EARTHQUAKE ENGINEERING & STRUCTURAL DYNAMICS

Earthquake Engng Struct. Dyn. (2013)

Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.2338

Effects of vertical irregularity and thickness of unreinforced

masonry infill on the robustness of RC framed buildings

Goutam Mondal and Solomon Tesfamariam*,†

School of Engineering, The University of British Columbia, Kelowna, BC V1V 1 V7, Canada

SUMMARY
Presence of irregularities in reinforced concrete (RC) buildings increases seismic vulnerability. During
severe seismic shaking, such buildings may suffer disproportionate damage or even collapse that can be
minimized by increasing robustness. Robustness is a desirable property of structural systems that can mitigate
susceptible buildings to disproportionate collapse. In this paper, the effects of vertical irregularity and
thickness of unreinforced masonry infill on the robustness of a six-story three-bay RC frame are quantified.
Nonlinear static analysis of the frame is performed, and parametric study is undertaken by considering two
parameters: absence of masonry infill at different floors (i.e., vertical irregularities) and infill thickness.
Robustness has been quantified in terms of stiffness, base shear, ductility, and energy dissipation capacity of
the frame. It was observed that the infill thickness and vertical irregularity have significant influence on the
response of RC frame. The response surface method is used to develop a predictive equation for robustness
as a function of the two parameters. The predictive equation is validated further using 12 randomly selected
computer simulations. Copyright © 2013 John Wiley & Sons, Ltd.

Received 8 October 2012; Revised 1 June 2013; Accepted 4 June 2013

KEY WORDS: robustness; vertical irregularity; infilled frame; unreinforced masonry (URM) frame;
pushover analysis; inter-story drift ratio

1. INTRODUCTION

Disproportionate collapse and damage of reinforced concrete (RC) buildings with unreinforced
masonry (URM) walls is reported from past earthquakes (Figure 1). In general, URM walls have
both beneficial and detrimental effects as a structural element. They increase initial stiffness and
decrease the natural period of the frame, which might be beneficial depending on the frequency of
earthquake motion. On the other hand, early brittle failure of infill wall and consequently formation
of soft-story mechanism and column shear failure are not desirable [1, 2]. In reality, URM walls are
often placed nonuniformly in different floors for functional reasons causing the RC buildings to
have vertical irregularity, such as stiffness irregularity (soft story), strength irregularity (weak story),
mass irregularity, and short-column effects [3, 4]. Because such walls are generally treated as
nonstructural elements and are not considered in the design process, thickness of these walls may
also vary for functional and architectural reasons [5].
Vulnerability of RC buildings in presence of these irregularity parameters has been studied by many
researchers. Al-Ali and Krawinkler [6] studied the effects of vertical irregularities in a 10-story
building model designed according to the strong-beam-weak-column (column hinge model)
philosophy. They found that the effect of strength irregularity is larger than that of stiffness

*Correspondence to: Solomon Tesfamariam, School of Engineering, The University of British Columbia, Kelowna,
BC V1V 1 V7, Canada.
†
E-mail: Solomon.Tesfamariam@ubc.ca

Copyright © 2013 John Wiley & Sons, Ltd.

 G. MONDAL AND S. TESFAMARIAM

(a) (b)

(c)

Figure 1. (a) Collapse of open ground story of two-story RC frame during 2004 Sumatra earthquake,
(b) in-plane failure, and (c) out-of-plane failure of masonry in RC frame during 2011 Sikkim Earthquake
(Photo courtesy: NICEE at IIT Kanpur).

irregularity. Chintanapakdee and Chopra [7] studied the effects of stiffness, strength, and combined
stiffness and strength irregularities on story drift demand and floor displacement responses.
Tesfamariam and Saatcioglu [4, 8] considered vertical irregularities for the vulnerability assessment
of RC building. However, in all the earlier studies, vertical irregularities mainly stem from
geometric irregularities of floors and framing members. In other words, vertical irregularities caused
by the improper application of nonstructural infill walls were not explicitly considered. Kappos et al.
[9] considered infill irregularities for the vulnerability assessment of RC building. Das and Nau [10]
investigated the effect of vertical irregularities (i.e., open ground story (OGS) and captive columns)
caused by nonstructural masonry infill. They observed that although the damage was very high in
infilled frames due to the formation of story mechanism in the OGS, it was not influenced
significantly by the strength and stiffness of the masonry infill.
In the present study, nonlinear static analysis (pushover analysis) of a six-story three-bay frame is
performed to study the effects of vertical irregularities and infill thickness on the response of RC
frame. Effects of these parameters are quantified through a robustness measure (see Section 2 for
detailed discussion). The robustness is quantified on the basis of energy dissipated by the frame with
respect to that of the bare frame. Response surface method (RSM) is used to develop a predictive
equation for robustness as a function of these response parameters. Optimal RSM approach is used
to generate an approximate relationship between the response and significant variables. For this, a
design of experiments (DOE) method, aimed at finding a functional description of how factors affect
the response, is considered. More specifically, the interest is in exploring the main and possible
interaction effects of the performance modifiers (also referred to as design factors) on a main

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2013)
DOI: 10.1002/eqe
 EFFECTS OF VERTICAL IRREGULARITY AND INFILL THICKNESS ON ROBUSTNESS

performance indicator (response) of an RC building under earthquake loads. It is believed that similar
applications of the DOE methods, especially once combined with computer experiments, can be
beneficial for quantifying robustness of structures. The concept of DOE has been previously used in
other earthquake engineering applications (e.g., [3, 11–13]).

2. ROBUSTNESS IN STRUCTURES

In literature, the definition of robustness varies significantly. In general, robustness is defined as the
insensitivity of a structure to local failure. It is a desirable property of structural systems, which helps to
mitigate their susceptibility to disproportionate collapse. It is strongly related to redundancy, ductility,
load distribution, joint behavior characteristics in structure, and the consequences of failure. Starossek
and Haberland [14] presented a list of such definitions available in literature and design standards.
Many modern building codes such as JCSS [15], National Building Code of Canada [16], Eurocode
[17], and ASCE standard [18] defined robustness and specified the need for the structural robustness.
Department of defense, USA provides framework with regard to design and assessment of military
buildings to resist disproportionate collapse [19]. An overview of code provisions on robustness of
structures was presented by Ellingwood [20]. However, most of these codes provide only qualitative
description of the robustness and do not specify the quantitative measurement of robustness in
structures and the minimum acceptable limit of robustness. Recently, several researchers [21, 22]
presented frameworks for the measurement of robustness. These methods can be broadly classified as
deterministic, probabilistic, and risk-based quantification approaches and are briefly presented in Table I.

Table I. Methods of quantifying robustness.

Robustness Index References

Residual strength factor (RSF) [29]

RSF ¼ RScc
where Rc is the characteristic value of the base shear capacity of an offshore platform and Sc is
the design load corresponding to ultimate collapse.
Stiffness-based index (Rs) [22]
det K
Rs ¼ min det K0j
j
where K0 and Kj are the active system stiffness matrix of the intact structure and of the structure
after removing a structural element or a connection j, respectively.
Damage-based index (Rd) [22]
Rd ¼ 1  pp
lim
where p denotes the maximum damage progression caused by the assumed initial damage and
plim is the acceptable damage progression.
ilim
Rd;int;lim ¼ 1  ilim :ð2i
2
∫ ½dðiÞ  idi
lim Þ 0
where i is the initial damage, ilim is the maximum extent of initial local damage, d(i) is the
maximum total damage resulting from and including the initial damage of extent i, both d(i),
and i are obtained by dividing the respective reference value (mass, volume, floor area, or
cost) by the corresponding value of undamaged structures.
Energy-based index (Re) [22]
E
Re ¼ 1  max Efr;j;k
j
where Er,j is the energy released during the initial failure of a structural element j and
contributing to damaging a subsequently affected structural element k, and Ef,k is the energy
required for the failure of the subsequently affected structural element k
Re = 1 perfect robustness
0 < Re < 1 acceptable to a lesser or greater degree
Re < 0 failure progression to complete collapse.
Risk-based index (IR) [21]
I R ¼ RDirRþR
Dir
Indir
where RDir and RIndir are the risks associated with direct and indirect consequences of failure,
respectively.

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2013)
DOI: 10.1002/eqe
 G. MONDAL AND S. TESFAMARIAM

Because redundancy is also closely related to the level of robustness, as redundant systems are
generally believed to be more robust, Frangopol et al. [23–25] proposed probabilistic measures of
structural redundancy, based on the relation between damage probability and system failure probability.
Lind [26, 27] proposed a measure of system damage tolerance based on the increase in failure
probability due to the occurrence of damage. Ben-Haim [28] quantified robustness using information-
gap theory. Baker et al. [21] proposed risk-based measurement of robustness by comparing risk
associated with direct consequence of potential damages to the system, and indirect consequences
corresponding to the increased risk of a damaged system. ISO-19902:2007 [29] specifies the
deterministic approach to obtain robustness for offshore platform. In this approach, the ratio of the base
shear capacity and design load corresponding to the ultimate collapse is used as the robustness
measure. In this approach, the base shear capacity of structure, with and without a particular structural
element, is compared to obtain the robustness index. Starossek and Haberland [22] proposed stiffness,
damage, and energy-based robustness indices. In stiffness-based approach, the stiffness matrices of the
undamaged structure and that after removal of a structural element are compared. In damage-based
approach, the maximum damage progression caused by the ‘assumed initial damage’ and ‘acceptable
damage progression’ are compared to obtain the robustness index [22]. Similarly, in energy-based
approach, the energy released by initial failure of a structural element and the energy required for the
failure of the subsequently affected structural element are used to quantify robustness of structure.
Izzuddin et al. [30] proposed a multilevel framework for progressive collapse assessment of building
structures due to sudden column losses. The framework employs three stages, namely, determination of
the nonlinear static response, simplified dynamic assessment, and ductility assessment.
The common desirable parameters of the design for earthquake load and for robust structure are the ductility
and good configuration of the structure to have alternate load path in damage state. Therefore, seismic design
of structure generally ensures the structure to be robust. The existing robustness methods are based on the
consideration of removing structural members (e.g., [22]) and are mainly developed for accidental loads,
blast loads, and so on. In the present study, robustness of the frame has been quantified for seismic loading
in terms of stiffness, strength, ductility, and energy dissipation capacity normalized to those of the bare frame.

3. OVERVIEW OF THE BUILDING FRAME

A six-story three-bay RC ductile moment-resisting frame of an office building located in Vancouver is

considered for the present study. The building is founded on very dense soil and soft rock and,
therefore, the frame is designed assuming it is fixed at ground level. Seismic design and detailing of
the frame was performed according to the National Building Code of Canada [31]. Figure 2 shows
the plan of the building, elevation of the frame considered in the study, and reinforcement details of
typical beam and column sections. The building is designed to carry live loads of 2.4 kN/m2 on
typical office floors, 4.8 kN/m2 on 6-m wide corridor bay, 2.2 kN/m2 on the roofs, and 1.6 kN/m2 on
6-m wide strip over corridor bay. Self-weight of RC was taken as 24 kN/m3, for the dead load
calculation. In addition, other dead loads considered are 1.0 kN/m2 partition loading and 0.5 kN/m2
mechanical services loading on all floors, and 0.5 kN/m2 roofing.
The size of interior columns is 500 × 500 mm and that of the exterior columns is 450 × 450 mm. The
beams are 400 × 600 mm for the first three stories and 400 × 550 mm for the top three stories. Normal


density concrete with cube strength f ′c of 30 MPa and modulus of elasticity (Ec) of 27,400 MPa, and
HYSD steel bars of the yield strength ( fy) of 400 MPa are used for design of the frame. As usually
followed in practice, the frame is designed without considering the stiffness and strength of masonry
infill. Further, details of the design are available in Canadian Concrete Design Handbook [32].
Details of the finite element modeling of the frame are described in the following section.

4. FINITE ELEMENT MODELING

Figure 3 shows the finite element model of the building frame considered in this study. The model is
prepared and analyzed in open source code OpenSees [33]. Details of the parameters used for modeling
RC members and infill wall are discussed in this section.

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2013)
DOI: 10.1002/eqe
 EFFECTS OF VERTICAL IRREGULARITY AND INFILL THICKNESS ON ROBUSTNESS

(b)

(a) (c) (d)

Figure 2. Details of the building frame considered in this study: (a) plan of the building, (b) elevation of
the frame considered, (c) reinforcement details of typical column section near beam–column joint, and (d)
reinforcement details of typical beam section near beam–column joint.

Rotational Elastic Rigid

spring element element
6 @ 3.65 m

Beam with
hinges elements
Inelastic
Mid-span fiber section hinf hcol
node with
OP mass
9m 6m 9m Elastic
y section
x
z Linf

Figure 3. OpenSees model of six-story three-bay fully infilled RC frame.

4.1. Modeling of RC frame

Beams and columns are modeled using linear elastic beam–column element. The nonlinear behavior of
the members is represented by nonlinear rotational springs assigned at a distance half of the average
plastic hinge length (Lp) (in meter) from each end of the elements. The parameter Lp has been
calculated using the following equation [34]:

Lp ¼ 0:08L þ 0:22db f y (1)

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2013)
DOI: 10.1002/eqe
 G. MONDAL AND S. TESFAMARIAM

where, L is the length of the member (in meter), db is the diameter of longitudinal reinforcement
(in meter), and fy is the yield strength of steel (in megapascal). The stiffness and the strain hardening
coefficient (the ratio of post-yield stiffness to elastic stiffness) of the plastic hinge are modified
following the approach described by Ibarra and Krawinkler [35] to match the nonlinear behavior of
the assembly with that of the actual frame member.
Typical moment–rotation (M–θ) relationship of nonlinear rotational springs is shown in Figure 4(a).
It is obtained from the moment–curvature (M–φ) relation of the critical section of the member
performing section analysis in OpenSees. For the M–φ analysis, the cross sections of these elements
are discretized into fibers of unconfined concrete (cover), confined concrete (core), and reinforcing
steel. The uniaxial Kent–Scott–Park constitutive model with unconfined properties of concrete is
used to model the concrete cover [36, 37]. Confinement model proposed by Braga et al. [38] is used
to model the concrete core because this model accounts for confinement effects due to different
arrangements of transverse reinforcement. Both the concrete models consider degraded linear
unloading reloading stiffness and no tensile strength of concrete. Giuffré–Menegotto–Pinto steel
constitutive model is used to represent the main reinforcing steel [39–41]. The effect of axial load
variation on the moment capacity of the section is ignored, and the M–φ relationship has been
obtained for the axial load corresponding to the design load. P–Δ effect is included in this analysis.
The Poisson's ratio and mass density of concrete are taken as 0.2, and 2500 kg/m3, respectively.

4.2. Modeling of infilled masonry

The element removal technique proposed by Kadysiewski and Mosalam [42] is used to model the masonry infill.
Salient features of this model are discussed in this section. Detailed discussion can be found elsewhere [42].
Among the different methods proposed in the recent literature to model infill panel, the most widely
adopted method is the single or multiple equivalent diagonal strut model [34, 43–50]. However, the
multiple equivalent struts are necessary if the local interaction between the infill panel and the RC
elements has to be modeled [45]. In this study, each infill panel is modeled as single equivalent
diagonal strut specified in FEMA-356 [51] and based on the work of Mainstone [46]. The diagonal
strut is comprised of two equal-size beam with hinges elements connected at the mid-point node
with out-of-plane (OP) mass as shown in Figure 3. The width of the strut is given by [51]:

w ¼ 0:175ðλI hcol Þ0:4 rinf (2)

h i0:25
E m tinf sin2θ
where, the coefficient λI ¼ 4Ec I col hinf and hcol and hinf are the height of the column and infill panel,
respectively (Figure 3), Icol denotes the moment of inertia of column, rinf is the diagonal length of infill
panel, tinf is the infill thickness, which is also the thickness of the diagonal strut (normal to the wall), the
angle θ = tan  1(hinf/Linf), and Linf is the length of the infill panel. In absence of actual material
properties, the modulus of elasticity of masonry (Em) can be taken as [52]:
E m ¼ kf ′m (3)
where, f ′m is the compressive strength of masonry and is taken as 17 MPa and factor k lies between 500 to
600. In the present study, k is taken as 550 that was found to be reasonable for infilled frame by others [53, 54].

Figure 4. Typical load–deflection relationship of (a) RC frame members and (b) infill wall.

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2013)
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 EFFECTS OF VERTICAL IRREGULARITY AND INFILL THICKNESS ON ROBUSTNESS

The inelastic fiber section is assigned to the ends of the elements connected to the mid-point node.
Elastic sections with very small moment of inertia are assigned to the ends attached to the surrounding
frame to simulate moment release. The hinge length near the mid-span node is selected as 1/10 of the
total length of the diagonal strut to produce a relatively sharp yield point for the element, while at the
same time providing a numerically stable solution.
This model is capable of considering the interaction of in-plane (IP) and OP effects. This interaction
is important because the load in the IP direction reduces the strength in the OP direction and vice versa
[55]. However, because in the present study, displacement controlled nonlinear static pushover
analysis is conducted in the IP direction only, the IP–OP interaction capability of the model is not
used and the OP mass is not necessary here. Therefore, for pure IP behavior, the model should be
provided with the axial load–deformation (P–Δ) curve for infill as shown in Figure 4(b). The
stiffness (kinf) and strength (Pu) of the strut are determined by the following equations [51]:

k inf ¼ atinf E m =Linf (4)

Pu ¼ V me An = cosθ (5)

where An is the net bed area of infill, and Vme is the expected shear strength of the masonry infill
obtained from the following equation in absence of experimental data [51]:
 
Pce
V me ¼ 0:75 V te þ =1:5 (6)
An

where Pce is the expected gravity compressive force applied on wall and Vte (considered here as 7 MPa)
is the average bed joint shear strength. Collapse of infill panel is modeled by the ‘collapse prevention’
limit states specified in FEMA 356 [51]. In this study, collapse of infill panels has been observed at
displacement ductility (Δu/Δy) of 4 to 6.

5. PARAMETRIC STUDY USING PUSHOVER ANALYSIS

Because the main objective of this study is to investigate the effects of infill thickness and vertical
irregularities on the seismic response of RC frame, thickness of infill, and absence of infill in
different stories are considered as parameters. Thickness of the wall depends on its function, and on
the size, orientation, and layout of the locally available brick units, which generally vary from place
to place. In reality, size of the wall may vary from 60 to 300 mm. The thin walls generally represent
partition walls, whereas the thick walls are mainly intended for external frames. The study has been
conducted on RC frame for four different thickness of infill (Tinfill) (75, 125, 175, and 250 mm) and
for absence of infill at first and subsequent stories. This constitutes a total 25 numerical analysis
cases as presented in Table II.
Pushover analysis is performed to obtain the post-yield behavior of the RC frame with 25 different
arrangements of infill wall. Collapse mechanism of infilled frame can easily be captured in this method
[56]. Displacement–controlled force profile similar to the fundamental mode shape of the frame is
applied with small increase in displacement (0.01 mm) in each step until the structure collapses. The
fundamental mode shape of the frame was obtained by carrying out modal analysis. The frame is
assumed to collapse if inter-story drift ratio (ISDR) exceeds a certain value. ISDR is defined as the
ratio of difference in lateral displacement of two consecutive floors to the height of the story.
Experimental and analytical studies have shown that the ISDR at collapse of moment-resisting RC
frame ranges from 6.5% to 10% [57–59]. Therefore, in this study, the frame is assumed to collapse
if the ISDR of any story exceeds 8%.
Effects of different parameters on the response of RC frame are discussed in the following sections
through pushover curves, displacement profile, ISDR at collapse, effective stiffness, maximum base
shear, ductility, and energy dissipation capacity of the frame. Base shear is plotted as a function of

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2013)
DOI: 10.1002/eqe
 Table II. Response and robustness index of different simulation cases.
Response parameters Robustness index

Simulation Tinfill No infill Fundamental Initial stiffness, Maximum base Energy,

case (mm) in story period, T1 Ki (kN/mm) shear, BS (kN) Ductility, μ E (×104 J) RKi RBS Rμ RE

1 1–6 0.92 7.7 818 9.6 53 1.00 1.00 1.00 1.00

2 75 — 0.42 11.8 850 9.3 46 1.54 1.04 0.96 0.87
3 1 0.54 13.3 822 12.2 45 1.74 1.01 1.27 0.85

Copyright © 2013 John Wiley & Sons, Ltd.

4 1–2 0.73 11.6 821 10.8 45 1.52 1.00 1.12 0.84
5 1–3 0.85 9.9 816 10.0 45 1.30 1.00 1.03 0.84
6 1–4 0.91 8.8 812 10.2 49 1.16 0.99 1.06 0.93
7 1–5 0.94 8.1 813 10.0 52 1.06 0.99 1.04 0.98
8 125 — 0.38 14.9 1064 10.7 52 1.95 1.30 1.11 0.98
9 1 0.52 14.1 885 7.7 35 1.84 1.08 0.80 0.66
10 1–2 0.74 13.6 847 10.0 36 1.78 1.04 1.04 0.67
11 1–3 0.87 10.8 807 10.4 44 1.41 0.99 1.08 0.83
12 1–4 0.94 9.1 805 10.1 49 1.19 0.98 1.05 0.91
13 1–5 0.95 8.2 810 10.2 52 1.08 0.99 1.06 0.97
14 175 — 0.35 17.6 1279 8.2 45 2.30 1.56 0.85 0.84
15 1 0.52 20.4 974 8.2 25 2.67 1.19 0.85 0.47
16 1–2 0.75 15.6 835 11.3 35 2.04 1.02 1.18 0.65
G. MONDAL AND S. TESFAMARIAM

17 1–3 0.89 11.6 796 11.0 43 1.51 0.97 1.14 0.81

18 1–4 0.96 9.6 798 10.6 48 1.25 0.98 1.10 0.90
19 1–5 0.97 8.3 806 10.1 51 1.08 0.99 1.04 0.96
20 250 — 0.33 21.6 1576 4.7 32 2.82 1.93 0.49 0.60
21 1 0.52 26.3 953 10.1 24 3.44 1.17 1.04 0.45
22 1–2 0.78 17.5 816 11.6 34 2.29 1.00 1.20 0.63
23 1–3 0.93 12.8 781 12.2 42 1.67 0.96 1.26 0.80
24 1–4 1.00 10.0 787 11.0 47 1.31 0.96 1.14 0.89
25 1–5 0.99 8.4 800 10.2 51 1.10 0.98 1.06 0.95

Earthquake Engng Struct. Dyn. (2013)

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 EFFECTS OF VERTICAL IRREGULARITY AND INFILL THICKNESS ON ROBUSTNESS

roof displacement to obtain the pushover curve (Figure 5). In typical pushover curves, characteristic
points (e.g., yield point and failure point) are not well-defined. Therefore, all the pushover curves
are idealized to obtain effective initial stiffness, yield point, failure point, and ductility. Figure 5
shows the idealized pushover curve of a bare frame and infilled frame. Pushover curve of the bare
frame is idealized as trilinear curve (Figure 5(a)). First two segments of this curve are generated
using equal energy concept (i.e., area under the actual curve is approximately same as that under
idealized curve) following the procedure specified in FEMA-400 [60]. The third segment is drawn
by joining end point of the second segment and a point at collapse displacement following the equal
energy concept for this segment.
Pushover curve for fully infilled frame is characterized by significant decrease in strength because of
damage to infill panel [61]. Therefore, frame where such degradation is observed is idealized as
multilinear curve with five linear segments as shown in Figure 5(b). First three segments are
idealized following the procedure proposed by Dolšek and Fajfar [61, 62]. The last two segments
are drawn such that the two segments meet at the maximum base shear of this portion, last segment
ends at the collapse displacement, and the areas under the idealized curve and actual curve for this
portion are approximately same.
Effective initial stiffness (Ki) is the slope of the initial segment of the idealized curve. Ductility (μ) is
determined by the ratio of ultimate displacement (Du), and yield displacement (Dy) (Figure 5).
Maximum base shear is obtained from the actual curve because it is clearly defined. Energy
dissipation capacity of the frame is determined by obtaining area under the idealized pushover curve
up to the collapse point.
Figure 6 shows the actual and idealized pushover curves of a fully infilled frame with infill
thickness of 250 mm (case 20). The response of the frame is characterized by a linear portion
followed by sudden drop of strength because of collapse of infill panel in the second story that makes
this story softer and weaker than the other stories. Further increase in lateral displacement causes
yielding of the columns in second story and ultimately the frame collapses because of plastic hinge
formation in the columns of this story. Table II summarizes the response of all the simulation cases in
terms of effective stiffness Ki, ductility μ, maximum base shear (BS), and energy (E) dissipation
capacity of the frame.
For infilled frame, the masonry panels also carry the loads until they fail although these are not
considered as structural elements during design of the frame. No significant damage is observed in
frame members till the failure of the masonry panel even in absence of infill in lower two stories. This
is because the frame is designed according to modern seismic code with strong-column-weak-beam
concept. However, for bare frame, the decrease in stiffness was observed at base shear of about
300 kN because of the damage in columns and beams.
As expected, stiffness and strength of infilled frame is significantly high as compared with the bare
frame (Table II). For example, in presence of infill walls with Tinfill = 125 mm in all the panels (case 8),
stiffness and strength of the frame increased by 95% and 30%, respectively. The beneficial effect of

(a) (b)

Figure 5. Idealization of pushover curves: (a) trilinear curve for bare frame and (b) multilinear curve for
infilled frame.

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2013)
DOI: 10.1002/eqe
 G. MONDAL AND S. TESFAMARIAM

Figure 6. Pushover curve with deformed shape of frame at key states for of fully infilled frame with infill
thickness 250 mm (case 20). Colors of the plastic hinges in the deformed shape of the frame indicate the
damage state of the plastic hinges as shown in Figure 4. The solid and dash diagonal lines indicate the
undamaged and collapse states of infill wall.

infill wall in increasing strength and stiffness of RC frame was also observed by others (e.g., [62–64]). In
case of fully infilled frame, the base shear attains the maximum value at a relatively small roof
displacement (=100 mm) as compared with the bare frame where the maximum base shear is reached
at roof displacement of about 600 mm. With a further increase in displacement, the masonry infill
walls at lower stories start to degrade and eventually collapse. Brittle failure of each of the walls is
associated with sudden drop in base shear as shown in Figures 6 and 7(a). In general, infill wall
reduces ductility and energy dissipation capacity of RC frame and the reduction depends on infill
thickness as discussed in the following section. For fully infilled frame with infill thickness
Tinfill = 250 mm (case 20), μ = 4.7 is the lowest among all the cases. This is because displacement of
the frame is caused by the deformation of columns at second story after the failure of infill panel in
that story and story mechanism occurs (Figure 6). This type of behavior occurs because infill wall is
not considered during the design of the frame. In such a case, increase of infill thickness beyond
certain limit may not be helpful from earthquake resistant design point of view. Formation of story
mechanism for RC framed buildings with uniformly distributed infill wall was also observed by
others (e.g., [1, 64, 65]).

5.1. Effect of infill thickness

Figures 7 and 8 show the effect of infill thickness on the response of infilled frames. It is observed that
Tinfill has significant effect on the response of fully infilled frames (cases 2, 8, 14, and 20). As expected,
both the strength and stiffness increase substantially because of increase in Tinfill (Table II and Figure 7
(a)). For example, 100% increase of Tinfill (from 125 (case 8) to 250 mm (case 22)) causes about 45%
and 48% increase in strength and stiffness, respectively. Conversely, μ and E dissipation capacity of
fully infilled frame decrease considerably with increase in infill thickness except for Tinfill = 125 mm
(case 8) (Table II). In general, an increase in infill thickness by 100% causes about 40% and 55%
decrease in μ and E dissipation capacity of the frame. This is because maximum roof displacement
decreases significantly, whereas yield displacement decreases slightly because of increase in Tinfill
(Figure 7(a)). Similarly, although the stiffness and strength increase, the maximum roof
displacement decreases so much that the area under the pushover curve and thus the E dissipation
capacity of the frames decreases with increase in Tinfill (Figure 7(a)). For case 8, roof displacement stems
almost equally from the story drift of first two stories, whereas for other cases (cases 2, 14, and 20) of
fully infilled frame, only one story (either story 1 or 2) contributes more than the other stories (Figure 8
(a)). In general, for low infill thickness, collapse of infill panels spreads from lower stories to the upper

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2013)
DOI: 10.1002/eqe
 EFFECTS OF VERTICAL IRREGULARITY AND INFILL THICKNESS ON ROBUSTNESS

Figure 7. Effect of infill thickness (Tinfill) on the response of infilled frames.

stories followed by damage to columns of these stories. However, as the infill thickness increases, chances
of infill failure restricted to a single story (mainly first or second) rendering this story softer than the other
stories. Therefore, thick nonstructural infill wall in ductile RC frame may not always be beneficial in
seismic condition. As compared with fully infilled frame, the effect of infill thickness on the response of
infilled frame with OGS (cases 3, 9, 15, and 21) reduces as shown in Figures 7(b) and 8(b). Infill
thickness has no significant effect on the response of RC frame with absence of infill in stories beyond
the ground story.
Figure 8 shows the effect of infill thickness on the story drift ratio of infill frames at collapse. In
general, the story drift attains its maximum value at ground story and decreases with increase story
level. This is because base shear decreases with height. However, for fully infilled frame with
Tinfill = 250 mm, the maximum drift occurs in the second story. Large thickness of infill makes the
infill stronger than the surrounding frame elements (beams and columns). For frame where infill
walls are absent in more than one story, drift mainly occurs only in the stories where infill walls are
absent. It can be observed that, the effect of infill thickness is insignificant for frames with absence
of infill beyond the ground story (Figures 7 and 8).

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2013)
DOI: 10.1002/eqe
 G. MONDAL AND S. TESFAMARIAM

Figure 8. Effect of infill thickness (Tinfill) on the story drift ratio of infilled frames.

5.2. Effect of vertical irregularities

For a particular infill thickness, stiffness of the frame in absence of infill in the first story is less than that
of the fully infilled frame (Table II). Absence of infill walls in the subsequent stories results in decrease in
stiffness of the frame. As the thickness of infill increases, vertical irregularity plays important role on the
response of frame. In general, lateral strength of frame decreases, and ductility and energy dissipation
capacity of the frame increase due to increase in vertical irregularity (i.e., more number of floors
without infill). Figure 8 shows that soft-story mechanism does not occur for frames with
Tinfill ≤ 125 mm. This is because infill walls in the upper stories gradually fail after the collapse of
those in the lower stories. Collapse of infill walls is followed by damage to beams and columns in
those stories. Therefore, displacement of the frame stems from the story drift of more than one story.
Hence, beam-sway mechanism prevails over the soft-story mechanism in these cases. On the other
hand, for frames with Tinfill > 125 mm, as the lateral load increases, the infill walls in the lower two
stories start to degrade until all the infill walls in one of the lower stories collapse. This is followed by
occurrence of plastic hinges in the columns in that story prior to the generation of sufficient number of
plastic hinges in beams of upper stories. Because thickness of the infill is more in these frames, infill
walls in upper stories do not collapse. This causes soft-story mechanism in these frames (Figure 8).

6. ROBUSTNESS OF RC FRAME

Robustness of the frame is obtained by four different robustness indices, namely, RKi, RBS, Rμ and
RE developed on the basis of response parameters, stiffness, strength, ductility, and energy

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2013)
DOI: 10.1002/eqe
 EFFECTS OF VERTICAL IRREGULARITY AND INFILL THICKNESS ON ROBUSTNESS

dissipation capacity, respectively. The values of the response parameters are normalized with that of a
bare frame to obtain the corresponding robustness index as shown in Table II. For example, RKi is
obtained by normalizing the initial stiffness of frame with that of the bare frame. Interaction
diagrams of these indices with the infill thickness are shown in Figure 9. From Figure 9(a) and 9(b),
it can be discerned that the robustness indices RKi and RBS increase with increase in infill thickness
and decrease with increase in vertical irregularity. The simulation cases 20 and 21 seem to be the
most robust structures as per RKi and RBS, respectively (Table II). However, these are not actually
robust structure because soft-story mechanism takes places in second and first stories, respectively,
because of the early brittle failure of infill in these stories (Figure 8). Therefore, RKi and RBS are not
good indicators of robustness of structure. Moreover, these indices do not directly reflect the
collapse of the structure. The robustness index Rμ decreases with increase in infill thickness and is
not affected significantly by vertical irregularities (Figure 9(c)). The index RE decreases with
increase in infill thickness but increases with vertical irregularities (Figure 9(d)). Both the indices
reflect the soft-story mechanism for simulation case 20. However, Rμ values are not unique because
it depends on the identification of yield and maximum displacements. On the other hand, the
energy-based index RE depends on the area under the pushover curve, which can be determined with
good precision. Therefore, it seems that RE is a good indicator as the robustness quantification of
RC infilled frame.

7. PREDICTIVE EQUATIONS USING RESPONSE SURFACE METHOD

On the basis of the results of 25 simulation cases considered so far, it is concluded that energy
dissipation capacity of the frame is a good indicator of the robustness of structure. In this section, a
model is proposed to predict the energy-based robustness index (RE) in terms of infill thickness
(Tinfill) and vertical irregularity. Table III presents the simulation cases considered for the prediction
model. Four different thickness of infill [Tinfill = 75, 125, 175, and 250 mm] are considered. Vertical
irregularity is indicated by the absence of infill wall in a single story (AIS). For example, case 28
represents simulation case with infill thickness 75 mm and without infill in story 4 (AIS = 4). For

Full Infill No Infill in Story 1 No Infill in Story 1-2

No Infill in Story 1-3 No Infill in Story 1-4 No Infill in Story 1-5
4 4

3 3
R Ki

R BS

2 2

1 1

(a) (b)
0 0

4 4

3 3
RE

2 2
R

1 1

(c) (d)
0 0
50 100 150 200 250 300 50 100 150 200 250 300

Infill Thickness, T infill (mm) Infill Thickness, T infill (mm)

Figure 9. Interaction diagrams of infill thickness (Tinfill) with robustness indices.

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2013)
DOI: 10.1002/eqe
 G. MONDAL AND S. TESFAMARIAM

Table III. Energy-based robustness index (RE) of the simulation cases used for the prediction
model.
Analysis cases Tinfill (mm) AIS RE
a
3 75 1 0.85
26a 2 0.86
27b 3 0.87
28a 4 1.02
29a 5 1.04
30b 6 0.97
2a 7 0.87
9a 125 1 0.66
31b 2 0.69
32a 3 0.94
33b 4 1.11
34a 5 1.18
35b 6 0.99
8a 7 0.98
15a 175 1 0.47
36a 2 0.79
37b 3 0.92
38a 4 1.05
39b 5 1.30
40b 6 0.86
14a 7 0.84
21a 250 1 0.45
41b 2 0.54
42b 3 0.83
43a 4 1.00
44b 5 1.20
45b 6 0.62
20a 7 0.60
a
Training data sets (16).
b
Validation data sets (12).

each Tinfill, six cases of vertical irregularity (AIS = 1 to 6) are possible. In addition, fully infilled
frame are considered for the prediction model and AIS = 7 is assigned to it. Table III presents
the energy-based robustness index RE obtained from the pushover analysis of 16 and 12 cases
used for the prediction and validation model, respectively (each set is denoted within the
footnote of Table III). Base on this result, a model is proposed using RSM to predict the
robustness of infilled frame.

7.1. Response surface method

Response surface method is a collection of mathematical and statistical techniques for solving
problems in which the goal is to optimize the response y of a system or process using n
independent variables, subject to observational errors [66]. Response surfaces are smooth
analytical functions and are most often approximated by linear function (first-order model) or
polynomial of higher degree (such as the second-order model). The second-order polynomial
response surface has the form:
n n n i
y ¼ β0 þ ∑ βi xi þ ∑ βii x2i þ ∑ ∑ βij xi xj (7)
i¼1 i¼1 i¼1 j¼1

where y is regression equation, and β0, βi, βii and βij are the regression coefficients. Estimates of the
coefficients β0, βi, βii and βij can be obtained by fitting the regression equation to the response
surface values observed at a set of data points. For a second-order response surface, (n + 1)(n + 2)/2
unknown regression parameters are present and to estimate these parameters, an equal number of
data points are needed. Different authors have reported generation of RSM in reliability
engineering [67–71].

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2013)
DOI: 10.1002/eqe
 EFFECTS OF VERTICAL IRREGULARITY AND INFILL THICKNESS ON ROBUSTNESS

To identify the individual effects of the parameters on the response RE, ‘main effect’ plots are
constructed as shown in Figure 10. In Figure 10(a), for example, coordinate of the four points are
found by collating response values (RE) provided in Table III and averaging them over the rows
for four infill thickness, Tinfill = 75 (i.e., cases 2, 3, and 27–30), 125 (i.e., cases 8, 9, and 31–35),
175 (i.e., cases 14, 15, 36, 37, 39, and 40), and 250 mm (i.e., cases 20, 21, and 41–45). Similar
procedure is applied to arrive at main effect plots for AIS. As a measure of parameter sensitivity,
the variation rate/slope of response with each of the individual factors may indicate the significance
of that factor. Consequently, between Tinfill and AIS, it appears that AIS has a higher effect on the
performance of the structure.
Optimal RSM approach in Design Expert V8 software [72] is used to generate an approximate
relationship between the response and significant variables [73, 74]. Through the optimal RSM
algorithm [72], the training data sets are selected as denoted in Table III. The trend of
variations in Figure 10(a) and 10(b) suggests that the first-order response surface for RE may
not be sufficient to capture the response of the structure over the range of the proposed factors'
levels. Therefore, a second-order polynomial model form has been chosen and indeed this
intuition was also validated using [72].
The regression coefficients shown in Equation (7) are generated using [72]. After investigating the normal
plot probability and F statistics, significant parameters, βi (i = 0, 1, 2) and βii (ii = 2), are summarized in Table IV.
Thus RE is quantified in terms of thickness of infill (Tinfill), and absence of infill in story (AIS) as:

RE ¼ β0 þ β1 T infill þ β2 AIS þ β22 AIS2 (8)

Results of the RSM are plotted in Figure 11(a). Figure 11(b) shows, the predicted and
calculated results of both the training and validation data sets shown in Table III. The
predicted and calculated RE values shown in Figure 11(b) are in good agreement, which shows
utility of the proposed RSM.

(a) (b)

Figure 10. Sensitivity of performance modifiers with respect to (a) infill thickness (Tinfill) slope and (b)
absence of infill in story (AIS).

Table IV. Coefficients of the prediction model for RE.

Coefficients Value
β0 0.33635
β1 9.31238 × 104
β2 0.39387
β22 0.045707

Copyright © 2013 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (2013)
DOI: 10.1002/eqe
 G. MONDAL AND S. TESFAMARIAM

1.2 1.4

1.2
1
1

Predicted RE
0.8
RE 0.8
0.6 0.6

0.4 0.4
Training data
0.2
Validation data
75 7 0
100 6
125 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4
150 4
175 3 AIS Calculated RE
200 2
Tinfill (mm) 225
250 1

(a) (b)

Figure 11. Results of response surface method: (a) response plot and (b) training and validation prediction
curves.

8. SUMMARY AND CONCLUSIONS

Robustness is a desirable property of structural systems that mitigates their susceptibility to disproportionate
collapse. In the paper, the effects of vertical irregularity and infill thickness on the robustness of RC infilled
frame have been studied. The robustness of the RC frame has been quantified on the basis of strength,
stiffness, ductility, and energy dissipation capacity of the RC frame normalized to these response
parameters of bare frame. Nonlinear pushover analysis is conducted and parametric study was performed
considering thickness of infill, and vertical irregularity as parameters. The RSM is used to develop a
predictive equation for robustness as a function of the parameters. The predictive equation is validated
through two randomly selected cases. The following conclusions have been drawn from the present study:
(1) Stiffness and strength of infilled frame is significantly high as compared with the bare frame. In
infilled frame, failure of each of the walls is brittle in nature and is, therefore, associated with
sudden drop in base shear.
(2) Strength and stiffness of RC frame increase because of increase in infill thickness. However,
ductility and the energy dissipation capacity of the frame decrease significantly with increase in
infill thickness. Therefore, very thick nonstructural infill wall may not be beneficial for RC frame.
(3) Infill thickness has no significant effect on the response of RC frame with absence of infill in
stories beyond the ground story.
(4) Infill thickness plays important role in collapse mechanism of RC frame under lateral load. As
the thickness of wall increases, tendency of occurrence of soft-story mechanism increases.
(5) The energy-based robustness index (RE) satisfactorily reflects the robustness of RC infilled frame.
(6) The predictive equation reasonably estimates the energy-based robustness index (RE). However,
it should be modified in future considering more number of parameters.

ACKNOWLEDGEMENT
The financial support from Natural Sciences and Engineering Research Council of Canada (NSERC) under
Discovery Grant Program is greatly acknowledged.

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