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7/25/2019 3)III Methods of Seismic Analysis
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METHODS OF
SEISMIC ANALYSIS
Luca Pel
Universidad Politcnica de Catalunya
ETS CAMINOS, CANALES Y PUERTOS DE BARCELONA
Diseo y Evaluacin Ssmica de Estructuras
mailto:[email protected]:[email protected]7/25/2019 3)III Methods of Seismic Analysis
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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.).
ETCECCPB UPC M th d f S i i A l i
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Examples of pushover analysis developed using FEM softwares
ETCECCPB UPC M th d f S i i A l i
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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.
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Examples of nonlinear dynamic 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.