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Slope and Displacement by the Moment area theorems

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Slope & Deflections of Beams

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Slope and Displacement by the Moment area theorems

1. 1. STRUCTURAL ANALYSIS - 1 Dr. OMPRAKASH 1
2. 2. Structural Analysis-I Code of the subject : CET-225 Lecture – 1 Dr.Omprakash Department of Civil Engineering Chandigarh University Dr. OMPRAKASH 2
3. 3. Slope and Displacement by the Moment area theorems Moment-Area Theorems is based on Two theorems of Mohr’s Dr. OMPRAKASH 4
4. 4. 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
5. 5. 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 doubleintegration method; consequently, it is widely used in practice. Dr. OMPRAKASH 6
6. 6. 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 Dr. OMPRAKASH 7
7. 7. Deflection Diagrams and the Elastic Curve ∆ = 0, Roller support Dr. OMPRAKASH 8
8. 8. Deflection Diagrams and the Elastic Curve ∆ = 0 pin Dr. OMPRAKASH 9
9. 9. Deflection Diagrams and the Elastic Curve ∆=0θ=0 fixed support Dr. OMPRAKASH 10
10. 10. 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 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). Dr. OMPRAKASH 11
11. 11. Moment Area Theorems • 1st - Theorem : • 2nd – Theorem : • • Gives Slope of a Beam and notation of slope by letter i Gives Deflection of a Beam (or) notation with letter and Y or Area of Bending moment diagram (A) Slope = Area of BMD (A) x Centeroidal distance (x) = Y= EI EI  Where EI is called Flexural Rigidity   E = Young's Modulus of the material, I = Moment of Inertia of the beam.  Expressed in M, CM, MM  Slope is expressed in radians. Dr. OMPRAKASH 12
12. 12. SLOPE & DISPLACEMENT BY THE MOMENT-AREA METHOD • Procedure for analysis :  1. Determine the support reactions and draw the beam’s bending moment diagram  2. Draw M/EI diagram  3. Apply Theorem 1 to determine the angle between any two tangents on the elastic curve and Theorem 2 to determine the tangential deviation. Dr. OMPRAKASH 13