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

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bENdINg
y
Compressive
bending stress
y
Tensile
bending stress
M
M
O
x
O
x
Tensile
bending stress
(a) Bending stresses caused by
positive M
Compressive
bending stress
(b) Bending stresses caused by
negative M
FIGURE 8.6 Relationship between bending moment M and bending stress.
Compression
+M +M
Tension
(a) Positive M
Tension
–M –M
Compression
(b) Negative M
FIGURE 8.7 Enhanced bendingmoment
sign convention.
In Chapter 7, a positive internal bending moment was defined as a moment that
• acts counterclockwise on the right-hand face of a beam; or
• acts clockwise on the left-hand face of a beam.
This sign convention can now be enhanced by taking into account the bending stresses
produced by the internal moment. The enhanced bending-moment sign convention is
illustrated in Figure 8.7.
A positive internal bending moment M causes
• compressive bending stresses above the neutral axis;
• tensile bending stresses below the neutral axis; and
• a positive curvature κ.
A negative internal bending moment M causes
• tensile bending stresses above the neutral axis;
• compressive bending stresses below the neutral axis; and
• a negative curvature κ
Maximum Stresses on a Cross Section
Since the intensity of the bending stress σ x varies linearly with distance y from the neutral
surface [see Equation (8.3)], the maximum bending stress σ max occurs on either the top or
the bottom surface of the beam, depending on which surface is farther from the neutral
surface. In Figure 8.5b, the distances from the neutral axis to either the top or the bottom of
the cross section are denoted by c top and c bot , respectively. In this context, c top and c bot are
taken as absolute values of the y coordinates of the top and bottom surfaces. The corresponding
bending stress magnitudes are given by the following equations:
σ
σ
max
max
Mctop
= =
I
z
Mcbot
= =
I
z
M
S
top
M
S
bot
forthe topsurfaceof thebeam
forthe bottomsurfaceof thebeam
(8.8)