Module 4. Analysis of Statically Indeterminate Structures by the Direct Stiffness Method. Version 2 CE IIT, Kharagpur

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1 Modle Analysis of Statically Indeterminate Strctres by the Direct Stiffness Method Version CE IIT, Kharagr

2 Lesson The Direct Stiffness Method: Trss Analysis (Contined) Version CE IIT, Kharagr

3 Instrctional Objectives After reading this chater the stdent will be able to. Transform member stiffness matri from local to global co-ordinate system.. Assemble member stiffness matrices to obtain the global stiffness matri.. Analyse lane trss by the direct stiffness matri.. Analyse lane trss sorted on inclined roller sorts.. Introdction In the revios lesson, the direct stiffness method as alied to trsses was discssed. The transformation of force and dislacement from local co-ordinate system to global co-ordinate system were accomlished by single transformation matri. Also assembly of the member stiffness matrices was discssed. In this lesson few lane trsses are analysed sing the direct stiffness method. Also the roblem of inclined sort will be discssed. Eamle. Analyse the trss shown in Fig..a and evalate reactions. Assme EA to be constant for all the members. Version CE IIT, Kharagr

4 The nmbering of joints and members are shown in Fig..b. Also, the ossible dislacements (degrees of freedom) at each node are indicated. Here lower nmbers are sed to indicate nconstrained degrees of freedom and higher nmbers are sed for constrained degrees of freedom. Ths dislacements, and are zero de to bondary conditions. First write down stiffness matri of each member in global co-ordinate system and assemble them to obtain global stiffness matri. Element : θ =, L =.9 m. Nodal oints k = EA () [ ] Element : θ = 9, L =. m. Nodal oints - Version CE IIT, Kharagr

5 k = EA (). [ ] Element : θ =, L =.9 m. Nodal oints k = EA () [ ] Element : θ =, L =. m. Nodal oints - k = EA (). [ ] Element : θ =, L =. m. Nodal oints - k = EA (). [ ] The assembled global stiffness matri of the trss is of the order. Now assemble the global stiffness matri. Note that the element k of the member stiffness matri of trss member goes to location (,) of global stiffness matri. On the member stiffness matri the corresonding global degrees of freedom are indicated to facilitate assembling. Ths, Version CE IIT, Kharagr

6 Version CE IIT, Kharagr [ ] = EA K () Writing the load-dislacement relation for the trss, yields = EA () The dislacements to are nknown. The dislacements = = =. Also = = = =. Bt kn =. = EA () Solving which, the nknown dislacements are evalated. Ths,

7 ..... = ; = ; = ; = ; = (9) Now reactions are evalated from eqation,.9 = EA EA.. () Ths, =. kn ; = ; =. kn. () Now calclate individal member forces. Member : l =. ; m =. ; L =. 9m. { ' } = [ l m l m ].9. ' =.. =. kn.9. { } [ ] () Member : l = ; m =. ; L =. m. { ' } = [ l m l m ].. ' = =. kn.9. { } [ ] () Member : l =. ; m =. ; L =. 9m. Version CE IIT, Kharagr

8 { ' } = [ l m l m ].9. ' =.... =. kn.9. { } [ ] () Member : l =. ; m = ; L =.. m. { ' } = [ l m l m ]. ' = =. kn.. { } [ ] () Member : l =. ; m = ; L =.. m. { ' } = [ l m l m ].. ' = =. kn.. { } [ ] () Eamle. Determine the forces in the trss shown in Fig..a by the direct stiffness method. Assme that all members have the same aial rigidity. Version CE IIT, Kharagr

9 Version CE IIT, Kharagr

10 The joint and member nmbers are indicated in Fig..b. The ossible degree of freedom are also shown in Fig..b. In the given roblem, and reresent nconstrained degrees of freedom and = = = = = de to bondary condition. First let s generate stiffness matri for each of the si members in global co-ordinate system. Element : θ =, L =. m. Nodal oints - [ k ] = EA. () Element : θ = 9, L =. m. Nodal oints - Version CE IIT, Kharagr

11 [ k ] = EA. () Element : θ =, L =. m. Nodal oints - [ k ] = EA. () Element : θ = 9, L =. m. Nodal oints - [ k ] = EA. () Element : θ =, L =. m. Nodal oints - [ k ].. = EA () Element : θ =, L =. m. Nodal oints - Version CE IIT, Kharagr

12 [ k ].. = EA () There are eight ossible global degrees of freedom for the trss shown in the figre. Hence the global stiffness matri is of the order ( ). On the member stiffness matri, the corresonding global degrees of freedom are indicated to facilitate assembly. Ths the global stiffness matri is, K = () [ ] The force-dislacement relation for the trss is,... = EA () The dislacements, and are nknowns. Here, = kn ; = ; = and = = = = =. Version CE IIT, Kharagr

13 ... = EA (9) Ths,. = () Solving which, yields. = ;.9 = ; = Now reactions are evalated from the eqation,.. = () =. kn ; =.9 kn ; =.9 kn ; =. kn ; =. kn In the net ste evalate forces in members. Element : θ =, L =. m. Nodal oints - Version CE IIT, Kharagr

14 { ' } = [ l m l m ].. ' = =. kn.. { } [ ] () Element : θ = 9, L =. m. Nodal oints - { ' } = [ l m l m ] ' = =.9kN.9 { } [ ] () Element : θ =, L =. m. Nodal oints - { ' } [ ]{} = = () Element : θ = 9, L =. m. Nodal oints - ' =. = () { ' } = [ l m l m ] { } [ ] { } Element : θ =, L =. m. Nodal oints - { ' } = [ l m l m ]. Version CE IIT, Kharagr

15 . ' =.. =. kn..9 { } [ ] () Element : θ =, L =. m. Nodal oints - { ' } = [ l m l m ]. { } [ ] { } ' =.. =. kn ().. Inclined sorts Sometimes the trss is sorted on a roller laced on an obliqe lane (vide Fig..a). At a roller sort, the dislacement erendiclar to roller sort is zero. i.e. dislacement along y" is zero in the resent case. Version CE IIT, Kharagr

16 If the stiffness matri of the entire trss is formlated in global co-ordinate system then the dislacements along y are not zero at the obliqe sort. So, a secial rocedre has to be adoted for incororating the inclined sort in the analysis of trss jst described. One way to handle inclined sort is to relace the inclined sort by a member having large cross sectional area as shown in Fig..b bt having the length comarable with other members meeting at that joint. The inclined member is so laced that its centroidal ais is erendiclar to the inclined lane. Since the area of cross section of this new member is very high, it does not allow any dislacement along its centroidal ais of the joint A. Another method of incororating inclined sort in the analysis is to sitably modify the member stiffness matri of all the members meeting at the inclined sort. Version CE IIT, Kharagr

17 Consider a trss member as shown in Fig... The nodes are nmbered as and. At, it is connected to a inclined sort. Let ' y' be the local co-ordinate aes of the member. At node, the global co-ordinate system y is also shown. At node, consider nodal co-ordinate system as " y", where y " is erendiclar to obliqe sort. Let ' and ' be the dislacements of nodes and in the local co-ordinate system. Let,v be the nodal dislacements of node in global co-ordinate system y. Let ", v" be the nodal dislacements along " - and y" - are in the local co-ordinate system " y" at node. Then from Fig.., ' This may be written as ' = + v cosθ sinθ " cosθ " + v" sin = θ (.) " ' cosθ sinθ v = ' cosθ " sinθ " " v" Denoting l = cosθ ; m = sinθ ; l" = cosθ " ; m" = sinθ " ' ' or { '} = [ T' ]{} l = m l" v m" " v" where [ T' ] is the dislacement transformation matri. (.) (.a) Version CE IIT, Kharagr

18 Similarly referring to Fig.., the force ' has comonents along and y aes. Hence = ' cosθ (.a) = ' sinθ (.b) Version CE IIT, Kharagr

19 Similarly, at node, the force ' has comonents along " and y" aes. " = ' cosθ (.a) " = ' sinθ (.b) The relation between forces in the global and local co-ordinate system may be written as, cosθ sinθ ' = " cosθ ' " sinθ (.) T { } [ T '] { ' } = (.) Using dislacement and force transformation matrices, the stiffness matri for member having inclined sort is obtained. Simlifying, T [] k = [ T '][ k' ][ T '] l m l m k l" L l" m" (.) m" [ ] = l lm ll" lm" EA lm m ml" mm" k = L " " " " " (.9) ll ml l l m lm" mm" l" m" m" [] If we se this stiffness matri, then it is easy to incororate the condition of zero dislacement erendiclar to the inclined sort in the stiffness matri. This is shown by a simle eamle. Version CE IIT, Kharagr

20 Eamle. Analyse the trss shown in Fig..a by stiffness method. Assme aial rigidity EA to be constant for all members. Version CE IIT, Kharagr

21 The nodes and members are nmbered in Fig..b. The global co-ordinate aes are shown at node. At node, roller is sorted on inclined sort. Hence it is reqired to se nodal co-ordinates " y" at node so that cold be set to zero. All the ossible dislacement degrees of freedom are also shown in the figre. In the first ste calclate member stiffness matri. Member : θ =., θ " =., L =. m. Nodal oints - l =. ; m =. ; l" =.99 ; m" =.. [ k ].. = EA () Version CE IIT, Kharagr

22 Member : θ =, θ " =, L =. m. Nodal oints - l = ; m = ; l" =. ; m" =.. Version CE IIT, Kharagr

23 [ k ].. = EA () Member : θ = 9, L =. m., l = ; m = Nodal oints -. [ k ] = EA. () For the resent roblem, the global stiffness matri is of the order ( ) global stiffness matri for the entire trss is. []. The k = EA () Writing load-dislacement eqation for the trss for nconstrained degrees of freedom,. = () Solving,... = ; = ; () = Version CE IIT, Kharagr

24 Now reactions are evalated from the eqation = () =. kn ; =.9 kn ; =. kn Smmary Sometimes the trss is sorted on a roller laced on an obliqe lane. In sch sitations, the direct stiffness method as discssed in the revios lesson needs to be roerly modified to make the dislacement erendiclar to the roller sort as zero. In the resent aroach, the inclined sort is handled in the analysis by sitably modifying the member stiffness matrices of all members meeting at the inclined sort. A few roblems are solved to illstrate the rocedre. Version CE IIT, Kharagr

Chapter 2 Introduction to the Stiffness (Displacement) Method. The Stiffness (Displacement) Method

Chapter 2 Introduction to the Stiffness (Displacement) Method. The Stiffness (Displacement) Method CIVL 7/87 Chater - The Stiffness Method / Chater Introdction to the Stiffness (Dislacement) Method Learning Objectives To define the stiffness matrix To derive the stiffness matrix for a sring element