Nonlinear Finite Element Analysis of Concrete Structures

- 63 -
*
dependent choices of a , it possesses only the trivial solution
given by the n equations
As this homogeneous equation holds for any of the n linear in-
"I-
B. . a a . dV - N. b. dV - N. t. dS - N. t r dS = 0
J IJCX
. dV -
xj 3
N. J ia la
b. dV
i - N.
J ia
t. dS -
i J ia i
S t
S
a = 1, 2
u
Use of the constitutive condition eq- (3) and eq. (11) finally
leads to
{ [B. . D. .. n B. , _ dv)a fl - |B. . D. .. . E?. dV + B. . a? . dV
\J lua ljkl klB / B J 13a 13k! kl J 13a IJ
- Ha *i dV "
a and 8 = 1, 2.
N. t. dS -
ia 1
S t
•f-
S
u
dS = 0
(4.1-15)
These are the equations that determine the unknown displacements
a . For every a-value (a = 1, 2 n) i.e. for every degree of
freedom there corresponds one equation. All terms are known except
the last one that represent the reaction forces corresponding
to the prescribed displacements. These displacements must be
dictated before eq. (15) can be solved. However, the last term
in eq. (15) contributes only to the equation in question provided
a prescribed displacement is related to the considered
degree of freedom. Therefore, a convenient way to dictate the
geometrical boundary conditions is simply to ignore the last
term in eq. (15) and then replace the effected equations by ones
that directly state the prescribed displacements. Thereby, rearrangements
of eq. (15) are avoided and symmetry of the equation
system is retained. The actual procedure is described in section
4.6. Therefore, in the following, the last term in eq. (15) will
be ignored as due regard to its influence will be taken at a
later stage.
Eq. (15) refers to the whole structure. Traditionally, however
a corresponding equation is set up for each element and then ap-