Nonlinear Finite Element Analysis of Concrete Structures

- p o ­
ized in this formulation for the following reasons: (1) it operates
directly with the differential equations in question and
no corresponding functional or potential function is needed opposed
to the Rayleigh-Ritz method; and (2) it demonstrates clearly
which equations are satisfied exactly and which only approximately.
The present section takes some advantage of the work of
Zienkiewicz (1977) pp. 42-92. Cartesian coordinates are assumed
and tensor notation is used for lower indices with Latin letters
ranging from 1 and 3 and Greek letters ranging from 1 to n or
from 1 to n .
Five basic equations define the response when a structure is
loaded. Three of these are field equations to be satisfied
throughout the whole volume of the structure while the last two
equations define the boundary conditions. Let us first consider
the field equations starting with the equilibrium equations
o. . ,. + b. =0 (4.1-1)
i] ] i
where o. . is the stress tensor and b. denotes the specified vol-
13 i
ume forces. Only static problems are considered. A tilda indicates
that the quantity in question is prescribed. The symmetry
of the stress tensor a.. = a., follows from equilibrium of moments;
this symmetry will be tacitly assumed in the following
therefore being exactly satisfied. Assuming small strains these
are defined by
e ij
= i
(u i,j
+u j,i>
(4 - 1 - 2)
where e.. is the symmetric strain tensor and u. denotes the displacements.
The stresses and strains are related through the
constitutive equation
°ij = D ijki < e ki -
e 2i> + o0 ij < 4 - 1 - 3 )
where D..,- is the elasticity tensor that might deppnd on stresses,
strains and time. The symmetry properties Z. .. , = D..., =
D. .,, follow from the symmetry of o. . and e. .. Moreover, to
achieve symmetric stiffness matrices for the finite elements,