Civil Engineering Reference
In-Depth Information
would suggest, therefore, that shear stresses have no effect on the failure of brittle
materials and that a direct relationship exists between the principal stresses at a point
in a brittle material subjected to a complex loading system and the failure stress in
simple tension or compression. This forms the basis for the most widely accepted
theory of failure for brittle materials.
Maximum normal stress theory
This theory, frequently attributed to Rankine, states that:
Failure occurs when one of the principal stresses reaches the value of the yield stress in
simple tension or compression.
For most brittle materials the yield stress in tension is very much less than the yield
stress in compression, e.g. for concrete
σ
Y
(compression) is approximately 20
σ
Y
(ten-
sion). Thus it is essential in any particular problem to know which of the yield stresses
is achieved first.
Suppose that a brittlematerial is subjected to a complex loading systemwhich produces
principal stresses
σ
I
,
σ
II
and
σ
III
as in Fig. 14.18. Thus for
σ
I
>σ
II
>σ
III
>
0 failure
occurs when
σ
I
=
σ
Y
(tension)
(14.57)
Alternatively, for
σ
I
>σ
II
>
0,
σ
III
<
0 and
σ
I
<σ
Y
(tension) failure occurs when
σ
III
=
σ
Y
(compression)
(14.58)
and so on.
A yield locus may be drawn for the two-dimensional case, as for the Tresca and von
Mises theories of failure for ductile materials, and is shown in Fig. 14.23. Note that
since the failure stress in tension,
σ
Y
(T), is generally less than the failure stress in
compression,
σ
Y
(C), the yield locus is not symmetrically arranged about the
σ
I
and
σ
II
σ
II
σ
II
σ
Y
(T)
σ
II
σ
Y
(T)
σ
Y
(T)
σ
I
0,
σ
II
0
σ
I
0,
σ
II
0
σ
I
σ
Y
(T)
σ
I
σ
Y
(C)
σ
Y
(T)
σ
I
σ
Y
(C)
σ
I
σ
Y
(C)
σ
I
σ
Y
(T)
σ
I
0,
σ
II
0
σ
I
0,
σ
II
0
F
IGURE
14.23
Yield locus for a
brittle material
σ
Y
(C)
σ
II
σ
Y
(C)
σ
II
σ
Y
(C)
