This document summarizes the principle of virtual work. It defines virtual displacement as any displacement consistent with structural constraints and boundary conditions. The principle of virtual work states that the virtual work of applied forces equals zero at mechanical equilibrium. Mathematically, this means the work done by forces must equal the strain energy stored in the system, as expressed by the equations presented. The principle is applicable to analyzing both linear and non-linear elastic materials and structures.

1. PURBANCHAL UNIVERSITY
KHWOPA ENGINEERING COLLEGE
SOLID MECHANICS
PRINCIPLE OF VIRTUAL WORK
REENA SUWAL (ME07814)
SACHIN POKHAREL (ME07815)
SAMI DANGOL (ME07816)
SUNDAR BARTAULA (ME07817)
Assoc. Prof. Dr. Manjip Shakya
Department of Earthquake Engineering
8th March 2023

2. Introduction
Virtual displacement and virtual Work
• Virtual work is the work done by a real force acting through a virtual displacement or a
virtual force acting through a real displacement.
• A virtual displacement is any displacement consistent with the constraints of the
structure, i.e., that satisfy the boundary conditions at the supports.
• A virtual force is any system of forces in equilibrium.
Principle of virtual Work
The principle of virtual work states that in equilibrium the virtual work of the forces
applied to a system is zero. Newton's laws state that at equilibrium the applied forces are
equal and opposite to the reaction, or constraint forces. This means the virtual work of the
constraint forces must be zero as well.

3. F1 = 1
F1
F3
F2 M1
M2
M3
F2 = 2
F3 = 3
Virtual linear displacement
Virtual Work

4. = 2
F1
F3
F2 M1
M2
M3
M2
M3
Virtual angular displacement
M1
= 3
= 1
Virtual Work

5. F1
F2
F3
M1
M2
M3 Total Virtual Work

6. F1
F2
F3
M1
M2
M3
R

7. F1
F2
F3
M1
M2
M3
R
Equilibrium
du = 0 If R = 0
= 0

8. Mathematical expression
• Consider an elastic system subjected to a number of forces (including
moments) F1, F2, . . . , etc. Let 𝛿1, 𝛿2, . . ., etc. be the corresponding
displacements.
• These are the work absorbing components (linear and angular
displacements) in the corresponding directions of the force as shown
in figure.

9. • Let one of the displacements 𝛿1 be increased by a small quantity
∆𝛿1. During this additional displacement, all other displacements
where forces are acting are held fixed, which means that additional
forces may be necessary to maintain such a condition.
• Further, the small displacement ∆𝛿 1 that is imposed must be
consistent with the constraints acting. For example, if point ‘I’is
constrained in such a manner that it can move only in a particular
direction, then ∆𝛿1 must be consistent with such a constraint.
• A hypothetical displacement of such a kind is called a virtual
displacement. In applying this virtual displacement, the forces F1, F2,
. . ., etc. (except F1) do no work at all because their points of
application do not move (at least in the work-absorbing direction).
• The only force doing work is F1 by an amount F1 ∆𝛿1. plus a fraction
of∆F1 ∆𝛿1. , caused by the change in F1. This additional work is
stored in strain energy ∆𝑈.

10. This is the principle of virtual work

11. The work done by the forces must be equal to the strain energy
which is stored up in the system. This fact can be expressed in
another form known as the principle of virtual work, i.e.

12. • In the equations above,
V denotes the material domain,
S the surface completely enclosing V,
and  the variational symbol signifies a virtual quantity
upon applying the divergence theorem


13. We know, WE = WI
Substituting the value of WE and WI
………(i)
………(ii) a
………(ii) b
Eqn (i) is a field eqn
Eqn (ii) specify boundary conditions.

14. Conclusion
This principle is applicable to any elastic body
• linear elastic materials
• Non-linear elastic materials

15. THANK YOU