Limit analysis of vaulted structures strengthened by an innovative technology in applying CFRP

Construction and Building Materials 145 (2017) 336–346

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Limit analysis of vaulted structures strengthened by an innovative technology in applying CFRP Laura Anania ⇑, Giuseppe D’Agata Department of Civil Engineering and Architecture, University of Catania, Italy

h i g h l i g h t s
A new approach to lower bound limit analysis of CFRP reinforced masonry arch is discussed.
The basic idea of the proposed approach is to carry out a series of lower bound limit analysis.
The shear strength is constant and given by the Mohr-Coulomb friction law from the previous step.
The methodology results in very good agreement with available original experimental data.

a r t i c l e

i n f o

Article history: Received 6 June 2016 Received in revised form 22 February 2017 Accepted 27 March 2017

Keywords: Arch Vaults Masonry structure Strengthening FRP Limit analysis Lower bound Seismic retrofitting Non-associative friction flow rule

a b s t r a c t Masonry vaults represent one of the most seismic vulnerable element in an ancient building. It also generally does not possess an adequate capability of redistribution of the seismic action among the walls of the buildings. Nowadays the preservation of the historic vaults is devolved to the application of advanced materials and new technologies on traditional structures. The evaluation of their effects has assumed a major relevance. From the analytical point of view, the plastic (limit) analysis methods are now commonly used to determine the ultimate load–carrying capacities of masonry arch The aim of the present paper is to discuss the efficiency of an analytical models validated by means of experimental investigations carried out on masonry arches reinforced with an innovative technology proposed by the same author and based on the use of CFRP strips, with a special configuration called as ‘‘X-wrap”. This configuration allows the resulting CFRP reinforced ribbed vault to assume the necessary strength and membranal and flexural rigidity so as to ensure the aforementioned seismic action redistribution capability and to avoid local collapse of the vault. A theoretical prediction of ultimate strength was derived in agreement with the occurrences observed during the experiments (masonry crushing, FRP rupture, debonding, sliding along the mortar joint). To this aim, a novel incremental step-by-step lower bound limit analysis approach was developed taking into account for the shear failure mechanism at each mortar joint. The shear strength is evaluated by the Mohr-Coulomb friction law for the mortar joint and by other nonlinear Italian Code relations for CFRP X-Wrap reinforcement. In the approximated incremental analysis process the current value of the shear strength, depend on the compressive stress resulting from the previous step. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Masonry vaults are usually highly vulnerable to seismic hazard, as demonstrated in the recent past by the collapses of many masonry churches during the Umbria Marche (1997–1998), l’Aquila (2009) and Amatrice (2016) earthquakes. It also generally does not possess an adequate capability of redistribution of the seismic action between the masonry piers of the buildings. ⇑ Corresponding author. E-mail address: [email protected] (L. Anania). http://dx.doi.org/10.1016/j.conbuildmat.2017.03.212 0950-0618/Ó 2017 Elsevier Ltd. All rights reserved.

So, nowadays, the theme of the seismic retrofitting of historical buildings, in presence of these structural curved components, is playing an increasing role, in the field of the civil engineering, The need of designing efficient and non-invasive strengthening techniques in high hazard area seems to be almost immediately clear to all technicians involved in the reconstruction. Recently, the use of innovative methods based on the use of light but high resistance materials (as FRP strips) appears a more powerful solution than the conventional retrofitting methods such as external reinforcement with steel plates, surface concrete coating and welded mesh, which have proven to be impractical, time expensive

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and add considerable mass to the structure. In this paper, only a brief resume of same relevant results obtained by means of a new methodology for approaching the limit analysis of the reinforced masonry vaults, limited to the case of the non-isolated XWRAP system, is reported. But, despite the always increasing diffusion of those innovative strengthening technique only few numerical method are nowadays available. Furthermore, when dealing with vaulted masonry structures, structural analysis becomes more complex because of the interaction between membrane and flexural actions. From the analytical point of view, with the development of computer based numerical methods, the FEM method is largely used to simulate the complex behaviour of the masonry arches and vaults even when associated to innovative material (Mahini et al. [1,2]). However, this kind of analysis is often very time expensive and the input parameters of the simplified theoretical constitutive models are difficult to calibrate experimentally. For this reason, other method is actually proposed in literature, for the structural analysis of the masonry vaults and arches. Among these, the one that is becoming increasingly important is the limit analysis (Heyman J. [3]). The method proposed by Livesley [4], is based on the resolution of the equations of static equilibrium for under a vertical, horizontal punctual load or a bending moment applied over each rigid voussoir of the arch. Later Sinha [5], Ferris and Tin-Loi [6], Orduna and Lourenço [7], Milani et al. [8], gave a more accurate contribute to limit analysis taking also into account the friction of the joint as well as including the approach based on the thrust network analysis (Block and Lachauer [9], Fraternali [10]). So, limit analysis with a strong reduction in the number of material parameters is capable of providing limit multipliers of loads, failure mechanisms. As a matter of fact, the reinforcement by means of FRP should require a non linear complex damaging models capable of taking into account both the typically brittle behaviour due to FRP delamination and the tensile cracking of the mortar joints, while limit analysis is based on the assumption of a perfect plasticity of the constituent compounds (Baratta & co. [11]). Anyway, limit analysis is an useful instrument for the design purpose as suggested by CNR-DT 200 Italian Code [12] and [13]. In the follow, an approximated methodology for approaching the lower bound limit analysis of masonry arch (and barrel vaults), reinforced by CFRP is illustrated. It is based on the central idea of carrying out a series of lower bound limit analyses. In each of them shear strength is assumed constant and given by the MohrCoulomb limit at the corresponding compression force, in each mortar joint, resulting from the previous step.

(a) Model of the barrel vault tested, dimensions in [mm];

337

In this way the associated flow rule holds at each step for the shear failure mechanism without dilatancy. The analytical approach is also capable of taking into account the shear mechanisms according to the dowel effect at the interface between masonry and CFRP. So, an incremental procedure was carried out in which for each single step the limit analysis problem was solved considering a constant value of the shear resistance derived taking into account the normal stress at the previous step according to Coulomb law. The proposed limit analysis method was implemented in Mathematica software to carry out numerical simulation of the experimental test conducts on the scaled barrel vault samples, described above (See Fig. 1). To this aim, a new technology for the retrofitting of masonry vaults by means of CFRP, proposed by Badalà et al. [14] and named X-WRAP is studied under analytical and experimental investigation. The basic idea of the ‘‘X-Wrap” system is to give high stiffness to the CFRP strips by wrapping it around a high resistance mortar core cast and molded in site (Fig. 1). This reinforcement, placed at the extrados of the vault, allows preserving the precious frescoes that commonly, in the historic masonry buildings, adorn the intrados. In addition, it presents a higher resistance against delamination respect to the canonic FRP – strip based reinforcement, as considered in the technical document CNR-DT 200/2004, for two orders of reasons: – The extrados curvature positively affects the delamination resistance due to the onset of compression normal stresses. See Basilio et al. [15], D’Ambrisi et al. [16], Malena & De Felice [17], Fagone et al. [18]; – The global bending stiffness of fiber-reinforced rib increases its resistance against shear induced out of plane peeling (Fig. 2); It is also under evaluation the effectiveness of another version of the proposed innovative reinforcement system, named Isolated-X-Wrap. This variant provides for the interposition of a neoprene bearing between the extrados of the vaults and the fiber reinforced concrete rib. This bearing, with a thickness of a few millimeters, continuously placed throughout the length of the leader of the vault and having the same width of the reinforcement, is expected to allow the vault to freely deformed under daily thermal-hygrometric gradient. On the contrary, remaining unchanged the sewing effect on the collapse hinges, and the cinematic confinement effect and thus the resistance against accidental and seismic loads. Some first results about this system is available in Anania [19].

(b) Ω-Wrap reinforcement

Fig. 1. a) Model of the barrel vault tested, dimensions in [mm]; b) X-Wrap reinforcement.

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The proposed limit analysis formulation relies on a number of assumptions necessary for the validity of the limit theorems:

Fig. 2. Shear induced out of plane reinforce peeling (by Valluzzi M.R., Modena C2001 [23]).

- No tensile strength for both the masonry and of the mortar rib of the X-Wrap, with rigid-perfectly plastic behaviour in compression; - Bricks have limited compressive strength; - No compression and flexional rigidity of the FRP reinforcement, with rigid-perfectly plastic behaviour in tension; - Transversal plane sections conservation; - Delamination of the CFRP is not considered explicitly. - Constant CFRP reinforcement ‘‘dowel effect” shear strength; - Plane geometry; It is well known, the interface CFRP-masonry behaviour is very far from being elastic-perfectly plastic, rather it’s strongly non linear with softening, as is typical for fracture cohesive mechanics. Therefore, the results obtained have to be considered only an approximation of the real load-bearing capacity of the masonry vaults reinforced by CFRP. In spite the necessary aforementioned considerations, the methodology results in very good agreement with available original experimental data.

Fig. 3. Static scheme of the barrel arch studied.

1.1. The geometry The assessment of the effectiveness of this new technique has implies the numerical and experimental investigation on several series of scaled model, as better illustrated in Anania et al. [14]. In this paper, the limit analysis procedure is referred only to the case of barrel vaults with constant thickness and circular generatrix, reinforced by the X-Wrap technique, although it is also valid for arch and vaults of generic generatrix and variable thickness. Further details of the application of this method to the case of unreinforced and classically CFRP strip reinforced vaults can be found in Badalà et al. [20,21]. The method is based on limit analysis and uses the static theorem approach to determine the ultimate capacity of the structures analysed by means of an optimization process.

The static scheme of the arch is described in Fig. 3. The geometry is described by decomposing the entire masonry vault (or arch) in a series of equally spaced short segments (or fictitious ‘‘voussoirs”) limited by sections oriented perpendicularly to the axis (Fig. 4). Fig. 5 shows the forces acting on the elementary fictitious voussoir; where N, V, M indicate the characteristics of internal forces, W is the weight of the voussoir (including X-Wrap system) and F[i] is the force acting on it, amplified by k and applied at the distance d [i] from the ‘‘y” axis of symmetry. There are accordingly 3(n + 1) internal forces and 3n equilibrium equations, being n the number of voussoir. With an appropriate choice of the three undetermined unknowns, the equations of equilibrium can be conveniently reformulated obtaining the internal forces as a function of them.

Fig. 4. Geometry discretization.

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 s s Niþ1  R þ þ hc þ yg  Ni  R þ þ hc  yg 2 2 þ ðkF i  R  sin v i Þ þ ðW tot  ni þ M iþ1  M i Þ ¼ 0

339

ð7Þ

1.2. The M-N domain The admissibility domain of interaction between bending moment and axial force (M-N domain) is calculated taking into account that the limit strength in tension of CFRP, assumed herein rigid-perfectly plastic as aforementioned, is evaluated with CNRDT200/2004 formulas. The M-N domain, for a rectangular section reinforced by the XWrap reinforcement, is derived considering four different mechanisms both for negative and positive bending moment, for a total of eight expressions. With reference to Fig. 5, we have: Fig. 5. Forces acting on the single fictitious voussoir of arch under vertical load.

s pr

ðNiþ1  sin v iþ1  Ni  sin v i Þ þ ðV iþ1  cos v iþ1  V i  cos v i Þ ¼ 0 i ¼ 1; 2; . . . n

ð1Þ

ðNiþ1  cos v iþ1  Ni  cos v i Þ  ðV iþ1  sin v iþ1 þ V i  sin v i Þ þ Wtot þ kF i ¼ 0 i ¼ 1; 2; . . . :n

ð2Þ





 s s Niþ1  R þ þ hc þ yg  Ni  R þ þ hc  yg 2 2 þ ðkF i  di Þ þ W  ni  Miþ1  M i Þ ¼ 0 i ¼ 1; 2; . . . n

hc, pc tf bf,sup a

ð3Þ

afcm rfmaxb

where yg defined by (4), represents the distance between the geometric center of the whole system to the estrados of the rib,

rfmaxc

yg ¼

h2c 2

 pc þ s  pr  ð2s þ hc Þ hc  pc þ s  pr

ð4Þ

In the case of the horizontal load the equilibrium Eqs. (1)–(3) must be re-written by referring to Fig. 6. In that case, the external load will appear in the horizontal equilibrium equation and the lever arm will be Rsinv(1) The following Eqs. (11)–(13) are obtained.

ðNiþ1  sin v iþ1  Ni  sin v i Þ þ ðV iþ1  cos v iþ1  V i  cos v i Þ  kF i ¼ 0 ð5Þ ðNiþ1  cos v iþ1  N i  cos v i Þ  ðV iþ1  sin v iþ1 þ V i  sin v i Þ þ Wtot ¼ 0

ð6Þ

afcc

thickness of masonry arch; effective width of the portion of masonry barrel vault collaborating with the reinforcement, evaluated according to CNR-DT200/2004 with relation to a reinforcement width given by pc + 2a; thickness and width of X-Wrap system; thickness of FRP reinforcement; width of FRP reinforcement at the top of X-Wrap system; width of the FRP reinforcement swaged at the extrados of the vault; compressive strength of masonry; tensile strength of C-FRP adherent to the masonry evaluated according to CNR-DT200/2004 (with account for delamination); tensile strength of C-FRP adherent to the mortar core of the X-Wrap system, evaluated according to CNRDT200/2004(with account for delamination); compressive strength of mortar core of the X-Wrap system;

Considering a positive bending moment as represented in Fig. 7, four mechanisms can be observed, according with the position of the neutral axes, four pair of parametric Eqs. (8)–(11) can be written. The bending moment is evaluated with regard to the point G, which is the geometric centre of section. In the mechanism no. 1, the position of the neutral axis varies in the range [0, hc]. So that part of the rib core is compressed, and

Fig. 6. Forces acting on the single fictitious voussoir of arch under vertical load.

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only part of the reinforcement that is attached to the mortar core is tense. The limit values of the axial force and bending moment are given by (8):

8 rfmaxb tf 2a  rfmaxc tf 2ðhc  xÞ >   < N 1 ¼ af cc pc x    hc þ x M 1 ¼ af cc pc x yg  2x  rfmaxb t f 2aðyg  hc Þ  rfmaxc t f 2ðhc  xÞ  yg  > 2 :

In mechanism no. 2, the position of the neutral axis position x varies in the range [hc, hc+s]. In this case the compression contributions of the mortar rib and of the masonry are active. The resultant expressions are (9):

(

N2 ¼ af cc pc hc þ af cm pr ðx  hc Þ    c  M2 ¼ af cc pc hc yg  h2c  af cm pr ðx  hc Þ xþh  yg 2

ð9Þ

In mechanism no.3 When the neutral axis is located at the upper edge of X-Wrap system (x = 0, Fig. 6), the mortar and the masonry do not carry any stress, rather the reinforcement attached to the masonry and to the lateral edges of the rib has reached its limit tensile stress. The reinforcement on the upper edge can carry any tensile stress the range [0,rfmaxc]. Therefore, there is a simple infinity of limit curvature, corresponding to a straight branch of the N-M domain. The expressions of the axial force and of the bending moment are given as parametric expression of the tensile stress in the upper reinforcement (10):

(

N3 ¼ rfc bf t f  rfmaxc t f 2hc  rfmaxb tf 2a   M3 ¼ rfc bf tf yg  rfmaxc tf 2hc yg  h2c  rfmaxb tf 2aðyg  hc Þ ð10Þ

In mechanism no. 4, the neutral axis is located at the bottom edge of the X-Wrap rib, (x = h_c), we have a similar situation of the mechanism no. 3, with the difference that the whole reinforcement carries zero stress, except the one attached to the masonry, whose tensile stress varies in the range [0,rfmaxb]. In this condition, the contribution of the compressed mortar core must be also considered. The expressions of the axial force and bending moment, parametric with the tensile stress of the reinforcement, are given by (11):

(

N4 ¼ af cc pc hc  rfb t f 2a   M4 ¼ af cc pc hc yg  h2c  rfb t f 2aðyg  hc Þ

ð11Þ

In a same way it is possible to determinate the other four pair of parametric Eqs. (12)–(15) of N-M domain for negative curvature by referring to Fig. 8. From each pair of parametric expressions it is easy to get the limit static admissibility conditions, for bending moment and axial force, in the form: M[Mi(N)], with i = 1,2,. . .8. The convex limit domain M-N envelopes of the eight functions Mi(N), obtained for the case of X-Wrap system reinforcement, is compared, in the Fig. 9, both with the unreinforced masonry vault and with the classical reinforcement. For this purpose only one strip of CFRP was considered, the same width of X-Wrap (200 mm) and the same total specific weight (300 g/m2) of uniaxial fabric of CFRP, attached at the extrados of the vault. 8 > < N5 ¼ af cc pc ðhc  xÞ þ af cm pr s  rfmaxc tf ð2x þ bf Þ  M 5 ¼ af cc pc ðhc  xÞ yg  h2c  2x  af cm pr s hc þ 2s  yg > : 
x  rfmaxc t f bf yg  rfmaxc tf 2x yg  2 8 N ¼ rfmaxc t f bf  rfmaxc tf 2hc  rfmaxb t f 2a þ af cm pr ðhc þ s  xÞ > < 6   M 6 ¼ rfmaxc tf bf yg  rfmaxc t f 2hc yg  h2c  rfmaxb t f 2aðyg  hc Þ >   :  yg af cm pr ðhc þ s  xÞ hc þsþx 2

(

N7 ¼ af cc pc hc þ af cm pr s  rfc t f bf M7 ¼ af cc pc hc ðyg  h2c Þ  af cm pr sðhc þ 2s  yg Þ  rfc t f bf yg

ð14Þ

8 N8 ¼ af cm pr s  rfmaxc t f ðbf þ 2hc Þ  rfb t f 2a > >     > < M 8 ¼ af cm pr s hc þ 2s  yg  rfmaxc t f bf yg  rfmaxc t f 2hc yg  h2c > > rfb tf 2aðyg  hc Þ > : The extreme mortar joints, near the supports, were implemented by a given infinite value of the resistant shear, in order to take account of the presence, in real structures, of the walls supporting the upper floors. Besides taking into account the presence of the sliding bearing in Section no.1 (Fig. 3), the appropriate M-N domain of Fig. 10 was determined in that section. The condition M1 = 0 and N1 = 0 constitutes a limit condition and M-N domain cross through the origin of the reference system. 1.3. Shear strength Sliding consists in relative movements of two parts along the mortar joint and it occurs when the friction mechanism is not capable of balancing the shear of the external load on the cross section with a corresponding shear force. The shear resistance of the XWrap reinforced masonry vault is the sum of three different contributions: The shear strength of the mortar joint, evaluated according to the Italian Code D.M. 14/01/2008 [22], with the Mohr–Coulomb friction low, with a friction coefficient of 0.4. As well known, this contribution is strongly compression normal stress dependent and causes the limit analysis to be rigorously considered with a non-associated flow – rule.

scm ¼ scm0 þ 0:4rcp

ð14Þ

where scmo is the shear stress resistance of the masonry in absence of normal stress rcp. The shear strength of the fiber reinforced concrete rib. It can be evaluated, in conformity with the Italian Code D.M. 14/01/2008 [22] and CNR-DT 200/200 [11] as a sum of only two contribution, because of the absence of steel shear reinforcement (see Eq. (14)): The shear resistance of the concrete (mortar) rib, evaluated with the Mohr –Coulomb friction low, with a friction coefficient of 0.15. The shear resistance of the fiber reinforcement, evaluated in conformity with the Italian Code D.M. 14/01/2008 [22]. In this case can be adopted the formulation valid for the X-wrap and corresponding to the Moersch’s truss model, assuming an inclination of the compressed rod of 45 degrees:

 V Rd ¼ min V Rd;ct þ V Rd;f ; V Rd;max

ð15Þ

where: VRd,ct is the contribution of the unreinforced concrete core evaluated as in (16):

V Rd;ct ¼ scc  hc  pc

ð16Þ

Being:

scc ¼ scc0 þ 0:15rcp

ð17Þ

where scco is the shear resistance of the core in absence of normal stress rcp. The value of scco is given by (18): 1=2 scc0 ¼ v min v min ¼ 0:035  k3=2  f ck ½N=mm2 

ð18Þ

Being:

k ¼ 1 þ ð200=dÞ

1=2

62

ð19Þ

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Fig. 7. Cross section of the X-Wrap system: Positive bending.

Fig. 8. Cross section of the X-Wrap system: Negative bending.

M [kNm] Extrados classic reinforcement with the same width of Ω-Wrap system and the same total uniaxial weight of the CFRP (200 mm, 300g/m2)

10

Ω-Wrap system reinforcement

5

Unreinforced vault

100

200

300

400

N [kN] –5

Fig. 9. Comparison of the interaction domain M-N for different reinforcing system of a masonry barrel vault.

where: d fck

is the effective height of the transversal section of the core of the ribbing (in mm); is the cylindrical characteristic compressive strength of the mortar or concrete contained in the core of the ribbing (in N/mm2);

Dowel-effect, Pd, of the portion of CFRP folded on the extrados of the vault. Refer to Badalà et al. [20] for more details. 1.4. Formulation of the optimization problem The lower bound limit analysis formulation leads to the follow optimization problem to finding the limit multiplier k of the applied load system:

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3) Repeated step 2) until result satisfies the two following conditions: A) No more change of the collapse mechanism; B) Convergence of the resulting limit load multiplier to the assumed final value, within a fixed numerical tolerance. At each step the average value of the normal stress acting on the rib and masonry sections, respectively, is computed in approximate way from the linear distribution obtained applying the Navier’s law to the homogenized cross section. The theoretical background on which the method stands is Radenkovic’s second theorem (Lubliner 1990 [24]), in conjunction with the assumption (numerically verified) that no reduction of compression stress in the interfaces can occur during the incrementing of the applied load. The aforementioned fundamental theorem assures the existence an uniqueness, at each step of the procedure, of a lower bound value for the effective limit load of the non-associative problem. The assumption of non-decreasing compressing stress at each interface allows to consider a shearnormal stress domain all included into the real one (Fig. 11).

Fig. 10. M-N domain in sect. no. 1 of X-Wrap vault.

kc ¼ maxk k;x 8 9 > > < M½i 6 M j ðN½iÞ; j ¼ 1; 2; 3; 4 = s:to M½i P Mk ðN½iÞ; k ¼ 5; 6; 7; 8 i ¼ 1; 2; . . . ; n þ 1 > > : ; jVj 6 V Rd;m þ V Rd;c þ Pd;a

ð20Þ

In this formulations with ‘‘i” is indicated each of the joints of the vault, being ‘‘n” the number of fictitious voussoirs, while ‘‘j” and ‘‘k” are the two series of the four relations for the M-N domain, in explicit form, for positive and negative sign of the rotation of the cross section, respectively. The vector X collects the three independent interfacial forces that determine all the other internal force in the vault. The functional dependence from the variables k and X is omitted for simplifying the notation. This formulation is valid for fixed values of the shear resistance in each section of the vaults, so we are within the framework of the validity of the Radenkovic’s second theorem for non-associated flow rules [21]. The proposed algorithm consists of the follows steps: 1) Solving the problem (20) for a value of the shear strength of each section of the vault corresponding to a null value of the axial force. 2) Solving again the same problem (20), considering a shear resistance corresponding to compressive stress found in conjunction with the limit load obtained from the previous problem resolution;

2. Experimental validation of the proposed analytical method 2.1. External vertical load To validate the proposed method, a series of experimental tests were carried out on several samples arranged like illustrated in Fig. 12, The vault specimen has a clear span of 150 cm and a clear rise of 50 cm, it is 55 cm wide and 7 cm thick. The calcareous bricks compound are squared geometry of 7  7  15 cm3 in volume, applied in a single layer, with 23 rows of blocks. Bricks are bonded by mortar of hydrated lime and cement of class M2.5 N/mm2 according to Italian code. The X-Wrap system nucleus dimensions are 5 cm in height and 10 cm in width, and it is casted on the extrados of the vault. The CFRP is placed around that core and swaged on the extrados of masonry for a width equal to 5 cm at each sides. Uniaxial sheet of C-FRP is applied in two layers around the concrete arranged at right angles to each other, so as to form a balanced biaxial reinforcement with equivalent thickness of 0.167 mm in each orthogonal directions. Fig. 12 shows also the experimental set-up for the X-WRAP-reinforced vaults. The vault’s restrains are a fixed abutment and a sliding one at the opposite side. To avoid unsuitable sliding improper hinges formation at the extremities near the supports, two steel profiles pieces capable

Fig. 11. Mohr-Coulomb interaction domain for shear and normal stress, and Radenkovic’s second theorem associated domains at each step of the incremental procedure.

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L. Anania, G. D’Agata / Construction and Building Materials 145 (2017) 336–346 Table 1 Mechanical properties of materials.

Fig. 12. Experimental set-up for the X-WRAP-reinforced vaults.

Property

Material

Value

Property

Material

Value

Compressive strength (cubic, average) [MPa]

Brick Mortar Masonry CFRP Mortar of the X-Wrap core

12.6 2.5 4.1 0 54.39

Young modulus (average, secant at break point) [MPa]

Brick Mortar Masonry CFRP Mortar of the X-Wrap core

9500 675 4100 230,000 2882

Tensile strength (average) [MPa]

Brick Mortar Masonry CFRP Mortar of the X-Wrap core

1.2 0.2 0 4830 3.87

Density (average) [kNm3]

Brick Mortar Masonry CFRP Mortar of the X-Wrap core

17.5 20.0 18.0 14.8 20.0

Initial shear strength (average) [MPa]

Masonry

0.1540.20

Overboard design tensile strength of CFRP with different materials (CNR-DT200/ 2004) [MPa]

Masonry

308.7

45 40

Load [kN]

35 30 25

Experimental

20

Analytical

15 10 5 0

Fig. 13. Shear restrain at the fixed abutment.

0

5

10

15

20

25

30

Displacements [mm] Fig. 15. Comparison between theoretical and experimental data-vertical load.

Fig. 16. Experimental collapse load-vertical load.

Fig. 14. Shear restrain at the mobile impost.

of simulating the presence in real structures, of the walls supporting the upper floors, were applied as shown in Figs. 13 and 14. Table 1 reports the main mechanical properties of the materials used, some of which were experimental determined, The design value of the intermediate delamination tensile strength of CFRP is determined multiplying by three the ones related to the overboard delamination, according to what prescribed by CNRDT200/2014 for concrete. In the first series of test, in order to have a very simple failure mode, the vertical load was applied at ¼ of the clear span of the

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Fig. 17. Comparison between theoretical and real damage in the vault-vertical load.

vault and transferred by means of a screw jack placed in series to a load cell of 250 KN capacity. The transmission of only the vertical component of the applied load related to the imposed displacement was ensured both by a ball joint and a mobile abutment. The applied load is distributed to all along the cross section of

the vault by a wood element, reinforced by a steel L 60  6 profile. The evaluation of the horizontal trust occurs throughout a special sliding bearing support, located on the impost at the opposite side to the loading point, connected to two cells of 25 kN capacity. Displacements and stresses are assessed by displacement transducers distributed along the directrix of the arch and strain gauges placed along the CFRP material, respectively [10].

Fig. 18. Load displacement curve.

Fig. 19. Theoretical collapse mechanism.

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(a) Sce. No 1

(b) sect. no. 23

345

(b) Sect. no 17

Fig. 20. Theoretical collapse mechanism for horizontal load at the sliding abutment.

The experimentally determined average load at the plateau was 38.0 kN (Fig. 15). The numerical incremental procedure gave a final result of 38.16 kN, with a very close accordance with the experimental data, also for the limit mechanism plotted in Figs. 16 and 17. 2.2. External horizontal load Other series of test was perform in a similar manner, but with an imposed displacement at the sliding abutment, imposed by means of a screw jack placed in series to a load cell of 25 KN capacity. In this case, no load is applied at the extrados of the vault, but a damage was considered in the section no. 17. The code consider the presence of the damage in section no. 17 by eliminating the resistance of C-FRP in that section. The bearing capacity given back by the limit analysis is equal to 8,59 kN (as shown in Fig. 18) in the case of damage in section no. 17. The collapse mechanism is represented by the formation of three hinges: one at the extrados at section no. 1, another at the intrados in section no. 23 and finally one at the extrados in section no. 18 (Fig. 19). The experimental test gave bake a collapse load of Fu = kcF = 8,50 kN very close to the analytical one. Also the collapse mechanism is in great accordance with the analytical one First hinge, indeed, appears at the extrados at section no. 1 (Fig. 20a), the secondo at the intrados in section no. 23 (Fig. 20b) and finally the last one at the extrados in section no. 18 (Fig. 20c) very close to damaged section, the most vunerable. Thanks to the perfect accordance between experimental and analytical data, it is possible to assess the collapse load of the any arch. For instance, in the specific case of not damaged arch a collapse load of Fu = kcF = 16.70 kN. Can be achieved, eight times greater than the corresponding values of 2 kN obtained for the unreinforced samples. 3. Conclusions An approximate methodology for approaching the lower bound limit analysis of masonry arch (and barrel vaults), reinforced by CFRP was been illustrated. The basic idea of the proposal methodol-

ogy is to carry out a series of lower bound limit analyses. In any one of which the shear strength of each fictitious sections is constant and given by the Mohr-Coulomb friction law (for the mortar joint) and other non linear Italian Code relations (for CFRP Omega-Wrap reinforcement) at a given level of normal compressive stress, resulting from the previous step. In this way the associated flow rule holds at each step for the shear failure mechanism also without dilatancy. In accordance with the Radenkovic’s second theorem, the current plastic domain of interaction between transversal and normal stress, with a constant value of shear strength at each step, is entirely enclosed into the real non-associative Mohr-Coulomb domain. That is true provided normal stress is not decreasing during the process of loading, as is confirmed in this case by the numerical results. Because of the non-linearity of the interacting domain ‘‘M-N” between axial force and bending moment, together with relatively small dimension of the example treated, the internal non linear optimization solver of the software MathematicaÒ was used, at each step of the iterative procedure. The results of the proposed formulations results are in very good agreement with available original experimental data, deriving from tests (with only two cycles of load and unload) carried out by the authors on calcareous voussoirs masonry barrel vaults, especially in terms of pick load and final collapse mechanism reached at the first cycle of loading. It’s interesting also to note that the limit load and mechanism provided by each intermediate step of this heuristic procedure seem to capture the evolution of the damage in the structure, reinterpreted in terms of sequence of different limit kinematic mechanisms, as was highlighted above. Moreover, the proposed analytical procedure is able to highlights all the capability of the new strengthening technique. In fact, the numerical M-N domains show the great improvement of bearing capacity in respect to no strengthened one. In fact, a gain of more than 52 times greater is obtained. References [1] S.S. Mahini, A. Eslami, H.R. Ronagh, Lateral performance and load carrying capacity of an unreinforced CFRP-retrofitted historical masonry vault – A case study, Constr. Build. Mater. 28 (2012) 146–156.

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