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A New Approach for the Performance Based Seismic Design of Structures

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### A New Approach for the Performance Based Seismic Design of Structures

### Performance Based Seismic Design

### Needs

### Objective

### Performance Based Seismic Design

### Performance Based Seismic Design

Instituto de Ingeniería

A Gustavo Ayala

September 2003

Conjunction of the design, construction and maintenance procedures necessary to reach, through engineering means, predictable performances for multiple design objectives.

- Its purpose is to minimize the economic losses after a seismic event during the useful life of the structures.

Background

- PBSD is not a new concept, however, with the current procedures of seismic design it is not possible to guarantee that the objectives of the design philosophy are satisfied.
- The application of the PBSD implies the use of methods and tools which emphasize a precise characterization of the structures and lead to predictions using a level of technology higher than that currently used.
- The Computational Mechanics group of the Institute of Engineering at UNAM has developed various procedures for the evaluation and design of structures using the philosophy of PBSD.

Procedures for the PBSD of structures validated with realistic performance indexes which guarantee for a given design level a better control of the performance objectives.

- Till now the design philosophy and the theoretical basis which regulate the PBSD of structure have been established. However, more work on the development of the procedures to implement the PBSD is required.

To develop a simplified method for the PBSD which implicitly involves in its formulation the non linear behaviour and be directly applicable to different criteria for the objectives of PBSD.

- Develop a methodology to determine design spectra based on the concepts of PBSD and the control of damage.
- Validate the simplified method of PBSD in plane frames, asymmetric buildings and bridges.

- Seismic design level.
- Seismic design objectives.

Expression the maximum acceptable damage in a structure subjected to earthquake action.

Seismic demand representing the hazard of a site where the structure would be located.

Union of a performance level and a level of seismic design.

- ATC-33
- FEMA – 273, ATC 40
- SEAOC- Vision 2000
- Euro Code 8
- Japanese code

EC8: Conventional Criterion

- Explicitly satisfy the level of performance “Life safety” under a design level “rare”
- Limit the economic losses through a check of the damage limits for a “frequent” demand
- Prevent the collapse under any imaginable demand through a “Capacity Design ”

Life safety

Collapse

prevention

Fully

operational

Operational

Frequent (43 years)

50% in 30 years

Ocassional (72 years)

50% in 50 years

Basic Objective

Rare (475 years)

10% in 50 years

Essential/Risk Objective

Critical Safety Objective

Very Rare (970 years)

10% en 100 years

Non acceptable performance in new construction

Seismic Design Level

Safety

Collapse

prevention

Fully

operational

Operational

Desempeño no aceptable en construcciones nuevas

Performance Level

SEAOC- Vision 2000

Performance level where essentially no damage occurs

Performance Level

SEAOC- Vision 2000

- General damage
- Vertical Elems.
- Horizontal Elems.
- Non structural Elems.
- Sanitary, electrical and mechanical systems
- Contents

Life

Safety

Collapse

prevention

operational

Desempeño no aceptable en construcciones nuevas

SEAOC- Vision 2000

- D max.
- Distortions 0.002-0.005
- Floor Accel. 0.10g
- Strength Rel. <1
- Non structural behaviour

Fully Operational

Performance level where essentially no damage occurs

Life

safety

Collapse prevention

Operational

Non acceptable performance for new construction

Extreme state of damage in which the capacity of the structure to sustain vertical loads is significantly diminished.

Performance Level

SEAOC- Vision 2000

- General damage
- Vertical Elems.
- Horizontal Elems.
- Non structural Elems.
- Sanitary, electrical and mechanical systems
- Contents

Life

safety

Fully

Operational

Operational

Non acceptable performance for new construction

Extreme state of damage in which the capacity of the structure to sustain vertical loads is significantly diminished.

Performance Level

SEAOC- Vision 2000

- D max
- Distortions 0.02-0.04
- Rotactions 0.02-0.05
- Floor Accel 1.5g
- Strength Rel. f(f,m)
- Ductility and dissipation of energy (Damage indexes)

Life

Safety

Fully

Operational

Operational

Non accepotable performance for new construction

de

vidas

Colapso

incipiente

Fully

Operational

Operational

Frequent (43 years)

50% in 30 years

Ocassional (72 years)

50% in 50 years

Objetivo Básico

Rare (475 years)

10% in 50 years

Objetivo Esencial/Riesgo

Objetivo Seguridad Crítica

Very Rare (970 years)

10% in 100 years

Design Level

- Location of epicentres and identification of seismic sources
- Frequency of events at each source
- Distribution of the magnitude of the events and their number
- Attenuation of seismic waves
- Effects of local soil conditions
- Determination of the seismic hazard

de

vidas

Colapso

incipiente

Completamente

Funcional

Funcional

Frequent (43 years)

50% in 30 years

Ocassional (72 years)

50% in 50 years

Objetivo Básico

Rare (475 years)

10% in 50 years

Objetivo Esencial/Riesgo

Objetivo Seguridad Crítica

Very Rare (970 years)

10% in 100 years

Design Level

de

vidas

Colapso

incipiente

Completamente

Funcional

Funcional

Frequent (43 years)

50% in 30 years

Ocassional (72 years)

50% in 50 years

Objetivo Básico

Rare (475 years)

10% in 50 years

Objetivo Esencial/Riesgo

Objetivo Seguridad Crítica

Very Rare (970 years)

10% in years

Design Level

de

vidas

Colapso

incipiente

Fully

Operational

Funcional

Frequent (43 years)

50% in 30 years

Ocassional (72 years)

50% in 50 years

Objetivo Básico

Rare (475 years)

10% in 50 years

Objetivo Esencial/Riesgo

Objetivo Seguridad Crítica

Very Rare (970 years)

10% in 100 years

Design Level

de

vidas

Colapso

incipiente

Fully

Opertional

Funcional

Frequent (43 years)

50% in 30 years

Ocassional (72 years)

50% in 50 years

Objetivo Básico

Rare (475 years)

10% in 50 years

Objetivo Esencial/Riesgo

Objetivo Seguridad Crítica

Very Rare (970 years)

10% in 100 years

Design Level

- Design process that relates a performance level with a seismic design level.

a) Displacements

Moehle 1992; Priestley 1998, 2000; Kowalsky 1994, 1997; Paulay 2000; Fajfar 1999, Calvi

b) Energy

Mander 1996

c) Distortions

Heidebrecht 2000

a),b) o c) +

d) Damage distribution

Ayala, Sandoval, Vidaud, Basilio, Torres and Avelar 1999->2002

Work Assumptions

- Based on concepts of structural dynamics extended to systems with non linear behaviour it is possible to transform the capacity curve in the behaviour curve of an equivalent SDFS.
- The behaviour curve of an equivalent SDFS can be idealized as bilinear.

Determine the elastic stiffness of the structure and transform it to the space Sa vs Sd

For an assumed damage distribution calculate the slope of the second branch of the behaviour curve

Define the demand spectrum for the target performance index

- Based on the stiffnesses for the elastic and ultimate state, calculate the strength spectrum corresponding to the chosen performance index.
- Relationship of the demand with the required state of functionality.

Define the strength spectrum for the target performance index

Uniqueness of the solution

Superpose the elastic and inelastic branches in the space of the demand spectrum

Superpose the elastic and inelastic branches in the space of the demand spectrum

Superpose the elastic and inelastic branches in the space of the demand spectrum

- Locus of the performance points which satisfy the target ductility
- Uniqueness of the solution

Translate the second branch to the point the demand spectrum satisfies the target performance index

Carry out a static analysis with a distribution of lateral forces equivalent to those acting on the structure under seismic conditions

Existing Approaches for the Design Level

Vision 2000:

- To use as seismic design level demands corresponding to intensities with a given probability of exceedence. It does not give information on the rate of exceedence of the performance level.

This work:

- To use seismic design objectives consisting in pairs of performance level versus seismic design level corresponding to an exceedence rate of the performance level.

Performance Based Design Spectra

Design Objective:

For an chosed design objective, spectra with a uniform rate of exceedence of the proposed performance level

Rate of exceedence of a performance level

Expected number of times per unit time in which the performance of the structure exceeds certain performance level when subjected to earthquakes of different magnitudes and seismic sources defining the seismic hazard of the site.

- Seismicity.

- Probability of exceedence of a performance level.

Considerations:

- Region under study, the lake zone of Mexico City

- The only source that contributes to the seismic hazards of Mexico City is the Guerrero gap.

- The probability that the structural system develops a ductility > 4 is equal to the probability that the system has a strength less than that required to reach such ductility.

Observation:It is necessary to checkthe uniqueness of the relationship strength-ductility.

- Evaluation of the seismic hazard

- Identifify the earthquake generating zones that affect an specific site.

- Evaluate the rate of seismic activity of the sourcers generators of earthquakes (rate of exceedence of magnitudes).

- Probability of exceedence of a performance level

- Response of a SDFS to a set of seismic events.

Performance level:Near to collapse, performance index μ = 4.

Design level:Very rare, rate of exceedence of the performance level of 1/1000.

Seismicity parameters for the subduction zone of Guerrero

T00 = 80 years(Elapsed time in years since the last occurrence of an earthquake with magnitude M>M0)

M0 = 7.0 (Threshold magnitude)

Mu = 8.4(Maximum magnitude)

D = 7.5

F = 0.0(D, F, Parameter defining the variation od expected magnitude with time)

σM = 0.27(Standard deviation of magnitudes)

To = 39.7 years(Median of the time between events of magnitude M>M0)

Expected magnitude value:

Exceedence rate of magnitudes λ(M)

Exceedence rate of an earhquake of magnitude M or higher λ(M), for the seismic source of Guerrero

Relationship of magnitude recurrence

Characteristic earthquake model

In the model of a characteristic earthquake the rate of exceedence of the magnitude changes as a function of tme and it is given by:

Probability of exceedence of a performance level

Registered 25 April 1989 at the SCT station in Mexico City

Earthquake

M = 6.9

Green Functions

Earthquake simulations

1000 simulations for each magnitude

7.2, 7.3, 7.4, 7.5, 7.6,7.7,

7.8, 7.9, 8.0, 8.1, 8.2

Simulated earthquakes

Probability density functions of strengths obtained for periods of 0.05 to 5 s and a M = 8.1 demand

Seismic design objective:performance level (μ = 4) and design level very rare (rate of exceedence 1/1000).

Ilustrative Example

SCT-EW, m=4, a=0.24

Force-Desplacements Spectra

(Constant Ductility)

Curva de

Comportamiento

Strength demand in structural elements

Fyi

Fui - Fyi

Lateral - elastic

Gravitational Loading

Lateral elastic

Illustrative Example

Proposed Damage Distribution

Obtained Damage Distibution

Evaluation Method

Obtained Damage Distibution

Step by Step Analysis

Design Forces

Wroof = 5.44 ton/m

wfloors =6.33ton/m

Base shear

V = 408.64 ton

Base shear

V = 313.57 ton

Gravitational Loading

Stage 1

Stage 2

Intersorey Drifts

Frame

Design A, Dynamic Analysis with100% in X and 30% in Y of SCT-EW

Design A, Dynamic Analysis with 30% in X and 100% in Y of SCT-EW

Evaluation II

Interstorey Drifts

Design B, Dynamic Analysis with100%in X of SCT-EW and 100% in Y of SCT-NS

Design B, Dynamic Analysis with100% in X of SCT-NS and 100% in Y of SCT-EW

Conclusions

- With this method it is possible to know if a given performance index can be reached for a structure and a seismic demand.
- With this method it is possible to control displacements and interstorey drifts and thus satisfy the design objectives.
- In general the method does not directly guarantee local performances e.g. it is not possible to control in a direct manner the magnitude of plastic rotations in elements, only their distribution within the structure.

Conclusions

- The results obtained with this method suggest the need to consider in the definition of design spectra the pos-yielding strength ratio of the capacity curve of the structure.
- As the nominal strengths obtained with this method need to be modified to standardize the design of a structure, it is necessary to check that the modified design satisfies the performance levels under these new conditions.

Conclusions

- The proposed method has the advantage to be able to control the damage in the structure. This characteristic makes it possible that a single design may satisfy different performance levels.
- The modal spectral version of the method can be applied to more general cases in which the contribution of higher modes is important.
- The recursive application of this method allows to control the economic implications of seismic design when varying the intensity and distribution of damage, balancing the initial costs with those of repairing the damage and colateral losses due to the lack of functionality after a design earthquake occurs.

Conclusions

- It is shown that it is possible to reach sismic design objectives considering as design level the corresponding to a rate of exceedence of a proposed performance level.
- Different damage configurations correspond to different slopes of the secon (inelastic) branchof the behaviour curve and, as a consequence, different design spectra.
- From a practical point of view it is not possible to exactly satisfy with a single design more than two design levels.

Recommendations

- Validate this method for other performance indexes, for which it is necessary to develop the design spectra for these performance indexes.
- Investigate further the definition and validation of the performance levels.
- Investigate seismic design levels with different probabilities of exceedence of other design levels.
- Considered the assumed relationship of the b parameter in the Park y Ang damage index with the stiffness degradation of the structure evaluate the range of values of this parameter in real structures.

- Investigate the relationship strength – ductility as it is possible that a given ductility is reached with more than one strength.
- Develop and validate a methodology which allows to satisfy with a single design different performance levels.
- Calculate design spectra for other ductilities and for other performance indexes.
- Obtain design spectra for different a values and from them reduction factors, funtion of a, to difine design spectra based on a reference nominal spectrum.

- Consider for the calculation of nominal design strengths for the elements realistic behaviour models for the concrete and steel.
- Whenever it is impossible to reach a performance index associated to the global behaviour of the structure, it is necessary to modify the structure accordingly and repeat the procedure.

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