Keywords: corotational method, Rigid-Bodies-Spring model, accuracy, geometrical nonlinearity

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Keywords: corotational method, Rigid-Bodies-Spring model, accuracy, geometrical nonlinearity

vo2=voi+(sina1+sina2)l/2+{f1(sin2a1+sin2a2) +Fz1(sina1cosai+sina2cosa2)}l/(2EA) woe=w01-l+(cosai+cosa2)l/2+{fy1(sinaicosai +sina2cosa2)+f21(cos2ai+cos2a2)}l/(2ea) a2=a1-{m1+(f1icosai-fz1sinal)l/2}/km -{(Fy12-Fz12)sin(2a1)/4 +FyiFzicos(2a1)/2}l/(EAKM) M2=M1+Fy1(l+wo2-wo1)-Fzi(vo2-vo1) (1-a-d) Vk Nk cosak-sinak sinakcosak Fyk FZk (k-1, 2) (3) vo2=vol+(sinai+sina2)l/2+[fy1{1-cos(al+a2)} +Fzisin(al+a2)]/(2KN) woe=wol-l+(cosai+cosa2)l/2+[f1sin(a1+a2) +F1{1+cos(al+a2)}]/(2KN) a2=al-{m1+(fyicosal-fzisfinal)l/2}/km -{(Fy12-F z12)sin(ai+a2)/4 +FyiFzicos(al+a2)/2}/(KNKM) M2=M1+Fy1(l+wo2-w01)-Ff1 (v02-voi) (2-a-d)

M1=4(EI/l)a1+2(EI/l)a2 -M2=2(EI/l)a1+4(EI/l) Fi1=Fz2=(EI/i)woe Fy1=Fy2=6(EI/l)(al+a2)/(l+woe) a2 (7-a-d) a2=-{2fz1(l2/ei)b4-8(23+o4)2(o4-1)} /{16Fz1(l2/Er)(2c3+c4)} M1=4(EI/l)53a1+2(EI/l)54a2 -M2=2(EI/l)b4a1+4(EI/l)g3a2 FZi=Fz2=(EA/l){woe+l(/31CY12+2/32a1a2+/ala2)) Fy1=Fy2=-(EI/l)(4b3+2b4)(a1+a2)/(l+wo2) 31={2Fz1(l2/Er)g4+8(23+54)2(54-1)} /{16Fz1(l2/Er)(2c3+54)} (g-a-d) (g-e,f)

4=rl(rl-sinrl)/(2qc) c=2-2cosrl-rlsinrl FZ1>0 yb3=rl(rlcoshrl-sinhrl)/(4c5t) 4=rl(sinhrl-TI)/(2br) qt=2-2coshrl+rlsinhrl r=if1l/ei (8-g-i) (g-j-1) (g-m) M1=4(EI/l)a1+2(EI/l)a2 +(EA/30)(4x1-a2)w02 +(EAl/280)(8x13-a23-3a12a2 +2t7122)x -M 2=2(EI/l)a2+4(EI/l)a2 -(EA/30)(a1-4x2)w 02 -(EAl/280)(x13-8x23-2a12a2 +3a1a22) FZl=Fz2=(EA/l){woe+1(2x12-ala2 +2a22)/30) Fy1=Fy2=-{(6EI/I+EAw02/10)(al+a2) +(EAl/280)(7a13+7x23 -a 12a2-a1a22)}/(l+woe) (9-a-d) Vk Nk COSa'k-S1IIak SinakCOSak Fyk Fzk (k=1, 2) (10) v02-v01 w02-w01+l a2-a1 cosr -sinr sinr0 cosr0 0 0 1 0 w02+l a2-al

V2 N2 cos(ai-a2) -sin(al-a2) sin(al-a2) cos (al-a2) Vi N1 (16)

2) Harrison, H. B.: Post-buckling Analysis of Non-Uniform Elastic Columns, Int. J. Numer. Methods Eng., Vol. 7, pp. 195-210, 1973. 3) Miller, R. E.: Numerical Analyses of a Generalized Plane Elastica, Int. J. Numer. Methods Eng., Vo1. 15, pp. 325-332, 1980. 4) El-Naschie, M. S., Wu, C. W. and Wifi, A. S.: A Simple Discrete Element Method for the Initial Post-Buckling of Elastic Structures, Int. J. Numer. Methods Eng., Vol. 26, pp. 2049-2060, 1988. 5) Coulter, B. A. and Miller, R. E.: Loading, Unloading and Reloading of a Generalized Plane Plastica, Vol. 28, pp. 1645. 1660, 1989.

12) Oran, C.: Tangent Stiffness in Plane Frames, Journal of Structural Division, ASCE, Vol. 99, No. ST 6, pp. 973 985, June, 1973. 13) Jennings, A.: Frame Analysis including Change of Geometry, Journal of Structural Division, ASCE, Vol. 94, No. ST 3, pp. 627-644, March, 1968. 15) Goto, Y., Yamashita, T. and Matsuura, S.: Elliptic Integral Solutions for Extensional Elastica with Constant Initial Curvature, Proc. of JSCE, No. 386, I-8, pp. 83-93, Oct. 1987. 16) Goto, Y., Yoshimitsu, T. and Obata, M.: Elliptic Integral Solutions of Plane Elastica with Axial and Shear Deformations, Int. J. Solids Structures, Vol. 26, No. 4, pp. 375-390, 1990. 17) Ai, M. and Nishino, F.: On Convergence of Geometrically Nonlinear Discretization at Limit Element Division, Proc. of JSCE, No. 374, 1-6, pp. 141-150, Oct. 1986. ACCURACY OF THE NUMERICAL METHODS FOR THE ANALYSIS OF PLANE FRAMES CONSIDERING FINITE DISPLACEMENTS AND FINITE STRAINS Yoshiaki GOTO, Tomoo YOSHIMITSU, Makoto OBATA It is accurate to use the theory of finite displacements and finite strains for the analysis of geometrical nonlinearity of structures. However, the governing equation for this theory becomes highly nonlinear and its direct use makes the solution procedure very much complicated. So approximate numerical methods, respectively referred to as the corotational method and the method with the Rigid Bodies-Spring Model, are often used to simplify the solution procedure for the framed structures. Herein, we precisely examine their theoretical accuracy as well as the convergence to the exact solutions. Based on this result, we further discuss an efficient method to be adopted in the analysis of frames considering finite displacements and finite strains. and Fumio NISHINO 76

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