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1 ª Œª Œ 7ƒ 3A Á 7 5œ pp. 77 ~ 89 ª w w w On the Finite Element Analysis of Shell Structures v Á x Lee, Phill-SeungÁNoh, Hyuk-Chun Abstract Based on recent research works, important concepts on the finite element analysis of shell structures and the relations among them are presented in this paper. We review the basic shell mathematical model, which is the underlying mathematical model of the continuum mechanics based shell finite elements. The asymptotic theory of shell structures then is reviewed and we present how to evaluate the asymptotic behavior in finite element solutions. S-norm is introduced as an error measure of finite element solutions and we show locking in the convergence curves of shell finite element solutions. We discuss the concept of uniform optimal convergence in finite element analysis of shells. We finally summarize requirements on ideal shell finite elements and propose how to perform benchmark tests of shell finite elements. Keywords : shell structures, finite elements, asymptotic behavior, uniform optimal convergence, benchmark tests m w w w w š w. w w» w r. ̃ ù ƒ w ({,, yw w wš w w mw ü. w w s-norm sƒw wš w w ½x w w š ùkù r. w w ³ w. w š w sƒ w w. w :, w w,, ³, x 1. vƒ ù j ƒ w. ƒ tvù Ë (shell structures w. ƒ š z w xk wš. ü wš š š w. w (finite element method x x w ƒ ù w w w ƒ y w š (Bathe, 1996;, ; Chapelle and Bathe, 3; Noh, 6. ù x,, w w p ̃ p w» w w w wƒ v (Chapelle and Bathe, 1998; Lee and Bathe, ; Chapelle and Bathe, 3. w w w w ̃ ùkù (asymptotic behavior w (Lee and Bathe, ; Bathe, Chapelle and Lee, 3. { (bending dominated, (membrane dominated, yw(mixed ù. œw(engineering t» w x(experiment (investigation m w r z w p / üš ƒ ƒ (assumptions w y g w (mathematical model. w w œw. w w z œ ( ( phillseung.lee@samsung.com z w ( cpebach@sejong.ac.kr 7ƒ 3A 7 5œ 77
2 w ù tš w ƒ w w w»ƒ ƒ w. ew (numerical analysis w w w (approximation w š x/ w w ù ew sƒw.,,, w, ew w., w w yw ww» w w w, w (mathematical shell model w w w w wƒ š w. ƒ w mw wƒ w w w w. ƒƒ ƒ w w» š (Lee and Bathe, ; Chapelle and Bathe, 3; Bathe, Chapelle and Lee, 3; Hiller and Bathe, 3; Lee and Bathe, 5 w š w š w w sƒ w. p w w w ww» (engineers ù w œ / w w / w w w w š w. w w z w w» w w w w w w w. w y(displacement based formulation w w (interpolation function (order { yw w w w ù kü (Bathe, 1996; Chapelle and Bathe, 3. ½x (locking phenomenon w z w w w wù. w ƒ w x ³ (uniform optimal convergence w w w. t w wš w» r z ý. w w w (error sƒw ³ r š k ½x w ³ w. w w sƒ w w. w ü (isotropic material w xk (linear elastic w.. w (mathematical shell model» w (basic shell mathematical model {(bending, ü(membrane, (transverse shearing y (coupling txw ƒ w 3 w l w (Ahmad, Irons and Zienkiewicz, 197; Bathe, 1996 w x w ƒ š (Chapelle and Bathe, 1998; Chapelle and Bathe, 3; Lee and Bathe, 5.,» w w w.»ww(differential geometry w x (geometry x (kinematics r š» w..1 x (shell geometry Ì(thicknessƒ 3. ̃ p x Ì w.»» w x w» ww mw. 1. x 78 ª Œª Œ
3 »ww w» w k wt ³ (Einstein summation convention w. α, β, λ, µ 1 ¾ w i, j, k 1 3¾ w (index. (midsurface 1 ω S (mapping ùkü w φ w. œ» l(covariant base vector ùkü. φ ( ξ 1,ξ aα = ξ α œ (covariant» l w» l (contravariant base vector w. α a aβ = δ α β α» δ β α βƒ 1 š Kronecker symbol. l œ (covariant» l l (vector product. a 1 a a 3 = a 1 a 3»wx tx. Φξ ( 1,ξ, ξ 3 = φ( ξ 1,ξ + ξ 3 a 3 ( ξ 1,ξ» ξ 1, ξ, ξ 3. Ω = ( ξ 1,ξ, ξ 3 R 3 ( ξ 1,ξ ω, ξ 3 t (,ξ ξ1 t( ξ 1,ξ» t Ì. w surface l (tensors w.» l D metric l œ x(covariant type. a αβ = aα a β x(contravariant type ùkù. a αβ = a α a β» l š (curvaturel š w š. œ x(covariant type. b αβ = a 3 aα,β w w x(mixedl. b β α = a αλ b λβ» l. c αβ = b α λ bλβ , (1 ( (3 (4 (5 (6 (7 (8 (9 (1 l w š w l œ (covariant derivative ùkù. w α β = w α,β Γ λ αβ wλ» Christoffel symbol. (4 l 3 œ (covariant» l. š l. (11 (1 (13 (14 (15 3 (contravariant» l. (16. x (shell kinematics x» ƒ x x w ù ù š, tx. (17» u( ξ 1,ξ (infinitesimal translation ùküš θ λ ( ξ 1,ξ z (infinitesimal rotation ùkü. θ λ λ a z l θ š ξ z l w 3 θ λ λ a ùkü. u (contravariant» l a 1, a, (= w a 3 a 3. xw w 3 Green-Lagrange œ x (covariant strain l x.» (18 (. (19 (14, (15 (17 (18 j x l œ (covariantw.» λ Γ αβ λ Γ αβ = aα,β aλ Φ( ξ 1,ξ, ξ 3 g i = ξ i gα = a α ξ 3 b α λ aλ g 3 = a 3 g i = δ j i g j U( ξ 1,ξ, ξ 3 = u( ξ 1,ξ + ξ 3 θ λ ( ξ 1,ξ λ a ( ξ 1,ξ 1 e = -- ij g i U,j + g j U,i ( U,i = U ξ1,ξ, ξ ξ i e αβ = γ αβ ( u + ξ 3 χ αβ ( u,θ ξ ( 3 κ αβ ( θ e = α3 ζ α ( u,θ e 33 = γ αβ ( u = χ αβ ( u,θ = κ αβ ( θ = ζ α ( u,θ = 1 -- ( u α β + u β α b αβ u θ θ λ ( α β + β α b βuλ α b α + c αβ u 3 λ ( θλ β 1 λ -- b βθλ α b α 1 -- ( θ α u 3,α b λ + + α uλ λ uλ β (a (b (c (1a (1b (1c (1d (isotropic material w s (plane 7ƒ 3A 7 5œ 79
4 stress condition w (σ 33 = x. σ αβ = C αβλµ e λµ σ α3 = 1 --D αλ e λ3 ( C αβλµ E = g αλ g βµ + g αµ g βλ + v ( 1 v v g αβ g λµ D αλ = E g αλ 1 + v (a (b (3a (3b» E k (elastic modulus š v s (Poisson's ratio g αβ (14 3 (contravariant» l metricl ( g αβ α = g β g. (rigid body motion ù w ( (3¾ w» w (basic shell mathematical model (variational equation. w w xw (test function V w (4 j ùkü. (4» F w (external loading xw g w. U C αβλµ e Ω αβ ( Ue λµ ( VdV + D αλ e Ω α3 ( Ue λ3 ( VdV = F VdV Ω V( ξ 1,ξ, ξ 3 = v( ξ 1,ξ + ξ 3 η λ ( ξ 1,ξ λ a ( ξ 1,ξ 3. (5 { ƒ w w yw(mixed w. x (geometry, (boundary condition, w (loading (Lee and Bathe, ; Bathe, Chapelle and Lee, 3; Chapelle and Bathe, (4 w x (variational form Ì(t w w t š w w yw ùký. Find U Ψ such that ε 3A b ( U, V + εa m ( U, V = F V (, V Ψ (6» ε Ì j» (t/l, ε Ab(, { 3, ε Ab(, w x (bilinear forms, U w, V xw, Ψ Sobolev œ (space 1, F ( w x (linear form ùkü. ̃ w ε Am w w š. ε r» w ε ρ (ρ = load scaling factorƒ w (scaled loading w. F( V = ε ρ G( V (7» G Ψ ( Ψ dual space w ρ. (6 ƒ w ε ε w ρ 3 1 j ù š 3 ù. {(bending, (membrane, (transverse shearing w w» (mechanism. w w {,, ü w. ̃ w j { w š w. ̃ p w w - { (bending dominated, (membrane dominated, yw(mixed - (asymptotic behavior w. ̃ { w w kw { (bending dominated shell w w kw (membrane dominated shell w. w ̃ 1 1k j Sobolev œ H 1 V i H ( Ω = V: V L ( Ω; L ( Ω; V x i = at prescribed displacement boundary j» L 3 ( Ω = V: (Bathe, ( Ω V dω<+ i i = 1 1 ρ 3 (8 œ (space w w w. Ψ = V Ψ A m ( V,V= (9 Ψ œ {(pure bending ùkü œ xk (patterns sww. œ (space w xk ƒ š { š w w { (inhibited shell w. ƒ xk(patternƒ { (mode ƒ š { (non-inhibited shell w. t». Ψ œ x F œ Ψ dual space( œ š Ψ t»w. 8 ª Œª Œ
5 표 쉘의 점근거동의 분류 하중 순수 휨을 유발하는 외력 1. 경우 순수 휨이 구속되지 않은 쉘 구조물, Ψ { } V Ψ such that G(V 순수 휨을 유발하지 않는 외력 (ii 불안정한 막지배 또는 혼합지배거동 G ( V, V Ψ Admissible membrane loading = 순수 휨이 구속된 쉘 구조물, Ψ { } = Ψ (3 순수 휨이 구속되지 않은 경우의 휨지배 상태는 하중이 휨 변위를 유발 시켜야만 일어날 수 있다. 만일 하중이 휨을 유발할 수 없다면 이론적인 점근거동은 휨이 구속된 경우와 같게 되나 이 경우 거동은 매우 불안정(unstable하다. 즉, 이런 경우 작은 하중의 변화로 쉘 구조물의 점근거동을 막 지배거동에서 휨지배거동으로 바꿀 수 있다. 순수 휨이 구속된 경우(즉, Ψ ={}, 적정한 ρ 값은 1이 며 막과 전단 에너지만을 유발시킬 수 있는 변위공간(space Ψm 에 의해 구조물의 거동이 표현될 수 있다. 그러므로 이 공간(space의 크기는 Ψ 보다 크다. 막지배거동의 한계문제 는 다음과 같이 표현된다. m such that Ψm 그림 쉘의 변형거동. 第7卷 第3A號 m 7年 5月 A m ( U, V = G ( V, V Ψ m G Ψm dual space (31 G ( V c A m ( V, V, V Ψ (3 여기서 외력 가 의 에 속해야만 막지배거 동의 한계문제가 잘 정의 되며 이 조건( G Ψm 은 다음 식과 동일하다. such that A b ( U, V = G ( V, V Ψ Find U (iv 혼합지배거동 G Ψm (iii 막지배거동 G Ψm Non-admissible membrane loading 동은 순수 휨이 구속되어 있는가/아닌가에 따라 변한다. 순수 휨이 구속되지 않은 경우(즉, Ψ {} 는 주로 쉘 구조의 휨지배 상태를 이끌어낸다. 이때 적당한 ρ 의 값은 3 이며 식 (6의 막 에너지 항이 사라지면서 이 경우 식 (6의 쉘 문제는 다음과 같이 표현될 수 있다. Find U 분류 (i 휨지배거동 여기서 c 는 상수이다. 위 식은 재하된 외력이 막응력만에 의하여 지지될 수 있다는 것을 뜻하며 이런 조건을 만족시 키는 외력을 admissible membrane loading 이라고 부른다. 만일 외력이 non-admissible membrane loading ( G Ψm 이라면 이 경우 막지배 문제는 정의 될 수 없으며 쉘의 점 근거동은 막과 휨의 혼합된 형태(mixed type를 띠게 된다. 표 1은 위에서 언급된 쉘 구조물의 점근거동을 정리/요약 하여 보여준다. 쉘 구조물의 설계 시 이러한 점근거동에 대 한 지식은 매우 유용하며 필수적이다. 주어진 하중에 대하여 쉘 구조물의 강성은 ε 에 비례하여 변한다. 즉, 쉘 구조물 의 거동이 휨에 의해 지배 받게 될 경우 구조물의 강성은 (t /L 에 비례 하게 되며 막거동에 의해 지배될 경우 강성은 t /L 에 비례하게 된다. 그러므로 효과적인 쉘 구조물은 휨거 동이 구속되어 막거동에 의해 지배되는 구조물이며 쉘 구조 물은 막지배거동을 하도록 설계하는 것이 바람직하다. 주어 진 외력에 대하여 쉘의 형상과 경계조건을 적절히 사용하여 최대의 기하학적 강성(geometrical rigidity을 갖을 수 있도 록 하여야 한다. ρ 3 그림 쉘의 변형형상과 유효응력분포에 나타난 경계층 3. (boundary layer 81
6 3. d p ¼ / x / yƒ ñ (smooth areas d (layers ù. d(layer š (curvature ù Ì y x y, ww (incompatible boundary condition, ³ew w (irregular loading w (Lee and Bathe,. d (layer / x / w w x ù. d p ¼ (L c, characteristic length Ì w Ì(t ¼ (L w ùkù. L c = ct 1 l L l (33» c l. š x(lee and Bathe, ̃ ùkù Scodelis-Lo roof shell problem d(boundary layer šs (hyperbolic paraboloid ü d(inner layer. š x(bathe, Chapelle and Lee, 3 xk d ̃ yw. 3 xx (deformed shape z (effective stress s ùkù d(boundary layer š. 3.3 w ü Lovadina w w w ù (Lovadina, 1. e w ü ƒ w w w mw ƒ Lee Bathe w (Lee and Bathe, ; Bathe, Chapelle and Lee, 3. ƒ wƒ w. ρ(load scaling factor w ý. Lee Bathe w ρ w. loge ε ε ρ ( + loge( ε log( ε + ε logε (34» E ε(=t/l w w w l w x (strain energy. ε w w w y ƒ yw ρ. ρ 1, 3 {, 1 3 { yw w. Lovadina (interpolation theory w w ρ ùkü w (Lovadina, 1. lim R ε ε ( ρ = (35» R(ε x w { x. R( ε = ε 3 A b ( U, U ε 3 A b ( U, U + ε A m ( U, U (36 R(ε w w ρ. Lee Bathe w sƒ(benchmarkw» w Scodelis-Lo roof shell problem w ρ w (Lee and Bathe,. z Lovadina w ρ w š ew. { w w š x(lee and Bathe,, Ì y ρ (fluctuationw w w w š x(bathe, Chapelle and Lee, w ½x ³ w w w ƒ j ½x (locking phenomenon. ½x ̃ w w (error ƒ k. w w k w ½x ³ w r. 4.1 w ½x w j s (flat shell w (degenerated shell or continuum mechanics based shell w ù (Ahmad, Iron and Zienkiewicz, 197; Bathe, 1996; Choi, Lee and Park, 1999;, ; Chapelle and Bathe, 3. s w sq w s w w w. š s ù txw w. w yƒ š üz (drilling degrees of freedom w 6 ƒ w 6 (beam w w r w. ù,» x s» w š xk x w txw» š w yw qw ƒ w ƒ š. 3 w w 5 ƒ š w w p w v w š w š x txw ý š yw ƒ. w, ƒ w (mathematical shell model» w w. w w y(formulation 8 ª Œª Œ
7 w j (displacement based method w w yw (mixed method w w ù. w w (displacement based shell finite element w. ù (interpolation order ̃ { (bending dominated yw (mixed w w w wƒ w e ƒ š ½(lockingx š w. (mesh w w w y ̃ ù x w ̃ w. ½x ½(membrane locking ½(shear lockingx ù ½x ù w w ƒ (9 { œ (space Ψ w»., ½x ƒ š. x,, w w» ½x w x,, w. ½x š ƒ wš s x w ù ½x š w. ½x { yw (1a (1d x w l w. š x(bathe, Lee and Hiller, 3 (Lee and Bathe, 5 ½x w w ùkù š ½x» wš. w w ½x w w w,, x x ùkù. ̃ w w w w w w w w ( sƒ w. w w y ƒ Ì w x ½x w w w ù» w ½x w w v. 4. w w d w w w w ƒ ƒw w w (mathematical shell model yw(exact solution w w ½x» w w w š (convergence curve w w.» w d w» w w ³ (norm w w. ³ (norm w p (point-wise value w š w w. w / / x ƒ š d w tw w w. x w ³ w y w w ƒ» y ³ v w. Hiller Bathe w s-norm( s w l» y (Hiller and Bathe, 3. U Uh s = Ω (37» U yw, Uh w w. ε σ ƒ t (global Cartesian coordinate system x l l. (38a (38b (37 x w yw w w (difference w. (39a (39b» C - x w (matrix š B x - (strain-displacement (operator. e l x x h ƒƒ (domain y w. l (injective mapping, Π w w. (4 w yw ü ƒ w w w w w yw š w s- norm w {. yw š w w Uref š w s-norm ùkü.» ε T σdω ε = [ ε xx, ε yy, ε zz, ε xy, ε yz, ε zx ] T σ = [ σ xx, σ yy, σ zz, σ xy, σ yz, σ zx ] T ε = ε σ = σ x = Π( x h Uref Uh s ε h = ε x σ h = σ x ( B h x h = Ωref ( Uh ( C h ( x h B h x h ( Uh ε T σdω ref, (41 ε = ε ref ε h = B ref ( x ref Uref B h ( x h Uh, (4a σ = σ ref σh = C ref ( x ref B ref ( x ref Uref C h ( x h B h x h x ref = Π( x h ( Uh, (4b (4c w Ì w w w y ƒ w w» w (relative error w w. (E h. Uref Uh s E h = Uref s (43 7ƒ 3A 7 5œ 83
8 4. : (a sq { (L=1., (b w šs (hyperbolic paraboloid (L=1. 5. Ì y 4 w š : (a ½x ú -QUAD4 (b ½x ù -MITC4 w w. kù š ³ (uniform optimal convergence w. E h ch k (44 q(plate š» c, h w j», k w ƒ w xk. 4(a w (displacement interpolation function sq {(plate bending. x(linear w w 3 w w w š r. k 4 w w k =1, , s.3, ¼ L=1., w (quadratic w w 6 9 w 9 Z w w. x, w k =. S-norm e w w, e w 4(a e (Lee, Noh and Bathe, 7. w. sq { t» w (displacement 4.3 š ³ based formulation w 4 w { mw (QUAD4 yw (mixed formulation w 4 ½x s-norm w w š ù w (MITC4 3 w (Ahmad, Irons and 3 MITC(Mixed Interpolation of Tensorial Components w n w MITCn w š., MITC4 4, MITC6 6 w. 84 ª Œª Œ
9 6. Ì y 6 w š : (a ½x ú -QUAD6 (b ½x y g - MITC6 Zienkiewicz, 197; Bathe and Dvorkin, 1989; Bathe, ƒƒ Ì y š ½x ùkù r. 5 ƒ 4 w w Ì y(t/l=1/1, 1/1, 1/1, 1/1 š (convergence curves.» h w j» ùkü w» w s- norm w. h log w (44 w š w. 5 ƒƒ v š»»(kƒ ̃. 5(a ½x ú x š. ̃ (relative errorƒ w. 5(b š ƒ Ì(t w š j» h w. w 5(b š»». 5(b xk w w xk ³ (uniform optimal convergence š w. 4(b w šs (hyperbolic paraboloid. (midsurface. X Y Z = L ξ 1 ; ( ξ 1, ξ ( ξ ( ξ 1 ξ , -- (45 X=.5 (selfweight Z w w. x Y= e 4(b e w. k. 1 11, s (Poisson's ratio.3, ¼ L=1. 8. { t» w (displacement based formulation w 6 ƒx w (QUAD6 yw (mixed formulation w 6 ƒ x w (MITC6 w (Ahmad, Irons and Zienkiewicz, 197; Bathe, 1996; Lee and Bathe, 4. 6 t/l 1/1, 1/1, 1/1 ƒ w 6 ƒx w š (convergence curves. ƒƒ v š»»(kƒ ̃. 6(a QUAD6 w ƒ ̃ w ƒ ù w,, ½x w. 6(b MITC6 w w ½x ù QUAD6 w w y ùkü.» ƒ w w š ½x w. w w ƒ ƒ w ½x j š w w ½x j š w., ƒ w ³ ½x j w w ½x w ½x j w w. w w w w š w š x ù Lee Bathe. w ½x» w w x ƒ { yw w 5 6 Ì y j x w w. w p ƒ { š w w. w x š w { ƒ w ƒ ƒ wš w yw w 7ƒ 3A 7 5œ 85
10 ., w ƒ w x w ³ w. 5. w sƒ w ùú j w š w sƒ š mw š. w ½x y j wš w sƒ w w. 5.1 w w w (variational form ùkü. Find U h Ψh such that A h ( Uh, Vh = F( Vh, Vh Ψh, (46» A h (, w y(discretization x (bilinear form š Ψh w Sobolev œ (space. Ψh Ψ w. T ( = Vh A h Uh, Vh B h TC h B h dω h Ω h Uh (47» U h w w(finite element solution, V h w xw, Ψh w œ, F ( ùkü x (linear form. w w (46 (6 xk tx. z w w w. w w. ù, w (spurious zero energy mode w. x w w 6 (zero energy mode ƒ w. ellipticity(k w. α> such that U h Ψh, A h ( U h, U h α U h 1 (48» α 1 Sobolev norm 4 1. w w (stiffness matrix š e (eigenvalues w š l (eigenvectors r x. k w x w. (48 j w w w x w ww w. ù, w w w» w w w w j»(hƒ ƒ ƒw w yw w w. consistency( w. lim U h = U h or lim A h ( U h, U h = A( U, U (49 h» A h (, w yw x (exact bilinear form U yw(exact solution. w w w w w w w. ù, w { yw w ³ (uniform optimal convergence w. j w Ì ½ ½ l w ƒ. 4 w x. yw w y (mixed formulation w w inf-sup condition (Bathe, 1996; Bathe, Iosilevich and Chapelle, b. ƒ w w w. w w y j «(Lee and Bathe, 4. (spurious zero energy mode (ellipticity Consistency q w ½ w ³ { yw w t/l (1/1~1/1 w x w z y 5. ½x ù w ½x w» w š. j ƒ ù. (47 x - B h xw x, ( reduced integration, ANS method, MITC method (47 B h w ƒw x œ (space, ( EAS method (46 w Uh œ ( Ψh, ( non-conforming method w ½x y j» w (categories w w w. ƒ (reduced integration w 4 1 Sobolev norm (Bathe, V 1 = Ω 3 V i i = 1 ( dω + 3 Ω V i dω x ij=1, j 86 ª Œª Œ
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13 š m w š w š w w sƒ w. t w {,, yw ù w» e ü. w { yw ùkù w ½x Ì y š mw š w. w ½x w š w sƒ w. w ww w w w» ù w w w yw w w. w w w w» w ½x w w v. mw w k w w w w w mw yw ww.» wš w ƒ e MIT(Massachusetts Institute of Technology Klaus-Jürgen Bathe KAIST(w w» Ì ¾. š x ( w. lj v. Ahmad, S., Irons, B.M., and Zienkiewicz, O.C. (197 Analysis of thick and thin shell structures by curved finite elements. International Journal for Numerical Methods and Engineering, Vol., pp Bathe, KJ. (1996 Finite Element Procedures. Prentice Hall: New Jersey. Bathe, K.J., Chapelle, D., and Lee, P.S. (3 A shell problem highly sensitive to thickness changes. International Journal for Numerical Methods and Engineering, Vol. 57, pp Bathe, K.J. and Dvorkin, E.N. (1989 A formulation of general shell elements - the use of mixed interpolation of tensorial components. International Journal for Numerical Methods and Engineering, Vol., pp Bathe, K.J., Iosilevich, A., and Chapelle, D. (a An evaluation of the MITC shell elements. Computers & Structures, Vol. 75, pp Bathe, K.J., Iosilevich, A., and Chapelle, D. (b An inf-sup test for shell finite elements. Computers & Structures, Vol. 75, pp Bathe, K.J., Lee, P.S., and Hiller, J.F. (3 Towards improving the MITC9 shell element. Computers & Structures, Vol. 81, pp Chapelle, D. and Bathe, K.J. (1998 Fundamental considerations for the finite element analysis of shell structures. Computers & Structures, Vol. 66, pp , pp.g Chapelle, D. and Bathe, K.J. (3 The finite element analysis of shells? fundamentals. Berlin:Springer-Verlag. Choi, C.K., Lee, P.S., and Park, Y.M. (1999 Defect-free 4-node flat shell element: NMS-4F element. Structural Engineering and Mechanics, Vol. 8, pp Hiller, J.F. and Bathe, K.J. (3 Measuring convergence of mixed finite element discretizations: an application to shell structures. Computers & Structures, Vol. 81, pp Lee, P.S. and Bathe, K.J. ( On the asymptotic behavior of shell structures and the evaluation in finite element solutions. Computers & Structures, Vol. 8, pp Lee, P.S. and Bathe, K.J. (4 Development of MITC isotropic triangular shell finite elements. Computers & Structures, Vol. 8, pp Lee, P.S. and Bathe, K.J. (5 Insight into finite element shell discretizations by use of the basic shell mathematical model. Computers & Structures, Vol. 83, pp Lee, P.S. Noh, H.C., and Bathe, K.J. (7 Insight into 3-node triangular shell finite elements: the effects of element isotropy and mesh patterns. Computers & Structures, Vol. 85, pp Lovadina, C. (1 Energy estimates for linear elastic shells. Computational Fluid and Solid Mechanics (Bathe KJ ed., pp , Elsevier Science. MacNeal, R.H. and Harder, R.L. (1985 A proposed standard set of problems to test finite element accuracy. Finite Element in Analysis and Design, Vol. 1, pp. 3-. Noh, H.C. (6 Nonlinear behavior and ultimate load bearing capacity of reinforced concrete natural draught cooling tower shell. Engineering Structures, Vol. 8, pp ( : / : / : ƒ 3A 7 5œ 89
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