Civil Engineering Reference
In-Depth Information
y
A
a
x
B
F
IGURE
P.14.4
P.14.5
A shear stress
τ
xy
acts in a two-dimensional field in which the maximum allow-
able shear stress is denoted by
τ
max
and the major principal stress by
σ
I
. Derive, using
the geometry of Mohr's circle of stress, expressions for the maximum values of direct
stress which may be applied to the
x
and
y
planes in terms of the parameters given.
τ
max
−
τ
max
−
Ans.
σ
x
=
σ
I
−
τ
max
+
τ
xy
σ
y
=
σ
I
−
τ
max
−
τ
xy
.
P.14.6
In an experimental determination of principal stresses a cantilever of hollow
circular cross section is subjected to a varying bending moment and torque; the inter-
nal and external diameters of the cantilever are 40 and 50mm, respectively. For a
given loading condition the bending moment and torque at a particular section of the
cantilever are 100 and 50Nm, respectively. Calculate the maximum and minimum
principal stresses at a point on the outer surface of the cantilever at this section where
the direct stress produced by the bending moment is tensile. Determine also the max-
imum shear stress at the point and the inclination of the principal stresses to the axis
of the cantilever.
The experimental values of principal stress are estimated from readings obtained from
a45
◦
strain gauge rosette aligned so that one of its three arms is parallel to and another
perpendicular to the axis of the cantilever. For the loading condition of zero torque
and varying bending moment, comment on the ratio of these strain gauge readings.
14
.
6N
/
mm
2
0
.
8N
/
mm
2
Ans
.
σ
I
=
σ
II
=−
7
.
7N
/
mm
2
13
.
3
◦
and
103
.
3
◦
.
τ
max
=
=−
−
θ
P.14.7
A thin-walled cylinder has an internal diameter of 1200mm and has walls
1
.
2 mm thick. It is subjected to an internal pressure of 0
.
7N
/
mm
2
and a torque, about
its longitudinal axis, of 500 kNm. Determine the principal stresses at a point in the
wall of the cylinder and also the maximum shear stress.
Ans.
466
.
4N
/
mm
2
,58
.
6N
/
mm
2
, 203
.
9N
/
mm
2
.
P.14.8
A rectangular piece of material is subjected to tensile stresses of 83 and
65N
/
mm
2
on mutually perpendicular faces. Find the strain in the direction of each
