Page 122 - Shigley's Mechanical Engineering Design

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Load and Stress Analysis 97
Table 3–2 Beam Shape Formula Beam Shape Formula

Formulas for Maximum V 3V V 2V
= =
avc A max = avc A max =
Transverse Shear Stress 2A A
from VQ/Ib
Rectangular
Hollow, thin-walled round
V 4V V
= A web .
avc A max = max =
3A A web
Structural I beam (thin-walled)
Circular

It is significant to observe that the transverse shear stress in each of these common
cross sections is maximum on the neutral axis, and zero on the outer surfaces. Since this
is exactly the opposite of where the bending and torsional stresses have their maximum
and minimum values, the transverse shear stress is often not critical from a design
perspective.
Let us examine the significance of the transverse shear stress, using as an example
a cantilever beam of length L, with rectangular cross section b h, loaded at the free end
with a transverse force F. At the wall, where the bending moment is the largest, at a dis-
tance y from the neutral axis, a stress element will include both bending stress and
transverse shear stress. In Sec. 5–4 it will be shown that a good measure of the com-
bined effects of multiple stresses on a stress element is the maximum shear stress.
Inserting the bending stress (My/I) and the transverse shear stress (VQ/Ib) into the
maximum shear stress equation, Eq. (3–14), we obtain a general equation for the max-
imum shear stress in a cantilever beam with a rectangular cross section. This equation
can then be normalized with respect to L/h and y/c, where c is the distance from the
neutral axis to the outer surface (h/2), to give

σ 3F 2
2
2
2
2
τ max = + τ = 4(L/h) (y/c) + 1 − (y/c) 2 (d)
2 2bh
To investigate the significance of transverse shear stress, we plot τ max as a function
of L/h for several values of y/c, as shown in Fig. 3–19. Since F and b appear only as
linear multipliers outside the radical, they will only serve to scale the plot in the verti-
cal direction without changing any of the relationships. Notice that at the neutral axis
where y/c 0, τ max is constant for any length beam, since the bending stress is zero at
the neutral axis and the transverse shear stress is independent of L. On the other hand,
on the outer surface where y/c 1, τ max increases linearly with L/h because of the
bending moment. For y/c between zero and one, τ max is nonlinear for low values of L/h,
but behaves linearly as L/h increases, displaying the dominance of the bending stress
as the moment arm increases. We can see from the graph that the critical stress element
(the largest value of τ max ) will always be either on the outer surface (y/c 1) or at the
neutral axis (y/c 0), and never between. Thus, for the rectangular cross section, the
transition between these two locations occurs at L/h 0.5 where the line for y/c 1
crosses the horizontal line for y/c 0. The critical stress element is either on the outer