Beams - Mechanics of Structures: Variational and Computational Methods - page 727

12.1.2 Kinematical Relationships

[ u x u y u z ] T are necessary to describe
the motion of an arbitrary point. More quantities are required to adequately describe the
spatial motion of a point on the axis of a bar. These are three translations

For a continuum (Chapter 1), three displacements u =

( u, v , w) , three

rotations (φ

x ,

θ

y ,

θ ) , and one warping parameter (ψ). Thus, the fundamental kinematic

z

variables are

Continuum Bar Axis

u
v
w
φ x
θ y
θ

Displacements

u x
u y
u z

⇒ (12.2)

Rotations

z
ψ

Warping parameter

In Chapter 1, Eq. (1.98), the axial displacement of a point on a cross-section due to bending
deformation in the xz plane was found on the basis of plane cross-sections remaining plane
to be

y (12.3)
where u was the axial displacement due to stretching at the centroid, z was measured from
the centroid, and

u x ( x, z ) = u ( x ) + z θ

y was the rotation about the y axis. It will be assumed that a bar in space
with bending, extension, and twisting responds in a similar fashion.
Assume that the displacement u x ,u y , and u z at some point P

θ

( y, z ) on the cross-section

( v w φ θ θ ψ)

are related to the bar axis variables u, , ,

x , y , z , and

at another point Q on the

cross-section by

u x = u − y θ + z θ + ω( y, z )ψ

z y

= v − φ

u y z (12.4)

x

= w + φ

u z y

x
where x, y, z are the coordinates of point P

( y, z ) relative to Q (Fig. 12.3). Observe that

= ( ) v = v( ) w = w( )

u u x , x , and x do not vary throughout a given cross-section. The

θ θ φ φ ω( )

terms y

z ,z y ,z x , and y x vary linearly. The function

y, z is referred to as the warping

function.
The relationships of Eq. (12.4) can be considered to be natural extensions of Eq. (12.3).
The geometrical explanations for the selection of these relationships follow.

Displacement u x
Axial Force
The contribution of the axial force N to the displacement u x is simply
u x

( N ) = u (12.5a)

= ( )

where u u x , the displacement of point Q , does not vary over the cross-section normal

to the x axis.

Bending
From Figs. 12.3a and b, the axial displacement for a point P at y, z in an element of length
dx is

=− θ + θ

du x yd zd

z y