Critical Excitation Methods in Earthquake Engineering

Izuru Takewaki

• Publisher: Butterworth-Heinemann
• Year: 2013
• Print ISBN: 9780080994369
• eBook ISBN: 9780080994291
• Edition: 2
• Topic: Tecnología e ingeniería
• Subtopic: Ingeniería civil

After the March 11, 2011, earthquake in Japan, there is overwhelming interest in worst-case analysis, including the critical excitation method. Nowadays, seismic design of structures performed by any seismic code is based on resisting previous natural earthquakes. Critical Excitation Methods in Earthquake Engineering, Second Edition, develops a new framework for modeling design earthquake loads for inelastic structures. The Second Edition, includes three new chapters covering the critical excitation problem for multi-component input ground motions, and that for elastic-plastic structures in a more direct way are incorporated and discussed in more depth. Finally, the problem of earthquake resilience of super high-rise buildings is discussed from broader viewpoints.- Solves problems of earthquake resilience of super high-rise buildings- Three new chapters on critical excitation problem for multi-component input ground motions- Includes numerical examples of one and two-story models

Chapter One

Overview of Seismic Critical Excitation Method

Contents

1.1. What is Critical Excitation?

1.2. Origin of Critical Excitation Method (Drenick’s Approach)

1.3. Shinozuka’s Approach

1.4. Historical Sketch in Early Stage

1.5. Various Measures of Criticality

1.6. Subcritical Excitation

1.7. Stochastic Excitation

1.8. Convex Models

1.9. Nonlinear or Elastic-Plastic SDOF System

1.10. Elastic-Plastic MDOF System

1.11. Critical Envelope Function

1.12. Robust Structural Design

1.13. Critical Excitation Method in Earthquake-Resistant Design

References

1.1 What is Critical Excitation?

It is natural to imagine that a ground motion input resonant to the natural frequency of the structure is a critical excitation. In order to discuss this issue in detail, consider a linear elastic, viscously damped, single-degree-of-freedom (SDOF) system as shown in Fig. 1.1. Let m, k, c denote mass, stiffness and viscous damping coefficient of the SDOF system. The time derivative will be denoted by over-dot in this book. The system is subjected to an external harmonic force

. The equation of motion of this system may be described as

(1.1)

By dividing both sides by m, Eq. (1) leads to

(1.2)

where

,

.

and

are the undamped natural circular frequency and the critical damping ratio.

Figure 1.1 Single-degree-of-freedom (SDOF) system subjected to external harmonic force p(t) = p₀ sin ωt.

Consider first the nonresonant case, i.e.

. The general solution of Eq. (1.1) can be expressed by the sum of the complementary solution of Eq. (1.1) and the particular solution of Eq. (1.1).

(1.3)

The complementary solution is the free-vibration solution and is given by

(1.4)

where

. On the other hand, the particular solution may be described by

(1.5)

The undetermined coefficients C and D in Eq. (1.5) can be obtained by substituting Eq. (1.5) into Eq. (1.2) and comparing the coefficients on sine and cosine terms. The expressions can be found in standard textbooks. On the other hand, the undetermined coefficients A and B in Eq. (1.4) can be obtained from the initial conditions

and

.

Consider next the resonant case, i.e.

. The solution corresponding to the initial conditions

can then be written by

(1.6)

Fig. 1.2 shows examples of Eq. (1.6) for several damping ratios.

Figure 1.2 Resonant response with various damping levels.

Consider the undamped and resonant case, i.e.

and

. As before, the general solution of Eq. (1.2) can be expressed by the sum of the complementary solution and the particular solution.

(1.7)

The complementary solution is the free-vibration solution and is given by

(1.8)

On the other hand, the particular solution may be described by

(1.9)

The final solution corresponding to the initial conditions

can then be written by

(1.10)

Fig. 1.3 shows an example of Eq. (1.10) for a special frequency

.

Figure 1.3 Resonant response of undamped model.

1.2 Origin of Critical Excitation Method (Drenick’s Approach)

Newton’s second law of motion may be described by

(1.11)

If the mass remains constant, the equation is reduced to

(1.12)

Consider the integration of Eq. (1.12) from time

through

.

(1.13)

where

and

. Assume here a unit impulse applied to a mass at rest

(1.14)

Then the change of velocity may be described as

(1.15)

Figure 1.4 Linear elastic, viscously damped SDOF system subjected to base motion.

Consider next a linear elastic, viscously damped SDOF system subjected to a base acceleration

as shown in Fig. 1.4. The equation of motion may be expressed by

(1.16)

By dividing both sides by m, Eq. (1.16) leads to

(1.17)

where

,

. The unit impulse response function can then be derived from Eqs. (1.4) and (1.15) as the free vibration response of the system at rest subjected to the unit impulse.

(1.18)

where

is the Heaviside step function. The displacement response of the system subjec...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Preface to the First Edition
• Preface to the Second Edition
• Permission details
• Chapter One. Overview of Seismic Critical Excitation Method
• Chapter Two. Critical Excitation for Stationary and Nonstationary Random Inputs
• Chapter Three. Critical Excitation for Nonproportionally Damped Structural Systems
• Chapter Four. Critical Excitation for Acceleration Response
• Chapter Five. Critical Excitation for Elastic-Plastic Response
• Chapter Six. Critical Envelope Function for Nonstationary Random Earthquake Input
• Chapter Seven. Robust Stiffness Design for Structure-Dependent Critical Excitation
• Chapter Eight. Critical Excitation for Earthquake Energy Input in SDOF System
• Chapter Nine. Critical Excitation for Earthquake Energy Input in MDOF System
• Chapter Ten. Critical Excitation for Earthquake Energy Input in Soil-Structure Interaction System
• Chapter Eleven. Critical Excitation for Earthquake Energy Input in Structure-Pile-Soil System
• Chapter Twelve. Critical Excitation for Earthquake Energy Input Rate
• Chapter Thirteen. Critical Excitation for Multi-Component Inputs
• Chapter Fourteen. Critical Excitation for Elastic-Plastic Response Using Deterministic Approach
• Chapter Fifteen. Earthquake Resilience Evaluation of Building Structures with Critical Excitation Methods
• Index