Structural Analysis of Historic Construction – D’Ayala & Fodde (eds)© 2008 Taylor & Francis Group, London, ISBN 978-0-415-46872-5

Modelling of damage accumulation in masonry subjected toa long-term compressive load

E. VerstryngeDepartment of Civil Engineering, KULeuven, Heverlee, Belgium

S. IgnoulTriconsult NV, Lummen, Belgium

L. Schueremans & D. Van GemertDepartment of Civil Engineering, KULeuven, Heverlee, Belgium

ABSTRACT: A rheological model is used to describe the long-term behaviour of masonry. Special attention ispaid to the evolution of damage and the description of the damage parameters, based on the continuum damagetheory.The parameters of the model are derived from a test program, including three types of mortar compositionsand three kinds of tests: monotonic compression, short-term and long-term creep tests. Furthermore, a range ofstress evolutions is simulated to evaluate the model and the results are compared with experimental test data.

1 INTRODUCTION

To describe the long-term behaviour of historicalmasonry, rheological models have been used dur-ing past researches. (Boukharov 1995, Anzani 2005)These models are phenomenological models, based onthe stress-strain curves and strain-time behaviour ofthe masonry during experimental research. In orderto link the model’s behaviour to specific types ofmasonry, the model’s parameters are deduced frommasonry characteristics, recorded during short-andlong-term testing on masonry.

Under high, sustained stresses, a load conditionwhich typically occurs in historical masonry towersand churches, the so-called brittle masonry exhibitsa ductile behaviour, showing an increasing visco-elastic deformation in time. Therefore, the Maxwell

Figure 1. Maxwell and Kelvin component.

model, with a combined spring-dashpot configuration,is often used to describe this creep effect. (Boukharov1995, Papa & Talierco 2005)

Experimental research has shown that masonryunder a high persistent loading follows a deformationpattern which can be described by the typical three-phase creep curve. It exhibits a primary creep phaseduring which the strain rate decreases in time, a sec-ondary phase or steady-state creep, with a constantstrain rate and a tertiary phase with an increasing strainrate, which leads to a sudden failure of the specimen(Binda 1993, Challamel 2005). In order to enable themodel to capture the primary creep phase, a Kelvincomponent is placed in series with the Maxwell com-ponent, forming a rheological model which is able tosimulate the first two stages of the creep curve.

Furthermore, damage parameters are added to themodel in order to describe the damage accumulationand loss of material cohesion, which will trigger theunstable strain increase of the tertiary creep phase andlead to failure of the masonry.

2 MODELLING OF DAMAGEACCUMULATION

There are different methods in describing the unsta-ble damage accumulation during the tertiary creepphase. Boukharov uses the Weibull cumulative distri-bution and experimentally defined strain values and

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failure times to describe the growth of ‘hot’ crackarrays up to a critical density. In the model presentedhere, a continuum damage mechanism will be fol-lowed, describing the overall damage throughout thematerial as a continuous phenomenon.

The theory of Kachanov (Kachanov 1985, Kra-jcinovic & Lemaitre 1987), which lies at the baseof continuum damage mechanics, poses to describethe absence of material deterioration by means of afield variable, the continuity.The more frequently useddamage parameter (D) can be seen as the comple-ment of this continuity. For a completely undamagedmaterial D equals 0, whereas D = 1 corresponds to acompletely deteriorated material. This model has beenextended in terms of the introduction of the effectivestresses, which act upon a decreasing surface area asa result of growing micro voids.

With s being the effective stress, which increaseswhen damage increases and σ the distributed stress.For an undamaged material, s equals σ.

The damage variable, which in general is describedas a tensor, can be reduced to a scalar variable incase of an isotropic material. The proposed modelis a time-dependent isotropic damage model, whichenables a simple scalar notation. The aim is to link theparameters of the model to specific material proper-ties of different types of masonry, in order to predictthe masonry’s behaviour during the experimental testsfrom which the parameters were deduced as well asto extend the model towards the prediction of real sit-uations at which high sustained loading acts duringperiods of hundreds of years. Therefore, the exper-imental program includes short-term and long-termcreep testing on masonry wallets, built with threetypes of typical mortar compositions: cement mor-tar, hydraulic lime mortar and cement-lime mortar.The complete set-up of the test program is discussedelsewhere (Verstrynge 2008).

The dilatant behaviour of the masonry is simulatedby including an increasing, damage-dependent Pois-son’s ratio. Previous studies have indicated that whena constant Poisson’s ratio is considered, the horizontaldeformation is highly underestimated.

Two types of damage are considered: the staticdamage, which influences the Young’s modulus in theelastic domain, and the viscous damage, which influ-ences the strain rate during steady-state creep andtriggers the tertiary creep phase.

2.1 Static damage

The static damage parameter (Ds) decreases theYoung’s modulus of the Maxwell spring as dam-age increases. This will result in higher strains at

Figure 2. Decrease of Young’s modulus due to damageincrease (from Hult 1987, p. 29).

stress increase. The amount of static damage can becalculated as proposed by Hult, see Figure 2.

From the relation

Follows:

D′ = 1 − E′/E

D′′ = 1 − E′′/E

. . .

With E the initial, maximum Young’s modulusThe evolution of theYoung’s modulus during short-

term creep test on masonry wallets (29 × 19 × 85 cm)was calculated according to Equation 2. The tests wereperformed by applying a constant load of 50% of themasonry compressive strength (fc) during a period of 2hours. Subsequently, the load was increased by 5% andagain kept at a constant level.This loading scheme wasfollowed until failure of the specimen. In total, 9 wal-lets were tested, of which 3 wallets were tested for eachof the three aforementioned mortar types respectively.

Figure 3 shows that the small stress increase andrelated very small strain increase for every load-stepresult into a rather large scatter on the evolution of theYoung’s modulus.

Therefore, extra short-term creep tests were per-formed on masonry cores with cement mortar (diame-ter 113 mm, height 10–18 cm), including an unloadingphase just before every stress increase in order toenlarge the interval over which Young’s modulus wascalculated. The load was kept constant during a periodof 30 minutes. Figure 4 shows a representative resultof this test series.

A permanent decrease of theYoung’s modulus is notfound, so assuming this would result in an overestima-tion of the elastic strain. Initially, an increasingYoung’smodulus is found as a consequence of the closure of

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Figure 3. Evolution of Young’s modulus in function ofstress at each step during a short-term creep test on a walletwith lime-cement mortar. Stress is formulated dimensionless(division by the maximum stress acquired during the test).

Figure 4. Evolution of Young’s modulus in function ofstress at each step during a short-term creep test on a masonrycore with cement mortar. Stress is formulated dimensionless(division by the maximum stress acquired during the test).

the cracks oriented parallel to the plane of maximumstress. After a maximum is reached, the static damageincreases due to the initiation and growth of cracks ori-ented perpendicular to the plane of maximum stress.When comparing the test specimen, there is a ratherlarge scatter on the stress level at which static damagebecomes important, sometimes this phenomenon evenonly appears during the last load step. Therefore, aninitial Young’s modulus (calculated between 10–30%of fc) will be taken to simulate small stress levels andan average of the higher values for each mortar typewill be used for the simulation of stress increases athigher stress levels (over 50% of fc).

2.2 Viscous damage

As the aim of the proposed model is to simulate long-term deformations of masonry, viscous damage, whichgoverns the strain rate at constant stress levels and ini-tiates the tertiary creep phase, has a highly importantrole. The viscous damage parameter (Dv) decreasesthe relaxation time of the Maxwell dashpot, therebyincreasing the strain rate during the steady-state creepand describing the accumulation of damage up to fail-ure. In order to describe the evolution of Dv in time,a twofold definition is considered: firstly, the viscousdamage parameter is a function of the stress level, as arelation between the strain rate and the stress level hasbeen observed during the current test program, as wellas in literature (Challamel 2005, Anzani 2000). Sec-ondly, a damage-rate formulation is adopted to enablethe simulation of damage increase in time when thestress level remains constant.

To establish a relation between the amount of vis-cous damage and the stress level, Dv was calculated foreach load step of the aforementioned short-term creeptests on masonry wallets, according to the equation fora Maxwell dashpot:

With �εsec the strain increase during a secondarycreep phase, τM the relaxation time, EM the elasticmodulus of the Maxwell element and�t the time incre-ment over which the strain increase is calculated. Foreach load step, τM(1-Dv) is calculated according toEquation 3. Initially, these calculated values remainconstant or increase slightly. After a few load steps, amaximum value is found, which is set as initial valuefor τM. The decrease of the relaxation time within thenext load steps is then calculated as an increase of DV.

When the stress is described in terms of percentageof the masonry’s compressive strength (fc), with fc thestrength during a short-term monotonic loading test,a linear relation is found between the viscous damageand the stress level (Fig. 5). This relation appears to besimilar for the different mortar types.

A threshold of around 70% of fc is found beneathwhich no viscous damage is detected. Below this stresslevel, the relaxation time (τM) remains almost con-stant or shows a slight increase. When the threshold iscrossed, the relaxation time decreases as the strain ratebecomes higher with each successive load step. Thislinear regression curve can be described by means oftwo parameters. Only the results for the short-termtests on lime and cement mortar are presented, as thetests on the lime-cement mortar included some unload-ing and reloading cycles, which had a corruptive effecton some of the results.

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Figure 5. Viscous damage parameter (DV) in function ofstress level from short-term creep tests on lime and cementmortar wallets. Stress is formulated dimensionless.

In function of stress, DV can be formulated asfollows:

With σ∗ being the dimensionless stress, acquired bydividing the absolute stress by the average compressivestrength (fc) of each mortar type, obtained during themonotonic compression tests.

Secondly, a damage-rate formulation is used toenable the simulation of damage increase in timewhen the stress level remains constant, according toKachanov (Kachanov 1985, Challamel 2005):

With s being the effective stress, as specified inEquation 1.

This simple formulation has the advantage thatonly two parameters have to be determined. Damageevolution is described using a power law equation.Alternatively, others have used an exponential equa-tion for damage development under constant stress(Anzani 2005, Bodner & Chan, 1986). The two param-eters, c and n, are determined by fitting the tertiarycreep curve. Unfortunately, experimental test data arelacking for two out of the three mortar types as tertiarycreep was not reached or only occurred very shortlyduring the short-term creep tests. Consequently, theparameters are fitted to the results of the masonry wal-lets built with hydraulic lime mortar, until more testdata are available from the long-term creep tests.

3 DETERMINATION OF THE PARAMETERS

The complete formula of the reological model, com-bining a Maxwell and a Kelvin component, can be

presented in a simple constitutive formulation, todescribe the vertical deformation, as follows:

With M as notation for the parameters of theMaxwell element and K for the material character-istics related to the Kelvin element. The parametersare calculated from monotonic compression tests andshort-term creep tests.

3.1 Experiments

As already mentioned, the test programme includedtests on masonry wallets (29 × 19 × 85 cm) with threetypes of typical mortar compositions: cement mortar,hydraulic lime mortar and a hybrid, cement-lime mor-tar. And an extension of the test programme towardsair hardening lime mortar has been planned.

On all masonry types, three kinds of tests have beenperformed (A more detailed discussion of the resultscan be found in Verstrynge 2008):

– Monotonic compression tests, under a deformationcontrolled environment;

– Short-term creep tests (also called acceleratedcreep tests or step-by-step tests), at which stressis increased in steps of 5% and in between keptconstant for a duration of 2 hours;

– Long-term creep tests, during which masonry wal-lets are loaded at different constant stress levels.

The long-term tests were set up in order to assessthe strain rate of the different mortar types at differentstress levels. These tests are still ongoing. A secondgoal of these tests is to assess the simulation of long-term behaviour of masonry, by means of the proposedmodel and the model parameters which are derivedfrom short-term tests. The model parameters werecalculated as an average for each mortar type. Atleast three test specimens were made for each mor-tar type and each type of test, which results in a totalof 27 specimens. The resulting model parameters aresummarized in Table 1.

The compressive strength of the masonry (fc)is obtained from monotonic compression tests. Thestrength of the cement mortar was remarkably low,possibly due to bad curing conditions and absorptionof the water out of the mortar by the bricks, leading toa weaker mortar.

The Poisson’s ratio (υ) is calculated from the com-pression tests, without considering the last part ofthe tests, where dilatation occurs and the Poisson’sratio suddenly increases. This dilatant behaviour of the

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Table 1. Overview of the parameters used during the sim-ulations with the rheological model. Presentation of results:average value (± standard deviation).

Parameters ofrheological Hydraulic Cement Hybridmodel lime mortar mortar

fc [MPa] 6.3 ± 0.9 5.7 ± 0.2 7.3 ± 0.3υ [−] 0.12 ± 0.05 0.16 ± / 0.29 ± 0.12τK [s] 300000 300000 300000EK [MPa] 1959 1764 2112τM [s] short-term 253200 491300 357200

long-term 200e6 200e6 200e6EM [MPa] 10–30% fc 2090 2390 2800

> 50% fc 4000 4000 4000DV A 0.52 0.57 0.45

B −2.24 −2.24 −2.24c 3e−15 3e−15 3e−15n 6 6 6

masonry is incorporated in the model by assuming adamage-dependent Poisson’s ratio, which increases asthe viscous damage increases.

The Kelvin relaxation time (τK) and the Kelvin elas-tic constant (EK) are derived by fitting the primarycreep phase of the short-term creep tests. These arenot discussed here in detail, as their importance forthe overall simulation is limited.

The initial Maxwell elastic constant (EM) is calcu-lated from the compression tests as secant modulusbetween 10–30% of fc. When the stress level exceeds50% of fc, a higher value is used during the simulationas a consequence of crack closure and compaction ofthe material, as discussed above. This value is calcu-lated as an average value of the different load increasesteps during the short-term creep tests.

The parameters A, B, c and n, which describe theevolution of the viscous damage (DV) are calculatedas mentioned above.

Two different values are mentioned for the Maxwellrelaxation time (τM), one value which is used to sim-ulate a short-term creep test and a higher value tosimulate long-term behaviour. The relaxation time isrelated to the “viscosity” (η) of the material, and isa measure for the strain rate during the steady-statecreep phase.

Initially, τM was taken to be constant with time, whichresulted in very high deformations and significantunderestimations of the time to failure. Comparisonof the steady state phase during short-term and long-term tests revealed that the assumed viscosity of thematerial is time dependent and increases in time. Cal-culation of the evolution of τM in time during the first

step of the short-term creep tests, revealed a powerlaw relation between the Maxwell relaxation time andthe time-interval during which the stress remainedconstant;

With α having a value of 0.7–0.8. Consequently, τM

will increase in time while the stress level remains con-stant, due to a strengthening effect which in rheology isdescribed as a time-dependent viscosity. On the otherhand, τM will also decrease in time as a consequenceof the damage accumulation due tot stress increase andtime-dependent deformations under constant stresslevel, which will lead to the tertiary creep phase.There-fore, the evolution of the viscosity was only assessedduring the first load step of the short-term creep test,where the viscous damage has a negligible influence.As Equation 8 was deduced from short-term tests, itis not fully able yet to simulate long-term behaviour.Therefore, a constant value will be taken for τM whichis based on experience from the ongoing long-termcreep tests.

4 SIMULATIONS

The constitutive equation describing the stress-strainrelationship in function of time has been implementedas Matlab code (The MathWorks, version R2007a).The model is evaluated in time increments, which arekept as small as possible. The size of the time incre-ments is chosen to be an optimum between the loss ofaccuracy and the reduction of calculation time whilethe size of the time increment increases. Therefore,smaller time increments are applied for the simulationof short-term tests. Each strain increment is com-posed of a linear strain increment, caused by the stressincrease (elastic strain) and a creep strain increment,caused by the initial stress at the beginning of the timestep.This initial stress is assumed to be constant withinthe interval of the specific time step.

4.1 A qualitative evaluation of the model

To evaluate the general behaviour of the model, strain-time curves are simulated for masonry at a rangeof different constant stress levels. Figure 6 shows asimulation of the strain evolution when a constantload of respectively 75, 80 and 85% of the compres-sive strength is applied on the tested masonry withhydraulic lime mortar.

Failure would occur after a period of respectively 23years, 3 years en 2.8 months. For a load level of 50%,a failure time of over a thousand years is calculated.

The three different creep phases are not clearly todistinguish, as the graph concerns a very large time

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85%

75%

80%

Figure 6. Simulation of strain evolution in time for masonrywith hydraulic lime mortar at a constant stress level of 75, 80and 85% of the compressive strength.

85%

80%

75%

Figure 7. Enlargement of the simulation of the primarycreep phase for masonry with hydraulic lime mortar at aconstant stress level of 75, 80 and 85% of the compressivestrength.

span. A closer look on the primary creep phase isshown in Figure 7, which is an enlargement of therectangle in Figure 6. The evolution of the viscousdamage in time is indicated in Figure 8.

Subsequently, the rheological model is used to sim-ulate a creep failure curve for each mortar type.Therefore, a series of constant stress levels are applied,ranging from 72 to 90% of fc, and the related verticalstrain at failure is simulated. The obtained creep fail-ure curves are presented in figure 9. The data pointrelated to the highest stress for every mortar type isthe strain at failure obtained during the experimentalcompression tests. These data appear to be well in linewith the others.

The creep failure curve of the strongest mortar issituated above the others. When a constant stress isapplied, the masonry with the weakest mortar will fail

75%

80%

85%

Figure 8. Evolution of the viscous damage parameter (DV)for masonry with hydraulic lime mortar at a constant stresslevel of 75, 80 and 85% of the compressive strength.

Figure 9. Simulation of creep failure curve for three differ-ent mortar types.

sooner and have a smaller vertical creep strain at fail-ure. This is plausible, considering that this stress levelwill be closer to the compressive strength of the weakmasonry. For the stronger masonry, this stress levelis quite low, leading to higher creep strains and longertimes to failure. When the same stress level in percent-age (relative to the compressive strength) is applied, theweaker mortars will have larger deformations beforefailure occurs.

The data point related to the lowest stress level isthe failure strain obtained at 72% of the masonry’scompressive strength. As the analysis of the vis-cous damage indicated that the threshold for viscousdamage was situated just below 70% of fc, an asymp-totic strain value can be expected here. This isin literature indicated as the viscosity limit curve(Bazant 1988).The simulations have shown that below70% of fc the failure time indeed increases rapidly.

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Figure 10. Experimental results and simulation ofshort-term creep test on hydraulic lime masonry, column 3.Evolution of horizontal en vertical strain in time (above) andvolumetric strain (below).

The existence of the viscosity limit curve, which canbe reached without failure of the specimen, has notbeen implemented here.

4.2 Comparison with experimental data

Subsequently, the model is used to simulate the short-term creep tests from the abovementioned test pro-gram. Two representative simulations are presentedbelow (Figs 10–11).

Here, the comment has to be made that some simu-lations showed a rather poor fit to the results. This wasnot due to an inadequacy of the model, but because ofthe unpredictable behaviour of some specimens. Afterall, masonry is a very inhomogeneous composite, witha rather large spread on its parameters. The parame-ters were calculated to fit a specific type of masonryrather than one specimen whose behaviour was to besimulated. Therefore, a stochastic description will bestudied in future analysis.

Figure 11. Experimental results and simulation ofshort-term creep test on hydraulic lime masonry, column 5.Evolution of horizontal en vertical strain in time (above) andvolumetric strain (below).

5 CONCLUSIONS

The long-term behaviour of masonry was describedusing a rheological model. The damage increase wasmodelled by means of a continuum theory and specialattention was paid to the evolution of the static andviscous damage.

The parameters of the model were obtained from atest program on new masonry, which included mono-tonic compression tests and creep tests. Parameterswere calculated as average characteristics for eachspecific mortar type rather than determined to fit a cer-tain specimen. Consequently, the spread on the meanparameter values could also be determined and con-firms a rather large scatter. Therefore, further analysiswill have to include the spread on the determinedparameters in order to create a reliability interval forthe simulated failure time. This issue lies outside thescoop of the present paper.

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The proposed straightforward model is able todescribe isotropic behaviour of masonry, includ-ing monotonic stress increases and high sustainedstresses. As other phenomena influence the stress-strain behaviour during short-term, monotonic load-ing, this loading condition is not considered here.A comparison with experimental data has shown asatisfying fit between the model and the experimen-tal results, but more experimental testing is necessaryin order to enlarge the amount of available data andimprove the fitted parameter values.

ACKNOWLEDGEMENTS

The authors express their thanks to the Flemish Fundfor Scientific Research (FWO) for the doctoral grant,offered to Els Verstrynge.

The authors highly appreciate the collaboration withthe Politecnico di Milano and the University of Minhoin this research area.

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