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212 Ketter Hall, North Campus, Buffalo, NY 14260 www.civil.buffalo.edu
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Control of Structural VibrationsLecture #4
MDOF Structures Analysis
Instructor:
Andrei M. Reinhorn P.Eng. D.Sc.Professor of Structural Engineering
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Multi Degree Of Freedom - MDOF
T1 T2 T3
Modal Analysis
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Review
Review of MDOF response using ortho-normalized modes
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MDOF Response
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Modal Quantities
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Modal AnalysisModal Response Calculations:_______________________________________________________
Spectral Displacement:
Spectral Acceleration:
_______________________________________________________
Modal Floor Displacement (u or )
Modal (Floor) Inertia Force
Modal Reaction- BASE SHEAR_______________________________________________________
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Modal SuperpositionTotal Response can be obtained from all modesby superposition:
SRSSn (xi) = ( SUMn (xi2))1/2
VBASE = srssk (Vi)
Ffloor J = srssk (FiJ)
Approximations in highly damped structures:1. Use of undamped mode shapes2. Use of proportional damping
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Capacity (Curve) of StructureCapacity is a function, not a single value, indicating the strength of the system at a deformed position of the structure
ElasticRange
MechanismRange
e
PartialFractureRange
CompleteCollapse
ProgressiveYieldingRange
EffectiveStructureYieldPoint
EffectiveStructureYieldStrength
EffectiveStructureYieldDisplacement DEFORMATION
STRE
NGTH
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Nonlinear Static Procedure
The static procedure is an equivalent instantaneous approximation of MDOF response to determine internal stresses and other …..
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Analysis of Yielding Storied Frame
Nonlinear Static Analysis (Procedure)Perform static analysis with increasing lateral forces, Fi,(applied at each floor) incrementally, proportional to the distribution of inertia forces which appear in the dynamicanalysis
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Analysis of Yielding Storied Frame
VB
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Analysis of Yielding Storied Frame
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Case Study Structure
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Case Study Structure
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Typical Capacity Curve
NS Direction0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.005 0.01 0.015 0.02 0.025 0.03
Shear wall cracks at the Base
Yielding of Shear Wall at the Base
Flexural Failure of Shear Wall at the Base
Cracking in Columns and Beams
First Yielding in Beams Observed
First Yielding in Columns Observed
First Column and Beam Failures Observed
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Typical Capacity Curve
NS Direction
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
k=0
k=1k=2
Failure
Dynamic
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Step-by-step DC Procedure[DC = Demand Capacity]
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Step-by-step DC Procedure[DC = Demand Capacity]
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“Spectral” Capacity
INCLUDING A NUMBER OF MODAL SHAPES:
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Spectral Evaluation of Response
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Equivalent Properties of Linearized System
T Teq 0
121 1
ln/
; for 1
eq
3
2
1 13
23
21 1
1 12
02 3 2
ln; for 1
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Spectral Evaluation - Linearized Procedure
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5
Sd (in)
Sa/g
Capacity Curve
Elastic Demand
Inelastic Response Estimate (=3)
Linearized procedures uses equivalent period and damping to evaluate response
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Spectral Evaluation - Inelastic Spectrum Procedure
Inelastic Spectrum Procedure uses no approximationsbut require inelastic spectrum to evaluate response
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5
Sd (in)
Sa/g
R=1
R=2
R=4
R=6
Capacity Curve
Elastic Demand
Inelastic Response Estimate (R=4)
Ve/W=0.4
Vy/W=0.1
R=0.4/0.1=4
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Response with Fluid Dampers
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Response with Friction Dampers
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Response with Wall Dampers
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Analysis and Experiments with Fluid
Dampers
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Analysis and Experiments with Friction Dampers
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Analysis and Experiments with Wall
Dampers
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Damped Inelastic Spectra
Note that it is necessary to adjust the inelastic spectra fro the increased damping:
