- Article
- Open Access
Effect of Constitutive Material Models on Seismic Response of Two-Story Reinforced Concrete Frame
- Md. Iftekharul Alam^{1} and
- Dookie Kim^{1}Email author
https://doi.org/10.1007/s40069-012-0010-3
© The Author(s) 2012
- Received: 26 March 2012
- Accepted: 30 May 2012
- Published: 1 July 2012
Abstract
This paper focuses on the finite element (FE) response sensitivity and reliability analyses considering smooth constitutive material models. A reinforced concrete frame is modeled for FE sensitivity analysis followed by direct differentiation method under both static and dynamic load cases. Later, the reliability analysis is performed to predict the seismic behavior of the frame. Displacement sensitivity discontinuities are observed along the pseudo-time axis using non-smooth concrete and reinforcing steel model under quasi-static loading. However, the smooth materials show continuity in response sensitivity at elastic to plastic transition points. The normalized sensitivity results are also used to measure the relative importance of the material parameters on the structural responses. In FE reliability analysis, the influence of smoothness behavior of reinforcing steel is carefully noticed. More efficient and reasonable reliability estimation can be achieved by using smooth material model compare with bilinear material constitutive model.
Keywords
- finite element analysis
- sensitivity analysis
- structural reliability
- constitutive models
- structural response
1 Introduction
The philosophy of performance-based earthquake engineering has gained recognition widely in the structural analysis and design field and has been included in many seismic design guidelines (ATC-55 2005; BSSC 2003) with noticeable advances in the field of structural reliability since last two decades. Several numerical and analytical methodologies have been developed considering the non-linear behavior, material parameters, and geometric uncertainties (Schueller et al. 2004). Der Kiureghian and Ke (1988) successfully implemented reliability methodology by using finite element (FE) method and first-order reliability method (FORM) (Ditlevsen and Madsen 1996). These methods were examined to investigate the influence of the correlation length of random property or load fields on the reliability of the subjects. The results have shown that the correlation length of property fields has influence on the displacement responses, but it may not be significant in stress limit states. The determination of design point(s) is considered as an essential step in gradient based reliability methodology. Structural response sensitivity is an important ingredient and a by-product of the design point search (Hohenbichler and Rackwitz 1986). For this purpose, the use of direct differentiation method (DDM) seems to be efficient and accurate approach to perform sensitivity analysis. A method has been presented to get key sensitive attributes in the material constitutive and discrete loading parameters of force-based FE frame systems (Conte et al. 2004). The material non-linearity was considered to get the static and dynamic responses using DDM. It was concluded that the developed procedure prone to sensitivity offers a powerful general tool for computing the responses. The analytical procedure and guidelines were developed in OpenSees (Mazzoni et al. 2005; Gu et al. 2010) framework to perform FE based reliability analysis (Haukaas and Der Kiureghian 2004). In this context, smoothed materials were modeled and the existing search algorithms were modified consequently. In case of non-linear beam column elements, response sensitivities have been evaluated with great precision using DDM (Scott et al. 2004; Scott and Filippou 2007). Similarly, the application of sensitivity and reliability analyses in the soil-foundation-structure-interaction systems has been reported in several research articles (Gu 2008; Gu et al. 2009a, b). Scott (2012) evaluated two existing formulations of force-based element response sensitivity and found the consistency of first formulation, while the second one proved to be inconsistent with a high condition number.
For gradient based methods, sensitivity analysis is performed to find the design point in reliability analysis. The continuity of the obtained response is essential for most of the algorithms. The discontinuity increases computational effort and due to using improper material model the process can be interrupted. Haukaas and Der Kiureghian (2004) performed static analysis on single degree of freedom systems to explain the discontinuity effects in the displacement sensitivity results using smooth and non-smooth material models. The results have shown the occurrence of discontinuity at yielding and unloading points in the bi-linear material. It has been suggested that by using the smooth material models such effect can be avoided. Another study was conducted focusing on the effect of gradient discontinuities caused by non-smoothness of the material models in the reliability context (Barbato and Conte 2006). A structural system was modeled using smooth and non-smooth reinforcing steels to compare response sensitivity and reliability analysis results. The consistency of the results was fairly verified with these obtained by previous researchers. Furthermore, study on the dynamic analysis concluded that the discontinuities can be effectively eliminated by providing smoothing effect in reinforcing steel and refining the time discretization of the equations of motion.
This paper presents the modeling and analysis of a reinforced concrete (RC) structure using different concrete and steel constitutive material models in the context of sensitivity based reliability analysis. DDM based response sensitivity analysis has been conducted for smooth and non-smooth material constitutive laws. Quasi-static cyclic analysis results show that discontinuities arise in the response sensitivities while using non-smooth material models. Moreover, this difficulty can be eliminated by using smooth constitutive material models. Later, the smoothness effect of reinforcing steel on failure probabilities of frame structure has been observed using structural reliability analysis.
2 Material Models
3 DDM Based Sensitivity Analysis
FE response sensitivity analysis is an important analytical tool in structural optimization, FE model updating, reliability analysis, and structural identification (Ditlevsen and Madsen 1996; Kleiber et al. 1997). The response sensitivity is a measure of the change in the response quantity due to a unit change in a system parameter. Cross-sectional geometry, materials properties, applied loads or nodal co-ordinates are considered as the key parameters. Generally, the structural response can be characterized in terms of deformations, forces, or integrated quantities, i.e., dissipated energy and accumulated damage (Haukaas and Der Kiureghian 2004). Such response can be computed by structural analysis tools such as OpenSees. The DDM method is well known for computing response quantities with high level of accuracy (Kleiber et al. 1997; Arora and Haug 1979; Zhang and Der Kiureghian 1993). In nonlinear DDM based analysis structural response is calculated in each time step after achieving the convergence of the response computation. Differentiation of the algorithm is required to obtain the response using specific sensitivity parameter.
In Eq. (5) \( \user2{K}_{T}^{dyn} = {1 \mathord{\left/ {\vphantom {1 {\beta (\Updelta t)^{2} }}} \right. \kern-\nulldelimiterspace} {\beta (\Updelta t)^{2} }}\user2{M} + {\alpha \mathord{\left/ {\vphantom {\alpha {\beta (\Updelta t)}}} \right. \kern-\nulldelimiterspace} {\beta (\Updelta t)}}\user2{C} + \left( {\user2{K}_{T}^{stat} } \right)_{n + 1} \) represents the tangent dynamic stiffness matrix, where, \( \left( {\user2{K}_{T}^{stat} } \right)^{i}_{n + 1} \) is the consistent tangent stiffness matrix defined by \( \left( {\user2{K}_{T}^{stat} } \right)_{n + 1} = {{\partial \user2{R}_{n + 1} } \mathord{\left/ {\vphantom {{\partial \user2{R}_{n + 1} } {\partial \user2{u}_{n + 1} }}} \right. \kern-\nulldelimiterspace} {\partial \user2{u}_{n + 1} }} \). The word ‘consistent’ emphasizes that the tangent operator is obtained through consistent linearization of the constitutive law integration scheme, which guarantees the quadratic rate of asymptotic convergence of the iterative solution strategies based on Newton’s method (Simo and Taylor 1985).
4 FE Structural Model
Parametric values of different concrete constitutive models.
Smoothed Popovics–Saenz concrete model | Kent–Scott–Park concrete model | ||||
---|---|---|---|---|---|
Parameter | Core concrete | Cover concrete | Parameter | Core concrete | Cover concrete |
f_{ c } (kPa) | 34473.8 | 27579.04 | f_{ c } (kPa) | 34485.6 | 27588.5 |
f_{ u } (kPa) | 25723.0 | 1000.0 | f_{ u } (kPa) | 20691.4 | 0.0 |
ɛ_{0} | 0.005 | 0.002 | ɛ_{0} | 0.004 | 0.002 |
ɛ_{ u } | 0.02 | 0.012 | ɛ_{ u } | 0.014 | 0.008 |
E_{ c } (kPa) | 2.7851 × 10^{7} | 2.4910 × 10^{7} | |||
η | 0.2 | 0.2 |
Parametric values of different steel constitutive models.
M–P model | J_{2} plasticity model | ||
---|---|---|---|
E (kPa) | 2.1 × 10^{8} | E (kPa) | 2.1 × 10^{8} |
f_{ y } (kPa) | 2.48 × 10^{5} | f_{ y } (kPa) | 2.48 × 10^{5} |
b | 0.02 | H_{ kin } (kPa) | 1.6129 × 10^{6} |
R _{0} | 20 | H _{ iso } | 0.0 |
cR _{1} | 0.925 | ||
cR _{2} | 0.15 | ||
a _{1} | 18.5 | ||
a _{2} | 0.15 |
Probabilistic properties of RC frame structure materials.
Concrete | Steel | ||||||
---|---|---|---|---|---|---|---|
Material | Core | Cover | Material | Mean | Coefficient of variation | ||
Mean | Coefficient of variation | Mean | Coefficient of variation | ||||
f_{ c } (kPa) | 34473.8 | 0.2 | 27579.04 | 0.2 | E (kPa) | 2.1 × 108 | 0.033 |
f_{ u } (kPa) | 25723.0 | 0.2 | 1000.0 | – | f_{ y } (kPa) | 2.48 × 105 | 0.106 |
ɛ_{0} | 0.005 | 0.2 | 0.002 | 0.2 | b | 0.02 | 0.2 |
ɛ_{ u } | 0.02 | 0.2 | 0.012 | 0.2 |
Correlation between the RVs.
RV | RV | Correlation |
---|---|---|
f _{c(core)} | f _{c(cover)} | 0.8 |
ɛ_{0(core)} | ɛ_{0(cover)} | 0.8 |
ɛ_{u(core)} | ɛ_{u(cover)} | 0.8 |
f _{c(core)} | f _{u(core)} | 0.8 |
ɛ_{0(core)} | ɛ_{u(core)} | 0.8 |
ɛ_{0(cover)} | ɛ_{u(cover)} | 0.8 |
ɛ_{0(core)} | ɛ_{u(cover)} | 0.64 |
f _{c(core)} | ɛ_{0(cover)} | 0.64 |
f _{u(core)} | f _{u(cover)} | 0.64 |
5 Results and Discussion
Parameter importance rankings for the inter-story drift of RC frame.
Response parameter | Concrete | Steel | |||||
---|---|---|---|---|---|---|---|
Core | Cover | ||||||
f _{ c } | ɛ_{ u } | f _{ c } | ɛ_{ u } | f _{ y } | E | b | |
u _{ drift } | 4 | 5 | 3 | 6 | 2 | 1 | 7 |
Reliability index β for FORM and important sampling of RC frame.
LSF | FORM | Important sampling |
---|---|---|
g _{1} | 3.19023 | 3.19118 |
g _{2} | 3.02357 | 3.02583 |
6 Conclusions
- 1.
In quasi-static FE analysis, response sensitivity discontinuities are observed in Popovics–Saenz concrete model without smoothing parameter, Kent–Scott–Park concrete model, and bi-linear steel model. Popovics–Saenz concrete with smoothing parameter and M–P (smooth model) steel models are used to avoid the occurrence of the discontinuity and get the continuous computation of response sensitivity. Therefore, for the design point search, using gradient based optimization algorithms, smooth material model plays an important role to get accurate and efficient computations of response sensitivities.
- 2.
DDM is very accurate and efficient method in computing the response sensitivities and also applicable to any material constitutive model. Good convergence of FFD results with DDM shows the accuracy in computation.
- 3.
Under dynamic motion, sensitivity result of the roof displacement with respect to the core concrete has shown minor effect of smoothing properties for two different concrete models. In dynamic analysis, the displacement response is smooth due to inertia effect of dynamic motion. Linear inertia in the equation of motion also shows considerable smoothing effects on the response sensitivity results of steel constitutive models along the pseudo-time axis. Moreover, a minor discontinuity effect has been found in non-smooth J_{ 2 } plasticity steel along the parametric axis. This effect can be eliminated by using smooth steel model (M–P).
- 4.
Structural reliability analysis shows the influence of smoothing on the computed probabilities of the RC structure. However, more efficient and reasonable reliability estimation can be achieved by using smooth material model compare with bilinear material constitutive model.
Declarations
Acknowledgments
This research was supported by the Ministry of Knowledge Economy and Korea Institute of Energy Technology Evaluation and Planning (KETEP) as a part of the Nuclear R&D Program (No. 20101620100230). The authors would like to express their appreciation for the financial support.
Authors’ Affiliations
References
- Arora, J. S., & Haug, E. J. (1979). Methods of design sensitivity analysis in structural optimization. AIAA Journal, 17, 970–974.MathSciNetView ArticleGoogle Scholar
- ATC-55. (2005). Evaluation and improvement of inelastic seismic analysis procedures. Advanced Technology Council, Redwood City, CA.Google Scholar
- Balan, T. A., Filippou, F. C., & Popov, E. P. (1997). Constitutive model for 3D cyclic analysis of concrete structures. Journal of Engineering Mechanics, 123(2), 143–153.View ArticleGoogle Scholar
- Balan, T. A., Spacone, E., & Kwon, M. (2001). A 3D hypoplastic model for cyclic analysis of concrete structures. Engineering Structures, 23, 333–342.View ArticleGoogle Scholar
- Barbato, M., & Conte, J. P. (2006). Finite element structural response sensitivity and reliability analyses using smooth versus non-smooth material constitutive models. International Journal of Reliability and Safety, 1(1–2), 3–39.View ArticleGoogle Scholar
- BSSC. (2003). The 2003 NEHRP recommended provisions for new buildings and other structures. Part 1: Provisions (FEMA 450). Building Seismic Science Council.Google Scholar
- Conte, J. P. (2001). Finite element response sensitivity analysis in earthquake engineering. In B. F. Spenser & Y. X. Hu (Eds.), Earthquake engineering frontiers in the new millennium (pp. 395–401). Lisse: Swets and Zeitlinger.Google Scholar
- Conte, J. P., Barbato, M., & Spacone, E. (2004). Finite element response sensitivity analysis using force-based frame models. International Journal of Numerical Methods in Engineering, 59(13), 1781–1820.View ArticleGoogle Scholar
- Conte, J. P., Vijalapura, P. K., & Meghella, M. (2003). Consistent finite-element response sensitivity analysis. Journal of Engineering Mechanics, 129(12), 1380–1393.View ArticleGoogle Scholar
- Der Kiureghian, A., & Ke, J. B. (1988). The stochastic finite element method in structural reliability. Probabilistic Engineering Mechanics, 3(2), 83–91.View ArticleGoogle Scholar
- Ditlevsen, O., & Madsen, H. O. (1996). Structural reliability methods. New York: Wiley.Google Scholar
- Filippou, F. C., Popov, E. P., & Bertero, V. V. (1983). Effects of bond deterioration on hysteretic behavior of reinforced concrete joints. Report EERC 83-19, Earthquake Engineering Research Center, University of California, Berkeley, CA.Google Scholar
- Gu, Q. (2008). Finite element response sensitivity and reliability analysis of soil-foundation-structure-interaction systems. Ph.D. dissertation, Department of Structural Engineering, University of California at San Diego, La Jolla, CA.Google Scholar
- Gu, Q., Barbato, M., & Conte, J. P. (2009b). Handling of constraints in finite-element response sensitivity analysis. Journal of Engineering Mechanics,135(12), 1427–1438.View ArticleGoogle Scholar
- Gu, Q., Conte, J. P., & Barbato, M. (2010). OpenSees command language manual response sensitivity analysis based on the direct differentiation method (DDM). Berkeley, CA: Pacific Earthquake Engineering Center, University of California.Google Scholar
- Gu, Q., Conte, J. P., Elgamal, A., & Yang, Z. (2009a). Finite element response sensitivity analysis of multi-yield-surface J2 plasticity model by direct differentiation method. Computer Methods in Applied Mechanics and Engineering, 198(30–32), 2272–2285.View ArticleGoogle Scholar
- Hasofer, A. M., & Lind, N. C. (1974). An exact and invariant first order reliability format. Journal of Engineering Mechanics, 100(12), 111–121.Google Scholar
- Haukaas, T., & Der Kiureghian, A. (2004). Finite element reliability and sensitivity methods for performance-based engineering. Report PEER 2003/14, Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA.Google Scholar
- Hohenbichler, M., & Rackwitz, R. (1986). Sensitivity and importance measures in structural reliability. Civil Engineering Systems, 3(4), 203–209.View ArticleGoogle Scholar
- Kleiber, M., Antunez, H., Hien, T.D., & Kowalczyk, P. (1997). Parameter sensitivity in nonlinear mechanics: Theory and finite element computations. New York: Wiley.Google Scholar
- Mahsuli, M., & Haukaas, T. (2010). Methods, models, and software for seismic risk analysis. In 9th US National & 10th Canadian Conference on Earthquake Engineering, Toronto, 25–29 July 2010.Google Scholar
- Mazzoni, S., McKenna, F., & Fenves, G. L. (2005). OpenSees command language manual. Berkeley, CA: Pacific Earthquake Engineering Center, University of California.Google Scholar
- Menegotto, M., & Pinto, P. E. (1973). Method for analysis of cyclically loaded reinforced concrete plane frames including changes in geometry and non-elastic behavior of elements under combined normal force and bending. In Proceedings, IABSE Symposium on‘Resistance and Ultimate Deformability of Structures Acted on by Well-Defined Repeated Loads’, Lisbon.Google Scholar
- Monti, G., & Nuti, C. (1992). Non-linear cyclic behavior of reinforcing bars including buckling. Journal of Structural Engineering, 118(12), 3268–3284.View ArticleGoogle Scholar
- Schueller, G. I., Pradlwarter, H. J., & Koutsourelakis, P. S. (2004). A critical appraisal of reliability estimation procedures for high dimensions. Probabilistic Engineering Mechanics, 19(4), 463–474.View ArticleGoogle Scholar
- Scott, M. H. (2012). Evaluation of force-based frame element response sensitivity formulations. Journal of Structural Engineering, 138(1), 72–80.Google Scholar
- Scott, M. H., & Filippou, F. C. (2007). Response gradients for nonlinear beam-column elements under large displacements. Journal of Structural Engineering, 133(2), 155–165.View ArticleGoogle Scholar
- Scott, M. H., Franchin, P., Fenves, G. L., & Filippou, F. C. (2004). Response sensitivity for nonlinear beam-column elements. Journal of Structural Engineering, 130(9), 1281–1288.View ArticleGoogle Scholar
- Scott, B. D., Park, R., & Priestley, M. J. N. (1982). Stress–strain behavior of concrete confined by overlapping hoops at low and high strain rates. ACI Journal, 79(1), 13–27.Google Scholar
- Simo, J. C., & Taylor, R. L. (1985). Consistent tangent operators for rate-independent elastoplasticity. Computer Methods in Applied Mechanics and Engineering, 48, 101–118.View ArticleGoogle Scholar
- Zhang,Y., Conte, J. P., Yang, Z., Elgamal, A., Bielak, J., & Acero, G. (2008). Two-dimensional nonlinear earthquake response analysis of a bridge-foundation-ground system. Earthquake Spectra, 24(2), 343–386.View ArticleGoogle Scholar
- Zhang, Y., & Der Kiureghian, A. (1993). Dynamic response sensitivity of inelastic structures. Computer Methods in Applied Mechanics and Engineering, 108(1–2), 23–36.View ArticleGoogle Scholar
- Zona, A., Barbato, M., & Conte, J. P. (2005). Finite element response sensitivity analysis of steel–concrete composite beams with deformable shear connection. Journal of Engineering Mechanics, 131(11), 1126–1139.View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.