Issue 29

A. Caporale et alii, Frattura ed Integrità Strutturale, 29 (2014) 19-27; DOI: 10.3221/IGF-ESIS.29.03 24 0,0 0,2 0,4 0,6 0,8 1,0 0,0 0,2 0,4 0,6 0,8 1,0     p     p    = 0.8, f v,eq = 30, f v,m = 10, 20    = 0.7, f v,m = 0, f v,eq = 30, 40, 50    = 0.8, f v,m = 0, f v,eq = 30, 40, 50 Figure 2 : Dimensionless stress-strain curves of concrete in uni-axial compression. C OMPRESSIVE FAILURE SURFACES he proposed model can also be used to determine the behavior of cement concrete in load cases of multi-axial compression. This is done in this work, where the failure surface of concrete subject to bi-axial or tri-axial compression is determined by using the previously described iterative procedure. The objective is to represent the compressive failure surface of cement concrete in the space of the eigenvalues of the average stress   c σ in concrete. The eigenvalues are the principal stresses in concrete and are contained in the vector       T 1 2 3 e e e e σ . The generic direction in the space of the eigenvalues of   c σ is represented by the unit vector        T 1 2 3 ˆ ˆ ˆ n . The proposed method determines the average strain   c ε in cement concrete subject to a prescribed stress       T 1 2 3 ˆ ˆ e e σ , where  is a positive parameter which increases during the loading process;  1 ˆ ,  2 ˆ and  3 e are constants with   1 ˆ 0 and   2 ˆ 0 . The principal directions of   c σ coincide with the coordinate axes, i.e.                     T T 1 2 3 11 22 33 ˆ ˆ c c c e e σ . In order to determine a point of the failure surface, the loading parameter  increases from zero up to the admissible maximum value  max . The above-mentioned iterative procedure is executed for each value of  . After the admissible values of  have been determined for given directions  1 ˆ and  2 ˆ and given stress  3 e , the stress       c ii ii for  1, 2 i can be plotted against the corresponding strain      c ii ii so as to obtain a    ii ii stress-strain curve in the i th direction. From a physical point of view, it is interesting to consider the    ii ii stress-strain curve when    ˆ ˆ j i with  i j , i.e.    ii jj ; this curve exhibits a maximum stress denoted by   , c ii p  and the tangent line to the curve at the maximum is about horizontal. The searched point of the failure surface of cement concrete in multi-axial compression is therefore given by       T T , max 1 max 2 3 11, 22, 3 ˆ ˆ c c e e p e p p e                σ σ (10) Next, the points , e p σ are estimated with the previously described iterative procedure and assuming       T 1 2 ˆ ˆ 0 n with   1 ˆ 0 and   2 ˆ 0 , i.e. a bi-axial compression is imposed to cement concrete. In this case, the points , e p σ represent a failure curve in the negative quadrant of the plane with  1 e - and  2 e -axes. In Fig. 3, the failure curves are illustrated for the cement concrete characterized by the geometric and mechanical properties reported in Tab. 1. The parameters of the evolution law (7) adopted for the curves of Fig. 3 have the same values assumed in Fig. 1: e.g., in the solid black curves of Fig. 3,     ,0 , 0 v v m f f ,  , 30, 40, 50 v eq f ,   0.7 . The failure curves exhibit an elliptical shape and the curves T