Instructional Objectives
After studying this lesson, the student will be able to:
1. Define Virtual Work.
2. Differentiate between external and internal virtual work.
3. Sate principle of virtual displacement and principle of virtual forces.
4. Drive an expression of calculating deflections of structure using unit load
method.
5. Calculate deflections of a statically determinate structure using unit load
method.
6. State unit displacement method.
7. Calculate stiffness coefficients using unit-displacement method.
5.1 Introduction
In the previous chapters the concept of strain energy and Castigliano’s theorems
were discussed. From Castigliano’s theorem it follows that for the statically
determinate structure; the partial derivative of strain energy with respect to
external force is equal to the displacement in the direction of that load. In this
lesson, the principle of virtual work is discussed. As compared to other methods,
virtual work methods are the most direct methods for calculating deflections in
statically determinate and indeterminate structures. This principle can be applied
to both linear and nonlinear structures. The principle of virtual work as applied to
deformable structure is an extension of the virtual work for rigid bodies. This may
be stated as: if a rigid body is in equilibrium under the action of a
system of
forces and if it continues to remain in equilibrium if the body is given a small
(virtual) displacement, then the virtual work done by the
F
−
F
−
system of forces as ‘it
rides’ along these virtual displacements is zero.
5.2 Principle of Virtual Work
Many problems in structural analysis can be solved by the principle of virtual work.
Consider a simply supported beam as shown in Fig.5.1a, which is in equilibrium
under the action of real forces
at co-ordinates
respectively.
Let
be the corresponding displacements due to the action of
forces
. Also, it produces real internal stresses
n
F
F
F
,.......
,
,
2
1
n
,.....
,
2
,
1
n
u
u
u
,......
,
,
2
1
n
F
F
F
,.......
,
,
2
1
ij
σ
and real internal
strains
ij
ε
inside the beam. Now, let the beam be subjected to second system of
forces (which are virtual not real)
n
F
F
F
δ
,......
,
,
2
1
in equilibrium as shown in
Fig.5.1b. The second system of forces is called virtual as they are imaginary and
they are not part of the real loading. This produces a displacement
Version 2 CE IIT, Kharagpur