96 Riccardo Rossi,Vitaliani Renato,and Eugenio Onate Considering that it is possible to write the cross product of two vectors in Voigt format as C 0 -a3 a2 c=ax b- C2 a3 -a1 b →{c}=[ax]{b} (49) C3 -a2a1 0 and taking in account (3)and(4)we obtain d(pN,{ge×gn}) (50) K11… [Kpr] Kk1... Kkk Kdl=(wk-n (51) d(oWpr)={ox)Kpr]{dx} (52) Linearized formulation The only step missing is to merge all the terms in (35)to find the final expression.The result of this operation is {ox)([Kgeo]+[Kmat]-[Kprl){dx}={ox)({fext}-{fint}) (53) invoking the arbitrariety of fox and introducing the definitions Ktan]Kgeo]+Kmat]-Kpr] (54) {R}={fext}-{fint} (55) the principle of virtual works gives for each element [Ktan]{dx}=(R) (56) 2.2 Solution Procedure As briefly outlined at the beginning of the section,membrane systems are possibly subjected to large rigid body motions which reflects in singular or ill-conditioned "static"stiffness matrices.In addition,convergence of the Newton-Raphson algorithm is often difficult as the final solution can be very "far"from the initial guess even for little variations of the applied loads. Dynamic solution techniques on the other hand are not affected by such problems.Mass and damping contributions remove the singularities from the system and generally provide a better conditioning to the problem.The intro- duction of dynamic terms provides as well an excellent source of stabilization for the solution (physically the solution can't change much in a small time), ending up with better convergence properties inside each solution step.96 Riccardo Rossi, Vitaliani Renato, and Eugenio Onate Considering that it is possible to write the cross product of two vectors in Voigt format as c = a × b → ⎛ ⎝ c1 c2 c3 ⎞ ⎠ = ⎛ ⎝ 0 −a3 a2 a3 0 −a1 −a2 a1 0 ⎞ ⎠ ⎛ ⎝ b1 b2 b3 ⎞ ⎠ → {c} = [a×] {b} (49) and taking in account (3) and (4) we obtain d (pNI {gξ × gη}) =  pNI ∂NJ ∂η [gξ×] − pNI ∂NJ ∂ξ [gξ×]  {dxJ} (50) [Kpr] = ⎛ ⎝ K11 ... K1k ... ... ... Kk1 . . . Kkk ⎞ ⎠ ; [KIJ] =  pNI ∂NJ ∂η [gξ×] − pNI ∂NJ ∂ξ [gξ×]  (51) d(δWpr) = {δx} T [Kpr] {dx} (52) Linearized formulation The only step missing is to merge all the terms in (35) to find the final expression. The result of this operation is {δx} T ([Kgeo] + [Kmat] − [Kpr]) {dx} = {δx}T ({fext} − {fint}) (53) invoking the arbitrariety of {δx} and introducing the definitions [Ktan] = [Kgeo] + [Kmat] − [Kpr] (54) {R} = {fext} − {fint} (55) the principle of virtual works gives for each element [Ktan] {dx} = {R} (56) 2.2 Solution Procedure As briefly outlined at the beginning of the section, membrane systems are possibly subjected to large rigid body motions which reflects in singular or ill-conditioned “static” stiffness matrices. In addition, convergence of the Newton–Raphson algorithm is often difficult as the final solution can be very “far” from the initial guess even for little variations of the applied loads. Dynamic solution techniques on the other hand are not affected by such problems. Mass and damping contributions remove the singularities from the system and generally provide a better conditioning to the problem. The intro￾duction of dynamic terms provides as well an excellent source of stabilization for the solution (physically the solution can’t change much in a small time), ending up with better convergence properties inside each solution step