Civil Engineering Reference
In-Depth Information
J
J
y
y
i
In 3D (multi-fiber beams):
H
H
y
MM
z
x
x
i
z
i
y
i
y
J
z
I
y
y
i
x
i
J
J
y
I
z
z
i
x
i
where H
x
is the average axial strain, y
i
is the distance between fiber i and the average
fiber, I
y
and I
z
are the bending curvatures (I
y
= dT
y
/dx and I
z
= dT
z
/dx), I
x
= dT
x
/dx
is the unit angle for torsion, and J
y
and J
z
are average shear strains.
The bending moments
M
y
,
M
z
and the normal force
N
x
are obtained by
integrating the axial strains
i
V
corresponding to the strains
i
H
:
³
³
³
[7.1]
N
V
dS
,
Mz
V
dS
and
My
V
dS
x
x
y
x
z
x
s
s
s
In the case of a Timoshenko beam element, shear loads
T
y
,
T
z
and the torsion
moment
M
x
are calculated by integrating the shear stresses
i
W
and
i
W
derived
from
i
J
and
i
J
:
³
³
T
W
,
dS
T
W
dS
[7.2]
y
y
z
z
S
S
³
M
y
W
z
W
dS
[7.3]
x
z
y
s
For a shear elastic material, the behavior law is written as follows:
W
DJ
G
and
W
DJ
G
[7.4]
y
y
y
z
z
z
i
i
i
i
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