Nonlinear Finite Element Analysis of Concrete Structures

-"* —
Use of Gauss's divergence theorem yields
K
J ij ri : ds " Rj : i3 dv + K b i dv = °
S v v
where dS denotes an infinitesimal surface. Use of eq. (2) and
the symmetry of •-. . gives
f * f * f *
J. . £. . dV - u- b. dV - u. _-. . n. dS = 0 (4.1-7)
J ID ID J i i J l 1] 3
v v s
* *
where the strain tensor e.- corresponds to the displacements u..
The last term can be split into integrations over S^ and S . In
region S
region S
the static boundary condition eq.(4) holds while in
the geometric boundary condition eq.(5) applies. These
latter prescribed displacements correspond to some tractions,
which however are unknown. Therefore in region S
we can write
G.. n. = t r on S (4.1-8)
13 j x u
where the index r suggests ti.at these tractions are the unknown
reaction forces. Integrating the last term in eq. (7) over S
and S
I'
and using eqs. (4) and (8) we derive
* f * " f * ~ f * r
.. £.. dV - u. b. dV - u. t. dS - u. t dS = 0 (4.1-9)
IJ 13 J i i J i i J i i
*
This equation states the principle of virtual work. Note that u.
are completely arbitrary displacements. In the derivation of the
virtual work equation use has been made of the equilibrium relation,
eq. (1), the definition of the strain tensor, eq. (2)
and the static boundary condition, eq. (4), so that these equations
may be replaced by the virtual work expression, eq. (9).
In the virtual work equation, no consideration has been taken to
the constitutive condition.
In a state of equilibrium given by the displacements u., the
stress tensor c.. depends on u. and satisfies the equilibrium
condition, eq. (1). Suppose now that an approximation u. to u.
is found where upper index a, in general, is related to approximative
quantities. Then a corresponding approximative strain