reliability analysis of a severely damaged rc building ...irf/proceedings_irf2016/... · the load...

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)

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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

Symposium_23: Structural Robustness

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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

Proceedings of the 5th International Conference on Integrity-Reliability-Failure

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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.

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