Professor Jorge Gutiérrez

Director, National Laboratory for Materials and Structural Models

LANAMME, School of Civil Engineering, University of Costa Rica,
San José, Costa Rica.
Chairman, Costa Rican Seismic Code Committee
P.O.Box 321-2010, San José, Costa Rica
jorgeg@lanamme.ucr.ac.cr
Earthquake resistant design of buildings, structural dynamics,
structural adequacy of historical and vernacular buildings

The name of Luis Esteva has been a personal referent since the time when, as a student, I learned
about the Esteva and Rosenblueth equation for estimating EQ’s peak ground accelerations. Later
on, as a young professor, I had the privilege of meeting him in several of the Latin American
Seminars on EQ Engineering that a group of colleagues, with more enthusiasm than financial
resources, organized in different countries since the late 70’s. From our very first meeting he
impressed me with his warm and friendly personality. Very soon I came to appreciate his vast
culture and enormous wealth of knowledge, permanently available for everyone. When it was the
time to organize the Seminar in Costa Rica, we invited him to deliver the inaugural lecture. He
delighted us with a profound dissertation on the intricacies of seismic design that is still
remembered as a masterpiece by all the privileged attendants.

In 1997, Luis and Haresh Shah were in Costa Rica, offering their advice on a seismic hazard
study for the earthquake prone Nicoya Peninsula. At that time, the Costa Rican Seismic Code
Committee was facing the challenge of drafting a new Code. We wanted it to belong to “the next
generation of codes”, including Performance Based Engineering PBE considerations. We invited
them to a meeting and, for several hours, we interchanged ideas and received their generous
advice. Our efforts culminated with the publication of the Costa Rican Seismic Code-2002,
where PBE considerations were included in a rational and practical way. For that reason, I
selected the subject for this important occasion, when Luis’s friends and colleagues offer a well
deserved tribute to a life full of generous contributions and meaningful achievements.

We all recognize Luis Esteva as a multitalented human being. What I personally admire the most
is his remarkable talent to conceal his huge talent.

Jorge Gutiérrez
August 28, 2005
 PERFORMANCE-BASED ENGINEERING IN THE COSTA RICAN SEISMIC CODE

ABSTRACT

In earthquake prone regions of the world, Performance-Based Engineering PBE

aims at the important and rational goal of managing seismic risk to minimize
social disruption and economic loss. As a result, there is a very strong
commitment among the academic and professional engineering communities to
develop a new generation of Seismic Codes where PBE will be explicitly
accounted for. However, this ambitious task will not be easily achieved as most
traditional Codes have evolved from empirical formulations where the nonlinear
inelastic response of the structure is accounted for by simple but conceptually
weak Reduction Factors R applied to an elastic design spectrum.

In contrast, the Costa Rican Seismic Code CRSC has implicitly considered
constant ductilitiy inelastic design spectra since it was first published in 1974.
The latest version, CRSC-2002 (CFIA 2003), introduces a conceptual framework
for PBE and explicitly considers the structural ductility in the inelastic design
spectra, allowing for a Capacity Spectrum Method CSM that has clear conceptual
and practical advantages over the methods being considered in the USA (ATC
1996; SEAOC 1999; ASCE 2000). This paper discusses the PBE philosophy of
the CRSC-2002 and describes the Capacity Spectrum Method CSM, a non linear
alternative for the structural analysis that allows the evaluation of the response
parameters required for the assessment of the building performance. The
Displacement-Based Plastic Design DBPD method, an effective and rational
procedure for EQ resistant design of buildings in complete agreement with the
PBE conceptual framework of the CRSC-2002, is also presented.

Introduction

Performance-Based Engineering, PBE, may be defined as the integrated effort to design,

construction and maintenance needed to produce engineered facilities of predictable
performance for multiple performance objectives (Fajfar and Krawinkler 1997). In earthquake
prone regions and countries, it aims at the ambitious and rational goal of managing seismic risk
to minimize social disruption and economic loss by allowing well informed decisions for the
selection of Performance Objectives PO comprehensively defined for different classes of built
structures according to their social and economic importance. For PBE to be implemented
(SEAOC 1995), the national and regional Seismic Codes and engineering standards of most
countries and regions must undergo radical changes, as most of them are essentially concerned
with the prevention of the catastrophic collapse of buildings and the subsequent large death toll
due to severe earthquakes. Furthermore, due to historic reasons, Seismic Codes were not initially
concerned with the inelastic response of the structures, limiting their considerations to the
definition of lateral forces accounting for the seismic effects, to be combined with gravity loads
in an structural design based on allowable stresses. As earthquake ground motions were
measured and dynamic methods developed to evaluate the structural response, Reduction Factors
R had to be introduced to match the empirically defined lateral forces representing the
earthquake effects upon the structure with the much larger theoretical values resulting from
elastic analyses. Eventually, codes explicitly recognized structural ductility and overstrength as
the main reasons for the R factors. However, as design seismic coefficients, obtained from elastic
spectra and reduced by R factors, do not correspond to actual constant ductility inelastic spectra,
the capacity to estimate the inelastic deformations of the structural elements and components,
which is essential to PBE, is severely limited. These limitations are readily apparent when trying
to implement conceptually sound procedures for PBE considerations, as evidenced in a vast
majority of seismic codes.

In contrast, since the first Costa Rican Seismic Code CRSC was drafted in 1973, it
included explicit considerations for ductility, inelastic deformations and displacements, ultimate
strength and capacity design concepts. Therefore, the CRSC-2002, the latest version of the Code
(CFIA 2003) incorporates PBE concepts and considerations in a rational, practical and
conceptually simple way. This paper comments those considerations and presents them in the
following sections: a) Performance Objectives; b) Definition of earthquake ground motion
demands; c) Procedures for estimation of parameters for structural response and related damage
limits and d) The Displacement-Based Plastic Design method as a design alternative for PBE.

Performance Objectives

The CRSC-2002 classifies buildings according to their importance and hazard level in
five categories: A.- Essential; B.- Hazardous: C.- Special: D.- Normal and E.- Miscellaneous. A
single Performance Objective is assigned to each category, defined in terms of a ground intensity
level and its corresponding building performance. Three ground shaking intensity levels are
defined: moderate, severe and extreme. Severe ground shaking corresponds to a Return Period
RP of 500 years, whereas moderate and extreme ground shaking are respectively defined as
having 0.75 and 1.5 times the intensity levels corresponding to severe ground shaking. These
three levels are accounted by means of an importance factor I, with values of 0.75, 1.0 and 1.5,
that directly affects the Seismic Coefficient C, as it will be commented in the next section.
Regarding building performance, two levels are defined: Immediate Occupancy IO and Life
Safety LS. When the simpler methods of linear elastic analysis are used, the basic parameter
related with these building performances is the inter-story drift; for the alternative non-linear
methods of analysis, more refined procedures for the verification of building performance, like
the inelastic internal deformations of the structural elements or components, may be used. The
Code Performance Objectives are summarized in Table 1.

The decision to consider only one Performance Objective for each Category of
Importance and Hazard Level may be considered as rather limited. Nevertheless, it recognizes
the present lack of experimental data required for reliable evaluations of building performances.
For the time being, the main objective was to introduce the concept of building performance in
the Code; the shift to multiple Performance Objectives, as well as the development of more
refined methods to evaluate building responses in terms of precise engineering parameters and
their related building performance, should be accomplished in a future revision of the Code.

Table 1. Building Performance Objectives according to Categories of Importance and

Hazard Level in the CRSC-2002.

Immediate Occupancy Life Safety

(IO) (LS)
Extreme A (Essential) B (Hazardous)
Ground (I = 1.5)
Intensity Severe C (Special) D (Normal)
Levels (I = 1.0)
Moderate E (Miscellaneous)
(I = .75)

Earthquake Ground Motion Demand

As mentioned, one of the main advantages of the CRSC for PBE considerations is the use
of Seismic Coefficients C obtained from design spectra that explicitly consider the structural
ductility instead of using the Reduction Factors R common in USA codes. Indeed, the Seismic
Coefficient is defined as:

C = aef I FED / SR (1)

where aef is the design effective peak acceleration; I is the already commented Importance
Factor. FED (for Spectral Dynamic Factor is Spanish) is an envelope function that depends on
the seismic zone, the ground site, the global structural ductility and the structural period. SR (for
Overstrength in Spanish) represents the ratio of actual to nominal structural strengths. Each term
requires further specific comments.

The design effective peak acceleration corresponds to a 500 year RP; that is, to severe
ground shaking, as commented in the previous section. Values for specific ground sites and
seismic zones are as follows:

Table 2. Design effective peak accelerations aef for severe ground shaking (RP= 500 years)

Ground site (*) Seismic Zone II Seismic Zone III Seismic Zone IV
S1 0.20 0.30 0.40
S2 0.24 0.33 0.40
S3 0.28 0.36 0.44
S4 0.34 0.36 0.36

* S1 = Rock; S2 = Firm soil; S3 = Medium dense soil; S4 = Soft soil.

Table 2 was adopted from SEAOC’s Blue Book (SEAOC 1999) with minor
modifications. The three seismic zones for the country (zones II, III and IV, corresponding to aef
rock values of 0.20, 0.30 and 0.40 of g respectively) are presented in the seismic zoning map of
Fig. 1 and were derived from the country’s most recent seismic hazard study (Laporte et al.
1994). As mentioned in the previous section, the Importance Factor I is used to scale up or down
those peak accelerations for extreme and moderate ground shaking intensity levels, respectively.

Figure 1. Seismic zones in the Costa Rican Seismic Code – 2002 (After CFIA 2003).

The FED envelope functions are normalized constant ductility design spectra. As seismic
zones and ground sites affect the ratios between maximum ground accelerations, velocities and
displacements, the resulting elastic design spectra (evaluated in all cases for a 5% of critical
damping) will be different for each one of the twelve combinations of Table 2, leading to twelve
different figures. In addition to the elastic spectrum, each figure contains constant ductility
spectra for five different structural ductility values (1.5, 2, 3, 4 and 6) obtained with well known
procedures (Newmark and Hall 1987; Newmark and Riddell 1980). For illustrative purposes,
two of these figures are presented next; note that for short periods both converge to 1.0 as they
are normalized spectra.

Fa c t or e s pe c t r a l di námi c o, FE D , pa r a s i t i os de r oc a , S1 - ZON A I I I Fa c t or e s pe c t r a l di námi c o, FE D , pa r a s i t i os de s ue l o S4 - ZON A I I I

( a mor t i gua mi e nt o ζ = 5 %; duc t i l i da de s µ= 1 , 1 . 5 , 2 , 3 , 4 , 6 ) ( a mor t i gua mi e nt o ζ = 5 %; duc t i l i da de s µ= 1 , 1 . 5 , 2 , 3 , 4 , 6 )

Elástico, µ=1.0 Elástico, µ=1.0

10. 000
µ=1.5 10, 000
µ=1.5
µ=2 µ=2
µ=3
2.50
1. 7 7
1. 4 4 µ=3 2,50
1. 12
1, 7 7 µ=4
µ=4
1. 000 0.94
0.75 1, 4 4
1, 12
µ=6
µ=6 1, 000 0,94
0,75

0. 100

0, 100

0. 010

0. 001 0, 010
0. 010 0. 100 1. 000 10. 000 0, 010 0, 100 1, 000 10, 000

P e r íodo ( s e g) P e r íodo ( s e g)

Figure 2. CRSC-2002 constant ductility FED functions for Seismic Zone III and Ground
Sites S1- Rock (left) and S4- Soft Soil (right) (After CFIA 2003).
 The ductility in the FED figures is called assigned global (i.e. structural, not local)
ductility and it represents a value likely to be developed by a particular structure. Its numerical
value is assigned from several considerations: the structural type (i.e. moment resisting frames,
dual, wall or cantilevered-column structural systems), the vertical and plan structural regularity
conditions and the local ductility of the structural elements and components, as defined from
their structural materials and detailing. Values range from 6.0 for regular (vertical and plan) steel
or concrete moment resisting frames with their elements and components detailed for optimum
local ductility to 1.0 for cantilevered column systems with moderate local ductility or structural
irregularities.

The last factor affecting the Seismic Coefficient C is the Overstrength SR which,
following Paulay (Fajfar and Paulay 1997), the code defines as “the ratio of the maximum
probable strength developed within a structure to an adopted reference (i.e. nominal) strength”.
It is obvious that, in order to compare actual ground shaking demands with maximum probable
strengths, this factor should not divide the Seismic Coefficient but multiply the nominal
structural strength; however, as both alternatives lead to identical results, SR was left in the
demand side of the equation. The main reason was “political” as the design effective peak
accelerations aef had been significantly increased from the values of the previous Code, which
did not consider Overstrength explicitly. The CRSC-2002 defines only two values of SR. A value
of 2.0 is defined for moment resisting frames, dual and wall systems whose seismic ground
shaking effects are defined in terms of lateral forces and linear elastic analysis; on the other
hand, a value of 1.2 is defined for cantilevered-column systems or for any structural system
analyzed with non linear methods of analyses, where the ultimate inelastic strengths of the
structure are calculated.

The use of Seismic Coefficients C that are derived from the actual inelastic response
spectra of a particular structural model, allows for the design of other particular structural
models as long as proper response spectra are derived for them. For instance, in frame or shear
wall buildings built with prefabricated concrete elements joined with debonded prestressed
tendons, it is possible to reach considerable interstory drifts with practically no structural
damage (Priestley and Tao 1993). However, the behavior of these structural systems is closer to
a non linear-elastic model rather than the usual elasto-plastic model. Non linear-elastic models
do not have hysteretic energy dissipation and therefore tend to produce larger displacements and
deformations than elasto-plastic models. To apply the CRSC-2002 in the design of this promising
type of prefabricated buildings, response spectra have been derived for non linear-elastic models
(Hernández 2003, Hernández and Gutiérrez 2005). The results, presented in the form of
incremental functions for the Seismic Coefficients C of the CRSC-2002, have already been used
in Costa Rica for the design of this promising type of prefabricated concrete buildings.

Most important, Seismic Coefficients C that are indeed constant ductility design spectra
derived from response spectra of specific structural models, give a tremendous advantage to the
CRSC-2002 for PBE as they avoid the need for the conceptually weak methodology of using an
elastic spectrum with an increased viscous damping to account for the nonlinear behavior, that
have been the trademark of most USA proposals (ATC 1996, ASCE 2000). Besides being
conceptually weak, this methodology leads to wrong results (Chopra and Goel 2000 and 2001b).
This important advantage will become apparent in the following section.

Structural Response and Related Performance

Once the earthquake ground motion demand has been defined, PBE requires the
evaluation of the structural response, which should be able to describe the building performance
in terms of such parameters as absolute and relative structural displacements, internal
deformations and cumulative damage of the structural members and components, absolute
accelerations, as well as its effects on architectonic and other non structural systems and
components. These parameters must be calculated by means of reliable analytical models of the
building and the input ground motions. They should be complete enough to allow for a
quantification of damage, providing the design engineer with a reliable estimation of the building
performance, which in turn should lead to well informed risk managing decisions by engineer
officials or decision makers.

The problem with the above procedure is that, as mentioned, most current seismic codes
(IAEE 1996 and 2000; ICC 2003) use force-based design procedures with linear elastic analyses
for lateral forces derived from an elastic design spectrum reduced by a force reduction factor to
account for inelastic behavior. Even if in the definition of the seismic lateral forces these codes
accept that buildings will deform beyond their linear elastic limits -inelastic response- the linear
elastic model is quite limited and the calculated parameters, basically internal forces and a crude
estimation of inelastic displacements and interstory drifts, do not allow for a precise estimation
of the building performance. The next subsections will discuss the four analytical methods
contained in the CRSC-2002, classified into the general categories of linear and non-linear
methods.

Linear Methods of Analysis

The CRSC-2002 maintains the so called Static and Dynamic Methods of analysis
included in previous CRSC as well as in most codes around the world. Essentially the Dynamic
Method corresponds to a linear elastic modal superposition method, whereas the Static Method
considers only a linear first mode approximation with the fundamental period obtained from
Rayleigh’s Method once the elastic displacements have been obtained. Due to these rather crude
approximations, this method is limited to regular buildings having heights of five stories or less.

The linear elastic Dynamic Method uses mode superposition with as many modes as
necessary to capture the most relevant features of the elastic response but its main conceptual
weakness is the use of inelastic design spectra, reduced by ductility considerations, either
implicit or explicit. In addition, for these elastic methods, the overstrength factor SR, that reduces
the Seismic Coefficient C and the corresponding lateral forces representing the earthquake
ground motions, is taken as 2.0 recognizing the presence of an important reserve strength, mostly
due to the plastic hinge formation process and the resulting inelastic redistribution of forces
characteristic of ductile structures.

It is clear that, due to ductility and overstrength considerations, the structural

displacements obtained from the elastic analysis should be much smaller than the inelastic
displacements likely to develop in the structure, which are the real indicators of the building
performance. To correct this situation the CRSC-2002 estimates the inelastic interstory drifts
multiplying the values obtained from the elastic analysis by the assumed structural ductility and
the overstrength:

∆(i) = µ SR ∆(e) (2)

where ∆(i) and ∆(e) are respectively the inelastic and elastic interstory drifts; µ is the assigned
structural ductility used for the calculation of the Seismic Coefficient C and SR is the structural
overstrength. Obviously, the maximum probable base shear strength is the elastic design base
shear –nominal strength- multiplied by the overstrength SR. These results are represented in the
following figure, which also contains a curve representing the actual maximum probable strength
and inelastic interstory drifts that these approximations are trying to match:

Base Equivalent Yield

shear Point Estimated Inelastic
Interstory drift ∆(i)
and Maximum
V(u) = SR V(e) probable strength V
(u)

“Actual” non-linear
Elastic response and “real”
design inelastic drift
V(e)
Base shear
(Nominal
strength)
Interstory drift

∆(e) SR ∆(e) ∆(i) =µ SR ∆(e)

Figure 3. Estimation of inelastic interstory drifts and maximum probable structural base
shear strength from the linear elastic Static or Dynamic Methods of CRSC-2002.

Surprisingly, at least for regular buildings, these rather crude approximations do produce
reasonably good results for inelastic interstory drifts when compared with the more refined non-
linear Capacity Spectrum Method CSM that will be presented in the next subsection (Bravo
2002; Ramirez 2002). However, the Static and Dynamic Methods of the CRSC-2002, or any of
the similar linear elastic methods presented in Seismic Codes, provide only an approximated
estimation of the structure inelastic interstory drifts, but do not provide the designer with any
information about the most important parameters that are essential for a proper evaluation of the
building performance, particularly internal deformations and cumulative damage of the structural
members. To evaluate these parameters, non-linear methods of analysis turn out to be essential.
Non-linear Methods of Analysis

For a non-linear analysis capable to provide all the significant structural parameters
required for PBE, the first obvious option is a Response History Analysis RHA, involving a time
step solution of the multi-degree-of-freedom equations of motion that represent the multistory
building (Chopra 2001). Although very sophisticated, this analysis is cumbersome and time
consuming. Furthermore, as the direct use of the design spectra is not possible, multiple analyses
for a family of accelerograms statistically related to the design spectra are required, followed by
a statistical analysis of all their responses. Nevertheless, the CRSC-2002 includes RHA as an
optional method of analysis.

As an intermediate alternative between the simple but incomplete linear elastic Static and
Dynamic Methods and the cumbersome RHA, the Capacity Spectrum Method CSM has been
proposed (Freeman 1998; ATC 1996; ASCE 2000). This is a non-linear static method that
performs a pushover analysis to determine the capacity curve, representing the lateral base force
versus a representative lateral displacement, usually at the roof. For each significant point in the
capacity curve, the state of absolute and relative displacements and internal deformations, as
well as their corresponding external and internal forces, are determined for the entire structure.
The earthquake ground motion demand is expressed in terms of the design spectra, represented
on a Sa-Sd plot with pseudo-acceleration Sa on the vertical axis, and inelastic displacement Sd on
the horizontal. The capacity curve is then expressed in the same Sa-Sd plot by simple scale
factors determined from well known principles of structural dynamics (Chopra 2001) and the
Performance Point PP is calculated. This point allows the evaluation of the peak lateral
displacements of the building relative to the ground, and the corresponding base shear,
associated with the design earthquake. For these values, the corresponding interstory drifts and
the corresponding inelastic deformations of all elements and components, necessary for the
evaluation of the Performance Objectives, can be evaluated.

As mentioned, the use of Seismic Coefficients C that are indeed constant ductility
inelastic design spectra, provide to the users of the CRSC-2002 a Capacity Spectrum Method that
has considerable conceptual and practical advantages (Chopra and Goel 1999) over the methods
currently proposed in USA (ATC 1996; FEMA 2000). The method is presented in the Code in a
very systematic way and will be described next:

a) For the particular Seismic Zone and Ground Site, a Sa-Sd plot is obtained with pseudo-
acceleration Sa on the vertical axis, and inelastic displacement Sd on the horizontal, through
the following equations:

Sa = C g (3)
Sd = µ (Sa / ω2) = µ (T/2π)2 Sa (4)

where the Seismic Coefficient C is calculated from Eq. 1 with SR = 1.2 instead of 2.0 as the
Capacity Spectrum Method is a non-linear method of analysis; µ is the specific global
ductility; ω the fundamental natural frequency and T the corresponding fundamental period.
A family of Sa-Sd spectra for different ductility is illustrated by the colour curves of Fig. 5.
b) Next, a non-linear pushover analysis is performed to the structure. For this analysis, the
CRSC-2002 recommends a distribution of lateral forces proportional to the fundamental
mode of vibration. From the resulting Capacity Curve, the intrinsic structural limit
displacement, corresponding to the point where the structure reaches its intrinsic or inherent
capacity, is determined. This value is defined as the displacement associated with those
internal deformations of the structural members or components that have been previously
defined as limiting values for a particular building performance, for instance immediate
occupation or life safety. From the Capacity Curve, an idealized bilinear response and its
corresponding Equivalent Yield Point are defined. Next, the structural global intrinsic
ductility, defined as the ratio of the intrinsic structural displacement to the Equivalent Yield
Point displacement, can be estimated. These concepts are illustrated in Fig. 4.

Base
shear Equivalent Capacity Curve
Yield Point

Structural
intrinsic
capacity

Global intrinsic ductility µGI = ∆IC / ∆YP

∆YP ∆IC

Roof displacement

Figure 4. Pushover Capacity Curve, Equivalent Yield Point and global intrinsic ductility
in the Capacity Spectrum Method of the CRSC-2002.

c) As the values of the structural Base shear and the Roof displacement are respectively
proportional to the spectral values Sa and Sd, the Capacity Curve of the previous step can be
converted to an Capacity Spectrum Curve, represented in a Sa-Sd plot; this is easily achieved
through well known relations from structural dynamics (Chopra 2001). This transformation
allows the Structural Response, now represented by the Capacity Spectrum Curve, and the
earthquake ground motion demand, also expressed in a in a Sa-Sd plot, to share a common
graph, as illustrated in Fig. 5. From this figure the structural Performance Point is readily
obtained as the point in the Capacity Spectrum Curve whose global ductility –ratio of Sd at
Performance Point to Sd corresponding to Equivalent Yield Point- approximates the ductility
interpolated from the Earthquake ground motion constant ductility demand curves. This
ductility is called required ductility and represents the structural ductility demanded by the
Earthquake ground motion to the particular structure; obviously the required ductility should
never exceed the intrinsic ductility, as this condition implies a transgression to the
 Performance Objectives. In the example of Fig. 5, the Performance Point corresponds to a
required ductility of 1.7, obtained simultaneously as the ratio of this point to the Equivalent
Yield Point and as the interpolated value between the µ = 1.5 and 2.0 constant ductility
curves.
9

µ =1
8
µ =1.5
Equivalent Yield Point
7
µ =2
µ =3
6 µ =4
Performance Point
µ =6
(Required ductility µ=1.7)
Sa (m/s2)

Capacity Spectrum Curve

4

0
0 0.0 0. 0.1 0. 0.2 0. 0.3 0. 0.4

Sd (m)
Figure 5. Graphic calculation of the Performance Point and its corresponding required
ductility according to the Capacity Spectrum Method CSM of the CRSC-2002.

d) From the Sa and Sd values corresponding to the Performance Point, the corresponding
structure Base shear and Roof displacement are easily obtained. The associated absolute
inelastic displacements and interstory drifts, as well as the internal deformations for all
elements and components, can also be obtained by interpolating the results from the two
steps of the pushover analyses flanking the Performance Point; this information allows the
evaluation of the building performance. If desired, the Maximum Probable Strength can be
obtained from the Base shear at the Performance Point multiplied by the overstrength factor
SR=1.2; however, this parameter is not as important for PBE as the inelastic displacements,
the interstory drifts or the internal deformations.

The Displacement-Based Plastic Design method

The Capacity Spectrum Method CSM presented in the previous section is a conceptually
simple and easy to apply methodology for the non-linear analysis of structures subjected to
seismic ground shaking. However, as a method of analysis, it is indeed a verification process, as
the structure needs to be already designed, with its structural layout and all its structural
elements well defined, as a step previous to the analysis. This is not a limitation for the
evaluation of existing structures, which was the initial motivation for PBE (ATC 1996; ATC
1997), but it is certainly a problem for new buildings that do require a previous design of the
structure, including a definition of the strengths and stiffness of all structural elements as well as
the force-deformation relationships of the potential plastic hinge sections as input data for the
analysis with the CSM. Surprisingly, the strengths and stiffness of the structural elements are
usually obtained through elastic methods of analysis (Priestley 2000), representing a severe
limitation for PBE as the structure is initially defined without any considerations to important
features as the inelastic redistribution of forces due to plastic hinge formation or the resulting
failure mechanism, which are totally absent from the analysis. In consequence, under severe
ground shaking, the structure may undergo uncontrolled severe inelastic deformations resulting
in an unexpected and undesirable performance.

For the sake of conceptual consistency and in order to be able to design and build a
structure that, when subjected to a non-linear analysis or, most important, to a real earthquake
ground shaking, will behave in a rather predictable way according to the desired Performance
Objectives, the definition of all element strengths and stiffness should be obtained trough plastic
design methods (Hodge 1981; Moy 1996). The author still remembers the passionate words of
Professor V. V. Bertero in his memorable lectures on plastic design: “with plastic design you tell
the structure what to do whereas with elastic design the structure is telling you what she wants
to do”; a crucial difference indeed.

For plastic design, a precise definition of the earthquake ground motion demand becomes
of paramount importance as it should consider the inelastic response characteristics of the
structure which, as already commented, is poorly represented by the elastic design spectra with
Reduction Factors R of most USA codes. To overcome this limitation, the Displacement-Based
Plastic Design DBPD method has been developed by the author; this conceptually simple
method is consistent with the Capacity Spectrum Method CSM of the CSCR-2002 described in
the previous section and can be summarized in the following six steps, also represented in Fig. 6.
A more detailed description as well as numerical examples can be found elsewhere (Gutiérrez
and Alpízar 2004):

Step 1. Initial dimensions and displacement shape.

Following Simple Plastic Theory SPT principles (Hodge 1981; Moy 1996) the
preliminary element strengths and their corresponding dimensions are initially defined as the
values required to resist the gravitational loads. Accordingly, the minimum strength of each
beam is selected to withstand the critical gravity load combination. In addition, at each structural
joint, the capacity design principle of strong columns-weak beams defines the minimum column
strength. Minimum code requirements must also be considered.

Next, a displacement shape is selected in agreement with the target interstory drift
corresponding to the building Performance Objective. This target drift may be defined from limit
internal deformations of critical structural elements or components or from specific
considerations for non-structural systems. A displacement shape proportional to the first mode of
the initially dimensioned structure, but scaled by a factor Ytar to satisfy the target design
interstory drift, is recommended (Fig. 6.1).
 1 Initial Data 2 Target Displacement
φ u N= Y φ N SDOF system
Sd tar.
uN
MN MN
(EI)Column
M *
ui
Mi Mi
Ui -Ui-1
(EI)Beam
=
∆
=
∆
K *
hi -hi-1 h h

hi
u1 max max tar
M1 M1

un Y tar.
Sd tar. = =
Geometry Displacement Limit Displacement L φN L
Shape Shape
M* M*

3 Inelastic Spectrum (Sa-Sd) 4 Distribution of Base Shear Force

Sa FN
Tel = 2 π Sdtar.
Sa µ G tar.
µ G tar. Vbase Fi

F1
Tel

Tsec
Sa

2
L Sa F = Sa L M
Sd Vbase =
Sd tar. M* M *

5 Plastic Design 6 Verification:

FN λ N FN Capacity-Demand-Diagram
Method
Fi θ
Sa
F1

Demand µ G tar.
λi FN λ 1 FN Capacity

λi Fi λ 1 Fi
µ GR

θ λ 1 F1

θ Sd
Sdy SdGR < Sd tar.
λN > λi > λ1 > 1

Figure 6. The six steps of the Displacement-Based Plastic Design DBPD method. (After
Gutiérrez and Alpízar 2004).
Step 2: Target design displacement for an equivalent SDOF system.

Once the displacement shape is defined, the target displacement Sd tar for the
corresponding single degree of freedom SDOF system is calculated from principles of dynamics
(Chopra 2001):
u Y L
S d tar = N = tar ; Γ= * (5)
Γφ N Γ M
For a system with lumped-masses mi at each level:
N
M * = ∑ miφi ; Generalized Mass
2
(6)
i =1
N
L = ∑ miφ i ; Participation factor (7)
i =1
where N is the number of stories.

Step 3: Earthquake ground motion demand.

The earthquake ground motion demand is obtained from the constant-ductility inelastic
design spectrum corresponding to the structure target ductility µG tar . This value is defined from
the selected Performance Objective as well as from the building structural classification, its plan
and vertical regularity conditions and the local ductility of its structural elements and
components.

As commented in the previous section, the constant-ductility spectrum for an elasto-

plastic system must be represented in a Sa-Sd plot with Sd corresponding to the inelastic peak
displacement. From this plot, the corresponding pseudo-acceleration Sa is obtained for the target
displacement Sd tar previously defined (Fig. 6.3). Furthermore, the expected elastic period of the
structure Tel can also be calculated with the following expression:
S d tar
Tel = 2π (8)
S a µG tar

If the calculated elastic period Tn , corresponding to the stiffness of the structure defined
in Step 1, is greater than the value calculated by Eq. 8, it would not be possible to fully reach the
global target ductility µG tar without exceeding its target displacements Sd tar . In this case, to be
able to satisfy both targets at the Performance Point, the stiffness of the structure should be
suitably increased.

Step 4: Distribution of base shear force.

Once Sa is obtained, the Base shear force can be calculated from simple principles of
dynamics (Chopra 2001):
L2
Vb = Sa (9)
M*
 This force is then vertically distributed in proportion to the masses and the selected
displacement shape, to obtain the forces at each level (Fig. 6.4):
L
F = Sa Mφ (10)
M*
where, in addition to the previously defined terms, F is the vector representing the forces at each
level, M is the Mass Matrix of the structure and φ its fundamental mode.

Step 5: Plastic design.

To define the strength of all the structural elements, a plastic design is performed
considering a series of partial collapse mechanisms, starting from the top story and descending to
the lower levels. To prevent these undesirable partial lateral collapse mechanisms, all of them
should have a safety factor greater than 1.0, say ≥1.05. In contrast, the desired complete collapse
mechanism, with plastic hinges forming at the base columns, should be equal to 1.0 to guarantee
that it will precede all undesired partial collapse mechanisms (Fig. 6.5). Once all the lateral
collapse mechanisms have been considered and the required strengths of the structural elements
have been defined, it is convenient to check for other possible mechanisms, such as soft-story.
Indeed, according to the Upper Bound Theorem of Plastic Theory, the calculated safety factor
may be on the unsafe side if an unforeseen mechanism, with a lower than unity safety factor,
precedes the desired collapse mechanism. This design method may also consider P-∆ effects,
overstrength of structural elements and rigid-finite-joint dimensions (Alpízar 2002).

Step 6: Verification procedure: Capacity Spectrum Method.

Finally, to validate the design procedure, the Capacity Spectrum Method CSM of the
CRSC-2002, described in the previous section, is applied (Fig. 6.6). As already explained, with
this procedure the Performance Point is readily determined as the point where the ductility,
determined from the Spectrum Capacity Curve and from the design spectra coincide (Fig. 5). For
this point, the required ductility, the structure inelastic absolute displacements and interstory
drifts, the internal inelastic deformations of the structural members and all other required
parameters necessary to determine the building performance, can be determined, as well as any
unforeseen undesirable collapse mechanism identified. In general terms, as the structural design
and the verification process using the CSM are actually based in the same concepts and
procedures, the latter usually becomes a self-fulfilling prophesy, with results very close to the
selected target values.

Obviously, the previously commented Response History Analysis RHA considered in the
CRSC-2002 can also be used for the verification procedure and, for particular ground motion
accelerograms, should produce more precise results than the CSM. In order to compare these two
alternative verification methods, analytical results have been obtained for a series of shear
building models having 5 and 10 stories in height, two different stiffness distributions and six
target ductility values. Each building model was designed for 4 actual ground motion
accelerograms, resulting in a total of 96 study cases. The results (Salas 2005; Salas and Gutiérrez
2005) show reasonable agreement between the CSM and RHA methods for the 5 story buildings.
For the 10 story buildings, as the higher mode effects ignored in the CSM tend to become
significant, more refined verification procedures that consider these effects, like the Modal
Pushover Analysis MPA (Chopra and Goel 2001a) are recommended over the CSM. Studies
involving RHA of more realistic building models, designed with the DBPD method, are being
carried out.

Concluding Remarks

By defining earthquake ground motion demands by means of Seismic Coefficients

derived from constant ductility inelastic design spectra and introducing non linear analysis
procedures such as the Capacity Spectrum Method CSM or the Response History Analysis RHA
method, the Costa Rican Seismic Code-2002 has been able to incorporate Performance Based
Engineering PBE concepts and considerations in a rational, practical and conceptually simple
way. Important practical limitations still remain, basically the lack of reliable data for building
performance determination. However, this can be easily accounted for in new revisions of the
Code without major modifications in the text and within the basic conceptual framework that has
been presented here.

The Displacement-Based Plastic Design DBPD method is a very effective tool for PBE
as it allows for the selection of the target global ductility and drift limits associated with the
desired performance of the building. In this method Plastic Theory is used to determine the
required strength of each structural element, necessary to produce a desired collapse mechanism
and to reach a selected Performance Point under the earthquake ground motion demand
represented by the constant ductility design spectra of the CRSC-2002. Hence, the DBPD method
is an explicit design procedure that uses the CSM of the CRSC-2002 as the analytical tool to
verify the design. For low rise buildings, where higher mode effects are unimportant, the DBPD
method leads to precise designs, able to control the inelastic response of buildings within the
established parameters corresponding to the selected Performance Objectives.

Acknowledgements

The author wishes to thank the Organizing Committee of the Luis Esteva Symposium
“Earthquake Engineering Challenges and Tendencies” for the opportunity to express his respect
and high esteem to Professor Luis Esteva in such a rich academic environment. He also
acknowledges and thanks his colleagues in the Costa Rican Seismic Code Committee and the
National Laboratory for Materials and Structural Models LANAMME, as well as his graduate
students at the University of Costa Rica, for all the fruitful discussions, support and feedback
that made this work possible.

References

Alpízar M. (2002). Idoneidad Sísmica de Edificios Prefabricados con Juntas Secas Tipo Híbrida (JSTH),
M.Sc. Thesis, University of Costa Rica. (In Spanish)
ASCE (2000). FEMA 356: Prestandard and Commentary for the Seismic Rehabilitation of Buildings,
American Society of Civil Engineers, prepared for the SAC Joint Venture, published by the
Federal Emergency Management Agency, Washington D.C.

ATC (1996). ATC-40: The Seismic Evaluation and Retrofit of Concrete Buildings, 2 volumes, Applied
Technology Council, Redwood City, CA.

ATC (1997). NEHRP Guidelines for the Seismic Rehabilitation of Buildings, FEMA 273, prepared by
ATC for the Buildings Seismic Safety Council, funded by the Federal Emergency Management
Agency FEMA, Washington, D.C.

Bravo, A. (2002). Relación entre Ductilidad Local y Ductilidad Global Asignada en Edificios de Concreto
Reforzado tipo Dual diseñados según la propuesta del Código Sísmico de Costa Rica 2002, M.Sc.
Thesis, University of Costa Rica. (In Spanish)

Chopra, A.K. (2001). Dynamics of Structures: Theory and Applications to Earthquake Engineering, 2nd
Edition, New Jersey: Prentice Hall.

Chopra A.K. and R.K. Goel (1999). Capacity-Demand-Diagram Methods Based on Inelastic Design
Spectrum, Earthquake Spectra; 15(4): 637-56.

Chopra A.K. and R.K. Goel (2000). Evaluation of NSP to Estimate Seismic Deformation: SDF Systems,
Journal of Structural Engineering, ASCE; 126(4): 482-90.

Chopra, A.K. and R.K. Goel (2001a). A Modal Pushover Procedure to Estimate Seismic Demands for
Buildings: Theory and Preliminary Evaluation., Report No. PEER-2001/03. Pacific Earthquake
Engineering Research Center, University of California, Berkeley.

Chopra A.K. and R.K. Goel (2001b). Direct Displacement-Based Design: Use of Inelastic vs. Elastic
Design Spectra, Earthquake Spectra; 17(1): 47-64.

CFIA (2003). Código Sísmico de Costa Rica – 2002, Colegio Federado de Ingenieros y de Arquitectos de
Costa Rica, Editorial Tecnológica de Costa Rica. (In Spanish)

Fajfar, P. and H. Krawinkler , Editors (1997). Seismic Design Methodologies for the Next Generation of
Codes, Balkema, The Netherlands, pg. XVIII.

Fajfar, P. and T. Paulay (1997). Notes on definitions of overstrength factors, in Fajfar, P. and H.
Krawinkler, Editors. Seismic Design Methodologies for the Next Generation of Codes, Balkema,
The Netherlands, 407-409.

Freeman, S.A. (1998). Development and Use of Capacity Spectrum Method, Paper 269, Proceedings 6th
U.S. National Conference on Earthquake Engineering, May 31-June 4, Seattle, Washington.

Gutiérrez, J. and M. Alpízar (2004). An Effective Method for Displacement-Based Earthquake Design of
Buildings. Paper 1512, Proceedings 13thWCEE, Vancouver, B.C., Canada, August 1-6, 2004.

Hernández, D. (2003). Respuesta Sísmica de Osciladores Simples con Comportamiento Elástico No-
Lineal. M.Sc. Thesis, University of Costa Rica. (In Spanish)
Hernández, D. and J. Gutiérrez (2005). Respuesta Dinámica de Osciladores Simples con comportamiento
Elástico No-Lineal: implicaciones para el Diseño Sismo-Resistente de Edificios Prefabricados de
Concreto. Accepted for publication, Proceedings VIII Seminario de Ingeniería Estructural y
Sísmica, San José, Costa Rica, September 21-23, 2005. (In Spanish)

Hodge, P.G. (1981). Plastic Analysis of Structures, McGraw-Hill.

IAEE (1996 and 2000). Regulations for Seismic Design - A World List – 1996 and Supplement – 2000,
International Association for Earthquake Engineering, Japan.

ICC (2003). International Building Code 2003, International Code Council Inc.

Laporte, M. et al. (1994). Seismic Hazard for Costa Rica, NORSAR Technical Report No. 2-14, Kjeller,
Norway.

Moy, .S.S.J. (1996). Plastic Methods for Steel and Concrete Structures, 2nd Edition, Great Britain:
Macmillan Press Ltd.

Newmark N.M. and W.J. Hall (1987). Earthquake Spectra and Design, Earthquake Engineering Research
Institute, California, USA.

Newmark N.M. and R. Riddell (1980). Inelastic Spectra for Seismic Design, Proceedings 7th WCEE,
Istanbul, Turkey; (4): 129-36.

Priestley, M.J.N. (2000). Performance Based Seismic Design. Proceedings 12thWCEE, New Zealand.

Priestley M.J.N and J.R. Tao (1993). Seismic Response of Precast Prestressed Concrete Frames with
Partially Debonded Tendons, PCI Journal, 38(1): 58-69.

Ramírez, H. (2002). Ductilidad Local y Ductilidad Global Asignada en Marcos Regulares de Concreto
Reforzado Diseñados según Propuesta del Código Sísmico de Costa Rica 2002, M.Sc. Thesis,
University of Costa Rica. (In Spanish)

Salas, L.D. (2005) Validación del Método de Diseño Plástico Basado en Desplazamientos (DPBD)
mediante Análisis No-Lineal de Respuesta en el Tiempo, M.Sc. Thesis, University of Costa Rica.
(In Spanish)

Salas, L.D. and J. Gutiérrez (2005). Validación del Método de Diseño Plástico Basado en
Desplazamientos mediante Análisis No-Lineal de Respuesta en el Tiempo, Accepted for
publication, Proceedings VIII Seminario de Ingeniería Estructural y Sísmica, San José, Costa
Rica, September 21-23, 2005. (In Spanish)

SEAOC (1995). VISION 2000: Performance Based Seismic Engineering of Buildings, Structural
Engineers Association of California, Sacramento, CA.

SEAOC (1999). Recommended Lateral Force Requirements and Commentary, 7th Edition, Seismology
Committee, Structural Engineers Association of California, Sacramento, CA.