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CSMM based seismic fragility analysis of shear dominant RC hollow rectangular bridge piers

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Abstract

An improved cyclic softened membrane model (ICSMM) based smeared reinforced concrete (RC) plane stress element has been developed in the present study. The newly developed two-dimensional (2D) element has been implemented in the OpenSEES nonlinear finite element analysis (NLFEA) framework. The effectiveness of the developed element has been thoroughly validated using experimental results of three RC panels tested under cyclic pure shear loading and three hollow rectangular RC (HRRC) bridge piers tested under static axial and reversed cyclic in-plane bending loads. The accuracy of the NLFEA model based on ICSMM based plane stress elements has been further verified with the results obtained from the simulation of the considered test specimens using the NLFEA model based on existing CSMM based plane stress elements. After thorough validation ICSMM based NLFEA model has been used to perform incremental dynamic analysis (IDA) and develop seismic fragility curves for the three considered HRRC bridge piers using maximum likelihood estimation (MLE) method. The robustness of the fragility curves, thus obtained have also been verified by comparing with the fragility curves of the same bridge piers developed using two commonly used one dimensional (1D) NLFEA models based on fiber-beam-column elements that can capture axial-flexure and axial-shear-flexure interactions. The advantages and disadvantages of using ICSMM based NLFEA model over 1D fiber based NLFEA models for seismic fragility assessments have also been presented. It has been concluded from the results obtained that the ICSMM based NLFEA model is very effective to assess the complete nonlinear response of HRRC bridge piers under both in-plane reversed cyclic and seismic loads compared to the existing CSMM based 2D model and two other widely used 1D simulation models.

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Abbreviations

a and b:

Stress reversal points in SteelZ01 material model.

C and D:

Structural capacity and seismic demand

\(D_{i}\) :

The seismic demand obtained from nonlinear time history analysis for a given intensity measure \(IM_{i}\)

\(E_{c}\), \(f^{\prime}_{c}\) and \(f_{t}\) :

Modulus of elasticity, compression and tensile strengths of unconfined concrete

\(\bar{E}_{1}^{C}\) and \(\bar{E}_{2}^{C}\) :

Uniaxial tangent moduli of concrete in the applied principal stress directions

\(\bar{E}_{{si}}\), \(f_{{si}}\) and \(\bar{\varepsilon }_{{si}}\) :

Uniaxial tangent modulus and smeared stress and smeared uniaxial strain in the direction of ith layer of steel rebars

\(f_{{ly}}\), \(f_{{ty}}\) :

Yield stress of main reinforcement in longitudinal and transverse directions

\(G_{{12}}^{C}\) :

Shear modulus of concrete in 1–2 plane

\(i_{\tau }^{{\min }}\) and \(i_{\tau }^{{\max }}\) :

Analysis step numbers at which the minimum and maximum shear stresses occurred

\(M_{{iter}}\) and \(C_{{iter}}\) :

Maximum and the current iteration numbers

m :

Total number of elements in the intensity measure (IM) vector

\(x_{{cr}}^{{ - ve}}\) and \(r_{c}\) :

Non-dimensional critical strain and shape parameters to control the compression envelope of concrete

\(x_{{cr}}^{{ + ve}}\) and \(r_{t}\) :

Non-dimensional critical strain and shape parameters to control the tension envelope of concrete

\(\left[ {T(\theta _{1} )} \right]\) and \(\left[ {T( - \theta _{1} )} \right]\) :

Coordinate transformation matrix from local coordinates to applied principal stress coordinates and vice versa

\(\left[ {T( - \theta _{{si}} )} \right]\) and \(\left[ {T(\theta _{{si}} - \theta _{1} )} \right]\) :

Transformation matrix for steel from the \(x_{{si}} - y_{{si}}\) to the local (x–y) coordinate systems and 1–2 to \(x_{{si}} - y_{{si}}\) coordinate systems.

\(\left[ B \right]\) and \([V]\) :

Strain–Displacement and Hsu/Zhu ratio matrices

\([k_{e} ]\) and \([K]\) :

Elemental and global stiffness matrices

\([\bar{D}_{C} ]\), \([\bar{D}_{S} ]\) and \(\left[ {\bar{D}} \right]\) :

Biaxial tangent stiffness matrices of concrete, steel and reinforced concrete

\(\left[ {D_{C} } \right]\) and \(\left[ {D_{{si}} } \right]\) :

Uniaxial tangent material constitutive matrices of concrete and steel rebars

\(\varepsilon _{1}\) and \(\bar{\varepsilon }_{1}\) :

Smeared biaxial and uniaxial normal strains in 1 direction.

\(\varepsilon _{2}\) and \(\bar{\varepsilon }_{2}\) :

Smeared biaxial and uniaxial normal strains in 2 direction.

\(\varepsilon ^{\prime}_{c}\) and \(\varepsilon _{t}\) :

Unconfined concrete strains at \(f^{\prime}_{c}\) and \(f_{t}\)

\(\varepsilon _{{cr}}^{{ + ve}}\) and \(\varepsilon _{{cr}}^{{ - ve}}\) :

Critical strains in tension and compression envelopes envelope

\(\varepsilon _{{crk}}\) and \(\varepsilon _{{sp}}\) :

Cracking and spalling strains of unconfined concrete

\(\varepsilon _{o}\) :

Initial strain on tension envelope

\(\left\{ {\varepsilon _{x} }\;{\varepsilon _{y} } \right\}\) and \(\left\{ {\sigma _{x} } \; {\sigma _{y} } \right\}\) :

Smeared biaxial normal strains and applied stresses in x–y directions

\(\delta _{{xy}}\) :

The displacement history applied on RC panels

\(\gamma _{{xy}}\) and \(\tau _{{xy}}\) :

Smeared shear strain and applied smeared shear stress in x–y directions

\(\gamma _{{12}}\) :

Smeared shear strain in the applied principal stress directions

\(\upsilon _{{12}}\), \(\upsilon _{{21}}\) :

Hsu/Zhu ratios, which represents the effect of strain in 1-direction due to strain in 2-direction and vice versa

\(\zeta\) and \(D_{c}\) :

Softening and damage coefficients in ConcreteZ01 material model

\(\theta _{1}\) and \(\theta ^{\prime}_{1}\) :

Applied principal stress orientation angle in the starting and ending of each trial

\(\theta _{{si}}\) :

Orientation of ith layer steel reinforcement with respect to local coordinate system.

\(\rho _{{si}}\) :

Ith layer steel reinforcement ratio used in the formulation of cyclic softened membrane model

\(\phi\) and \(\rho _{l}\) :

Diameter and ratio of steel reinforcement in longitudinal direction

\(\varphi\) and \(\rho _{t}\) :

Diameter and ratio of steel reinforcement in transverse direction

\(\sigma _{1}^{C}\), \(\sigma _{2}^{C}\) and \(\tau _{{12}}^{C}\) :

Smeared normal and shear stresses of concrete along the applied principal stress directions

\(\Phi ( \bullet )\) :

Standard normal cumulative distribution function.

\(\{ \Delta F\}\) and \(\{ \Delta f_{e} \}\) :

Global and elemental force increment vectors.

\(\{ \Delta R\}\) and \(\{ \Delta R^{\prime}\}\) :

Residual force vector evaluated at the starting an iteration

\(\{ \Delta R^{\prime}\}\) and \(\left\| {\Delta R^{\prime}} \right\|\) :

Residual force vector and its norm evaluated at the ending of an iteration

\(\left\| {\Delta R^{\prime}} \right\|_{{final}}\) :

It is the norm of the residual force vector (\(\left\| {\Delta R^{\prime}} \right\|\)) evaluated at the end of a successful iteration or when \(M_{{iter}}\) is reached

\(\{ u\}\) and \(\{ \Delta u\}\) :

Nodal displacement and displacement increment vectors

\(\{ \sigma \}\) and \(\{ \varepsilon \}\) :

Applied stress and strain vectors i.e. \(\left\{ {\sigma _{x} ,\sigma _{y} ,\tau _{{xy}} } \right\}^{T}\) and \(\left\{ {\varepsilon _{x} ,\varepsilon _{y} ,0.5\gamma _{{xy}} } \right\}^{T}\)

\((\varepsilon _{{yp}} ,\sigma _{{yp}} )\) and \((\varepsilon _{n} ,\sigma _{n} )\) :

Yield strain–stress pairs of bare steel bar and the steel bar embedded in concrete

\((\varepsilon _{a} ,\sigma _{b} )\) and \((\varepsilon _{b} ,\sigma _{b} )\) :

Representative strains and stresses at the locations of unloading and reloading

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Acknowledgements

The authors would like to thank Industrial Research and Consultancy Centre (IRCC) at IIT Bombay for financially supporting this research project through Grant No. 14IRTAPSG011. The RC bridge pier specimens analyzed in this manuscript have been tested at the National Center for Research in Earthquake Engineering, Taiwan. The authors sincerely thank Dr. Y.L. Mo for providing the test results.

Funding

Industrial Research and Consultancy Centre (IRCC) at IIT Bombay, Grant No. 14IRTAPSG011.

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  1. Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, India

    Vijay Kumar Polimeru & Arghadeep Laskar

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Appendix

Appendix

Three additional shear critical HRRC bridge pier specimens tested by Calvi et al. (2005) (specimens S250 and S500) and Cassese (2017) (specimen P4) under static axial loading and in-plane reversed cyclic lateral loading conditions has been analysed to study the efficacy of the CSMM and ICSMM based finite element models in predicting the pinching effect of HRRC bridge piers. The geometric and material properties, applied reversed cyclic load histories, experimental load displacement curves and failure modes of these bridge piers has been obtained from Calvi et al. (2005), Cassese et al. (2017) and Cassese et al. (2019). The hysteretic loops of the three bridge pier specimens obtained from the finite element models developed using CSMM and ICSMM based elements and their comparison with the test results are presented in Figs. 25, 26, 27. The original cyclic constitutive model of concrete as proposed by Mansour (2001) is also presented in Fig. 28.

Fig. 25
figure 25

Comparison of hysteretic load—deflection curves of pier P4

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Fig. 26
figure 26

Comparison of hysteretic load—deflection curves of pier S250

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Fig. 27
figure 27

Comparison of hysteretic load—deflection curves of pier S500

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Fig. 28
figure 28

The original cyclic constitutive relationship of concrete proposed by Mansour (2001) (redrawn from Mansour (2001))

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Polimeru, V.K., Laskar, A. CSMM based seismic fragility analysis of shear dominant RC hollow rectangular bridge piers. Bull Earthquake Eng 19, 5051–5085 (2021). https://doi.org/10.1007/s10518-021-01151-8

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