a new displacement-based seismic design method for

XX CONGRESO NACIONAL DE INGENIERÍA SÍSMICADEL 24 AL 27 DE NOVIEMBRE DE 2015, ACAPULCO, GUERRERO, GRAND HOTEL
SOCIEDAD MEXICANA DE INGENIERÍA SÍSMICA A. C.
A NEW DISPLACEMENT-BASED SEISMIC DESIGN METHOD FOR BUILDING
FRAMES CONSIDERING DYNAMIC INSTABILITY INDUCED BY P-DELTA
EFFECTS
Saúl E. López Ríos (1), A. Gustavo Ayala Milián (2)
1Posdoctorante, Instituto de Ingeniería, Universidad Nacional Autónoma de México, Circuito Escolar s/n, Ciudad Universitaria,
Ciudad de México, C.P 04510. [email protected] 2Profesor Investigador, Instituto de Ingeniería, Universidad Nacional Autónoma de México, Circuito Escolar s/n, Ciudad
Universitaria, Ciudad de México, C.P. 04510. [email protected]
RESUMEN
Se presenta un método de diseño sísmico basado desplazamientos para marcos planos regulares orientado a la
prevención de colapso incremental. Este método, el cual se basa en la caracterización de un sistema de múltiples grados
de libertad mediante un sistema de un grado de libertad, permite diseñar estructuras con “rigidez negativa” causada
por efectos P-Delta para satisfacer un umbral de deriva de entrepiso y / o prevenir inestabilidad dinámica. El
procedimiento de diseño requiere el uso de análisis elásticos y espectros particulares de colapso o ductilidad constantes.
Para ilustrar el potencial del método propuesto, se presenta su validación mediante análisis dinámicos incrementales.
ABSTRACT
A displacement based seismic design method for regular frame structures aimed at sidesway-collapse prevention is
presented. This method, based on the characterization of a multiple degree of freedom system by means of a single
degree of freedom system, allows the design of structures with a P-Delta induced negative stiffness to satisfy an
interstorey drift threshold and/or prevent dynamic instability. The design procedure requires the use of elastic analysis
and particular collapse or constant ductility spectra. To illustrate the potential of the method proposed, its validation
via incremental dynamic analysis is shown.
INTRODUCCIÓN
The current approach for the seismic design of structures is performance oriented since it establishes that a building
structure should be able to exhibit adequate behaviour when subjected to ground motions induced by seismic events
that may occur during its entire lifespan. In accordance with this approach, adequate performance of a structure is
deemed as the accomplishment of prescribed limit states, measured with a performance index or indices, e.g.,
interstorey drift, plastic rotation of structural components, when subjected to seismic design intensities. The set of limit
states to be satisfied for a given set of demands is known as performance objective, PO, which is defined according to
the function, type and importance of a building.
For low probability seismic events that induce high intensity ground motions, the current seismic design approach
includes in its basic PO a collapse-prevention limit state. This limit state considers that a building structure should be
able to maintain global stability, i.e., avoid collapse, accepting the possibility of severe strength-stiffness degradation
of its structural elements, such that substantial safety for its occupants under this scenario is provided.
Structural collapse is the local or global failure of a system due to the severe reduction or complete loss of its load
carrying capacity. Several studies carried out recently (e.g., Ibarra and Krawinkler (2005), Haselton et al. (2009),
Lignos and Krawinkler (2008)) have been focused primarily on the global building failure under earthquakes of the
XX Mexican Congress of Earthquake Engineering Acapulco, 2015
type known as sidesway-collapse, consequence of the severe deterioration of lateral shear stiffness when subjected to
large displacements, usually caused by the destabilizing effect of gravity loads and/or in-cycle deterioration (FEMA
2009). Sidesway-collapse may occur when a structure subjected to seismic loading exhibits an instability condition,
i.e., a segment with negative stiffness on its load-deformation relationship. Particular concern has recently emerged
for this type of collapse as the current trend towards more ductile and lighter structures may lead to such condition
(Naeim 2001).
Sidesway-collapse of frame structures with high ductility capacity is governed by P-Delta effects (Adam and Jäger
(2012b). For this reason, several authors, e.g., Bernal (1998) and Adam and Jäger (2012b), have proposed simplified
design procedures that provide a sufficient approximation of the collapse capacity of structures associated with P-Delta
induced instability, as an alternative to rigorous collapse assessment methods. As it is well known, the latter are not fit
for practical seismic design applications since they require refined structural models and extensive non-linear time
history analyses and, consequently, are computationally expensive and time consuming.
However, few efforts in the development of simplified design methods based on performance principles that seek to
appropriately consider P-Delta induced instability have been carried out. This circumstance has motivated the authors
of this paper to develop and validate a new simplified design method that can be used to design instability prone
structures due to P-Delta effects, not only for deformation control corresponding to a near collapse limit state as
currently established in several design codes, but also for actual sidesway-collapse prevention.
The displacement based seismic design method described in this work is an evolution of the original method proposed
by Ayala et al. (2012). The design approach consists on the definition of a bilinear or trilinear behaviour curve, i.e.,
spectral displacement, Sd, vs. pseudo-acceleration, Sa, of a reference SDOF system corresponding to the fundamental
mode of a structure, defined in such a way that a given PO, is satisfied. The yield and maximum displacements of the
aforementioned curve are defined in such a way that a given interstorey drift threshold and/or structural stability is
achieved under the design demands corresponding to the considered limit states. From such curve and a criterion to
account for higher mode contribution, which involves modal spectral analysis, the corresponding design storey shears
are defined for which the structural components are designed.
The method proposed allows the design of regular framed structures that are potentially unstable due to the effects of
gravity loads, i.e., P-Delta induced negative post-yield stiffness, under severe seismic demands. Even though instability
due to gravity loads of a designed structure is an undesirable condition, in many cases it is not practical or economically
feasible to avoid completely such condition (Fenwick et al. 1992). Currently, the method proposed addresses only the
design of non-degrading structures with P-Delta induced instability; in-cycle degradation and cyclic deterioration of
structures are not considered in this study as it is a subject of current investigations. Nonetheless, ductile structures
need stringent detailing requirements, hence, moderate levels of deterioration can be expected.
This paper presents a brief review on the influence of P-Delta effects in structural response followed by a detailed, yet
brief, description of the fundamentals of the method proposed for frame structures and a step-by-step design procedure.
Subsequently, it shows the results of the validation of example applications of 8-,12-,16- & 20 storey frames via
incremental dynamic analysis (IDA), using a record set of far-source earthquakes recorded at soft soil types of the
Valley of Mexico. In the final part of the paper a discussion of the results and conclusions regarding this investigation
is given.
P-Delta effects in a SDOF system
Second order effects, commonly known to as P-Delta effects, is the name given to the amplification of demands of a
structure subjected to lateral displacements due to the action of vertical loads over its deformed shape. To attain an
insight of P-Delta effects in structures, consider the simple case of a SDOF system of height H consisting of a rigid
column attached at its base to a flexural spring, whose behaviour is defined by a bilinear backbone of an elastic stiffness
DEL 24 AL 27 DE NOVIEMBRE DE 2015, ACAPULCO, GUERRERO, GRAND HOTEL
KE and a post-yield stiffness, KD=αKE; a mass, m, and a viscous damper with a damping coefficient c. Such system is
subjected to a lateral load V and a vertical load P (fig. 1).
Figure 1 SDOF system subjected to lateral dynamic loading and vertical force
As may be inferred from the preceding figure, the vertical load generates an additional moment at its base due to the
displacement of the system, Δ, thus, the total moment, M, is:
= + (1)
The increase in flexural demand with respect to the first order response can be interpreted as a decrease of the system´s
lateral strength and stiffness, which can be characterized by a geometric transformation. Under such interpretation, the
stiffness decrease due to gravity load can be represented via the geometric stiffness (eq. 2) and, consequently, the
effective tangent stiffness of the SDOF system in either its elastic or inelastic stage of behaviour, Kt', i.e., second order
tangent stiffness, can be defined by eq. 3, in which Kt denotes the first order tangent stiffness.
=
′ = − (3)
The parameter commonly used to quantify the influence of P-Delta effects in the lateral stiffness of a structure is the
so-called stability coefficient θ, defined as the ratio of stiffness decrease to first order elastic stiffness, KE. For the
SDOF system shown in fig. 1, such parameter, which is the same in any state of the structure, is given by:
= − ′

(4)
The decrease in strength and stiffness of the structure due to gravity loads may be interpreted as rotation of its load-
deformation capacity relationship in function of the stability coefficient (fig. 2). Accordingly, the effective elastic
stiffness of the system can be written in terms of θ as,
′ = (1 − ) (5)
Consequently, the effective period of vibration, i.e., second order period, of the SDOF system, T´, considering the
influence of gravity loads is,
′ =
√1 − (6)
Furthermore, the effective post-yield stiffness in terms of KE, the first order post-yield stiffness ratio, α, and θ is
′ = ( − ) (7)
Evidently, this “shearing” effect of the load displacement relationship has an influence in the response of SDOF
systems under dynamic loading. The consequences of stiffness modification due to gravity loads in the elastic stage of
behaviour arise from the period lengthening that influences dynamic response (eq. 6). However, it is in the inelastic
stage where P-Delta effects influence significantly the system’s response as the first order tangent stiffness in such
instance (KD) is considerably less than that of the elastic stage, hence, the effect of gravity loads in the lateral stiffness
of the system is more severe. In fact, the stiffness reduction may be of such magnitude that the post-yield stiffness
becomes negative (α’ < 0), leading to a static instability condition which, in turn, may induce sidesway-collapse due
to dynamic instability, defined as a disproportionate response of a system subjected to dynamic loading for a relatively
small variation of its intensity in a lapse of time (fig. 3).
Figure 2 Shearing of load-deformation relationship due to P-Delta effects (Adam and Jagger 2012b)
A necessary condition for such phenomenon to occur is a static instability condition, however, it is not sufficient, as
the inertial and damping forces may provide a stabilizing effect in the response (Bernal 1998). Therefore, the intensity
and frequency content of the input are essential factors in the occurrence of dynamic instability. Consequently, non-
linear dynamic analysis is necessary to identify sidesway-collapse for a particular dynamic loading. Assuming that the
system is modelled correctly, dynamic instability is considered equivalent to numerical instability, i.e, disproportionate
displacements in a transient analysis (Vamvatsikos and Cornell 2002).
A particular feature of systems with negative post-yield stiffness is that, even though the displacement and,
accordingly, the ductility associated with dynamic instability are infinite by definition, its displacement response prior
to sidesway-collapse is bounded by the static (monotonic) collapse ductility µcst given by eq. 8 (Jäger and Adam 2013),
defined as the ductility for which the system reaches zero ultimate strength under monotonic loading. Therefore,
collapse ductility can be conveniently defined as µcst.
= 1 −
(8)
The slope of the post-yield stiffness (stability coefficient) gives an insight of the instability potential of the system. In
an SDOF system with a given yield strength, the larger this coefficient, the more likely dynamic instability will occur.
Furthermore, the yield strength of the system also plays a significant role in dynamic stability, since the decrease of
strength in the post-yielding stage necessary to attain null ultimate strength is larger as the yield strength is increased.
Hence, systems with larger stability coefficients require larger lateral strength to avoid dynamic instability.
Moreover, dynamic instability depends on the characteristics of the hysteretic model that rules the response of the
system. Several studies show that systems with bilinear behaviour are more susceptible to dynamic instability than
peak-oriented models (Rahnama and Krawinkler 1993, Pettinga and Priestley 2007, Adam and Jäger 2012a).
According to such studies, this is due to the fact that the response of bilinear systems remains, in larger and more lapses
of time, in a negative segment, thus, the displacement increase in a single direction is more severe than that of systems
with peak-oriented behaviour.
Figure 3 Dynamic instability of an SDOF system (FEMA 2009)
P-Delta effects in MDOF system
Second order effects in MDOF structures can be characterized using an approach similar to the aforementioned for
SDOF systems via the geometric stiffness matrix, [Kg], which represents the stiffness decrement due to gravity loading
of the vertical load carrying components of the structure in terms of their ratios of axial load to storey height.
Accordingly, the effective tangent stiffness matrix of the system, [Kt′], i.e., second order stiffness matrix, of an MDOF
structure in any stage of behaviour can be defined as:
[′] = [] − [] (9)
where [Kt] is the first order tangent stiffness matrix.
A measure of the effect of gravity loads in the global lateral stiffness of a MDOF structure subjected to earthquake
loading for a particular damage state may be attained from the comparison of the results of first and second order
modal analyses, expressed by the following equations,
([] − []){} = {0}
(10)
([′] − ′[]){′} = {0} (11)
where, [M] is the mass matrix, λ and λ’ are the first and second order eigen-values respectively; {} and {′} are their
corresponding eigen-vectors, i.e., modal shapes.
It is evident from inspection of the preceding equations that the stiffness decrease due to gravity loads leads to a
modification of all eigen-values and eigen-vectors. For lower modes such reduction is higher since the eigen-values
are smaller. For this reason, the influence of P-Delta in the tangent global stiffness of a MDOF structure in a certain
damage state may be measured by the stability coefficient θt associated with the properties of the first mode of the
structure, which may be defined as:
= 1
1 1
(12)
where λ1 E is the first order eigen-value of the fundamental mode in the elastic stage and MPR1
E the corresponding
modal mass participation ratio; λ1 t is the first order eigen-value of the fundamental mode in any damage state and
MPR1 t the associated modal mass participation ratio, and MPR1
t′ and λ1 t′ are the corresponding second order properties.
In the aforementioned equation, the stability coefficient with subindex t to denote the tangent stiffness associated to a
particular damage state, as in MDOF structures this parameter varies from one damage state to another, unlike SDOF
systems in which the stability coefficient is the same regardless of the damage state of the structure. Such difference
between stability coefficients in MDOF structures is a consequence of the difference between modal properties from
one damage state to another. Moreover, as the first and second order dynamic properties of a particular damage state
are different, the stability coefficient given by eq. 12 is normalized to the first order modal participation factor to
provide a reference with respect to the first order modal stiffness.
If at least one negative eigen-value occurs, the structure presents a static instability condition for the associated modal
shape. In a similar manner as in SDOF systems, the fundamental eigen-value provides insight of the instability potential
of the structure; the larger the negative eigen-value, the more susceptible the structure is to fail in a sidesway mode.
The magnitude of the eigen-value increases as the damage state of the structure is more severe since the elements of
the geometric stiffness matrix are small.
Due to their characteristics, multi-storey frame structures are particularly susceptible to P-Delta induced dynamic
instability. In fact, P-Delta effects govern sidesway-collapse of flexible frames with gravity load induced negative
stiffness, to such an extent that the influence of cyclic deterioration is not significant (Adam and Jäger 2012b; and
Ibarra and Krawinkler 2005). Furthermore, several studies show that prediction of P-Delta induced sidesway-collapse
may be carried out with a sufficient degree of approximation by means of an inelastic SDOF system characterizing the
fundamental mode of vibration whose properties are defined from pushover analysis using a first mode modal load
pattern (Adam and Jäger 2012a). The approach followed in the design method proposed relies on a similar approach,
but uses elastic analyses only, and, as shall be shown in the following, allows the seismic design of instability prone
frame structures due to P-Delta effects.
FUNDAMENTALS OF THE METHOD PROPOSED
Reference SDOF system and design approach
The main basis of the method proposed is the assumption that it is possible to approximate the maximum displacement
response of an inelastic MDOF structure by means of an inelastic SDOF oscillator whose properties are consistent
with those of the fundamental (first) mode of the structure. In the method proposed, this SDOF system is termed as
reference SDOF system, RSDOF, (Ayala 2001) and, accordingly, the main tool used to estimate the maximum response
of a given structure is the backbone curve of spectral displacement, Sd, vs. spectral pseudo-acceleration (strength per
unit mass), Sa, i.e., the so-called behaviour curve of the reference SDOF oscillator.
Accordingly, the design approach followed in the method proposed is the definition of a design behaviour curve that
provides the stiffness and strength required by the structure to satisfy a given performance objective (PO). To design
a structure for a PO comprised of a serviceability limit state (SLS) and a collapse prevention limit state (CPLS), such
as that established in the Mexico City Building Code (GDF 2004), a design bilinear behaviour curve is built (fig. 4).
The characteristic points that define this curve are: origin (0, 0), yield (Sdy, Say) and ultimate (Sdu, Sau).
The first branch of this curve characterizes the elastic stage of behaviour of the structure. Its slope, λE, is limited in
such a way that the interstorey drift thresholds for both the service, SLS, or ultimate, ULS, limit states are not exceeded
for the demand levels given by the corresponding design spectrum. The demand associated to the SLS is given by the
point (Sds, Sas). The second branch of the curve portrays the structural post-yielding behaviour, whose slope, αλE, is a
function of the design damage state, e.g., strong column-weak beam, at maximum inelastic response under the design
demand considered for the ULS.
The yield displacement, Sdy depends on the material and geometric properties of the structural elements; the ultimate
displacement, Sdu, is set so that the interstorey drift threshold of the ULS is not exceeded for the corresponding demand
level. The yield spectral pseudo-acceleration, Say, is directly related to the design forces of the structural elements that
are expected to yield in accordance with the considered design damage state, whereas the ultimate spectral pseudo-
acceleration, Sau, is associated with the force levels of the structural components that should remain elastic or develop,
at most, limited inelastic behaviour. The remainder of Say and Sau is referred to as post-yield spectral pseudo-
acceleration, Sapy, which is related to the redistribution of forces up to the point of maximum inelastic response.
Figure 4 Design behaviour curve of the RSDOF system for a two-limit state performance objective
Elastic and “damaged” model
To define the design behaviour curve, a structural model is built considering a preliminary proposal of structural
element sizes, as it is done in any conventional design procedure. Evidently, the dynamic properties of the RSDOF
structure in the elastic stage can be calculated in a straightforward manner using modal analysis. To estimate the
dynamic properties associated with the inelastic branch of the RSDOF, it is considered that modal analysis of an elastic
model whose stiffness is representative of the stiffness of the MDOF system under the assumed damage state can
provide a congruent estimation of such properties. For such purpose, a replica of the elastic model is created, in which
the plastic hinges corresponding to the design damage state are characterized by rotational springs whose stiffnesses
match the post-yield stiffnesses of the structural components; such replica is termed as damaged model (fig. 5).
Figure 5 Damaged model representing a strong column-weak beam damage distribution From the modal analysis of the aforementioned structural models, the eigen-values and eigen-vectors, i.e., modal
stiffnesses and shapes, are obtained. As the models characterize the stiffness of the structure in its elastic and inelastic
behaviour stages, from structural dynamics concepts regarding the modal superposition approach, the slopes of the Sd
vs. Sa relations of each mode j in the spectral space are related to the eigen-values λE j and λD
j, respectively (Ayala
2001). Accordingly, the second modal post-yield stiffness ratios can be defined by eq. 13,
=

where ′ and
′ are the second order modal mass participation ratios of mode j of the elastic and damaged
models, respectively.
The post-yield stiffness ratio of the RSDOF system, α, is equal to α1 of the corresponding MDOF structure (fig. 4). In
the following, the variables corresponding to the parameters and demands of the RSDOF system are denoted without
a sub-index to differentiate them from those of higher modes of the actual MDOF structure where the mode numbers
are indicated.
Consideration of P-Delta effects
P-Delta effects are considered in a straightforward manner by performing modal analysis using the geometric stiffness
matrix formulation. If at least a negative eigen-value is attained, that for regular frame structures will likely be that of
the first mode, the slope of the second branch of the design behaviour curve is negative, thus, indicating that the
structure is prone to be dynamically unstable for the design seismic demand associated with the ULS. If such is the
case, and it is not considered feasible, economically-wise, to avoid such condition, the objective of the design is to
ensure dynamic stability of the system for the design demand considered. For such purpose, it is necessary to use an
artifice to appropriately account for second order effects in the representation of the MDOF structure via an SDOF
system, the so-called auxiliary SDOF system. Moreover, the design demands are defined from a particular type of
spectra built for SDOF systems with negative post-yield stiffness, which is related with the level of axial load present
in the system.
Auxiliary SDOF system
The auxiliary SDOF system, (Ibarra and Krawinkler 2005), is an artifice that has been employed extensively by several
authors to approximate the collapse capacity of a MDOF structure with P-Delta induced negative stiffness via an
equivalent SDOF system. It consists on the definition of a SDOF system whose second order backbone matches the
corresponding of the equivalent SDOF system (fig. 6). It is necessary to use this artifice since the stability coefficients
in the elastic and inelastic stages of behaviour in MDOF systems are usually different (Medina and Krawinkler 2003),
whereas in SDOF systems the stability coefficient is the same regardless of its damage state. In the method proposed,
the elastic and inelastic stability coefficients of the MDOF structures are estimated via the following equations:
= MPR − MPR ′′
MPR (15)
where λE’ and MPRE’, λE, and MPRE, are the elastic fundamental eigen-values and modal mass participation ratios
with and without P-Delta effects, respectively; λD’ and MPRD’, λD, and MPRD are the inelastic fundamental eigen-
values and modal mass participation ratios of the structure with and without P-Delta effects, respectively. The period
and stability coefficient of the auxiliary SDOF system are calculated by means of Eqs. 16 and 17 (Ibarra and Krawinkler
2005).
a) b)
Figure 6 First and second order design SDOF systems: a) Reference SDOF system; b) Auxiliary SDOF system
= 1√ 1 −
1 − −+
Definition of design displacement of RSDOF
As the elastic and damage models are representative of the stiffness properties of the structure, a reasonable
approximation of the design drift and displacement shape associated with a given limit state may be estimated with the
fundamental mode shape obtained from the modal analyses of such models. This can be carried out by equating a given
interstorey drift threshold to the maximum modal interstorey drift: the critical storey of such shape is where the latter
occurs (fig. 7). Accordingly, for the serviceability limit state considered herein, where elastic behaviour of the structure
is a requirement, Sds, is calculated with the following expression:
=
(18)
where IDRs denotes the interstorey drift threshold for the SLS, k is the critical storey in the elastic stage; Hk is the
height of the critical storey; ΦE k 1′ and ΦE
k-1 1′ are the fundamental modal coordinates of the critical storey and preceding
storey obtained from modal analysis (second order) of the elastic model, respectively; ΓE 1′ is the fundamental modal
participation factor.
The design yield displacement, Sdy, can be defined in a similar manner as:
=
(19)
where IDRy is the yield interstorey drift which can be estimated with eq. 20 (Priestley et al. 2007),
=
(20)
in which βm is a constant that depends on the material and structural type, L is the beam span and hb is the beam depth.
To calculate the ultimate design displacement, Sdu, associated with inelastic behaviour, it is assumed that the
displacement shape of a bilinear MDOF system subjected to severe earthquake excitation is a linear combination of
the eigen-vectors corresponding to the elastic and inelastic stages. Therefore, the design spectral displacements of the
reference SDOF system may be estimated by means of the following equations:
=
(22)
′] (23)
where IDRu denotes the interstorey drift ratio threshold of the ULS; k, is the critical storey in the inelastic stage; Hk is
the height of the critical storey; ΓE 1′ and ΓD
1′ identifies the modal participation factor of the first mode attained from
the modal analyses (second order) of the elastic and inelastic model, respectively; ΦE i 1′ and ΦD
i 1′are the corresponding
elastic and inelastic modal shapes; ΦD*i 1′ identifies the design modal shape for the ULS; ΦD*k 1′ and ΦD*k-1 1′ denote
the coordinates of the modal shape corresponding to the critical storey and the preceding storey, respectively. It should
be noted that the critical storey k in the inelastic stage of the structure is not necessarily the same critical storey in the
elastic stage. In frames that develop a strong column-weak beam damage state with yielding at the bases of the first
floor columns, the maximum inelastic interstorey drift usually occurs at the base storey or at its vicinity. Therefore, k
in Eqs. 18 and 19 will likely be different than in eq. 22.
Consequently, design displacement profiles (first mode only), Ds, Dy and Du, corresponding to the SLS, yield and ULS
of the structure, respectively, can be defined via the following equations.
= Γ1 ′ 1
′
′
∗′
(26)
Figure 7 Design displacement shapes and definition of design displacement of reference SDOF system
Constant ductility spectra and constant ductility spectra of unstable SDOF systems due to P-Delta effects
The design demands used in the method proposed are extracted from constant ductility spectra and collapse spectra
built from SDOF systems that exhibit a P-Delta induced negative post-yield stiffness. The parameters that define such
type of spectra are an effective post-yield stiffness ratio, i.e., the difference of stability coefficient and hardening ratio,
θ – α; a damping ratio, ζ; and a hysteresis rule (Adam and Jäger 2012a). The former are used for design applications
considering a near-collapse limit state in which “collapse” is deemed by the exceedance of an interstorey drift and/or
a ductility thresholds. Collapse spectra are used when the design application is oriented towards actual sidesway-
collapse.
Regardless of the collapse definition employed, spectra in terms of yield pseudo-acceleration and ultimate
displacement are necessary. In order to provide straightforwardness to the application of the method, the abscissas of
the spectra should be the second order periods of the SDOF systems. Even though the displacement response associated
with collapse is infinite by definition, ultimate displacement spectra can be built considering the static collapse
ductility, μcst (eq. 8). In fact, it can be fairly argued that collapse spectra is a constant ductility spectra associated to μcst
(Lopez et al. 2015).
In this study, constant ductility spectra and collapse spectra, corresponding to 16%, 50%, 84% response (non-
exceedance), were built for a set of records used in Miranda and Ruiz-Garcia (2002). Such record set is comprised of
100 actual earthquake accelerograms, recorded at soft soil sites at the Valley of Mexico, denoted as VM set in the
remainder of this paper. The ductility values considered were 3 to 10 in increments of one. Both types of spectra were
calculated considering aleatory uncertainty, i.e., record to record variability only. Figs. 8 and 9 depicts a set of collapse
spectra (µ = µcst) and constant ductility spectra (µ = 4) of the aforementioned set of records in terms of median yield
pseudo-acceleration and median ultimate displacement for various θ – α values and ζ = 0.02.
a) b)
Figure 8 Median spectra in terms of ultimate displacement of SDOF systems with P-Delta induced negative post-yield stiffness of the VM record set for various θ-α values: a) constant ductility (μ = 4); b) collapse (μ = μcst)
a) b)
Figure 9 Median spectra in terms of yield pseudo-acceleration of SDOF systems with P-Delta induced negative post- yield stiffness of the VM record set for various θ-α values: a) constant ductility (μ = 4); b) collapse (μ = μcst)
Definition of modal force demands
According to Sullivan et al. (2008) the contribution of higher modes to force demands may be significant in frame
structures in comparison with the displacement response, thus, it is necessary to take them into account in its design.
For the sake of simplicity, each modal yield pseudo-acceleration, Sa′y j, including that of the first mode, is defined
directly from the yield pseudo-acceleration spectrum corresponding to the properties of the RSDOF system, assuming
implicitly that such spectrum is representative of the demand of higher modes up to the yield point of the structure.
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00
0.05
0.10
0.15
0.20
T (s)
U lt
0.15
0.30
0.45
0.60
0.10
0.20
0.30
0.40
T (s)
Y ie
ld p
ra ti
o n
S a
y ( m
/s 2 )
0.10
0.20
0.30
0.40
ra ti
o n
S a
y ( m
/s 2 )
The maximum inelastic demands are defined assuming that higher modes develop the same ductility as the first mode
and that inelastic coupling is negligible. Hence, the ultimate pseudo-acceleration of the j-th mode, Sa′u j, including that
of the RSDOF system, is given by the following equation:
′ = ′ [1 +∝ ′( − 1)] (27)
where α′j is the second order post-yield stiffness ratio calculated with eq. 13 considering the second order modal
properties, λE a′, MPRE
Modal spectral analysis
In accordance with the assumptions regarding the contribution of higher modes, the design demands for the elastic and
inelastic stages of behaviour are obtained via modal spectral analysis of the simplified models using conventional
modal combination rules, such as SRSS, CQC, as applicable. The considered inelastic strength per unit mass spectrum
is employed in the modal spectral analysis of both simplified models. As mentioned in the foregoing section, the modal
demands of the elastic model are obtained directly from the ordinates of the considered spectrum. For the damaged
model, the modal demands are defined by the post-yield strength, Sapy j, which is given by
′ = ′ ∝ ′( − 1) (28)
The ULS design forces are calculated via the modal combination of the maximum inelastic responses, global and local,
of all modes. If the SRSS rule is employed, the design forces, Fk, of each structural component, i.e., design moments,
shears and axial forces; are estimated with the following equation,
= √∑( +
)2

(29)
where FE k j denotes the demand of element k corresponding to mode j obtained from the modal spectral analysis of the
elastic model; FD k j is the demand of element k associated with mode j attained from the modal spectral analysis of the
damaged model and n is the number of modes considered.
Furthermore, for the sake of consistency, the design displacement shape may be calculated by means of modal
combination considering the aforementioned assumptions regarding higher mode contribution. This can be carried out
by calculating the displacement shapes for each mode via Eqs. 30 to 32, using the demands obtained from the design
spectrum associated with the ductility calculated previously considering the fundamental mode only, and then
combining them through the selected modal combination rule. Such adjustment of the displacement shape is usually
not necessary, but it may be used to attain a rough estimation of higher mode contribution for long period structures
for which it may be significant.
Δ = √∑ [βΓ ′(
′ − −1 ′)]
2 =1
2 =1
2 =1
(32)
where:
∗′ =
′]
(33)
and βj is the displacement of mode j normalized by the displacement of the RSDOF system:
β =
DESIGN PROCEDURE
The application of the method proposed to design an instability prone frame due to P-Delta effects aimed to satisfy a
two-limit state PO, considering actual collapse prevention or deformation control associated with a near collapse
condition, can be summarized in the following steps.
1. Pre-dimensioning of the structural elements based on designer’s experience or a rough force-based design.
Consequently, an elastic model is built in a structural analysis software.
2. Modal analysis of the elastic model without considering P-Delta effects. From the first order elastic properties, the
required SLS displacement, Sds, is calculated using eq. 18
3. The required first order period Ts corresponding to Sds is obtained from the design displacement spectrum associated
with the SLS. Subsequently, the elastic model is modified so that its fundamental period matches Ts.
4. Gravity load and modal analyses of the elastic model considering P-Delta effects. From the results the stability
coefficient, θE, is calculated with eq. 19.
5. Definition of the design damage distribution corresponding to strong-column weak-beam behaviour with inelastic
action in first storey column bases for the ULS and construction of the “damaged model”.
6. Modal analyses of the “damaged model” considering P-Delta effects. If a negative eigen-value is attained, modal
analysis without P-Delta effects is carried out. Subsequently, the period of the auxiliary SDOF system related to the
required stiffness for the SLS, TAUXS, and the corresponding stability coefficient, θAUX, are calculated via Eqs. 14 to
17.
7. For a deformation control-based design, the yield and ultimate displacements are calculated with Eqs. 19 to 23,
hence, the design ductility µ is defined. If the application is oriented towards sidesway-collapse control, µ may be
taken as µcst.
8. From the ULS design ultimate displacement spectrum corresponding to µ and θAUX – α, the required auxiliary period
for such limit state, TAUXU, is obtained.
9. Definition of the final design auxiliary period, Tdes, as the smaller value of TAUXS and TAUXU. If the latter is the
smallest value and is significantly different than TAUXS, recalculate the effective negative post-yield stiffness, θAUX –
α, using the first mode eigen-value calculated in step 5 and repeat steps 6 to 8 until a sufficient approximation of Tdes
is attained.
10. Modal spectral analysis of the simplified models and calculation of design forces of structural elements with eq.
29. It is recommended that in this step the design displacement is calculated using Eqs. 30 to 34, since an estimate of
higher mode demands is possible at this instance.
11. Design and detailing of structural elements with appropriate criteria regarding the behaviour of materials and
structural types according to building codes or other accepted design provisions.
VALIDATION OF THE METHOD PROPOSED
Overview of the case studies considered and validation approach
In order to validate the method proposed, design applications aimed at theoretical P-Delta induced collapse, were
carried out for 8-,12-, 16- and 20-storey non-deteriorating generic frames, regular in elevation. The seismic demands
considered were a set of 100 real earthquake records from soft soil sites at the Valley of Mexico (Miranda and Ruiz-
Garcia 2002). Each frame was designed for different magnitudes of axial load corresponding to θAUX – α values equal
to 0.025 to 0.10 in increments of 0.025.
To allow flexibility in the validation of the method proposed for various levels of axial load and ductility values,
strength and stiffness independency in structural components is considered in these case studies. Accordingly, a fixed
fundamental period was considered in such case studies; the periods were estimated using empirical expressions given
by Chopra and Goel (2000), which provide mean period values from measurements of actual structures in function of
their height and material. The design was carried out assuming that such fundamental period is that required by the
SLS and that the stiffness requirement of the structure is governed by such limit state in accordance with step 3 of the
design procedure. Subsequently, the strength of structural components was provided according to the design targets of
the ULS in conformance with the rest of the design procedure.
The validation of the design applications was carried out using incremental dynamic analysis, IDA, (Vamvatsikos and
Cornell 2002); the intensity measured considered was the elastic pseudo-acceleration of the corresponding linear
system. Collapse assessment was carried out by means of the IM-approach (Vamvatsikos and Cornell 2002; Ibarra and
Krawinkler 2002). In accordance with this approach, from each single-record IDA, the individual intensities associated
with collapse were attained. Subsequently, to characterize structural response in probabilistic terms, the 16%, 50% and
84% collapse intensities were calculated via counted statistics and were compared with the corresponding spectral
percentile intensities. The latter were calculated as the product of the relative intensity attained from the corresponding
spectrum by the design yield strength of the auxiliary SDOF system in accordance with the simplified collapse
assessment method proposed by Adam and Jäger (2012a). Moreover, to prove that the method proposed provides a
good approximation of structural response at a near-collapse limit state, the intensities associated with interstorey drift
thresholds calculated with Eqs. 21 to 26, considering fixed ductility values ranging from 3 to 8 were attained from IDA
and the same assessment approach was used.
Characteristics of example frames
All case studies considered, share the following characteristics: first storey height of 5.00 m, and 3.50 m elsewhere;
three spans of 10.00 m; uniform column to beam stiffness ratio of 1.5 at joints. The stiffness of beams and columns
decreases 25% every four stories and the masses are distributed uniformly along the height of the frame. The
fundamental period considered for the flexible set of frames, representative of steel structures, was obtained from eq.
35 (Chopra and Goel 2000).
1 = 0.09050.80 (35)
The stiffness values of the structural components and the mass of the system are such that the fundamental period (first
order) of the frame is approximately the same as that given by the preceding equation. Gravity nodal loads are applied
in leaning columns; they are uniformly distributed along the height of the structure and its magnitude is consistent with
the predefined effective post-yield stiffness values considered, θAUX – α; each frame is designed for these levels of
axial load. Structural components exhibit bilinear non-deteriorating behaviour; the post-yield stiffness of all beams
and columns is 0.02. Furthermore, axial-flexure interaction in columns is neglected and interdependency between
strength and stiffness is not considered, for the reasons mentioned at the beginning of this section.
Design demands and performance targets
The goal of the design applications is that dynamic instability occurs for 50% of the record set at the design target in-
tensity. Median yield pseudo-acceleration spectra and median ultimate-displacement spectra are employed to design
these case studies. Such spectra were scaled in each application in such a way that the intensity of each frame matches
the spectral pseudo-acceleration value at period TAUX of the response spectrum of the East-West component of the
Michoacán Earthquake of 1985, recorded at the SCT station in Mexico City. Such intensity is denoted in the following
as Satar.
For such purpose, median collapse spectra in terms of relative intensity, R, of the set of records considered was used.
R is the ratio of Sae/Say, where Sae is the spectral pseudo-acceleration of the corresponding elastic SDOF system. Such
type of spectra is frequently referred to as collapse capacity spectra (Adam and Jagger 2012a). According to Ibarra and
Krawinkler (2005), the collapse intensity in terms of pseudo-acceleration can be calculated by simply de-normalizing
the R value attained from collapse capacity spectra by the yield strength on the system. Hence, for each frame, the R
ordinate at TAUX was attained from the corresponding collapse capacity spectrum (fig. 10.a). Subsequently, a median
collapse yield strength was calculated by dividing Satar by R, and the scaling factor for the design spectra used was
defined by dividing such collapse yield strength by the ordinate at Taux of the corresponding yield pseudo-acceleration
spectrum (fig. 10.b).
The design damage states considered for all frames are consistent with the strong column-weak beam criterion. Since
in the method proposed the design displacement shapes are a function of the design damage state, slightly different
damage configurations were tested. The 8-storey frame was designed expecting that all beams yield barring those of
the roof-level, and the 12- and 16- and 20-storey frames were designed in such a way that only the beams of the last
two floors remain elastic. Yielding at column bases was considered in the design damage states of all case studies.
a) b)
Figure 10 Attainment of R factor and scaling of yield pseudo-acceleration spectra Incremental dynamic analysis of designed frames
IDAs of all designed frames were performed with OpenSees (McKenna et al. 2004). Plastic hinges were modelled as
zero length rotational springs with bilinear behaviour without considering stiffness deterioration. Second order
analyses were performed via the geometric matrix formulation. The non-linear step-by-step dynamic analyses of the
designed structure were carried out using the Rayleigh damping approximation with damping ratios ζ=0.05, for the
first and third modes. The integration of the non-linear equations of motion was carried out using Newmark’s Beta
method with parameters γ = 0.5 and β = 0.25 together with the Newton-Raphson method. IDA was performed using
an initial intensity, Sae, of 0.20 m/s2 with increments of 0.20 m/s2 up to sidesway collapse, i.e., numerical instability.
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00
1.00
2.00
3.00
4.00
5.00
0.10
0.20
0.30
0.40
0.50
Evaluation of results
From the IDA of the designed frames, the intensity steps corresponding to the onset of near-collapse or dynamic
instability were identified for each record, from which the 16%, 50% and 84% intensities, Saana, were calculated via
counted statistics. The interstorey drift thresholds associated with near-collapse were defined using Eqs. 22 and 32 for
ductility values of 3 to 8 in increments of one. The attainment of numerical instability in the analyses of the designed
frames is interpreted as dynamic instability, i.e., flattening of the IDA curves and, thus, as sidesway-collapse. In order
to quantify the uncertainty of Saana, confidence intervals associated with a 0.95 confidence level were estimated using
the bootstrap method (DiCiccio and Efron 1996), generating 3000 bootstrap samples. For this number of samples,
stability in the values of the confidence limits was assured in all case studies.
The target intensity, Satar, of each percentile and ductility value was estimated in an analogous manner as it is carried
out in the collapse capacity spectrum method (Adam and Jäger 2012b). In such method, the capacity curve of the given
structure is obtained from pushover analysis, from which an auxiliary SDOF system is defined. Subsequently, the
collapse intensity is obtained by de-normalizing the ordinate of the corresponding collapse capacity spectrum at period
TAUX by the yield strength of such SDOF system.
However, in this study, the yield strength of the auxiliary SDOF system used for assessment was calculated for each
ductility and percentile value with the SRSS rule, considering the spectral shape of the corresponding yield pseudo-
acceleration spectrum and taking the design base shear as a fixed value. The design base shear is that obtained directly
from the design application using the median collapse spectrum. The aforementioned scheme was used to account for
the possible variation of modal yield strengths under different intensities of the seismic demand. The target intensity
for each ductility and percentile was then calculated by de-normalizing the corresponding relative intensity by the
calculated yield shear. Evidently, the median collapse intensity attained with such scheme is the original Satar for which
the spectra used to design the frames was scaled.
Individual comparisons between Satar and Saana for all percentiles and ductilities were carried out. In general, a good
approximation of the target intensities was attained in all case studies considered and, in most of the cases, the target
intensities fell within the confidence interval. Figure 11 depicts the IDA curves, in terms of maximum interstorey drift,
IDRmax vs Sae, of the 16-storey frame with θAUX-α=0.05, along with the computed (blue) and target (red) 16%, 50%
and 84% intensities associated with both the non-exceedance of µ=4 and the onset of dynamic instability, and the
corresponding confidence interval [LCL, UCL] where the good approximation can be observed; the frame shown in
this figure can be considered representative of the results of this investigation.
To give a global perspective of the good approximations attained with the method proposed, a statistical analysis of
the errors between target and analysis percentile intensities was carried out for all ductility values. Figure 12 illustrates
the results of such analysis. From left to right, the first subplot depicts the median (m) and standard deviation (σ) of
the relative errors (Er) between Satar and Saana, in colors blue and red, respectively. The second subplot illustrates the
relative frequency (fr) of the location of the design intensity with respect to the confidence interval, in which BCI, WCI
and ACI denote “below”, “within” and “above” the confidence interval, respectively.
As can be readily observed in Figure 12, the means and standard deviations of the relative errors between target and
analysis percentile collapse intensities are low, thus, indicating that, in general, the correspondence was good in all of
the case studies considered. Moreover, the target intensities fell within the confidence interval in most of the cases.
This statistical analysis also shows that the correspondence between target and actual intensities was better as the
ductility value was larger, which may be, in part, a consequence of the variability of the yield point of the structure
under different earthquake records. The reason behind this opinion is that the error in the estimation of such point
would be less significant as the ductility value increases, since the ratio of yield displacements to total displacement
would decrease accordingly.
a) b)
Figure 11 IDA curves and 16%, %50 and %84 collapse intensities of 16-storey frame with θAUX-α=0.05: a) constant ductility (μ = 4); b) collapse ductility (μ = μcst)
0 0.005 0.010 0.015 0.020 0.025 0.030
5.00
10.00
15.00
ra ti
o n
5.00
10.00
15.00
20.00
25.00
30.00
2.00
4.00
6.00
8.00
10.00
12.00
ra ti
o n
5.00
10.00
15.00
20.00
2.00
4.00
6.00
8.00
ra ti
o n
2.00
4.00
6.00
8.00
10.00
12.00
a) b) Figure 12 Comparison of analysis vs. target 16%, %50 and %84 collapse intensities of frames designed for the VM set of records: a) means and standard deviations of relative error; b) location of design intensity with respect to the confidence interval [LCL, UCL]
Furthermore, since the method proposed relies on a displacement-based approach, a comparison between the target
displacement and the interstorey drift profiles (eq. 32) and the actual percentile profiles corresponding to the last non-
collapse intensity step attained from IDA, was carried out. The profiles corresponding to actual collapse design
applications were estimated considering µ=µcst. Figure 13 shows the envelope profile comparisons of the 16-storey
frame with 16-storey frame with θAUX-α=0.05. In such figure, IDR denotes the interstorey drift ratio, NDR denotes the
displacement normalized to the total height of the structure.
3 4 5 6 7 8 C -25
-20
-15
-10
-5
0
5
10
15
20
25
ductility
20
40
60
80
100
ductility
-20
-15
-10
-5
0
5
10
15
20
25
ductility
20
40
60
80
100
ductility
-20
-15
-10
-5
0
5
10
15
20
25
ductility
20
40
60
80
100
ductility
µ c
a) b) Figure 13 50%Displacement and 50%interstorey drift profiles of 16 storey frame T1=2.00 s and θAUX-α=0.05: a) constant
ductility (μ = 4); b) collapse ductility (μ = μcst)
The correspondence between target and analysis profiles was measured via the modal assurance criterion (MAC),
given by eq. 36. In such equation, VEC denotes the response vector, sub-indexes dem and tar indicate the demand and
target, respectively. A MAC value of 1.00 implies total correspondence between the compared shapes.
= ({} ⋅ {})2
({} ⋅ {})({} ⋅ {}) (36)
Statistical analysis of the MAC results shows good correspondence between the target shapes and the percentile shapes
attained from IDA. For all percentile and ductility values considered, the median MAC values were in the range of
0.98 to 1.00 for the displacements, and 0.90 to 1.00 for interstorey drifts: the associated standard deviations values
were in the range of 0 to 0.02, and 0.03 to 0.08, respectively. Evidently, the good agreement between the target and
analysis percentile intensities is a consequence of the good matching between the shapes. However, being this a
simplified method, there were some cases were the shape correspondence was not that good, nonetheless, the results
obtained show that the interstorey drift calculated with the method proposed was, in the vast majority of cases, close
to the maximum interstorey drift attained from IDA.
CONCLUSIONS
The results obtained in this investigation demonstrate that the method proposed by the authors allows to approximate
sufficiently the seismic performance of framed structures subjected to seismic loading that exhibit P-Delta induced
negative stiffness since the design RSDOF is defined directly from the dynamic properties of the structure. Therefore,
this method can be used to design instability prone structures due to P-Delta effects for either, near collapse and
sidesway-collapse limit states. The application of the method requires the use of elastic analysis and a set of design
spectra corresponding to SDOF systems with a P-Delta induced negative post-yield stiffness, hence, it does not require
non-linear dynamic analysis and can be carried out using commercial software that performs modal spectral analysis.
Up to this point, the method proposed has been extensively validated using regular planar frames. The results obtained
encourage the continued development of this displacement based approach for both assessment and design purposes,
thus, applications of the method proposed in regular and irregular 3-D buildings is currently underway.
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ACKNOWLEDGEMENTS
The authors thank the Institute of Engineering, UNAM for the financial support awarded to the first author of this
paper to complete his doctorate and the sponsorship with the Faculty of Engineering of the project “Multi-Level
Performance- Based Seismic Evaluation and Design Considering Collapse Prevention”, and to the National Council
of Science and Technology, CONACyT, the doctoral fellowship of the first author and the sponsorship of the project
No 221526 "Development and Validation of a New Approach to the Multilevel Seismic Assessment and Design of
Structures Based on Displacements and Damage Control".

a new displacement-based seismic design method for

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