Slope and Displacement by the Moment area theorems �

Slope & Deflections of Beams

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1. STRUCTURAL ANALYSIS - 1 1 Dr. OMPRAKASH

2. Structural Analysis-I Code of the subject : CET-225 Lecture – 1 Dr.Omprakash Department of Civil Engineering Chandigarh University 2 Dr. OMPRAKASH

3. Unit -1 Topics Part – A Methods of Calculation for Deflections and Rotations for Beams 1.Moment area method 2.Conjugate Beam Method 3.Unit Load Method 4.Strain energy Method 5.Maxwell’s reciprocal theorem Part – B Thin Cylinders and spherical shell 3 Dr. OMPRAKASH

4. Slope and Displacement by the Moment area theorems Moment-Area Theorems is based on Two theorems of Mohr’s 4 Dr. OMPRAKASH

5. Introduction The moment-area method, developed by Otto Mohr in 1868, is a powerful tool for finding the deflections of structures primarily subjected to bending. Its ease of finding deflections of determinate structures makes it ideal for solving indeterminate structures, using compatibility of displacement . Mohr’s Theorems also provide a relatively easy way to derive many of the classical methods of structural analysis. For example, we will use Mohr’s Theorems later to derive the equations used in Moment Distribution. The derivation of Clayperon’s Three Moment Theorem also follows readily from application of Mohr’s Theorems. 5 Dr. OMPRAKASH

6. AREA‐MOMENT METHOD The area-moment method of determining the deflection at any specified point along a beam is a semi graphical method utilizing the relations between successive derivatives of the deflection y and the moment diagram. For problems involving several changes in loading, the area-moment method is usually much faster than the double-integration method; consequently, it is widely used in practice. 6 Dr. OMPRAKASH

7. Deflection of Beams Slope and Displacement by the Moment area theorem Assumptions: Beam is initially straight, Is elastically deformed by the loads, such that the slope and deflection of the elastic curve are very small, and Deformations are caused by bending. S 7 Dr. OMPRAKASH

8. Deflection Diagrams and the Elastic Curve ∆ = 0, Roller support 8 Dr. OMPRAKASH

9. ∆ = 0 pin 9 Deflection Diagrams and the Elastic Curve Dr. OMPRAKASH

10. ∆ = 0 θ = 0 fixed support 10 Deflection Diagrams and the Elastic Curve Dr. OMPRAKASH

11. Mohr’s Theorems - 1 & 2 Theorem 1 The angle between the tangents at any two points on the elastic curve equals the area under the M/EI diagram between these two points. Theorem 2 The vertical deviation of the tangent at a point ( A ) on the elastic curve w.r.t . the tangent extended from another point ( B ) equals the moment of the area under the ME/I diagram between these two pts ( A and B ). 11 Dr. OMPRAKASH

12. Moment Area Theorems 1 st - Theorem : Gives Slope of a Beam and notation of slope by letter i ( or) q Area of Bending moment diagram (A) Slope = q = EI Where EI is called Flexural Rigidity E = Young's Modulus of the material, I = Moment of Inertia of the beam. Slope is expressed in radians. 2 nd – Theorem : G ives Deflection of a Beam and notation with letter Y or Area of BMD (A) x Centeroidal distance (x) Y = EI Expressed in M, CM, MM 12 Dr. OMPRAKASH