Dynamic Analysis of Loads Moving Over Structures

Transcription

1 h Inernaional ongress of roaian ociey of echanics epember, 18-, 3 Bizovac, roaia ynamic nalysis of Loads oving Over rucures Ivica Kožar, Ivana Šimac Keywords: moving load, direc acceleraion mehod 1. Inroducion Increase in raffic inensiy and speed requires more complex analysis of srucures han i was case before; oday i is necessary o consider dynamic behaviour of srucures ha has been induced by loads moving over a srucure. The simples case of a moving load dynamic analysis is he case of a simple beam over which a concenraed load is moving, ha is represened wih a h order parial differenial equaion. P..E. for moving load has been solved numerically wih many benefis over closed soluion various boundary condiions, inroducion of damping and discree elemens like springs and dashpos, addiional suppors and many more. verage acceleraion mehod has been employed since direc use of finie differences had shown as being pracically unusable. Numerical and analyical soluions have been compared. On he basis of he above numerical soluion he procedures for finie elemen analysis have been developed. The resul is a F.E. compuer program for dynamical analysis ha is especially suiable for moving load analysis. Numerical model and measuremens on a real bridge example have been successfully compared.. escripion of he problem Problem of a mass less load moving on a beam is described wih he known parial differenial equaion [] u u 1 EI m P x nalyical soluion can be obained mos easily hrough use of harmonic series 5 P 1 x u x, sink v sin k sink L 1 k1 k k L k L where modal frequencies are EI k k L and parameer v L 1 Graphical represenaion of he funcion shows us vibraions of he oal beam in ime 1 3

2 Figure 1. Beam displacemen in ime 3. Numerical soluion of he governing P..E. imple ranslaion of he equaion 1 ino finie differences does no work well since convergence and accuracy are very poor. One approach o he above problem is o consider he discree version of he problem one wih finie number of masses. iscree form of he above equaion in marical noaion is 5 If we observe he equaion as incremenal in ime and given he iniial displacemens and velociies we can wrie he iniial acceleraion vecor as follows 1 6 ssuming ha our acceleraion is consan wihin he ime inerval average acceleraion mehod [1] han we obain his incremenal equaion 7.a In his equaion we have unknown incremenal acceleraions, incremenal speeds and incremenal displacemens bu inroducing he above assumpions heir values can be deduced and subsiued ino eq.7.a wha finally gives us R Q 7.b where 7.c R Q 7.d Equaion 7.b is solved for unknown incremenal displacemen vecor and incremenal velociies and incremenal acceleraions are hen R 7.e Q 7.f wih

3 Q R 7.g We solve for incremenal acceleraions, incremenal speeds and incremenal displacemens, bu heir oal values are also needed, so hey are calculaed 8.a 1 8.b 1 8.c 1 verage acceleraion assumpion can be replaced wih he assumpion ha acceleraion wearies linearly wihin he ime inerval, in which case we obain linear acceleraion mehod [1] wih somewha faser convergence and slighly beer accuracy. On he oher hand his mehod is only condiionally sable while he average acceleraion mehod is uncondiionally sable and is he mehod of choice for all subsequen numerical analysis. pecial care should be aken in discreisaion in ime of he exernal load since i is of grea influence on he convergence of he numerical procedure. Based on he above equaions exising finie elemen compuer program OKVIRW has been exended o accommodae moving load analysis.. Examples.1 imple beam example 7.h P=1 kn v=, m/s EI=1 knm =1 /m x L=1 m m=5; =L/m*v; n=5; x=l/n Figure. imple beam example The above example has been used o compare numerical and analyical resuls ha are presened in a shape of ime-displacemen diagram:..5. displacemen zz n.15 g n ime Figure 3. id-poin displacemen in ime g = analyical, z=numerical resuls s i can be seen here is excellen agreemen beween analyical and numerical soluion. 3

4 . imple beam wih springs and damping P=1 kn v=, m/s EI=1 knm =1 /m =1+; k=kl=k L=1 m x m=5; =L/m*v; n=5; x=l/n Figure. imple beam wih damping and various suppors 1 = srucural damping, = dashpos under suppors k = spring siffness, = mass marix, = siffness marix 1 K 9 Wih siff suppors and no srucural damping we obain he same resuls as wih analyical soluion, and wih =,1, K=, i,i=,,=,, n,n=, kl=k=k=5. kn/m his is he comparison of he mid-poin displacemen in ime..5. zz n.15 g n Figure 5. omparison of mid-poin displacemens in ime for g = undamped analyical, z=damped numerical resuls. Periodic force influence If he force is changing in ime sable numerical mehods could sill give good answers. In his example here is no damping and load is varying by 5% eigh imes during is movemen over he bema span analysis is coninued afer he load had ransferred he beam. For numerical purposes concenraed force is represened as irak funcion varying in space and ime 1 P Psin 8 T x x i if v x x i v x i oherwise Varying force is bes described graphically

5 Figure 7. Load inensiy varying over he beam s can be seen only 5% change in inpu load inensiy gave us abou 3% change in resuling beam forces since inpu load frequency is very close o he firs eigenfrequency of he beam zz o g o Figure 8. omparison of he resuling displacemen wih he one produced by a consan force.3 Two span beam We can easily calculae dynamic behaviour due o a moving load of a wo span beam Figure 6. Two span beam displacemen in ime 5

6 5. onclusion s i can be seen hrough examples numerical approach o he problem of a moving load is quie suiable for engineering purposes: soluions are accurae and procedure based on average acceleraion is robus. Furher benefi of he numerical formulaion is ha various boundary condiions, damping, various ways of suppors, changing forces can all be easily aken ino analysis. cknowledgemen Work presened in his paper has been parially financed hrough echnological proec TP-/11- financed by he inisry of cience and Technology. References [1] Weaver, W., Johnson,P. R.: rucural ynamics by Finie Elemens, Prenice-Hall, Inc., Englewood liffs, New Jersey [] Inglis,.E., ahemaical Treaase on Vibraion in Railway Bridges, ambridge Universiy Press, London UK, 193. [3] Kožar, I.: Kompleksno operećeni šapovibeam-columns, FRK, 18/19, prosinac [] Šimac, I., NLIZ OTOVKIH KONTRUKIJ POBUĐENIH POKRETNO O, agisarska radna, Zagreb, 3 Prof.dr.sc. Ivica Kožar Faculy of ivil Engineering/ Universiy of Rieka, V.. Emina 5, 51 Rieka, roaia, , , ivicak@gradri.hr mr.sc. Ivana Šimac Faculy of ivil Engineering/ Universiy of Rieka, V.. Emina 5, 51 Rieka, roaia, ,