Civil Engineering Reference
In-Depth Information
For an axial load
P
P
A
=
P
(
π/
4)
70N
/
mm
2
σ
x
=
=
(Eq. (7.1))
50
2
×
so that
P
=
137
.
4kN
Also for the torque
T
and using Eq. (11.4) we have
Tr
J
=
T
×
25
29
.
7N
/
mm
2
τ
xy
=
=
50
4
(
π/
32)
×
which gives
T
=
0
.
7kNm
Note that
P
could have been found directly in this case from the axial strain
ε
a
. Thus
from Eq. (7.8)
10
−
6
70N
/
mm
2
σ
x
=
E
ε
a
=
70 000
×
1000
×
=
as before.
14.10 T
HEORIES OF
E
LASTIC
F
AILURE
The direct stress in a structural member subjected to simple tension or compression
is directly proportional to strain up to the yield point of the material (Section 7.7). It
is therefore a relatively simple matter to design such a member using the direct stress
at yield as the design criterion. However, as we saw in Section 14.3, the direct and
shear stresses at a point in a structural member subjected to a complex loading system
are not necessarily the maximum values at the point. In such cases it is not clear how
failure occurs, so that it is difficult to determine limiting values of load or alternatively
to design a structural member for given loads. An obvious method, perhaps, would be
to use direct experiment in which the structural member is loaded until deformations
are no longer proportional to the applied load; clearly such an approach would be
both time-wasting and uneconomical. Ideally a method is required that relates some
parameter representing the applied stresses to, say, the yield stress in simple tension
which is a constant for a given material.
In Section 14.3 we saw that a complex two-dimensional stress system comprising direct
and shear stresses could be represented by a simpler system of direct stresses only, in
other words, the principal stresses. The problem is therefore simplified to some extent
since the applied loads are now being represented by a system of direct stresses only.
Clearly this procedure could be extended to the three-dimensional case so that no
matter how complex the loading and the resulting stress system, there would remain at
