Program: RFEM 5, RSTAB 8, RF-DYNAM Pro, DYNAM Pro. Category: Isotropic Linear Elasticity, Dynamics, Member

Transcription

1 Verificaion Example Program: RFEM 5, RSTAB 8, RF-DYNAM Pro, DYNAM Pro Caegory: Isoropic Linear Elasiciy, Dynamics, Member Verificaion Example: 0104 Canilever Beam wih Periodic Exciaion 0104 Canilever Beam wih Periodic Exciaion Descripion A canilever I-beam of lengh L, heigh h, and widh b wih a nodal mass m on is free end is considered, he self-weigh is negleced. This single-degree-of-freedom sysem (SDOF) is excied by a periodic oscillaion F() = F 0 sin(ω) wih an angular frequency Ω a is free end. The deflecion () of he beam is deermined. Maerial Srucure Isoropic Linear Elasic Canilever Beam Cross Secion IPE 80 Modulus of Elasiciy Shear Modulus E GPa G GPa Lengh L m Load F kn Deph d m Widh b m Web Thickness Flange Thickness w m f m Radius r mm Momen of Ineria I y m 4 SDOF Sysem Nodal Mass m kg Damping Lehr's Damping D Periodic Exciaion Frequency Ω rad/s Ampliude F kn Iniial Condiions Displacemen u m Velociy v m/s w F 0 Z X L Z Y d b f Figure 1: Problem skech

2 Analyical Soluion The following erms and relaions are based on he work of J. W. Tedesco [1]. The siffness k of he canilever beam is equal o k = 3EI y L kn/m (104 1) from where he circular frequency of he undamped SDOF sysem is hen calculaed as ω = k rad/s (104 2) m which corresponds o he naural frequency f = ω 2π Hz (104 3) SDOF Sysem wihou Damping The moion of a freely vibraing SDOF sysem is described by he homogeneous second-order ordinary differenial equaion m ü h + k u h = 0 (104 4) he general soluion of which is u h () = A sin(ω) + B cos(ω) (104 5) where he consans A and B are deermined from he iniial condiions. The equaion of moion for an SDOF sysem under forced vibraion, excied by a harmonic funcion wih frequency Ω, is given by m ü p + k u p = F 0 sin(ω) (104 6) The resuling paricular soluion for his differenial equaion is u p () = F 0 /k 1 (Ω/ω) 2 sin(ω) = u s sin(ω) (104 7) 1 η2 where u s = F 0 /k is he equivalen saic deflecion ha would resul from a force F 0, u s = F 0 k mm (104 8)

3 and η = Ω/ω is he so-called frequency raio, η = Ω ω (104 9) Recall ha he dynamic response facor R d, which is he raio beween he saic and dynamic displacemen, is for he undamped SDOF sysem defined as R d = 1 1 η 2 (104 10) The complee soluion of his SDOF sysem is he sum of he homogeneous and paricular soluion, in paricular, he soluion for displacemen u(), velociy u() and acceleraion ü() is given by u s u() = A sin(ω) + B cos(ω) + sin(ω) 1 η2 (104 11) u() = A ω cos(ω) B ω sin(ω) + u s Ω cos(ω) 1 η2 (104 12) ü() = (A ω 2 sin(ω) + B ω 2 cos(ω) + u s 1 η 2 Ω 2 sin(ω)) (104 13) The consans A and B from he homogeneous par of he soluion are deermined from he iniial condiions u(0) = u 0 and u(0) = v 0, namely A = η u s 1 η 2 (104 14) B = 0 (104 15) Insering A and B ino (104 11) (104 13), he final soluion for his SDOF sysem reads as u() = u s [sin(ω) η sin(ω)] (104 16) 1 η2 u() = Ω u s [cos(ω) cos(ω)] (104 17) 1 η2 ü() = Ω u s [ω sin(ω) Ω sin(ω)] (104 18) 1 η2 SDOF Sysem wih Viscous Damping The equaion of moion for a freely vibraing damped SDOF sysem is given by m ü h + c u h + k u h = 0 (104 19) where c is he viscous-damping coefficien. The relaion beween Lehr's damping D and he viscous damping coefficien c reads as

4 D = c C c = c 2 m ω (104 20) where C c = 2m ω is he so-called criical-damping consan. The presened SDOF sysem exhibis subcriical damping, as D < 1. Then, he soluion o (104 19) is given by u h () = e Du [A sin(ω d ) + B cos(ω d )] (104 21) where ω d is he damped circular frequency, ω d = 1 D 2 ω rad (104 22) The specific values of he consans A and B follow again from he iniial condiions. The equaion of moion damped SDOF sysem under forced vibraion, excied by a harmonic funcion wih frequency Ω, is given by m ü p + c u p + k u p = F 0 sin(ω) (104 23) A paricular soluion of his differenial equaion is u p () = R d u s sin(ω Ψ) (104 24) where u s is he equivalen saic deflecion (104 8), and η he frequency raio (104 9). The dynamic response facor R d is given by R d = 1 (1 η 2 ) 2 + (2Dη) 2 (104 25) which represens he raio beween saic and dynamic ampliude, for furher deails see [2]. Noe ha for D = 0 he equaion is idenical o (104 10). The phase angle Ψ represens he lag of he response behind he periodic exciaion, Ψ = an 1 ( 2Dη 1 η 2 ) rad (104 26) The complee soluion of his SDOF sysem is he sum of he homogeneous and paricular soluion, namely

5 u() = e Du [A sin(ω d ) + B cos(ω d )] + R d u s sin(ω Ψ) (104 27) u() = e Du [ [A D ω + B ω d ] sin(ω d ) + [A ω d B D ω] cos(ω d )] + R d u s Ω cos(ω Ψ) (104 28) ü() = e Du [[ A(D 2 ω 2 + ω 2 d ) + 2 B D ω ω d] sin(ω d ) [2 A D ω ω d B(D 2 ω 2 + ω 2 d )] cos(ω d )] R d u s Ω 2 sin(ω Ψ) (104 29) where A = R d u s ω d [D ω sin( Ψ) + Ω cos( Ψ)] (104 30) B = R d u s sin( Ψ) (104 31) are again deermined from he iniial condiions u(0) = u 0 and u(0) = v 0. Resuls In RSTAB DYNAM Pro and RFEM RF-DYNAM Pro, boh direc inegraion and modal analysis are available. The values of he displacemen and he acceleraion ü z are compared wih he analyical soluion, separae for he undamped and he SDOF sysem wih viscous damping, in he ables below. RFEM 5 and RSTAB 8 Seings Modeled in version RFEM and RSTAB The member is no divided ino finie elemens (RFEM) nor inernal nodes (RSTAB) Linear dynamic analysis is considered, modal analysis and direc inegraion (Newmark mehod) are used The ime incremen is Δ = s for he implici Newmark mehod Isoropic linear elasic maerial model is used Shear siffness of members is deacivaed Analyical soluions are compared wih he resuls of direc inegraion and modal analysis in boh RFEM and RSTAB. The displacemens and he acceleraions a ime seps where he maximum displacemens a he free end of he canilever beam occur are compared. The ime sep Δ = s for he implici Newmark mehod has been chosen wih recommendaions given in [3], Δ = 1 20 f = s (104 32)

6 Resuls of he SDOF Sysem wihou Damping Srucure File Program Analysis Mehod Dynamic Load Case RFEM 5 RF-DYNAM Pro Modal Analysis DLC RFEM 5 RF-DYNAM Pro Direc Inegraion DLC RSTAB 8 DYNAM Pro Modal Analysis DLC RSTAB 8 DYNAM Pro Direc Inegraion DLC2 As seen from he following comparisons, excellen agreemens of he analyical and numerical soluions were achieved. Time Analyical Soluion RFEM 5 - Modal Analysis Δ = s ü z ü z Time Analyical Soluion RFEM 5 - Direc Inegraion Δ = s ü z ü z In RSTAB, he modal analysis provides he exac analyical soluion, and also he direc inegraion provides very good resuls. Time Analyical Soluion RSTAB 8 - Modal Analysis Δ = s u Z

7 Time Analyical Soluion RSTAB 8 - Direc Inegraion Δ = s u Z All resuls, achieved in RFEM and RSTAB are compared graphically wih he analyical soluion, he difference can be hardly seen. Figure 2: Displacemen u Z versus ime, he analyical soluion compared wih RFEM and RSTAB. The differences can be hardly seen, he curves are on op of each oher. Figure 3: Acceleraion versus ime, he analyical soluion compared wih RFEM and RSTAB. The differences can be hardly seen, he curves are on op of each oher

8 Resuls of he Damped SDOF Sysem Srucure File Program Analysis Mehod Dynamic Load Case RFEM 5 Modal Analysis DLC RSTAB 8 Modal Analysis DLC1 As can be seen from he following comparisons, good agreemens of analyical soluions wih numerical oupus were achieved in RFEM. In case of a damped sysem, a smaller ime sep would increase he accuracy furher. Time Analyical Soluion RFEM 5 - Modal Analysis Δ = s u Z In RSTAB, he modal analysis provides he exac analyical soluion, and also he direc inegraion provides good resuls. To increase he accuracy even furher a smaller ime sep would be required. Time Analyical Soluion RSTAB 8 - Modal Analysis Δ = s u Z All resuls, achieved in RFEM and RSTAB are compared graphically wih he analyical soluion, he difference can be hardly seen

9 Figure 4: Displacemen u Z versus ime, he analyical soluion compared wih RFEM and RSTAB. The differences can be hardly seen, he curves are on op of each oher. Figure 5: Acceleraion versus ime, he analyical soluion compared wih RFEM and RSTAB. The differences can be hardly seen, he curves are on op of each oher. References [1] TEDESCO, J., MCDOUGAL, W. and ROSS, C. Srucural Dynamics - Theory and Applicaions. Addison-Wesley. [2] CHOPRA, A. K. Dynamics of Srucures- Theory and Applicaions o Earhquake Engineering. Prenice Hall, [3] STELZMANN, U., GROTH, C. and MÜLLER, G. FEM für Prakiker - Band 2: Srukurdynamik. Exper Verlag