Forced Vibration Analysis Of Rectagular Plates Usinig Higher Order Finite Layer Method

Transcription

1 Abdul-Razzak : Forced Vibraion Analysis Of Recagular Plaes Usinig Higher Order Forced Vibraion Analysis Of Recagular Plaes Usinig Higher Order Finie Layer Mehod A. A. Abdul-Razzak J. H. Haido Assisan Professor Civil Engineering Deparmen Mosul Universiy, Mosul, Iraq Assisan Lecurer Civil Engineering Deparmen Dohuk Universiy, Dohuk, Iraq Absrac In his paper a higher order finie layer formulaion based on he auxiliary nodal surface (AN) echnique for a forced vibraion analysis of recangular plaes is presened. The forced vibraion analysis has been performed using he Newmark inegraion mehod for invesigaing he vibraion characerisics and finding he response of recangular plaes under he acion of dynamic loads. The forced vibraion of he recangular plaes subjeced o moving and impac loads has been sudied. everal examples have been sudied o show he good performance of he higher order finie layer wih one AN for forced vibraion analysis of plae. Keywords: Finie layer, Forced Vibraion, Thick Plae. / / (Newmark).... Received 19 July 007 Acceped 10 Dec

2 Al-Rafidain Engineering Vol.16 No.5 Dec. 008 Noaion: a, b Lengh and widh of plae [B] rain marix [C] Coefficien marix for displacemen funcion c Thickness of plae [D] Elasiciy marix DAF Dynamic Amplificaion Facor E Modulus of elasiciy {f} Displacemen funcion h Thickness of layer m, n Paricular harmonic number in X and Y-direcion [M] Mass marix r, s pecified number of harmonic erms in x and y-direcions [] iffness marix...,, Vecors of nodal displacemens, velociies and Acceleraions x, y Global coordinae in lengh wise and widh-wise direcion Xm, Yn Harmonic funcion in x and y-direcions z Global coordinae in hickness -wise direcion { } Eigen vecor Mass per uni volume Poisson's raio Naural frequency (rad/sec) Inroducion: Recangular plaes are he main pars in various srucures as bridges, hydraulic, waer anks. ec; in hese srucures he plae is an imporan componen in carrying direcly he applied loads. The dynamic behavior of he srucure is defined by a special frequency specrum consising of an infinie number of naural frequencies and mode shapes which can be found by knowing he geomerical shape, mass disribuion, siffness and boundary condiions of he plaes [1]. I is necessary o undersand he dynamic response of differen ype of recangular plaes such as isoropic plaes, orhoropic plaes, sandwich plaes and laminaed plaes because of greaer increase in he dynamic deformaion compared o he saic deformaion. Plaes may be subjec o dynamical effecs in he form of loads, which lead o vibrae he plae. These loads may be moving forces on he surface of he plae, is locaion may be fixed on he plae bu is magniude or is direcion is changing wih he ime such as he impac loads and explosions. There are many cases of he applied loads on plaes as uniformly disribued loads, pach loads and concenraed loads. ec. I is of necessiy o consider he dynamic effec of hese loads on plaes in he srucural analysis and design because of prevening he effecs of he applied loads o be accompanied wih any damages which may lead o failure. Many researchers have sudied he effecs of some parameers on he vibraion of plaes subjeced o dynamic loads. In 198 [], analyical resuls are found for he dynamic ineracion of an elasic flexural plae and an elasic half-space subjeced o harmonic seismic waves. Displacemens and conac sresses are found for square, massless plaes having a 44

3 Abdul-Razzak : Forced Vibraion Analysis Of Recagular Plaes Usinig Higher Order paricular range of flexural siffness and subjeced o inciden waves oriened parallel o eiher an edge or a diagonal of he plae. I is found ha he plae exhibis addiional resonance. A he beginning of nineies of he las cenury and especially in 1990 [3] an algorihm based on a finie elemen approach has been developed by Mechael and Ting o sudy he ransien response of recangular plaes having arbirary boundary condiions and subjeced o a moving force. Thin plae heory is used for he plae model wih no resricion is placed on he loading condiions. The algorihm accouns for he complee dynamic ineracions beween he moving loads and he plae. Therefore, he mehod can be applied o he general moving mass problems and also o he simplified moving force and saic problems. The dynamic response of isoropic recangular plaes, composie laminaed plaes and shell srucures under he acion of low velociy impacs was invesigaed by Liu e al. in 1997 [4] who used nine-node finie elemens. The hisograms were drawn for he conac sress, displacemen, and conac force. The researches found ha he low velociy impac phenomena can be represened by using he finie elemen model. Iu e al. performed a sudy in 1998 [5] on he non linear vibraion analysis of hin plaes in presence of iniial sress, by using spline finie srip mehod. The forced vibraion has been sudied under exisence of iniial sress and damping of he plae. The deflecion response of plae lied on an elasic foundaion and subjeced o an effec of a moving acceleraed loads has been sudied by Huang and Thambiranam in 001 [6] by using he finie srip mehod in analysis. This sudy refers o ha when he load conacs he surface of plae hen he srucure behaves such as i is subjeced o a suddenly applied load. The dynamic behavior of orhoropic recangular plaes under he acion of a moving force has been sudied by Law e al. in 003 [7]. The plae was suppored on wo parallel edges, he analysis was based on Lagrange equaion. The auhors sudied he effecs of some parameers on he dynamic response of he plaes as vehicle velociy, dynamic amplificaion facor, plae lengh and he locaion of he moving force. In he presen sudy, a higher order finie layer wih a second order polynomial has been used for he force vibraion analysis of plaes. Effecs of many ype of dynamic loads on he response of a recangular plae has been sudied, such as moving loads and impac loads. Finie layer mehod: The finie srip mehod pioneered in 1968 by Y.K. Cheung is an efficien ool for analyzing srucures wih regular geomeric plaform and simple boundary condiions. Basically, he finie srip mehod reduces a wo-dimensional problem o a one-dimensional problem. By selecing funcions saisfying he boundary condiions in wo direcions, he philosophy of he finie srip mehod can be exended o layer sysems. The resuling mehod is called he finie layer mehod (FLM). The mehod was firs proposed by Cheung and Chakrabari (1971)[8]. The finie layer mehod is useful for layered maerials, recangular in planform. The higher order layer can be produced by inroducing an auxiliary nodal surface a mid-disance beween he upper and he lower surface of he lower order layer[9]. To illusrae he mehod, (Fig. 1) is considered. (Fig. 1) Finie Layer Mehod 45

4 Al-Rafidain Engineering Vol.16 No.5 Dec. 008 Le z z / h hen he laeral displacemen componen of he higher order layer, can be expressed as [9, 10]: r s w [ c1. w1 c. w c3. w3 ] Xm( x). Yn ( y) where m1n 1 c1 z 3z 1 c 4z 4z...1 c3 z z... w w 1, and w 3 are displacemen parameers for surface 1, auxiliary nodal surface and surface, respecively. The funcions X m and Y n are aken as erms in a rigonomeric series, saisfying he boundary condiions. In his manner a hree-dimensional problem is reduced o one-dimensional problem wih considerable saving in compuer sorage and compuaional ime [8]. By linear heory, he (x, y) displacemen componens (u, v) are linearly relaed o he derivaives of w ; ha is w w u A, v B...3 x y Thus, as wih Eq. (1), hen u v r s [ c1. u1 c. u c3. u3 ] X ( x). Y ( y) m1n 1 r s [ c1. v1 c. v c3. v3 ] X ( x). Y ( y) m1n 1 m m n n 4 X and in which m n displacemen vecor is u r s f v [ N m 1n 1 w Y are he firs derivaives of ] [ N ] where X m and T [ u 1, v1, w1, u, v, w, u3, v3, w3 ] The srain-nodal displacemen relaionship is obained as[11]:...5 Y n respecively. The

5 Abdul-Razzak : Forced Vibraion Analysis Of Recagular Plaes Usinig Higher Order xx yy xy yz zx u / x v / y w / z 1 1 ( vz wy) 1 ( w x u z) zz ( uy vx) [ B] [ B] The sress-srain relaionship is r s m1 n1 T [,,,,, ] [ D][ B]{ } xx yy zz xy yz zx By applying he minimum oal poenial energy, he siffness marix can be derived in he following form abc T [ ] pq [ B] [ D][ B] pq dx. dy. dz where... 1s [ ]... s... rs [ ] pq is he generalized siffness marix, which has he following expanded form: s1 11 s1 rs s 11s 1s1s 11s s1s rs1s s1 11 s1 rs1 Consisen mass marix of he layer: s 1s 1ss 1s ss rss rs 1rs 1srs 1rs srs rsrs..10 udies have clarified ha he consisen mass marix is preferable in dynamic analysis problems because i is gives a more accurae resuls han he lumped mass marix [1]. If he mass is disribued uniformly hrough each layer hen each acceleraion leads o generae a disribued ineria force which is [13] d { f } { ( )} { q} [ N]...11 d 47

6 Al-Rafidain Engineering Vol.16 No.5 Dec. 008 where is he mass per uni volume, he general displacemen f depends on ime. Corresponding o naural vibraions, all poins of a vibraion sysem move in he same phase or in ordinary mode hen [14, 15]: { ( )} { }.sin( ) ubsiuing Eq. (1) in Eq. (11), hen { q} [ N ]. { }.sin( ) By dropping sin( ) from Eq. (13)[16], hen { q} [ N]. { } By applying he principle of virual work he nodal forces which are equivalen o any uniformly disribued load as {q} can be found as follows[17]: { F} T [ N ].{ q}. d( vol.) vol [ M ]{ } T M ] pq.[ N ].[ N ] pq. d ( vol) vol T [ N ] [ N ]{ } d( vol) vol [ Forced vibraion analysis of plaes: When he plae is exposed o he dynamical loads hen he plae movemen can be expressed by a differenial equaion called forced vibraion equaion and has he following form [ M ].{ } [ C].{ } [ ].{ } { R} where [M ], [C], [] and {R} are he consisen mass marix, damping marix, siffness marix and vecor of exernal applied forces on he plae. The damping marix can be found in a simplified manner as i is given by (Rayleigh), he suiable mahemaical shape of calculaing he damping marix for a plae is as follows [18, 19] [ C ].[ ].[ M where ], and and ( ) is he damping raio and ( 1, and ) are he firs and he second naural frequencies of he plae and given in (rad/sec). Eq. (17) can be solved by various mahemaical mehods which may be explici or implici and each of hem includes ime inegraion manner [0]. The implici inegraion manner is sable and uncondiionally; he Newmark mehod is one of hem and used in his sudy. Thus i can be express he relaion beween values of derivaives of vecor {} a he ime ( ) and he values a he ime ( ) as: 48

7 Abdul-Razzak : Forced Vibraion Analysis Of Recagular Plaes Usinig Higher Order.. { }. a { } a { } a { } { } { } a { } a { } To guaranee he suiabiliy and accuracy of he Newmark mehod he values of is parameers (, and ) are aken as follows: 0.5(0.5 ) and he bes value of hese parameers are [1] ( 0. 5, 0. 5 ). Values of variables ( a0 o a 7 ) depend on and. Two cases of applied loads are considered which are given as follows: 1- Moving force: I is a simple model of load which consiss of a lonely single force moving on he surface of a plae. The force may represen a vehicle or any moving hings ha have a mass effecing on he plae and producing a dynamic vibraion of i. If ( m ) is he moving mass on plae and ( g ) is he graviy acceleraion hen he moving force (weigh) can be wrien as [3, ] p m. v g Thus he force vecor for a each layer is T { R } {0, 0, p, 0, 0, 0, 0, 0, 0}... - Time dependen load: The plae may be exposed o ime-varying loads as impac loads, explosions and seismic forces. These loads are represened by uniformly disribued loads or pach loads in his sudy. The used vecor for dynamic load in he finie layer mehod is given as: T { R } [ N ].{ q}. d ( area)...3 Area Where ( q ) is he value of he uniformly disribued or pach load. eps for analyzing forced vibraion: The forced vibraion can be analyzed by finding he soluion for he Eq. (17), by aking he value for { } as he change in value(ascending or descending) of he vecor {} and his change occurs during he ime sep from ( ) o ( ). By using Newmark inegraion mehod hen he relaion can be expressed beween he values of derivaives of vecor {} a a ime ( ) and he values a a ime ( ) as follows: } a 0{ } a { } a { } } { } a { } a { } { 3..4 { 6 7 by using he parameers (, and ), he variables in Eq. (3) can be expressed as follows [3]: 1 1 a a a 1 1,, a3 1 0, ( ).. 49 v

8 Al-Rafidain Engineering Vol.16 No.5 Dec. 008 a4 1, a5 ( ), a6 (1 50 ), a7. The values for iniial velociy and displacemen is assumed equal o zero hus he values for iniial acceleraion vecors a ime ( 0 ) are as follows: { 1 } 0 [ M] ({ R} 0 [ C]{ } 0 [ ]{ } 0).. 6 Then he applied siffness marix can be formulaed as [ ] [ ] a0[ M] a1[ C].. 7 Thus i is possible a any ime sep o calculae he dynamic response in he following seps: 1- Calculae he applied force vecor a ime ( ) as follows: [ R] [ R] [ M]( a0{ } a [ C]( a { } a { } a { } ) { } a { } ) Calculae he values of displacemens a ime ( ) as follows: [ ]{ } { } { F} [ ] 1 { F} Calculae values of vecors of acceleraions and velociies a ime ( ) by Eq. (4). In he problems which are relaed o plaes excied by moving forces he locaion of he moving force on he surface of he plae is found a each ime sep by finding he (x, y) coordinaes for applied poin of force as follows: x y x 1 x x v.. v.( ) y 1 y y v.. v.( ) where ( x, and y ) are he coordinaes of he force conac locaion on he surface of x x he plae. ( v, and v ) represen he acceleraion and he velociy of he moving force in X- y y direcion, ( v, and v ) represen he acceleraion and he velociy in he Y-direcion. Numerical applicaions: A- Analysis of plae excied by suddenly applied uniformly disribued load: The response of a simply suppored square plae has been invesigaed, he maerial and geomerical properies of he plae are as follows : Plae lengh (a) = 10 in = 0.54 m Plae hickness (c) = 0.5 in =0.017 m Densiy () = 0.59 x 10-3 lb.sec/in 4 = 4.8x10-11 kg.sec/m 4 Modulus of elasiciy (E) = 10 x 10 6 psi = kn/m Poisson's raio ( ) = 0.3 The uniformly disribued applied load is shown in Fig. (). The higher order finie layer mehod is applied for analysis of he plae and he resul is compared wih ha found by (Huang) [4] who used he rule of Gauss-Legendre 3x3 poins in his analysis, he comparison

9 Abdul-Razzak : Forced Vibraion Analysis Of Recagular Plaes Usinig Higher Order is shown in Fig. (3). I is found ha he finie layer resuls are in good agreemen wih he finie elemen resuls. Fig. () uddenly applied uniformly disribued load (Example-A) Cenral Deflecion (in x 1E-1) =(mm*5.4e-1) Presen sudy Ref. [4] Time (sec x 1E-3) Fig. (3) Hisogram of cenral response of plae (Example-A) B- Analysis of plae excied by suddenly applied pach load: The plae shown in Fig. (4) is analyzed dynamically by he higher order finie layer mehod, he plae is subjeced o a pach load of magniude of ( psi) and he geomerical and maerial dimensionless properies are given in Fig. (4). The presen resuls are compared wih he previous resuls as given by (Huang) [4] in Fig. (5). I is found ha he finie layer resuls are in good agreemen wih he resuls of reference[4]. E=1 =0.3 h=0. =1 =0.1 Fig. (4) uddenly applied pach load (Example-B) 51

10 Al-Rafidain Engineering Vol.16 No.5 Dec. 008 O Presen sudy ---- Ref. [4] Cenral Deflecion Non-Dimensional Time Fig. (5) Hisogram of cenral response of plae (Example-B) C- Forced vibraion analysis of plae subjeced o a moving single force: This example includes he analysis of undamped forced vibraion of a square plae subjeced o moving force ( lb). The force is moving wih a velociy (34 mile/ hour (mph)) on he cenral line of he plae as in Fig. (6). The dimensions for he plae are given as follows: 6 a 4 in, b 4 in, c 0.1 in, E 3010 psi lb.sec / in 4, 0.3 Figs. (7) and (8) show he comparison of he resuls found by using (4) higher order layers wih he resuls found by (Ting e al.) [3] who used he manner of srucural impedance, he resuls are represened by dynamic magnificaion facor (DAF) and normalized displacemens. Fig. (6) quare plae subjeced o a moving force 5

11 Abdul-Razzak : Forced Vibraion Analysis Of Recagular Plaes Usinig Higher Order Presen sudy Ref. [3] Fig. (7) Normalized displacemens for a plae subjeced o moving force DAF Normalized Displacemen Normalized Force Locaion (x/a) Presen sudy Ref. [3] Force Velociy (in/sec) Fig. (8) Relaion beween dynamic amplificaion facor wih force's velociy D- Parameric sudy: The effecs of damping of plae on he response of i has been considered in his example, his is done by considering he plae properies in example (C) bu wih using differen damping raios. The relaions beween he normalized displacemens and he velociy of he moving force wih he damping raio have been found by making diagrams of Figs. (9-11). 53

12 Al-Rafidain Engineering Vol.16 No.5 Dec DAF Damping Raio Damping Raio Fig. (9) Relaion beween DAF and damping raio DAF Damping Raio=0.01 Damping Raio=0.0 Damping Raio=0.03 Damping Raio=0.04 Normalized Displacemen Force Velociy (mph) Force Velociy (mile/hour) Fig. (10) Relaion of moving force's velociy and DAF wih damping raio 1.8 Damping Raio = Dampling Raio= Damping Raio=0.0 Damping Raio= Damping Raio= Normalized Force Locaion (x/a) Fig. (11) Effec of damping on he normalized displacemens of he plae 54

13 Abdul-Razzak : Forced Vibraion Analysis Of Recagular Plaes Usinig Higher Order Conclusion: In he presen sudy, he higher order finie layer mehod was used for dynamic analysis of simply suppored plaes ha have a single span. According o his sudy he following poins are concluded: 1- In he case of sudying he vibraion of a plae subjeced o moving loads i is clear ha by changing he value of he damping raio of he plae he criical velociy remains consan, he criical velociy is he moving force velociy a which he magniude of he dynamic amplificaion facor is a maximum. - The normalized displacemen is increased by increasing he moving disance of he moving force on he surface of he plae unil i reaches a specific limi (approximaely (0.7) of he pah of he moving force) afer ha i begins in decreasing. 3- By increasing he damping raio of he plae, he normalized displacemen will be decreased. 4- The dynamic amplificaion facor increased by increasing he velociy of he moving force unil he velociy reach he limi of criical velociy hen he curve of dynamic amplificaion facor will be descending. 5- The relaion beween he dynamic amplificaion facor and he damping raio is in reverse relaion. This shows ha by increasing he damping raio he dynamic amplificaion facor will be decreased and also he normalized displacemen will be decreased. References: 1. Chen H.L., pyrakos C.C. and Venkaesh G. Evaluaing srucural deerioraion by dynamic response, Journal of rucural Engineering. Vol. 11, No. 8, Aug. 1995, pp Whiaker W.L. and Chrisiano P. Response of a plae and elasic half-space o harmonic waves, Earhquake Engineering and rucural Dynamics. Vol. 10, 198, pp Mechael R.T. and Ting E.C. Dynamic response of he plae o moving loads; finie elemen mehod, Compuers and rucures. Vol. 34, No. 3, 1990, pp Zishun Liu and omsak waddiwudhipong Response of he plae and shell srucures due o velociy impac, Journal of engineering Mechanics. Vol. 13, No. 1, Dec. 1997, pp Cheung Y.K. Zhu D.. and Iu V.P. Non-linear vibraion of hin plaes wih iniial sress by spline finie srip mehod, Thin-Walled rucures. Vol. 3, 1998, pp Huang M.H. and Thambiranam Deflecion response of plae on Winkler foundaion o moving acceleraed loads, Engineering rucures. Vol. 3, 001, pp Zhu X.Q. and Law.. Dynamic behavior of orhoropic recangular plaes under moving loads, Journal of Engineering Mechanics. Vol. 19, No. 1, Jan. 003, pp Cheung Y.K. and Tham L.G. (1998) The finie srip mehod, CRC Press, Boca Raon, FL. 9. Majeed A.. "Analysis of hick recangular plaes by he finie layer mehod", M.c. Thesis, Dep. of Civil Eng., Universiy of Mosul, Mosul, Haido J.H. "Dynamic analysis of recangular plaes using higher order finie layer", M.c. Thesis, Dep. of Civil Eng., Universiy of Mosul, Mosul,

14 Al-Rafidain Engineering Vol.16 No.5 Dec Boresi A.P. and Chong K.P. (000) Elasiciy in engineering mechanics, Wiley, New York. 1. Farfard M., Laflamme M., avard M. and Bennur M. Dynamic analysis of exising coninuous bridge, ACE, J. Bridge Engineering. Vol. 3, No.1, 1998, pp Cook R.D., Malkus D.. and Plesha M.E. (1989) Conceps and applicaions of finie elemen analysis, Wiley, New York. 14. Miquel C.J., uarez B. and Onae E. Dynamic analysis of srucures using Reissner- Mindlin finie srip formulaion, Compuers and rucures. Vol. 31, No. 6, 1989, pp Zhou D., Lo.H., Au F.T.K. and Cheung Y.K. Vibraion analysis of recangular Mindlin plaes wih inernal line suppors using saic Timoshenko beam funcion, Inernaional Journal of Mechanical cience. Vol. 44, Nov. 00, pp Cheng Y.K. (1976) Finie srip mehod in srucural analysis, Pergamon Press, Oxford. 17. Abdul-Razzak A.A. and Haido J.H. "Free Vibraion o Recangular Plaes using Higher Order Finie Layer", AL-RAFIDAIN Engineering Journal. Vol. 15, 007, pp mih L.M. and Griffihs D.V. (1998) Programming he finie elemen mehod, Third Ediion. 19. Tedescon J.W., Mcdougal W.G. and Ross C.A. (1999) rucural dynamic: Theory and Applicaions, Addiional Wesley Longman, Inc. California. 0. Yang Y.B. and Yau J.D. Vehicle-Bridge ineracion elemen for dynamic analysis, Journal of rucural Engineering, ACE. Vol. 13, No.11, 1997, pp Yang Y.B., Yau J.D. and Hsu L.C. Vibraion of simple beams due o rains moving a high speed, Engineering rucures. Vol. 19, No. 11, 1997, pp Green M.F. and Cebon D. Dynamic ineracion beween heavy vehicles and highway bridges, Compuers and rucures. Vol. 6, No., 1997, pp Wu Y.. and Yang Y.B. A versaile elemen for analyzing vehicle-bridge ineracion response, Engineering rucures. Vol. 3, 001, pp Hou-Cheng Huang (1988) aic and dynamic analysis of plaes and shells heory, sofware and applicaions. wansea, UK. The work was carried ou a he college of Engg. Universiy of Mosul 56