3)iii methods of seismic analysis

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METHODS OF

SEISMIC ANALYSIS

Luca Pel

Universidad Politcnica de Catalunya

[email protected]

ETS CAMINOS, CANALES Y PUERTOS DE BARCELONA

Diseo y Evaluacin Ssmica de Estructuras
mailto:[email protected]:[email protected]


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ETCECCPB-UPC Methods o f Seismic Analysis

CONTENTS

1. INTRODUCTION

TYPES OF ANALYSIS

2. RESPONSE SPECTRUM ANALYSIS

3. MULTI-MODAL RSA FOR BUILDINGS WITH SYMMETRIC PLAN

4. EQUIVALENT STATIC ANALYSIS OF LINEAR MDOF SYSTEMS

5. PUSHOVER ANALYSIS (INTRODUCTION)

4. NONLINEAR DYNAMIC ANALYSIS (INTRODUCTION)


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1. INTRODUCTION

Because elastic and damping forces are related to the relative displacementsand velocities between the masses and the base, respectively, and inertial

forces are related to the total accelerations of the masses, the equation of

motion of a MDOF system subject to an earthquake reads

The vector utis given by

By substitution one gets


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The modal expansion of displacements can be expressed as

Modal equations:

the system of coupled modal equations is transformed into a system of

uncoupled equations.


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The n-th modal equation can be finally expressed as:


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TYPES OF ANALYSIS

Depending on the structural characteristics of the system, the following types ofanalysis are used:

1.Linear elastic analysis

a) response history analysis: calculation of structural response as a

function of time when the system is subjected to a given groundacceleration.

b) response spectrum analysis: computation of the peak response of

a structure during an earthquake directly from the earthquake

response or design spectrum.

c) equivalent static analysis: single-mode spectral and uniform-load

method.

2. Nonlinear methods

d) nonlinear static analysis (pushover analysis)

e) nonlinear time-history analysis (dynamic analysis)


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2. RESPONSE SPECTRUM ANALYSIS (RSA)

Structural design is usually based on the peak values of forces anddeformations over the duration of the earthquake-induced response.

For SDOF systems, the peak response can be determined directly from the

response spectrum for the ground motion.

The modal equations of motion of a damped MDOF system subjected to

earthquake are given by:

Therefore, the maximum value of qn(t), denoted as Qn, is given by spectralacceleration


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The maximum modal response un(t), i.e. the maximum displacement for mode

n, reads

Summary of maxima values per mode n and component j


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Once evaluated the peak modal responses un,max for each node, it is not

possible to determine the exact peak value of the total response umax, because

in general the modal responsesu

n(t), or qn(t), attain their peaks at different timeinstants.

The rules suggested by EC8 to compute the peak total response are the

square root of sum of squares (SRSS), if natural frequencies are well-

separated.

or the complete quadratic combination (CQC)

mnis the correlation coefficient for the m-th and n-th modes


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MODAL PARTICIPATION FACTORS AND EFFECTIVE MODAL MASS

The effective modal mass provides a method for judging the significance of avibration mode. Modes with relatively high effective masses can be readily

excited by base excitation. Using modal analysis, it is possible to define:

The base shear of a SDOF system with mass Mneff and frequency n is the

same as the nth mode base shear in a multi-storey system with mass

distributed along the height.

The sum of the effective modal masses over all the

N modes is equal to the total mass of the system:


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For the 3D analysis of buildings with in-plane rigid floor diaphragms, the influence

vector can have different forms:

Therefore, the modal participation factor and the effective modal mass has to be

defined by mode and by direction

total mass

Within the modal analysis, the contribution of all modes must be included to

obtain the exact value of the response. The EC8 suggests to include in modalanalysis all the modes with effective modal masses greater than 5% of the total

mass or to sum the effective modes until reaching at least 90%of the total mass.

5%effjM M


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Example

1 2 3

eff eff

structure

M M m M+ = =


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3. MULTI-MODAL RSA FOR BUILDINGS WITH SYMMETRIC PLAN

Multistory buildings with symmetric plan:- rigid floor diaphragms, plans with 2 orthogonal axes of symmetry

- subjected to horizontal ground motion along one of axis of symmetry

maximum modalbase shear

maximum modal

horizontal force


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Procedure:

1) Solve eigenvalue problem (N*N) to obtain a number p


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Example: 2DOF shear frame

2


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Modal masses and participation factors:

Maximum modal displacements and forces:


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4. EQUIVALENT STATIC ANALYSIS OF LINEAR MDOF SYSTEMS

Conditions of applicability according to EC8 - Regularity in plan and elevation

1. With respect to the lateral stiffness and mass distribution, the structure is

approximately symmetrical in plan with respect to two orthogonal axes

2. The plan configuration is compact, i.e., each floor shall be delimited by a

polygonal convex line.

3. In-plan stiffness of the floors shall be sufficiently large in comparison with thelateral stiffness of the vertical structural elements, so that the rigid diaphragm

condition is satisfied.

4. The slenderness =Lmax/Lminof the building in plan shall be not higher than 4,

where Lmaxand Lminare respectively the larger and smaller in plan dimension of

the building, measured in orthogonal directions.

1. All lateral load resisting systems, such as cores, structural walls, or frames,

shall run without interruption from their foundations to the top of the building.

2. The lateral stiffness and the mass of the individual storeys shall remain

constant or reduce gradually from the base to the top of the building.

When setbacks are present, there are special additional conditions (made

available to limit the unfavourable effects of setbacks) that must be satisfied.


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The method assumes that seismic load can be considered as an equivalent

static horizontal force applied to an individual frame.

The method is based on the natural period of a SDOF system (normally the

fundamental period) and code-specified response or design spectra.

Procedure:

1. Estimate the period of the fundamental mode T1 usually by some simplified

approximate method. In EC8:

2. Find the corresponding spectral acceleration Sa from the design response

spectra.

3. Calculate the base shear in the dominant mode:


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4. Distribute the horizontal load up the building in proportion to mass mj and

estimated mode shape j

if the mode shape is estimated as s straight line,

zjis the height of thejth storey above the base

5. Calculate member forces and displacements de by static analysis. If the

forces were calculated assuming a structure ductility q, then the actual

structural displacements are

(Single-mode

distribution of

loads)

(Linear

distribution

of loads)


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EXAMPLE: 2DOF shear frame

Single-mode distribution of loads


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Linear distribution of loads

ETCECCPB UPC M th d f S i i A l i


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5. PUSHOVER ANALYSIS (INTRODUCTION)

The pushover analysis (nonlinear static analysis) evaluates the capacity of the

structure to resist a system of lateral forces, that simulates the actions during

the earthquake.

The lateral forces are increased monotonically until reaching the failure of the

model, that includes the description of nonlinearities (material, geometric, etc.).



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Examples of pushover analysis developed using FEM softwares



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6. NONLINEAR DYNAMIC ANALYSIS (INTRODUCTION)

The nonlinear dynamic analysis is a step-by-step analysis in the time domain,

that evaluates the response at each step of the structure subjected to a time-

dependent action.

The structure is analysed in the time domain including the inertial effects.

The input earthquake is defined by a time-history function (accelerogram).The model includes the material nonlinearity, the geometric nonlinearity, the

hysteretic behaviour of structural members, etc.

It requires sophisticated computational programs (e.g. FEM).

The computational cost is high.

ETCECCPB UPC Methods of Seismic Analysis


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Examples of nonlinear dynamic analysis

ETCECCPB UPC Methods of Seismic Analysis


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REFERENCES

Chopra A.K. Dynamics of Structures. Prentice Hall, 1995.

Petrini L.; Pinho R.; Calvi G. M. Criteri di progettazione antisismica degli edifici

(in Italian). IUSS Press, 2004.

EN 1998-1. Eurocode 8: Design of structures for earthquake resistance - Part 1:

General rules, seismic actions and rules for buildings.

3)iii methods of seismic analysis

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