Verification of Shell Elements by Eigenanalysis of Vibration Problems

Transcription

1 YOKOHAMA National University Initiative for Global Arts & Sciences Verification of Shell Elements by Eigenanalysis of Vibration Problems Takahiro Yamada, Yokohama National University, Yokohama Kazumi Matsui, Yokohama National University, Yokohama Kenjiro Terada, Tohoku University, Sendai Masami Sato, Mechanical Design & Analysis Corporation, Chofu, Tokyo Takaya Kobayashi, Mechanical Design & Analysis Corporation, Chofu, Tokyo Makoto Tsukino, Quint Corporation, Fuchu Keizo Ishii, Quint Corporation, Fuchu Sayaka Endoh, JSOL Corporation, Tokyo Yasuyoshi Umezu, JSOL Corporation, Tokyo Takahiko Miyachi, JSOL Corporation, Tokyo

2 What is JANCAE? The Japan Association for Nonlinear CAE a non-profit organization established in 2001 Mission to promote professional development in practical use of the nonlinear computational mechanics and the CAE technologies Members academia software venders simulation consulting firms industries Activities 4 day training course held twice a year Working group for special research

3 Motivation Engineers experience Different commercial codes give slightly different results. Difference comes from Element library Solution procedure (nonlinear iterative procedure, time integration) Benchmark tests for finite elements to understand their characteristics Solid element for incompressible materials Shell element

4 Benchmark tests for shell element P1(Problem 1) : standard static analysis Pinched cylinder with diaphragm P2(Problem 2) : eigenanalysis of free vibration 1 Circular cylinder clamped at both ends P3(Problem 3) : eigenanalysis of free vibration 2 Complete spherical shell Remarks: These are not original and can be found in literatures. Results by quadratic 4 node element will be shown today.

5 P1: Pinched cylinder with diaphragm z Problem definition y Analysis domain 1/8 symmetric portion x Rigid diaphragm D R A F C t B Radius R=300 Length L=600 Thickness h=3 Young s modulus E= Poisson s ratio ν=0.3 Concentrated load F=1 Rigid diaphragm L/2 L/2 F Boundary Conditions Diaphragm Symmetric condition D-A: v=w= x =0 A-B: v= x =0 B-C: u= t =0 C-D: w= x =0

6 P1: Problem features Feature Classical benchmark problem reported first in 1969 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 the National Aeronautical Establishment, National Research Council of Canada, Vol. 4, 1969 Analytical solution W. Flugge, Stresses in Shells, 2nd edition, Springer, 1973 Comparison with analytical solution Displacement at loading point Membrane stress Remarks: Analytical solution reported in the classical literatures is not accurate enough for fine mesh Membrane stress distribution of analytical solution

7 P1: Comparison of displacements Error analytical solution

8 P2: Eigenanalysis of cylindrical shell Definition clamped R Features L clamped Radius R=0.1 Length L=0.4 Thickness t= This problem is reported in ADINA news. Analytical solutions can be found in the literatures. e.g. W. Soedel: Vibrations of Shells and Plates, Third Edition, Marcel Dekker 2004 Single curvature Bending-dominated and membrane-dominated solutions can be given by controlling shell thickness Young s modulus E= Poisson s ratio ν=0.3 Density ρ=7800

9 P2: Natural frequency and mode Low frequency does not correspond to low circumferential wave number t= t=

10 P2: Natural modes Representative natural modes (t=1 10-3) Circumferential wave number M Logitudinal half wave number N Mode 5 (M=3, N=1) Mode 9 (M=5, N=2) Mode 13 (M=7, N=1) Mode 15 (M=7, N=2) Mode 17 (M=4, N=2) Mode 19 (M=2, N=1) 12/05/04 ASME V&V 2012 Abaqus S4

11 P2: Mesh Three types of meshes Number of circumferential division x Number of longitudinal division Coarse Moderate Fine

12 P2: Element characteristic (1) ADINA MITC4(Assumed strain, consistent mass) Coarse mesh leads to stiff results wrong wave number

13 P2: Element characteristic (2) ANSYS SHELL63(noncomforming, consistent mass) Coarse mesh leads to soft results wrong wave number

14 Comparison of codes(fine mesh) Comparison of natural frequency for the same mode (Hz) Circumferential wave number [C] consistent mass [L] lumped mass

15 P3: Eigenanalysis of spherical shell Definition of problem Geometry Radius Thickness r = 100mm t = 0.25 mm Material properties Young s modulus E=1.0 MPa Poisson s ratio ν=0.3 Density ρ= kg/mm 3 Boundary condition None (floating) t = 0.25 mm r = 100 mm Mesh Coarse Fine Number of elements1,620 Number of nodes 1,622 FineNumber of elements13,216 Number of nodes 3,218 domain of shellas

16 P3: Problem feature Feature Double curvature Rigid body modes Degenerate eigenvalues due to symmetry Membrane-dominated Analytical solution can be found in the literatures e.g. W. Soedel: Vibrations of Shells and Plates, Third edition, Marcel Dekker, 2004 Comparison with analytical solution Eigenvalue Mode shape n overall view mode shape sectional view Frequency [ Hz ]

17 P3: Mode and Frequency (default option, coarse mesh) Extracted ill mode Unreasonable mode shapes associated with low frequency appear! Analytical solution S4 S4R CQUAD4 el18 el16 el2 Abaqus NX Nastran LS-DYNA MITC4 ADINA CQUAD4 CQUADR MD Nastran Shell 63 Shell 63 ANSYS

18 P3: Mode and Frequency (default option, fine mesh) Extracted ill mode Unreasonable mode shapes associated with low frequency appear! Analytical solution S4 S4R CQUAD4 el18 el16 el2 Abaqus NX Nastran LS-DYNA MITC4 ADINA CQUAD4 CQUADR MD Nastran Shell 63 Shell 63 ANSYS

19 P3: Mode and Frequency (cosistent mass option, fine mesh) Extracted ill mode Unreasonable mode shape vanishes in almost codes! Analytical solution S4 S4R CQUAD4 el18 el16 el2 Abaqus(Lumped NX Nastran LS-DYNA MITC4 CQUAD4 CQUADR ADINA MD Nastran Shell 63 Shell 181 ANSYS 9 DKQ CAE-Linux

20 Summary Benchmark tests are carried out to understand behaviors of shell element in various commercial codes. Shell elements in commercial codes are verified by comparison with analytical solutions. Eigenanalysis gives more information of element characteristics than static analysis and it is an important tool for verification. Shell elements are characterized by eigenanalysis of free vibration as follows: Fine meshes are required to express mode shapes associated with even low frequency Lumped mass may lead to unreasonable solutions. Collaboration among researchers, software vendors and engineers is very helpful to carry out benchmark tests and a non-profit organization is one of effective platforms to discuss this kind of issue.