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Chapter I2 A  Theory of Nonlinear Finite Element Analysis              229







                                                                                      (12.1)





                          I
                  where ( ) = d/ds and  s denotes the axial coordinate of the element.
                  A generalized elastic stress vector { CT } is expressed as:
                       w = [DE l@4


                       b>= bX Fy F, Mx My M,)T                                        (12.2)
                       [DE]= [EAx GAY GA, GZx GZ, GZ,J
                  where E is Yong’s modulus ,G the shear modulus, A,  denotes the area of the cross-section,
                  A,  , and  A,  denote the effective shear areas, I, and  I, are moments of inertia, and I, denotes
                  the torsional moment of inertia.
                  Applying a virtual work principle, we obtain:
                       I..erccrI+ &I)=  p4w+                                          (12.3)


                  where L is the length of the element, { uc } is the elastic nodal displacement vector, and the
                  external load vector is  v). Substituting the strains and stresses defined in Eqs.  (12.1) and
                  (12.2) into Eq.(12.3),  and omitting the second order terms of the displacements, we get (Bai
                  and Pedersen, 1991):
                       [kEl(dU‘J=                                                     (12.4)
                  where,

                                                                                      (12.5)
                       [kE 1 = [kL I+ [k, I+ kD1
                  and
                       @I  = VI+ k!f>  - @L  I+ [k, Dbe 1                             (12.6)

                  The  matrix  [kL] is  a  standard  linear  stiffness  matrix  (Przemieniecki,  1968),  [k,] is  a
                  geometrical  stiffness  matrix  (Ancher,  1965), and  [kD] is  a  deformation stiffness  matrix
                  (Nedergaard and Pedersen, 1986).

                  12.3  The Plastic Node Method

                  12.3.1  History of the Plastic Node Method
                  The Plastic Node Method was named by Ueda et a1 (1979). It is a generalization of the Plastic
                  Hinge Method developed by Ueda et  a1  (1967) and others,  Ueda and Yao (1980) published
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