Lecture7 Pure Bending

Pure

Ben

ding

4 -3

Pur

e B

endi

ng

Pure

Ben

ding

: Pr

ism

atic

mem

bers

su

bjec

ted

to e

qual

and

opp

osite

cou

ples

ac

ting

in th

e sa

me

long

itudi

nal p

lane

4 -4

Oth

er L

oadi

ng T

ypes

•Pr

inci

ple

of S

uper

posi

tion:

The

nor

mal

st

ress

due

to p

ure

bend

ing

may

be

com

bine

d w

ith th

e no

rmal

stre

ss d

ue to

ax

ial l

oadi

ng a

nd sh

ear s

tress

due

to

shea

r loa

ding

to fi

nd th

e co

mpl

ete

stat

e of

stre

ss.

•Ec

cent

ric

Load

ing:

Axi

al lo

adin

g w

hich

do

es n

ot p

ass t

hrou

gh se

ctio

n ce

ntro

id

prod

uces

inte

rnal

forc

es e

quiv

alen

t to

an

axia

l for

ce a

nd a

cou

ple

•Tr

ansv

erse

Loa

ding

: C

once

ntra

ted

or

dist

ribut

ed tr

ansv

erse

load

pro

duce

s in

tern

al fo

rces

equ

ival

ent t

o a

shea

r fo

rce

and

a co

uple

4 -5

Sym

met

ric M

embe

r in

Pur

e B

endi

ng

MdA

yM

dAz

MdA

F

xz

xy

xx

00

•Th

ese

requ

irem

ents

may

be

appl

ied

to th

e su

ms

of th

e co

mpo

nent

s and

mom

ents

of t

he st

atic

ally

in

dete

rmin

ate

elem

enta

ry in

tern

al fo

rces

.

•In

tern

al fo

rces

in a

ny c

ross

sect

ion

are

equi

vale

nt

to a

cou

ple.

The

mom

ent o

f the

cou

ple

is th

e se

ctio

nbe

ndin

g m

omen

t.•

From

stat

ics,

a co

uple

M c

onsi

sts o

f tw

o eq

ual

and

oppo

site

forc

es.

•Th

e su

m o

f the

com

pone

nts o

f the

forc

es in

any

di

rect

ion

is z

ero.

•Th

e m

omen

t is t

he sa

me

abou

t any

axi

s pe

rpen

dicu

lar t

o th

e pl

ane

of th

e co

uple

and

ze

ro a

bout

any

axi

s con

tain

ed in

the

plan

e.

4 -6

Ben

ding

Def

orm

atio

ns Bea

m w

ith a

pla

ne o

f sym

met

ry in

pur

e be

ndin

g:•

mem

ber r

emai

ns sy

mm

etric

•be

nds u

nifo

rmly

to fo

rm a

circ

ular

arc

•cr

oss-

sect

iona

l pla

ne p

asse

s thr

ough

arc

cen

ter

and

rem

ains

pla

nar

•le

ngth

of t

op d

ecre

ases

and

leng

th o

f bot

tom

in

crea

ses

•a

neut

ral s

urfa

cem

ust e

xist

that

is p

aral

lel t

o th

e up

per a

nd lo

wer

surf

aces

and

for w

hich

the

leng

th

does

not

cha

nge

•st

ress

es a

nd st

rain

s are

neg

ativ

e (c

ompr

essi

ve)

abov

e th

e ne

utra

l pla

ne a

nd p

ositi

ve (t

ensi

on)

belo

w it

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -7

Stra

in D

ue to

Ben

ding

Con

side

r a b

eam

segm

ent o

f len

gth

L.

Afte

r def

orm

atio

n, th

e le

ngth

of t

he n

eutra

lsu

rfac

e re

mai

ns L

. A

t oth

er se

ctio

ns,

mx

mmx

cy

cc

yy

L

yy

LL

yL

or

linea

rly)

ries

(stra

in v

a

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -8

Stre

ss D

ue to

Ben

ding

•Fo

r a li

near

ly e

last

ic m

ater

ial,

linea

rly)

var

ies

(stre

ssm

mx

x

cy

Ecy

E

•Fo

r sta

tic e

quili

briu

m,

dAy

c

dAcy

dAF

m

mx

x 0

0

Firs

t mom

ent w

ith re

spec

t to

neut

ral

plan

e is

zer

o. T

here

fore

, the

neu

tral

surf

ace

mus

t pas

s thr

ough

the

sect

ion

cent

roid

.

•Fo

r sta

tic e

quili

briu

m,

IMy

cySM

IMc

cI

dAy

cM

dAcy

ydA

yM

x

mx

m

mm

mx

ngSu

bstit

uti

2

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -9

Bea

m S

ectio

n P

rope

rties

•Th

e m

axim

um n

orm

al st

ress

due

to b

endi

ng,

mod

ulus

sect

ion

iner

tiaof

mom

ent

sect

ion

cISI

SMIMc

m

A b

eam

sect

ion

with

a la

rger

sect

ion

mod

ulus

w

ill h

ave

a lo

wer

max

imum

stre

ss

•C

onsi

der a

rect

angu

lar b

eam

cro

ss se

ctio

n,

Ahbh

hbhcI

S61

361

3121

2

Bet

wee

n tw

o be

ams w

ith th

e sa

me

cros

s se

ctio

nal a

rea,

the

beam

with

the

grea

ter d

epth

w

ill b

e m

ore

effe

ctiv

e in

resi

stin

g be

ndin

g.

•St

ruct

ural

stee

l bea

ms a

re d

esig

ned

to h

ave

a la

rge

sect

ion

mod

ulus

.

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -1

0

Pro

perti

es o

f Am

eric

an S

tand

ard

Sha

pes

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -1

1

Def

orm

atio

ns in

a T

rans

vers

e C

ross

Sec

tion

•D

efor

mat

ion

due

to b

endi

ng m

omen

t M is

quan

tifie

d by

the

curv

atur

e of

the

neut

ral s

urfa

ce

EIMIMc

EcEc

cm

m1

1

•A

lthou

gh c

ross

sect

iona

l pla

nes r

emai

n pl

anar

whe

n su

bjec

ted

to b

endi

ng m

omen

ts, i

n-pl

ane

defo

rmat

ions

are

non

zero

,y

yx

zx

y

•Ex

pans

ion

abov

e th

e ne

utra

l sur

face

and

co

ntra

ctio

n be

low

it c

ause

an

in-p

lane

cur

vatu

re,

curv

atur

ec

antic

last

i1

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -1

2

Sam

ple

Pro

blem

4.2

A c

ast-i

ron

mac

hine

par

t is a

cted

upo

n by

a 3

kN

-m c

oupl

e. K

now

ing

E=

165

GPa

and

neg

lect

ing

the

effe

cts o

f fil

lets

, det

erm

ine

(a) t

he m

axim

um

tens

ile a

nd c

ompr

essi

ve st

ress

es, (

b)

the

radi

us o

f cur

vatu

re.

SOLU

TIO

N:

•B

ased

on

the

cros

s sec

tion

geom

etry

,ca

lcul

ate

the

loca

tion

of th

e se

ctio

n ce

ntro

id a

nd m

omen

t of i

nerti

a.2 d

AI

IAAy

Yx

•A

pply

the

elas

tic fl

exur

al fo

rmul

a to

fin

d th

e m

axim

um te

nsile

and

co

mpr

essi

ve st

ress

es.

IMc

m

•C

alcu

late

the

curv

atur

e

EIM1

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -1

3

Sam

ple

Pro

blem

4.2

SOLU

TIO

N:

Bas

ed o

n th

e cr

oss s

ectio

n ge

omet

ry, c

alcu

late

th

e lo

catio

n of

the

sect

ion

cent

roid

and

m

omen

t of i

nerti

a.

33332

1011

430

0010

4220

1200

3040

210

9050

1800

9020

1m

m,

mm

,m

mA

rea,

AyA

Ayy

mm

3830

001011

43

AAyY

49-

3

23

1212

3121

23

1212

m10

868

mm

1086

8

1812

0040

3012

1800

2090

I

dA

bhd

AI

I x

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -1

4

Sam

ple

Pro

blem

4.2

•A

pply

the

elas

tic fl

exur

al fo

rmul

a to

find

the

max

imum

tens

ile a

nd c

ompr

essi

ve st

ress

es.

49

49

mm

1086

8m

038

.0m

kN3m

m10

868

m02

2.0

mkN3

IcMIc

MIMc

BB

AAm

MPa

0.76

A

MPa

3.13

1B

•C

alcu

late

the

curv

atur

e

49-

m10

868

GPa

165

mkN3

1EIM

m7.

47

m10

95.20

11-

3

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -1

5

Ben

ding

of M

embe

rs M

ade

of S

ever

al M

ater

ials

•C

onsi

der a

com

posi

te b

eam

form

ed fr

om

two

mat

eria

ls w

ith E

1an

dE 2

.

•N

orm

al st

rain

var

ies l

inea

rly.

yx

•Pi

ecew

ise

linea

r nor

mal

stre

ss v

aria

tion.

yE

Ey

EE

xx

22

21

11 Neu

tral a

xis d

oes n

ot p

ass t

hrou

gh

sect

ion

cent

roid

of c

ompo

site

sect

ion.

•El

emen

tal f

orce

s on

the

sect

ion

are

dAyE

dAdF

dAy

EdA

dF2

22

11

1

121

12

EEn

dAny

EdAy

nEdF

•D

efin

e a

trans

form

ed se

ctio

n su

ch th

at

xx

x

nIMy

21

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -1

6

Exa

mpl

e 4.

03

Bar

is m

ade

from

bon

ded

piec

es o

f st

eel (

E s=

29x1

06ps

i) an

d br

ass

(Eb

= 15

x106

psi).

Det

erm

ine

the

max

imum

stre

ss in

the

stee

l and

br

ass w

hen

a m

omen

t of 4

0 ki

p*in

is

app

lied.

SOLU

TIO

N:

•Tr

ansf

orm

the

bar t

o an

equ

ival

ent c

ross

se

ctio

n m

ade

entir

ely

of b

rass

•Ev

alua

te th

e cr

oss s

ectio

nal p

rope

rties

of

the

trans

form

ed se

ctio

n

•C

alcu

late

the

max

imum

stre

ss in

the

trans

form

ed se

ctio

n. T

his i

s the

cor

rect

m

axim

um st

ress

for t

he b

rass

pie

ces o

f th

e ba

r.

•D

eter

min

e th

e m

axim

um st

ress

in th

e st

eel p

ortio

n of

the

bar b

y m

ultip

lyin

g th

e m

axim

um st

ress

for t

he tr

ansf

orm

ed

sect

ion

by th

e ra

tio o

f the

mod

uli o

f el

astic

ity.

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -1

7

Exa

mpl

e 4.

03

•Ev

alua

te th

e tra

nsfo

rmed

cro

ss se

ctio

nal p

rope

rties

4

3121

3121

in06

3.5

in3in

.25.2

hb

IT

SOLU

TIO

N:

•Tr

ansf

orm

the

bar t

o an

equ

ival

ent c

ross

sect

ion

mad

e en

tirel

y of

bra

ss.

in25.2

in4.0in

75.093

3.1

in4.0

933

.1ps

i10

15ps

i10

2966

T

bs

b

EEn

•C

alcu

late

the

max

imum

stre

sses

ksi

85.11

in5.

063

in5.1in

kip

404

IMc

m

ksi

85.11

933

.1m

ax

max

ms

mb

nks

i22

.9

ksi

85.11

max

max

sb

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -1

8

Rei

nfor

ced

Con

cret

e B

eam

s•

Con

cret

e be

ams s

ubje

cted

to b

endi

ng m

omen

ts a

re

rein

forc

ed b

y st

eel r

ods.

•Th

e st

eel r

ods c

arry

the

entir

e te

nsile

load

bel

ow

the

neut

ral s

urfa

ce.

The

uppe

r par

t of t

he

conc

rete

bea

m c

arrie

s the

com

pres

sive

load

.

•In

the

trans

form

ed se

ctio

n, th

e cr

oss s

ectio

nal a

rea

of th

e st

eel,

A s, i

s rep

lace

d by

the

equi

vale

nt a

rea

nAs

whe

ren

= E

s/Ec.

•To

det

erm

ine

the

loca

tion

of th

e ne

utra

l axi

s,

0

02 2

21d

An

xA

nx

b

xd

An

xbx

sss

•Th

e no

rmal

stre

ss in

the

conc

rete

and

stee

l

xs

xcx

nIMy

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -1

9

Sam

ple

Pro

blem

4.4

A c

oncr

ete

floor

slab

is re

info

rced

with

5/

8-in

-dia

met

er st

eel r

ods.

The

mod

ulus

of

ela

stic

ity is

29x

106p

si fo

r ste

el a

nd

3.6x

106p

si fo

r con

cret

e. W

ith a

n ap

plie

d be

ndin

g m

omen

t of 4

0 ki

p*in

for 1

-ft

wid

th o

f the

slab

, det

erm

ine

the

max

imum

st

ress

in th

e co

ncre

te a

nd st

eel.

SOLU

TIO

N:

•Tr

ansf

orm

to a

sect

ion

mad

e en

tirel

yof

con

cret

e.

•Ev

alua

te g

eom

etric

pro

perti

es o

f tra

nsfo

rmed

sect

ion.

•C

alcu

late

the

max

imum

stre

sses

in

the

conc

rete

and

stee

l.

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -2

0

Sam

ple

Pro

blem

4.4 SO

LUTI

ON

:•

Tran

sfor

m to

a se

ctio

n m

ade

entir

ely

of c

oncr

ete.

22

854

66

in95.4

in2

06.8

06.8ps

i10

6.3ps

i10

29

s

cs

nA

EEn

•Ev

alua

te th

e ge

omet

ric p

rope

rties

of t

he

trans

form

ed se

ctio

n.

42

23

31in4.

44in

55.2in

95.4in

45.1in

12

in45

0.1

04

95.42

12 I

xx

xx

•C

alcu

late

the

max

imum

stre

sses

.

42

41

in44

.4in

55.2in

kip

4006.8

in44

.4in

1.45

inki

p40

IM

cn

IMc

scks

i30

6.1

c

ksi

52.18

s

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -2

1

Stre

ss C

once

ntra

tions

Stre

ss c

once

ntra

tions

may

occ

ur:

•in

the

vici

nity

of p

oint

s whe

re th

e lo

ads a

re a

pplie

d

IMc

Km

•in

the

vici

nity

of a

brup

t cha

nges

in

cro

ss se

ctio

n

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -2

2

Pla

stic

Def

orm

atio

ns •Fo

r any

mem

ber s

ubje

cted

to p

ure

bend

ing

mx

cyst

rain

var

ies l

inea

rly a

cros

s the

sect

ion

•If

the

mem

ber i

s mad

e of

a li

near

ly e

last

ic m

ater

ial,

the

neut

ral a

xis p

asse

s thr

ough

the

sect

ion

cent

roid

IMy

xan

d

•Fo

r a m

ater

ial w

ith a

non

linea

r stre

ss-s

train

cur

ve,

the

neut

ral a

xis l

ocat

ion

is fo

und

by sa

tisfy

ing

dAy

MdA

Fx

xx

0

•Fo

r a m

embe

r with

ver

tical

and

hor

izon

tal p

lane

s of

sym

met

ry a

nd a

mat

eria

l with

the

sam

e te

nsile

and

co

mpr

essi

ve st

ress

-stra

in re

latio

nshi

p, th

e ne

utra

l ax

is is

loca

ted

at th

e se

ctio

n ce

ntro

id a

nd th

e st

ress

-st

rain

rela

tions

hip

may

be

used

to m

ap th

e st

rain

di

strib

utio

n fr

om th

e st

ress

dis

tribu

tion.

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -2

3

Pla

stic

Def

orm

atio

ns •W

hen

the

max

imum

stre

ss is

equ

al to

the

ultim

ate

stre

ngth

of t

he m

ater

ial,

failu

re o

ccur

s and

the

corr

espo

ndin

g m

omen

t MU

is re

ferr

ed to

as t

he

ultim

ate

bend

ing

mom

ent.

•Th

em

odul

us o

f rup

ture

in b

endi

ng, R

B, is

foun

d fr

om a

n ex

perim

enta

lly d

eter

min

ed v

alue

of M

U

and

a fic

titio

us li

near

stre

ss d

istri

butio

n.

Ic

MR

UB

•R B

may

be

used

to d

eter

min

e M

Uof

any

mem

ber

mad

e of

the

sam

e m

ater

ial a

nd w

ith th

e sa

me

cros

s sec

tiona

l sha

pe b

ut d

iffer

ent d

imen

sion

s.

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -2

4

Mem

bers

Mad

e of

an

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stop

last

ic M

ater

ial

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ecta

ngul

ar b

eam

mad

e of

an

elas

topl

astic

mat

eria

l

mom

ent

elas

ticm

axim

umY

YY

m

mY

x

cIM

IMc

•If

the

mom

ent i

s inc

reas

ed b

eyon

d th

e m

axim

um

elas

tic m

omen

t, pl

astic

zon

es d

evel

op a

roun

d an

el

astic

cor

e.

thic

knes

s-

half

core

elas

tic1

22

3123

YY

Yy

cyM

M

•In

the

limit

as th

e m

omen

t is i

ncre

ased

furth

er, t

he

elas

tic c

ore

thic

knes

s goe

s to

zero

, cor

resp

ondi

ng to

a

fully

pla

stic

def

orm

atio

n.

shap

e)se

ctio

ncr

oss

onon

ly(d

epen

dsfa

ctor

shap

e

mom

ent

plas

tic23 Yp

Yp

MMk

MM

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -2

5

Pla

stic

Def

orm

atio

ns o

f Mem

bers

With

a

Sin

gle

Pla

ne o

f Sym

met

ry•

Fully

pla

stic

def

orm

atio

n of

a b

eam

with

onl

y a

verti

cal p

lane

of s

ymm

etry

.

•R

esul

tant

sR1

and

R 2of

the

elem

enta

ry

com

pres

sive

and

tens

ile fo

rces

form

a c

oupl

e.

YY

AA

RR

21

21

The

neut

ral a

xis d

ivid

es th

e se

ctio

n in

to e

qual

ar

eas.

•Th

e pl

astic

mom

ent f

or th

e m

embe

r,

dA

MY

p21

•Th

e ne

utra

l axi

s can

not b

e as

sum

ed to

pas

s th

roug

h th

e se

ctio

n ce

ntro

id.

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -2

6

Res

idua

l Stre

sses

•Pl

astic

zon

es d

evel

op in

a m

embe

r mad

e of

an

elas

topl

astic

mat

eria

l if t

he b

endi

ng m

omen

t is

larg

e en

ough

.

•Si

nce

the

linea

r rel

atio

n be

twee

n no

rmal

stre

ss a

nd

stra

in a

pplie

s at a

ll po

ints

dur

ing

the

unlo

adin

g ph

ase,

it m

ay b

e ha

ndle

d by

ass

umin

g th

e m

embe

r to

be

fully

ela

stic

.

•R

esid

ual s

tress

es a

re o

btai

ned

by a

pply

ing

the

prin

cipl

e of

supe

rpos

ition

to c

ombi

ne th

e st

ress

es

due

to lo

adin

g w

ith a

mom

ent M

(ela

stop

last

icde

form

atio

n) a

nd u

nloa

ding

with

a m

omen

t -M

(ela

stic

def

orm

atio

n).

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e fin

al v

alue

of s

tress

at a

poi

nt w

ill n

ot, i

n ge

nera

l, be

zer

o.

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -2

7

Exa

mpl

e 4.

05, 4

.06 A

mem

ber o

f uni

form

rect

angu

lar c

ross

sect

ion

is

subj

ecte

d to

a b

endi

ng m

omen

t M =

36.

8 kN

-m.

The

mem

ber i

s mad

e of

an

elas

topl

astic

mat

eria

l w

ith a

yie

ld st

reng

th o

f 240

MPa

and

a m

odul

us

of e

last

icity

of 2

00 G

Pa.

Det

erm

ine

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he th

ickn

ess o

f the

ela

stic

cor

e, (b

) th

e ra

dius

of c

urva

ture

of t

he n

eutra

l sur

face

.

Afte

r the

load

ing

has b

een

redu

ced

back

to z

ero,

de

term

ine

(c) t

he d

istri

butio

n of

resi

dual

stre

sses

, (d

) rad

ius o

f cur

vatu

re.

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -2

8

Exa

mpl

e 4.

05, 4

.06

•Th

ickn

ess o

f ela

stic

cor

e:

666

.0m

m60

1m

kN28

.8m

kN8.36

1

22

3123

22

3123

YY

Y

YY y

cycy

cyM

M

mm

802

Yy

mkN

8.28

MPa

240

m10

120m

1012

0

1060

1050

36

36

23

332

232

YY

cIM

mm

bccI

•M

axim

um e

last

ic m

omen

t:•

Rad

ius o

f cur

vatu

re: 33

3

96

102.1

m10

40102.1

Pa10

200

Pa10

240

YYYY

YY

yyE

m3.33

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -2

9

Exa

mpl

e 4.

05, 4

.06

•M

= 36

.8 k

N-m

MPa

240m

m40

YYy

•M

= -3

6.8

kN-m

Y36 2

MPa

7.30

6m

1012

0m

kN8.36

IMc

m

•M

= 0

6

3

6

96

105.

177

m10

40105.

177

Pa10

200

Pa10

5.35

core

,el

astic

the

ofed

geA

t the

xYxx

yE

m22

5

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -3

0

Ecc

entri

c A

xial

Loa

ding

in a

Pla

ne o

f Sym

met

ry•

Stre

ss d

ue to

ecc

entri

c lo

adin

g fo

und

by

supe

rpos

ing

the

unifo

rm st

ress

due

to a

cen

tric

load

and

line

ar st

ress

dis

tribu

tion

due

a pu

re

bend

ing

mom

ent

IMy

APx

xx

bend

ing

cent

ric

•Ec

cent

ric lo

adin

g

PdM

PF

•V

alid

ity re

quire

s stre

sses

bel

ow p

ropo

rtion

al

limit,

def

orm

atio

ns h

ave

negl

igib

le e

ffec

t on

geom

etry

, and

stre

sses

not

eva

luat

ed n

ear p

oint

s of

load

app

licat

ion.

MEC

HA

NIC

SO

F M

ATE

RIA

LSB

eer

•Jo

hnst

on •

DeW

olf

4 -3

1

Exa

mpl

e 4.

07

An

open

-link

cha

in is

obt

aine

d by

be

ndin

g lo

w-c

arbo

n st

eel r

ods i

nto

the

shap

e sh

own.

For

160

lb lo

ad, d

eter

min

e (a

) max

imum

tens

ile a

nd c

ompr

essi

ve

stre

sses

, (b)

dis

tanc

e be

twee

n se

ctio

nce

ntro

id a

nd n

eutra

l axi

s

SOLU

TIO

N:

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nd th

e eq

uiva

lent

cen

tric

load

and

be

ndin

g m

omen

t

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perp

ose

the

unifo

rm st

ress

due

to

the

cent

ric lo

ad a

nd th

e lin

ear s

tress

du

e to

the

bend

ing

mom

ent.

•Ev

alua

te th

e m

axim

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nsile

and

co

mpr

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ve st

ress

es a

t the

inne

r an

d ou

ter e

dges

, res

pect

ivel

y, o

f the

su

perp

osed

stre

ss d

istri

butio

n.

•Fi

nd th

e ne

utra

l axi

s by

dete

rmin

ing

the

loca

tion

whe

re th

e no

rmal

stre

ss

is z

ero.

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -3

2

Exa

mpl

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07

psi

815

in19

63.0

lb16

0in

1963

.0

in25.0

20

2

22

APcA

•N

orm

al st

ress

due

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ce

ntric

load

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uiva

lent

cen

tric

load

an

d be

ndin

g m

omen

t

inlb

104

in6.0lb

160

lb16

0Pd

MP

psi

8475

in10

068

.in

25.0in

lb10

4in

1006

8.3

25.0

43

43

441

441 IM

ccI m

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orm

al st

ress

due

to

bend

ing

mom

ent

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -3

3

Exa

mpl

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07

•M

axim

um te

nsile

and

com

pres

sive

st

ress

es

8475

815

8475

815 00

mc

mt

psi

9260

t

psi

7660

c

•N

eutra

l axi

s loc

atio

n

inlb

105

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068

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5

0

43

0

0

MIAP

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IM

yAP

in02

40.0

0y

MEC

HA

NIC

SO

F M

ATE

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LS

4 -3

4

Sam

ple

Pro

blem

4.8

The

larg

est a

llow

able

stre

sses

for t

he c

ast

iron

link

are

30 M

Pa in

tens

ion

and

120

MPa

in c

ompr

essi

on.

Det

erm

ine

the

larg

est

forc

eP

whi

ch c

an b

e ap

plie

d to

the

link.

SOLU

TIO

N:

•D

eter

min

e an

equ

ival

ent c

entri

c lo

ad a

nd

bend

ing

mom

ent.

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alua

te th

e cr

itica

l loa

ds fo

r the

allo

wab

le

tens

ile a

nd c

ompr

essi

ve st

ress

es.

From

Sam

ple

Prob

lem

2.4

,

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e la

rges

t allo

wab

le lo

ad is

the

smal

lest

of

the

two

criti

cal l

oads

.4

923

m10

868

m03

8.0

m10

3

IYA

•Su

perp

ose

the

stre

ss d

ue to

a c

entri

c lo

ad a

nd th

e st

ress

due

to b

endi

ng.

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -3

5

Sam

ple

Pro

blem

4.8 •

Det

erm

ine

an e

quiv

alen

t cen

tric

and

bend

ing

load

s.

mom

ent

bend

ing

028

.0load

cent

ricm

028

.001

0.0

038

.0

PPd

MPd

•Ev

alua

te c

ritic

al lo

ads f

or a

llow

able

stre

sses

.

kN6.79

MPa

120

1559

kN6.79

MPa

3037

7

PP

PP

BA

kN0.

77P

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e la

rges

t allo

wab

le lo

ad

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perp

ose

stre

sses

due

to c

entri

c an

d be

ndin

g lo

ads P

PP

IM

cAP

PP

PI

Mc

AP

AB

AA

1559

1086

802

2.0

028

.010

3

377

1086

802

2.0

028

.010

3

93

93

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -3

6

Uns

ymm

etric

Ben

ding

•A

naly

sis o

f pur

e be

ndin

g ha

s bee

n lim

ited

to m

embe

rs su

bjec

ted

to b

endi

ng c

oupl

es

actin

g in

a p

lane

of s

ymm

etry

.

•M

embe

rs re

mai

n sy

mm

etric

and

ben

d in

th

e pl

ane

of sy

mm

etry

.

•W

ill n

ow c

onsi

der s

ituat

ions

in w

hich

the

bend

ing

coup

les d

o no

t act

in a

pla

ne o

f sy

mm

etry

.

•Th

e ne

utra

l axi

s of t

he c

ross

sect

ion

coin

cide

s with

the

axis

of t

he c

oupl

e

•C

anno

t ass

ume

that

the

mem

ber w

ill b

end

in th

e pl

ane

of th

e co

uple

s.

•In

gen

eral

, the

neu

tral a

xis o

f the

sect

ion

will

no

t coi

ncid

e w

ith th

e ax

is o

f the

cou

ple.

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -3

7

Uns

ymm

etric

Ben

ding

•

neut

ral a

xis p

asse

s thr

ough

cen

troid

dAy

dAcy

dAF

mx

x

0or0

•

defin

es st

ress

dis

tribu

tion

iner

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mom

ent

II

cI

dAcy

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M

zm

mz

Mor

Wis

h to

det

erm

ine

the

cond

ition

s und

er

whi

ch th

e ne

utra

l axi

s of a

cro

ss se

ctio

n of

arb

itrar

y sh

ape

coin

cide

s with

the

axis

of t

he c

oupl

e as

show

n.

•Th

e re

sulta

nt fo

rce

and

mom

ent

from

the

dist

ribut

ion

of

elem

enta

ry fo

rces

in th

e se

ctio

nm

ust s

atis

fyco

uple

appl

ied

MM

MF

zy

x0

•

coup

le v

ecto

r mus

t be

dire

cted

alo

ng

a pr

inci

pal c

entro

idal

axi

s

iner

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prod

uct

IdA

yz

dAcy

zdA

zM

yz

mx

y

0or0

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -3

8

Uns

ymm

etric

Ben

ding Su

perp

ositi

on is

app

lied

to d

eter

min

e st

ress

es in

th

e m

ost g

ener

al c

ase

of u

nsym

met

ric b

endi

ng.

•R

esol

ve th

e co

uple

vec

tor i

nto

com

pone

nts a

long

th

e pr

inci

ple

cent

roid

al a

xes.

sin

cos

MM

MM

yz

•Su

perp

ose

the

com

pone

nt st

ress

dis

tribu

tions

yy

zzx

Iy

MI

yM

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long

the

neut

ral a

xis,

tan

tan

sin

cos

0

yz

yz

yy

zzx

IIzy

Iy

MI

yM

Iy

MI

yM

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -3

9

Exa

mpl

e 4.

08SO

LUTI

ON

:

•R

esol

ve th

e co

uple

vec

tor i

nto

com

pone

nts a

long

the

prin

cipl

ece

ntro

idal

axe

s and

cal

cula

te th

e co

rres

pond

ing

max

imum

stre

sses

.si

nco

sM

MM

My

z

•C

ombi

ne th

e st

ress

es fr

om th

e co

mpo

nent

stre

ss d

istri

butio

ns.

yy

zzx

Iy

MI

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A 1

600

lb-in

cou

ple

is a

pplie

d to

a

rect

angu

lar w

oode

n be

am in

a p

lane

fo

rmin

g an

ang

le o

f 30

deg.

with

the

verti

cal.

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erm

ine

(a) t

he m

axim

um

stre

ss in

the

beam

, (b)

the

angl

e th

at th

e ne

utra

l axi

s for

ms w

ith th

e ho

rizon

tal

plan

e.

•D

eter

min

e th

e an

gle

of th

e ne

utra

l ax

is.

tan

tan

yz IIzy

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -4

0

Exa

mpl

e 4.

08•

Res

olve

the

coup

le v

ecto

r int

o co

mpo

nent

s and

cal

cula

teth

e co

rres

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ing

max

imum

stre

sses

. psi

5.60

9in

9844

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75.0in

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0

alon

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curs

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est

ress

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larg

est t

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e

psi

6.45

2in

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75.1in

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The

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42

41

43

121

43

121

yy

z

zz

z

yz

yz

Iz

MAD

M

Iy

MAB

M

IIMM

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sile

stre

ss d

ue to

the

com

bine

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curs

at A

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609

6.45

22

1m

axps

i10

62m

ax

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -4

1

Exa

mpl

e 4.

08

•D

eter

min

e th

e an

gle

of th

e ne

utra

l axi

s.

143

.3

30ta

nin

9844

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359

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nta

n44

yz II o 4.72

MEC

HA

NIC

SO

F M

ATE

RIA

LS

4 -4

2

Gen

eral

Cas

e of

Ecc

entri

c A

xial

Loa

ding

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onsi

der a

stra

ight

mem

ber s

ubje

ct to

equ

al

and

oppo

site

ecc

entri

c fo

rces

.

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e ec

cent

ric fo

rce

is e

quiv

alen

t to

the

syst

em

of a

cen

tric

forc

e an

d tw

o co

uple

s.

PbM

PaMP

zy

forc

ece

ntric

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y th

e pr

inci

ple

of su

perp

ositi

on, t

he

com

bine

d st

ress

dis

tribu

tion

is

yy

zzx

Iz

MI

yM

AP

•If

the

neut

ral a

xis l

ies o

n th

e se

ctio

n, it

may

be

foun

d fr

om

APz

IMy

IMyy

zz