# An Impulse and Earthquake Energy Balance Approach in Nonlinear Structural Dynamics

Motivation of the proposed approachSimplification of near-fault pulse-type ground motionResonant response in nonlinear structural dynamics and earthquake-resistant designDouble impulse and corresponding one-cycle sine wave with the same frequency and same maximum Fourier amplitudeEnergy balance under earthquake ground motion and impulseUndamped modelDamped modelCritical input timing of second impulse in double impulseComparison of conventional methods and the proposed method for nonlinear resonant analysisOutline of this bookSummariesReferencesCritical earthquake response of an elastic–perfectly plastic SDOF model under double impulse as a representative of near-fault ground motionsIntroductionDouble impulse inputSDOF systemMaximum elastic-plastic deformation of SDOF system to double impulseAccuracy investigation by time-history response analysis to corresponding one-cycle sinusoidal inputDesign of stiffness and strength for specified velocity and period of double impulse and specified response ductilityApplication to recorded ground motionsSummariesReferencesA. Appendix 1: proof of critical timing of second impulseB. Appendix 2: derivation of critical timingCritical earthquake response of an elastic–perfectly plastic SDOF model under triple impulse as a representative of near-fault ground motionsIntroductionTriple impulse inputSDOF systemMaximum elastic-plastic deformation of SDOF system to triple impulseCASE 1CASE 2CASE 3CASE 3-1CASE 3-2CASE 4Accuracy investigation by time-history response analysis to corresponding three wavelets of sinusoidal wavesDesign of stiffness and strength for specified velocity and period of triple impulse and specified response ductilityApproximate prediction of response ductility for specified design of stiffness and strength and specified velocity and period of triple impulseComparison between maximum response to double impulse and that to triple impulseApplication to recorded ground motionsSummariesReferencesA. Appendix 1: Proof of critical timingB. Appendix 2: Upper bound of maximum response via relaxation of timing of third impulseC. Appendix 3: Triple impulse and corresponding 1.5-cycle sine wave with the same frequency and same maximum fourier amplitudeCritical input and response of an elastic–perfectly plastic SDOF model under multi-impulse as a representative of long-duration earthquake ground motionsIntroductionMultiple impulse inputSDOF systemMaximum elastic-plastic deformation of SDOF system to multiple impulseNon-iterative determination of critical timing and critical plastic deformation by using modified input sequenceDetermination of critical timing of impulsesCorrespondence of responses between input sequence 1 (original one) and input sequence 2 (modified one)Accuracy investigation by time-history response analysis to corresponding multi-cycle sinusoidal inputProof of critical timingSummariesReferencesA. Appendix 1: Multi-impulse and correspondingmulti-cycle sine wave with thesame frequency and samemaximum fourier amplitudeCritical earthquake response of an elastic–perfectly plastic SDOF model with viscous damping under double impulseIntroductionModeling of near-fault ground motion with double impulseElastic–perfectly plastic SDOF model with viscous dampingElastic-plastic response of undamped system to critical double impulseLinear elastic response of damped system to critical double impulseElastic-plastic response of damped system to critical double impulseApproximate critical response of the elastic-plastic system with viscous damping based on the energy balance lawCASE 1: Elastic response even after second impulseCASE 2: Plastic deformation only after the second impulseCASE 3: Plastic deformation, even after the first impulseMaximum deformation under the critical double impulse with respect to the input velocity levelAccuracy check by time-history response analysis to one-cycle sinusoidal waveApplicability of proposed theory to actual recorded ground motionSummariesReferencesAppendix 1: Critical impulse timing for linearelastic system with viscousdampingAppendix 2: Velocity at zero restoring force after attaining umax1 in case 3Critical steady-state response of a bilinear hysteretic SDOF model under multi-impulseIntroductionBilinear hysteretic SDOF systemClosed-form expression for elastic-plastic steady-state response to critical multi-impulseCASE 1: Impulse in unloading processCASE 2: Impulse in loading process (second stiffness range)Results in numerical exampleDerivation of critical impulse timingConvergence of critical impulse timingAccuracy check by time-history response analysis to corresponding multi-cycle sinusoidal waveProof of critical timingApplicability of critical multi-impulse timing to corresponding sinusoidal waveAccuracy check by exact solution to corresponding multi-cycle sinusoidal waveSummariesReferencesAppendix 1: Time-history response to criticalmulti-impulse and derivation ofcritical time intervalAppendix 2: Adjustment of input level ofmulti-impulse and correspondingsinusoidal waveCritical earthquake response of an elastic–perfectly plastic SDOF model on compliant ground under double impulseIntroductionDouble impulse inputDouble impulse inputClosed-form critical elastic-plastic response of SDOF system subjected to double impulse (summary of results in Chapter 2)Maximum elastic-plastic deformation of simplified swaying-rocking model to critical double impulseSimplified swaying-rocking modelEquivalent SDOF model of simplified swaying-rocking modelCritical elastic-plastic response of simplified swaying-rocking model subjected to double impulseNumerical exampleApplicability of critical double impulse timing to corresponding sinusoidal waveToward better correspondence between double impulse and sinusoidal inputApplicability to recorded ground motionsSummariesReferencesClosed-form dynamic collapse criterion for a bilinear hysteretic SDOF model under near-fault ground motionsIntroductionDouble impulse inputDouble impulse inputPrevious work on closed-form critical elastic–perfectly plastic response of SDOF system subjected to double impulseMaximum elastic-plastic deformation and stability limit of SDOF system with negative post-yield stiffness to critical double impulsePattern 1: Stability limit after the second impulse without plastic deformation after the first impulsePattern 2: Stability limit after the second impulse with plastic deformation after the first impulsePattern 3: Stability limit after the second impulse with closed-loop in restoring-force characteristicAdditional Pattern 1: Limit after the first impulseAdditional Pattern 2: Limit without plastic deformation after the second impulseResults for numerical exampleDiscussionApplicability of critical double impulse timing to corresponding sinusoidal waveApplicability to recorded ground motionsSummariesReferencesAppendix 1: Maximum elastic-plastic deformationof sdof model with negativepost-yield stiffness to double impulseAppendix 2: Maximum elastic-plasticdeformation of sdof model withpositive post-yield stiffness todouble impulseClosed-form overturning limit of a rigid block as a SDOF model under near-fault ground motionsIntroductionDouble impulse inputMaximum rotation of rigid block subjected to critical double impulseLimit input level of critical double impulse characterizing overturning of rigid blockNumerical examples and discussionSummariesReferencesAppendix 1: Verification of critical timing ofdouble impulse for various inputlevelsCritical earthquake response of a 2DOF elastic–perfectly plastic model under double impulseIntroductionDouble impulse inputTwo-DOF system and normalization of double impulseDescription of elastic-plastic response process in terms of energy quantitiesUpper bound of plastic deformation in first story after second impulseMaximization ofMaximization of Δ E (minimization of Δ E in addition)Minimization of (maximization of in addition)Upper bound of plastic deformation in the first story after the second impulseNumerical Examples of Critical ResponsesUpper bound of critical responseInput level for tight upper boundInput level for loose upper boundVerification of criticalityApplication to recorded ground motionsSummariesReferencesAppendix 1: Adjustment of amplitudes ofdouble impulse and correspondingone-cycle sinusoidal waveAppendix 2: Upper and lower bounds of plasticdeformation in first story aftersecond impulse (case of elasticresponse in first story after firstimpulse)Appendix 3: Comparison of time histories to double impulse and corresponding sinusoidal waveAppendix 4: Input level of double impulse for characterizing critical response close to upper or lower boundAppendix 5: Effect of viscous dampingOptimal viscous damper placement for an elastic–perfectly plastic MDOF building model under critical double impulseIntroductionInput ground motionProblem of optimal damper placement and solution algorithmThree models for numerical examplesDynamic pushover analysis for increasing critical double impulse (DIP: Double Impulse Pushover)Numerical examplesExamples for Problem 1 using Algorithm 1Examples for Problem 2 using Algorithm 2Examples for Mixed Problem (Problem 3) of Problem 1 and 2 using Algorithm 3Comparison of IDA (Incremental Dynamic Analysis) and DIPSummariesReferencesFuture directionsIntroductionTreatment of noncritical caseExtension to nonlinear viscous damper and hysteretic damperTreatment of uncertain fault-rupture model and uncertain deep ground propertyApplication to passive control systems for practical tall buildingsStopper system for pulse-type ground motion of extremely large amplitudeRepeated single impulse in the same direction for repetitive ground motion inputRobustness evaluationPrinciples in seismic resistant designReferences