Nonlinear Finite Element Analysis of Concrete Structures

- 51 -
By use of eq. (3) we have then derived the final incremental
equations
deP . d£P llil (3_ €)
J
e
where de is given by eq. (2) and a by eq. (1).
If increasing proportional loading is considered it follows
that
J
where the ratio s. ./a
lj e
that
e
is constant. From this equations follows
^
£P= U £?. ,*.)* (3-6)
\3 13 13/
Eqs. (5) and (6) hold exactly for increasing proportional loading.
It is now assumed that they also apply to nonproportional
loadings. However, while in the finite element program the reinforcement
stresses are not directly determined, the total reinforcement
strains are known as these are assumed to be identical
to the concrete strains. It is therefore advantageous to derive explicitly
the relation between reinforcement stresses and total
strains. Noting that total strains e.. are composed of elastic
strains e . . and plastic strains e?. i.e.
e. . = e e . + e?. (3-7)
13 13 13
and working only with principal stresses and strains which is
allowable here inasmuch as the corresponding principal directions
always are assumed to coincide, we therefore write
1 e P
e 1 = - (o x - u(o 2 + o 3 )) + 35- (2 0;L - o 2 - o 3 )
e
e 2 = | (o 2 - u«^ + o 3 )) + fj- (2G 2 - o 1 - o 3 >
e
1 £ P
c 3 = s (o 3 - .j( 0l + o 2 )) + jo - (2o 3 - °1 " a 2>
e