Beams - Mechanics of Structures: Variational and Computational Methods - page 729

Thus, warping contributes

u x ( M

ω ) = ω(

y, z )ψ (12.5c)

to the axial displacements, where u x is indicated to be due to a warping moment M which

will be considered later. This relationship is similar to that of Chapter 1, Eq. (1.141) for
simple torsion. The kinematic parameter for warping,

ω

ψ , is a function of the x (and not y, z )

coordinate, i.e., ψ = ψ( x ) .

Total Displacement
The sum of the contributions of Eqs. (12.5a, b, and c) leads to the total displacement u x as
given in Eq. (12.4), i.e.,

= − θ + θ + ω( )ψ

u x u y z y, z

z y

Displacements u y , u z
Suppose a particular cross-section is not distorted in its own plane, i.e., the shape of a cross-
section does not change while the bar is being bent. Furthermore, suppose twisting occurs
about point Q . If the displacements of Q along the y, z axes are v , w , then point P ( y, z )

displaces (Fig. 12.5)

u y = v − z φ

x

(12.6)

u z = w + y φ

x

where, again, y and z are the coordinates of point P relative to point Q , and φ x = φ is the

angle of twist.

FIGURE 12.5
Twisting.