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Calculation of Shear Areas and Torsional Constant using the Boundary Element Method with Scada Pro
6
The calculation of the warping function
S
is achieved by solving the following
boundary value problem [2.2, 2.3]
2
2
2
2
2
0
S
S
S
y
z
in
Ω
(2.3a)
S
y
z
zn yn
n
on
Γ
(2.3b)
where
cos ,
/
y
n
y n dy dn
and
sin
,
/
z
n
z n dz dn
are the directional cosines of the
external normal vector
n
to the boundary of the cross-section. 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 (2.3a,b) for the
evaluation of the warping function
S
, is accomplished employing a pure BEM approach
[2.4], that uses only boundary discretization. Finally, since the uniform torsion problem is
solved by the BEM, the domain integral in equation (2.2) is converted to boundary line
integral in order to maintain the pure boundary character of the method [2.3]. Thus, once the
aforementioned warping function is established along the boundary, the torsional constant
t
I
is evaluated using only boundary integration.
BIBLIOGRAPHY
[2.1]. Saint–Venant B. (1855) “Memoire sur la torsion des prismes”,
Memoires des Savants
Etrangers
, 14, 233-560.
[2.2]. Sapountzakis E.J.
(2000) “Solution of Nonuniform Torsion of Bars by an Integral
Equation Method”,
Computers and Structures
, 77, 659-667.
[2.3]. Sapountzakis E.J. and Mokos V.G. (2004) “3-D Elastic Beam Element of Composite
or Homogeneous Cross Section Including Warping Effect with Applications to
Spatial Structures”, Τechnika Chronika, Scientific Journal of the TCG, Section I,
Civil Engineering, Rural and Surveying Engineering, 24(1-3), 115-139.
[2.4]. Katsikadelis, J.T (2002)
Boundary Elements: Theory and Application
, Elsevier,
Amsterdam-London.