SEC 8 2 OPERATION ON THE PARTITIONED EQUATIONS P1 B1F P2 B2 F F Fi kAUt 8 Displacement Method Ideal Truss Fi k eo A2 U2 GENERAL The basic equations defining the behavior of an ideal truss consist of force is linear We obtain a set of nd equations relating the n displacement unknowns U by substituting for F in a The resulting matrix equation has the form canspreduce the relations One v u u and force displacement nltins liter ilm ol i UI ia i 8 2 OPERATION ON THE PARTITIONED EQUATIONS The governing partitioned equations for an ideal truss are developed in Sec 6 7 For convenience we summarize these equations below 178 P BIFi 8 1 We solve 8 1 for U 1 determine F from c and P2 from b The coefficient matrix for U1 is called the system stiffness matrix and written as K1 B1kA 1 Ut UtllllI system to a set of equations involving only the unknown joint displacements by substituting the force displacement relations into the force equilibrium equations This particular method of solution is called the displacement or stiffhess method Alternatively one can by eliminating the displacements reduce the governing equations to a set of equations involving certain bar forces The latter procedure is referred to as the fJrce or flexibility method We emphasize that these two methods are just alternate procedures for solving the same basic equations The displacement method is easier to automate than the force method and has a wider range of application However it is a computer based method i e it is not suited for hand computation In contrast the force method is more suited to hand computation than to machine computation In what follows we first develop the equations for the displacement method by operating on the governing equations expressed in partitioned form We then describe a procedure for assembling the necessary system matrices using only the connectivity table This procedure follows naturally ifone first operates on the unpartitioned equations and then introduces the displacement restraints The remaining portion of the chapter is devoted to the treatment of nonlinear behavior We outline an incremental analysis procedure apply the classical stability criterion and finally discuss linearized stability analysis a b c d The unknowns are the m bar forces F the r reactions P2 and the nd joint displacements U One can consider F to represent the initial bar forces that is the bar forces due to the initial elongations and support movements to Us When the with U1 0 The term kA 1UI represents the bar forces due r material is linearelastic k and eo are constant Also Aj B when the geometry B1kA 1 U 8 1 nd eqs r eqs m eqs 179 8 2 One can interpret B1 Fi as representing the initial joint forces due to the initial elongations and support movements with U1 0 Then P1 B1 Fi represents the net unbalanced joint forces When the geometry is linear K1 1 reduces to K11 BlkBT ATfkA 1 8 3 If the material is linear k is constant and positive definite for real materials Then the stiffness matrix for the linear case is positive definite when the system is initially stable that is when r B nd t Conversely if it is not positive definite the system is initially unstable If the material is nonlinear k and e depend on e We have employed a piecewise linear representation for the force elongation curve which results in linear relations However one has to iterate when the limiting elongation for a segment is exceeded The geometrically nonlinear case is more difficult since both A and B depend on U 1 One can iterate on 8 1 but this requires solving a nonsymmetrical system of equations It is more efficient to transform 8 1 to a symmetrical system by transferring some nonlinear terms to the right hand side Nonlinear analysis procedures are treated in Sec 8 4 Even when the behavior is completely linear the procedure outlined above for generating the system matrices is not efficient for a large structure since t See Prob 2 14 180 DISPLACEMENT METHOD IDEAL TRUSS CHAP 8 it requires the multiplication of large sparse matrices For example one obtains the system stiffness matrix by evaluating the triple matrix product K 11 AkA 1 O PI P2 Pj Uj U2 2 t x 1 ii x 1 as the general external joint force and joint displacement matrices Now we write the complete system of ij joint force equilibrium equations expressed in terms of the displacements as Asf I 0P o J 8 7 matrix The elements of 0 are the required joint forces due to the initial elongations and Sq represents the required joint forces due to the joint We start with 6 37 the force displacement relation for bar n kJt7llnun We have defined We refer to f which is of order ij x ij as the unrestrained system stiffness THE DIRECT STIFFNESS METHOD F Fo 181 THE DIRECT STIFFNESS METHOD a One can take account of symmetry and the fact that k is diagonal but Al is generally quite sparse Therefore what is needed is a method of generating K which does not involve multiplication of large sparse matrices A method which has proven to be extremely efficient is described in the next section 8 3 SEC 8 3 kl 7YnUn a F0 n k eo n where n n denote the joints at the positive and negative ends of bar n One can consider F0 as the bar force due to the initial elongation with the ends fixed u un 0 Now we let p p be the external jointforce matrices required to equilibrate the action of F Noting 6 43 we see that Fn r Pn Pn Pn 8 4 displacements We assemble A and Po in partitioned form working with successive mem bers The contributions for member n follow directly from 8 6 0o Partitioned Form Is j x 1 oF I n o Pit kn k kn k b Pn Pn One can interpret b as end action joint displacement relations since the elements of F pT are the components of the bar force with respect to the basic frame Continuing we let kn kp n 8 8 f Partitioned Form Is j X j Substituting for F 8 4 expands to Pn IiFo 0 kn P YnUn kPnYt un in row n in row n in in in in row row row row n column n column n column n column n n n n 8 9 Example 8 1 The connectivity table and general form of Yf and two for the numbering shown in Fig E8 I are presented below 8 5 Fig E8 1 Note that kn is of order i x i where i 2 or 3 for a two or three dimensional truss respectively When the geometry is linear I Yn eOn and k is sym metrical With this notation b takes a more compact form kUn k u PnFo Pn Pn TFo l kull knUn 8 6 We refer to k as the bar stiffness matrix Equation 8 …