editionfourthmechanics of materials beer • johnston • dewolf

MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Design of a Transmission Shaft

• If power is transferred to and from the shaft by gears or sprocket wheels, the y g p ,shaft is subjected to transverse loading as well as shear loading.

• Normal stresses due to transverse loads may be large and should be included in d t i ti f i h idetermination of maximum shearing stress.

• Shearing stresses due to transverse loads are usually small and contribution to maximum shear stresscontribution to maximum shear stress may be neglected.

© 2006 The McGraw-Hill Companies, Inc. All rights reserved. 8 - 1


FourthEdition


Design of a Transmission Shaft• At any section,

MMMI

Mczym +==σ 222where

JTcI

m

zym

=τ

• Maximum shearing stress,

( )22

22 TcMc ⎞

⎜⎛⎞

⎜⎛⎞

⎜⎛σ ( )2max

2 section,-crossannular or circular afor

22

JI

JTc

IMc

mm

=

⎠⎞

⎜⎝⎛+

⎠⎞

⎜⎝⎛=+

⎠⎞

⎜⎝⎛= τστ

22max TM

Jc

+=τ

• Shaft section requirement,

TMJ max22 ⎟⎠⎞⎜

⎝⎛ +

⎞⎜⎛


allc τmax

min

⎠⎝=⎠⎞

⎜⎝⎛


FourthEdition


Sample Problem 8.3

SOLUTION:

• Determine the gear torques and corresponding tangential forces.

i d i d• Find reactions at A and B.

• Identify critical shaft section from d b di di

Solid shaft rotates at 480 rpm and transmits 30 kW from the motor to

torque and bending moment diagrams.

• Calculate minimum allowable shaft ditransmits 30 kW from the motor to

gears G and H; 20 kW is taken off at gear G and 10 kW at gear H. Knowing that σ = 50 MPa determine the

diameter.

that σall = 50 MPa, determine the smallest permissible diameter for the shaft.



FourthEdition


Sample Problem 8.3SOLUTION:

• Determine the gear torques and corresponding tangential forces.

( ) mN597Hz82

kW302

⋅===E fPT

ππ ( )

kN73.3m0.16

mN597Hz822

=⋅

==E

EE r

TF

f ππ

( )kW10

kN63.6mN398Hz82

kW20=⋅== CC FT

π

( ) kN49.2mN199Hz82

kW10=⋅== DD FT

π

• Find reactions at A and BFind reactions at A and B.

kN90.2kN80.2

kN22.6kN932.0

==

==

zy

zy

BB

AA


zy


FourthEdition


Sample Problem 8.3• Identify critical shaft section from torque and

bending moment diagrams.

( )mN1357

5973731160 222max

222

⋅=

++=++ TMM zy



FourthEdition


Sample Problem 8.3• Calculate minimum allowable shaft diameter.

222 ++ TMMJ

36m101427mN 1357 −×=⋅

=

++=

all

zy TMMcJ

τ

m1014.27MPa50

×==

For a solid circular shaft,

m1014.272

363 ×== −ccJ π

For a solid circular shaft,

mm7.512 == cd

m25.85m02585.0 ==c



FourthEdition


Stresses Under Combined Loadings

• Wish to determine stresses in slender structural members subjected to jarbitrary loadings.

• Pass section through points of interest. g pDetermine force-couple system at centroid of section required to maintain equilibriumequilibrium.

• System of internal forces consist of three force components and threethree force components and three couple vectors.

D t i t di t ib ti b• Determine stress distribution by applying the superposition principle.



FourthEdition



• Axial force and in-plane couple vectors contribute to normal stress distribution in the section.

• Shear force components and twisting• Shear force components and twisting couple contribute to shearing stress distribution in the section.



FourthEdition



• Normal and shearing stresses are used to determine principal stresses, maximum p p ,shearing stress and orientation of principal planes.

• Analysis is valid only to extent that conditions of applicability of superpositionconditions of applicability of superposition principle and Saint-Venant’s principle are met.



FourthEdition


Sample Problem 8.5

SOLUTION:

• Determine internal forces in Section EFG.

• Evaluate shearing stress at H

• Evaluate normal stress at H.

• Calculate principal stresses and maximum shearing stress

• Evaluate shearing stress at H.

Three forces are applied to a short steel post as shown. Determine the principle stresses, principal planes and

maximum shearing stress. Determine principal planes.

principle stresses, principal planes and maximum shearing stress at point H.



FourthEdition


Sample Problem 8.5SOLUTION:

• Determine internal forces in Section EFG.

( )( ) ( )( )m200.0kN75m130.0kN50

kN75kN50kN 30

−=

−==−=

x

zx

M

VPV

( )( ) mkN3m100.0kN300

mkN5.8

⋅===

⋅−=

zy MM

Note: Section properties,

( )( ) 23106514000400 −A ( )( )

( )( ) 463121

23

m1015.9m140.0m040.0

m106.5m140.0m040.0−×==

×==

xI

A

( )( ) 463121 m10747.0m040.0m140.0 −×==zI



FourthEdition


Sample Problem 8.5• Evaluate normal stress at H.

−++= xzy

bMaMPσ

( )( )m107470

m020.0mkN3m105 6

kN504623- ×

⋅+

×=

++

−

xzy IIA

σ

( )( )m1015.9

m025.0mkN5.8m10747.0m105.6

46×

⋅−

××

−

( ) MPa66.0MPa2.233.8093.8 =−+=

• Evaluate shearing stress at H.g

( )( )[ ]( )

m10585

m0475.0m045.0m040.036

11

×=

==

−

yAQ

( )( )( )( )m040.0m1015.9

m105.85kN75

m105.85

46

36

×

×==

×

−

−

tIQV

x

zyzτ


( )( )MPa52.17

0 0.005.9

=

x


FourthEdition


Sample Problem 8.5• Calculate principal stresses and maximum

shearing stress. i i i l lDetermine principal planes.

=+== MPa4.3752.170.33 22max Rτ

−=−=−=

=+=+=

MPa4.74.370.33

MPa4.704.370.33

min

max

ROC

ROC

σ

σ

°=

°===

9813

96.2720.33

52.172tan pp

p

CDCY

θ

θθ

98.13pθ

=

MP470

MPa4.37maxτ

°

−=

=

9813

MPa4.7

MPa4.70

min

max

θ

σ

σ


°= 98.13pθ

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