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Calculation of Shear Areas and Torsional Constant using the Boundary Element Method with Scada Pro
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On the contrary, in most cases either arbitrary torsional boundary conditions are
applied at the edges or concentrated twisting body forces at any other interior point of the
bar due to construction requirements. This bar under the action of general twisting loading is
leaded to
nonuniform
torsion, while the angle of twist per unit length is no longer constant
along it (Img.2.2).
The uniform torsion (
Saint Venant
torsion) is characterized by the torsional constant of
the section
t
I
. More specifically, the above-applied constant along the axis of the element
torque
t
M
is obtained from the equation
t
t
x
M GI
(2.1)
where
x
stands for the axis of the member,
G
is the shear modulus of the material of the
bar,
/
x
x
d dx
denotes the rate of change of the angle of twist
θ
and it can be regarded as
the torsional curvature, while the variable
t
I
is called torsional moment of inertia according
to
Saint Venant
or
torsional constant
and is
calculated from the equation
2 2
S
S
t
I
y z y
z
d
z
y
(2.2)
(a)
(b)
Img. 2.3. Warping function
S
for (a) standard UPE-100 and (b) Box shaped bar cross-
sections.
where
,
S
y z
is the (torsional) warping function with respect to the shear center
S
of the
bar’s cross-section (Img.2.3). The warping function
S
expresses the warping (longitudinal
displacement) which is the result of single-unit relative angle of twist (
1
x
), while, as the
same definition introduces, it depends only from the geometry of the section, i.e. it’s its
independent of the coordinate
x
parameter. Finally, the quantity
t
GI
is called
torsional
rigidity
of the cross-section. In the previous, we have consider a bar with constant (along the
longitudinal axis of the bar) cross-section with an arbitrarily shaped occupying the two-
dimensional simply or multiply connected region
Ω
of the
y; z
plane bounded by the curve
Γ
.