584 GENERAL FORMULATION LINEAR SYSTEM I I Y Rsse CHAP 17 in a and 17 131 IIp v T V 1 P BU o Tk Y H3 17 132 A1BU A1 H3 A2U 2 The variation of HIp considering U as the independent variable is drfp AUT BT P P BTATk AIB U BTAk A 1 H3 A 2 l 2 g AUT BTKlB U BTH 4 Requiring I to be stationary for arbitrary AU results in 17 126 Note that we could have used the reduced form for V i e equation d Also we still have to determine the constraint forces REFERENCES 1 FENVES S J and F H BRANIN JR Network Topological Formulation of Struc tural Analysis J Structural Div A S C E Vol 89 No ST4 August 1963 pp 483 514 2 DIMAGrIO F L and W R SPILLARS Network Analysis of Structures J Eng Mech Div A S C E Vol 91 No EM3 June 1965 pp 169 188 3 ARGYRIS J H The Matrix Analysis of Structures with Cut Outs and Modifica tions Proc Ninth InternationalCongress Appl Mech Vol 6 1957 pp 131 142 18 Analysis of Geometrically Nonlinear Systems 18 1 INTRODUCTION In this chapter we extend the displacement formulation to include geometric nonlinearity The derivation is restricted to small rotation i e where squares of rotations are negligible with respect to unity We also consider the material to be linearly elastic and the member to be prismatic The first phase involves developing appropriate member force displacement relations by integrating the governing equations derived in Sec 13 9 We treat first planar deformation since the equations for this case are easily integrated and it reveals the essential nonlinear effects The three dimensional problem is more formidable and one has to introduce numerous approximations in order to generate an explicit solution We will briefly sketch out the solution strategy and then present a linearized solution applicable for doubly symmetric cross sections The direct stiffness method is employed to assemble the system equations This phase is essentially the same as for the linear case However the governing equations are now nonlinear Next we described two iterative procedures for solving a set of nonlinear algebraic equations successive substitution and Newton Raphson iteration These methods are applied to the system equations and the appropriate re rrence relations are developed Finally we utilize the classical stability criterion to investigate the stability of an equilibrium position 18 2 MEMBER EQUATIONS PLANAR DEFORMATION Figure 18 1 shows the initial and deformed positions of the member The centroidal axis initially coincides with the X direction and X2 is an axis of symmetry for the cross section We work with displacements ul u2 c3 585 MEMBER EQUATIONS PLANAR DEFORMATION SEC 18 2 FR ANALYSIS OF GEOMETRICALLY NONLINEAR SYSTEMS CHAP 18 587 Boundary Conditions F2 M713 referred to the distributed external force b2 and end forces F1 the chord is denoted by of rotation The frame member X X initial X 1 2 3 by displacements end p3 and is related to the UA2 tB 2 18 1 L P3 For x 0 Al or IF FA U2 UA2 or IF2 Fu 2 0o FA2 0 A 3 Or IMIo UB2 or or U c MA3 For x L The governing equations follow from 13 88 For convenience we drop the subscript on x1 and M 3 03 i3 Also we consider hb m3 0 U2 d F1 U2 xL FB2 IMIL or OB3 CO IF IL FB IF 2 MB3 Integrating a leads to x2 FI FB1 P F2 PU2 PC 2 fX b2 dx M 3 Pu2 C3P C 2 Px fx Sx b 2 dx dx e where C2 C3 are integration constants We include the factor P so that the dimensions are consistent Thc axial displacement ilt is determined from the first equation in a rv 182 PL I u1 UA1 u 2 AE 2 18 2 dx Combining the remaining two equations in a we obtain G2 xx G A M EI1 b2 f Finally the governing equation for u2 follows from the third equation in e 2 U2 xx t where 2 2 u C 2 Lb2 A C3 P b 2 dX dd 18 3 where Fig 18 1 Notation for planar bending P 2 t1 EI Equilibrium Equations GA2 F 1 0 a dx F2 b2 0 Ft2 F2 The solutions for u2 and M are C 2 C COS sx C 5 sinx C2x Mx CO I l GA x F FE Fi2 F2 GA2 M t u1 u1X X GA 2 U2b x where Ut2 denotes the particular solution due to b2 If b2 is constant CO U2b O 18 4 2 U2 b U2 X b2 dx C2 GA 2 2 U2b Cs COSx C4 sin Force DisplacemenltRelations 3 GA 2 P GA2 x 2 t c 12 18 5 588 ANALYSIS OF GEOMETRICALLY NONLINEAR SYSTEMS CHAP 18 Enforcing the boundary conditions on U2 co at x 0 L leads to four linear equations relating C2 C5 When the coefficient matrix is singular the member is said to have buckled In what follows we exclude member buck ling We also neglect transverse shear deformation since its effect is small for a homogeneous cross section We consider the case where the end displacements are prescribed The net displacements are lnet u Onet I O C tC C4 C4 C5 1 cos L sin JtL 2 u u 2 1 18 10 2 B3 In order to evaluate the stiffness coefficients P has to be known If one end say B is unrestrainedwith respect to axial displacement there is no difficulty since FBI is now prescribed The relative displacement is determined from P FI PL 18 7 L CL o UA1 A Ler a U2 2 dx er D 2 1 cos uL ML sin I1 L Note that D 0 as FtL buckling load 2 MA3 co o It sin tL oL sin bL u1 Cs bL A2 FB2 18 6 5 C 3 UA 589 The i functions were introduced by Livesley Ref 7 and are plotted in Fig 18 2 They degenerate rapidly as ylL 2 The initial end forces depend on the transverse loading b2 If b2 is constant U2b x x O L Evaluating 18 4 with A2 oo we obtain C2 01 MEMBER EQUATIONS PLANAR DEFORMATION B3 L 2 u2b O L SEC 18 2 This defines the upper limit on P i e the member 2er 5 C 2 4 e jL uA2 uB OA COB t2 i6 cJB3 J A 3 C5 2 47r EI PInax 18 8 L2 The end forces can be obtained with c e We omit the algebraic details since they are obvious and list the final form below MA3 M 3 02 UA2 r r MB 3 FA 2 MB F FB2 F 3 2 2 I 3 LLEI3FL blB3 b4 20A 3 B3 O A3 UB2 2 CB3 CA3 L2 UB I P FeI PL j T I2 u 2 UA2 P fo where 6 UA2 1 U 2 2 x dx IA PL 18 11 2 C 3b3 …