MECHANICS OF

MATERIALS

Fourth Edition

Ferdinand P. Beer, E. Russell Johnston, Jr., John T. DeWolf

CHAPTER

Analysis and Design

of Beams for Bending

Lecturer: Nazarena Mazzaro, Ph.D.

Aalborg University

Denmark

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Analysis and Design of Beams for Bending

Introduction

Shear and Bending Moment

Diagrams

Example 5.01

Sample Problem 5.1

Sample Problem 5.2

Relations Among Load, Shear, and

Bending Moment

Example 5.03

Sample Problem 5.3

Sample Problem 5.5

Part 1: 45 min

Design of Prismatic Beams for

Bending

Sample Problem 5.8

Singularity Functions

Example 5.05

Part 2: 30 min

Exercises

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Introduction

•Beams-structural members supporting loads at

various points along the member

•Objective -Analysis and design of beams

•Transverse loadings of beams are classified as

concentratedloads or distributedloads

•Applied 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

is often the critical design criteria

S

M

I

cM

I

My

mx

==−=

σσ

Requires determination of the location and

magnitude of largest bending moment

Elastic Flexure

Formulas

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Introduction

Classification of Beam Supports

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Shear and Bending Moment Diagrams

•Determination of maximum normal and

shearing stresses requires identification of

maximum internal shear force V and

bending couple M.

•Shear force and bending couple at a point are

determined by passing a section through the

beam and applying an equilibrium analysis on

the beam portions on either side of the

section.

•Sign conventions for shear forces Vand V’

and bending couples Mand M’-> positive

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Example 5.01

Draw the shear and bending-moment diagrams for a simply

supported beam AB of span L subjected to a single concentrated

load P at it midpoint C

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Example 5.01

•Determination of the reactions RA=RB=1/2P

•Cut the beam at D and plot free body diagrams with

positive V and M. Equilibrium equations:

2/)(;0)(

2/1;0;0

2/;0;0

2/1;0;0

xLPMMxLRM

PRVVRFy

PxxRMMxRM

PRVVRFy

B

E

BB

AAD

AA

−==−−=

−=−==+=

===+−=

===−=

∑

∑

∑

∑

V is constant between concentrated loads and M

varies linearly

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Sample Problem 5.1

For the timber beam and loading

shown, draw the shear and bend-

moment diagrams and determine the

maximum normal stress due to

bending.

SOLUTION:

•Treating the entire beam as a rigid

body, determine the reaction forces

•Identify the maximum shear and

bending-moment from plots of their

distributions.

•Apply the elastic flexure formulas to

determine the corresponding

maximum normal stress.

•Section the beam at points near

supports and load application points.

Apply equilibrium analyses on

resulting free-bodies to determine

internal shear forces and bending

couples

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Sample Problem 5.1

SOLUTION:

•Treating the entire beam as a rigid body

•Section the beam and apply equilibrium analyses

on resulting free-bodies

()()

00m0kN200

kN200kN200

111

11

==+

∑

=

−

=

=

−

−

∑

=

MMM

VV

F

y

()()

mkN500m5.2kN200

kN200kN200

222

22

⋅−==+

∑

=

−

=

=

−

−

∑

=

MMM

VV

F

y

0kN14

mkN28kN14

mkN28kN26

mkN50kN26

66

55

44

33

=−=

⋅+=−=

⋅+=+=

⋅

−

=

+

=

MV

MV

MV

M

V

46;14

1453405,2200

6040200

==

=⇒×+×−×==

−=⇒+−+−==

∑

∑

BD

DDB

BDDB

RR

RRM

RRRRFy

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Sample Problem 5.1

•Identify the maximum shear and bending-

moment from plots of their distributions.

mkN50kN26

⋅

=

=

=

Bmm

MMV

•Apply the elastic flexure formulas to

determine the corresponding

maximum normal stress.

(

)

(

)

36

3

36

2

6

1

2

6

1

m1033.833

mN1050

m1033.833

m250.0m080.0

−

−

×

⋅×

==

×=

==

S

M

hbS

B

m

σ

Pa100.60

6

×=

m

σ

V1=-20, M1=0; V2=-20, M2

=-50; V3= 26, M3=-50;

V4=26, M4= 28; V5=-14, M5=28, V

6=-14, M

6=0

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Sample Problem 5.2

The structure shown is constructed of a

W10x112 rolled-steel beam. (a) Draw

the shear and bending-moment diagrams

for the beam and the given loading. (b)

determine normal stress in sections just

to the right and left of point D.

SOLUTION:

•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.

•Section the beam at points near the

support and load application points.

Apply equilibrium analyses on

resulting free-bodies to determine

internal shear forces and bending

couples.

•Apply the elastic flexure formulas to

determine the maximum normal

stress to the left and right of point D.

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Sample Problem 5.2

SOLUTION:

•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-bodies.

()

()

ftkip5.1030

kips3030

:

2

2

1

1

⋅−==+

∑

=

−==−−

∑

=

xMMxxM

xVVxF

C

to

A

From

y

()()

ftkip249604240

kips240240

:

2

⋅−==+−

∑

=

−==−−

∑

=

xMMxM

VVF

D

to

C

From

y

()

ftkip34226kips34

:

⋅−=−=xMV

Bto

D

From

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Sample Problem 5.2

•Apply the elastic flexure formulas to

determine the maximum normal stress to

the left and right of point D.

From Appendix C for a W10x112 rolled

steel shape, S= 126 in3

about the X-Xaxis.

3

3

in126

inkip1776

:

in126

inkip2016

:

⋅

==

⋅

==

S

M

DofrighttheTo

S

M

D

o

f

le

ft

theTo

m

m

σ

σ

ksi0.16

=

m

σ

ksi1.14

=

m

σ

•Concentrated loads: V constant

M varies linearly

•Distributed load: V varies linearly

M parabola

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Relations Among Load, Shear, and Bending Moment

(

)

xwV

xwVVVFy

∆−=∆

=

∆

−

∆

+

−

=

∑

0:0

∫

−=−

−=

D

C

x

x

CD

dxwVV

w

dx

dV

•Relationship between load and shear:

()

()

xwV

x

M

xwxVM

x

xwxVMMMM

C

∆−=

∆

∆

⇒∆−∆=∆

=

∆

∆+∆−−∆+=

∑

′

2

1

0

2

:0

2

2

1

∫

=−

=→∆

D

C

x

x

CD

dxVMM

V

dx

dM

x;0

•Relationship between shear and bending

moment:

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Example 5.03

Draw the shear and bending moment diagrams for the

simply supported beam and determine the maximum value

of the bending moment.

SOLUTION:

•RA=RB=wL/2

•Determination of V and M at any distance from A:

8

)(

2

1

)

2

1

(

0;.

)

2

1

(

2

1

.

2

max

2

0

0

0

wL

MxLxwdxxLwM

MdxVMM

xLwwLwLwxVV

wxdxwVV

x

A

x

A

A

x

A

⇒−=−=

==−

−=−=−=

−=−=−

∫

∫

∫

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Sample Problem 5.3

Draw the shear and bending

moment diagrams for the beam

and loading shown.

SOLUTION:

•Taking the entire beam as a free body,

determine the reactions at Aand D.

•Apply the relationship between shear and

load to develop the shear diagram.

•Apply the relationship between bending

moment and shear to develop the bending

moment diagram.

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Sample Problem 5.3

SOLUTION:

•Taking the entire beam as a free body, determine the

reactions at Aand D.

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

kips18

kips12kips26kips12kips200

0F

kips26

ft28kips12ft14kips12ft6kips20ft240

0

y

=

−+−−=

=

∑

=

−−−=

=

∑

y

y

A

A

A

D

D

M

•Apply the relationship between shear and load to

develop the shear diagram.

dxwdVw

dx

dV

−=−=

-zero slope between concentrated loads

-linear variation over uniform load segment

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Sample Problem 5.3

•Apply the relationship between bending

moment and shear to develop the bending

moment diagram.

dxVdMV

dx

dM

==

-bending moment at Aand Eis zero

-total of all bending moment changes across

the beam should be zero (108-16-140+48=0)

-net change in bending moment is equal to

areas under shear distribution segments

-bending moment variation between D

and Eis quadratic

-bending moment variation between A, B,

C and Dis linear

dxVMM

D

C

CD

.

∫

=−

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Sample Problem 5.5

Draw the shear and bending moment

diagrams for the beam and loading

shown.

SOLUTION:

•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.

•Apply the relationship between

bending moment and shear to develop

the bending moment diagram.

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Sample Problem 5.5

SOLUTION:

•Taking the entire beam as a free body,

determine the reactions at C.

⎟

⎠

⎞

⎜

⎝

⎛

−−=+

⎟

⎠

⎞

⎜

⎝

⎛

−==

∑

=+−==

∑

33

0

0

0

2

1

0

2

1

0

2

1

0

2

1

a

LawMM

a

LawM

awRRawF

CC

C

CCy

Results from integration of the load and shear

distributions should be equivalent.

•Apply the relationship between shear and load

to develop the shear diagram.

()

curveloadunderareaawV

a

x

xwdx

a

x

wVV B

a

a

AB

−=−=

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−−=

∫

⎟

⎠

⎞

⎜

⎝

⎛

−−=−

0

2

1

0

2

0

0

0

2

1

-No change in shear between Band C.

-Compatible with free body analysis

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Sample Problem 5.5

•Apply the relationship between bending moment

and shear to develop the bending moment

diagram.

2

0

3

1

0

32

0

0

2

0

622

awM

a

xx

wdx

a

x

xwMM

B

a

a

AB

−=

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−−=

∫

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−−=−

(

)

()

()

⎟

⎠

⎞

⎜

⎝

⎛

−=−−=

−−=

∫

−=−

32

3

0

0

6

1

0

2

1

0

2

1

a

L

wa

aLawM

aLawdxawMM

C

L

a

CB

Results at Care compatible with free-body

analysis

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Design of Prismatic Beams for Bending

•Among beam section choices which have an acceptable

section modulus, the one with the smallest weight per unit

length or cross sectional area

will be the least expensive

and the best choice.

•The largest normal stress is found at the surface where the

maximum bending moment occurs.

S

M

I

cM

m

maxmax

==

σ

•A safe design requires that the maximum normal stress be

less than the allowable stress for the material used. This

criteria leads to the determination of the minimum

acceptable section modulus.

al

l

allm

M

S

σ

σ

σ

max

min

=

≤

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Design of Prismatic Beams for Bending

•Determine σall

(Ifσall

is thesame for tension and compresionthen

follow1,2,3 –Otherwiseconsider4 in step 2)

•1-Draw shearand bending-momentdiagrams and determine|M|

max

•2-Determine S

min

•3-Find thedimentionsofthebeamto use: b, h for S>Smin

•4-Select thebeamsectionso thatσm<= σall for tensileand

compressivestresses.

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Sample Problem 5.7

A 3.6 m-long overhanging timber

beam AC is to be designed to support

the distributed and concentrated loads

shown. Knowing that timber of 100-

mm nominal width (90-mm actual

width) with a 12 MPaallowable stress

is to be used, determine the minimum

required depth h of the beam.

SOLUTION:

•Considering the entire beam as a free-

body, determine the reactions at Aand

B.

•Develop the shear diagram for the

beam and load distribution. From the

diagram, determine the maximum

bending moment.

•Determine the minimum acceptable

beam section modulus. Find the value

for the h.

60 kN/m

20 kN/m

90 mm

2.4 m

1.2 m

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Sample Problem 5.7

•Considering the entire beam as a free-body

(

)

(

)

(

)

(

)

(

)

A

VkN8.2

kN20kN4.14kN2.370

VckN2.37

m6.3kN20m2.1kN4.14m4.20

=−=

−−+==

==

−−==

∑

∑

y

yy

A

A

AF

B

BM

•Plot shear diagram and determine M

max.

(

)

kN2.17

2.174.148.2414

4.14)4.2)(/6(

−=

−=−−=−−=

−

=

−

=

−

=

−

B

AB

AB

V

kNkNkNVV

kNmmkNcurveloadunderareaVV

•M

A=Mc=0, Between A and B, M decreases an

amount equal to area VAB, and between B and

C in increases the same amount. Thus the

|M|max= 24 kN.m

mmhmmhmmSbh

mm

MPa

mkN

M

S

all

2.365102)90(

6

1

6

1

102

12

.24

362

min

2

36

max

min

≥⇒×≥⇒≥

×===

σ

14.4 kN

20 kN

2.4 m

1.2m

20

kN

(+24)

(+24)

-17.2 kN

-2.8

kN

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