Slope & Deflections of Beams 

1. STRUCTURAL ANALYSIS - 1
Dr. OMPRAKASH
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2. Structural Analysis-I
Code of the subject : CET-225
Lecture – 1
Dr.Omprakash
Department of Civil Engineering
Chandigarh University
Dr. OMPRAKASH
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3. Slope and Displacement by the Moment area theorems
Moment-Area Theorems
is based on Two theorems of Mohr’s
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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.
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Dr. OMPRAKASH

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
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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
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7. Deflection Diagrams and the Elastic Curve
∆ = 0, Roller support
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8. Deflection Diagrams and the Elastic Curve
∆ = 0 pin
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9. Deflection Diagrams and the Elastic Curve
∆=0θ=0
fixed support
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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).
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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.
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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.
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