8
374
Mechanics of Materials: Stress Transformation
M. Vable
Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm
January, 2010
CHAPTER EIGHT
STRESS TRANSFORMATION
Learning objectives
1.
Learn the equations and procedures of relating stresses in different coordinate systems (on different planes) at a point.
2.
Visualize planes passing through a point on which stresses are given or are being found, in particular the planes of
maximum normal and shear stress.
_______________________________________________
Figure 8.1 shows failure surfaces of aluminum and cast iron members under axial and torsional loads. Why do different mate-
rials under similar loading produce different failure surfaces? If we had a combined loading of axial and torsion, then what
would be the failure surface, and which stress component would cause the failure? The answer to this question is critical for
the successful design of structural members that are subjected to combined axial, torsional, and bending loads. In Chapter 10
we will study combined loading and failure theories that relate maximum normal and shear stresses to material strength. In
this chapter we develop procedures and equations that transform stress components from one coordinate system to another at
a
given point
.
Stress transformation can also be viewed as relating stresses on different planes that pass through a point. The outward
normals of the planes form the axes of a coordinate system. Thus relating stresses on different planes is equivalent to relating
stresses in different coordinate systems. We will use both viewpoints in this chapter of stress transformation.
P
P
xx
xx
x
x
T
T
Cast iron
Aluminum
(b)
(a)
Figure 8.1
Failure surfaces. (
a
) Axial load. (
b
) Torsional load. (Specimens courtesy Professor J. B. Ligon.)

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8
375
Mechanics of Materials: Stress Transformation
M. Vable
Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm
January, 2010
8.1
PRELUDE TO THEORY: THE WEDGE METHOD
In this chapter we will study three methods of stress transformation. The
wedge method
, described in this section, is used to
derive stress transformation equations in the next section. The stress transformation equations are then manipulated to gener-
ate a graphical procedure called
Mohr’s circle
, which is described in Section 8.3.
Two coordinate systems will be used in this chapter. First, the entire problem is described in a fixed reference coordinate
system called the
global coordinate system
. We usually relate internal forces and moments to external forces and moments in
the global coordinate system. The internal quantities are then used to obtain stresses, such as axial stress, torsional shear
stress, and bending normal and shear stresses. And second, a
local coordinate system
that can be fixed at any point on the
body. The orientation of the local coordinate system is defined with respect to the global coordinate. In all two-dimensional
problems in this book, the local coordinate system will be the
n
,
t
,
z
coordinate system.

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