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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 73
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON CIVIL AND STRUCTURAL ENGINEERING COMPUTING
Edited by: B.H.V. Topping
Paper 53

Efficient Procedure for Stress Integration in Concrete Sections using a Gauss-Legendre Quadrature

J.L. Bonet+, P.F. Miguel+, M.A. Fernandez+ and M.L. Romero*

+Construction Engineering and Civil Engineering Projects Department, Polytechnic University, Valencia, Spain
*Department of Technology, University Jaime I, Castellon, Spain

Full Bibliographic Reference for this paper
J.L. Bonet, P.F. Miguel, M.A. Fernandez, M.L. Romero, "Efficient Procedure for Stress Integration in Concrete Sections using a Gauss-Legendre Quadrature", in B.H.V. Topping, (Editor), "Proceedings of the Eighth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 53, 2001. doi:10.4203/ccp.73.53
Keywords: non-linear, reinforced concrete, stress integration, cross-section analysis, Gauss-Legendre quadrature, biaxial bending.

Summary
The nonlinear analysis reinforced concrete framework programs and, particularly, which are based on the finite element method, require the computation of the constitutive equation of the cross-section and the evaluation of the internal forces through the integration of the stress fields. This operation is performed by those programs many times, hence its optimisation conduces to an important reduction of the computing time consumed.

The runtime is mainly consumed during the evaluation of the constitutive stiffness matrix and the internal equilibrium forces of the reinforced concrete element, Bonet [1] and Romero [2].

This paper deals with the implementation of a very accurate and efficient procedure for the resolution of nonlinear reinforced concrete frame structures. It has been developed a method based on a Gauss-Legendre cuadrature (Carnahan [3]) for sections with axial and unsymmetrical bending forces, and with any geometric form..

The integration procedure of the proposed section, can be divided in two phases or steps: decomposition of the integration area in quadrilaterals or triangles and stress integration of each quadrilateral. The quadrilaterals are decomposed can be transformed in a square of 2x2 units by side, Smith [4] and Miguel [5], using a coordinate transformation.

The sign of the Jacobean states if the quadrilateral is solid or empty. With this purpose the vertex will be enumerated in clockwise direction (positive) or anti- clockwise (negative). The proposed algorithm implements this enumeration automatically.

The stresses are obtained by addition of each stresses acting over the quadrilaterals. Also the constitutive matrix of the sections is obtained by composition of the quadrilaterals.

The proposed method has been tested in six different sections and 390 cases are analysed. Three alternatives to the proposed method has been used: fixed Gauss Points, non-fixed Gauss Points and parallel "thick layers". Also the fibber decomposition approach has been implemented. The comparison among the different methods has been performed versus the time and the accuracy obtained.

If the definition of the integration area does not follow the non-zero section stress field (fixed Gauss Points), the optimisation of the Gauss points location is not achieved, and it is impossible to assure an improvement in the accuracy of the results if an increase the density of the Gauss points in each integration quadrilateral is performed. Hence, it is not recommended the implementation of this integration method.

For the case that the integration area follows the non-zero stress section fields (non-fixed Gauss Points), can be noted a good improvement regarding the obtained error, meanwhile the computation runtime are similar to the previous method. In this case, if the Gauss points density is increased, the accuracy of the method is guaranteed.

Comparing the "thick layers" method with the fibber decomposition method, , for the same accuracy, the computation time of the fibber method is larger than the obtained with the proposed method.

In conclusion, the proposed "thick layers" method, regarding the accuracy, efficiency, and continuity in the stress field integration it is advisable for the implementation in nonlinear reinforced concrete frameworks programs, with a number of Gauss points equal to 4x4.

References
1
J.L.Bonet: "Método simplificado de cálculo de soportes esbeltos de hormigón armado de sección rectangular sometidos a compresión y flexión biaxial", PhD Thesis , Civil Engineering Dept. , Universidad Politécnica de Valencia, Mayo 2001
2
M.L. Romero. "Computación de Altas Prestaciones en el análisis No Lineal de Estructuras de Hormigón Armado para Edificación", PhD Thesis, Civil Engineeering Dept., Universidad Politécnica de Valencia, 1999.
3
B. Carnahan, Luther H.A and Wilkes J.O., "Numerical analysis: Methods and Aplications", Editorial Rueda, Madrid, España, 1979
4
I.M. Smith, "Programing the finite element mehod: with application to geomechanichs", John Wiley, England, 1982
5
P.F. Miguel, J.L. Bonet, M.A. Fernandez, "Integracion de tensiones en secciones de hormigon sometidas a flexocompresion esviada", Revista Internacional de Metodos Numericos para el cálculo y diseño en ingenieria, V.16, No. 2, pp 209-225, 2000

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