Structural Dynamics and Earthquake Engineering

Transcription

1 Srucural Dynamics and Earhquae Engineering Course 1 Inroducion. Single degree of freedom sysems: Equaions of moion, problem saemen, soluion mehods. Course noes are available for download a hp:// Dynamics of srucures deerminaion of response of srucures under he effec of dynamic loading Dynamic load is one whose magniude, direcion, sense and poin of applicaion changes in ime Dynamics of srucures p() p() p() ü g () 1

2 Single degree of freedom sysems Simple srucures: mass m siffness Single degree of freedom sysems (SDOF) Objecive: find ou response of SDOF sysem under he effec of: a dynamic load acing on he mass a seismic moion of he base of he srucure The number of degree of freedom (DOF) necessary for dynamic analysis of a srucure is he number of independen displacemens necessary o define he displaced posiion of masses wih respec o heir iniial posiion m Single degree of freedom sysems One-sorey frame = mass componen siffness componen damping componen Number of dynamic degrees of freedom = 1 Number of saic degrees of freedom =? 2

3 Force-displacemen relaionship Force-displacemen relaionship Linear elasic sysem: elasic maerial firs order analysis fs = u Inelasic sysem: plasic maerial S Firs-order or second-order analysis S (, ) f = f u u 3

4 Damping force Damping: decreasing wih ime of ampliude of vibraions of a sysem le o oscillae freely Cause: hermal effec of elasic cyclic deformaions of he maerial and inernal fricion Damping in real srucures: Damping fricion in seel connecions opening and closing of microcracs in r.c. elemens fricion beween srucural and non-srucural elemens Mahemaical descripion of hese componens impossible Modelling of damping in real srucures equivalen viscous damping 4

5 Damping Relaionship beween damping force and velociy: c - viscous damping coefficien unis: (Force x Time / Lengh) Deerminaion of viscous damping: free vibraion ess forced vibraion ess Equivalen viscous damping modelling of he energy dissipaed by he srucure in he elasic range f D = c u Equaion of moion for an exernal force Newon s second law of moion D'Alamber principle Siffness, damping and mass componens 5

6 Equaion of moion: Newon s 2nd law of moion Forces acing on mass m: exernal force p() elasic (or inelasic) resising force f S damping force f D u ( ) Exernal force p(), displacemen u(), velociy and acceleraion u ( ) are posiive in he posiive direcion of he x axis Newon s second law of moion: p f f = mu S D mu + fs + fd = p mu + cu + u = p( ) Equaion of moion: D'Alamber principle Inerial force equal o he produc beween force and acceleraion acs in a direcion opposie o acceleraion D'Alamber principle: a sysem is in equilibrium a each ime insan if al forces acing on i (including he ineria force) are in equilibrium f I + fs + fd = p f I = mu mu + f + f = p S D mu + cu + u = p( ) 6

7 Equaion of moion: siffness, damping and mass componens Under he exernal force p(), he sysem sae is described by displacemen u() velociy u ( ) acceleraion u ( ) Sysem = combinaion of hree pure componens: siffness componen fs = u damping componen f D = c u f I + fs + fd = p mass componen f I = mu Exernal force p() disribued o he hree componens SDOF sysems: classical represenaion 7

8 Equaion of moion: seismic exciaion Dynamics of srucures in he case of seismic moion deerminaion of srucural response under he effec of seismic moion applied a he base of he srucure Ground displacemen u g Toal (or absolue) displacemen of he mass u Relaive displacemen beween mass and ground u u ( ) = u( ) + u ( ) g Equaion of moion: seismic exciaion D'Alamber principle of dynamic equilibrium Elasic forces relaive displacemen u f S = u Damping forces relaive displacemen u f D = c u Ineria force oal displacemen u f I = mu fi + fs + f D = mu + cu + u = u ( ) = u( ) + u ( ) mu + cu + u = mu g g 8

9 Equaion of moion: seismic exciaion Equaion of moion in he case of an exernal force mu + cu + u = p( ) Equaion of moion in he case of seismic exciaion mu + cu + u = mu g Equaion of moion for a sysem subjeced o seismic moion described by ground acceleraion u g is idenical o ha of a sysem subjeced o an exernal force mu g Effecive seismic force p ( ) = mu ( ) eff g Problem formulaion Fundamenal problem in dynamics of srucures: deerminaion of he response of a (SDOF) sysem under a dynamic exciaion a exernal force ground acceleraion applied o he base of he srucure "Response" any quaniy ha characerizes behaviour of he srucure displacemen velociy mass acceleraion forces and sresses in srucural members 9

10 Deerminaion of elemen forces Soluion of he equaion of moion of he SDOF sysem displacemen ime hisory u( ) Displacemens forces in srucural elemens Imposed displacemens forces in srucural elemens Equivalen saic force: an exernal saic force f S ha produces displacemens u deermined from dynamic analysis f ( ) = u( ) s Forces in srucural elemens by saic analysis of he srucure subjeced o equivalen seismic forces f S Combinaion of saic and dynamic response Linear elasic sysems: superposiion of effecs possible oal response can be deermined hrough he superposiion of he resuls obained from: saic analysis of he srucure under permanen and live loads, emperaure effecs, ec. dynamic response of he srucure Inelasic sysems: superposiion of effecs NOT possible dynamic response mus ae accoun of deformaions and forces exising in he srucure before applicaion of dynamic exciaion 1

11 Soluion of he equaion of moion Equaion of moion of a SDOF sysem mu ( ) + cu ( ) + u( ) = p( ) differenial linear non-homogeneous equaion of second order In order o compleely define he problem: iniial displacemen u() u () iniial velociy Soluion mehods: Classical soluion Duhamel inegral Numerical echniques Classical soluion Complee soluion u() of a linear non-homogeneous differenial equaion of second order is composed of complemenary soluion u c () and paricular soluion u p () u() = u c () +u p () Second order equaion 2 inegraion consans iniial condiions Classical soluion useful in he case of free vibraions forces vibraions, when dynamic exciaion is defined analyically 11

12 Classical soluion: example Equaion of moion of an undamped (c=) SDOF sysem excied by a sep force p()=p, : mu + u = p Paricular soluion: Complemenary soluion: where A and B are inegraion consans and The complee soluion ( ) p u p = Iniial condiions: for = we have u () = and u () = A p = B = he eq. of moion uc ( ) = Acosωn + B sinωn ω = p u( ) = Acosωn + Bsinωn + n m p u( ) = (1 cos ωn) Duhamel inegral Basis: represenaion of he dynamic exciaion as a sequence of infiniesimal impulses Response of a sysem excied by he force p() a ime sum of response of all impulses up o ha ime 1 u( ) = p( τ )sin[ ωn ( τ )] dτ mω n Applicable only o "a res" iniial condiions u () = u () = Useful when he force p() is defined analyically is simple enough o evaluae analyically he inegral 12

13 Duhamel inegral: example Equaion of moion of an undamped (c=) SDOF sysem, excied by a ramp force p()=p, : mu + u = p τ = 1 p cos ωn ( τ ) u( ) = p sin[ ωn ( τ )] dτ mω = = n mω n ωn τ = p = (1 cos ωn) Equaion of moion u p ( ) = (1 cos ωn) 13