Civil Engineering Reference
In-Depth Information
Sections through membrane
Rectangular
cross-section
z
Membrane
contours
y
t
t
b
b
(a) Transversely loaded membrane
(b) Distribution of shear stress
Figure 10.5
Prandtl's membrane analogy for uniform torsion.
Change in warping
displacement due
to shear strain
= (
yx
/
G
)δ
y
Longitudinal axis
x
of member
Total change in
warping displacement δ
u
z
y
δ
z
yx
Change in warping
due to twisting
δ
y
xy
xy
Rotation of longitudinal
fibres due to twisting
yx
Thin element
δ
y
× δ
z
× δ
x
δ
x
Figure 10.6
Warping displacements.
trajectories, the slope of the membrane is proportional to the shear stress, and the
volumeunderthemembraneisproportionaltothetorque.Theshearstrainscaused
by these shear stresses are related to the twisting and warping deformations, as
indicated in Figure 10.6.
The stress distribution can be found by solving the equation [2]
∂
2
θ
∂
y
2
+
∂
2
θ
∂
z
2
=−
2
G
d
φ
d
x
,
(10.1)
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