# Numerical Modelling and Validation of the Response of Masonry Infilled RC Frames Using Experimental Testing Results

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## Abstract

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## 1. Introduction

## 2. Research Methodology

- Classify the typologies of masonry infills representative of the typical configurations adopted in Italy and other Mediterranean countries based on experimental data. The results of in-plane pseudostatic cyclic tests, carried out on single-storey and single-bay masonry-infilled RC frames, with and without openings, are analysed to identify the most common configurations. The masonry infill typologies are defined based on the masonry infills strength; this parameter was found to be the most suitable for future analytical applications on the seismic performance assessment of masonry infilled RC building portfolios accounting for infill variability [20];
- Define the most accurate numerical modelling approach to simulate the experimental lateral response of masonry-infilled RC frames. If the frames are not designed according to modern seismic provisions, then the numerical modelling should also be able to account for the typical phenomena observed in existing buildings, such as material and geometrical nonlinearity, bar slippage, joint flexibility, behaviour of poorly detailed and non-ductile RC frame members, among others;
- The main parameters affecting the numerical modelling of masonry infills are investigated. The hysteretic behaviour of the masonry infill panel depends on several parameters, such as the strut width, reduction coefficient to account for the presence of openings, failure mechanism model or formulations to define the backbone curve. For each of these parameters, the main formulations available in the literature are analysed to undertake parametric static pushover analysis;
- In order to define the most reliable numerical modelling approach, for each masonry infill typology identified in point 1, a set of parametric static pushover analyses are carried out combining all the formulations defined in point 3. The comparison is then performed in terms of capacity curves. In specific, given a selected backbone curve, different models could be employed to predict the failure mechanism and strut width; hence, for each of the selected strength models, the impact and accuracy of all the strut width equations is investigated and the same procedure is repeated for all the parameters investigated;
- Finally, once the most accurate numerical model is identified, cyclic pushover analysis (according to the loading protocol used for the corresponding testing) is performed to investigate the effectiveness of the proposed numerical models, when it comes to predicting the hysteretic response of the masonry infilled RC frames.

## 3. Numerical Modelling and Structural Response of Masonry Infills

#### 3.1. Macro-Modelling Approaches

#### 3.1.1. Approaches for Full-Height Solid Infill Panels

_{w}and d

_{w}are the strut width and length, respectively.

_{wϑ}is the elastic modulus of masonry panel in the diagonal direction, t

_{w}is the thickness of the infill panel, E

_{c}I

_{c}is the flexural stiffness of the columns of the surrounding frame, h

_{w}is the panel’s height and ϑ is the angle related with the aspect ratio of the panel $({h}_{w}/{L}_{w})$ and defined according to Equation (3):

_{w}is the length of infill panel. Higher values of $\vartheta $ indicate that the surrounding frame is less stiff than the infill panel.

_{1}and K

_{2}), which are defined as a function of λh.

#### 3.1.2. Approaches for Infill Panels with Openings

_{l}is 100 multiplied by the ratio between the length of the opening (l

_{p}) and the length of the panel (L

_{w}), expressed in percentage; α

_{a}is 100 multiplied by the ratio between the product of the opening’s dimensions (l

_{p}·h

_{p}) and the product of the panel’s dimensions (Lw·hw), expressed in percentage.

#### 3.1.3. Infill-Frame Contact Length

_{c}and Z

_{b}(for columns and beams respectively) are defined by means of $\lambda $; Z

_{c}can be assessed through Equation (2), whereas Z

_{b}is assessed replacing EcIc with E

_{b}I

_{b}in Equation (2) (where E

_{b}I

_{b}is the flexural stiffness of the beam). Consequently, depending on the selected macro-modelling approach, the location of the off-diagonal struts varies between z/3 (for the two-strut model) and z/2 (for the three-strut model). Alternatively, according to Al-Chaar [47], the location of the off-diagonal struts may be assessed using two non-dimensional parameters C

_{d}and C

_{od}; the expressions in Equation (17) provide ${z}_{c}$ and ${z}_{b}$, where the off-diagonal struts are positioned, from the beam-column joints:

_{d}and C

_{od}are set equal to 0.50 and 0.25, respectively (which correspond to the portion of the horizontal stiffness assigned to each strut).

#### 3.2. Failure Modes and Backbone Curves

**Surrounding frame**: this failure mode is associated with the development of plastic hinges in the RC elements. The collapse mechanism could be due to flexure, shear, beam-column joint failure or high axial load. The location of flexural plastic hinges is strongly related to the features of the frame-infill systems and may occur (very rarely) in the beams and/or columns, where the maximum bending moment demand is reached. Shear failure in the columns is due to high shear stress in the contact length zones and depends on the amount of transverse reinforcement, concrete strength and efficiency of the concrete confinement. Especially in existing RC frames built according to old codes and prescriptions, the panel may cause wide diagonal cracks along the beam-columns joints and, consequently, their failure. Finally, even though it is very rare, due to concrete strength effect, an axial failure might take place as consequence of high axial load transmitted by a truss mechanism;**Shear sliding**: this mode produces horizontal sliding failure through several bed joints; it is related to the aspect ratio of the masonry units and the infill panel, as well as the poor mechanical properties of the mortar in the bed joints. This failure mode is associated with a strong frame and weak mortar joints. The crack pattern starts a few courses beneath the upper loaded corner and continues along the diagonal direction until reaching the centre of the panel, where finally the cracks spread horizontally;**Corner crushing**: this failure mode produces compression failure (due to a biaxial compression state) of the infill panels with crushing of the units near the beam-column joints; later on, it might produce out-of-plane (OOP) failure and eventually collapse. It normally occurs if the contact length is very or the contact length may be reduces increasing the lateral displacement and the infilled frame is characterized by weak infill panel, combined with strong columns/beams and weak joints;**Diagonal compression**: it is another compression failure mode however, in contrast with the previous failure mode, the crushing of masonry units appears in the centre of the panel. This failure mode is due to the geometry of panel, that is, when the infill is slender, with a subsequent OOP failure;**Diagonal tension or cracking**: this is related to the failure of the compressed diagonal strut, which consists of widespread cracking along the panel; as highlighted in El-Dakhakhni [15], this failure mode occurs when the RC frame is weak or is characterized by weak joints and strong elements, combined with a rather strong infill.

_{max}), the cracking point (0.8F

_{max}), the residual strength (0.35 F

_{max}), the cracking-to-peak stiffness and the residual-to-peak strength stiffness, the latter being derived though the secant stiffness (−0.02K

_{sec}). Both cracking-to-peak and softening-to-peak stiffness ratios are defined according to De Sortis et al. [52].

_{cr}and the maximum shear strength F

_{max}. As opposed to the previous model, the first parameter depends on the actual cracking shear strength, whereas the second parameter is set as a function of F

_{cr}(${F}_{max}=1.3{F}_{cr}$). The elastic stiffness depends on the shear modulus of the panel and on the geometrical properties, whereas the secant-to-peak stiffness is derived according to Mainstone [36]. The degrading branch, up to the attainment of a residual strength, has a slope that depends on the elastic stiffness (K

_{deg}= αK

_{el}) − α is in the range [0.005%, 0.1%] according to Reference [19]. Finally, the residual strength is defined by a multiplier of the maximum shear strength. The residual-to-maximum shear strength ratio β is suggested as 0.01–0.02. It is worth noting that this model relies on the definition of only two parameters, G

_{m}and τ

_{cr}, with significant benefits for what concerns its calibration; the former parameter is related to the modulus of elasticity (E

_{m}= 0.4G

_{m}), whereas the latter can be obtained as 0.275 times the square root of the comprehensive strength of a masonry prism (f

_{mv}).

_{r}is the lateral displacement at a given storey, L is the frame centreline span and h is the centreline storey height.

_{cr}, F

_{peak}, K

_{cr}, K

_{sec}and K

_{deg}) needed to define the Panagiotakos and Fardis backbone curve. This modification of the methodology proposed by Panagiotakos and Fardis [19] was made because the CoV values calculated in the comparison with the experimental tests were lower than those related to the model by Bertoldi et al. [18]. After a disaggregation of the collected data by considering horizontal holes bricks and tests with vertical holes bricks separately, De Risi et al. [14] pointed out that the CoV values are lower if the test results involving horizontally hollow bricks are compared with those provided by the Panagiotakos and Fardis backbone [19].

## 4. Classification of Masonry Infills According to Test Data

_{w}) of the masonry panels varies between 494 and 8140 MPa, the shear modulus (G

_{w}) is in the range 130–2547 MPa, whereas the cracking (F

_{cr}) and peak (F

_{peak}) loads are in the range of [5 kN, 270 kN] and [10 kN, 350 kN], respectively. Comparing these ranges with those of the five masonry infill typologies selected, it can be concluded that the latter are representative of the ranges of variation of the main mechanical properties evinced from the database, as can be gathered from Table 3.

## 5. Numerical Modelling Results and Validation with Experimental Data

_{p}, in the case that the masonry infills present openings. This further test specimen belongs to the medium-strong masonry infill typology. The numerical modelling validations were carried out using the FE software OpenSees [60].

#### 5.1. Bare Frames

^{2}and eight (3 bottom + 2 middle + 3 top) 22 mm diameter rebars were provided as longitudinal reinforcement. The transverse reinforcement, consisting of 8 mm diameter bars spaced at 9 cm, was uniformly distributed. Also, the beam had a 35 × 35 cm

^{2}cross section, in which eight (4 bottom + 4 upper) 14 mm and two 10 mm (in the middle) longitudinal rebars were located. Moreover, 8 mm diameter stirrups spaced at 20 cm and 7 cm were lodged in the central and end regions, respectively.

#### 5.2. Masonry-Infilled Frames

_{v0}—according to the analytical and experimental results presented in Morandi et al. [5]—provides the best estimation of the lateral strength. Finally, the relationships proposed by Bertoldi et al. [18] were selected to calculate all the points defining the backbone curve. A comparison of experimental and numerical results is provided in Figure 10a.

_{p}, to account for the opening. This reduction coefficient is applied to both strength and stiffness. The formulation selected to evaluate r

_{p}is the one proposed by Decanini et al. [46], since the parametric analysis shows it to be in very good agreement with the experimental response. Nonetheless also the reduction coefficient r

_{p}proposed by Imai and Miyamot [42], Tasnimi and Mohebkah [43] and Dawe and Seah [45] provided good results. Compared to the fully-infilled configuration, the match between the numerical prediction and the experimental results is slightly worse, confirming that the presence of openings leads to high variability and modelling issues. Moreover, in this particular case, an additional challenging is introduced by the opening size and the force-displacement curves that shown an unsymmetrical behaviour, although almost the same values of force were achieved for both loading directions. In spite of these issues, the modelling approach provides results in good agreement with the experimental ones: the strength is very well predicted, the stiffness in the positive loading direction is captured, whilst the stiffness in the negative direction of loading is slightly underestimated. After a diagonal crack formed at 0.4–0.5% drift in the left and right panel, a stepwise crack pattern developed starting from the bottom left corner and reaching up to mid-height of the right panel at 0.6% drift. At the end of the test (1% drift), the diagonal crack on the left panel widened further (this panel was already more damaged than the right one), while the failure mechanism was not completely formed in the right pane.

_{i}to be assigned to the struts, have been modified with respect to the ones reported in Table 3. In particular, ε

_{m}(peak response) and ε

_{u}(collapse) are set equal to 0.0012 and 0.044, respectively. The values of the strains ε

_{i}to be assigned at the struts are in good agreement with Morandi et al. [72], who considered the same specimen as benchmark test for numerical validation purposes. For this reason, in addition to the numerical results obtained in this study, the numerical validation proposed by Morandi et al. [72] is presented in Figure 11.

_{i}. Moreover, the numerical results of this study are in good agreement with those provided in Morandi et al. [72], although the macro-modelling of the infill panel and RC elements are different.

_{c}) is 2500 MPa and the material properties of the masonry infills are listed in Table 3.

## 6. Discussion and Influence of Modelling Assumptions

_{max}) of the backbone curve be derived using the cracking stress (τ

_{cr}) of the masonry (measured in a diagonal compression test), rather than with one of the relationships proposed in Table 2. Although the backbone curves proposed by Panagiotakos and Fardis [19] and De Risi et al. [14] are based on very few parameters, the obtained response is still affected by high dispersion, caused by the effect of the cracking stress, which can be estimated through a series of vertical compression tests or through the expression $0.275\sqrt{{f}_{mv}}$ proposed by Jeon et al. [73]; in both cases, the dispersion surrounding the material properties is generally high.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Different failure modes of reinforced concrete (RC) infilled frames: (

**a**) corner crushing and diagonal compression, (

**b**) diagonal tension and shear sliding (adapted from Asteris et al. [48]).

**Figure 5.**Numerical modelling approach for masonry infill, frame elements and beam-column joints, implemented in OpenSees [60].

**Figure 6.**Comparison between numerical and experimental test results by Calvi and Bolognini [4].

**Figure 7.**Numerical versus experimental results of the GLD specimen by Verderame et al. [7] for (

**a**) modelling option 1 (rigid beam-column joints) and

**(b**) modelling option 2 (with shear springs).

**Figure 8.**Comparison between numerical and experimental results of the SLD specimen by Verderame et al. [7].

**Figure 9.**Comparison between numerical and experimental results of the test by Morandi et al. [5].

**Figure 10.**Numerical versus experimental test results for masonry infill type 4 [5]: (

**a**) without openings and (

**b**) with openings.

**Figure 11.**Numerical versus experimental test results for masonry infill type 1 [4].

**Figure 13.**(

**a**) Backbone curves by Bertoldi et al. [18] varying the strut width and the strength models; (

**b**) pushover curves resulting from each backbone curve versus experimental test results.

**Table 1.**Formulations to evaluate the width of the strut and reduction coefficients of stiffness and strength due to the presence of the openings.

Diagonal Infill Strut Width According to Different Models | ||
---|---|---|

Bertoldi et al. [18] | $\frac{{b}_{w}}{{d}_{w}}=\frac{{K}_{1}}{\lambda h}+{K}_{2}$ $\left\{\begin{array}{c}{K}_{1}=1.300,{K}_{2}=-0.178;if\lambda h3.14\\ {K}_{1}=0.707,{K}_{2}=-0.010;if3.14\lambda h7.85\\ {K}_{1}=0.470,{K}_{2}=-0.040;if\lambda h7.85\end{array}\right\}$ | $\lambda =\sqrt[4]{\frac{{E}_{w\vartheta}{t}_{w}\mathrm{sin}\left(2\vartheta \right)}{4{E}_{c}{I}_{c}{h}_{w}}}$ (Stafford Smith [33]) ${E}_{w\vartheta}$: elastic modulus of masonry (inclined direction) ${E}_{c}$: elastic modulus of concrete ${I}_{c}$: elastic modulus of concrete ${t}_{w}$: thickness of the infill ${b}_{w}$: strut width of the infill ${d}_{w}$: diagonal length of the infill |

Paulay and Priestley [39] | $\frac{{b}_{w}}{{d}_{w}}=0.25$ | |

Holmes [26] | $\frac{{b}_{w}}{{d}_{w}}=0.33$ | |

Liauw and Kwan [37] | $\frac{{b}_{w}}{{d}_{w}}=\frac{0.95\mathrm{sin}\left(2\vartheta \right)}{2\sqrt{\lambda h}}$ | |

Mainstone [36] | $\frac{{b}_{w}}{{d}_{w}}=0.175{\left(\lambda h\right)}^{-0.4}$ | |

Stafford Smith [33] | $0.1<\frac{{b}_{w}}{{d}_{w}}<0.25$ | |

Decanini and Fantin [38] | Uncracked: $\left\{\begin{array}{c}\left(0.085+\frac{0.748}{\lambda}\right){d}_{w};if\lambda h\le 7.85\\ \left(0.130+\frac{0.393}{\lambda}\right){d}_{w};if\lambda h7.85\end{array}\right\}$ Cracked: $\left\{\begin{array}{c}\left(0.010+\frac{0.707}{\lambda}\right){d}_{w};if\lambda h\le 7.85\\ \left(0.040+\frac{0.470}{\lambda}\right){d}_{w};if\lambda h7.85\end{array}\right\}$ | |

Papia et al. [40] | $\frac{{b}_{w}}{{d}_{w}}=\frac{kc}{z{\left({\lambda}^{*}\right)}^{\beta}}$ c, β: accounting for Poisson’s ratio k: accounting for vertical load z: geometrical parameter | ${\lambda}^{*}=\frac{{E}_{w\vartheta}{t}_{w}h}{{E}_{c}{A}_{c}}\left(\frac{{h}^{2}}{{L}^{2}}+\frac{{A}_{c}L}{4{A}_{b}h}\right)$ According to Papia et al. [40] L: frame centreline span h: centreline storey height. ${A}_{c}$: column cross section ${A}_{b}$: beam cross section |

Reduction Coefficients of Stiffness and Strength due to the Presence of Openings | ||

Dawe and Seah [45] | ${r}_{p}=1-\frac{1.5{\alpha}_{l}}{100};{\alpha}_{l}66\%$ | ${\alpha}_{a}=100\frac{{l}_{p}{h}_{p}}{{L}_{w}{h}_{w}}$ ${\alpha}_{l}=100\frac{{l}_{p}}{{L}_{w}}$ ${l}_{p}$: opening length ${h}_{p}$: opening height ${L}_{w}$: infill length ${h}_{w}$: infill height |

Imai and Miyamoto [42] | ${r}_{p}=\mathrm{min}\left(1-0.01{\alpha}_{l};1-0.1{\alpha}_{l}{}^{0.5}\right)$ | |

Tasnini and Mohebkhan [43] | ${r}_{p}=1-2.238\left(\frac{{\alpha}_{a}}{100}\right)+1.49{\left(\frac{{\alpha}_{a}}{100}\right)}^{2};{\alpha}_{a}40\%$ | |

Decanini et al. [46] | ${r}_{p}=0.55{e}^{\left(-0.035{\alpha}_{a}\right)}+0.44{e}^{\left(-0.025{\alpha}_{l}\right)}$ | |

Asteris [44] | ${r}_{p}=1-2{\left(\frac{{\alpha}_{a}}{100}\right)}^{0.54}+{\left(\frac{{\alpha}_{a}}{100}\right)}^{1.14}$ |

Infill Strength According to Different Models | ||
---|---|---|

Paulay and Priesley [39] | ${V}_{W}=min\left({V}_{s};{V}_{c}\right)$ Sliding shear failure: ${V}_{s}=\frac{{f}_{v0}{t}_{w}{L}_{w}}{1-\mu \frac{h}{L}}$ Compression failure: ${V}_{c}=\frac{2}{3}z{t}_{w}{f}_{lat}$ | $z=\frac{\pi}{2}{\lambda}^{-1}$ ${f}_{v0}$: initial shear strength of bed-joints; $\mu :$friction coefficient ($\mu =0.3)$; $\lambda $: according to (Stafford Smith [33]. |

Bertoldi et al. [18] | ${f}_{m}^{\u2019}=\mathrm{min}\left({\sigma}_{w,cc};{\sigma}_{w,corn};{\sigma}_{w,\mathrm{ss}};{\sigma}_{w,\mathrm{sd}}\right)$ Compression at the centre: ${\sigma}_{w,cc}=\frac{1.16{f}_{vert}tan\vartheta}{{K}_{1}+{K}_{2}\lambda h}$ Compression at the corners: ${\sigma}_{w,\mathrm{ccorn}}=\frac{1.2{f}_{vert}sin\vartheta cos\vartheta}{{K}_{1}{\left(\lambda h\right)}^{-0.12}+{K}_{2}{\left(\lambda h\right)}^{0.88}}$ Sliding shear failure: ${\sigma}_{w,\mathrm{ss}}=\frac{\left(1.2sin\vartheta +0.45cos\vartheta \right){f}_{v0}+0.3{\sigma}_{v}}{\frac{{b}_{w}}{{d}_{w}}}$ Diagonal cracking: ${\sigma}_{w,\mathrm{sd}}=\frac{0.6{f}_{t}+0.3{\sigma}_{v}}{\frac{{b}_{w}}{{d}_{w}}}$ | ${f}_{t}$: shear strength under diagonal compression; ${f}_{vert}:$compression strength in vertical direction; ${f}_{v0}$: initial shear strength of bed-joints; ${\sigma}_{v}:$vertical stress; $\lambda $: according to Stafford Smith [33]. ${b}_{w}$: according to Bertoldi et al. [18]. |

EC6/EC8 [49,50] | ${V}_{W}={f}_{v}{t}_{w}{L}_{w}$ | ${f}_{v}={f}_{v0}+0.4{\sigma}_{v}$ ${f}_{v0}$: initial shear strength of bed-joints; ${\sigma}_{v}:$vertical stress. |

FEMA 306 [34] | ${V}_{mf}\u2a7d{V}_{W}=min\left({V}_{s};{V}_{c};{V}_{cr}\right)\u2a7d{V}_{mi}$ Sliding shear failure: ${V}_{s}=\left({f}_{v0}+\mu {\sigma}_{v}\right){t}_{w}{L}_{w}$ Compression failure: ${V}_{c}={t}_{w}{b}_{w}{f}_{lat}cos\theta $ Diagonal cracking failure: ${V}_{cr}=\frac{2\sqrt{2}{t}_{w}{L}_{w}{\mathsf{\sigma}}_{cr}}{\frac{{L}_{w}}{{h}_{w}}+\frac{{h}_{w}}{{L}_{w}}}$ | $Vmi=2\sqrt{0.0069{f}_{vert}}{t}_{w}{L}_{w}$ ${V}_{mf}=0.3{V}_{mi}$ ${f}_{v0}$: initial shear strength of bed-joints; ${\mathsf{\sigma}}_{cr}$: cracking strength of masonry; $\mu :$friction coefficient ($\mu =0.4)$; ${\sigma}_{v}:$vertical stress; ${f}_{lat}:$compression strength in horizontal direction of masonry; ${f}_{vert}:$compression strength in vertical direction; ${b}_{w}$: according to Mainstone [36]; $\lambda $: according to Stafford Smith [33]. |

Backbone Curve according to Different Models | ||

Bertoldi et al. [18] | cracking strength: ${F}_{cr}=0.8{F}_{max}$ residual strength: ${F}_{res}=0.35{F}_{max}$ elastic stiffness: ${K}_{fc}=4{K}_{sec}$ softening-to-peak stiffness: ${K}_{deg}=-0.02{K}_{sec}$ | ${F}_{max}$: peak strength, defined according to the selected infill strength model; ${K}_{sec}:$ secant stiffness according to Mainstone [36]. |

Panagiotakos and Fardis [19] | cracking strength: ${F}_{max}=1.3{F}_{cr}$ residual strength ${F}_{res}=\mathsf{\beta}{F}_{max}$ elastic stiffness ${K}_{fc}=4{K}_{sec}$ softening-to-peak stiffness ${K}_{deg}=-\mathsf{\alpha}{K}_{sec}$ | ${F}_{cr}$: cracking strength, defined according to the selected infill strength model; α: [0.5%, 10%]; β: [1%, 2%]; ${K}_{sec}:$ secant stiffness according to Mainstone [36]. |

De Risi et al. [14] | cracking strength: ${F}_{cr}=0.7{F}_{max}$ residual strength: ${F}_{res}=0$ elastic stiffness: ${K}_{el}=2.8{K}_{MS}$ secant stiffness: ${K}_{sec}=0.8{K}_{MS}$ softening-to-peak stiffness: ${K}_{deg}=-0.1{K}_{MS}$ | ${F}_{max}$: peak strength, defined according to the selected infill strength model; ${K}_{sec}:$ secant stiffness according to Mainstone (${K}_{MS})$ [36]. |

Sassun et al. [51] | Backbone according to Bertoldi et al. [18] modified with prefixed values of drift capacity ϑ (or equivalently in terms of strain capacity ε [55]) | DS1 (Operational): $\vartheta =0.18\%$ DS2 (Damage Limitation): $\vartheta =0.46\%$ DS3 (Life Safety): $\vartheta =1.05\%$ DS4 (Ultimate): $\vartheta =1.88\%$ |

References | Type | Macro Classification | t_{w} [mm] | E_{wv} [MPa] | E_{wh} [MPa] | G_{w} [MPa] | f_{wv} [MPa] | f_{wlat} [MPa] | f_{wu} [MPa] |
---|---|---|---|---|---|---|---|---|---|

Calvi and Bolognini [4] | 1 | Weak | 80 | 1873 | 991 | 1089 | 2.02 | 1.18 | 0.44 |

Hak et al. [55] | 2 | Weak-Medium | 240 | 1873 | 991 | 1873 | 1.5 | 1.11 | 0.25 |

Hak et al. [55] | 3 | Medium-Strong | 300 | 3240 | 1050 | 1296 | 3.51 | 1.5 | 0.3 |

Morandi et al. [5] | 4 | Medium-Strong | 350 | 5299 | 494 | 2120 | 4.64 | 1.08 | 0.359 |

Cavaleri and Di Trapani [6] | 5 | Strong | 150 | 6401 | 5038 | 2547 | 8.66 | 4.18 | 1.07 |

_{w}: thickness; E

_{wv}: elastic modulus vertical direction; E

_{wh}: elastic modulus horizontal direction; G

_{w}: shear modulus; f

_{wv}: vertical strength; f

_{wlat}: lateral strength; f

_{wu}: shear sliding strength.

**Table 4.**Summary of the best numerical modelling approach adopted to define the strut’s hysteretic behaviour for each masonry infill typology.

Type | Macro Classification | Strut Width | Reduction Coefficient | Failure Mechanism | Backbone Curve |
---|---|---|---|---|---|

1 | Weak | Bertoldi et al. [18] | - | Bertoldi et al. [18] | Sassun et al. [51] (modified) |

2 | Weak-Medium | Bertoldi et al. [18] | - | Bertoldi et al. [18] | Sassun et al. [51] (modified) |

3 | Medium-Strong | Bertoldi et al. [18] | - | Bertoldi et al. [18] | Sassun et al. [51] (modified) |

4 | Medium-Strong | Mainstone [36] | Decanini et al. [46] | Paulay and Priestley [39] | Bertoldi et al. [18] |

5 | Strong | Mainstone [36] | - | Paulay and Priestley [39] | Bertoldi et al. [18] |

**Table 5.**Summary of the best numerical models for the strong masonry infill typology, as function of strut width and strength formulations.

Numerical Modelling ID | Backbone Curve | Strut Width | Strength Model | μ | σ |
---|---|---|---|---|---|

Model-1 | Bertoldi et al. [18] | Mainstone [36] | Bertoldi et al. (f_{wu}/2) [18] | 1.00 | 2.46 |

Model-2 | Bertoldi et al. [18] | Stafford Smith [33] | Bertoldi et al. (f_{wu}/2) [18] | 1.02 | 2.64 |

Model-3 | Bertoldi et al. [18] | Bertoldi et al. [18] | Paulay and Priestley [39] | 0.86 | 1.29 |

Model-4 | Bertoldi et al. [18] | Mainstone [36] | Paulay and Priestley (f_{wu}/2) [39] | 1.00 | 2.46 |

Model-5 | Bertoldi et al. [18] | FEMA 306 [34] | Bertoldi et al. [18] | 0.93 | 1.35 |

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**MDPI and ACS Style**

Mucedero, G.; Perrone, D.; Brunesi, E.; Monteiro, R.
Numerical Modelling and Validation of the Response of Masonry Infilled RC Frames Using Experimental Testing Results. *Buildings* **2020**, *10*, 182.
https://doi.org/10.3390/buildings10100182

**AMA Style**

Mucedero G, Perrone D, Brunesi E, Monteiro R.
Numerical Modelling and Validation of the Response of Masonry Infilled RC Frames Using Experimental Testing Results. *Buildings*. 2020; 10(10):182.
https://doi.org/10.3390/buildings10100182

**Chicago/Turabian Style**

Mucedero, Gianrocco, Daniele Perrone, Emanuele Brunesi, and Ricardo Monteiro.
2020. "Numerical Modelling and Validation of the Response of Masonry Infilled RC Frames Using Experimental Testing Results" *Buildings* 10, no. 10: 182.
https://doi.org/10.3390/buildings10100182