8 principal stresses-mechanics of materials

8/20/2019 8 Principal Stresses-Mechanics of materials

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MECHANICS OF

MATERIALS

Third Edition

Ferdinand P. Beer

E. Russell Johnston, Jr.

John T. e!ol"

Le#ture Notes$

J. !alt Oler

Te%as Te#h &ni'ersit(

CHAPTER

© 2002 The McGraw-Hill Companies, Inc. All rights

8Principle StressesUnder a Given

Loading


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© 2002 The McGraw-Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALST h i r d Beer ) Johnston ) e!ol"

Principle Stresses Under a Given Loading

Introduction

Principle Stresses in a Beam

Sample Problem 8.1

Sample Problem 8.2

Design of a Transmission Shaft

Sample Problem 8.3

Stresses nder !ombined "oadings

Sample Problem 8.#


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MECHANICS OF MATERIALST h i r d Beer ) Johnston ) e!ol"

Introduction

$ In !haps. 1 and 2% &ou learned ho' to determine the normal stress due

to centric loadsIn !hap. 3% &ou anal&(ed the distribution of shearing stresses in a

circular member due to a t'isting couple

In !hap. )% &ou determined the normal stresses caused b& bending

couples

In !haps. # and *% &ou e+aluated the shearing stresses due totrans+erse loads

In !hap. ,% &ou learned ho' the components of stress are transformed

b& a rotation of the coordinate aes and ho' to determine the

principal planes% principal stresses% and maimum shearing stress

at a point.

$ In !hapter 8% &ou 'ill learn ho' to determine the stress in a structural

member or machine element due to a combination of loads and

ho' to find the corresponding principal stresses and maimum

shearing stress


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MECHANICS OF MATERIA ST


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MECHANICS OF MATERIALSTh i r d Beer ) Johnston ) e!ol"


MECHANICS OF MATERIALST


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$ !ross-section shape results in large +alues of τ xy

near the surface 'here σ x is also large.

$ σ max ma& be greater than σ m



7/25© 2002 The McGraw-Hill Companies, Inc. All rights reserved.


Sample Problem !"

/ 1*0- force is applied at the end

of a 200#2 rolled-steel beam.

eglecting the effects of fillets and

of stress concentrations% determine

'hether the normal stresses satisf& a

design specification that the& be

e4ual to or less than 1#0 5Pa at

section A-A’.

S6"TI67$ Determine shear and bending

moment in Section A-A’

$ !alculate the normal stress at top

surface and at flange-'eb unction.

$ +aluate the shear stress at flange-

'eb unction.

$ !alculate the principal stress at

flange-'eb unction





Sample Problem !"

S6"TI67

$ Determine shear and bending moment inSection A-A’

( ) ( )

1*0

m-*0m3,#.01*0

===

A

A

V

M

$ !alculate the normal stress at top surface

and at flange-'eb unction.

( )

5Pa9.102

mm103

mm).905Pa2.11,

5Pa2.11,

m10#12

m*03*

=

===

×

⋅== −

c

yσ

S

M

bab

Aa

σ

σ





Sample Problem !"

$ +aluate shear stress at flange-'eb unction.

( )

( ) ( )( )( )

5Pa#.9#

m00,9.0m10,.#2

m10*.2)81*0

m10*.2)8

mm10*.2)8,.9**.1220)

)*

3*

3*

33

=

×

×==

×=×=×=

−

−

−

It

QV

Q

Abτ

$ !alculate the principal stress at

flange-'eb unction

( )

( )

( )5Pa1#05Pa9.1*9

#.9#2

9.102

2

9.102 22

22

21

21

ma

>=

+

+=

++= bbb τ σ σ σ

Design specification is not satisfied.





Sample Problem !#

The o+erhanging beam supports a

uniforml& distributed load and a

concentrated load. :no'ing that for

the grade of steel to used σ all ; 2) si

andτ all ; 1).# si% select the 'ide-

flange beam 'hich should be used.

S6"TI67

$ Determine reactions at A and D.

$




Sample Problem !#

$ !alculate re4uired section modulusand select appropriate beam section.

section beam*2select 21

in,.119si2)

inip2) 3mamin

×

=⋅

==all

M S

σ

S6"TI67

$ Determine reactions at A and D.

ips)10

ips#90

=⇒=∑

=⇒=∑

A D

D A

R M

R M

$ Determine maimum shear and bending

moment from shear and bending moment

diagrams.

ips)3

ips2.12'ithinip).239

ma

ma

=

=⋅=

V

V M





Sample Problem !#

$




$esign o% a Transmission S&a%t

$ If po'er is transferred to and from the

shaft b& gears or sprocet 'heels% theshaft is subected to trans+erse loading

as 'ell as shear loading.

$ ormal stresses due to trans+erse loads

ma& be large and should be included indetermination of maimum shearing

stress.

$ Shearing stresses due to trans+erse

loads are usuall& small andcontribution to maimum shear stress

ma& be neglected.




MECHANICS OF MATERIALSThi r d Beer ) Johnston ) e!ol"

$esign o% a Transmission S&a%t

$ /t an& section%

J

Tc

M M M I

Mc

m

z ym

=

+==

τ

σ

222

'here

$ 5aimum shearing stress%

( )

22ma

222

2

ma

2section%-crossannularorcircularafor

22

T M

J

c

J I

J

Tc

I

Mcm

m

+=

=

+

=+

=

τ

τ σ

τ

$ Shaft section re4uirement%

all

T M

c

J

τ

ma

22

min

+

=

MECHANICS OF MATERIALST h




Sample Problem !'

Solid shaft rotates at )80 rpm and

transmits 30 from the motor to

gears G and H = 20 is taen off at

gear G and 10 at gear H .:no'ing that σ all ; #0 5Pa% determine

the smallest permissible diameter for

the shaft.

S6"TI67

$ Determine the gear tor4ues and

corresponding tangential forces.

$




Sample Problem !'

S6"TI67

$ Determine the gear tor4ues and correspondingtangential forces.

( )

( )

( ))9.2m 199

>(802

10

*3.*m 398>(802

20

,3.3

m0.1*

m #9,

m #9,>(802

30

2

=⋅==

=⋅==

=⋅

==

⋅===

D D

C C

E

E E

E

T

T

!

T

"

# T

π

π

π π

$



MECHANICS OF MATERIALShi r d Beer ) Johnston ) e!ol"

Sample Problem !'$ Identif& critical shaft section from tor4ue and

bending moment diagrams.

( )

m 13#,

#9,3,311*0 222

ma

22

⋅=

++=

+T M





Sample Problem !'

$ !alculate minimum allo'able shaft diameter.

m2#.8#m02#8#.0

m101).2,2

shaft%circularsolida




Stresses Under Combined Loadings

$ ish to determine stresses in slender

structural members subected toarbitrar& loadings.

$ Pass section through points of interest.

Determine force-couple s&stem at

centroid of section re4uired to maintain

e4uilibrium.

$ S&stem of internal forces consist of

three force components and three

couple +ectors.

$ Determine stress distribution b&

appl&ing the superposition principle.


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Sample Problem !(

Three forces are applied to a short

steel post as sho'n. Determine the principle stresses% principal planes and

maimum shearing stress at point H.

S6"TI67

$ Determine internal forces in Section

EG.

$ !alculate principal stresses and

maimum shearing stress.

Determine principal planes.

$ +aluate shearing stress at H .

$ +aluate normal stress at H .





Sample Problem !(

S6"TI67

$ Determine internal forces in Section EG.

( ) ( ) ( ) ( )

( ) ( ) m3m100.0300

m#.8

m200.0,#m130.0#0

,##030

⋅===

⋅−=

−=

−==−=

z y

x

z x

M M

M

V # V

ote7 Section properties%

( ) ( )

( ) ( )

( ) ( ) )*3121

)*3121

23

m10,),.0m0)0.0m1)0.0

m101#.9m1)0.0m0)0.0

m10*.#m1)0.0m0)0.0

−

−

−

×==

×==

×==

z

x

I

I

A




MECHANICS OF MATERIALSi r d Beer ) Johnston ) e!ol" Sample Problem !(

$ +aluate normal stress at H .

( ) ( )

( ) ( )

( ) 5Pa**.05Pa2.233.8093.8

m101#.9

m02#.0m#.8

m10,),.0

m020.0m3

m10#.*

#0

)*

)*23-

=−+=×

⋅−

×

⋅+

×=

−++=

−

−

x x

z z y

I

b M

I

a M

A

# σ

$ +aluate shearing stress at H .

( ) ( )[ ]( )

( ) ( )( )( )

5Pa#2.1,

m0)0.0m101#.9

m10#.8#,#

m10#.8#

m0),#.0m0)#.0m0)0.0

)*

3*

3*

11

=

×

×==

×=

==

−

−

−

t I

QV

y AQ

x

z yz τ


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8 principal stresses-mechanics of materials

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