Principle Of Superposition

Transcription

1 ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give poit i the system caused by two or more loads is the sum of the resposes which would have bee caused by each load idividually. ice the additio fuctio is preserved this is sometimes referred to as a additive map. Cosider a liear sprig where K ad K is the liear sprig costat.

2 or the iitial load o the sprig ecture 5: PREIMINRY CONCEP O RUCUR NYI K Now icrease the deflectio o the system by a amout. he secod additioal load o the sprig is K he fial force o the sprig is from the additioal deflectio is with ad K K K ( )

3 ecture 5: PREIMINRY CONCEP O RUCUR NYI the K his result, although deceptively obvious, idicates that for a liear sprig system the deflectio caused by a force ca be added to the deflectio caused by aother force to obtai the deflectio resultig from both forces beig applied; the order of loadig is ot importat ( or could be applied first); he is the Priciple of uperpositio or a structure with a liear respose, the load effects caused by two or more loads are the sum of the effects caused by each load applied separately. or the priciple to be applicable to structural aalysis the material the structure is fabricated from must be liear elastic. o guaratee this we typically require the structure to udergo small deformatios.

4 ecture 5: PREIMINRY CONCEP O RUCUR NYI Cosider the beam i figure (a) subject to exteral actios ad. hese actios produce various reactios ad displacemets throughout the structure. Reactios are developed at the supports. displacemet is produced at the mid spa. he effects of ad are show separately i (b) ad (c). sigle prime is associated with ad a double prime with. rom the figure it becomes obvious that the followig equatios ca be developed through the use of superpositio: R R R M M M RB R B R B

5 ecture 5: PREIMINRY CONCEP O RUCUR NYI Next cosider the same beam subjected to displacemets, i.e., the support at B is traslated dow a amout ad rotated couterclockwise a amout θ. gai various reactios ad displacemets are id iduced di the structure. Reactios ad displacemets with sigle primes are associated with. hose with double primes are associated with θ. uperpositio ca be ivoked for a liear system if the iput variable is a actio, or if the iput variables are displacemets.

6 ecture 5: PREIMINRY CONCEP O RUCUR NYI ctio d isplacemet Equatios he relatioship betwee actios ad displacemets play a importat role i structural aalysis. coveiet way to see this relatioship is through a liear, elastic sprig he actio will compress (traslate) the sprig a amout. his ca be expressed through the simple expressio: I this equatio is the flexibility of the sprig, ad this quatity is defied as the displacemet produced by a uit value of the actio.

7 ecture 5: PREIMINRY CONCEP O RUCUR NYI his relatioship ca also be expressed as Here (earlier it was K) is the stiffess of the sprig ad is defied as the actio required to produce a uit displacemet i the sprig. he flexibility ad stiffess of the sprig are iverse to oe aother.

8 ecture 5: PREIMINRY CONCEP O RUCUR NYI he relatioship that holds for a sprig holds for ay structural compoet. Cosider the simple beam subjected to a actio that produces a traslatio. he actio ad displacemet equatio holds if the flexibility ad stiffess are determied as show. he actio ad displacemet equatio give o the previous slide is valid oly whe oe actio is preset ad we are lookig for oe displacemet withi the structure. More tha oe actio ad oe displacemet requires a matrix format. 48EI ( ) 48 EI ( ) ( 48EI)

9 ecture 5: PREIMINRY CONCEP O RUCUR NYI et s cosider a geeral example where a beam is subjected to three actios, i.e., two forces ( ad ) ad a momet ( ). he directios i for the actios are assumed positive. he deflected shape is give i figure (b) ad displacemets, ad correspod dto,, ad. By usig superpositio each displacemet ca be expressed as the sum of displacemets due to actios through I a similar maer expressios for ad are

10 ecture 5: PREIMINRY CONCEP O RUCUR NYI With deflectio at caused by deflectio at deflectio at caused by caused by ad the fact that is directly proportio al to oly is directly proportio al to oly is directly proportio al to oly the

11 We ca express the equatios for the deformatios, ad as ecture 5: PREIMINRY CONCEP O RUCUR NYI Each term o the right-had side of the equatios is a displacemet writte i the form of a coefficiet times the actio that produces a deformatio represeted by the coefficiet. he coefficiets are called flexibility coefficiets. he physical sigificace of the flexibility coefficiets are depicted i figures (c), (d) ad (e) ll the flexibility coefficiets i the figures have two subscripts ( ij ). he first subscript idetifies the displacemet ( i ) associated with a actio ( j ). he secod subscript deotes where the uit actio is beig applied. igure (c) is associated with actio, figure (d) is associated with actio, ad figure (e) is associated with actio. lexibility coefficiets are take as positive whe the deformatio represeted by the coefficiet is i the same directio as the i th actio.

12 ecture 5: PREIMINRY CONCEP O RUCUR NYI Istead of expressig displacemets i terms of actios, it is possible to express actios i terms of displacemets, i.e., his system of equatios ca be obtaied from the displacemet system of equatios uder suitable coditios. Here is a stiffess coefficiet ad represets a actio due to a uit displacemet. o impose these uit displacemets requires that artificial restraits must be provided. hese restraits are show i the figure by simple supports correspodig to actios, ad.

13 ecture 5: PREIMINRY CONCEP O RUCUR NYI Each stiffess coefficiet is show actig i its assumed positive directio, which is the same directio as the correspodig actio. If the actual directio of oe of the stiffess coefficiets is opposite to that assumptio, the the stiffess coefficiet will have a egative value. he calculatios of the stiffess coefficiets for the beam show ca be quite legthy. However, aalyzig a beam like the oe show previously by the stiffess method ca be expedited by utilizig a special structure where all the joits of the structure are restraied. We will get ito the details of this i the ext sectio of otes. he primary purpose of this discussio is for the studet to visualize what flexibility ad stiffess coefficiets represet physically.

14 ecture 5: PREIMINRY CONCEP O RUCUR NYI lexibility ad tiffess Matrices We ca ow geeralize the cocepts itroduced i the precedig sectio. If the umber of actios applied to a structure is, the correspodig equatios for displacemets are: M M I matrix format these equatios become M M M M M or { } [ ]{ }

15 ecture 5: PREIMINRY CONCEP O RUCUR NYI he actio equatios with actios applied to the structure are M M I matrix format these equatios become M M M M M or { } [ ]{ } { } [ ]{ }

16 ecture 5: PREIMINRY CONCEP O RUCUR NYI ice the actios i ad displacemets i correspod to oe aother i both formats, it follows the flexibility matrix ij ad the stiffess matrix ij are related to each other. akig the matrix iverse of yields { } [ ]{ } { } [ ] { } With the { } [ ]{ } [ ] [ ]

17 ecture 5: PREIMINRY CONCEP O RUCUR NYI I a similar fashio oe ca show that [ ] [ ] hus the stiffess matrix is the iverse of the flexibility matrix ad vice versa provided that the same set of actios ad displacemets are beig cosidered i both equatios Note that a flexibility matrix or stiffess matrix is ot a array that is determied by the geometry of the structure oly. he matrices are directly related to the geometry ad the set of actios ad displacemets uder cosideratio.

18 ecture 5: PREIMINRY CONCEP O RUCUR NYI Example he catilever beam show i the figure below is subjected to a force ( ) ad momet ( ) at the free ed. evelop the flexibility matrix ad the stiffess matrix for assumig displacemets ad are of iterest.

19 Makig use of Case #7 ad Case #8 from the followig table ecture 5: PREIMINRY CONCEP O RUCUR NYI

20 (cotiued) ecture 5: PREIMINRY CONCEP O RUCUR NYI

21 ecture 5: PREIMINRY CONCEP O RUCUR NYI he the flexibility coefficiets are as follows: EI EI EI he displacemets are EI EI EI EI he flexibility matrix becomes EI EI EI EI [ ]

22 ecture 5: PREIMINRY CONCEP O RUCUR NYI I order to develop the stiffess matrix cosider the followig beam reactios due to applied displacemets:

23 ecture 5: PREIMINRY CONCEP O RUCUR NYI he the stiffess coefficiets are as follows: EI 6EI 4EI he actios are EI 6EI EI 4EI he stiffess matrix becomes [ ] EI 6EI 6EI 4EI

24 ecture 5: PREIMINRY CONCEP O RUCUR NYI Whe the flexibility matrix ad the stiffess matrix are multiplied together, the result is the idetity matrix: [ ][ ] EI 6EI EI EI 6EI 4EI EI EI 00 0 his ifers but does ot prove that the two matrices are iverses of oe aother.

25 ecture 5: PREIMINRY CONCEP O RUCUR NYI Equivalet Joit oads he calculatios of displacemets i larger more extesive structures by the meas of the matrix methods derived later requires that the structure be subject to loads applied oly at the joits. hus i geeral, loads are categorized ito those applied at joits, ad those that are ot. oads that tare ot applied to joits must tbe replaced with statically ti equivalet loads. Cosider the statically idetermiate beam with a distributed load betwee joits ad B, ad a poit load betwee joits B ad C:

26 ecture 5: PREIMINRY CONCEP O RUCUR NYI irst oe must idetify joits, ad here we select poits, B, ad C for coveiece. By superpositio the beam ca be separated ito two beams, oe with loads located at the joits, ad a secod with loads betwee the joits:

27 ecture 5: PREIMINRY CONCEP O RUCUR NYI o trasfer the loads that act o the members to the joits, the joits of the structure are restraied agaist all displacemets. his produces two fixed ed beams: Whe these fixed ed beams are subjected to the member loads, a set of fixed ed actios is produced. d he same fixed-ed d actios are show i the followig figure where they are depicted as restrait actios i the restraied structure.

28 ecture 5: PREIMINRY CONCEP O RUCUR NYI

29 ecture 5: PREIMINRY CONCEP O RUCUR NYI If the restrait actios are reversed i directio, they costitute a set of forces ad couples that are statically equivalet to the member loads. hese equivalet joit loads, whe added to the actual joit loads produce the combied joit loads show i the ext figure. he combied loads are the used i carryig out the structural aalysis. Will the ukow displacemets at odes be correct usig this statically equivalet system? Will the displacemets computed alog segmet B i the figure above be equivalet to the displacemets alog segmet B give the origial load cofiguratio?

30 ecture 5: PREIMINRY CONCEP O RUCUR NYI Reciprocal heorems If the loads o a structure are zero ad gradually icrease such that all loadigs hit peak values at the same time, the work doe durig this period of time will be the average, hece ( ) W I a matrix format but both ad are colum vectors by defiitio so to perform this matrix multiplicatio we must use the traspose of oe or the other colum vectors. hus ( ){ } { } ( ){ }{ } W Recall that { } [ ]{ } Now, substitute this i the above equatio.

31 ecture 5: PREIMINRY CONCEP O RUCUR NYI his substitutio leads to I additio the followig relatioship holds from matrix algebra { } { } { } [ ]{ } W I additio, the followig relatioship holds from matrix algebra { } [ ]{ } ( ) ubstitutig this relatioship i the equatio from the previous slide yields [ ] { } ubstitutig this relatioship i the equatio from the previous slide yields { }{ } { }[ ] { } W { }{ } { }[ ] { } W

32 Equatig the two equatios for work we obtai ecture 5: PREIMINRY CONCEP O RUCUR NYI { } [ ]{ } { }[ ] { } { } [ ]{ } { }[ ] { } Multiplyig both sides by ({} ) - ad {} - we obtai ({ } ) { } { } [ ]{ } { } ({ } ) { } ( { } { } )[ ] { } ( ) { } { }[ ] { } ( ) { } { } { } [ I ][ I ][ ] [ I ][ I ][ ] [ ] [ ] ( )[ ]

33 ecture 5: PREIMINRY CONCEP O RUCUR NYI hus the flexibility matrix must be symmetric. o prove the stiffess matrix is symmetric recall that ubstitutig this i the equatio for work { } [ ]{ } W { }{ } [ ]{ }{ } I additio, the followig relatioship holds from matrix algebra { } ([ ]{ } ) { } [ ] { }

34 ecture 5: PREIMINRY CONCEP O RUCUR NYI ubstitutig this i the above equatio for work { } { } [ ] { } { } Equatig these two relatioships for work { } { } [ ] { } { } W [ ]{ }{ } [ ] { } { } Multiplyig both sides by [] - ad [ ] - we obtai [ ]{ }{ } [ ] { } { } { } ( ) { } { } [ ]{ } { } ( ) { } { }[ ] { }

35 urther maipulatio yields ({ } ) { } ( { } { } )[ ] { } ecture 5: PREIMINRY CONCEP O RUCUR NYI ( ) { } { } { } [ I ][ I ][ ] [ I ][ I ][ ] [ ] [ ] ( )[ ] Hece the stiffess matrix is symmetric. Of course the fact that the stiffess matrix is symmetric could have bee cocluded from the fact that the flexibility matrix is symmetric ad the stiffess matrix is the iverse of the flexibility matrix. But this has ot bee formally prove.