© 2009 The McGraw-Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Ed

ition

Beer • Johnston • DeWolf • Mazurek

4- 1

Sample Problem 4.2SOLUTION:

• Based on the cross section geometry, calculate the location of the section centroid and moment of inertia.

2dAIIA

AyY x

• Apply the elastic flexural formula to find the maximum tensile and compressive stresses.

I

Mcm

• Calculate the curvature

EI

M

1

A cast-iron machine part is acted upon by a 3 kN-m couple. Knowing E = 165 GPa and neglecting the effects of fillets, determine (a) the maximum tensile and compressive stresses, (b) the radius of curvature.

© 2009 The McGraw-Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Ed

ition

Beer • Johnston • DeWolf • Mazurek

4- 2

Sample Problem 4.2SOLUTION:

Based on the cross section geometry, calculate the location of the section centroid and moment of inertia.

mm 383000

10114 3

A

AyY

3

3

3

32

101143000

104220120030402

109050180090201

mm ,mm ,mm Area,

AyA

Ayy

49-43

2312123

121

231212

m10868 mm10868

18120040301218002090

I

dAbhdAIIx

© 2009 The McGraw-Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Ed

ition

Beer • Johnston • DeWolf • Mazurek

4- 3

Sample Problem 4.2

• Apply the elastic flexural formula to find the maximum tensile and compressive stresses.

49

49

m10868

m038.0mkN 3m10868

m022.0mkN 3

I

cM

I

cMI

Mc

BB

AA

m

MPa 0.76A

MPa 3.131B

• Calculate the curvature

49- m10868GPa 165

mkN 3

1

EI

M

m 7.47

m1095.201 1-3

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MECHANICS OF MATERIALS

Fifth

Ed

ition

Beer • Johnston • DeWolf • Mazurek

4- 4

Bending of Members Made of Several Materials

• Normal strain varies linearly.

y

x

• Piecewise linear normal stress variation.

yE

EyE

E xx2

221

11

Neutral axis does not pass through section centroid of composite section.

• Elemental forces on the section are

dAyE

dAdFdAyE

dAdF

222

111

• Consider a composite beam formed from two materials with E1 and E2.

xx

x

nI

My

21

1

2112 E

EndAn

yEdA

ynEdF

• Define a transformed section such that

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MECHANICS OF MATERIALS

Fifth

Ed

ition

Beer • Johnston • DeWolf • Mazurek

4- 5

Example 4.03

SOLUTION:

• Transform the bar to an equivalent cross section made entirely of brass

• Evaluate the cross sectional properties of the transformed section

• Calculate the maximum stress in the transformed section. This is the correct maximum stress for the brass pieces of the bar.

• Determine the maximum stress in the steel portion of the bar by multiplying the maximum stress for the transformed section by the ratio of the moduli of elasticity.

Bar is made from bonded pieces of steel (Es = 200 GPa) and brass (Eb = 100 GPa). Determine the maximum stress in the steel and brass when a moment of 4.5 KNm is applied.

© 2009 The McGraw-Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Ed

ition

Beer • Johnston • DeWolf • Mazurek

4- 6

Example 4.03

• Evaluate the transformed cross sectional properties

46

3

1213

121

mm 1096875.1

mm 75mm 56

hbI T

SOLUTION:

• Transform the bar to an equivalent cross section made entirely of brass.

mm 56mm 10mm 182mm 10

0.2GPa100

GPa200

T

b

s

b

E

En

• Calculate the maximum stresses

MPa 7.85

m 101.96875

m .03750Nm 450046-

I

Mcm

MPa 7.852max

max

ms

mb

n

MPa .4711

MPa 7.85

max

max

s

b

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MECHANICS OF MATERIALS

Fifth

Ed

ition

Beer • Johnston • DeWolf • Mazurek

4- 7

Reinforced Concrete Beams• Concrete beams subjected to bending moments are

reinforced by steel rods.

• To determine the location of the neutral axis,

0

022

21

dAnxAnxb

xdAnx

bx

ss

s

• The steel rods carry the entire tensile load below the neutral surface. The upper part of the concrete beam carries the compressive load.

• In the transformed section, the cross sectional area of the steel, As, is replaced by the equivalent areanAs where n = Es/Ec.

• The normal stress in the concrete and steel

xsxc

x

nI

My

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MECHANICS OF MATERIALS

Fifth

Ed

ition

Beer • Johnston • DeWolf • Mazurek

4- 8

Sample Problem 4.4

SOLUTION:

• Transform to a section made entirely of concrete.

• Evaluate geometric properties of transformed section.

• Calculate the maximum stresses in the concrete and steel.

A concrete floor slab is reinforced with 16-mm-diameter steel rods. The modulus of elasticity is 200 GPa for steel and 25 GPa for concrete. With an applied bending moment of 4.5 kNm for 0.3 m width of the slab, determine the maximum stress in the concrete and steel.

© 2009 The McGraw-Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Ed

ition

Beer • Johnston • DeWolf • Mazurek

4- 9

Sample Problem 4.4

• Evaluate the geometric properties of the transformed section.

46223

31 mm108.12mm2.63mm3216mm8.36mm300

mm8.36010032162

300

I

xxx

x

SOLUTION:

• Transform to a section made entirely of concrete.

22

4 mm3216mm1620.8

0.8GPa 52

GPa 002

s

c

s

nA

E

En

• Calculate the maximum stresses.

46-2

46-1

m108.21

m0632.0Nm45000.8

m108.21

m0368.0Nm4500

I

Mcn

I

Mc

s

c

MPa9.12c

MPa8.177s

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MECHANICS OF MATERIALS

Fifth

Ed

ition

Beer • Johnston • DeWolf • Mazurek

4- 10

Stress Concentrations

Stress concentrations may occur:

• in the vicinity of points where the loads are applied

I

McKm

• in the vicinity of abrupt changes in cross section

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MECHANICS OF MATERIALS

Fifth

Ed

ition

Beer • Johnston • DeWolf • Mazurek

4- 11

Plastic Deformations• For any member subjected to pure bending

mx c

y strain varies linearly across the section

• If the member is made of a linearly elastic material, the neutral axis passes through the section centroid

I

Myx and

• For a member with vertical and horizontal planes of symmetry and a material with the same tensile and compressive stress-strain relationship, the neutral axis is located at the section centroid and the stress-strain relationship may be used to map the strain distribution from the stress distribution.

• For a material with a nonlinear stress-strain curve, the neutral axis location is found by satisfying

dAyMdAF xxx 0

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MECHANICS OF MATERIALS

Fifth

Ed

ition

Beer • Johnston • DeWolf • Mazurek

4- 12

Plastic Deformations• When the maximum stress is equal to the ultimate

strength of the material, failure occurs and the corresponding moment MU is referred to as the ultimate bending moment.

• RB may be used to determine MU of any member made of the same material and with the same cross sectional shape but different dimensions.

• The modulus of rupture in bending, RB, is found from an experimentally determined value of MU and a fictitious linear stress distribution.

I

cMR U

B

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MECHANICS OF MATERIALS

Fifth

Ed

ition

Beer • Johnston • DeWolf • Mazurek

4- 13

Members Made of an Elastoplastic Material• Rectangular beam made of an elastoplastic material

moment elastic maximum

YYYm

mYx

c

IM

I

Mc

• If the moment is increased beyond the maximum elastic moment, plastic zones develop around an elastic core.

thickness-half core elastic 12

2

31

23

Y

YY y

c

yMM

• In the limit as the moment is increased further, the elastic core thickness goes to zero, corresponding to a fully plastic deformation.

shape)section crosson only (dependsfactor shape

moment plastic 23

Y

p

Yp

M

Mk

MM