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

The masonry vaults: Geometry definition and possible approachesto the static analysis

O. CorbiDepartment Structural Engineering, University of Naples Federico II, Naples, Italy

ABSTRACT: The paper focuses on the analysis of masonry vaults: on one side it addresses some of thegeometrical features involved in the treatment of shells, which usually implies not negligible difficulties inhandling the vaults’ problem due to the differential nature of their geometry, and, on the other side, it outlinespossible approaches to the static analysis of masonry vaults. To this regard, the overall static approach shouldbe conceived in such a way to allow the selection of membrane stress surfaces able to equilibrate the appliedloads and to satisfy the admissibility conditions of the masonry material. The paper, then, deals with these twofeatures, outlining a general approach for analytically handling the problem of static analysis of vaults.

1 INTRODUCTION

The forum about the preservation of the architecturalheritage has always been object of a big interest frommany fields; in the last decades many disciplines havedifferently contributed to the topic.

According to the commonly agreed opinion fromthe scientific community, any intervention for therehabilitation of an historical monumental buildingcannot be outlined autonomously; the safeguard objec-tive makes impossible to separate the architecturalfrom the constructive-structural feature.

Any preservation strategy requires the deep under-standing of the behaviour of the structure, usuallymade of masonry material, and of its past history.

Figure 1. Correspondence between the two metrics for representing the mid-surface of a shell of general form relevant to thereference systems (Ox1,x2,x3) and (Ou1,u2).

This is why it would be of fundamental relevanceto have at one’s disposal proper analytic tools for per-forming reliable static analyses of masonry structures.In the following one reports some discussion aboutthe still open research problem of masonry vaults,outlining possible future developments.

2 GEOMETRY OF SHELLS OF GENERALFORM

2.1 Shells of general form

A shell of general shape may be represented as a reg-ular surface in the space (Fig. 1), and described as asubset of points P whose position is identified by any

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single-valued and continuously differentiable vectorfunction r(u1,u2) depending on two variables u1 andu2 belonging to a bi-dimensional space U.

The components of r(u1,u2) are the coordinatesof the point P with respect to any reference frameOx1x2x3.

By denoting the first derivative of r(u1,u2) withrespect to ui by ri(u1,u2), and, marking by “×” thevector product, it is assumed that

The equation of the surface, as regards to cartesian co-ordinates xi in the reference Ox1x2x3, can be expressedin the parametric form

After suppressing the parameters (u1,u2), the surfaceequation can be put in the implicit form

One can observe that the vectors r1 and r2 are parallelto the tangents (characterized by unit vectors t1 andt2) of the coordinate curves u1 = const and u2 = constrespectively.The normal line to the surface at the point(u1,u2) of the surface (with unit normal vector N),whose direction is given by the vector product (r1 ×r2), has the following equation (Fig. 2)

where λ is a variable parameter defining points on thenormal, µ the length of the normal and N the unitsurface normal vector at P.

Figure 2. Surface normal vector N and tangent vector t of the surface curve � in P.

Substitution of ui = ui(t), in Eq. (2) givesr = r[u1(t), u2(t)], which is the equation of a curve �on the surface in Eq.(2), as shown in Figure 2. Thevector

coincides with the tangent to this surface curve .The tangents to all surface curves (Mitrinovic

& Ulcar 1969), which pass through a fixed point(u1,u2) of the surface, lie in the tangent plane(Fig. 3) to the surface at that point, which con-tains r1(u1,u2) and r2(u1,u2). After denoting by R theposition vector of any point other than P on the tan-gent plane, the equation of the tangent plane is thefollowing

Figure 3. Tangent plane to the surface at P.

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The tangent surface of a curve �, r = r(s), is the surfacegenerated by lines tangent to �. It has the equation

where λ is a parameter and t is the unit vector tangentto �. The moving of a straight line (generator) in anydirection along a curve (directrix) generates a ruledsurface. In the case when every tangent plane to asurface is tangent to a generator, the surface is calleda developable surface.

The differential of arch length ds of a curve ui = ui(t) (i = 1, 2) on the surface r = r(u1,u2) may be cal-culated by means of the first fundamental form of thesurface, as follows

where gik = ri · rk.Using the summation convention, the expression of

the first fundamental form of the surface (Mitrinovic& Ulcar 1969) can be written in the form

For the surface in the form x3 = z(x1,x2), assumingu1 = x1, u2 = x2, one has that

The arch length s of the curve ui = ui(t) (i = 1, 2)between the points t1 and t2 is then given by definiteintegration of ds in Eq. (9) between the two bounds t1and t2, as

On a surface with the first fundamental formgikduiduk, let consider the two curves u = u(t) andv = v(t). If they intersect at a point, then the tangentsto these two curves at the point of intersection definean angle α between the unit tangent vectors tu and tv,which is defined by (Fig. 7)

where

is the determinant of the metric tensor g containing thegik components, which is a symmetric second-ordertensor of the type

thus representing the first fundamental tensor of thesurface.

Let consider the net formed on the surface by thecoordinate curves u1 = const and u2 = const (Fig. 4).

One can evaluate the area dA of the surface elementsd� as

Thereafter the area A of the region �(u1,u2) of thesurface r = r(u1,u2) can be reassembled and definedin the form

representing the double integral on � of the elementsd � with area dA.

Figure 4. The net formed on the surface by the coordinatecurves u1 = const and u2 = const.

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Figure 5. Normal section curve �n and surface curve �:relevant normal vectors and curvatures.

Since the normal vector N to the surfacer = r(u1,u2) has the direction of the vector r1 × r2,one has

Let consider now a normal section �n of the sur-face at P (Fig. 5), i.e. the intersection of a surfacewith a plane which passes through the surface nor-mal at a point P of the surface (thus containing theunit vector normal N to the surface at that point). Thenormal curvature of the normal section at P is denotedby kn = 1/ρn with ρn the radius of curvature.

According to the Meusnier Theorem, one can relatethe radius of curvature ρ of the generic surface curve�[ui = ui(t) (i = 1,2)] with curvature k = 1/rρ at thepoint P and the angle θ between the normal vectors nand N at P as follows

where hik = r1 × r2 · rik = − ri · Nk,with rik = ∂2r/∂ui∂uk and Nk = ∂N/∂uk.The quadratic differential form hikduiduk in Eq. (19)

is the second fundamental form of the surface andthe symmetric second-order tensor formed on thehikcomponents

represents the second fundamental tensor of the sur-face, with determinant h = det(h) = h11h22 − (h12)2.

At the a certain point, i.e. the umbilical point, onehas that g = λh, and the curvature is the same for all

Figure 6. Principal curvatures of the surface at P.

normal sections. If this is not the case, then the normalcurvature kn at the point P has two extreme valueswhich determine two tangents at P.

The stationary values of kn are the principal curva-tures of the surface at P and the corresponding tangentsare the principal directions at P.

The principal curvatures are denoted by k1 and k2and are the roots of the equation

Denoting by φ the angle between any direction at Pand the principal direction at P corresponding to “1”(Fig. 6), the normal curvature kn in the given directionat P is defined by Euler’s formula, as

Actually by considering the product, K = k1k2 = h/gof the principal curvatures, which represents the Gaus-sian curvature of the surface at P, one may check if thesurface is developable, since, in this case, the Gaussiancurvature is zero at ali points of the surface.

2.2 Shells of revolution

With reference to shells of revolution (Baratta & Corbi2007b), one may refer to the generic element ABCDidentified by the cut of the surface along vertical planespassing through two couples of adjacent parallels andmeridians, as shown in Fig. 7. The generic point onthe shell may be located by its Cartesian coordinatesx1,x2 in the reference system (O x1,x2,x3), whilst thesurface equation is x3= z(x1,x2).

After denoting by r the vector detecting the positionof the generic point of the shell, whose projection onthe x1x2 plane is denoted by r, one can move from

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Figure 7. Shell of revolution: Cartesian and polarco-ordinates.

Cartesian coordinates x1,x2 to polar coordinates θ, r inthe horizontal plane, leaving <z> as the ordinate axis,as shown in Figures 7 and 8, where the position of theedge A of the element ABCD of the shell is identified.Moreover a point on the meridian is located by theangle ϕ.

Therefore the following relations exist betweenpolar and Cartesian coordinates at the generic point

where the variable z depends on the shape of the merid-ian line of the shell, and it is thus dependent on r, i.e.z = z(r).

After calculating the first order derivatives of r, onecan evaluate the first fundamental form of the surfacegik dxi dxk (with dxi, dxk the variables), which involvesthe coefficients gik = ri · rk and, with reference to thegeneric surface element, calculate the length of thearch curves along the parallel dsθ and meridian dsrdirections respectively, and its area dA (Fig. 9) asfollows

By the second fundamental form of the surface(Ugural 1999) hik dxi dxk (with dxi, dxk the variables),involving the coefficients hik = r1 × r2 · rik√

g , where in the

Figure 8. Normal section of a shell of revolution andprincipal curvatures.

Figure 9. Generic shell element of dimension dA.

specific case r1 = rr , r2 = rθ, after some algebraicoperations and further developments, one may obtain,according to the Meusnier theorem (Mitrinovic &Ulcar 1969, Ugural 1999), the principal curvatures k1and k2 solving the equation |h − k g| = 0 with respectto the curvature k, as

where ρ1 and ρ2 are the principal radii of curvature.Since the normal vector N of the surface at point P

has the direction of the vector rr × rθ, one has

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whence, after some algebraic operations, one gets

In the case of shell of revolution the local principalradii of curvatures on the surface can be immediatelyidentified. With reference to Figure 8, at some pointon the shell the meridian has a radius of curvatureρ1, which is the radius of curvature of the small archlength ds.

The patch of the shell cut out by two meridians andtwo parallel circles has, however, a second radius ofcurvature ρ2; the normals at adjacent points on twomeridians intersect on the axis of the shell, and ρ2 isthe length of the normal from the point on the shell tothe axis.

Thus the shell surface is described by the fourparameters θ, ϕ, ρ1, and ρ2, but these quantities arenot all independent and, for example, with referenceto Figure 8, r = ρ2 sin ϕ.

Figure 9 shows the small element of theshell and it is seen that the dimensions of theelement are

which can be shown to be in agreement to whatpreviously found in Eq. (24).

By comparison with Eq. (24) one has

After some further developments, one gets

Figure 10. Local equilibrium of the generic shell element.

whence

which yield the same result as in Eq.(25).

3 POSSIBLE APPROACHES TO THE STATICANALYSIS

3.1 Set up of the problem

When addressing problems relevant to the static anal-ysis of masonry vaults one should handle the twoproblems of equilibrium with applied loads an admis-sibility with respect to the masonry material which,as well known, exhibits a very reduced capacity ofresisting tensile stresses (Baratta 1984,1991, Baratta &Corbi 2005, 2006, 2007).

So the above reported treatment can be used, onone side, for describing the real vault geometry, i.e.the geometry of its extrados and intrados surfaces z1and z2 bounding its thickness and the geometry ofits mid-surface, and, on the other side, for describingthe geometry of its membrane surface z, envelop-ing the local membrane stress resultants in equilib-rium with the applied loads (Baratta & Corbi 2006a,2007).

In this latter case both the equilibrated local stressresultants and the membrane surface z are unknown.

So the approach for the static study of the vaultcould consist of searching for the membrane surfaceboth satisfying equilibrium and material admissibility.

As regards to equilibrium, considerations aboutstresses and external loads acting on the element

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of Figure 10 yield, as usual, three equations ofequilibrium.

The external load which may be acting on the ele-ment has been resolved into the three perpendicularcomponents pϕ, pθ, pn., while stresses are those shownin the figure, yielding, after some algebraic develop-ments, the three equilibrium conditions respectively inthe tangential direction (direction of θ), in the tangen-tial direction (direction of ϕ), and in the radial direction(i.e. normal direction), as follows

In the case when one considers an axial-symmetricloading the shear stress resultant Nθϕ is zero every-where and one has

When referring to the equilibrium conditions undernot axial-symmetric Eqs (32) and axial-symmetricload Eqs (33), one should remember that, as men-tioned in the above, the involved geometric quantities(angles and radii of curvature) depend on the variablez expressing the equation of the membrane stress sur-face.Therefore, besides the unknown membrane stressresultants, one also should search for the membranesurface expressions both satisfying equilibrium (in theforms of Eqs (32) or (33) and material admissibility(which implies that the surface must be contained inthe vault profile z1 ≤ z ≤ z2).

After individuating the set of admissible membranesurfaces (in terms of admissibility and equilibrium),a possible approach in order to find the solution interms of stresses is to set up a Complementary Energyproblem to be formulated as a kind of extension tomasonry vaults of the classical analogous energeticapproach for linearly elastic structures.

This kind of approach has been already successfullyapplied to masonry arches modeled by the Not Resist-ing Tension (NRT) assumption (Baratta 1984,1991,

Baratta & Corbi 2005a, 2006, 2007, Heyman 1977),showing to produce in its numerical implementationresults in excellent agreement with experimental data(Baratta & Corbi 2005b, 2006b).

In order to undertake this approach the expressionof the Complementary Energy embedded in a masonryvault element should be evaluated. In case of NRTassumption, one should consider that the generic ele-ment appears to be partially resistant; this means that,denoting by u the distance of the membrane surfacefrom the upper profile of the vault element, the posi-tion of the neutral surface (i.e. characterized by nullstresses) is located at a distance of 3u from the extradosof the element.

For a bi-linear distribution of the normal stress onthe volume element, when neglecting the shear stresscomponent, the elastic energy can be calculated asfollows

with Ar, Aθ the areas of the compressed part of thevolume element respectively in the two cross sectionscontaining dsr and dsθ, er and dGr respectively theeccentricity of the solicitation centre with respect tothe mass centre Gr, and the distance of the mass cen-tre from the neutral plane in the two cross sections,and E and ν respectively the Joung modulus and thePoisson coefficient.

The final Complementary Energy functional Cexpression is given by adding to the elastic energyterm L, the energy related to the work developed bythe constraint reactions R, as C = L + R.

The solution in terms of stresses can then besearched for by numerically implementing the min-imization of the Complementary Energy functionalunder the condition that the solution itself is respectfulof the above individuated equilibrium equations and ofadmissibility.

As an alternative an inverse approach can be suc-cessfully outlined also leading to the analytical expres-sions of the membrane stress functions for the differentvault typologies. In this case the spatial surface equi-librium problem can be coupled to the one projectedin a selected plane according to the classical Pucker’sapproach. One can then refer to the 2D-projected prob-lem, and introduce a stress function which is built up insuch a way to a priori satisfy some of the equilibriumconditions.

The key of the problem lays then in the coupling ofthe stress function with the membrane function.

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So the stress function should be modeled in such away to recognize, after imposing the remaining equi-librium conditions and admissibility, the membranestress functions that can be assumed as the membranesurfaces relevant to different vault geometries.

Some practical applications relevant to the casesof the barrel vaults with indefinite length, the barrelvaults confined at their extremity cross-sections, andthe spherical domes are developed in details in papersby the author (Baratta & Corbi 2007a, 2007b; Barattaet al. 2008).

To this regard, one should emphasize that this isnot at all a trivial objective to be pursued. Even sim-ple applications, which means the specialization of thedescribed approach to vaults of simple shapes, requirea pretty hard work for setting up and checking theexpressions of the relevant stress functions.

As a counterpart, this approach allows to obtainsolutions in analytical form, which represents its majorresult.

Thereafter the solution for each case can be com-pleted by setting up the energetic approach as men-tioned in the above.

Definitively this approach is pretty complex sinceit requires to hypothesize and test a number of analyt-ical functions for any vault typology, which representsa consistent effort even for simple vaults geometries;anyway it has the big advantage and original researchresult, never available in literature before, to producethe explicit analytical expressions of the stress andmembrane functions for the single cases, i.e. to givethe solutions in analytical form.

4 CONCLUSIONS

The paper presents some of the fundamental featuresinvolved in the study of masonry vaults.

The first part of the paper is devoted to show themain difficulties encountered in the description of thevaults geometries and to introduce the differential toolsfor handling this problem.

In the second part one outlines some discussion onpossible approaches to the analysis of masonry vaults,also as regards to the possibility of obtaining analyti-cal expressions of the solutions in terms of membranesurfaces for different vault typologies.

REFERENCES

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Baratta, A. & Corbi, O. 2005a. On variational approachesin NRT continua. J. of Solids and Structures. 42:5307–5321.

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Baratta, A. & Corbi, O. 2006a. Analysis of masonry vaultedsystems: the barrel vaults. In Structural Analysis of His-torical Constructions, New Delhi.

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Baratta, A., Corbi, I. & Corbi, O. 2008. Stress Analysis ofMasonry Structures: arches, walls, and vaults. SAHC08,VI International Conference on Structural Analysis ofHistorical Constructions, Bath, UK.

Heyman, J. 1977. Equilibrium of shell structures, OxfordUniversity Press 1977.

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