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

CHAPTER9
Shear Stress In beams
9.1 Introduction
For beams subjected to pure bending, only tensile and compressive normal stresses are
developed in the flexural member. In most situations, however, loadings applied to a beam
create nonuniform bending; that is, internal bending moments are accompanied by internal
shear forces. As a consequence of nonuniform bending, shear stresses as well as normal
stresses are produced in the beam. In this chapter, a method will be derived for determining
the shear stresses produced by nonuniform bending. The method will also be adapted to
consider beams fabricated from multiple pieces joined together by discrete fasteners.
9.2 Resultant Forces produced by Bending Stresses
Before developing the equations that describe beam shear stresses, it is instructive to
consider in more detail the resultant forces produced by bending stresses on portions of
the beam cross section. Consider the simply supported beam shown in Figure 9.1, in
which a concentrated load P = 9,000 N is applied at the middle of a 2 m long span. The
shear-force and bending-moment diagrams for this span and loading are shown.
For this investigation, we will arbitrarily consider a 150 mm long segment BC of the
beam. The segment located 300 mm from the left support, as shown in the figure. The beam is
made up of two wooden boards, each having the same elastic modulus. The lower board will
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