Development DKMQ Shell Element with Five Degrees of Freedom per Nodal
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1 International Journal of Mechanical Engineering and Robotics Research Vol. 6, No., May 7 Development DKMQ Shell Element with Five Degrees of Freedom per Nodal Herry Irpanni, Irwan Katili, and Imam J. Maknun Civil Engineering Department Universitas Indonesia Depok, Indonesia herry.irpanni@gmail.com U i u p, Ni, Vi i W i bstract In this paper we develop element DKMQ five Degrees of Freedom (DOF) per nodal with curvilinear approach. The advantage of this element that it is not required fictitious rigidity and can be used to analyze thick and thin structural problem without shear locking. s numerical result, DKMQ five DOF are able to give good performance. β, Ni, βi Pk, β sk t sk Index Terms DKMQ, five DOF, convergence test βi θi ni i k 5 β Ni ξ,η RN i I. INTRODUCTION i Finite Element Method has grown rapidly because it is powerful tool to solve the various problems in engineering. The formulation of finite element which is formulated with using Reissner-Mindlin hypothesis has problem in alleviate shear locking. Many methods have been introduced, such as reduced integration [], selective integration [], integration technique [] and ssumed Natural Strain (NS) which is applied in MITC []. With NS method, Katili has proposed Discrete Kirchhoff Mindlin technique and use shear coefficient factor in DKMT [5] and DKMQ [6] elements to maintain their consistency in thick to thin plate bending condition. These elements free shear locking and passed the patch test. Katili et al. develop of DKMQ plate element to shell element [7]. In this paper, DKMQ shell element with Five DOF () are formulated and consequently the fictitious stiffness and MacNeal stabilization as it used in is vanished. In RN i uq,, z u p, z, with. n formulations of element [7], represents the normal vector. However, In The formulation of membrane e t (5) and curvature deformation is the same as [7]. [7] we have the following equation: Db s, ss k sk Ds h Db with k Ds K sk (6) The factor k (shear coefficient factor) maintains the consistency of element in thin and thick problem. We apply the Kirchhoff-Mindlin method [5], [6] on element for getting and as function of nodal degrees of freedom. From [7] and applying Hu-Washizu functional, we have: The sk sk ds (7) From [7] and (6) we have s u p, s. n. t sk ; sk k sk ~k ~ ~ ~ () () Therefore where: Manuscript received January, 7; revised pril, 7. 7 Int. J. Mech. Eng. Rob. Res. doi:.7/ijmerr θi Pk tsk β sk () θ i i 5 RN i RN i RN i t With defining rotation in local coordinate system, is quadrilateral element which has five degrees of freedom for each node i Ui,Vi,Wi, i, i following system. (), RN i can be deduced as follow: II. SHELL ELEMENT the global and local coordinates displacement uq is defined [7]-[9]: () u p, s n tsk k sk ds= (9)
2 International Journal of Mechanical Engineering and Robotics Research Vol. 6, No., May 7. Parabolic-Hyperboloid Shell The case is shallow shell which has data; a = 5 cm; c = cm; h =. cm; E = 5 kg/cm; υ =. and fz =. kg/cm. Symmetrical conditions are U=V=W = θ = θ = at the side of BCD. solution [] in the middle surface is Wo =.6 x -. This forms shell has a negative Gaussian curvature with boundary conditions in the form of a straight line (Fig. -Fig. 7). where n is a unit normal vector from one point s on the side k of element. On each side k [7], From [7] we have: u ui ; j s s.tsk i j tsk s s sk L k u p,s () TBLE I. DEFLECTION WO (WO = WO X ) HYPERBOLIC PRBOLOID SHELL. x x 6 x 6 x [] On each side, the value nk : nk ni n j () Equivalently, we have:.6 Convergence of deflection Wo tsk. t i + tsk. t i + Wo x k sk nk 5 Finally, we get: u ji 5 tsk ti tsk t i xi 5 () 6 Figure. Convergence deflection at Parabolic-Hyperbolic Shell. xj B. Scordelis - Lo Roof Problem cylindrical panel with free boundary conditions along the cylinder and on the other side in form of rigid diaphragm and is used to compare the element performance of flat facet and curved shell. Due to small deformation transversal shear deformation is neglected. The membrane deformation is more dominant than bending deformation. The structure has data: L = 6 m; R = m; h =. m; ϕ = ; E = Pa; υ = ; fz =.65 x Pa. Symmetrical condition, U = θ = on the side CD and V = θ = on the side CB. freedom un can be deduced as in [7]. We neglected fictitious stiffness and the stiffness matrix [k] is the sum of the stiffness matrices due to membrane, bending, and transversal shear. k km kb ks If () is applied on all four sides, then the independent parameter n is a function of degrees of III u j ui nk i j tsk k sk () () CONVERGENCE TEST Convergence test is done by comparing with [7]. Shells structure have a constant thickness and known boundary conditions and do not have initial deformation C B free C W D b free D L Meshing x a E O R Diaphragms φ Figure. Scordelis - Lo roof problem. B Boundary condition: U = W = θ = on the side D. solution [] (theory of deep shell), WB = -.6 Figure. Parabolic-Hyperboloid shell. 7 Int. J. Mech. Eng. Rob. Res. 9
3 International Journal of Mechanical Engineering and Robotics Research Vol. 6, No., May 7 cm and WC =.5 cm. solution [] (theory of shallow shell), WB = -.7 cm and WC =.55 cm. Deflection in point B (WB) and point C (Wc) are presented in Table II and Table III. In this problem gives the same convergence rate as. TBLE II. DEFLECTION WB FOR THE SCORDELIS - ROOF PROBLEM. x x x 6 x 6 x Deep Shell [] Shallow Shell [] Figure 6. Pinched cylindrical shell with end diaphragms. TBLE IV. DEFLECTION WC (R/H =) FOR PINCHED CYLINDRICL x x x 6 x 6 x [] Convergence of deflection WB WB Shallow Shell -. Deep Shell For R/h =, analytical solution was given by G.M. Lindberg et al [] and for R/h =, analytical solution considering transversal shear effects that given by K. Bhaskar dan T.K. Varadan []. The structures has data, L = 6 m; R = m; h =. m dan. m. Symmetrical conditions; V = θ = on the side BC; U = θ = on the side CD and W = θ = on the side B. Boundary condition, U = W = θ = on the side D. solution for thin shell [] (R/h=): WC = -Eh Wc/P = 6.. solution for thick shell [] (R/h=); Wc= -Eh Wc / P =.5. In this test, Table V and Fig., we can see the difference between and, since mesh x, displacement between and are become slightly different due to fictitious rigidity Figure. Convergence of deflection WB for the Scordelis - Lo roof problem. TBLE III. DEFLECTION WC FOR THE SCORDELIS - ROOF PROBLEM. x x x 6 x 6 x Deep Shell [] Shallow Shell [] Convergence of deflection Wc (Thin Shell, R/h = ) Convergence of deflection Wc E h Wc / P Wc Shallow Shell - Deep Shell Figure 7. Convergence of deflection Wc thin pinched cylindrical shell. Figure 5. Convergence of deflection WC for the Scordelis - Lo problem. TBLE V. DEFLECTION WC (R/H =) FOR PINCHED CYLINDRICL SHELL C. Pinched Cylindrical Shell with end Diaphragms pinched cylinder with two concentrated loads located in opposite direction and a rigid diaphragm at both ends. This problem is difficult for the shell element which is indicating the complex deformation of the membrane where most of the structure (under concentrated load) is dominated by bending effect ±7%. The numerical test presented in two thickness ratio, R/h = and. 7 Int. J. Mech. Eng. Rob. Res. x x x x x [] 5 -.5
4 International Journal of Mechanical Engineering and Robotics Research Vol. 6, No., May 7 we can conclude formulation rotation in local coordinate does not affect the convergences rate. Convergence of deflection Wc (Thick Shell, R/h = ) - TBLE VI. DEFLECTION U FOR H =. OF TWISTED BEM E h Wc / P Load Px = N N=NELT 5.96 Pz = Figure. Convergence of deflection Wc thick pinched cylindrical shell D. Twisted Beam Introduced by R.J. MacNeal dan R.L. Harder [] and become one of major test to validate a new shell element. The structures model with helicoidally shaped with one side clamped and a concentrated load is applied on the other side (see Fig. 9). The structure has data 9 rotation at point O and, L = m; b =. m; h =. m, E = 9 6 Pa; υ =., Boundary condition is U = V = W = θ = θ =. Geometrical surface is defined with s and θ parameter X = s cos θ; Y = θ L/π; Z = s sin θ, with range θ π/; b/ s +b/. This case will give difficulty for element formulated with facet-shell approach in quadrilateral as well as triangular shape. The reference solution is based on the beam theory (deformation in transversal direction is ignored) with or without taking into account the transversal shear deformation. [] TBLE VII. DEFLECTION W FOR H =. OF TWISTED BEM Load N N=NELT Px =.6.6 Pz = [].75 Convergence of deflection W, h = W O L.6.6 PZ.6 NELT Mesh PX Figure. Convergence of deflection W Twisted beam h =.. b IV. Figure 9. Twisted beam. has more advantages than because it does not require fictitious rigidity which means it need less time computation. Moreover, has the same convergence rate as. In future we will study the measurement convergence of shell structures behavior by using S-Norm method [5]. Convergence of deflection U, h =. 5.5 U REFERENCES 5. [] 5.5 [] 5. NELT [] Figure. Convergence of deflection U Twisted beam h =.. The advantages of this test is to see a couple energy between bending and membrane, In Fig. and Fig., 7 Int. J. Mech. Eng. Rob. Res. CONCLUSIONS [] 5 O. C. Zienkiewicz, R. L. Taylor, and J. M. Too, Reduced integration technique in general analysis of plates and shells, International Journal for Numerical Methods in Engineering, vol., pp. 75-9, 97. E. I. W.. R. L. T. W. P. Doherty, Stress analysis of axisymmetric solids utilizing higher order quadrilateral finite elements, structural engineering laboratory, University of California Berkeley, California 969. R. H. Macneal, Derivation of element stiffness matrices by assumed strain distributions, Nuclear Engineering and Design, vol. 7, pp. -, June 9. K. J. Bathe and E. N. Dvorkin, formulation of general shell elements the use of mixed interpolation of tensorial
5 International Journal of Mechanical Engineering and Robotics Research Vol. 6, No., May 7 []. C. Scordelis and K. S. Lo, Computer analysis of cylindrical shells, Journal Proceedings, vol. 6, May 96. [] G. M. Lindberg, M. D. Olson, and G. R. Cowper, New developments in the finite element analysis of shells, Q. Bull. Div. Mech. Eng. and Nat. eronautical Establishment, 969. [] K. Bhaskar and T. K. Varadan, nalytical solution for a pinched laminated cylinder, Journal of pplied Mechanics, vol. 57, pp. -, 99. [] R. H. Macneal and R. L. Harder, proposed standard set of problems to test finite element accuracy, Finite Elements in nalysis and Design, vol., pp. -, pril 95. [5] J. F. Hiller and K. J. Bathe, Measuring convergence of mixed finite element discretizations: n application to shell structures, Computers & Structures, vol., pp , 5//. components, International Journal for Numerical Methods in Engineering, vol., pp , 96. [5] I. Katili, new discrete kirchhoff-mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fields Part I: n extended DKT element for thick-plate bending analysis, International Journal for Numerical Methods in Engineering, vol. 6, pp. 59-, 99. [6] I. Katili, new discrete Kirchhoff-Mindlin element based on Mindlin Reissner plate theory and assumed shear strain fields Part II: n extended DKQ element for thick-plate bending analysis, International Journal for Numerical Methods in Engineering, vol. 6, pp. 5-9, 99. [7] I. Katili, J. L. Batoz, I. J. Maknun,. Hamdouni, and O. Millet, The development of DKMQ plate bending element for thick to thin shell analysis based on the Naghdi/Reissner/Mindlin shell theory, Finite Elements in nalysis and Design, vol., pp. 7, 5. [] I. Katili, Formulation et évaluation des Nouveaux éléments finis pour l analyse linéaire des plaques et coques de forme quelconque, 99. [9] I. Katili, Rancang bangun elemen unggulan dan pengembangannya pada analisa struktur pelat dan cangkang dengan metode elemen hingga, Universitas Indonesia995. [] C.. Brebbia and J. J. Connor, Fundamentals of Finite Element Techniques for Structural Engineers, Wiley, Int. J. Mech. Eng. Rob. Res. Herry Irpanni is with the Civil Engineering Department, Universitas Indonesia, Depok, Indonesia 5
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