Nonlinear Finite Element Analysis of Concrete Structures

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propriate assembly rules usin? the superposition principle are
applied to obtain the total equation system. In the follov.'ing,
this approach will be adopted.
Within each element eq. (10) degenerates to
u ae = N G a e o=l, 2 n e (4.1-16)
i ia a '
where the upper index e expresses that an element is considered.
Tne element possesses an n e -degree of freedom represented by the
nodal displacements a
of the element. As adjacent elements most
often share nodal points some of the nodal displacement appear
in different a - vectors.
a
Corresponding to eqs. (2) and (16) we have
e ae = B e . a e a = 1, 2 n e (4.1-17)
lj lja a
Ignoring tr>e last term in eq. (15) and carrying out the integrations
element by element assuming appropriate smoothness of the
involved functions we then derive
) !YfB e . D e M1 B® dvVo " |B e - D 6 .,. e° dV +
a_, [VJ ija .i.jkl klø J 6 J ija i]kl kl
" x e e
v
v
f B e . a 0 . dV -|N e b. dV - IN? t.dsl = 0
J lja lj J la i J ia i J
e e ce
v v S t
a and M 1, 2 n e (4.1-18)
where m is the number of elements. This equation in itself also
contains the assembly rules for connection of all the elements.
For each element the equation yields
K^ a^ = F^ + F? e + F te + F E ° e + F 0 ° e
a P P a a a a a
a and 3 = 1/ 2 n e (4.1-19)