Finite Element Analysis of Membrane Structures Robert L.Taylor*1,Eugenio Onate2 and Pere-Andreu Ubach2 1 Department of Civil and Environmental Engineering University of California at Berkeley,USA rltOce.vulture.berkeley.edu 2 International Center for Numerical Methods in Engineering Edificio C1,Gran Capitan s/n 08034 Barcelona-Spain onate@cimne.upc.es Summary.This paper summarizes the development for a large displacement for- mulation of a membrance composed of three-node triangular elements.A formulation in terms of the deformation gradient is first constructed in terms of nodal variables. In particular,the use of the right Cauchy-Green deformation tensor is shown to lead to a particulary simple representation in terms of nodal quantities.This may then be used to construct general models for use in static and transient analyses. Key words:Membrane structures,finite element 1 Introduction The behavior of curved,thin bodies can be modeled by a membrane theory of shells. In such a theory only the in-plane stress resultants are included.The deformation state for a membrane may be represented by the position of points on the two- dimensional surface.General theories for shells may be specialized to those for a membrane by ignoring the resultant couples and associated changes in curvature deformations as well as any transverse shearing effects.A numerical approximation to the shell may then be constructed using a finite element approach.Examples for general shell theory and finite element solution may be found for small deforma- tions in standard books.]Theory for large deformation can proceed following the presentations of Simo et al.23.4.5]or Ramm et al.6.7.8.9.10.11,12.13 For the simplest shape finite element composed of a 3-node triangular form with displacement parameters at each vertex (a 9-degree of freedom element)it is far simpler to formulate the membrane behavior directly.This is especially true for large displacement response.Here the initially flat form of a simple triangular response is maintained throughout all deformation states.Consequently,one may proceed Visiting Professor,CIMNE,UPC,Barcelona,Spain. 47 E.Onate and B.Kroplin (eds.).Textile Composites and Inflatable Structures,47-68. 2005 Springer.Printed in the Netherlands
Finite Element Analysis of Membrane Structures Robert L. Taylor1, Eugenio O˜nate2 and Pere-Andreu Ubach2 1 Department of Civil and Environmental Engineering University of California at Berkeley, USA rlt@ce.vulture.berkeley.edu 2 International Center for Numerical Methods in Engineering Edificio C1, Gran Capit´an s/n 08034 Barcelona - Spain onate@cimne.upc.es Summary. This paper summarizes the development for a large displacement formulation of a membrance composed of three-node triangular elements. A formulation in terms of the deformation gradient is first constructed in terms of nodal variables. In particular, the use of the right Cauchy-Green deformation tensor is shown to lead to a particulary simple representation in terms of nodal quantities. This may then be used to construct general models for use in static and transient analyses. Key words: Membrane structures, finite element 1 Introduction The behavior of curved, thin bodies can be modeled by a membrane theory of shells. In such a theory only the in-plane stress resultants are included. The deformation state for a membrane may be represented by the position of points on the twodimensional surface. General theories for shells may be specialized to those for a membrane by ignoring the resultant couples and associated changes in curvature deformations as well as any transverse shearing effects. A numerical approximation to the shell may then be constructed using a finite element approach. Examples for general shell theory and finite element solution may be found for small deformations in standard books.[1] Theory for large deformation can proceed following the presentations of Simo et al. [2, 3, 4, 5] or Ramm et al. [6, 7, 8, 9, 10, 11, 12, 13] For the simplest shape finite element composed of a 3-node triangular form with displacement parameters at each vertex (a 9-degree of freedom element) it is far simpler to formulate the membrane behavior directly. This is especially true for large displacement response. Here the initially flat form of a simple triangular response is maintained throughout all deformation states. Consequently, one may proceed Visiting Professor, CIMNE, UPC, Barcelona, Spain. 47 E. Oñate and B. Kröplin (eds.), Textile Composites and Inflatable Structures, 47–68. © 2005 Springer. Printed in the Netherlands
48 Robert L.Taylor,Eugenio Onate and Pere-Andreu Ubach directly with the construction of the kinematic behavior,even in the presence of large strains.This approach is followed in the present work. The loading of a membrane is often by pressures which remain normal to the surface throughout all deformations.Such follower loading generally leads to a form which yields an unsymmetric tangent matrix.Such formulation has been presented in works by Schweizerhof and Ramm4l and by Simo et al The general approach presented in the last work is used for the special case of the flat triangular element used in this work. The formulation included in the present study includes inertial and damping terms based on second and first time derivatives of the motion.These are discretized in time using standard techniques (e.g.,the Newmark method6,171).Both explicit and implicit schemes are presented together with all linearization steps needed to implement a full Newton type solution.The inclusion of the damping term permits solution of the first order form in order to obtain a final static solution.Generally, the first order form is used until the final state is reached at which point the rate terms are deleted and the full static solution achieved using a standard Newton iterative method. The work presented is implemented in the general purpose finite element solution system FEAPlsl and used to solve example problems.The solution to some basic example problems are included to show the behavior of the element and solution strategies developed. 2 Governing Equations Reference configuration coordinates in the global Cartesian frame are indicated in upper case by X and current configuration in lower case by c.The difference between the coordinates defines a displacement u. Using standard interpolation for a linear triangle positions in the element may be specified as X=Eax (1) in the reference configuration and =ata (2) for the current configurations.If necessary,the displacement vector may be deduced as u =faua (3) In the above x,ua denote nodal values of the reference coordinates,current coordinates and displacement vector,respectively.Furthermore,the natural (area) coordinates satisfy the constraint 51+51+53=1 (4) It is convenient to introduce a surface coordinate system denoted by Yi,Y2 with normal direction N in the reference state and yi,y2 with normal direction n in the current state (see Fig.1)
48 Robert L. Taylor, Eugenio O˜nate and Pere-Andreu Ubach ˜ directly with the construction of the kinematic behavior, even in the presence of large strains. This approach is followed in the present work. The loading of a membrane is often by pressures which remain normal to the surface throughout all deformations. Such follower loading generally leads to a form which yields an unsymmetric tangent matrix. Such formulation has been presented in works by Schweizerhof and Ramm[14] and by Simo et al. [15] The general approach presented in the last work is used for the special case of the flat triangular element used in this work. The formulation included in the present study includes inertial and damping terms based on second and first time derivatives of the motion. These are discretized in time using standard techniques (e.g., the Newmark method[16, 17] ). Both explicit and implicit schemes are presented together with all linearization steps needed to implement a full Newton type solution. The inclusion of the damping term permits solution of the first order form in order to obtain a final static solution. Generally, the first order form is used until the final state is reached at which point the rate terms are deleted and the full static solution achieved using a standard Newton iterative method. The work presented is implemented in the general purpose finite element solution system FEAP[18] and used to solve example problems. The solution to some basic example problems are included to show the behavior of the element and solution strategies developed. 2 Governing Equations Reference configuration coordinates in the global Cartesian frame are indicated in upper case by X and current configuration in lower case by x. The difference between the coordinates defines a displacement u. Using standard interpolation for a linear triangle positions in the element may be specified as X = ξα X˜ α (1) in the reference configuration and x = ξα x˜α (2) for the current configurations. If necessary, the displacement vector may be deduced as u = ξα u˜ α (3) In the above X˜ α , x˜α, u˜ α denote nodal values of the reference coordinates, current coordinates and displacement vector, respectively. Furthermore, the natural (area) coordinates satisfy the constraint ξ1 + ξ1 + ξ3 = 1 (4) It is convenient to introduce a surface coordinate system denoted by Y1, Y2 with normal direction N in the reference state and y1, y2 with normal direction n in the current state (see Fig. 1)
Finite Element Analysis of Membrane Structures 49 t y 1 2 Fig.1.Description of coordinates for triangular element Placing the origin of the Yi and yi coordinates at nodal location and respectively,the unit base vectors may be constructed from the linear displacement triangle by aligning the first vector along the 1-2 side.Accordingly,we define the first unit vector as 元2-龙1 4221 v1=2--A2可 (5) where and lall -(a'a)"3. Next a vector normal to the triangle is constructed as v3=△元21×△231 (6) and normalized to a unit vector as U3 n=Tosll () The vector v3 plays a special role in later development of nodal forces for follower pressure loading as it is twice the area of the triangle times the unit normal vector n. Finally,a second orthogonal unit vector in the plane of the triangle is be com- puted as 02=n X U1 (8) The above developments have been performed based on the current configura- tion.However,reference quantities may be deduced by replacing lower case letters by upper case ones (e.g.,vi -Vi,etc.)
Finite Element Analysis of Membrane Structures 49 Fig. 1. Description of coordinates for triangular element Placing the origin of the YI and yi coordinates at nodal location X˜ 1 and x˜1, respectively, the unit base vectors may be constructed from the linear displacement triangle by aligning the first vector along the 1 − 2 side. Accordingly, we define the first unit vector as v1 = x˜2 − x˜1 x˜2 − x˜1 = ∆x˜21 ∆x˜21 (5) where ∆x˜ij = x˜i − x˜j and a = aT a 1/2 . Next a vector normal to the triangle is constructed as v3 = ∆x˜21 × ∆x˜31 (6) and normalized to a unit vector as n = v3 v3 (7) The vector v3 plays a special role in later development of nodal forces for follower pressure loading as it is twice the area of the triangle times the unit normal vector n. Finally, a second orthogonal unit vector in the plane of the triangle is be computed as v2 = n × v1 . (8) The above developments have been performed based on the current configuration. However, reference quantities may be deduced by replacing lower case letters by upper case ones (e.g., v1 → V 1, etc.). x2 x3 x1 1 2 3 y1 y2 n
50 Robert L.Taylor,Eugenio Ofate and Pere-Andreu Ubach With the above base vectors defined,positions in the plane of the triangle may be given directly as 班=(x-)·v (9) In general,an interpolation may be given as y=￡a9 (10) We note from Eq.(9)that is identically zero hence Eq.(10)reduces to y=522+33. (11) However,the constraint (4)still restricts the admissible values for 2 and 3. 2.1 Deformation Gradient From the above description of the motion of the triangle it is now possible to deduce the deformation gradient in the plane of the triangle as F= dy Ψ =1+ (12) Using the parametric representations(11)we can compute the deformation gradient from ay ay aY ay Oy OE -FOEE (13) If we define the arrays J and j as J= ay and y OE j=0 (14) then the deformation gradient is given by F=jJ-1 (15) In the above J is the Jacobian transformation for the reference frame and j that for the current frame.Expanding the relations for each Jacobian we obtain (16) and j= (△221)Tu1(421)T1 (Ai2)Tv2 ()Tv2 (17) By noting that is orthogonal to V2 and similarly for the current configu- ration that 2-is orthogonal to v2 and in addition using the definition for Vi
50 Robert L. Taylor, Eugenio O˜nate and Pere-Andreu Ubach ˜ With the above base vectors defined, positions in the plane of the triangle may be given directly as yi = x − x˜1 · vi (9) In general, an interpolation may be given as y = ξαy˜α (10) We note from Eq. (9) that y˜1 is identically zero hence Eq. (10) reduces to y = ξ2 y˜2 + ξ3 y˜3 . (11) However, the constraint (4) still restricts the admissible values for ξ2 and ξ3. 2.1 Deformation Gradient From the above description of the motion of the triangle it is now possible to deduce the deformation gradient in the plane of the triangle as F = ∂y ∂Y = 1 + ∂u ∂Y (12) Using the parametric representations (11) we can compute the deformation gradient from ∂y ∂Y ∂Y ∂ξ = F ∂Y ∂ξ = ∂y ∂ξ (13) If we define the arrays J and j as J = ∂Y ∂ξ and j = ∂y ∂ξ (14) then the deformation gradient is given by F = j J−1 (15) In the above J is the Jacobian transformation for the reference frame and j that for the current frame. Expanding the relations for each Jacobian we obtain J = ⎡ ⎣ ∆X˜ 21T V 1 ∆X˜ 31T V 1 ∆X˜ 21T V 2 ∆X˜ 31T V 2 ⎤ ⎦ (16) and j = ∆x˜21T v1 ∆x˜31T v1 ∆x˜21T v2 ∆x˜31T v2 (17) By noting that X˜ 2 − X˜ 1 is orthogonal to V 2 and similarly for the current configuration that x˜2 − x˜1 is orthogonal to v2 and in addition using the definition for V i
Finite Element Analysis of Membrane Structures 51 and v;the above simplify to - (18) and -[.a2a (19) lv3/川△金川 Using these definitions,the right Cauchy-Green deformation tensor may be ex- panded as C=FTF=J-TjTjJ-1=GTgG (20) where G is used to denote the inverse of J.In component form we have 1 2201911912]「J2-☑2 c=-加hi]9e9e0hi (21) in which the terms in the kernel array involving j may be expressed in the particu- larly simple form 911= =(4221)T4221 912-j12i1=(4221)T△231 (22) 922=+品2=(4元31)T4立31 2.2 Material Constitution-Elastic Behavior In the present work we assume that a simple St.Venant-Kirchhoff material model may be used to express the stresses from the deformations.Stresses are thus given by S=DE (23) where D are constant elastic moduli.and the Green-Lagrange strains E are given in terms of the deformation tensor as E=(C-1) (24) In each triangular element the deformation may be computed from (19)to (22), thus giving directly the stress. 3 Weak Form for Equations of Motion A weak form for the membrane may be written using a virtual work expression given by 6= δri po h:hd2+ 6xi co iid+ 6EISIs hds 2 (25) xibi dw- 6xi tidy=0
Finite Element Analysis of Membrane Structures 51 and vi the above simplify to J = ⎡ ⎣ ∆X˜ 21 , ∆X˜ 21T ∆X˜ 31 /∆X˜ 21 0 , V 3/∆X˜ 21 ⎤ ⎦ (18) and j = ∆x˜21 , ∆x˜21T ∆x˜31 /∆x˜21 0 , v3/∆x˜21 (19) Using these definitions, the right Cauchy-Green deformation tensor may be expanded as C = F T F = J −T jT jJ−1 = GT gG (20) where G is used to denote the inverse of J. In component form we have C = 1 J 2 11J 2 22 J22 0 −J12 J11 g11 g12 g12 g22 J22 −J12 0 J11 (21) in which the terms in the kernel array involving j may be expressed in the particularly simple form g11 = j2 11 = ∆x˜21T ∆x˜21 g12 = j12j11 = ∆x˜21T ∆x˜31 g22 = j2 12 + j2 22 = ∆x˜31T ∆x˜31 (22) 2.2 Material Constitution - Elastic Behavior In the present work we assume that a simple St.Venant-Kirchhoff material model may be used to express the stresses from the deformations. Stresses are thus given by S = D E (23) where D are constant elastic moduli. and the Green-Lagrange strains E are given in terms of the deformation tensor as E = 1 2 (C − I) . (24) In each triangular element the deformation may be computed from (19) to (22), thus giving directly the stress. 3 Weak Form for Equations of Motion A weak form for the membrane may be written using a virtual work expression given by δΠ = Ω δxi ρ0 h x¨i h dΩ + Ω δxi c0 x˙ i dΩ + Ω δEIJ SIJ h dΩ − Ω δxibi dω − γt δxi t ¯i dγ = 0 (25)
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)
Finite Element Analysis of Membrane Structures 55 3 Fig.2.Cable reinforced membrane element at(i-j) 4 Reinforcement Cables It is common to place reinforcing cables in membranes to provide added strength or shape control.The cables are generally very strong in axial load capacity (generally tension)and weak in bending.Accordingly,they may be modeled by a truss type member.In the form admitted here it is not necessary for the reinforcement to be placed at the edges of membrane elements-they may pass through an element as shown in Fig 2. The ends of a typical reinforcement are denoted as i and j in the figure and have reference coordinates Xi and X,respectively.These points may be referred to the nodal values of the membrane by computing the values of the natural coordinates so that Xko for k=i,j (49) The solution for the two points may be trivially constructed from linear interpolation on the edges.The results are (for the points intersecting the edges shown in the figure) 货=IK-x X2-X ；i=1-；结=0 X3-X川 X3-X i=1-结；经=0 (50) Using these values the deformed position of the reinforcement cable may be written as {}-{升” (51)
Finite Element Analysis of Membrane Structures 55 Fig. 2. Cable reinforced membrane element at(i-j) 4 Reinforcement Cables It is common to place reinforcing cables in membranes to provide added strength or shape control. The cables are generally very strong in axial load capacity (generally tension) and weak in bending. Accordingly, they may be modeled by a truss type member. In the form admitted here it is not necessary for the reinforcement to be placed at the edges of membrane elements - they may pass through an element as shown in Fig 2. The ends of a typical reinforcement are denoted as i and j in the figure and have reference coordinates Xi and Xj ,respectively. These points may be referred to the nodal values of the membrane by computing the values of the natural coordinates so that Xk = ξk α Xα for k = i, j (49) The solution for the two points may be trivially constructed from linear interpolation on the edges. The results are (for the points intersecting the edges shown in the figure) ξi 2 = Xi − X1 X2 − X1 ; ξi 1 = 1 − ξi 2 ; ξi 3 = 0 ξj 3 = Xj − X1 X3 − X1 ; ξi 1 = 1 − ξi 3 ; ξi 2 = 0 (50) Using these values the deformed position of the reinforcement cable may be written as % xi xj & = % ξi α I ξj α I & x˜α (51) x2 x3 x1 1 2 3 i j y1 y2 n
56 Robert L.Taylor,Eugenio Ofate and Pere-Andreu Ubach 4.1 Deformation of Cable The deformation of the reinforcement cable may be expressed in terms of the Green- Lagrange strain given by =”=乐-可 (52) where Xi=X-X'.The variation of the strain is then expressed as 6E,=(5x3-P(4ry (53) △X2 4.2 Material Constitution For simplicity we again assume that the material is elastic and may be represented by a one-dimensional form of a St.Venant-Kirchhoff model expressed as Sij =EEij (54) where Sij is the constant second Piola-Kirchhoff stress in the cable and E is an elastic modulus. 4.3 Weak Form for Reinforcement A weak form for an individual reinforcement cable in an element may be written as 6Π，=ix(Mk'+Ck-6ESAL;k,l=i,j(55) where Ai;is the cross sectional area of the reinforcement;Li;the length of the cable (i.e.,X);Mk is the mass matrix;and C is the damping matrix. The variation of the strain is rewritten from Eq.(53)as 6E=[6zr (56) Equation(55)is appended to the other terms from the membrane by replacing variations of end displacements and the rate terms by their representation in terms of the membrane nodal parameters as given by Eq.(51).The result is: .={M+c-P》 (57) where P=Ac Sy Au (58)
56 Robert L. Taylor, Eugenio O˜nate and Pere-Andreu Ubach ˜ 4.1 Deformation of Cable The deformation of the reinforcement cable may be expressed in terms of the GreenLagrange strain given by Eij = 1 2 (xj − xi ) T (xj − xi ) (Xj − Xi )T (Xj − Xi ) − 1 = 1 2 ∆xji2 ∆Xji2 − 1 (52) where ∆Xji = Xj − Xi . The variation of the strain is then expressed as δEij = (δxj − δxi ) T (∆xji) ∆Xji 2 (53) 4.2 Material Constitution For simplicity we again assume that the material is elastic and may be represented by a one-dimensional form of a St.Venant-Kirchhoff model expressed as Sij = E Eij (54) where Sij is the constant second Piola-Kirchhoff stress in the cable and E is an elastic modulus. 4.3 Weak Form for Reinforcement A weak form for an individual reinforcement cable in an element may be written as δΠr = δxk Mklx¨l + Cklx˙ l − δEij Sij Aij Lij ; k, l = i, j (55) where Aij is the cross sectional area of the reinforcement; Lij the length of the cable (i.e., ∆Xji ); Mkl is the mass matrix; and Ckl is the damping matrix. The variation of the strain is rewritten from Eq. (53) as δEij = δxi,T δxj,T ⎧ ⎪⎪⎨ ⎪⎪⎩ −∆xji L2 ij ∆xji j L2 ij ⎫ ⎪⎪⎬ ⎪⎪⎭ (56) Equation (55) is appended to the other terms from the membrane by replacing variations of end displacements and the rate terms by their representation in terms of the membrane nodal parameters as given by Eq. (51). The result is: δΠr = δx˜α,T ' Mαβ r x¨˜ β + Cαβ r x˜˙ β − P α r ( (57) where P α r = ∆ξji α ∆ξji β x˜β Sij Aij Lij ; (58)