94 Riccardo Rossi,Vitaliani Renato,and Eugenio Onate Linearization Equation (16)is nonlinear,its practical use needs therefore its lineariza- tion.The best rate of convergence is theoretically given by Newton-Raphson technique which guarantees quadratical convergence to the solution.Defining =6Wint -oWert -oWpr each Newton-Raphson step takes the form d亚+亚=0 (35) The term can be explicitated using expression(30)(34)we therefore miss only the differential dy that can be evaluated from the linearization of the different contributions Linearization of internal work The term connected to the internal works can be linearized as follows d(Wint)= ( 6C:S 名 =空a:s+人cds6 the first terms gives,by using(22) 告46c.s=空/(g）= afog)d(x)(s) 8{x} (37) now it can be seen that a(（行og1r）=(dgcdg,gdgs+gds,)问= =(s16ge·dges226gn·dgns12(⑥gn·dge+ge·dgn) (38) substitution of the shape functions gives immediately a set of equalities in the form 2NN,d=u2086网 s1dge·dg=9110诞0c which makes possible to write On on 512 (40)94 Riccardo Rossi, Vitaliani Renato, and Eugenio Onate Linearization Equation (16) is nonlinear, its practical use needs therefore its lineariza￾tion. The best rate of convergence is theoretically given by Newton-Raphson technique which guarantees quadratical convergence to the solution. Defining Ψ = δWint − δWext − δWpr each Newton–Raphson step takes the form dΨ + Ψ = 0 (35) The term Ψ can be explicitated using expression (30)(34) we therefore miss only the differential dΨ that can be evaluated from the linearization of the different contributions Linearization of internal work The term connected to the internal works can be linearized as follows d (Wint) = d h0 2  Ω δC : S  == h0 2  Ω d (δC) : S + h0 2  Ω δC : d (S) (36) the first terms gives, by using (22) h0 2  Ω d (δC) : S = h0 2  Ω d  {δg}T  {s} = = h0 2  Ω d  ∂ {δg}T ∂ {x} d {x} {s} (37) now it can be seen that d 1 2 {δg} T  =  δgξ • dgξ δgη • dgη δgη • dgξ + δgξ • dgη  {s} = =  s11δgξ • dgξ s22δgη • dgη s12 (δgη • dgξ + δgξ • dgη)  (38) substitution of the shape functions gives immediately a set of equalities in the form s11δgξ • dgξ = s11 ∂NI ∂ξ ∂NJ ∂ξ δijδxI • dxjJ = s11 ∂NI ∂ξ ∂NJ ∂ξ δijδxiIdxjJ (39) which makes possible to write d 1 2 {δg} T  =  s11 ∂NI ∂ξ ∂NJ ∂ξ + s22 ∂NI ∂η ∂NJ ∂η + s12 ∂NI ∂η ∂NJ ∂ξ + ∂NI ∂ξ ∂NJ ∂η  δij δxiIdxjJ (40)