modal damping ratios for irregular in...

MODAL DAMPING RATIOS FOR IRREGULAR IN HEIGHT CONCRETE / STEEL STRUCTURES

Athanasios Papageorgiou, Charis Gantes National Technical University of Athens, School of Civil Engineering, Athens,

Greece, [email protected]

ABSTRACT

The present work deals with the dynamic response of structures that are irregular in height consisting of two parts. The first part is made of concrete, is founded on the ground and has a damping ratio equal to 5%, while the second one is made of steel, is resting on the first one and has a damping ratio equal to 2%. Current regulations do not cover the design of such structures, particularly the cases where the steel part has considerable mass compared to that of the concrete part. The only analyti-cal solution existing is in complex form, not suitable for every day use. Time history analyses of 2-DOF systems are carried out, in order to evaluate the error occurring by the use of these complex modal characteristics in real valued modal analysis. The results are shown in contour plots of modal equivalent damping ratio as a function of the dynamic characteristics of the two parts of the structure. These modal damping ratios are then applied for the analysis of a real 2-DOF structure.

KEYWORDS

Vertical irregularity, complex modes, time history analyses, equivalent modal damping, application in frame structure.

1 INTRODUCTION

Aim of this work is to deal with the seismic response of irregular structures, con-sisting of two parts, a lower part, called primary structure or substructure, founded on the ground, and an upper part, usually referred to as secondary structure or superstructure, resting on the primary one. The primary structure is usually denoted by the letter p and the secondary structure by the letter s. These two parts are characterized by different dynamic response. One reason for that can be the mate-rial distribution over the height of the structure, for example primary structure

158 A. Papageorgiou, C. Gantes

made of concrete and secondary one made of steel, leading to different damping properties of the two parts. Another reason could be the different lateral stiffness systems and/or energy dissipation mechanisms, e.g. a primary structure with brac-ings or bearing walls and a secondary structure with frames, or both. Thus, the overall dynamic response of the combined structural system, when it is subjected to earthquake excitation, can be very complex. Such structures are, for example, frequently encountered in stadiums like in Figure 1, where the spectators’ seats are usually configured on concrete frames, used also to house auxiliary facilities beneath the seats, while cover of the seats is often provided by steel frames or trusses supported by the concrete substructure. Other possible applications include the addition of relatively light steel frames on top of existing reinforced concrete buildings, in an effort to reduce the additional dead weight, or for speed of construction. In accordance with several practical applica-tions, such as the ones mentioned above, the particular case of structures where the lower part is made of reinforced concrete and has a damping ratio equal to 5% (typical for reinforced concrete structures) and the upper part is made of steel with damping ratio equal to 2% (typical for welded steel structures) is investigated here. The seismic design of such structures is not satisfactorily covered by the analysis methods suggested by current design codes (EC8), especially when the mass of the secondary system is of the same order as the mass of the primary system. If one decides to carry out a full time history analysis of the irregular structure, a damping matrix that will account for the different levels of damping must be created. This kind of analysis is not usual for every day design practice and can not be handled by existing commercial software. Another alternative is to separate the structure in its eigenmodes and carry out a modal time history analysis or a modal spectral analysis, but the resulting eigenmodes will be complex. None of the above alterna-tives is appealing for every day use, as they require significant computational cost as well as dealing with complex numbers, which is not generally implemented for

Fig.1: Stadium with spectators’ seats

Modal damping ratios for irregular in height concrete / steel structures 159

design in civil engineering structures. Thus, a simple yet conservative approach is usually adopted in practice, namely that the structure has an overall damping ratio equal to 2%. This makes possible the use of a code-based response spectrum to evaluate the maxima of the design quantities, e.g. absolute accelerations and/or shear forces, or even the execution of a modal time history analysis, without the drawback of complex eigenmodes. If the use of complex modes is finally accepted as a means of analysis, applying a response spectrum method is possible. As shown by Igusa and der Kiureghian [1], the mass, stiffness and damping matrices are easily transformed in the state – space formulation and then the modal characteristics can be estimated, e.g. modal fre-quencies, modal damping and eigenvectors. In the case where there are irregulari-ties in one of the aforementioned matrices, the resulting modal characteristics will be in complex form. Then, one can carry out a modal time history analysis using Duhamel’s integral, or proceed with a response spectrum method, appropriately modified to allow for the complex form of the modal characteristics. Another way of approaching the analysis of such irregularly damped structures is that of decoupling them in two parts, each one with normal damping distribution. The primary structure is analyzed first, and its response is then induced in the secondary structure as a new fictitious excitation, introducing thus an error in predicting the response of each part. Provided that the resulting error is adequately small, such a procedure is appealing for the case of concrete / steel irregular struc-tures investigated here, since it allows different design teams to deal with each part. Gupta and Tembulkar [2] have investigated the effects of decoupling of multiply connected secondary structures on the response of the supporting primary structure through an analytical procedure. Chen and Wu [3] also use analytical expressions to predict the error response of the secondary structure when decoupling occurs. Initially they study SDOF / SDOF structures and the results they obtain, are used to predict the decoupling error if the primary structure has more than one degrees of freedom. The merit of their approach is that it is not restricted to cases where the secondary structure has a considerably smaller mass when compared with the primary structure. Expanding the work of Chen and Wu [3], Papageorgiou and Gantes [4] performed time history analyses of SDOF / SDOF irregularly damped structures and investi-gated the error in the prediction of the response of the secondary structure. Their work expands also in MDOF / SDOF structures following that of Chen and Wu [3] and then in MDOF / MDOF structures. Furthermore, a modification to the decoup-ling procedure is introduced by adding the mass of the secondary structure to the one of the primary structure, in an effort to take it into consideration and reduce the decoupling error. Other decoupling methods include iterations in the analysis of the two parts. Adam and Fotiu [5] introduce two such analysis procedures to study the dynamic behav-iour of elastoplastic primary / secondary irregular structures. The result of the


iterations is that the error initially introduced is minimized and the obtained re-sponse characteristics are practically identical to the ones of the coupled procedure, where the structure is analyzed as a whole, while typically, the number of iterations is adequately small. Many investigators have proposed methods for simulating irregular damping distributions with more usual ones, mostly regarding cases of structures with added viscous dampers, due to their widespread implementation in buildings and the subsequent need for providing solutions for issues pertaining to their design. Per-haps the most commonly used method of that type is the so called modal strain energy method. The basic assumption of the method is that the actual complex mode shapes are replaced by real ones that correspond to the undamped structure. Then, the ratio of the energy dissipated by the damping devices to the total strain energy of the structure is calculated for each mode shape and the equivalent damp-ing for the specific mode is extracted. Based on the modal strain energy method, other methods have been developed in order to enhance its results. Bilbao et al. [6] simulate the effect of the added damp-ers in structures with a Rayleigh damping matrix that becomes diagonal with modal transformation using real modes. Lee et al. [7] arrive at closed - form analytical solutions for the estimation of the equivalent damping of structures equipped with dampers. They apply state - space transformation of the equation of motion of the irregularly damped system, and through a procedure of computations where the eigenmodes of the system are in complex form, they finally arrive at proposing real valued damping ratios for the structure, accounting also for the case of non linear damping devices. Shen et al. [8] conduct scale experiments on multi - storey RC frames. After the frames are damaged by the ground motion they add dampers, monitor their new response characteristics and derive analytical solutions for the equivalent damping of the system with the added dampers, by introducing a modi-fication in the modal strain energy method. Similarly Chang et al. [9] study steel multi - storey frames experimentally, and also propose a modification to the modal strain energy method to simulate the irregular damping distribution with equivalent modal damping. The aforementioned methods can be implemented in irregularly damped concrete / steel frames by appropriately simulating the damping of each storey with an equivalent damper and then use each proposal to evaluate the equivalent damping. A different approach is attempted by Huang et al. [10]. They examine a multi degree of freedom (MDOF) structure comprised of two parts, each with different damping ratio and also seek to extract conclusions for the equivalent modal damp-ing of the structure using a numerical and an analytical approach. In the numerical – exact one, they firstly simulate the structure with the exact damping distribution and then proceed with a new simulation where the damping is arbitrarily set to specific ratios. The error between the two analyses is then calculated and the equivalent modal damping is elected as the one that minimizes the error. In the


analytical – approximate method, they simulate each of the two discrete parts of the MDOF system with their first mode oscillator, and thus arrive at a 2-DOF system and they assume that the orthogonality condition applies for the damping matrix of the irregularly damped 2-DOF structure. From the normalized values of the damp-ing matrix they extract analytical expressions for the equivalent modal damping of the simplified structure, and compare it to the results obtained by the numerical solution with satisfactory accuracy. In the present work, an attempt is made to use the complex modal damping ratios resulting from the modal analysis of an irregularly damped system in their real value form, and conduct a modal time history analysis. The incentive for this is to investigate the error resulting from disregarding the complex values of the system response characteristics, in order to make it more appealing for every day design practice.

2 EQUIVALENT MODAL DAMPING RATIOS

A procedure combining that of Huang et al. [10] and the complex modal character-istics is attempted. Simple, elastic 2-DOF structures are considered, as the one shown in Figure 2, where iM and iK , ,i s p= denote the mass and stiffness of each part. The coefficients m

iC and kiC denote the Rayleigh mass and stiffness proportional

damping coefficients of each part of the structure. Their evaluation in the case of structures with uniform damping is a quite simple procedure. In the case studied here, with ζ p = 5% and ζ p = 2% a slight modification in their calculation has to be made, as described by Clough and Penzien [11]. For the SDOF-SDOF studied here, the damping matrix C is composed by two separate damping matrices sC and

pC , each one corresponding to one of the two different parts of the structure. First though, the stiffness and mass matrices have to be divided in two submatrices iM and iK , ,i s p= , as described by equations (1) and (2):

Fig.2: 2-DOF irregular structure


ss MM

=

00 0

, p

pM

M

=

0 00

(1)

s ss

s s

K KK

K K−

= − , p

pK

K

=

0 00

(2)

Then, the two eigenfrequencies ω1 and ω2 of the complete structure have to be evaluated through a classical eigenvalue analysis. Next, the Rayleigh coefficients for the two damping submatrices can be evaluated as follows:

,

,

i i

i

aa

ωωζω ω

=

+

0 1 2

1 1 2

21

, ,i s p= (3)

And then the two damping submatrices are extracted:

, ,i i i

i iC a M a K= +0 1 , ,i s p= (4)

Finally, the damping matrix is computed as the sum of the two damping subma-trices:

i

i

C C=∑ , ,i s p= (5)

The coefficients mic are calculated by the mass proportional part of the total damp-

ing matrix, e.g. ,i

ia M0 and the coefficients kic by the stiffness proportional part,

e.g. ,i

ia K1 , ,i s p= . In order to characterize the response of the system depending on the properties of the two parts, the eigenfrequency Rω and the mass ratio mR of secondary to pri-mary part are defined:

s

p

Rωωω

= , sm

p

MRM

= (6)

For the analyses purposes, the primary system is selected to have an eigenperiod equal to 0.1 sec and a mass equal to 1000 Mgr. A wide range of eigenfrequency and mass ratios is examined and for each ratio pair, and given the characteristics of the primary system, the complete 2 – DOF structure can be formed. First, for each pair of eigenfrequency and mass ratios, an exact time history analysis of the 2 – DOF structure is carried out as described by the following equation:

[ ]{ } [ ]{ } [ ]{ } [ ]{ } gM y C y K y M r x+ + = −&& & && (7)


Where [ ]M , [ ]C , and [ ]K are the mass, damping and stiffness matrices of the

structure, { }y is the vector of relative displacements of the degrees of freedom of the structure with respect to its base and ( .... )T

nr =

11 1 , for the general case of an n

degree of freedom structure. The damping matrix is formed as described above, allowing for the different damping ratios along the height of the structure. The ground motion gx&& is selected to be harmonic in resonance with the first mode of the 2-DOF structure. The results obtained are the total accelerations at each level e.g. { } { } { } gy y r x= +&& && && , and the storey shears { }V .

Following, the equation (7) is transformed into its state – space form:

gAu Bu Fx+ =& && (8)

where: M

AM C

=

0,

MB

K−

=

00

, FMr

= −

0 and

yu

y

=

& (9)

The new eigenvalue equation is:

i i iB s Aφ φ= − , i = 1, 2 (10)

where the eigenvalues is and eigenvectors iφ are complex valued.

The resulting modal damping ratio for each complex mode will be equal to:

)Re( ii

i

ss

ζ−

= , i = 1, 2 (11)

The plot of the resulting modal damping ratios of the two modes over the ( )mR Rω − plane is shown in Figure 3. The modal damping ratios ζ1 and ζ2 for each pair of eigenfrequency and mass ratios are used in a real valued modal time history analysis of the 2-DOF structure under the same excitation gx&& and the new approximate results obtained, are in terms of

total accelerations at each level{ } { } { } gy y r x′ ′= +&& && && and storey shears { }V ′ . Now,

each mode is analyzed as a SDOF oscillator with damping ratio ζ resulting from Fig. 3 according to the dynamic characteristics of the initial 2 – DOF system:

i i i i i i i gq q q xζ ω ω+ + = −Γ22&& & && (12)

And the final structural response is then calculated: y q′ = Φ (13)


(a) (b)

Fig. 3: Distribution of modal damping ratios, (a) first mode and (b) second mode

The maxima of the two responses, exact and approximate, are obtained and the error at each level between the two analyses is calculated as in equation (14). The plots of the errors are shown in Figure 4 and Figure 5.

( ) ( )( )

max max

maxaccl

y ye

y

′−=

&& &&

&&,

( ) ( )( )

max max 'maxshear

V Ve

V−

= (14)

As can be seen from Figure 4 and Figure 5, the final error from applying complex valued modal damping ratios in real valued modal analysis are very small, allowing thus their application in real structures. The error levels are adequately small, so that even the sum of the absolute values of the individual errors or their Euclidean norm is not at levels that are prohibitive to their application.

3 APPLICATION TO REAL STRUCTURES

The proposed equivalent modal damping ratios are next applied to a real two-storey, one-bay frame with realistic dimensions, shown in Fig. 6, in order to examine the accuracy of the proposed approach in a very simple structure. The frame has a concrete lower storey with damping ratio equal to 5% and a steel upper storey with damping ratio equal to 2%. The Young’s modulus of concrete is assumed to be equal to 27.5 GPa, while the corresponding modulus for the steel part is 210 GPa. The concrete columns have a rectangular cross-section of 50 cm by 50 cm, and the steel column has a HEB320 cross-section. The slabs are considered sufficiently stiff to ensure diaphragm action at both levels. The mass of the p-system is 20 Mgr and the one of the s-system is 16 Mgr. The dynamic characteristics of the structure are shown in Table 1 and the deformed shapes of each mode are shown in Figure 7.


(a) (b)

Fig. 4: Error in terms of total acceleration, (a) primary structure and (b) secondary structure

(a) (b)

Fig. 5: Error in terms of storey shear, (a) primary structure and (b) secondary structure

These values correspond to a mass ratio equal to 0.8 and an eigenfrequency ratio of 0.75. According to Figure 3, the resulting equivalent damping ratio is 2.5% for the first mode and 3.5% for the second mode. For the structure described above, subjected to a sinusoidal excitation in resonance with its first mode, an exact time history analysis is carried out, accounting for the real distribution of the damping, as well as three approximate equivalent ones, one with the use of the resulting modal damping of 2.5% and 3.5% for the two modes, one with the use of the conservative damping consideration of 2% and one consid-ering an overall damping equal to 5%. Also, spectral analyses are carried out using the CQC modal combination rule for the above damping ratios.


Fig. 6: Frame structure under consideration

Fig. 7: 1st and 2nd mode deformed shapes

Table 1: Dynamic characteristics of frame structure

Mode Period (s) Modal mass participation factor 1 0.14 0.84 2 0.06 0.16

From each analysis the absolute maxima of the total accelerations at each level of the structure were obtained. The error that each of the analyses with a uniform damping ratio exhibits, in comparison to the analysis with the actual damping distribution, is calculated as indicated next, in equation (15), and the resulting errors for each level of the structure are shown in Figure 8, for both time-history analysis and spectral analysis.

( ) ( )( )

*max max

maxex appr

ex

y ye

y

−=

&& &&

&& (15)


(a) (b)

Fig. 8: Error in terms of total acceleration, (a) primary structure and (b) secondary structure

The differences between the exact analysis and the ones with the arbitrarily se-lected damping confirm that for the structure under consideration the modal damp-ing ratio equal to 2.5% for the first mode and 3.5% for the second mode gives response properties that are significantly closer to the ones of the same structure with the exact damping distribution. Furthermore, as expected, the assumption of a 2% damping ratio gives conservative results, while 5% underestimates the response at both levels. The use of the harmonic excitation in resonance with the first mode that has a quite large participation ratio, as shown in Table 1, is rather conservative, which means that for an actual seismic record the resulting errors will be smaller. Thus, an upper limit for the estimation of the error occurring by the use of the equivalent modal damping ratios is provided.

4 CONCLUSIONS

Structures that have irregular damping configuration over their height are exam-ined. For a wide range of dynamic characteristics, the complex modal damping ratios are computed. Then they are used in real valued modal analysis and the occurring errors are estimated. An example of a real two-storey, one-bay frame with different damping ratios over its height is analyzed with its exact damping distribution, with the proposed one and with the damping ratios corresponding to each part. The proposed uniform ratio gave response characteristics that were close to the ones of the actual structure, while the other two damping ratios gave signifi-cant errors. Future extension of this work will include (i) evaluation of the pro-posed modal damping ratios for multi-story frame structures, (ii) expansion to more complex structural systems, and (iii) evaluation for the case of seismic input mo-tion, using actual accelerograms.


ACKNOWLEDGMENT

This research project is co-financed by E.U.-European Social Fund (75%) and the Greek Ministry of Development-GSRT (25%).

REFERENCES

[1] Igusa T., Der Kiureghian A., Response Spectrum Method for Systems with Non-Classical Damping, Proceedings 4th Engineering Mechanics Division Specialty conference, 1983.

[2] Gupta A.K., Tembulkar J.M., Dynamic Decoupling of Multiply Connected MDOF Secondary Systems, Nuclear Engineering and Design, 81, North Hol-land, 1984.

[3] Chen G., Wu J., Transfer-Function-Based Criteria for Decoupling of Secon-dary Structures, Journal of Engineering Mechanics, 125(3), ASCE, 1999.

[4] Papageorgiou A.V., Gantes C.J., Decoupling Criteria for the Dynamic Re-sponse of Primary/Secondary Structural Systems, Proceedings of the 4th Euro-pean Workshop on the Seismic Behaviour of Irregular and Complex Structures, 2005.

[5] Adam C., Fotiu P.A., Dynamic Analysis pf Inelastic Primary-Secondary Systems, Engineering Structures, 22, Elsevier, 2000.

[6] Bilbao A., Avilẻs R., Agirrebeitia J., Ajuria G., Proportional damping ap-proximation for structures with added viscoelastic dampers, Finite Elements in Analysis and Design, 42, Elsevier, 2005.

[7] Lee, S. H., Min, K. W., Hwang, J. S., Kim, J., Evaluation of equivalent damp-ing ratio of a structure with added dampers, Engineering Structures, 26, El-sevier, 2003.

[8] Shen K.L., Soong T.T., Chang K.C., Lai M.L., Seismic behaviour of rein-forced concrete frame with added viscoelastic dampers, Engineering Struc-tures, 17, Elsevier, 1995.

[9] Chang K.C., Soong T.T., Oh S.T., Lai M.L., Seismic behaviour of steel frame with added viscoelastic dampers, Journal of Structural Engineering, 121, ASCE, 1995.

[10] Huang B.C., Leung A.Y.T., Lam K.M., Cheung Y.K., Analytical determina-tion of equivalent modal damping ratios of a composite tower in wind induced vibrations, Computers & Structures, 59, Elsevier, 1994.

[11] Clough R.W., Penzien J., Dynamics of Structures, McGraw Hill, 1993.

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