Nonlinear Finite Element Analysis of Concrete Structures

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tensor c.. follows which in turn determines an approximative
stress tensor o^ through the constitutive equation- It is obvious,
however, that this stress tensor will not in all points
of the structure satisfy the equilibrium condition, eq. (1), as
no means have been taken for this purpose. However, the method
of Galerkin ensures an approximative satisfaction of these equilibrium
equations. Let us now consider this procedure in some
detail.
The true displacements u. are approximated by
u. = u a = N. a a = 1, 2 n (4.1-10)
i i la a
where the tensor N.
depends on position and is assumed to be
known while the coefficients a are to be determined. It is cona
venient to consider these coefficients as displacements of some
points distributed all over the structure. These points are
termed nodal displacements. Obviously, to obtain an accurate
approximation by the available n degrees of freedom, the nodal
points should be distributed closely where large changes in the
displacement field are expected- It is a crucial feature of the
finite element method that the approximative displacement functions
given by the tensor N.
and, in a finite element context,
termed shape functions are not the same all through the structure,
but render different expressions for each subdomain or
element, the total of which covers the whole structure. Moreover,
the finite element method assumes that within each element, the
approximative displacements can be expressed solely by the nodal
displacements located within or on the boundary of the element
in question. However, with these remarks in mind we will retain
the formulation given by eq. (10).
The approximative strain tensor follows from eqs. (2) and (10)
i.e.
4j = B ija a a
a - !' 2 n (4.1-11)
where the tensor B .. depends on position and follows from knowledge
of N i(j . Through the constitutive condition eq. (3) , the