Potential energy of a structure. Vforce. joints j

Transcription

1 Potntial nrgy of a strctr P y P x y x P y P x ij ij ij V K ( ) Vforc P V ij ij j j K ( ) P [ K] r Elastic Potntial nrgy of mmbr ij Potntial nrgy of applid forc at joint i i i i mmbrs joints j Find th st of joint displacmnts that minimizs V. Ths minimizing displacmnts ar thos attaind by th strctr in static qilibrim [ K] r Strain Enrgy for a singl mmbr: EA EA b a ab V ( ) n L L n n n n n n ab ab ab ab ab ab ab ab ab ab ab ab n n n n n n ab ab ab ab ab ab n n n n n n ab ab ab ab ab ab a a n n n n n n EA a a b b V b L b V K K and ar th lmnt stiffnss matrix and lmnt displacmnt vctor.

2 (b) (Sam as it vr was ) (a) V EA EA L L b a ab ( ) n K EA L F H G I K J L c h M NM V K c h n ab ab ab ab ab n n n n n ab ab ab ab n n n n n ab ab ab ab ab n n n n n ab ab ab c h n n cn ab h abo ab c h ab P QP n ab ab ab ab ab n n n n n ab ab c h c h n n n ab ab ab ab ab c h n n n n n c h, L NM a a a b b b O QP Bttr to writ mmbr nrgis in trms of th global displacmnt vctor: V V K K K K () () () () () () 4 () () () () () () () K K K K4 () () () () () () K K K K4 () () () () () () K4 K4 K4 K44 K K K K 0 0 () () () () () () 4 () () () () () K K K4 0 0 () () () () () K K4 0 0 () () () K () () 0 0 () () 0 V V K K K K () () () () () () 4 () () () () () () () K K K K4 () () () () () () K K K K4 () () () () () () K4 K4 K4 K () () () () () () () () () () () K K K K 4 () () () () () K K K4 () () () () K K 4 () () () K 44 4

3 Total Elastic Enrgy is assmbld asily V V V () () () () () () () () K K K K4 0 0 () () () () () K K K4 0 0 () () () () () () () K K K4 K K K () 4 () () () () () () K44 K K K4 () () () () K K 4 () () () K44 V K [K] is th Global Stiffnss Matrix. It is th sm of all th lmnt stiffnss matrics. 5 n = (a) Positioning th lmnt stiffnss componnts in th global stiffnss matrix n = (b) L N M ( a ) ( a ) ( b) ( b) K K K K 4 K K K4 K K4 K44 O Q P ( a ) ( a ) ( b) ( b) K K, K K, K K K K a, a a, a a, b 4 a, b K K, K K K K a, a a, b 4 a, b K K K K b, b 4 b, b K K 44 b, b K K ( i) ii,( j) jj ( ni) ii,( n j) jj i, and j, for th st and nd nods for th mmbr. Nod #s n ii, and jj, for th - and - dirctions of displacmnt i 6

4 Enrgy D to Extrnal Loads: (a) (b) P (b) ( b) ( b) Vloads P r Loadd Nods P ( b) i r( b ) i Total Potntial Enrgy for th Systm: Tot V V Q [ ] [ ] [ ] [ K] r N N N K r i ij j i i i j i 7 Minimizing th Potntial Enrgy W hav st p th following xprssion for th potntial nrgy of a trss V[ ] [ K] r N N N K r i ij j i i i j i Now, minimiz V : V K K r K r 0 tot N N N i ik kj j k kj j k k i j j or, in matrix notation: [ K] r 9 4

5 W hav not takn into accont prscribd displacmnts! (so dt[k]=0) To nforc, say () w cold modify th finit lmnt qations as follows: Bttr: Prsrvs th all-important symmtry of [K]! 0 NOW SOLVE FOR : ( ) n / L ab b a ab ab [ K] r Onc yo hav, th mmbr strains F and forcs ar ab ab EA b a ab F H G I L K J ( ) n (b) (a) F ab >0: tnsion, F ab <0: comprssion 5

6 Mapl Program Trss Rads in gomtric and matrial data dscribing a trss, its spports, and its loads from an inpt fil Shows graphically th ndformd and dformd strctr with mmbr forcs Writs nodal displacmnts and mmbr forcs to an otpt fil Abot th Inpt/Otpt fils:. Both inpt and otpt fils ar txt fils, and ar opnd with th Notpad ditor. Yo can jst click on th fil Icon.. If th inpt fil is calld rats.inp.txt, an otpt fil calld rats.ot.txt. It will b in th sam dirctory and foldr as th inpt fil.. To viw th otpt fil: Opn it with Notpad. 6

7 Ttorial Trss P.0 45 o 45o Lt EA=, L =L = Ttorial Inpt Fil Titl Nmbr of nods x y for nod x y for nod x y for nod Nmbr of lmnts Nod # nod # EA for lmnt Nod # nod # EA for lmnt Nod # nod # EA for lmnt Nmbr of constraints (fixd displacmnts) Nod # dirction val of fixd displacmnt Nod # dirction val of fixd displacmnt Nod # dirction val of fixd displacmnt Nmbr of loadd nods Nod # P P L=.0 45 o 45o EA= 7

8 To Rn: > rstart: # Fint Elmnt Trss Analysis Program # #********** GIVE THE NAME OF THE INPUT FILE IN THE LINE BELOW *********** # Us frontslash/ in plac of backslash\ # Enclos th nam in singl backqots` # End th lin with a colon: # infilnam:=`c:/docmnts and Sttings/laptop06/Dsktop/tandm.inp.txt`: # #*************************** THEN HIT RETURN************************* Pictrs of th loadd, constraind trss appar on scrn Hit Entr again if ths look right to yo. And that s all! 8

9 Scrn Otpt: Dformation Displacmnts scald by:, Scrn Otpt: Mmbr Forcs 9

10 Otpt Fil: Problm Statmnt Ttorial Nmbr of Nods and Elmnts: Nod# x x El# Nod Nod EA Fixd Nod# DOF Val Loadd Nod# P P Otpt Fil: Rslts 45 o 45o Nodal Displacmnts: Nod Strains and Forcs: Elmnt Strain Forc

11 Compting procdr. Rad inpt data. Loop ovr lmnts, calclat lmnt stiffnss mx, and add componnts to th global stiffnss mx [K]. Assmbl th rsidal r vctor from th applid forcs 4. Modify [K] and r to accont for prscribd displacmnts 5. Solv for th global displacmnts = [K] - r 6. Calclat lmnt strains and forcs. 7. Print rslts to a fil. Mapl Cod: Inpt Variabls Variabl Typ Dscription Titl String Rn titl nnod intgr # of nods coord(i,j) Array: nnod x coord(i,)=x coord of nod # i coord(i,)=y coord of nod # i nlm intgr # of lmnts connct array: nlm x connct(i,)= st nod no. of lmnt i connct(i,)= nd nod no. of lmnt i connct(i,)=ea for lmnt i nfix intgr # of prscribd displacmnts fixnods array: nfix x fixnods(i,)=nod nmbr with th i th fixd displ fixnods(i,)=dirction ( or ) of th i th fixd displ fixnods(i,)=val of th i th fixd displ nload intgr # of nods with xtrnal loads loads array: nloadx loads(i,)=nod nmbr of th i th loadd nod loads(i,)=forc on nod # loads(i,) in th x-dirction loads(i,)= forc on nod # loads(i,) in th y-dirction