Nonlinear Finite Element Analysis of Concrete Structures

- bZ -
tensor c., determines the approximative stress tensor o... ConiD
a
1]
sider now eq. (6) when use is being made of the a.. tensor i.e.
13
fu* (a* . + b.) dV = 0 (4.1-12)
j j- ij t j 1
As a.. . + b. in general differs from zero, this equation cannot
!D»3 1 *
be satisfied for any displacements u.. However, if we consider
a finite set of functions for u. only, then eq. (12) may be
satisfied. It also follows that certain continuity restrictions
* a
have to be placed on u. and u. enabling the integral to be evaluated,
cf. Zienkiewicz (1977) pp. 46-47 and pp.63-65. The term
a *
a.. . + b. defines a residual and as u. serves the r purpose of
13,3 1 1
weighting functions for the residuals, a method based on the approximative
satisfaction of the equilibrium equation envisaged
by eq. (12) is often termed a method of v/eighted residuals. The
Galerkin method consists of a particular choice for the weighting
functions namely
u. = N. a* a = 1, 2 n (4.1-13)
1 ia a
where the tensor N.„ is the same as that for used for the approxiia
+
mative displacements, cf. eq. (10). The n coefficients a are
completely arbitrary, but as only n linear independent choices
for a
exist, eq. (13) determines n linear independent functions.
Corresponding to eq. (13) we have
e*. = B.. a* a = 1, 2 n (4.1-14)
13 13a a
Inserting eqs. (13) and (14) into the virtual work equation
given by eq. (9) and utilizing also the approximative stress
tensor, we derive
B. . a a?. dV -
LDa a 13
N ia a a h i
dV
"
N. a t, dS
ia a 1
S t
- N. a* t r dS = 0
I ia a r
ex = 1, 2
n