8 Displacement Method-- Ideal Truss 8-1. GENERAL The basic equations defining the behavior of an ideal truss consist of force-u u v;liter;ilm .... nltins and force-displacement relations. One can sp reduce the Ut;UtllllI i ia .UI-i ol 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. 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 SEC. 8-2. OPERATION ON THE PARTITIONED EQUATIONS 179 P1 = B1F (nd eqs.) (a) P2 = B2 F (r eqs.) (b) F = Fi + kAUt (m eqs.) (c) Fi = k(-eo + A2 U2 ) (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 with U1 = 0. The term kA1UI represents the bar forces due to Us. When the material is linearelastic, k and eo are constant. Also, Aj = Br when the geometry is linear. We obtain a set of 'ndequations relating the n displacement unknowns, U, by substituting for F in (a). The resulting matrix equation has the form (B1kA 1)U = 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 (8-2) One can interpret B1Fi as representing the initial joint forces due to the initial elongations and support movements with U1 = 0. Then P1 -B1Fi represents the net unbalanced joint forces. When the geometry is linear, K11 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 U1. 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, = AkA 1 (a)K11 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. THE DIRECT STIFFNESS METHOD We start with (6-37), the force-displacement relation for bar n: F, = Fo, + kJt7llnun -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 Pn+ + Fnr (8-4) Pn_-= Pn+ Substituting for F,, (8-4) expands to 0, , + kn,P YnUn+ -kPnYt,,un (b)Pn+ = IiFoPn_ = -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 (8-5) 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+Pn+ = ±PnFo, -k,,u (8-6) Pn = --TFo,,, l-kull,,+ + knUn We refer to k as the bar stiffness matrix. Equation (8-6) defines the joint forces required for bar n. The total joint forces required are obtained by summing over the bars. SEC. 8-3. THE DIRECT STIFFNESS METHOD 181 We have defined = {PI, P2,..., Pj} (t x 1) O&= { U2,2 , Uj} (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 J. = .Asf,I + 0P'o (8-7) We refer to ,f[, which is of order ij x ij, as the unrestrained system stiffness matrix. The elements of -. 0 are the required joint forces due to the initial Sq!elongations and represents the required joint forces due to the joint displacements. We assemble ,A' and Po in partitioned form, working with successive mem-bers. The contributions