Reliability Analysis of Unstabilized Rammed Earth Under Seismic Loads using Monte Carlo Simulation
Reliability Analysis of Unstabilized Rammed Earth Under Seismic Loads using Monte Carlo Simulation |
||
|
||
© 2024 by IJETT Journal | ||
Volume-72 Issue-12 |
||
Year of Publication : 2024 | ||
Author : Chaymae Salhi, Mouna El Mkhalet, Nouzha Lamdouar |
||
DOI : 10.14445/22315381/IJETT-V72I12P134 |
How to Cite?
Chaymae Salhi, Mouna El Mkhalet, Nouzha Lamdouar, "Reliability Analysis of Unstabilized Rammed Earth Under Seismic Loads using Monte Carlo Simulation," International Journal of Engineering Trends and Technology, vol. 72, no. 12, pp. 397-417, 2024. Crossref, https://doi.org/10.14445/22315381/IJETT-V72I12P134
Abstract
Rammed earth is experiencing a resurgence of interest due primarily to its environmental properties. This study analyzes the reliability of unstabilized rammed earth in a one-story building under seismic loads, an aspect never addressed in previous research, focusing on out-of-plane bending moments according to the Moroccan Seismic Regulation for Earth Constructions (RPCTerre 2011) using Monte Carlo simulation. Load and resistance parameters were considered dom variables, with additional parameters varying. A Python code was developed for the simulations. The results provide recommendations on wall dimensions and minimum compressive strength for different seismic zones. These recommendations were compared to existing guidelines with seismic provisions, revealing some commonalities. However, the comparison also highlights suboptimal thickness values for low and low seismicity zones and insufficient compressive strength recommendations. The study concludes that higher seismicity levels and wider walls require greater minimum thicknesses. This study addresses a crucial gap in the literature by offering a probabilistic analysis of rammed earth structures under seismic loads, providing insights to enhance design guidelines and improve the safety and sustainability of earth construction in seismic regions.
Keywords
Monte Carlo simulation, Seismic loads, Structural reliability analysis, Unstabilized rammed earth.
References
[1] J.E. Aubert, et al. “An Earth Block with a Compressive Strength Higher than 45MPa!” Construction and Building Materials, vol. 47, pp. 366‑369, 2013.
[CrossRef] [Google Scholar] [Publisher Link]
[2] Luisa María Gil-Martín, Manuel Alejandro Fernández-Ruiz, and Enrique Hernández-Montes, “Mechanical Characterization and Elastic Stiffness Degradation of Unstabilized Rammed Earth,” Journal of Building Engineering, vol. 56, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[3] Mohamad M. Hallal, Salah Sadek, and Shadi S. Najjar, “Evaluation of Engineering Characteristics of Stabilized Rammed-Earth Material Sourced from Natural Fines-Rich Soil,” Journal of Materials in Civil Engineering, vol. 30, no. 11, 2018.
[CrossRef] [Google Scholar] [Publisher Link]
[4] Lassana Bakary Traoré et al., “Experimental Assessment of Freezing-Thawing Resistance of Rammed Earth Buildings,” Construction and Building Materials, vol. 274, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[5] Matthew Hall, and Youcef Djerbib, “Rammed Earth Sample Production: Context, Recommendations and Consistency,” Construction and Building Materials, vol. 18, no. 4, pp. 281‑286, 2004.
[CrossRef] [Google Scholar] [Publisher Link]
[6] Luís Mateus et al., “Mineralogical and Mechanical Characterization of Rammed Earth External Renderings of the South of Portugal “, Construction and Building Materials, vol. 225, pp. 1160‑1169, 2019.
[CrossRef] [Google Scholar] [Publisher Link]
[7] Quou Bao Bui et al., “Compression Behaviour of Non-Industrial Materials in Civil Engineering by Three Scale Experiments: The Case of Rammed Earth,” Materials and Structures, vol. 42, pp. 1101‑1116, 2009.
[CrossRef] [Google Scholar] [Publisher Link]
[8] Nowamooz Hossein, and Cyrille Chazallon, “Finite Element Modelling of a Rammed Earth Wall,” Construction and Building Materials, vol. 25, no 4, pp. 2112‑2121, 2011.
[CrossRef] [Google Scholar] [Publisher Link]
[9] Tiegang Zhou et al., “Investigation of Intralayer and Interlayer Shear Properties of Stabilized Rammed Earth by Direct Shear Tests,” Construction and Building Materials, vol. 367, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[10] Pierre Gerard et al., “A Unified Failure Criterion for Unstabilized Rammed Earth Materials Upon Varying Relative Humidity Conditions,” Construction and Building Materials, vol. 95, pp. 437‑447, 2015.
[CrossRef] [Google Scholar] [Publisher Link]
[11] Vasilios Maniatidis, and Peter Walker, A Review of Rammed Earth Construction, Paper presented at DTI Project Report, Bath, 2003.
[Google Scholar] [Publisher Link]
[12] T.-T. Bui et al., “Failure of Rammed Earth Walls: From Observations to Quantifications,” Construction and Building Materials, vol. 51, pp. 295‑302, 2014.
[CrossRef] [Google Scholar] [Publisher Link]
[13] Fernando Ávila, Esther Puertas, and Rafael Gallego, “Characterization of the Mechanical and Physical Properties of Unstabilized Rammed Earth: A Review, Construction and Building Materials, vol. 270, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[14] Lorenzo Miccoli, Urs Müller and Patrick Fontana, “Mechanical Behaviour of Earthen Materials: A Comparison Between Earth Block Masonry, Rammed Earth and Cob,” Construction and Building Materials, vol. 61, pp. 327‑339, 2014.
[CrossRef] [Google Scholar] [Publisher Link]
[15] Fernando Ávila, Esther Puertas, and Rafael Gallego, “Characterization of the Mechanical and Physical Properties of Stabilized Rammed Earth: A Review,” Construction and Building Materials, vol. 325, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[16] Johan Vyncke, Laura Kupers, and Nicolas Denies, “Earth as Building Material - an Overview of RILEM Activities and Recent Innovations in Geotechnics,” 2nd International Congress on Materials & Structural Stability (CMSS-2017), vol. 149, 2018.
[CrossRef] [Google Scholar] [Publisher Link]
[17] Maria Idália Gomes, Teresa Diaz Gonçalves, and Paulina Faria, “Unstabilized Rammed Earth: Characterization of Material Collected from Old Constructions in South Portugal and Comparison to Normative Requirements,” International Journal of Architectural Heritage, vol. 8, no. 2, pp. 185‑212, 2014.
[CrossRef] [Google Scholar] [Publisher Link]
[18] Ehsan Kianfar, and Vahab Toufigh, “Reliability and Uncertainty in Analysis of Rammed Earth Walls,” Proceedings of the 3rd World Congress on New Technologies (NewTech'17), Rome, Italy, 2017.
[CrossRef] [Google Scholar]
[19] Juan C. Reyes et al., “Shear Behavior of Adobe and Rammed Earth Walls of Heritage Structures,” Engineering Structures, vol. 174, pp. 526‑537, 2018.
[CrossRef] [Google Scholar] [Publisher Link]
[20] Luisa María Gil-Martín et al., “Mechanical Characterization and Elastic Stiffness Degradation of Unstabilized Rammed Earth,” Journal of Building Engineering, vol. 56, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[21] Alexandra H. Meek, Christopher T.S. Beckett, and Mohamed Elchalakani, “Alternative Stabilised Rammed Earth Materials Incorporating Recycled Waste and Industrial By-Products: Durability with and without Water Repellent,” Construction and Building Materials, vol. 265, 2020.
[CrossRef] [Google Scholar] [Publisher Link]
[22] R. Sri Bhanupratap Rathod, and B.V. Venkatarama Reddy, “Behaviour of Plain and Fibre Reinforced Cement Stabilised Rammed Earth Under Compression, Tension and Shear,” Construction and Building Materials, vol. 344, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[23] R. El Nabouch et al., “Numerical Modeling of Rammed Earth Structures: Analyses and Recommendation,” Academic Journal of Civil Engineering, vol. 33, no. 2, pp. 72-79, 2015.
[CrossRef] [Google Scholar] [Publisher Link]
[24] Thi-Loan Bu et al., “Out-of-Plane Behavior of Rammed Earth Walls Under Seismic Loading: Finite Element Simulation,” Structures, vol. 24, pp. 191‑208, 2020.
[CrossRef] [Google Scholar] [Publisher Link]
[25] Ana Perić et al., “Experimental Campaigns on Mechanical Properties and Seismic Performance of Unstabilized Rammed Earth-A Literature Review,” Buildings, vol. 11, no. 8, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[26] Moein Ramezanpour, Abolfazl Eslami, and Hamid Ronagh, “Seismic Performance of Stabilised/Unstabilised Rammed Earth Walls,” Engineering Structures, vol. 245, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[27] Mehmet Emin Arslan, Mehmet Emiroğlu, and Ahmet Yalama, “Structural Behavior of Rammed Earth Walls Under Lateral Cyclic Loading: A Comparative Experimental Study,” Construction and Building Materials, vol. 133, pp. 433‑442, 2017.
[CrossRef] [Google Scholar] [Publisher Link]
[28] Tiegang Zhou et al., “Seismic Performance Test of Rammed Earth Wall with Different Structural Columns” Advances in Structural Engineering, vol. 24, no. 1, pp. 107‑118, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[29] Sima Samadianfard, and Vahab Toufigh, “Stabilization Effect on the Hygrothermal Performance of Rammed Earth Materials,” Construction and Building Materials, vol. 409, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[30] Alessia Emanuela Losini et al., “Extended Hygrothermal Characterization of Unstabilized Rammed Earth for Modern Construction,” Construction and Building Materials, vol. 409, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[31] Miguel Rocha, Paulina Faria, and António Sousa Gag, “Conservation of Defensive Military Structures Built with Rammed Earth,” Buildings, vol. 14, no. 1, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[32] Wenting Chen et al., “An In-Situ Conservation Method of the Rammed Earth Sites Using a New Silica Protective Agent,” Construction and Building Materials, vol. 452, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[33] Ehsan Kianfar, and Vahab Toufigh,“Reliability Analysis of Rammed Earth Structures,” Construction and Building Materials, vol. 127, pp. 884‑895, 2016.
[CrossRef] [Google Scholar] [Publisher Link]
[34] Ministry of National Territorial Planning, Urban Planning, Housing and City Policy, Ministry of National Territorial Planning, Urban Planning, Housing and City Policy, 2021. [Online]. Available: https://www.mhpv.gov.ma/fr/5121-2/ [35] Vikas Khare et al., Chapter 6 - Reliability Assessment Model, Tidal Energy Systems, Design, Optimization and Control, pp. 295‑330, 2019.
[CrossRef] [Google Scholar] [Publisher Link]
[36] Zhibao Zheng, Hongzhe Dai, and Michael Beer, “Efficient Structural Reliability Analysis Via a Weak-Intrusive Stochastic Finite Element Method,” Probabilistic Engineering Mechanics, vol. 71, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[37] Carl Rollenhagen, “Chapter 13 - Safety Culture and Safety Quality,” Radioactivity in the Environment vol. 19, pp. 215‑237, 2013.
[CrossRef] [Google Scholar] [Publisher Link]
[38] Paolo Gardoni, Risk and Reliability Analysis, Theory and Applications Springer Series in Reliability Engineering, Springer, Cham, pp. 3 -24, 2017.
[CrossRef] [Google Scholar] [Publisher Link]
[39] Peng Huang et al., “Positioning Accuracy Reliability Analysis of Industrial Robots Based on Differential Kinematics and Saddlepoint Approximation,” Mechanism and Machine Theory, vol. 162, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[40] Jinhui Wu, Yourui Tao, and Xu Han,” Polynomial Chaos Expansion Approximation for Dimension-Reduction Model-Based Reliability Analysis Method and Application to Industrial Robots,” Reliability Engineering & System Safety, vol. 234, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[41] Dequan Zhang et al., “Kinematic Trajectory Accuracy Reliability Analysis for Industrial Robots Considering Intercorrelations Among Multi-Point Positioning Errors,” Reliability Engineering & System Safety, vol. 229, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[42] Yang Liu et al., “Reliability Analysis for Stall Warning Methods in an Axial Flow Compressor,” Aerospace Science and Technology, vol. 115, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[43] Hong Zhang et al., “Active Extremum Kriging-Based Multi-Level Linkage Reliability Analysis and Its Application in Aeroengine Mechanism Systems,” Aerospace Science and Technology, vol. 131, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[44] Jianqiao Liu et al., “A Study on Assigning Performance Shaping Factors of the SPAR-H Method for Adequacy Human Reliability Analysis of Nuclear Power Plants,” International Journal of Industrial Ergonomics, vol. 81, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[45] Giovanni Roma et al., “A Bayesian Framework of Inverse Uncertainty Quantification with Principal Component Analysis and Kriging for the Reliability Analysis of Passive Safety Systems,” Nuclear Engineering and Design, vol. 379, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[46] Bowen Zou et al., “Reliability Analysis and Allocation: Development of A Hierarchical Structure Modeling Platform in I&C system Software Life Cycle,” Nuclear Engineering and Design, vol. 328, pp. 345‑352, 2018.
[CrossRef] [Google Scholar] [Publisher Link]
[47] Asle Natskår, and Torgeir Moan, “Structural Reliability Analysis of a Seafastening Structure for Sea Transport of Heavy Objects,” Ocean Engineering, vol. 235, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[48] Fanglong Yin et al., “Non-Probabilistic Reliability Analysis and Design Optimization for Valve-Port Plate Pair of Seawater Hydraulic Pump for Underwater Apparatus,” Ocean Engineering, vol. 163, pp. 337‑347,2018.
[CrossRef] [Google Scholar] [Publisher Link]
[49] B. Zhu, H. Pei, et Q. Yang, “Reliability Analysis of Submarine Slope Considering the Spatial Variability of the Sediment Strength Using Random Fields,” Applied Ocean Research, vol. 86, pp. 340‑350, 2019.
[CrossRef] [Google Scholar] [Publisher Link]
[50] Changqi Luo et al., “An Enhanced Uniform Simulation Approach Coupled with SVR for Efficient Structural Reliability Analysis,” Reliability Engineering & System Safety, vol. 237, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[51] Mariano Angelo Zanini, and Lorenzo Hofer, “Center and Characteristic Seismic Reliability as New Indexes for Accounting Uncertainties in Seismic Reliability Analysis,” Soil Dynamics and Earthquake Engineering, vol. 123, pp. 110‑123, 2019.
[CrossRef] [Google Scholar] [Publisher Link]
[52] Haiyang Song, and Jian Zhang, “Structural Reliability Analysis Based on Interval Analysis Method in Statistical Energy Analysis Framework,” Mechanics Research Communications, vol. 117, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[53] Changqi Luo et al., “Hybrid Enhanced Monte Carlo Simulation Coupled with Advanced Machine Learning Approach for Accurate and Efficient Structural Reliability Analysis,” Computer Methods in Applied Mechanics and Engineering, vol. 388, 2022. [CrossRef] [Google Scholar] [Publisher Link] [54] Haibin Li and Xiaobo Nie, “Structural Reliability Analysis with Fuzzy Random Variables Using Error Principle,” Engineering Applications of Artificial Intelligence, vol. 67, pp. 91‑99, 2018.
[CrossRef] [Google Scholar] [Publisher Link]
[55] Chao Dang et al., “Structural Reliability Analysis by Line Sampling: A Bayesian Active Learning Treatment,” Structural Safety, vol. 104, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[56] Chao Dang et al., “Parallel Adaptive Bayesian Quadrature for Rare Event Estimation,” Reliability Engineering & System Safety, vol. 225, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[57] Robert L. Harrison, “Introduction to Monte Carlo Simulation,” AIP Conference Proceedings, vol. 1204, no. 1, pp. 17‑21, 2010.
[CrossRef] [Google Scholar] [Publisher Link]
[58] Ansel C. Ugural, Plates and Shells, Theory and Analysis, 4th ed., Boca Raton, 2017.
[CrossRef] [Google Scholar] [Publisher Link]
[59] Rui Li et al., “A Unified Analytic Solution Approach to Static Bending and Free Vibration Problems of Rectangular Thin Plates,” Sciemtific Reports, vol. 5, 2015.
[CrossRef] [Google Scholar] [Publisher Link]
[60] Madhujit Mukhopadhyay, “A General Solution for Rectangular Plate Bending,” Forschung im Ingenieurwesen A, vol. 45, pp. 111‑118, 1979.
[CrossRef] [Google Scholar] [Publisher Link]
[61] Ömer Civalek, “Harmonic Differential Quadrature-Finite Differences Coupled Approaches for Geometrically Nonlinear Static and Dynamic Analysis of Rectangular Plates on Elastic Foundation,” Journal of Sound and Vibration, vol. 294, no. 4, pp. 966‑980, 2006.
[CrossRef] [Google Scholar] [Publisher Link]
[62] M.-H. Huang, and D.P. Thambiratnam, “Analysis of Plate Resting on Elastic Supports and Elastic Foundation by Finite Strip Method,” Computers & Structures, vol. 79, no. 29, pp. 2547‑2557, 2001.
[CrossRef] [Google Scholar] [Publisher Link]
[63] H. Nguyen-Xuan et al., “A Smoothed Finite Element Method for Plate Analysis,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 13, pp. 1184‑1203, 2008.
[CrossRef] [Google Scholar] [Publisher Link]
[64] Linxiong Hong et al., “Portfolio Allocation Strategy for Active Learning Kriging-Based Structural Reliability Analysis,” Computer Methods in Applied Mechanics and Engineering, vol. 412, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[65] João B. Cardoso et al., “Structural Reliability Analysis Using Monte Carlo Simulation and Neural Networks,” Advances in Engineering Software, vol. 39, no. 6, pp. 505‑513, 2008.
[CrossRef] [Google Scholar] [Publisher Link]
[66] K. Fedra, Environmental Modeling Under Uncertainty: Monte Carlo Simulation, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1983.
[Google Scholar] [Publisher Link]
[67] Eduard Ventsel, and Theodor Krauthammer, Thin Plates and Shells: Theory: Analysis, and Applications, 1st ed., Boca Raton, 2001.
[CrossRef] [Google Scholar] [Publisher Link]
[68] Rudolph Szilard, Theories and Applications of Plate Analysis: Classical, Numerical and Engineering Methods, John Wiley & Sons, 2004.
[CrossRef] [Google Scholar] [Publisher Link]
[69] David Thompson, Charles Augarde, and Juan Pablo Osorio, “A Review of Current Construction Guidelines to Inform the Design of Rammed Earth Houses in Seismically Active Zones,” Journal of Building Engineering, vol. 54, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[70] M. H. Faber, and J.D. Sørensen, “Reliability-Based Code Calibration: The JCSS Approach,” 9th International Conference on Applications of Statistics and Probability in Civil Engineering - San Francisco, Californien, United States, vol. 2, pp. 927-935, 2003.
[Google Scholar] [Publisher Link]
[71] Standards New Zealand, “NZS 4297: 2020 Engineering Design of Earth Buildings,” Standards New Zealand, 2020.
[Google Scholar] [Publisher Link]
[72] Peter Walker, The Australian Earth Building Handbook, Standards Australia International, 2001.
[Google Scholar] [Publisher Link]
[73] Issuing Agency: Construction Industries Division of The Regulation and Licensing Department, Title 14 - Housing and Construction, Chapter - 7 Building Codes General, Part 4 - New Mexico Earthen Building Materials Code, 2016. [Online]. Available : https://www.srca.nm.gov/parts/title14/14.007.0004.html
[74] Eliana Baglioni, Luisa Rovero, and U. Tonietti, The Moroccan Drâa Valley Earthen Architecture: Pathology and Intervention Criteria, CRC Press, 2012.
[Google Scholar] [Publisher Link]
[75] BABET - Office of ParaSeismic Studies, French Standard NF P 06-001, 2024. [Online]. Available : http://babet.sarl.free.fr/surcharges.htm
[76] Vittoria Strazzeri, and Ali Karrech, “Qualitative and Quantitative Study to Assess the Use of Rammed Earth Construction Technology in Perth and the South-West of Western Australia,” Cleaner Materials, vol. 7, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[77] Ranime El-Nabouch et al., “Assessing the In-Plane Seismic Performance of Rammed Earth Walls by Using Horizontal Loading Tests,” Engineering Structures, vol. 145, pp. 153‑161, 2017.
[CrossRef] [Google Scholar] [Publisher Link]
[78] Guidelines For Earthquake Resistant Design, Construction, And Retrofitting of Buildings in Afghanistan (English), United Nations Centre for Regional Development, 2003. [Online]. Available: https://uncrd.un.org/content/pub-guidelines-earthquake-resistant-design-construction-afghanistan
[79] Indian Standard: Improving Earthquake Resistance of Earthen Buildings - Guidelines, Bureau of Indian Standards, The National Standards Body of India, IS 13837: 1993, 2023. [Online]. Available: https://www.services.bis.gov.in/php/BIS_2.0/bisconnect/knowyourstandards/Indian_standards/isdetails/
[80] Guidelines For Earthquake Resistant Building Construction: Earthen Building (Eb), Nepal National Building Code NBC 204: 2015. [Online]. Available: http://www.moud.gov.np/search-page?search-field=Guidelines+for+Earthquake+Resistant+Building+Construction%3A+Earthen+Building
[81] Reza Allahvirdizadeh, Daniel V. Oliveira and, Rui A. Silva, “Numerical Modeling of the Seismic Out-Of-Plane Response of a Plain and TRM-Strengthened Rammed Earth Subassembly,” Engineering Structures, vol. 193, pp. 43‑56, 2019.
[CrossRef] [Google Scholar] [Publisher Link]
[82] Quoc-Bao Bui, Ali Limam, and Tan-Trung Bui, “Dynamic Discrete Element Modelling for Seismic Assessment of Rammed Earth Walls,” Engineering Structures, vol. 175, pp. 690‑699, 2018.
[CrossRef] [Google Scholar] [Publisher Link]
[83] Alric Lucas, Guy Pluvinage, and Capelle Julien, “Reliability Index of a Pipe Transporting Hydrogen Submitted to Seismic Displacement,” International Journal of Pressure Vessels and Piping, vol. 208, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[84] Mark G. Stewart, and Stephen Lawrence, “Structural Reliability of Masonry Walls in Flexure,” Masonry International, vol. 15, no.2, pp. 48-52, 2002.
[Google Scholar] [Publisher Link]
[85] Optimal Number of Trials for Monte Carlo Simulation, Valuation Research Corporation (VRC), 2017. [Online]. Available: https://www.valuationresearch.com/insights/special-report-optimal-number-trials-monte-carlo-simulation/ [86] Sheldon M. Ross, Chapter 2 - Descriptive Statistics, In Introduction to Probability and Statistics for Engineers and Scientists (Fourth Edition), pp. 9‑53, 2009.
[CrossRef] [Publisher Link]
[87] Smail Mahdi and Myrtene Cenac, “Estimating Parameters of Gumbel Distribution Using the Methods of Moments, Probability Weighted Moments and Maximum Likelihood,” Mathematics Magazine: Theory and Applications, vol. 12, no. 1‑2, pp. 151‑156, 2012.
[CrossRef] [Google Scholar] [Publisher Link]
[88] Numpy.random.lognormal - NumPy, 2024. [Online]. Available: https://numpy.org/doc/stable/reference/random/generated/numpy.random.lognormal.html
[89] Robert Kissell, and Jim Poserina, Chapter 4 - Advanced Math and Statistics, Optimal Sports Math, Statistics, and Fantasy, pp. 103‑135, 2017.
[CrossRef] [Publisher Link]