sustainable technology 2 - part 3

8/10/2019 Sustainable Technology 2 - Part 3

1/42

ELASTIC INSTABILITY

(or

Buckl ing Fai lure)

Examples of buckl ing

fa i lure

r

t

"ttot"

,

n

0r^1

"n".t

.- -

*"-6",-

:"'\a't

-to cr

6;'p'zc'i'rr'

trn*L

''

c"U8 .0'' stvv'i ' .

trut5,

u'*' il"lz c"""

:ecti'^s

co'ioJ

"""tT,

,""""D

^r

r".",."jdt-

l*9^;_

c..*il*

E--""1.- ' ; f

i l , , , c,,, , ,^LIO

I

l

c,o"j.r.

u., \^

|efalds

O^

ft1

l l l

J

sq,êler ' "s ,

"* i "

o- t

$"

l I l /

, " f . ;

t \ " .

( ,16,- . . ,

ee\on

e.J

l; i1

GJrio^"

/ l j

l /

"**ni=

*-

,11o

c.-q*J$

,

,,i

I li: l.t l

{1""

"*b."

.-

$ J".-- 1 \J

;+t

+l*,

Q1"rth

fe;.J{

J.^ JL

u^}e,

-lo^p,,o''"

n

".+"e+"

o^,(

tll.

'^

Yl

"

U r r r^^--- t i^" 1 I i i I , , . - .v,kn..

sQ"êler'.s"

Yd*io

o-l

il"-

I

Aj*Una

md dlnrtinr

ol tlE

qltndrtqt

rrtt

of . c.n

utbj.ct d ro

conlr.tdlr

for...

3) Eidelrays buckling of beans

,t

).

sQ"^["X1]-o tf,q

-"*r^",.

",,q

I

t

t

S"".^l\

o^ .

1.1

..\rL.n

.-et1

\L*",

r\: c"_?*,,,"-

vo.Jl

2l buckting of thin

,.rat

ed sections

strerlth

ts

veôlneJ

'

( ) Io6ronol

8 .tllng

(b)

Loc6lBucrlng

'rN


2/42

4)

buckl ing

produced

by

torsron

'*o-1..GlW*,

T,i.r.b.r


3/42

lji, l;.:li,1 l

-"

:Lf^:::^ :n_"_:.: :-.

gl

r he

.cro

s

sect

on.

He ce.

ol loh'

tubular

st-rucrules

are m6r; ; :^; ; ; , : - - ;" :^ i : rv l r '

' 'ence'

olumns

than

sol id

di mensions

wI-flJ-TEEEf

n

e

cross

secr ional

EEEc6FETa

n

-

ncreases

iTe-EucF[fn

ecause

r

is

inciEaie[

process

has

a pract ica

wa] I i tsel f beComesunstable.

l r t , however becauEE-Eie tu a tv the

Then

local

ised

buckl ing

occuls

in

the form

of

smal l

colrugat ions

or

wrint l is.

I f -a

strut

or

column

1s

free

to buckle

in

wi. I l

bend

about

the axi .s

wi th

the

smal le i

any

direct ion,

then

it

seconal

mornent

of

area.

aqua

e

enatqesmal@

Effect of lateral restraint

(5)

( ( )

c)

rhe

dlagram

(b)

above

shows.

-how,. i f

the

strut

or

colurnn

ig

aterally

restrained

from

bu.ckling

i"

ti,.

.iaij ,

i 'Iarger

load

ay

be

carr ied

but

evenrual ty

ai

a

roia

' i - . i i ' i i . i . "

the

f

j . rsr

buckl lng

load

then

buckl ing

ir i t t

o""o,

i l r r -Trr .- ' .J in. ,

"norn.

imi lar ly,

1f

that

buckl i lq

is

p, iveq ga

trr"n

1o" i r ing

in

the

: l t l ld .Tode-.as

hown

in

diagian

-(c)

wi l l

o""oi - . i ' "

road

nine

tr.mes

the

flrs

Ioad.

rhe

Euler

vaiue

ig

tfriJit-rni"n

buckling

would

first

occur

for

an

unrestrained

strut.---

-- -"-


4/42

8f. f

ect

of

large

def lect ions.

imperfect ions

and

rnelast i .q

betraviour

The

Euler

equat ion

for

the buck- l ing

load

was

derived

tor

an

ideal

strut

or

co Iumn

in whicb.the

aef

I ect ions

ire

smaf

f

,

tne

construct ion

is

perfect

and

t-he--mater ia l

fo l lows

f loof l ,"

lu*.

e"

a consequence, the sidewavs def lect ion is undef ined in che Euler

folmuta.

rhus

rhe

srrui

ft.ay

h^-r;

;.y

-s;; i i

-jeiie

it

ron,

wrricir

rnay

be.ver j . f i -ed

expel imental ly

and

is

shown

in

the

diagram

below

by ho izonta

I

l ine

'A' .

g.lhen

deflectionE

become

larger

a more

exact

1 l i rgel

and

larger

load

is

necesary

to

cause

qe recttons

(as

shown

by l ine

.8.)

I f

the

strut

is not

Derfect ly

stra_ight

etc

then

smal l

def lect ions

: : : r , ,

- .^ :

"

f rom

rhe onser-

of

rr ; l i ; ; - ; ; ' : i i : i i "c.

wir t

be

Final ly.- i f_we consider irhat happens when

the

stresses

exceed

the

etasric

linit

and

rhe

nateriar,ir-o

r

ongei-'ro-ii.;;

-H;;k;.;- i;;:

i;i;

curve

for

inelastic

behaviour

,0,

depJiis

iio.

ii,"--"i""tic

curve

continues

upwald,

reaches

a

maximum

rd

-ar;;;";;;i.

The.

detai led.

shape

_of

th-e

curves

depends

on

che

mater

j .aI

propert j .es

and

the

column

dimensions

bui

the g"n"r i t

natu e

of

the

behaviour

is

as

shown.

The basic Euler fornula assumes ,rp innedo ends but in p act ice

1,"11?-th::_:gl9itions

are possi-bre

""n

""

:1."

.iiJ,'tixed

enas,

erastlc

supports

etc.

The

diagrarns

belorr,

shor^,

'he

four

main

support

conditions

aasuted

in

practice.

analysis

shows

that

an

increase

in

the

I

l

r

,P

lo_

I l

i-i

F

I

I

P

I i

-fi-i'

I

I l r

l r

t i

-Lr_t'\F

(4)


5/42

va

I idit

The

Euler

f

ormul

a

(L/k

is

known

as

the

S1(e9s

F_rom

equat ion

(2)

i . t

nay

be

r

a

curve

may

be

drawn

of

.seen

that

for

a part icular

value

of

oucKl

rng

stress

against

slenderness

l in i t

fo i

Euler 's

theor

nay

be

wri t ten

as

le

=

Cg_

A tL/k)\

s lenderness

rat

io

)

_z_ _

-?-

tr- EAk'

It^',

I

J.e.r.1

Lc,s.":

v.o

-r,.""r.F

+l,a_

("-1re*.v*

.{ .- . . , t

-

Tl"

-r tvvr cu.\ \(n(L

i . \$^ - .

th.

.cafrca,,dq

i t l l , "

. , .*J", . . , ,

r"aro

(L-g

)

ts

leu>

tl^,.r,Lo,

L

/t '

However,

i t wi l l

be

aDDarent

that

for

smal l

values

of

L,/k

the

y:] : :

: f

the

Euler

_Uuci i ing

stress

wi l t

U".o..

"" l iono*ical

and

ence

must

be

inappl iqabte.

lh-" . ,"""on- i " ,

o- f

-o i ' r "" ,

an"t

to,

ny particular

materiat

rhe

buckling

";.;'";

n"J#r

exceed

the

naximun

compressive

stress

and

i."

f;c;

i;

;;h1'noi'io

.x.eea

r"

latsic

1j .n i t

stress

for

a sare

qestgn.

I r r t refaqt ic-e- i t gr l l b9 helpf r l

arr

matert

a ls

L,/k

values

belot

dangerous

to

use.

Rankine

fornula

Because

.of

the

l imited

range-

of

the

Euler

equatron

and

its

nsuitabilit.y

with

snalI

L/k

v-aIu:_s,

ilprijl l i

r=o".*.li 'u"

ave

been

:J:- l " id ,ro

sat is fy

at I values

ol

L/k:

-

r#re-- . r"

many

of

iii"'"'rrT*"rf"t"rn"'o?1""',i:.it'rT ?". ,-ri';*."*"il,.tirli*:;

should

be

a

conpromise

bet

wourd

ause

;ii";-;;;

;;'J;?.t1'.."n1t"'"iJ:.u

i:u

the

roaa

hai

as a guide to

100

wi

l1

make

iildnF"o&'#il

"_, \

. r l

-'1"-'

I


6/42

The

forrnula

becoll |es:

where

dl

Tt say

/il

=

*

then

\

=

LJ,.

.-81{)

and the solut ion becomes x

=

Ae

6"

(-Fr*)t

+ Beur^(-F,-{) t

?\ TF A=I

sher

Xr

=

x. ' - -Fa, ' .

i .F ' . ,nE(A+B)

:

The__r.esul t in9

motion is again

aperiodic

bu

the

damping

coeff ic ient

now has

the least value which

wi l l produce

uch

not ion,

The dahping

j .n

th is case

is said

to be

cr i t ica l .

A

cr j . t j .ca l ly

damped systen wi l l

have

the

smal lest

dampinq

lequired

fo aperiodic

motion and

hence

the mass wi l l

return

t6

the

posi t ion

of

rest in the

sholtest

possib le

t ine without

overshootjng.

-

.The

propelty

of

critical damping

is u6ed

in nany

pract ical

appl icat ions.

For

example,

in large guns

have

'dashpots'

r i th

a cr i t ica l

danping value,

so

that

ihey reEuln

o

their original

poEition

after

recoj.l in

the rniniftu

time wj.thout

vj .brat ing. I f the daloping provided was nore than the cr i t ica l

value

some delay

would be

caused before

the next

f , i r ing.

Sincep

=

115.tr

q

fo cr i t ica l .

danping

=

2nur^

qt^pti

t \r.e


29/42

3)

rf

B


30/42

Iogarithnic

decrenEnt

f t

can

be

sholrn

that

the

anplitude

is

given

by

-

rat io

of

the

in i t ia l

to

the

.nth.

F-".

tn

value

of

p

ryPe

of

danping

danped

freguencv

natural

fleouencv

0,67

{structural

danping

(vehiqle

danping

(instrunent

dehping

I t t

290

where

t^= interraL

betweea lhe first

and

the

nth

anplltudes.

Sin_ce

pqJ^t^ is

the logari thn

(to

base_e)

of

th ls

rat io

i t

is

ca1led

the loqarirhmic-decreirenr.

l-.,r"

F;^;:

-__'

3t

-, '

--

This provides

a

conveni;nt-

e*p.tir.n.h

nethod

for

a"F...i"rrn

the

danpLng

ratio

in

any slngle-

degree

of

fieiaom

syit"r.

p

--

(

+_ ) i"

cat

led

the

-damptng

atlo

and

usual

y

I ies

\

2'{,.'r.

/

between

0 and

I in

nost practical

sysrehs.

I

-- . - l

I

r

I


31/42

P\nor"wc'sg

\ , /n,

"-"'-

'latYue

|,r^on

n/t'br',tio''

Vv

N,\

lnei-rr

"^-

o

^

I

YNX

\_

:

CJryh stl1-z{

rF)

CX

--'+

uet'l

n

, I tlp

---'+

hstn

vP

oal'-

t,

,----,

ql"'sco"do|

hoi

e

l1t'rcvnaktcf

Dqn4v".4s

3

LJ

rn

/t|flwe^e"r x

Vel*''t1

i

Ac.QZo*in

il

V--

\z-*r

-

A,r\

=

A"

=i.

tz-

tr

At

d,F

At-+"

*=y+-#:*;#=

*FvlU

l-

m'i+ei+r

t-

r

ep"itA

.=

sd-(4r"-rt

+*"-

s-*-

Cee{+;"-

Forc.


32/42

('":.*Y=

{

- :

* .*

+Kx:f

L-*

.-

Kx--tr

vnlo-n4

tn-fu/

F = Fo Gs Lc^tt)

0. r

f o

--

A

mgl+t'-de

?t ,

I

=

,T-*z

)ve7o"> ml.

=-KX

-

f , ro

t l

Vr

U"ation

t l

Crh,,,ltar

x=s

.\

*

F--

, " .

AJI=

@2..=

L-

'-

).X

J7

L*e*)

rT:,

(

fp^\

_

l rv lz

' ---6-

"*,

o:1

h^t"t*l-

*ry,^U

fvee

-vnJqnrpeo[

=

t

-'Yl

+K{a- o

f

vee-

Jn*g-L

-t

,.^^

+

Ci +Kx

=- o

(o'

"J

-u.

J**poJ.*'

^'{

t

Kx

--

F

{"'"J

unh^7rl

rAbroton

'e

c,

y

1'1

K

Fdcoia..[-

^".eu-'\*-

n

T-+ci

+KX--F


33/42

tl2)

l-tLn,

-r=

-T;

Nt=

loor:

fPm

7{2=

Looo

\r-

n,=

o'5

K3

V

=

72-po //n

:\l_o

,-rLx

\oo

o

fJ

6o


34/42

$D

5'1;Jt

9Jtut"';

UNIVERS]TYF

WOLVERHAMPTON

School f Engineeringnd he Bui l tEnvironment

PHASE

ESTPAPER

ModuleCode:ET2006

Page1 of

5

ET2006

SUSTAINABLEECHNOLOGY

Date f Examination: 2 June2005

Timeol Examinatlon:

10:00

Examiner:

Dr Kadd6 ahiaoui

Examiner(s)oomi

SC113

TimeAllowed:

2 hours

Number f

Ouestions

o beattempted:

Answer ll

questions

Additlonal aterials

equired:

.

One

ight

age

nswer ook

lnstructionso Studenls:

.

Answer

questions.

.

You hould riteegiblyn black rblue

nk.

.

Check

our

work arefully

or naccuratepelling,

unctuation

nd

grammar.

.

Youmaynot eave heexamination

oom

ntilafter he

irst

hourof

he

examination.

.

An

announcement

illbemade henhere re30minutesf heexam

time eft.

.

THISSAN'OPEN OOK"

SSESSMENT


35/42

Module

ode: T2006

page

2 ot

s

Ql

(a)

Briefly

xplain hat s meant

y he ollowing

erms:

(i)

Bi-axialtate

of stress

-

4

(ii)

A state f

pure

hear

t . - - , ; , - l

..-i

t

(iii)

PrinciPal

tresses

l ) r l I Ao 9tye3,

51steh

tt--_--)

v +r

'4-l

,, l----[ .

j i l

No,^oI

i tYcts:a

IL-l

u"--"

oJ-

rsr"

iii)

^^,

oJ

;.

rt".'''

a.til

o-

l\o"

witL

^"

glta^'

su':s'

(b)

Mohr's

ircle or stresss a

grapiical

tress

nalysis

method hat is commonly

sed by the

designer.

Describe ow hiscircle an

be constructed

or he

case of a 2D stress

systemand

nam6 lhree

important

iaces

f informationhat

can

be readily

obtainedromsuch onstruction.

i"

Nafet

s'v.

(c)

A rotating

haft arries omeheavy

quipmenthat

induces

dirctstressof 10

MN/mz,t also

transmits torquehat

convertso a

shear tress f

5 MN/m'.Draw he Mohr's

Circle or this

caseand

determinehe maximum

hear tress hat he

shaft

isexperiencing.

A material

point

at

the

cylindrical hell under

shown o be subjected

state,

intsrnal

pressure

an be

to the

following

D stress

IOL

4l^L5

=a.O7

-

or

=

Pci/2t

tro=Pd/4t

Where

P is

the

applied

ressure,

the

diamter nd

t the wall hickness.

lf ths shellhas a diameter-to-thickness

atio

d/t)

of

50 and s madeof a materialwith

a

yield

strongth

f

240 MN/m2,

stlmatehe

ailur

prssure

sing:

oS:,*,*J

,,ii\

(i)

Thgmaximum

rincipal

treqs riterion Ra"r,L

)

fke

^eo1

t5

dppkr\ti^tch

G**

tr

Ogt

ir""

*l

b.,*r-

(iD TheTresca ieldcriterion

-

hcr6\ds

(iiD

The Von-Mises

ield

criterion

(:;:,

nc)

u3{ .e-t:rt

y'(6"-3f*+'r*

=


36/42

N,4odule

odetET2006

Page

of5

By consideration

of

the

complexity

of the

calculationsnvolved

and the results

obtained,

discuss the relativemerits

of each of

the above

threecriteria.

:-

?tv'

1oo%

Q3

(a)

Briefly xplainhe ollowingerms

s applido

,dL,vvo

uclqvrvur,

(i)

Stress ycle

(ii)

l\4eantress

(iiD

Alternatingstress

30vo

(b)

Sketch

generalised

atigue, r S-Ncurve

ora

structural aterial

hat

hasa tensile trength

f 750

MN/m2,nd rom

his

qrye stimate:

*

"ah

(i)

Themaximum

lternatinghesshat he

material ansustain

ithout

atigueailure

Y.-t.

(ii)Themaximumlternatingtressf failure s not

to

occur elow 00000cycles

n:1

(ii i)

The ifeof a component.ade

rom his

material, nd stressedo i575 MN/mz

45'/.

(

c) List ive actors hatcan nfluenche

tatigue

behaviourf a component,

25%

Q4

(a)

For each

of

the ollowing

ituations,ndicatehe

main ype of loading nd dentify

he caseswhere

the bucklingmodeof failure an

be of concern:

i. A cable upporting

chandelierhat

hangsroma ceiling

ii. The egof a tall barstool

iii. An empty

plastic

ottle

usedby someone


37/42

Module

Code:8T2006

Page4

of 5

to support

book helf

The wingof

an aircraftn flight

Theouter hell

f a submarine

An deal lender

olumns 5 m ong

and

pinned

t

bothends. t s made

of hollow

quare ection

aluminium ith

a

Young's

odulus

f 70

GPa.The

outer ength f hesection

s 100mm

and he

wall

thickness

s2 mm,as shown

n he

diagram

elow.

CalculatetsEuler

ucklingoad.

Pcr

(c)

Howwould he

Euler riticaload

vary f the

end

conditionsere o bechangedrompinned-pinned

to the

ollowing?

i.

Fixedjixed

ii. Fixedjree

iii. Fixed-pinned

Which ndJixingondition

ives

he

strongest

arrangemenl s regards

uckling?

(b)

25%

50./"

25"/.

Give hree easonswhyvibrations

an

sometimes

be seriously ndesirablend

brieflyifdescribe

wo

situations here ibrations re

actuallywanted.

(b)

A designer as

developed n impressive

hild's oy

withgreatmarket otential.t isa two-speed adget

that

can rotateat 1000and2000

pm.Detailed

25%


38/42

Llodule

Code:

ET2006

analysis

asshownhis oy o

have

an overall

mass

of 0.5kg

anda stiffnessf 7200

NUm.

s a feltow

designer,

hohasbeen

ntrusted

ithsome

checks,

ou

are equiredo:

i. Calculatehnaturalrequencyl the

gaoger.

ii.

Calculatehe orcing

requencies

correspondingo the

wo

operatingpesds.

Explain

hy h6 ower

peeds

acceptable

while he

higher ne

s not.

Suggesl ays o help

mprove

hedesign.

757c


39/42

lvlZWlTiVii.nnvproN

School

of

Engineering

nd

he

Bui l t

Environment

PHASE

EST

PAPER

ET2006

Sustainable

echnology

Oale

f Examination;

Tuesday

0

N/ay

006

Time

f Examination:

j

O.OO

m

Examlner:

K

yahiaoui

Examination

oom:

MC401

TimeAllowed:

2 hours

Number

f

Questionso beatlemptd:

Attempt

llS

questions

All

questlons

are

qualty

weighted

unless

otherwise

stated)

and are

markod

on the gradepoini syslem.

Materlals

equired:

.

Oneeight

age

nswer

ook

.

Copies

of approved

exts

Instructions

o

Studonts:

.

Programmable

alculators

renot

permitted

n the

exam.

. Answer ll lyg questionsallquestionsreequally

eighted).

.

You

shouldwrite

egjbly

n black

or blue

nk.

.

You

will receive

redit

or appropriale

eferences

nd

quotations.

.

Check

our

workcafefully

or naccurate

pelling,

unctuation

nd

grammar.

.

Youmay

not eave

he examination

oom

untilafter

he irst

hourof the

examination.

.

An

announcement

ill

be made

when here

are 30 minutes

f the

exam

ime

eft.


40/42

dJ

t .

+Lry

T--rr


41/42

*

situation?

dentify

the

locations

ofmardmum

sh,ess

ndmaximum

deflection

and

calculate

hemaximum

moment

xperienced

y the

overhang

fit

is 30

m

long

andcaries

a concrete

lockhaying

a

mass fone

metric orine.

Qoya

c) Wlat *ould be heminimum equired eco[d

moment

farea

fthe

maximum

ending

tress

s not

o exceed

00

Mpa?

you

may

assume

hat,

at

the

ocation

ofinterest,

he

distalce

rom

he

neutral xis

o the

outer

ibres

of

the

section

s 0.5

m.

strength

f 1200

MPa.

(30%)

nv

/ ._

-,/eq,

t

/?

d) Calculate

he

deflection

t he

oaded

nd f the

material

as

a

youne,s

modulus

f2l0

CPa.

r30l"r

.

a)

Under what

loading

conditions

would the

atigue

mode

of failEe

be

of

concem

o designersrd engineers?n particular, iscusshe ]?esofsh.esses

that

areofrolevance

o fati8u

ailue

arallsis.

-

(15%\

b)

Construct

genemlised

-Ncurve or

a material

with

anultimate

ensile

c)

For a machine art,

made

fthe above

matcrial,

Calculate;

a) The

stess imit

below

which

atigue

ailure

will

not occw

(1

0%)

b) The

stlessimit,

which

must

not be exceeded,

f the

Dart

were

o

Iast or

10,000

ycles

e5%\

c) Theexpecredatigueife of rhepafl f ir were o operate ror

belowa

sress evel

of

760

MNrm2

e5%)

d)

Give

briefviews

on what

elfect would

the

following

have

on fatigue

ife

exDectancv:

(20%)

(s%)

a)

Surfaceinish

b) Material

density

c)

Component

ize

d) Stress

aisers

e)

Appeara.nce

Q5:

Using he

usualnotatioq

he

general

quation

fmotion

ofa

second

rder

vibration

system

s written

as:

nLi

+ cx + lat

=

fo

cosot

IdentiS/

each

ofthe

four forces

nvolved

n this

equation.

(20%)

Whatare

he

SI system

nitsofeach

oftbe

variables

nvolved

n this

)

equation?

(20%)


42/42

c)

vd)

Calculate

heundanryed

sh[al

ftcquency

fa vibrating

system

hat

can

b:gescriblg

y

8tre4uarion

fthis

rypewhen

he

massis

O kg,

the

stiftiess

kl.{/m

and he

damping

oefficient

NVm.

-

(2OyO

Explainwhat

would

happeDfthis

systcmwere

o

be shake

at aftequcocyf l.59IIz.?

eO%\

By how

muchwill

the

natunl

frquacy

bc changcd

fboth

&e mass

nd

stifftiess

fthe

sy-stern

crc o

bedoubled?

(2OyO

Y\"r"J

["5.,""%= t,J -

r

sustainable technology 2 - part 3

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