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Dynamic analysis of buildings for earthquakeresistant design1
Murat Saatcioglu and JagMohan Humar
Abstract: The proposed 2005 edition of the National Building Code of Canada specifies dynamic analysis as the preferred method for computing seismic design forces and deflections, while maintaining the equivalent static force
method for areas of low seismicity and for buildings with certain height limitations. Dynamic analysis procedures are
categorized as either linear (elastic) dynamic analysis, consisting of the elastic modal response spectrum method or the
numerical integration linear time history method, or nonlinear (inelastic) response history analysis. While both linear
and nonlinear analyses require careful analytical modelling, the latter requires additional considerations for proper simulation of hysteretic response and necessitates a special study that involves detailed review of design and supporting
analyses by an independent team of engineers. The paper provides an overview of dynamic analysis procedures for use
in seismic design, with discussions on mathematical modelling of structures, structural elements, and hysteretic response. A discussion of the determination of structural period to be used in association with the equivalent static force
method is presented.
Key words: dynamic analysis, earthquake engineering, elastic analysis, fundamental period, hysteretic modelling, inelastic analysis, National Building Code of Canada, seismic design, structural analysis, structural design.
Résumé : L’édition 2005 du Code National du Bâtiment du Canada spécifie l’analyse dynamique comme méthode de
calcul de préférence afin de déterminer les forces de conception sismiques et les déflexions, tout en maintenant la méthode de force statique équivalente pour les régions à faible sismicité et pour les bâtiments ayant certaines limitations
en hauteur. Les procédures d’analyse dynamique sont catégorisées comme étant linéaire (élastique) avec des méthodes
de spectre de réponse modale élastique et des méthodes temporelles d’intégration numérique linéaire, ou des méthodes
temporelles non-linéaires (inélastiques). Bien que les deux types d’analyse (linéaire et non-linéaire) requièrent une modélisation analytique soignée, cette dernière requière des considérations additionnelles afin de simuler correctement la
réponse d’hystérésis et nécessite une étude spéciale, qui comporte une révision détaillée de la conception et des analyses secondes provenant d’équipes indépendantes d’ingénieurs. Cet article procure un survol des procédures d’analyse
dynamique utilisées pour la conception sismique, avec des discussions sur les modèles mathématiques des structures,
éléments structurels et réponse d’hystérésis. Une discussion sur la détermination de la période structurelle a être utilisée
en association avec la méthode de force statique équivalente est présentée.
Mots clés : analyse dynamique, génie sismique, analyse élastique, période fondamentale, modèle d’hystérésis, analyse
inélastique, Code National du Bâtiment du Canada, conception sismique, analyse structurale, conception structurale.
[Traduit par la Rédaction]
Saatcioglu and Humar
Introduction
Structural response to earthquakes is a dynamic phenomenon that depends on dynamic characteristics of structures
Received 7 March 2002. Revision accepted 5 December
2002. Published on the NRC Research Press Web site at
http://cjce.nrc.ca on 23 April 2003.
M. Saatcioglu.2 Department of Civil Engineering, The
University of Ottawa, Ottawa, ON K1N 6N5, Canada.
J. Humar. Department of Civil and Environmental
Engineering, Carleton University, Ottawa, ON K1S 5B6,
Canada.
Written discussion of this article is welcomed and will be
received by the Editor until 31 August 2003.
1
This article is one of a selection of papers published in this
Special Issue on the Proposed Earthquake Design
Requirements of the National Building Code of Canada,
2005 edition.
2
Corresponding author (e-mail: murat@eng.uottawa.ca).
Can. J. Civ. Eng. 30: 338–359 (2003)
359
and the intensity, duration, and frequency content of the exciting ground motion. Although the seismic action is dynamic in nature, building codes often recommend equivalent
static load analysis for design of earthquake-resistant buildings due to its simplicity. This is done by focusing on the
predominant first mode response and developing equivalent
static forces that produce the corresponding mode shape,
with some empirical adjustments for higher mode effects.
The use of static load analysis in establishing seismic design
quantities is justified because of the complexities and difficulties associated with dynamic analysis. Dynamic analysis
becomes even more complex and questionable when nonlinearity in materials and geometry is considered. Therefore,
the analytical tools used in earthquake engineering have
been a subject for further development and refinement, with
significant advances achieved in recent years.
The seismic provisions of the 1995 edition of the National
Building Code of Canada (NBCC 1995) are based on the
equivalent static load approach with dynamic analysis permitted for obtaining improved distribution of total static base
doi: 10.1139/L02-108
© 2003 NRC Canada
Saatcioglu and Humar
shear over the height, and for special cases where a better
assessment of building response is required near its ultimate
state. This is especially beneficial for irregular buildings
with significant plan and elevation irregularities, buildings
with setbacks, significant stiffness tapers, and mass variations. It is also beneficial for buildings with significant torsional eccentricities. Dynamic analysis in the 1995 NBCC is
not, however, intended for independent determination of the
design base shear. The base shear obtained by the equivalent
static force approach is specified as minimum base shear,
with implications that those obtained by dynamic analyses
should be scaled up to the static value when lower. It is possible to obtain substantially lower design base shear forces
through dynamic analysis, depending on the assumptions
made in structural and behavioural models and the ground
motion records used. Engineers have been discouraged from
using dynamic analysis as a means of establishing seismic
design force levels because of its sensitivity to the characteristics of ground motions selected and engineering assumptions made, which in turn are dependent on the experience
and judgment of the analyst. Furthermore, the restrictions of
available computer software, sometimes very severe, are not
always clear to the users, raising concerns over the accuracy
of results. Studies in the past have shown that distinctly different results could be obtained from analyses of the same
building conducted by different analysts. Therefore, dynamic analysis procedures were regarded as unsafe, unless
conducted by experienced and knowledgeable engineers.
Consequently, the 1995 NBCC limited the computation of
design base shear to the equivalent static load approach,
which had been calibrated based on previous experience to
provide an acceptable level of protection.
Despite the aforementioned concerns over the use of dynamic analysis in seismic design, it is used in practice to
carry out special studies of tall buildings and irregular structures because of its superiority in reflecting seismic response
more accurately, when used properly. These studies often include a large number of analyses under different ground motion records and different structural parameters to provide
insight into the structural behaviour.
With the advent of personal computers and the subsequent
evolution in information technology, coupled with extensive
research in nonlinear material modelling, more reliable computational tools have become available for use in design of
buildings. The proposed 2005 edition of the National Building Code of Canada recognizes dynamic analysis as a reliable design tool. In fact, dynamic analysis is specified as the
preferred procedure for structural analysis. The application
of the technique and various types of dynamic analysis
methodologies for seismic analysis of buildings and some
aspects of mathematical modelling are discussed in this paper.
The 2005 NBCC permits the use of the equivalent static
load method of analysis for buildings satisfying certain
height and regularity restrictions and (or) located in areas of
low seismicity. For such buildings the design base shear may
be obtained from the uniform hazard spectrum (UHS) specified for the site (Adam and Atkinson 2003; Humar and
Mahgoub 2003). Such a spectrum provides the spectral
acceleration Sa(T) for a reference ground condition. The
spectral acceleration measured from the UHS, corresponding
339
to the fundamental period of building, is multiplied by the
weight of building and then adjusted for higher mode effects
to determine the base shear. The fundamental period can be
determined by using the empirical expressions provided in
the 2005 NBCC. Alternatively, it may be determined from
standard methods of engineering mechanics, including the
Rayleigh method. The empirical expressions for the calculations of fundamental period and the restrictions associated
with the use of methods of engineering mechanics are discussed in the following sections.
Period determination
The fundamental period is an important design parameter
that plays a significant role in the computation of design
base shear. The 2005 NBCC provides approximate empirical
expressions to estimate the fundamental period. Although
the use of more accurate methods of mechanics is permitted
in the code, it is specified that the value obtained by such
methods must not exceed 1.5 times the value determined by
the empirical expressions. This limit may be justifiable from
the point of view of safety for three reasons: (i) uncertainties
associated with the participation of nonstructural elements,
whose effects may not have been considered in period determination and on the seismic response; (ii) possible inaccuracies in analytical modelling when applying the more
accurate methods of mechanics; and (iii) potential differences between the design and as-built conditions, especially
in terms of structural stiffness and mass. It may be noted
that the limit of 1.5 times the value determined by the empirical expression may not apply to shear wall structures. As
shown later in the paper, for such structures the period calculated by methods of mechanics is close to the measured
value.
Frame buildings
The empirical expressions given in the 2005 NBCC for
the calculation of fundamental period Ta of moment resisting
frames are given as follows: for concrete frames,
[1]
Ta = 0.075(hn)3/4
for steel frames,
[2]
Ta = 0.085(hn)3/4
and for all other moment frames,
[3]
Ta = 0.1N
where N is the total number of stories above grade and hn is
the total height of building above the base in metres. The
same expressions are also specified in the 1995 NBCC, except that eq. [3] can be used for any moment frames.
The periods of actual concrete buildings recorded during
past earthquakes are compared in Fig. 1 with periods calculated using eq. [1]. The database for these buildings includes
a total of 91 cases, consisting of 44 actual buildings in two
orthogonal directions. In some cases the measured periods
are two to three times larger than the code-computed values.
The scatter of data indicates that the empirical expression
given in the 2005 NBCC provides approximate estimates of
actual building periods. A similar comparison is made in
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Can. J. Civ. Eng. Vol. 30, 2003
Fig. 1. Comparison of measured periods with those calculated using eq. [1] for reinforced concrete moment resisting frame buildings.
Fig. 2. Comparison of measured periods with those calculated using eq. [2] for steel moment resisting frame buildings.
Fig. 2 for 53 steel frame buildings in two orthogonal directions, providing measured data for 103 cases.
It has been observed during recent earthquakes that nonstructural elements in existing buildings are often not well
separated from the structural framing system and they do
participate in seismic response at varying degrees, even
though they were not designed as seismic-resisting elements
(Saatcioglu et al. 2001). Although the 2005 NBCC clearly
calls for the separation of these nonstructural components
from lateral force resisting systems and otherwise requires
their consideration in seismic design, unintended participation of these elements may explain the scatter of data observed in Fig. 1. The effects of unintended participation of
nonstructural elements were assessed by Morales and
Saatcioglu (2002), who compared fundamental periods of reinforced concrete frame buildings with different heights and
degrees of lateral bracing provided by nonstructural masonry
walls. Figure 3 illustrates the variation of computed periods
with the amount of lateral bracing expressed in terms of the
ratio of wall cross-sectional area to floor area. When the
presence of nonstructural masonry walls was ignored, the
fundamental periods for 5-, 10-, and 15-storey bare frame
buildings, with reduced stiffnesses due to cracking, were
computed as 1.7, 3.7, and 5.7 s, respectively. The same
buildings were computed to have fundamental periods of
0.7, 1.2, and 1.6 s, respectively, based on eq. [1] (the 2005
NBCC equation). The analytically computed values were
2.4–3.6 times the values computed by the expression given
in the 2005 NBCC, indicating that analytically computed periods could be significantly longer. The results also indicate
that the computed values approach those empirically determined by eq. [1] when the bracing effects of nonstructural
infill walls are considered. It was found that a few
nonstructural masonry infill walls were sufficient to significantly shorten the fundamental period of low- to mediumrise frame buildings to approximately the levels given by
eq. [1] when the walls were not well separated from the lateral force resisting system. Therefore, designers should exercise caution when using bare frame models for the
computation of fundamental period and ensure that the
structure is not braced by nonstructural components that
were not considered in structural design, after it is built.
Braced frame and shear wall buildings
The overall stiffness and period of a structure are often
dominated by the characteristics of braced frames and shear
© 2003 NRC Canada
Saatcioglu and Humar
341
Fig. 3. Effects of nonstructural masonry infills on fundamental periods of 5-, 10-, and 15-storey reinforced concrete frame buildings
when not properly isolated from the frames.
Fig. 4. Comparison of measured periods with those calculated using eq. [4] for reinforced concrete shear wall buildings.
walls in the structure. The 1995 NBCC specifies the fundamental period of such structures as a function of the length
of braced frames or shear walls and building height as follows:
[4]
T =
0.09hn
Ds
where hn and Ds are building height and wall length in
metres, respectively. Ds is defined as the length of the primary lateral load resisting wall or braced frame, although
this is often not very clear when more than one bracing element is used in the structure, as is typically the case in most
practical applications. When Ds cannot be defined clearly,
the entire building length in the direction of analysis (D) is
used instead. This results in a conservative estimate of period.
Data on measured periods of shear wall buildings, obtained during previous earthquakes, are plotted in Fig. 4.
The data show a scatter similar to that shown in Fig. 1 for
frame buildings, with eq. [4] providing conservative values
when D is substituted in place of Ds. Hence, it may be
argued that the implied accuracy of eq. [4] may not be justifiable. Instead, an expression similar to those recommended
for frame buildings may be used for shear wall and braced
frame buildings as a function of building height only (Goel
and Chopra 1997; Morales and Saatcioglu 2002). Figure 5
shows the correlation of measured periods with the following equation, which has been adopted by UBC (1997), IBC
(2000), and the 2005 NBCC:
[5]
T = 0.05(hn)3/4
Among the instrumented buildings, 10 buildings had shear
walls in two orthogonal directions with well-identified geometry. These buildings were modelled and analyzed using
computer software SAP2000 to determine their periods of
vibration analytically. The flexural rigidities were reduced to
account for cracking in concrete. Accordingly, 70 and 35%
of uncracked rigidities were used for vertical elements (columns and walls) and beams, respectively. The results presented in Fig. 6 indicate that reasonably accurate values of
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Can. J. Civ. Eng. Vol. 30, 2003
Fig. 5. Comparison of measured periods with those calculated using eq. [5] for reinforced concrete shear wall buildings.
Fig. 6. Comparison of computed periods (by SAP2000) and measured periods for shear wall buildings.
seismicity with IEFaSa(0.2) ≥ 0.35; and (iii) all buildings that
have rigid diaphragms and are torsionally sensitive.
Dynamic analysis is conducted to obtain either a linear
(elastic) or a nonlinear (inelastic) structural response. When
elastic analysis is conducted, an empirical assessment of inelastic response is made, since the design philosophy is
based on nonlinear behaviour of buildings under strong
earthquakes. This does not, however, deter engineers from
preferring elastic dynamic analysis because of its simplicity
and direct correspondence to the design response spectra
provided in building codes.
The first step in dynamic analysis is to devise a mathematical model of the building, through which estimates of
strength, stiffness, mass, and inelastic member properties (if
applicable) are assigned. Detailed discussion of mathematical modelling is presented later in the paper in the section titled Mathematical modelling.
fundamental period can be computed for shear wall buildings analytically. The 2005 NBCC allows the computation
of fundamental periods of shear wall buildings by accepted
methods of mechanics without an upper limit based on the
empirical expression.
Linear (elastic) dynamic analysis
The modal response spectrum method or the numerical integration linear time history method is used to conduct linear
analysis. Spectral analysis is intended for the computation of
maximum structural response. The spectral analysis is conducted for a single-degree-of-freedom structure or for a
building that can be approximated to behave in its first mode
response by selecting the value corresponding to its period,
directly from the design response spectrum. Correct assessment of the fundamental period becomes important in obtaining spectral values, as discussed in the section Period
determination. In the 2005 NBCC, the design response spectrum is based on the UHS for the site as adjusted for the
ground condition (Humar and Mahgoub 2003). A UHS provides the maximum spectral acceleration that a singledegree-of-freedom system with 5% damping is likely to experience with a given probability of exceedance and reflects
the seismicity of the region. Such spectra have been developed for use with the 2005 NBCC (Adam and Atkinson
2003).
A water tank supported by a single column (or a truss system) with a concentrated mass at the top represents a typical
single-degree-of-freedom structure with a corresponding
mode shape and a period. On the other hand, a typical frame
Methods of dynamic analysis
Dynamic analysis is specified in the 2005 NBCC as a
general method of analysis to compute design earthquake actions. The equivalent static force procedure is permitted for
buildings in low seismic regions, regular buildings below a
certain height limit, and short buildings with certain irregularities. It may be preferred by designers because of its simplicity when dynamic analysis is not mandatory. In the 2005
NBCC, dynamic analysis is mandatory for the following
classifications of buildings: (i) regular structures that are
60 m or taller or have fundamental period greater than or
equal to 2.0 s and are located in areas of high seismicity
with IEFaSa(0.2) ≥ 0.35, where IE is the moment of inertia,
Fa is an acceleration-based site coefficient, and Sa(0.2) is the
spectral response acceleration for a period of 0.2 s; (ii) irregular buildings that are 20 m or taller or have a fundamental
period of 0.5 s or longer and are located in areas of high
© 2003 NRC Canada
Saatcioglu and Humar
building with rigid floors develops as many modes as the
number of floors (with concentrated mass at floor levels).
The treatment of a multistory building as a single-degree-offreedom structure may be possible on the basis of its dominant first mode response but provides only an approximation
of its complete structural response.
In general, for a multistory building it is necessary to take
into account contributions from more than one mode. Each
mode has its own particular pattern of deformation. For
building applications, the dominant first mode shape resembles the flexural deformation of a cantilever beam. The contribution of higher modes diminishes very quickly, and it is
nearly always sufficient to consider the first three modes of
vibration to obtain reasonably accurate results for most
short- to medium-rise buildings. For high-rise buildings, it
may be necessary to consider more than three modes. The
significant modes that contribute to response may be determined by selecting the number of modes such that their
combined participating mass is at least 90% of the total effective mass in the structure.
Once the number of significant modes is established, the
response is computed as the superposition of responses of all
the contributing modes. This implies that the response of the
entire structure can be modelled as a number of singledegree-of-freedom responses with their respective modal
properties and contributions to overall response. This approach, known as modal analysis, is a useful design tool
where spectral values for each of the participating modes
can be obtained separately from UHS and superimposed
with due considerations given to the participation of each
mode. The use of UHS as the design response spectrum for
a multi-degree-of-freedom system is conservative but provides reasonably accurate results (Humar and Mahgoub
2003).
The method of modal analysis is described in standard
texts (Humar 1990; Clough and Penzien 1993; Chopra 2001)
and will not be discussed in detail. It provides the elastic
base shear Ve and the elastic storey shears, storey forces,
member forces, and deflections. The design base shear Vd is
determined by dividing the elastic base shear Ve by RoRd to
allow for overstrength and ductility and multiplying by the
importance factor IE:
[6]
Vd =
Ve
IE
RoRd
where Rd is a ductility-related force modification factor that
reflects the capability of a structure to dissipate energy
through inelastic behaviour, and Ro is an overstrength-related
force modification factor that accounts for the dependable
portion of reserve strength in a structure designed in accordance with the 2005 NBCC. If the base shear Vd obtained
from eq. [6] is less than 80% of the lateral earthquake design
force, V, established by the equivalent static force procedure,
Vd is replaced by 0.8V, except for irregular structures requiring dynamic analysis as previously specified, in which case
Vd is taken as the larger of Vd determined by eq. [6] and
100% of V.
The elastic quantities obtained from modal analysis are
multiplied by Vd/Ve to obtain design elastic storey shears,
storey forces, member forces, and deflections.
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As stated earlier, the restrictions imposed on the design
values obtained from a dynamic analysis are justified by the
uncertainties involved in the modelling, so it is not considered prudent to deviate too far from the design values obtained by the equivalent static load procedure. Compared
with the 1995 NBCC, however, the limits prescribed in the
2005 NBCC are more liberal. Thus under the provisions of
the 2005 NBCC, if the period determined from a dynamic
analysis is longer than that obtained from the empirical expressions, such a longer period, up to a limit of 50% longer
than the empirical period, may be used in calculating the
static base shear V. The limit of 80% (or 100%) is then related to the shear calculated from the longer period.
The linear time history method is employed when the entire time history of the elastic response is required during the
ground motion of interest. In this method, the dynamic response of the structure is computed at each time increment
under a specific ground motion. The structure is first modelled with due considerations given to stiffness and mass
distributions and other factors as outlined in the section
Mathematical modelling. It is then subjected to pairs of
ground motion time history components that are compatible
with design response spectra specified in building codes. It
is usually a requirement to conduct time history analysis for
an ensemble of ground motion records that represent magnitudes, fault distances, and source mechanisms that are consistent with those of the design earthquakes used to generate
design response spectra. This is done either by using previously recorded ground motions, scaled to match the design
response spectrum, or by using artificially generated records
with similar properties. Depending on the number of ground
motion records considered, either the maximum or the average of each design parameter obtained from different analyses is used in design. When the analysis conforms to the
2005 NBCC, the ground motion records should be compatible with the response spectrum constructed from the specified design spectral acceleration values, S(T), which are the
product of spectral acceleration Sa(T) and site coefficients Fa
or Fv.
The base shear obtained from a linear time history analysis has to be divided by Ro and Rd to allow for the reductions
associated with overstrength and ductility in the system and
multiplied by the importance factor IE to arrive at the design
base shear level Vd. If the base shear Vd obtained is less than
80% of the static base shear V established by the equivalent
static force procedure, Vd is taken as 0.8V, except for irregular structures requiring dynamic analysis as previously specified, in which case Vd is taken as the larger of Vd determined
by linear dynamic analysis and 100% of V. If the design base
shear is taken to be larger than that computed by the elastic
time history analysis, all other design quantities, including
storey forces, storey shears, member forces, and deflections,
should be proportionately increased before they are used in
design.
Nonlinear (inelastic) response history analysis
Nonlinear time history analysis involves the computation
of dynamic response at each time increment with due consideration given to the inelasticity in members. Nonlinear
analysis allows for flexural yielding (or other inelastic actions) and accounts for subsequent changes in strength and
© 2003 NRC Canada
344
stiffness. Hysteretic behaviour under cyclic loading is evaluated. Softening caused by inelasticity of deformations during
loading, unloading, and reloading is computed. This implies
that the interaction between changing dynamic characteristics of structures due to inelasticity, such as lengthening of
period, and the exiting ground motion is accounted for, improving the accuracy of dynamic response.
The most important distinction between linear and nonlinear time history analyses is the inelastic hysteretic behaviour
of elements that make up the structure. This behaviour is incorporated into analysis computer software through
hysteretic models. Therefore, mathematical modelling of
structures gains a new dimension, with new difficulties and
challenges introduced. The hysteretic behaviour depends on
the characteristics of structural materials, design details, and
a large number of design parameters, as well as the history
of loading. It is a rather complex behavioural phenomenon
that has to be understood clearly by the analyst before such
an analysis is attempted. A section is presented later in the
paper under Mathematical modelling to highlight salient features of selected hysteretic models.
A nonlinear time history analysis provides the maximum
ductility demands in members and the maximum deflections
experienced by the structure. If the ductility demands are
less than the ductility capacities and the deflections are
within acceptable limits, the design is satisfactory. The results of nonlinear time history analysis directly account for
reduction in elastic forces associated with inelasticity. The
structural overstrength can also be accounted for directly
through appropriate modelling assumptions. The analysis results need not therefore be modified by Rd and Ro. The importance factor IE can be accounted for either by scaling up
the design ground motion histories or by reducing the acceptable deflection and ductility capacities.
Inelastic time history analysis has been adopted by the
2005 NBCC with a special study clause. Accordingly, when
nonlinear time history analysis is used to justify a structural
design, a special study is required, consisting of a complete
design review by a qualified independent engineering team.
The review is to include ground motion time histories and
the entire design of the building, with emphasis placed on
the design of lateral force resisting system and all the supporting analyses.
Mathematical modelling
Mathematical modelling of a structure, for the purpose of
structural analysis, can be done in a variety of different ways
depending on the method of analysis adopted. In a general
sense, modelling involves the representation of a structure
by elements to which physical and material characteristics
are assigned and the appropriate loading is applied or transmitted. These elements involve different degrees of discretization of the structure (or structural component).
Perhaps the most common forms of structural analysis involve modelling by finite elements or by line elements.
When dynamic inelastic response history analysis is considered, it is usually more convenient and reliable to represent
the behaviour of an entire structural element or component
mathematically. Furthermore, the analysis results should be
Can. J. Civ. Eng. Vol. 30, 2003
in an easily interpretable form that can be related to the design parameters used in practice. Therefore, in this section a
common form of mathematical modelling involving line elements is used to illustrate important aspects of modelling for
dynamic analysis, while also highlighting some of the pitfalls that should be avoided to improve the accuracy of results.
There are three stages of modelling that an analyst has to
go through prior to performing a dynamic analysis: (i) structural modelling, (ii) member modelling, and (iii) hysteretic
modelling (for nonlinear analysis).
Structural modelling
The first stage in mathematical modelling involves the
representation of the entire structure by elements to which
physical and material properties can be assigned. Figure 7 illustrates the modelling of common forms of building structures with line elements. In frame and frame–wall interactive
systems the vertical elements such as columns and flexuredominant structural walls are represented by vertical line elements, and the horizontal elements, such as beams, and slab
diaphragms are represented by horizontal line elements
(Figs. 7a, 7e). Sometimes shear panels, like shear walls and
infill panels, as well as bracing elements are modelled by diagonal struts and ties (Figs. 7b–7d).
An important aspect of structural modelling is the selection of correct boundary conditions that represent supports
and connections with appropriate end restraints. Another important aspect is the consideration of finite widths of members and corresponding stiffness variations. This becomes
especially important in modelling wide columns and structural walls for which the segments of elements that are integral with the adjoining members can be represented with
infinite stiffness. Figures 7f and 7g illustrate the member-end
regions, which are sometimes referred to as member-end eccentricities in certain computer programs. The analyst must
be aware of the procedure used for considering finite widths
of members in the computer software employed and incorporate these regions properly.
Seismic analysis is conducted in two orthogonal directions, separately and independently. When a building with
rigid diaphragms is torsionally irregular, a three-dimensional
(3-D) analysis must be carried out. Even when the building
is determined to have no torsional irregularity, accidental
torsional eccentricity will still cause torsional response, and
again a 3-D analysis is required.
The 2005 NBCC defines a torsional sensitivity index B as
a measure of the susceptibility of a building to large torsional motion during an earthquake (Humar et al. 2003). The
index B is determined by calculating the ratio Bx for each
level x according to the following equation for each orthogonal direction determined independently:
[7]
Bx =
δ max
δ ave
where δmax is the maximum storey displacement at the extreme points of the structure at level x in the direction of the
earthquake, and δave is the average of the displacements at
the extreme points. Displacements δmax and δave are automatically obtained when a 3-D dynamic analysis is carried out.
© 2003 NRC Canada
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345
Fig. 7. Structural modelling.
The index B is taken as the maximum of all values of Bx in
both orthogonal directions.
Buildings for which B is greater than 1.7 are considered to
be torsionally sensitive and must be designed with extra
care. For such buildings the effect of accidental torsion is accounted for by combining the static effects of torsional moments produced by the application of a force of ±0.1DnxFx at
each level x with the effects determined by dynamic analysis, where the forces Fx may be taken as those determined
from a dynamic analysis or the equivalent static analysis.
When B is less than 1.7, accidental torsion can be accounted for by carrying out a set of 3-D dynamic analyses
with the centres of mass shifted by distances of –0.05Dnx
and +0.05Dnx.
Buildings with rigid diaphragms have three degrees of
freedom at each floor, two translational and one rotational.
The inertia masses at the floor levels are assigned to these
degrees of freedom. The inertia mass may be computed as
being equal to the entire dead load and a portion of the live
load, depending on the occupancy of the building.
Member modelling
Force–deformation characteristics of individual members,
essential for structural analysis, are specified through member models. In a linear dynamic analysis, members are modelled by elastic line elements. In a nonlinear analysis,
member modelling can be done in a number of different
ways. A convenient procedure is to idealize each member as
an elastic line element with inelastic springs at the ends. The
springs account for potential plastic hinges at member ends.
This model, known as single-component model, was developed by Giberson (1967). According to the singlecomponent model, inelastic member-end deformation at one
end is directly related to the member-end force at the same
end, making it a simple and convenient model for structural
analysis. Inelasticity along the member is lumped at the
springs whose characteristics are determined by assuming
deformed shapes for members. Double curvature can be assumed for columns and beams with a fixed point of
contraflexure, as illustrated in Fig. 8. Single curvature can
be assumed for walls, with some moment gradient along the
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Fig. 8. Single-component element model.
Fig. 9. Dual-component element model.
member. The single-component model can also have more
than one spring at each end, where each spring represents a
separate deformation component, as in the case of flexure
and shear. Similarly, a third spring may be added to simulate
potential softening of fixity at member ends, as in the case
of anchorage slip deformations in reinforced concrete elements. The main disadvantage of the single-component
model is the assumption that fixes the location of the point
of contraflexure for the purpose of inelastic deformation
computations. During response, the points of contraflexure
in structural members move continually, and the memberend rotations become a function of member-end forces at the
two ends. The single-component model is therefore regarded
as an approximate and yet reasonably accurate member
model.
A dual-component model was suggested for elastoplastic
analysis of structures (Clough et al. 1965). In this model,
two parallel line elements, one representing perfectly elastoplastic behaviour to mark the yield point and the other to introduce the post-yield stiffness of the member, are connected
at their ends so that the two parallel elements have the same
member-end deformation. This simplifies the formulation of
the stiffness matrix. Member-end deformations are related to
member-end forces at both ends. The dual-component
model, however, is applicable to structural members that exhibit elastoplastic behaviour without any stiffness degradation. This model can be extended into a multicomponent
model where each parallel element represents different characteristic features of hysteretic response. Figure 9 illustrates
the dual-component model.
Another form of member modelling is to divide the member into segments. This provides a convenient approach to
account for the spread of inelasticity more accurately. In this
case, the behaviour of each subelement can be represented
by a separate inelastic spring. The resultant model is termed
a multiple-spring model (Takayanagi and Schnobrich 1976).
In other similar approaches, a variable length of inelastic
zones was considered in segmenting the member (Roufaiel
and Meyer 1983; Chung et al. 1988; Keshavarzian and
Schnobrich 1984). The variation of rigidity along the member length can also be represented by a continuous function
(Umemura et al. 1974).
Members can be modelled by dividing into segments not
only along the length but also across the cross section. Sections of a member can be divided into layers, and sectional
response is computed using material constitutive models.
This type of model is known as a layered model. Although
the approach followed in a layered model is more rational,
the computational effort involved makes the approach prohibitive for general-purpose use.
Keshavarzian and Schnobrich (1985a) have presented a
comprehensive review of member models. The first task in
any member modelling is the computation of elastic member
properties. This can be done using the principles of mechanics with due considerations given to member geometry and
material behaviour. Areas and moments of inertia for elements with well-defined cross-sectional dimensions are easy
to compute. Sometimes, however, the members represent
portions of 3-D elements where concentrations of internal
flow of forces occur. Portions of concrete slabs in monolithic construction that are effective as beam flanges and diagonal compression struts and tension ties in walls can be
given as examples of this type of member for which the analyst has to select an “effective member width” and define the
sectional geometry. The effective width is defined in Canadian Standards Association Standard A23.3 (CSA 1994) for
reinforced concrete slabs as illustrated in Fig. 10a. When a
column – flat plate system is used, the column strip of slab
may be used as the effective width of a beam element. Similarly, various approximate methods have been proposed to
compute the geometric properties of struts in wall panels.
One such method proposed by Stafford-Smith (1966) on the
basis of contact lengths is illustrated in Fig. 10b. Holmes
(1963) recommended that the diagonal width could be taken
as one third of the diagonal panel length. The New Zealand
Code (Standards Association of New Zealand 1990) specifies the effective wall width as one quarter of the wall
length.
Once the sectional geometry is established, element properties are computed analytically. Flexural rigidity EI, shear
rigidity GA, and axial rigidity AE can be determined from
cross-sectional properties (A and I, where A is the crosssectional area and I is the moment of inertia) and material
moduli (E and G, where E is the modulus of elasticity and G
is the shear modulus). This is especially true for steel structures where material characteristics are well defined. Although elastic rigidities of steel structures can be computed
on the basis of elastic material properties, inelastic behaviour leads to analysis techniques that may require different
levels of sophistication. Figure 11 illustrates typical behaviour of a cantilever steel beam with a plastic end region. The
sectional behaviour, incorporating post-yield response, can
be computed by establishing the plastic hinge length and the
variation of curvatures within the hinging region. For engineering applications, however, this level of sophistication is
often not warranted and a rigid–plastic hinge model can be
employed with zero plastic hinge length and bilinear
moment–curvature idealization (Bruneau et al. 1998). For reinforced concrete structures, where significant cracking is
expected beyond a relatively low initial resistance, the analyst may have to conduct a moment–curvature analysis to
obtain the flexural rigidity. While a standard plane-section
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347
Fig. 10. Effective widths of members: (a) effective flange width in beams, (b) formation of diagonal struts, and (c) equivalent braced
frame.
analysis is sufficient for this purpose, it is important to incorporate strain hardening in steel and confinement in concrete for improved accuracy of results in assessing the postyield rigidity. Figure 12 depicts a typical moment–curvature
relationship for a reinforced concrete section. The relationship consists of three segments: (i) the elastic region prior to
cracking, (ii) the post-cracking segment between the cracking and yield points, and (iii) the post-yield segment beyond
yielding. These regions can be idealized by a trilinear relationship. Bilinear relationships are often used in practice for
convenience, however, with the first segment representing
“the effective elastic rigidity” including the effects of cracking. Therefore, the elastic rigidity used in elastic dynamic
analysis must include the effects of cracking. Post-cracking
and post-yield rigidities depend on many parameters, including cross-sectional shape, amount and arrangement of reinforcement, material properties, and level of accompanying
axial force. The effective elastic rigidity varies between approximately 30% and 50% of the rigidity based on gross,
uncracked section properties (EIg, where Ig is the moment of
inertia based on gross uncracked properties), for most
beams. Wall sections with boundary elements show similar
behaviour, although the rigidity can be substantially lower
for walls with relatively low percentage of distributed reinforcement, especially under a low level of axial compression. In the latter case the yielding of low-percentage steel
occurs one row at a time, as the depth of the neutral axis becomes smaller very quickly, resulting in a rapid degradation
of flexural rigidity upon cracking. It has been the practice to
use 50% EIg for beams and 100% EIg for columns in static
gravity load analysis. This was justified because the beams
crack even under their own weight, whereas the columns un-
der axial compression remain mostly uncracked. It was believed that the 1:2 ratio used in the reduction of column and
beam stiffnesses, respectively, would result in a reasonable
distribution of forces at beam column connections. Furthermore, the overestimation of column rigidities by ignoring
cracking would result in higher design forces for columns,
which are the critical elements. The same justification cannot be used for lateral seismic analysis where the columns
are subjected to significant bending and cracking. Both the
ACI Committee 318 (2002) and the CSA (1994) recommend
effective elastic flexural rigidities to be 35% and 70% of
their gross, uncracked values for beams and vertical members (columns and walls), respectively. For shear wall structures, if moments exceed the modulus of rupture, then the
analysis should be repeated with wall stiffnesses further reduced to 35% of elastic rigidities. The use of elastic rigidities, based on gross sectional properties may result in
substantial overestimation of member stiffnesses and the
corresponding shortening in fundamental period.
Different approaches may be used to idealize moment–
curvature relationships depending on their application. For
elastic analysis, the only parameter required is the initial
slope of the relationship, which defines the effective elastic
rigidity. This can be achieved by drawing a horizontal line
parallel to the curvature axis at nominal flexural capacity.
Another line can be drawn to connect the origin and a point
on the ascending branch of the curve corresponding to 75%
of the nominal capacity (Park and Paulay 1975). The intersection of the two lines gives the yield point, with the slope
of the ascending line representing the effective elastic rigidity. This is illustrated in Fig. 12b. In this approach, the analyst need not consider the strain hardening of steel and the
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Can. J. Civ. Eng. Vol. 30, 2003
Fig. 11. Formation of plastic hinge in a cantilever steel beam and associated deformations (adopted from Bruneau et al. 1998).
confinement of concrete in computing the actual moment–
curvature relationship, and the accuracy of the post-yield
branch is of little interest. For nonlinear dynamic analysis,
the post-yield slope becomes important. In such analysis it is
usually required to compute the member behaviour rather
than the sectional behaviour. Post-yield slope of the
moment–rotation or moment–drift (chord angle) relationship
may have to be specified to define the hysteretic model. This
requires the construction of curvature distribution along the
length of the member and modelling of the potential plastic
hinge region. It is convenient to model a plastic hinge at ultimate load by a fully developed hinge with constant curvature. In this case, an empirically suggested plastic hinge
length can be used with constant curvature equal to the ultimate curvature, as illustrated in Fig. 12c. The area under the
curvature diagram provides member rotation, with the moment of the area giving member displacement. The ultimate
member deformation obtained in this manner can be used
along with yield deformation to establish the second slope of
the force–deformation relationship. It is also possible to
compute the entire post-yield region by an incremental moment–rotation analysis. In this case two line segments can
idealize the moment–curvature relationship such that the
area between the actual and idealized curves is approximately equal, as illustrated in Fig. 12d. The resulting idealization may have a nonzero post-yield slope and can be used
in integrating curvatures within the plastic hinge region.
The aforementioned approaches result in the ascending
slope of the force–deformation relationship that ignores
gradual strength degradation with increased inelasticity. Al© 2003 NRC Canada
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349
Fig. 12. Moment–curvature idealizations for reinforced concrete elements: (a) trilinear idealization, (b) bilinear idealization, (c) idealized plastic hinge length, and (d) idealization based on equal areas.
though they can be used until the onset of strength decay,
computer programs equipped with such models do not flag
this limit. The analyst should be aware of this hidden restriction and should ignore the analysis results beyond a realistic
level of inelastic capacity. If, however, the analysis program
is equipped with a hysteretic model that allows strength degradation, it is possible to compute the progression of plastic
hinging and the accompanying strength degradation by an
incremental force–deformation analysis (Razvi and Saatcioglu 1999).
Sound seismic design practice calls for proper design and
detailing of beam–column joints. This does not, however,
ensure full rigidity at joints during seismic response. Additional deformations may occur due to the softening in joints,
connections, and anchorage regions. In steel-frame buildings, distortions of the beam–column panel zones can contribute to overall softening of the structure, even if they are
well designed to prevent column web yielding and crippling,
flange distortions, and the panel zone failure. Tests of steel
beam – column subassemblages by Krawinkler et al. (1971,
1975) showed that panel zones can develop significant shear
distortions with the ability to dissipate seismic-induced energy. Figure 13 shows shear distortions of a panel zone and
an analytical model proposed by Krawinkler et al. (1971),
consisting of an elastoplastic column segment enclosed by
rigid elements connected by inelastic springs.
In reinforced concrete, flexural reinforcement anchored in
adjoining members may develop significant extensions due
to the penetration of yielding into the adjacent members.
These deformations are not accounted for in flexural analysis. Although anchorage failure can be prevented by proper
design, the extension of properly anchored reinforcement in
tension cannot be avoided if the critical section is located
near the interface of two adjacent members. The extension
and (or) slippage of reinforcement produce a rigid-body rotation at member end. This type of deformation may be negligible prior to yielding of longitudinal reinforcement and
hence may be conveniently ignored. The penetration of
yielding into the anchorage zone, however, especially beyond the strain hardening of reinforcement, can be significant and lead to inelastic deformations whose magnitudes
approach that caused by flexure. Sometimes the force–
deformation relationship for flexure is softened by the analyst to account for the combined effects of bar extension and
flexure. Figure 14a illustrates the strain profile in an embedded reinforcing bar, the integration of which gives the exten© 2003 NRC Canada
350
Fig. 13. (a) Panel zone deformations (Krawinkler et al. 1971).
(b) Modelling panel zones in steel beam – column joints
(Krawinkler et al. 1978).
sion of reinforcement due to yield penetration. Figure 14b
shows displacement caused by the extension of reinforcement in an adjoining member (Alsiwat and Saatcioglu
1992).
In most practical applications it is sufficient to consider
deformations associated with flexure alone. Short and stubby
members that are subjected to significant shear stresses develop additional deformations due to shear. It may be sufficient to use cross-sectional area and shear modulus to
establish elastic shear rigidity for elastic dynamic analysis.
Computation of inelastic shear effects may become necessary for certain members, with a complete shear force –
shear deformation relationship assigned to shear springs of
member models, if available.
Another type of commonly encountered structural element
to model is steel bracing. Lateral bracing in steel structures
results in concentrically braced frames (CBF) or eccentrically braced frames (EBF), as illustrated in Fig. 15. These
Can. J. Civ. Eng. Vol. 30, 2003
braces are axially loaded members and are modelled by
specifying their elastic and post-yield properties under axial
force. The elastic force – deformation relationship is specified in terms of axial rigidity AE. The tensile strength is
governed by axial yielding and the compressive strength is
governed by buckling and post-buckling residual compression force, which may be taken as approximately 20% of the
buckling load (NEHRP 1997). The EBF is a hybrid framing
system that may possess both increased lateral stiffness and
good ductility characteristics, both achieved through the link
beams. It becomes important to assess the flexural and shear
rigidities of link beams before they can be modelled. Short
links dissipate seismic energy primarily through inelastic
shearing, whereas long beams develop flexural hinging at
the ends. Figure 16 shows a bilinear idealization of the
force–deformation relationship of a link beam. The plastic
rotation capacity of an adequately stiffened short link may
be taken to be approximately equal to 0.12 rad (NEHRP
1997).
Hysteretic modelling
Dynamic inelastic response history analysis of reinforced
concrete structures requires realistic conceptual models that
can simulate strength, stiffness, and energy-dissipation characteristics of members. The current state of knowledge may
not be sufficient to model every aspect of hysteretic response. Significant advances have been made in recent
years, however, that enable analysts to obtain reasonably accurate results from nonlinear dynamic analyses. A large
number of hysteretic models have been proposed for seismic
evaluation of structures. These models are often based on
specific test data. Therefore, their applicability to other cases
requires careful examination of their features and limitations.
Structural members exhibit certain hysteretic features that
are common to members of similar properties, which are associated with well-established design parameters. Some of
the unfavourable features of hysteretic response, for example, early strength decay caused by lack of proper design
and detailing practices, can be prevented and need not be
considered in hysteretic modelling. Other features of
hysteretic response, however, some of which are outlined in
the following paragraphs, may have to be considered depending on specific applications. Special care should be exercised to make sure that the computed response could be
attained with the design and detailing techniques employed.
Most hysteretic models consist of a primary curve (backbone curve), which can be computed analytically by wellestablished procedures of mechanics, and a set of empirical
rules that define the branches of loading, unloading, and reloading under reversed cyclic loading. The primary curve provides an envelope of hysteresis loops of force–deformation
relationships. Therefore, it provides a convenient means of
defining the strength boundary for modelling purposes. Primary curves used for hysteretic modelling are generally in the
form of moment–curvature, moment–rotation, shear force –
shear distortion, moment–slip, and force–displacement relationships. They can be computed as discussed earlier in the
section Member modelling.
Hysteretic rules within the strength boundary (primary
curve) simulate the actual member behaviour, often as ob© 2003 NRC Canada
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351
Fig. 14. Deformations caused by anchorage slip in reinforced concrete structures (Alsiwat and Saatcioglu 1992).
Fig. 15. (a) Cross-braced CBF; (b) diagonally braced CBF; (c) inverted, V-braced CBF; (d) split K-braced EBF; (e) D-braced EBF;
and (f) V-braced EBF.
served during tests. Structural steel elements exhibit elastoplastic behaviour, with unloading and reloading branches of
hysteresis loops parallel to the initial elastic branch. The
post-yield slope of the primary curve may reflect steel strain
hardening with a post-yield rigidity approximately equal to
0.5–5% of the elastic rigidity. The Bauschinger effect, in the
form of rounded yield regions upon load reversals, as opposed to the sharp initial yield, can be considered as refinement. This is illustrated in Fig. 17, which shows different
variations of the elastoplastic hysteretic model. Zero postyield slope of the primary curve may also be used for simplicity, though this may occasionally lead to numerical problems in some computer software. Steel structures do not
exhibit significant stiffness degradation in flexure and hence
are relatively simple to model by means of elastoplastic
models.
Hysteretic response of steel bracing elements under axial
force reversals is different from that exhibited by frame elements in flexure. Figure 18 illustrates a typical response of a
steel brace under reversed cyclic loading and a hysteretic
model proposed by Jain and Goel (1978). The model exhibits higher capacity in tension, as governed by yielding, and a
lower capacity in compression, dictated by buckling and a
subsequent residual compressive capacity.
Panel zones in beam–column joints of moment resisting
steel frames show shear distortions with stable, well-rounded
© 2003 NRC Canada
352
Fig. 16. Typical force–deformation relationship of a link.
loops as illustrated in Fig. 19. If these regions are to be
modelled in structural analysis, the elastoplastic hysteretic
models shown in Fig. 17 may be employed with shear stiffness, for improved accuracy of overall frame response.
Unlike structural steel elements, reinforced concrete develops stiffness degradation under inelastic deformation reversals. The degradation of stiffness takes place during
reloading and unloading as evidenced by reductions in the
slopes of corresponding force–deformation hysteresis loops.
The degradation in stiffness increases at a varying rate, usually as a function of the number of cycles and the level of
peak deformations attained. Figure 20 shows the degradation
of stiffness (reductions in the slopes of the force–
deformation relationship) recorded during a wall test. A
simple hysteretic model was developed by Clough (1966),
incorporating stiffness degradation into a perfectly elastoplastic response. Accordingly, the reloading branches of hysteresis loops are aimed at previous maximum response
points, thereby simulating stiffness degradation. The reloading slope is decreased with increasing maximum response
deformation. Unloading slopes remain parallel to the effective elastic stiffness, which implies that the model does not
recognize the stiffness degradation during unloading. A bilinear primary curve is used to set the limitations for
strength, with a nonzero post-yield slope. Figure 21 illustrates the basic features of the Clough model. The model is
simple to use and suitable for modelling stable hysteresis
loops, typically observed in flexural response. It was reported (Saiidi 1982; Clough 1966) that the response waveforms of the Clough model were significantly different from
those of the elastoplastic model and showed better correlation with data obtained from tests of reinforced concrete elements.
An improved degrading stiffness model was developed by
Takeda et al. (1970) with a trilinear primary curve. The reloading points in the model are aimed at the response point
at previous maximum deformations. Unloading slopes are
reduced as a function of the previous maximum deformation, hence the stiffness degradation is introduced in a more
refined manner as compared to the Clough model. Energy
dissipation during small amplitudes is considered. Figure 22
illustrates the basic features of the Takeda et al. model,
which is applicable to reinforced concrete members with stable hysteresis loops under constant axial load. The Takeda et
Can. J. Civ. Eng. Vol. 30, 2003
Fig. 17. Elastoplastic hysteretic model for steel structures:
(a) zero post-yield rigidity; (b) with strain hardening; and
(c) with optional strain hardening and Bauschinger effects.
al. model was later simplified by Otani and Sozen (1972)
and Powell (1975), who used a bilinear primary curve. Riddell and Newmark (1979) introduced improvements relative
to small cycle reversals. The model was further simplified
by Saiidi and Sozen (1979), who incorporated a bilinear primary curve and simpler rules for the unloading slope. Another degrading trilinear model was developed by Fukuda
(1969) for a predominantly flexural response of reinforced
concrete, consisting of a trilinear primary curve with unloading point considered as the new yield point.
An important feature of hysteretic response is strength decay. Structural members exhibit progressive loss of strength
under relatively high levels of inelastic deformation cycles.
Figure 23 illustrates a strength decay obtained in a concrete
column under reversed cyclic loading. The degree of
strength decay depends on many parameters, including the
governing deformation mode, concrete confinement, shear
strength, loading history, and level of axial load. The envelope of hysteresis loops in such a member cannot be obtained by bilinear or trilinear idealizations discussed earlier.
Early and rapid strength decay can have a very significant
effect on structural response and, if not considered in
hysteretic modelling, can lead to significant errors in dynamic analysis.
Hysteresis loops of both steel and reinforced concrete
members generally show a marked change in slope during
reloading. In steel members this may be attributed to the
slippage of a bolt or a rivet until the force is completely reversed. The change in slope in concrete elements is associated with opening and closing of cracks under cyclic
loading. Following a crack caused by a load in one direction,
reversed loading in the opposite direction meets with little
© 2003 NRC Canada
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353
Fig. 18. (a) Typical hysteretic force–displacement relationship of a steel brace (NEHRP 1997) (1 kip = 4.45 kN). (b) Hysteretic model
for steel brace members by Jain and Goel (1978).
initial resistance as the crack closes. Subsequently, the
cracked surfaces come in full contact, increasing the load resistance. This phenomenon is reflected in the force–
deformation relationship as a change in slope during reloading, and is known as pinching of hysteresis loops. If the
cracks are inclined diagonal tension cracks, as in the case of
shear response, some sliding occurs between the cracked
surfaces before they come in full contact. Also, the slippage
of reinforcing bar in the vicinity of a crack results in increased deformations with very low resistance in the opposite direction as the reinforcement slips back to its previous
position before the crack is completely closed and full resistance is attained. Therefore, pinching is more prevalent in
shear force – shear distortion and force – bar slip relation-
ships and may have to be considered in the hysteretic models used for analysis of structures where these deformation
components are significant. Strength decay and pinching of
hysteresis loops were modelled by Takayanagi and
Schnobrich (1976), who modified the Takeda et al. model to
incorporate features that are prevalent in shear and bar slip,
as illustrated in Fig. 24. Ozcebe and Saatcioglu (1989) developed a hysteretic model for inelastic shear effects that
considers the interaction between flexural and shear yielding
and the pinching of hysteresis loops as shown in Fig. 25.
Other hysteretic models exhibiting strength and stiffness
degradation and pinching were developed by others for analysis of reinforced concrete structures (Roufaiel and Meyer
1983, 1987; Banon et al. 1981).
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Can. J. Civ. Eng. Vol. 30, 2003
Fig. 19. Experimentally obtained steel panel zone hysteretic relationship (Krawinkler et al. 1971).
Fig. 20. Stiffness degradation recorded during a reinforced concrete wall test (Oesterle et al. 1979).
The pinching of hysteresis loops is a dominant feature in
reinforcement bar slippage. A number of hysteretic models
were developed specifically to simulate this phenomenon.
Fillipou et al. (1983a, 1983b) developed an analytical approach to describe the hysteretic bond–slip relationship.
Bond deterioration and associated bar slip were evaluated by
dividing the reinforcement into three segments. The method
was later simplified (Fillipou 1985) and used to describe the
response of an anchored bar to generalized excitation.
Morita and Kaku (1983) proposed a hysteretic model for the
moment–slip rotation relationship. It is based on linear stress
and strain distributions along the reinforcement and uses an
average bond stress observed experimentally. Pinching of
hysteresis loops is introduced empirically. Saatcioglu et al.
(1992) developed a hysteretic model for the moment –
Fig. 21. Stiffness degrading model by Clough (1966).
anchorage slip rotation relationship. The primary curve is
computed by establishing a nonlinear strain distribution
along the bar, with pinching of hysteresis loops considered,
as illustrated in Fig. 26. Others approximated bar–slip deformations through additional softening in response, without
the pinching effect. Otani (1974) modified the Takeda et al.
model to introduce the additional softening by assuming
constant bond stress within the development length.
Soleimani et al. (1979) modelled bar slip deformations by
means of a rotational spring. The elastic rigidity of the
moment – bar slip rotation relationship was taken as one
third of the flexural rigidity. The hysteretic rules proposed
by Clough (1966) were followed without the pinching of
hysteresis loops.
Important changes may occur in the hysteretic response of
vertical members due to the interaction of axial forces with
© 2003 NRC Canada
Saatcioglu and Humar
Fig. 22. Stiffness degrading model by Takeda et al. (1970).
355
Fig. 25. Hysteretic shear model (Ozcebe and Saatcioglu 1989).
Fig. 26. Moment – anchorage slip model.
Fig. 23. Strength decay in a column (Ozcebe and Saatcioglu
1989).
Fig. 24. Hysteretic model incorporating pinching and strength
decay (Takayanagi and Schnobrich 1976).
flexure and shear during seismic response. Variable axial
forces can be induced during seismic response in columns of
frame structures and in coupled walls as a result of the coupling action of the linking beams (Saatcioglu et al. 1983;
Abrams 1987). Figure 27 illustrates the behaviour of a column specimen tested under simultaneous variable axial force
and lateral deformation reversals. The increase and decrease
in strengths accompanied by axial compression and tension,
respectively, can be observed in Fig. 27. It is also evident in
the figure that the presence of axial compression reduces
ductility and accelerates strength decay. The effect of variable axial force was shown to be especially significant on
coupled walls (Takayanagi and Schnobrich 1976; Saatcioglu
and Derecho 1980; Saatcioglu et al. 1983; Keshavarzian and
Schnobrich 1985b). The effect of axial force on flexural
yield level was considered by Mahin and Bertero (1976) and
Aktan and Bertero (1982) by modifying the elastoplastic
model. Takayanagi and Schnobrich (1976) modified the
Takeda et al. model to include the effects of axial force flexure interaction. Saatcioglu et al. (1980, 1983) modified
Powell’s (1975) version of the Takeda et al. model to incorporate the axial force – flexure interaction during response.
The model, shown in Fig. 28, is based on updating member
stiffnesses for the subsequent time increment based on axial
force computed during the current time step.
© 2003 NRC Canada
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Fig. 27. Effect of variable axial force on hysteretic behaviour
(Saatcioglu and Ozcebe 1989).
Can. J. Civ. Eng. Vol. 30, 2003
sponse. This is especially true for flexure-dominant buildings where hysteretic response is dominated by wellrounded stable hysteresis loops, a feature that is addressed
by the majority of available computer software. In most
cases it is sufficient to analyze these buildings with the use
of an elastoplastic model for steel structures and a stiffness
degrading model for reinforced concrete structures. Special
care should be exercised, however, to make sure that the
computed response can be attained with the design and detailing practices employed.
References
Fig. 28. Axial force – moment interaction model (Saatcioglu et
al. 1983).
Conclusions
Dynamic analysis of buildings requires careful structural
modelling, appropriate selection of ground motion records,
and thorough knowledge and familiarity of the analyst with
the procedures and computer software employed. Nonlinear
time history analysis continues to provide challenges. Significant advances have been made in recent years, however, in
developing reliable analysis tools. Therefore, it is possible to
attain reasonably accurate assessment of inelastic seismic response of buildings through dynamic analysis, which can be
used in earthquake-resistant design of buildings. Although
the issues to be confronted appear to be numerous, often it is
possible to eliminate many of the modelling parameters discussed in the paper with appropriate justifications and simplify the process considerably. Design and detailing
provisions of current building codes often ensure the elimination of undesirable features of response that translate into
premature strength decay, excessive pinching, and inelastic
behaviour of brittle members or deformation modes. In such
buildings it is possible to ignore features of hysteretic response intended to model these aspects of structural re-
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List of symbols
A cross-sectional area
B maximum value of Bx
Bx ratio at level x used to determine torsional sensitivity and defined in eq. [7]
c neutral axis depth, measured from the extreme
compression fibre
d effective depth, measured from the extreme compression fibre to the centroid of tension steel
db depth of structural steel beam section
dc depth of structural steel column section
dm diagonal length of masonry strut
D dimension of the building in a direction parallel to
the applied forces
Dnx plan dimension of the building at level x perpendicular to the direction of seismic loading being
considered
Ds dimension of wall or braced frame that constitutes
the main lateral load resisting system in a direction parallel to the applied forces
E modulus of elasticity of material
(EI)sh flexural rigidity in steel strain hardening region
F lateral force
Fa acceleration-based coefficient, as defined in the
2005 NBCC
Fx lateral force applied to level x
G shear modulus of material
h height of infill wall
hn total height of building above the base in metres
H height of column
I moment of inertia
Ie earthquake importance factor of the structure
Ig moment of inertia based on gross (uncracked) sectional properties
i, j member ends of an element
K, k stiffness (slope of force–deformation relationship)
Ke elastic stiffness
Ks spring stiffness
l, l1, l2 member length
L length of infill wall
Lo length of overhang of an effective beam width
Lp, lp plastic hinge length
M bending moment
Mc, Mcr cracking moment at which the first cracking occurs
Mi, Mj moment at ends i and j
Mn nominal moment capacity
Mo moment at which unloading slope changes
Mp plastic moment
My yield moment
N total number of stories above grade
P vertical force
P1, P2, P3, P4 axial force levels where P1 > P2 > P3 > P4
Pyn, Pync, Pyp tension and compression yield force levels in a
steel brace, as illustrated in Fig. 18b
Q force in steel link
Qce elastic limit of steel link force
Rd ductility-related force modification factor that reflects the capability of a structure to dissipate energy through inelastic behaviour
Ro overstrength-related force modification factor that
accounts for the dependable portion of reserve
strength in a structure designed in accordance with
the 2005 NBCC
S(T) design spectral response acceleration for a period
of T
Sa(T) 5% damped spectral response acceleration for a
period of T as defined in the 2005 NBCC
t thickness of infill wall
T fundamental lateral period of vibration of the
building or structure in seconds in the direction
under consideration
ue elastic bond stress
uf frictional bond stress
V lateral earthquake design force at the base of the
structure, as determined by the equivalent static
force procedure
Vc shear force at which the first diagonal tension
crack occurs
Vd lateral earthquake design force at the base of the
structure as determined by dynamic analysis
Ve lateral earthquake elastic force at the base of the
structure as determined by dynamic analysis
Vy shear force at which shear yielding occurs
w width of equivalent wall strut
α angle of strut with the horizontal
α h length of the contact surface between equivalent
strut and column
α L length of the contact surface between equivalent
strut and beam
δ axial displacement of a steel brace
δa.s. extension of reinforcement in the adjoining member due to anchorage slip
δmax maximum storey displacement at the extreme
points of the structure at level x in the direction of
earthquake
δave average of the displacements at the extreme points
at level x
∆ displacement
∆a.s. displacement caused by anchorage slip
∆TIP tip deflection of a cantilever member
∆y yield deflection
© 2003 NRC Canada
Saatcioglu and Humar
εs
ε sh
εy
φ
φu
φy
γ
steel strain
strain at the onset of steel strain hardening
yield strain of steel
curvature
ultimate curvature
yield curvature
link distortion
359
γp
γ av
p
γy
θ
θa.s.
θP
plastic link distortion
average panel zone distortion
yield link distortion
member rotation
member-end rotation caused by anchorage slip
plastic rotation
© 2003 NRC Canada
Murat Saatcioglu