[ths]2012 defl-01

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DEFLECTIONS of STRUCTURES

Tharwat Sakr

Definition, Importance and Definition, Importance and Causes of DeflectionsCauses of Deflections

Derivation of Moment – Derivation of Moment – Curvature RelationCurvature Relation

The Double Integration The Double Integration MethodMethod

1

3

2

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■ ■ DefinitionDefinitionDeflection is defined as the displacement of various points of the structure from their original positions including

linear deformations of points.

rotational deformations of lines (slopes) from their

original position.

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■ ■ Importance of DeflectionsImportance of Deflections

3

Bad appearance and uncomforting of occupants

Cracking of plaster.

Drainage problems.

Damage of walls and non structural Elements.

■ ■ All Codes and Standards specify limits of deflection All Codes and Standards specify limits of deflection as excessive deflections lead to problems as:as excessive deflections lead to problems as:

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■ ■ Importance of DeflectionsImportance of Deflections

■ ■ In Addition the analysis of indeterminate In Addition the analysis of indeterminate structuresstructures requires the calculation of deflections as in advance requires the calculation of deflections as in advance stepstep

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■ ■ Sources of DeflectionsSources of Deflections

The external loads.

Temperature variation

Differential settlements between supports cause

various deformations.

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■ ■ Factors Affecting DeflectionsFactors Affecting Deflections

Span and Configuration

The Applied Loads

Material of the structure

Cross section of the Structure

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■ ■ Methods to calculate DeflectionsMethods to calculate Deflections

1. The Double Integration Method.1. The Double Integration Method.

5. The Real Work Method.5. The Real Work Method.

2. The Moment Area Method.2. The Moment Area Method.

3. The Elastic Load Method.3. The Elastic Load Method.

4. The Conjugate Beam Method.4. The Conjugate Beam Method.

6. The Virtual Work Method.6. The Virtual Work Method.

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1. Plane section before deformation remains plane after

deformations (Bernoulli's law )

THE DIFFERENTIAL EQUATION OF THE ELASTIC LINE

Main Assumptions

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2. Stress is Proportional to Strain (Elastic Material) (Hook’s Low)


Main Assumptions

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3. The Depth – Span Ration is very small ( Slender Members)


Main Assumptions

Slender BeamSlender Beam Deep BeamDeep Beam

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3. The Deflection is very – small compared to Span (Small deflection)


Main Assumptions

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Curvature of elastic line Curvature of elastic line


Curvature of Elastic Line (Property of Curves)



d

R

R

1

2

ds

1 2

d 12

Y

X

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To be in x, yTo be in x, y d R

1

2

ds

1 2

d 12

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As As is so small is so small



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AsAs

ThenThen



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Lead toLead to



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For Very small values of For Very small values of deflection (y)deflection (y)

Is the equation of radius of curvature of any curve and Is the equation of radius of curvature of any curve and its simplification for very small values of yits simplification for very small values of y



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Moment - Curvature Moment - Curvature RelationshipRelationship


Moment-Curvature Relationship

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MM

dh.

d

d

dh.

1

1 1

1

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Elongation at any fiberElongation at any fiber

Strain can be defined as Strain can be defined as

From Hook’s LawFrom Hook’s Law dh.

d

dh.

1

1 1

1

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From the stress formulaFrom the stress formula

By SubstitutionBy Substitution

FromFrom

Is the differential equation of the elastic line subjected Is the differential equation of the elastic line subjected to momentto moment

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Sign Convention

DeflectionDeflection

2

2

dxyd

dxdy

y

SlopeSlope

CurvatureCurvature

Is the rate of change of the displacementIs the rate of change of the displacement

Is the rate of change of the slopeIs the rate of change of the slope

Represents the displacement of the structureRepresents the displacement of the structure

Investigating the moment – deflection signInvestigating the moment – deflection sign

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Sign ConventionPositive MomentPositive Moment

Slope decreased with xSlope decreased with xNegative MomentNegative Moment

Slope Increased with xSlope Increased with x

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Is the Relation between Deflection and MomentIs the Relation between Deflection and Moment


Sign Convention


Concepts

Deflection at any pointDeflection at any point

Slope angel at any pointSlope angel at any point

Curvature at any pointCurvature at any point

Moment at any pointMoment at any point

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The method is based on the direct application of the The method is based on the direct application of the differential equation of elastic linedifferential equation of elastic line

THE DOUBLE INTEGRATION METHOD

Theoretical Bases


Basic Procedure

Derive an Equation of the Moment M as function of Derive an Equation of the Moment M as function of position variable x position variable x (M(x))(M(x))

Apply the differential equation of elastic Apply the differential equation of elastic line and integrate twice line and integrate twice

Apply the Boundary and Continuity conditions to obtain the Apply the Boundary and Continuity conditions to obtain the integration constantsintegration constants

Substitute with the integration constants into the deflection Substitute with the integration constants into the deflection and slope equationsand slope equations


Boundary Conditions

At Roller supportAt Roller support

At Hinged supportAt Hinged support

At Fixed supportAt Fixed support


Continuity Conditions

At Any intermediate pointAt Any intermediate point

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Standard Cases of Beam Deflection

Tharwat Sakr

[ths]2012 defl-01

Technology