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

374
SHEAR STRESS IN bEAMS
The exact location of the
shear center for thick-walled
unsymmetrical cross
sections is difficult to obtain
and is known only for a few
cases.
where the transverse loads should be placed to avoid twisting of the cross section. In other
words, transverse loads applied through the shear center cause no torsion of the beam.
Determination of the shear center location has important ramifications for beam design.
Beam cross sections are generally configured to provide the greatest possible economy
of material. As a result, they are frequently composed of thin plates arranged so that
the resulting shape is strong in flexure. Wide-flange and channel shapes are designed with
most of the material concentrated at the greatest practical distance from the neutral axis.
This arrangement makes for an efficient flexural shape because most of the beam material
is placed in the flanges, which are locations of high flexural stress. Less material is used in
the web, which is near the neutral axis, where flexural stresses are low. The web serves
primarily to carry shear force while also securing the flanges in position. An open cross
section that is made up of thin plate elements may be strong in flexure, but it is extremely
weak in torsion. If a beam twists as it bends, torsional shear stresses will be developed in
the cross section, and generally, these shear stresses will be quite large in magnitude. For
that reason, it is important for the beam designer to ensure that loads are applied in a manner
that eliminates twisting of the beam. This aim can be accomplished when external loads
act through the shear center of the cross section.
The shear center of a cross section is always located on an axis of symmetry. The
shear center for a beam cross section having two axes of symmetry coincides with the centroid
of the section. For cross sections that are unsymmetrical about one axis or both axes,
the shear center must be determined by computation or observation. The method of solution
for thin-walled cross sections is conceptually simple: We will first assume that the
beam cross section bends, but does not twist. On the basis of this assumption, the resultant
internal shear forces in the thin-walled shape will be determined by a consideration of the
shear flow produced in the shape. Equilibrium between the external load and the internal
resultant forces must be maintained. From this requirement, the location of the external
load necessary to satisfy equilibrium can then be computed.
Shear center for a channel Section
Consider the thin-walled channel shape used as a cantilever beam, as shown in Figure
9.24a. A vertical external load P that acts through the centroid of the cross section will
P
P
(a) Vertical load P acting through centroid
FIGURE 9.24 Bending and twisting of the cantilever beam.
(b) Bending and twisting in response to the
applied load