Civil Engineering Reference
In-Depth Information
The maximum and minimum values of the shear stress
τ
occur when Q
coincides with
F and D at the lower and upper extremities of the circle. At these points
τ
max,min
are
clearly equal to the radius of the circle. Hence
(
σ
x
−
1
2
σ
y
)
2
4
τ
xy
τ
max,min
=±
+
(see Eq. (14.11))
The minimum value of shear stress is the algebraic minimum. The planes of maxi-
mum and minimum shear stress are given by 2
θ
=
β
+
π/
2 and 2
θ
=
β
+
3
π/
2 and are
inclined at 45
◦
to the principal planes.
E
XAMPLE
14.4
Direct stresses of 160N
/
mm
2
, tension, and 120N
/
mm
2
, com-
pression, are applied at a particular point in an elastic material on two mutually
perpendicular planes. The maximum principal stress in the material is limited to
200N
/
mm
2
, tension. Use a graphical method to find the allowable value of shear
stress at the point.
τ
Q
2
(120 N/mm
2
, 112 N/mm
2
)
B
s
O
P
2
P
1
C
s
1
(200 N/mm
2
)
Q
1
(160 N/mm
2
,
112 N/mm
2
)
F
IGURE
14.12
Mohr's circle of
stress for Ex. 14.4
First, axes O
στ
are set up to a suitable scale. P
1
and P
2
are then located corre-
sponding to
σ
x
=
120N
/
mm
2
, respectively; the centre C of the
circle is mid-way between P
1
and P
2
(Fig. 14.12). The radius is obtained by locating
B(
σ
1
=
160N
/
mm
2
and
σ
y
=−
200N
/
mm
2
) and the circle then drawn. The maximum allowable applied shear
stress,
τ
xy
, is then obtained by locating Q
1
or Q
2
. The maximum shear stress at the
point is equal to the radius of the circle and is 180N
/
mm
2
.
14.5 S
TRESS
T
RAJECTORIES
We have shown that direct and shear stresses at a point in a beam produced, say,
by bending and shear and calculated by the methods discused in Chapters 9 and 10,
respectively, are not necessarily the greatest values of direct and shear stress at the
point. In order, therefore, to obtain a more complete picture of the distribution,
