Beams - Mechanics of Structures: Variational and Computational Methods - page 746

= A I z ω = A ω = A I y ω = A ω

in which [Goodier, 1938] I ω z ω =

r p ds . The quantities I ω y and I ω z have the same meaning here as they do in Eq. (12.16). The
coordinates y S and z S are obtained by equating to zero the coefficients of V y and V z . Thus,

d z dA, I ω y d y dA, and d

− −

I yy I ω z I yz I ω y I zz I ω y I yz I ω z

= − = −

y S y P , z S z P (12.57)

I yz I yz

I yy I zz − I yy I zz −

These give the coordinates of the shear center relative to an arbitrary point P , which often
is the location of the origin of the coordinate system. Also, it follows that when P and S
coincide

I yy I − I yz I I zz I − I yz I

ω z ω y ω z ω y

I yz = I yz =

0 (12.58)

I yy I zz − I yy I zz −

or I ω y = I ω z = 0. This conforms with Eq. (12.20a).

12.2.2 Finite Element Analysis for Cross Sections of Arbitrary Shape

Some of the equations for the stresses in Section 12.2.1 can be solved analytically only for
cross-sections of regular shape such as a circle or a rectangle. The calculation of the normal
stress in Eq. (12.39) is straightforward for many shapes. However, the shear stresses for
bars of arbitrary cross-sectional shapes are exceedingly difficult to calculate accurately. For
example, the shear stress of Eq. (12.43) is based on the assumption that the shear stress
does not vary along the width b More accurate analyses show that this is a questionable

assumption and that theory of elasticity equations need to be employed. For the accurate
calculations of the shear stress on cross-sections of arbitrary shape, computational tech-
niques can be employed. The commonly used numerical procedures in the cross-sectional
analyses are the finite element method and the boundary solutions, including the boundary
element and the direct boundary integration methods. The boundary solution methods are
introduced in Chapter 9, with example problems concerning beam cross-sectional analyses.
Here, finite element analyses of the beam cross-sectional problems will be discussed briefly.
User-friendly postprocessors for calculating the distribution of cross-sectional stresses are
now available for general purpose structural analysis software programs. For all but the
simplest calculations, e.g., normal stresses for solid cross-sections, these postprocessors give
more accurate distributions than the traditional analytical formulas. In particular, shear and
thermal stresses should be calculated using these postprocessors. Furthermore these same
postprocessors are useful as preprocessors to calculate cross-sectional constants that are
needed as input to general purpose structural analysis programs. A typical finite element
formulation will be introduced here. Detailed formulations and software are given in Pilkey
(2002). See the website http://www.mae.virginia.edu/faculty/software/pilkey.php for the
software.

.

Computation of the Warping Function ω and the Related Stresses
Recall the formulations for torsion of Chapter 1, Section 1.9, where the shear stresses

τ

xy

and

xz are expressed using both displacement and force formulations. The displacement
formulation involves the warping function

τ

ω which satisfies [Chapter 1, Eq. (1.151)]

2 2

∂

y 2 + ∂

ω ω

2

z 2 =

0or ∇ ω = 0 (12.59)

∂ ∂

with the boundary condition
∂ω
∂

y a z +

∂ω
∂

z a y =

z − y +

0 (12.60)