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

9.3 The Shear Stress Formula
325
THE SHEAR STRESS FORMuLA
In this section, a method for determining the shear stresses produced in a prismatic beam
made of a homogeneous linear-elastic material will be developed. Consider the beam
shown in Figure 9.5a. The beam is subjected to various loadings, and its cross section is
shown in Figure 9.5b. In the development that follows, particular attention is focused on
one portion of the cross section. Call that portion area A′.
Now consider a free-body diagram of the beam with cross section of length Dx and
located some distance x from the origin (Figure 9.6a). The internal shear force and bending
moment on the left side of the free-body diagram (section a–b–c) are designated as V and M,
respectively. On the right side of the free-body diagram (section d–e–f ), the internal shear
force and bending moment are slightly different: V + DV and M + DM. Equilibrium in the
horizontal x direction will be considered here. The internal shear forces V and V + DV and
the distributed load w(x) act in the vertical direction; consequently, they will have no effect
on equilibrium in the x direction, and they can be omitted in the subsequent analysis.
The normal stresses acting on this free-body diagram (Figure 9.6b) can be determined
by the flexure formula. On the left side of the free-body diagram, the bending stresses due
to the internal bending moment M are given by My/I z , and on the right side, the internal
bending moment M + DM creates bending stresses given by (M + DM)y/I z . The signs
associated with the bending stresses will be determined by inspection. Above the neutral
Area A′
y
P 0 P 1
w(x)
y 2
t
y
y 1
A
a d
c f
M 0 M 1
B
x
z
x ∆x
(a) Beam loading
FIGURE 9.5 Prismatic beam subjected to nonuniform bending.
∆x
(b) Cross section
w(x)
a
b
V
y
e
d
V+∆V
a d
M y
(M+∆M)y
σ x =
σ
b e
x =
I z I z
y
x
x
M
c
∆x
FIGURE 9.6
f
M+∆M
(a) Free-body diagram
Free-body diagrams of beam segment.
M
c
∆x
f
M+∆M
(b) Bending stresses due to internal
bending moments