6.0 6.0 ELASTIC DEFLECTION OF BEAMSELASTIC DEFLECTION OF BEAMS

6.1 Introduction

6.2 Double-Integration Method

6.3 Examples

6.4 Moment Area Method

6.5 Examples

IntroductionxP P

y

Elastic curve

The deflection is measured from the original neutral axis to the neutral axis of the deformed beam.

The displacement y is defined as the deflection of the beam.

It may be necessary to determine the deflection y for every value of x along the beam. This relation may be written in the form of an equation which is frequently called the equation of the deflection curve (or elastic curve) of the beam

Importance of Beam Deflections

A designer should be able to determine deflections, i.e.

In building codes ymax <=Lbeam/300

Analyzing statically indeterminate beams involve the use of various deformation relationships.

Methods of Determining Beam Deflections

a) Double-Integration Method

b) Moment-Area Method

c) Elastic Energy Methods

d) Method of singularity functions

Double-Integration MethodThe deflection curve of the bent beam is M

dx

ydEI

2

2

In order to obtain y, above equation needs to be integrated twice.

y

Radius of curvature

y

x

)(Curvature1

EI

MEIM

An expression for the curvature at any point along the curve representing the deformed beam is readily available from differential calculus. The exact formula for the curvature is

2

32

2

2

1

dxdy

dxyd

small is dx

dy2

2

dx

yd M

dx

ydEI

2

2

The Integration Procedure

Integrating once yields to slope dy/dx at any point in the beam.

Integrating twice yields to deflection y for any value of x.

The bending moment M must be expressed as a function of the coordinate x before the integration

Differential equation is 2nd order, the solution must contain two constants of integration. They must be evaluated at known deflection and slope points (i.e. at a simple support deflection is zero, at a built in support both slope and deflection are zero)

Sign Convention

Positive Bending Negative Bending

Assumptions and Limitations

Deflections caused by shearing action negligibly small compared to bending

Deflections are small compared to the cross-sectional dimensions of the beam

All portions of the beam are acting in the elastic range

Beam is straight prior to the application of loads

ExamplesLx

x

y

P

PL

P

PxPLM

Mdx

ydEI

2

2

@ x PxPLdx

ydEI

2

2

Integrating once 1

2

2c

xPPLx

dx

dyEI

@ x = 0 0

2

0000 11

2

ccPPLEIdx

dy

Integrating twice 2

32

62c

xP

PLxEIy

@ x = 0 0

6

00

200 22

32 ccP

PLEIy

62

32 xP

PLxEIy

@ x = L y = ymax

EI

PLy

PLLP

LPLEIy

3662

3

max

332

max

EI

PL

3

3

max

Lx

x

y

WL

2

2xL

WM

Mdx

ydEI

2

2

@ x 2

2

2

2xL

W

dx

ydEI

Integrating once

1

3

32c

xLW

dx

dyEI

@ x = 0 63

0

200

3

11

3 WLcc

LWEI

dx

dy

W N per unit length

2

2WL

66

33 WL

xLW

dx

dyEI

24624

434 WL

xWL

xLW

EIy

Max. occurs @ x = L

EI

WLy

WLWLLWEIy

88246

4

max

444

max

EI

WL

8

4

max

Integrating twice

2

34

646cx

WLxLWEIy

@ x = 0 24

064

0

600

4

22

34 WLcc

WLLWEIy

Example

L

x

y x

2

WL

2

WL

22

xWxx

WLM

22

2

2

2 xWx

WL

dx

ydEI

Integrating 1

32

3222c

xWxWL

dx

dyEI

Since the beam is symmetric 02

@ dx

dyLx

1

32

32

222

20

2@ c

LW

LWL

EIL

x24

3

1

WLc

2464

332 WLx

Wx

WL

dx

dyEI

Integrating2

343

244634cx

WLxWxWLEIy

@ x = 0 y = 0 2

343

0244

0

63

0

40 c

WLWWLEI 02 c

xWL

xW

xWL

EIy242412

343

Max. occurs @ x = L /2 384

5 4

max

WLEIy

EI

WL

384

5 4

max

Exampley

20for

22

2 Lxx

P

dx

ydEI

Integrating 1

2

22c

xP

dx

dyEI

Since the beam is symmetric 02

@ dx

dyLx

1

2

22

20

2@ c

LP

EIL

x16

2

1

PLc

164

22 PLx

P

dx

dyEI

xP

ML

x22

0for

L/2

x

x

2

P

2

P

P

L/2

Integrating2

23

1634cx

PLxPEIy

@ x = 0 y = 0 2

23

0163

0

40 c

PLPEI 02 c

xPL

xP

EIy1612

23

Max. occurs @ x = L /2 48

3

max

PLEIy

EI

PL

48

3

max

Moment-Area MethodFirst Moment –Area Theorem

Tangent at A

The first moment are theorem states that: The angle between the tangents at A and B is equal to the area of the bending moment diagram between these two points, divided by the product EI.

B

A

dxEI

M

The second moment area theorem states that: The vertical distance of point B on a deflection curve from the tangent drawn to the curve at A is equal to the moment with respect to the vertical through B of the area of the bending diagram between A and B, divided by the product EI.

dxEI

MxB

A

A B

Tangent at B

d

d

xdx

ds

d

dsdds

EIM dx with ds replace sdeflection lateral small isit ds

EI

Md

dxEI

Mddx

EI

Md

B

A give willgintegratin

B

A

dxEI

Mxdx

EI

Mxxd

M

The Moment Area Procedure

1. The reactions of the beam are determined

2. An approximate deflection curve is drawn. This curve must be consistent with the known conditions at the supports, such as zero slope or zero deflection

3. The bending moment diagram is drawn for the beam. Construct M/EI diagram

4. Convenient points A and B are selected and a tangent is drawn to the assumed deflection curve at one of these points, say A

5. The deflection of point B from the tangent at A is then calculated by the second moment area theorem

Comparison of Moment Area and Double Integration Methods

If the deflection of only a single point of a beam is desired, the moment-area method is usually more convenient than the double integration method.

If the equation of the deflection curve of the entire beam is desired the double integration method is preferable.

Assumptions and Limitations

Same assumptions as Double Integration Method holds.

Examples

P

PL L

P

A

B

Tangent at A

Tangent at B

PL

M

33

2

2

3PLLPL

LEI

EI

PL

3

3

PLL

EI 2 EI

PL

2

2

WL2

2WL

Tangent A

L

A W N per unit length

B

= ?

2

2WL

xL

WLA

23

1 2

Lx4

3

84

3

23

42 WL

LLWL

EI

EI

WL

8

4

Example

L

aP

aP

P P

aaL

2

Pa

Tangent A

A = ?

aPaa

aaL

aL

PaEI3

2

2242

322

32448a

PaLaLaLPa

3

3332 43

2468 L

a

L

aPLPaPaL

3

33 43

24 L

a

L

a

EI

PL