Meccanica DOI 10.1007/s11012-016-0533-9

Equilibrium formulation of masonry helical stairs A. Gesualdo . C. Cennamo . A. Fortunato . G. Frunzio . M. Monaco . M. Angelillo

Received: 8 January 2016 / Accepted: 7 September 2016 Ó Springer Science+Business Media Dordrecht 2016

Abstract In this paper the lower bound theorem of limit analysis for No-Tension materials is applied to study the equilibrium of spiral vaults, modeled as continuous unilateral membranes. The most efficient approach to the equilibrium of a thin shell is the covariant representation proposed by Pucher and adopted in the present study. Statically admissible singular stresses in the form of line or surface Dirac deltas and lying inside the masonry, are taken into account. The unilateral restrictions require that the Airy stress function representing the stress, be

concave. The case study is a helical stair with a central pillar in Sanfelice Palace in Naples, whose structure is a tuff masonry spiral vault. The maps of the stress corresponding to two different stress functions and the safety factors in the two cases are provided. Keywords Masonry  Vaults  Unilateral materials  Airy’s stress function

1 Introduction A. Gesualdo Department of Structures for Engineering and Architecture, University of Naples Federico II, Naples, Italy e-mail: [email protected] C. Cennamo  G. Frunzio  M. Monaco Department of Architecture and Industrial Design, 2nd University of Naples, Aversa, CE, Italy e-mail: [email protected] G. Frunzio e-mail: [email protected] M. Monaco e-mail: [email protected] A. Fortunato  M. Angelillo (&) Department of Civil Engineering, University of Salerno, Salerno, Italy e-mail: [email protected] A. Fortunato e-mail: [email protected]

The present paper is concerned with the application of the unilateral No-Tension model to masonry vaults. The unilateral model for masonry, that, though in a mathematically unconscious way, has been around since the nineteenth century (see Moseley [1]), first rationally introduced by Heyman [2], was divulgated and extended in Italy, thanks to the effort of Salvatore Di Pasquale [3] and other members of the Italian school of Structural Mechanics, such as Romano and Romano [4], Baratta [5], Del Piero [6], Como [7], Angelillo [8]. Due to the analytical complexities involved and the development of computer capacity at the time, this particular constitutive model for masonry structures has been initially applied to the analysis of masonry panels loaded in their plane [6], and several applications based mainly on the fundamental previous papers [9, 10], were proposed. Subsequent papers

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considered numerical analyses in which constitutive models were based on classical failure criteria; a thorough discussion of this issue can be found in Gesualdo and Monaco [11]. Only lately, when the improvements of computing performances could produce significant results implementing the model based on the no-tension model, both research on the development [12] and implementation [13] of the theoretical model and research on the analysis of case studies were done. A research group of the University of Salerno (in which one of the present authors has played a leading part) has contributed to the debate on the unilateral models for masonry and, more generally, to the modelling of the onset of fractures, with a number of recent articles: Fortunato [14] and Angelillo et al. [15]. The formulation of the boundary value problem for unilateral masonry materials, that is Rigid No-Tension materials for which the latent strains (fractures) satisfy a normality condition with respect to the admissible stresses, can be found in the recent paper [16]. Indeed the more efficient tool that can be introduced for applying the unilateral No-Tension model to masonry structures is the systematic use of singular stress and strain fields, within the framework defined by the two theorems of Limit Analysis (see Angelillo et al. [17], for applications of the safe theorem and Angelillo et al. [18], for applications of the kinematic theorem to walls). For what concerns the application of the unilateral model to vaults, the existing more recent literature is rather vast; apart from the production of the school of Salerno, originated by the paper on the Lumped Stress Method [19], and applied to vaults in the papers by Fraternali et al. [20], Angelillo & Fortunato [21], Fraternali [22], and recently by Angelillo et al. in [23], we recall the pioneering work by O’Dwyer [24], and the works by Block [25], Block et al. [26], Vouga et al. [27], De Goes et al. [28], Block and Lachauer [29] and Miki et al. [30]. The case of spiral stairs, treated with a classical elastic membrane model by Calladine in [31], was also considered with the unilateral model by Block in his dissertation [25], by Angelillo in [32] and by Angelillo et al. in [33]. Here, the case study of the stair in Palazzo Sanfelice in Naples, a helical barrel vault made of tuff cut stone with a central circular pillar, is considered as an interesting application of the method.

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2 Analysis 2.1 Geometry With reference to Fig. 1, a vault can be described by its intrados and extrados surfaces, by the geometry of the filling, and eventually by the form and dimension of the top enlargements of the piers sustaining the vault. With the present model, it is assumed that the load applied to the vault is carried by a membrane structure S of thickness s. The geometry of the membrane S is not fixed, in the sense that it can be displaced and distorted, provided that it lays inside the masonry. The surface S is continuous but not necessarily smooth. For the shell surface S a Monge representation is considered. The position x on the middle surface S has Cartesian coordinates given by: x ¼ fx1 ; x2 ; f ðx1 ; x2 Þg;

fx1 ; x2 g 2 X;

ð1Þ

where: X is a two-dimensional connected domain whose boundary oX is composed of a finite number of closed curves, of outer normal n, called the planform of S, fx1 ; x2 g; the curvilinear coordinates on S, are the Cartesian coordinates of S in the planform X, x3 ¼ f ðx1 ; x2 Þ is the rise of the membrane with f 2 C o ðXÞ. The unit vectors associated to the Cartesian reference system are denoted fe1 ; e2 ; e3 g. A three-dimensional view of S is shown in Fig. 1. A few coordinate lines x1 constant and x2 constant are indicated in Fig. 1, whilst on the right a magnified view is shown of a differential element of the shell, this element being bounded by arcs of coordinate lines. The membrane is loaded by the external forces q, given per unit area of S and balanced by the membrane stresses T whose components are depicted in Fig. 1. The natural or covariant base vectors tangent to S, associated to the curvilinear coordinates fx1 ; x2 g, are [21]: a1 ¼ f1; 0; f;1 g; a2 ¼ f0; 1; f;2 g; 1 a3 ¼ ff;1 ; f;2 ; 1g; J

ð2Þ

in which the comma followed by an index, say i, denotes differentiation with respect to xi , and the unit normal m to S is coincident with a3 . J ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ f;12 þ f;22 is the Jacobian determinant, that is

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(a)

(b) dx2 x2 const P

(c)

dx1 st

on x1 c

nst

S

o

e e

2

1

12

S22

T 12

T 11

T 22 q3

T 21

q2

2

t ns co

3

a

S11

S

S21

2

x2

e

a 1

3

x

S12

S11

o x 1 c a3 P

x

S22

p3 S21 p2 p1

x

q1 T

T 22

1

T 12

21

T 11

Fig. 1 Membrane and surface stresses (a), covariant basis (b), shell element and stresses (c)

the ratio between the differential surface area on S and its projection on the planform. Notice that, the basis fa1 ; a2 g is neither unit nor orthogonal, and the reciprocal, or contravariant, base vectors are: 1 f1 þ f;22 ; f;1 f;2 ; f;1 g; J2 1 a2 ¼ 2 ff;1 f;2 ; 1 þ f;12 ; f;2 g: J

a1 ¼

ð3Þ

S11;1 þ S12;2 þ p1 ¼ 0; S21;1 þ S22;2 þ p2 ¼ 0; Sab f;ab  pc f;c þ p3 ¼ 0;

2.2 Membrane equilibrium in Pucher form Here we follow essentially the developments presented in [21], repeating the essential ingredients of the analysis only for completeness. The generalized membrane stress on S is defined by the surface stress tensor T, represented in the covariant base (2) as follows: T ¼ T ab aa  ab ;

ð4Þ

T ab being the contravariant components of T and where summation convention over repeated Greek indices: a; b; c. . . ¼ 1; 2 has been adopted. Some manipulations are needed to transform the contravariant components of stress into Cartesian ones, since contravariant components of stress are useful and convenient, but are non-physical. In equilibrium the divergence of the generalized surface stress T balances the load q ¼ f q1 ; q2 ; q3 g, defined per unit surface area on S: o ðT ab aa  ab Þac þ q ¼ 0: oxc

The most efficient way to describe membrane equilibrium of a thin shell under a load q is due essentially to Pucher [34]. The generalized contravariant stress components T ab on the membrane surface are transformed into projected stress components Sab ¼ JT ab in the planform, so that Eq. (5) projected into the nonorthonormal system fe1 ; e2 ; m ¼ a3 g, after some algebra, becomes:

ð5Þ

ð6Þ

where p ¼ Jq is the load per unit projected area. Using the projected stresses the first two equilibrium equations are identical to those of the plane stress problem. In the case of pure vertical loading, say p ¼ f0; 0; pg, the problem may be solved introducing a continuous Airy stress function Fðx1 ; x2 Þ in the form: S11 ¼ F;22 ;

S22 ¼ F;11 ;

S12 ¼ S21 ¼ F;12 :

ð7Þ

Transverse equilibrium corresponds to the balance of the vertical component of the force p3 ¼ p with the scalar product of the Pucher stress matrix times the Hessian of the function f in the covariant form (63). Adopting the Airy’s stress function, (63) can be rewritten in the form F;22 f;11 þ F;11 f;22  2F;12 f;12 ¼ p:

ð8Þ

2.3 Unilateral membranes A rigid No-Tension material in the sense of Heyman is assumed, so that the following material restrictions are

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imposed: the generalized stress T is negative semidefinite and does no work for the corresponding strain E, that is positive semi-definite: T 2 Sym ;

E 2 Symþ ;

TE¼0

ð9Þ

The first application of Pucher’s transformation for NT masonry vaults can be found in Angelillo and Fortunato [21], where it is shown that, due to the NT constraint, both the surface stress tensor and the matrix of the projected stresses must be negative semidefinite. In terms of the stress function F, this condition can be written: F;11 þ F;22  0;

2 F;11 F;22  F;12  0;

ð10Þ

tractions, on a beam structure with shape oX. In many cases of interest, such as those concerning domes and spiral stairs based on circular planforms, the Monge description of the surface S based on polar coordinates fh1 ¼ r; h2 ¼ hg is more convenient: x ¼ fr cos h; r sin h; f ðr; hÞg;

With Pucher’s approach the generalized stress on the membrane surface is transformed into projected stress in the planform, so that in the covariant base associated to the polar reference system: b1 ¼ fcos h; sin hg;

k1 ¼ fcos h; sin hg; 2.4 Singular stress and equilibrium in the membrane If F is only continuous, it may present folds. In this case the projected stress is a line Dirac delta with support along the projection C of the fold. The Hessian H of F is singular transversely to C, namely it has a uniaxial singular part parallel to the unit vector h normal to C. Correspondingly the directional derivative of F in the direction of h, called Fh , presents a jump. Therefore, the singular part of the Hessian H of F can be written as: ð11Þ

dðCÞ being the unit line Dirac delta on C and DFh the jump of slope along the direction h. Analogously the singular part of the corresponding projected stress (7) is a Dirac delta on C: Ss ¼ dðCÞDFh k  k;

ð12Þ

where k is the unit vector tangent to C. The concavity of F implies the concavity of the fold C. Then DFh is negative and the corresponding projected singular stress concentrated on C is compressive. The equilibrium problem for the unilateral membrane S, in pure vertical loading, consists in finding a concave stress function Fðx1 ; x2 Þ satisfying Eq. (8), with the boundary conditions: dF ¼ hðx1 ; x2 Þ on oX; Fðx1 ; x2 Þ ¼ gðx1 ; x2 Þ or dn ð13Þ being gðx1 ; x2 Þ and hðx1 ; x2 Þ the contact internal moment and shear force produced by the allied

123

ð14Þ

b2 ¼ fr sin h; r cos hg;

ð15Þ

to which is associated the variable orthonormal base:

hence Fðx1 ; x2 Þ is concave.

Hs ¼ dðCÞDFh h  h;

fx1 ; x2 g 2 X:

k2 ¼ f sin h; cos hg;

ð16Þ

the projected stress S can be represented as follows: S ¼ Sab ba  bb ¼ rab ka  kb ;

ð17Þ

where Sab , r11 ¼ rrr , r22 ¼ rhh , r12 ¼ r21 ¼ rhr , are the contravariant components and the physical components of the projected stress in the polar reference system. The equilibrium equations in X are those of the plane stress problem. In particular for the general case of distributed loading defined per unit projected area: p ¼ p1 b1 þ p2 b2  pe3 ¼ pr k1 þ ph k2  pe3

ð18Þ

they are 12 1 S11 =1 þ S=2 þ p ¼ 0;

22 2 S21 =1 þ S=2 þ p ¼ 0;

ð19Þ

where =a stands for covariant derivative with respect to ha . In the case of pure vertical loading the stress components in (19) can be written in terms of the stress function F, as follows:  1 1 S ¼ F;1 þ F;22 ; r  r2  1 1 F;2 ; S12 ¼  r r ;1 11



S22 ¼

1 F;11 ; r2

ð20Þ

where a comma followed by a denotes differentiation with respect to ha . The transverse equilibrium equation reads: Sab f=ab  pc f;c  p ¼ 0;

ð21Þ

and can be rewritten in terms of physical stress components:

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S11 ¼ rrr ;

S22 ¼

1 rhh ; r2

1 S12 ¼ rrh ; r

ð22Þ

with   1 1 f=11 ¼ f;11 ; f=22 ¼ r 2 f;1 þ 2 f;22 ; r r   1 f=12 ¼ r F;2 : r ;1

ð23Þ

Taking into account (21), (22) and (23), the equilibrium Eqs. (19), (21) can be rewritten as: 1 oðrrrr Þ 1 orrh rhh þ  þ pr ¼ 0; r or r oh r 1 oðr 2 rhr Þ 1 orhh þ þ ph ¼ 0; r 2 or r oh    1 of 2 2  o r oh o f 1 of 1of þ rrr 2 þ rhh þ 2rrh or r or r 2 oh2 or of 1 of ¼ p:  pr  ph or r oh

ð24Þ

For pure vertical loading, taking into account that:   1 oF 1 o2 F o2 F rrr ¼ þ 2 2 ; rhh ¼ 2 ; r or  r  oh or ð25Þ o 1r oF oh ; rrh ¼  or Equation (243) in polar coordinates becomes:

    1 oF 1 o2 F o2 f o2 F 1 of 1 o2 f þ 2 2 þ þ r or r oh  or2 or 2 r or r 2 oh2   1 of o 1 oF o r oh  2 r oh or or ¼ p:

ð26Þ

As above mentioned, the projected stresses must be negative semi-definite and the corresponding Airy stress function be concave in case of NT materials. Some equilibrium solutions for axially symmetric domes can be found in a recent paper on masonry vaults [23] and for some special spiral stairs in [32, 33].

3 The case study As a case study we consider the helical stair in Sanfelice Palace in Naples shown in Fig. 2. This double circular staircase, whose structure consists of a tuff masonry barrel vault, connects the courtyard with the middle and the first floor of the building. The helical vault of the stair has a semi-circular cross section. The axisymmetric surfaces, from which the spiral surfaces of the intrados and extrados of the vault can be generated, are approximated by the graphs of the functions:

Fig. 2 The helical stair in Sanfelice Palace

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The function f describing the spiral surface is then:

yintr ¼ r cos he1 þ rsenhe2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ R2s  ðRp  Rs þ rÞ2 e3 ;

f ðr; hÞ ¼

yextr ¼ r cos he1 þ rsenhe2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ðRs þ sÞ2 ðRp  Rs þ rÞ2 e3 ; ð27Þ while the simple equation that we choose to describe the axisymmetric generator of the membrane surface contained inside the vault is: y ¼ r cos he1 þ rsenhe2  h  1 2 þ R d  R2p  2Rp ðRs  rÞ 1 R2s s i  þ2Rs r  r 2 ðRs  d1 þ d2 Þ e3 ;

ð28Þ

where r and h are the polar coordinates with respect to the center of the stair, Rp is the internal radius of the stair, Rs is the half span of the cross section of the vault, d1 and d2 are the distances of the membrane surface from intrados at springing and key respectively, and s is the thickness of the vault structure (Fig. 3). We point out that the real geometry of the problem requires to add the term Hh 2p to the third component of y in (28).

Fig. 3 Geometry of the helical vault, sectioned axonometric view (a), section and geometrical parameters (b)

(a)

1 2 ½R d1  ðR2p  2Rp ðRs  rÞ þ 2Rs r R2s s h  r 2 ÞðRs  d1 þ d2 Þ þ H ; 2p

H being the inter-floor height (Fig. 4). The physical components of the Hessian of f ðr; hÞ, appearing in (26) are: o2 f R s  d1 þ d2 ¼ 2 ; 2 or R2s 1 of 1 o2 f ðRs  d1 þ d2 ÞðRp þ Rs  rÞ þ 2 2¼2 ; r or r oh R2s r   o 1 of H : ¼ or r oh 2pr Two distinct stress functions F 1 ðr; hÞ and F 2 ðr; hÞ have been considered as solutions of the equilibrium problem, defined by Eq. (26), in case of constant vertical load per unit projected area. Considering the simple case of constant load per unit projected area, and assuming the separation of variables in the expression of the stress function F (that is on assuming that Fðr; hÞ ¼ aðrÞbðhÞ) the partial differential Eq. (26) can be solved symbolically. By solving with the DSolve routine of Mathematica, two types of solutions are obtained: F1 ¼ p

R2s ðRp þ Rs Þr þ 12 R2s r 2 ; 4ð Rs  d 1 þ d 2 Þ

ð29Þ

(b) Central pillar

Lateral pillar

y

spiral membrane spiral vault

δ2

s r

o

123

δ1

θ Rp

Rs

x

Meccanica Fig. 4 Membrane surface (a) and its superposition with extrados and intrados (b), particular (c)

R2s ðRp þ Rs Þr þ 12 R2s r 2 4ð Rs  d 1 þ d 2 Þ HR2s # 2pðRs  d1 þ d2 Þr 2 2pðRs d 2 1 þd2 Þr ; e k HR2s

F2 ¼ p

ð30Þ

where, in the solution type F 2 ðr; hÞ, an integration constant k appears. A discussion on boundary conditions, on the choice of the constant k; and of its influence on the vault thrust is given in the final part of this section. A picture of the generator of the membrane surface is reported in Fig. 5, together with the stress functions defined in (29) and (30). For the geometrical parameters of the Sanfelice stair we have (see Fig. 6a): Rp ¼ 0:25 m; Rs ¼ 0:90 m; H ¼ 5:70 m; s ¼ 0:15 m; d1 ¼ 0:30 m; d2 ¼ 0:135 m; ð31Þ

where the parameters d1 ; d2 are chosen in geometrical manner. A parabola, dashed in red in Fig. 6a, contained in the spiral vault, is considered in such a way to leave a minimum distance from the intrados and the extrados equal to 1.5 cm. On assuming a uniform stress distribution across the thickness, a shell of minimum thickness 3 cm, internal to the vault, is considered. On adopting the values given in (31), the stress functions (29) and (30) become respectively, see Fig. 5b, c:   F1 ¼ 0:3401p 0:9315r þ 0:405r 2 ; ð32Þ h  i h F2 ¼  1:0003kr2 er2 þ 0:3401p 0:9315r þ 0:405r 2 ;

ð33Þ where the k and p are expressed in MN m and MPa respectively (notice that if p [ 0 the load is downward), r in meters, so that the dimension of the stress functions is MN m. The stresses associated to the function F1 ðr; hÞ are:

Fig. 5 Axisymmetric generator surface of the membrane surface (a), stress function F1 ðr; hÞ (b) and F2 ðr; hÞ (c)

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considered spiral vault

δ2

s

y

s

y

15 cm

4.2 cm

δ2

9.5 cm

15 cm

30 cm

δ1

o

x 90

25

Heyman vault

smin

13.5 cm

δ1

o

(b)

25

42.5 cm

(a)

x 90

Fig. 6 Considered spiral vault (a) and Heyman vault with parameter for the Heyman safety factor (b)

R2 ðRp þ Rs þ rÞ ; rrr ðF 1 Þ ¼ p s 4ðRs  d1 þ d2 Þr 2 Rs ; rhh ðF 1 Þ ¼ p 4ð Rs  d 1 þ d 2 Þ

rrh ðF 1 Þ ¼ 0; ð34Þ

and numerically, using the data (31) and putting p ¼ 2:33  102 MPa (chosen in such a way that a maximum compressive membrane stress of 0:02 MPa m and an associate stress of 0:67 MPa are obtained, as shown in Eq. (36)) relations (34) become: 0:00642ð1:15 þ rÞ ; r rhh ðF 1 Þ ¼ 0:00642; rrh ðF 1 Þ ¼ 0;

rrr ðF 1 Þ ¼ 

with r expressed in meters and stresses in MPa m. While for the stress function F2 ðr; hÞ the related stress components are: rrr ðF 2 Þ ¼ 

  HR2 sh pR2s Rp þ Rs þ r  ke2pðRs d1 þd2 Þr2 4ðRs  d1 þ d2 Þr

H 2 R4s þ 4pHR2s ðRs  d1 þ d2 Þhr 2 þ 8p2 ðRs  d1 þ d2 Þ2 r 4 ; 2pH R2s ðRs  d1 þ d2 Þr 4 2

rhh ðF 2 Þ ¼ 

H 2 R4s h2 þ pHR2s ðRs  d1 þ d2 Þhr 2 þ 2p2 ðRs  d1 þ d2 Þ2 r 4 ; pH R2s ðRs  d1 þ d2 Þr 4 

HR2s h 2pðRs d1 þd2 Þr 2

HR2s h  pðRs  d1 þ d2 Þr 2 ; pðRs  d1 þ d2 Þr 4

ð35Þ adopting the data (31), the load p ¼ 2:33  102 MPa of the previous case and k ¼ 1:45  103 MN m, the stresses in (35) are:

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where r is in meters and stresses in MPa m. The level curves and 3D stress maps are reported in the following Figs. 7, 8 and 9 for both stress functions examined. It must be noted that in the cases represented by F 1 ðr; hÞ and F 2 ðr; hÞ, the maximum compressive stress is 0:02 and 0:018 MPa m (maximum compressive principal Cauchy stress). On assuming that the stress is approximately constant over a shell whose thickness doubles the minimal distance from the extrados (see Fig. 6 a), the stress per unit area can be estimated as follows: rrr jminF1 0:02 ¼ 0:67 MPa ¼ 2ðs  d2 Þ 2ð0:015  0:0135Þ rrr jminF2 0:018 ¼ 0:60 MPa; ¼ ¼ 2ðs  d2 Þ 2ð0:015  0:0135Þ ð36Þ

 F1 ¼ rj

HRs h pR2s  2ke2pðRs d1 þd2 Þr2 4ðRs  d1 þ d2 Þ

rrh ðF 2 Þ ¼ ke

7:38 rrr ðF 2 Þ ¼ 103 6:42  r  

1:45 h  h2 þe r 2:9 þ 4 þ 2:9 2 ; r r  

hðr2 þ 2hÞ  h2 3 rhh ðF 2 Þ ¼ 10 6:42 þ 2:9e r 1 þ ; r4 2h  r2  h2 rrh ðF 2 Þ ¼ 1:45  103 e r: r4

 F2 rj

then, taking conservatively as the yield compressive stress of the tuff material:rY ¼ 2:5 MPa, the strength safety factors (that we may call the Galilean safety factor in the sense of Huerta [35]), related to the same external load p ¼ 2:33  102 MPa, are:

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Fig. 7 Stresses [MPa m] rrr (left), rhh (center) and rmin (right) for the stress function F 1 ðr; hÞ

Fig. 8 Stresses [MPa m] rrr (left), rhh (center) and rrh (right) for the stress function F 2 ðr; hÞ

rY 2:5 ¼ 3:75; ¼  F1 0:67 rj rY 2:5 ¼ 4:17 sG jF2 ¼ ¼  F2 0:6 rj

sG j F 1 ¼

ð37Þ

The geometrical safety factor, in Heyman’s sense (see [36]), can be calculated by finding the ratio between the thickness of the real vault and the minimal thickness of a homothetic fictitious vault containing

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Meccanica Fig. 9 Stress rmin [MPa m] related to stress function F 2 ðr; hÞ

Fig. 10 Axisymmetric generator surface of the membrane surface (a), stress function F1 ðr; hÞ (b) and F2 ðr; hÞ (c)

the equilibrium membrane structure S, as shown in Fig. 6b. For both cases above considered, the geometrical safety factor is (see Fig. 6b): sH ¼

s smin

¼

15 ¼ 3:57: 4:2

ð38Þ

The different value of the safety factors in relations (36) provides a first simple motivation to take the more complex stress function F 2 ðr; hÞ. Other issues are the fact that the function F 1 ðr; hÞ gives to a stress state and to a thrust, both constant along the spiral vault, while the ones related to function F 2 ðr; hÞ are variable, in the central pillar (r ¼ 0:25 m) and in the external curved wall (r ¼ 0:90 m), with the spiral angle h, as shown in Fig. 10a. The same picture also indicates that, for the chosen value of the integration constant k ¼ 1:45  103 MN m, the effect on the thrust is lower for the upper part of the staircase. Figure 10b reports the shear stress at the internal and external boundary which is associated to the aforesaid

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reduction of the thrust force: a torsional effect is produced in the upper part of the well. In Fig. 11 the contour plot of the principal stresses due to stress function F2 ðr; hÞ are depicted in order to verify the relevant constraint (91) is satisfied: The proposed method can also be applied for more complex combined loads using, for example, a class of bounding theorems as reported in [37] and used a s a support for the monitoring of existing structures (see [38]).

4 Final remarks An application of the lower bound theorem of limit analysis for No-Tension materials to spiral vaults has been presented. Statically admissible singular stress fields in the form of line or surface Dirac deltas are considered. The equilibrium is expressed as an extension of the Pucher’s method, so that convenient

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Fig. 11 Principal stresses due to Airy function F2 ðr; hÞ: rmax ðF2 Þ (a), rmin ðF2 Þ (b)

systems of coordinates for the formulation of the stress problem and a concave stress function are assumed. The case study is an helical stair with a central pillar in Sanfelice Palace in Naples, whose structure is a tuff masonry barrel vault. The maps of the stress corresponding to two different stress functions and the safety factors in the two cases are provided.

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