Civil Engineering Reference
In-Depth Information
E
XAMPLE
14.7
The state of stress at a point in a structural member is defined by
a two-dimensional stress system as follows:
σ
x
=+
140N
/
mm
2
,
σ
y
=−
70N
/
mm
2
and
60N
/
mm
2
. If the material of the member has a yield stress in simple tension of
225N/mm
2
, determine whether or not yielding has occurred according to the Tresca
and von Mises theories of elastic failure.
τ
xy
=+
The first step is to determine the principal stresses
σ
I
and
σ
II
. From Eqs (14.8)
and (14.9)
(140
1
2
(140
1
2
70)
2
60
2
σ
I
=
−
70)
+
+
+
4
×
i.e.
155
.
9N
/
mm
2
σ
I
=
and
(140
1
2
(140
1
2
σ
II
=
−
70)
−
+
70)
2
+
4
×
60
2
i.e.
85
.
9N
/
mm
2
σ
II
=−
=
Since
σ
II
is algebraically less than
σ
III
(
0), Eq. (14.42) applies.
Thus
241
.
8N
/
mm
2
σ
I
−
σ
II
=
225N
/
mm
2
) so that according to the Tresca theory
This value is greater than
σ
Y
(
=
failure has, in fact, occurred.
Substituting the above values of
σ
I
and
σ
II
in Eq. (14.55) we have
(155
.
9)
2
85
.
9)
2
+
(
−
−
(155
.
9)(
−
85
.
9)
=
45 075
.
4
The square root of this expression is 212
.
3N
/
mm
2
so that according to the von Mises
theory the material has not failed.
E
XAMPLE
14.8
The rectangular cross section of a thin-walled box girder (Fig. 14.22)
is subjected to a bending moment of 250 kNm and a torque of 200 kNm. If the allow-
able equivalent stress for the material of the box girder is 180N
/
mm
2
, determine
whether or not the design is satisfactory using the requirement of Eq. (14.56).
The maximum shear stress in the cross section occurs in the vertical walls of the section
and is given by Eq. (11.22), i.e.
10
6
T
max
2
At
min
=
200
×
80N
/
mm
2
τ
max
=
10
=
2
×
500
×
250
×
