# Assessment of Existing Masonry Resistance Using Partial Factors Approaches and Field Measurements

^{*}

## Abstract

**:**

## 1. Introduction

_{L}= 5.5 in Zagreb and M

_{L}= 6.3 in Petrinja, there is, more than ever, a need for a more reliable assessment of existing masonry structures. Such an assessment will provide important information essential for further decision-making and have a positive long-term effect on the safety and economic aspects of existing building stock management. Hence, different reliability approaches for resistance assessment of existing masonry are presented through a real case study. Such case studies are generally rapidly evaluated immediately after an earthquake to determine usability according to the EMS-98 scale [11]. Afterward, more detailed inspection and in situ tests are conducted to determine the current state of the geometric and mechanical properties as well as the detailing of the structure. The results obtained from the investigative work are essential since they serve as input data for further analysis [12]. Furthermore, input data and knowledge essential for post-earthquake assessment can be updated through novel methods such as value of information (VoI) analysis. Also, the use of machine-learning techniques can be useful in the seismic assessment of existing structures [13]. Various methods are used for investigative work. Some are non-destructive, such as ultrasound, sonic pulse velocity, operational modal analysis, photogrammetry and close-range remote sensing using unmanned aerial vehicles. Common semi-destructive methods are flat-jacks, shear tests, compression tests and core sampling. Some of the mentioned methods are explained in more detail in the following papers [14,15]. The typical in-plane failure modes used in this research as well as out-of-plane failure modes of URM walls, are explained in more detail in [16,17]. During earthquakes, structures made of different materials and built in various periods are damaged. Therefore, they should be approached properly [18,19]. For example, vernacular architecture combines a cultural tradition and adaptation, resulting in resilient and sustainable systems [20,21]. Hence, it should be strengthened accordingly. On the other hand, more modern structures should be strengthened with a modern solution, but again, their cultural value should be preserved. Therefore, a less invasive and reversible strengthening technique like textile-reinforced mortar (TRM) [8] can be used.

## 2. Input Data and Methods

#### 2.1. Structural Reliability Methods

_{k}and R

_{k}are the main groups of characteristic value of basic variables, while γ

_{R}and γ

_{E}are partial factors for resistance and loads, respectively [40]. Basic variables are all random values described by distribution functions. For example, a Gaussian or lognormal distribution can be assumed for geometric and material properties [41].

_{f}tell us whether the structure is safe, $\beta ={\Phi}^{-1}\cdot \left(Pf\right)$. We can calculate the probability of failure P

_{f}, which can be expressed with the limit state function g (or performance function) and the vector of base variables X. The limit state function g(X) = 0 divides the total space, described by X, into safe and unsafe areas. The mentioned failure probability P

_{f}can be expressed as P

_{f}= P(R − E < 0) where R is the resistance function and E is the load function. The uncertainty and randomness of these events are fundamental principles in the structural reliability theory. These principles are the leading cause of the gradual change in structural reliability verification from a deterministic approach to a more complete probabilistic approach.

#### 2.2. Procedures in the Assessment of Existing Masonry Structures and Target Reliability Levels

#### 2.3. DVM and APFM

#### 2.3.1. Design Value Method (DVM)

_{M}is the final partial factor for resistance, γ

_{m}the partial factor accounting for material variability, γ

_{Rd1}the partial factor accounting for model uncertainty and γ

_{Rd2}the partial factor accounting for geometrical uncertainties [25]. Partial factors accounting for the variability of the material can be determined by Equation (3):

_{k}is the characteristic value for material properties, X

_{d}the design value for material properties, µ

_{x}the mean value of variable X, β the reliability index and V

_{x}the coefficient of variation of variable X. Partial factor accounting for model uncertainty can be determined by Equation (4):

_{ϴR}is the mean value of variable ϴ, ϴ

_{Rd}the random variable, α

_{R}the sensitivity factor for resistance (0.32 according to [49]) and V

_{ϴR}the coefficient of variation of variable ϴ.

_{Rd2}= 1.0 if measurements of a structure indicate insignificant variability of geometrical properties.

#### 2.3.2. Adjusted Partial Factor Method (APFM)

_{x}is the partial factor for resistance, w

_{y}the adjustment factor and γ

_{x,new}the partial factor for resistance for new structures.

_{Rd}is the partial factor accounting for model and geometrical uncertainty (same as in the DVM method), β′ the reliability index for new structures, β″ the reliability index for existing structures, α

_{R}the sensitivity factor for resistance and V

_{x}the coefficient of variation of variable X.

#### 2.4. Failure Modes of URM According to the Current and New Proposal of EN 1998-3

## 3. Case Study

#### 3.1. Case Study Information

#### 3.2. Results and Discussion

_{Rd2}= 1.0 is assumed.

_{M}and the confidence factor CF is used, while in the new version, these two factors are combined in the form of γ

_{Rd}. A more pronounced difference can be seen in the case of bending failure mode. This difference can be explained by the different positions of the partial safety factors within the same formula. In the first case, the partial safety factor reduces the compressive strength of the masonry, while in the second case, it reduces final resistance, resulting in a much smaller value. The values of the partial safety factors also differ slightly between the mentioned standards, but they are not the main reason for this difference in the final results.

## 4. Conclusions

- Considering that the results of the mechanical properties vary considerably in relation to the recommended values, in situ testing is desirable for all structures, especially for more important structures such as critical infrastructure and cultural heritage, for which testing should be mandatory;
- The calculated resistances increase with the increasing complexity of the methods used;
- The observed differences in the results for the resistance of analyzed masonry wall obtained with different methods can vary from 48% to 161% depending on the failure mode;
- Methods such as DVM and APFM can contribute to a more efficient and high-quality renovation of numerous existing masonry structures in earthquake-affected areas without unreasonably greater design efforts.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Gu, J.-B.; Tao, Y.; Xin, R.; Yang, Z.; Shi, Q.-X. Seismic Performance of Multistorey Masonry Structure with Openings Repaired with CFRP Grid. Adv. Civ. Eng.
**2018**, 2018, 4374876. [Google Scholar] [CrossRef] - Canditone, C.; Diana, L.; Formisano, A.; Rodrigues, H.; Vicente, R. Failure Mechanisms and Behaviour of Adobe Masonry Buildings: A Case Study. Eng. Fail. Anal.
**2023**, 150, 107343. [Google Scholar] [CrossRef] - Bernardo, V.; Sousa, R.; Candeias, P.; Costa, A.; Campos Costa, A. Historic Appraisal Review and Geometric Characterization of Old Masonry Buildings in Lisbon for Seismic Risk Assessment. Int. J. Archit. Herit.
**2022**, 16, 1921–1941. [Google Scholar] [CrossRef] - Blagojević, P.; Brzev, S.; Cvetković, R. Seismic Retrofitting of Mid-Rise Unreinforced Masonry Residential Buildings after the 2010 Kraljevo, Serbia Earthquake: A Case Study. Buildings
**2023**, 13, 597. [Google Scholar] [CrossRef] - Bilgin, H.; Leti, M.; Shehu, R.; Özmen, H.B.; Deringol, A.H.; Ormeni, R. Reflections from the 2019 Durrës Earthquakes: An Earthquake Engineering Evaluation for Masonry Typologies. Buildings
**2023**, 13, 2227. [Google Scholar] [CrossRef] - Kržan, M.; Bosiljkov, V. Compression and In-Plane Seismic Behaviour of Ashlar Three-Leaf Stone Masonry Walls. Int. J. Archit. Herit.
**2023**, 17, 829–845. [Google Scholar] [CrossRef] - Šipoš, T.K.; Hadzima-Nyarko, M. Seismic Risk of Croatian Cities Based on Building’s Vulnerability. Teh. Vjesn.
**2018**, 25, 1088–1094. [Google Scholar] [CrossRef] - Kišiček, T.; Stepinac, M.; Renić, T.; Hafner, I.; Lulić, L. Strengthening of Masonry Walls with FRP or TRM. Gradjevinar
**2020**, 72, 937–953. [Google Scholar] - Wilson, R.; Szabó, S.; Funari, M.F.; Pulatsu, B.; Lourenço, P.B. A Comparative Computational Investigation on the In-Plane Behavior and Capacity of Dry-Joint URM Walls. Int. J. Archit. Herit.
**2023**, 2023, 2209776. [Google Scholar] [CrossRef] - Tomić, I.; Penna, A.; DeJong, M.; Butenweg, C.; Correia, A.A.; Candeias, P.X.; Senaldi, I.; Guerrini, G.; Malomo, D.; Wilding, B.; et al. Shake-Table Testing of a Stone Masonry Building Aggregate: Overview of Blind Prediction Study. Bull. Earthq. Eng.
**2023**, 1–43. [Google Scholar] [CrossRef] - EMS Comision Sismologica Europea. Escala Macro Sísmica Europea EMS-98; European Seismological Commission: Luxembourg, 1998; Volume 15. [Google Scholar]
- Hogan, L.S.; Giongo, I.; Walsh, K.Q.; Ingham, J.M.; Dizhur, D. Full-Scale Experimental Pushover Testing of an Existing URM Building. Structures
**2018**, 15, 66–81. [Google Scholar] [CrossRef] - Xu, J.; Feng, D.; Mangalathu, S.; Jeon, J. Data-driven Rapid Damage Evaluation for Life-cycle Seismic Assessment of Regional Reinforced Concrete Bridges. Earthq. Eng. Struct. Dyn.
**2022**, 51, 2730–2751. [Google Scholar] [CrossRef] - Krolo, J.; Damjanović, D.; Duvnjak, I.; Smrkić, M.F.; Bartolac, M.; Košćak, J. Methods for Determining Mechanical Properties of Walls. Gradjevinar
**2021**, 73, 127–140. [Google Scholar] [CrossRef] - Ortega, J.; Stepinac, M.; Lulić, L.; García, M.N.; Saloustros, S.; Aranha, C.; Greco, F. Correlation between Sonic Pulse Velocity and Flat-Jack Tests for the Estimation of the Elastic Properties of Unreinforced Brick Masonry: Case Studies from Croatia. Case Stud. Constr. Mater.
**2023**, 19, e02467. [Google Scholar] [CrossRef] - Tomazevic, M. Earthquake-Resistant Design of Masonry Buildings; Imperial College Press: London, UK, 1999; Volume 1, ISBN 978-1-86094-066-8. [Google Scholar]
- Albanesi, L.; Morandi, P. Lateral Resistance of Brick Masonry Walls: A Rational Application of Different Strength Criteria Based on In-Plane Test Results. Int. J. Archit. Herit.
**2023**, 17, 846–867. [Google Scholar] [CrossRef] - Romão, X.; Bertolin, C. Risk Protection for Cultural Heritage and Historic Centres: Current Knowledge and Further Research Needs. Int. J. Disaster Risk Reduct.
**2022**, 67, 102652. [Google Scholar] [CrossRef] - D’Ayala, D.; Speranza, E. Definition of Collapse Mechanisms and Seismic Vulnerability of Historic Masonry Buildings. Earthq. Spectra
**2003**, 19, 479–509. [Google Scholar] [CrossRef] - Khafizova, A. Vernacular Architectural Preservation of Material and Spiritual Interconnected Cultural Heritage. J. Contemp. Urban Aff.
**2018**, 2, 10–19. [Google Scholar] [CrossRef] - Ramírez Eudave, R.; Ferreira, T.M.; Vicente, R.; Lourenco, P.B.; Peña, F. Parametric and Machine Learning-Based Analysis of the Seismic Vulnerability of Adobe Historical Buildings Damaged After the September 2017 Mexico Earthquakes. Int. J. Archit. Herit.
**2023**, 2023, 2200739. [Google Scholar] [CrossRef] - Vrouwenvelder, T.; Scholten, N. Assessment Criteria for Existing Structures. Struct. Eng. Int.
**2010**, 20, 62–65. [Google Scholar] [CrossRef] - Szabó, S.; Funari, M.F.; Lourenço, P.B. Masonry Patterns’ Influence on the Damage Assessment of URM Walls: Current and Future Trends. Dev. Built Environ.
**2023**, 13, 100119. [Google Scholar] [CrossRef] - Caspeele, R.; Sykora, M.; Allaix, D.L.; Steenbergen, R. The Design Value Method and Adjusted Partial Factor Approach for Existing Structures. Struct. Eng. Int.
**2013**, 23, 386–393. [Google Scholar] [CrossRef] - Fib. Bulletin 80: Partial Factor Methods for Existing Concrete Structures; Recommendation Task Group 3.1; Fib: Lausanne, Switzerland, 2016. [Google Scholar]
- Sousa, H.; Sørensen, J.; Kirkegaard, P. Reliability Analysis of Timber Structures through NDT Data Upgrading Short Term Scientific Mission, COST E55 Action; Aalborg University: Aalborg, Denmark, 2010. [Google Scholar]
- Mohsenian, V.; Padashpour, S.; Hajirasouliha, I. Seismic Reliability Analysis and Estimation of Multilevel Response Modification Factor for Steel Diagrid Structural Systems. J. Build. Eng.
**2020**, 29, 101168. [Google Scholar] [CrossRef] - Androić, B.; Dujmović, D.; Lukačević, I. Razlike u Procjeni Pouzdanosti Uobičajenih i Iznimnih Konstrukcija. J. Croat. Assoc. Civ. Eng.
**2009**, 61, 943–953. [Google Scholar] - Domański, T.; Matysek, P. The Reliability of Masonry Structures—Evaluation Methods for Historical Buildings. Tech. Trans.
**2018**, 115, 91–108. [Google Scholar] [CrossRef] - Schueremans, L.; Van Gemert, D. Reliability Analysis in Structural Masonry Engineering. In Proceedings of the IABSE Colloquium—Saving Buildings in Central and Eastern Europe, Berlin, Germany, 4–5 June 1998. [Google Scholar]
- Sykora, M.; Holicky, M. Probabilistic Model for Masonry Strength of Existing Structures. Eng. Mech.
**2010**, 17, 61–70. [Google Scholar] - Skrzypczak, I.; Kujda, J.; Buda-Ożóg, L. The Use of Probabilistic Methods in Assessing the Reliability of Masonry Structures. Procedia Eng.
**2017**, 193, 160–167. [Google Scholar] [CrossRef] - Lara, C.; Tanner, P.; Zanuy, C.; Hingorani, R. Reliability Verification of Existing RC Structures Using Partial Factor Approaches and Site-Specific Data. Appl. Sci.
**2021**, 11, 1653. [Google Scholar] [CrossRef] - Holický, M.; Jung, K. Reliability Verification of an Existing Reinforced Concrete Slab. TCES
**2019**, 18, 11–14. [Google Scholar] [CrossRef] - Grubišić, M.; Ivošević, J.; Grubišić, A. Reliability Analysis of Reinforced Concrete Frame by Finite Element Method with Implicit Limit State Functions. Buildings
**2019**, 9, 119. [Google Scholar] [CrossRef] - Skejić, D.; Dokšanović, T.; Čudina, I.; Mazzolani, F.M. The Basis for Reliability-Based Mechanical Properties of Structural Aluminium Alloys. Appl. Sci.
**2021**, 11, 4485. [Google Scholar] [CrossRef] - Rücker, D.W.; Hille, D.-I.F.; Rohrmann, D.-I.R. F08a Guideline for the Assessment of Existing Structures; Federal Institute of Materials Research and Testing (BAM): Berlin, Germany, 2006. [Google Scholar]
- EN 1990; Eurocode—Basis of Structural Design. European Committee for Standardization: Brussels, Belgium, 2002.
- Dujmović, D.; Lukačević, I.; Androić, B. Design of Structures According to EN 1990: Theory and Worked Examples; IA Projektiranje: Zagreb, Croatia, 2020. [Google Scholar]
- Diamantidis, D. Reliability Assessment of Existing Structures. Eng. Struct.
**1987**, 9, 177–182. [Google Scholar] [CrossRef] - Tanner, P.; Lara, C.; Bellod, J.L.; Sanz, D. “The Plastic Cathedral”: Innovation to Extend the Service Life of a Heritage Structure. Struct. Concr.
**2020**, 21, 1425–1440. [Google Scholar] [CrossRef] - Borri, A.; Corradi, M.; Castori, G.; De Maria, A. A Method for the Analysis and Classification of Historic Masonry. Bull. Earthq. Eng.
**2015**, 13, 2647–2665. [Google Scholar] [CrossRef] - Croatian Parliment. Law on the Reconstruction of Earthquake-Damaged Buildings in the City of Zagreb, Krapina-Zagorje County and Zagreb County (NN 102/2020); Croatian Parliament: Zagreb, Croatia, 2020.
- ISO 13822; Bases for Design of Structures—Assessment of Existing Structures. ISO: Geneva, Switzerland, 2010.
- ISO 2394; General Principles on Reliability for Structures. ISO: Geneva, Switzerland, 2015.
- CEN/TS 17440:2020; Assessment and Retrofitting of Existing Structures. CEN: Brussels, Belgium, 2020.
- Diamantidis, D. JCSS: Probabilistic Assessment of Existing Structures; RILEM: Paris, France, 2001. [Google Scholar]
- EN 1990-2 CEN/TC 250/SC 10; “EN 1990-2 Basis of Structural and Geotechnical Assessment of Existing Structures” Working Draft. CEN: Brussels, Belgium, 2022.
- Konig, G.; Hosser, D. The Simplified Level II Method and Its Application on the Derivation of Safety Elements for Level I, CEB, Bulletin No. 147; Fib: Lausanne, Switzerland, 1982. [Google Scholar]
- HRN EN 1998-3:2011; Eurocode 8: Design of Structures for Earthquake Resistance—Part 3: Assessment and Retrofitting of Buildings (EN 1998-3:2005+AC:2010). NSAI: Dublin, Ireland, 2005.
- EN 1998; CEN/TC 250/SC 8, Final Document EN1998-3 NEN SC8 PT3, Working Draft. European Committee for Standardization: Brussels, Belgium, 2018.
- Joanna, K. Analysis of Limit State of Load Resistance and Reliability of Masonry Structures Made of AAC Blocks. MATEC Web Conf.
**2019**, 262, 02001. [Google Scholar] [CrossRef] - Lulić, L.; Stepinac, M.; Bartolac, M.; Lourenço, P.B. Review of the Flat-Jack Method and Lessons from Extensive Post-Earthquake Research Campaign in Croatia. Constr. Build. Mater.
**2023**, 384, 131407. [Google Scholar] [CrossRef] - Stepinac, M.; Lulić, L.; Damjanović, D.; Duvnjak, I.; Bartolac, M.; Lourenço, P.B. Experimental Evaluation of Unreinforced Brick Masonry Mechanical Properties by the Flat-Jack Method—An Extensive Campaign in Croatia. Int. J. Archit. Herit.
**2023**, 2023, 2208542. [Google Scholar] [CrossRef] - HRN EN 1996-1-1:2012; Eurocode 6: Design of Masonry Structures—Part 1-1: General Rules for Reinforced and Unreinforced Masonry Structures (EN 1996-1-1:2005+A1:2012). CEN: Brussels, Belgium, 2005.
- Schueremans, L.; Van Gemert, D. Probability Density Functions for Masonry Material Parameters—A Way to Go? In Proceedings of the Structural Analysis of Historical Constructions—Possibilities of Numerical and Experimental Techniques; Lourenco, P.B., Roca, P., Modena, C., Agrawal, S., Eds.; Macmillan India Ltd.: New Delhi, India, 2006; pp. 921–928. [Google Scholar]
- Graubner, C.A.; Brehm, E. JCSS Probabilistic Model Code Part 3: Resistance Models. In Joint Committee 12th Draft on Structural Safety; JCSS: Zurich, Switzerland, 2011. [Google Scholar]
- Madrigal, J.M.P. Some Notes about Architecture, Urbanism and Economy. J. Contemp. Urban Aff.
**2018**, 2, 1–11. [Google Scholar] [CrossRef] - Churilov, S.; Dumova-Jovanoska, E. In-Plane Shear Behaviour of Unreinforced and Jacketed Brick Masonry Walls. Soil Dyn. Earthq. Eng.
**2013**, 50, 85–105. [Google Scholar] [CrossRef] - Hafner, I.; Kišiček, T.; Gams, M. Review of Methods for Seismic Strengthening of Masonry Piers and Walls. Buildings
**2023**, 13, 1524. [Google Scholar] [CrossRef] - Acikgoz, E.K. Catching Up With BIM: A Curriculum Re-Design Strategy. J. Contemp. Urban Aff.
**2018**, 2, 40–48. [Google Scholar] [CrossRef] - Sassu, M.; Stochino, F.; Mistretta, F. Assessment Method for Combined Structural and Energy Retrofitting in Masonry Buildings. Buildings
**2017**, 7, 71. [Google Scholar] [CrossRef] - Valluzzi, M.R.; Saler, E.; Vignato, A.; Salvalaggio, M.; Croatto, G.; Dorigatti, G.; Turrini, U. Nested Buildings: An Innovative Strategy for the Integrated Seismic and Energy Retrofit of Existing Masonry Buildings with CLT Panels. Sustainability
**2021**, 13, 1188. [Google Scholar] [CrossRef] - D’Urso, S.; Cicero, B. From the Efficiency of Nature to Parametric Design. A Holistic Approach for Sustainable Building Renovation in Seismic Regions. Sustainability
**2019**, 11, 1227. [Google Scholar] [CrossRef] - Privitera, R.; La Rosa, D. Reducing Seismic Vulnerability and Energy Demand of Cities through Green Infrastructure. Sustainability
**2018**, 10, 2591. [Google Scholar] [CrossRef]

**Figure 3.**Flat-jack test setup for determining (

**a**) vertical stress state, (

**b**) modulus of elasticity and (

**c**) shear strength.

**Figure 4.**Emergency interventions taken to ensure the stability of (

**a**) damaged vaults and (

**b**) staircases.

**Figure 6.**Results for URM wall resistance according to the current EN 1998-3 and the new proposal for EN 1998-3.

**Table 1.**Structural reliability methods adopted from [37].

Method | Approach |
---|---|

Allowable stress | Deterministic |

Plastic design | |

Partial safety factors | Semi-probabilistic |

Analytical and numerical | Probabilistic |

Simulation |

**Table 2.**Structural assessment levels for existing (masonry) structures adopted from [37].

Level | Reliability Class |
---|---|

1 | only visual assessment of damage and methods like MQI [42] |

2 | simple hand calculations |

3 | simple small-scale model-based assessment with assumed mechanical properties from the literature (no in situ tests) |

4 | more detailed large-scale model-based assessment with real mechanical properties obtained through detailed in situ tests |

5 | assessment considering target reliability and modified parameters of the structure through VoI analysis or Bayesian updating |

6 | probabilistic assessment through full probabilistic analysis (in situ tests) |

**Table 3.**Target reliability indices for ultimate limit state adopted from [37].

Service Life | β | Consequences of Failure |
---|---|---|

50 years | 2.3 | very low |

50 years | 3.1 | low |

50 years | 3.8 | medium |

50 years | 4.3 | high |

**Table 4.**Target reliability indices adopted from [38].

Service Life | β | Reliability Class |
---|---|---|

50 years | 3.3 | 1 |

50 years | 3.8 | 2 |

50 years | 4.3 | 3 |

EN 1998-3 [50] | New Proposal EN 1998-3 [51] | ||
---|---|---|---|

${f}_{\mathrm{v}}={f}_{\mathrm{v}0}+\mu \cdot {\sigma}_{\mathrm{d}}$ | (7) | ${V}_{1,\mathrm{Rd}}=\frac{{f}_{\mathrm{v}}}{{\gamma}_{\mathrm{Rd}}}\cdot {L}_{\mathrm{c}}\cdot {t}_{\mathrm{w}}$ | (9) |

${V}_{1,\mathrm{Rd}}=\frac{{f}_{\mathrm{v}}}{CF\cdot {\gamma}_{\mathrm{M}}}\cdot {L}_{\mathrm{c}}\cdot {t}_{\mathrm{w}}$ | (8) | ||

${f}_{\mathrm{d}}=\frac{{f}_{\mathrm{m}}}{CF\cdot {\gamma}_{\mathrm{M}}}$ | (10) | $\nu =\frac{N}{L\cdot {t}_{\mathrm{w}}\cdot {f}_{\mathrm{m}}}$ | (13) |

$\nu =\frac{N}{L\cdot {t}_{\mathrm{w}}\cdot {f}_{\mathrm{d}}}$ | (11) | ||

${V}_{2,\mathrm{Rd}}=\frac{L\cdot N}{2\cdot {h}_{0}}\cdot \left(1-1,15\cdot \nu \right)$ | (12) | ${V}_{2,\mathrm{Rd}}=\frac{1}{{\gamma}_{\mathrm{Rd}}}\cdot \frac{L\cdot N}{2\cdot {h}_{0}}\cdot \left(1-1,15\cdot \nu \right)$ | (14) |

${V}_{3,\mathrm{Rd}}=L\cdot {t}_{\mathrm{w}}\cdot \frac{{f}_{\mathrm{t}}}{{\gamma}_{\mathrm{M}}\cdot CF\cdot b}\cdot \sqrt{1+\frac{{\gamma}_{\mathrm{M}}\cdot CF\cdot {\sigma}_{0}}{{f}_{\mathrm{t}}}}$ | (15) | ${V}_{3,\mathrm{Rd}}=L\cdot {t}_{\mathrm{w}}\cdot \frac{{f}_{\mathrm{t}}}{{\gamma}_{\mathrm{Rd}}\cdot b}\cdot \sqrt{1+\frac{{\sigma}_{0}}{{f}_{\mathrm{t}}}}$ | (16) |

/ | ${V}_{4,\mathrm{Rd}}=\frac{L\cdot {t}_{\mathrm{w}}}{{\gamma}_{\mathrm{Rd}}\cdot b}\cdot \left(\frac{{f}_{\mathrm{v}0}}{(1+{\mu}_{\mathrm{j}}\cdot \varphi )}+\frac{{\mu}_{\mathrm{j}}}{1+{\mu}_{\mathrm{j}}\cdot \varphi}\cdot {\sigma}_{0}\right)$ | (17) | |

where: f _{v}—shear strength of masonry;f _{v0}—initial shear strength of the masonry (independent of vertical force);μ—coefficient of friction (tanφ), where φ is the angle of internal friction; σ _{d}—design vertical compressive stress;γ _{M}—partial safety factor for the material;CF—confidence factor; L _{c}—length of the masonry in compression;t _{w}—wall thickness;L—wall length (total); N—vertical compressive force; h _{0} = h/2—the height of the inflexion point (fixed-fixed boundary conditions assumed);h—height of the wall; ν—normalized axial force; f _{d}—design compressive strength of masonry;f _{m}—compressive strength of masonry;f _{t}—tensile strength of masonry;σ _{0}—average vertical compressive stress (over the entire surface of the wall);b—geometry factor (b = h/L, but in between 1 and 1.5); γ _{Rd}—partial safety factor;µ _{j}—local coefficient of friction of the joint (can be taken as 0.6);Ø—clamping coefficient. |

Assumed Properties | Measured Properties | |
---|---|---|

h [m] | 5.25 | 5.25 |

h_{0} [m] | 2.63 | 2.63 |

L [m] | 11 | 11 |

L_{c} [m] | 11 | 11 |

b [/] | 1 | 1 |

t_{w} [m] | 0.45 | 0.45 |

N [kN] | 2200 | 2970 |

e [m] | 1.1 | 1.1 |

M [kNm] | 2420 | 3267 |

V [kN] | 920 | 1244 |

f_{vo} [MPa] | 0.2 (0.16 *) | 0.22 |

μ [/] | 0.4 (0.5 *) | 0.45 |

σ_{d} [MPa] | 0.45 | 0.6 |

CF [/] | 1.35 | 1.0 |

γ_{M} [/] | 1.5 | 1.5 |

γ_{Rd} [/] | ** | ** |

K [/] | 0.45 | 0.45 |

f_{b} [MPa] | 10 | 11 |

f_{m} [MPa] | 2.5 | 1.93 |

f_{t} [MPa] | 0.15 (0.114 *) | 0.15 |

μ_{j} [/] | 0.6 | 0.45 |

Ø [/] | 1 | 1 |

**Table 7.**Statistical information on the basic variables obtained by on-site testing and factors obtained with DVM.

Variable | Distr. | Mean | CoV | γ_{m} | γ_{Rd} | γ_{M} = γ_{m} × γ_{Rd} |
---|---|---|---|---|---|---|

f_{b} [MPa] | Lognormal | 11 | 0.44 | 1.17 | 1.04 | 1.22 |

f_{m} [MPa] | Lognormal | 1.93 | 0.16 | 1.06 | 1.04 | 1.10 |

f_{v0} [MPa] | Lognormal | 0.22 | 0.29 | 1.14 | 1.25 | 1.43 |

μ [/] | Lognormal | 0.45 | 0.38 | 1.11 | 1.25 | 1.39 |

**Table 8.**Statistical information on the basic variables obtained by on-site testing and factors obtained with APFM.

Variable | Distr. | Mean | CoV | γ_{Rd}(β′) | γ_{Rd}(β″) | w_{y} |
---|---|---|---|---|---|---|

f_{b} [MPa] | Lognormal | 11 | 0.44 | 1.06 | 1.04 | 0.62 |

f_{m} [MPa] | Lognormal | 1.93 | 0.16 | 1.06 | 1.04 | 0.83 |

f_{v0} [MPa] | Lognormal | 0.22 | 0.29 | 1.44 | 1.25 | 0.59 |

μ [/] | Lognormal | 0.45 | 0.38 | 1.44 | 1.25 | 0.64 |

Phase I. | EN 1998-3 [50] | New Proposal of EN 1998-3 [51] |
---|---|---|

Shear sliding | 929 kN | 1147 kN |

Bending | 3003 kN | 1822 kN |

Diagonal flat | 975 kN | 810 kN |

Diagonal stepped | / | 783 kN |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lulić, L.; Lukačević, I.; Skejić, D.; Stepinac, M.
Assessment of Existing Masonry Resistance Using Partial Factors Approaches and Field Measurements. *Buildings* **2023**, *13*, 2790.
https://doi.org/10.3390/buildings13112790

**AMA Style**

Lulić L, Lukačević I, Skejić D, Stepinac M.
Assessment of Existing Masonry Resistance Using Partial Factors Approaches and Field Measurements. *Buildings*. 2023; 13(11):2790.
https://doi.org/10.3390/buildings13112790

**Chicago/Turabian Style**

Lulić, Luka, Ivan Lukačević, Davor Skejić, and Mislav Stepinac.
2023. "Assessment of Existing Masonry Resistance Using Partial Factors Approaches and Field Measurements" *Buildings* 13, no. 11: 2790.
https://doi.org/10.3390/buildings13112790