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Keywords: failure surface, multi-axial loads, high-strength concrete, steel fibre-reinforced concrete.

Summary

It is observed through a survey of existing literature that researchers use different failure criteria for concrete under different loading conditions. In addition, research to develop a failure surface for high-strength and steel fibre-reinforced concrete under multi-axial loads is still relatively new. In this paper, a unified, 5-parameter failure criterion for plain, high strength and steel fibre-reinforced concrete is proposed for concrete with strength ranging from 20MPa to 165MPa. It can be used to determine the state of stress at failure in both confined and unconfined concrete subjected to multi-axial loads. To facilitate the prediction of stresses at failure, a method to obtain a closed-form solution is also developed. This new formulation allows the constitutive model to be conveniently implemented in a finite element package, thus enabling the engineer to model, analyse and design concrete structures under complex states of stress.

The proposed failure surface for concrete is described by a tensile meridian, $\rho_t$ , where the angle of similarity, =0^o, and a compressive meridian, $\rho_c$ , where =60^o. An elliptical curve is used to interpolate between $\rho_t$ and $\rho_c$ for other angles of similarity between 0^o and 60^o. Due to the six-fold symmetry of the surface, these three equations are sufficient to define the entire failure surface. The coefficients required to define the tensile and compressive meridians are obtained through the regression of 296 experimental data points of cubes and cylinders under multiaxial stresses for both normal and high strength plain concrete failing on $\rho_t$ and $\rho_c$ , respectively. A coefficient is also introduced to rotate the tensile meridian to account for the beneficial effects of fibre addition. This allows the proposed failure surface to be adjusted according to the strength of fibre, volume fraction, as well as the aspect ratio of the fibre added to the plain concrete matrix. The proposed surface is then verified against an additional 178 experiments of plain and fibre-reinforced concrete cubes under triaxial loads, plates under biaxial loads and concrete confined with steel tubes.

As concrete exhibits a non-linear behaviour under load, its stress-strain curve may be approximated by a step-by-step incremental approach using constitutive models for plain and fibre-reinforced concrete developed by Tho et al. [1] and Mattar [2], respectively. Knowledge of the stresses and strains at failure are required to calculate the incremental tangential modulus at each step. To facilitate the implementation of the proposed failure surface into a Finite Element package, a closed-form solution for predicting the state of stress in concrete under monotonically increasing, proportional, multi-axial loads is presented in this paper. The state of stress at failure is given by the point whereby a line drawn through the origin and the current state of stress pierces the failure surface. Since $\rho_t$ and $\rho_c$ are quadratic curves, and $\rho(\xi,\theta)$ is interpolated from $\rho_t$ and $\rho_c$ , it is postulated that $\rho(\xi,\theta)$ is also a quadratic curve. Through equating the equation for $\rho(\xi,\theta)$ with that of the load path, a closed-form solution is developed to predict the state of stress in concrete at failure.

The proposed failure surface and constitutive models are implemented through a user subroutine in a well-established commercial Finite Element software, ABAQUS. Experiments of FRC plates under biaxial loads tested by Yin et al. [3] and a plain concrete deep beam under two-point loading by Kong et al. [4] are modelled. The analytical results show a good fit to the experimental results and demonstrate the ability of the proposed failure surface to be applied to various types of concrete under different load conditions. Together with the closed-form solution for determining the stress at failure in concrete, this failure criterion can be conveniently and efficiently implemented in a Finite Element analysis, thus aiding the engineer in the analysis and design of concrete under complex stress conditions.

References

1: K.K. Tho, P.E.C. Seow, S. Swaddiwudhipong "Numerical Method for Analysis of Concrete Under Multi-axial Loads", Magazine of Concrete Research (submitted for review). doi:10.1680/macr.55.6.537.37598
2: S.N.H. Mattar, "Behaviour of Fibre Reinforced Concrete Structures Using Finite Element Method", B.Eng Thesis, National University of Singapore, Singapore, 2003.
3: W.S. Yin, E.C.M. Su, M.A. Mansur, T.T.C. Hsu, "Biaxial Tests of Plain and Fibre Concrete", ACI Materials Journal, 86, 236-243, 1989.
4: F.K. Kong, P.J. Robins, D.F. Cole, "Web reinforcement effects on deep beams", Journal of the American Concrete Institute, 67 (73), 1010-1017, 1970.

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 77 PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON CIVIL AND STRUCTURAL ENGINEERING COMPUTING Edited by: B.H.V. Topping Paper 50 A Unified Failure Criterion for Finite Element Analysis of Concrete Structures P.E.C. Seow, S. Swaddiwudhipong and K.K. Tho Department of Civil Engineering, National University of Singapore, Singapore doi:10.4203/ccp.77.50 purchase the full-text of this paper Full Bibliographic Reference for this paper P.E.C. Seow, S. Swaddiwudhipong, K.K. Tho, "A Unified Failure Criterion for Finite Element Analysis of Concrete Structures", in B.H.V. Topping, (Editor), "Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 50, 2003. doi:10.4203/ccp.77.50 Keywords: failure surface, multi-axial loads, high-strength concrete, steel fibre-reinforced concrete. Summary It is observed through a survey of existing literature that researchers use different failure criteria for concrete under different loading conditions. In addition, research to develop a failure surface for high-strength and steel fibre-reinforced concrete under multi-axial loads is still relatively new. In this paper, a unified, 5-parameter failure criterion for plain, high strength and steel fibre-reinforced concrete is proposed for concrete with strength ranging from 20MPa to 165MPa. It can be used to determine the state of stress at failure in both confined and unconfined concrete subjected to multi-axial loads. To facilitate the prediction of stresses at failure, a method to obtain a closed-form solution is also developed. This new formulation allows the constitutive model to be conveniently implemented in a finite element package, thus enabling the engineer to model, analyse and design concrete structures under complex states of stress. The proposed failure surface for concrete is described by a tensile meridian, $\rho_t$ , where the angle of similarity, =0^o, and a compressive meridian, $\rho_c$ , where =60^o. An elliptical curve is used to interpolate between $\rho_t$ and $\rho_c$ for other angles of similarity between 0^o and 60^o. Due to the six-fold symmetry of the surface, these three equations are sufficient to define the entire failure surface. The coefficients required to define the tensile and compressive meridians are obtained through the regression of 296 experimental data points of cubes and cylinders under multiaxial stresses for both normal and high strength plain concrete failing on $\rho_t$ and $\rho_c$ , respectively. A coefficient is also introduced to rotate the tensile meridian to account for the beneficial effects of fibre addition. This allows the proposed failure surface to be adjusted according to the strength of fibre, volume fraction, as well as the aspect ratio of the fibre added to the plain concrete matrix. The proposed surface is then verified against an additional 178 experiments of plain and fibre-reinforced concrete cubes under triaxial loads, plates under biaxial loads and concrete confined with steel tubes. As concrete exhibits a non-linear behaviour under load, its stress-strain curve may be approximated by a step-by-step incremental approach using constitutive models for plain and fibre-reinforced concrete developed by Tho et al. [1] and Mattar [2], respectively. Knowledge of the stresses and strains at failure are required to calculate the incremental tangential modulus at each step. To facilitate the implementation of the proposed failure surface into a Finite Element package, a closed-form solution for predicting the state of stress in concrete under monotonically increasing, proportional, multi-axial loads is presented in this paper. The state of stress at failure is given by the point whereby a line drawn through the origin and the current state of stress pierces the failure surface. Since $\rho_t$ and $\rho_c$ are quadratic curves, and $\rho(\xi,\theta)$ is interpolated from $\rho_t$ and $\rho_c$ , it is postulated that $\rho(\xi,\theta)$ is also a quadratic curve. Through equating the equation for $\rho(\xi,\theta)$ with that of the load path, a closed-form solution is developed to predict the state of stress in concrete at failure. The proposed failure surface and constitutive models are implemented through a user subroutine in a well-established commercial Finite Element software, ABAQUS. Experiments of FRC plates under biaxial loads tested by Yin et al. [3] and a plain concrete deep beam under two-point loading by Kong et al. [4] are modelled. The analytical results show a good fit to the experimental results and demonstrate the ability of the proposed failure surface to be applied to various types of concrete under different load conditions. Together with the closed-form solution for determining the stress at failure in concrete, this failure criterion can be conveniently and efficiently implemented in a Finite Element analysis, thus aiding the engineer in the analysis and design of concrete under complex stress conditions. References 1 K.K. Tho, P.E.C. Seow, S. Swaddiwudhipong "Numerical Method for Analysis of Concrete Under Multi-axial Loads", Magazine of Concrete Research (submitted for review). doi:10.1680/macr.55.6.537.37598 2 S.N.H. Mattar, "Behaviour of Fibre Reinforced Concrete Structures Using Finite Element Method", B.Eng Thesis, National University of Singapore, Singapore, 2003. 3 W.S. Yin, E.C.M. Su, M.A. Mansur, T.T.C. Hsu, "Biaxial Tests of Plain and Fibre Concrete", ACI Materials Journal, 86, 236-243, 1989. 4 F.K. Kong, P.J. Robins, D.F. Cole, "Web reinforcement effects on deep beams", Journal of the American Concrete Institute, 67 (73), 1010-1017, 1970. purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £123 +P&P)
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