Nonlinear Finite Element Analysis of Concrete Structures

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1^
1 = 2A 4T
b.
i
c.
i
b 5
c .
a. c. z a. c.z a cz
_Ln,.+_L_ o -i + b.+-J- O -^+b +-Sr
i r r i r r m r
b.
i
c.
m
m
m
(4.2-7)
O
b
m
It appears that all strains except the tangential strain are constant
within an element. However, in the present report this
variation of the tangential strain is ignored and instead, as an
approximation, the value at the centroid of the element is applied.
The stresses are given by the vector.
a =
hi
°R
a e
(4.2-8)
T RZ
with obvious notation. The usual elastic constitutive equation
is assumed to hold, i.e.
a = D (e - e )
o
(4.2-9)
where D is a (4x4) symmetric matrix termed the constitutive or
material matrix and é
o
is a vector containing initial strains
due to temperature. As the strains within an element are constant
the same follows for the stresses. It is here assumed that
the D-matrix may depend on the stress state and time. For an
isotropic material we have
J
D = (1 + v) (1 - 2v)
1-v v v
v 1-v v
V V 1-V
0 0 0
0
0
0
1 - 2v
(4.2-10)
where E and v are Young's modulus and Poisson's ratio, respectively,
that might depend on the stress state and time. The change