Exact Solutions for Fixed-Fixed Anisotropic Beams under Uniform Load by Using Maple

Transcription

1 Eact Solution for Fied-Fied Aniotropic Beam under Uniform Load b Uing Maple Department of Civil Engineering, Ho Ci Min Cit Univerit of Arcitecture, Vietnam ABSTRACT Te approimate olution of tree and diplacement were obtained for fied-fied aniotropic beam ubjected to uniform load. A tre function involving un-known coefficient wa contructed, and te general epreion of tre and diplacement were obtained b mean of air tre function metod. Two tpe of te decription for te fied end boundar condition were conidered. Te introduced unknown coefficient in tre function were determined b uing te boundar condition. Te approimate olution for tree and diplacement were finall obtained. Numerical tet ow tat te olution agree wit te FEM reult. Tee olution are acieved b uing Maple oftware. KEYWORDS: Fied-fied beam, Aniotrop, Stre function, Approimate olution, Maple. INTRODUCTION Te plane tre problem of beam i a claical ubject in te elaticit teor and i alo frequentl encountered in practical cae. Iotropic beam ave been invetigated b Timoenko and Goodier (970) for man cae, uc a tenion, earing, pure bending, bending of a cantilever ubjected to a tranvere load at te end, bending of a impl upported beam under uniform load and oter cae of continuoul loaded beam. Leknitkii (968) tudied te deformation of aniotropic beam including tenion, earing, pure bending, bending of a cantilever loaded at te end, bending of impl upported beam and cantilever beam under uniform load or linearl ditributed load. For uniforml loaded, bot end fied beam, Gere and Timoenko (98) preented te epreion of deflection and tre b emploing Euler-Bernoulli beam teor. Approimate olution of fied-fied aniotropic beam ubjected to uniform load are reported in ti paper. Baic Equation In - plane, te baic function for aniotropic material can be epreed a: / / O Figure : Fied-fied beam ubjected to a uniform load u = v = 6 6 u v = q l τ τ τ () Accepted for Publication on 5// JUST. All Rigt Reerved.

2 u, v,,, τ are te component of diplacement and tre, repectivel, are elatic ij compliance contant, wit: = ij. Stre function Φ ji Φ Φ = = in wic te tre function Φ mut atif te following compatibilit equation: τ Φ = Φ Φ Φ Φ Φ 6 ( 66 ) = 0 (3) () Approimate Solution for Fied-Fied Aniotropic Beam b Uing Maple Software Conider a fied-fied beam wit rectangular croection ubjected to a uniform load q a own in Figure. Suppoe tat te widt of te beam i unit, and te lengt and eigt are, repectivel, l and. Te tre function i recommended in a polnomial form a: Φ = A B C D E F G H L J K () > pi:=a*^5b**^c*^d*^*^3e**^3f*^3g**^h*^l *^*J**K*^; φ := A 5 B C D 3 E 3 F 3 G H L J K A, B, C, D, E, F, G, H, L, J and K are unknown contant. Subtituting Eq.() into Eq.(3) ield: > piub:=*(diff(pi,,,,))- *6*(diff(pi,,,,))*(66)*(diff(pi,,,,))- *6*(diff(pi,,,,))*(diff(pi,,,,))=0; piub := ( 66 ) D 6 ( B D 6 E ) ( 0 A B C ) = 0 Te parameter mut be atified tat piup i equal to 0 for te arbitrar value,. Tree equation can be derived a: > reult:=diff(piub,); reult := ( 66 ) D 8 6 B 0 A = 0 > reult:=diff(piub,); reult := 6 D B = 0 (5) > reult3:=ub(=0,=0,piub); reult3 := 6 E C =

3 Eact Solution for Te ubtitution of Eq.() into Eq.() give te epreion of te tre a: > igma:=diff(pi,,); igma := 0 A 3 B C 6 D 6 E 6 F G H > igma:=diff(pi,,); igma := D 3 L K (6) > to:=-diff(pi,,); to := B 3 6 D 3 E G L J B ubtitution of Eq.(6) into Eq.() and integration, te epreion of te diplacement are ten obtained a: > dud:=*igma*igma6*to; dud := ( 0 A 3 B C 6 D 6 E 6 F G H ) ( D 3 L K) 6 ( B 3 6 D 3 E G L J ) > dvd:=*igma*igma6*to; dvd := ( 0 A 3 B C 6 D 6 E 6 F G H ) ( D 3 L K) 6 ( B 3 6 D 3 E G L J ) > dudpludvd:=6*igma6*igma66*to; dudpludvd := 6 ( 0 A 3 B C 6 D 6 E 6 F G H ) 6 ( D 3 L K) 66 ( B 3 6 D 3 E G L J)

4 and trialu (, ) := ( 5 6 A ( 66 ) B 6 D ) ( ( 0 A 6 B D ) 6 C ( 66 ) E) 3 ( 3 ( B 6 D ) 3 ( C 6 E ) 3 6 F ( 66 ) G 6 L ) ( D 3 3 E ( 6 F L 6 G ) ) ( G 6 L ) ( H 6 J K) w u0 trialv (, ) := ( 5 A.5 D 6 B ) ( ( B 6 D ) C 6 E ) 3 ( 3 ( D E F) 6 G L ) ( ( G 6 L ) H 6 J K).5 D E 3 ( 3 F 6 G ( 66 ) L ) ( 6 H 6 K 66 J) w v0 (7) were uo, vo and ω are arbitrar contant. So, te diplacement component involve undetermined contant. Te boundar condition BC for te Timoenko teor in (Timeoenko and Goodier, 970) can be repreented a: = / = 0 q = / = = ± / τ = 0 v v = 0, = 0, u = 0, v = 0, = 0 = l, = 0, u = 0, v = 0, = 0 and = l/, = 0 u = 0 We obtain equation equal to 0: > equation:=ub(=/,igma); equation := D 3 L K > equation:=ub(=-/,igmaq); equation := D 3 L K q

5 Eact Solution for > equation3:=ub(=/,to); equation3 := B 3 3 D 3 E G L J > equation:=ub(=-/,to); equation := B 3 3 D 3 E G L J > equation5:=ub(=0,=0,trialu(,)); equation5 := u0 > equation6:=ub(=0,=0,trialv(,)); equation6 := v0 > equation7:=ub(=0,=0,diff(trialv(,),)); equation7 := 6 H 6 K 66 J w > equation8:=ub(=l,=0,trialu(,)); equation8 := ( G 6 L ) l ( H 6 J K) l u0 > equation9:=ub(=l,=0,trialv(,)); equation9 :=.5 D l E l 3 ( 3 F 6 G ( 66 ) L ) l ( 6 H 6 K 66 J ) l w l v0 > equation0:=ub(=l,=0,diff(trialv(,),)); equation0 :=.0 D l 3 3 E l ( 3 F 6 G ( 66 ) L ) l 6 H 6 K 66 J w > equation:=ub(=l/,=0,trialu(,)); equation := ( G 6 L ) l ( H 6 J K ) l u0-78 -

6 Togeter wit te 3 equation in (5), we get equation to determine unknown contant: G = 6 q C = 6 l q E =. l q q q {,,, v0 = 0., u0 = 0., D =., L =, K = q, w = q ( l l 6 ), J = q (. 3. l ) H = q ( l 6, ), A = ( ) q, 3 F = q ( l ), 3 B =. 6 q 3 } Te component of tre: igmaub ( ) q 3. 6 q := q l q q l q ( l ) q ( l 6 ) 6 q (8a) igmaub := q 3 q q (8b)

7 Eact Solution for 6 q 3 q 6 q l 6 q toub := q (. 3. l ) q (8c) Te component of diplacement: u := ( ) q. ( 66 ) 6 q q ( ) q 3 6 q l ( ) q l q 3 6 q 6 6 q l q ( l ) 3 ( 66 ) 6 q 6 q q 3 6 q l q ( l ) 3 6 q q ( l 6 ) q 3 q q (. 3. l) q q ( l l 6 ) /( ) (9a)

8 v ( ) q.0 q. 6 6 q := q 6 q. 6 q l 6 q l q q l q ( l ) q q 6 q 6 q q ( l 6 ) q (. 3. l ) q.0 q 3 (9b) q l q ( l ) q ( 66 ) q 6 q ( l 6 ) q q (. 3. l ) q ( l l 6 ) /( ) Te boundar condition BC for te Goodier teor in (Timoenko and Goodier, 970) can be repreented a: = / = 0 q = / = = ± / τ = 0 u u = 0, = 0, u = 0, v = 0, = 0 = l, = 0, u = 0, v = 0, = 0 and = l/, = 0, u = 0 Similarl, we will ave te reult of tree and diplacement

9 Eact Solution for Eample and comparion of te reult of BC, BC and FEM Suppoe tat te geometric parameter of te beam are: pan 0 m, eigt m and widt unit. Te uniform load intenit i q =0 7 N/m. Te material propertie are:.60 =,, 6 =.87 0, 6 =.7 0, =.557 0, = (unit: 66 = m.n - ). Figure ow te curve of diplacement component v at =0 (te deflection of te neutral ai) and Figure 3 ow te curve of diplacement component u at = /, for BC, BC and FEM finite element metod. Te FEM reult are acieved b ABAQUS. Te boundar condition for FEM are treated a: (i) = 0, l, - / /, u = v = 0; (ii) = /, 0 l, τ = = 0, (iii) = -/, 0 l, = 0 7 Pa, τ = 0. Te Quad element of 0.0 m 0.0 m i emploed and te total element for te wole beam are 000. Figure : Dimenionle diplacement component v at =0 Figure 3: Dimenionle diplacement component u at = / CONCLUTION Te approimate olution for fied-fied aniotropic beam ubjected to uniform load are preented in ti paper. Te olution uppl a claical eample for te elaticit teor. Numerical tet ow tat te olution agree wit te FEM reult. Te approimate olution of te two tpe of decription for fied-end boundar provide a teoretical range for FEM reult. REFERENCES Gere, J.M. and Timoenko, S.P. 98. Mecanic of Material, nd Ed., PWS-KENT Publiing Compan, Boton. Leknitkii, S.G Aniotropic Plate, Gordon and Breac, New York. Timoenko, S P. and Goodier, J.N Teor of Elaticit, 3 rd Ed., McGraw-Hill, New York