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Engineering Beam Theory
1.56 According to the engineering beam theory, the stresses in a beam of rectangular cross-
section are
h
2
4
−
z
2
M
I
V
2
I
σ
=
z,
σ
=
0
,
σ
=
0
,
τ
=
x
y
z
xz
where
z
is measured from the neutral axis (a centroidal axis) and
h
is the height of
the cross-section. Do these stresses satisfy equilibrium and boundary conditions for
a cantilevered beam with
(a) a uniform load along the beam?
(b) a concentrated force at the free end?
1.57 Suppose that the stress function for a uniformly loaded (
p
z
)
beam of rectangular
cross-section of depth
h
is given by
p
z
x
2
z
h
3
z
h
5
z
h
3
3
4
z
h
−
1
4
h
2
5
1
2
ψ
=
−
−
−
Note that the plane of interest is now
xz
and not
xy
.
(a) Find the stresses in the beam.
(b) Compute the resultant moment and shear on the boundaries at
x
=
0
,x
=
L
.
Can
you determine what the boundary conditions are?
1.58 Use the stress formulas of Probl
em
1.56 to determine the distribution of stresses in a
cantilevered beam with a force
P
applied at the free end in the
xz
plane at an angle
of
with the beam axis (
x
axis).
1.59 Verify that the displacement formulation governing equations for a shear beam are
given by Eqs. (1.130).
α
Torsion
1.60 Show that the stress functions defined by Eq. (1.155) satisfy the equations of equilib-
rium.
1.61 Show that for a bar undergoing torsion, the resultant forces
A
τ
A
τ
xy
dy dz,
xz
dy dz
are zero.
Hint:
To prove the first integral is zero, use
∂ω
∂
z
∂ω
∂
z
y
∂
y
2
2
2
z
2
+
∂
ω
ω
y
−
=
y
−
+
∂
∂
y
∂ω
∂
z
y
∂ω
∂
y
=
∂
∂
+
∂
∂
y
−
z
+
y
z
Then use Green's theorem and Eq. (1.153).
1.62 Utilize the force formulation to solve the torsion problem of a solid shaft of circular
cross-section. Use a stress function
r
2
a
2
,
where
a
is the radius of the outer
surface of the shaft. Verify that your solution agrees with that given in an elementary
treatment of strength of materials.
ψ
=
c
(
−
)
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