4.4 1) 단순지지된깊은보 선형동적해석검증예제 ANALYSIS REFERENCE. REFERENCE NAFEMS 1 Beam elements, solid elements

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1 그림 가진방향에따른응답변화예시 Reaction Sum. Total Strain Energy Excitation ngle 4.4 선형동적해석검증예제 1) 단순지지된깊은보 REFERENCE NFEMS 1 ELEMENTS Beam elements, solid elements MODEL FILENME LinearDynamic01.mpb 아래그림은단순지지된깊은보모델로등분포하중을받고있는구조이다. 시간에따른가진 형태를다르게했을때, 지점에서의변위와응력을구하는문제이다. Figure Simply supported deep beam model Z F 0 =10 6 N/m Units : m X Section 4. 동적응답해법 171

2 Material data Elastic modulus Poisson's ratio Mass density Mass proportional damping Stiffness proportional damping Modal damping E = 200 GPa ν = 0.3 ρ = 8000 kg/m 3 α = 5.36 sec -1 β = sec ξ = 0.02 Section property Squre cross-section 2 m x 2 m Forcing functions (1) Harmonic F = F sinωt 0 ( where (2) Periodic w= 2π f ) F = F0 ( sinωt sin 3ωt) (3) Transient ( where w = 2 π f, f = 20Hz ) F = F 0, t > 0 * Rayleigh damping coefficients, α and β are chosen to give 0.02 damping in the dominant first mode. 172 Section 4. 동적응답해법

3 Table Peak responses and frequency of beam subjected to harmonic load Peak u [mm] Peak Z σ [MPa] Peak Frequency [Hz] Element type Reference Number of elements Direct Modal Direct Modal Direct Modal BEM HEX PENT Table Peak responses of beam subjected to periodic forcing function Peak u [mm] Peak Z σ [MPa] Element type Reference Number of elements Direct Modal Direct Modal BEM HEX PENT- 6 10x Section 4. 동적응답해법 173

4 Table Peak responses of beam subjected to transient step load Peak u Z [mm] Peak time [sec] σ Peak [MPa] Static u Z [mm] Element type Reference Number of elements BEM HEX PENT- 6 Direct Modal Direct Modal Direct Modal Direct Modal x Section 4. 동적응답해법

5 2) 단순지지된얇은정사각형판 REFERENCE NFEMS 1 ELEMENTS Shell elements MODEL FILENME LinearDynamic02.mpb 아래그림은단순지지된얇은정사각형판모델로면에수직방향으로등분포압력이작용하는 구조이다. 시간에따른가진형태를다르게했을때, 지점에서의변위와응력을구하는 문제이다. Figure Simply supported thin plate model Y 10.0 F 0 =100 N/m 2 x = y = Rz = 0 at all nodes Rx = Ry = 0 along 4 edges z X Units: m 10.0 Material data Section property Elastic modulus Poisson's ratio Mass density Mass proportional damping Stiffness proportional damping Modal damping Thickness E = 200 GPa ν = 0.3 ρ = 8000 kg/m 3 α = sec -1 β = 1.339x10-3 sec ξ = 0.02 t = 0.05 m Section 4. 동적응답해법 175

6 Forcing (1) Harmonic F = F sinωt 0 ( where w= 2π f ) functions (2) Periodic F = F0 ( sinωt sin 3ωt) ( where w = 2 π f, f = 1.2Hz ) (3) Transient F = F 0, t > 0 * Rayleigh damping coefficients, α and β are chosen to give 0.02 damping in the dominant first mode. Table Peak responses and frequency of thin plate subjected to harmonic load Peak u [mm] Peak Z σ [MPa] Peak Frequency [Hz] Element type Reference Number of elements Direct Modal Direct Modal Direct Modal QUD TRI Table Peak responses of thin plate subjected to periodic forcing function Peak u [mm] Peak Z σ [MPa] Element type Reference Number of elements Direct Modal Direct Modal QUD TRI Table Peak responses of thin plate subjected to transient step load 176 Section 4. 동적응답해법

7 Peak u [mm] Peak time [sec] Peak Z σ [MPa] Static u Z [mm] Element type Reference Number of elements Direct Modal Direct Modal Direct Modal Direct Modal QUD TRI Section 4. 동적응답해법 177

8 3) 단순지지된두꺼운정사각형판 REFERENCE NFEMS 1 ELEMENTS Shell elements, solid elements MODEL FILENME LinearDynamic03.mpb 아래그림은단순지지된두꺼운정사각형판모델로면에수직방향으로등분포압력이작용하는구조이다. 시간에따른가진형태를다르게했을때, 지점에서의변위와응력을구하는문제이다. Figure Simply supported thick plate model Y z X F 0 =10 6 N/m 2 x = y = Rz = 0 at all nodes Rx = Ry = 0 along 4 edges Units: m Material data Elastic modulus Poisson's ratio Mass density Mass proportional damping Stiffness proportional damping E = 200 GPa ν = 0.3 ρ = 8000 kg/m 3 α = sec -1 β = 6.929x10-5 sec ξ = Section 4. 동적응답해법

9 Modal damping Section Thickness t = 1 m property Forcing (1) Harmonic F = F sinωt 0 ( where w= 2π f ) functions (2) Periodic F = F0 ( sinωt sin 3ωt) ( where w = 2 π f, f = 1.2Hz ) (3) Transient F = F 0, t > 0 * Rayleigh damping coefficients, α and β are chosen to give 0.02 damping in the dominant first mode. Table Peak responses and frequency of thick plate subjected to harmonic loads Peak u [mm] Peak Z σ [MPa] Peak Frequency [Hz] Element type Reference Number of elements Direct Modal Direct Modal Direct Modal QUD TRI HEX Table Peak responses of thick plate subjected to periodic forcing function Peak u [mm] Peak Z σ [MPa] Element type Reference Number of elements Direct Modal Direct Modal QUD TRI HEX Section 4. 동적응답해법 179

10 Table Peak responses of thick plate subjected to transient step load Peak u [mm] Peak time [sec] Peak Z σ [MPa] Static u Z [mm] Element type Reference Number of elements Direct Modal Direct Modal Direct Modal Direct Modal QUD TRI HEX Section 4. 동적응답해법

11 4) 단순지지된보의응답스펙트럼해석 REFERENCE Biggs, J.M. 2 ELEMENTS Beam elements, shell elements, solid elements MODEL FILENME LinearDynamic04.mpb 아래그림은직사각형단면을가지는단순지지된보모델을나타낸다. 스펜중앙에서의변위, 모멘트, 응력을응답스펙트럼해석을통해구하는문제이다. Figure Simply supported beam model Z Y B X Z Y x=y=z=rx=0 at, y=z=0 at B 37.0 Units : mm Material data Section property cross- Elasitic modulus Mass density Rectangular section E = GPa ρ = kg/m mm 37.0 mm Mass option Lumped mass Section 4. 동적응답해법 181

12 Table Response spectra definition (unit: m) Frequency [Hz] Period [sec] Type (scale factor) Displacement(1.0) Velocity(1.0) cceleration(0.5) Table Response spectrum analysis results obtained using beam elements Result at mid-span Displacement [mm] Stress [MPa] Moment x10 5 [Nm] Reference Element type Spectra type Displacement BEM-2 Velocity cceleration Table Response spectrum analysis results obtained using shell elements Result at mid-span Displacement [mm] Stress [MPa] Reference Element type QUD-4 Spectra type Displacement Velocity Section 4. 동적응답해법

13 cceleration Displacement TRI-3 Velocity cceleration Table Response spectrum analysis results obtained using solid elements Result at mid-span Displacement [mm] Stress [MPa] Reference Element type Spectra type Displacement HEX-8 Velocity cceleration Section 4. 동적응답해법 183

14 5) 트러스구조의선형동해석비교 REFERENCE Chopra,.K. 3 ELEMENTS Truss elements, shell elements, solid elements MODEL FILENME LinearDynamic05.mpb 아래그림은축방향거동만을받는트러스구조를나타낸다. 지점의가속도나 D 지점의하중이가해졌을때여러가지선형동해석결과를비교하는문제이다. Figure Simply supported beam model Z Y X B C 30 X=y=z=Rx=0 at, y=z=0 at B/C/D D Units : m Material data Section property nalysis condition Elastic modulus Mass density E = 5 Pa ρ = 1/90 kg/m 3 Cross-section rea = 2.0 m 2 Modal transient with tip load Modal transient with base acceleration Modal frequency with tip load Response spectrum F = 10 N, 10% damping xr =1.0m/sec 2,10% damping F = 10 N, 10% damping displacement spectra, 2% damping 184 Section 4. 동적응답해법

15 Figure Displacement response spectra Pseudo Displacement (m) Frequency Table Displacement and acceleration at point D using modal transient analysis with tip load Result type Displacement [m] cceleration [m/sec 2 ] Time step [sec] Reference Element type TRUSS QUD HEX Table Total displacement at point D using modal transient analysis with base acceleration Result type Displacement [m] Time step [sec] Reference 0.244x x x10-4 Element type TRUSS x x x10-4 Section 4. 동적응답해법 185

16 QUD x x x10-4 HEX x x x10-4 Table Stress of element 1 and reaction force at point using modal frequency analysis with tip load Result type Displacement [m] cceleration [m/sec 2 ] Time step [sec] Reference Element type TRUSS QUD HEX Table Peak displacement at point D using response spectrum analysis with 2% modal damping ratio Result type Combination method Displacement [m] BS SRSS TENP NRL CQC Reference Element type TRUSS QUD HEX Section 4. 동적응답해법

17 6) 지진하중을받는기둥 REFERENCE Hilber, H.M. et al 4, Hurty, W.C. et al 5 ELEMENTS Truss elements, shell elements, solid elements MODEL FILENME LinearDynamic06.mpb 아래그림은고정단에지진하중모사를위한지반가속도를강제가속도로주어진기둥모델을나타내다. 시간이력해석의최대값을기준값으로보고, 응답스펙트럼해석의조합방법에따른결과차이를비교하는문제이다. 또한, 지반가속도의기준선조정방법을적용하기전후의결과양상을알아본다. Figure Column model Y 7620 X Z Y X Units : mm Material data Elastic modulus Mass density E = GPa ρ = 7780 kg/m 3 Section property Rectangular cross-section 50.8 mm x 25.4 mm Section 4. 동적응답해법 187

18 Figure El Centro N-S acceleration history Normalized acceleration Time [sec] Figure Displacement spectra for the period range 0.03~10 sec 1.0E E+00 Displacement [in] 1.0E E E E E E E+01 Period [sec] 188 Section 4. 동적응답해법

19 Figure Velocity spectra for the period range 0.03~10 sec 1.0E+02 Velocity [in/sec] 1.0E E E E E E E+01 Period [sec] Figure bsolute displacement of the cantilever s tip with and without baseline correction Total displacement [mm] corrected original Time [sec] Section 4. 동적응답해법 189

20 Figure Base displacement with and without baseline correction Displacement [mm] corrected original Time [sec] Table Maximum displacement and velocity at the top of the column provided by transient analysis (relative to base) Result type Displacement [mm] Velocity [m/sec] Reference nalysis type Direct transient Modal transient Number of elements Section 4. 동적응답해법

21 Table Maximum displacement and velocity at the top of the column provided by response spectrum analysis (relative to base) Result type Displacement [mm] Velocity [m/sec] Reference Number of elements Spectrum type Displacement Velocity Displacement Velocity Displacement Velocity Combination method BS SRSS BS SRSS BS SRSS BS SRSS BS SRSS BS SRSS Section 4. 동적응답해법 191

22 7) 잔류모드를이용한모드기반주파수응답 REFERENCE Dickens et al 6 ELEMENTS Elastic link elements, mass elements MODEL FILENME LinearDynamic07.mpb 아래그림은모드기반주파수응답해석에서잔류모드를이용한모드중접의정확도향상 효과를보기위한스프링 - 질량시스템이다. 총 4 개 DOF 이기때문에, 4 개모드를다사용한 경우의응답을기준값으로보고결과향상정도를파악한다. Figure Spring-mass system u 1 u 2 F u 3 u 4 k k k k k m m m 0.5m Material data Lumped mass Link stiffness Modal damping m = 1.0 kg k = N/m ξ = Section 4. 동적응답해법

23 Figure Displacement amplitude response for DOF 3 Displacement response 1.E-02 1.E-03 1.E-04 1.E-05 ll modes with residual modes without residual modes 1.E Frequency [Hz] Figure cceleration amplitude response for DOF 1 1.E+02 1.E+01 ll modes with residual modes without residual modes cceleration response 1.E+00 1.E-01 1.E Frequency [Hz] Section 4. 동적응답해법 193

24 Table Displacement and percentage error at 3 Hz DOF 1 DOF 2 DOF 3 DOF 4 Reference 4.52E E E E-05 ll Modes % error 4.52E % 8.89E % 1.29E % 6.53E % with residual modes % error 4.54E % 8.89E % 1.29E % 6.51E % without residual modes % error 6.60E % 1.05E % 1.01E % 5.65E % Table cceleration and percentage error at 3 Hz DOF 1 DOF 2 DOF 3 DOF 4 Reference 1.61E E E E-02 ll Modes % error 1.61E % 3.16E % 4.60E % 2.32E % with residual modes % error 1.61E % 3.16E % 4.60E % 2.31E % without residual modes % error 2.35E % 3.74E % 3.61E % 2.01E % 194 Section 4. 동적응답해법

25 8) 판요소의정상상태응답 REFERENCE Thomson 7 ELEMENTS Shell elements MODEL FILENME LinearDynamic08.mpb 아래그림은왼쪽변이구속된 2 차원판모델의오른쪽변에면내방향압력이주어진모델을 나타낸다. 주파수응답의두가지기법인직접법과모드법의결과연속성을확인하기위한 문제이다. Figure D steady state dynamics model D D C C L = 1.0 m F = N/m B B L = 1.0 m Material data Section property Elastic modulus Poisson s ratio Mass Density Mass proportional damping Stiffness proportional damping Thickness E = Pa v = 0.0 ρ = 8000 kg/m 3 α = 5.36 sec -1 β = sec t = 1m Section 4. 동적응답해법 195

26 Table Preak displacement and stress at resonant frequency Peak u [mm] Peak σ [MPa] Frequency [Hz] Element type Reference Number of elements Direct Modal Direct Modal Direct Modal QUD TRI-3 2 (2 2) Section 4. 동적응답해법

27 9) 집중질량이있는타워 REFERENCE Paz 8 ELEMENTS Bar elements MODEL FILENME LinearDynamic09.mpb 아래그림은꼭대기에질량이있는타워모델을나타낸다. 타워부는간략화과정을통해유효한 강성을가지는보요소로, 상부의질량은집중질량으로모델링하였다. 시간에따른타워 꼭대기의횡변위를구하는문제이다. Figure tower model subjected to sinusoidal force m m Ft ( ) = Fsinωt 5 F = 10 lbf o o K L =100 in E 1 1 Units: in Model data Equivalent material data Spring constant Mass Elastic modulus Poisson s ratio K = Pa m = 100 lbm E = psi v = 0.3 Section property Mass option Square cross-section = 1.0 In 2 Coupled mass Section 4. 동적응답해법 197

28 Table Horizontal displacement at t=0.1, 0.2 and 0.3 seconds Time Reference Method Direct Modal Table Horizontal velocity at t=0.1, 0.2 and 0.3 seconds Time Reference Method Direct Modal Table Horizontal acceleration at t=0.1, 0.2 and 0.3 seconds Time Reference Method Direct Modal References 198 Section 4. 동적응답해법

29 1 NFEMS, "Selected Benchmarks for Forced Vibration", Ref. R0016, NFEMS, Glasgow, Biggs, J.M. Introduction to Structural Dynamics, McGraw-Hill, Inc., New York, Chopra,.K., Dynamics of Structures: Theory and pplications to Earthquake Engineering, Prentice-Hall, Englewood Cliffs, N.J., Hilber, H.M., Hughes T.J.R. and Taylor R.L., Improved Numerical Dissipation of Time Integration lgorithms in Structural Dynamics, Earthquate Engineering and Structural Dynamics, Vol. 5, pp , Hurty, W.C. and Rubinstein M.F., Dynamics of Structures, Prentice-Hall, Englewood Cliffs, N.J., Dickens, J.M., Nakagawa and Wittbrodt, M.J., Critique of Mode cceleration and Modal Trucation ugmentation Methods for Modal Response nalysis, Computers & Structures, Vol. 62, pp , Thomson, W.T., Theory of Vibration with pplication, 4 th Edition, Prentice-Hall, Englewood Cliffs, N.J., Paz, M., Structural Dynamics: Theory and Computation, 4 th Edition, Chapman & Hall, International Thomson Publishing, 1997 Section 4. 동적응답해법 199

Prob. 1 SDOF Structure subjected to Ground Shaking

Prob. 1 SDOF Structure subjected to Ground Shaking What is the maximum relative displacement and the amplitude of the total displacement of a SDOF structure subjected to ground shaking? magnitude of ground