Civil Engineering Reference
In-Depth Information
Similarly resolving parallel to AB
21
.
2 BC cos 60
◦
−
28
.
3 BC sin 60
◦
+
28
.
3AC cos 60
◦
τ
AB
=−
so that
21
.
2 sin 60
◦
cos 60
◦
−
28
.
3 sin
2
60
◦
+
28
.
3 cos
2
60
◦
τ
=−
from which
23
.
3N
/
mm
2
τ
=−
acting in the direction AB.
14.3 P
RINCIPAL
S
TRESSES
Equations (14.5) and (14.6) give the direct and shear stresses on an inclined plane at
a point in a structural member subjected to a combination of loads which produces
a general two-dimensional stress system at that point. Clearly for given values of
σ
x
,
σ
y
and
τ
xy
, in other words a given loading system, both
σ
n
and
τ
vary with the angle
θ
and will attain maximum or minimum values when d
σ
n
/
d
θ
=
0 and d
τ/
d
θ
=
0. From
Eq. (14.5)
d
σ
n
d
θ
=−
2
σ
x
cos
θ
sin
θ
+
2
σ
y
sin
θ
cos
θ
−
2
τ
xy
cos 2
θ
=
0
then
−
(
σ
x
−
σ
y
) sin 2
θ
−
2
τ
xy
cos 2
θ
=
0
or
2
τ
xy
σ
x
−
tan 2
θ
=−
(14.7)
σ
y
Two solutions,
π/
2, satisfy Eq. (14.7) so that there are two mutually
perpendicular planes on which the direct stress is either a maximum or a minimum.
Furthermore, by comparison of Eqs (14.7) and (14.6) it can be seen that these planes
correspond to those on which
τ
−
θ
and
−
θ
−
=
0.
The direct stresses on these planes are called
principal stresses
and the planes are called
principal planes
.
From Eq. (14.7)
2
τ
xy
σ
x
−
σ
y
(
σ
x
−
sin 2
θ
=−
(
σ
x
−
cos 2
θ
=
σ
y
)
2
+
4
τ
xy
σ
y
)
2
+
4
τ
xy
