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212 Ketter Hall, North Campus, Buffalo, NY 14260 www.civil.buffalo.edu

Fax: 716 645 3733 Tel: 716 645 2114 x 2400

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: