Nonlinear Finite Element Analysis of Concrete Structures

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where the (2 x 6) matrix w contains the shape functions. As linear
shape functions are assumed we have
N =
a.+b.r+c.z 0 a.+b.r+c.z 0 a +b r+c z 0
i l l j j j m m m
0 a.+b.r+c.z
i l l
a.+b.r+c.z 0 a +b r+c z
j j j m m m
(4.2-4)
where the coefficients are given by
a. =
r-z - r z.
Dm m 3
b. = z. - z
J m
c. = r - r.
i m j
and a., b., c- and a , b,# c are obtained using J
j j D m m m
cyclic
-•
permu-
c
tations of i, j and m. The term 2A denotes twice the area of the
triangle and we have
2A = a. + a. + a„
i j m
The strains are given by the vector
e =
V
£ R
e e
>z
(4.2-5)
containing the vertical, radial, circumferential and the engineering
shearing strain, respectively. Differentiating eq. (3) and
using eq. (4) yield
e = B a (4.2-6)
where th