15. Nonlinear Static Analysis of MDOF Structures• Monotonic Pushover AnalysisMonotonic lateral load pattern slowly increased–Monotonic lateral load pattern slowly increased until lateral capacity of the structure is reached– Seismic code equivalent static or first mode loading pattern often used– Capacity of the structure depends on loading patternCIE 619 Chapter 4 – Seismic Analysis1pattern• Monotonic Pushover Analysis5. Nonlinear Static Analysis of MDOF StructuresCodeVCIE 619 Chapter 4 – Seismic Analysis22• Monotonic Pushover AnalysisE l (C 2004)5. Nonlinear Static Analysis of MDOF Structures–Example (Carr, 2004)NZS 4203: 1992 and Outward ParabolicCIE 619 Chapter 4 – Seismic Analysis3• Adaptive Monotonic Pushover AnalysisL di tt fl t th d f ti tt f5. Nonlinear Static Analysis of MDOF Structures–Loading pattern reflects the deformation pattern of the structure at end of each loading step– Structural capacity independent of initial loading pattern– Several procedures developedCIE 619 Chapter 4 – Seismic Analysis43• Adaptive Monotonic Pushover AnalysisE l P d d b S t (1998)5. Nonlinear Static Analysis of MDOF Structures–Example: Procedure proposed by Satyarno (1998)CIE 619 Chapter 4 – Seismic Analysis5• Adaptive Monotonic Pushover AnalysisElPdd b R i h 19975. Nonlinear Static Analysis of MDOF Structures–Example: Procedure proposed by Reinhorn, 1997– (197)Satyarno (1998)CIE 619 Chapter 4 – Seismic Analysis64• Cyclic Pushover AnalysisClild dil thit lidf5. Nonlinear Static Analysis of MDOF Structures–Cyclic load or displacement history supplied for specified degrees of freedom.22teqDeqkEt50/300 03060ad (kN)teqkEquivalent viscous damping ratio:CIE 619 Chapter 4 – Seismic Analysis7FiWihiV-60-30-100 -50 0 50 100Displacement (mm)LoattDE• Capacity Spectrum AnalysisCompares structural capacity (pushover curve) with5. Nonlinear Static Analysis of MDOF Structures–Compares structural capacity (pushover curve) with structural demands (response spectrum). – Base shear and roof displacement from a non-linear pushover curve converted into equivalent spectral accelerations and displacements.– Graphical intersection of the two curves approximates CIE 619 Chapter 4 – Seismic Analysis8peak structural response.– Non-linear inelastic behavior of structural system accounted for by effective viscous damping values applied to linear-elastic response spectrum.5Sa• Capacity Spectrum Analysis5. Nonlinear Static Analysis of MDOF Structuresroofd1ΔSΓSa1d1effaWSΓkS roof= Roof displacement1= First modal participation factorCIE 619 Chapter 4 – Seismic Analysis9SdfactorW1= First modal weightkeff = Effective secant stiffnessNote: Fundamental mode shape normalized to unity at roof level5. Nonlinear Static Analysis of MDOF StructuresCIE 619 Chapter 4 – Seismic Analysis1065. Nonlinear Static Analysis of MDOF StructuresCIE 619 Chapter 4 – Seismic Analysis115. Nonlinear Static Analysis of MDOF StructuresCIE 619 Chapter 4 – Seismic Analysis1275. Nonlinear Static Analysis of MDOF StructuresCIE 619 Chapter 4 – Seismic Analysis135. Nonlinear Static Analysis of MDOF StructuresApproximated Simplifiedpp o ated S p edCIE 619 Chapter 4 – Seismic Analysis141.0 for 0.5seckT2.0 for 2.5seckT0.51 for otherwise2Tk85. Nonlinear Static Analysis of MDOF StructuresApproximated Multi-mode1kiiibNklllWhFVWhApproximated MultimodeCIE 619 Chapter 4 – Seismic Analysis151.0 for 0.5seckT2.0 for 2.5seckT0.51 for otherwise2Tk5. Nonlinear Static Analysis of MDOF StructuresApproximated Linear First k=11iiibNlllWhFVWhCIE 619 Chapter 4 – Seismic Analysis161.0 for 0.5seckT2.0 for 2.5seckT0.51 for otherwise2Tk95. Nonlinear Static Analysis of MDOF StructuresApproximated Uniform k=0biVFNCIE 619 Chapter 4 – Seismic Analysis171.0 for 0.5seckT2.0 for 2.5seckT0.51 for otherwise2Tk5. Nonlinear Static Analysis of MDOF StructuresApproximated Uniform k=0biVFNCIE 619 Chapter 4 – Seismic Analysis181.0 for 0.5seckT2.0 for 2.5seckT0.51 for otherwise2Tk105. Nonlinear Static Analysis of MDOF StructuresAdaptive – Instantaneous Mode111oldiiibiNlllWFVFW CIE 619 Chapter 4 – Seismic Analysis195. Nonlinear Static Analysis of MDOF StructuresAdaptive – Instantaneous Modes1/ 2211/2211nmiijjjoldibiNnmlljjljWFVFW CIE 619 Chapter 4 – Seismic Analysis20115. Nonlinear Static Analysis of MDOF StructuresKFPPPPF    KuFP P P P Ft V FR HY IW corr errcwhere tK is the tangent structural stiffness; u is the increment vector of lateral displacements; F is the increment vector of lateral forces; VP , FRP , HYP , and IWP are the increment vectors of forces in viscous dampers, friction dampers, CIE 619 Chapter 4 – Seismic Analysis21IWp, p,hysteretic dampers, and infill panels respectively; and errF is the vector of the unbalanced forces in the structure; corrc is a correction coefficient (usually taken as one). 5. Nonlinear Static Analysis of MDOF StructuresCIE 619 Chapter 4 – Seismic Analysis22125. Nonlinear Static Analysis of MDOF StructuresIt is possible to convert the axes of the capacity diagram to It is possible to convert the axes of the capacity diagram to Proportional spectral values based on the previous relationsCIE 619 Chapter 4 – Seismic Analysis235. Nonlinear Static Analysis of MDOF StructuresSPECTRAL CAPACITYCIE 619 Chapter 4 – Seismic Analysis24135. Nonlinear Static Analysis of MDOF StructuresComposite Acceleration Displacement SpectraCIE 619 Chapter 4 – Seismic Analysis255. Nonlinear Static Analysis of MDOF StructuresINELASTIC SPECTRA -CIE 619 Chapter 4 – Seismic Analysis26May use NSPECTRA from http://civil.eng.buffalo.edu/nspectra145. Nonlinear Static Analysis of MDOF StructuresComposite Acceleration Displacement SpectraElastic and InelasticCIE 619 Chapter 4 – Seismic Analysis275. Nonlinear Static Analysis of MDOF StructuresEvaluation of Inelastic Response Using Spectral Capacity and Inelastic