Section 3.4
Solid Mechanics Part I
Kelly
49
3.4 Equilibrium of Stress
Consider two perpendicular planes passing through a point
p
.
The stress components
acting on these planes are as shown in Fig. 3.4.1a.
These stresses are usually shown
together acting on a small material element of finite size, Fig. 3.4.1b.
It has been seen
that the stress may vary from point to point in a material but, if the element is very small,
the stresses on one side can be taken to be equal to the stresses acting on the other side.
Figure 3.4.1: stress components acting on two perpendicular planes through a point;
(a) two perpendicular surfaces at a point, (b) small material element at the point
It will be shown below that the stress components acting on
any
other plane through
p
can
be evaluated from a knowledge of only these stress components.
3.4.1
Symmetry of the Shear Stress
Consider the material element shown in Fig. 3.4.1b, reproduced in Fig. 3.4.2a below.
The
element has dimensions is
y
x
Δ
×
Δ
and is subjected to arbitrary uniform stresses over its
sides.
The resultant forces of the stresses acting on each side of the element act through
the side-centres, and are shown in Fig. 3.4.2b.
The stresses shown are positive, but note
how positive stresses can lead to negative forces, depending on the definition of the
y
x
−
axes used.
The resultant force on the complete element is seen to be zero.
yy
σ
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σ
xx
σ
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σ
yx
σ
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σ
xy
σ
xy
σ
)
a
(
)
b
(
x
y
•
p
xx
σ
yx
σ
xx
σ
yx
σ
yy
σ
xy
σ
yy
σ
xy
σ

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