Civil Engineering Reference
In-Depth Information
function, we compute the stress components as
G
∂
z
u
xy
=
G
xy
=
y
−
∂
(9.140)
G
∂
y
u
xz
=
G
yz
=
z
+
∂
and note that
G
∂
2
u
∂
xy
∂
=
z
−
z
∂
y
∂
(9.141)
G
∂
2
u
∂
xz
∂
=
z
+
y
∂
y
∂
Combining the last two equations results in
∂
xy
∂
2
2
−
∂
xz
∂
∂
+
∂
=
=−
2
G
(9.142)
y
2
z
2
z
y
∂
∂
as the governing equation for Prandtl's stress function. As with the fluid formu-
lations of Chapter 8, note the analogy of Equation 9.142 with the case of heat
conduction. Here the term
2
G
is analogous to internal heat generation
Q
.
9.9.1 Boundary Condition
At the outside surface of the torsion member, no stress acts normal to the surface,
so the resultant of the shear stress components must be tangent to the surface.
This is illustrated in Figure 9.13c showing a differential element d
S
of the surface
(with the positive sense defined by the right-hand rule). For the normal stress to
be zero, we must have
xy
sin
−
xz
cos
=
0
(9.143)
or
d
z
d
s
−
xz
d
y
d
s
=
xy
0
(9.144)
Substituting the stress function relations, we obtain
∂
d
z
d
z
d
s
+
d
d
y
d
y
d
s
=
d
d
s
=
0
(9.145)
which shows that the value of the stress function is constant on the surface. The
value is arbitrary and most often taken to be zero.
9.9.2 Torque
The stress function formulation of the torsion problem as given previously does
not explicitly include the applied torque. To obtain an expression relating the
