lamis theorem.pptx

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 ∑ F x = Algebraic sum of all the forces and their components acting along horizontal direction ∑ F y = 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 Mechanics

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 Mechanics

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 Mechanics

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 Mechanics

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 20 ⁰ with the wall. Determine the reaction of the wall and tension induced in the string . Answer: Draw FBD Apply law of transmissibility Draw skeleton forces Represent concurrent force system Now apply Lami’s Theore m 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 Mechanics

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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, 14 th 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