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International Journal of Architectural HeritageConservation, Analysis, and Restoration

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Discrete Element Analysis of the MinimumThickness of Oval Masonry Domes

József Simon & Katalin Bagi

To cite this article: József Simon & Katalin Bagi (2016) Discrete Element Analysis of theMinimum Thickness of Oval Masonry Domes, International Journal of Architectural Heritage,10:4, 457-475, DOI: 10.1080/15583058.2014.996921

To link to this article: http://dx.doi.org/10.1080/15583058.2014.996921

Accepted author version posted online: 31Dec 2014.Published online: 31 Dec 2014.

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Discrete Element Analysis of the Minimum Thickness of Oval Masonry DomesJózsef Simona and Katalin Bagib

aDepartment of Structural Engineering, Budapest University of Technology and Economics, Budapest, Hungary; bDepartment of StructuralMechanics, Budapest University of Technology and Economics, Budapest, Hungary

ABSTRACTThis study focuses on domes the ground plan of which, instead of the more common circularshape, is an oval, and aims at finding the minimally necessary uniform wall thickness for domes ofdifferent geometries loaded by their selfweight. The discrete element code 3DEC was appliedbecause of its capability of simulating the collapse mechanisms of masonry structures. Results onthe minimal wall thickness, corresponding masonry volume and failure mechanisms for differentdome geometries are presented. Three ranges of the friction coefficient were found. For very lowfrictional resistance collapse happens with pure frictional sliding, for any arbitrarily large wallthickness. In the range of relatively high (i.e., realistic) friction coefficients the structure collapseswithout any sliding if the wall is not sufficiently thick, and in the observed range of the frictioncoefficient the necessary wall thickness is nearly insensitive to its value (collapse initiates withhinging cracks only). Between the two domains an intermediate behavior was found: combinedcracking and sliding collapse modes occur for insufficient wall thickness, and the minimal thick-ness strongly depends on the friction coefficient. The critical and transitional friction coefficientsseparating the failure modes were determined for different eccentricities of the groundplan.

ARTICLE HISTORYReceived 28 August 2014Accepted 07 December 2014

KEYWORDS3DEC; brick; distinctelement; stone; structuralstability

1. Introduction

While mathematicians use the term “oval” in a verygeneral sense (e.g., Descartes or Cassini ovals, Laméellipsoids; see for instance in Lawrence, 1972), in archi-tecture an oval is usually understood as a closed, convex,smooth curve having two axes of symmetry. A thoroughoverview was given by Huerta (2007) on the history ofthe application of oval shapes in different structural rolesin architecture from the first civilizations until theBaroque. In this article, the role of ovals will be toprovide the ground plan of a dome.

Although most of the existing domes of our architec-tural heritage have a circular plan, oval plans are morecommon than they are usually thought to be. As empha-sized by Huerta (2007), already in the Middle Ages hun-dreds of Romanesque churches were built with oval plan(Chappuis, 1976); then in the Renaissance and in theBaroque the idea became even more popular and quicklyspread all over Europe. In spite of this, their mechanicalbehavior (particularly the similarities to and differencesfrom spherical domes) is hardly analyzed and hencepoorly understood.

Masonry domes—similarly to masonry arches andvaults—are constructions of separate voussoirs supporting

each other basically by compression and friction. Even ifsome kind of mortar is applied in the joints, its tensilestrength is negligibly small compared to the compressivestrength of the bricks or stones, particularly in the case ofhistorical buildings with centuries or millennia old mortarin the joints. The compressive strength of the voussoirs canpractically be considered infinite. (Of medium sandstone, a2 km high column can be built without crushing at its basedue to its own weight. For granite this height is 10 kmaccording to Heyman, 2001, and even for different bricksthis height is hundreds of meters, while the height of exist-ing masonry domes and vaults does not exceed a fewdozens of meters.) The frictional resistance is usually ratherhigh even for dry joints (the angle of friction is at leastapproximately 30°–50°); indeed, sliding failure under staticloads is not characteristic for domes and vaults. Hence, thebasic question of the safety analysis of such a dome iswhether the structure with its given geometry is able toequilibrate its self weight and perhaps certain live loadswithout cracking collapse. This is a question of stabilityrather than a problem of strength: “Equilibrium is achievedby geometry” (Huerta, 2001).

Keeping these specific features in mind, Heyman(1967, 1995) derived the minimal thickness of spherical

CONTACT Katalin Bagi kbagi@mail.bme.hu Department of Structural Mechanics, Budapest University of Technology and Economics, H-1111Budapest, Műegyetem rkp. 3. K.mf. 63, Hungary.Color versions of one or more of the figures in this article can be found online at www.tandfonline.com/uarc.

INTERNATIONAL JOURNAL OF ARCHITECTURAL HERITAGE2016, VOL. 10, NO. 4, 457–475http://dx.doi.org/10.1080/15583058.2014.996921

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domes (see Figure 1). The problem he solved can beformulated as follows.

● Consider a spherical dome with a vertical axis ofsymmetry, characterized by the angle φ shown inFigure 1, with uniform thickness t.

● The material obeys the following basic assump-tions (e.g., Boothby, 2001):○ The blocks are perfectly rigid.○ The blocks have infinite compressive strength.○ The joints have infinite frictional resistance.○ The joints have zero tensile strength.

● No translations are allowed at the supports at thebase of the dome.

● The aim is to find the smallest thickness, tmin, forwhich the dome is still in equilibrium under itsown weight.

Heyman solved the problem using the Static Methodof plastic limit analysis (e.g., Heyman, 1995; Huerta,2001), and for φ = 90° friction angle he found that thisminimal thickness is 0.042 times the radius of thespherical surface.

Heyman made no distinction between the radii of theintrados, extrados, and middle surface. Indeed, tmin is sosmall in comparison with the radius of the dome that foran engineer the difference may seem to be negligible.Since in his derivations of the minimum thickness forarches (e.g., Heyman, 1969) the radius was defined to bemeasured to the center line of the arch, the same can beassumed for his studies on domes, which is in agreementwith the usual convention in membrane theory wherethe middle surface gives the basis of the calculations.Consequently, the above value 0.042 is understood as theratio of the minimal wall thickness vs. the radius of themiddle surface.

An important improvement of Heyman’s result wasprovided by Lau (2006), who, by combining membranetheory with the Static Method and hence taking intoconsideration hoop forces as well, received a slightlydifferent value, 0.041 times the radius in the case of ahemispheral dome.

The safety of an analyzed dome can then be estimatedby using the “factor of geometrical safety”, invented alsoby Heyman (1969), originally introduced for arches. Thisfactor compares the actual thickness of the structure totheminimal thickness which can still carry its ownweight.(When live loads have to be considered, the minimalthickness belonging to the worst position of the loadshas to be found. However, the dominant load on adome is usually its self weight.) Although the assumptionof uniform thickness is not always realistic for existingstructures, even in these cases the geometrical factor ofsafety provides an intuitive measure for the engineer ofhow safe an actual structure is. The dome of the RomanPantheon has, for instance, a wall thickness varying fromtop to bottom between about 5.5–30% of the radius; thesevalues are far above the necessary minimal thickness. Thethickness of the oval dome of the sanctuary of Vicoforte,as another example, is significantly smaller (and does notvary thismuch). Comparing to the equivalent groundplanradius, i.e., to the radius of the circle having the same areaas the groundplan of the dome, the thickness is approxi-mately 7–12% (Aoki et al., 2003). The knowledge of thenecessary minimum thickness helps the engineer inappraising the safety of the structure. So the basic aim ofthe present study was to extend Heyman’s problem todomes with oval plans (see the applied restrictions on thedome geometry in Section 2.1), and to find the minimalthickness for those cases.

To provide a possible method for the analysis of ovaldomes, Huerta (2010) called the attention on the trans-formation theorem of Rankine: “If a structure of agiven figure have stability of position under a systemof forces represented by a given system of lines, thenwill any structure, whose figure is a parallel projectionof that of the first structure, have stability of positionunder a system of forces represented by the corre-sponding projection of the first system of lines.” Ingeneral, this theorem could be useful for predictingthe stability of an oval dome constructed by an affinetransformation from a spherical dome. Obviously, it isnot reliably applicable to the problem of the presentstudy for several reasons:

tmin

R

0,02

0,04

51,8°

60° 70° 80° 90° ϕ

ϕR

Figure 1. Minimal thickness for spherical domes, after Heyman (1967).

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● the plan composed of circular arcs is not an affinetransformation of a single circular plan;

● the “stability of friction” of the transformed struc-tures is not ensured by the theorem, i.e., the forces onthe joints between individual blocks may or may notbe inside the friction cone independently of the sta-bility of friction of the original spherical dome; and

● the uniform thickness of a spherical dome wouldbe distorted by an affine transformation.

Different recent numerical methods can be found inthe literature to solve the problem of minimal wallthickness. The most important approaches are shortlyoverviewed below.

→ 3D Thrust Network Analysis (TNA, see Blockand Ochsendorf 2007) and its extension in Block andLachauer (2014) are based on the Static Theorem ofHeyman’s classical theory of masonry structures. Themethod applies three-dimensional graphic statics tofind a thrust surface. The power of this method isconvincingly demonstrated by Block et al. (2010) withthe help of 3D-printed structural models. (Its disadvan-tage is that the possibility of sliding is not excludedwith sufficient safety.)

→ With an improvement over membrane theory byallowing hoop and meridional forces to deviate fromthe middle surface, an iterative computation is used byZessin et al. (2010) to analyze cracked masonry domes.

→ D’Ayala and Casapulla (2001) derived a proofthat under a symmetry requirement for the loads, slid-ing mechanisms may be included in the limit stateanalysis of masonry domes without loss of uniqueness.A lower-bound computer method is proposed by theauthors who apply their method for Heyman’s problemextended to the effect of different friction coefficients inthe joints. An interesting result is that if the frictioncoefficient is between 0.12 – 0.7, the minimal thicknesscan be smaller than derived by Heyman. Based on thesuccess of the method, D’Ayala and Tomasoni (2011)presented a computational procedure and convincinglyapply it for pavilion vaults, also giving a comparison toFEM results. The method has the potential to analyzeproblems with general loads and constraints, recognizesliding as well as hinging collapse mechanisms, andincorporate the three-dimensional effects for a widerange of vault types.

→ Another alternative tool which is also capable ofmodeling 3D effects and frictional sliding is theDiscrete Element Method. This numerical techniquewas first introduced for the simulation of fracturedrocks by Cundall (1971), and has been applied in theengineering practice approximately from the 1990s,when computer hardware became powerful enough to

simulate realistic problems on average PCs. A discreteelement model considers the structure to be a collectionof separate blocks, “discrete elements,” each of which isable to move and—in most software—to deform inde-pendently of each other. The blocks may come intocontact with each other, where distributed forces canbe transmitted from one block to another, causingstresses and deformations in the blocks. According tothe criteria formulated by Cundall and Hart (1992), anumerical technique is a discrete element model if (1)the elements are able for finite (i.e., large) translationsand rotations; and (2) complete detachment as well asformulation of new contacts are allowed and automa-tically followed.

The second criterion means two important differ-ences from FEM: there are no continuity conditions atthe common points of the contacting elements, and theelements are continuously checked throughout the cal-culations whether they get into contact with each other.

The large displacements are usually followed with thehelp of some kind of a time-stepping scheme: mostDEM codes determine the characteristic motions of theanalyzed system along a series of small but finite timeintervals, applying Newton’s laws of motion. UsingDEM, a simulated structure may split into pieces (e.g.,a vault may fall into masonry voussoirs) which may evenbounce into each other on the ground forming a heapbalanced under its own weight. There are innumerabledifferent versions of discrete element techniques—theelements may be rigid or deformable, spherical, polyhe-dral, or irregular, the time integration may be explicit orimplicit, or may be replaced by a quasi-static method,etc. A helpful introduction is given by O’Sullivan (2011)on the most important techniques. (A particular value ofthis book is that the issue of numerical stability, which isparticularly important in the stability analysis ofmasonry vaults and domes, is discussed in detail.) Anexcellent overview is given by Lemos (2007) on thedifferent mathematical and practical approaches tosimulate masonry structures with the help of DEM,including practical engineering applications as well.The method has been successfully applied in practice-inspired investigations like Alexakis and Makris (2013)or Isfeld and Shrive (2015). In relation to the subject ofthe present paper, the study of Rizzi et al. (2014) has tobe referred to in which the authors successfully apply themethod for the analysis of the Couplet-Heyman pro-blem, taking into account the effect of the frictionalresistance between voussoirs.

Despite the few doubts regarding its usage (e.g.,Huerta 2008), the capability to simulate block separa-tion and contact sliding makes a suitably chosen andcarefully calibrated DEM model a powerful tool for

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the collapse analysis of masonry structures. Becauseof its ability to follow frictional sliding, failure pro-cesses and collapse histories in detail, DEM has beenthe basic tool of the investigations of the authorsrelated to different masonry mechanics problems,including the subject of the present paper. Detailsof the applied numerical technique are introducedin Section 3. It has to be emphasized that themodel verification and within this the careful calibra-tion of model parameters is a crucial issue in DEMmodeling; this is why Section 3 puts a particularlystrong emphasize on the question.

Before turning to the details of the analysis, twoimportant aspects regarding Heyman’s problem haveto be underlined. First, Heyman assumed fixed sup-ports under the domes. In reality, the supports alwaysmove: deformations of the underlying walls, soil sink-ing, etc. are inevitable. As a result, the masonry dome—whose voussoirs have only rather low if not negligibledeformability—adjusts itself to the new position of thesupports by cracking leading to the rearrangement ofthe internal force system. Consequently, the limit thick-ness for a dome with deforming supports will differfrom Heyman’s solution for the same dome. The ana-lysis of the effect of support flexibility is out of thescope of the present study; it is left for further researchand the present study is restricted to the original pro-blem with fixed supports. However, sliding between thedome blocks and the fixed supports is allowed accord-ing to the Coulomb model.

Second, in reality the voussoirs are never infinitelyrigid, so if the dome is built with the help of centering,a slight deformation always occurs at de-centering,hence modifying the shape of the dome. This effectmay also lead to deviations of the minimal thicknessfrom that of Heyman. In the DEM investigations thedeformability of the voussoirs was not taken intoaccount: the blocks were perfectly rigid, and the numer-ical control parameters of the model were carefullyverified to match the result of Heyman’s value fororange-sliced hemispheral domes. This calibratedmodel was then used for the simulation of oval domes.

This article is organized as follows. The analyzedgeometries are specified in Section 2. Section 3describes the most important characteristics of the uti-lized discrete element code (3DEC), and introduces thecalibration tests which were done to ensure the relia-bility of the results; these numerical experiences may behelpful for researchers simulating other types ofmasonry vaults and domes with the help of DEM.The simulation results on oval domes are presentedand discussed in Section 4. Finally, the most importantconclusions are summarized in Section 5.

2. Preparation of the model geometry

2.1. Definition of the ground plan

The aim of this section is to introduce the mathematicaldefinition of the ovals serving as plans for the domes tobe analyzed. The most important versions, i.e., theEgyptian oval and its generalized versions which wereapplied in the present study, are described here indetail; other ovals being out of the scope of the presentarticle are only shortly mentioned.

The Egyptian oval is based on a right-angled trianglewhose sides are 3, 4, and 5 units. The quarter of theoval can be drawn, as shown in Figure 2, with the helpof two pegs fixed at points A and B, and with an 8-unitlong string whose beginning point is attached to thepeg at point A.

Starting from point A’, draw a circular arc (with 8-unitradius) by moving the endpoint of the string upwards,until the string is stopped by the peg at point B. The end ofthe string is now at point C. Continue moving this end-point of the string further upwards; because of the peg atB, the radius of this new arc (from C to D) is only 3 units.The two arcs drawn this way will have a common tangentat point C, so the curve is smooth The rest of the oval canbe prepared in the same way.

Right-angled triangles with other side ratios can also beapplied as the basis of an oval: different generalizedEgyptian ovals can be received this way. Figure 3a showsthat applying a right triangle with catheti a and b, and astring of length l = 2a, a similar oval will be producedwhose arcs have the radii l ¼ 2a and r ¼ 2a� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ b2p

.Obviously, since a and b have to satisfy the relation

a �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ b2p

� 2a;

the length b must be

0 � b � affiffiffi

3p

:

AA'

B

C

D

a = 4 a = 4

b = 3

Figure 2. The Egyptian oval.

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The case b → 0 means that the oval degenerates intoa circle with a radius a (Figure 3b). The other extreme,i.e., b ! a

ffiffiffi

3p

; gives the greatest possible deviationfrom the circle, however, this is not a smooth curveany more (Figure 3c), so this extreme cannot be con-sidered as an oval. (The analyzed groundplans inSection 4 varied between these two extremes.)

The deviation from the circle can be characterized bythe eccentricity angle β shown in Figure 3a: β = 0corresponds to the circle, while the other extreme isβ = 60° (Figure 3c) for which the smaller radius rvanishes.

The groundplans of all domes analyzed in the pre-sent paper were generalized Egyptian ovals. The area ofthe plan was the same in all cases, while they differed intheir eccentricity angle β. The aim was to investigatethe effect of the deviation from the circle on the mini-mal thickness of the dome.

Ovals can be constructed in several other ways. Onepossibility—often seen in Gothic architecture—is toplace the basic point A outside the curve (seeFigure 4). Such a solution was applied in certain histor-ical structures (see Huerta (2007) citing, e.g., Koepf(1969), Bucher (1972), or Vandelvira (1580)). Other

possibilities arise when more than two different arcsare used to compose the oval; or different constructionmethods of ellipses are applied, etc. However, the pre-sent investigation is restricted to the most prevalentversions, i.e., the generalized Egyptian ovals only.

2.2. The surface of the intrados

After drawing the groundplan, the builders of a domehad to define the 3D geometry of the internal surface(“intrados”) in order to prepare the centering for thevoussoirs, or—in case the dome was to be built withoutcentering—a light guiding scaffold to prescribe thegeometry of the intrados. According to Huerta (2007),the most common solution was to define the intradosas a surface of revolution, being rotated either about thelonger axis of symmetry of the plan (e.g., San Andreadel Quirinale), or about the short axis (e.g., CesareanLibrary in Vienna). In the present study both possibi-lities were considered (see Figure 5): the first optionwill be referred to as Type 1 surface (“flat” domes),while Type 2 will denote the second option (“high”domes).

2.3. The complete dome

Now the 3D body of the dome is, in principle, easy todefine mathematically: from every point of the intrados,perpendicularly to the surface, the uniform thicknesshas to be measured outwards, to receive the points ofthe extrados. In the DEM simulations the task is morecomplicated: the continuous domain between the intra-dos and extrados has to be constructed as a collectionof 3D brick-like discrete elements the length, l, andheight, h, of which are aimed to be approximatelyequal along the complete dome, and their thickness tto be uniform. This was achieved in the 3DEC modelsin the following way.

(a) (b) (c)

aal

AC A'≡

D

Ba

b

a

l

r B

AA'

C

D

a

b

a

l

C B≡

AA'

Figure 3. (a) The generalized Egyptian oval; (b) the extreme case β = 0°; and (c) the extreme case β = 60°.

AA'

C

D

B

b

aa

l

r

Figure 4. Oval with A positioned outside the curve.

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Since the intrados is a surface of revolution, itssections perpendicular to the axis of revolution arecircular arcs. The composition of the discretizedmodel of a dome starts by considering the circular arcwith the largest radius perpendicular to the axis ofrotation. This arc is shown in Figure 6. Obviously,Rint is the half of the other axis of the plan beingperpendicular to the axis of revolution. Measure thedesired brick height, h, starting from the lowest pointof the arc proceeding upwards as many times aswanted. (An opening for an oculus may be left out atthe top in the model, while if such a hole is not desired,the small domain, where a full brick height cannot beplaced, will be filled at the end of the constructionprocedure by a single closing element.) Through eachpoint received this way (see the dark dots in Figure 6),imagine a horizontal cutting plane. The intersection ofthe i-th plane with the intrados will serve as the i-th“master curve,” i.e., the basis of preparing the i-thhorizontal ring of discrete elements. (Note that themaster curves are not similar to each other.)

Figure 7 shows how the inner-lower sides of theelements are placed along the master curve. The pro-cess starts at a randomly chosen point, P1. The bricklength, l, should be measured along the curve, as manytimes as possible. However, this way the master curve isnot followed smoothly by the brick elements, and it

may lead to sharp corners particularly in those parts ofthe dome where the curvatures are high. Thus, thecurvature of the master curve is taken into account inthe algorithm by dynamically changing the brick lengthalong the curve as shown in Figure 7 (imitating amason who uses smaller stones for the strongly curvedparts of the structure). Finally, truncated or elongatedlengths are applied to close the ring, similarly to realmasonry constructions.

The k-th discrete element of the ring is formed with thehelp of its two already existing nodes, Pk and Pk+1. Bymeasuring the uniform thickness perpendicularly to theintrados in the outwards direction, two additional nodesare received. The intersection of the plane being perpen-dicular to the master curve in point Pk with the i+1-thmaster curve and a similar intersection point of the per-pendicular plane at Pk+1 with the i+1-th master curve willgive two more nodes. From these two new nodes thethickness is measured outwards again to get the twomissing nodes of the k-th element. (Note that the anglesdetermined by the edges of such an element slightlydeviate from being perpendicular.) After finishing thering, a next layer can be prepared in a similar manner.

Since the master curve is horizontal and the top facesof the blocks forming a ring are determined by the out-wards normal vector sweeping along the master curve, theunion of the top faces is not planar. Consequently, in a

h

h

h

1.

2.

. . . .i

i+1.

h

Rint

Figure 6. Largest vertical cross-section perpendicularly to the axisof rotation: definition of the vertical positions of the master curves.

Figure 5. The intrados as a surface of revolution: (a) about the longer axis, Type 1 (flat) dome; and (b) about the shorter axis, Type 2(high) dome.

Figure 7. Horizontal view: The i-th master curve, with dynami-cally shortening brick lengths where the curvature is large.

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mathematical sense, the bottom corners of the next ring ofblocks do not perfectly fit to the top of the blocks belowthem. However, the discrepancy was found to be negligi-ble for those dimensions applied in the simulations, nearthe range of numerical rounding errors, hence it did notmake any mechanical effect.

Similarly to real masonry structures, truncated orslightly elongated elements are applied not only toclose the horizontal rings of voussoirs but also toavoid having interstices just above each other.

Figure 8 shows two geometrical models constructedin the above manner: a generalized Egyptian oval withβ = 50° is applied to generate a Type 1 and a Type 2dome.

Finally, a remark has to be made on the issue ofusing uniform thickness in the applied geometricalmodels. In many real domes, especially of largishradius, the thickness is not uniform along the meridian:smaller around the top and larger near the bottom. Thischange in thickness results in a specific distribution ofloads along the meridian, and can make the dome morestable. Consequently, the results in this article can beconsidered as a conservative approximation of thebehavior of domes with varying thickness.

3. Numerical modeling issues

3.1. 3DEC

The commercial DEM code 3DEC was applied in thesimulations. UDEC (“Universal Distinct ElementCode”), the 2D ancestor of 3DEC, was originally devel-oped for the modeling of fractured rocks (Cundall, 1971),but today its 2D and 3D versions are widely used in theengineering practice also for masonry structures.

The discrete elements in 3DEC may have any poly-hedral shape, and may either be perfectly rigid, or theycan be made deformable in such a way that they are

divided into simplexes (tetrahedra in 3D) which serveas uniform-strain finite elements. This article appliedperfectly rigid elements. In this case each rigid elementhas a reference point whose displacements (translationsand rotations) during a small finite Δt time interval aredetermined with the help of Newton’s laws of motion,taking into consideration the inertia of the whole ele-ment. Forces acting on the elements may be eitherexternal loads or exerted by the neighboring elementsthrough the joints. The mechanical model of the jointbehavior is a crucial issue, and will be discussed inSection 3.2 in detail. An explicit time integrationscheme, based on central differences, is used for simu-lating the mechanical behavior (motions, changes ofcontact forces, etc.) over time: the simulation of thestate changing under a given loading process isachieved by step-by-step calculation of the motions ofthe reference points.

3.2. Calibration: Simulations of spherical domes

3.2.1. Introductory remarksThe key of composing a reliable discrete element modelfor a structure is the careful calibration of the modelproperties, i.e., the necessary numerical parameters aswell as those controlling the numerical time integration,method of loading, boundary conditions, etc. The modelproperties in the present study were calibrated with thehelp of the analysis of Heyman’s original problem, i.e., ahemispherical dome being cut into orange-slice-like par-titions (see Figure 9). The following characteristics hadto be suitably set so that the minimal wall thicknesswould be the same as that of Heyman:

(i) Contact parameters. The contacts in all simula-tions were frictional which means that the follow-ing three contact parameters had to be chosenproperly. The normal stiffness (“penalty

Figure 8. Discrete element model of the geometry of: (a) a Type 1 and (b) a Type 2 dome based on a generalized Egyptian oval (β=50°).

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stiffness,” kN) expresses how difficult it is to pressthe boundary point of an element into the inter-ior of its neighbor: it is the intensity of the dis-tributed compression force occurring at unitdepth of the penetration between the two ele-ments. The second parameter, the shear stiffnessof the contact, ensures that when a slight relativedisplacement increment occurs in the tangentialdirection between two blocks, a tangential force—equal to the displacement increment times thetangential stiffness—arises in the opposite direc-tion, and prevents the contact from unrestrictedsliding. The third parameter is the coefficient ofthe classical Coulomb-friction.The joints between the elements in a DEM modelmay have different physical meanings. The jointsmay represent some kind of mortar layer (or, e.g.,clay layer in fractured rocks) having a finite thick-ness in reality between deformable blocks repre-senting the voussoirs. If using perfectly rigiddiscrete elements (like in the present study), thejoints in the model should also reflect the deform-ability of the voussoirs: in this case the materialparameters of the joints express the total deform-ability of the blocks–and–joints system, therefore anormal and a tangential stiffness have to be prop-erly verified along with either some fracture criteria(normal and shear strength) and/or a friction coef-ficient in order to define the conditions when thejoints fail. In the present study the friction coeffi-cient gave the only limit for the failure of thecontact, and no fracture criteria were applied.Note that for dry joints between blocks havingirregular (nonsmooth) surfaces the material para-meters of the joints should also include the possi-bility of the contacting blocks to get closer to eachother under large pressure and to resist tangentialrelative translation due to the shear resistance ofthe small irregularities on the contact surface.Since these phenomena related to nonsmooth

surfaces are too complex, their quantitative analy-sis was not considered in the present study.From a computational point of view, the larger arethe applied contact stiffnesses in amodel, the smal-ler is themaximally allowed time step in the explicitsolver, so the calculations become more time-con-suming with increasing stiffnesses. On the otherhand, too low stiffnesses may lead to unrealisticresults: the contacting voussoirs may significantlypenetrate into each other leading to a globalmechanical behavior similar to a collection of softblocks, and may even cause the numerical collapseof the simulations, while a rigid-block system withincreasing stiffness tends to the theoretical modelof Heyman’s classical theory. So in the calibrationtests special attentionwas paid to find the regime ofcontact penalty parameters for which the behavioris already acceptable regarding convergence toHeyman’s classical results to the accuracy intendedto reach. The results were then compared to mor-tared joints with realistic elastic characteristics.These tests are introduced in Section 3.2.2. Thesecond parameter, the tangential stiffness, kS, waschosen to be 1/10 of the normal stiffness.The third parameter, the friction coefficient (f),should be infinitely large according to Heyman’sconditions. In the calibration tests where the aimwas to reproduce Heyman’s results the frictionangle was indeed set to 90°.After the calibration tests, in the analysis of ovaldomes in Section 4, Coulomb friction with f = 0.7was applied first. Then coefficients ranging from0.3–0.7 were considered to carry out sensitivityanalysis regarding the contact properties. Finally,friction coefficient was decreased further, withvery small steps (determining the minimal wallthickness for each value), to find that value forevery intrados where the purely hinging collapsemechanism turned into a combined hinging-slid-ing mode for insufficient wall thickness, and that

Figure 9. Calibration of the element size: (a) The analyzed densities; (b) collapse of a 50-segment spherical dome (colors indicatedisplacement magnitude).

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value where the mixed collapse mechanism turnedinto pure sliding failure mode so that the structurewas unable to equilibrate its weight with any wallthickness.

(ii) Element size. The elements in a DEM modelmay either directly correspond to the individualvoussoirs, or each element models a collection ofseveral bricks or stones. While the computa-tional time decreases with the decreasing num-ber of elements, due to the bending resistance ofthe individual blocks the overall load bearingcapacity of the structure may be overestimatedif the subdivision is not dense enough. So it wasalso an open modeling issue how densely adome should be divided into discrete elements.Section 3.2.2 deals with this question as well.

(iii) Loading technique: method of de-centering. In aDEM simulation the gravity can either be“switched on” in a single step (as if dropping theself weight suddenly on the whole structure), orthe structure can initially be fixed at all nodeswhile the gravity is already acting and then thenodes are gradually released, starting from the topand proceeding downward in several steps, findingthe equilibrium in each step before releasing thenext set of nodes (as if decentering a real dome).With a series of calibration tests it was checkedwhether the difference in the loading techniquemade an influence on the results. Details will begiven in Section 3.2.3.

(iv) Parameters controlling the numerical timeintegration. Since the basis of the presentstudy was to check whether different domeswith given geometries can find their equilibriumstate without collapse (hence the exact history ofhow the blocks find their equilibrium position isunimportant), in all tests the 3DEC default para-meters for static analysis were applied: “auto”damping with a coefficient of 50% (half of thechange of kinetic energy was dissipated in eachtime step), and the time step length offeredautomatically by 3DEC was accepted (see Itasca(2007) or Cundall (1982) for detailed explana-tions). Note that these parameters would not besuitable when simulating a dynamic process,e.g., earthquake analysis.

(v) Boundary conditions. The lowest elements ofthe domes were supported from below by a ringof perfectly fixed elements having the same con-tact parameters as those blocks forming thedomes. So the lowest voussoirs of the domeshad an elastic–frictional, Coulomb-type contactwith this “foundation”.

3.2.2 Calibration of the element size and the contactnormal stiffnessFigure 9a illustrates the analyzed geometries: the dis-crete elements were defined by slicing the dome volumeby dividing its base perimeter into 10, 20, . . ., 50 sec-tions, and applying the same density of discretizationfor each slice. Note that the vertical interstices betweenthe elements are just above each other in the calibrationtests, corresponding to the slicing technique presentedby Heyman. This very special arrangement was appliedonly in the calibration tests where Heyman’s theoreticalprediction was intended to reach; then in the simula-tions in Section 4 the vertical joints between stoneblocks were always shifted with respect to each other.

For each geometry shown in Figure 9a, a sequence ofdifferent penalty stiffnesses was tried, and the minimallynecessary wall thickness was determined for each penaltystiffness in the following way. For a given middle sur-face, kN and density of subdivision, several differentdomes were prepared which differed only in their wallthickness. Unlike in the simulations of oval domes(whose geometry were prepared according to themethod explained in Section 2), for the calibration teststhe middle surface of all domes was the same hemi-sphere with unit radius R, in order to follow Heyman’sanalysis as closely as possible. Half of the thickness wasmeasured outward and the other half was measuredinwards along the normal vector of the middle surface.The applied thicknesses differed by 0.1% of the middleradius, e.g., t = 0.039 R, 0.040 R, 0.041 R, etc. wereapplied. Then for every case the selfweight was put onthe dome, time stepping was started, and continued untileither collapse or finding a balanced state. The collapsewas recognized by monitoring the vertical displacementsof the closing element on the top of the structure. If thisdisplacement was greater than 5% of the height of thedome, collapse was recognized and the simulationstopped. On the other hand, balanced state was validatedby checking the displacement and velocity of the closingelement on the top of the dome and also the unbalancedforces in the system. The balanced state was assuredunder the following condition: the displacement, velocityand unbalanced force had to be lower than 10−6 m, 10−7

m/s and 10−5 N, respectively. To illustrate this, Figure 9bshows the collapse mechanism for the 50-segment domewith a thickness slightly under the critical value: theperfectly axisymmetric geometry of the collapse mechan-ism can clearly be observed.

Finally, the smallest thickness for which the domedid not collapse was selected; this thickness is shownon the vertical axis of Figure 10. The thickness isnormalized with the radius of the middle surface R.The results show that as the contact stiffness and

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subdivision density increase, the minimal thicknesstends to Heyman’s prediction. The stiffness value kN= 1011 – 1012 N/m3 and subdivision of the perimeterinto 30 sections (brick length l = 0.209 R) are alreadysufficient to get a good agreement with Heyman’sresult: the minimal thickness for which the domesdid not collapse approached 0.042 times the radius ofthe middle surface. Based on this, the characteristiclength of the bricks applied in the simulations inSection 4 were around 40 cm (l = 0.08 R), which isclose to realistic sizes, and definitely smaller thanwhat would be necessary to capture Heyman’s resulton minimal wall thickness of spherical domes.

Real contacts and real voussoirs have finite stiff-nesses, consequently according to the results shown inFigure 10, real domes require bigger thickness to bestable than predicted by Heyman’s rigid block theory.However, the results also show that if the subdivision isdense enough (i.e. the perimeter is subdivided into atleast 30 elements), then for kN being above 2 GPa/m theminimal wall thickness does not significantly differfrom Heyman’s prediction: it remains under 105% ofthe theoretical value, and decreases with increasing kN.For kN being above 10 GPa the difference is negligible.So the question is what is the realistic range of jointstiffness. To approximate this range, consider twoblocks shown in Figure 11a, both having the samethickness lb = 0.40 m, and a mortar layer of lm =1.5 cm (the two contact surfaces have unit area).Assuming unrestricted crosswise extension, the kNjoint stiffness of the equivalent rigid block—deformablejoint—rigid block system is

kN ¼ EbEmlbEm þ Eblm

;

where Eb and Em denote the Young-modulus of theblock and the mortar respectively. Typical Young-moduli Eb are between 5 GPa (shale or mudstone)and 100 GPa (exceptionally stiff marble or granite).For mortar Em = 1 – 5 GPa can be estimated. Theequivalent joint stiffness for the very small values Eb =5 GPa and Em = 1 GPa becomes kN = 10,53 GPa/m forwhich the minimal wall thickness is 102% ofHeyman’s prediction; for more realistic Young modulithe equivalent joint stiffness is an order of magnitudehigher so the deviation from Heyman’s prediction isnegligible. (For instance, for marble and a strongermortar, Eb = 50 GPa and Em = 2 GPa, the equivalentstiffness becomes kN = 64,52 GPa/m, for whichFigure 10 shows that the deviation from Heyman’sprediction is negligible.) Note that for restricted cross-wise deformations the equivalent stiffness becomeshigher because of the Poisson-effect, and dry contactscan probably be approximated with even higher kNthan mortared joints, around 1011–1012 Pa/m.Similar values for kN were also found in discrete ele-ment simulations of two deformable blocks with athird discrete element between them representing amortar layer between the two voussoirs, submitted tocompression as shown in Figure 11b. DifferentYoung-moduli, Poisson ratios, and friction coeffi-cients were tried for both unrestricted and restrictedcrosswise deformations. The simulation results con-firmed the above values for kN.

3.2.3. The role of the loading methodConstruction with centering was simulated and tested inthe following way. Two spherical domes were preparedwith the same intrados and a chosen thickness (thearrangement of the discrete elements was realistic now,

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30 segments 40 segments

50 segments Theoretical value - Heyman

Figure 10. Dependence of the minimal wall thickness on the density of subdivision and on the contact penalty stiffness fororange-slice-like spherical domes.

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i.e., the elements were of the same size apart from thosefew which had to be truncated, so that the interstices werenot above each other). At the beginning, the gravity wasalready acting while all nodes of the elements were stillfixed against translations. Then in the first case the ele-ments were gradually released starting from the top of thedome and proceeding downwards, as if removing a cen-tering, in ten consecutive steps. The structure was equili-brated after every step. For the other dome the wholecentering was removed (all the elements were released)in one single step. Using the same intrados with severaldifferent wall thicknesses, the minimally necessary thick-ness was then determined for each case in the samemanner as explained in Section 3.2.2.

The results on the minimal thickness showed nonoticeable differences between the different de-centeringtechniques. So the simplest possible technique can reliablybe used in the computer simulations, hence the rest of thetests were done by decentering the domes in a single step.It is important to emphasize, however, that this statement

has been checked only for the loading case which isrelevant for the present study, i.e., only for pure self-weight. In other problems where the effect of a graduallyincreasing live load is analyzed special attention has to bepaid if an adaptive global damping technique is used: thedamping parameter has to be returned to the original highvalue before a new load increment is applied.

4. Simulation results

4.1. The analyzed domes

The applied ground plans are shown in Figure 12; theirdeviation from the perfect circle is quantified with thehelp of the eccentricity angle β (see Section 2.1). Allground plans had equal internal area, the same as thatof a circle with a 5 m radius (i.e., the equivalent innerradius, Req, was 5 m for each oval). (Note that the ratioof minimal thickness to inner radius does not dependon the actual radius size, except that the penalty

(a) (b)

Block Mortar Block

reference points of the blocks

lb lm lb

Figure 11. Mortared joint between deformable blocks: (a) geometry of the block – joint – block system; and (b) the systemsubmitted to compression.

(a) (b) (c) (d)

(e) (f) (g)

(a) ; (b) ; (c) ; (d) ; (e) ; (f) ; (g)

Figure 12. Ground plans of the analyzed domes.

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stiffness should be recalibrated if the size of the dome(thus the internal forces) change with orders of magni-tude in comparison to the calibrated model.)

Rotating each groundplan about its longer or shorteraxis of symmetry, a Type 1 or a Type 2 intrados wasreceived, respectively. Then—according to the methoddescribed in Section 2.3—several complete domes wereconstructed for each intrados, differing in their wallthicknesses but covering the same net area. (Thismeans that the area of the groundplan defined by theintrados was the same in every model, but the equiva-lent radius of the middle surface also increased with thethickness.) The characteristic lengths for the definitionof the elements were

l = 0.40 mh = 0.40 mand the thickness, t, was changed from simulation to

simulation in steps of 0.1% of the equivalent innerradius Req.

For every geometry the material density 2500 kg/m3

and the usual gravitational acceleration, 9.81 m/sec2

was applied in the calculation of the weight of theblocks. The same contact characteristics were used inevery model: normal stiffness kN = 1012 N/m3, tangen-tial stiffness kS = 1011N/m3.

4.2. Results for f = 0.7 friction coefficient

Since most of the constructional stones and brickshave at least or about this frictional capacity, forevery intrados the minimal wall thickness was firstdetermined for 35° friction angle (0.7 friction coeffi-cient). Figure 13 summarizes the most importantresults. The horizontal axis measures the eccentricitywith the angle β: the deviation from the circle

increases from left to right. The vertical axis showsthe minimally necessary thickness, tmin, normalized byReq, and expressed in [%].

In the case of a circular ground plan and rigid ele-ments tmin = 0.037 R was received, smaller than the value0.041 found by Lau (2006) by taking into account hoopresistance. The result naturally differed from the out-come of the calibration tests and Heyman’s result: thosedomes in the calibration and in Heyman’s derivationhad an orange-slice structure, while the analysis inSection 4 was based on realistically, irregularly arrangedvoussoirs where the frictional resistance on the horizon-tal contact surfaces and the tension resistance of theindividual blocks provided a hoop resistance. This isexplained in Figure 14: the separation of the two blocksin the middle ring is encumbered by the frictional forcesexpressed by the upper and lower blocks. A possibleexplanation of the deviation from Lau (2006) is sug-gested by the characteristic collapse mode shown inFigure 10 for the orange-slice-like arrangements andalso found for the realistically placed voussoirs. Unlikein the case of the continuum model applied by Lau(2006), in a DEM model the tangential stiffness kS isfinite so the elements may slightly translate elasticallyalong each other in the plane of the contact, even with-out frictional sliding. The typical collapse mechanism ofa hemisphere includes that the lowest part of the domerotates outwards about the external perimeter of thegroundplan (again, see Figure 10). A slight outwardstranslation of the lowest blocks shifts this external sup-porting ring of hinges into a more favorable position,and contributes to the resistance against collapse.

The results in Figure 13 show that with significant devia-tions from the spherical shape, the minimum thicknessdiffers only 25% of the value required for a hemispheral

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wal

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]

Eccentricity angle β [°]

Type 1 - f = 0.7

Type 2 - f = 0.7

Figure 13. The necessary minimal wall thickness normalized by the equivalent radius (expressed in %) as the function of theeccentricity of the groundplan.

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dome with the same net groundplan area in case of Type 1(flat) domes, but this difference can be as high as 50% if theresults of the Type 2 (high) domes are considered.

Type 2 (“high”) geometry appears to be morefavourable from mechanical point of view than a sphe-rical intrados: apart from the range of nearly circularground plans, smaller wall thickness is sufficient tobalance the own weight of the structure. This is notthe case for the Type 1 geometries: the necessary wallthickness is greater for oval ground plans than for aspherical dome except for domes with very high eccen-tricity. This difference may be explained by the fact thatin case of Type 2 geometries the longitudinal crosssections are half-circles and the transverse cross-sec-tions are vertically elongated ovals (somewhat resem-bling to a pointed arch); for Type 1 geometries thetransverse cross sections are the half-circles and thelongitudinal cross-sections are horizontally elongatedovals whose resistance against the self weight is poorer.

Figure 15 shows the total volume of the elementsassuming that the minimal wall thickness is appliedeverywhere. The horizontal axis measures the eccentri-city β again. For medium to large eccentricities (β =

20°–55°) Type 2 domes are more favourable in thesense that they require less building material thaneither the spherical or the Type 1 domes; however, inthe case of extreme eccentricity (β = 60°) Type 1 domesneed the smaller amount of masonry.

It is interesting to compare how the Type 1 andType 2 domes collapse if their wall thickness is toosmall. For low eccentricities the collapse mechanismsare similar to that of a hemisphere (see Figures 16a and16b): two intermediate rings of hinges form the lowerof which moving outward, the upper moving inward.The rings are not perfectly horizontal: in all cases theyslightly descend towards the two ends of the longeraxis. As the eccentricity increases, a characteristic dif-ference can be noticed between the collapse mechan-isms. A Type 1 dome is a flat, elongated structure: inthis case the rings of hinges deviate more and morefrom the horizontal plane with increasing eccentricity,and a kind of bulge can be noticed where the outwardsring approaches the base level near the narrow ends ofthe oval base (see Figure 17a). The collapse of Type 2domes with large eccentricities happens in nearly thesame way (Figure 17b), the only difference is that since

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0.40

0.50

0 10 20 30 40 50 60

Ele

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t vol

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[m3 /m

2 ]

Eccentricity angle β [°]

Type 1 Type 2

Figure 15. Total volume of masonry normalized with groundplan area when applying the minimally necessary thickness, expressedin m3/m2 on the vertical axis, as the function of the eccentricity of the groundplan.

Figure 14. Explanation of hoop resistance: red arrows represent frictional forces, not allowing the blocks of the middle layer toseparate.

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the strongly curved parts of the dome form a relativelyrigid vertical circular arch-like strip overarchingbetween the two sharp ends of the groundplan abovethe longer axis of symmetry, such a bulge like the oneseen in Figure 17a cannot occur and the lower parts ofthe strongly curved arch-like strip simply buckleoutward.

Domes are often built with an oculus, i.e., a roundopening (circular or oval) around the top of the dome.For spherical domes a small circular oculus does notinfluence the minimally necessary wall thicknessaccording to Heyman’s theory, and the same can beexpected for oval domes with small eccentricities. Toanalyze the question, additional simulations were donefor the same intrados geometries as before, but with anoculus of the size 10% of the longer axis of the ground-plan, and checked the collapse modes for f = 0.7 andslightly insufficient wall thickness. The only differencecould be detected in the collapse modes of Type 2domes with large eccentricities (see Figure 18). AType 2 structure becomes higher with increasing eccen-tricity; the vertical circular ach-like strip becomes more

and more dominant along the longitudinal cross sec-tion, and the two sides of the dome flatten out as theirsurface curvatures decrease. For eccentricities about β >45° the failure was initiated by the flattened parts fallinginwards because of their low resistance for bending andbecause of the missing part of the masonry shell aroundthe top (Figure 18).

4.3. The effect of contact friction coefficient

In order to analyze the effect of frictional resistance onthe minimal wall thickness and collapse modes, everyintrados applied in Section 4.2 was simulated withseveral different friction coefficients and wall thick-nesses. The simulation results are summarized inFigure 19. Every line in the diagram belongs to aspecific intrados, and shows how the minimal wallthickness depends on the value of the friction coeffi-cient f between the voussoirs. (The minimal wall thick-ness is normalized by the equivalent radius Req.)Collapse modes of the domes with t being slightly

Figure 16. Hinging collapse modes: (a) Type 1 (flat) dome and (b) Type 2 (high) dome, both with β = 30° ground plan (color scaleindicates displacement magnitude).

Figure 17. Hinging collapse modes: (a) Type 1 (flat) dome and (b) Type 2 (high) dome, both with β = 50° ground plan (color scaleindicates displacement magnitude).

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below the minimally necessary value were alsoobserved. According to the experiences, three domainsof the friction coefficient can be identified for eachintrados.

(1) For f above approximately 0.2–0.3 the minimalwall thickness is not sensitive to f in the observed range(up to f = 0.7). In these cases collapse occurring forinsufficient wall thickness happens with a hingingmechanism, without sliding. The lower boundary ofthis domain of f, the transitional friction coefficientftr, depends on the geometry of the intrados.

(2) Below ftr but above the value of critical frictioncoefficient fcr, a combined collapse mode containingsliding and rotational hinging occurs if the wall thick-ness is too small. Such a collapse mode can be seen inFigure 20a for a Type 1 dome and in Figure 20b for aType 2 dome. In this range the value of tmin stronglydepends on f. The smaller is the friction coefficient, thelarger is the necessary wall thickness to balance the selfweight, and sliding becomes more and more dominantin the collapse mechanisms.

(3) Below the fcr critical friction coefficient belongingto the analyzed intrados, the structure cannot be inequilibrium for any thickness. In this case a pure slid-ing mechanism can be observed. Failures of this typeare shown in Figures 21a and 21b.

These three domains were also found by D’Ayala andCasapulla (2001) for hemispheral domes with finite fric-tion.While for Heyman’s orange-slice-like cracked domesthe two special values of f separating the three domains arefcr = 0.20 and ftr = 0.25 in the DEM simulations, accordingto their theory D’Ayala and Casapulla (2001) found thatfor membranes with fixed supports against any transla-tion, and considering their “x+z” curvewhich correspondsto removing a fictitious constraint on the position of thethrust surface, their values were 0.136 and 0.25, respec-tively. For the hemispheric dome with brick orderingproviding hoop resistance 0.20 and 0.23 is given by theDEM simulations (For semicircular arches the same threedomains were recognized and determined by Sinopoliet al. (1997) and Gilbert et al. (2006), with the separatorvalues f = 0.31 and 0.395. The existence of the threecollapsemodes was also confirmed by the discrete elementsimulations of Rizzi et al. (2014) for different arches.)

Figure 22 summarizes the fcr and ftr values dependingon the eccentricity of the groundplan. The critical frictionvalues (two curves with solid markers) indicate that Type1 (“flat”) domes are more sensitive to sliding failure, whilein the case of Type 2 (“high”) domes the critical frictioncoefficients are always lower than that of the sphericaldome (which was found in the present study to beapproximately 0.20). The highest and lowest critical fric-tion coefficients are 0.24 (for Type 1) and 0.14 (for Type2), and both values were obtained at very high eccentricity(angle β around 50°). The two curves with the emptymarkers represent the transitional friction coefficient.For a hemisphere it was 0.23 and for different oval shapesits value varied between 0.31 and 0.16.

Figure 18. Hinging collapse mode of a Type 2 (high) dome withan oculus, with β = 55° ground plan (color scale indicatesdisplacement magnitude).

0

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8

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12

14

0.10 0.20 0.30 0.40 0.50 0.60 0.70

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wal

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ss (

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) [%

]

Friction coefficient f [-]

Type1 - β = 10° Type1 - β = 20° Type1 - β = 30°Type1 - β = 40° Type1 - β = 50° Type1 - β = 60°Type2 - β = 10° Type2 - β = 20° Type2 - β = 30°Type2 - β = 40° Type2 - β = 50° Type2 - β = 60°β = 0°

Figure 19. The necessary minimal wall thickness normalized by the equivalent radius (expressed in %) as the function of the appliedfriction coefficient (solid lines: Type 1 domes; broken lines: Type 2 domes).

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Consequently, if the friction coefficient is aboveapproximately 0.3, every dome is in the range of purelyrotational hinging collapse modes, and tendencies simi-lar to those observed in the case of 0.7 friction coeffi-cient can be recognized. This can be seen in Figure 23,where the minimal wall thickness (normalized by the

equivalent radius) as the function of the eccentricityangle β is shown for different friction coefficients ran-ging from 0.3–0.7. Curves belonging to Type 1 domeswith different friction angles (circular markers) aresimilar to each other. They indicate that slightlyeccentric domes need higher thickness, and very

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-]

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Type 1 - f_transitional Type 2 - f_transitional

Type 1 - f_critical Type 2 - f_critical

Figure 22. Critical and transitional frictional coefficient as the function of the eccentricity of the groundplan.

Figure 20. Mixed collapse modes: (a) Type 1 (flat) dome with β = 50° ground plan and (b) Type 2 (high) dome with β = 30° groundplan (color scale indicates displacement magnitude).

Figure 21. Sliding collapse modes: (a) Type 1 (flat) dome and (b) Type 2 (high) dome, both with β = 50° ground plan (color scaleindicates displacement magnitude).

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eccentric Type 1 domes are more favourable than sphe-rical or slightly eccentric Type 1 domes. The curvesbelonging to the Type 2 domes (diamond markers)are also similar to each other; they show that domeswith large eccentricities need smaller wall thicknessthan spherical or slightly eccentric Type 2 domes.

Since the friction coefficient can safely be assumedwell above 0.3 in all practical situations (apart fromvery special extreme cases), the diagrams in Figure 23offer a possibility for the practicing engineer to assessthe geometrical safety of an actual oval dome. Thethick broken lines (the upper and the lower of whichbelonging to the Type 1 and to the Type 2 domes,respectively) give a conservative estimation of thenecessary wall thickness, depending on the eccentri-city of the groundplan. This estimated minimal wallthickness can be compared to the actual thickness ofthe analyzed real dome to appraise the safety of thestructure.

5. Conclusions

After a careful calibration procedure, several 3DECsimulations were done on dome models with differentintrados shapes and wall thickness. The main advantageof using a discrete element code instead of developingan ad-hoc theoretical model like those in some of thequoted references is that the discrete element simula-tions could serve as virtual experiments: the historyleading to failure or reaching the equilibrium could befollowed with a step-by-step time integration scheme.The main disadvantage of using DEM is that the calcu-lations are very time-consuming: on an average PCseveral hours may be needed for a given dome tocollapse or to become equilibrated.

The following conclusions can be drawn from theresults.

● In order to receive a close approximation ofHeyman’s classical theory with rigid blocks, thenumerical control parameter kN should be at least1011–1012 N/m3 and the perimeter of the ground-plan should be subdivided at least into 30 sections.(The value of kN should be rescaled for modeldimensions differing by orders of magnitude fromthose applied in this article.) Models with smaller kNrequire slightly larger wall thicknesses to be stable;however, results based on realistic contact stiffnessesof mortared joints (kN = 1010–1011 N/m3) differ onlywithin a few percentage from the results belongingto theoretically infinite kN. Domes with smaller kNrequire larger wall thickness to be stable: the classicaltheory is slightly on the unsafe side.

● The simulations on hemispherical domes withrigid elements confirmed the conclusion of Lau(2006) that with hoop forces (caused by the fric-tional resistance along the horizontal joints) theminimal wall thickness is definitely smaller thansuggested by Heyman. The discrete element simu-lations gave an even smaller minimum wall thick-ness than the value found by Lau: 0.037 times theradius was received for hemispheral domes. Notethat this value was measured for 35° frictionalangle (f = 0.7). In the analyzed range of the fric-tion coefficient between ftr ≤ f ≤ 0.7 the value of fonly weakly affects tmin.

● Type 1 oval domes need larger wall thicknesses thanhemispherical domes except for geometries withvery high eccentricity. On the other hand, apartfrom the region of low eccentricities (i.e., nearly

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Type 1 - f = 0.3 Type 2 - f = 0.3Type 1 - f = 0.4 Type 2 - f = 0.4Type 1 - f = 0.5 Type 2 - f = 0.5Type 1 - f = 0.6 Type 2 - f = 0.6Type 1 - f = 0.7 Type 2 - f = 0.7

Figure 23. The necessary minimal wall thickness normalized by the equivalent radius (expressed in %) as the function of theeccentricity of the groundplan.

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hemispheral domes) Type 2 domes can balance theirweight with smaller thicknesses than hemisphericaldomes. Depending on the eccentricity of thegroundplan, both Type 1 and Type 2 domes mayrequire a lower total volume of masonry than ahemispherical dome to cover the same area.

● While in most cases it was found that for realisticfrictional resistance the hinging failure mechanismwas basically the same as for hemispherical domes,in the case of Type 2 domes having an oculus anew hinging failure mode was revealed for largeeccentricities of the groundplan.

● For low frictional resistance (f being below ftr whosevalue was found to be approximately 0.16–0.31depending on the shape of the intrados), combinedsliding-hinging collapse modes occur and the mini-mal wall thickness becomes strongly sensitive to f.Decreasing the frictional resistance even further, acritical value fcr is reached which varies from 0.14–0.24, depending on the exact geometry of the intra-dos: at this critical value the necessary minimal wallthickness tends to infinity and a pure sliding failureoccurs for any wall thickness.

● Conservative approximations on the necessary wallthickness were given in Figure 23 for the practicingengineer, to provide a tool for appraising the geome-trical safety of an actual oval dome. After measuringthe eccentricity of the groundplan and decidingwhether the intrados has a Type 1 (flat) or Type 2(high) geometry, the wall thickness of the analyzedreal dome can be compared to the approximatednecessary value.

Funding

This research was partially supported by the OTKA(Hungarian Research Fund) under grant no. 100770.Software 3DEC was provided by Itasca CG under theEducation Partnership Program for which the authorsalso express their gratitude.

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