Beams - Mechanics of Structures: Variational and Computational Methods - page 737

Note that satisfies Eqs. (1.151) and (1.153), so that the integrals on the right-hand side of

the above equation vanish. This leads to

ω

y 2 dA

z ∂ω
∂

y ∂ω
∂

z 2

= = + +

y −

J t J (12.24)

z

A

The forces and displacements for the beam will be referred to these special axes. Use
of these assumptions will simplify the relationships derived thus far. The internal virtual
work of Eq. (12.16) now reduces to

− δ W i =

x (

EA δ + EI yy κ z δκ z + EI zz κ y δκ y + EI

ωω κ ω δκ ω +

GJ t γ tP δγ tP ) dx (12.25)

Bending
about the
z axis

Bending
about the
y axis

Axial
Extension

Warping
Torsion

Primary
Torsion

where, for convenience, the superscripts have been dropped from the coordinates. If dis-
placements are introduced using u , κ z = θ z = v , κ y = θ y =− w , κ ω = ψ =− φ x , and ε =

γ tP = φ x , and u, v , and w are referred to the shear center, then

x (

EAu S δ + v S δκ w S δκ

ωω φ S δκ ω +

φ S δγ − δ W i = EI yy − EI zz − EI GJ t ) dx

z y tP

x (

EAu S δ u S +

v S δv S + w S δw S +

EI ωω φ S δφ S +

φ S δφ S )

=

EI yy EI zz GJ t dx (12.26)

where φ S = (φ x ) S , and the subscripts indicate that the quantities are with respect to the

shear center. Remember that Bernoulli's and Wagner's hypotheses still apply. Comparison
of Eqs. (12.14) and (12.26) shows that the net internal forces and displacements are related by

EA u =

N

EI yy v = M z

EI zz w = − M y (12.27)

ωω φ =

− EI M

ω

GJ t φ =

M t

where N, M z , and M y are referred to the centroidal principal axes, and M tP = M t is referred

to the principal center of twist (shear center). The S subscripts have been dropped. The
cross-sectional property I is often represented by and is called the warping constant.

ωω

12.1.5 External Virtual Work

Consider a cross-section with loads applied at point P as shown in Fig. 12.9. An expression
for the external virtual work is given as the final integral of Eq. (12.1). For the beam, we
will form the external virtual work as the load times the virtual displacement of the point
of application of the load in the direction of the load. This displacement will be measured
relative to the shear center.
The displacements v P , w P of the point ( P ) of application of the loading can be expressed as

v = v − z S φ

P S

(12.28)

w = w +

y S φ

P S