Page 120 - Shigley's Mechanical Engineering Design

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Load and Stress Analysis 95
w(x)
y My x My dMy
x
I
I
I
c
y 1 x
V M dM
M x
V dV
dx
x dx
(b)
(a)
A
c
y F dM y
Figure 3–17 dx y 1 I
b
Beam section isolation. Note:
x
Only forces shown in x
direction on dx element in (b). (c)

where, for the isolated area y 1 to c, ¯y is the distance in the y direction from the neutral

plane to the centroid of the area A . With this, Eq. (3–29) can be written as
VQ
τ = (3–31)
Ib
This stress is known as the transverse shear stress. It is always accompanied with bend-
ing stress.
In using this equation, note that b is the width of the section at y = y 1 . Also, I is
the second moment of area of the entire section about the neutral axis.
Because cross shears are equal, and area A is finite, the shear stress τ given by

Eq. (3–31) and shown on area A in Fig. 3–17c occurs only at y = y 1 . The shear stress

on the lateral area varies with y, normally maximum at y = 0 (where ¯y A is maximum)

and zero at the outer fibers of the beam where A 0.
The shear stress distribution in a beam depends on how Q/b varies as a function
of y 1 . Here we will show how to determine the shear stress distribution for a beam with
a rectangular cross section and provide results of maximum values of shear stress for
other standard cross sections. Figure 3–18 shows a portion of a beam with a rectangu-
lar cross section, subjected to a shear force V and a bending moment M. As a result of
the bending moment, a normal stress σ is developed on a cross section such as A–A,
which is in compression above the neutral axis and in tension below. To investigate the
shear stress at a distance y 1 above the neutral axis, we select an element of area dA at
a distance y above the neutral axis. Then, dA = bdy, and so Eq. (3–30) becomes

c c 2 c
by b 2 2
Q = yd A = b ydy = = c − y 1 (b)
2 2
y 1 y 1 y 1
Substituting this value for Q into Eq. (3–31) gives
V 2 2
τ = c − y 1 (3–32)
2I
This is the general equation for shear stress in a rectangular beam. To learn something
about it, let us make some substitutions. From Table A–18, the second moment of area
3
for a rectangular section is I = bh /12; substituting h = 2c and A = bh = 2bc gives
Ac 2
I = (c)
3