Bridging Program- Lecture Note 01

7/23/2019 Bridging Program- Lecture Note 01

http://slidepdf.com/reader/full/bridging-program-lecture-note-01 1/51

12/15/15

Prepared by: Eng. Chamil DumindaMahagamage

1n"erna"ional College o# Business and$echnology

Philosophy of EngineeringDesign



Engineering Materials-Classifcation

12/15/15

Prepared by: Eng. Chamil Duminda Mahagamage B.Sc.Eng (Hons), C Eng,


Ductile,Malleable

Brittle

Elastic

Plastic

Isotropic

Anisotropicrt!otropic

De%ni"ion

& ma"erial is said "o be duc"ile i#i" is capable o# 'i"hs"anding larges"rains under sus"ainable loadbe#ore #rac"ure occurs.

$hese large s"rains areaccompanied by a isible changein cross sec"ional dimensions and

"here#ore gie 'arning o#impending #ailure

E: mild s"eel, aluminium, copper, polymer

De%ni"ion

Bri""leness deno"es rela"ielyli""le or no elonga"ion or increasein leng"h a" #rac"ure. $he s"rainnormally being belo' *+. &ma"erial "ha" ehibi"s bri""lenessis called a bri""le ma"erials.

Bri""le ma"erials "here#ore may

#ail suddenly 'i"hou" isible'arning.

E: concre"e, cas" iron, glass,"imber, ceramic



Properties o" EngineeringMaterials

12/15/15


#n"erna"ional College o# Business and$echnology

P!$sical C!e%icalMec!anical

$hose deried #rom "he proper"ies o# ma""er or

a""ribu"ed "o "he physicals"ruc"ure

densi"y

-oid con"en"

$e"ure

Color

Shape Permeabili"y

$hose per"aining "o"he composi"ion and

po"en"ial reac"ion o#a ma"erial

&cidi"y

&lalini"y

/esis"ance "o

corrosion

Proper"ies 'hich rela"e "o

"he behaiour o# "hema"erial 'hen sub0ec"ed"o ac"ing loads.Mechanical proper"ies areusually epressed in "ermso# 1uan"i"ies "ha" are

primarily #unc"ion o#stress & strain.

Stress

Strain

S"reng"h S"i2ness Elas"ici" y

Plas"ici" y

Duc"ili"y Bri""leness

Hardness

Endurance



Hooke’s Law(1678)

12/15/15


&n"erna"ional College o# Business and$echnology

'

(')

(e)

Cross section area o" t!e steel ire * Ariginal lengt! o" t!e ire * l+



Stress (σ)

n"ensi"y o# "he in"ernally dis"ribu"ed #orces "ha" resis" a change in

"he #orm o# "he body. " is a measuremen" o# densi"y o# #orces,de%ned as #orce per uni" area o# cross sec"ion.

3ni" 45m6 (Pa), 45mm6 (MPa)

7 8 P5&

Strain ()

Propor"ional de#orma"ion produced in a ma"erial under "hein9uence o# s"ress

4umerical /a"io : E"ended or shor"ened leng"h5riginal !eng"h

; 8 e5l<

!o"#l#s of Elasti$ity (E)

$he ra"io o# s"ress "o s"rain in "he linear region is no'n asmodulus o# elas"ici"y.

E * /

12/15/15





Stress%Strain &#r'e

12/15/15


.n"erna"ional College o# Business and$echnology

+ to a - inear, obe$ing 0oos a

a - i%it o" proportionalit$b - Elastic i%it (%a3i%u% stress t!at can be

applie4 to a %aterial it!out pro4ucing aper%anent plastic 4e"or%ation )

c - pper $iel4 point4 - oer 6iel4 point4 to " - strain increases at a roug!l$ constant 7alueo" stress

" to g - Increase in stress acco%panie4 b$ a largeincrease in

straing - lti%ate stress



Proles

=.& s"eel bar o# rec"angular cross sec"ion, ><mm by 6<mm carries

an aial load o# ?<4. Es"ima"e "he aerage "ensile s"ress oer anormal cross sec"ion o# "he bar.

6.& s"eel bol", 6*mm in diame"er, carries a "ensile load o# ?<4.Es"ima"e "he aerage "ensile s"ress a" "he shan and a" "hescre'ed sec"ion, 'here "he diame"er a" "he roo" o# "he "hread is6=mm.

>.& cylindrical bloc o# ><<mm long has a circular cross sec"ion,=<<mm in diame"er. " carries a "o"al compression load o# =<<4and under "his load i" con"rac"s by <.6mm. Es"ima"e "he aeragecompressie s"ress oer a normal cross sec"ion and "hecompressie s"rain.

12/15/15





?. & "ensile "es" is carried ou" on a bar o# mild s"eel o# diame"er6<mm. $he bar yields under a load o# @<4. " reaches amaimum load o# =*<4 and breas %nally a" a load o# A<4.

Es"ima"e

i). $he "ensile s"ress a" "he yield poin"

ii). $he ul"ima"e "ensile s"ress

iii). $he aerage s"ress a" "he breaing poin", i# "he diame"ero# "he #rac"ure nec is =<mm

*. & circular me"al rod o# diame"er =<mm is loaded in "ension.

hen "he "ensile load is *4, "he e"ension o# a 6*<mmleng"h is measured accura"ely and #ound "o be <.66Amm .Es"ima"e "he modulus o# elas"ici"y o# "he me"al.

. & mild s"eel column is hollo' and circular in cross sec"ion 'i"han e"ernal diame"er o# >*<mm and an in"ernal diame"er o#><<mm. " carries a compressie aial load o# 6<<<4.De"ermine "he direc" s"ress in "he column and also "heshor"ening o# "he column i# i"s ini"ial heigh" is *m. $ae E86<*45mm6

12/15/15





12/15/15


:n"erna"ional College o# Business and$echnology

*elationship of stress an" strain for'ario#s aterials

(a) Bri""le

(b) !inear elas"ic 'i"h dis"inc" propor"ional limi" (e: lo' carbon

s"eel or mild s"eel)

(c) !inear elas"ic 'i"h an indis"inc" propor"ional limi" (e:aluminium)

(d) 4one linear (e: concre"e)



+"eali,e" short ter stress%strain $#r'efor "esigns=. 4ormal 'eigh" concre"e

I%portant properties to be non "or basic 4esigns are,

a); C!aracteristic co%pressi7e strengt! (" c)

b); Mo4ulus o" elasticit$ (Ec%)

c); <ensile strengt! (" ct) - about one tent! o" " c

12/15/15


1+n"erna"ional College o# Business and$echnology

Varies with theconcrete grade



12/15/15



6. /ein#orcemen" S"eel


a); C!aracteristic $iel4 strengt!(" $) * &++ - .++ =/%%2

b); Mo4ulus o" elasticit$ (Es) * 2++ =/%%2



12/15/15



>. S"ruc"ural S"eel

S"ressS"rain cure similar "o i"em (6) aboe


a); =o%inal 7alue o" $iel4 strengt! (" $)- >aries it! t!e steel

gra4e an4 t!e t!icness

b); Mo4ulus o" elasticit$ (E) * 21+ =/%%2

c); Poissons ratio (?) * +;#+

4); @!ear Mo4ulus () * E/2(1?)

e); Coecient o" linear t!er%al e3pansion (F) * 12 3 1+-./+C



12/15/15


1#n"erna"ional College o# Business and$echnology

?. $imber



12/15/15


1&n"erna"ional College o# Business and$echnology

-asi$ Properties of Soe !aterials



12/15/15





12/15/15


1.n"erna"ional College o# Business and$echnology



12/15/15



S.*/&./*0L .HE*2

Classifcation base4 on t!e nature o" internal "orces

=. &ial #orce member

& member "ha" is sub0ec"ed ei"her "o aial "ensile or aial compressie#orce is re#erred "o as an aial #orce member. E: a cable and "russmember

6. Bending and shear resis"ing member

Members are sub0ec"ed predominan"ly "o bending or 9eural ac"ions.E: a member sub0ec"ed "o loads "ranserse "o i"s leng"h.

>. Members sub0ec"ed "o "orsion

Members are sub0ec"ed predominan"ly "o "orsion or "'is"ing ac"ions. E: a sha#" "ransmi""ing mo"ion #rom one sha#" "o o"her.

?. Members sub0ec"ed "o a combined ac"ion

Members are sub0ec"ed "o any combina"ion o# aial #orce, bendingmomen", shear #orce and "orsion



12/15/15



Basic rigi4 ele%ents

=. Beams and Columns

Beams are generally horion"al, 'hich carry loads applied "ransersely

"o "heir leng"hs and "rans#er "hem "o "he suppor"ing er"ical columns oro"her suppor"s. $he beams bend under "ranserse loads and are said "ocarry loads by bending. $he elemen"s carrying aial compressie #orces"ermed s"ru"s, 'hen er"ical "hey are "ermed columns.

6. rames

& #ramed ob0ec" or s"ruc"ure is made by assembling beam and column

elemen"s 'i"h rigid 0oin"s.>. $russes

$he "russ is composed o# shor" and s"raigh" discre"e elemen"sarranged in "o "riangula"ed pa""erns. $he "russ is nonrigid, bu" i"main"ains i"s shape as a resul" o# "he eac" 'ay "he indiidual lineelemen"s are posi"ioned rela"ie "o one ano"her.

?. &rches

&n arch is a cured line#orming s"ruc"ural member spanning be"'een"'o poin"s and carry "he loads "o "he suppor"s 'hile being sub0ec"ed

predominan"ly "o aial compression.

*. alls and Pla"es

$hese are rigid sur#ace elemen"s. & loadbearing 'all can "ypicallycarry bo"h er"ically and la"erally ac"ing loads along i"s leng"h.



12/15/15


1:n"erna"ional College o# Business and$echnology

Internal 'orces

=. $ension #orce

$ension #orce "end "o pull an elemen" apar". $he s"reng"h o# a "ension

member is generally independen" o# i"s leng"h and "ension s"resses areuni#ormly dis"ribu"ed across "he cross sec"ion o# "he member.

6. Compression #orces

Compression #orces "end "o crush or bucle "he elemen". Shor"members "end "o crush and hae higher s"reng"h compared "o a "ensionmember. $he load carrying capaci"y o# a long member, ho'eer,

decreases 'i"h "he increase in "he leng"h. $he long compression membersmay become uns"able and may suddenly snap ou" #rom benea"h "he loada" cer"ain cri"ical load leels. $his phenomenon is called bucling.Because o# "his bucling phenomenon, long compression members areno" capable o# carrying ey high loads.

>. Bending #orce

Bending #orce is a #orce s"a"e associa"e 'i"h bending o# a member. $hebending ac"ion causes %bres on one #ace o# "he member "o elonga"e, andhence are in "ension, and %bres on "he opposi"e #ace "o compress.

?. Shearing #orce

Shearing #orce is a #orce s"a"e associa"ed 'i"h "he ac"ion o# opposing#orces "ha" "end "o cause one par" o# "he member "o slide 'i"h respec" "o

"he ad0acen" par".



12/15/15


2+n"erna"ional College o# Business and$echnology

*. $orsion

$orsion is a "'is"ing ac"ion. Bo"h "ension and compression s"resses arenormally deeloped in "he member sub0ec"ed "o "orsion.

. Bearing s"ressesBearing s"resses eis" a" "he in"er#ace be"'een "'o members 'hen

#orces are "rans#erred #rom one member "o ano"her. $hey ac" perpendicular "o con"ac" sur#aces. $he bearing s"resses are alsodeeloped a" "he ends o# beams 'here "hey res" on "he 'alls.

I4ealiGation o" @tructures "or Anal$sis

$he primary aim o# "he analysis is "o de"ermine "he reac"ions, in"ernal#orces and de#orma"ion a" any poin" o# "he gien body caused by appliedloads and #orces. $o achiee "his ob0ec"ie, i" becomes necessary "oidealie a body in a simpli%ed #orm emendable "o analysis procedure. $hemembers are normally represen"ed by "heir cen"roidal ais.



12/15/15



S#pport &on"itions

Pinned Connec"ion

n "his connec"ion 0oin" allo's a""ached member "o ro"a"e #reely bu" doesno" allo' "ransla"ion in any direc"ion. Conse1uen"ly, "he 0oin" canno"

proide momen" resis"ance bu" can proide #orce resis"ance in anydirec"ion.

/oller Connec"ion

n addi"ion "o ro"a"ion, "his connec"ion also allo's a""ached member "o"ransla"e #reely parallel "o "he sur#ace o# "he suppor", i.e. does no" proideany #orce resis"ance parallel "o "he sur#ace o# "he suppor". Ho'eer, "he

0oin" resis"s "ransla"ions in "he direc"ion perpendicular "o "he sur#ace o#"he suppor".

ied Connec"ion

$his connec"ion comple"ely res"rains ro"a"ions and "ransla"ions o# "hea""ached members in any direc"ion. Conse1uen"ly, "he 0oin" can proidemomen" and #orce resis"ances in any direc"ion.

mpor"an": or an ob0ec" "o be s"able in e1uilibrium, "he suppor"s mus" becapable o# proiding speci%c minimum number o# #orce res"rain"s. E: #ora simple beam sub0ec"ed "o "he er"ical and horion"al #orces, "he suppor"mus" proide "hree #orce res"rain"s #or i"s e1uilibrium corresponding "o"hree condi"ions o# e1uilibrium namely,

F 8 <, F y 8 <, FM 8 <



12/15/15





12/15/15


2#n"erna"ional College o# Business and$echnology

E3#iliri# an" &opatiility

EHuilibriu% an4 co%patibilit$ are use4 !en tr$ing to fn4 t!e unnon"orces in a structure (bea%, "ra%e or truss); Be"ore eHuilibriu% orco%patibilit$ can be use4 on a structure t!e 4eter%inac$ o" t!e structure

%ust be "oun4; I" t!e structure to be anal$se4 is 4eter%ine4 t!e unnon"orces can be "oun4 using onl$ t!e eHuilibriu% eHuation "or eit!er a 2-D or#-D bo4$; I" t!e structure is "oun4 to be in4eter%inate t!en co%patibilit$con4ition %ust be use4 in a44ition to t!e eHuilibriu% con4itions;

Con4itions o" eHuilibriu%

& s"ruc"ure in general is sub0ec"ed "o a se" o# #orces 'hich include

e"ernal or applied #orces, in"ernal #orces or reac"ions "ha" are deeloped'i"hin "he body a" connec"ion poin"s and grai"y #orces caused by "hemass o# "he elemen"s. $he s"ruc"ure mus" be in "he s"a"e o# s"a"ice1uilibrium 'i"h respec" "o "hese #orces.

i) $ransla"ional e1uilibrium ΣF x=0, ΣF y=0, ΣF z =0

ii) /o"a"ional e1uilibrium ΣM x=0, ΣM y=0, ΣM z =0

Compa"ibili"y

Compa"ibili"y is used 'hen soling inde"ermina"e members because "hee1ua"ion o# e1uilibrium do no" allo' "o sole #or all o# "he unno'ns 'i"hina sys"em. Compa"ibili"y is a me"hod used "o proide e"ra e1ua"ion 'hen"rying "o %nd "he unno'n in an inde"ermina"e member. $his is done byrela"ing "he geome"ry o# "he de#ormed member 'i"h "he unno'n #orces

in "he s"ruc"ure. $his me"hod allo's one or more o# "he unno'n #orces "obe sho'n as a #ac"or o# ano"her unno'n #orce resul" in a remoal o# one



12/15/15


2&n"erna"ional College o# Business and$echnology

0nalysis of Str#$t#res

0nalyti$al !etho"s$here are "hree approaches "o "he analysis

1. Mechanic of Materials (Strength of materials)

6. Elas"ic "heory (Special case o# "he more general %eld o#con"inuum mechanics)

>. ini"e elemen"Mec!anic o" Material Approac!

Assu%ptions

a) $he ma"erials in 1ues"ion are elas"ic, "ha" s"ress is rela"edlinearly "o s"rain

b) Ma"erial (bu" no" "he s"ruc"ure) behaes iden"ically regardlesso# direc"ion o#

"he applied load.

c) &ll de#orma"ions are small.

d) Beams are long rela"ie "o "heir dep"h.



12/15/15


25

n"erna"ional College o# Business and$echnology

Stati$ally "eterinate an" in"eterinate str#$t#res

# #or "he s"able s"ruc"ure i" is possible "o %nd "he in"ernal #orcesin all "he members cons"i"u"ing "he s"ruc"ure and suppor"ing reac"ions a"

all "he suppor"s proided #rom s"a"ical e1ua"ions o# e1uilibrium only, "hes"ruc"ure is said "o be s"a"ically de"ermina"e.

Sys"em #or 'hich "he principles o# s"a"ical e1uilibrium areinsuGcien" "o de"ermine suppor" reac"ions and5or in"ernal #orcedis"ribu"ions, i.e. "here are grea"er number o# unno'ns "han "he numbero# e1ua"ions o# s"a"ical e1uilibrium, are no'n as s"a"ically inde"ermina"eor hypers"a"ic sys"ems.

S"ruc"ural sys"ems may be

=. E"ernally inde"ermina"e bu" in"ernally de"ermina"e

6. E"ernally de"ermina"e bu" in"ernally inde"ermina"e

>. E"ernally and in"ernally inde"ermina"e

?. E"ernally and in"ernally de"ermina"e

& sys"em 'hich is e"ernally and in"ernally de"ermina"e is said "o bede"ermina"e sys"em.



12/15/15


2.


S"a"ical De"erminacy in #rames

!e": m8 "o"al number o# members in a "russ

08 number o# connec"ions

Minimum number o# members re1uired #or "he geome"ric or con%gura"ions"abili"y

m = 2j-3

# mI 60> s"ruc"ure is uns"able

mJ 60> s"ruc"ure 'i"h redundan" members

$he epression seres as an indica"or 'he"her or no" "he in"ernal#orces in a s"ruc"ure can be calcula"ed by "he e1ua"ions o# s"a"ics.

Since all "he 0 0oin"s o# "he s"ruc"ure are in e1uilibrium and "here are "'oe1uilibrium e1ua"ions inoling "he summa"ion o# #orces in K and Ldirec"ions, namely F8< and Fy8< a" each 0oin", 60 e1ua"ions o# s"a"ice1uilibrium are aailable #or "he en"ire s"ruc"ure "o compu"e "he suppor"

reac"ions and in"ernal #orces in all "he members.!e": m8 unno'n member #orces

r8 unno'n suppor" reac"ions

or "he ade1uacy o# "he number o# "he aailable e1uilibrium e1ua"ions "ocompu"e "he suppor" reac"ions and in"ernal #orces in all "he members,

2j=(m+r) or m=2j-r - staticall determinate



12/15/15


28


# 60 I mr, "here are more unno'ns "han "he number o# e1uilibriume1ua"ions, "he s"ruc"ure is s"a"ically inde"ermina"e. $he degree o#inde"erminacy n=(m+r)-2j

# 60 J mr, "here are more e1uilibrium e1ua"ions aailable "han "henumber o# unno'ns, such a s"ruc"ure is a mechanism and al'ays!nsta"le. $he s"ruc"ure does no" hae uni1ue solu"ion. Eis"ence o# more"han one solu"ion indica"es ins"abili"y

E:= De"ermine 'he"her "he "russes sho'n in #ollo'ing %gures are s"able.# s"able, "hen %nd 'he"her "hey are s"a"ically de"ermina"e or

inde"ermina"e.



12/15/15


29


Sign Conen"ion

3se normal sign conen"ion adop"ed #or a >dimensional righ"handed sys"em o# Car"esian or rec"angular coordina"e aes K, L and

N 'i"h origin on "he e"reme le#" o# "he s"ruc"ure. $he #orcesmeasured #rom "he origin "o'ards posi"ie direc"ions o# aes are al'ays posi"ie. /o"a"ional momen"s epressed in ec"or #orm poin"ing "o'ards posi"ie direc"ions o# "he aes are posi"ie. $hus, momen"s "ha" "end "o produce coun"ercloc'ise ro"a"ions are considered posi"ie and "hose"end "o produce cloc'ise ro"a"ions are considered nega"ie.

ree Body Diagrams

or s"a"ic analysis o# bodies sub0ec"ed "o e"ernal loads,analy"ical diagrams "ha" illus"ra"e "he #orce sys"ems ac"ing on "he ob0ec"sare called e1uilibrium or #reebody diagrams. 3sing e1uilibrium concep"s,"he numerical alues o# reac"ions "ha" occur a" suppor"s and hencein"ernal #orces, i.e. aial #orces, shear #orces and bending momen"s canbe de"ermined. $he ma0or applica"ion o# e1uilibrium analysis is in "heealua"ion o# reac"ions and in"ernal #orces by represen"ing an ob0ec" by aseries o# #ree body diagrams.



12/15/15


2:




12/15/15


#+


Shear 4or$e an" -en"ing !oent

/ela"ionships among loading, shear #orce, and bendingmomen"

& small elemen" o# "he beam sho'n in %gure aboe (le#") is "aen a" adis"ance #rom end =. $he #orces ac"ing on "he elemen" are sho'n in%gure aboe (righ").

/esoling #orces er"ically,

- 8 (-O-)'O and O-5O 8 '

!imi"ing condi"ion

dV#d$ = -w indica"es "ha" "he slope o# "he shear #orcediagram, a" any sec"ion, e1uals "he in"ensi"y o# loading a" "ha" sec"ion.

>

>>



12/15/15


#1


Alternatively, since

dV = -wdx

dV = -wdx, and ʃ ʃ

V 2-V 1 = ʃ x1

x2 -wdx where,

V 1= Shear force in the ea! at x=x1 , V 2 = shear force in the ea! at x=x2 and the chan"e in

shear force etween the two sections e#$als the area of the load intensity dia"ra! etween

the two sections%

&a'in" !o!ents ao$t the lower ri"ht corner of the ele!ent "ives the ex(ression

M = )M*+M- V+x * w )+x 2 2

.e"lectin" the s!all val$e )+x 2 ,

+M+x = V

&he li!itin" condition is

dM/dx = V indicates that the slo(e of the endin" !o!ent dia"ra! at any section e#$als the shear force at that section%



12/15/15


#2n"erna"ional College o# Business and$echnology

Alternatively, since

dM = Vdx

dM = Vdx, and ʃ ʃ

M 2-M 1 = ʃ x1

x2 Vdx where,

M 1= /endin" !o!ent in the ea! at x=x1 , M 2 = /endin" !o!ent in the ea! at x=x2 and

the chan"e in endin" !o!ent etween the two sections e#$als the area of the shear force

dia"ra! etween the two sections%



12/15/15


##n"erna"ional College o# Business and$echnology

Analysis of statically determinate structures by Virtual Work and Energy

Theorem

These methods are perfectly adequate for the comparatively simple problems to which they

have been applied. However, other more powerful methods of analysis are required for morecomplex structures which may possess a high degree of statical indeterminacy. These

methods will, in addition, be capable of providing rapid solutions for some statically

determinate problems, particularly those involving the calculation of displacements.

The methods fall into two categories and are based on two important concepts; the first, the

principle of virtual work, is the most fundamental and powerful tool available for the

analysis of statically indeterminate structures and has the advantage of being able to dealwith conditions other than those in the elastic range, while the second, based on strain

energy, can provide approximate solutions of complex problems for which exact solutions

may not exist. The two methods are, in fact, equivalent in some cases since, although the

governing equations differ, the equations themselves are identical.

or'

Before we consider the principle of virtual work in detail, it is important to clarify exactlywhat is meant by work. The basic definition of work in elementary mechanics is that work is

done when a force moves its point of application!.



12/15/15


#&n"erna"ional College o# Business and$echnology

"or case #a$,

%" & "#'cos($ or %" & #" cos($ '

"or case #b$,

%c & "#a)*$+ "#a)*$+ & "a+

"or case #c$,

%- & -+



12/15/15



ote/0 The force, ", and moment, - are in position before the displacements take place and

are not the cause of them. 1lso in 2ase #a$, the component of ' parallel to the direction of "

is in the same direction as "; if it had been in the opposite direction the work done would

have been negative. The same argument applies to the work done by the moment, -, where

we see in 2ase #3$ that the rotation, +, is in the same senses as -. ote also that if thedisplacement, ', had been perpendicular to the force, ", no work would have been done by ".

"inally it should be remembered that work is a scalar quantity since it is not associated with

direction. Thus the work done by a series of forces is the algebaic sum of the work done by

each force.

rinci(le of Virt$al or'

4n figure above a particle, 1, is acted upon by a number of concurrent forces, "5, "*, 66,"k ,6.,"r ; the resultant of these forces is 7. 8uppose that the particle is given a small arbitrary

displacement, 'v, to 1! in some specified direction; 'v is an imaginary or virtual displacement

and is sufficiently small so that the directions of "5, "*, etc., are unchanged. 9et +7 be the

angle that the resultant, 7, of the forces makes with the direction of 'v and +5, +*,6, +k ,6..,

+r the angles that "5, "*,6,"k ,6, "r make with the direction of 'v, respectively.



12/15/15


#.n"erna"ional College o# Business and$echnology

Then the total virtual work, %", done by the forces " as the particle moves through the

virtual displacement, 'v , is given by

%" & "5 'v cos+5 "* 'v cos+* 6 "k 'v cos+k 6 "r 'v cos+r

Thus

4f the particle, 1, is in equilibrium under the action of the forces, "5, "*,6., "k,6, "r, the

resultant, 7 of the force is :ero. 4t follows from the above equation that the virtual work done

by the forces, ", during the virtual displacement, 'v , is :ero.

4t can be stated the principle of virtual work for a particle

If a particle is in equilibrium under the action of a number of forces the total work done

by the forces for a small arbitrary displacement of the particle is zero.

12/15/15



12/15/15



4t is possible for the total work done by the forces to be :ero even though the particle is not

in equilibrium if the virtual displacement is taken to be in a direction perpendicular to their

resultant, 7. %e cannot, therefore, state the converse of the above principle unless we specify

that the total work done must be :ero for any arbitrary displacement.

Thus;

A particle in equilibrium under the action of a system of forces if the total work done by

the forces is zero for ant irtual displacement of the particle.

.ote- &he 3v is a ($rely i!a"inary dis(lace!ent and is not related in anyway to the (ossile

dis(lace!ent of the (article $nder the action of the forces F% 3v has een introd$ced ($rely

as a device for settin" $( the wor'-e#$iliri$! relationshi(% &he forces, F, therefore re!ain

$nchan"ed in !a"nit$de and direction d$rin" this i!a"inary dis(lace!ent4 this wo$ld not

e the case if the dis(lace!ent were real%

rinciple of <irtual work for a 7igid Body

12/15/15



12/15/15



2onsider the rigid body shown in figure above, which is acted upon by a system of external

forces, "5, "*,6., "k ,6, "r . These external forces will induce internal forces in the body,

which may be regarded as comprising an infinite number of particles; on ad=acent particles,

such as 15 and 1*, these internal forces will be equal and opposite, in other words self/

equilibrating. 8uppose now that the rigid body is given a small, imaginary, that is virtualdisplacement, 'v #or a rotation or a combination of both$, in some specified direction. The

external and internal forces then do virtual work and the total virtual work done , %t, is the

sum of the virtual work, %e, done by the external forces and the virtual work, %i, done by

the internal forces.

t = e * i

8ince the body is rigid, all the particles in the body move through the same displacement, 'v,

so that the virtual work done on all the particles is numerically the same. However, for a pair

of ad=acent particles, such as 15 and 1* in above figure, the self equilibrating forces are in

opposite directions, which means that the work done on 15 is opposite in sign to the work

done on 1*. Thus the sum of the virtual work done on 15 and 1* is :ero. The argument can

be extended to the infinite number of pairs of particles in the body from which we conclude

that the internal virtual work produced by a virtual displacement in a rigid body is :ero.

%t & %e

8ince the body is rigid and the internal virtual work is therefore :ero, we may regard the

body as a large particle. 4t follows that if the body is in equilibrium under the action of set of

forces, "5, "*,6., "k ,6, "r , the total virtual work done by the external forces during an

arbitrary virtual displacement of the body is :ero.

12/15/15<irtual work in a deformable body



12/15/15


#:n"erna"ional College o# Business and$echnology

<irtual work in a deformable body

4n structural analysis we are not generally concerned with forces acting on a rigid body.

8tructures and structural members deform under load, which means that if we assign a virtual

displacement to a particular point in a structure, not all points in the structure will suffer the

same virtual displacement as would be the case if the structure were rigid. This means that

the virtual work produced by the internal forces is not :ero as it is in the rigid body case,

since the virtual work produced by the self/equilibrating forces on ad=acent particles does not

cancel out. The total virtual work produced by applying a virtual displacement to a

deformable body acted upon by a system of external forces is

t = e * i

4f the body is in equilibrium under the action of the external force system then every particle

in the body is also in equilibrium. Therefore, from the principle of virtual work, the virtualwork done by the forces acting on the particle is :ero irrespective of whether the forces are

external or internal. 4t follows that, since the virtual work is :ero for all particles in the body,

it is :ero for the complete and

e * i = >

ote that in the above argument only the conditions of equilibrium and the concept of work

are employed. Thus the above equation does not require the deformable body to be linearlyelastic #i.e.it need not obey Hooke!s law$ so that the principle of virtual work may be applied

to any body or structure that is rigid, elastic or plastic. The principle does require that

displacements, whether real or imaginary, must be small, so that we may assume that external

and internal forces are unchanged in magnitude and direction during the displacements.

12/15/154n addition the virtual displacements must be compatible with the geometry of the structure



12/15/15


&+n"erna"ional College o# Business and$echnology

4n addition the virtual displacements must be compatible with the geometry of the structure

and the constraints that are applied, such as those at a support .

roblems

2alculate the support reactions in the simply supported beam shown in figures below.

5. 3.

*. ?.

12/15/15



12/15/15


&1n"erna"ional College o# Business and$echnology

Loa"s a$ting on str#$t#resDea" Loa"s (DL) %Peranent a$tions

$he loads 'hich cons"an" in magni"ude and %ed in loca"ion "hrough ou""he li#e"ime o# "he s"ruc"ure. Ma0or par" is "he 'eigh" o# s"ruc"ure i"sel#and all "he o"her permanen" cons"ruc"ion including serices o# a

permanen" na"ure.

$he charac"eris"ic dead loads can be es"ima"ed using schedule o# 'eigh"so# building ma"erials gien in %S ' (a"le 2.1)* enold,s and"ooor manu#ac"urers li"era"ure.

@$%bols

g -J ni"or%l$ 4istribute4 c!aracteristic 4ea4 loa4s

-J <otal c!aracteristic 4ea4 loa4s

12/15/15



12/15/15


&2


12/15/15



12/15/15


&#


Li'e loa"s(LL)5 +pose" loa"s (+L) % ariale a$tions

$he load assumed "o be produced by "he in"ended occupancy or use,including "he 'eigh" o# moable par"i"ions, dis"ribu"ed, concen"ra"ed,

impac" Q iner"ia loads. $heir magni"ude and dis"ribu"ion any gien "imeare uncer"ain and een "heir maimum in"ensi"ies "hroughou" "he li#e"imeo# "he s"ruc"ure are no" no'n 'i"h precision.

@$%bols

H -J ni"or%l$ 4istribute4 c!aracteristic li7e loa4s

K -J <otal c!aracteristic li7e loa4s

K

12/15/15



12/15/15


&&


En'ironental Loa"s % ariale a$tions

Consis" mainly Sno' load, ind pressure and suc"ion, Ear"h1uae load(i.e iner"ia #orces caused by ear"h1uae mo"ions), Soil pressures on

subsur#ace por"ions o# s"ruc"ure, loads o# possible ponding o# rain 'a"eron 9a" sur#aces and #orces caused by "empera"ure di2eren"ials. $heseloads are uncer"ain bo"h magni"ude Q dis"ribu"ion.

in" Loa"s (L)

ind pressure can ei"her add "o "he o"her grai"a"ional #orces ac"ing on"he s"ruc"ure or e1ually 'ell, eer" suc"ion or nega"ie pressures on "he

s"ruc"ure. 3nder par"icular si"ua"ions, "he la""er may 'ell lead "o cri"icalcondi"ions and mus" be considered in "he design.

$he charac"eris"ic 'ind loads ac"ing on a s"ruc"ure can be assessed inaccordance 'i"h "he recommenda"ions gien in /03 /hater V 0art2 142 5ind 6oads or %S 3 0art 2 14 /ode of 0ractice for5ind 6oads.

@$%bols -J ni"or%l$ 4istribute4 c!aracteristic in4 loa4s

L -J <otal c!aracteristic in4 loa4s

12/15/15



12/15/15



DES+9 !E.HDS

/ela"ionship be"'een s"ress and s"reng"h

1: Perissile stress "esignn permissible s"ress design, some"imes re#erred "o as modular ra"io orelas"ic design, "he s"resses in "he s"ruc"ure a" 'oring loads are no"allo'ed "o eceed a cer"ain propor"ion o# "he yield s"ress o# "hecons"ruc"ion ma"erial, i.e. "he s"ress leels are limi"ed "o "he elas"ic range.

;: Loa" fa$tor "esign

!oad #ac"or or plas"ic design 'as used "o "ae accoun" o# "he behaiour o#"he s"ruc"ure once "he yield poin" o# "he cons"ruc"ion ma"erial had beenreached. $his approach inoled calcula"ing "he collapse load o# "hes"ruc"ure. $he 'oring load 'as deried by diiding "he collapse load by aload #ac"or.

12/15/15



12/15/15


&.n"erna"ional College o# Business and$echnology

<: Liit State Design

$he limi" s"a"e design can be seen as a compromise be"'een "he permissible and load #ac"or me"hods. " is in #ac" a more comprehensieapproach 'hich "aes in"o accoun" bo"h me"hods in appropria"e 'ays.Mos" modern s"ruc"ural codes o# prac"ice are no' based on "he limi" s"a"eapproach such as,

BS @==< #or concre"e BS *R*< #or s"ruc"urals"eel'or

BS *?<< #or bridges BS *6@ #or masonry

Code o# prac"ice #or design in "imber, BS*6@ and old s"ruc"ural s"eel'orcode, BS ??R are based on permissible s"ress designs.

/ltiate liit state

$he 'hole s"ruc"ure or i"s elemen"s should no" collapse, oer"urn orbucle 'hen sub0ec"ed "o "he design loads. Considera"ions are,

S"reng"h

$he s"ruc"ure mus" be designed "o carry "he mos" seere combina"ion o#loads "o 'hich i" is sub0ec"ed. $he sec"ions o# "he elemen"s mus" becapable o# resis"ing aial loads, shears and momen"s deried #rom "heanalysis. $he design is made #or ul"ima"e loads and design s"reng"hs o#ma"erials 'i"h par"ial sa#e"y #ac"ors applied "o loads and ma"erials"reng"hs.

12/15/15



12/15/15



S"abili"y

$he layou" should be such as "o gie a s"able and robus" s"ruc"ure. eralls"abili"y should ensure compa"ibili"y o# design and de"ails o# par"s andcomponen"s. $he s"ruc"ure should be such as "o "ransmi" all loads, dead,

imposed and 'ind, sa#ely "o "he #ounda"ions.

/obus"ness

Damage "o a small area or #ailure o# a single elemen" should no" causecollapse o# a ma0or par" o# a s"ruc"ure. $his means "ha" "he design shouldbe resis"an" "o progressie collapse.

Ser'i$eaility liit state$he s"ruc"ure should no" become un%" #or use due "o ecessie de9ec"ion,cracing or ibra"ion. Considera"ions are,

De9ec"ion

$he de#orma"ion o# "he s"ruc"ure should no" adersely a2ec" i"s eGciencyor appearance.

Cracing

Cracing should be ep" 'i"hin reasonable limi"s by correc" detailing

12/15/15.1+2BE- ecture =oteJ



Design 7alues o"actions

12/15/15


&9


+1

In general, t!e 4esign 7alue o" an action is obtaine4 b$ %ultipl$ing

t!e representati7e 7alue b$ t!e appropriate partial sa"et$ "actor "oractions;

<!e %a3i%u% 7alues o" partial sa"et$ "actors "or per%anent an47ariable actions reco%%en4e4 in EC1 are 1;#5 an4 1;5 respecti7el$;

<!e co%parable 7alues in B@ 911+ are 1;& an4 1;.;

It can also be seen t!at t!e partial sa"et$ "actors "or actions 4epen4

on a nu%ber o" ot!er aspects inclu4ing t!e categor$ o" li%it stateas ell as t!e eect o" t!e action on t!e 4esign situation un4erconsi4eration;




12/15/15


&:


+1




/ /


5+


+1

N+ - acco%pan$ing "actor "or co%bination 7alue

N1 - acco%pan$ing "actor "or "reHuent 7alue

N2 - acco%pan$ing "actor "or Huasi-per%anent7alue

O - e4uction "actor "or un"a7orable per%anentactions reco%%en4e4 7alue is +;:25 in Q national

Anne3

12/15/15



/ /

Loa" .ransfer

ig: Se1uence o# load "rans#er be"'een elemen" o# a s"ruc"ure

Bridging Program- Lecture Note 01

Documents

Transcript of Bridging Program- Lecture Note 01