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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 Vibrations Lecture #4 MDOF Structures Analysis Instructor: Andrei M. Reinhorn P.Eng. D.Sc. Professor of Structural Engineering.

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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 Vibrations

Lecture #4

MDOF Structures Analysis

Instructor:

Andrei M. Reinhorn P.Eng. D.Sc.

Professor of Structural Engineering


Multi degree of freedom mdof
Multi Degree Of Freedom - MDOF

T1

T2

T3

Modal Analysis


Review
Review

Review of MDOF response

using ortho-normalized modes




Modal analysis
Modal Analysis

Modal Response Calculations:

_______________________________________________________

Spectral Displacement:

Spectral Acceleration:

_______________________________________________________

Modal Floor Displacement (u or  )

Modal (Floor) Inertia Force

Modal Reaction- BASE SHEAR

_______________________________________________________


Modal superposition
Modal Superposition

Total Response can be obtained from all modes

by superposition:

SRSSn (xi) = ( SUMn (xi2))1/2

VBASE = srssk (Vi)

Ffloor J = srssk (FiJ)

Approximations in highly damped structures:

1. Use of undamped mode shapes

2. Use of proportional damping


Capacity (Curve) of StructureCapacity is a function, not a single value, indicating the strength of the system at a deformed position of the structure


Nonlinear static procedure
Nonlinear Static Procedure

  • The static procedure is an equivalent instantaneous approximation of MDOF response to determine internal stresses and other …..


Analysis of yielding storied frame
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 dynamic

analysis








Step by step dc procedure dc demand capacity
Step-by-step DC Procedure[DC = Demand Capacity]


Step by step dc procedure dc demand capacity1
Step-by-step DC Procedure[DC = Demand Capacity]


Spectral capacity
“Spectral” Capacity

INCLUDING A NUMBER OF MODAL SHAPES:




Spectral evaluation linearized procedure
Spectral Evaluation - Linearized Procedure

Linearized procedures uses equivalent period

and damping to evaluate response


Spectral evaluation inelastic spectrum procedure
Spectral Evaluation - Inelastic Spectrum Procedure

Inelastic Spectrum Procedure uses no approximations

but require inelastic spectrum to evaluate response








Damped inelastic spectra
Damped Inelastic Spectra

Note that it is necessary to adjust the inelastic spectra fro the

increased damping:


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