editionfourth mechanics of materials beer • johnston...

MECHANICS OF MATERIALS

FourthEdition

Beer • Johnston • DeWolf

Sample Problem 4.2SOLUTION:

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

( )∑ +∑ 2dAIIAyY ( )∑ +=∑∑= ′

2dAIIAyY x

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

Mc

A cast-iron machine part is acted upon by a 3 kN-m couple. Knowing E = 165

IMc

m =σ

• Calculate the curvatureby a 3 kN m couple. Knowing E 165 GPa and neglecting the effects of fillets, determine (a) the maximum tensile and compressive stresses (b)

Calculate the curvature

EIM

=ρ1

© 2006 The McGraw-Hill Companies, Inc. All rights reserved. 4 - 1

tensile and compressive stresses, (b) the radius of curvature.


FourthEdition


Sample Problem 4.2SOLUTION:

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

32 mmmmmmArea Ayy

×=××=×

3

3

3

104220120030402109050180090201

mm,mm,mm Area, Ayy

10114 3×∑ Ay

∑ ×==∑ 3101143000 AyA

mm383000

10114=

×=

∑∑=

AAyY

( ) ( )2312( ) ( )( ) ( )23

12123

121

231212

18120040301218002090 ×+×+×+×=

+=+= ∑∑′ dAbhdAIIx


49-43 m10868mm10868 ×=×=I


FourthEdition


Sample Problem 4.2

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

m022.0mkN 3 ×⋅

=

cMI

Mc

A

m

σ

σ

MPa076+=Aσ

49

49

m10868m038.0mkN 3

m10868

−

−

××⋅

−=−=

×==

IcM

I

BB

AA

σ

σ MPa0.76+Aσ

MPa3.131−=Bσm10868×I

• Calculate the curvature

mkN 3

1

⋅

=EIM

ρ

m1095201 1-3×= −

( )( )49- m10868GPa 165 ×=

m 7.47

m1095.20

=

×=

ρρ



FourthEdition


Bending of Members Made of Several Materials• Consider a composite beam formed from

two materials with E1 and E2.

• Normal strain varies linearly.

ρε y

x −=

• Piecewise linear normal stress variation.

ρεσ

ρεσ yEEyEE xx

222

111 −==−==

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

• Elemental forces on the section are

dAyEdAdFdAyEdAdFρ

σρ

σ 222

111 −==−==

ρρ

( ) ( ) 211 EdAyEdAynEdF

• Define a transformed section such thatx

nI

My

σσσσ

σ

==

−=


( ) ( )1

2112 E

ndAnydAydF =−=−=ρρ

xx nσσσσ == 21


FourthEdition


Example 4.03SOLUTION:

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

• Evaluate the cross sectional properties of• Evaluate the cross sectional properties of the transformed section

C l l t th i t i th• Calculate the maximum stress in the transformed section. This is the correct maximum stress for the brass pieces of

Bar is made from bonded pieces of steel (Es = 29x106 psi) and brass

the bar.

• Determine the maximum stress in the ( s p )(Eb = 15x106 psi). Determine the maximum stress in the steel and brass when a moment of 40 kip*in

steel portion of the bar by multiplying the maximum stress for the transformed section by the ratio of the moduli of


brass when a moment of 40 kip in is applied.

section by the ratio of the moduli of elasticity.


FourthEdition


Example 4.03SOLUTION:• Transform the bar to an equivalent cross section

made entirely of brassmade entirely of brass.

933.1psi1015psi1029

6

6=

×

×==

b

sEEn

• Evaluate the transformed cross sectional properties

in 25.2in 4.0in 75.0933.1in 4.0 =+×+=Tb

( )( )4

31213

121

in.063.5

in. 3in. 25.2

=

== hbI T

• Calculate the maximum stresses( )( ) ksi85.11in.5.1in.kip 40

4 =⋅

==Mc

mσ in.5.063 4Im

( )( ) k i85119331

max = mb σσ ( )( ) k i22 9

ksi 85.11max =bσ


( ) ksi85.11933.1max ×== ms nσσ ( ) ksi22.9max =sσ


FourthEdition


Reinforced Concrete Beams• Concrete beams subjected to bending moments are

reinforced by steel rods.

• 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 area

concrete beam carries the compressive load.

nAs where n = Es/Ec.

• To determine the location of the neutral axis,( ) ( ) 0dAxb( ) ( )

0

022

21 =−+

=−−

dAnxAnxb

xdAnbx

ss

s

• The normal stress in the concrete and steel

x IMyσ −=


xsxc nI

σσσσ ==


FourthEdition


Sample Problem 4.4

SOLUTION:

• Transform to a section made entirely• Transform to a section made entirely of concrete.

• Evaluate geometric properties of• Evaluate geometric properties of transformed section.

• Calculate the maximum stresses

A t fl l b i i f d ith

• Calculate the maximum stresses in the concrete and steel.

A concrete floor slab is reinforced with 5/8-in-diameter steel rods. The modulus of elasticity is 29x106psi for steel and 3.6x106psi for concrete. With an applied bending moment of 40 kip*in for 1-ft width of the slab, determine the maximum


stress in the concrete and steel.


FourthEdition


Sample Problem 4.4SOLUTION:• Transform to a section made entirely of concrete.

6

( ) 225

6

6

in954in2068

06.8psi 106.3psi1029

⎥⎤

⎢⎡×

=×

×==

π

c

s

nA

EEn

( )85

4 in95.4in206.8 =⎥⎦⎤

⎢⎣⎡×= π

snA

• Evaluate the geometric properties of the t f d titransformed section.

( )

( )in450.10495.4

212 ==−−⎟

⎠⎞

⎜⎝⎛ xxxx

( )( ) ( )( ) 422331 in4.44in55.2in95.4in45.1in12 =+=I

• Calculate the maximum stresses.

2

41

in55.2inkip40068

in44.4in1.45inkip40

×⋅

×⋅==

Mc

IMc

cσ ksi306.1=cσ

k i5218


42

in44.4in55.2inkip4006.8==

IMcnsσ ksi52.18=sσ


FourthEdition


Stress Concentrations

Stress concentrations may occur:

• in the vicinity of points where theI

McKm =σ

in the vicinity of points where the loads are applied

• in the vicinity of abrupt changes


y p gin cross section


FourthEdition


Plastic Deformations• For any member subjected to pure bending

mx cy εε −= strain varies linearly across the sectionc

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

IMy

x −=σand

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

∫ −=∫ == dAyMdAF xxx σσ 0

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


strain relationship may be used to map the strain distribution from the stress distribution.


FourthEdition


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

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

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

IcMR U

B =

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



FourthEdition


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

=≤ mYx IMcσσσ

moment elastic maximum === YYYm cIM σσσ

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

thickness-half core elastic 1 2

2

31

23 =⎟

⎟⎠

⎞⎜⎜⎝

⎛−= Y

YY y

cyMM

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

moment plastic 23 ==

p

Yp

M

MM


shape)section crosson only (dependsfactor shape==Y

pM

k

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