Page 178 - Marks Calculation for Machine Design

P1: Shibu
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January 4, 2005
Brown.cls
Brown˙C04
STRENGTH OF MACHINES
160
Stress Element. The general stress element shown in Fig. 4.2 becomes the stress element
shown in Fig. 4.11, where the normal stress (σ xx ) is a combination of the axial stress and
bending stress, the normal stress (σ yy ) is zero, and the shear stress (τ xy ) is the shear stress
due to bending.
s yy 0
t xy t xy
t
t xy xy
P My
s xx s xx s = – + –
xx
→ A I
s xx
t xy t = VQ
xy
Ib
t xy t xy
s yy 0
FIGURE 4.11 Stress element for axial and bending loads.
As shown in Fig. 4.11, the normal stress (σ xx ) has two terms, one due to the compressive
axial load (P) that is constant across the cross section of the beam and is always negative,
and the other term is due to the bending moment (M) in the beam and will be positive on one
side of the neutral axis and negative on the other side. It will always be zero at the neutral
axis. For the particular loading shown in Fig. 4.10, the top of the beam is in compression
and the bottom is in tension.
The shear stress due to bending (τ xy ) is shown downward on the right edge of the stress
element because the shear force (V ) will be downward at the right side of the cross section
of the beam. This shear stress due to bending will be maximum at the neutral axis and zero
at the top and bottom of the beam.
As a consequence of what has just been said about the normal and shear stresses, there
are actually two stress elements to consider. One is a stress element at the top or the bottom
of the beam where the bending stress is maximum and the shear stress zero; and the other
is a stress element at the neutral axis where the shear stress is maximum and the bending
stress zero. The axial stress will be the same on both these elements. Figure 4.12 shows an
element at the top of the beam and the element at the neutral axis.

0 0
t
0 max
0 t max
P My P
s s = – – s s = –
xx
xx
A I xx xx A
VQ max
0 t max =
Ib
0 t max
Top of beam Neutral axis
0 0
FIGURE 4.12 Special elements for axial and bending loads.
Notice that the element at the top of the beam, as well as the one that would be at the
bottom, are uniaxial stress elements. The element at the neutral axis is a general stress