Topic03-StructuralDynamicsofSDOFSystemsNotes
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Topic03-StructuralDynamicsofSDOFSystemsNotes

Course Number: CE 7119, Fall 2009

College/University: U. Memphis

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Structural Dynamics of Linear Elastic Single-Degree-of-Freedom (SDOF) Systems Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 1 This set of slides covers the fundamental concepts of structural dynamics of linear elastic single-degree-of-freedom (SDOF) structures. A separate topic covers the analysis of linear elastic multiple-degree-of-freedom (MDOF) systems. A separate topic...

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Dynamics Structural of Linear Elastic Single-Degree-of-Freedom (SDOF) Systems Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 1 This set of slides covers the fundamental concepts of structural dynamics of linear elastic single-degree-of-freedom (SDOF) structures. A separate topic covers the analysis of linear elastic multiple-degree-of-freedom (MDOF) systems. A separate topic also addresses inelastic behavior of structures. Proficiency in earthquake engineering requires a thorough understanding of each of these topics. FEMA 451B Topic 3 Notes Slide 1 Structural Dynamics Equations of motion for SDOF structures Structural frequency and period of vibration Behavior under dynamic load Dynamic magnification and resonance Effect of damping on behavior Linear elastic response spectra Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 2 This slide lists the scope of the present topic. In a sense, the majority of the material in the topic provides background on the very important subject of response spectra. FEMA 451B Topic 3 Notes Slide 2 Importance in Relation to ASCE 7-05 Ground motion maps provide ground accelerations in terms of response spectrum coordinates. Equivalent lateral force procedure gives base shear in terms of design spectrum and period of vibration. Response spectrum is based on 5% critical damping in system. Modal superposition analysis uses design response spectrum as basic ground motion input. Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 3 The relevance of the current topic to the ASCE 7-05 document is provided here. Detailed referencing to numbered sections in ASCE 7-05 is provided in many of the slides. Note that ASCE 7-05 is directly based on the 2003 NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures, FEMA 450, which is available at no charge from the FEMA Publications Center, 1-800-480-2520 (order by FEMA publication number). FEMA 451B Topic 3 Notes Slide 3 Idealized SDOF Structure F(t) Mass Damping Stiffness F ( t ), u ( t ) t u(t) t Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 4 The simple frame is idealized as a SDOF mass-spring-dashpot model with a time-varying applied load. The function u(t) defines the displacement response of the system under the loading F(t). The properties of the structure can be completely defined by the mass, damping, and stiffness as shown. The idealization assumes that all of the mass of the structure can be lumped into a single point and that all of the deformation in the frame occurs in the columns with the beam staying rigid. Represent damping as a simple viscous dashpot common as it allows for a linear dynamic analysis. Other types of damping models (e.g., friction damping) are more realistic but require nonlinear analysis. FEMA 451B Topic 3 Notes Slide 4 Equation of Dynamic Equilibrium f I (t ) fD (t ) F (t ) 0 .5 f S ( t ) 0 .5 f S ( t ) F (t ) f I (t ) fD (t ) fS (t ) = 0 fI (t ) + fD (t ) + fS (t ) = F (t ) Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 5 Here the equations of motion are shown as a force-balance. At any point in time, the inertial, damping, and elastic resisting forces do not necessarily act in the same direction. However, at each point in time, dynamic equilibrium must be maintained. FEMA 451B Topic 3 Notes Slide 5 Observed Response of Linear SDOF Applied Force, kips 40 0 -40 0.00 0.20 0.40 0.60 0.80 1.00 Displacement, in 0.50 0.00 -0.50 0.00 0.20 0.40 0.60 0.80 1.00 15.00 0.00 Velocity, in/sec -15.00 0.00 400.00 0.00 -400.00 0.00 0.20 0.40 0.60 0.80 1.00 Acceleration, in/sec2 0.20 0.40 0.60 0.80 1.00 Time, sec Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 6 This slide (from NONLIN) shows a series of response histories for a SDOF system subjected to a saw-tooth loading. As a result of the loading, the mass will undergo displacement, velocity, and acceleration. Each of these quantities are measured with respect to the fixed base of the structure. Note that although the loading is discontinuous, the response is relatively smooth. Also, the vertical lines show that velocity is zero when displacement is maximum and acceleration is zero when velocity is maximum. NONLIN is an educational program for dynamic analysis of simple linear and nonlinear structures. Version 7 is included on the CD containing these instructional materials. FEMA 451B Topic 3 Notes Slide 6 Observed Response of Linear SDOF (Development of Equilibrium Equation) Spring Force, kips 30.00 Damping Force, Kips 4.00 50.00 Inertial Force, kips 15.00 2.00 25.00 0.00 0.00 0.00 -15.00 -2.00 -25.00 -30.00 -0.60 -0.30 0.00 0.30 0.60 -4.00 -20.00 -10.00 0.00 10.00 20.00 -50.00 -500 -250 0 250 2 500 Displacement, inches Velocity, In/sec Acceleration, in/sec Slope = k = 50 kip/in Slope = c = 0.254 kip-sec/in Slope = m = 0.130 kip-sec2/in f S ( t ) = k u( t ) & f D ( t ) = c u( t ) && f I ( t ) = m u( t ) SDOF Dynamics 3 - 7 Instructional Material Complementing FEMA 451, Design Examples These X-Y curves are taken from the same analysis that produced the response histories of the previous slide. For a linear system, the resisting forces are proportional to the motion. The slope of the inertial-force vs acceleration curve is equal to the mass. Similar relationships exist for damping force vs velocity (slope = damping) and elastic force vs displacement (slope = stiffness). The importance of understanding and correct use of units cannot be over emphasized. FEMA 451B Topic 3 Notes Slide 7 Equation of Dynamic Equilibrium f I (t ) fD (t ) F (t ) 0 .5 f S ( t ) 0 .5 f S ( t ) f I (t ) + fD (t ) + fS (t ) = F (t ) && & m u( t ) + c u( t ) + k u( t ) = F ( t ) Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 8 Here the equations of motion are shown in terms of the displacement, velocity, acceleration, and force relationships presented in the previous slide. Given the forcing function, F(t), the goal is to determine the response history of the system. FEMA 451B Topic 3 Notes Slide 8 Properties of Structural Mass Mass Internal Force M 1.0 Acceleration Includes all dead weight of structure May include some live load Has units of force/acceleration Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 9 Mass is always assumed constant throughout the response. Section 12.7.2 of ASCE 7-05 defines this mass in terms of the effective weight of the structure. The effective weight includes 25% of the floor live load in areas used for storage, 10 psf partition allowance, operating weight of all permanent equipment, and 20% of the flat roof snow load when that load exceeds 30 psf. FEMA 451B Topic 3 Notes Slide 9 Properties of Structural Damping Damping Force Damping C 1.0 Velocity In absence of dampers, is called inherent damping Usually represented by linear viscous dashpot Has units of force/velocity Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 10 Except for the case of added damping, real structures do not have discrete dampers as shown. Real or inherent damping arises from friction in the material. For cracked concrete structures, damping is higher because of the rubbing together of jagged surfaces on either side of a crack. In analysis, we use an equivalent viscous damper primarily because of the mathematical convenience. (Damping force is proportional to velocity.) FEMA 451B Topic 3 Notes Slide 10 Properties of Structural Damping (2) Damping Force AREA = ENERGY DISSIPATED Damping Displacement Damping vs displacement response is elliptical for linear viscous damper. Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 11 The force-displacement relationship for a linear viscous damper is an ellipse. The area within the ellipse is the energy dissipated by the damper. The greater the energy dissipated by damping, the lower the potential for damage in structures. This is the primary motivation for the use of added damping systems. Energy that is dissipated is irrecoverable. FEMA 451B Topic 3 Notes Slide 11 Properties of Structural Stiffness Spring Force Stiffness K 1.0 Displacement Includes all structural members May include some seismically nonstructural members Requires careful mathematical modelling Has units of force/displacement Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 12 In this topic, it is assumed that the force-displacement relationship in the spring is linear elastic. Real structures, especially those designed according to current seismic code provisions, will not remain elastic and, hence, the force-deformation relationship is not linear. However, linear analysis is often (almost exclusively) used in practice. This apparent contradiction will be explained as this discussion progresses. The modeling of the structure for stiffness has very significant uncertainties. Section 12.7.3 of ASCE 7-05 provides some guidelines for modeling the structure for stiffness. FEMA 451B Topic 3 Notes Slide 12 Properties of Structural Stiffness (2) Spring Force AREA = ENERGY DISSIPATED Stiffness Displacement Is almost always nonlinear in real seismic response Nonlinearity is implicitly handled by codes Explicit modelling of nonlinear effects is possible Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 13 This is an idealized response of a simple inelastic structure. The area within the curve is the inelastic hysteretic energy dissipated by the yielding material. The larger hysteretic energy in relation to the damping energy, the greater the damage. In this topic, it is assumed that the material does not yield. Nonlinear inelastic response is explicitly included in a separate topic. FEMA 451B Topic 3 Notes Slide 13 Undamped Free Vibration Equation of motion: Initial conditions: Assume: Solution: && m u( t ) + k u( t ) = 0 & u0 u0 B = u0 u ( t ) = A sin( t ) + B cos( t ) A= & u0 = k m u (t ) = & u0 sin( t ) + u 0 cos( t ) SDOF Dynamics 3 - 14 Instructional Material Complementing FEMA 451, Design Examples In this unit, we work through a hierarchy of increasingly difficult problems. The simplest problem to solve is undamped free vibration. Usually, this type of response is invoked by imposing a static displacement and then releasing the structure with zero initial velocity. The equation of motion is a second order differential equation with constant coefficients. The displacement term is treated as the primary unknown. The assumed response is in terms of a sine wave and a cosine wave. It is easy to see that the cosine wave would be generated by imposing an initial displacement on the structure and then releasing. The sine wave would be imposed by initially shoving the structure with an initial velocity. The computed solution is a combination of the two effects. The quantity is the circular frequency of free vibration of the structure (radians/sec). The higher the stiffness relative to mass, the higher the frequency. The higher the mass with respect to stiffness, the lower the frequency. FEMA 451B Topic 3 Notes Slide 14 Undamped Free Vibration (2) & u0 Displacement, inches 3 2 1 0 -1 -2 -3 0.0 T = 0.5 sec 1.0 u0 0.5 1.0 Time, seconds 1.5 2.0 Circular Frequency (radians/sec) Cyclic Frequency (cycles/sec, Hertz) Period of Vibration (sec/cycle) = k m f = 2 T = 1 2 = f SDOF Dynamics 3 - 15 Instructional Material Complementing FEMA 451, Design Examples This slide shows a computed response history for a system with an initial displacement and velocity. Note that the slope of the initial response curve is equal to the initial velocity (v = du/dt). If this term is zero, the free vibration response is a simple cosine wave. Note also that the undamped motion shown will continue forever if uninhibited. In real structures, damping will eventually reduce the free vibration response to zero. The relationship between circular frequency, cyclic frequency, and period of vibration is emphasized. The period of vibration is probably the easiest to visualize and is therefore used in the development of seismic code provisions. The higher the mass relative to stiffness, the longer the period of vibration. The higher the stiffness relative to mass, the lower the period of vibration. FEMA 451B Topic 3 Notes Slide 15 Approximate Periods of Vibration (ASCE 7-05) Ta = Ct hnx Ct = Ct = Ct = Ct = 0.028, x = 0.8 0.016, x = 0.9 0.030, x = 0.75 0.020, x = 0.75 for steel moment frames for concrete moment frames for eccentrically braced frames for all other systems Note: This applies ONLY to building structures! T = 0.1N a For moment frames < 12 stories in height, minimum story height of 10 feet. N = number of stories. Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 16 One of the first tasks in any seismic design project is to estimate the period of vibration of the structure. For preliminary design (and often for final design), an empirical period of vibration is used. Section 12.8.2 of ASCE 705 provides equations for estimating the period. These equations are listed here. FEMA 451B Topic 3 Notes Slide 16 Empirical Data for Determination of Approximate Period for Steel Moment Frames 0 Ta = 0.028hn .8 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 17 Ta is based on curve-fitting of data obtained from measured response of California buildings after small earthquakes. As will be seen later, the smaller the period, the larger the earthquake force that must be designed for. Hence, a lower bound empirical relationship is used. Because the empirical period formula is based on measured response of buildings, it should not be used to estimate the period for other types of structure (bridges, dams, towers). FEMA 451B Topic 3 Notes Slide 17 Periods of Vibration of Common Structures 20-story moment resisting frame 10-story moment resisting frame 1-story moment resisting frame 20-story braced frame 10-story braced frame 1-story braced frame Gravity dam Suspension bridge T = 1.9 sec T = 1.1 sec T = 0.15 sec T = 1.3 sec T = 0.8 sec T = 0.1 sec T = 0.2 sec T = 20 sec SDOF Dynamics 3 - 18 Instructional Material Complementing FEMA 451, Design Examples This slide shows typical periods of vibration for several simple structures. Engineers should develop a feel for what an appropriate period of vibration is for simple building structures. For building structures, the formula T = 0.1 in is the simplest reality check. The period for a 10-story building should be approximately 1 sec. If a computer analysis gives a period of 0.2 sec or 3.0 sec for a 10-story building, something is probably amiss in the analysis. FEMA 451B Topic 3 Notes Slide 18 Adjustment Factor on Approximate Period (Table 12.8-1 of ASCE 7-05) T = Ta Cu Tcomputed SD1 > 0.40g 0.30g 0.20g 0.15g < 0.1g Cu 1.4 1.4 1.5 1.6 1.7 Applicable ONLY if Tcomputed comes from a properly substantiated analysis. Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 19 In some cases, it is appropriate to remove the conservatism from the empirical period formulas. This is done through use of the Cu coefficient. This conservatism arises from two sources: 1. The lower bound period was used in the development of the period formula. 2. This lower bound period is about 1/1.4 times the best-fit period. The empirical formula was developed on the basis of data from California buildings. Buildings in other parts of the country (e.g., Chicago) where seismic forces are not so high will likely be larger than those for the same building in California. It is important to note that the larger period cannot be used without the benefit of a properly substantiated analysis, which is likely performed on a computer. FEMA 451B Topic 3 Notes Slide 19 Which Period of Vibration to Use in ELF Analysis? If you do not have a more accurate period (from a computer analysis), you must use T = Ta. If you have a more accurate period from a computer analysis (call this Tc), then: if Tc > CuTa use T = CuTa if Ta < Tc < TuCa use T = Tc if Tc < Ta use T = Ta SDOF Dynamics 3 - 20 Instructional Material Complementing FEMA 451, Design Examples This slide shows the limitations on the use of CuTa. ASCE-7-05 will not allow the use of a period larger than CuTa regardless of what the computer analysis says. Similarly, the NEHRP Recommended Provisions does not require that you use a period less than Ta. FEMA 451B Topic 3 Notes Slide 20 Damped Free Vibration Equation of motion: && & m u( t ) + c u( t ) + k u( t ) = 0 & Initial conditions: u0 u0 st Assume: u ( t ) = e & u + u 0 u ( t ) = e t u 0 cos( D t ) + 0 sin( D t ) D Solution: = c c = 2m cc D = 1 2 SDOF Dynamics 3 - 21 Instructional Material Complementing FEMA 451, Design Examples This slide shows the equation of motion and the response in damped free vibration. Note the similarity with the undamped solution. In particular, note the exponential decay term that serves as a multiplier on the whole response. Critical damping (cc) is defined as the amount of damping that will produce no oscillation. See next slide. The damped circular frequency is computed as shown. Note that in many practical cases (x < 0.10), it will be effectively the same as the undamped frequency. The exception is very highly damped systems. Note that the damping ratio is often given in terms of % critical. FEMA 451B Topic 3 Notes Slide 21 Damping in Structures = c c = 2m cc cc is the critical damping constant. is expressed as a ratio (0.0 < < 1.0) in computations. Sometimes is expressed as a% (0 < < 100%). Displacement, in Time, sec Response of Critically Damped System, =1.0 or 100% critical Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 22 The concept of critical damping is defined here. A good example of a critically damped response can be found in heavy doors that are fitted with dampers to keep the door from slamming when closing. FEMA 451B Topic 3 Notes Slide 22 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. Spring Force, kips 30.00 Damping Force, Kips 4.00 50.00 Inertial Force, kips 15.00 2.00 25.00 0.00 0.00 0.00 -15.00 -2.00 -25.00 -30.00 -0.60 -0.30 0.00 0.30 0.60 -4.00 -20.00 -10.00 0.00 10.00 20.00 -50.00 -500 -250 0 250 2 500 Displacement, inches Velocity, In/sec Acceleration, in/sec Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 23 An earlier slide is repeated here to emphasize that damping in real structures is NOT viscous. It is frictional or hysteretic. Viscous damping is used simply because it linearizes the equations of motion. Use of viscous damping is acceptable for the modeling of inherent damping but should be used with extreme caution when representing added damping or energy loss associated with yielding in the primary structural system. FEMA 451B Topic 3 Notes Slide 23 Damped Free Vibration (2) Displacement, inches 3 2 1 0 -1 -2 -3 0.0 0.5 1.0 Time, seconds 1.5 2.0 0% Damping 10% Damping 20% Damping Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 24 This slide shows some simple damped free vibration responses. When the damping is zero, the vibration goes on forever. When the damping is 20% critical, very few cycles are required for the free vibration to be effectively damped out. For 10% damping, peak is approximately of the amplitude of the previous peak. FEMA 451B Topic 3 Notes Slide 24 Damping in Structures (2) Welded steel frame Bolted steel frame Uncracked prestressed concrete Uncracked reinforced concrete Cracked reinforced concrete Glued plywood shear wall Nailed plywood shear wall Damaged steel structure Damaged concrete structure Structure with added damping Instructional Material Complementing FEMA 451, Design Examples = 0.010 = 0.020 = 0.015 = 0.020 = 0.035 = 0.100 = 0.150 = 0.050 = 0.075 = 0.250 SDOF Dynamics 3 - 25 Some realistic damping values are listed for structures comprised of different materials. The values for undamaged steel and concrete (upper five lines of table) may be considered as working stress values. FEMA 451B Topic 3 Notes Slide 25 Damping in Structures (3) Inherent damping is a structural (material) property independent of mass and stiffness Inherent = 0.5 to 7.0% critical Added damping C is a structural property dependent on mass and stiffness and damping constant C of device Added = 10 to 30% critical Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 26 The distinction between inherent damping and added damping should be clearly understood. FEMA 451B Topic 3 Notes Slide 26 Measuring Damping from Free Vibration Test 1 For all damping values u1 0.5 Amplitude ln u2 u3 u1 = u2 2 1 2 0 -0.5 u0 e t 0.50 1.00 1.50 Time, Seconds 2.00 2.50 3.00 For very low damping values -1 0.00 u1 u 2 2 u2 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 27 One of the simplest methods to measure damping is a free vibration test. The structure is subjected to an initial displacement and is suddenly released. Damping is determined from the formulas given. The second formula should be used only when the damping is expect to be less than about 10% critical. FEMA 451B Topic 3 Notes Slide 27 Undamped Harmonic Loading Equation of motion: & m u&( t ) + k u ( t ) = p 0 sin( t ) = frequency of the forcing function T = 150 100 50 0 -50 -100 -150 0.00 2 T = 0.25 sec po=100 kips Force, Kips 0.25 0.50 0.75 1.00 Time, Seconds 1.25 1.50 1.75 2.00 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 28 The next series of slides covers the response of undamped SDOF systems to simple harmonic loading. Note that the loading frequency is given by the omega term with the overbar. The loading period is designated in a similar fashion. FEMA 451B Topic 3 Notes Slide 28 Undamped Harmonic Loading (2) && Equation of motion: m u ( t ) + k u ( t ) = p 0 s in ( t ) Assume system is initially at rest: Particular solution: u ( t ) = C s in ( t ) Complimentary solution: u( t ) = A sin(t ) + B cos(t ) Solution: u (t ) = p0 1 k 1 ( / ) 2 sin( t ) sin( t ) SDOF Dynamics 3 - 29 Instructional Material Complementing FEMA 451, Design Examples This slide sets up the equation of motion for undamped harmonic loading and gives the solution. We have assumed the system is initially at rest. FEMA 451B Topic 3 Notes Slide 29 Undamped Harmonic Loading Define = Loading frequency Structures natural frequency Transient response (at structures frequency) Dynamic magnifier u( t ) = 1 p0 (sin( t ) sin( t ) ) k 1 2 Steady state response (at loading frequency) SDOF Dynamics 3 - 30 Static displacement Instructional Material Complementing FEMA 451, Design Examples Here we break up the response into the steady state response (at the frequency of loading) and the transient response (at the structures own natural frequency). Note that the term po/k is the static displacement. The dynamic magnifier shows how the dynamic effects may increase (or decrease) the response. This magnifier is a function of the frequency ratio . Note that the magnifier goes to infinity if the frequency ratio is 1.0. This defines the resonant condition. In other words, the response is equal to the static response, times a multiplier, times the sum of two sine waves, one in phase with the load and the other in phase with the structures undamped natural frequency. FEMA 451B Topic 3 Notes Slide 30 = 4 rad / sec = 2 rad / sec Loading (kips) 200 100 0 -100 -200 0.00 = 0.5 uS = 5.0 in. 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Steady state response (in.) 10 5 0 -5 -10 0.00 10 5 0 -5 -10 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Transient response (in.) 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Total response (in.) 10 5 0 -5 -10 0.00 sp ace e t, 0.25 0.50 0.75 1.00 Time, seconds 1.25 1.50 1.75 2.00 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 31 This is a time-history response of a structure with a natural frequency of 4 rad/sec (f = 2 Hz, T = 0.5 sec), and a loading frequency of 2 rad/sec (f = 1 Hz, T = 1 sec), giving a frequency ratio of 0.5. The harmonic load amplitude is 100 kips. The static displacement is 5.0 inches. Note how the steady state response is at the frequency of loading, is in phase with the loading, and has an amplitude greater than the static displacement. The transient response is at the structures own frequency. In real structures, damping would cause this component to disappear after a few cycles of vibration. FEMA 451B Topic 3 Notes Slide 31 4 rad / sec = 4 rad / sec Loading (kips) 150 100 50 0 -5 0 -1 0 0 -1 5 0 0 .0 0 = 0.99 u S = 5.0 in. 0 .2 5 0 .5 0 0 .7 5 1 .0 0 1 .2 5 1 .5 0 1 .7 5 2 .0 0 5 00 Steady state response (in.) 2 50 0 -2 50 -5 00 0 .0 0 0.25 0.50 0 .7 5 1.00 1.25 1 .5 0 1.7 5 2.00 500 250 Transient response (in.) 0 -250 -500 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 80 , p 40 0 -40 -80 0.00 Total response (in.) 0.25 0.50 0.75 1.00 T ime, seconds 1.25 1.50 1.75 2.00 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 32 In this slide, has been increased to 4 rad/sec, and the structure is almost at resonance. The steady state response is still in phase with the loading, but note the huge magnification in response. The transient response is practically equal to and opposite the steady state response. The total response increases with time. If one looks casually at the steady state and transient response curves, it appears that they should cancel out. Note, however, that the two responses are not exactly in phase due to the slight difference in the loading and natural frequencies. This can be seen most clearly at the time 1.75 sec into the response. The steady state response crosses the horizontal axis to the right of the vertical 1.75 sec line while the transient response crosses exactly at 1.75 sec. In real structures, the observed increased amplitude could occur only to some limit and then yielding would occur. This yielding would introduce hysteretic energy dissipation (apparent damping), causing the transient response to disappear and leading to a constant, damped, steady state response. FEMA 451B Topic 3 Notes Slide 32 Undamped Resonant Response Curve 80 40 Displacement, in. 2 uS 0 -40 Linear envelope -80 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Time, seconds Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 33 This is an enlarged view of the total response curve from the previous slide. Note that the response is bounded within a linear increasing envelope with the increase in displacement per cycle being 2 times the static displacement. FEMA 451B Topic 3 Notes Slide 33 4 rad / sec = 4 rad / sec Loading (kips) 150 100 50 0 -5 0 -1 0 0 -1 5 0 0 .0 0 500 = 1.01 u S = 5.0 in. 0 .2 5 0 .5 0 0 .7 5 1 .0 0 1 .2 5 1 .5 0 1 .7 5 2 .0 0 Steady state response (in.) 250 0 -2 5 0 -5 0 0 0 .0 0 0 .2 5 0 .5 0 0 .7 5 1 .0 0 1 .2 5 1 .5 0 1 .7 5 2 .0 0 500 Transient response (in.) 250 0 -2 5 0 -5 0 0 0 .0 0 80 , 0 .2 5 0 .5 0 0 .7 5 1 .0 0 1 .2 5 1 .5 0 1 .7 5 2 .0 0 p Total response (in.) 40 0 -4 0 -8 0 0 .0 0 0 .2 5 0 .5 0 0 .7 5 1 .0 0 T im e , s e c o n d s 1 .2 5 1 .5 0 1 .7 5 2 .0 0 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 34 In this slide, the loading frequency has been slightly increased, but the structure is still nearly at resonance. Note, however, that the steady state response is 180 degrees out of phase with the loading and the transient response is in phase. The resulting total displacement is effectively identical to that shown two slides back. FEMA 451B Topic 3 Notes Slide 34 = 4 rad / sec Loading (kips) 150 100 50 0 -50 -100 -150 0.00 6 = 8 rad / sec = 2.0 uS = 5.0 in. 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Steady state response (in.) 3 0 -3 -6 0 .0 0 0 .2 5 0 .5 0 0 .7 5 1 .0 0 1 .2 5 1 .5 0 1 .7 5 2 .0 0 6 3 Transient response (in.) 0 -3 -6 0 .0 0 6 , 3 0 p -3 -6 0 .0 0 0 .2 5 0 .5 0 0 .7 5 1 .0 0 1 .2 5 1 .5 0 1 .7 5 2 .0 0 Total response (in.) 0 .2 5 0 .5 0 0 .7 5 1 .0 0 T im e , s e c o n d s 1 .2 5 1 .5 0 1 .7 5 2 .0 0 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 35 The loading frequency is now twice the structures frequency. The important point here is that the steady state response amplitude is now less than the static displacement. FEMA 451B Topic 3 Notes Slide 35 Response Ratio: Steady State to Static (Signs Retained) 12.00 Magnification Factor 1/(1- 2) 8.00 In phase 4.00 Resonance 0.00 -4.00 -8.00 180 degrees out of phase 0.50 1.00 1.50 2.00 2.50 3.00 -12.00 0.00 Frequency Ratio Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 36 This plot shows the ratio of the steady state response to the static displacement for the structure loaded at different frequencies. At low loading frequencies, the ratio is 1.0, indicating a nearly static response (as expected). At very high frequency loading, the structure effectively does not have time to respond to the loading so the displacement is small and approaches zero at very high frequency. The resonance phenomena is very clearly shown. The change in sign at resonance is associated with the inphase/out-of-phase behavior that occurs through resonance. FEMA 451B Topic 3 Notes Slide 36 Response Ratio: Steady State to Static (Absolute Values) 12.00 Resonance Magnification Factor 1/(1- 2) 10.00 8.00 6.00 4.00 Slowly loaded 2.00 1.00 0.00 0.00 0.50 1.00 1.50 Frequency Ratio 2.00 Rapidly loaded 2.50 3.00 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 37 This is the same as the previous slide but absolute values are plotted. This clearly shows the resonance phenomena. FEMA 451B Topic 3 Notes Slide 37 Damped Harmonic Loading Equation of motion: && & m u ( t ) + cu ( t ) + k u ( t ) = p 0 sin( t ) T = 150 100 50 0 -50 -100 -150 0.00 2 = 0.25 sec po=100 kips Force, Kips 0.25 0.50 0.75 1.00 Time, Seconds 1.25 1.50 1.75 2.00 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 38 We now introduce damping into the behavior. Note the addition of the appropriate term in the equation of motion. FEMA 451B Topic 3 Notes Slide 38 Damped Harmonic Loading && & m u ( t ) + cu ( t ) + k u ( t ) = p 0 sin( t ) Assume system is initially at rest Particular solution: Equation of motion: u ( t ) = C sin( t ) + D cos( t ) Complimentary solution: u ( t ) = e t [ A sin( D t ) + B cos( D t ) ] Solution: = c 2 m u ( t ) = e t [ A sin( D t ) + B cos( D t ) ] + C sin( t ) + D cos( t ) Instructional Material Complementing FEMA 451, Design Examples D = 1 2 SDOF Dynamics 3 - 39 This slide shows how the solution to the differential equation is obtained. The transient response (as indicated by the A and B coefficients) will damp out and is excluded from further discussion. FEMA 451B Topic 3 Notes Slide 39 Damped Harmonic Loading Transient response at structures frequency (eventually damps out) u ( t ) = e t [ A sin( D t ) + B cos( D t ) ] + C sin(t ) + D cos(t ) Steady state response, at loading frequency C= po 1 2 k (1 2 ) 2 + (2 ) 2 D= po 2 k (1 2 ) 2 + (2 ) 2 SDOF Dynamics 3 - 40 Instructional Material Complementing FEMA 451, Design Examples This slide shows the C and D coefficients of the steady state response. Note that there is a component in phase with the loading (the sine term) and a component out of phase with the loading (the cosine term). The actual phase difference between the loading and the response depends on the damping and frequency ratios. Note the exponential decay term causes the transient response to damp out in time. FEMA 451B Topic 3 Notes Slide 40 Damped Harmonic Loading (5% Damping) BETA=1 (Resonance) Beta=0.5 Beta=2.0 50 Displacement Amplitude, Inches 40 30 20 10 0 -10 -20 -30 -40 -50 0.00 1.00 2.00 3.00 4.00 5.00 Time, Seconds Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 41 This plot shows the response of a structure at three different loading frequencies. Of significant interest is the resonant response, which is now limited. (The undamped response increases indefinitely.) FEMA 451B Topic 3 Notes Slide 41 Damped Harmonic Loading (5% Damping) 50 40 Displacement Amplitude, Inches 30 20 10 0 -10 -20 -30 -40 -50 0.00 1 Static 2 1.00 2.00 3.00 4.00 5.00 Time, Seconds Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 42 For viscously damped structures, the resonance amplitude will always be limited as shown. FEMA 451B Topic 3 Notes Slide 42 Harmonic Loading at Resonance Effects of Damping 200 Displacement Amplitude, Inches 150 100 50 0 -50 -100 -150 -200 0.00 1.00 2.00 3.00 4.00 5.00 Time, Seconds 0% Damping %5 Damping Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 43 A comparison of damped and undamped responses is shown here. The undamped response has a linear increasing envelope; the damped curve will reach a constant steady state response after a few cycles. FEMA 451B Topic 3 Notes Slide 43 14.00 Resonance 12.00 Dynamic Response Amplifier 0.0% Damping 5.0 % Damping 10.0% Damping 25.0 % Damping 10.00 8.00 RD = 1 (1 2 ) 2 + ( 2 ) 2 6.00 4.00 2.00 Slowly loaded Rapidly loaded 1.00 1.50 2.00 2.50 3.00 0.00 0.00 0.50 Frequency Ratio, Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 44 This plot shows the dynamic magnification for various damping ratios. For increased damping, the resonant response decreases significantly. Note that for slowly loaded structures, the dynamic amplification is 1.0 (effectively static). For high frequency loading, the magnifier is zero. Note also that damping is most effective at or near resonance (0.5 < < 2.0). FEMA 451B Topic 3 Notes Slide 44 Summary Regarding Viscous Damping in Harmonically Loaded Systems For systems loaded at a frequency near their 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. Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 45 A summary of some of the previous points is provided. FEMA 451B Topic 3 Notes Slide 45 Summary Regarding Viscous Damping in 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 (1/2) times the static displacement. Damping is most effective for systems loaded at or near resonance. Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 46 Summary continued. FEMA 451B Topic 3 Notes Slide 46 CONCEPT of ENERGY STORED and Energy DISSIPATED F Energy Stored F u 1 Energy Dissipated 2 1 u YIELDING Total Energy Dissipated LOADING Energy Recovered F 2 u UNLOADING F 3 u 4 3 UNLOADED Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 47 It is very important that the distinction between stored energy and dissipated energy be made clear. (Note that some texts use the term absorbed energy in lieu of stored energy.) In the first diagram, the system remains elastic and all of the strain energy is stored. If the bar were released, all of the energy would be recovered. In the second diagram, the applied deformation is greater than the elastic deformation and, hence, the system yields. The energy shown in green is stored, but the energy shown in red is dissipated. If the bar is unloaded, the stored energy is recovered, but the dissipated energy is lost. This is shown in Diagrams 3 and 4. FEMA 451B Topic 3 Notes Slide 47 General Dynamic Loading F(t) Time, T Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 48 The discussion will now proceed to general dynamic loading. By general loading, it is meant that no simple mathematical function defines the entire loading history. FEMA 451B Topic 3 Notes Slide 48 General Dynamic Loading Solution Techniques Fourier transform Duhamel integration Piecewise exact Newmark techniques All techniques are carried out numerically. Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 49 There are a variety of ways to solve the general loading problem and all are carried out numerically on the computer. The Fourier transform and Duhamel integral approaches are not particularly efficient (or easy to explain) and, hence, these are not covered here. Any text on structural dynamics will provide the required details. The piecewise exact method is used primarily in the analysis of linear systems. The Newmark method is useful for both linear and nonlinear systems. Only the basic principles underlying of each of these approaches are presented. FEMA 451B Topic 3 Notes Slide 49 Piecewise Exact Method F ( ) = Fo + Fo dF dt dF dt dt Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 50 In the piecewise exact method, the loading function is broken into a number of straight-line segments. In a sense, the name of the method is a misnomer because the method is not exact when the actual loading is smooth (like a sine wave) because the straight line load segments are only an approximation of the actual load. When the actual load is smooth, the accuracy of the method depends on the level of discretization when defining the loading function. For earthquake loads, the load almost is always represented by a recorded accelerogram, which does consist of straight line segments. (There would be little use in trying to interpolate the ground motion with smooth curves.) Hence, for the earthquake problem, the piecewise exact method is truly exact. FEMA 451B Topic 3 Notes Slide 50 Piecewise Exact Method Initial conditions & uo ,0 = 0 uo ,0 = 0 Determine exact solution for 1st time step u1 = u ( ) & & u1 = u ( ) && && u1 = u ( ) Establish new initial conditions u o ,1 = u ( ) u 2 = u ( ) & & u 0 ,1 = u ( ) LOOP Obtain exact solution for next time step & & u 2 = u ( ) && && u 2 = u ( ) SDOF Dynamics 3 - 51 Instructional Material Complementing FEMA 451, Design Examples The basic idea of the piecewise exact method is to develop a solution for a straight line loading segment knowing the initial conditions. Given the initial conditions and the load segment, the solution at the end of the load step is determined and this is then used as the initial condition for the next step of the analysis. The analysis then proceeds step by step until all load segments have been processed. FEMA 451B Topic 3 Notes Slide 51 Piecewise Exact Method Advantages: Exact if load increment is linear Very computationally efficient Disadvantages: Not generally applicable for inelastic behavior Note: NONLIN uses the piecewise exact method for response spectrum calculations. Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 52 It should be noted that the piecewise exact method may be used for nonlinear analysis in certain circumstances. For example, the fast nonlinear analysis (FNA) method developed by Ed Wilson and used in SAP 2000 utilizes the piecewise exact method. In FNA, the nonlinearities are right-hand sided, leaving only linear terms in the left-hand side of the equations of motion. FEMA 451B Topic 3 Notes Slide 52 Newmark Techniques Proposed by Nathan Newmark General method that encompasses a family of different integration schemes Derived by: Development of incremental equations of motion Assuming acceleration response over short time step Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 53 The Newmark method is one of the most popular methods for solving the general dynamic loading problem. It is applicable to both linear and nonlinear systems. It is equally applicable to both SDOF and MDOF systems. The Newmark method is described in more detail in the topic on inelastic behavior of structures. FEMA 451B Topic 3 Notes Slide 53 Newmark Method Advantages: Works for inelastic response Disadvantages: Potential numerical error Note: NONLIN uses the Newmark method for general response history calculations Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 54 The advantages and disadvantages of the Newmark method are listed. The principal advantage is that the method may be applied to inelastic systems. The method also may be used (without decoupling) for multiple-degree-offreedom systems. FEMA 451B Topic 3 Notes Slide 54 Development of Effective Earthquake Force 0.40 GROUND ACC, g 0.20 0.00 -0.20 -0.40 0.00 1.00 2.00 3.00 TIME, SECONDS 4.00 5.00 6.00 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 55 In an earthquake, no actual force is applied to the building. Instead, the ground moves back and forth (and up and down) and this movement induces inertial forces that then deform the structure. It is the displacements in the structure, relative to the moving base, that impose deformations on the structure. Through the elastic properties, these deformations cause elastic forces to develop in the individual members and connections. FEMA 451B Topic 3 Notes Slide 55 Earthquake Ground Motion, 1940 El Centro 0.4 Ground Acceleration (g's) 0.3 0.2 0.1 40 0 -0.1 -0.2 -0.3 0 10 20 30 Time (sec) 15 Ground Velocity (cm/sec) 30 20 10 0 -10 -20 -30 0 10 20 30 Time (sec) 40 50 60 40 50 60 Ground Displacement (cm) 10 5 0 -5 -10 -15 0 10 20 30 Time (sec) 40 50 60 Many ground motions now are available via the Internet. SDOF Dynamics 3 - 56 Instructional Material Complementing FEMA 451, Design Examples Earthquake ground motions usually are imposed through the use of the ground acceleration record or accelerogram. Some programs (like Abaqus) may require instead that the ground displacement records be used as input. FEMA 451B Topic 3 Notes Slide 56 Development of Effective Earthquake Force && ug && ut && ur Ground Acceleration Response History 0.40 GROUND ACC, g 0.20 0.00 -0.20 -0.40 0.00 1.00 2.00 3.00 TIME, SECO D NS 4.00 5.00 6.00 && && & m[ug ( t ) + ur ( t )] + c ur ( t ) + k ur ( t ) = 0 && & && mur ( t ) + c ur ( t ) + k ur ( t ) = mug ( t ) Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 57 In this slide, it is assumed that the ground acceleration record is used as input. The total acceleration at the center of mass is equal to the ground acceleration plus the acceleration of the center of mass relative to the moving base. The inertial force developed at the center of mass is equal to the mass times the total acceleration. The damping force in the system is a function of the velocity of the top of the structure relative to the moving base. Similarly, the spring force is a function of the displacement at the top of the structure relative to the moving base. The equilibrium equation with the zero on the response history spectrum (RHS) represents the state of the system at any point in time. The zero on the RHS reflects the fact that there is no applied load. If that part of the total inertial force due to the ground acceleration is moved to the right-hand side (the lower equation), all of the forces on the left-hand side are in terms of the relative acceleration, velocity, and displacement. This equation is essentially the same as that for an applied load (see Slide 8) but the effective earthquake force is simply the negative of the mass times the ground acceleration. The equation is then solved for the response history of the relative displacement. FEMA 451B Topic 3 Notes Slide 57 Simplified form of Equation of Motion: && & && mur (t ) + cur (t ) + kur (t ) = mu g (t ) Divide through by m: && ur (t ) + k c & && ur (t ) + ur (t ) = u g (t ) m m k =2 m Make substitutions: c = 2 m Simplified form: && & && ur (t ) + 2 ur (t ) + 2ur (t ) = u g (t ) Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 58 In preparation for the development of response spectra, it is convenient to simplify the equation of motion by dividing through by the mass. When the substitutions are made as indicated, it may be seen that the response is uniquely defined by the damping ratio, the undamped circular frequency of vibration, and the ground acceleration record. FEMA 451B Topic 3 Notes Slide 58 For a given ground motion, the response history ur(t) is function of the structures frequency and damping ratio . Structural frequency & & & u&r (t ) + 2 u r (t ) + 2 u r (t ) = u&g (t ) Damping ratio Ground motion acceleration history Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 59 This restates the point made in the previous slide. A response spectrum is created for a particular ground motion and for a structure with a constant level of damping. The spectrum is obtained by repeatedly solving the equilibrium equations for structures with varying frequencies of vibration and then plotting the peak displacement obtained for that frequency versus the frequency for which the displacement was obtained. FEMA 451B Topic 3 Notes Slide 59 Response to Ground Motion (1940 El Centro) 0.4 Ground Acceleration (g's) 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 0 10 20 30 Time (sec) 40 50 60 Excitation applied to structure with given and SOLVER 6 Structural Displacement (in) 4 2 0 -2 -4 Computed response Change in ground motion or structural parameters and requires recalculation of structural response Peak displacement -6 0 10 20 30 40 50 60 Time (sec) Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 60 The next several slides treat the development of the 5% damped response spectrum for the 1940 El Centro ground motion record. The solver indicated in the slide is a routine, such as the Newmark method, that takes the ground motion record, the damping ratio, and the system frequency as input and reports as output only the maximum absolute value of the relative displacement that occurred over the duration of the ground motion. It is important to note that by taking the absolute value, the sign of the peak response is lost. The time at which the peak response occurred is also lost (simply because it is not recorded). FEMA 451B Topic 3 Notes Slide 60 The Elastic Displacement Response Spectrum An elastic displacement response spectrum is a plot of the peak computed relative displacement, ur, for an elastic structure with a constant damping , a varying fundamental frequency (or period T = 2/ ), responding to a given ground motion. 5% damped response spectrum for structure responding to 1940 El Centro ground motion 16 DISPLACEMENT, inches 12 8 4 0 0 2 4 6 8 10 PERIOD, Seconds Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 61 This slide is a restatement of the previous point. FEMA 451B Topic 3 Notes Slide 61 Computation of Response Spectrum for El Centro Ground Motion 0.08 Displacement, Inches 0.06 0.04 0.02 0.00 -0.02 -0.04 -0.06 -0.08 Computed response 0 1 2 3 4 5 6 Time, Seconds 7 8 9 10 11 12 10.00 Elastic response spectrum 8.00 Displacement, Inches = 0.05 T = 0.10 sec Umax= 0.0543 in. 6.00 4.00 2.00 0.00 0.00 0.50 1.00 Period, Seconds 1.50 2.00 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 62 Here, the first point in the response spectrum is computed. For this and all subsequent steps, the ground motion record is the same and the damping ratio is set as 5% critical. Only the frequency of vibration, represented by period T, is changed. When T = 0.10 sec (circular frequency = 62.8 radians/sec), the peak computed relative displacement was 0.0543 inches. The response history from which the peak was obtained is shown at the top of the slide. This peak occurred at about 5 sec into the response, but this time is not recorded. Note the high frequency content of the response. The first point on the displacement response spectrum is simply the displacement (0.0543 inches) plotted against the structural period (0.1 sec) for which the displacement was obtained. FEMA 451B Topic 3 Notes Slide 62 Computation of Response Spectrum for El Centro Ground Motion 0.40 Displacement, Inches 0.30 0.20 0.10 0.00 -0.10 -0.20 -0.30 -0.40 Computed response 0 1 2 3 4 5 6 Time, Seconds 7 8 9 10 11 12 10.00 Elastic response spectrum 8.00 = 0.05 T = 0.20 sec Umax = 0.254 in. Displacement, Inches 6.00 4.00 2.00 0.00 0.00 0.50 1.00 Period, Seconds 1.50 2.00 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 63 Here the whole procedure is repeated, but the system period is changed to 0.2 sec. The computed displacement history is shown at the top of the slide, which shows that the peak displacement was 0.254 inches. This peak occurred at about 2.5 sec into the response but, as before, this time is not recorded. Note that the response history is somewhat smoother than that in the previous slide. The second point on the response spectrum is the peak displacement (0.254 inch) plotted against the system period, which was 0.2 sec. FEMA 451B Topic 3 Notes Slide 63 Computation of Response Spectrum for El Centro Ground Motion 0.80 Displacement, Inches 0.60 0.40 0.20 0.00 -0.20 -0.40 -0.60 -0.80 Computed response 0 1 2 3 4 5 6 Time, Seconds 7 8 9 10 11 12 10.00 Elastic response spectrum 8.00 = 0.05 T = 0.30 sec Umax = 0.622 in. Displacement, Inches 6.00 4.00 2.00 0.00 0.00 0.50 1.00 Period, Seconds 1.50 2.00 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 64 The third point on the response spectrum is the peak displacement (0.622 inch) plotted against the system period, which was 0.3 sec. Again, the response is somewhat smoother than before. FEMA 451B Topic 3 Notes Slide 64 Computation of Response Spectrum for El Centro Ground Motion 1.20 Displacement, Inches 0.90 0.60 0.30 0.00 -0.30 -0.60 -0.90 -1.20 Computed response 0 1 2 3 4 5 6 Time, Seconds 7 8 9 10 11 12 10.00 Elastic response spectrum 8.00 = 0.05 T = 0.40 sec Umax = 0.956 in. Displacement, Inches 6.00 4.00 2.00 0.00 0.00 0.50 1.00 Period, Seconds 1.50 2.00 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 65 The fourth point on the response spectrum is the peak displacement (0.956 inch) plotted against the system period, which was 0.40 sec. FEMA 451B Topic 3 Notes Slide 65 Computation of Response Spectrum for El Centro Ground Motion 2.40 Displacement, Inches 1.80 1.20 0.60 0.00 -0.60 -1.20 -1.80 -2.40 Computed response 0 1 2 3 4 5 6 Time, Seconds 7 8 9 10 11 12 10.00 Elastic response spectrum 8.00 = 0.05 T = 0.50 sec Umax = 2.02 in. Displacement, Inches 6.00 4.00 2.00 0.00 0.00 0.50 1.00 Period, Seconds 1.50 2.00 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 66 The next point on the response spectrum is the peak displacement (2.02 inches) plotted against the system period, which was 0.50 sec. FEMA 451B Topic 3 Notes Slide 66 Computation of Response Spectrum for El Centro Ground Motion Displacement, Inches 3.20 2.40 1.60 0.80 0.00 -0.80 -1.60 -2.40 -3.20 0 1 2 3 4 5 6 Tim Seconds e, 10.00 7 8 9 10 11 12 Computed response Elastic response spectrum 8.00 = 0.05 T = 0.60 sec Umax= -3.00 in. Displacement, Inches 6.00 4.00 2.00 0.00 0.00 0.50 1.00 Period, Seconds 1.50 2.00 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 67 The next point on the response spectrum is the peak displacement (3.03 inches) plotted against the system period, which was 0.60 sec. Note that only the absolute value of the displacement is recorded. The complete spectrum is obtained by repeating the process for all remaining periods in the range of 0.7 through 2.0 sec. For this response spectrum, 2/0.1 or 20 individual points are calculated, requiring 20 full response history analyses. A real response spectrum would likely be run at a period resolution of about 0.01 sec, requiring 200 response history analyses. FEMA 451B Topic 3 Notes Slide 67 Complete 5% Damped Elastic Displacement Response Spectrum for El Centro Ground Motion 12.00 10.00 Displacement, Inches 8.00 6.00 4.00 2.00 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Period, Seconds Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 68 This is the full 5% damped elastic displacement response spectrum for the 1940 El Centro ground motion. Note that the spectrum was run for periods up to 4.0 sec. This spectrum was generated using NONLIN. Note also that the displacement is nearly zero when T is near zero. This is expected because the relative displacement of a very stiff structure (with T near zero) should be very small. The displacement then generally increases with period, although this trend is not consistent. The reductions in displacement at certain periods indicate that the ground motion has little energy at these periods. As shown later, a different earthquake will have an entirely different response spectrum. FEMA 451B Topic 3 Notes Slide 68 Development of Pseudovelocity Response Spectrum 35.00 5% damping 30.00 25.00 20.00 15.00 10.00 5.00 0.00 0.0 1.0 2.0 Period, Seconds Pseudovelocity, in/sec PSV (T ) D 3.0 4.0 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 69 If desired, an elastic (relative) velocity response spectrum could be obtained in the same way as the displacement spectrum. The only difference in the procedure would be that the peak velocity computed at each period would be recorded and plotted. Instead of doing this, the velocity spectrum is obtained in an approximate manner by assuming that the displacement response is harmonic and, hence, that the velocity at each (circular) frequency is equal to the frequency times the displacement. This comes from the rules for differentiating a harmonic function. Because the velocity spectrum so obtained is not exact, it is called the pseudovelocity response spectrum. Note that it appears that the pseudovelocity at low (near zero) periods is also near zero (but not exactly zero). FEMA 451B Topic 3 Notes Slide 69 Development of Pseudoacceleration Response Spectrum 400.0 350.0 300.0 250.0 200.0 150.0 100.0 50.0 0.0 0.0 1.0 2.0 Period, Seconds 5% damping Pseudoacceleration, in/sec 2 PSA (T ) 2 D 3.0 4.0 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 70 The pseudoacceleration spectrum is obtained from the displacement spectrum by multiplying by the circular frequencies squared. Note that the acceleration at a near zero period is not near zero (as was the case for velocity and displacement). In fact, the pseudoacceleration represents the total acceleration in the system while the pseudovelocity and the displacement are relative quantities. FEMA 451B Topic 3 Notes Slide 70 Note About the Pseudoacceleration 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. 400.0 350.0 300.0 250.0 200.0 150.0 100.0 50.0 0.0 0.0 1.0 2.0 Period, Seconds 5% damping Peak ground acceleration Pseudoacceleration, in/sec 2 3.0 4.0 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 71 For very rigid systems (with near zero periods of vibration), the relative acceleration will be nearly zero and, hence, the pseudoacceleration, which is the total acceleration, will be equal to the peak ground acceleration. FEMA 451B Topic 3 Notes Slide 71 && ug && ut PSA is TOTAL Acceleration! && ur Ground Acceleration Response History 0.40 GROUND ACC, g 0.20 0.00 -0.20 -0.40 0.00 1.00 2.00 3.00 TIME, SECO D NS 4.00 5.00 6.00 && && & m[ug ( t ) + ur ( t )] + c ur ( t ) + k ur ( t ) = 0 && & && mur ( t ) + c ur ( t ) + k ur ( t ) = mug ( t ) Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 72 This slide explains why the pseudoacceleration is equal to the total acceleration. The relative displacement is multiplied by omega to get pseudovelocity. The pseudovelocity then is multiplied by omega to get the total acceleration. FEMA 451B Topic 3 Notes Slide 72 Difference Between Pseudo-Acceleration and Total Acceleration (System with 5% Damping) 350.00 Acceleration (in/sec ) 300.00 250.00 200.00 150.00 100.00 50.00 0.00 0.1 1 Period (sec) Total Acceleration Pseudo-Acceleration 10 2 Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 73 This plot shows total acceleration and pseudoacceleration for a 5% damped system subject to the El Centro ground motion. Note the similarity in the two quantities. The difference in the two quantities is only apparent at low periods. The difference can be much greater when the damping is set to 10%, 20%, or 30% critical, and the differences can appear in a wider range of periods. FEMA 451B Topic 3 Notes Slide 73 Difference Between Pseudovelocity and Relative Velocity (System with 5% Damping) 40 35 Ve lo city (in/se c) 30 25 20 15 10 5 0 0.1 1 Period (sec) Relative Velocity Pseudo-Velocity SDOF Dynamics 3 - 74 10 Instructional Material Complementing FEMA 451, Design Examples This plot shows relative velocity and pseudovelocity for a 5% damped system subject to the El Centro ground motion. Here, the differences are much more apparent than for pseudoacceleration, and the larger differences occur at the higher periods. The differences will be greater for systems with larger amounts of damping. FEMA 451B Topic 3 Notes Slide 74 Displacement Response Spectra for Different Damping Values 25.00 Displacement, Inches Damping 0% 5% 10% 20% 20.00 15.00 10.00 5.00 0.00 0.0 1.0 2.0 3.0 4.0 5.0 Period, Seconds Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 75 The higher the damping, the lower the relative displacement. At a period of 2 sec, for example, going from zero to 5% damping reduces the displacement amplitude by a factor of two. While higher damping produces further decreases in displacement, there is a diminishing return. The % reduction in displacement by going from 5 to 10% damping is much less that that for 0 to 5% damping. FEMA 451B Topic 3 Notes Slide 75 Pseudoacceleration Response Spectra for Different Damping Values Damping 4.00 Pseudoacceleration, g 0% 5% 10% 20% 3.00 2.00 1.00 0.00 Peak ground acceleration 0.0 1.0 2.0 3.0 4.0 5.0 Period, Seconds Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 76 Damping has a similar effect on pseudoacceleration. Note, however, that the pseudoacceleration at a (near) zero period is the same for all damping values. This value is always equal to the peak ground acceleration for the ground motion in question. FEMA 451B Topic 3 Notes Slide 76 Damping Is Effective in Reducing the Response for (Almost) Any Given Period of Vibration An earthquake record can be considered to be the combination of a large number of harmonic components. Any SDOF structure will be in near resonance with one of these harmonic components. Damping is most effective at or near resonance. Hence, a response spectrum will show reductions due to damping at all period ranges (except T = 0). Instructional Material Complementing FEMA 451, Design Examples SDOF Dynamics 3 - 77 Damping is generally effective at all periods (except at T = 0). The reason for this is that ground motions consist of a large number of harmonics, each at a different frequency. When a response spectrum analysis is run for a particular period, there will be a near resonant response at that period. Damping is most effective at resonance and, hence, damping will be effective over the full range of periods for which the response spectrum is generated. FEMA 451B Topic 3 Notes Slide 77 Damping Is Effective in Reducing the Response for Any Given Period of Vibration Amplitude 4.00 2.00 0.00 -2.00 -4.00 0.0 6.0 ` 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60.0 66.0 72.0 78.0 84.0 90.0 Time (sec) Example of an artificially generated wave to resemble a real time ground motion accelerogram. Generated wave obtained by combin...
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