evaluation and improvement of calculation methods for large ......collins and mitchell (collins and...

Proc. of the 11th fib International PhD Symposium in Civil Engineering Aug 29 to 31, 2016, The University of Tokyo, Tokyo, Japan

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Evaluation and Improvement of Calculation Methods for Large-Scale Concrete Structures in Service Limit States Reignard Tan1,3, Max Hendriks1,2, Mette Geiker1, Dan-Evert Brekke3, Terje Kanstad1 1Department of Structural Engineering, Norwegian University of Science and Technology, Rich. Birkelandsvei 1A, 7491 Trondheim, Norway 2Faculty of Civil Engineering & Geosciences, Delft University of Technology, Stevinweg 1, 2628CN Delft, The Netherlands 3Multiconsult ASA, Nedre Skøyen Vei, 0276 Oslo, Norway

Abstract Concrete structures shall be designed and constructed to limit cracking and crack widths for durability, functionality and aesthetic reasons. Current design methods and requirements are, however, only in a limited manner verified for large-scale concrete structures and long service life, as well as for new binder and concrete types. To facilitate an improved design basis for large-scale reinforced concrete structures, the present project on evaluation and improvement of calculation methods for large-scale concrete structures in Service Limit State has been initiated. Initially, the occurrence of shear cracks and excessive deformations in concrete cantilever bridges has been investigated. A calculation model based on the Modified Compression Field Theory was established under the assumption that creep in principal compression direction may cause the occurrence of diagonal shear cracks in webs of the cross section. In a shear cracked state, the shear stiffness will be significantly reduced, which further results in increase of shear deformations. The calculation model was applied to, and verified on a real segmen-tally cast cantilever bridge: the Sålåsund Bridge with main span L = 120 m, where this type of cracking was observed.

1 Introduction Shear cracks and excessive deformations tend to appear in segmental in-situ cast concrete cantilever bridges (CCB), and these problems have drawn great attention among bridge designers past decades (Tan 2013). The occurrence of excessive deformations is a well-known problem in CCB with hinge at mid-span, but has also been observed in continous CCB without hinge recently. The occurrence of shear cracks mainly concern durability aspects of the structure, but is considered to be harmless as long as the crack width is within an acceptable nominal value (EN-1992-1-1 2004). However, shear cracks may also lead to reduction of shear stiffness, which further results in increase of shear deformations (Park 1975). Combined with the occurance of excessive deformations, this may lead to reduced service life, aesthetics and functionality for the structure. This research was carried out as an initial part of a PhD study entitled evaluation and improvement of calculation methods for large-scale concrete structures in Service Limit State.

It is common to distinguish between uncracked and cracked state when designing CCB. Conducted traditional analyses for CCB often show that the increase of total deformations due to shear defor-mations in uncracked state is negligible (Takács 2002). Thus, the designed pre-camber is in most cases based on long-term effects caused by bending deformations, prestressing, creep, shrinkage and relaxa-tion of prestressed steel only. Furthermore, stress levels obtained in traditional analyses are in most cases below the tensile strength of the utilized concrete quality due to extensive prestressing, which implies an uncracked state during the service life of the structure. However, since the occurrence of shear cracks and excessive deformations actually do appear within the service life of the structure, it is evident that these problems are not identified when conducting traditional design analyses of concrete cantilever bridges. Traditional design analyses are primarily executed using linear 3D-beam elements

11th fib International PhD Symposium in Civil Engineering

2 Structural Analysis and design (Evaluation and Improvement of Calculation Methods for Large-Scale Concrete Structures in Service Limit State)

in a program formulation which include the ability of simulating the segmentally development of can-tilevers during construction phase, as well as the long-term behavior of the structure after completion.

2 Sålåsund Bridge Sålåsund Bridge exhibit the same problems described above and is used as a case study. The structure is a prestressed box-girder bridge consisting of deck slabs in the sidespans from axis A1 to A3 and A6 to A7, and a CCB part from axis A3 to A6, see Fig. 1. The total length of the bridge is 348 m with a main span of 120 m. The CCB part is continuous without a hinge in the mid-span. The cross-section is a hollow box-girder type and varies in height from 2.0 m at mid-span to 7.0 m at piers for the CCB part. The structure is situated in the county of Møre- and Romsdal in Norway and was opened in 1977. A distinct deformation at mid-span can be observed in Fig. 2 (left), while Fig. 2 (right) depict diagonal cracks in the web of segment 142. The Norwegian Public Roads Administration (NPRA) conducted a thorough maintenance inspection in 2010 and the report clearly states pronounced deformations at mid-span and diagonal cracks with varying intensity in all segments for the CCB part (BRUTUS 2010).

Fig. 1 Segmental division of Sålåsund Bridge and spans.

Fig. 2 Photo of Sålåsund Bridge (left) and shear cracks in web at segment 142 (right).

In order to simulate the deformation history of the structure, a traditional design analysis of Sålåsund Bridge has been conducted. The model includes the historical segmental construction phase and the long-term material behavior according to Eurocode 2 (EN-1992-1-1 2004). Results obtained from the analysis show small deviations in long-term deformations when compared to the designed pre-camber for Sålåsund Bridge, see Fig. 3. Furthermore, stress levels are observed to be below the tensile strength of the utilized concrete quality in the structure. These results confirm that the occurance of shear cracks and excessive deformations usually are not identified when conducting traditional design analysis.

Preparation of papers for the 11th fib International PhD Symposium in Civil Engineering (Evaluation and Improvement of Calculation Methods for Large-Scale Concrete Structures in Service Limit States)

Reignard Tan, Max Hendriks, Mette Geiker, Dan-Evert Brekke, Terje Kanstad 3

Fig. 3 Calculated long-term deformations and designed pre-camber for Sålåsund Bridge.

3 Hypothesis As the problems are not identified in traditional design analyses, it is natural to point suspiciousness towards possible flaws in the construction phase. According to a routine inspection shortly after opening of the bridge, no visible cracks or distinct deformations were observed in any part of the structure. However, a remarkable observation from later inspections showed that shear cracks seemed to have appeared a couple of years after opening, before growing in quantity and magnitude with time. A sim-ilar behavior was observed for the distinct deformations at mid-span. These observations, and knowing that diagonal cracking reduces the shear stiffness which further results in increase of shear defor-mations, leads to an assumption that the problems may be related to each other as a function of time rather than as a consequence of possible flaws occurring in the construction phase. A more controversial assumption is to claim that creep in the direction of minimum principal strains (maximum compression) may cause the occurrence of shear cracking. Nevertheless, the latter statement may be justified by con-sidering equilibrium of slender webs exposed to relatively large shear stresses.

The panel in Fig. 4 (left) is considered a part of an uncracked web in a concrete cantilever bridge, subjected to a large constant shear stress over time that initially generates a strain state as illustrated. The minimum principal strains ε2,0 are now assumed exposed to creep deformations due to the presence of a large minimum principal stress. The assumption postulates an increase of minimum principal strains with time, even though the shear stress is constant. As the minimum principal strains grow with time, it is reasonable to assume that the maximum principal strains similarly grow with time under the assumption of continuing simple shear. From a physical point of view, the assumption is claiming that a shortness in minimum principal strain direction leads to an elongation in maximum principal strain direction. Diagonal cracking occur when the maximum principal strains reach the tensile strain capacity of concrete, see Fig. 4 (right).

Fig. 4 Slender webs exposed to shear stresses and creep in minimum principal strains.



Shortly summarized, the hypothesis for shear cracking in CCB is based on an assumption that the in-fluence of creep on the minimum principal strains causes the occurrence of diagonal cracking. As crack-ing grows, the shear stiffness reduces and results in increase of shear deformations.

4 Calculation method As the occurrence of shear cracks and excessive deformations are not identified when conducting tra-ditional design analyses, a more proper calculation method is required. Collins and Mitchell (Collins and Mitchell 1991) offers an analytical model based on the Modified Compression Field Theory (MCFT) developed by Vecchio and Collins (Vecchio and Collins 1986). The model has the ability of predicting non-linear response in reinforced concrete cross sections subjected to the combined effect of moment, shear and axial forces, both in the pre-cracked and post-cracked state. With a proper algorithm, the calculation model yields relevant data such as average stress and strain states, and average crack angle and crack widths over the cross-section height. Tan (Tan 2013) gives a thorough description of the calculation model and the utilized algorithm.

As mentioned earlier, there were observed cracks with varying quantity and magnitude in every segment of the CCB part. However, segments with most cracks and largest crack widths were observed in segment 142 and adjacent segments (and symmetric for segments to the adverse cantilever), see Fig. 1. The magnitude of the observed crack widths was in the range of w = 0.3 mm – 0.6 mm and might according to Eurocode 2 (EN-1992-1-1 2004) pose a potential threat to the durability of the structure. Segments in this area lie in a typical inflection point, where there are relatively large shear forces com-pared to shear capacities. This area often tend to yield problems with respect to traditional shear design according to standard and codes. The discussion further on is limited to the behavior of segment 142.

The cross-section geometry and corresponding reinforcement is depicted in Fig. 5. The average cross-section height and average bottom girder thickness is set to 2984 mm and 281 mm respectively, while the thickness of the webs is 250 mm. There are in total 2x9 prestressed tendons symmetrically distributed in the top girder (total Ap = 21312 mm2), where the strain difference due to prestressing is set to Δεp = 6.0·10-3. The distance between web reinforcement is 200 mm and the average shear height is set to 2573 mm. The concrete is assumed to crack when a tensile strain of εcr = fcr/Ec = 0.068·10-3 is reached. The yield strength of reinforced- and prestressed steel is set to 400 MPa and 1550 MPa re-spectively.

Fig. 5 Averaged cross-section of segment 142.

5 Shear-deformation response In order to predict the shear-deformation response of segment 142, sectional forces are required. Gen-erally, sectional forces vary with respect to time due to the time-dependent effects of creep and shrink-age in concrete and relaxation of prestressed steel. Shear forces however, alter to a minimal extent with respect to time and are assumed constant in the segment. Thus, only moments and axial forces are required averaged with respect to time in segment 142. These are averaged from results obtained in the traditional design analysis and have the following values M = 27.3 MNm (moment, averaged value)

(horizontal)

(vertical)



N = 0.83 MN (axial compression force, averaged value) V = 2.28 MN (shear force, constant value)

These forces include the total effect of prestressing. If the averaged values for moment and axial forces are held constant, the complete shear-deformation response may be predicted, see Table 1. Each row in the table represent a stress and strain state in equilibrium with the applied shear force V in column 10. Furthermore, each parameter in the table is an averaged value over the effective shear height according to MCFT for cross-sections (Collins and Mitchell 1991), where ε1 is the maximum principal strain ε2 is the minimum principal strain θ is the angle of the minimum principal strain with respect to longitudinal direction εx is the strain in longitudinal direction γxy is the shear strain fc2 is the minimum principal stress fc2max is the characteristic compressive cylinder strength in biaxial stress conditions fv is the steel stress in web reinforcement w is the crack width V is the applied shear force

A plot of shear strain γxy versus shear force V in Table 1 is depicted in Fig. 6. Three important stages are noticed as the shear force evolves. The first stage being an uncracked state, where the segment shows a linear behavior and large stiffness. The second stage shows behavior in the post-cracking state, where a non-linear behavior is observed. The shear stiffness is now considerably reduced as opposed to the uncracked state which is expected. In the last stage, yielding of web reinforcement occurs and the segment capacity is now limited by the ability of cracks transmitting shear stresses through aggre-gate locking (Walraven 1981; Vecchio and Collins 1986). At this stage, crushing of concrete is also imminent.

First row in Table 1 represents the strain and stress state in equilibrium with actual sectional forces in the segment. It is noticed that the maximum principal strain ε1 is below the crack strain of εcr = 0.068·10-3, which implies an uncracked state and corresponds with observations according to routine inspections shortly after opening of the bridge. Furthermore, it is observed in row four that a shear force of V = 5.94 MN must be attained in order for the crack strain to be reached. Moreover, it can be shown that an applied traffic load according to the Norwegian code (Håndbok-185 2009) would lead to a superimposed shear force of V ≈ 3.93 MN in the segment, which is still not sufficient to generate cracks. Based on these observations, it can be concluded that the level of loads is not sufficient for shear cracks to evolve.

Table 1 Table for shear-deformation response in segment 142.

ε1 [10-3]

ε2 [10-3]

θ [°]

εx [10-3]

γxy [10-3]

fc2 [MPa]

fc2max [MPa]

fv [MPa]

w [mm]

V [MN]

0.01 -0.41 8.75 -0.40 0.13 11.1 30 0.04 0.00 2.28

0.02 -0.42 12.2 -0.39 0.18 11.2 30 0.06 0.00 3.13

0.05 -0.43 18.6 -0.38 0.29 11.6 30 0.12 0.01 5.04 0.068 -0.44 21.2 -0.38 0.35 11.9 30 0.13 0.02 5.94

0.1 -0.44 19.8 -0.38 0.35 11.8 30 7.48 0.03 5.46

0.5 -0.49 20.9 -0.36 0.66 12.9 30 74.7 0.16 6.05

1.0 -0.56 22.6 -0.33 1.11 14.4 30 153 0.31 7.07 1.5 -0.66 24.2 -0.30 1.62 15.7 28.4 227 0.46 8.01

2.0 -0.79 25.5 -0.27 2.17 16.8 26.3 296 0.61 8.84

2.5 -0.95 26.7 -0.25 2.77 17.8 24.5 360 0.76 9.60

3.0 -1.08 26.7 -0.25 3.27 18.0 22.9 400 0.92 9.58 3.5 -1.19 26.5 -0.25 3.75 18.0 21.5 400 1.06 9.49



As mentioned earlier, crack widths were observed in the interval of 0.3 – 0.6 mm for segment 142. Row 7 and 8 in Table 1 imply that a shear force of V = 7.07 – 8.01 MN must be attained in order for crack widths of the observed magnitude to evolve. Thus, these observations justify the credibility of the for-mulated hypothesis additionally as it already may be disregarded that the load level in the segment would generate shear cracks.

If the formulated hypothesis shall be realistic, minimum principal stress fc2 in the segment has to be of a significant magnitude. It may be observed from row 1, which is the state of equilibrium with actual sectional forces, that the minimum principal stress is fc2 = 11.1 MPa. This corresponds to ap-proximately 29% of the mean compressive strength fcm of the concrete quality used for Sålåsund Bridge. One of the requirements for using creep models according to Eurocode 2 (EN-1992-1-1 2004) and Model Code 1990 (CEB-FIP 1993) is that stress levels must be below 40% of the mean compressive strength (Bažant 1988). This elucidates the large minimum principal stress level in the segment and may amplify the assumption in the formulated hypothesis. Another important observation is that the value of the longitudinal strain εx is in compression for the whole segment, mainly due to the extensive prestressing. This combined with relatively large shear stresses are the main reasons for the presence of large minimum principal stress in the segment

Fig. 6 Shear-deformation response of segment 142.

6 Shear-deformation response with time dependent analyses The formulated hypothesis has also been attempted to be investigated explicitly by conducting time dependent analyses of segment 142. The influence of creep on the minimum principal strains is imple-mented in the calculation model of MCFT thru a simplified approach proposed by Tan (Tan 2013). In this approach, a constitutive relation between shear strains and creep strains is formulated based on compatibility conditions for a cracked element according to Vecchio and Collins (Vecchio and Collins 1986). The creep model utilized is according to Eurocode 2 (EN-1992-1-1 2004) where the effective cross section thickness h0 is assumed related to the effective shear area only. Sectional forces are applied with same magnitude as earlier and are held constant. The creep curve is linearly interpolated over logarithmically discretized time steps. Results obtained from the analyses are depicted in Fig. 7 and show that the time dependent behavior of segment 142 corresponds fairly well with the discussed stages in Fig. 6 and observations from routine inspections.

7 Discussion The simplified approach proposed by Tan (Tan 2013) is based on MCFT, which is an analytical model for predicting the non-linear reponse of an arbitrary cross section subjected to sectional forces (Collins and Mitchell 1991). Thus, the implementation of creep in MCFT is questionable, as it originally is not intended for conducting time-dependent analyses. Furthermore, important assumptions regarding the constitutive relation between shear strains and creep strains, and the utilized creep model, influence results significantly. The intention however, is initially to investigate the time dependent behavior of the segment in a relatively simplified manner, which seem to correspond fairly well with the formulated hypothesis and observations from routine inspections. Still, a more refined calculation method should be conducted. A possible method is to conduct non-linear finite element analyses with viscoelastic



material features as a virtual experimental approach, to increase the understanding and further verify the acclaimed behavior based on results from the simplified approach. Furthermore, it is also important to be aware of that the concrete tensile strain capacity (εcr) is both time- and scale-dependent.

Fig. 7 Time dependent behaviour of segment 142 with simplified approach.

8 Conclusions The working hypothesis for this research, which basically is based on observations from routine inspec-tions on Sålåsund Bridge, was that diagonal cracking in CCB may be explained by the influence of creep on the minimum principal strains. The formulated hypothesis is attempted justified by conducting a complete shear deformation response of segment 142 based on MCFT. An interestering observation from the obtained results is that sectional forces alone do not seem to cause occurrence of cracks. An-other prominent observation is the presence of a significantly large minimum principal stress in the segment, caused by relatively high shear stresses and extensive prestressing. The latter observation enables the possibility of creep influencing minimum principal strains. Furthermore, results obtained from time dependent analyses correspond fairly well with observations from routine inspections. How-ever, the calculation method for predicting the time dependent behavior of segment 142 is based on a simplified approach by implementing creep to MCFT. Since MCFT is not intended for conducting time dependent analyses, the approach is questionable. Thus, a more refined calculation method, by for in-stance conducting non-linear finite element analyses with viscoelastic features, will be utilized to in-crease the understanding and further verify results based on the simplified approach as a part of the ongoing PhD study.

Acknowledgement The work presented in this paper is part of an ongoing PhD study funded by the Norwegian Public Roads Administration as a part of the Coastal Highway Route E39 project. Reignard Tan would like to express his outmost gratitude to the supervisors and colleagues at Multiconsult ASA for contributions and making this PhD study possible.

References Book

Bažant, Zdeněk P. 1988. Mathematical Modelling of Creep and Shrinkage of Concrete. John Wiley & Sons Ltd.

Collins, Michael P., and Denis Mitchell. 1991. Prestressed concrete structures. Prentice Hall. Park, Robert., and Thomas Paulay 1975. Reinforced Concrete Structures. John Wiley & Sons, Inc.

Inspection reports BRUTUS. 2010. “Inspeksjonsrapport: Sållåsundbrua”. Oppdragsnr. 15-1729. Maintenance system

for existing road structures, the Norwegian Puclic Roads Administration. Journal article

Cracking



Vecchio, Frank J., and Michael P. Collins. 1986. “The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear.” ACI Journal, Vol. 83, No. 2, Mar.-Apr., pp. 219–231.

Walraven, Joost C. 1981. “Fundamental Analysis of Aggregate Interlock.” Proceedings, ASCE, Vol. 107, No. 11, November, pp. 2245–2270. Thesis

Takács, Peter F. 2002. “Deformations in Concrete Cantilever Bridges: Observations and Theoreti-cal Modelling.” PhD Thesis, Norwegian University of Science and Technology, Department of Struc-tural Engineering.

Tan, Jesus R. M. 2013. “Utilsiktede deformasjoner og skjærriss i fritt frambygg bruer.” M.Sc The-sis, University of Oslo, Faculty of Mathematics and Natural Sciences. Standard and Codes

CEB-FIP. 1993. “fib Model Code for Concrete Structures 1990”. Lausanne, Switzerland. EN-1992-1-1. 2004. “Eurocode 2, Design of concrete structures – Part 1-1: General rules and rules

for buildings”. Brussels, Belgium: CEN European Committee for Standardization. Håndbok-185. 2009. “Bruprosjektering”. Statens vegvesen Vegdirektoratet, Teknologiavdelingen,

Bruseksjon.

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