Method of Moment Area Equations

Transcription

1 Noe proided b JRR Page-1 Noe proided b JRR Page- Inrodcion ehod of omen rea qaions Perform deformaion analsis of flere-dominaed srcres eams Frames asic ssmpions (on.) No aial deformaion (aiall rigid members) no change in lengh of all members o Proide eqaions o deermine isplacemen Roaion eformed sae Undeformed sae eformed sae Undeformed sae o Undeformed sae eformed sae asic ssmpions Small displacemen (,) and small roaion () /, /, << 1; ~ characerisic dimension of he srcre Roaion is approimaed b d/ rare is approimaed b κ d / Kinemaics of he cross secion Plane secion remains plane No shear deformaion Plane secion alwas normal o Neral is aerial behaior inearl elasic, i.e. linear sress-srain relaion Isoropic, i.e. maerial properies are direcional independen σ ong odls 1 ε N Undeformed sae N eformed sae qilibrim of he srcre qilibrim eqaions are se p on ndeformed sae

2 Noe proided b JRR Page- Noe proided b JRR Page-4 omen rea qaions Firs omen rea qaion asic qaions Kinemaics κ d d d omen-rare Relaionship Plane secions remain plane ε κ d hange of angle oer he porion aerial behaior σ ε Graphical Inerpreaion qilibrim of he cross secion κ Saic eqilibrim dv d q ; V I is he momen of ineria of he cross secion in erms of applied loads d omen-rare Relaion / diagram d d

3 Noe proided b JRR Page-5 Noe proided b JRR Page-6 Firs omen rea qaion (on.) Second omen rea qaion d hange of angle oer he porion d omen-rare Relaionship ssme no disconini of slope wihin porion (e.g. no hinge) d Toal change of angle oer he porion d hange of angle oer he porion ssme small roaion / rea / rea nder / diagram oer porion 1 s ONT R QUTION d ( ) d ( ) d deiaion of elemen measred on a normal line a poin Toal change of angle oer he porion Graphical Inerpreaion Sign onenion and Remarks Graphical Inerpreaion -, are posiie when he are measred W from ndeformed sae d d / / is posiie when i is measred W from angen line from Sign conenion of bending momen follows he local coordinae - / diagram + / diagram - -

4 Noe proided b JRR Page-7 Noe proided b JRR Page-8 Second omen rea qaion (on.) lernaie Second omen rea qaion d ( ) d ( ) eiaion of elemen measred on a normal line a poin d d eiaion of elemen measred on a normal line a poin ssme no disconini of slope and displacemen (e.g. hinge&shear release) ssme no disconini of slope and displacemen (e.g. hinge&shear release) d ( ) Sm deiaion measred on a normal line a poin oer he porion d Sm deiaion measred on a normal line a poin oer he porion omen of area abo poin of / diagram oer a porion omen of area abo poin of / diagram oer a porion / ( rea/ ) nd ONT R QUTION / ( rea/ ) nd ONT R QUTION Relaie disance of poin relaie o Relaie disance of poin relaie o Graphical Inerpreaion Sign onenion and Remarks Graphical Inerpreaion Sign onenion and Remarks enroid / diagram / +, are posiie when he direc in posiie -direcion / is posiie when ecor emanaing from poin on he angen line a o poin direcs in posiie -direcion Sign conenion of bending momen and he roaion follows he same conenion described aboe is he disance along he -ais measred from poin o he cenroid of / diagram /,, and are relaed o / ( + ) / enroid / diagram / is posiie when ecor emanaing from poin on he angen line a o poin direcs in posiie -direcion Sign conenion of bending momen follows same conenion shown aboe is he disance along he -ais measred from poin o he cenroid of / diagram /,,, and are relaed o / ( ) Sign conenion of,, follows he same conenion shown aboe

5 Noe proided b JRR Page-9 Noe proided b JRR Page-1 engh onsrain qaion No ial eformaion Small isplacemen and Roaion No hange in engh of ember pplicaion of omen rea and engh onsrain qaions Smmar of qaions independen eqaions per member / rea / ongidinal isplacemen are onsan / / ( + ) ( rea/ ) ( ) ( rea/ ) 1 s omen rea qaion / / diagram enroid / Graphical Inerpreaion Sign onenion and Remarks nd omen rea qaion is he lengh of he member measred in he longidinal direcion of he ndeformed sae of he member is he longidinal componen of he displacemen a an poin wih he member engh onsrain qaion Remark: The sperscrip or sbscrip is sed o emphasize ha qaniies are associaed wih he member,, and are posiie when he direc in posiie -direcion The real lengh of he deformed sae can be approimaed b he projeced lengh proided ha displacemen and roaion of he member are small Kinemaical Unknowns nknowns a poin : nknowns a poin :,, Toal 6 nknowns per member, and, and The roaions {, } and he ranserse componens of he displacemen {, } are relaed b he 1 s and nd momen area eqaions The longidinal componens of he displacemen {, consrain eqaion } are relaed b lengh

6 Noe proided b JRR Page-11 Noe proided b JRR Page-1 Usefl Remarks Remark1: If one of {, consrain eqaion } is known, he oher can be obained from lengh One of {, } is known engh onsrain qaion Usefl Remarks (on.) Remark5: If here eiss a poin wihin he srcre where he roaion and he wo componens of he displacemen are known (e.g. a poin a a fied sppors or a poin where heir roaion and displacemen were alread comped), he roaions and displacemens a all oher poins can be deermined from 1 s and nd momen area eqaions and he lengh consrain eqaion Remark: If one of {, } and one of {, } are known, he oher wo of {,,, } can be comped from he 1 s momen area eqaion and hen follow b he nd momen area eqaions One of {, One of {, } is known } is known 1 s & nd omen rea qaions Remark: If {, } are known, one of roaions {, } is comped from nd momen area eqaion and he oher roaion is obained from 1 s momen area eqaions Remark6: member or a segmen sed in he calclaion ms no conain hinge ecep a is ends {, } are known nd omen rea qaions 1 s omen rea qaions Remark4: If all hree nknowns are known a one end, oher hree nknowns a he oher end of he member can be comped from 1 s and nd momen area eqaions and he lengh consrain eqaion F {,, } are known 1 s & nd omen rea qaions or {,, } are known engh onsrain qaion Segmens and (conain hinge inside) are no allowed Segmens,,, and (conain hinge a he ends) can be sed

7 Noe proided b JRR Page-1 Noe proided b JRR Page-14 ample 1 Gien a saicall deerminae frame sbjeced o a concenraed load P as shown in he figre. The ong s modls and momen of ineria of he cross secion are consan and denoed b and I, respeciel. eermine he displacemens and roaions a,,,, and. P Obain and / diagram based on he local coordinae ssems R Skech Qaliaie lasic re R P/ P -P/ -P/ 4P/ P/ R P/ Solion efine local coordinae ssems for he segmens, and, Sar a segmen since qaniies are alread known ( & ) engh onsrain qaion nd omen rea qaion ompe sppor reacions from saic eqilibrim eqaions (i.e. ΣF ΣF Σ ) P R R P/ R P/ / 1 s omen rea qaion / / ( + ) ( rea/ ) 1 4P ( + ( ) ( ) ) + 1 P 7P + ( ) 7P 7P 9 9 7P 1 4P 1 P / rea ( ) ( ) / + 9 1P 18 P

8 Noe proided b JRR Page-15 Noe proided b JRR Page-16 oe o segmen displacemen and roaion a poin are alread known 7P 9 engh onsrain qaion / nd omen rea qaion / engh onsrain qaion 1 s omen rea qaion / 5P 9 14P 9 P P rea ( ) / P nd omen rea qaion / / / 1 s omen rea qaion 7P 1 4P ( + ) ( rea/ ) ( ) + 9 ( ) P P 8P 9 oe o segmen displacemen and roaion a poin are alread known P 5P 9 / 7P 9 5P rea / 1 4P ( ) 9 4P / 5P P P ( ) ( rea/ ) ( ) 9 ( ) 19P 19P oe o segmen displacemen and roaion a poin are alread known 19P 18 P 14P / / 9 engh onsrain qaion 19P 19P s omen rea qaion / 14P 9 7P 18 1 rea P / ( ) nd omen rea qaion / P ( ) ( rea/ ) P 14P 9 ( ) P 9 P 9 P P ( )

9 Noe proided b JRR Page-17 Noe proided b JRR Page-18 ample Gien a saicall deerminae beam sbjeced o eernal loads as shown in he figre. The ong s modls is consan and he momen of ineria of he cross secion is denoed b I for a segmen and b I for segmens and. eermine he relaie roaion a he hinge and he displacemens a he end poin. Skech Qaliaie lasic re q q Sar a segmen displacemen and roaion a poin are alread known Solion ocal coordinae ssems for all segmens are he same as he global coordinae ssem ompe sppor reacions from saic eqilibrim eqaions (i.e. ΣF ΣF Σ ) q q q R R q/ R q/ Obain and / diagram based on he local coordinae ssems q q q R R q/ R q/ -q / -q / -q / nd omen rea qaion / / / ( + ) ( rea/ ) engh onsrain qaion 1 s omen rea qaion + ( ) ( / 1 q ( ) ( ) ) 4 4q 1 q 5q 5q q + 4 4q rea / 1 q ( ) + ( ) 4 ( )

10 Noe proided b JRR Page-19 Noe proided b JRR Page- oe o segmen qaniies a boh ends of segmen are alread known 4 4q oe o segmen displacemen and roaion a poin are alread known / / engh onsrain qaion nd omen rea qaion / ( + ) ( rea/ ) / / engh onsrain qaion nd omen rea qaion nd omen rea qaion 4 4q + q 1 q ( ) ( ) / ( + ) ( rea ) 1 q 4 4 q q / ( + ( ) ( ) ) q 1 q / rea/ ( ) Relaie roaion a hinge nd omen rea qaion 1 q q q / rea/ ( ) 5q 6 q 11q 6

11 Noe proided b JRR Page-1 Noe proided b JRR Page- ample Gien a saicall deerminae frame sbjeced o eernal loads as shown in he figre. The ong s modls and momen of ineria of he cross secion are consan and denoed b and I, respeciel. eermine he displacemens and roaions,,, and. P P Obain and / diagram based on he local coordinae ssems 4P/ P/ P/ Skech Qaliaie lasic re Solion efine local coordinae ssems for he segmens, and ompe sppor reacions from saic eqilibrim eqaions (i.e. ΣF ΣF Σ ) P P R P Two componens of displacemen a poin and he ranserse componen of displacemen a poin are prescribed eqal o zero while he displacemen and roaion a poin and are sill nknowns. Ths, he segmens and conain oo man nknowns o be soled b he momen area and lengh consrain eqaions. appling Remark1 o he segmen, we obain engh onsrain qaion R -P R -P

12 Noe proided b JRR Page- Noe proided b JRR Page-4 oe o segmen The ranserse componens of displacemen a boh ends of his segmen are alread known. sing Remark, we obain nd omen rea qaion / ( ) ( rea/ ) / / 1 s omen rea qaion / 8P 14P rea 1 P 14P / ( ) P / / 1 s omen rea qaion 1 4P ( ( ) ( ) ) 4P 1 4P 4P / rea/ ( ) 8P 8P Rern o segmen Now, for kinemaical qaniies of he segmen are alread known. 4P 8P nd omen rea qaion ( + ) ( rea/ ) / + ( ) ( ) 1 P 4P Rern o segmen The longidinal componen of he displacemen a poin of he segmen is alread known; hs, he longidinal componen of he displacemen a poin can be readil comped from he lengh consrain eqaion. 8P 14P 8P engh onsrain qaion 8P 8P

13 Noe proided b JRR Page-5 Finall moe o segmen The displacemen and roaion a poin are alread known 8P 8P engh onsrain qaion / / 8P nd omen rea qaion / ( ) ( rea/ ) 8P P ( ) 11P P ( ) 1 s omen rea qaion 8P P P / rea/ ( ) 14P 14P