Keywords: concrete, damage model, non-linear analysis, degrading function, secant stiffness.

The analysis of concrete structures is made by the use of numerical models in order to define its behaviour in terms of ultimate loads and service conditions. The constitutive equation of concrete under compression in the design codes is defined by a parabola equation, as a function of two parameters, that are the concrete strength at the peak stress and the corresponding strain. Other equations can be used and the CEB-FIP Model Code 1990, from the Committee Euro-International du Beton (MC90) indicates a more complex non-linear equation dependant also on the tangent modulus of elasticity at the origin. This equation is more appropriate to approximate the behaviour of high strength classes of concrete in the ultimate design or in service conditions. In the present work an elastic-degrading model is used in the analysis of concrete under three dimension states of stress.

In the model developed in the present work there is a degrading function indicating whether the incremental process corresponds to loading or unloading. Two degrading functions are used, that are the Von Mises and the Drucker Prager criteria. New loading produces a degradation of the stiffness matrix while unloading is made with the current secant stiffness. The incremental strain is divided into an elastic part and a degrading strain. The elastic strain is computed with the secant stiffness and the degrading strain is calculated by the use of a flow rule. The formulation of the model is similar to the elasto-plastic model.

The elasto-plastic model has been used in the non-linear analysis of concrete structures. The initiation of plastic deformations in the elasto-plastic model is controlled by a yield criterion defined according to the behaviour of the material. Drucker-Prager criterion, described in Owen et al. [1], is used in concrete. In the present work the elastic degrading model is developed in a similar way to the elasto- plastic model.

The use of the elasto-plastic model with the non-linear constitutive equation of the MC90 is made in Reference [2]. The development of the elastic degrading model is made in References [3,4] is applied in the present work.

Acknowledgement

This work was performed under the financial support from the Portuguese Minister of Science and Technology, FCT, by Programa Operacional do Quadro Comunitário de Apoio III (POCTI) and by FEDER, with grant POCTI/ECM/12126/1998, Fase II.

1

Owen, D.R.J., and Hinton E. (1980). Finite Elements in Plasticity: Theory and Practice. Pineridge Press Ltd. Swansea UK.

2

Barros, M.H.F.M. (2001), "Elasto-plastic modelling of confinement on concrete columns: comparison to MC90 equations." Int. J. of Eng. Struct 23, Elsevier, 311-318. doi:10.1016/S0141-0296(00)00056-0

3

Carol, I., Rizzi, E., and Willam, K. (1994). "A unified theory of elastic degradation and damage based on a loading surface." Int. J. Solids Struct., 31(20), 2835-2865. doi:10.1016/0020-7683(94)90072-8

4

Rizzi, E., Carol, I., and Willam, K. (1995). "Localization analysis of elastic degradation with application to scalar damage." J. Engrg. Mech. Div., ASCE, 121(4), 541-554. doi:10.1061/(ASCE)0733-9399(1995)121:4(541)

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 £125 +P&P)