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# Lesson 5 Structural Dynamics

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1. Lesson 5Structural Dynamics

2. Lesson Objectives Upon conclusion, participants should have: • Aclear understanding of seismic structural response in terms of structural dynamics • An appreciation that code-based seismic design provisions are based on the principles of structural dynamics

3. Part I Linear Single Degree of Freedom Systems

4. Structural Dynamics of SDOF Systems: Topic Outline • Equations of Motion for SDOF Systems • Structural Frequency and Period of Vibration • Behavior under Dynamic Load • Dynamic Amplification • Effect of Damping on Behavior • Linear Elastic Response Spectra

5. Idealized Single Degree of Freedom System F(t) Mass t Damping Stiffness u(t) t

6. Equation of Dynamic Equilibrium

7. Properties of Structural MASS MASS INERTIAL FORCE M 1.0 ACCELERATION • Includes all dead weight of structure • May include some live load • Has units of FORCE/ACCELERATION

8. Properties of Structural DAMPING DAMPING FORCE DAMPING C 1.0 VELOCITY • In absence of dampers, is called Natural Damping • Usually represented by linear viscous dashpot • Has units of FORCE/VELOCITY

9. Properties of Structural STIFFNESS SPRING FORCE STIFFNESS K 1.0 DISPLACEMENT • Includes all structural members • May include some “seismically nonstructural” members • Has units of FORCE/DISPLACEMENT

10. Properties of Structural STIFFNESS (2) SPRING FORCE STIFFNESS AREA = ENERGY DISSIPATED DISPLACEMENT • Is almost always nonlinear in real seismic response • Nonlinearity is implicitly handled by codes • Explicit modelling of nonlinear effects is possible

11. Equation of Motion: Undamped Free Vibration Initial Conditions: Solution:

12. Undamped Free Vibration (2) T = 0.5 sec 1.0 Cyclic Frequency (cycles/sec, Hertz) Period of Vibration (sec/cycle) Circular Frequency (radians/sec)

13. Damped Free Vibration Equation of Motion: Initial Conditions: Solution:

14. Damping in Structures x = Damping ratio Whenx = 1.0, the system is called critically damped. Displacement, inches Time, seconds Response of Critically Damped System, x = 1.0or 100% critical

15. Damped Free Vibration

16. Damping in Structures True damping in structures is NOT viscous. However, for low damping values, viscous damping allows for linear equations and vastly simplifies the solution. • Damping Force, Kips • Velocity, in/sec.

17. Damping in Structures (2) Welded Steel Frame x = 0.010 Bolted Steel Frame x = 0.020 UncrackedPrestressed Concrete x = 0.015 Uncracked Reinforced Concrete x = 0.020 Cracked Reinforced Concrete x = 0.035 Glued Plywood Shear wall x = 0.100 Nailed Plywood Shear wall x = 0.150 Damaged Steel Structure x = 0.050 Damaged Concrete Structure x = 0.075 Structure with Added Damping x = 0.250

18. Damping in Structures (3) Natural Damping is a structural (material) property, independent of mass and stiffness Supplemental Damping is a structural property, dependent on mass and stiffness, and damping constant C of device C

19. Undamped Harmonic Loading Equation of Motion: = Frequency of the Forcing Function = 0.25 sec po=100kips

20. Undamped Harmonic Loading (2) Equation of Motion: Assume system is initially at rest: Solution:

24. Undamped Resonant Response Curve Linear Envelope

30. Summary Regarding Viscous Dampingin Harmonically Loaded Systems • For system loaded at a frequency less than √2 or 1.414 times its natural frequency, the dynamic response exceeds the static response. This is referred to as Dynamic Amplification. • An undamped system, loaded at resonance, will have an unbounded increase in displacement over time.

31. Summary Regarding Viscous Dampingin Harmonically Loaded Systems • Damping is an effective means for dissipating energy in the system. Unlike strain energy, which is recoverable, dissipated energy is not recoverable. • A damped system, loaded at resonance, will have a limited displacement over time, with the limit being (1/2x) times the static displacement. • Damping is most effective for systems loaded at or near resonance.

32. Concept of Energy Absorbed and Dissipated ENERGY DISSIPATED F F ENERGY ABSORBED u u LOADING YIELDING TOTAL ENERGY DISSIPATED ENERGY RECOVERED F F u u UNLOADING UNLOADED

33. Development of Effective Earthquake Force • Unlike wind loading, earthquakes do not apply any direct forces on a structure • Earthquake ground motion causes the base to move, while masses at the floor levels try to stay in their places due to inertia. • This creates stresses in the resisting elements.

34. Development of Effective Earthquake Force • ug = Ground displacement • ur = Relative displacement • ut = Total displacement • = ug + ur • ut = Total acceleration • = ug + ur : : :

35. Development of EffectiveEarthquake Force • Inertia force depends on the total acceleration of the masses. • Resisting forces from stiffness and damping depend on the relative displacement and velocity. • Thus, the equation of motion can be written as:

36. Earthquake Ground Motion - 1940 El Centro Many ground motions now available via the Internet

37. “Simplified” form of Equation of Motion: Divide through bym: Make substitutions: Simplified form:

38. “Simplified” form of Equation of Motion: • For a given ground motion, the response history ur(t) is a function of the structure’s frequency w and damping ratio x Structural frequency Damping ratio Ground motion acceleration history

39. Response to Ground Motion (1940 El Centro) Excitation applied to structure with given x and w SOLVER Computed Response Change in ground motion or structural parameters x and w requires re-calculation of structural response Peak Displacement

40. The Elastic Response Spectrum An Elastic Response Spectrum is a plot of the peak computed relative displacement, ur, for an elastic structure with a constant damping xand a varying fundamental frequency w(or period T=2p/w), responding to a given ground motion. 5% Damped Response Spectrum for Structure Responding to 1940 El Centro Ground Motion PEAK DISPLACEMENT, inches

41. Complete 5% Damped Elastic Displacement Response Spectrum for El Centro Ground Motion

42. Note about the Response Spectrum The Pseudoacceleration Response Spectrum represents the TOTAL ACCELERATION of the system, not the relative acceleration. It is nearly identical to the true total acceleration response spectrum for lightly damped structures. 5% Damping

43. Difference Between Pseudo-Acceleration and Total Acceleration • System with 5% Damping

44. Pseudoacceleration Response Spectrafor Different Damping Values Damping

45. Use of an Elastic Response Spectrum Example Structure K = 500 kips/in. W = 2,000 kips M = 2000/386.4 = 5.18 kip-sec2/in. w = (K/M)0.5 =9.82 rad/sec T=2p/w = 0.64 sec 5% Critical Damping @T=0.64 sec, Pseudoacceleration = 301 in./sec2 Base Shear = MxPSA = 5.18(301) = 1559 kips

46. Displacement, in. Acceleration, g 10.0 10.0 1.0 1.0 0.10 0.1 0.01 0.01 0.001 0.001 Four-Way Log Plot of Response Spectrum

47. Displacement, in. Acceleration, g 10.0 10.0 1.0 1.0 0.10 0.1 0.01 0.01 0.001 0.001 1940 El Centro, 0.35 g, N-S For a given earthquake, small variations in structural frequency (period) can produce significantly different results.