Chapter 14 Truss Analysis Using the Stiffness Method

Transcription

1 Chapter 14 Truss Analsis Using the Stiffness Method Structural Mechanics 2 ept of Arch Eng, Ajou Univ

2 Outline undamentals of the stiffness method Member stiffness matri isplacement and force transformation matri Member global stiffness matri Truss stiffness matri Application of the stiffness method for truss analsis odal coordinates Trusses having thermal changes & fabrication errors Space-truss analsis

3 14-1 undamentals of the Stiffness Method Subdivides the structure into a series of discrete finite elements represented b members and joints The force-displacement relations Equilibrium condition at each node Stiffness matri Unknown displacement vector Given loading Unknown eternal and internal forces

4 14-2 Member Stiffness Matri + =

5 14-2 Member Stiffness Matri or +ve displacement, d in local and coordinates, the forces developed at the ends of the members are q' q' AE d L AE d L

6 14-2 Member Stiffness Matri or +ve displacement, d at the far end, keeping the near end pinned, results in member forces q'' q'' AE d L AE d L

7 14-2 Member Stiffness Matri B superposition, the resultant forces caused b both displacements are q AE L d AE L d q AE L d AE L d

8 14-2 Member Stiffness Matri These load-displacement equations ma be written in matri form as matri stiffness : ' ' L AE k d k q d d L AE q q

9 14-3 isplacement and orce Transformation Matrices Transformation of member forces q and displacement d defined in local coordinates to global coordinates Global coordinates convention: +ve to the right and +ve upward

10 14-3 isplacement & orce Transformation Matrices direction cosines ) ( ) ( cos ) ( ) ( cos L L

11 14-3 isplacement & orce Transformation Matrices isplacement Transformation Matri d cos cos

12 14-3 isplacement & orce Transformation Matrices isplacement Transformation Matri d cos cos

13 14-3 isplacement & orce Transformation Matrices isplacement Transformation Matri T d d d d d In matri form, ; cos ; cos Let [T] transforms the 4 global displacement into 2 local displacement

14 14-3 isplacement & orce Transformation Matrices orce Transformation Matri q q Q q q Q cos cos q q Q q q Q cos cos

15 14-3 isplacement & orce Transformation Matrices orce Transformation Matri In matri form q T Q q q Q Q Q Q T [T] T transforms the 2 local forces q into 4 global force components Q

16 14-4 Member Global Stiffness Matri Since q q k d and d T kt Also since, Q T T q T Q T kt k where k orce - isplacement is obtained as T T kt,

17 14-4 Member Global Stiffness Matri Performing the matri operation ields

18 14-5 Truss stiffness matri Stiffness matri [K] for entire truss can be obtained b assembling all member stiffness matrices [k] in global coordinates The 4 code numbers to identif the 2 global degrees of freedom at each end of a member Appropriate for analsis b computer programming

19 Eample 14.1 etermine the structure stiffness matri. Use constant AE. (Constrained) () Two unknown displacement () (Constrained) () Member 1 Stiffness matri: irection cosines: 3 1; 3 3

20 Eample 14.1 etermine the structure stiffness matri. Use constant AE. (Constrained) () Two unknown displacement () (Constrained) () Member 2 Stiffness matri: irection cosines: 3.6;

21 Eample 14.1 etermine the structure stiffness matri. Use constant AE. Algebraicall added to form structure stiffness matri

22 14-6 Application of the Stiffness Method for Truss Analsis The relationship between global force components Q acting on a truss and its global displacements Q K The structure stiffness equation

23 14-6 Application of the Stiffness Method for Truss Analsis Epanding the structure stiffness equation Q Q k u K K Often k = since the supports are not displaced, then Qk K11 u Solving unknown displacement 1 K Then unknown reactions at supports u u K K u 11 Q k K Q u 21 u k k

24 14-6 Application of the Stiffness Method for Truss Analsis The member forces can be determined using q k T Epanding this epression Since with q = -q for equilibrium

25 Eample 14.3 etermine the force in each member. AE is constant [1] [2] Since 3=4=5=6= and Q1=, Q2=-2k, k k Q Structure stiffness equation:

26 Eample solution Separating and solving unknown displacements AE ; 2 AE AE Substituting for unknown reactions

27 Eample solution Then reactions at supports are Q.5k ; Q k ; Q 1.5k ; Q 2. k The force in each member or member 1, 1, or member 2,.6,, L 3m q.8, L 5m 1 1.5k q 2 2.5k

28 14-7 odal Coordinates A truss supported b a roller placed on an incline The condition of zero displacement at node 1 is defined onl along the ais, while the displacement along the ais will have displacement components along both global coordinates aes and

29 14-7 odal Coordinates Consider truss member 1 having a global coordinate sstem, at the near node and a nodal coordinate sstem, at the far node

30 14-7 odal Coordinates " " cos cos cos cos " " d d " " d d

31 14-7 odal Coordinates " " cos cos cos cos " " q Q q Q q Q q Q q q Q Q Q Q " " " "

32 14-7 odal Coordinates Member stiffness matri k T T k' T

33 Eample 14.6 etermine the support reactions

34 Eample 14.6 etermine the support reactions Member 1 1,, ".77, ".77

35 Eample 14.6 etermine the support reactions Member 2, 1, ".77, ".77

36 Eample solution Member 3.8,.6

37 Eample solution Assembling the matrices to determine the structure stiffness matri AE ; AE ; AE Q4 31.8k ; Q5 7.5k ; Q k

38 14-8 Trusses having thermal changes and fabrication errors Superposition method Calculate fied end forces to prevent movement of the nodes b temperature or fabrication Calculate the displacement due to equal but opposite forces placed on the truss at the nodes etermine the actual forces in the members and the reactions b superposing the results

39 14-8 Trusses having thermal changes and fabrication errors or staticall indeterminate truss, increment in length of a member due to T is L = TL A decrease in length due to a compressive force q is L' = q L/AE Equating the two gives q = AE T

40 14-8 Trusses having thermal changes and fabrication errors This force will hold the nodes of the member fied ( q ) AET ( q ) AET

41 14-8 Trusses having thermal changes and fabrication errors If a truss member is made too long b an amount L before it is fitted into a truss, the force q needed to keep the member at its design length L is q = AEL /L ( q ) AEL L ( q ) AEL L

42 14-8 Trusses having thermal changes and fabrication errors If the member is too short, then L becomes negative and these forces will reverse In global coordinates, these forces are

43 14-8 Trusses having thermal changes and fabrication errors The initial force-displacement relationship due to temperature changes and fabrication errors Q K Q Where Q o is the initial fied-end forces caused b temperature changes and fabrication errors

44 14-8 Trusses having thermal changes and fabrication errors Carring out the multiplication on the right hand side, Q K K Q k 11 u Q u K21 u K22 k Qk 12 Using the superposition procedure, the unknown displacements are determined from the first equation b subtracting K 12 k and (Q k ) from both sides and then solving for u k k

45 14-8 Trusses having thermal changes and fabrication errors The member forces are determined b superposition q k T q The force at the far end of the member

46 Eample 14.7 etermine the force in member 1 and 2 if member 2 was made.1 m too short before it was fitted into place. Use AE = 81 3 k. Since L = -.1m, m orces due to shortage in global coordinates 4 m

47 Eample 14.7 etermine the force in member 1 and 2 if member 2 was made.1 m too short before it was fitted into place. Use AE = 81 3 k. Structure stiffness matri:

48 Eample 14.7 etermine the force in member 1 and 2 if member 2 was made.1 m too short before it was fitted into place. Use AE = 81 3 k. Partitioning the matrices to obtain the equation for the unknown displacement m.284m

49 Eample 14.7 etermine the force in member 1 and 2 if member 2 was made.1 m too short before it was fitted into place. Use AE = 81 3 k. Member 1 q 1, 5.56k 1, L 3m, AE 81 3 k Member 2.8,.6, L 5m, AE 81 3 k q k

50 14-9 Space-truss analsis To account for the 3- aspects of the problem, additional elements must be included in the transformation matri [T]

51 14-9 Space-truss analsis The direction cosines ) ( ) ( ) ( cos z z L ) ( ) ( ) ( cos ) ( ) ( ) ( cos z z z z z z L z z z z L

52 14-9 Space-truss analsis The transformation matri in 3 Member stiffness matri

53 14-9 Space-truss analsis Member stiffness matri in global coordinates