Beams - Mechanics of Structures: Variational and Computational Methods - page 748

In a computational solution, typically the stresses are calculated initially at the Gaus-
sian integration points. These stresses can be multiplied by a smoothing matrix to find the
element nodal stresses. The stresses from adjacent elements can be averaged at the element
nodes.
If the cross-section is thin, it may be convenient to model it using line elements. Line
elements can be obtained by collapsing the isoparametric elements into very thin (line)
elements [Surana, 1979]. For example, in the limit as the thickness is reduced, the nine-
node isoparametric element becomes a three-node line element.

Transverse Shear Loads Related Properties and Stresses
This topic is covered extensively in Pilkey (2002), where finite element computer programs
are provided for all of the cross-sectional properties and stresses discussed here.
In Section 12.2, the expressions for the shear stresses due to transverse shear forces are
obtained from the assumption that the plane of the cross-section remains plane after defor-
mation. In reality, when shear stresses are present, the cross-section cannot remain plane
and, as a consequence, some error is incurred in evaluating the shear stresses. Better solu-
tions for the shear stresses can be obtained from the theory of elasticity. Exact solutions for
the shear stresses are available only for a few beams with particular boundary conditions
and loading, e.g., for a beam with the left end clamped and a transverse tip-load applied
at the right end. It is also known that the distribution of the transverse shear stresses for a
beam with uniformly distributed transverse applied loading is the same as for a tip-loaded
beam [Mason and Herrmann, 1968]. Thus, it is reasonable to assume that the functional
relationship between the internal shear force and shear strains for the tip-loaded beam
applies to other cases and can form the basis for the analysis of the shear stresses on a
cross-section of arbitrary shape. This leads to a warping function that can be computed
using a finite element analysis. This distribution over the cross section of the warping
function, which is not related to the warping function for torsion, can be computed and

then used to calculate several shear-related cross-sectional properties, as well as the shear
stresses due to the transverse shear loads. Brief discussions of two shear related properties
follow.

ω

Shear Center
As mentioned previously, the shear center is the point on the cross section through which
the resultant shear force should pass if there is to be no torsion. For a cross section with
two axes of symmetry, the shear center is at the centroid of the cross section. If there is
one axis of symmetry, the shear center falls on this axis. If the cross section consists of two
intersecting flanges, the shear center is at the intersection point.
The most common shear center equations, which are based on the theory of elasticity, are
functions of a material constant, usually Poisson's ratio. Thus, in this case the shear center
is not a purely geometric property of the cross section.
An alternative definition of a shear center by Trefftz (1936) does not involve the de-
pendence on Poisson's ratio. These shear center equations are particularly suitable for
thin-walled beams.

Shear Deformation Coefficients
Shear deformation coefficients have been employed for many years to improve beam de-
flections. In this topic, the shear stiffness factor k s was introduced in Eq. (1.109). A brief
history of the various definitions of the shear coefficients is provided by Hutchinson (2001).
The approach in Pilkey (2002) is to define the coefficients by equating the strain energy
for a beam based on the theory of elasticity to the strain energy for a beam represented by