Civil Engineering Reference
In-Depth Information
PROBLEMS
P.14.1
At a point in an elastic material there are two mutually perpendicular planes,
one of which carries a direct tensile stress of 50N
/
mm
2
and a shear stress of 40N
/
mm
2
while the other plane is subjected to a direct compressive stress of 35N
/
mm
2
and a
complementary shear stress of 40N
/
mm
2
. Determine the principal stresses at the
point, the position of the planes on which they act and the position of the planes on
which there is no direct stress.
65
.
9N
/
mm
2
,
θ
21
.
6
◦
50
.
9N
/
mm
2
,
θ
111
.
6
◦
.
Ans.
σ
I
=
=−
σ
II
=−
=−
No direct stress on planes at 27
.
1
◦
and 117.1
◦
to the plane on which the 50N
/
mm
2
stress acts.
P.14.2
One of the principal stresses in a two-dimensional stress system is 139N
/
mm
2
acting on a plane A. On another plane B normal and shear stresses of 108 and
62 N
/
mm
2
, respectively, act. Determine
(a) the angle between the planes A and B,
(b) the other principal stress,
(c) the direct stress on the plane perpendicular to plane B.
Ans.
(a) 26
◦
34
, (b)
16N/mm
2
, (c) 15N/mm
2
.
−
P.14.3
The state of stress at a point in a structural member may be represented by
a two-dimensional stress system in which
σ
x
=
100N
/
mm
2
,
σ
y
=−
80N
/
mm
2
and
45N
/
mm
2
. Determine the direct stress on a plane inclined at 60
◦
to the pos-
itive direction of
σ
x
and also the principal stresses. Calculate also the inclination of
the principal planes to the plane on which
σ
x
acts. Verify your answers by a graphical
method.
τ
xy
=
16N
/
mm
2
110
.
6N
/
mm
2
90
.
6N
/
mm
2
13
.
3
◦
Ans.
σ
n
=
σ
I
=
σ
II
=−
θ
=−
and
103
.
3
◦
.
−
P.14.4
Determine the normal and shear stress on the plane AB shown in Fig. P.14.4
when
60
◦
,
σ
x
=
54N
/
mm
2
,
σ
y
=
30N
/
mm
2
,
τ
xy
=
5N
/
mm
2
;
(i)
α
=
120
◦
,
σ
x
=−
60N
/
mm
2
,
σ
y
=−
36N
/
mm
2
,
τ
xy
=
5N
/
mm
2
.
(ii)
α
=
52
.
3N
/
mm
2
,
τ
7
.
9N
/
mm
2
;
Ans
. (i)
σ
n
=
=
58
.
3N
/
mm
2
,
τ
7
.
9N
/
mm
2
.
(ii)
σ
n
=−
=
