Engineering Beam Theory for the First Order Analysis
with Finite Element Method
1998 Winter, Kikuchi
Slender structures whose length is much larger than the size of the cross section, are
called beams. In such structures, deformation may be decomposed into
1.
axial deformation
2.
bending deformation in two directions
3.
torsional deformation
In order to describe the engineering beam theory, we shall introduce the Cartesian
coordinate system (x,y,z), where the z axis coincides with the beam axis define by the
line formed by the centroid of the cross section, while the x and y axes are the
principal axes of the cross section. The origin is set up at the centroid of the left edge
cross section of the beam. Based on
a)
the cross section does not deform, i.e.,
ε
x
=ε
y
= γ
xy
=
0
b)
Bernoulli-Euler assumption that no shearing strain is generated by pure
bending
c)
Saint-Venant torsion theory
the displacement of an arbitrary point (x,y,z) of the beam is approximated by
Ux
,
y
,
z
()
=
u
s
z
()−
y
−
y
s
θ
z
Vx
,
y
,
z
=
v
s
z
()+
x
−
x
s
θ
z
Wx
,
y
,
z
=
wz
xu
s
'
z
yv
s
'
z
()+ω
ns
x
,
y
θ
'
z
where g’ is the derivative of g in z,
u
s
z
,
v
s
s
transverse deflections of the shear center
x
s
,
y
s
in the x and y
direction, respectively,
average axial displacement
θ
z
angle of twist of the cross section at z
ω
ns
x
,
y
normalized Saint-Venant warping function when torque is
applied about the shear center axis ( that is the line passing through the
shear center
x
s
,
y
s
), and is defined a property attached to the cross
section such that