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Calculation of Shear Areas and Torsional Constant using the Boundary Element Method with Scada Pro
8
called shear deformation coefficient
1
a
, while with the aid of these coefficients the
shear areas of the cross-section of the beam can be easily derived, as will occur in the next.
More specifically, the shear areas of the beam cross-section loaded by constant shear
force (
y
Q
,
z
Q
components, along y, z axis, respectively) are given by the
equations
y
y
y
A
A A
,
z
z
z
A
(3.1a,b)
while, the cross-section shear rigidities of the Timoshenko’s flexural beam theory are
identified as
y
y
y
GA
GA GA
,
z
z
z
G
G
GA
(3.2a,b)
where, the shear deformation coefficients
y
,
z
are determined by the equations [3.1-3.3]
2
2
2
2
cy
cy
y
y
AG
d
y
z
Q
(3.3a)
2
2
2
2
cz
cz
z
z
AG
a
d
y
z
Q
(3.3b)
In equations (3.3a,b) the
cy
,
cz
are the (shear) warping functions resulting from solving
the following boundary value problem [3.1-3.3]
2
2
2
2
2
1
,
c
c
c
z zz
y zz
yy zz
y z
Q I
z Q I y
GI I
y
z
in
Ω
(3.4a)
0
c
n
on
Γ
(3.4b)
for cases
0
y
Q
,
0
z
Q
and by defining
,
cy
y z
as the resulting warping function
0
y
Q
,
0
z
Q
and by defining
,
cz
y z
as the resulting warping function
In the previous, we have consider a beam with constant (along the longitudinal axis of the
beam) 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
. The aforementioned
boundary value problem arises from the equation of equilibrium of the three-dimensional
theory of elasticity neglecting the body forces and the physical consideration that the
traction vector in the direction of the normal vector
n
vanishes on the free surface of the bar.
The numerical solution of the boundary value problem stated above (3.4a,b) for the
evaluation of the (shear) warping functions
cy
and
cz
is accomplished employing a pure
BEM approach [3.4], that uses only boundary discretization. Finally, since the torsionless
bending problem (transverse shear loading problem) of beams is solved by the BEM, the