Civil Engineering Reference
In-Depth Information
The elastic matrix
D
is one single constant as
E
. The elastic stiffness element
can be derived as
EA
l
−
EA
l
0
0
0
0
EI
EI
l
−
EI
EI
l
12
6
12
6
0
0
l
3
2
l
3
2
EI
l
EI
l
−
EI
EI
l
6
4
6
2
0
0
2
2
2
l
K
0
=
(3.46)
−
EA
l
EA
l
0
0
0
0
−
EI
−
EI
EI
−
EI
12
6
12
6
0
0
l
3
l
2
l
3
l
2
EI
l
EI
l
−
6
EI
EI
l
6
2
4
0
0
2
2
2
l
where:
A
denotes the cross-sectional area of an element
=
∫
2
denotes the moment inertia to the neutral axis of the cross
section
I
y dA
When geometric nonlinearity is considered, the axial strain will be coupled
with bending deflection. Equation 3.42 will become
2
du
dx
dv
dx
y
d v
dx
1
2
+
=
ε
ε
a
ε
e
=
(3.47)
2
b
−
2
Following similar procedures, the initial stress matrix of a frame element
can be derived by Equation 3.39:
0
0
0
0
0
0
0
36
3
l
0
−
−
36
3
l
2
2
0
3
l
4
l
0
3
l
−
l
F
x
30
K
σ
=
(3.48)
l
0
0
0
0
0
0
0
−
36
−
3
l
0
36
−
3
l
2
2
0
3
l
−
l
0
−
3
l
4
l
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