MECHANICS OFFourth Edition

MECHANICS OF MATERIALS

CHAPTER

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

Lecture Notes:

pUnder a Given LoadingJ. Walt Oler

Texas Tech UniversityLoading

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MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Principle Stresses Under a Given Loading

IntroductionIntroduction

Principle Stresses in a Beam

Sample Problem 8 1Sample Problem 8.1

Sample Problem 8.2

D i f T i i Sh ftDesign of a Transmission Shaft

Sample Problem 8.3

Stresses Under Combined Loadings

Sample Problem 8.5

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MECHANICS OF MATERIALS

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Beer • Johnston • DeWolf

Introduction• In Chapter 1 and 2, you learned how to determine the normal stress due to

centric loads.

• In Chapter 3, you analyzed the distribution of shearing stresses in a circular member due to a twisting couple.

• In Chapter 4, you determined the normal stresses caused by bending couples.

• In Chapters 5 and 6, you evaluated the shearing stresses due to transverse loads.

• In Chapter 7, you learned how the components of stress are transformed by a rotation of the coordinate axes and how to determine the principal planes, principal stresses, and maximum shearing stress at a point.

• In Chapter 8, you will learn how to determine the stress in a structural member or machine element due to a combination of loads and how to find the

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corresponding principal stresses and maximum shearing stress.

MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Principle Stresses in a Beam

• Prismatic beam subjected to transverse loadingg

VQVQIMc

IMy

mx =−= σσ

ItVQ

ItVQ

mxy =−= ττ

P i i l t d t i d f th d• Principal stresses determined from methods of Chapter 7

• Can the maximum normal stress within the cross-section be larger than

MIMc

m =σ

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MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Principle Stresses in a Beam

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MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Principle Stresses in a Beam

• Cross-section shape results in large values of τxynear the surface where σ is also largenear the surface where σx is also large.

• σ may be greater than σ• σmax may be greater than σm

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MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Sample Problem 8.1

SOLUTION:

• Determine shear and bending moment in Section A-A’

• Calculate the normal stress at top surface and at flange-web junction.

A 160-kN force is applied at the end of a W200x52 rolled-steel beam.

• Evaluate the shear stress at flange-web junction.

Neglecting the effects of fillets and of stress concentrations, determine

• Calculate the principal stress at flange-web junction

whether the normal stresses satisfy a design specification that they be equal to or less than 150 MPa at

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qsection A-A’.

MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Sample Problem 8.1SOLUTION:

• Determine shear and bending moment in Section A-A’

( )( )kN160

m-kN60m375.0kN160 ==AVM

kN160=AV

• Calculate the normal stress at top surface• Calculate the normal stress at top surface and at flange-web junction.

mkN60 ⋅MA

mm490MPa2.117

m10512 36

=×

== −

y

SA

aσ

( )

MPa9.102mm103mm4.90MPa2.117

=

==cyσ b

ab σ

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MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Sample Problem 8.1• Evaluate shear stress at flange-web junction.

( ) mm106.2487.966.12204 33×=×=Q

( )( )( )

m106.248kN160

m106.24836

36

×==

×=−

−

QVAbτ ( )( )

MPa5.95m0079.0m107.52 46

=×

== −Itbτ

• Calculate the principal stress at flange-web junction

( )

( )5959.1029.102 22

2221

21

max

+⎞⎜⎛+

++= bbb τσσσ

( )

( )MPa 150MPa9.159

5.9522>=

+⎠

⎜⎝

+=

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Design specification is not satisfied.

MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Sample Problem 8.2

SOLUTION:

• Determine reactions at A and D.

• Determine maximum shear and

The overhanging beam supports a

bending moment from shear and bending moment diagrams.

The overhanging beam supports a uniformly distributed load and a concentrated load. Knowing that for h d f l d 24 k i Fi d i l

• Calculate required section modulus and select appropriate beam section.

the grade of steel to used σall = 24 ksi and τall = 14.5 ksi, select the wide-flange beam which should be used. • Find maximum shearing stress.

• Find maximum normal stress.

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MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Sample Problem 8.2SOLUTION:

• Determine reactions at A and D.

kips410kips590

=⇒=∑

=⇒=∑

AD

DARMRM

• Determine maximum shear and bending moment from shear and bending moment diagramsdiagrams.

kips43

kips 2.12withinkip4.239max=

=⋅=

V

VM

• Calculate required section modulus and select appropriate beam section

pmax

and select appropriate beam section.

in7.119ksi24

inkip24 3maxmin =

⋅==

all

MS

σ

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sectionbeam62select W21×

MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Sample Problem 8.2• Find maximum shearing stress.

Assuming uniform shearing stress in web,

ksi14.5ksi 12.5in 8.40

kips432

maxmax <===

webAVτ

• Find maximum normal stress.

ksi6.22inkip602873 3max =

⋅==a

Mσ

( ) ksi3.215.10

88.9ksi6.22

27in1 3

=== bab

a

cyσ

S

σ

ksii45.1in8.40kips 2.12

5.10

2b ===webAV

c

τ

( )ksi45.12

ksi3.212

ksi3.21 22

max +⎟⎠⎞

⎜⎝⎛+=σ

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ksi24ksi4.21 <=