Principle of virtual work

Transcription

1 Ths prncple s the most general prncple n mechancs Prncple of vrtual work There s Equvalence between the Prncple of Vrtual Work and the Equlbrum Equaton You must know ths from statc course and dynamcs of rgd body moton (teachers J. Hartkanen before 217 and Susanne et. al. after) and statc Rgd body moton New term for you W nt W, u A concse presentaton: ( ) ext

2 VRtual dsplacements Equlbrum poston Real dsplacement Vrtual dsplacements Vrtual dsplacements are the all magnable nfntesmal change n the poston coordnates from the equlbrum poston Real dsplacements In addton, the vrtual dsplacements whch satsfy the knematc constrants are called cnematcally admssble and they are generally the one meant n the vrtual dsplacement prncple. VRtual dsplacements

3 Prncple of Vrtual Work Ths s the Prncple Example of pure bendng: M κ( v ) dx q v dx Av vdx, v V - knematcally admssble W nt Wext Wacc v : vrtual dsplacement (small perturbaton, varaton) κ : nduced vrtual curvature κ [ v] v ( x) Vrtual dsplacements: u u, v,

4 Prncple of Vrtual Work Example of pure bendng: v κ M κ( v ) dx q v dx Av vdx, W nt W ext? : vrtual dsplacement (small perturbaton, varaton) : nduced vrtual curvature Vrtual dsplacements: 1. Axal dsplacement: Aksaalnen srtymä 2. Deflecton and rotaton: Tapuma ja kertymä W ext u u, v, - Vrtual work of external forces: n( x) u( x)dx Wacc v V j Pu ( x - knematcally admssble Wext q( x) v( x)dx M j( x j ) Fkv ) k ( x k ) 3. Twstng: Vääntö W ext mdx M ( x m ) m m

5 Prncple of vrtual Work - Vrtual work of nternal forces: General 3-D form Wext Wacc σ : ε dv t uds f u dv u udv, u V W nt... Vrtual dsplacements: Ndx M dx 1. Axal Stretch: Veto/purstus (venytys) 2. Flexural/Bendng: Tavutus 3. Shearng: ekkaus 4. Torsonal/Twstng: Vääntö t u W nt Mdx... W W W Qdx Vrtual strans: nt nt nt Ndx d Mdx Qdx M dx t ε u - knematcally admssble Vrtual deformatons of an element of a structure. d Shear correcton factor Pokklekkauksen srtymäkerron u

6 1) Derve the equlbrum equatons and the mechancal boundary condton startng from the prncple of vrtual dsplacements. 2) Assume lnear elastcty and small strans snce and wrte the equlbrum equaton n terms of dsplacements then solve t! u(x) : s an equlbrum poston W W I1 N( x) u( x)dx N( x) ( x)dx u - vrtual stran Nu N udx N( ) u( ) ( N q) udx d N( x) dx u( x) d x N( ) u( ) N() u() Nu dx arbtrary vrtual dsplacement q( x) u( x)dx,, u( x) V q q ga knematcally admssble arbtrary vrtual dsplacement u ~ () u() u() u( ) N( ) u( ) Nu dx Equlbrum equatons

7 2) Assume lnear elastcty and small strans snce and wrte the equlbrum equaton n terms of dsplacements. q near elastcty N( x) EA ( x) EAu( x) N q N( ) u( ) Mechancal boundary condton [ EAu( x)] q u( ) u( ) knematcal boundary condton q ga Dsplacement formulaton Force formulaton Analytcal soluton: EAu( x) gax1 x 2

8 Applcaton 2: the prncple of vrtual dsplacements as a method for fndng approxmate solutons 1: we approxmaton for actual dsplacements: u( x) c1 1( x) c1x, c1 R q 2: we chose vrt. dsplacement * a R ( a ) Snce the vrtual u( x) a 1( x) a x dsplacement s arbtrary and knematcally admssble u( ) a knematcally admssble 3: Apply vrtual work prncple W EAu( x) u( x)dx c ( 1 1 x ) a q( x) u( x)dx a x, u If: u( x) c x c x Q: what happens? a EAc dx q xdx, a 1 q c 1 2E u( x) 1 2 qx

9 Prncple of Vrtual Force Vrtuaalsten vomen peraate Ths prncple s the bascs for the unt- (dummy) force method and the general force method

10 Prncple of vrtual forces The Prncple developed by John Bernoull (1717) and s also called often the unt-load method 1-kkö-vomamenetelmä The prncple s the bass for: unt-load method general method for determnng dsplacements General force method for analyss of structures Readngs: Parnes, Chap clear dervaton of the prncple Hbbeler, Chap. 9.3 (an (over) smplfed presentaton but t s enough for the requrements of ths course)

11 Prncple of complementary vrtual Work - Prncple of vrtual forces The Prncple developed by John Bernoull (1717) and s also known as the unt-load method 1-kkö-vomamenetelmä The deformaton of the system s compatble and dsplacement boundary condtons hold on u The prncple s the bass for: unt-load method general method for determnng dsplacements General force method for analyss of structures The sum of the external and the nternal complementary vrtual work vanshes for any self-equlbratng vrtual force system (and satsfyng homogeneous force boundary condtons on ) t * * W nt W, σ, t σ : εdv t vds b vdv : statcally admssble vrtual stress feld: b f body forces: ext Vrtual forces Real dsplacements σ, t n σ vrtual forces satsfy equlbrum condtons statcally admssble vrtual stress feld

12 Vrtual and real deformaton of a structure Generalzed deformatons: Axal Flexural Shearng Torsonal vrtual real u d u d d d 12

13 Complementary Internal Vrtual Work: 1. Axal Stretch: Veto/purstus (venytys) 2. Flexural/Bendng: Tavutus 3. Shearng: ekkaus 4. Torsonal/Twstng: Vääntö Arbtrary materal behavor near elastc materal Notaton: M M Q Q N N M t M t - Shear correcton factor 5. Generalzed sprng: Ylestetty jous Vrtual nternal forces Real deformatons Example: near elastc materal Frame Kehä Grder Arna Rstkko Truss

14 Unt-dummy-force method or smply unt-force method: A general and systematc method for computng dsplacements n structures ykskkövomamenetelmä

15 Unt-dummy-force method M M * * W W, nt ext M Example of bendng: Vrtual forces Real state M κds 1 Real dsplacements & deformatons M κ( x) κ m κ κ EI real curvature 1 1 N ds N N ( x) m EA A ntal stran case of elastcty F 1, 1 Unt-dummy force method: Internal stress resultant nduced by the unt-load: M M, N N,... Q Q,... b A M (x) P F v (A) b A Vrtual state - appled vrtual force - nduced vrtual moment N Real state Vrtual state

16 Prncple of vrtual forces M M(x) Applcaton n Structural Analyss: dsplacements? Vrtual forces M κds 1 Real deformatons & dsplacements M ( x) κ( x) κ ( x ) EI real curvature Unt-dummy force method: 1 Internal stress resultant nduced by the unt-load: A F M M (x), Example of bendng: A b A Real state 1 Vrtual state For only mechancal loadng, the curvature s ( x) For only mechancal loadng, the Intal curvature s zero: κ ( x) For any ntal curvature as for nstance from non-unform thermal gradent κ ( x) κ th ow up M For nstance, n the homework for problem III b) th ( ) / h and κ ( x) th th m EI κ M m / EI

17 Vrtuaalsen työn peraate Prncple of Vrtual Work * Vrtuaalsten srtymen peraate esm. Puhdas tavutus * Vrtuaalsten vomen peraate Todellnen tla Vrtuaalnen tla Kun Hooke n lak pätee: elastnen käyrstymä (tavutuksen ahettama) Alkukäyrstymä 17 (alkujänntykset, lämpötlan garadentt,...)

18 Unt-dummy-force method - Examples 18

19 Unt-dummy-force method Cantlever beam Problem: determne tp rotaton and dsplacement of the cantlever beam accountng for: a) bendng only b) bendng & shear deformatons a) : 1 3 P EI v A 3 A 1 2 P EI 2 M κds Q ds 1v M κds v A, M M κds (M) v A -bendng b): v ( Q) A (Q) v A -shear 1 F A :1 unts N 1 A, M A :1 A (Q) v A b): unts N.m Q ds 3 3 ( M ) P P P EI va 1 2 3EI GA 3EI GA NB. n whch cases ths shear-effect term becomes mportant? Q 1 ds GA P GA

20 Unt-dummy-force method - Trusses Problem: Determne the vertcal dsplacement of pn V. The axal rgdty s constant and equal for all members. N 1 N EA N 1V S, Real state 1 N dx N, 1 1 V N ( EA) Vrtual state F V :1 Apply vrtual unt-load to V N N ( EA) N N 2

21 Unt-dummy-force method - Trusses Problem: Determne the vertcal dsplacement of pn V. The axal rgdty s constant and equal for all members. N 1 N EA N 1V S, F V :1 Newton Real state Vrtual state F V :1 21

22 Unt-dummy-force method - Trusses Problem: a) determne the dsplacement components of pn B due to mechancal load P Vrtual state Real state Horzontal and vertcal dsplacements dem. 22

23 Problem: b) determne the dsplacement components of pn B due to the temperature change T of n member AB. Sauvojen jäykkyys EA ja ptuuden lämpötlakerron α T ovat vakot. Vrtuaalnen tla Todellnen tla sauva AB saa lämpötlan muutoksen T. b) lämpötlan muutos 23

24 Unt-dummy-force method - Frames 24

25 Example: Determne the rotaton at secton C under constant dstrbuted load q. EI = constant Vrtual state True state M -kuvo ykskkövomamenetelmässä vodaan löysäst ajatella, että vrtuaalset ylestetyt vomat δf 1 25

26 Example: Determne rotaton at secton C when the dfference n temperatures between lower and upper sdes of the beam s T n the span A-B (span BC remans at constant temperature). EI = constant, h- heght of the secton and thermal expanson coeffcent s α T [1/K]. Vrtual state T = T -T U True state near temperature dstrbuton across the heght h T = T -T U thermal gradent beam AB ntal curvature: T T U 26

27 A challenge to students: Determne the reacton at the support B. Hnt: uses what you have learnt tll now Suggestons? 27

28 Homework: dummy unt load method n computng dsplacements 28