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

outer faces of the flanges. Load P is known to be 35 kips, and the
strain in gage H is measured as ε H = 120 × 10 −6 in./in. Determine
(a) the magnitude of load Q.
(b) the expected strain reading in gage K.
P
a
y
a
Q
p8.59 A short length of a rolled-steel [E = 29 × 10 3 ksi]
column supports a rigid plate on which two loads P and Q are
applied as shown in Figure P8.58a/59a. The column cross section
(Figure P8.58b/59b) has a depth d = 8.0 in., an area A = 5.40 in. 2 ,
and a moment of inertia I z = 57.5 in. 4 . Normal strains are measured
with strain gages H and K, which are attached on the centerline
of the outer faces of the flanges. The strains measured in the
two gages are ε H = −530 × 10 −6 in./in. and ε K = −310 × 10 −6 in./in.
Determine the magnitudes of loads P and Q.
z
x
Gage H
d
Gage K
Gage H
x
FIGURE p8.58a/59a
z
Cross section
FIGURE p8.58b/59b
8.8 Unsymmetric Bending
In Sections 8.1 through 8.3, the theory of bending was developed for prismatic beams.
In deriving this theory, beams were assumed to have a longitudinal plane of symmetry
(Figure 8.2a), termed the plane of bending. Furthermore, loads acting on the beam, as
well as the resulting curvatures and deflections, were assumed to act only in the plane
of bending. If the beam cross section is unsymmetric or if the loads on the beam do not
act in the plane of bending, then the theory of bending developed in Sections 8.1 through
8.3 is not valid.
To see why, consider the following thought experiment: Suppose that the unsymmetric
flanged cross section shown in Figure 8.15a (termed a zee section) is subjected to equalmagnitude
bending moments M z that act as shown about the z axis. Suppose further that the
beam bends only in the x–y plane in response to M z and that the z axis is the neutral axis for
bending. Then, if the latter supposition is correct, then the bending stresses shown in
Figure 8.15b will be produced in the zee section. Compressive bending stresses will occur
above the z axis, and tensile bending stresses will occur below the z axis.
Next, consider the stresses that act in the flanges of the zee section. Bending
stresses will be uniformly distributed across the width of each flange. The internal resultant
force of the compressive bending stresses acting in the top flange will be termed
F C (Figure 8.15c). Its line of action passes through the midpoint of the flange (in the
horizontal direction) at a distance z C from the y axis. Similarly, the internal resultant
force of the tensile bending stresses in the bottom flange will be termed F T , and its line
of action is located a distance z T from the y axis. Since these two forces are equal in
magnitude, but act in opposite directions, they form an internal couple that creates a
bending moment about the y axis. This internal moment about the y axis
(i.e., acting in the x–z plane) is not counteracted by any external moment (since the
applied moments M z act about the z axis only); therefore, equilibrium is not satisfied.
Consequently, bending of the unsymmetric beam cannot occur solely in the plane of the
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