Finite Element Method FEM FOR FRAMES

Transcription

1 Finit Ent Mthod FEM FOR FRAMES

2 CONENS INROUCION FEM EQUAIONS FOR PLANAR FRAMES Equtions in oc coordint sst Equtions in gob coordint sst FEM EQUAIONS FOR SPAIAL FRAMES Equtions in oc coordint sst Equtions in gob coordint sst REMARKS

3 INROUCION for i nd trnsvrs. It is cpb of crring both i nd trnsvrs forcs, s w s onts. Hnc cobintion of truss nd b nts. Fr nts r ppicb for th nsis of skt tp ssts of both pnr frs ( frs) nd spc frs ( frs). Known gnr s th b nt or gnr b nt in ost corci softwr.

4 FEM EQUAIONS FOR PLANAR FRAMES Considr pnr fr nt d d u d v dipcnt coponnts t nod d d4 u d 5 v dipcnt coponnts t nod d, V, v,, u nod (u, v, ) nod (u, v, ) =, U 4

5 Equtions in oc coordint sst Cobintion of th nt trics of truss nd b nts d u v u v russ B Fro th truss nt, k (Epnd to ) truss d u d u 4 AE AE d u AE d u s. 4 5

6 Equtions in oc coordint sst Fro th b nt, k b d ( v ) d ( ) d ( v ) d ( ) 5 EI EI EI EI d v EI EI EI d EI EI s. d5 v EI d (Epnd to )

7 7 Equtions in oc coordint sst. s AE AE AE truss k. EI EI EI EI EI EI EI b EI EI EI s k + EI EI EI AE EI EI EI EI EI EI EI AE AE s. k

8 8 Equtions in oc coordint sst Siir so for th ss tri nd w gt s A And for th forc vctor, s f s s s f s s f f f f f f f f f

9 9 Equtions in gob coordint sst Coordint trnsfortion d whr j j j i i i, i - i- j - j u u f s gob nod j oc nod gob nod i oc nod o v v j j - i

10 Equtions in gob coordint sst irction cosins in : cos(, ) cos cos(, ) sin j j i i gob nod i oc nod cos(, ) cos(9 ) sin cos(, ) cos j i f s o v j i- i i u i - j v j j - u j - gob nod j oc nod ( ) ( ) j i j i (Lngth of nt)

11 Equtions in gob coordint sst hrfor, K k M F f

12 FEM EQUAIONS FOR SPAIAL FRAMES Considr spti fr nt v d d u d v d w d4 d5 d d u 7 d8 v d 9 w d d d ispcnt coponnts t nod ispcnt coponnts t nod v w u u w

13 k Equtions in oc coordint sst Cobintion of th nt trics of truss nd b nts u v w u v w AE AE EI EI EI EI EI EI EI EI s. GJ GJ EI EI EI EI EI EI AE EI EI EI EI GJ EI EI v w u v u w

14 4 Equtions in oc coordint sst r s r r A whr A I r

15 Equtions in gob coordint sst j-4 j- d8 d d7 d d d9 j- j-5 i-4 j i- d d5 d j- d d4 d i- i-5 i i- 5

16 Equtions in gob coordint sst Coordint trnsfortion d whr j j j j j j i i i i i i , n n n

17 Equtions in gob coordint sst irction cosins in cos(, ), cos(, ), n cos(, ) cos(, ), cos(, ), n cos(, ) cos(, ), cos(, ), n cos(, ) 7

18 Equtions in gob coordint sst Vctors for dfining oction nd orinttion of V V V fr nt in spc k k k k k k k, =,, V V V V V V V V V V V ( V V ) ( V V ) V V 8

19 Equtions in gob coordint sst Vctors for dfining oction nd orinttion of fr nt in spc (cont d) V V ) V V cos(, ) cos(, ) n cos(, ) ( V V ) ( V V ) ( V V ) ( V V ) ( A {( ) ( ) ( V V V V V ( V V ) ( V V ) ) } V V A ( ) ( ) ( ) 9

20 Equtions in gob coordint sst Vctors for dfining oction nd orinttion of fr nt in spc (cont d) V V V V ) ( ) ( V V V V V V V ) ( ) ( ) ( A n A A n n n n n

21 Equtions in gob coordint sst hrfor, K k M F f

22 REMARKS In prctic structurs, it is vr rr to hv b structur subjctd on to trnsvrs oding. Most skt structurs r ithr trusss or frs tht crr both i nd trnsvrs ods. A b nt is ctu vr spci cs of fr nt. h fr nt is oftn convnint cd th b nt.

23 CASE SU Finit nt nsis of bicc fr

24 CASE SU oung s oduus, E GP Poisson s rtio, nts (7 nods) Ensur connctivit 4

25 CASE SU Constrints in dirctions Horiont od 5

26 CASE SU M =

27 CASE SU Ai strss P P P P P P -.4 P 7