Engineering mechanics lami's theorem

1. U18CE105 Engineering Mechanics
Classroom Discussion Topic
W2 – L4-CDT7
Topic: PRINCIPLES OF EQUILIBRIUM, LAMI’S
THEOREM
Dept of CIVIL ENGINEERING
Kakatiya Institute of Technology and Science, Warangal, TS, India

2. Engineering Mechanics
LO1: Force systems and their applications
• Laws of Mechanics: Parallelogram law of forces, triangle law of forces, Newton's law of
gravitation, law of superposition and transmissibility of forces.
• Force Systems: Types of forces, co-planar, concurrent and parallel forces, moment and
couple, free body diagram, resultant of force systems, resolution of forces, composition
of forces, equilibrium equations of forces, Lami’s theorem, Varignon’s theorem, moment
equilibrium equations, types of supports, beams and loadings, statically determinate
structures, resultant and equilibrium of general force system
• CO1 : Articulate various force systems and their applications
U18CE105 Engineering Mechanics

3. After completion of this lecture, students will be able to…
LO1. Recite Principles of Equilibrium and develop equations of equilibrium
LO2. Devise methods to solve a rigid body in equilibrium
U18CE105 Engineering Mechanics
Lecture Outcomes (LOs)

4. Equilibrium of coplanar force system
U18CE105 Engineering Mechanics
• When some external forces (which may be concurrent or parallel) are
acting on a stationary body, the body may start moving or may start
rotating about any point.
• But if the body does not start moving and also does not start rotating
about any point, then the body is said to be in equilibrium.

5. Principle of Equilibrium
U18CE105 Engineering Mechanics
• The principle of equilibrium states that, a stationary body which is
subjected to coplanar forces (concurrent or parallel) will be in
equilibrium if the algebraic sum of all the external forces is zero and
also the algebraic sum of moments of all the external forces about
any point in their plane is zero.
• Mathematically, it is expressed by the equations,
∑ 𝑭 = 𝟎(Force Law of Equilibrium)
∑ 𝑴 = 𝟎(Moment Law of Equilibrium)
• The forces are generally resolved into horizontal and vertical
components.

6. Conditions of Equilibrium
∑Fx =0
∑Fy=0 and
∑M=0
Where ∑Fx = Algebraic sum of all the forces and their components acting
along horizontal direction
∑Fy = Algebraic sum of all the forces and their components acting
along vertical direction
And ∑M= Algebraic sum of moments of forces taken at a moment centre.
If the above conditions are satisfied the rigid body is said to be in static
equilibrium.
U18CE105 Engineering

7. Reflection Spot - 1
Having discussed some content, here is another reflection spot
Question-1:
•People sitting in a moving bus experience the speed and forces on them,
can we call a person sitting steadily in a moving bus as in the state of
equilibrium?
U18CE505 Soil Mechanics
U18CE105 Engineering Mechanics 7

8. Principles of Equilibrium
Though there are many principles of equilibrium, yet the
following three are important from the subject point of view:
1. Two force principle. As per this principle, if a body in
equilibrium is acted upon by two forces, then they must be
equal, opposite and collinear.
2. Three force principle. As per this principle, if a body in
equilibrium is acted upon by three forces, then the resultant
of any two forces must be equal, opposite and collinear with
the third force.
3. Four force principle. As per this principle, if a body in
equilibrium is acted upon by four forces, then the resultant of
any two forces must be equal, opposite and collinear with the
resultant of the other two forces.
U18CE105 Engineering

9. METHODS FOR THE EQUILIBRIUM OF COPLANAR FORCES
METHODS FOR THE EQUILIBRIUM OF COPLANAR FORCES
Though there are many methods of studying the
equilibrium of forces, yet the following are important
from the subject point of view:
1. Analytical method.
2. Graphical method.
U18CE105 Engineering

10. ANALYTICAL METHOD FOR THE EQUILIBRIUM OF COPLANAR FORCES
• The equilibrium of coplanar forces may be studied, analytically, by Lami’s
theorem as discussed below:
LAMI’S THEOREM:
• It states, “If three coplanar forces acting at a point be in equilibrium, then each
force is proportional to the sine of the angle between the other two.”
Mathematically,
where, P, Q, and R are three forces and α, β, γ are the angles as shown in Fig.1
U18CE105 Engineering

11. Reflection Spot - 2
Having discussed some content, here is another reflection spot
Question-3:
A horizontal cylinder of weight 240 kN is held against a wall with the help of string as shown in Fig.2.10. The
string makes an angle 20⁰ with the wall. Determine the reaction of the wall and tension induced in the string.
Answer:
1. Draw FBD
2. Apply law of transmissibility
3. Draw skeleton forces
4. Represent concurrent force system
Now apply Lami’s Theorem
U18CE105 Engineering Mechanics 11

12. • Determine the tension in the string OA and reaction of inclined plane when a roller of weight
600 N is supported as shown in Fig. 2.12.
Answer
U18CE105 Engineering

13. U18CE105 Engineering

14. Lecture Outcome Revisited
Having completed the discussion on Equilibrium and conditions of
equilibrium, now, students should be able to…
LO1: Resolve the force into components
LO2. Reframe the equations of equilibrium for the given system of forces
U18CE105 Engineering Mechanics

15. Further Reading
• Textbook: Tayal A.K., Engineering Mechanics: Statics and Dynamics, 14th
ed. New Delhi: Umesh Publishers, 2014
• Reading:
• Chapter -2;
• Problem solving:
• Chapter 2 Problems-
• Reference Book: Timoshenko S., Young D.H., Rao J.V., and Sukumar Pati,
Engineering Mechanics in SI units, 5th ed. New Delhi: McGraw Hill
Education Pvt. Ltd., 2013.
• Reading:
• Chapter -2; Topic-
• Problem solving:
• Chapter-1; Additional Problems-
• Weekly Tutorial Sheet
U18CE105 Engineering Mechanics