Nonlinear Finite Element Analysis of Concrete Structures

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circular shape with increasing hydrostatic pressure. The criterion
has been demonstrated to be in good agreement with experimental
results for different types of concrete and covers a
wide range of stress states including those where tensile stresses
iccur. The formulation in terms of one function for all stress
states facilitates its use in structural calculations and it has
been shown that a sufficiently accurate calibration of the parameters
in the criterion is obtained by knowledge of the uniaxial
compressive strength c and the uniaxial strength a alone.
As mentioned previously, the other failure criterion implemented
in the finite element program is the classical Coulomb criterion
with tension cut-offs which consist of a combination of the Coulomb
criterion suggested in 1773 and the maximum tensile stress
criterion often attributed to Rankine, 1876. This dual criterion
was originally proposed by Cowan (1953) but using the terminology
of Paul (1961), it is usually termed the modified Coulomb
criterion. It reads,
ma, - o~. = o
6 C
°1 = °t
(2.1-9)
where, as previously, a, - o - o and tensile stress is considered
positive. The criterion contains three parameters and it
includes a cracking criterion given by the second of the above
two equations. The coefficient m is related to the friction angle
ip by m = (1 + sinip)/(l - sinip) . Different m-values have been proposed
in the past, but here we adopt the value
m = 4 (2.1-10)
corresponding to a friction angle eoual to 37 . This value has
been proposed both by Cowan (1953) and by Johansen (1958, 1959)
and is applied almost exclusively in the Scandinavian countries.
As shown in fig. 8 the modified Coulomb criterion corresponds to
an irregular hexagonal pyramid with straight meridians and with
tension cut-offs. The trace in the deviatoric plane is shown in
fig. 8 together with the other criterion implemented in the fin-