The Finite Element Method

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Εἰλείθυια Μανωλάς
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1 Th Finit Elmnt Mthod Plan (D) Truss and Fram Elmnts Rad: Sctions 4.6 and 5.4 CONTENTS Rviw of bar finit lmnt in th local coordinats Plan truss lmnt Rviw of bam finit lmnt in th local coordinats Plan fram lmnt Numrical ampls JN Rdd

2 FINITE ELEMENT ANALYSIS OF PLANE TRUSSES AND FRAMES JN Rdd Trusss and Frams:

3 REVIEW OF THE BAR ELEMENT Linar bar lmnt in th lmnt coordinat sstm Ku F Q f Q f h Horizontal K EA fh Q F + h Q JN Rdd E A h K D F ū v ū v F 0 F 0 v F u v h θ Inclind F u Trusss & Frams: 3

4 Bar Elmnt in Global Coordinats Bar lmnt in th lmnt coordinats E A h K ū v ū v D F F 0 F 0 v h Transformation rlations btwn th two coordinat sstms cos θ + sin θ ȳ sin θ + cos θ cos θ sin θ ȳ sin θ cos θ F u v P θ ( ) ( ) θ F u JN Rdd Trusss & Frams: 4

5 Bar Elmnt in Global Coordinats Transformation rlations btwn th displacmnts of th two coordinat sstms u cosθ sinθu sin cos v θ θ v u cosθ sinθu sin cos v θ θ v v u F F v v h F u F θ v u F F u ū v ū v cos θ sin θ 0 0 sin θ cos θ cosθ sin θ 0 0 sin θ cos θ u v u v 0 0 { } [T ]{ } { F } [ T ]{ F } JN Rdd Trusss & Frams: 5

6 Bar Elmnt in Global Coordinats: Truss K D [ K ][T ]{ } [T ]{F } [T ] [T ] T [T ] T [ K ][T ]{ } {F } or [K ]{ } {F } [K ] EA h {F } [K ][T ] T [ K ][T ] {F } [T ] T { F } F F F 3 F 4 F cos θ sin θ cos θ sin θ sin θ sin θ sin θ sin θ cos θ sin θ cos θ sin θ sin θ sin θ sin θ sin θ P cos θ f P cos θ sin θ P cos θ + f sin θ f P cos θ sin θ f sin θ JN Rdd Trusss & Frams: 6

7 EXAMPLE Givn truss C P 00 kn P 00 kn L A B L F ( ) F Finit lmnt discrtization P 00 kn 45 3(56) (34) F F P 00 kn 90 Elmnt Global Gom. Matr. Orint. numbr nods prop. prop. 3 A h L E θ 90 3 A h L E θ 45 JN Rdd Trusss & Frams: 7

8 EXAMPLE (continud) Th lmnt stiffnss matrics ar [/( ) ] P 00 kn [K ] EA [ K ] L F ( ) F 45 3(56) (34) F F P 00 kn 90 [[K K 3 ]] EA L JN Rdd Trusss & Frams: 8

9 EXAMPLE (continud) Connctivit arra for DoF pr nod truss [B] () () 3 4 é ù ê 5 6ú ë û F ( ) () 34 () P 00 kn 45 () 356 () P 00 kn 34 () () F Assmbld stiffnss cofficints K K K K K 0 () () 3 () () () () () K 0 K K K K K K K K K K K K F K K K K K K () () () () () () () () () () () () () () () () () K K K K K K K K K K K K K K K K K K K F K K K K K K JN Rdd Trusss & Frams: 9

10 EA L EXAMPLE (continud) Assmbld sstm of uations of th truss smm Th displacmnt continuit conditions ar u u 3 U v v 3 V u u U v v V u u 3 U 3 v v 3 V 3 ìu ü ìïf üï V F U F ï í ï V ý ï í ï F ý ï ï ï ï ï U ï ïf ï 3 3 V F ïî 3 ïþ ï 3ï î þ JN Rdd Trusss & Frams: 0

11 EXAMPLE (continud) whr th global forcs and displacmnts ar JN Rdd F + F 3 F F 3 + F F F3 + F3 3 F 3 U V U { } {F } V U 3 V 3 Boundar conditions F + F 3 F F 4 + F F F4 + F4 3 F 3 F + F 3 F + F 3 F3 + F F4 + F F3 + F3 3 F4 + F4 3 F F F F F 3 F 3 U V U V 0 F 3 P F 3 P U F () F V kn 3(56) (34) F U V F V 00 kn U 3 90 Trusss & Frams:

12 EXAMPLE (continud) Solution EA L F F F EA L F U 3 (3+ ) PL EA U3 V PL EA P P U 3 V 3 V 3 3PL EA U (m) F () F U kn 3(56) (34) F V V F U 00 kn U 3 90 (P 0 5 N EA 0 8 N) F P F P F F ( ) - F ( ) 3P; F ( ) F ( ) -F 3P P (N) JN Rdd Trusss & Frams:

13 EXAMPLE (continud) Post-computation of mmbr displacmnts and strsss P ū v ū v P σ P P A A A E h ū ū cos θ sin θ 0 0 sin θ cos θ cosθ sin θ 0 0 sin θ cos θ U u v u v F () F U kn 3(56) (34) F V V F U 00 kn U 3 90 W hav PL 3PL u v u v 0; u u U3 ( 3+ ) v v V3 - AE AE JN Rdd Trusss & Frams: 3

14 EXAMPLE (continud) Post-computation of mmbr displacmnts and strsss 3PL u U3cos+ V3sin V3 - A u U cos + V sin U + V ( ) P - P 3P P - P - P s 3P P - s A A ( ) ( ) JN Rdd Trusss & Frams: 4

15 PLANE FRAME STRUCTURES Displ. dgrs of frdom in th lmnt coordinats Forc dgrs of frdom in th lmnt coordinats u w S w h S u F3 Q3 + F Q + f F Q + h F5 Q5 + 3 F6 Q6 + 4 F4 Q4 + f u z w z α S S w u u S w z α w S u Displ. dgrs of frdom in th local coordinats Displ. dgrs of frdom in th global coordinats JN Rdd Trusss & Frams: 5

16 Fram Elmnt in Global Coordinats u cosα sinα 0u u cosα sinα 0u w sinα cos α 0 sin cos 0 w w α α w θ θ θ θ u u cosα sinα 0 w sinα cosα 0 0 w θ 0 0 θ cos sin 0 u α α u 0 sin cos 0 w α α w 0 0 θ θ D T D [ K ][T ]{ } [T ]{F } [T ] [T ] T [T ] T [ K ][T ]{ } {F } or [K ]{ } {F } JN Rdd [K ][T ] T [ K ][T ] {F } [T ] T { F } Trusss & Frams: 6

17 Fram Elmnt in Global Coordinats (th Eulr-Brnoulli bam fram lmnt) JN Rdd Trusss & Frams: 7

18 AN EXAMPLE OF A FRAME STRUCTURE A B a b C F Givn structur P A (456) B (3) (456) a (3) (3) z P C 3(789) (456) F Finit lmnt discrtization b Connctivit arra for 3 DoF pr nod [B] () () é ù ê ú ë û JN Rdd a 0 a 90 or

19 ASSEMBLY OF ELEMENT MATRICES Global stiffnss cofficints in trms of lmnt stiffnss cofficints K K K K... K K () () () 6 6 () () () K K K K K K K K K K K K K K () () () () () () () () K K K K K K K K + K K K + K () () () () K K + K K K + K () () () () K K K K K K () () () K K K K K K () () () K K K K K K () () () K K K K K K () () () K K K K K K. 6 (3) (3) [B] () () () JN Rdd -D Problms: 9 () () a b F (456) (456) (3) P (789) (456) é ù ê ú ë û

20 P 7 lb/in. P a 90 z a EXAMPLE (from th ttbook) 4P 7 in. B a in. a tan Th -ais is into th plan of th papr and is masurd countrclockwis from -ais to -ais A 0 in. I 0 in. 4 E psi 08 in. 44 in. ( ) 3 4 (456) (3) (456) Q + z 5 3 Q + Q + 4 f Q + f Q Q + 3 Q Q 3 JN Rdd Q P 4P z Q 5 Q 6 Q 4 K K + K K K + K () () () () () () () () () () () () () () () () 5 5 () () K K K K K K K K K K K K F F F P F F + F (- P) + (- P) -4P F F + F 4P- 7P -48P a K K K U F K54 K55 K 56 U5 F5 K64 K65 K66 U6 F6 Th rst of th calculations can b found on pp of th Book.

21 SUMMARY In this lctur w hav covrd th following topics: Plan truss lmnt in th local and global coordinats Transformation of lmnt uations from lmnt coordinats to global coordinats Eampls illustrating assmbl application of boundar conditions and calculation of lmnt forcs and strsss Plan fram lmnt in th local and global coordinats Transformation of lmnt uations from lmnt coordinats to global coordinats JN Rdd Trusss & Frams:

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