Nonlinear Finite Element Analysis of Concrete Structures

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As the total strains are composed of elastic and plastic strains
it follows from eq. (11) that
1
E
r
1
-u
"I
-u
1
r
a oi
a 02
(3-13)
r
Equation (11) and (12) form the basis for the initial stress
method employed in the program for consideration to nonlinearities
in membrane reinforcement, cf. section 4.3.2.
However, some furthe.- derivations are necessary as the finite
element program directly determines only the total strains and
also because the only quantities that are stored from the previous
loading stage are the initial stresses. Parameters R and
S present in eq. (12) and defined by eq. (10), however, require
knowledge of the equivalent stress a and the equivalent plastic
strain e both corresponding to the actual total strains. Using
the plastic incompressibility, eq. (6) can be written
£ P =
VT V/E?2 + e5 ? + e? e? (3-14)
Through eqs. (13) and (14) the equivalent plastic strain e p
tuereby also the equivalent stress a
can be determined, both
corresponding to the previous loading stage. Obviously, an iteration
sequence is necessary to determine the present values of
e p and a and different iteration schemes can be employed for
e
this purpose. Here we make use of the proposal of Mendelson
(1968) which has the advantage of quick convergence and applicability
even in the case of ideal plasticity. In essence this proposal
given below enables one to compute plastic strains from
total strains without recourse to stresses.
Letting e ±j = c ±j - §
and
.., e kk and ej., = z e ±. - ± 6. . ej fc denote
the deviatoric total strain and the deviatoric elastic strain,
respectively, and noting that the plastic strain tensor e?. is
purely deviatoric, eq. (7) yields