Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

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STRESS TRANSFORMATIONS
such as the normal stresses produced by axial loads or bending moments, or the shear
stresses produced by torsion or transverse shear in beams. Before proceeding to the methods
used for stress transformations, it is helpful to consider how the analyst determines the
stresses that appear on a stress element. This section focuses on solid components in which
several internal loads or moments act simultaneously on a member’s cross section. The
method of superposition will be used to combine the various stresses acting at a particular
point, and the results will be summarized on a stress element.
The analysis of the stresses produced by multiple internal loads or moments that act
simultaneously on a member’s cross section is usually referred to as combined loadings.
In Chapter 15, combined loadings will be examined more completely. For instance, structures
with multiple external loads will be considered, along with solid components that
have three-dimensional geometry and loadings. Stress transformations will also be incorporated
into the analysis in that discussion. The intent of this section is simply to introduce
the reader to the process of evaluating the state of stress at a specific point. The process of
putting together the stress element is demonstrated with the use of geometrically simple
components and basic loadings.
ExAmpLE 12.1
A vertical pipe column with an outside diameter D = 114 mm and an
inside diameter d = 102 mm supports the loads shown. Determine the
normal and shear stresses acting at point H, and show these stresses on
a stress element.
Column cross-sectional
dimensions.
Plan the Solution
The cross-sectional properties will be
computed for the pipe column. Each
of the applied loads will be considered
in turn. The normal and/or shear
stresses created by each at point H will
be computed. Both the magnitude and
direction of the stress must be evaluated
and shown on the proper face of a
stress element. By the principle of superposition,
the stresses will be combined
appropriately so that the state of
stress at point H is summarized succinctly
by the stress element.
SolutioN
Section Properties
The outside diameter of the pipe is D = 114 mm, and the inside diameter is d = 102 mm.
The area, the moment of inertia, and the polar moment of inertia for the cross section
are, respectively, as follows:
π
A D d
4 [ 2 2 ] π
2 2 2
= − = [(114 mm) − (102 mm) ] = 2,035.752 mm
4
π
I D d
64 [ ] π
= − = [(114 mm)
64
− (102 mm) ] = 2,977,287 mm
π
J D d
32 [ ] π
= − = [(114 mm)
32
− (102 mm) ] = 5,954,575 mm
4 4 4 4 4
4 4 4 4 4