Disclaimer
These lecture slides were prepared and used by me to conduct lectures for first-year B. Tech. Students as
part of the course ‘Engineering Mechanics’ (XEC 01) at the National Institute of Technology Durgapur.
Theories, problems, figures, and concepts used in the slides to fulfil the course requirements are primarily
taken from the following textbooks and PowerPoint slides available on the internet. The material is used
purely for educational purposes. Students are instructed to use it for their reading. Don’t distribute as the
presentation slides contain copyright materials. Despite my best efforts, some of the content may contain
errors. Students are requested to rectify when using the same, and I invite them to write to me about the
mistakes to ranajnkumar.mitra@me.nitdgp.ac.in. I thank the following authors for making their books and
lecture notes available for reference.
• Vector Mechanics for Engineer: Statics, 9th edition, Ferdinard P. Beer and E. Russell Johnston,
Jr., McGraw-Hill
• Vector Mechanics for Engineer: Statics, 10th edition, Ferdinard P. Beer and E. Russell Johnston,
Jr., McGraw-Hill
• Lecture notes on Engineering Mechanics: Statics, J. Walt Oler, Taxas Tech University, 9th edition,
McGraw-Hill Companies, Inc.
• Lecture notes on Engineering Mechanics: Statics, John Chen, California Polytechnic State
University, McGraw-Hill Companies, Inc.
Dr. Ranjan Kumar Mitra
Brief Outline of the Lecture
• Introduction
• Sample Problem 2
• Work of a Force
• Sample Problem 3
• Work of a Couple
• Work of a Force During a Finite
• Principle of Virtual Work
• Applications of the Principle of
Virtual Work
• Real Machines. Mechanical
Efficiency
Displacement
• Potential Energy
• Potential Energy and Equilibrium
• Stability and Equilibrium
• Sample Problems 4
• Sample Problem 1
10 - 3
Application
In certain cases, for example the analysis of a system of connected rigid bodies, the
method of virtual work is a more efficient method than applying equilibrium conditions.
10 - 4
What is the work of a Force?


dU = F  dr


= work of the force F corresponding to the displacement d r
dU = F ds cos
 = 0, dU = + F ds
 =  , dU = − F ds
 = 2 , dU = 0
dU = Wdy
10 - 5
What is virtual work?
▪ Consider the static equilibrium position of a particle
determined by the forces acting on it.
▪ Any assumed and arbitrary small displacement 𝛿 𝒓 away
from this neutral position and consistent with the system
constraints is called a virtual displacement.
▪ The term virtual is used to indicate that the displacement
does not really exists but only is assumed to exists so that we
may compare various possible equilibrium positions to
determine the correct one.
• The work done by any force 𝑭 acting on the particle during virtual displacement is called virtual work 𝛿 𝑈=
𝑭. 𝛿𝒓
or
𝛿 𝑈 = 𝐹𝛿𝑠 cos 𝛼
• 𝑑𝒓 refers to an actual infinitesimal change in position and can be integrated.
10 - 6
What is the work of a Force?
•𝛿𝒓refers to an infinitesimal virtual or assumed movement and can not be integrated.
• Mathematically both quantities are first order differentials.
•A virtual displacement may also be rotation 𝛿𝜽of a body; 𝛿 𝑈 = 𝑴. 𝛿𝜽
•𝑴 denotes moment of a couple acting on a body
𝛿𝑈 = 𝑴. 𝛿𝜽
Sample Problem 1
Sample Problem 1
Sample Problem 1
Sample Problem 1
Sample Problem 2
Sample Problem 2
Sample Problem 2
Sample Problem 3
Sample Problem 3
Sample Problem 3
Sample Problem 4
Sample Problem 4
Sample Problem 4
Sample Problem 4
Sample Problem 5
Determine the magnitude of the couple M required to maintain the
equilibrium of the mechanism.
SOLUTION:
• Apply the principle of virtual work
U = 0 = U M + U P
0 = M + PxD
xD = 3l cos
xD = −3l sin 
0 = M + P(− 3l sin  )
M = 3Pl sin 
• Note that no support reactions were needed to solve the problem, nor was it
necessary to take apart the machine at any connection. A clear and
accurate FBD is still highly recommended, however.
10 - 26
Sample Problem 6
Determine the expressions for  and for the tension in the spring which
correspond to the equilibrium position of the spring. The unstretched length of the
spring is h and the constant of the spring is k. Neglect the weight of the
mechanism.
SOLUTION:
• Apply the principle of virtual work
U = U B + U F = 0
0 = Py B − FyC
y B = l sin 
y B = l cos
yC = 2l sin 
F = ks
= k ( yC − h )
yC = 2l cos
= k (2l sin  − h )
0 = P(l cos ) − k (2l sin  − h )(2l cos )
P + 2kh
4kl
F = 12 P
sin  =
10 - 27
Sample Problem 7
SOLUTION:
• Create a free-body diagram for the platform and linkage.
A hydraulic lift table consisting of two identical
linkages and hydraulic cylinders is used to raise a 1000kg crate. Members EDB and CG are each of length 2a
and member AD is pinned to the midpoint of EDB.
Determine the force exerted by each cylinder in raising
the crate for  = 60o, a = 0.70 m, and L = 3.20 m.
• Apply the principle of virtual work for a virtual
displacement  recognizing that only the weight and
hydraulic cylinder do work.
U = 0 = QW + QFDH
• Based on the geometry, substitute expressions for the virtual
displacements and solve for the force in the hydraulic
cylinder.
10 - 28
Sample Problem 7
SOLUTION:
• Create a free-body diagram for the platform.
• Apply the principle of virtual work for a virtual displacement 
U = 0 = QW + QFDH
0 = − 12 Wy + FDHs

• Based on the geometry, substitute expressions for the virtual
displacements and solve for the force in the hydraulic cylinder.
y = 2a sin 
y = 2a cos
s 2 = a 2 + L2 − 2aL cos
2ss = −2aL(− sin  )
aL sin 

s
aL sin 
0 = (− 12 W )2a cos + FDH

s =

s
s
FDH = W cot 
L
FDH = 5.15 kN
10 - 29
Thank you!