Civil Engineering Reference
In-Depth Information
BIAXIAL STRESS SYSTEM
Amore complex stress systemmay be produced by a loading system such as that shown
in Fig. 14.4 where a thin-walled hollow cylinder is subjected to an internal pressure,
p
.
The internal pressure induces circumferential or hoop stresses
σ
y
, given by Eq. (7.63),
on planes parallel to the axis of the cylinder and, in addition, longitudinal stresses,
σ
x
, on planes perpendicular to the axis of the cylinder (Eq. (7.62)). Thus any two-
dimensional element of unit thickness in the wall of the cylinder and having sides
perpendicular and parallel to the axis of the cylinder supports a biaxial stress system
as shown in Fig. 14.4. In this particular case
σ
x
and
σ
y
each have constant values
irrespective of the position of the element.
Let us consider the equilibrium of a triangular portion ABC of the element as shown
in Fig. 14.5. Resolving forces in a direction perpendicular to AB we have
=
+
σ
n
AB
σ
x
BC cos
θ
σ
y
AC sin
θ
or
BC
AB
cos
θ
AC
AB
sin
θ
σ
n
=
σ
x
+
σ
y
which gives
σ
x
cos
2
θ
σ
y
sin
2
θ
σ
n
=
+
(14.3)
Internal pressure
, p
s
x
s
y
F
IGURE
14.4
Generation of a
biaxial stress
system
s
x
s
y
s
y
B
B
s
n
u
u
s
x
s
x
s
x
τ
F
IGURE
14.5
Determination of
stresses on an
inclined plane in a
biaxial stress
system
A
C
A
C
s
y
s
y
(a)
(b)
