Deflection of Beams

MECHANICS OF MATERIALS

Fourth Edition

Ferdinand P. BeerE. Russell Johnston, Jr.John T. DeWolf

Lecture Notes:J. Walt OlerTexas Tech University

CHAPTER

© 2006 The McGraw-Hill Companies, Inc. All rights reserved.

9 Deflection of Beams


MECHANICS OF MATERIALSFourthEdition

Beer • Johnston • DeWolf

9 - 2

Deflection of Beams

Deformation of a Beam Under Transverse Loading

Equation of the Elastic Curve

Direct Determination of the Elastic Curve From the Load Di...

Statically Indeterminate Beams

Sample Problem 9.1

Sample Problem 9.3

Method of Superposition

Sample Problem 9.7

Application of Superposition to Statically Indeterminate ...

Sample Problem 9.8

Moment-Area Theorems

Application to Cantilever Beams and Beams With Symmetric ...

Bending Moment Diagrams by Parts

Sample Problem 9.11

Application of Moment-Area Theorems to Beams With Unsymme...

Maximum Deflection

Use of Moment-Area Theorems With Statically Indeterminate...




9 - 3

Deformation of a Beam Under Transverse Loading• Relationship between bending moment and

curvature for pure bending remains valid for general transverse loadings.

EIxM )(1

• Cantilever beam subjected to concentrated load at the free end,

EIPx

1

• Curvature varies linearly with x

• At the free end A, AA

ρρ

,01

• At the support B, PLEI

BB

,01




9 - 4

Deformation of a Beam Under Transverse Loading• Overhanging beam

• Reactions at A and C

• Bending moment diagram

• Curvature is zero at points where the bending moment is zero, i.e., at each end and at E.

EIxM )(1

• Beam is concave upwards where the bending moment is positive and concave downwards where it is negative.

• Maximum curvature occurs where the moment magnitude is a maximum.

• An equation for the beam shape or elastic curve is required to determine maximum deflection and slope.




9 - 5


• From elementary calculus, simplified for beam parameters,

2

2

232

2

2

1

1dx

yd

dxdy

dxyd

• Substituting and integrating,

2100

10

2

21

CxCdxxMdxyEI

CdxxMdxdyEIEI

xMdx

ydEIEI

xx

x




9 - 6


2100

CxCdxxMdxyEIxx

• Constants are determined from boundary conditions

• Three cases for statically determinant beams,

– Simply supported beam0,0 BA yy

– Overhanging beam0,0 BA yy

– Cantilever beam0,0 AAy

• More complicated loadings require multiple integrals and application of requirement for continuity of displacement and slope.




9 - 7

Direct Determination of the Elastic Curve From the Load Distribution

• Equation for beam displacement becomes

xwdx

ydEIdx

Md 4

4

2

2

432

2213

161 CxCxCxC

dxxwdxdxdxxyEI

• Integrating four times yields

• For a beam subjected to a distributed load,

xwdxdV

dxMdxV

dxdM 2

2

• Constants are determined from boundary conditions.




9 - 8

Statically Indeterminate Beams• Consider beam with fixed support at A and roller

support at B.• From free-body diagram, note that there are four

unknown reaction components.• Conditions for static equilibrium yield

000 Ayx MFF

The beam is statically indeterminate.

2100

CxCdxxMdxyEIxx

• Also have the beam deflection equation,

which introduces two unknowns but provides three additional equations from the boundary conditions:

0,At 00,0At yLxyx




9 - 9

Sample Problem 9.1

ft 4ft15kips50

psi1029in7236814 64

aLP

EIW

For portion AB of the overhanging beam, (a) derive the equation for the elastic curve, (b) determine the maximum deflection, (c) evaluate ymax.

SOLUTION:

• Develop an expression for M(x) and derive differential equation for elastic curve.

• Integrate differential equation twice and apply boundary conditions to obtain elastic curve.

• Locate point of zero slope or point of maximum deflection.

• Evaluate corresponding maximum deflection.




9 - 10

Sample Problem 9.1SOLUTION:

• Develop an expression for M(x) and derive differential equation for elastic curve.

- Reactions:

LaPR

LPaR BA 1

- From the free-body diagram for section AD,

LxxLaPM 0

xLaP

dxydEI 2

2

- The differential equation for the elastic curve,




9 - 11

Sample Problem 9.1

PaLCLCLLaPyLx

Cyx

61

610:0,at

0:0,0at

113

2

• Integrate differential equation twice and apply boundary conditions to obtain elastic curve.

213

12

61

21

CxCxLaPyEI

CxLaP

dxdyEI

xLaP

dxydEI 2

2

32

6 Lx

Lx

EIPaLy

PaLxxLaPyEI

Lx

EIPaL

dxdyPaLx

LaP

dxdyEI

61

61

3166

121

3

22

Substituting,




9 - 12

Sample Problem 9.1• Locate point of zero slope or point

of maximum deflection.

32

6 Lx

Lx

EIPaLy

LLxL

xEI

PaLdxdy

mm 577.0

331

60

2

• Evaluate corresponding maximum deflection.

32

max 577.0577.06

EI

PaLy

EIPaLy6

0642.02

max

46

2max

in723psi10296in180in48kips500642.0

y

in238.0max y




9 - 13

Sample Problem 9.3

For the uniform beam, determine the reaction at A, derive the equation for the elastic curve, and determine the slope at A. (Note that the beam is statically indeterminate to the first degree)

SOLUTION:

• Develop the differential equation for the elastic curve (will be functionally dependent on the reaction at A).

• Integrate twice and apply boundary conditions to solve for reaction at A and to obtain the elastic curve.

• Evaluate the slope at A.




9 - 14

Sample Problem 9.3• Consider moment acting at section D,

LxwxRM

MxLxwxR

M

A

A

D

6

032

1

0

30

20

LxwxRM

dxydEI A 6

30

2

2

• The differential equation for the elastic curve,




9 - 15

Sample Problem 9.3

LxwxRM

dxydEI A 6

30

2

2

• Integrate twice

215

03

14

02

12061

2421

CxCL

xwxRyEI

CLxwxREI

dxdyEI

A

A

• Apply boundary conditions:

01206

1:0,at

0242

1:0,at

0:0,0at

214

03

13

02

2

CLCLwLRyLx

CLwLRLx

Cyx

A

A

• Solve for reaction at A

0301

31 4

03 LwLRA LwRA 010

1




9 - 16

Sample Problem 9.3

xLwL

xwxLwyEI

3

05

030 120

112010

161

xLxLxEIL

wy 43250 2120

• Substitute for C1, C2, and RA in the elastic curve equation,

42240 65120

LxLxEIL

wdxdy

EILw

A 120

30

• Differentiate once to find the slope,

at x = 0,




9 - 17

Method of Superposition

Principle of Superposition:

• Deformations of beams subjected to combinations of loadings may be obtained as the linear combination of the deformations from the individual loadings

• Procedure is facilitated by tables of solutions for common types of loadings and supports.




9 - 18

Sample Problem 9.7

For the beam and loading shown, determine the slope and deflection at point B.

SOLUTION:

Superpose the deformations due to Loading I and Loading II as shown.




9 - 19

Sample Problem 9.7

Loading I

EI

wLIB 6

3

EIwLy IB 8

4

Loading II

EI

wLIIC 48

3

EIwLy IIC 128

4

In beam segment CB, the bending moment is zero and the elastic curve is a straight line.

EI

wLIICIIB 48

3

EI

wLLEI

wLEI

wLy IIB 3847

248128

434




9 - 20

Sample Problem 9.7

EI

wLEI

wLIIBIBB 486

33

EI

wLEI

wLyyy IIBIBB 3847

8

44

EIwL

B 487 3

EIwLyB 384

41 4

Combine the two solutions,




9 - 21

Application of Superposition to Statically Indeterminate Beams

• Method of superposition may be applied to determine the reactions at the supports of statically indeterminate beams.

• Designate one of the reactions as redundant and eliminate or modify the support.

• Determine the beam deformation without the redundant support.

• Treat the redundant reaction as an unknown load which, together with the other loads, must produce deformations compatible with the original supports.

Date post:	09-Feb-2016
Category:	Documents
Upload:	ely
View:	271 times
Download:	6 times

Deflection of Beams

Documents