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### 20060821_Kalkan_EquivalentSDOFSystems_Presentation

Course: CE 7137, Fall 2009
School: U. Memphis
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Word Count: 740

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SDOF Equivalent Systems to Simulate MDOF System Behavior Erol Kalkan PEER-GMSM, Berkeley Aug-21 2006 PEERAug- 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 Capacity Curve Plastic Hinge Development A sequence of a typical...

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SDOF Equivalent Systems to Simulate MDOF System Behavior Erol Kalkan PEER-GMSM, Berkeley Aug-21 2006 PEERAug- 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 Capacity Curve Plastic Hinge Development A sequence of a typical pushover analysis PEER GMSM 2 1 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 3 Typical Pushover (Capacity) Curve Global Yielding Point PEER GMSM 4 2 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 5 Understanding Modal Patterns The dynamic load can be expressed in terms of a spatial distribution (independent of time) & a time-varying function: .. . & & m u + cu + mu&+ cu + f s (u ) = p f f (t ) p f = p n = n mn For a given response spectrum, resulting forces at level `i' for mode `j' Fij = j mi ij S a ( j ) Select which modes are being considered: > > for low to mid-rise: 1st 2 modes for taller structures: 1st 3 or 4 modes PEER GMSM 6 3 Height-wise load distributions (sn = mn) 6 3.00 6 1.13 6 0.63 5 3.51 5 0.67 5-0.25 4 2.94 -0.41 4 -0.82 4 3 2.28 -1.08 3 -0.31 3 2 1.55 -1.25 2 2 0.52 1 0.96 -0.96 1 1 0.72 0 0.00 0 0.00 0 0.00 s1 = m1 s2 = m2 s3 = m3 Invariant modal load vectors for 6-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 4 Step-by-Step Procedure 1. Determine mode shapes n and frequencies. 2. Perform pushover analysis using mn as force pattern 3. Idealize pushover curve and determine properties of equivalent inelastic SDF system: Fsn(y)/Ln and Dn(y) 4. Calculate D(t), Dno (initial assumption in pushover). 5. The same procedure is applied for the other modes. Note: The is procedure not always converge. PEER GMSM 9 Performance Assessment based on ESDOF systems ESDOF systems are the fundamentals of current performance evaluation methods including: ATC-40 Capacity Spectrum Method (Relies on first mode response only) FEMA-440 (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. 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 contributions at mid and upper story levels. PEER GMSM 10 5 Case Study: 6-Story Building Ground motion record used is the Kobe JMA record which has distinct forward directivity pulse in the velocity time series. Step-1: Determine mode shapes n and frequencies. 6 6 6 5 5 5 4 Storey Level Storey Level 4 Storey Level 4 3 3 3 2 2 2 1 1st Mode 1 2nd Mode 0 1 3rd Mode 0 0 0.5 1 -1 -0.5 0 0.5 1 0 0 0.2 0.4 0.6 -0.5 PEER GMSM 6-Story Building Modal Periods (sec), Tn Modal Participation Factors, n Mass Participation Factors, n Mode-1 1.39 2.58 0.85 Elastic Modes Mode-2 0.51 0.96 0.12 Mode-3 0.31 0.46 0.03 11 Step 2. Perform pushover analysis using mn as force pattern 6 3.00 6 1.13 6 0.63 5 3.51 5 0.67 5-0.25 4 2.94 -0.41 4 -0.82 4 3 2.28 -1.08 3 -0.31 3 2 1.55 -1.25 2 2 0.52 1 0.96 -0.96 1 1 0.72 0 0.00 0 0.00 0 0.00 PEER GMSM 12 6...

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