley 1996] and Eurocode 8 [EC8 1994] proposed a duration of thestrong pulse phase (tsp in gure 5.3) obtained by reducing the total signal lengthup to 20 % (see gure 5.3 below), which is adopted here; tsp = 0.2ttotal = 4 s. Thisvalue is strongly related to DSR,5−75, but they are based on dierent denitions;typically the proposed records present DSR,5−75 ≈ 7 s. A critical discussion aboutDSR,5−75 and tsp is included in appendix E.

5.4.2 Calculation scheme

An ad hoc modied scheme based on the proposed algorithm of Levy & Wilkinson[Levy 1975] (expression (5.2)) has been implemented9 in order to generate syntheticrecords. This algorithm has been selected due to its simplicity, retaining a funda-mental concept; the seismic signal may be decomposed as a sum of harmonics:

9The code employed in the implementation is Matlab [Matlab 2011].

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5.4. Synthetic accelerograms description 113

xg (t) = F (t)

Nm∑i=1

(−1)i Sai sin (2πfit+ ϕp) (5.2)

Where xg (t) is the synthetic accelerogram, F (t) is a modulating function whichdescribes the envelope of ground accelerations recorded during an earthquake, Nm isthe number of vibration modes included in the signal generation, whilst Sai and fiare respectively the spectral acceleration associated with i-mode and its frequency.

ϕp is a phase angle, randomly varied to make several independent synthetic ac-celerograms. According to Eurocode 8 specications, the three-axial seismic loadingshould be completely independent, in fact all seismic codes prohibit the use of thesame accelerogram in two orthogonal directions. The independence of the horizontal(longitudinal and transverse) records, which match the same spectrum, is ensuredby considering dierent phase angles (ϕp).

The envelope function10 F (t) adopted is presented in gure 5.3 and is takenfrom Priestley's recommendations [Priestley 1996] and Eurocode 8 (part 2, bridges)[EC8 1994]. Iterations are performed in order to obtain acceptable records match-ing the target code spectra, using in each i-frequency the ratio between the spectralacceleration obtained from previous iteration (Sai) and the target spectral accel-eration. The spectrum of the trial accelerogram is obtained in each iteration, in-tegrating the SDOF dierential equation of dynamics with a method based on theinterpolation of the excitation [Chopra 2007], considering a step-time equal to 0.01

s (which allows accurate calculations of responses with frequencies below 20 Hz).

1.00.050.1 0.3

0.35 0.5

Str

ong

puls

e

in

terv

al

F=1.0

F=0.75

F=0.5

F=0.25

Figure 5.3.Modulating envelope function, F (t), employed in the generation of the syn-

thetic accelerograms. Proposed by [Priestley 1996] [EC8 1994].

Eurocode 8 (part 2, bridges) [EC8 2005a] states that the average spectrumshould be above 1.3 times11 the target design spectrum, which in this case would

10The modulating envelope function F (t) is obtained as the ratio between the envelope absoluteground acceleration (xg,max(t)) and the PGA. Consequently F (t) ranges from 0 to 1.

11Such high safety factor is probably considered in Eurocode 8 (part 2) [EC8 2005a] to coveruncertainties due to the use of only three records, however, more restrictive conditions are appliedhere (e.g. 12 records are employed).

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114 Chapter 5. Seismic action

lead to unreasonably high seismic intensities.Following Eurocode 8 (part 1) [EC8 2004] specications, the average spectrum

resulting from the accelerograms should be above 90 % of the target spectral ac-celeration12 in the interval of contributing periods; dened as [0.2T1, 2T1], where T1

is the fundamental period of the structure in the direction where the accelerogramwill be applied. In this work, a more restrictive solution is adopted; the spectrum ofeach record, not only the average value, must be inside the tolerance range markedby 90 % and 115 % of each spectral acceleration13.

The t of the spectral accelerations inside this period range particularized forthe specic vibration periods (obtained prior to the earthquake) of all the proposedcable-stayed bridges was deemed important. It is not possible to consider all theperiods of each structure due to the large number of required modes (chapter 4)and studied bridges, which would cause convergence problems in the generationalgorithm. Instead, the vibration periods of all the models are grouped and sorted,and nally, around 100 modes have been selected in the interval of contributingperiods (Nm ≈ 100), distributed so that there is more density of samples in therange of governing periods.

In the signal generation, T1 is the largest fundamental period from all the stud-ied cases in any principal direction (corresponding to the models with LP = 600

m), in order to cover the whole set of important modes in all the proposed struc-tures with the same seismic action. The upper limit (2T1) yields too large periodsconsidering such exible structures (up to 11 s), therefore, the indication of NCSP[NCSP 2007] seems more reasonable; the upper limit of the contributing interval isassumed 1.2T1 ≈ 6.6 s 14.

With respect to the lower limit of the discussed interval of important periods,bridges with shorter spans control this value (because they are stier); followingEurocode 8 provisions, the result is 0.2T1 ≈ 0.4 s considering the rst global verticalmode in bridges with 200 m main span, but it has been veried that the rst verticalmode of the towers clearly contributes to the seismic axial loads with periods around0.25 and 0.083 s (i.e. 4 and 12 Hz). NSCP points that the lower limit should be1.25flim, where the concept of flim was introduced in section 4.3, taking values inthis case between 5 and 15 Hz (gure 4.6). Therefore, seismic regulations may leadto neglect important vibration modes in the generation of synthetic accelerograms,and hence to introduce misleading actions. On the other hand, the step-time of theanalysis is ∆t = 0.01 s (chapter 6) and hence frequencies higher than 10 Hz areltered15, but the range between this value and 20 Hz is reasonably well captured.

12Eurocode 8 [EC8 2004] states that the target spectral acceleration is the one dened by theelastic design spectrum with ξ = 5 %, however it has been considered here more reasonable toadopt the design spectra included in gure 5.1 with ξ = 4 %, which is on the safe side.

13Here, not only the minimum but also the maximum spectral accelerations are limited in orderto reduce the possibility of numerical problems associated with unreasonable strong shaking.

14Adopting 1.2T1 ≈ 6.6 s, the period elongation in bridges with LP = 600 m may be not includedin the denition of the seismic action, however, models with large spans do not present relevantinelastic seismic demands in comparison with shorter models (chapter 8).

15To study properly a vibration mode with period Ti; ∆t/Ti < 0.1 should be satised.

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5.4. Synthetic accelerograms description 115

According to Chopra [Chopra 2007], the contribution of frequencies above 33 Hz ismerely static. Taking into account this information, it was decided to consider thelower limit as 35 Hz (0.028 s), but reducing the number of frequencies in which theirspectral acceleration is to be matched to the target in the range between 12 and 35Hz (this may be observed in the right horizontal part of the spectrum in gure 5.5).

In conclusion, the interval of vibration periods which is adopted in the genera-tion of the articial records is [6.6, 0.029] s ([0.15, 35] Hz) regardless of the studiedstructure and direction. Around 100 vibration modes selected among all the pro-posed bridges have been considered, distributed along this period range in order toensure not only the proper spectral acceleration of the governing modes in all thedirections, but also the broadband frequency content of the signals. Furthermore,the possibility of period elongation due to damage in the structure is also coveredby the expansion of the maximum period included (1.2T1).

The step-time of the obtained accelerogram is 0.01 s, a normal value in earth-quake engineering.

Damping is an important factor within the implementation scheme to obtain thesynthetic records, which may lead to confusion depending on the analysis procedureemployed, as it is discussed in the following lines.

5.4.3 MRHA accelerograms: constant damping

In modal dynamics (MRHA), the integration of the system of dynamics (2.7) isdirectly based on the vibration modes and allows the denition of constant dampingin all the frequencies: ξ = constant = 4 %. Therefore, the same constant dampingis clearly applied to each SDOF system in the frequency range, when the spectrumof the articial record is obtained with the procedure proposed above. Figure 5.4graphically describes the framework.

Figure 5.5 shows the horizontal design and synthetic record spectra correspond-ing to soft soil (TD) in terms of the frequency, expressed in semi-logarithmic coor-dinates in order to highlight the t at higher modes. The spectrum of each articialaccelerogram is observed to fall within the tolerance range (in the interval between12 and 35 Hz this condition has been relaxed as it was already discussed), and henceit is ensured that the average value is compliant with Eurocode 8. The procedure hasbeen repeated for the set of records obtained from the horizontal and vertical spec-trum for soft and rocky soils. Each set of accelerograms have 12 samples obtainedby modifying ϕp in expression (5.2), this number is justied in section 5.4.5.

5.4.4 NL-RHA accelerograms: Rayleigh damping

Unfortunately, analysis procedures based on the direct integration of the coupledsystem of dynamics (2.7), with integration schemes like HHT [Hilber 1977], do notallow the denition of a constant damping ratio for all the vibration modes, andnormally Rayleigh damping distributions in terms of the frequency are considered[Abaqus 2010]:

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116 Chapter 5. Seismic action

Figure 5.4. Calculation of accelerograms for analysis procedures with constant damping

(modal dynamics, MRHA). Spectrum corresponding to synthetic accelerogram No. 1 of the

set associated with the horizontal target spectrum for soft soil (TD).

ξn (fn) =a0

2ωn+a1ωn

2=

a0

4πfn+ a1πfn (5.3)

Where ωn = 2πfn is the angular frequency and a0 is a coecient which estab-lishes the proportionality of the damping matrix (c) with the mass matrix (m) ofthe structure, whilst a1 determines the relationship between the damping matrixand the stiness matrix (k). Both coecients are obtained by imposing the desiredcritical damping ratio ξ = 4 % in the rst and the last frequencies of interest (ωiand ωj), resulting the expression:

a0 = ξ2ωiωjωi + ωj

(5.4a)

a1 = ξ2

ωi + ωj(5.4b)

The maximum frequency of interest in the selection of Rayleigh damping is notconsidered 35 Hz as in the case of the accelerogram generation, since this wouldproduce very low damping ratios in several important modes around 2 and 15 Hz.Instead, the upper limit is assumed 20 Hz, and hence the vibration modes above

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5.4. Synthetic accelerograms description 117

Figure 5.5. Synthetic accelerogram spectra tted to the design horizontal spectrum for

modal dynamics (MRHA). Soil type D.

this limit (which may introduce numerical instabilities) are more dampened. Thus,ωi = 2π0.15 = 0.94 rad/s and ωj = 2π20 = 125.66 rad/s are considered, whichintroduced in expression (5.4) yields; a0 = 7.48 · 10−2 and a1 = 6.32 · 10−4.

Therefore, the `environment' where the accelerograms are applied in direct dy-namics (linear (DRHA) or nonlinear (NL-RHA)) has a damping variable with thefrequency (gure 5.6). Considering a SDOF structure with natural frequency (fi)within the range [0.15,20] Hz and analyzed employing direct dynamics, its Rayleighdamping is lower than the reference value (ξ(fi) < 4 %) as it is shown in the up-per box of gure 5.6 and, consequently, if one of the accelerograms obtained in theprevious section with constant damping (ξ = 4 %) is imposed, the maximum to-tal acceleration recorded (uT ) is larger than the spectral acceleration of the targetspectrum16 for such frequency. Analogous considerations lead to lower spectral ordi-nates in the vibration frequencies outside the interval where the reference dampingis imposed in the calculation of Rayleigh coecients. This is the reason behindthe increased spectrum resulting when synthetic accelerograms obtained with con-stant damping are employed in direct dynamics (DRHA and NL-RHA) with variabledamping, which is presented in gure 5.6.

It is noteworthy that, in light of gure 5.6, synthetic accelerograms obtainedwith constant damping (section 5.4.3) may be employed in direct dynamics withRayleigh damping, being compliant with Eurocode 8, since they fall on the safeside along the range of governing modes, producing larger spectra than the target

16The dierent between pseudo-acceleration and acceleration is ignored in civil engineering ap-plications [Chopra 2007], hence Sa = ω2Sd = max(|uT |). The term pseudo, referred to the accel-eration spectrum, is usually removed in this thesis.

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118 Chapter 5. Seismic action

Figure 5.6. Rayleigh damping considered in direct dynamics (DRHA or NL-RHA) (upper

box); and average spectrum produced by synthetic accelerograms obtained with constant

damping (intended for modal dynamics (MRHA)) in both constant and variable damping

analysis, along with the target design spectrum (lower box).

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5.4. Synthetic accelerograms description 119

design ones. However, the real seismic action imposed to the structure should beborne in mind from the outset and may cause numerical problems in nonlinearanalysis due to its increased power. On the other hand, one of the purposes of thisthesis is to compare the results obtained with modal and direct dynamics (chapter6), and it is thus essential to impose equivalent seismic actions. Consequently,the accelerograms employed in direct dynamics with variable damping are obtainedfollowing a modied procedure described below.

The idea proposed now is to employ dierent damping ratios to dene the action;in this direction, Luco and Cornell [Luco 2007] suggested alternative ground-motionintensity measures based on the spectral acceleration with several damping ratios,which are intended for use in assessing the seismic performance of the structure.Here, the accelerograms proposed for analysis based on direct integration are ob-tained by introducing the specic Rayleigh damping frequency dependence directlyin the denition of each SDOF, which is integrated numerically in order to obtainthe spectrum of the trial accelerogram in each iteration of the generation scheme.Therefore, the nal resulting accelerogram yields the target spectrum in the `en-vironment' with the specied Rayleigh damping. Figure 5.7 aims to clarify theprocedure employed.

Figure 5.7. Calculation of accelerograms for analysis procedures with variable Rayleigh

damping (DRHA or NL-RHA). Spectrum corresponding to synthetic accelerogram No. 1

of the set associated with horizontal target spectrum for soft soil (TD).

The eect of damping reduction in the SDOF systems representing the vibrationmodes is clearly seen in their total acceleration response; comparing gures 5.4 and

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120 Chapter 5. Seismic action

5.7 it is observed that the extreme absolute total acceleration (uT ), which governsthe spectra, is nearly the same in both procedures as it was intended in order tomatch the design spectrum, however, the instant when this maximum occurs isdelayed if the damping is reduced.

Repeating the detailed process, appropriate sets of synthetic records are obtainedin the horizontal and vertical directions both for rocky and soft soils, being theirspectra in calculations with Rayleigh damping quite similar to the target one. Theresults of the t, analogous to gure 5.5, are not presented for the sake of brevity.

In order to verify the procedure proposed, a model of a simple 2D elastic can-tilever beam (gure 5.8(a)) is subjected to the following excitation; (i) one accelero-gram obtained with constant damping and analyzed with modal dynamics (MRHA),considering the same damping; and (ii) one record with the same spectrum than theprevious accelerogram, but obtained with Rayleigh damping and calculating withdirect dynamics (DRHA). Figure 5.8(b) presents the drift of the top section in thecantilever, observing that the extreme response is nearly the same with both proce-dures, despite the time-history is dierent as it was expected. Extended validationanalyses were conducted in other (more complex) structures, like the model rep-resenting only the `H'-shaped tower of a cable-stayed bridge, concluding that therecords generated with variable damping are appropriate for direct dynamics, inorder to compare the extreme seismic response with procedures in which the damp-ing is constant. However, it should be admitted that only the maximum responsewas considered here, and no attempt has been proposed to take into account theequivalence of the response history, which may be important if variables related tothe energy balance in nonlinear analysis are of interest.

(a) Problem description

0 5 10 15 20

-0.4

-0.2

0.0

0.2

0.4MRHA

NL-RHA

0.29 m 0.28 m

Time; t [s]

Rel

ativ

e dis

pla

cem

ent;

x [m

]

(b) Seismic response

Figure 5.8. Benchmark problem studied to validate the synthetic accelerograms obtained

for direct dynamic analysis (DRHA) with variable damping. Records matched to the hori-

zontal design spectrum associated with rocky soil (TA). Elastic 2D cantilever.

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5.4. Synthetic accelerograms description 121

Considerations about the applicability in inelastic dynamics

In addition to the dierences in the time-history of the response, introduced by theaforementioned procedure to generate accelerograms from variable damping `en-vironments', two features arise if the behaviour is inelastic; (i) the vibration fre-quencies could change in some extent due to material degradation, aecting thecorresponding Rayleigh damping (this is expected to have a reduced inuence); and(ii) the hysteretic dissipation increases the associated damping, being this eectvariable with the vibration mode, hence the accelerograms are generated with lowerdamping levels (the Rayleigh distribution) than the real ones, which is a safe as-sumption. The use of an inelastic spectrum [Chopra 2007] seems the solution to thissecond problem, but it is not considered here because this work is interested in thecomparison between the elastic and inelastic responses 8, and thus imposing iden-tical records is advisable; the same accelerograms obtained with variable dampingfrom elastic design spectra are employed in direct dynamics both with elastic andinelastic response, which is on the safe side.

5.4.5 Number of required records

Guidance on the selection of accelerograms is generally rather poor; Bommer andRuggeri [Bommer 2002] reviewed more than thirty seismic building codes and ob-served that the minimum number of required records is usually three or four, butresearchers agree that this limit is too low. A common requirement of regulationslike Eurocode 8 is to use a minimum of three ground motions but then take themaximum structural response, or consider 7 or more and then base the design onthe average behaviour. The number of required records depends primarily on the re-sponse parameter being measured, but also on the quality in the t between recordedand target spectra, the type of structure and the inelastic seismic demand, amongothers [Shome 1998] [Hancock 2008].

Specic studies about the optimum number of time-history analyses were con-ducted in this work in order to nd a compromise situation between the robustness ofthe results and the calculation agility. Despite the dierence between the spectrumof each record and the target never exceeds 15 % (section 5.4.2), it has been veriedthat the extreme responses may present signicant variations from one analysis toanother.

Twenty four modal dynamic elastic analyses were performed in one of the pro-posed cable-stayed bridges, and the results concluded that the average response of12 records leads to accurate and robust solutions. Figure 5.9 illustrates one of theseresults, remarking that employing only three records17 may lead to considerableunder-predictions in the axial load18 (up to 20 %), even if the extreme envelope re-

17Obtained with a more restrictive procedure than the one suggested by Eurocode 8, since notthe average value, but each synthetic spectra is matched to the target one.

18Here, the axial load is the governing response in terms of the number of required analyses,probably due to the contribution of vertical sti modes of the towers. These modes sometimespresent frequencies around 12 Hz, where the t with the target spectrum is worst (section 5.4.2).

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122 Chapter 5. Seismic action

sponse is assumed. Modern seismic codes propose a number of time-history analyseswhich may be unsafe in light of the results, in close agreement with other researchworks. If 12 analyses are considered to obtain the mean response, the standarddeviation of any seismic force or bending moment is typically 10 % along the tower(see section 6.3.5.1).

A) Torre en “H” B) Torre en “Y” invertida C) Torre en “Y” invertidacon diamante

D) Torre en “A” E) Torre en “A”con diamante

Extreme seismic transverse shear force; [MN]

Singular record resultAverage: 3 recordsAverage: 6 recordsAverage: 12 recordsAverage: 24 records

H-LCP. Soil TA. H-LCP. Soil TA.

Dim

ensi

onle

ss t

ow

er

heig

ht;

Singular record resultAverage: 3 recordsAverage: 6 recordsAverage: 12 recordsAverage: 24 records

Extreme seismic axial load; [MN]

Dim

ensi

onle

ss t

ow

er

heig

ht;

Figure 5.9. Extreme seismic forces in terms of the number of analyses considered to obtain

the average; (left) transverse shear VY , (right) axial load N . H-LCP model with LP = 200

m and θZZ xed. Rocky soil (TA).

5.5 Synthetic accelerograms validation

As it was introduced in section 5.3, many authors discourage the use of syntheticaccelerograms due to their unrealistic energy content. The root of the problem isthe shape of the target design spectrum to which the accelerograms are matched,obtained as the envelope of earthquakes with very dierent sources, leading to an un-realistic curve. An extensive validation of the synthetic accelerograms employed hasbeen moved to appendix E, here just the validation procedure and some importantconclusions are included.

The synthetic accelerograms are compared with natural ground motions corre-sponding to PEER-NGA database by means of typical seismologic features; PeakGround Acceleration (PGA), Arias Intensity (expression (5.1)), signicant durations(DSR,5−75 and DSR,5−95, section 5.4.1) and the Husid plot, which depicts the cu-mulative relative Arias Intensity along the record. The recommended procedure bythe earthquake engineering community, rst denes the earthquake scenario fromthe disaggregation of a Probabilistic Seismic Hazard Analysis (PSHA) made forthe specic location of the structure19, then the realistic seismological features and

19The earthquake scenario is basically dened by the combination of moment magnitude (MW )and rupture distance to the site, which is more consistent with the seismologic characteristics (typeof faulting, topography, sediment depth, etc.) of the location and the design return period.

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5.5. Synthetic accelerograms validation 123

earthquake spectra may be dened by means of empirical models already available.Unfortunately, the earthquake scenario is not established in this study due to itsgenerality, and only the target design spectra are known. As a reasonable approach,the inverse procedure is suggested for verication purposes; the earthquake scenarioshave been dened so that the empirical associated spectra match the target ones (asfar as possible) in the range of dominant modes (its unreasonable shape complicatesthe procedure), and based on this information, the empirical models predict dier-ent features of the natural earthquakes associated with these scenarios, which arenally compared with the synthetic accelerograms (more information is included inappendix E).

Figure 5.10 presents the average Husid plot from the set of 12 synthetic ac-celerograms obtained with the proposal of this thesis (section 5.4.2), or using a wellknown software for the generation of articial records; Simqke [Gasparini 1999]. TheHusid plot predicted for natural ground motions employing the empirical model ofStaord et al [Staord 2009] is also included. The algorithm proposed here, incomparison with Simqke, yields accelerograms with more realistic input energy dis-tribution along the signal, but it is still somewhat dierent than the prediction innatural records. This is likely due to the length of the strong pulse interval, which islarger than the predictions despite it has been obtained following the specicationsof Eurocode 8 [EC8 1994] and Priestley [Priestley 1996] (section 5.4.1).

Figure 5.10. Average Husid plot obtained with the proposal of this thesis for constant

damping, or using Simqke [Gasparini 1999]. The empirical prediction based on natural

earthquakes is included [Staord 2009] (red solid line). Soft soil (type D).

Figure 5.11 includes, for one specic set of accelerograms, the comparison be-tween the mentioned features in synthetic records and the predictions for naturalearthquakes. Again, the results of Simqke are worst in contrast with the ones ob-

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124 Chapter 5. Seismic action

tained with the proposed methodology. It is clear that the energy content (ariasintensity) is higher in the synthetic records, in agreement with other research works[Naeim 1995] [Buratti 2009], but the reason may be the larger value of DSR,5−75.This duration is strongly related to the strong pulse interval in the modulatingfunction (tsp), which in turn is proposed by the Eurocode 8 (section 5.4.1). It wasdecided to maintain tsp, supported by the conclusions of previous research, whichstated that the structural response is not particularly sensitive to the envelope mod-ulating function used, as long as it appropriately reects the energy content of theground shaking [Jangid 2004] [Quek 1990].

PGA [g] Ia [m/s] DSR,5−75[s] DSR,5−95[s]0

1

2

3

4

5

Sim

ulat

ed/P

redi

cted

0.6 2.6 5.0 11.60.7

10.4

8.3

13.80.7

11.8

10.3

14.1

MRHA accelerograms. Soil D

Predicted natural recordProposed in this thesis. SyntheticSimqke. Synthetic

(a) Accelerograms obtained with constantdamping, or using Simqke [Gasparini 1999]

PGA [g] Ia [m/s] DSR,5−75[s] DSR,5−95[s]0

1

2

3

4

5

Sim

ulat

ed/P

redi

cted

0.6 2.6 5.0 11.60.7

5.3

7.9

13.5

NL-RHA accelerograms. Soil D

Predicted natural recordPropoposed in this thesis. Synthetic

(b) Accelerograms obtained with variable damp-ing

Figure 5.11. Average value of the PGA, Arias intensity Ia and signicant relative du-

rations (DSR,5−75 and DSR,5−95). The values are expressed in ordinates as their ratio

with the predicted result in natural ground motions employing the model of Staord et al

[Staord 2009]. Soft soil (type D).

The algorithm proposed here is more appropriate than Simqke [Gasparini 1999]for the purpose of this thesis; the strong pulse interval tsp may be easily modiedthrough the modulating function (gure 5.3) and dierent damping may be con-sidered in the integration of the SDOF response, which is important in order toobtain adequate ground motions to be employed in analysis with Rayleigh damping(section 5.4.4).

The energy content is on the safe side for the proposed accelerograms, but itdiers somewhat from the predictions based on natural records. Nevertheless, goodagreement with other empirical predictions are observed, especially in the signals ob-tained with Rayleigh damping employed in direct dynamics (compare gures 5.11(a)and 5.11(b)). The energy content of the record is only of a concern in nonlinear anal-ysis due to the energy dissipation, hence the improve in the prediction of the energyin the accelerograms employed in these calculations is undoubtedly a good result.On the other hand, articial accelerograms for rocky soil (TA) are closer to theempirical predictions than the records used for soft soil (TD).

It is noteworthy that the t between the empirical natural spectrum and thetarget design one employed for soft soil (TD) is far from being perfect due to the

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5.6. Basic assumptions 125

unrealistic shape of the design spectrum and, therefore, the predicted features innatural earthquakes associated with this scenario should be quoted (more informa-tion is included in appendix E).

To sum up, the synthetic accelerograms are validated to be used in this thesisas the seismic input in elastic and inelastic seismic analysis.

5.6 Basic assumptions

This chapter is nished with the summary of the main assumptions introduced inthe seismic action considered hereinafter:

• Eurocode 8 three-axial design spectra with ag = 0.5 g (representative in seis-mic prone areas worldwide) and ξ = 4 % are considered as the reference seismicaction. The rotations imposed in the foundations by the ground motions arenot considered in this work (section 5.2).

• Taking into account the pros and cons of synthetic and natural accelerograms,and after a thorough validation process, articial records have been adoptedto conduct analysis based on the history of the response (section 5.3). Thestep-time of the accelerograms is 0.01 s.

• Far-fault earthquakes have been considered; no attempt has been made toinclude the eects associated with near-eld records in the generation of syn-thetic accelerograms, neither in the denition of design spectra.

• The sensitivity of the response to variations in the signicant earthquake du-rations is not considered in this thesis; reasonable values (suggested by seismicregulations) are adopted for the strong pulse interval in the modulating func-tion (tsp = 4 s) and the total duration (ttotal = 20 s) of the record, which aremaintained constant for all the generated signals (section 5.4.1).

• The specic orientation of the structure with respect to the wave-train is notstudied.

• The spatial variability of the seismic action is neglected, as it was discussedin chapter 3.

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