O metodě konečných prvků Lect_5.ppt Pitfalls of FE Computing M. Okrouhlík Ústav termomechaniky, AV ČR, Praha Plzeň, 2010 N u m erica l m eth od s in F E A E qu ilib rium p ro b lem s K (q ) q Q so lu tion o f alg eb raic eq u ation s S tead y -state v ib ratio n p rob lem s K Ω 2 M q 0 g en eralized eig en v alu e p rob lem P rop ag ation p ro b lem s F Mq int F ext , F int B σ dV T V step b y step in teg ratio n in tim e In lin ear cases w e h av e C q Kq P (t ) Mq Now a few examples from • Statics • Steady-state vibration • Transient dynamics using • Rod elements • Beam elements • Bilinear (L) and biquadratic (Q) plane elements S tatic loa d in g o f a ca n tilev er b ea m b y a v ertica l fo rce a ctin g at th e free en d . B eam 4 -dof elem en ts (E u ler-B erno u lli) are u sed. 2 4 1 3 P 1 2 3 k max 6 2 EI k 3 l sym. 3l 2l 2 6 3l 6 3l 2 l 3l 2 2l Kq P v tip 1 10 PL 3 ‘Exact’ formula for thin beams 3 EI 1 .9 047 619 047 619 0 5 e-00 3 1 .9 047 61 904 761 90 4e-003 1 .90 476 19 047 6 0 43 e-00 3 1.9 04 761 904 76 1 904 e-0 03 A thin cantilever beam, vertical point load at the free end, four-node plane stress elements FE technology Default in ANSYS L -3 6 Reduced integration in Marc maximum displacements for kmax = 10 x 10 5 4 3 "EXACT" 2 Data obtained with 10 elements 1 KHGS 0 0 Exact integration 1 2 0 1 2 3 4 5 6 3 4 5 case 6 7 8 9 k B T 10 EB d V V full integration 1 … anal, 2 … Gauss q., different types of underintegration Exact integration does not yield ‘exact’ results Gauss quadrature and underintegrated elements Summary_2 Isoparametric approach Four-node bilinear element, plane stress, full integration Four-node bilinear element, plane stress, full integration Comparison of beam and L elements used for modelling of a static loading of a cantilever beam • Beam … even one element gives a negligible error • L … too stiff in bending, tricks have to be employed to get “correct” results A single four-node plane stress elem ent G eneralized eigenvalueproblem F ull integration ( K M )q 0 1 2 3 1 2 3 4 5 6 4 5 6 7 8 7 8 4 3 x 10 GEIVP GEIVP 15000 2 10000 1 5000 0 0 5 10 consistent 0 0 5 diagonal 10 Free transverse vibration of a thin elastic cantilever beam FE vs. continuum approach (Bernoulli_Euler theory) Generalized eigenvalue problem ( K M )q 0 1 2 3 Two cases will be studied - L (bilinear, 4-node, plane stress elements) - Beam elements kmax ax ial, bending and F Eversus frequenccounter ies [H z ] Frequencies in [Hz] 10000 Analytical axial 8000 L Analytical bending 6000 4000 FE frequencies 2000 0 0 1 2 3 4 5 6 7 8 9 10 diagonal Natural frequencies and modes of a cantilever beam … diagonal mass m. Four-node plane stress elements, full integration 1, 100.506 2, 586.4532 3, 1299.7804 4, 1501.8821 5, 2651.6171 6, 3862.638 7, 3931.881 8, 5258.5331 9, 6313.8768 10, 6567.2502 11, 7802.0473 12, 8571.0469 13, 8907.7934 14, 9804.6267 15, 10394.9981 16, 10520.9827 Sixth FE frequency is the second axial Seventh FE frequency is the fifth bending This we cannot say without looking at eigenmodes Higher frequencies are useless due to discretization errors axial, bending and FE frequencies [Hz] 10000 8000 6000 4000 2000 0 0 1 2 3 4 5 6 consistent 7 8 9 10 Natural frequencies and modes of a cantilever beam … consistent mass m. Four-node plane stress elements, full integration 1, 101.2675 2, 615.4501 3, 1302.6992 4, 1663.1767 5, 3135.596 6, 3941.6886 7, 4996.7233 8, 6682.2631 9, 7215.1641 10, 9593.4882 11, 9739.1512 12, 12430.9913 13, 12740.3253 14, 14990.9545 15, 16164.1614 16, 16950.4893 Eigenfrequencies of a cantilever beam Four-node bilinear element, plane strain Relative errors [%] of FE frequencies x … axial – continuum, o … bending – continuum, * … FE frequencies frekvence analyticke axialni (x), ohybove (o) a MKP frekvence (*) frekvence analyticke axialni (x), ohybove (o) a MKP frekvence (*) 10000 10000 8000 8000 6000 6000 4000 4000 2000 2000 0 0 1 2 3 4 5 6 7 8 9 10 relativni chyba v % pro axialni (x) a ohybove (o) frekvence 25 0 0 1 2 3 4 5 6 7 8 9 10 relativni chyba v % pro axialni (x) a ohybove (o) frekvence 20 Relative errors for axial (x) and bending (o) freq. 20 10 15 0 10 -10 5 0 Relative errors for axial (x) and bending (o) freq. 1 1.5 2 2.5 3 3.5 konsistentni formulace 4 4.5 5 -20 1 1.5 2 2.5 3 3.5 diagonalni formulace 4 4.5 5 Free transverse vibration of a thin elastic cantilever beam FE (beam element) vs. continuum approach (Bernoulli_Euler theory) 1 2 2 3 kmax 4 1 156 lA m 420 sym. 22 l 4l 2 54 13 l 156 13 l 2 3l 2 22 l 2 4l 3 6 2 EI k 3 l sym. ( K M )q 0 3l 2l 2 6 3l 6 3l 2 l 3l 2 2l Free transverse vibration of a thin elastic beam ANALYTICAL APPROACH The equation of motion of a long thin beam considered as continuum undergoing transverse vibration is derived under Bernoulli-Euler assumptions, namely - there is an axis, say x, of the beam that undergoes no extension, - the x-axis is located along the neutral axis of the beam, - cross sections perpendicular to the neutral axis remain planar during the deformation – transverse shear deformation is neglected, - material is linearly elastic and homogeneous, - the y-axis, perpendicular to the x-axis, together with x-axis form a principal plane of the beam. These assumptions are acceptable for thin beams – the model ignores shear deformations of a beam element and rotary inertia forces. For more details see Craig, R.R.: Structural Dynamics. John Wiley, New York, 1981 or Clough, R.W. and Penzien, J.: Dynamics of Structures, McGraw-Hill, New York, 1993. T he equation is usually presented in the form v v EI A p ( x , t ) 2 2 2 x x t 2 2 2 w here x is a longitudinal coordinate, v is a transversal displacem ent of the beam in y direction, w hich is perpendicular to x, t is tim e, E is the Y oung’s m odulus, I is the planar m om ent of inertia of the cross section, A is the cross sectional area and is the density. O n the right hand side of the equation there is the loading p ( x , t ) - generally a function of space and tim e - acting in the xy plane.For free transverse vibrations w e have zero on the right-hand side of E q. (1). If the bending stiffness E I is independent of tim e and space coordinates w e can w rite v 4 x A v 2 4 EI t 2 0 . (4 a) A ssum ing the steady state vibration in a harm onic form v ( x , t ) V ( x ) cos( t ) w e get 4 d V (x) dx 4 V (x) 0 4 (4 b ) w here w e hav e introduced an aux iliary variable b y A /( EI ) 4 2 (5) T he gen eral solution of E q. (4) can be assum ed (see K re ysig, E .: A dvanced E ngineerin g M athem atics, John W iley & S ons, N ew Y ork, 1993 ) in the form V ( x ) C 1 sinh x C 2 cosh x C 3 sin x C 4 cos x (6) w here constants C 1 to C 4 depend on boun dary conditions. A pplyin g boundary cond itions for a thin cantilever beam (clam ped – free) w e get a freq u en cy d eterm in an t [0, [lam, [sinh(lam*L)*lam^2, [cosh(lam*L)*lam^3, 1, 0, cosh(lam*L)*lam^2, sinh(lam*L)*lam^3, 0, 1 ] lam, 0 ] -sin(lam*L)*lam^2, -cos(lam*L)*lam^2] -cos(lam*L)*lam^3, sin(lam*L)*lam^3] F ro m th e co nd itio n th at th e freq u en cy d eterm in an t is eq u al to zero w e g et th e frequen cy equ ation in h e form cosh L cos L 1 0 R oots of this equation can only be found num erically, D enoting x i i L w e get th e natural frequencies in the form i xi L 2 I E A i 1, 2 , 3 , ... C om p arison of an alytical an d F E resu lts counter 1 2 3 4 5 6 7 8 9 continuum frequencies 5.26650 4690912090e+002 3.300 462151726965e+003 9.24 1389593048039e+003 1.81 0943523875022e+004 2.99 3619402962561e+004 4.4 71949023233439e+004 6. 245945376065551e+004 8. 315607746908118e+004 1.0 68093617279631e+005 5 9 x 10 o - FEM, x - analytical FE frequencies 5.26650 9194371887e+002 3.300 571391657554e+003 9.24 3742518773286e+003 1.81 2669270993247e+004 3.00 1165614576545e+004 4.4 96087393371327e+004 6. 308228786109306e+004 8. 451287572802173e+004 1.0 92740977881639e+005 relative errors for FE frequencies [%] 2.5 8 2 angular frequencies 7 6 FE computation with 10 beam elements Consistent mass matrix Full integration Are analytically computed frequencies exact to be used as an etalon for error analysis? 1.5 5 To answer this you have to recall the assumptions used for the thin beam theory 4 1 3 2 0.5 1 0 0 5 10 counter 15 20 0 0 5 counter 10 Comparison of Beam and Bilinear Elements Used for Cantilever Beam Vibration • 10 beam elements … the ninth bending frequency with 2.5% error • 10 beam elements … this element does not yield axial frequencies • 10 bilinear elements … the first bending frequency with 20% error, the errors goes down with increasing frequency counter • 10 bilinear elements … the errors of axial frequencies are positive for consistent mass matrix, negative for diagonal mass matrix • Where is the truth? T ran sien t p ro b lem s in lin ear dyn am ics, no d am p in g Kq P t Mq M o d elling th e 1 D w av e equ atio n u 2 t 2 u 2 c 2 0 x 2 eps t = 1.6 dis 1.5 1.4 1.2 1 1 0.5 0.8 0 -0.5 0.6 0 0.5 1 0.4 vel 100 1 50 0.5 0 0 -50 0 0.5 1 L1 cons 100 elem Houbolt (red) 0.5 1 acc 1.5 -0.5 0 -100 0 0.5 1 Newmark (green), h= 0.005, gamma=0.5 Rod elements used here, the results depend on the method of integration Example of a transient problem C lassical L am b’s problem L or Q 1 m ra d ia l Loading a point force equiv. pressure 1 m B Elements L, Q, full int. Consistent mass axisymmetric Mesh Coarse 20x20 Medium 40x40 Fine 80x80 Newmark a xia l C A p0 t T im p Axial displacements for point force loading Fine, Q Fine L and medium Q Medium L and coarse Q Coarse, L Axial displacements for pressure loading Point and pressure loading of the solid cylinder by a rectangular pulse in time 12000 P-L coarse P-Lmedium P-Lfine P-Q coarse P-Qmedium P-Qfine Total energy 10000 point Q fine Total energy [J] 8000 FEM-models with the concentrated point-load 6000 point Q medium 4000 point L fine point Q coarse 2000 point L medium point L coarse For all FEM-models with distributed loads 0 0 1 Time [s] 2 x 10 -4 Pollution-free energy production by a proper misuse of FE analysis