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STRUCTURAL ANALYSIS - 1

Dr. OMPRAKASH

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Structural Analysis-ICode of the subject : CET-225

Lecture – 1

Dr.Omprakash

Department of Civil Engineering

Chandigarh University

Dr. OMPRAKASH

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Slope and Displacement by the Moment area theorems

Moment-Area Theorems is based on Two theorems of Mohr’s

Dr. OMPRAKASH

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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.

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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.

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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

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Deflection Diagrams and the Elastic Curve

∆ = 0, Roller support

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∆ = 0 pin

Deflection Diagrams and the Elastic Curve

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∆ = 0 θ = 0fixed support

Deflection Diagrams and the Elastic Curve

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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).

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Moment Area Theorems

• 1st - 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.

• 2nd – Theorem :

• Gives 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

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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