54 Robert L.Taylor,Eugenio Onate and Pere-Andreu Ubach A residual form for each element may be written as S11 hA B S22 (41) S12 where [M and where [C are the element mass and damping matrices given by M11M12M13 C11C12C137 [Me]= M21 M22 M23 and [C.]= C21C22C23 (42) M31M32M33 C31C32C33 with M8 Po hEo EadI and Ca8= co hEa Ea dn I (43) 3.1 Pressure Follower Loading For membranes subjected to internal pressure loading,the finite element nodal forces must be computed based on the deformed current configuration.Thus,for each triangle we need to compute the nodal forces from the relation 6ia.Tfo=oaa.T Ea (pn)dw (44) For the constant triangular element and constant pressure over the element,denoted by pe,the normal vector n is also constant and thus the integral yields the nodal forces f°=3 pen Ae (45) We noted previously from Eq.(6)that the cross product of the incremental vectors Ai21 with Ai resulted in a vector normal to the triangle with magnitude of twice the area.Thus,the nodal forces for the pressure are given by the simple relation 1 f°=i:421×A21 (46) Instead of the cross products it is convenient to introduce a matrix form denoted by 421×4元31=2]3 (47) where 0 -△ 金]- 0-49 (48) -△ △ 054 Robert L. Taylor, Eugenio O˜nate and Pere-Andreu Ubach ˜ A residual form for each element may be written as ⎧ ⎨ ⎩ R1 R2 R3 ⎫ ⎬ ⎭ = ⎧ ⎨ ⎩ f 1 f 2 f 3 ⎫ ⎬ ⎭ − [Me] ⎧ ⎪⎨ ⎪⎩ x¨˜ 1 x¨˜ 2 x¨˜ 3 ⎫ ⎪⎬ ⎪⎭ − [Ce] ⎧ ⎪⎨ ⎪⎩ x˜˙ 1 x˜˙ 2 x˜˙ 3 ⎫ ⎪⎬ ⎪⎭ − h A [B] T ⎧ ⎨ ⎩ S11 S22 S12 ⎫ ⎬ ⎭ (41) where [Me] and where [Ce] are the element mass and damping matrices given by [Me] = ⎡ ⎣ M11 M12 M13 M21 M22 M23 M31 M32 M33 ⎤ ⎦ and [Ce] = ⎡ ⎣ C11 C12 C13 C21 C22 C23 C31 C32 C33 ⎤ ⎦ (42) with Mαβ =  Ω ρ0 h ξα ξβ dΩ I and Cαβ =  Ω c0 h ξα ξβ dΩ I (43) 3.1 Pressure Follower Loading For membranes subjected to internal pressure loading, the finite element nodal forces must be computed based on the deformed current configuration. Thus, for each triangle we need to compute the nodal forces from the relation δx˜α,T f α = δx˜ α,T  ω ξα (p n) dω (44) For the constant triangular element and constant pressure over the element, denoted by pe, the normal vector n is also constant and thus the integral yields the nodal forces f α = 1 3 pe n Ae (45) We noted previously from Eq. (6) that the cross product of the incremental vectors ∆x˜21 with ∆x˜31 resulted in a vector normal to the triangle with magnitude of twice the area. Thus, the nodal forces for the pressure are given by the simple relation f α = 1 6 pe ∆x˜21 × ∆x˜31 (46) Instead of the cross products it is convenient to introduce a matrix form denoted by ∆x˜21 × ∆x˜31 =  ∆ $x˜ 21 ∆x˜31 (47) where  ∆ $x˜ ij  = ⎡ ⎣ 0 −∆x˜ij 3 ∆x˜ij 2 ∆x˜ij 3 0 −∆x˜ij 1 −∆x˜ij 2 ∆x˜ij 1 0 ⎤ ⎦ . (48)