221

J

8Deflections of Beams:Geometric Methods6.' D,Nerenlrat Equallon 'or Beam Deflectron6.2 Drectlnlegrallon Meltlod6.3 Supelllos~JOn Meltlod6.4 Moment·Area Method6.5 Bending Momenl Dragrams by Parts6.6 Conjugate Beam Method

SummaryProbtems

Strut:lUfC like ,III \lthcr ph} \11.:,11 bodic!'>. defonn and change shape\\h"o suh)ltCtcd to fon.:c\_ Olher common causes of deformations of'JtrUl.:lUn:, mdude Icmpa,llurc changes and support settlements. If thedeltHJndllons t.I1\.lrrcar .md the 'itructure regains its original shape whenthe action ... l.:.1U mg the deformations are removed, the deformations areh:nncd deWit ,/(/omwflofh. The permanent deformations of structuresare referred h) ..... mt/mlll or ,,!mli£ cJelormutions. In this text \\e WIllIlx:us our alh.:nUon on l/nwr t/tlllic tIt1uT/1wtimlJ, Such deformations\011") Imearly "lth dpphcd loads rfor instance. if the magnitudes of theload altlflg on the lrul.:ture ar.: dl)Ubled. its deformations are alsodoubled. and so forth Rel:alllr\)m Scl,;lion 3.6 that in order for a struc.lure to rc.: pond Imearh 10 Jpphed load'\.. It musl be composed of lInearcia til. matenal ilnd II mu t undergo ~mall deformations. The pnoclplef UJlt.iTk.'SHlon I \ahd lor ut:h trudures.

h r rno I trueture e;\l.:\." (ve deformations are undesirable athey ma~ lmp.ur the tructurc s abiht~ 10 '!oCl"\e its intended purposef or example a high n bUlldmg rna) bt= perll.'\:tl) safe in the sen thatthe II )v-able Ir re dft a n I,",eec ed )CI useless unlX"Cupied If It ....

51 Ie\: I I) due (0 \\md l.:au 109 uack in the \\alls and windowruclUre re u uallv de d h

d Ign 0 I at their deflections under DOrmalm;~ n III n \\111 not e l.:eed the aJlo\\able \alues specified tn buiJd.

SECTION 1.1 DI""..... ' ............. Dsn •

6.1 DIFFERENTIAL EQUATION FOR BEAM DEFLECTION

•

221

J

8Deflections of Beams:Geometric Methods6.' D,Nerenlrat Equallon 'or Beam Deflectron6.2 Drectlnlegrallon Meltlod6.3 Supelllos~JOn Meltlod6.4 Moment·Area Method6.5 Bending Momenl Dragrams by Parts6.6 Conjugate Beam Method

SummaryProbtems

Strut:lUfC like ,III \lthcr ph} \11.:,11 bodic!'>. defonn and change shape\\h"o suh)ltCtcd to fon.:c\_ Olher common causes of deformations of'JtrUl.:lUn:, mdude Icmpa,llurc changes and support settlements. If thedeltHJndllons t.I1\.lrrcar .md the 'itructure regains its original shape whenthe action ... l.:.1U mg the deformations are removed, the deformations areh:nncd deWit ,/(/omwflofh. The permanent deformations of structuresare referred h) ..... mt/mlll or ,,!mli£ cJelormutions. In this text \\e WIllIlx:us our alh.:nUon on l/nwr t/tlllic tIt1uT/1wtimlJ, Such deformations\011") Imearly "lth dpphcd loads rfor instance. if the magnitudes of theload altlflg on the lrul.:ture ar.: dl)Ubled. its deformations are alsodoubled. and so forth Rel:alllr\)m Scl,;lion 3.6 that in order for a struc.lure to rc.: pond Imearh 10 Jpphed load'\.. It musl be composed of lInearcia til. matenal ilnd II mu t undergo ~mall deformations. The pnoclplef UJlt.iTk.'SHlon I \ahd lor ut:h trudures.

h r rno I trueture e;\l.:\." (ve deformations are undesirable athey ma~ lmp.ur the tructurc s abiht~ 10 '!oCl"\e its intended purposef or example a high n bUlldmg rna) bt= perll.'\:tl) safe in the sen thatthe II )v-able Ir re dft a n I,",eec ed )CI useless unlX"Cupied If It ....

51 Ie\: I I) due (0 \\md l.:au 109 uack in the \\alls and windowruclUre re u uallv de d h

d Ign 0 I at their deflections under DOrmalm;~ n III n \\111 not e l.:eed the aJlo\\able \alues specified tn buiJd.

SECTION 1.1 DI""..... ' ............. Dsn •

6.1 DIFFERENTIAL EQUATION FOR BEAM DEFLECTION

•

6.4 MOMENT-AREA METHOO

MDiapm- __-'--..J EJ

lECTJONu --- mM, P, P,

wlx)

A BeamI--x I /Ix I

T.....'"B

in wblcb fl. and fI, are Ibe slopes of tbe ela tiC curve al pomt A and BrespectIvely. wltb respect 10 tbe axis of tbe beam m tb und ~ rmOO(borizontal stale fllU denotes tbe angle bet\\een tbe t ng n t Ielasllc curve at A and Band f: \I £1 d< repre n th undoM EI diagram between polDIS A and B

Equallon 6.12 represents Ibe malbern IIcalmomtnl-a"o Ih "m wblcb can be stated a ~ II

The chan In 51 pc bet\\een !be la

polDl I equalt tbe a... under ,he \I EI ~:::,:proVided that the elastIC curve I Un LU

fIG. 6.4

J'M

Eldxo,

or

- -- - --------

The momt'n;~~re.1 method for computing ...Iopes and def1ectio~sof beams\\3" de\dopcd 1:1) Charlc... E Greene m 187J, The me,thod IS based 011t\\O thcorem .... called the mOn/tIlt-area ,I"ore",.\'. reiatlOg the geometryof the ela... lic cune of a ocam to 1" \f EI diagram. \\ hich is constructedb) di\ iding Ihe ordmate... of the bendmg moment diagram by the ftexu­ral rigid II} £1. The method uuhle~ graphical mterpretations of integra)imohed in Ihe \DIUlion of the deflection dllTcrentJal equation (Eq. 69in term... of the ilreas and the moments of arcas of the .\1/El diagram.Therefore. II is more comcnient to use for beams with loading dlscootinuitle... and the \ariahle EI as compared to the direct integratlODmethod de-.cribcd pre\iou ..l)

To derive the moment-area theorems, consider a beam subjectedto an arbitrar} loading as sho",n in Fig. 6.4. The clastic curve and the.\1 £1 diagram for the beam arc also shown in the figure. FOCUSing ourattention on a differential clement d\ of the beam. \\ e recall from thepre\iou... ~ctlon .IEq. (6.10)) that dO, which represents the change mslope of the e1a!lllc curve mer the differential length dx. is given by

Ifdl/ EI dl 611

Note thai the lerm \1 £1 dx repre\ents an infinitesimal area underM Ef dIagram. as ho\\nin Fig. 6.4. To determine the change ID slopebet\\een (\\0 arbitrary pomts A and 8 on the beam we integrate6.11 from A to B 10 obtam '

rdll r~d<

DetIectionI of Beams: Geometric Methods. lIlJl\ idu.dl~ on the beam. The slope and

to ""h.h \'1 Ihe I(lad .ll.:tdmg·d

.1 kllJ C,1O Ix l,:oll1pulcd by using elt"-. d tel -il HI 1\ I u.· _ ICT

dl,;lkxllI.m Ut' ll.l h' d' 'Tlocd prC\10U'iJ) or one of the 0'''--I In mel llU l.... ~

Ihe Jlftx:t mh:gr.t It b' 'Ill ~dlon' Aho. man) struclural tnDlh d d · J III ..u -.t:qUt; -' eo

Illet i..' I"l.-U 'l II nlUlI ofS(I.'/ ('Of1\trlU fUm p.ubhshed by .L_. - h ndh:ll1k o! ,1 _ 'UlI:

nC\:nng a" ~ I (QmITUl 1/ n wllta," ddkClIon formulas for""til TIl all Inwfllt: ,: ..'1 ~,f lo.ld...and \Upport condltion\. ," hich can bet'leam.. fN \.trI(lU ., rx

s. -h fonnula ... for ,Io~' and deflections of ....._-­

d tor Ihl'" rurpo"C: .. m: . . .-msu . 11 Illd .tOd ,upport conditio"" are gl\en IDslClefor \,.lmt' l,.'omnll1n t~ pt: \. l • , .~ .

_ '·,h. bOl1k for clm\t'Olcnt rekren\,;t'the tronl l:l" er 0 ..

CHAPTEIII

6.4 MOMENT-AREA METHOO

MDiapm- __-'--..J EJ

lECTJONu --- mM, P, P,

wlx)

A BeamI--x I /Ix I

T.....'"B

in wblcb fl. and fI, are Ibe slopes of tbe ela tiC curve al pomt A and BrespectIvely. wltb respect 10 tbe axis of tbe beam m tb und ~ rmOO(borizontal stale fllU denotes tbe angle bet\\een tbe t ng n t Ielasllc curve at A and Band f: \I £1 d< repre n th undoM EI diagram between polDIS A and B

Equallon 6.12 represents Ibe malbern IIcalmomtnl-a"o Ih "m wblcb can be stated a ~ II

The chan In 51 pc bet\\een !be la

polDl I equalt tbe a... under ,he \I EI ~:::,:proVided that the elastIC curve I Un LU

fIG. 6.4

J'M

Eldxo,

or

- -- - --------

The momt'n;~~re.1 method for computing ...Iopes and def1ectio~sof beams\\3" de\dopcd 1:1) Charlc... E Greene m 187J, The me,thod IS based 011t\\O thcorem .... called the mOn/tIlt-area ,I"ore",.\'. reiatlOg the geometryof the ela... lic cune of a ocam to 1" \f EI diagram. \\ hich is constructedb) di\ iding Ihe ordmate... of the bendmg moment diagram by the ftexu­ral rigid II} £1. The method uuhle~ graphical mterpretations of integra)imohed in Ihe \DIUlion of the deflection dllTcrentJal equation (Eq. 69in term... of the ilreas and the moments of arcas of the .\1/El diagram.Therefore. II is more comcnient to use for beams with loading dlscootinuitle... and the \ariahle EI as compared to the direct integratlODmethod de-.cribcd pre\iou ..l)

To derive the moment-area theorems, consider a beam subjectedto an arbitrar} loading as sho",n in Fig. 6.4. The clastic curve and the.\1 £1 diagram for the beam arc also shown in the figure. FOCUSing ourattention on a differential clement d\ of the beam. \\ e recall from thepre\iou... ~ctlon .IEq. (6.10)) that dO, which represents the change mslope of the e1a!lllc curve mer the differential length dx. is given by

Ifdl/ EI dl 611

Note thai the lerm \1 £1 dx repre\ents an infinitesimal area underM Ef dIagram. as ho\\nin Fig. 6.4. To determine the change ID slopebet\\een (\\0 arbitrary pomts A and 8 on the beam we integrate6.11 from A to B 10 obtam '

rdll r~d<

DetIectionI of Beams: Geometric Methods. lIlJl\ idu.dl~ on the beam. The slope and

to ""h.h \'1 Ihe I(lad .ll.:tdmg·d

.1 kllJ C,1O Ix l,:oll1pulcd by using elt"-. d tel -il HI 1\ I u.· _ ICT

dl,;lkxllI.m Ut' ll.l h' d' 'Tlocd prC\10U'iJ) or one of the 0'''--I In mel llU l.... ~

Ihe Jlftx:t mh:gr.t It b' 'Ill ~dlon' Aho. man) struclural tnDlh d d · J III ..u -.t:qUt; -' eo

Illet i..' I"l.-U 'l II nlUlI ofS(I.'/ ('Of1\trlU fUm p.ubhshed by .L_. - h ndh:ll1k o! ,1 _ 'UlI:

nC\:nng a" ~ I (QmITUl 1/ n wllta," ddkClIon formulas for""til TIl all Inwfllt: ,: ..'1 ~,f lo.ld...and \Upport condltion\. ," hich can bet'leam.. fN \.trI(lU ., rx

s. -h fonnula ... for ,Io~' and deflections of ....._-­

d tor Ihl'" rurpo"C: .. m: . . .-msu . 11 Illd .tOd ,upport conditio"" are gl\en IDslClefor \,.lmt' l,.'omnll1n t~ pt: \. l • , .~ .

_ '·,h. bOl1k for clm\t'Olcnt rekren\,;t'the tronl l:l" er 0 ..

CHAPTEIII

-SECT10IIu _

ProeedUJlI for Analysis

61

6

Sub-.tllution of

dO

( \~) ,",U

dl

.11

8 t' [he ckment lh"here J the dl 1.101,.·..: In1m l

ft.!1 mIl) Eq "I ~ \Idd:lo

r: d\f'1 Hit,U

"r

fitAs< ,d,,E/

in \\hich ~'H fl:prc..enh the IIl1/e/l'II.';O/ dlrwtioll of Bfrom the tanat 4 \\ hlch ... the deflection of rOlll! B In the dlrl:~t1on perpendlto the undcfonncd ,1\1'" l,f the beam from the tangcnt at pomt Ar:, \1 f1jrdr Tepre..."nh lhe moment of the Jrea undcr the M £1gram bctwt.-cn poinh 4 itnd B about pomt B

Equation (6.15) rcpre~nh the mathematical expression of Ihond m01lUnr·/lHU Ihl oft m, \\hil.:h can he stated as folloYos.

Tht' lang~nllal dC\lalion In Ih~ dtf('(tl\ln pcrlll:ndil.:ular to the undefialll of Ihe tlcam tIt a POint on the (I,...u!.: ("une fwm the laD DtcIa tiC tune Jt .molher pt1lnll" equal {{l lhl: moment or the area\11:/ diagram hel\\.een (hI; 1\\0 pl1mt.. Jboul Ihe pomt al ",hi htlon I d m:J pr l\u,k"11lhal the cia Ih,; I.un I l;llntmUtlU betweenpomts

It is Imponant 10 nole the ord~r of Ihe subscripts used for6 15 The fir,t !lu~npt denutes the pt.)!nt "here the devlaUoD

temuned and ahout "hl..:h the muments are c\aluated whereond ubs4;npt denote the poml \I, here the tangent 10 the elastICdra\l,n Also In the dlstanl.:e an Eq. 6.1<; i alway takentl\e the IgD or ~B I Ihe same as thaI of Ihe area of the Mgram bel\\ecn A and B If ,h area "f 'he If Ef dlagram beltwe_and B 1 positne then ~B IS also po Ilne and pomt B Ithe po5ll1 direct hIOn t e tang ot to the cia tl4; cune at polo

ICe 'rna

. on the nl!ht~h.:md side of Eq. 6.14 representote Lhal Ihl.: h;~ • 1m 11 .trC.1 u"rf\: pendmg to til: llbout B

n1C'mcntEcJ t~el~n ~~~lcen an ... (\\(l arhllraf) point -f and B on

granng q.•' .bc:J.m \\e: obtain

DeflectionS of Bums: Geometric Methods _

J I t) thl,.' unddomll.:d aXIs of theI ........qxllllU.lfl

ek'l11l n1 d, llO I 1Jll ,.

lrllm a roml 81' gl\l;ll t'l~

CHAPTtR 8

-SECT10IIu _

ProeedUJlI for Analysis

61

6

Sub-.tllution of

dO

( \~) ,",U

dl

.11

8 t' [he ckment lh"here J the dl 1.101,.·..: In1m l

ft.!1 mIl) Eq "I ~ \Idd:lo

r: d\f'1 Hit,U

"r

fitAs< ,d,,E/

in \\hich ~'H fl:prc..enh the IIl1/e/l'II.';O/ dlrwtioll of Bfrom the tanat 4 \\ hlch ... the deflection of rOlll! B In the dlrl:~t1on perpendlto the undcfonncd ,1\1'" l,f the beam from the tangcnt at pomt Ar:, \1 f1jrdr Tepre..."nh lhe moment of the Jrea undcr the M £1gram bctwt.-cn poinh 4 itnd B about pomt B

Equation (6.15) rcpre~nh the mathematical expression of Ihond m01lUnr·/lHU Ihl oft m, \\hil.:h can he stated as folloYos.

Tht' lang~nllal dC\lalion In Ih~ dtf('(tl\ln pcrlll:ndil.:ular to the undefialll of Ihe tlcam tIt a POint on the (I,...u!.: ("une fwm the laD DtcIa tiC tune Jt .molher pt1lnll" equal {{l lhl: moment or the area\11:/ diagram hel\\.een (hI; 1\\0 pl1mt.. Jboul Ihe pomt al ",hi htlon I d m:J pr l\u,k"11lhal the cia Ih,; I.un I l;llntmUtlU betweenpomts

It is Imponant 10 nole the ord~r of Ihe subscripts used for6 15 The fir,t !lu~npt denutes the pt.)!nt "here the devlaUoD

temuned and ahout "hl..:h the muments are c\aluated whereond ubs4;npt denote the poml \I, here the tangent 10 the elastICdra\l,n Also In the dlstanl.:e an Eq. 6.1<; i alway takentl\e the IgD or ~B I Ihe same as thaI of Ihe area of the Mgram bel\\ecn A and B If ,h area "f 'he If Ef dlagram beltwe_and B 1 positne then ~B IS also po Ilne and pomt B Ithe po5ll1 direct hIOn t e tang ot to the cia tl4; cune at polo

ICe 'rna

. on the nl!ht~h.:md side of Eq. 6.14 representote Lhal Ihl.: h;~ • 1m 11 .trC.1 u"rf\: pendmg to til: llbout B

n1C'mcntEcJ t~el~n ~~~lcen an ... (\\(l arhllraf) point -f and B on

granng q.•' .bc:J.m \\e: obtain

DeflectionS of Bums: Geometric Methods _

J I t) thl,.' unddomll.:d aXIs of theI ........qxllllU.lfl

ek'l11l n1 d, llO I 1Jll ,.

lrllm a roml 81' gl\l;ll t'l~

CHAPTtR 8

IECT10ll u .....--. DID........ " .....

,,....---(r-Ft6. 6.10

-

c

., ...ms: GeOmetric M...odaDeRictiN.ClIAl"IER •

1Ilk ~ klftA B Ie

Hill

c::r:=Q!~C .;i I c + A ....I t

Jk Jk J I t4k 16k30 k. 3D k

JOft I ID ft

225

169

8 C +

CIIA II

h6k 46kt120

I JOft I lOft

120

fa (b, (e)

225

d)

120

Ill. ..9

6.-5-8E-N-D-IN-G-M-O-:-M=E:::N::T-=D~IA:'G:;R:'A~M;;;S;B;Y;-PIA~TS. . IpplKallon of Ihe moment

J n' '(tll'O.•\ ... Jllu..tr.llw 10 Ihl: prc.."'(c I)~ of the ;In:.'" and mome~ts of areas

h _.1 nH11\L'''' l,.'llmrut.lll~ m It \\ ill ~ ..hown In the fol~01('( lOU I . \I f I dlagr.1 . ,. d fl

nolrthlO' 1.." the - lh ld for detcnmnmg e eCliona\anou" I"~ • be 1m me l - . h '-__-I n that the (cmJugah:- < - I" thc: ...e quanuues. W en a~

-.ex 10 mputatu1n 0 . . f d..... 11"'0 n:ljum:' \.'t

1. J m.:h as a combmatlon 0 lStI'i.......tm.. • . 01 loa ... ... .

..ubJccted to Jltfl.'renl t~ rc~ dc{ermin.Hllm of the properties of theut·d and COl1l.:Cnlralcd loa h' 'ombmcd effect of all the loads,

t: - due to I I: r..: oded b..ultant \I 1::/ tlJagrJnl. Th- difficult\ can be 3\01 Y con tl'UCtba:ome d fomlldabk t.l"~ ,... .

IECT10ll u .....--. DID........ " .....

,,....---(r-Ft6. 6.10

-

c

., ...ms: GeOmetric M...odaDeRictiN.ClIAl"IER •

1Ilk ~ klftA B Ie

Hill

c::r:=Q!~C .;i I c + A ....I t

Jk Jk J I t4k 16k30 k. 3D k

JOft I ID ft

225

169

8 C +

CIIA II

h6k 46kt120

I JOft I lOft

120

fa (b, (e)

225

d)

120

Ill. ..9

6.-5-8E-N-D-IN-G-M-O-:-M=E:::N::T-=D~IA:'G:;R:'A~M;;;S;B;Y;-PIA~TS. . IpplKallon of Ihe moment

J n' '(tll'O.•\ ... Jllu..tr.llw 10 Ihl: prc.."'(c I)~ of the ;In:.'" and mome~ts of areas

h _.1 nH11\L'''' l,.'llmrut.lll~ m It \\ ill ~ ..hown In the fol~01('( lOU I . \I f I dlagr.1 . ,. d fl

nolrthlO' 1.." the - lh ld for detcnmnmg e eCliona\anou" I"~ • be 1m me l - . h '-__-I n that the (cmJugah:- < - I" thc: ...e quanuues. W en a~

-.ex 10 mputatu1n 0 . . f d..... 11"'0 n:ljum:' \.'t

1. J m.:h as a combmatlon 0 lStI'i.......tm.. • . 01 loa ... ... .

..ubJccted to Jltfl.'renl t~ rc~ dc{ermin.Hllm of the properties of theut·d and COl1l.:Cnlralcd loa h' 'ombmcd effect of all the loads,

t: - due to I I: r..: oded b..ultant \I 1::/ tlJagrJnl. Th- difficult\ can be 3\01 Y con tl'UCtba:ome d fomlldabk t.l"~ ,... .