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## Lesson 5 Structural Dynamics

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**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**Part I**Linear Single Degree of Freedom Systems**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**Idealized Single Degree of Freedom System**F(t) Mass t Damping Stiffness u(t) t**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**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**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**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**Equation of Motion:**Undamped Free Vibration Initial Conditions: Solution:**Undamped Free Vibration (2)**T = 0.5 sec 1.0 Cyclic Frequency (cycles/sec, Hertz) Period of Vibration (sec/cycle) Circular Frequency (radians/sec)**Damped Free Vibration**Equation of Motion: Initial Conditions: Solution:**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**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.**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**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**Undamped Harmonic Loading**Equation of Motion: = Frequency of the Forcing Function = 0.25 sec po=100kips**Undamped Harmonic Loading (2)**Equation of Motion: Assume system is initially at rest: Solution:**Undamped Harmonic Loading**LOADING FREQUENCY Define Structure’s NATURAL FREQUENCY Transient Response (at structure’s frequency) Dynamic Magnifier, The Steady State Response is alwaysat the structure’s loading frequency Static Displacement,**LOADING,kips**STEADY STATE RESPONSE, in. TRANSIENT RESPONSE, in. TOTAL RESPONSE, in.**LOADING, kips**STEADY STATE RESPONSE, in. TRANSIENT RESPONSE, in. TOTAL RESPONSE, in.**Undamped Resonant Response Curve**Linear Envelope**LOADING, kips**STEADY STATE RESPONSE, in. TRANSIENT RESPONSE, in. TOTAL RESPONSE, in.**LOADING, kips**STEADY STATE RESPONSE, in. TRANSIENT RESPONSE, in. TOTAL RESPONSE, in.**Response Ratio: Steady State to Static(Absolute Values)**Resonance Slowly Loaded Rapidly Loaded 1.00**Resonance**Slowly Loaded Rapidly Loaded**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.**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.**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**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.**Development of Effective Earthquake Force**• ug = Ground displacement • ur = Relative displacement • ut = Total displacement • = ug + ur • ut = Total acceleration • = ug + ur : : :**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:**Earthquake Ground Motion - 1940 El Centro**Many ground motions now available via the Internet**“Simplified” form of Equation of Motion:**Divide through bym: Make substitutions: Simplified form:**“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**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**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**Computation of Deformation (or Displacement) Response**Spectrum**Complete 5% Damped Elastic Displacement Response Spectrum**for El Centro Ground Motion**Development of PseudoaccelerationResponse Spectrum**5% Damping**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**Difference Between Pseudo-Acceleration and Total**Acceleration • System with 5% Damping**Pseudoacceleration Response Spectrafor Different Damping**Values Damping**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**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**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.