1CIE 619Chapter 6 – Energy ConceptsChapter 6Energy Concepts in Earthquake Engineering2CIE 619Chapter 6 – Energy ConceptsCONTENT1. Introduction2. Rain Flow Analogy3. Energy Balance Equation4. Examples of Energy Computation3CIE 619Chapter 6 – Energy Concepts1. Introduction• Seismic energy formulation natural way to understand effect of supplemental damping and seismic isolation systems• Main advantages of energy formulation:– replacement of vector quantities (displacements, velocities and accelerations) by scalar energy quantities– flow of energy quantities can be tracked during seismic response4CIE 619Chapter 6 – Energy Concepts2. Rain Flow AnalogyDuring seismic shaking5CIE 619Chapter 6 – Energy Concepts2. Rain Flow AnalogyAt the end of seismic shaking6CIE 619Chapter 6 – Energy Concepts3. Energy Balance Equation• Derivation7CIE 619Chapter 6 – Energy Concepts3. Energy Balance Equation8CIE 619Chapter 6 – Energy Concepts3. Energy Balance Equation9CIE 619Chapter 6 – Energy Concepts3. Energy Balance Equation10CIE 619Chapter 6 – Energy Concepts• Relative and absolute input energy– Equation (3.10) based on equivalent seismic load applied to a rigid based structure • i.e. “relative” energy formulation– Rigid body translation of structure not considered– “Absolute” energy formulation required• Can be derived from equations of motion• Relative energy formulation can be transformed into absolute energy formulation3. Energy Balance Equation11CIE 619Chapter 6 – Energy Concepts3. Energy Balance Equation12CIE 619Chapter 6 – Energy Concepts3. Energy Balance Equation13CIE 619Chapter 6 – Energy Concepts3. Energy Balance Equation14CIE 619Chapter 6 – Energy Concepts3. Energy Balance Equation• Remarks on Relative and Absolute Energy Formulations– Absolute input energy = base shear * ground displacement: true physical meaning – Both formulations are mathematically equivalent– Sums of kinetic and input energy equal in both formulations– Ground displacement required in absolute formulation– Uang and Bertero (1990) showed that relative and absolute input energies equal for wide period range (0.1 sec to 5.0 sec)15CIE 619Chapter 6 – Energy Concepts3. Energy Balance Equation• Discrete energy expressions– Required for time-integration schemes– Kinetic energy obtained directly at each time– Other energy quantities require time integration– e.g. Trapezoidal rule:16CIE 619Chapter 6 – Energy Concepts3. Energy Balance Equation• Energy Balance– All energy terms computed individually– Energy Balance Error (EBE) can be calculated at each time-step– Tolerance can be set on EBE– EBE can be normalized as follows:17CIE 619Chapter 6 – Energy Concepts4. Examples of Energy Computation• Scope– Ensemble of six two-storey frames– Idealized as 2-DOF systems– Three frames incorporate supplemental damping and seismic isolation systems– Energy computation for various earthquake ground motions18CIE 619Chapter 6 – Energy Concepts4. Examples of Energy Computation• Structural Models– Moment Resisting Frame (MRF)– Braced Moment Resisting Frame (BMRF)• Elastic braces added to MRF– Soft Storey Frame (SSF)• First floor braces removed from BMRF– Base Isolated Frame (BIF)• Laminated rubber bearings installed under BMRF• Bearings modeled as elastic springs• Supplemental DOF considered above bearings– Friction Damped Braced Frame (FDBF)• Friction dampers installed in braces of BMRF• Dampers modeled as Coulomb friction elements– Viscously Damped Braced Frame (VDBF)• Linear viscous dampers installed in braces of BMRF• Braces connected to dampers assumed rigid• Damping constant based on 23% damping in first mode19CIE 619Chapter 6 – Energy Concepts4. Examples of Energy Computation• Structural Models• 2% structural damping in each mode20CIE 619Chapter 6 – Energy Concepts4. Examples of Energy Computation• Structural Models21CIE 619Chapter 6 – Energy Concepts4. Examples of Energy Computation• Earthquake Ground Motions– Records:• 1940 Imperial Valley earthquake recorded at El Centro (S00E)• 1988 Saguenay earthquake recorded at Chicoutimi (TRAN)• 1977 Romania earthquake recorded at Budapest (North-South)– All records scaled to PGA of 0.50 g– First 15 seconds of each records considered22CIE 619Chapter 6 – Energy Concepts4. Examples of Energy Computation• Earthquake Ground Motions23CIE 619Chapter 6 – Energy Concepts4. Examples of Energy Computation• Earthquake Ground Motions24CIE 619Chapter 6 – Energy Concepts4. Examples of Energy Computation• Peak Transient Displacement Ductility Ratios25CIE 619Chapter 6 – Energy Concepts4. Examples of Energy Computation• Energy Time-Histories26CIE 619Chapter 6 – Energy Concepts4. Examples of Energy Computation• Comparison of Relative Input EnergyBIFFDBFBMRFVDBF27CIE 619Chapter 6 – Energy Concepts4. Examples of Energy Computation• Comparison of First Floor Absorbed Energy28CIE 619Chapter 6 – Energy Concepts4. Examples of Energy Computation• Fraction of Seismic Input Energy Absorbed29CIE 619Chapter 6 – Energy