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

8.7 Bending Due to an Eccentric Axial Load
As discussed in Chapters 1, 4, and 5, an axial load whose line of action passes through the
centroid of a cross section (termed a centric axial load) creates normal stress that is uniformly
distributed across the cross-sectional area of a member. An eccentric axial load is
a force whose line of action does not pass through the centroid of the cross section. When
an axial force is offset from a member’s centroid, bending stresses are created in the member
in addition to the normal stresses caused by the axial force. Analysis of this type of
bending, therefore, requires a consideration of both axial stresses and bending stresses.
Many structures, including common objects such as signposts, clamps, and piers, are subjected
to eccentric axial loads.
Suppose that the normal stresses acting on the section containing the C are to be
determined for the object shown in Figure 8.13a. The analysis presented here assumes that
the bending member has a plane of symmetry (see Figure 8.2a) and that all loads are
applied in that plane.
The line of action of the axial load P does not pass through the centroid C; therefore,
this object (between points H and K) is subjected to an eccentric axial load. The eccentricity
between the line of action of P and the centroid C is denoted by the symbol e.
The internal forces acting on a cross section can be represented by an internal
axial force F acting at the centroid of the cross section and an internal bending moment
M acting in the plane of symmetry, as shown on the free-body diagram cut through C
(Figure 8.13b).
Both the internal axial force F and the internal bending moment M produce normal
stresses (Figure 8.14). These stresses must be combined in order to determine the complete
stress distribution at the section of interest. The axial force F produces a normal stress σ x =
F/A that is uniformly distributed over the entire cross section. The bending moment M
produces a normal stress, given by the flexure formula σ x = −My/I z , that is linearly distributed
over the depth of the cross section. The complete stress distribution is obtained by
superposing the stresses produced by F and M as
F My
σ x = − (8.19)
A I
The sign conventions for F and M are the same as those presented in previous chapters.
A positive internal axial force F produces tensile normal stresses. A positive internal bending
moment produces compressive normal stresses for positive values of y.
An axial force whose line of action is separated from the centroid of the cross section by
an eccentricity e produces an internal bending moment M = P × e. Thus, for an eccentric
axial force, Equation (8.19) can also be expressed as
z
F ( Pe)
y
σ x = − (8.20)
A I
z
M = Pe
y
C
H
K
e
P
P
x
P
y
C
e
x
F
(a) Configuration of bending member
(b) Free-body diagram
FIGURE 8.13 Bending due to an eccentric axial load.
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