Beams - Mechanics of Structures: Variational and Computational Methods - page 738

FIGURE 12.9
Applied loads.

where w S are the displacements of the shear center, and y S , z S are the coordinates of

point P relative to the shear center. The external virtual work is

v S ,

]

b
a

[ M z δv S − M y δw S −

ω δφ +

δ W e =

x (

p x δ u + p y δv P + p z δw P + m t δφ) dx + M M t δφ

P ]

b
a

+ δ + δv + δw .

[ N u V y V z (12.29)

P

This expression contains the vir tu al work due to the distributed forces p x , p y , and p z , as
well as the distributed moment m t .

If concentrated applied forces are present, the appro-
priate integrals would be replaced by summations. Terms for concentrated applied forces
on the boundaries at a and b are also given in Eq. (12.29). In terms of virtual displacements
of the shear center, Eq. (12.29) becomes

δ W e = [ p x δ u + p y δ(v S − z S φ) + p z δ(w S + y S φ) + m t δφ ] dx

x

δv S − δw S − M ω δφ +

+ [ M z M y N δ u + V y δv + V z δw

S S

]

b
a .

+ ( M t − V y z S + V z y S )δφ (12.30)

12.1.6 The Complete Virtual Work

From Eqs. (12.26) and (12.30), with subscript S ignored, the total virtual work appears as

− δ W =− δ( W i + W e ) = 0

EAu )δ u + ( w )δw + ( v )δv + ( EI ωω φ )δφ + ( φ )δφ ] dx = [ ( EI zz EI yy GJ t

x

− [ p x δ u + p y δv + p z δw + ( m t − p y z S + p z y S ) δφ ] dx

x

m φ

δv − δw − M ω δφ + − [ M z M y N δ u + V y δv + V z δw

]

b
a =

+ ( M t − V y z S + V z y S ) δφ 0 . (12.31)

M φ