52 Robert L.Taylor,Eugenio Ofate and Pere-Andreu Ubach in which po is mass density in the reference configuration,co is a linear damping coef- ficient in the reference configuration,h is membrane thickness,Sr are components of the second Piola-Kirchhoff stress,bi are components of loads in global coordi- nate directions (e.g.,gravity),and ti are components of specified membrane force per unit length.Upper case letters refer to components expressed on the reference configuration,whereas,lower case letters refer to current configuration quantities. Likewise,and w are surface area for the reference and current configurations, respectively.Finally,Yt is a part of the current surface contour on which traction values are specified. The linear damping term is included only for purposes in getting initially stable solutions.That is,by ignoring the inertial loading based on only first derivatives of time will occur.This results in a transient form which is critically damped-thus permitting the reaching of a static loading state in a more monotonic manner. We note that components for a normal pressure loading may be expressed as bi=pni (26) where p is a specified pressure and na are components of the normal to the surface. Writing Eq.(20)in component form we have CIJ Gil gij Gjj for i,j=1,2 I,J=1,2 (27) where 1 1 -J;Ga1=0 Gu=Gaa=Ga (28) The integrand of the first term in(25)may be written as 8CIJ SIJ=Gil 6gij GjJ SIJ=6gis Sij (29) where the stress like variable sij is defined by Sij Gil GiJ SIJ (30) The transformation of stress given by (30)may be written in matrix form as s=QS (31) in which Qab-Gil GjJ where the index map is performed according to Table 1,yielding the result Gh 00 Q= G12 G22G12G22 (32) 2G11G120G11G22 Since the deformation tensor is constant over each element,the results for the stresses are constant when h is taken constant over each element and,thus,the surface integral for the first term leads to the simple expression h6EIJ SiJdn )56gdn-9ys号A83 h
52 Robert L. Taylor, Eugenio O˜nate and Pere-Andreu Ubach ˜ in which ρ0 is mass density in the reference configuration, c0 is a linear damping coef- ficient in the reference configuration, h is membrane thickness, SIJ are components of the second Piola-Kirchhoff stress, bi are components of loads in global coordinate directions (e.g., gravity), and t ¯i are components of specified membrane force per unit length. Upper case letters refer to components expressed on the reference configuration, whereas, lower case letters refer to current configuration quantities. Likewise, Ω and ω are surface area for the reference and current configurations, respectively. Finally, γt is a part of the current surface contour on which traction values are specified. The linear damping term is included only for purposes in getting initially stable solutions. That is, by ignoring the inertial loading based on x¨ only first derivatives of time will occur. This results in a transient form which is critically damped - thus permitting the reaching of a static loading state in a more monotonic manner. We note that components for a normal pressure loading may be expressed as bi = p ni (26) where p is a specified pressure and ni are components of the normal to the surface. Writing Eq. (20) in component form we have CIJ = GiI gij GjJ for i, j = 1, 2 I, J = 1, 2 (27) where G11 = 1 J11 ; G22 = 1 J22 ; G12 = −J12 J11 J22 ; G21 = 0 (28) The integrand of the first term in (25) may be written as δCIJ SIJ = GiI δgij GjJ SIJ = δgij sij (29) where the stress like variable sij is defined by sij = GiI GjJ SIJ (30) The transformation of stress given by (30) may be written in matrix form as s = QT S (31) in which Qab ← GiI GjJ where the index map is performed according to Table 1, yielding the result Q = ⎡ ⎣ G2 11 0 0 G2 12 G2 22 G12 G22 2 G11 G12 0 G11 G22 ⎤ ⎦ (32) Since the deformation tensor is constant over each element, the results for the stresses are constant when h is taken constant over each element and, thus, the surface integral for the first term leads to the simple expression Ω h δEIJ SIJ dΩ = Ω h 2 δCIJ SIJ dΩ = Ω h 2 δgij sij dΩ = h 2 δgij sij A (33)
Finite Element Analysis of Membrane Structures 53 Table 1.Index map for Q array Indices Values a 123 1J1,12,21,2&2,1 b12 3 ij1,12,21,2&2,1 where A is the reference area for the triangular element. The variation of gis results in the values 6g11=2(62-6)T△221 6g12=(62-6)T△231+(63-6m)T△元21 (34) 6g22=2(63-621)T△231 At this stage it is convenient to transform the second order tensors to matrix form and write S11 7=6JSJ=[6D1 S22 -6ETS (35) S12 or for the alternative form 吉yw=吉[m n2ne】 811 (36) 2 822 812 Using(34)we obtain the result directly in terms of global cartesian components as gs-[eyr6eyey]rs =[6()T(2)T6(3)T]Q'S=6Es (37) where the strain-displacement matrir b is given by -(4221)T (△221)T0 b= -(4231)T 0 (431)T (38) -(△221+△231)T(4281)T(4221)T 3×9 Thus,directly we have in each element 6E=Qb62=80 (39) where denotes the three nodal values on the element.It is immediately obvious that we can describe a strain-displacement matrix for the variation of E as B=Qb (40)
Finite Element Analysis of Membrane Structures 53 Table 1. Index map for Q array Indices Values a 12 3 I,J 1,1 2,2 1,2 & 2,1 b 1 2 3 i,j 1,1 2,2 1,2 & 2,1 where A is the reference area for the triangular element. The variation of gij results in the values δg11 = 2 δx˜2 − δx˜1T ∆x˜21 δg12 = δx˜2 − δx˜1T ∆x˜31 + δx˜3 − δx˜1T ∆x˜21 δg22 = 2 δx˜3 − δx˜1T ∆x˜31 (34) At this stage it is convenient to transform the second order tensors to matrix form and write 1 2 δCIJ SIJ = δEIJ SIJ = δE11 δE22 2 δE12 ⎡ ⎣ S11 S22 S12 ⎤ ⎦ = δET S (35) or for the alternative form 1 2 δgij sij = 1 2 δg11 δg22 2 δg12 ⎡ ⎣ s11 s22 s12 ⎤ ⎦ = 1 2 δgT s (36) Using (34) we obtain the result directly in terms of global cartesian components as 1 2 δgT s = δ(x˜1) T δ(x˜2) T δ(x˜3) T [b] T s = δ(x˜1) T δ(x˜2) T δ(x˜3) T [b] T QT S = δET S (37) where the strain-displacement matrix b is given by b = ⎡ ⎣ −(∆x˜21) T (∆x˜21) T 0 −(∆x˜31) T 0 (∆x˜31) T −(∆x˜21 + ∆x˜31) T (∆x˜31) T (∆x˜21) T ⎤ ⎦ 3×9 (38) Thus, directly we have in each element δE = Q b δx˜ = 1 2 δC (39) where x˜ denotes the three nodal values on the element. It is immediately obvious that we can describe a strain-displacement matrix for the variation of E as B = Q b (40)
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) -△ △ 0
54 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)