Element Morphing 17 1

Transcription

1 17 Element Morphing 17 1

2 Chpter 17: ELEMENT MORPHING TABLE OF CONTENTS Pge 17.1 Introduction Plte in Plne Stress Source Element 1: 4-Node Rectngulr Pnel Rectngulr Pnel Templte Morphing to Br Morphing to Simpl Supported Bem Morphing to Spr BenchmrkPlne Bem Solutions Smmetric Solution Antismmetric Elsticit Solution Antismmetric Timoshenko Solution Speciliztion to Rectngulr Cross Section

3 17.1 INTRODUCTION Introduction The suject of this Chpter is finite element friction, testing nd optimiztion method clled morphing. The generic mening of this word is to undergo trnsformtion (Merrim-Wester dictionr; From Gr. morphos: shpe). The term is nowds used in imge processing: mpping the imge of n oject into tht of nother one mens of computer softwre, process sometimes endowed with nimtion. But the ide preceeds the computer s illustrted severl well known woodcuts of M. C. Escher, for instnce Figure Figure Escher s Sk nd Wter: morphing s shpe trnsformtion. In the finite element field, the uthor introduced the term morphing to identif the mpping of source element or mcroelement to trget element through kinemtic constrints. (A mcroelement is n sseml of few elements tht m e view s repeting mesh unit.) The trget element hs smller numer of degrees of freedom nd often smller dimensionlit. The process ws originll clled retrofitting of prent element to child element in 43. This terminolog hs een chnged ecuse child element hs een widel dopted since in seprte topic: dptive mesh refinement processes. The feedck process, which m e clled demorphing fits ppropritel under cse elow. Wht re the resons for morphing? Improving element performnce is the dominnt motive. Two cses m e considered: Improving the source. The source model contins djustle prmeters tht survive in the trget element. Improving the performnce of the ltter m ields vlues for those prmeters, which re fed ck. 17 3

4 Chpter 17: ELEMENT MORPHING Source element or mcroelement Morphing kinemtic trnsformtion(s) Modif or specilize source element Trget element Figure 17.. Element morphing nd demorphing processes. Deriving specil elements. The source element is fied. Morphing is used to produced functionll specilized instnces; for emple solid shell element from generl isoprmetric rick. The trnsformtion m introduced free prmeters in the trget. These prmeters re chosen to improve the performnce of the trget element. In turn this m pose constrints in the trnsformtion. The process is est eplined through specific emples tht conve the scope of wht cn e ccomplished Plte in Plne Stress Severl of the following emples pertin to thin plte in plne stress (lso known s memrne) stte. Nottion nd sign conventions for displcements, strins, stresses nd forces in this mthemticl model re collected in Figure Assumptions ssocited with plne stress re discussed in detil in Chpter 14 of 57. Here we simpl recll the in-plne governing equtions e / 0 u e = Du or e = 0 /, u e / / σ E11 E 1 E 13 e σ = Ee or σ = E 1 E E 3 e, (17.1) D T σ + = 0 or σ E 13 E 3 E 33 σ σ σ / 0 / 0 / / + e = 0. 0 As regrds stresses in the z direction, one ssumes σ zz = σ z = σ z = 0. The strins {e zz, e z, e z } re determined from the three-dimensionl strin-stress (complince) constitutive equtions. If the plte mteril is isotropic with elstic modulus E nd Poisson s rtio ν, the mtri constitutive eqution σ = Eend its inverse e = E 1 σ ecome σ e 1 ν 0 σ ν 1 0 σ 0 0 (1+ν) σ = E 1 ν 0 e ν 1 0 σ 1 ν ν e, e = 1. e e E σ (17.) For the isotropic cse e z = e z = 0, wheres the thickness strin is e zz = ν ( ) ν ( ) σ +σ = e +e. (17.3) E 1 ν 17 4

5 17.3 SOURCE ELEMENT 1: 4-NODE RECTANGULAR PANEL Thin plte in plne stress z In-plne internl forces h p p p d d + sign conventions for internl forces, stresses nd strins d d h In-plne stresses σ σ = σ σ In-plne od forces h In-plne strins In-plne displcements h e e e = e h u u Figure Nottion for displcements, strins, stresses nd forces in thin plte in plne stress stte Source Element 1: 4-Node Rectngulr Pnel The 4-node, 8-DOF rectngulr element modeling elstic thin plte in plte stress, sketched in Figure 17.4(), is used s source element for the following emples. The element is clled rectngulr pnel for revit. As 40 nrrtes, this is one of the two oldest continuum structurl elements, hving een developed in the 1950s to model wing skin pnels. This geometr is simple enough to e menle to complete nlticl development, ut it is not trivil. Thus it provides good illustrtion of the morphing process s well s of its strengths nd pitflls Rectngulr Pnel Templte The rectngulr pnel geometr is defined in Figure 17.4(). Aes nd re ligned with the sides for convenience. The element hs constnt thickness h nd elsticit mtri E. The 8 displcement degrees of freedom nd ssocited node forces re collected in the vectors u RP = u 1 u 1 u u u 3 u 3 u 4 u 4 T, f RP = f 1 f 1 f f f 3 f 3 f 4 f 4 T. (17.4) The element stiffness mtri m e epressed in templte form s refs the sum of sic stiffness 17 5

6 Chpter 17: ELEMENT MORPHING () Thin plte in plne stress z 4 Constnt plte thickness h nd elsticit mtri E. Aes nd lie in the plte midplne 3 1 () Rectngulr pnel element Figure The four-node, 8DOF rectngulr pnel modeling thin plte in plne stress. K nd higher order stiffness K h : K = K + K h = V Hc T EH c + V Hh T WT RWH h, H c = , H h = , / 0 R11 R W =, R = 1, V = h. 0 1/ R 1 R (17.5) Here R 11, R 1 nd R re three free prmeters with dimension of elstic moduli. The list {R 11, R 1, R } is clled the templte signture. All possile elements of this tpe tht pss the Individul Ptch Test (IET) of Bergn nd Hnssen re instnces of (17.5). Note tht the sic stiffness K is the sme for ll possile elements wheres the higher order stiffness K h depends on the signture {R 11, R 1, R }. The former specifies the element response to rigid od modes nd constnt strin sttes, wheres the ltter chrcterizes the response to ending deformtions. A prticulr templte instnce is defined specifing its signture. Hence the onl difference etween elements of this tpe is their response to ending modes Morphing to Br The morphing of the rectngulr pnel to -node, prismtic, -ligned r element is digrmmed in Figure The nd direction re clled the il nd trnsverse directions, respectivel. The mteril is ssumed isotropic, oeing the constitutive equtions (17.). The trget is the -node r element shown in Figure 17.5(d), with node displcements nd forces u r = ui u j fi, f r =. (17.6) f j 17 6

7 17.3 SOURCE ELEMENT 1: 4-NODE RECTANGULAR PANEL () 4 Rectngulr pnel of thickness h 3 () Assume smmetric deformtionl motion w.r.t. nd es d / 1 B linking d to d, (c) collpse dimension to r longitudinl () is i j (d) Br cross-section re A = h f i,u i f j,u i Br j L = d / ; j Figure Morphing of the rectngulr pnel of Figure 17.4 (source element) to -node prismtic r element in the il direction. The morphing process is crried out in two steps for convenience. First, the pnel is ssumed to deform smmetricll with respect to es nd plced long the rectngle medins, s sketched in Figure 17.5(). The il nd trnverse elongtions re clled d nd d, respectivel, collected in rr d RP. B inspection u 1 = u 4 = 1 d, u = u 3 = 1 d, u 1 = u = 1 d nd u 3 = u 4 = 1 d. Those reltions re collected in the trnsformtion u RP = T d d RP = 1 T d RP (17.7) Appling (17.7) to the pnel gives the deformtionl stiffness equtions 1 ν Eh d 1 ν ν = 1 f + f 3 f 1 f 4 = d f 3 + f 4 f 1 f f f. (17.8) Note tht the higher order stiffness prmeters {R 11, R 1, R } re gone from (17.8) ecuse HT d = 0. Net, to get rid of the dimension it is necessr to link d nd d, ringing the D nture of the prolem into pl. If the motion is unrestrined, Poisson s effect ss tht d = νd /, which m lso e otined solving the second of (17.8) for f = 0. On the other hnd, d = 0 if tht motion is precluded. The degree of trnsverse restrint cn e prmetrized tking d = (1 α) ν d /, where α vries from 0 for unconstrined to 1 for full constrined. Grouping tht reltion with d = u j u i gives the trnsformtion to the r freedoms: d RP = T u r = d d = ui u j. (17.9)

8 Chpter 17: ELEMENT MORPHING Appling (17.9) to (17.8) nd replcing h A nd L to pss to the more usul r nottion, gives the r stiffness equtions s ÊA 1 1 ui f + (1 α)ν = L f fi =, (17.10) L 1 1 u j f (1 α)ν L f f j in which Ê = 1 (1 α )ν E. (17.11) 1 ν is n effective il modulus tht reduces to E if either ν = 0orα = 0. If the trnsformtions in (17.7) nd (17.9) re comined one gets 1 T R = T Rd T d = T (17.1) Appling T R to the pnel produces (17.10) in one shot, ut the two-step morphing process is more illuminting. One could dd rigid od motions of u = 1 nd u = 1 to the columns of T R, respectivel, to produce more zero entries ut the morphed r eqution would e the sme. The corresponding process for the non-isotropic cse, which is not covered here, is more involved unless mteril tensor smmetries llow smmetries to e introduced in the first step. Otherwise one hs to strt with uniform stress ssumptions, convert to strins, integrte for displcements nd evlute t nodes Morphing to Simpl Supported Bem We net consider morphing to prismtic, -node, simpl-supported, ligned em element s digrmmed in Figure The mteril is ssumed to e isotropic. The trget is the -node Bernoulli-Euler em element shown in Figure 17.6(d), with node displcements nd forces u em = θzi θ zj, f em = Mzi M zj. (17.13) Here θ z is the cross section rottion out z, positive CCW nd M z the corresponding generlized force. The element cn tke onl constnt ending moment. As in the previous emple the process is crried out in two steps. First the deformtionl motion sketched in Figure 17.6() is ssumed. This motion is smmetric out nd ntismmetric out. Deformtionl displcements d nd d re defined in tht figure. The resulting freedom trnsformtion is u RP = T d d RPend = 1 The trnsformed stiffness equtions re hr d 0 0 T d. (17.14) d d = 1 f f 3 f 1 + f 4 = f 1 + f + f 3 + f f f. (17.15)

9 17.3 SOURCE ELEMENT 1: 4-NODE RECTANGULAR PANEL () 4 Rectngulr pnel of thickness h 3 () Assume deformtionl ending motion smmetric w.r.t. nd ntismmetric w.r.t. d / d / (c) 1 Directl collpse dimension to em longitudinl () is i j (d) Bem cross-section nd moment 3 inerti w.r.t. N.A. I = h/1 M i,θi M j,θ Bem i j L = d / zz ; j d / Figure Morphing of the rectngulr pnel of Figure 17.4 (source element) to -node, prismtic, simpl-supported Bernoulli-Euler em element in the il direction. Although the trnsverse () motion pictured in Figure 17.6() is deformtionl, rising from Poisson s rtio effects, it ppers s rigid od to the pnel freedoms since u 1 = u = u 3 = u 4 = 1 d. This eplins the disppernce of d in (17.15). The deformtion d m e epressed in term of em rottionl DOF s d / = (θ zj θ zi ) /, or in mtri form θzi d SSem = d = 1 1 = T θ d θ. (17.16) zj Trnsforming (17.15) s per (17.16), nd setting h I zz nd L to pss to stndrd em nottion, ields 4R 11 I zz 3L 1 1 θi = θ f1 f + f 3 f 4 Mzi =. (17.17) j f 1 + f f 3 + f 4 M zj To mke (17.17) gree with the correct Bernoulli-Euler em stiffness equtions, it is necessr to tke R 11 = 1 E. (17.18) 3 This result ws estlished in 45,48 with other methods. The nisotropic cse is treted there Morphing to Spr We net consider morphing to prismtic, -node, ligned spr element s digrmmed in Figure The mteril is ssumed to e isotropic. The trget is the -node spr element shown 17 9

10 Chpter 17: ELEMENT MORPHING () 4 Rectngulr pnel of thickness h 3 () Assume ntismmetric deformtionl motion w.r.t. is, none in d / 1 (c) i Directl collpse dimension to the spr longitudinl () is j (d) i Spr effective sher re A = 5h/6 s f,u f i i j,u Spr j L = ; j Figure Morphing of the rectngulr pnel of Figure 17.4 (source element) to -node, prismtic, spr element in the il direction. in Figure 17.6(d), with node displcements nd forces u u spr = i, f spr = u j fi f j. (17.19) The spr element cn tke onl constnt shr force. As in the previous emples the process is crried out in two steps. First the deformtionl motion sketched in Figure 17.7() is ssumed. This motion is ntismmetric out nd zero long, since the element deforms onl in sher. The deformtionl displcement d is defined in tht figure. The resulting freedom trnsformtion is u RP = T d d RPspr = T d. (17.0) The trnsformed stiffness equtions re E (1 + ν) 1 d = 1 f + f 3 f 1 f 4 = f. (17.1) The deformtion d m e epressed in term of spr trnsltions u d = 1 1 i = T ds u spr. (17.) Appling this trnsformtion, nd pssing to the usul spr element nottion: E/((1 + ν)) G, h A, L,gives GA 1 1 u i fi =. (17.3) L 1 1 u j f j θ j 17 10

11 17.4 BENCHMARKPLANE BEAM SOLUTIONS (I) Smmetric (S) lod cse M S z L (II) Antismmetric (A) lod cse 1 M A V A z L M S A A V = M /L 1 (III) Generl em cross section Neutrl is ; z A M (negtive) (IV) Speciliztion to rectngulr cross section ; z H h Figure Benchmrk em solution used for dvnced morphing emples. The result (17.3) grees with the spr stiffness equtions derived with Mechnics of Mterils methods (for emple, in Chpter 5 of the Introduction to Finite Elements Notes) ecept for one detil: the trnsverse pnel re A = hshould e replced the effective sher re. For nrrow rectngle with h <<, A s = 5h/6. This discrepnc does not reflect, however, flw in the source element. The prolic sher stress distriution tht produces tht A s ssumes tht σ vnishes t the top nd ottom spr surfces =± 1. This will not generll e the cse when the pnel is used in D mesh Benchmrk Plne Bem Solutions To support more dvnced morphing emples it is convenient to hve enchmrk em solution tht is ect in the sense of plne stress isotropic elsticit, s well s tht provided the sher-fleile Timoshenko em model. The prolem is illustrted in Figure The em hs spn L nd prismtic cross section. Aes {,, z} re chosen s shown. Equtions re initill derived for generl cross section sketched in Figure 17.8(III) nd lter specilized to the rectngulr cross section of height H nd width h illustrted in Figure 17.8(IV). For generl cross section, the moment of inerti with respect to the neutrl is z is I zz, the cross section re is A. The internl ending moment is M z nd the trnsverse (resultnt) sher force is V, with the usul sign conventions. The mteril is isotropic with elstic modulus E, sher modulus G nd Poisson s rtio ν nd sher modulus G = E/((1 + ν)). The in-plne displcements re {u, u } nd the infinitesinl rottion out z is θ z = 1 ( u / u / ). The effective sher re is A s = k s A, in which k s is Timoshenko s sttic sher correction coefficient. 1 This coefficient is defined the energ eqution du S d = 1 σ γ da def = 1 σ γ da= 1 σ σ k s G da= V Gk s A. (17.4) Here σ nd γ re the ctul sher stresses nd strins respectivel, over the em cross section, σ = V /A is the verge sher stress nd γ = σ /(k s G) n energ-equivlent men sher strin. The following smols re introduced for lter use: r Gz = I zz A, = 1 EI zz GA s L = 1 Er Gz Gk s L = 4(1 + ν)r Gz k s L. (17.5) 1 Some uthors use the inverse vlue: e.g., k = 1/k s in?, p. 177 nd m = 1/k s.in?, p

12 Chpter 17: ELEMENT MORPHING (I) Smmetric deformtion: elsticit solution (sme s BE em theor) (II) Antismmetric deformtion: elsticit solution (III) Antismmetric deformtion: Timoshenko model solution (IV) Difference etween (II) nd (III) mgnified 0 Figure Deformed shpes for enchmrk solutions. Bem of rectngulr cross section with L = 5, H =, h = 1, E = 1, ν = 1/4, κ S = 1 nd κ A = 3. Here r Gz denotes the rdius of grtion of the cross section out the neutrl is z while is dimensionless mesure of the section-verged sher rigidit of the element, which is used in the Timoshenko model. The lst form of given in (17.5) ssumes n isotropic mteril. Two lod sstems pictured in Figure 17.8(I,II) re considered: (S) Smmetric: pure ending produced equl nd opposite pplied end moments M S. The ending moment M z = M zs is constnt long the em wheres the trnsverse sher force V is zero. (A) Antismmetric: linerl vring ending produced equl nd opposite sher forces V A lnced end moments with totl resultnt M A = V L. The internl ending moment M z = M A /L vries linerl from + 1 M A t = 1 L through 1 M A t = 1 L. The trnsverse sher force V is constnt nd equl to V A Smmetric Solution The smmetric displcement solution is u S = κ S, u S = 1 κ S ( + ν), θ S z = 1 ( ) u S us = κ S. (17.6) in which κ S = Mz S/(EI zz) = u S / is the constnt curvture produced M S. The deformed shpe, pictured in Figure 17.9(I), is smmetric out nd. The ssocited strins nd stresses re e S = κ S, e S = νe, γ S = 0, σs = Eκ S, σ S = 0, σs = 0. (17.7) The stresses (17.7) stisf identicll the plne stress differentil equilirium equtions for n plnesmmetric cross section. The internl energ stored in the em is U = 1 σ e dv = 1 ELκ S da= 1 EI zz κ S L = 1 M S κ S L. (17.8) V A 17 1

13 17.4 BENCHMARKPLANE BEAM SOLUTIONS Antismmetric Elsticit Solution The ntismmetric displcement solution for n elstic isotropic mteril is u A = κ A ( 3 ( + ν) + 18 r Gz 6L (1 + ν)), u A = κ A 6L ( + 3ν ), θ A z = κ A ( (17.9) ( ) + 3r Gz L (1 + ν)) The deformed shpe is pictured in Figure 17.9(II) for rectngulr cross section with the vlues given there. The ssocited strins nd stresses re e A = κ A L, e A = νe, γ A = κ A (1 + ν) (3r Gz L ), σ A = E κ A L, σ A = 0, σ A = E κ (17.30) A L (3r Gz ) It is esil verified tht σ A A da = EI zz κ A (/L) = M A. To show tht σ A A da = V A, integrte over A, use I zz = A da = ArGz nd V A = M A /L = EI zz κ A /L. The elsticit equilirium equtions re lso stisfied identicll. As regrds stress oundr conditions, note tht σ A vnishes t =± 3r Gz. These re not the top nd ottom em fiers unless the cross section is rectngulr. The internl energ sored em of rectngulr cross section is U A = U A + U A s = 1 V σ A e A dv + 1 V σ A γ A dv = κ A EH3 L h 88L where U nd U s denote the ending nd sher energies, respectivel Antismmetric Timoshenko Solution + κ A EH5 h (1 + ν). (17.31) 10L The solution provided the Timoshenko em model, which ssumes constnt verge sher over the cross section is u Timo = κ ( A 3 + ν + 1 r ) Gz (1 + ν) = κ A ( 6 + ν + L ), 6 L k s 1L u Timo = κ A 6L ( + 3ν ), (17.3) θ Timo z = κ ( A + ν + r ) Gz (1 + ν) = κ A L 4L (1 + 1 ν + L ). k s The deformed shpe is pictured in Figure 17.9(III) for rectngulr cross section with the sme vlues used for (II). There is no discernile difference etween the elsticit nd Timoshenko solutions t the plot scled used there. The difference is shown in Figure 17.9(IV) with displcements mgnified 0. While verticl displcements re the sme, the elsticit solution ccounts for cross section wrping. The ssocited strins nd streses re e Timo = κ A L, etimo = νe Timo, γ Timo = γ = κ A L = V 1 GA s σ Timo = κ A L E, σtimo = 0, σ Timo = σ = κ A GksL = V (17.33) A. Note tht for this model σ is not Gγ ut Gk s γ ; this the plne stress constitutive eqution is djusted. The internl energ sored em of rectngulr cross section is the sme s given in (17.31). This is no ccident, since the Timoshenko model is sed on sher energ mtch Speciliztion to Rectngulr Cross Section For the H h rectngulr cross section, A = Hh, I zz = H 3 h/1, r Gz = H /1, 17 13