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Simulation of post-tensioned bridge columns under reversed-cyclic loads

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Abstract

Thin-walled prestressed concrete structures such as bridge columns can be subjected to large magnitudes of dynamic loads. It is essential to develop a robust model to accurately predict the nonlinear behavior of these structures and thereby ensure their structural integrity under extreme loading conditions. A research project was performed to study the properties of prestressed concrete (PC) elements under shear loads using the Universal Panel Tester built at the Structural Engineering Lab of University of Houston. The test results were used to develop the softened membrane model for prestressed concrete (SMM-PC) (Wang, Constitutive relationships of prestressed concrete membrane elements, 2006). SMM-PC has been extended to the cyclic softened membrane model for prestressed concrete (CSMM-PC) by using the cyclic softened membrane model (CSSM) for reinforced concrete (Mansour and Hsu, J Struct Eng 131:44–53, 2005, J Struct Eng 131:54–65, 2005) CSMM-PC has been implemented into a finite element program to simulate the non-linear behavior of PC structures under cyclic loads. The program thus developed was named simulation of concrete structures (SCS). SMM-PC has been previously validated by tests on prestressed concrete beams under monotonic loading (Laskar et al., 12th international conference on engineering, science, construction, and operations in challenging environment, earth and space, 2010). In this paper CSMM-PC incorporated in SCS has been validated by tests on precast post-tensioned axisymmetric bridge columns under reversed cyclic loading. The analysis results of the bridge columns using SCS showed good agreement with the test results in terms of the primary backbone curves and the hysteretic loops including the energy dissipation and the strength degradation in the post-peak region. The damage and residual drift of the specimens were also closely predicted from the analysis results. Based on the accuracy of the analytical results it can be concluded that CSMM-PC implemented in SCS is an effective tool to simulate the cyclic non-linear behavior of thin-walled prestressed concrete structures and can be used for simulation of 3D actions of these type of structures through proper implementation.

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Abbreviations

1:

Direction of applied principal tensile stress

2:

Direction of applied principal compressive stress

x :

local coordinate of prestressed concrete element

y :

local coordinate of prestressed concrete element

X si :

local coordinate of prestressing steel layer in ith direction, where X si is the direction of the steel layer

Y si :

local coordinate of prestressing steel layer in ith direction

d max :

Maximum displacement of the loop

d min :

Minimum displacement of the loop

D :

Damage factor of concrete due to cyclic shear

D y :

Yield drift at the first yielding of the ED bars

D yi :

Idealized yield drift

D u :

Ultimate drift prior to fracture of ED bars

\(\overline{E}_{1}^{\text{c}}\) :

Secant stiffness of uniaxial moduli of concrete in 1 direction at a given stress/strain state

\(\overline{E}_{2}^{\text{c}}\) :

Secant stiffness of uniaxial moduli of concrete in 2 direction at a given stress/strain state

E c :

Slope of concrete stress–strain curve in tension prior to cracking

E cc :

Slope of concrete stress–strain curve during unloading and reloading

E c0 :

Slope of linear compressive stress–strain curve of concrete

E D :

Energy dissipation for a cycle of loading, equal to area of hysteresis loop corresponding to that cycle

E ps :

Elastic modulus of prestressing steel

\(E_{\text{ps}}^{{\prime \prime }}\) :

Modulus of prestressing tendons, used in plastic region

\(\overline{E}_{{{\text{s}}i}}\) :

Uniaxail secant modulus of prestressing tendons

f + :

Force at maximum displacement

f − :

Force at minimum displacement

f cr :

Cracking tensile stress of concrete

\(f_{\text{c}}^{'}\) :

Cylinder compressive strength of concrete

f ps :

Smeared stress in embedded prestressing steel

f pu :

Ultimate strength of prestressing steel

\(f_{\text{pu}}^{{\prime }}\) :

Revised strength of prestressing steel

f si :

Smeared stress in embedded steel

\(F_{y}\) :

Yield force corresponding to yield drift D y

F p :

Plastic force corresponding to idealized yield drift D yi

F u :

Ultimate peak lateral strength of specimens

\(G_{12}^{\text{c}}\) :

Shear modulus of concrete in (1–2) coordinate, \(G_{12}^{\text{c}} = \frac{{\sigma_{1}^{\text{c}} - \sigma_{2}^{\text{c}} }}{{\varepsilon_{1} - \varepsilon_{2} }}\)

k :

Constant to lower upper slope of concrete compressive stress–strain curve

K eff :

Effective stiffness

n :

Prestressing force applied on bridge column finite element model

N :

Summation of prestressing and applied load on bridge column finite element model

[B]:

Matrix representing the shape function of elements

[D]:

Secant material stiffness matrix

[D c]:

Concrete secant uniaxial stiffness matrix

[D si ]:

Steel secant uniaxial stiffness matrix

[k]:

Element stiffness matrix

[K]:

Global stiffness matrix

\([ {T(\theta _{1} )}]\) :

\(\left[ {\begin{array}{*{20}l} {\cos ^{2} \theta _{1} } & {\sin ^{2} \theta _{1} } & {2\sin \theta _{1} \cos \theta _{1} } \\ {\sin ^{2} \theta _{1} } & {\cos ^{2} \theta _{1} } & { - 2\sin \theta _{1} \cos \theta _{1} } \\ { - \sin \theta _{1} \cos \theta _{1} } & {\sin \theta _{1} \cos \theta _{1} } & {\cos ^{2} \theta _{1} - \sin ^{2} \theta _{1} } \\ \end{array} } \right]\)

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

Transformation matrix from the 1–2 coordinate to the x–y coordinate \(\left[ {\begin{array}{*{20}l} {\cos ^{2} \theta _{1} } & {\sin ^{2} \theta _{1} } & { - 2\sin \theta _{1} \cos \theta _{1} } \\ {\sin ^{2} \theta _{1} } & {\cos ^{2} \theta _{1} } & {2\sin \theta _{1} \cos \theta _{1} } \\ {\sin \theta _{1} \cos \theta _{1} } & { - \sin \theta _{1} \cos \theta _{1} } & {\cos ^{2} \theta _{1} - \sin ^{2} \theta _{1} } \\ \end{array} } \right]\)

\(\left[ {T( - \theta_{{{\text{s}}i}} )} \right]\) :

Transformation matrix from the x si –y si coordinate to the x–y coordinate \(\left[ {\begin{array}{*{20}l} {\cos ^{2} \theta _{{{\text{s}}i}} } & {\sin ^{2} \theta _{{{\text{s}}i}} } & { - 2\sin \theta _{{{\text{s}}i}} \cos \theta _{{{\text{s}}i}} } \\ {\sin ^{2} \theta _{{{\text{s}}i}} } & {\cos ^{2} \theta _{{{\text{s}}i}} } & {2\sin \theta _{{{\text{s}}i}} \cos \theta _{{{\text{s}}i}} } \\ {\sin \theta _{{{\text{s}}i}} \cos \theta _{{{\text{s}}i}} } & { - \sin \theta _{{{\text{s}}i}} \cos \theta _{{{\text{s}}i}} } & {\cos ^{2} \theta _{{{\text{s}}i}} - \sin ^{2} \theta _{{{\text{s}}i}} } \\ \end{array} } \right]\)

\(\left[ {T(\theta_{{{\text{s}}i}} - \theta_{1} )} \right]\) :

\(\left[ {\begin{array}{*{20}l} {\cos ^{2} (\theta _{{{\text{s}}i}} - \theta _{1} )} & {\sin ^{2} (\theta _{{{\text{s}}i}} - \theta _{1} )} & {2\sin (\theta _{{{\text{s}}i}} - \theta _{1} )\cos (\theta _{{{\text{s}}i}} - \theta _{1} )} \\ {\sin ^{2} (\theta _{{{\text{s}}i}} - \theta _{1} )} & {\cos ^{2} (\theta _{{{\text{s}}i}} - \theta _{1} )} & { - 2\sin (\theta _{{{\text{s}}i}} - \theta _{1} )\cos (\theta _{{{\text{s}}i}} - \theta _{1} )} \\ { - \sin (\theta _{{{\text{s}}i}} - \theta _{1} )\cos (\theta _{{{\text{s}}i}} - \theta _{1} )} & {\sin (\theta _{{{\text{s}}i}} - \theta _{1} )\cos (\theta _{{{\text{s}}i}} - \theta _{1} )} & {\cos ^{2} (\theta _{{{\text{s}}i}} - \theta _{1} ) - \sin ^{2} (\theta _{{{\text{s}}i}} - \theta _{1} )} \\ \end{array} } \right]\)

[V]:

Hsu/Zhu matrix \(\left[ {\begin{array}{*{20}l} {\frac{1}{{1 - \nu _{{12}} \nu _{{21}} }}} & {\frac{{\nu _{{12}} }}{{1 - \nu _{{12}} \nu _{{21}} }}} & 0 \\ {\frac{{\nu _{{21}} }}{{1 - \nu _{{12}} \nu _{{21}} }}} & {\frac{1}{{1 - \nu _{{12}} \nu _{{21}} }}} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\)

{∆f}:

Element resisting force increment vector

{∆F}:

Global resisting force increment vector

\(\left\{ {\Delta R} \right\}\) :

Residual force vector

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

Residual force vector

\(\left\{ {\Delta u} \right\}\) :

Nodal displacement increment vector

\(\left\{ u \right\}\) :

Nodal displacement vector

\(\left\{ \varepsilon \right\}\) :

Applied strain vector of the plane stress element defined as \(\left\{ {\begin{array}{*{20}l} {\varepsilon_{x} } & {\varepsilon_{y} } & {0.5\gamma_{xy} } \\ \end{array} } \right\}\)

\(\left\{ \sigma \right\}\) :

Applied stress vector of the plane stress element defined as \(\left\{ {\begin{array}{*{20}l} {\sigma_{x} } & {\sigma_{y} } & {\tau_{xy} } \\ \end{array} } \right\}\)

\(W_{\text{p}}\) :

Prestress factor in concrete softening coefficient

β :

Deviation angle of the direction of concrete principal compressive stress and the applied principal compressive stress direction

\(\bar{\varepsilon }\) :

Uniaxial strain

ε 0 :

Concrete cylinder strain corresponding to peak cylinder strength \(f_{\text{c}}^{'}\)

ε 1 :

Smeared biaxial strain in the 1- direction

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

Smeared uniaxial strain in the 1-direction

ε 2 :

Smeared biaxial strain in the 2-direction

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

Smeared uniaxial strain in the 2-direction

ε c :

Smeared concrete strain

\(\varepsilon_{\text{c}}^{{\prime }}\) :

Maximum compression strain normal to the direction being considered and occurring in previous loading cycles

ε cr :

Strain corresponding to cracking stress of concrete

\(\bar{\varepsilon }_{i}\) :

Uniaxial smeared concrete strain at the start of unloading or reloading

\(\bar{\varepsilon }_{i + 1}\) :

Uniaxial smeared concrete strain at the end of unloading or reloading

\(\bar{\varepsilon }_{\text{s}}\) :

Uniaxial smeared strain in prestressing steel embedded in concrete

\(\varepsilon_{{{\text{s}}i}}\) :

Smeared biaxial strain of prestressing steel in ith direction

\(\varepsilon_{{{\text{s}}i^{{\prime }} }}\) :

Smeared biaxial strain of steel perpendicular to ith direction

\(\varepsilon_{\text{T}}^{{\prime }}\) :

Uniaxial tensile strain normal to the compression direction being considered

\(\varepsilon_{x}\) :

Smeared biaxial strain in the x-direction

\(\varepsilon_{y}\) :

Smeared biaxial strain in the y-direction

\(\gamma_{12} ,\gamma_{21}\) :

Smeared shear strains in the 1–2 coordinate system

\(\gamma_{{{\text{s}}i}}\) :

Smeared shear strain in the ith direction coordinate system

\(\gamma_{xy}\) :

Smeared shear strain in the x–y coordinate system

\(\theta_{1}\) :

Angle between the (x–y) coordinate system and (1–2) coordinate system

\(\theta_{1}^{{\prime }}\) :

Trial angle between the (x–y) coordinate system and (1–2) coordinate system

\(\theta_{{{\text{s}}i}}\) :

Angle between the (x–y) coordinate system and (x si –y si ) coordinate system

\(\rho_{si}\) :

Prestressing steel ratio in the “ith” direction

\(\nu_{12}\) :

Hus/Zhu ratio (effect of strain in 2-direction on strain in 1-direction)

\(\nu_{21}\) :

Hsu/Zhu ratio (effect of strain in 1-direction on strain in 2-direction)

\(\sigma_{1}\) :

Applied principal stress in the 1-direction

\(\sigma_{2}\) :

Applied principal stress in the 2-direction

\(\sigma_{1}^{\text{c}}\) :

Concrete smeared stress in the 1-direction

\(\sigma_{2}^{\text{c}}\) :

Concrete smeared stress in the 2-direction

\(\sigma^{\text{c}}\) :

Concrete smeared stress

\(\sigma_{i}^{\text{c}}\) :

Concrete smeared stress at the start of unloading and reloading

\(\sigma_{i + 1}^{c}\) :

Concrete smeared stress at the end of unloading and reloading

\(\sigma_{x}\) :

Applied normal stress in the x-direction

\(\sigma_{y}\) :

Applied normal stress in the y-direction

\(\tau_{12}\) :

Applied shear stress in the 1–2 coordinate system

\(\tau_{12}^{\text{c}}\) :

Concrete smeared shear stress in the 1–2 coordinate system

\(\tau_{xy}\) :

Applied shear stress in the x–y coordinate system

\(\rho_{{{\text{s}}i}}\) :

Reinforcing bar or prestressing steel ratio of the steel layer in the ith direction

\(\zeta\) :

Softened coefficient of concrete in compression when the peak stress-softened coefficient is equal to strain-softened coefficient

\(\zeta_{\text{eq}}\) :

Equivalent viscous damping ratio

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Acknowledgments

This research project is supported by the Texas Department of Transportation. The materials presented are the research findings by the authors, and are not necessarily expressed for the funding agency’s opinion.

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Authors and Affiliations

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

    Arghadeep Laskar

  2. Department of Civil and Environmental Engineering, University of Houston, 4800 Calhoun, Houston, 77204, USA

    Y. L. Mo & Thomas T. C. Hsu

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  1. Arghadeep Laskar
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Laskar, A., Mo, Y.L. & Hsu, T.T.C. Simulation of post-tensioned bridge columns under reversed-cyclic loads. Mater Struct 49, 2237–2256 (2016). https://doi.org/10.1617/s11527-015-0646-y

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