Laboratoire de Mecanique des Terrains, INERIS - Ecole des
Mines, Pare Saurupt, 54042 Nancy Cedex, France
Abstract
The distinct element method is presented as an efficient tool for stabilityanalysis and understanding of failure mechanisms of old masonry structuresbuilt with stones and mortars. Quite different from the continuous approachof other methods, it deals with discontinuous media and can be used for thestudy of a wide range of structures composed of discrete bodies. Dynamicsimulations of masonry constructions are proposed and illustrate theefficiency of the method
1 Introduction
The increasing interest of studying masonry buildings in seismic regions is dueto the widespread use of them all over the world and to the fact that most ofthe damages caused by earthquakes, from the point of view of fatalities andof historical heritage, are associated with them.
In spite of the popular usage of the finite element method to study themechanical behaviour of this type of edifices, the authors propose the use ofthe Distinct Element Method (DEM) which is originally dedicated to analysisof fractured rock masses. Because fractured rock masses and masonrybuildings are similar in nature of materials, existence of discontinuities andtypical non-linear behaviour, the use of the DEM is justified. Furthermore, theDEM can easily take into account the decaying state of materials and differenttypes of loading forces.
Consequently, the DEM is used here to analyse the behaviour of oldmasonry structures under earthquake loads. Dynamic response of a simplemasonry wall, an arch and an old masonry building are discussed.
2 Modelling methods and masonry structures
At present, there are many attempts to model the behaviour of masonrystructures using available techniques and specially the Finite Element (FEM),
the Finite Difference (FDM) and the Boundary Element Methods (BEM). Forexample, Alessandri et al [1] have modelled masonry structures subjected to astatic load by the boundary element method. In particular, they developpedan iterative incremental process based on the initial stress method with linearboundary elements. Using the finite element method, Pietruszczak [2],Benedetti et al. [3], Gentil [4], Leftheris et al. [5] and many others havemodelled masonries as either linear elastic media, linear continuous equivalenthomogeneised media or as non-linear media. Nevertheless, if modellingmasonries as discrete blocks is possible with some FEM, FDM or BEMnumerical codes (ANSYS, ADINA, FLAG, etc...) it is not an easy task and it isusually very time-consuming.
Moreover, it is apparent that nature and orientation of discrete blocksplay an important role in the performance of masonry structures duringearthquakes. This is why the representation of individual blocks offered by theDEM as well as its capacity to model a large number of them (as it is inmasonry structures) certainly provide more efficiency than other methods.
3 The Distinct Element Method
The DEM, proposed by Cundall in 1971 [6], has been developed to studyjointed media subjected to quasi-static and dynamic loads. The method wasfirstly dedicated to fractured rock masses analysis but it can now be used inmany other fields. The main feature of the DEM consists in the representationof the studied medium by an assemblage of discrete blocks (rigid ordeformable) which interact through contact points. The discontinuities arethus regarded as boundary interactions between these blocks and the methodhas the capability to take into account a large number of them (hundreds...).
The interaction forces and relativedisplacements between blocks are obtained byapplying joint constitutive relations in anincremental form. Large translations androtations of blocks are thus allowed. Newcontacts are also automatically recognised.
To illustrate the performance of the DEMand the importance of a real representation ofdiscontinuities on failure mechanism offractured media, the example of anunderground opening has been modelled withthe code UDEC (Itasca Consulting Group[7]).The possible collapse of the opening due to theorientation of discontinuities takes place by the Figure 1. A geotechnicaldetachment of the roof blocks under gravity case modelled by the DEMforces as clearly shown on figure 1. Suchproblem could not be easily modelled by means of other numerical methods.
4 The advantages of the DEM in historical masonry modelling.
The DEM has many advantages in modelling historical structures compared toother methods : the representation of contacts by means of normal and shearstiffnesses, combined to available failure criteria (Mohr-Coulomb, tension cut-off, etc.), makes it possible to model the assembly of stones bonded bymortars in a realistic manner (figure 2); the soil-structure interactions can be
Figure 2. Keywords in masonrymodelling with the DEM
modelled in a physical way ; thedegradation of stones and mortars canbe easily simulated by changing theirmechanical characteristics during thecalculations ; the method gives bothstatic and dynamic solutions ; it ispossible to study some reinforcementschemes by introducing bolts elementsor structural supporting systems in themodel; the method was firstly set upfor 2-D problem analysis, but it hasalso been developed for 3-D problem analysis (Cundall [8]).
5 Particularity of historical structures
Originally, masonry structures are built of individual blocks of stones orbricks, assembled together to form a whole stable edifice. These blocks can bewell cut or unworked, placed with mortar or roughly in contact withoutmortar. In fact, the particular nature of old structures that use stone masonrywalls is found in many factors : the materials used have a brittle behaviour ;they are usually decayed ; the weight of these structures is excessive (overdimensioned) ; there is no continuity (not monolithic) and most of oldstructures have been built without any anti-seismic design philosophy.
For old degraded masonry, the situation can be very complex. In thatcase, it is necessary to identify and to analyse the degradation processes. Inmost cases, the problems are coming from : an eccentricity of loads caused byan unsuitable design or changes in the structure with the consequentialpossibility of an increase in tensile stresses ; deterioration of mortars, stonesor bricks with a consequent progressive reduction of the bearing capacity ofblocks ; the underground water table variations leading to the deterioration offoundation blocks or to changes in soil deformability ; an increase of staticloads by the addition of new storeys, restoration works, etc ; exteriorexcessive loads such as earthquakes...
6 Masonry structures under earthquakes loads
Earthquakes damages have a tendency to concentrate on the weak points ofmasonry structures. Because of their nature, masonries have a large number ofpotential weak points compared to other types of structures, specially whenthere are not reinforced. The particular pathologies of historical structures mayalso interact with the seismic effects worsening their conservation, even duringsmall earthquakes. Cracking and other disruption of the structure will changeits natural frequency and may divide the structure into various smaller parts(with their own dynamic characteristics) which will vibrate in different waysand act upon each other in a destructive manner.
It has also been observed that a masonry building can survive a fewshocks of great intensity, but shakings of long duration are more damaging.
Observation of earthquake damages into masonry structures shows thatthere are three main factors affecting the earthquake performance of masonrybuildings : status or quality of mortars, quality of workmanship, and directionof the seismic waves.
In most cases, one or two sets of diagonal cracks are observed on masonry
walls which have been subjected to earthquakes. This means that the shearstrength is critical for this type of buildings, as well as cohesion and tensilestrength of mortars which play the role of bonding the masonry blocks togetherand may control sliding phenomena.
7 Modelling an old masonry wall with the DEM
The first case we modelled is amasonry wall subjected to aseismic load in its plan. The wallis 30 m long and 10 m high asshown on figure 3.
As far as material propertiesare concerned, the question thatarises when modelling an oldmasonry structure is whichmechanical characteristics ofstones and mortars will be usedfor calculations in the lack ofexperimental data. To overcomethis difficulty, it has been decidedto refer to the experimentsrealised by Sheppard et al. [9] ondifferent types of old masonrystructures. The analysis startswith mortars of good quality (hightensile strength and cohesion) andundecayed stones. Then, defor-mability of stones and resistanceof mortars are decreased tosimulate the decay of thosematerials. Six simulations havebeen carried out as shown ontable 1. All chosen mechanicalcharacteristics of stones andmortars are given in table 1.
The earthquake load consistsof a harmonic shear wave for thefirst five models and acombination of compression (P)and shear (S) waves for the sixthone, applied at the base of themodel and propagating upward.To prevent the downwardreflected waves to return upwardin the model, viscous boundariesare introduced at the base of themodel. Amplitude, frequency andduration of the input groundmotion are indicated in table 2.Futhermore, Raleigh damping isused in the calculations.
- determination of the funda-mental frequency of the model bygravity load, without damping. Thisleads to the free vibration of themodel from which the fundamentalfrequency can be found;
- set-up of the initial state ofstress (before application of seismicloads) by gravity loading with anautomatic damping scheme. Thisstage provides the initial stressdistribution which can influence thepropagation of seismic waves in thefollowing dynamic stage.
- dynamic modelling by means ofan harmonic wave applied at thebase.
The response of each model toearthquake loading is shown onfigures 4 to 9. Each one illustrates thedeformation of the wall duringdynamic loading, at different timeintervals. These deformations havebeen magnified to illustrate themovement of blocks more clearly.
Figure 4 shows the deformation ofmodel n°l under a shear waveloading. The wave tends to shake thewall horizontally from left to rightand from right to left. Because of thevery high strength of the joints, thereis no dislocation in the model. For allthe other models, as usually inmasonry structures, ruptures occurprimarily along mortar joints.
The response of model n°2,shown on figure 5, illustrates a typicaldiagonal failure as a result of theshear wave load.
In the case of model n°3, shownon figure 6, there is no joint cohesion.So, the wall can not resist to theapplied shear stress. Accordingly, asliding of the first row of blocks istaking place without affecting upperrows. In this case, the whole energy ofseismic wave is dissipated throughthis sliding.
In addition to that, for the case ofmodel n°4 where joints have no tensilestrength (figure 7), it can be observed
Figure 4 : Results of model n°l(displacements are very magnified)
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disorders and a dislocation ofmasonry blocks which are not movinganymore as a single entity. Diagonalfailure also appears with an opening ofmany joints all over the wall.
Moreover, in the case where jointshave no tensile strength and nocohesion at the same time as it is in themodel n°5 (figure 8), damages aretaking the form of blocks detachmentsat lateral sides of the model. A slidingis also observed between the first rowof blocks and the upper part of model.Opening of joints by horizontalextension as a result of the shear wave
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shaking is unavoidable when there is no joint tensile strength.Finally, in the case of combination of the two waves as for model n°6, the
figure 9 clearly shows the contribution of each waves. Here, both shear andcompression waves have been given the same amplitude. The devastatingaction of this combination is characterized by two diagonal openings. Inaddition to that, the uplifting vertical displacement induced by thecompression wave propagating vertically is observed.
8 Modelling an old masonry arch with the DEM
Archs are very frequent in historical building architecture. They are generallymade of stones or bricks and there are used for the construction of arcades orentrances. Isolated arches are also used often as doorways in boundary walls.
A 2-D model has been used to represent the behaviour of masonry archssubjected to an earthquake load in its plane. The possibility offered by theDEM to represent a detailed configuration of individual blocks, allows abetter understanding of the scenario of failure for such type of structures. Themechanical properties of the model are the same as the model n°l of the wall(table 2) with a joint tensile strength and a joint cohesion equal to 1 MPa. Theresults are shown on figure 10 where it can be seen that the arch remainsglobally stable during seismic loading. Nevertheless, opening of joints areclearly visible at some locations of the arch where cracks are usually observedin real cases. This illustrates the capacity of the DEM to simulate the effects ofdynamic loads for such typical architectural forms.
Figure 10. Behaviour af a roman arch subjected to a shear wave load appliedat its base. The time is increasing from left to rigth. Deformations aremagnified by 20.
The case study modelled here represents an old housedated to the XVI th century, studied (with theboundary element method) by Alessandri et al. (1993).The geometry of the model is shown on figure 11. Theconstruction is built entirely with stones. Thenumerical model took into account neither thefoundation nor the soil under the structure because theinterest of this study is only to emphasize theinfluence of mechanical characteristics on the dynamicbehaviour of such type of structures.
The mechanical properties used represent a oldmasonry structure with a very degraded mortar havingno cohesion and no tensile strength and decayedstones. The earthquake load consists of an harmonicshear wave applied at the base of the structure. Astatic analysis has been carried out before applyingthe dynamic load in order to compute the initial stressdistribution due to the weigth of the building.
Figure 11: Geometryof the model of anold building facade.
L1UU.U.U11 UUC IU uic vvt.J.gi.J.1 v/i niv, i M.j.i«.i ij.j.Under such conditions, the scenario leading to damages or collapse of any
masonry building generally consists in three steps : cracking, dislocation ofstones or partial collapse and final total collapse. With the model proposed,shaking action of shear waves is very dramatic as shown on figure 12 : cracksoccur primarily along mortar joints and at weak points of the structure(between and around the openings). Then, these cracks widely open and acollapse of the right side of the fagade is taking place as a result of the nonsymmetry of the facade. The failure mechanism ends with failure of the wholestructure. Because of the very weak material properties, the structure did notprovide any significant resistance and did not withstand earthquake effects.
Openings of mortar joints Beginning of the dislocation
Figure 12. Behaviour of a masonry facade under an earthquake load asmodelled with the DEM. There is no magnification of displacements.
After using the DEM in modelling masonry structures, it has been found thatthis method may considerably improve understanding of unstabilityphenomena. The concept of interacting deformable or rigid blocks separatedby joints which may be given realistic deformation properties, seems to be verywell adapted to the study of ancient masonry structures (walls, archs,columns). The ability of the DEM and specially of UDEC to take into accountthe typical non-linear behaviour of such structures is of very high importancein the analysis of ancient monuments. Consequently, the DEM is supposed toanswer to many engineers and researchers who always face this problem inusing finite element methods.
In the case of dynamic problems, the DEM also offers a suitable mannerof modelling because of its physically-based solution scheme (like the oneimplemented in UDEC code). Multi-step modelling must be carried out to takeinto account the history of those structures and many parameters have to becontrolled simultaneously. This is why several trials are usually necessarybefore succeeding a simulation. Nevertheless, modelling of architectural unitsas complex as archs or complete building facades is practically feasible withthis method with a reasonable computer time consumed.
References
1. Alessandri C. & Brebbia C.A. Strength of masonry wall under statichorizontal loads : boundary element analysis and experimental tests,Proceedings of the 3rd Int. Con/, on Structural studies, repairs and maintenanceof historical buildings, STREMA 93, Bath, UK, 16-18 June 1993,
2. Pietruszczak S. On mechanics of jointed media. Masonry and relatedproblems, pp 407-413, Computer Methods and Advanceds in Geomechanics,Beer, Booker & Carter (eds), Balkema, Rotterdam, 1991.
3. Benedetti D. & Benzoni G. Identification of structural schemes for masonrystructures from recorded accelerations, pp 449-456, Structural Dynamics,Kratzig et al (eds), Balkema, Rotterdam, 1990.
4. Gentile C. Parametric identification of equivalent models for masonrystructures, pp 457-464, Structural Dynamics, Kratzig et al (eds), Balkema,Rotterdam., 1990.
5. Leftheris B., Providakis C.P. Nonlinear analysis of masonry walls :parametric studies, pp 399-406, Proceedings of the 3rd Int. Conf. onStructural studies, repairs and maintenance of historical buildings, STREMA 93,Bath, UK, 16-18 June 1993,
6. Cundall P.A. A computer model for simulating progressive large-scalemovements in blocky rock systems, paper II-8, Proceedings of the Symp. ofthe Int. Soc. of Rock Mechanics on Rock Fracture, Nancy, France, 1971.
7. Itasca Consulting Group, UDEC 2.00 Manual, Minneapolis, USA, 19938. Cundall P.A. Formulation of a three dimensional distinct element model.
Part 1. A scheme to detect and represent contacts in a system composed ofmany polyhedral blocks, Int. Journal on Rock Mechanics & Mining Sciences,1988, 25, 107-116.
9. Sheppard P.E. In situ test of the shear strength and deformability of a 18thcentury stone and brick masonry wall, 7th Int. Brick Masonry Conference,February 1985, Melbourne, Autralia.
