Experimental investigation for the friction evaluation in

Int. J. Masonry Research and Innovation, Vol. 1, No. 1, 2016

Experimental investigation for the friction evaluation in the masonry structures R.S. Olivito* Department of Civil Engineering, University of Calabria – 87036, Rende (CS), Calabria, Italy Email: [email protected] *Corresponding author

M. Esposito Department of Civil Engineering, University of Calabria, via G. Amendola, 25 – 88831, Scandale (KR), Calabria, Italy Email: [email protected]

N. Totaro Department of Civil Engineering, University of Calabria – 87036, Rende (CS), Calabria, Italy Email: [email protected] Abstract: In the scientific literature on historic static masonry, the term ‘friction’ is sporadically mentioned to justify certain phenomena: “removing friction, neglecting friction,…”. Recent studies have shown that in some static situations the friction factor can play a crucial role in the equilibrium, considering the intrinsic value of the gravitational force of traditional vector systems. In this regard, the thesis and the experiments in De Zarlo (2008) and in Lanzino (2008) are an interesting attempt to highlight the importance of the role of friction in masonry structures. This work wants to study both from the point of view of experimental and theoretical the assessment of the friction forces occurring in the equilibrium of the masonry elements. Only in the arches and domes made with the technique of overhanging, the frictional forces play a hidden but decisive role. Starting by some heuristic experiments this paper advances a series of results and observations leading to the investigation of a theory on static overhanging domes that is independent of the membrane theory. This paper points out the complex problem of friction in masonry structures having recourse to simple experimental models aiming to the determination of friction, also using the Coulomb theory on friction. Keywords: experimental test; friction; masonry structures. Reference to this paper should be made as follows: Olivito, R.S., Esposito, M. and Totaro, N. (2016) ‘Experimental investigation for the friction evaluation in the masonry structures’, Int. J. Masonry Research and Innovation, Vol. 1, No. 1, pp.27–47. Copyright © 2016 Inderscience Enterprises Ltd.

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R.S. Olivito et al. Biographical notes: Renato Olivito is a Full Professor of Structural Engineering (ICAR/08) in the Department of Civil Engineering at the University of Calabria. He was a member of the following associations: AIMETA, AIAS, IABSE, SEM; and the AIAS Management Committee. He is the national co-coordinator of a CNR research plan entitled Mechanical Behaviour of Monumental Constructions. His main research interests are Mechanics of Materials and Structures, with particular reference to Composite Materials. Further research interests concern the study of the mechanical behaviour of masonry structures subjected to in-plane and out-of-plane static loads and to cyclic loads; the study of durability of masonry and reinforced concrete structures reinforced by FRP materials, delamination of FRP materials from masonry structures, monitoring and control techniques of FRP applications to existing structures. Another area of research is the study and analysis of historical and monumental buildings with specific reference to the recovery and structural strengthening. Mario Esposito completed his graduation with honours in Civil Engineering from the University of Calabria. He has a good knowledge of graphic design that he acquired from the university and through his practical work experiences. He worked in civil engineering especially in the field of masonry buildings by exploiting the knowledge university. His area of interest is in the experimental aspects of masonry structures. Nicola Totaro retired as an Associate Professor of Structural Engineering from the University of Calabria. His main research interests were structural safety, non-linear analysis of concrete structures, mechanics of masonry, and structural problems of rehabilitation. He has released a lot of effort in the testing of masonry structures for the understanding of some aspects of structural mechanics.

1

Introduction

Masonry behaviour is characterised by poor tensile strength (Como, 2010; Olivito, 2003; Di Pasquale, 1996; Sparacio, 1998; Heyman, 1982). While for one-dimensional structures, such as arches, friction does not contrast the opening of the cracks between the joints, the opposite occurs for vaulted and domed masonry structures. In general, arches crack because of a relative rotation between two segments arranged across a lesion (Figure 1). Through the cracked section, therefore, the axial force continues to be transmitted and, consequently, the creep resistance remains at a high shear. In the case of a domed or vaulted structure, cracking occurs displaying different features. It may occur in a dome in solid bricks, and similarly in a vaulted structure, under the form of meridian fractures; these are a consequence of the overcoming of tensile strength by the stresses of hoop agents along the parallels in the lower area of the dome. Upon cracking the bricks must slide one against the other (Figure 2); this sliding is thwarted by the resistance to friction which develops on the horizontal beds subject to sundial compression (Giuffrè, 1986; Timoshenko, 1940). The sliding implies a tensile strength that is neither dependent on the adhesion strength between mortar and bricks, nor on the tensile strength of the mortar. Cracking can, therefore, only take place once the frictional strength has been overcome: this occurs, as a rule, due to the spread of moisture within the body wall that gradually reduces the frictional strength in the masonry, and depletes the mortars, but it

Experimental investigation for the friction evaluation

29

may also be caused by the onset of sudden dynamic actions. This explains why in the vaulted and /or domed structures, cracking occurs, as a rule, in a sudden way or over a longer period from the acting loads, and only with a permanent tensile stress. In the scientific literature on historic static masonry, the term ‘friction’ is sporadically mentioned, to justify certain phenomena, “removing friction, neglecting friction,”. The same modern formula of the shear strength of the sections in bending (NTC 2008) (14 gennaio, 2008): fvk = fvko + 0.4 σn

(1)

cautiously refers to friction, where 0.4 is a mean value of safety, unusually independent from masonry types. This paper points out the complex problem of friction in masonry structures through simple experimental models aiming to the determination of friction. It also emphasises the theory of Coulomb on friction for a better understanding of this issue. Figure 1

The opening of a lesion between two sections of an arch retains sliding strength

Figure 2

Meridian compression contrasts meridian cracking due to friction among brick rows (see online version for colours)

2

The law of Coulomb friction

The classical theory of elastic contact surface was advanced by Hertz. The analysis of Hertz is valid for points of linear contacts, starting from some hypotheses such as the homogeneity and isotropy of the solids in contact, elastic deformations within certain limits, and in the absence of sliding friction forces between two solids. Only the action of the normal force allows establishing the size of the deformed area and the value of the crushing. It also allows inferring the law of distribution of the pressures in the contact zone (Tabor, 1975; Greenwood, 1992). The friction between solid bodies is referred to as Coulomb friction (Coulomb, 1821). According to Coulomb, the friction factor f depends on the features of the materials in contact and on the conditions of the surfaces in contact. It neither depends on normal forces, nor on the extension of the contact surfaces or on the shape of the conjugate surfaces. It does not even depend on the relative sliding velocity. The phenomenon of friction is extremely complex from a physical point of view and,

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therefore, it recoils from accurate modelling. For example, experimental results showed that the friction coefficient is not independent from velocity (as it is assumed in Coulomb’s model), but it evolves, as in Figure 3, after a sharp decrease (Coulomb, 1821), from zero velocity (friction static) to a very limited velocity; the friction coefficient rises with the increase of velocity. For velocities greater than a given value, the friction coefficient remains initially constant, and subsequently it tends to decrease along with velocity. Figure 3

Friction trend as a function of velocity

The resistance to the relative motion of two surfaces in contact is due to a set of interacting phenomena. Thus, the tangential component of the contact force is given by the sum of several factors (Tabor, 1975; Greenwood, 1992; Coulomb, 1821): Ft = Ft1 + Ft2 + Ft3 + Ft4

(2)

where: Ft1 is the force required to counteract the adhesion bonds (micro-junctions), Ft2 is the force required to produce viscoelastic deformation, Ft3 is the force required to remove geometrically interfering asperities, Ft4 is the force required to produce plastic grooves. The value of each component is a function of the materials in contact, the characteristics of the surfaces, the presence or absence of lubricants, and operating conditions. The contributions to the frictional force by Ft3 and Ft4 are mainly due to surface roughness, resulting from technological processing. Experimental tests have shown that the friction factor also varies considerably, under equal physical-chemical conditions of the surfaces, depending on: contact time and sliding velocity, thus highlighting that the coefficient of dynamic friction between surfaces in relative motion, is generally lower than the static friction. The friction factor also depends on mean contact pressure, temperature of the interface, lubrication conditions, and atmosphere, especially in the case of polymers (Tabor, 1975). As concerns the physical explanation of friction, Coulomb states that there are two possible causes for friction: “the interlocking of surface asperities, which can be released only by bending, breaking, lifting one over the other, or we must suppose that the molecules of the two surfaces in contact exercising, because of their closeness, a cohesion that must be overcome to produce movement: only experience will allow us to decide on the reality of these different causes” (Coulomb, 1821). Coulomb concludes that only the former cause may well explain experimental results. Cohesion has a very limited impact on friction because it acts proportionally to “the number of contact points or area of surfaces”, while friction is


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almost always independent from the area. Coulomb, therefore, proposed an interpretation of friction in terms of surface asperities - which interlock, to a greater or lesser extent, depending on the load - and which deform laterally under tangential stress (Figure 4). Coulomb compares these asperities to the roughness of two horse hair brushes and the relationship between them, ascribing everything to the shape of the particles forming the surfaces and to the inclination of the tangent plane in the contact points. Coulomb admits that there is a failure plane along which, under extreme conditions, a portion of the axially loaded pillar slides with respect to the other, and solves the problem of defining the position of the plane. The force opposing the sliding movement is formed by two components, namely the cohesion of the material, assumed to be constant, and the internal friction, which is proportional to the normal force. In terms of stresses: |τ| = c + σ tan φ

(3)

where τ is the shear stress acting on the sliding surface at the time of rupture, and σ is the corresponding normal stress; constants c and φ being the cohesion and angle of friction of the material, respectively. The friction constant n, hypothesised by Coulomb (Coulomb, 1821) is the result of: tan φ = 1/n Figure 4

(4)

Model of roughness according to Coulomb

Source: Coulomb (1821)

3

Instability of the masonry system

Starting from the arch examined in the experiment carried out by Lanzino (Lanzino, 2008) according to which the stability analysis of a 45° angle section on a 1:50 scale of the Treasury of Atreus, the analysis pointed out that the equilibrium in this system cannot occur without the contribution of friction forces. In fact, the various blocks are not glued together, but they are simply placed one upon another (Figure 5). This was the starting point for the intuition that the masonry cell could contain frictional forces. (Figure 6). The concepts of stabilising moment (Mstab) and overturning moment (Mrib), for the equilibrium condition are introduced, thus having: Mstab > Mrib

(5)

From Figure 6 it could be inferred that the stabilising moment is equal to: Mstab = H h + γ h α L L/2

(6.1)

where it is considered unit thickness of the block, γ = 19 Kg/cm3 is the specific weight, h = 10 cm is the height, and L = 25 cm is the length. Following of that, it is possible to obtain the total weight of the block. Mstab = H h + (αW) αL/2

(6.2)

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R.S. Olivito et al.

while the overturning moment is equal to: ⎡ L (1 − α ) ⎤ M rib = β Wd olim + ⎡⎣W (1 − α ) ⎤⎦ ⎢ ⎥ 2 ⎣ ⎦

(7)

where dolim is the equilibrium break-even point. Figure 5

Stability analysis of a 45° angle section on a 1:50 scale of the Treasury of Atreus (see online version for colours)

Figure 6

Scheme of masonry cell (see online version for colours)

By making reference to Figure 6, L is the length of the block; αL is the overlapping portion of the two blocks in contact; x is the position of load application; and H = γW; P = βW; γ and β are the variable parameters measuring P and H with respect to the weight of the block. The condition that it wants to focus on is that of the indifferent equilibrium obtained from the equality between these two moments, which also represents the equilibrium to the rotation around the point P1, namely: Mstab. = Mrib Hh + (αW )

(8)

αL 2

⎡ L (1 − α ) ⎤ = βWd olim + ⎣⎡W (1 − α ) ⎦⎤ ⎢ ⎥ 2 ⎣ ⎦

(9)


33

This leads to an equation in which the weight W can be elided, thus obtaining a formula which may be found in the framework of the ‘Theory of Proportions’.

γh+

α 2L 2

= β d olim +

L (1 − α )

2

2

(10)

Explaining γ as a function of β, the following equation is obtained:

γ =β

2 2 d olim L ⎡ (1 − α ) − α ⎤ + ⎢ ⎥ h h ⎢⎣ 2 ⎥⎦

(11)

Explaining γ in function of β, H increases in function of F, i.e.:

γ= β

d olim L ⎡ 1 ⎤ + ⎢ −α ⎥ h h ⎣2 ⎦

(12)

which can be assimilated to the formula: y = mx + n

(13)

When parameter α varies the load laws γ = f(β) are in a bundle of parallel lines, illustrated in Figure 7. For α < 1/2 the block tends to tip over, and, therefore, the H reaction is also required when the applied force F is void, while for α > ½, the block slightly protrudes, and, therefore, it can bear a maximum load Fo = βoW before the H reaction takes place. Figure 7

Horizontal load as function of parameter α (see online version for colours)

To determine the system’s limit state of sliding, it is defining the quantity η as the ratio:

η=

Strength frictional sliding υa (W + βW ) = H γW

(14)

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R.S. Olivito et al.

finally obtaining:

η=

υa (1+ β ) γ

Condition η ≥ 1 must be satisfied; therefore where γ =

(15)

υa (1 + β ) : with η = 1 a straight η

line is obtained

γ = υ a (1 + β ) = υa + υa β

(16)

on the same plane. The non-indefinite overcoming of the limit state of sliding is that there are no intersections between straight lines (12) and (16), namely that: i.e. dolim < υ0 h is sufficient, where υ0 is the limit value of the friction coefficient in the condition of limit state of sliding. However, the same result can be obtained through a different process. In order to reach the condition of equilibrium described by Equation (7), referring to Figure 6, it has: Px ≤ H x h + WdG

(17)

where dG is the distance from the barycentre of the upper block than the inner edge P1. The Hx reaction is obtained through the sliding condition: Hx < (P + W)v. Substituting Hx < (P + W)v in the previous equation and assuming the load P as a multiple of the weight W, it is resulted: P = αW

(18)

αW ( x −ν h ) ≤ W (ν h − dG )

(19)

By highlighting x, it is obtained: d 1⎞ ⎛ x ≤ ⎜ 1 + ⎟ (ν h ) − G α α ⎝ ⎠

(20)

The latter formula shows that for x ≤ d olim , where x is the point of load application, the d term ± G , – where the sign ± takes into account whether the centre of gravity of the

α

block top is internal or external with respect to the edge P1 of the lower block – is neglected in so far as values of α → ∞ vanish, and, therefore, it is understood that for x ≤ d olim the load P grows indefinitely. Variable α represents the load multiplier. Therefore, dox ≤ d olim represents the condition of non- overcoming of the limit state of sliding. The present study aims to analyse the maximum overhanging between two consecutive blocks, in order to better understand the building technique of overhanging constructions. Starting from Figure 6, the different steps leading to the calculation of the cantilever are highlighted. The first step is to write the equilibrium equation with respect to the point P1, thus obtaining the Hx reaction: Hx = P

d olim h

(21)


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We can summarise that x ≤ dolim represents the condition of non-overcoming of the limit state of sliding. At this point it is worth investigating the maximum overhanging between two consecutive blocks (Figure 8a). Writing the equilibrium equation with respect to the point P1, thereby obtaining the Hx reaction, followed by the equation of equilibrium to the horizontal translation from which it obtained the maximum value of Hx that is equal to: H(x,max) = Pν

(22)

By equating the two formulas, it is found that the condition satisfying the equilibrium is: ν ≥ dolim/h

(23)

from which: dolim < νh

(24)

identifying what can be considered as a limit value of do:dolim = ν h. Thus, for values of x < dolim, the weight P is not determining, provided that it is elided, while for values of x > dolim, for any value of P there is a component Hx which balances the system (Figure 8b). This is how it was possible to obtain the limit overhanging value of an upper block with respect to the lower one. If it is considered the case of three blocks in which the length of the overhanging is always lower than dolim, it had that - when the two blocks are considered as a whole: with two blocks B and C considered as a whole adding the piece D, and the block A in lug xA < dolim - the system is resistant to an infinite force P (Figure 9). When the three blocks are analysed separately, starting from block A and moving downwards, and considering that the different swings are always lower than dolim, also taking into account the possible force applicable to block B, following that the applied load is always infinite because xi < dolim (Figure 12). Since it is not possible to assume a force P = ∞; firstly it is necessary to find a preload force Po corresponding to a block of weight Wp which, in turn, corresponds to a horizontal force Hx that under the weight of Wp the entire system of blocks can be kept under an equilibrium condition and then continue the loading phase. Figure 8

Maximum overhanging between two consecutive blocks

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R.S. Olivito et al.

Figure 9

Blocks B and C considered as a whole (see online version for colours)

The horizontal force Hx corresponds to a shear that is transmitted along the entire system of blocks (Figure 11). Considered, at this point, the case in which xi > dolim. Hx for xi > dolim has already reached its maximum value equal to Hx, max = P ν, and substituting Hx with Pv in the equation of equilibrium to the rotation around the point P1, it is possible obtained: P x = P ν h, then: x = ν h. But x can also be seen as x = dolim + dx then: x = d o lim + d x = ν h and remembering however, that dolim = ν h, finally: dx = 0. In this case, the contribution of the force caused by the weight of the block has been neglected, which now will be taken into consideration. In case of x > dolim and dG > 0, where dG represents the distance of the barycentre of the upper block with respect to the point P1. From the equation of equilibrium to the rotation around the point P1 (Figure 8), the stabilising moment Mstab and the overturning moment Mrib are obtained. The maximum value of Hx is equal to: H x ,max = ( P + W ) *ν

(25)

From the indifferent equilibrium one can obtain: P* ( x − ν*h ) = W*( ν*h ± d G )

(26)

inferred as a ratio of the applied load: P=

W * (ν * h ± dG )

( x − = *h )

(27)

Knowing that x = dolim + dx, and substituting it in the previous equation:

P=

W (ν h ± dG )

(28) dx By plotting the external load P as a function of dx the diagram shown in Figure 10 was obtained.

Experimental investigation for the friction evaluation Figure 10 Three blocks are considered separate

Figure 11 Constant shear transmitted along the entire system of blocks

Figure 12 Load P as function of dx (see online version for colours)

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R.S. Olivito et al.

It is possible to infer that, in order for the value of the external load P to be positive, one d should check that ν * h − dG > 0 i.e. h > G if dG > 0; or, in the case of dG < 0, a positive

ν

value of P (P > 0) is always found. Taking into consideration the condition of indefinite stability of masonry overhanging arch, it will be found that, given the position of the external load (Figure 8), if d0 < ν0 * h there is a position of the load P such that it can grow indefinitely without any alteration of the equilibrium of the system.

4

Materials and experimental equipment

To perform the laboratory tests, an electromechanical testing machine performing a compression test (Figure 13a) was employed. The compression test was performed under displacement control obtaining the load-displacement diagram (Figure 13b). The lava stone specimen consists of two 600 × 30 × 150 mm blocks placed at the basis of the system fixed at the ends by HEA 200 metal profiles and two pairs of 250 × 100 × 100 mm blocks symmetrically placed opposite one another in projection over a length of 105 mm (1–3, 2–4). The experimental tests were performed by placing the centre of gravity of the lower blocks three and four at a distance of 20 mm from the inner edge of the underlying building blocks (one and two) (Figure 14). In order to apply the load, two stainless steel rods with a 6 mm diameter (Ф) were placed on the upper part of the system, glued in different positions by means of a two-component glue. The choice of applying the load along a line was dictated by the fact that, as previously noted, friction does not depend on surfaces contact area. This also enabled to better observe the raising of the blocks. The latter parameter was chosen as a reference to analyse and monitor the evolution of the system during the application of the load. For each interpretation of the raising points A and B, carried out by centesimal comparators (Figure 15), the values recorded during the increasing phase were examined, up to a maximum value of 5 mm. Another 250 × 100 × 100 mm block, together with an HEA 100 steel beam, was placed above the system, to exert a preload of 120 N to the specimen and to constitute a rigid element to uniformly transmit the load. The test was conducted with a load velocity equal to 0.5 mm/min, and for each test the maximum load reached was detected. Figure 16 shows the values of the load recorded during testing by placing the rods at a distance of 70 mm from the edge of the lower block. These tests have allowed to determine an initial value of the friction coefficient of the system of blocks, and by means of the Equation (27) it was possible to plot the theoretical curve of the external load as a function of position x (Figure 16). For each position of the applied load a minimum of two tests, and a maximum of three, were carried out. For the position dx equal to 70 mm and 950 mm, tests two and three were carried out, while for the remaining positions only test two was conducted. All these tests allowed the determination of the corresponding value of the friction coefficient. Table 1 shows the values of the friction coefficients experimentally obtained with position x = 70 mm. If the external load P is applied to x < ν h, the system is theoretically able to withstand an infinite load. To this end, a test was carried out by placing the hinge load 60 mm from the edge and its values are shown in Figure 17. Figure 18 shows some images of the test where the rotation of the blocks following the application of the load can be observed. During the test, with increasingly higher values of the load, a progressive breaking of the edge was observed between the two blocks in


39

the key (Figure 19); the breaking occurred for a value of the load equal to 50,000 N. The breaking of the edge was subsequently tested for axial compression on two specimens taken from the 100 × 100 × 100 mm cracked block and two extremely high values of Rc compressive strength, i.e. 144,0 and 111,4 N/mm2 were obtained (Figure 20). Assuming a width of the contact area equal to 1÷1,5 mm, the area of the contact surface will be equal to 100 to 150 mm2, corresponding to a stress equal to σn = 100 ÷ 150 N/mm2, which is in line with the results obtained through the experimental tests. Figure 13 a) Testing machine; b) load-displacement diagram (see online version for colours)

Figure 14 a) Scheme test pattern; b) different position of applied load (see online version for colours)

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Figure 15 Centesimal comparator for reading of raising (see online version for colours)

Figure 16 Diagram load - displacement (see online version for colours)

x = 7 cm

Table 1

Calculation of the friction coefficient

Test

Pbl Kg

P cell Kg

Preloading Kg

Ptot Kg

Ptot/2 Kg

dx cm

dG cm

h cm

Hx Kg

Ν

P.1

7

143

11,66

154

77

7

2

10

52,6

0,625

P.2

7

129

11,66

141

70

7

2

10

47,9

0,619

P.3

7

189

11,66

200

100

7

2

10

68,7

0,641

Experimental investigation for the friction evaluation Figure 17 Load-displacement diagram for x = 6 cm (see online version for colours)

Note:

Values negative of displacement and load are due to the phase of adjustment at the beginning of each test of electromechanical testing machine

Figure 18 Rotation between the blocks (see online version for colours)

Figure 19 Breaking of the edge in key (see online version for colours)

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Figure 20 - Axial compression test (see online version for colours)

5

Results and conclusions

The experimentally obtained values of the friction coefficients range between 0.597 and 0.705. Therefore, the theoretical curves of the loads to be applied were plotted, in order to obtain the specific friction coefficient using Equation (27). Figure 21 shows: Figure 21 Diagram load - displacement for fixed values of the friction coefficient and experimental results (see online version for colours)


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1

The theoretical curves corresponding to the geometric set up of the experiment, considering the friction coefficient as a variable, with values 0.6, 0.625, 0.65, 0.7.

2

The experimental results relating to the positions of the load on the overhanging at x = 70, 80, 90, 95 mm.

3

The result out of the frame P = 50116 N, obtained in the test with x = 60 mm, where the theoretical value is P = ∞.

From Figure 21 a satisfactory agreement between the theoretical curves and the experimental data is clearly noticeable. In particular, the following was observed: a

the hyperbolic load - span overhanging trend, testifying a drastic reduction in the vertical load bearable at the increase in distance x

b

the asymptotic soaring of the potential load when the approaching value ν h from the right-hand side

c

the theoretical possibility of endless loads for spans lower than ν h.

It is striking that for x = 60 mm (slightly lower than the limit value estimated at 65 mm with ν = 0.65) a load of 50,000 N is reached. A hidden horizontal force, obtained by the equilibrium to the rotation, of over 15,020 N was transmitted along the edge; hence a great guarantee is obtained in these overhanging structures, by limiting the number of projections to xi ≤ νhi. From the experimental stage, it is possible to draw several conclusions, such as the existence of frictional forces of great intensity. The experiment demonstrates the existence of equilibrium configurations of masonry blocks systems, which imply the presence of other types of forces, beyond traditional gravitational, frictional forces that, under certain circumstances, can reach extremely high values (Figure 22). Each of the two blocks has a rotation axis with respect to the inner edge, the vertical reaction will cross the centre of the edge and will be equal to 25,06 KN plus the weight of the block (70 N). An overturning moment equal to 150,220 KN cm is then quantified. The equilibrium to the rotation can only be restored by a 10 cm torque arm produced by a contact compression force and by a reaction of friction equal and opposite to the value15,020 KN. Since the horizontal translation equilibrium is respected in the 15020 N = 0.5977 as it was indeed assumed, experiment, it must also be true that ν real > 25130 N through the other tests. The transmission among successive blocks of the friction forces is also assumed. A frictional force developed by symmetry or contrast to a certain height, is transmitted constantly to the ground (Figure 23). This phenomenon is equivalent to a hidden shear stress. Approaching the ground, supposing the friction coefficient to be constant, the normal stress necessarily grows; and friction resistance above the shear to the various sliding planes occurs. Finally, the existence of critical overhanging ν hi was assumed, below which extremely large vertical loads may be exerted; and large friction forces may be developed simultaneously, the former being tilting, and the latter being stabilising, with respect to a given rotation hinge (Figure 24). By integrating our results with other results from the traditional static theory, it is possible to develop a general theory to understand and check the equilibrium of false arches, as well as to design new overhanging arches. Few theories and static interpretations deal with overhanging arches, often referred to as ‘false arches,’ (Heymann, 1999; Heyman, 1976; AA. VV, 1990; Como, 2005; Orlandos, 1966–1968; Como and Grimaldi 1983; De Zarlo, 2008;

44

R.S. Olivito et al.

Cavanagh and Laxton, 1981); such theories are all based on the balance of gravitational forces, and there is no reference to attractive forces. In Figure 25 the equilibrium of the arch is ensured if R falls within the support base (AA. VV, 1990). Some treatises on the analysis deal with issues where the numerical series, applied to the problem of overhanging buildings, lead to striking results (i.e. covering long spans) which, however, are in contrast with evidence from theoretical and experimental statics. The previous analysis enabled to infer the stability conditions for a local and general isolated semiarch: the condition of local stability can be expressed by affirming that each added dressed stone must be in equilibrium with respect to the one below, i.e. its centre of gravity must fall within the area of support on the underlying dressed stone. This condition is necessary but not sufficient. As for the condition of General Stability, the centre of gravity of the entire progressive semi-arch must fall within the support foot. Each added overhanging dressed stone which causes the centre of gravity of the entire underlying solid to move upwards and approach the vertical to the edge of the supporting foot. This second criterion greatly limits the possibility of exceeding greater spans, contrary to the indications in the harmonic series (Figure 26). The existence of the ν h critical overhanging allows one to derive the minimum height for a false arch of a given span (L) (Figure 27). If the projection of each block is brought to the value ν h, the result for simple similar triangles is the following: * : hi :ν *hi = htot

L 2

(29)

from which: * = htot

L 2*ν

(30)

It follows that the ‘false arch’ cannot be lower than h*tot, as in this case, at least one projection will exceed the value ν*hi for geometric reasons. Therefore, it may bear very small loads in the edge, as the experiments have proven. The collapse during the construction phase of a false arch, designed according to these principles, is inevitable. Therefore, a necessary, but not sufficient condition, to achieve a false arch is that the height h must be greater than or equal to h*tot. The above-mentioned condition is not sufficient because, in most geometric cases, the principle of global equilibrium for gravitational forces is not respected. In these cases the stabilising moment produced by the contact force between the two segments at the top is essential. In order to achieve overhanging arches that do not meet the second criterion, when the centre of gravity of the figure falls outside the support of the basic building block, the balance must be restored by a horizontal force in key. The value of this horizontal force Hx, achievable by means of a pre-load plus friction affecting the two touching blocks in key, can be obtained through the balance, being Hx h the stabilising moment, while the overturning moment is given by the total weight of the figure multiplied by its torque arm in relation to the dressed stone at the bottom. M stab = H x *h M rib = Wtot *dG

Experimental investigation for the friction evaluation so that H x ≥ Wtot *

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dG . h

Hx is the minimum value that allows the equilibrium. This is the prerequisite for the transmission of the Hx force as a constant sheer, along the whole height of the arch, because crisis of the element does not occur due to the scrolling. With the same friction coefficient along the sliding surfaces, if no sliding below the segments in key takes place, sliding never occurs because the limit friction force rises along with the increase of the load above, since normal stress increases when going through the keystones. Therefore, at the end of this work, the intended aim was to achieve and determine the friction coefficient experimentally. Figure 22 a) System block; b) the equilibrium of the block key (see online version for colours)

Figure 23 Equilibrium at the sliding of the two protruding semi-arches (see online version for colours)

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Figure 24 Critical overhanging ν hi, below with extremely large vertical loads may be exerted and large friction forces may be developed simultaneously

Figure 25 Gravitational forces (see online version for colours)

Figure 26 Condition of global stability


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Figure 27 Minimum height for a false arch of a given span (L)

References 14 gennaio, D.M. (2008) Nuove Norme Tecniche per le Costruzioni, par. 4.5.6.1. AA, VV . (1990) Architettura in Pietra a Secco, Schena editore, Fasano, BR. Cavanagh, W.G. and Laxton, R.R. (1981) ‘The structural mechanics of the Mycenaean Tholos Tomb’, BSA, Vol. 76, p.109. Como, M. (2010) ‘Statica delle costruzioni storiche in muratura’, in Arane editore, Roma. Como, M. and Grimaldi, A. Analisi limite di pareti murarie sotto spinta; Quaderni di teoria e tecnica delle strutture, Ist. di Tecnica delle Costruzioni, Università di Napoli, p.546. Como, M.T. (2005) L’architettura delle tholoi micenee: analisi degli aspetti compositivi, costruttivi e statici, Tesi di dottorato, Università degli Studi Suor Orsola Benincasa, Napoli. Coulomb, C.A. (1981) ‘Théorie des machines simples’, Memoire de Mathematique et de Physique de l’Academie Royale, 1785, 2nd edition. De Zarlo, E. (2008) ‘Avanzamenti per una teoria delle cupole litiche in aggetto – Tesi di Laurea’, Università degli Studi della Calabria – Relatore Prof. N. Totaro. Di Pasquale, S. (1996) L’arte del costruire, Tra conoscenza e scienza-Marsilio, Venezia. Giuffrè, A. (1986) La meccanica nell’architettura, La statica – NIS, Roma. Greenwood, J.A. (1992) ‘Contact of rough surface’, in Singer, I.L. and Pollock, H.M (Eds.): Fundamental of Friction: Macroscopic and Microscopic Processes, Kluwer Academic Publishers, University of Cambridge, pp.37–56. Heyman, J. (1976) The Stone Skeleton, Cambridge University Press, Cambridge. Heyman, J. (1982) The Masonry Arch, Ellis Horwood Publisher, Chichester. Heymann, J. (1999) Il saggio di Coulomb sulla statica, Hevelius Edizioni, Benevento, pp.121–136. Lanzino, F. (2008) Sperimentazione in scala ed indagini archeostoriche sulle grandi cupole murarie in aggetto per l’avanzamento di una nuova teoria interpretativa – Tesi di Laurea, Università degli Studi della Calabria – Relatore prof. N. Totaro. Olivito, R.S. (2003) Statica e Stabilità delle Costruzioni Murarie, Pitagora Editrice, Bologna. Orlandos, A.K. (1966–1968) Les matériaux de construction et les techniques architecturales des anciensGrecs, De Boccard, Paris. Sparacio, R. (1998) La Scienza e i Tempi del Costruire, UTET, Torino. Tabor, D. (1975) ‘Interactions between surfaces: adhesion and friction’, in Blakey, J.M. (Ed.): Surface Physics of Materials, Vol. II, Academic Press, New York, pp.475–529. Timoshenko, S. (1940) Theory of Plates and Shell, MacGraw-Hill, New York, p.80.