Reliability Analysis in Structural Masonry Engineering

Luc SCHUEREMANS Dionys VAN GEMERTCivil Engineer Civil EngineerK.U. Leuven K.U.LeuvenLeuven, Belgium Leuven, Belgium

Luc Schueremans, born 1970, graduated as a civil engineer at KU Leuven in 1995. Since 1995, he is a researcher at

the Building Materials Division of the Civil Engineering Department of KU Leuven. His research focuses on the

developmen t of a reliability method for the structural evaluation of ancient mason ry structures.

Dionys van Gemert, born 1948, graduated as a civil engineer at KU Leuven in 1971. He obtained PhD at KU

Leuven in 1 976. Sinc e 1978 he is memb er of the acad emix staff of the C ivil Engineerin g Depa rtment of K U Leuve n.

He is presid ent of W TA-Inter national for R estoration. H e is head of the Reyntjens L aborato ry for Mate rials Testing.

His researc h concern s repair and strengthening o f constructions a nd concr ete polyme r compo site.

Summary

This paper presents a probabilistic method to evaluate the reliability of structural masonryelements. This methodology is meant as a decision tool in the restoration process to determinewhether or not structural strengthening by grouting or other methods is needed. The proposedreliability analysis method will be illustrated on three structural masonry problems. The firstillustration focusses on the theoretical aspects and calculates the local probability of failure of amasonry shear walls. Two case-studies deal with the practical goals in masonry engineeringproblems: the reliability analysis of a masonry sewer system and the reliability of the facade ofthe St.-Amandus chapel. This research is part of a complete program on structural strengtheningof ancient masonry and grout design, in which risk analysis, grouting and non-destructive testsare considered.

1. Introduction

Research in the Reyntjens Laboratory on ancient masonry deals with different aspects ofrestoration and renovation of masonry structures. An extensive database is being built up whichdescribes, explains and helps to diagnose the damage at historical brick-masonry structures [1]. A non-destructive technique, based on electrical resistivity measurements, is being developed toevaluate the extent of deterioration at the inside of massive masonry elements [2]. The resistivitymethod enables to make an image of the internal geometrical and physical situation of themasonry. Eventually, it can be used to compare the situation before and after a consolidation orstrengthening treatment. Concerning consolidation and strengthening, the Reyntjens Laboratoryfocuses on grouting with mineral and polymer grouts, or a combination of both [3,4].However, consolidation and strengthening of ancient masonry are always expensive procedures. Therefore it is of utmost importance to decide whether or not the masonry has to be strengthenedor consolidated. Such a decision must be based on safety or risk considerations. This will be illustrated on tested shear walls, reported in literature [5,6]. Attention is paid to: thelocal probability of failure, the different failure modes and corresponding limit state functionsand the probability distributions for the basic variables in these limit state functions. Theprobability of failure is calculated by means of a FORM-algorithm. A simplified analysis ispresented to calculate the probability of failure of a masonry sewer system [7]. In the thirdexample, the global probability of failure of an out of plumb standing facade is calculated.

Fig.1: The limit state function andthe FORM-principle

2. Local probability of failure of masonry shear panels

2.1. Structural reliability analysis, the FORM-method

In the most general case, the state of a structure is described by a structural model that ischaracterized by a number of stochastic and/or deterministic variables X (the stresses in themasonry panel Fy, Jxy, the material properties, ...), that can vary as a function of someindependent parameters t (time or place). Moreover, design parameters p (the choice of the usedmaterial, dimensions, ...), chosen by the designer, can have an influence on the probabilitydistribution and/or deterministic value of X. Probabilistic design means that a structure isdesigned so that the probability of failure pf does not exceed a certain threshold pfs during aprescribed period tL (life cycle) : pf < pfs.

Usually more than one failure mode has to be reviewedin determining the safety of a structure . Failure mode ican be described mathematically in a limit state function:gi(X,p,t). Failure is defined as exceeding a certain limitstate function, resulting in the following 3 situations(figure 1): 1. gi(X,p,t) < 0 failure in failure mode i,2. gi(X,p,t) = 0 the structure is in a critical situation,3. gi(X,p,t) > 0 a certain safety margin does exist withregard to failure mode i.For the limit state function under consideration and apreset reference period, the probability of failure iscalculated as: pf,i = P[gi(X,p) <0]. To perform thecalculation a FORM- algorithm is used. The algorithm,implemented in MATLAB, calculates the probability offailure pf or the reliability index $ ($= -M-1(pf), in whichM is the standardized cumulative normal distribution) asa direct measure for the reliability of the structure. Thisrelationship is outlined in table 1.

pf 10-1 10-2 10-3 10-4 10-5 10-6 10-7

$ 1,3 2,3 3,1 3,7 4,2 4,7 5,2

Table 1: The relation between $ en pf

The probability of failure pf,i, calculated for the limit state function gi(X,p), needs to be combinedwith the other possible failure modes of the structural element. The theory of system reliabilityprovides a method to obtain a sharp lower- and upper bound for the reliability $ or probability offailure pf. The method used in this paper has been outlined elsewhere [8,9].

2.2. Failure criteria for masonry shear walls

Using the compressive strength of masonry as a threshold for a compression failure criterion isan approximation which is believed to be too narrow [10], to model the failure of a masonrystructure. To evaluate the different failure modes in structural masonry, shear walls are aninteresting subject. Tests on shear panels in laboratory, show that 3 types of failure modes areactivated, which mainly describe the real failure process. The 3 failure modes are represented by following failure criteria (limit state functions), [10]:

- (mode I) shear failure of bed joints (1),- (mode II) compressive strength of masonry (2),- mode (III) tensile strength of the masonry (3).

2.3. Material properties

The limit state functions require information on the following material properties of the masonry:tensile strength (ft), compressive strength (f’m) and shear strength as defined by the adhesioncoefficient (c) and the friction coefficient (:). In addition, the stress-strain relationship of themasonry has to be known in order to evaluate the stresses in the finite-element model. Thus, fora linear elastic analysis, the modulus of elasticity (E) and the Poisson’s ratio (<) are required. Asnone of these characteristics is known exactly, a probabilistic distribution has to be defined andproper values for the parameters of the distribution must be estimated. For the applicationpresented here, the results of test data, from in literature are used [6]. The way in which theseresults are obtained is outlined elsewhere [11]. The mean value and the coefficient of variationof the different variables are listed in table 2. A lognormal distribution is used to model thematerial strengths [8].

modelparameter

mean value c.o.v. [%] modelparameter

mean value c.o.v. [%]

f’m 8.9 N/mm2 20 : 0.83 9.5

ft 0.13 N/mm2 44.1 Fy FEM-analysis 30

c 0.65 N/mm2 12 Jxy FEM-analysis 30

Table 2: mean values and coefficient of variation of the model parameters

In the linear elastic finite element calculation of the stresses in the masonry shear panels, themean values of the modulus of elasticity (E) and Poisson’s ratio (<) are used.As masonry only behaves approximately linear elastic until 1/3th of the ultimate strength, themodel is only representative for low stresses. Reaching the ultimate strength, the calculatedstresses will not correspond with the actual stresses. In addition, there is uncertainty in the usedmodulus of elasticity (E) and Poisson’s ratio (<). As a consequence, uncertainty in themathematical model has to be defined, which is introduced as an extra basic variable on thestresses calculated in the finite element analysis, as outlined in [11]. The coefficient of variation(c.o.v.) of these variables is listed in table 2.

2.4. Graphical interpretation of the probability of failure

The probability of failure pf or the reliability index $, is calculated using the method describedabove at different points of a regular raster on the rectangular wall panel, that is coincident withthe raster used in the finite element analysis. The found results are plotted in contour graphs, sothat a visual interpretation of the local probability of failure or the reliability index is possible.Table 3 summarizes the result for wall panel V3D. The different columns correspond to: thetested shear panel [6], the imposed loads (H,V) as known from the test results, the calculatedprobability of failure and the corresponding reliability in 3 regions that are mainly dominated bydifferent failure modes: tension (two opposite corners submitted to tensile stresses), shear (in themid region) and compression ( the other two opposite corners). For the shear panel V3D thecontour plots are shown in figure 2.

Fig. 3: Pressure lines under diffe-rent loading conditions

panel H [kN] V[kN] tensile shear compression

V3D 62 95 pf

$0.9991-3.14

0.07551.44

0.06881.48

Table 3: pf and $ at the ultimate loading

Figure2: contour maps of the probability of failure pf and the reliability index $

3. Sewer System

The method was applied in a restoration project tocalculate a first value for the safety level of a masonrysewer system, figure 3. In this application, reaching thecompressive strength of the masonry was the only ultimatelimit state considered: g(x1,x2) = x1-x2. The compressivestrength of the masonry was obtained from test results on12 core samples. The mean value and standard deviationare::x1=18,1 N/mm2, Fx1=5.5 N/mm2. For thecompressive stresses in the masonry, as obtained from afinite element analysis [7], figure 3, following meanvalues and variances were found: :x2=1 N/mm2, Fx2=0.5N/mm2. A normal distribution is accepted for bothvariables. The reliability index and corresponding

probability of failure are: ,

pf=M(-$)–10-3.Whether such a safety level is acceptable or not is a socio-economical problem. Theprobabilistic analysis provides a quantitative measure of the safety that can be used to comparedifferent alternatives.

4. The facade of the St.-Amandus chapel - local versus global probability offailure

The system behaviour of a structure is important as system failure is the worst consequence ofstructural failure. In the case study of the Saint-Amandus chapel at Erembodegem (B), figure 3,the global probability of failure of an out of plumb standing wall is calculated. To save theauthenticity of the chapel, it was decided to minimize the (semi)-destructive test program.

Figure 3: Front view and cross section of the facade

Figure 5 : Reliability index as a function of the measured eccentricities

Because of the leaning forward of the facade, it was decided to monitor the evolution of thecracks, deformations and eccentricities. Supplementary, a reliability analysis of this structuralelement was performed, to assess the remaining safety. Therefor, it is required to be able toevaluate the reliability based on a few data, i.e., the geometry and the measured eccentricities. This reliability assessment allowed to conclude that the remaining safety margin is large enough,so that no further structural or technical interventions are needed and the authenticity of thebuilding will be maximally maintained, which was one of the preset objectives in this restorationproject.

To be able to evaluate the remainingsafety on the basis of a restrictedmeasuring campaign, three simplemodels were built, as outlined elsewhere[11]:- controlling the turn over equilibrium(overall stability): g(G1,G2,e0,e1,e2) =G1(e0-e1)+(e0-e2),- controlling the middle third,- control via NTM (Non-Tension-Material)-elements.These models were kept that simple sothat in a first evaluation the eccentricitymeasurements and global estimates of thestrength of the masonry would besufficient.

The results aresummarized infigure 4. Followinginformation isoutlined: theeccentricities asmeasured at aheight of 2.72 m(height of thecenter of gravity)on the x-axis, thereliability index $on the y-axis, thecalculatedreliability indexusing the modelof the overallstability (withoutincorporation of

the model uncertainty, with incorporation of the model uncertainty as an extra stochastic variable(+2), or as a correction coefficient (*2)) and the calculated reliability index $ using the modelbased on NTM-element

.This figure enables one to judge the remaining safety, using a single input parameter. Modeluncertainty is incorporated to correct the idealized model of the global stability with the resultsobtained from the model based on the NTM-element [11].

5. Conclusions

A method to evaluate the structural reliability of historical masonry buildings is described.Using a FORM-algorithm, the local probabilities of failure of masonry shear walls are calculated,and a first value of the remaining safety of two practical masonry engineering problems isperformed. The method is meant as a judgmental instrument. It provides a support to furthersteps in the restoration process.

6. References

[1] Van Balen K., Binda L., Van Hees R., Franke L., e.a,”Expert system for the evaluation of thedeterioration of ancient brick structures”, report Ec-environment,EV5V-CT92-0108,1996.[2] Vendrickx K., Van Gemert D., “Geo-electrical evaluation of historical masonry forconsolidation purposes”, 4. International Kolloquium Werkstofffwissenschaften undBausanierung, Proceedings, Band II, pp. 1017-1026, Aedificatio Publ. Fraunhofer IRB Verlag,1996.[3] Toumbakari E., Van Gemert D., “Lime-pozzolane-cement injection grouts for the repair andstrengthening of three-leaf masonry structures”, Proceedings 4th International Symposium on theConservation of Monuments in the Mediterranean, Rhodos 6-11 May 1997, Vol. 3, pp. 385-394.[4] Van Gemert D., Ladang C., Carpentier L., Geltmeyer B., “Consolidation of the tower of St.Mary’s Basilica at Tongeren”, Internationale Zeitshrift für Bauinstandsetzen, Vol.1, no 5, Sep.1995, pp. 371-392.[5] CUR 171, "Constructief metselwerk : een experimenteel/numerieke basis voor praktischerekenregels", Civieltechnisch Centrum Uitvoering Research en Regelgeving, 1994.[6] Vermeltfoort, A. Th, Raijmakers, T.M.J., "Vervormingsgestuurde meso- schuifproeven opmetselwerk", TNO-rapport B-92-1156, 1991. [7] Van Gemert D., Herroelen B., Van Mechelen D, Heinsrich F, “Scientific and economicalapproach of the rehabilitation of old masonry sewer systems, 4th International ColloquiumMaterials Science and Restoration, Ostfildern, Workshop 8, 1996. [8] Melchers, R.E., "Structural reliability, analysis and prediction", Ellis Horwood Series in CivilEngineering, 1987.[9] Schueremans L. and Van Gemert D., "A probabilistic model for reliability evaluation ofhistorical masonry", 4th International Colloquium Materials Science and Restoration, Ostfildern,1996, pp.703-715 [10] Lourenço, P.B., "Computational strategies for masonry structures", doctoral thesis, TNODelft, 1996.[11] Schueremans L., Schaerlaekens Steven, “Restauratie van buitenmuren : Typologie enprocedures, deel 3 : Standzekerheid van historische structuren, IWT-project GI/95/01,Eindverslag, Vol. 1 , 1997.