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

The sense of σ x (either tension or compression) is dictated by the sign of the bending
moment. The quantities S top and S bot are called the section moduli of the cross section, and
they are defined as
S
top
Iz
I
= and Sbot
=
c
c
top
z
bot
(8.9)
245
NORMAL STRESSES IN bEAMS
The section modulus is a convenient property for beam design because it combines two
important cross-sectional properties into a single quantity.
The beam cross section shown in Figure 8.5 is symmetric about the y axis. If a beam
cross section is also symmetric about the z axis, it is called a doubly symmetric cross
section. For a doubly symmetric shape, c top = c bot = c and the bending stress magnitudes at
the top and bottom of the cross section are equal and given by
Mc M
Iz
σ max = = where S =
I S
c
z
(8.10)
Again, Equation (8.10) gives only the magnitude of the stress. The sense of σ x (either tension
or compression) is dictated by the sense of the bending moment.
Nonuniform Bending
The preceding analysis assumed that a slender, homogeneous, prismatic beam was subjected
to pure bending. If the beam is subjected to nonuniform bending, which occurs when a
transverse shear force V exists, then the shear force produces out-of-plane distortions of the
cross sections. Strictly speaking, these out-of-plane distortions violate the initial assumption
that cross-sectional surfaces that are planar before bending remain planar after bending.
However, the distortion caused by transverse shear forces is not significant for common
beams, and its effect may be neglected. Therefore, the equations developed in this section
may be used to calculate flexural stresses for beams subjected to nonuniform bending.
Summary
Bending stresses in a beam are evaluated in a three-step process.
Step 1—Determine the Internal Bending moment M: The bending moment may be
specified, but more typically, it is determined by constructing a shear-force and
bending-moment diagram.
Step 2 — calculate properties for the cross Section of the Beam: The location of
the centroid must be determined first, since the centroid defines the neutral surface for pure
bending. Next, the moment of inertia of the cross-sectional area must be calculated about
the centroidal axis that corresponds to the bending moment M. For example, if the bending
moment M acts about the z axis, then the moment of inertia about the z axis is required.
(Appendix A presents a review of area geometric properties, such as centroids and moments
of inertia.) Finally, bending stresses within the cross section vary with depth. Therefore,
the y coordinate at which stresses are to be calculated must be established.
Step 3 — Use the Flexure Formula to calculate Bending Stresses: Equations (8.7)
and (8.10) for bending stresses were derived:
σ
x
=− My
I
z
Mc
σ x = =
I
z
M
S