Equivalent SDOF Systems to Simulate MDOF System Behavior

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Equivalent SDOF Systems to Simulate MDOF System Behavior Erol Kalkan PEER-GMSM, Berkeley Aug- How to find equivalent SDOF systems via Pushover analysis: MDOF system seismic behavior can be

1 Equivalent SDOF Systems to Simulate MDOF System Behavior Erol Kalkan PEER-GMSM, Berkeley Aug- How to find equivalent SDOF systems via Pushover analysis: MDOF system seismic behavior can be approximated with certain accuracy by equivalent SDOF systems whose properties are computed by conducting pushover analyses. Base Shear Plastic Hinge Development Capacity Curve A sequence of a typical pushover analysis PEER GMSM

2 Pushover Curve is a plot of base shear versus roof displacement shows nonlinear behavior of the building is usually idealized by bilinear curve is strongly dependent on the shape of loading vector Note that global yield point not the same as first local yield point PEER GMSM Typical Pushover (Capacity) Curve Global Yielding Point PEER GMSM

3 Equivalent inelastic single-degreeof-freedom (SDF) system Force-displacement relation of SDF system is determined from pushover curve (base shear-roof displacement) Assumptions The response of the multi-degree-of-freedom (MDF) structure can be related to the response of an equivalent SDF system, implying that the response is controlled by a single mode and this mode shape remains unchanged even after yielding occurs The invariant lateral force distribution can represent and bound the distribution of inertia forces during an earthquake Modal responses are assumed to be uncoupled similar to elastic case. PEER GMSM Understanding Modal Patterns The dynamic load can be expressed in terms of a spatial distribution (independent of time) & a time-varying function: mu.. && + cu &. + fs ( u) = p f f ( t) p f = pn = For a given response spectrum, resulting forces at level i for mode j Fij ΓnmΦn = Γ j mφ ( ) i ij Sa j Select which modes are being considered: > for low to mid-rise: st modes > for taller structures: st or modes PEER GMSM

4 Height-wise load distributions (s n = mφ n ) s = mφ s = mφ s = mφ Invariant modal load vectors for -story building PEER GMSM 7 Converting Capacity Curves to Capacity Spectrum in ADRS format The smooth capacity curve is idealized by bi-linear curve by the assumption that area (energy) underneath both curves becomes the same. MDOF system Level Backbone curve of ESDOF system PEER GMSM 8

5 Step-by-Step Procedure. Determine mode shapes φ n and frequencies.. Perform pushover analysis using mφ n as force pattern. Idealize pushover curve and determine properties of equivalent inelastic SDF system: Fs n(y) /L n and D n(y). Calculate D(t), D no (initial assumption in pushover).. The same procedure is applied for the other modes. Note: The procedure is not always converge. PEER GMSM 9 Performance Assessment based on ESDOF systems ESDOF systems are the fundamentals of current performance evaluation methods including: ATC- Capacity Spectrum Method (Relies on first mode response only) FEMA- (Modified version of Displacement Coefficient Method) MPA or Modified MPA (Multi-Modal Pushover Method) In all these methods, the primary objective is the accurate prediction of the target displacement to be used in the pushover analysis. On the other hand there common limitation is the use of invariant load vectors. These methods in general effective in predicting the seismic response if the structure is mostly behave in first mode. If there is significant higher mode contributions, they tend to underestimate the enhanced demands at mid and upper story levels. PEER GMSM

6 Case Study: -Story Building Ground motion record used is the Kobe JMA record which has distinct forward directivity pulse in the velocity time series. Step-: Determine mode shapes φn and frequencies. Storey Level Storey Level Storey Level st Mode... nd Mode -.. rd Mode Elastic Modes -Story Building Mode- Mode- Mode- Modal Periods (sec), Tn.9.. Modal Participation Factors, Γn.8.9. PEER GMSM Mass Participation Factors, αn.8.. Step. Perform pushover analysis using mφ n as force pattern PEER GMSM

7 Step-: Idealize pushover curve and determine properties of equivalent inelastic SDF system: Fsn(y)/Ln and Dn(y) Base Shear Roof Displacement (inch), urn Fsn/Ln = Sa 8 Dn = Sd MDOF System Capacity Curve ESDOF System Capacity Spectrum By computing the yield displacement and yield force and also post yield stiffness ratio, one can conduct inelastic SDOF analysis using the acceleration time-series to find the target spectral displacement (D n ). Step-: PEER GMSM ESDOF and MDOF Peak Displacement Values and Resultant Response Profiles ESDOF system D on values:.,.,. inches for first, second and third modes. Corresponding MDOF target displacement values (u rno ) are obtained as:.8,.,.8 inches for first, second and third modes pushover analyses. MPA NTH Story Level Target Drift.... Interstory Drift Ratio 8 Ductility.... Roof Drift Ratio PEER GMSM 7

Non-Linear Seismic Analysis of Multi-Storey Building

Non-Linear Seismic Analysis of Multi-Storey Building Uzair Khan 1 1 M.Tech (Structural & Construction Engineering) Department of Civil Engineering, National Institute of Technology, Jalandhar. Hina Gupta