MECHANICS OFFourth Edition

MECHANICS OF MATERIALS

CHAPTER

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

Analysis and DesignJohn T. DeWolf

Lecture Notes:of Beams for Bending

J. Walt OlerTexas Tech University

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MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Analysis and Design of Beams for Bending

IntroductionShear and Bending Moment DiagramsSample Problem 5.1

l blSample Problem 5.2Relations Among Load, Shear, and Bending MomentSample Problem 5 3Sample Problem 5.3Sample Problem 5.5Design of Prismatic Beams for BendingSample Problem 5.8

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MECHANICS OF MATERIALS

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Beer • Johnston • DeWolf

Introduction

• Beams - structural members supporting loads at

• Objective - Analysis and design of beams

Beams structural members supporting loads at various points along the member

• Transverse loadings of beams are classified asTransverse loadings of beams are classified as concentrated loads or distributed loads

• Applied loads result in internal forces consistingApplied loads result in internal forces consisting of a shear force (from the shear stress distribution) and a bending couple (from the normal stress distribution)normal stress distribution)

• Normal stress is often the critical design criteriaMMMSM

IcM

IMy

mx ==−= σσ

Requires determination of the location and

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magnitude of largest bending moment

MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Introduction

Classification of Beam Supports

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MECHANICS OF MATERIALS

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Beer • Johnston • DeWolf

Shear and Bending Moment Diagrams

• Determination of maximum normal and shearing stresses requires identification of maximum internal shear force and bending couple.

• Shear force and bending couple at a point are determined by passing a section through the beam and applying an equilibrium analysis on pp y g q ythe beam portions on either side of the section.

• Sign conventions for shear forces V and V’and bending couples M and M’

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MECHANICS OF MATERIALS

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Beer • Johnston • DeWolf

Sample Problem 5.1SOLUTION:

• Treating the entire beam as a rigid g gbody, determine the reaction forces

• Section the beam at points near psupports and load application points. Apply equilibrium analyses on resulting free-bodies to determine

For the timber beam and loading shown, draw the shear and bend-

t di d d t i th

resulting free bodies to determine internal shear forces and bending couples

moment diagrams and determine the maximum normal stress due to bending.

• Identify the maximum shear and bending-moment from plots of their distributionsdistributions.

• Apply the elastic flexure formulas to determine the corresponding

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determine the corresponding maximum normal stress.

MECHANICS OF MATERIALS

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Beer • Johnston • DeWolf

Sample Problem 5.1SOLUTION:

• Treating the entire beam as a rigid body, determine eat g t e e t e bea as a g d body, dete ethe reaction forces

∑ ∑ ==== kN14kN46:0 from DBBy RRMF

• Section the beam and apply equilibrium analyses on resulting free-bodies

kN200kN200 11 −==−−∑ = VVFy

( )( ) 00m0kN200 111

11

==+∑ =

∑

MMMy

kN200kN200 22 −==−−∑ = VVFy

( )( ) mkN500m5.2kN200 222

22

⋅−==+∑ =

∑

MMMy

mkN50kN26 33 ⋅−=+= MV

mkN28kN14

mkN28kN26

55

44

33

⋅+=−=

⋅+=+=

MV

MV

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0kN14 66 =−= MV

MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Sample Problem 5.1• Identify the maximum shear and bending-

moment from plots of their distributions.

mkN50kN26 ⋅=== Bmm MMV

• Appl the elastic fle re form las to• Apply the elastic flexure formulas to determine the corresponding maximum normal stress.

( )( )36

2612

61

m1033.833

m250.0m080.0−×=

== hbS

36

3

1033833mN1050

−⋅×

==S

M Bmσ 36 m1033.833 ×S

Pa100.60 6×=mσ

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MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Sample Problem 5.2SOLUTION:

• Replace the 10 kip load with an equivalent force-couple system at D. Find the reactions at B by considering the beam as a rigid body.g y

• Section the beam at points near the support and load application points.

The structure shown is constructed of a W10x112 rolled-steel beam. (a) Draw th h d b di t di

pp pp pApply equilibrium analyses on resulting free-bodies to determine internal shear forces and bendingthe shear and bending-moment diagrams

for the beam and the given loading. (b) determine normal stress in sections just

internal shear forces and bending couples.

• Apply the elastic flexure formulas toto the right and left of point D. • Apply the elastic flexure formulas to determine the maximum normal stress to the left and right of point D.

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MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Sample Problem 5.2SOLUTION:

• Replace the 10 kip load with equivalent force-couple system at D. Find reactions at B.

• Section the beam and apply equilibrium analyses on resulting free-bodiesanalyses on resulting free bodies.

kips3030:

−==−−∑ = xVVxFCtoAFrom

y

( )( ) ftkip5.1030 221

1 ⋅−==+∑ = xMMxxMy

:DtoCFrom

( ) ( ) ftkip249604240

kips240240

2 ⋅−==+−∑ =

−==−−∑ =

xMMxM

VVFy

( ) ftkip34226kips34:

⋅−=−= xMVBtoDFrom

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MECHANICS OF MATERIALS

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Beer • Johnston • DeWolf

Sample Problem 5.2• Apply the elastic flexure formulas to

determine the maximum normal stress to th l ft d i ht f i t Dthe left and right of point D.

From Appendix C for a W10x112 rolled steel shape S = 126 in3 about the X-X axissteel shape, S = 126 in about the X-X axis.

inkip2016:

⋅MDoflefttheTo

k i0163

iki1776:in126p

==

MDofrighttheTo

Smσ ksi0.16=mσ

3in126inkip1776 ⋅

==SM

mσ ksi1.14=mσ

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MECHANICS OF MATERIALS

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Relations Among Load, Shear, and Bending Moment

( ) xwVVVFy =∆−∆+−=∑ 0:0• Relationship between load and shear:

xwV ∆−=∆

−= wddV

∫−=−D

C

x

xCD dxwVV

dx

x∆

• Relationship between shear and bending moment:

( )

( )221

02

:0

xwxVM

xxwxVMMMMC

∆−∆=∆

=∆

∆+∆−−∆+=∑ ′

dM

∫

=

DxdxVMM

Vdx

dM

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∫=−Cx

CD dxVMM

MECHANICS OF MATERIALS

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Beer • Johnston • DeWolf

Sample Problem 5.3

SOLUTION:

T ki th ti b f b d• Taking the entire beam as a free body, determine the reactions at A and D.

• Apply the relationship between shear and load to develop the shear diagram.

Draw the shear and bending moment diagrams for the beam

• Apply the relationship between bending moment and shear to develop the bending moment diagramand loading shown. moment diagram.

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MECHANICS OF MATERIALS

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Sample Problem 5.3SOLUTION:

• Taking the entire beam as a free body, determine the reactions at A and D.

( ) ( )( ) ( )( ) ( )( )ft28kips12ft14kips12ft6kips20ft2400

−−−==∑ A

DM

( ) ( )( ) ( )( ) ( )( )

0Fkips26

ft28kips12ft14kips12ft6kips20ft240

y =∑

==

DD

kips18

kips12kips26kips12kips200

=

−+−−=

y

y

A

A

• Apply the relationship between shear and load to develop the shear diagram.

ddVdV dxwdVwdx

−=−=

- zero slope between concentrated loads

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- linear variation over uniform load segment

MECHANICS OF MATERIALS

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Beer • Johnston • DeWolf

Sample Problem 5.3• Apply the relationship between bending

moment and shear to develop the bending dimoment diagram.

dxVdMVdx

dM==

- bending moment at A and E is zero

- bending moment variation between A, B,

- bending moment variation between D d E i d i

C and D is linear

- net change in bending moment is equal to areas under shear distribution segments

and E is quadratic

- total of all bending moment changes across the beam should be zero

areas under shear distribution segments

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t e bea s ou d be e o

MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Sample Problem 5.5

SOLUTION:

T ki h i b f b d• Taking the entire beam as a free body, determine the reactions at C.

• Apply the relationship between shear and load to develop the shear diagram.

Draw the shear and bending moment diagrams for the beam and loading

• Apply the relationship between bending moment and shear to develop the bending moment diagramshown. the bending moment diagram.

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MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Sample Problem 5.5SOLUTION:

• Taking the entire beam as a free body, determine the reactions at C.

⎞⎛⎞⎛

=+−==∑ 0 021

021

aaawRRawF CCy

⎟⎠⎞

⎜⎝⎛ −−=+⎟

⎠⎞

⎜⎝⎛ −==∑

330 02

102

1 aLawMMaLawM CCC

Results from integration of the load and shear distributions should be equivalent.

• Apply the relationship between shear and load t d l th h dito develop the shear diagram.

axxwdx

axwVV

aa

AB⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠

⎞⎜⎜⎝

⎛−−=∫ ⎟

⎠⎞

⎜⎝⎛ −−=−

20

00 2

1

( )curveloadunderareaawV

aa

B −=−=

⎥⎦⎢⎣ ⎠⎝⎠⎝

021

00 2

- No change in shear between B and C

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No change in shear between B and C.- Compatible with free body analysis

MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Sample Problem 5.5• Apply the relationship between bending moment

and shear to develop the bending moment didiagram.

320

0

20 622 a

xxwdxa

xxwMMa

aAB

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠

⎞⎜⎜⎝

⎛−−=∫ ⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎜⎝

⎛−−=−

203

100 622

awM

aa

B −=

⎥⎦⎢⎣ ⎠⎝⎠⎜⎝ ⎠⎝

( )L( ) ( )

( ) ⎞⎜⎛ −=−−=

−−=∫ −=−

3 00

1

021

021

aLwaaLawM

aLawdxawMM

C

L

aCB

( )⎠

⎜⎝

==32

306 LaLawMC

Results at C are compatible with free-body anal sisanalysis

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