Parametric Investigation on the Effectiveness of FRM-Retrofitting in Masonry Buttressed Arches
Abstract
:1. Introduction
2. Methods
2.1. Limit Analysis of Buttressed Arches
- 1.
- Masonry does not have any tensile strength;
- 2.
- Masonry has infinite compressive strength;
- 3.
- Masonry has infinite shear strength, i.e., sliding between masonry parts cannot occur.
- (I)
- Local mechanism (L), with the formation of four hinges A, B, C and D within the thickness of the arch, whose locations are identified through the angles αA, αB, αC and αD, measured with respect the horizontal line;
- (II)
- Global mechanism (G), characterised by the presence of two hinges at the pier basis A (internal) and D (external) and two hinges B and C within the thickness of arch, whose positions are defined by the angles αB and αC, respectively;
- (III)
- Semi-global mechanism (S); characterised by the presence of one hinge D at the base of the pier (external) and three hinges A, B and C within the thickness of the arch, whose positions are identified by the angles αA, αB and αC.
- Elastic behaviour. In this case, the internal work reads where is the domain volume. Since at collapse plastic deformations are usually much larger than elastic, this contribution on the overall energy is usually disregarded and the blocks are assumed to be rigid.
- Perfectly plastic hinges. In this case a constant distribution of stresses is assumed along the crack, equal to tensile strength (Figure 2a); the displacement profile is linear and equal to , where is the maximum crack opening, with angular opening of the hinge, t the member thickness and x the local axis parallel to the crack. The internal work of the i-th hinge becomes:
- Not resisting hinge. This is the usual case in the analysis of masonry structures at collapse. The internal work of the i-th hinge is null, .
- Perfectly plastic reinforcement at intrados or extrados (Figure 2b). In this case, it is possible to assume that upon crack opening the reinforcement is able to provide a force equal to with tensile strength of the reinforcement and its thickness. The virtual work is if the hinge opens on the side of the reinforcement, zero otherwise, since the contribution of FRM reinforcement can usually be neglected in compression.
2.2. An Automatic Tool for Limit Analysis of Buttressed Arches
- the geometrical and material features of the structure including the reinforcement, in the form of a perfectly plastic layer at intrados, extrados or both sides. It is possible to define the tensile strength, the thickness and the ultimate displacement of the reinforcement layer, whose contribution is evaluated by introducing the relevant internal work in Equation (3) as per Section 2.1.
- the type of hinge: Not resisting or Fully plastic. Following the discussion in Section 2.1, the hinge constitutive behaviour modifies the internal work contribution;
- the type of variable horizontal forces: Concentrated in a point or Proportional to masses;
- optional additional static vertical forces;
- the type of analysis: Evaluate, to evaluate a multiplier for a specific hinge position; Minimise, to evaluate the collapse multiplier solving the minimisation problem (4);
- the discretisation Δα for the hinges, used in the collapse multiplier exploration;
- the user-defined keystone displacement;
2.3. Parametric Analysis Settings
- -
- span (S), or, alternatively, half-span (s);
- -
- arch thickness (t);
- -
- eccentricity (e);
- -
- arch rise (h);
- -
- pillars base (B);
- -
- pillar height (H).
- -
- e/s = 0 (Semicircular arch);
- -
- e/s = 0.5 (Drop pointed arch);
- -
- e/s = 1 (Equilateral pointed arch);
- -
- e/s = 1.5 (Lancet pointed arch).
- -
- t/s = 0.125;
- -
- t/s = 0.15;
- -
- t/s = 0.175;
- -
- t/s = 0.2.
- -
- B/s = 0.4;
- -
- B/s = 0.5;
- -
- B/s = 0.6;
- -
- h/H = 0.5;
- -
- h/H = 0.75;
- -
- h/H = 1.
3. Results
3.1. Validation of the Automatic Tool
- specific weight w;
- elastic behaviour: Young’s modulus E, shear modulus G;
- shear behaviour of the block: elastic-plastic Turnsek–Cacovic criterion with shear strength τ0 and ultimate shear strain γu (Figure 9b);
- sliding behaviour: elastic-plastic Mohr–Coulomb criterion with cohesion c, friction coefficient μ and fracture energy Gf, assumed very high to simulate the elastic-perfectly plastic behaviour as an approximation of the frictional behaviour of the joints (Figure 9c).
3.2. Unreinforced Buttressed Arches
3.3. Retrofitted Buttressed Arches
4. Conclusions
Supplementary Materials
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Masonry Type | w (kN/m3) | E (Mpa) | G (Mpa) | fc (Mpa) | Gc (N/mm) | τ0 (Mpa) | γu (-) | c (Mpa) | μ (-) | Gf (N/mm) |
---|---|---|---|---|---|---|---|---|---|---|
Weak | 19 | 690 | 230 | 1.0 | 3.0 | 0.018 | 0.005 | 0.018 | 0.6 | 2.0 |
Strong | 19 | 1800 | 600 | 4.6 | 3.0 | 0.13 | 0.005 | 0.13 | 0.6 | 2.0 |
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Chisari, C.; Cacace, D.; De Matteis, G. Parametric Investigation on the Effectiveness of FRM-Retrofitting in Masonry Buttressed Arches. Buildings 2021, 11, 406. https://doi.org/10.3390/buildings11090406
Chisari C, Cacace D, De Matteis G. Parametric Investigation on the Effectiveness of FRM-Retrofitting in Masonry Buttressed Arches. Buildings. 2021; 11(9):406. https://doi.org/10.3390/buildings11090406
Chicago/Turabian StyleChisari, Corrado, Daniela Cacace, and Gianfranco De Matteis. 2021. "Parametric Investigation on the Effectiveness of FRM-Retrofitting in Masonry Buttressed Arches" Buildings 11, no. 9: 406. https://doi.org/10.3390/buildings11090406