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

a. The normal stresses caused by F and M act in the longitudinal direction of the beam,
either in tension or in compression.
b. The proper direction for the shear stress τ produced by nonuniform bending is sometimes
challenging to establish. Determine the direction of the shear force V acting on
a transverse plane at the point of interest (Figure 15.2). The transverse shear stress
τ acts in the same direction on this plane. After the direction of the shear stress has
been established on one face of the stress element, the shear stress directions on all
four faces are known.
c. It is generally more reliable to use inspection to establish the directions of normal
and shear stresses acting on the stress element. Consider the positive internal
shear force V shown in Figure 15.2. (Recall that a positive V acts downward on
the right-hand face of a beam segment and upward on the left-hand face.)
Although the shear force V is positive, the corresponding shear stress τ xy is
considered negative according to the sign conventions used in the stress transformation
equations.
5. Once all stresses on orthogonal planes through the point are known and summarized on
a stress element, the methods of Chapter 12 can be used to calculate the principal
stresses and the maximum shear stresses at the point.
Examples 15.2 and 15.3 illustrate the procedure.
+V
y
x
Transverse
plane
H
H
τ xy
+V
FIGURE 15.2 Correspondence
between direction of V and
direction of τ.
ExAmpLE 15.2
The simply supported wide-flange beam supports the loadings shown. Determine the
principal stresses and maximum shear stress at points H and K. Show these stresses on a
properly oriented stress element.
200 mm
16 mm
y
80 kN/m
A H
K
1 m 1 m
3 m 1.5 m
60 kN
1.5 m
B
x
75 mm
35 mm
K
z
H
y
9 mm
350 mm
Plan the Solution
The moment of inertia of the wide-flange section will be computed from the crosssectional
dimensions. The shear-force and bending-moment diagrams will be constructed
for the simply supported beam. From these diagrams, the internal shear force and the
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