Nonlinear Finite Element Analysis of Concrete Structures

The characteristics of the failure surface given by eqs. (5) and
(8) are: (1) only four parameters used; (2) use of invariants
makes determination of the principal stresses unnecessary; (3)
the surface is smooth and convex with the exception of the vertex;
(4) the meridians are parabolic and opens in the direction of
the negative hydrostatic axis; (5) the trace in the deviatoric
plane changes frcm nearly triangular to circular shape with increasing
hydrostatic pressure; (6) it contains several earlier
proposed criteria as special cases, in particular, the criterion
of Drucker and Prager (1952) for A = 0, A = 'constant, and the
von Mises criterion for A = B = 0 and X = constant.
In evaluating the four parameters A, B, K and K use has been
made of the biaxial tests of Kupfer et al. (1969, 1973) and the
triaxial results of Balmer (1949) and Richart et al. (1928). The
parameters are determined so as to represent the following three
failure states exactly: (1) uniaxial compressive strength o ;
(2) biaxial compressive strength •; , = 1.16 a corresponding to
the tests of Kupfer et al. (1969, 1973) and (3) uniaxial tensile
strength a given by the o /a -ratio (dependence on this ratio
is illustrated in tables 2 and 3). Finally,the method of least
squares has been used to obtain the best fit of the compressive
meridian for f,/a - - 5.0 to the test results of Balmer (1949)
c
and Richart et al. (1928), cf. fig. 5. The compressive meridian
is hereby found to pass through the point (F,/a , p/o ) = (-5.0,
c c
4.0). The foregoing procedure implies values of the parameters
as given in table 2. The values of K, and K- correspond to the
those of A,
t
and A
c
found in table 3.
Table 2.1-2: Parameter values and their dependence on the o./a -
ratio.
1
°t /o c
A
B
K l
« 2 !
' 0.08
0.10
0.12
1.8076
1.2759
0.9218
4.0962
3.1962
2.5969
14.4863
11.7365
9.9110
0.9914
0.9801
0.9647