Keywords: nonlinear spatial beam, reinforced-concrete, analytical stress field integration.

The finite-element analysis of reinforced-concrete beams requires the integration of stresses and tangent material moduli over the cross-sections. Since the governing equations of the structure are non-linear and must therefore be solved iteratively, the integrals over the cross-sections need to be evaluated many times. Thus, it is of great importance to be able to evaluate cross-sectional integrals as efficiently as possible.

A number of numerical methods have been proposed in order to make the integrations more efficient, see, e.g. Bonet et al. [1], Fafitis [2], Rasheed and Dinno [3]. The methods presented by Bonet et al. and Fafitis are particularly convenient for the cases where the stress field varies in only one direction. Their methods use Green's Theorem and transform the area integral into the boundary integral, which is then integrated numerically. While such an approach is more efficient than the one using the area integrals, it is still not optimal due to the fact that the numerical integration is used, for the numerical integration introduces an integration error and is too time-consuming. The error, however, can be made smaller by increasing the number of integration points, but this unfortunately increases the computational time and consequently reduces the time-efficiency of the overall finite-element algorithm. To make the integration time-effective and exact, we have developed an analytical integration scheme over the boundary of a reinforced-concrete cross-section.

For the deduction of an analytical integration algorithm, the constitutive law of concrete needs to be prescribed in the exact analytical form. We assume the constitutive law proposed by Desayi and Krishnan [4] for the concrete in compression, and that of Bergan and Holand [5] for the concrete in tension. The stress-strain relationship is thus given by a known strain function, smooth almost everywhere, with an only exception of a finite number of discrete points. The next assumption concerns the strain distribution over the cross-section. We follow the standard approach and assume the linear strain distribution (see, e.g. El-Metwally et al. [6]). For the linear strain distribution, it is easy to find a constant strain direction, which results in a constant stress field in that direction. With the help of some change of integration variables and by the use of Green's Theorem, we obtain the path integrals of known, analytically integrable functions. If the cross-section is approximated by a polygon (which is often the case in practice), an efficient formula for the analytical integration is obtained.

Because the integration is performed analytically and the results are thus exact, the convergence studies of the accuracy of the integration method are not needed. Therefore only the efficiency of the proposed method is shown, together with a number of numerical simulations of the behaviour of spatial reinforced-concrete frames, where the present integration method combined with the finite-element method of Zupan and Saje [7] is employed. The results of our numerical simulations are compared to the experimental results presented by Espion [8], Drysdale and Huggins [9], Kim and Lee [10], and Ferguson and Breen [11].

1

J. L. Bonet, P. F. Miguel, M. L. Romero, M. A. Fernández, "A Modified Algorithm for Reinforced Concrete Cross Section Integration", In Proceedings of the Sixth International Conference on Computational Structures Technology, B.H.V. Topping (editor), Civil-Comp Press, article 120, 2002. doi:10.4203/ccp.75.120

2

A. Fafitis, "Interaction Surfaces of Reinforced-Concrete Sections in Biaxial Bending", Journal of Structural Engineering, ASCE 127, 840-846, 2001. doi:10.1061/(ASCE)0733-9445(2001)127:7(840)

3

H. A. S. Rasheed, K. S. Dinno, "An efficient nonlinear analysis of RC sections", Comput. Struct. 53, 613-623, 1994. doi:10.1016/0045-7949(94)90105-8

4

P. Desayi, S. Krishnan, "Equation for the stress-strain curve of concrete", Journal of American Concrete Institute 61, 345- 350, 1964.

5

P. G. Bergan, I. Holand, "Nonlinear finite element analysis of concrete structures", Comput. Methods Appl. Mech. Eng. 17/18, 443-467, 1979. doi:10.1016/0045-7825(79)90027-6

6

S. E. El-Metwally, A. M. El-Shahhat, W. F. Chen, "3-D nonlinear analysis of R/C slender columns", Comput. Struct. 37, 863- 872, 1990. doi:10.1016/0045-7949(90)90114-H

7

D. Zupan, M. Saje, "Rotational invariants in finite-element formulation of three-dimensional beam theories", In Proceedings of the Sixth International Conference on Computational Structures Technology,Barry H. V. Topping (editor), Stirling, Civil-Comp Press, article 16, 2002. doi:10.4203/ccp.75.16

8

B. Espion, "Benchmark examples for creep and shrinkage analysis computer programs", Creep and shrinkage of concrete, RILEM TC 114 subcommittee 3, 1993.

9

R. G Drysdale, M. W. Huggins, "Sustained biaxial load on slender concrete columns", Journal of Structural Engineering, ASCE 97, 1423-1443, 1971.

10

S. Kim, S. Lee, "The behavior of reinforced concrete columns subjected to axial force and biaxial bending", Eng. Struct. 23, 1518-1528, 2000. doi:10.1016/S0141-0296(99)00090-5

11

P. M. Ferguson, J. E. Breen, "Investigation of the long concrete column in a frame subject to lateral loads", Symp. On Reinforced Concrete Columns, ACI SP-13, 75-119, 1966.

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