Practical Seismic Design Criteria and Life Cycle Optimization for

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Earthquake Engineering and Engineering Seismology
                         Earthquake Engineering and Engineering Seismology                 13
Volume 1, Number 1, September 1999, pp. 13–26

          Practical Seismic Design Criteria and
       Life-Cycle Optimization for Structures with
          Hysteretic Energy-Dissipating Devices

                Luis Esteva1)           Orlando Díaz1)           Jaime García1)

      1) Professor, Institute of Engineering, National University of Mexico.


         A review is presented of several topics related to seismic design criteria,
      response and damage estimation methods and life-cycle optimization
      analysis of structural frames provided with hysteretic energy dissipating
      devices. A ductility-based design method is proposed, adequate for
      practical applications.      Stiffness-and-strength degrading hysteretic
      moment-rotation models are used to represent the behavior of critical
      sections of reinforced concrete beams and columns. These models are
      used to study the process of damage accumulation, as well as the influence
      of some seismic-design parameters on the performance and reliability of
      multistory frames.

           INTRODUCTION                             ever, many of the concepts and criteria
                                                    presented are directly applicable to
   One approach of modern earthquake                systems provided with other types or
engineering to the control of structural            devices, or may be easily adapted to their
damage produced by earthquakes is the               specific conditions.
use of energy-dissipating devices (EDD’s).              Up to now, a large portion of the
These are elements capable of undergoing            applications of EDD’s in earthquake
a large number of high amplitude                    engineering problems have been ad-
deformation cycles without experiencing             dressed to the strengthening or retro-
any significant degradation in the                  fitting of existing structures, including
strengths and stiffnesses of their cons-            both cases, when the system considered
titutive functions (stress-strain or force-         is in an undamaged state and when it
deformation curves). Our discussion will            shows some initial damage. The latter
be oriented to those devices that base              may arise as a result of previous earth-
their energy-dissipation capability on the          quakes or of other perturbations, such as
hysteretic behavior of the materials or             differential settlements or excessive
elements that constitute them. How-                 stresses produced by gravitational loads.
14               Earthquake Engineering and Engineering Seismology, Vol. 1, No. 1

Here we devote attention to this type of              The need for a simple and realistic,
applications, as well as to those when the        performance-based, seismic design cri-
use of EDD’s is considered at the design          terion is obvious. The approach pro-
stage of a new construction as a means            posed here is an attempt to contribute to
for attaining an efficient structural             satisfy this need. It is based on repre-
system from a life-cycle perspective. For         senting the EDD’s as conventional
this purpose, the mentioned devices must          structural members, characterized by
be considered as ordinary structural              their constitutive functions for alternating
elements, described in terms of their             load cycles. The main frame system is
cyclic constitutive functions. In addi-           described also in terms of its constitutive
tion, the decisions regarding design              functions, expressed in terms of either the
values and repair and maintenance                 base shear vs. top displacement curves or
policies must be made in terms of a cost-         of the shear vs. drift at each story. In
benefit analysis that accounts for the            both cases, these curves correspond to
manners in which the initial costs and            monotonic load application, and are
those expected to be generated in the             modified by simplified rules that account
future may be affected by those decisions.        for stiffness and strength degradation
In order to permit the comparison of              resulting from damage accumulation.
initial costs with those that may be                 In order to establish optimum design
generated at unknown times in the                 criteria for systems of the type described
future, the latter must be transformed            above, it is necessary to formulate the
into their equivalent values at the mo-           problem within a life-cycle optimization
ment for which initial costs are estimated.       framework that permits the comparison
   The main benefits of the use of EDD’s          of alternatives regarding design require-
in structures exposed to severe earth-            ments and maintenance policies. This
quakes arise from (a) their capability to         creates the need for damage accumu-
control damage on the structural and              lation models and indices for a given
non-structural elements of the main               system, as well as of simple relations that
system, and (b) the possibility of being          can be readily used to transform those
easily replaced when damage accu-                 indices into the corresponding variations
mulated on them reaches unacceptable              in the mechanical properties and relia-
levels. However, these objectives are not         bility functions of the system. These
explicitly considered in quantitative terms       problems are the subject of the last
in the conventional practice of earth-            sections of this paper.
quake resistant design, which empha-
sizes the additional damping supplied by
EDD’s and the local ductility demands                DUCTILITY-BASED DESIGN
experienced by them, as well as by the
members of the main structural system at             The design criterion that follows is
their critical sections. But talking about        applicable to cases where the energy-
additional equivalent damping ratio when          dissipation system (EDS) is assumed to
dealing with hysteretic dampers may be            act in parallel with the conventional frame
inadequate, or at least ambiguous, since          (CF) at each story. Thus, the design
that value is a function of the response          parameters are the contributions of both
amplitudes and frequencies, which are in          systems to the story strengths and
turn correlated.                                  stiffnesses (Rd , Rc , Kd and Kc , res-
              Esteva, Díaz, García: Practical seismic design criteria and life-cycle optimization   15

pectively).     These are conveniently
expressed in terms of the resulting story
strengths and stiffnesses (R and K), and of
the ratios a = Kd / Kc and b = Rd / Rc . The
design conditions are expressed in terms
of md*, mc* and d*, the allowable values of
the story ductility demands md and mc , for
the energy-dissipating and conventional
systems, and of the story drift d,
respectively. As a first approximation for
preliminary design, a set of values K, R, a
                                                         Fig. 1 Idealized shear vs. drift function
and b are chosen so that the combined                           for a story of a building frame
system (CS) satisfies the conditions
mentioned above for the set of nonlinear
response spectra of the design earth-                    the ordinates of the acceleration response
quake for different ductility values. For                spectra are assumed to be S(T, m). The
this purpose, the global properties of the               ratio S(T, 1) /S(T, m) will be denoted by
CS (K and R at each story, as well as the                Q, and the ratio Q / m will be denoted by g.
allowable ductility value, m* ) are ex-                  Design proceeds in accordance with the
pressed in terms of the parameters                       following steps:
defined above.       For an elasto-plastic                 1. Target values of mc* and md* are
system, the following relations are                           established. In some cases, these
obtained from Fig. 1:                                         values are established independently
                                                              of each other; that is, there are no
           d               d               d                  restrictions to the possible values of e.
mc =          ,   md =        ,     m=
         d yc            d yd             dy                  In many cases, either Kc / Rc or
                                         (1a, 1b, 1c)         Kd / Rd , or both, can only adopt
                                                              values contained within narrow
Here, dyc = Rc / Kc , dyd = Rd / Kd and dy =                  intervals, which imposes restrictions
R / K are the yield deformations of the CF,                   to the possible values of e. Under
                                                              these conditions, a new set of
the EDS and the CS, respectively. It is
                                                              allowable ductility values mc¢ < mc*, and
convenient to define a new parameter e =
                                                              md¢ < md* is adopted, such the e lies
a / b, which is equal to md* / mc* = dyc / dyd .
                                                              within the range of its acceptable
    From the foregoing concepts it is easy                    values.
to show the following relation between the
                                                           2. In any of the previous cases, the
ductility ratios developed by the CS and
                                                              design is started, for instance, by
the CF:
                                                              assuming reasonable values for Kc
          1+ a
                                                              and Kd . This determines a and T.
m = mc                                            (2)      3. Once a practically feasible value of e
                                                              has been adopted, b is taken equal to
    The design earthquake is supposed                         a / e.
to be defined by a set of elasto-plastic                   4. Taking into account Eq. (2), the target
response spectra for different ductility                      ductility value for the CS is equal to m¢
ratios. For the linear-behavior natural                       = mc¢ (1 + a ) / (1 + b ). The required
period of the system to be designed,                          base-shear ratio is determined from
16               Earthquake Engineering and Engineering Seismology, Vol. 1, No. 1

     the ordinate of the design response          different ductility levels, by a system with
     spectrum corresponding to this               an equivalent ductile capacity; second,
     ductility value. This determines the         the strength and stiffness degradation
     required lateral strength R at each          properties of members of the CF differ
     story.                                       from those typical of the EDD’s.
 5. The peak story displacement d is                   The calibration proposed is underway.
    obtained as m¢ R / K, and compared            It has consisted in comparing the ratios of
    with d*. If d < d*, the design has been       actual ductility demands (computed by
    completed at the level of global system       means of step-by-step integration) to their
    parameters; otherwise, a new set of           target values, for both, conventional
    tentative parameters must be inves-           frames and systems provided with EDD’s,
    tigated.                                      assuming they are all designed on the
                                                  basis of allowable values of story ductility
    In ordinary multi-story frames, EDD’s
are connected to the beams or to the
                                                       Figures 2 and 3 show ductility spectra
beam-column joints by means of diagonal
                                                  for systems designed in accordance with
members. The axial forces acting on the
                                                  the algorithm described above for
columns will be the result of the story
                                                  prescribed values of the allowable
shear forces taken by both, the CF and
                                                  ductilities, mc¢ and md¢, for different values
the EDS.
                                                  of a and e. The case a = 0 corresponds to
                                                  the ordinary frame without EDD’s.
                                                  These curves are representative of a
     CALIBRATION OF DESIGN                        number of curves that were obtained for
           CRITERION                              systems with different values of those
                                                  parameters. Each curve corresponds to a
    The design criterion sketched above is
consistent with modern ideas about
performance-based      design   of    con-
ventional systems. Because the EDD’s
are explicitly dealt with as structural
elements with known force-deflection
functions, peak story drifts and ductility
demands on those elements and on the
CF are estimated with the same tools
applied in the design of conventional
systems. However, before adopting this
criterion for routine practical design, it
seems convenient to calibrate it,
comparing the safety levels attained
through its use with those implicitly
accepted    for   conventional    systems
according to current design procedures.
The need to do this arises from two main
considerations. First, an approximation
is implied in the replacement of the CS,          Fig. 2 Ductility demand for 1dof con-
made of elements capable of developing                   ventional system
           Esteva, Díaz, García: Practical seismic design criteria and life-cycle optimization   17

                                                      ductility m¢ that were studied, the ductility
                                                      demands on the CF are largest for aT = 0.4
                                                      and bT = 0.33. Here, aT and bT are the
                                                      parameters of Takeda’s model, such that
                                                      aT = 0 means that the stiffness of the
                                                      unloading branch is not reduced, while b
                                                      = 1.0 corresponds to the case where the
                                                      re-loading branch reaches the original
                                                      force-deflection curve at the yield-point
                                                      for the first load cycle. For m¢ = 2, the use
                                                      of EDD’s reduces in 44 percent the
                                                      maximum ductility demand, which
                                                      occurs for a natural period equal to 2.5s.
                                                      Also, for both values of m¢ the use of EDD’s
                                                      causes an increase of the response for
                                                      systems with aT = 0, regardless of the
                                                      value of bT , for natural periods shorter
                                                      than 1.5s. Thus, the change in dynamic
                                                      properties resulting from the use of EDD’s
                                                      is not always beneficial to the expected
                                                      behavior of a structure. More systematic
                                                      studies are necessary to clarify this point.
                                                           The results discussed in the foregoing
                                                      paragraphs correspond to deterministic
                                                      systems subjected to random earthquake
Fig. 3 Ductility demand for 1dof system               ground motion of a given intensity. In
       with EDD; a = 0.5, e = 2.0                     reality, the design conditions are
                                                      established in terms of nominal values of
value of m¢, the target ductility demand for          the design variables (loads, strengths,
the CS. The ordinates show the expected               stiffnesses, safety factors, etc.).    The
values of the ductility demands on the CF             expected or the most likely value of the
for systems subjected to a sample of                  base-shear lateral capacity of a given
simulated acceleration time histories with            structure are usually significantly greater
statistical properties equal to those of              than the value implied by the nominal
the EW component of the SCT (Mexico                   base-shear coefficient used in design.
City) record of September 19, 1985                    Therefore, before using graphs similar to
(SCT850919EW). The behavior of the                    those included in Figs. 2 and 3 for the
EDD’s was assumed to be elasto-plastic,               purpose of estimating expected ductility
while the CF was assumed to behave in                 demands of actual systems, adequate
accordance with a Takeda model, with                  criteria must be developed to transform
parameters representative of those                    the nominal values of lateral strengths
obtained in laboratory tests of ductile               into their expected or most likely values.
moment-resisting reinforced concrete                  Table 1 gives an idea of the order of
frame components [1].                                 magnitude of the ratios between expected
    A comparison of Figs. 2 and 3 shows               and nominal lateral strength coefficients
that, for the two values of the target                for some typical multistory frames
18               Earthquake Engineering and Engineering Seismology, Vol. 1, No. 1

designed in accordance with the Mexico            concentrated at the critical sections
City seismic regulations of 1993. The             located at their member ends. For each
variables considered are N, the number of         of these sections, the plastic component
stories, and the nominal value of the             of the curvature varies with the acting
base-shear coefficient used in design.            internal moment as shown in Fig. 4. The
The results show that the ratios between          constitutive law shown [2] was derived
expected and nominal values of lateral            starting from one previously proposed by
capacities, measured by base-shear                Wang and Shah [3], which appeared to
coefficients, are ordinarily greater than 2,      predict excessive stiffness-and-strength
and can be very high for low values of the        degradation.      In its present version,
nominal design coefficient.                       moments vary within two enveloping
                                                  straight lines that intersect the vertical
                                                  axis at the values of the corresponding
Table 1 Comparison of nominal and
                                                  yield moments. A damage index D is
        expected values of base shear
                                                  defined, which is associated with the
        ratios (c , c )
                                                  residual stiffness and capacity of the
                                                  critical    section    in  each   loading
      N              c              c /c
                                                  direction. This index is obtained as a
                    0.05            3.88          conventional fatigue index: D = S (q /qF),
      5             0.10            2.42          where q and qF are respectively the
                    0.15            2.07          plastic-hinge rotation and the rotation at
                    0.05            3.10          failure (the point where the lateral load
      10            0.10            2.37          reaches its maximum) under monotonic
                    0.15            2.31          loading conditions.        The reloading
                    0.05            2.52
                                                  branch in a given direction is a straight
      15                                          line     that   intersects  the   vertical
                    0.10            2.32
                                                  corresponding to the maximum rotation
                    0.05            2.46
      20                                          amplitude previously reached in that
                    0.10            2.38
                                                  direction at an ordinate M ¢ smaller than


     The behavior models described in the
following paragraphs were used in the
study on ductility and damage distri-
bution in multistory frames presented
below. They are representative of cases
frequently found in engineering practice.

Behavior Models for Members of
Conventional Frames
   The nonlinear behavior of the                  Fig. 4 Stiffness-degrading functions for
conventional frames is assumed to be                     plastic-hinge rotation
           Esteva, Díaz, García: Practical seismic design criteria and life-cycle optimization   19

that associated with the virgin moment-               the initial damage values on both groups
rotation curve for monotonic load                     of elements.       Thus, it was deemed
application (M). The original and the                 convenient to obtain curves displaying
reduced values are related by the                     the expected final value of the damage
equation M ¢ = M ž (1 – l), where l = 1 –             index Dk as a function of the initial value,
exp (– k D). Moment-rotation curves of                for different intensities [9]. Figures 5
this form are used in this paper. A value             and 6 show some of those curves for
of k equal to 0.0671 was adopted, after               systems with natural periods equal
fitting the mathematical models to some               respectively to 1.0 and 1.5s, built on the
experimental results reported by Ma, et               soft soil area in Mexico City, and designed
al., [4], Wang and Shah [3], Townsend                 according to the current seismic code for
and Hanson [5], Scribner and Wight [6]                a nominal base shear ratio of 0.10. The
and Uzumeri [7].                                      stiffness and strength ratios, a and b, are
                                                      both equal to 1.0. The intensity of the
Behavior Models for EDD’s                             excitation is represented by its normalized
    The constitutive functions proposed
for EDD’s are based on some derived from
the results of laboratory tests carried out
by Aguirre and Sánchez [8] in U-shaped
hysteretic EDD’s. A non-degrading bi-
linear moment-rotation curve is assumed.
Failure of one of these elements is
assumed to occur when the fatigue index
DF exceeds unity, where DF = S Ni–1, and Ni
is the expected number of constant-
amplitude deformation cycles that lead to
fatigue failure. This number is given by
the equation Ni = exp (121 (x –0.02 – 1) ),
where xi is the ratio of the amplitude of the
                                                      Fig. 5 Cumulative damage functions for
deformation cycle to the deformation at                      frame with EDD; T = 1.0s, rk = 0.5
failure under monotonic load.

Story Damage Index
    At any individual story, the damage
index Dk in a given load direction is
defined as the ratio (K0 – Ks) / K0, where Ks
is the secant stiffness associated with the
maximum story drift in that direction,
and K0 is the corresponding initial
tangent stiffness for small deformations.
The life-cycle optimization formulation
presented at the end of this paper makes
use of conditional probability distri-
butions of the damage on both, the CF
and the EDD’s, at the end of an                       Fig. 6 Cumulative damage functions for
earthquake, in terms of the intensity and                    frame with EDD; T = 1.5s, rk = 0.5
20               Earthquake Engineering and Engineering Seismology, Vol. 1, No. 1

value, y / ym , with respect to that of           Table 2          Properties of systems studied
SCT850919EW. Use was made of the
modified Wang and Shah model, with                            Stories
                                                  System       with      T (s)     rk         y     m    r
strength-degradation parameters ob-                           EDD’s
tained from the results published by them.          a         None       1.41      ¾          ¾     4   ¾
Because these parameters seem to                    b           all      1.33     0.25       0.50   4   ¾
predict excessively high degradation                c           all      1.38     0.25       0.50   5   ¾
values, the damage indexes shown in Figs.
                                                    d           all      1.37     0.50       0.50   4   ¾
5 and 6 may be too high.
                                                    e           all      1.45     0.50       0.50   5   ¾
                                                    f           all      1.34     0.50       1.00   5   ¾
   DUCTILITY AND DAMAGE                                       1~4        1.36     0.50       1.00   5   1.0
                                                    h         1~4        1.34     0.50       1.00   5   1.1
                                                     i        1~4        1.34     0.50       1.00   5   1.2
    A study on the spatial distributions of
                                                  members of the lowest stories (see over-
ductility demands and damage indexes
                                                  strength factor r, in the last column in
was carried out in a group of several
                                                  Table 2). In all cases, the excitation was
building frames, including one that did
                                                  represented by a sample of simulated
not have EDD’s (see Table 2, taken from
                                                  acceleration time histories with statistical
Campos [10]). As in the sdof systems
                                                  properties     equal     to    those      of
studied in the previous section, the
                                                  SCT850919EW [11], normalized to the
responses computed for the system
                                                  same spectral intensity I :
without EDD’s serve as a basis of
comparison for the responses of the other                1         T1
cases. All the frames studied in this             I =
                                                         2p    ò   0
                                                                        T × S a (T , z) dT                   (3)
series were fourteen-story high, with their
fundamental periods of vibration ranging               Figure 7 shows the average ordinates
between 1.34 and 1.41s. The variables             of the elasto-plastic response spectra for
studied were the nominal value m of the           different values of the ductility value.
target design ductility for the combined          The cyclic behavior of the EDD’s was
system, the ratio y of the yield deflection       assumed to be bilinear, with a post-
of the EDD to that of the CF at each story,       yielding stiffness equal to 3.2 percent of
and the ratio rk between the stiffnesses of       that corresponding to the linear range.
the EDD and the CS at each story. The             The nonlinear behavior of the conven-
last three cases in Table 2 correspond to         tional frame was assumed to be con-
systems where EDD’s were placed only in           centrated at plastic hinges forming at its
the first four stories, with the objective of     member ends, which were represented by
concentrating all energy dissipation in           stiffness-and-strength-degrading elements
those stories. In case g, the members in          following the model proposed by Shah
the stories above the first four were             and Wang [3], modified by Esteva and
designed for the same load factor used for        Díaz [2], and summarized above. The
the lower ones. In cases h and i the load         results of the analysis are summarized in
factors used for the design of the upper          Figs. 8 and 9, which represent the mean
stories were respectively 1.1 and 1.2             values of the story damage indexes and of
times those applied for the design of the         the story ductility demands, respectively.
           Esteva, Díaz, García: Practical seismic design criteria and life-cycle optimization   21

Fig. 7 Mean response spectra for simu-                Fig. 8 Story damage indexes in multi-
       lated accelerograms                                   story frames

    The results show that story ductility
demands never exceeded significantly of
3.0, and were usually substantially
smaller, in spite of the fact that the target
design ductilities for the CS were equal to
4.0 or 5.0. This is the result of two
effects that tend to compensate each
other.    On one hand, the degrading
behavior of the CF tends to produce larger
ductility demands than expected for the
non-degrading bilinear model; on the
other, nominal values of design loads and
member capacities lie on the conservative
side of their most probable values. This
leads to larger expected ductility de-
mands than their nominal values
assumed in design.
    The beneficial influence of the non-              Fig. 9 Story ductility demand in multi-
                                                             story frames
degrading behavior of the EDD’s, as
compared with the degrading behavior of
the CF, is clearly shown in Figs. 6 and 7.            keeping target ductility values for the CS
Thus, a comparison of curves corres-                  equal to 4.0. By means of curves a, c, e
ponding to systems a, b and d show that               and f a comparison is made of the
mean story damage indexes and ductility               behavior of conventional frames designed
demands are significantly reduced when                for a target ductility value of 4.0 with that
fractions of story stiffnesses equal to 0.25          of systems with EDD’s that had been
and 0.50 are provided by EDD’s, while                 designed for a system ductility of 5.0. It
22               Earthquake Engineering and Engineering Seismology, Vol. 1, No. 1

can be appreciated that damage indexes            stand for expected value and standard
and ductility demands are lower along a           deviation, respectively. For frame a, bC
large portion of the structure’s height for       was equal to 3.18. The largest value
the systems with EDD’s than for the               adopted by any of the systems with EDD’s
conventional frame system.                        corresponded to systems d, f and i (4.76,
    Frames g, h and i were studied in             7.88 and 5.81, respectively), while the
order to obtain some information about            lowest values corresponded to systems e
the efficiency of EDD’s used only in the          and h (3.62 and 3.27, respectively).
lowest stories of a frame. The design
values of the lateral shear forces acting on
the stories of the CS were derived from the
design spectrum corresponding to a                    DAMAGE ACCUMULATION
nominal target ductility of 5.0. However,                UNDER RANDOM
an additional load factor greater than               EARTHQUAKE SEQUENCES
unity was applied to the design of the
structural members in stories above the
                                                       For simplicity, the discussion that
fourth one for systems h and i. Thus, the
                                                  follows refers to a single-story system
stories having EDD’s were conceived as
                                                  with one EDD. Just after the occurrence
subsystems       providing    partial   base
                                                  of the j-th earthquake, the damage
isolation. The figures show that for the
                                                  accumulated on that element is equal to
lowest four stories the damage indexes
                                                  Ddj , while that affecting the conventional
and ductility demands were lower for the
                                                  structural frame is equal to Dcj . After the
systems with EDD’s than for the
                                                  (j + 1)-th event, these values become
conventional frame designed for a
                                                  respectively Dd (j+1) = Ddj + dd (j+1) and Dc (j+1)
nominal target ductility value of 4.0. At
                                                  =Dcj + dc (j+1), where dd (j+1) and dc (j+1) are the
the upper stories, ductility demands in
systems g and h were significantly larger         corresponding damage increments. If Dcj
than for frame a, but remained in general         exceeds a given threshold, designated
below the maximum value experienced by            here as Drc , the frame is repaired in such
the conventional frame anywhere along             a manner as to eliminate the damage
its height.                                       accumulated, thus restoring its initial
                                                  strength and stiffness, Rc and Kc . It is
    The response studies of the systems
                                                  assumed that the damage level on the
in Table 2 were also used to assess the
                                                  frame can be assessed from observation
influence of the EDD’s on the structural
                                                  of the evidences of physical deterioration,
reliability levels. For this purpose, the
                                                  while that on the EDD’s is inferred from
reliability for the family of ground motion
records used to represent the random              the estimated value of the low-cycle-
excitation for a given intensity was              fatigue index. This information is used
measured by means of index bC (Cornell’s          to implement the preventing strategy of
beta). This was defined in terms of the           replacing the EDD after the occurrence of
random variable Z, equal to the natural           a number of high-intensity earthquakes,
logarithm of the minimum value, along             on the basis of a threshold value Drd ,
the system’s height, of the ratio of the          defined within the framework of a life-
story     available    ductility   to     the     cycle optimization approach.
corresponding ductility demand that                   Whether the process of occurrence of
results from the dynamic response                 earthquake ground motions with different
analysis: bC = E(Z) / sZ , where E( ž ) and s     characteristics involves some kind of
            Esteva, Díaz, García: Practical seismic design criteria and life-cycle optimization   23

correlation with previous history or is                objective function must be minimized:
independent from it, the levels of damage
                                                                 é¥        ù
accumulated Dcj and Ddj , j = 1, ¼, ¥, at              U = C + E êå L-q Ti ú
                                                                       i                          (4)
the end of the j-th earthquake occur as                          ëi =1     û
events of a Markov process.                 The
                                                       Here, E[ × ] stands for expected value, and
transition probabilities from (Dcj, Ddj ) to
                                                       q is an adequate discount rate.
(Dc (j+1), Dd(j+1)) are obtained from the proba-
bility density functions of dc (j+1) and dd (j+1),
which depend on Dcj and Ddj, as well as on
the probability density function of Yj +1,
                                                                       CASE STUDY
the intensity of the (j + 1)-th event.                      Esteva, et al., [9] applied the optimi-
     In order to determine the conditional             zation analysis described above to three
probability density functions of Dc (j+1) and          reinforced concrete buildings: five, ten
Dd (j+1), given the values corresponding to            and fifteen-story high, respectively. The
the end of the j-th earthquake, it is                  following paragraphs make a brief
necessary both, to calculate the joint                 description of the method used and
probability density function of the                    present the main results for the tallest
intensity of the (j + 1)-th and the waiting            building. It has a square plan area of
time to its occurrence, and to determine               346m2, and a fundamental vibration
the damage states Dci¢ and Ddi¢ of the                 period of 1.5s. It is assumed that it will
system’s components after carrying out                 be built at a soft soil site in the Valley of
the       operations   of   repairing     the          Mexico, where local soil conditions are
conventional frame members and/or                      similar to those of the recording site of the
replacing the EDD’s. The conditional                   accelerogram SCT850919EW mentioned
probability functions obtained in this                 above. The seismic hazard at the site
manner are integrated recursively in                   was expressed by an intensity-recurrence
order to obtain the marginal probability               curve previously obtained.
distributions of all Dcj and Ddj . Details                  The behavior of the reinforced
are given by Díaz and Esteva [2]. In the               concrete members was represented by
life-cycle optimization studies reported in            Wang and Shah’s model [3].               The
this paper, the direct approach described              modifications proposed by Díaz and
above was replaced by a Monte Carlo                    Esteva [2] were not included. Therefore,
simulation procedure described by Esteva,              according to the comments presented in
et al. [9].                                            the section on damage accumulation
                                                       models, it is thought that the predicted
                                                       values of the stiffness and strength
   LIFE-CYCLE OPTIMIZATION                             degradation of those members are
                                                       excessively large, thus leading to
    Let Ci be the initial construction cost            excessively large responses and damage
of a system of interest, Ti , i = 1, ¼, ¥ the          levels.   The case study is presented,
(random)     times    of    occurrence     of          however, because of the value of the
earthquakes that may affect it, and Li , i =           qualitative conclusions extracted from its
1, ¼, ¥ the losses associated with those               results.
earthquakes; they include damage and                        On the basis of some approximate
failure consequences, as well as repair                estimates of initial construction costs in
and maintenance actions. The following                 terms of the seismic design coefficient, c,
24               Earthquake Engineering and Engineering Seismology, Vol. 1, No. 1

it was concluded that the initial con-                 The estimation process starts by
struction cost C of the frame system              simulating a number of seismic histories,
varies with that coefficient according to         each consisting of the intensities and the
the equation C = C0 (1 + 0.14 × Nc1.5 ) 0.4,      times of occurrence of the corresponding
where N is the number of stories and C0 is        events. Damage levels on the CF and the
the value of C for c = 0. The cost of             EDS at the end of each event are obtained
installing an EDD of the type considered,         by Monte Carlo simulation, taking into
with a lateral yield strength P, is equal to      account the intensity of the ground
6.17 P 0.68, where P is expressed in kilo-        motion and the damage levels on both
grams and the cost is expressed in US             elements at the beginning of the
dollars.                                          earthquake.      Repair or replacements
   The repair-cost of the structural and          actions are taken, if adequate, after
non-structural elements at a given story          comparing the damage levels with the
was assumed to vary linearly with the             pre-established threshold values. The
damage index Dk at that story.                    initial damage conditions for the next
    The variables considered in the study         earthquake are established.            The
of the building selected were the seismic         corresponding costs are then obtained
design coefficient and the pre-established        and used to calculate the value of the
threshold values of the story damage              term inside the parenthesis in Eq. (4).
levels whose exceedance would lead to             After doing this for a sufficiently large
repair and/or replacement actions (Drc            sample of seismic histories, the expected
and Drd , for elements of the CF and of the       value appearing in that equation can be
EDS, respectively). At each story, the            obtained. The results for the system
contribution of the EDS, both to the              considered are shown in Table 3. There,
lateral stiffness and to the lateral strength     C is the initial construction cost for the
of the CS, is equal to 75 percent. This           structure; it is a function of the seismic
value is higher than those found in typical       design coefficient c. C1 is the initial
practical cases.                                  construction cost for a reference system,
     The mechanical properties of the             which in this case was a conventional
structure and the gravitational loads             frame, without EDD’s, designed for c = 0.1.
acting on it were taken equal to their            The figures reported in the table are
expected values. Once these values were           values of U / C1. The minimum values of
derived from the nominal design para-             this variable are shown in bold type. It is
meters and the assumed statistical mo-            easy to see that the normalized utility is
dels and safety factors, an equivalent sdof       not very sensitive to the repair and
system was assumed to represent the real          replacement thresholds. It is interesting
system for the purpose of performing a            to see that the optimum Drc is very high,
life-cycle optimization analysis. The free        which may be a consequence of the fact
variables    are   the    seismic   design        that the contribution of the CF to the
coefficient c and the threshold values for        lateral stiffness and strength of the
repair and replacement of the CF and the          system is relatively low (0.25).
EDS: Drc and Drd . For each combination               This example is only intended to show
of these variables an estimate is made of         an approach to the life-cycle optimization
the negative utility U, given by Eq. (4).         of a system with EDD’s on the basis of
The estimate is obtained by Monte Carlo           expected utilities.    Another approach
simulation.                                       deserving attention is that of decision
           Esteva, Díaz, García: Practical seismic design criteria and life-cycle optimization   25

making on the basis of tolerable risk                       Because the reductions in response
criteria.                                               and damage achieved by the use of EDD’s
                                                        have a cost, it may happen that a stronger
                                                        conventional frame, designed for the
Table 3   Influence of c, Drd on C/C1 and
                                                        same tolerable lateral drifts as one with
                                                        EDD’s, has a cost lower than that of the
  c                               Drc                   latter.  Optimum decisions in these
 C/C1             0.2      0.4           0.6     0.8    cases have to be made under a life-cycle
 0.05     0.2   1.470     1.493         1.508   1.507   framework that accounts for the process
1.168     0.4   1.472     1.493         1.510   1.501   of damage accumulation.
          0.6   1.470     1.489         1.500   1.502
 0.10     0.2   1.280     1.282         1.287   1.285
1.194     0.4   1.281     1.288         1.286   1.290
          0.6   1.281     1.283         1.285   1.289
 0.15     0.2   1.257     1.252         1.250   1.252
1.2259    0.4   1.255     1.252         1.251   1.250      The authors express their gratitude to
          0.6   1.255     1.252         1.251   1.250   Danto Campos, Miguel Ruiz and
                                                        Leonardo Veras, for their valuable colla-
                                                        boration in some of the studies reported

     Significant reductions in the expected
seismic response of conventional struc-
tural frames may be obtained by replacing
a portion of their lateral strength and
                                                        1. Bertero, V.V. and Popov, E.P. (1975).
stiffness with that provided by hysteretic
                                                           “Hysteretic    behavior    of    ductile
energy-dissipating devices.      This is a
                                                           moment-resisting reinforced concrete
consequence of the capacity of the latter
                                                           frame components,” Earthquake Engi-
elements to sustain large numbers of                       neering Research Center, University of
large-amplitude deformation cycles with-                   California, Berkeley, Report EERC 75-
out suffering significant degradation of                   16.
their mechanical properties. The best                   2. Díaz, O.J. and Esteva, L. (1993).
way to estimate their possible influence                   “Seismic damage indexes in decisions
on the response of specific nonlinear                      related to structural safety,” Proc. 7th
degrading systems is by dealing with both,                 IFIP   WG7.5     Working    Conference,
EDD’s and conventional frame members,                      University of Colorada, Boulder.
as with nonlinear elements, each charac-                3. Wang, M.L. and Shah, S.P. (1987).
                                                           “Reinforced concrete hysteresis model
terized by a constitutive function that
                                                           based on the damage concept,”
accounts for the corresponding stiffness-
                                                           Earthquake Engineering and Structural
and-strength degrading properties.
                                                           Dynamics, Vol. 15, pp. 993–1003.
     For systems where the EDD’s act in                 4. Ma, S.M., Bertero, V.V. and Popov, E.P.
parallel with the conventional frame                       (1976). “Experimental and analytical
members, an approach based on global                       studies on the hysteretic behavior of
ductilities can be used in practical design.               reinforced     concrete     rectangular
Simplified approaches applicable to other                  T-beams,” Earthquake Engineering
configurations need to be developed.                       Research Center, Report EERC 76-2,
26              Earthquake Engineering and Engineering Seismology, Vol. 1, No. 1

   University of California at Berkeley.          9. Esteva, L., Díaz, O.J. and García, J.
5. Townsend, W.H. and Hanson, R.D.                   (1998). “Optimization analysis in the
   (1977). “Reinforced concrete connection           earthquake-resistant design of struc-
   hysteresis loops,” ACI Special Publi-             tures with energy-dissipating devices,”
   cation SP53-13, pp. 351–370.                      draft of a paper reporting work in
6. Scribner, C.F. and Wight, J.K. (1978).            progress at the Institute of Engineering,
   “Delaying shear-strength decay in                 National University of Mexico.
                                                 10. Campos, D. (1998).          “Optimization
   reinforced concrete flexural members
                                                     criteria for the design of buildings with
   and large load reversals,” Report UMEE
                                                     hysteretic energy-dissipating devices,”
   78R2, Department of Civil Engineering,
                                                     doctoral thesis in progress, National
   University of Michigan.
                                                     University of Mexico.
7. Uzumeri, S.M. (1977). “Strength and           11. Alamilla, J., Esteva, L., García, J. and
   ductility of cast-in-place beam-column            Díaz, O.J. (1998). “Evolutionary pro-
   joints,” ACI Special Publication SP53-            perties of stochastic models of strong
   13, pp. 283–350.                                  ground motion: their dependence on
8. Aguirre, M. and Sánchez, R. (1992). “A            magnitude and distance,” manuscript
   structural seismic damper,” ASCE,                 submitted for publication, Institute of
   Journal of Structural Engineering, Vol.           Engineering, National University of
   118, No. 5, pp. 1158–1171.                        Mexico.

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