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
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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
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Typical Pushover (Capacity) Curve
Global Yielding Point
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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.
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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
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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
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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
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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.
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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.
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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
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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
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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
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