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Proceedings of the 5th International Conference on Integrity-Reliability-Failure, Porto/Portugal 24-28 July 2016
Editors J.F. Silva Gomes and S.A. Meguid
Publ. INEGI/FEUP (2016)
-1375-
PAPER REF: 6370
RELIABILITY ANALYSIS OF A SEVERELY DAMAGED RC
BUILDING, CONSIDERING THE EFFECT OF NON-STRUCTURAL
MASONRY WALLS
Mariana Barros1(*)
, Eduardo Cavaco2, Luís Neves
3, Eduardo Júlio
4
1Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Portugal
2CEris, ICIST, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Portugal
3Centre for Risk and Reliability Engineering, University of Nottingham, Faculdade de Ciências e Tecnologia,
Universidade Nova de Lisboa, Portugal 4CEris, ICIST, Instituto Superior Técnico, Universidade de Lisboa, Portugal
(*)Email:[email protected]
ABSTRACT
This paper presents a reliability analysis of a reinforced concrete building severely damaged
due to the destructions of three outer columns, caused by a landslide. The main objective of
this study is to evaluate the relevance of the masonry walls to mitigate the probability of
failure of the damaged building. A nonlinear finite element analysis (FEA) was performed to
simulate the structural behaviour of the damaged building with and without the contribution
of the masonry walls. An Artificial Neural Network (ANN) was also defined and calibrated in
order to approach the structural behaviour depending on the possible range values of the
significant random variables. Monte Carlo Simulation (MCS), based on the ANN, was finally
used to assess the probability of failure of the damaged building considering or not the effect
of masonry walls.
Keywords: Reliability, unforeseen event, RC building, damage.
INTRODUCTION
Recent structural failures, such as the Bad Reichnhall Ice-Arena or the Towers of the World
Trade Center, which have exhibited an extension of direct and indirect consequences clearly
disproportionate relatively to the original damage (Pearson and Delatte 2005; Andersen and
Dietsch 2011; Baker et al. 2008; Cavaco 2013), have increased the interest in defining
methods to design robust structures capable of supporting severe damage without collapsing.
On the other hand, examples showing structures surviving extreme actions with limited
consequences show that some properties of structures, disregarded in design, can have
fundamental impact of performance under extreme events. Tiago and Júlio (2010) studied a
frame reinforced concrete (RC) building that withstood the failure of several columns at the
base level without collapsing. Analysis of this building shows that infill walls have a crucial
role on the robustness of reinforced concrete (RC) frames, subjected to severe structural
damages, as the failure of one or more columns (Tiago and Júlio 2010; Cachado et al. 2011;
Farazman et al. 2013; Xavier et al. 2014; Sasani 2008; Helmy et al. 2015, Sattar and Liel,
2010).
The purpose of the study described in this paper is to determine the contribution of masonry
walls, usually not considered as structural elements, to the reliability of damaged RC framed
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structures under extreme events. A real case study is investigated and the probability of
having a global failure is provided considering or neglecting the effect of non-structural
masonry walls. Achieved results are of paramount importance regarding structural robustness
and the likelihood of a collapse during the following repair and strengthening works.
CASE STUDY
Accident description
In 2000, in Coimbra, Portugal, a landslide caused severe damages to the RC structure of a
residential building of 16 stories (Fig. 1a). As a result of this extreme event, the first two
levels of three exterior columns were completely destroyed and the rear body of the building
supported by these, with a dimension in plant of 9.5 × 6.7 m2, became a 7.0 meters span
cantilever with 12 stories (Fig. 1b) (Cachado et al. 2011). After the inspection, all debris were
removed and the retrofitting works started (Fig. 1b).
Conclusions from the structural analysis of the damaged building (Tiago and Júlio, 2010)
suggest that the progressive collapse was prevented due to the contributions of non-structural
masonry walls. The later allowed the development of a new load path to support the gravity
forces initially equilibrated by the outer columns. These forces were transferred to the
masonry walls by compressions stresses (struts) and to the slabs by tensile forces (ties).
(a) (b)
Fig 1 - Severely damaged RC building due to a land slide: (a) Rear façade of the building after the accident.
(b) Cleaning and debris removal from the accident site.
Structural characterization
The building’s main structure is composed of RC frames, orthogonally disposed and settled
on direct foundations. The floorings are composed of ceramic blocks and precast pre-stressed
joists topped by a cast-in-place concrete layer. This type of flooring is usually only adopted
for housing or low-rise buildings, which is not the case, due to poor diaphragm performance
under seismic loads. Pre-stressed joist are supported by edge beams sustained by the columns.
The cantilevered beams developed after the accident have a cross section of 0.30 × 0.35 m2
with 4 top reinforcement bars with 12 mm of diameter and 4 bottom reinforcement bars with
10 mm of diameter. The edge beams (V1) have a cross section of 0.45 × 0.60 m2 with 4 top
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and bottom reinforcement bars with 12 mm of diameter. The corner columns have a cross
section of 0.30 × 0.60 m2 and 8 reinforcement bars with 16 mm of diameter. The middle
columns have a cross section of 0.30 × 0.70 m2 with 10 reinforcement bars with 16 mm of
diameter. The concrete and reinforcing bars (rebars) are of the C20/25 and S400 grades,
respectively.
The façades and partition walls are composed of ceramic bricks connected and overlaid with a
cement mortar. The first and the latter have a 300mm and 150mm thickness, respectively and
2300mm height. On the two façade walls over the cantilevered beams a set of two openings
exist per façade with the following dimensions: 2.10 × 1.00 m2 plus 1.10 × 1.00 m
2; and 2.10
× 1.00 m2
plus 2.10 × 1.00 m2.
NUMERICAL MODEL
RC frames
A finite element model of both the intact and the damage structure was built using the
OpenSees software (Mazzoni et al. 2015). Force-based finite elements with distributed
plasticity and physical non-linear behaviour were used for the generality of beams and
columns. Exception was made to the edge beams of the damaged model of the structure,
where linear elastic elements have been adopted, since no plastic hinges were expected to
develop in this these structural elements.
The Opensees built-in “Concrete02” and “Steel01” constitutive relations were adopted for the
structural materials, respectively concrete and steel rebars (FEDEAS, 2015; Karsan & Jirsan,
1969; Mazzoni et al. 2015; Yassin, 1994). The concrete model is defined introducing the
parameters related with the tension and compression behaviour. The tension behaviour is
characterized by a bi-linear branch defined by the tensile strength and the tension softening
stiffness, the compression behaviour is nonlinear and described by the concrete compressive
strength at 28 days, concrete strain at maximum strength, concrete crushing strength, concrete
strain at crushing strength and ratio between unloading slope and initial slope. The steel
model has a plastic behaviour with hardening, defined by the steel yield strength, the initial
Young’s modulus and the strain-hardening ratio.
The structural effect of the flooring system was considered as an equivalent tie at the
cantilevers beams level. The self-weight and the intermittent load of the floorings were
applied directly on the edge beams.
Masonry walls
The structural contribution of the masonry walls considering by introducing an eccentric strut
model suggested by Al-Chaar (2002), developed having in mind horizontal loads in the plane
of the wall. Therefore, and for this case study, it is necessary to adapt the model for vertical
loads as suggested in the UFC manual (US Department of Defence, 2013). It is assumed that
the load is redistributed to the frame beams, instead of the columns, as foreseen in the
undamaged model. The equivalent strut must be anchored to the beams at a distance bL
measured from the edge of the columns. This length corresponds to the plastic hinge
development and, according to Al-Chaar (2002), should be modelled as a rigid element
(Figure 3).
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l
h
Lb
P
θ b
a
H
Fig. 2 - Eccentric diagonal strut to vertical loads (adapted from Al-Chaar 2002)
The equivalent strut width, a, is given by eq. (1) and depends on the relative bending stiffness
between the beams and the masonry panel, λL, given by eq. (2):
( ) 4.0175.0
−××= LDa λ (1)
( )4
4
2sin
lIE
tELL
beamc
bm
×××××
×=θ
λ (2)
where L is the distance between the columns midlines, l is the masonry panel width, t is the
panel thickness, Em refers to the Young’s modulus of masonry, Ec represents the Young’s
modulus of concrete, Ibeam is the second moment of inertia of the beams and D is the diagonal
length of the panel. The lengths of formation of plastic hinges, Lb and Lc, are assessed
according to the following equations:
( )c
c
aL
θcos= (3)
( )b
b
aL
θsin= (4)
where:
( )tanc
h Lc
lθ
−= (5)
( )tanb
h
l Lbθ =
− (6)
The effect of the openings is taken into account by reducing the strut width to ared, obtained
by product of initial width by a reduction factor R1 dependent on the ratio between the area of
the openings and the total area of the panel (Al-Chaar, 2002):
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1Raared ×=
(7)
panel
open
panel
open
A
A
A
AR 6.16.0
2
1 −
= (8)
The failure of the masonry panel can occur due to material crushing, in the direction of higher
compression stresses, or if the shear strength is exceeded. Therefore, the strength of the
equivalent strut, Rstrut, corresponds to the minimum of eq. (9) and eq.(10) (Al-Chaar, 2002):
'
'/ cos cos
cr m
strut
shear strut n v strut
R a t fR
R A fθ θ
= × × =
= × (9)
2tan c
strut
h L
lθ
−=
(10)
where θstrut is the angle of the eccentric strut with respect to the horizontal plane, Rcr and
Rcr/cosθstrut are the compressive and shear strength of the equivalent strut, respectively, and
f’m and f’v are the compressive and shear strength of the masonry, respectively.
Numerical analysis
Three different models of the structure were subjected to a static pushdown analysis: the
intact structure; the damaged structure neglecting the effect of the masonry walls (Fig. 2); and
the damaged structure considering the effect of the masonry walls according to Al-Chaar
(2002) model. The effect of the masonry walls was also neglected in the numerical model of
the intact structures, as this is the usual procedure during the design stage.
a) b) c) d)
Fig. 3 - 3D Schematic drawing of numerical models: a) damaged structure with masonry struts; b) cantilever Plan; c) damaged structure without masonry struts d) intact structure.
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RELIABILITY ANALYSIS
Random variables characterization
The number of random variables used in the reliability study was limited to the most
important, due to demands of the numerical analysis in terms of computational resources. In
what concerns the resistance, the random variables related to the steel yielding strength fy, the
concrete compressive strength, fc, and the masonry compressive strength, fm, were considered.
The steel yielding strength was defined according to the Probabilistic Model Code
(Vrouwenvelder, 1997). A normal distribution with a mean value equal to Snom+2σ was
considered, where Snom is the nominal value of steel grade, and σ is the standard deviation (30
MPa). Thus, for an S400 steel grade, the yielding stress is defined as N(460,30).
The compressive strength of a of C20/25 concrete grade is defined by a lognormal distribution
with a mean value equal to 28 MPa, and a coefficient of variation of 15% (Cavaco, 2013),
which results in a standard deviation equal to 4.2MPa.
The compressive strength of the masonry wall, fm, according the Probabilistic Model Code is
defined by:
'
1m mf f Y= ×
(11)
where Y1 is a random variable defined by a lognormal distribution with an average of 1.0 and
a coefficient of variation equal to 17%. According to Cachado (2010) the mean value of the
compressive strength for this type of masonry can be considered equal to 13 MPa and the
Young’s modulus equal to 10 GPa. This value respects to the effective compressive strength
of the masonry wall, therefore considering only the effective thickness of the ceramic bricks
plus mortar and not the real thickness of the wall.
In relation to the acting loads, four additional random variables were considered. For the
permanent loads related to the structural elements, a normal distribution was assumed for the
concrete self-weight, γc, with a mean value of 25.0 kN/m3 and a standard deviation of
0.75 kN/m3 (Vrouwenvelder, 1997). The self-weight of clay masonry walls, γm, is defined
with a normal distribution with a mean value of 2.9 kN/m2 and coefficient of variation equal
to 5%.
Two types of live loads were defined according to the Probabilistic Model Code considering a
residential occupancy (Vrouwenvelder, 1997): the sustained live load qs, with a mean value of
0.30 kN/m2 and a standard deviation of 0.31kN/m
2, and an average renewal rate of one in
each 7 years; and the intermittent live load qi with a mean value of 0.30 kN/m2 and a standard
deviation of 0.36 kN/m2, and an average renewal time of 1 year and the duration of 1 day.
Therefore two loading combinations are possible: the two live loads acting at the same time
(Case 1) with an average occurrence of 1 day per year; and the sustained load acting alone
(Case 2) respecting to the remaining 364 days of the same year.
Finally, and in order to take into account a certain level of uncertainty related with the
resistance and action models, two more random variables were considered: the uncertainty
related to the strength model, θR, with a lognormal distribution with mean of 1.2 and standard
deviation equal to 0.15; and the uncertainty of the loading models, θE, assumed also with a
lognormal distribution with mean 1.0 and standard deviation equal to 0.1 (Cavaco, 2013;
Vrouwenvelder, 1997).
The remaining properties and loads were assumed as deterministic. The slab self-weight was
taken as 3.5 kN/m2, and the interior masonry walls were assumed with a self-weight of
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1.8 kN/m2. Table 1 summarizes the random variables and the respective distributions, used in
is study. Table 1 - Random variables distributions and parameters
Random
Variable Distribution
Mean
value
Standard
deviation
cf (MPa) lognormal 28.0 4.2
yf (MPa) normal 460.0 30.0
mf (MPa) lognormal 13.0 2.21
cγγγγ (kN/m3) normal 25.0 0.25
mγγγγ (kN/m2) normal 2.9 0.15
sq (kN/m2) gamma 0.30 0.31
iq (kN/m2) gamma 0.30 0.36
Rθθθθ lognormal 1.2 0.15
Eθθθθ lognormal 1.0 0.1
Methodologies
Monte Carlo Simulation was used to proceed with the reliability analysis and to determine the
probability of failure and the reliability index of the building prior and after the accident. In
the first stage, the samples of the variables were randomly generated. In the second step, the
limit state function was evaluated and the probability of failure was determined. The limit
state function is defined as follows:
R EG α θ θ= × − (12)
where α corresponds to ratio between the acting and resisting loads obtained through the non-
linear static pushdown analysis. Due to high number of structural analysis required for the
Monte Carlo simulation, an Artificial Neural Network (ANN) was used to approach the
results of the pushdown analysis. The probability of failure, Pf, is assessed calculating the
number of failures Nf, defined by G<0, for the N simulations.:
f
f
NP
N= (11)
The reliability index β, is obtained according to the following expression:
1[ ]fPβ −= −Φ (12)
Artificial neural network definition and training
Different multi-layer feedforward Artificial Neural Networks (ANN) were defined to
approximate the response of the intact and damaged structures of the building, in terms of the
α coefficient. For the damaged structure, several samples of the input random variables were
generated, and the respective structural analysis was performed, in order to train the ANN. In
the first stage, a 1000 size sample was randomly generated according to the distribution of the
input random variables. This allowed achieving an adequate training of the ANN in a domain
of the variables values likely to be generated during the Monte Carlo simulation. Then, a 576
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size sample was also generated, this time neglecting the contribution of the masonry wall in
the structure analysis. MATLAB (Mathworks, 2010) was used to construct the ANN and the
Levenberg-Marquardt (Mathworks Inc., 2015) back propagation algorithm was used
considering 70% of the total data set to train the ANN, 15% to validate the training and the
remaining 15% to test the network. The ANN training was performed with 20 neurons in the
hidden layer, which revealed to be sufficient to ensure an adequate fit, resulting in a mean
square error of 9.76×10-4
and 7.54×10-4
for the validation and test data sets, respectively.
In the case of the intact structure, a 200 size sample of the input random variable was
sufficient to train an ANN with 20 neurons in the hidden-layer and to obtain a good
approximation of the structural response. The obtain mean square error was about 5.801×10-6
for the validation set and 6.07×10-6
for the test set.
RESULTS
As referred to, the probability of failure of the intact and damaged structure with masonry
walls was assessed using to Monte Carlo simulation based on the trained ANN.
The accuracy of MC method depends on the number of simulations (nMCS) and can be
measured by coefficient of variance of the probability of failure (COV(Pf)) given by
(Chojaczyk et al. 2014):
( )11( )
f f
f
f MCS
P PCOV P
P n
− ×= × (13)
For the damaged structure with masonry walls, 107 simulations were used and the obtained
COV(Pf) was 0.03 and 0.01 for Cases 1 and 2, respectively. For the intact structure, 18×107
MCS were made for the Case 1 and 8×107 for the Case 2. The obtained COV(Pf) was less than
0.5 for both cases. The values obtained are sufficient to ensure a good level of accuracy.
Table 2 presents the probabilities of failure and reliability index to the intact structure and
damaged structure with masonry walls, for the Case 1 and 2. For the Case 1 and 2, the
damaged structure is considered as unsafe.
Despite the damaged building being unsafe due to the low reliability index, the probability of
failure diminished significantly when compared with the case of the damage building without
considering the masonry wall contribution. The damaged model without the contribution of
masonry walls do not support any value of live load resulting in a probability of failure
approximately of 100% (Pf ≈ 100%). In short, if the contribution of the masonry wall is
considered, the building is unsafe and must be repaired, but collapse is not eminent.
Table 2 - Probability of failure and reliability index for the damaged structure and intact structure.
Model Case 1 Case 2 Case 1+2
fP ββββ fP ββββ fP ββββ
Damaged Structure
Without Infill Walls z100% - z100% -
z100% -
Damaged Structure
With Infill Walls 7.45 1.44 6.07 1.55 6.07 1.55
Intact Structure 0.0014 4.19 6.2×10-6
5.29 1.0×10-5
5.20
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CONCLUSION
The safety of a severely damaged building was analyzed in a reliability framework,
considering a combination of Monte-Carlo simulation and Artificial Neural Networks.
The building was analyzed under three scenarios: no damage, damaged without masonry
walls and damaged with masonry walls. For the damaged building and considering the
masonry infills, the reliability analysis yielded a probability of failure of 6.05%. This is a high
probability of failure, showing that remedial actions must be taken, but also showing that the
building collapse is likely to be avoided.
This is in extreme contrast with the analysis of the building disregarding the influence of
masonry walls, when a probability of failure of 99.9% was computed.
These results clearly state that in this case the masonry walls were determinant for ensuring
the required reliability, and thus survival, of the damaged building.
In the case of the simultaneous application of the two live loads (Case 1), a probability of
failure of 0.02% shows a good level of safety independently of the level of stated damages.
Finally, it is concluded that, without the use of ANN for the approximation of the building
response, it would not be possible to execute the Monte Carlo simulation due to the high
number of simulations necessary for an acceptable precision.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the funding by Ministério da Ciência, Tecnologia e
Ensino Superior, FCT, Portugal, under grants of PTDC/ECM-COM/2911/2012.
REFERENCES
[1]-Al-Chaar GK. Evaluating Strength and Stiffness of Unreinforced Masonry Infill
Structures, US Army Corps of Engineering, 2002.
[2]-Baker JW, Schubert M, Faber MH. On the assessment of robustness. Journal of Structural
Safety, 2008, 30, p. 253–267.
[3]-Cachado A, Grilo I, Júlio E, Neves L. Use of Non-Structural Masonry Walls as
Robustness Reserve. In 35th Annual Symposium of IABSE, 2011.
[4]-Cavaco ESRG. Robustness of corroded reinforced concrete structures. PhD Thesis.
Universidade Nova de Lisboa, 2013.
[5]-Chojaczyk AA, Teixeira AP, Neves LC, Cardoso JB, Soares CG. Review and application
of Artificial Neural Networks models in reliability analysis of steel structures. Structural
Safety, 2015, 52, p. 78-89.
[6]-Farazman, S., Izzuddin, B.A. & Cormie, D., 2013. Influence of unreinforced masonry
infill panels on the robustness of multistory buildings. Journal of Performance of Constructed
Facilities, 27, pp.673–682.
[7]-FEDEAS. http://www.ce.berkeley.edu/~filippou/Research/Fedeas/material.html, 2015.
[8]-Helmy H, Hadhoud H, Mourad S. Infilled masonry walls contribution in mitigating
progressive collapse of multistory reinforced concrete structures according to UFC guidelines.
International Journal of Advanced Structural Engineering, 2015, 7(3), p. 233-347.
Symposium_23: Structural Robustness
-1384-
[9]-Karsan ID, Jirsan JO. Behavior of Concrete Under Compressive Loadings. Journal of the
Structural Devision, 1969, 95, p. 2543–2563.
[10]-Mathworks, 2010. Matlab.
[11]-Mathworks Inc. http://www.mathworks.com/.2015.
[12]-Mazzoni, S. et al., Opensees command language manual, Berkeley: Pacific Earthquake
Engineering Research Center.
[13]-Munch-Andersen J, Dietsch P. Robustness of large-span timber roof structures - Two
examples. Engineering Structures, 2011, 33, p. 3113–3117.
[14]-Pearson C, Delatte N. Ronan Point Apartment Tower Collapse and its Effect on
Building Codes. Journal of Performance of Constructed Facilities, 2005, 19, p. 172–177.
[15]-Sasani M. Response of a reinforced concrete infilled-frame structure to removal of two
adjacent columns. Engineering Structures, 2008, 30, p. 2478–2491.
[16]-Sattar S, Liel AB. Seismic performence of reinforced concrete frame structures with and
without mansory infill walls. In 10th Canadian Conference on Earthquake Engineering, 2010,
Toronto.
[17]-Tiago P, Júlio E. Case study: Damage of an RC building after a landslide—inspection,
analysis and retrofitting. Engineering Structures, 2010, 32(7), p. 1814–1820.
[18]-US Department of Defence. Unified Facilities Criteria ( UFC ) Design of buildings to
resist progressive collapse approved, , 2013.
[19]-Vrouwenvelder T. The JCSS probabilistic model code. Structural Safety, 1997, 19, p.
245–251.
[20]-Xavier FB, Macorini L, Izzuddin BA. Robustness of Multistory Buildings with Masonry
Infill. Journal of Performance of Constructed Facilities, 2015, 29.
[21]-Yassin MHM. Nonlinear Analysis of Prestressed Concrete Structures under Monotonic
and Cycling Loads. PhD Thesis. University of California, 1994.