## Strengthofmaterial parameters

The governing material parameter in SOM-based fracture propagation models is either critical stress or strain. From a rigorous study with some 12 000 published test results, Raphael (1984) proposed the following relationship between tensile and compressive strengths of concrete under static loading:

where f'c and at are respectively static compression and tensile strengths of concrete in MPa. In absence of experimentally determined values, the above equation can be applied as an approximation to the expected static tensile strength of concrete.

Tensile strength of concrete increases significantly with increasing rate of applied loading, but the failure strain remains more or less unchanged under varying load rate. In the limited dynamic tests performed on mass concrete, the dynamic load rate effect has been observed to be higher than that in usual structural concrete. The selection of dynamic magnification of the concrete tensile strength is not very evident from the literature. Raphael (1984), from his study with the published data, proposed a dynamic magnification factor of 1.50 resulting in the dynamic tensile strength of concrete:

Briihwiler and Wittmann (1990) observed a dynamic magnification of up to 80% for the investigated strain rates between 10-5 and 10-2 per second, and this magnification decreased significantly due to compression preloading on the tested specimens. There is a controversy as to whether the static tensile strength or the dynamic strength should be used in seismic response analysis of concrete dams. Under alternating tensile and compressive loadings, substantial micro- level damage may take place in the material, resulting in reduced dynamic tensile strength. Nevertheless, 50% dynamic magnification proposed by Raphael (1984) seems to have received wide recognition (Chopra 1988; Kollgaard 1987; NRC 1990).

Figure 5.7: Strength-of-material-based failure criterion

A confusion, however, exists about the interpretation of tensile stresses computed from finite element analysis. Since the pre-peak stress-strain relationship is assumed to be linear elastic in most analyses, some investigators have suggested to compare the predicted tensile stresses with the apparent strength of the material (Figure??-a). Experimental evidences seem to support an apparent static tensile strength, aa, about 30% higher than the value given by equation 5.5. Under dynamic loading, the near-peak stressstrain nonlinearity decreases substantially (Briihwiler 1990; Hatano 1960). Recognizing the fact that the initial tangent modulus does not increase at the same rate as the tensile strength under dynamic loading, the Canadian Electrical Association (1990) report on safety assessment of concrete dams recommended the following relationship between Ej and Ej, the dynamic and static initial tangent moduli respectively:

Assuming that the failure strain remains the same under static and dynamic loadings, above relation gives the apparent dynamic tensile strength, o'a, to be 25% higher than the apparent static strength, oa. Thus,

Comparing equation 5.6 and equation 5.7, the apparent dynamic tensile strength is only about 80% higher than the actual dynamic strength, instead of 30% as proposed by Raphael (1984). Equation 5.8 respects the experimental evidences of rate insensitive failure strain, £t, and reduced near-peak o - £ nonlinearity under dynamic loading. The apparent tensile strength, however, may have to be lowered by about 10%o - 20% due to the existence of relatively weak construction lift joints in dams, if such joints are not modelled explicitly (NRC 1990).

Apparently, no empirical formula is readily available in the literature for dam concrete to select the initial static tangent modulus,Ej, from the static compressive strength, fCof concrete. Many more experimental investigations are required to establish an acceptable mathematical model for the effects of pre-compression load and the fatigue behaviour under cyclic loading. In the experimental study by Mlakar et al. (1985), the tensile strength of concrete under tension- compression loading was observed to increase with increasing rate of loading; the failure strain showed the general tendency of rate in-sensitivity. The behaviour under biaxial loading can be expressed using the standard failure envelopes in principal stress space (Figure??-b).

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