Point Load Test Research Articles

Assessment of rock slopes stability using stereographic projection along Wadi Shorshirin in Zurbatiya region/ east of Iraq

Wadi Shorsirin is located in the Wasit governorate east of Iraq, it is considered a promising tourist site for the people who live close to this area. The research aims to study the rocky slopes and assess their stability, along the two banks of Wadi Shorhirine, by choosing five stations distributed where the failure occurs and to determine the expected rock failure along this valley. In addition, determine the factors that affect stability, such as discontinuities, rock resistance, and weathering to take them into account when establishing engineering projects such as roads, and tourist complexes. The stereoscopic projection technique was used to analyze the stability of these rocky slopes and showed there are two types of slopes which are oblique lateral and orthogonal slopes. Toppling and rockfall were found within the study area, in addition to the rolling. Differential eroded in the foot slope was the main reason for the failure, moreover, the existence of joint sets. The rocks of the slopes were evaluated based on the unconfined compressive strength deduced from the point load test into the moderately strong and strong.

A NOVEL WAY TO CLASSIFY THE SLAKE DURABILITY INDEX OF ROCKS

Slaking, the disintegration of rocks upon exposure to moisture and air, poses significant challenges in engineering applications, such as in slope stability and embankment design. This study improves the slake durability index test by introducing several enhancements to better reflect the field conditions of rocks. A new test methodology is developed which includes a revised slake durability classification table, a novel Discard Method for calculating weight retention in the test drum, and an updated classification chart. The improved methodology considers parameters such as porosity and strength, evaluated through index, Schmidt hammer rebound and point load tests. The new classification table categorizes rocks into durability classes according to the field observations and laboratory test results. The Discard Method improves accuracy by eliminating broken pieces from the durability calculation, addressing the limitations in the suggested method. A Configured Discard Method equation further defines the durability estimation by combining porosity, strength and slake durability data. Field investigations and laboratory tests conducted with various rock types demonstrate that the new classification system provides more accurate and practical assessments of rock durability. This study highlights the importance of effective porosity and strength impact which enhance the reliability of slake durability evaluations, offering a revised classification framework that simplifies field assessments while maintaining the accuracy. Future research should focus on developing these methods and expanding their applicability to diverse geological contexts by including more material to test.

Research on resin modified gelatin as environmental-friendly high temperature resistant wellbore stabilizer

In the development of deep shale oil and gas, the working temperature of the water-based drilling fluids is relatively high, which can cause wellbore instability due to the poor temperature resistance of the traditional environmental-friendly additives. To address this issue, this study presents a wellbore stabilizer called PMG, prepared by modifying gelatin (GT) with a resin synthesized by polyvinyl alcohol (PVA) and methacrylic acid (MA) through hydrothermal free radical polymerization, and it can withstand temperature up to 200 °C. The synthesis and thermal stability of PMG were confirmed by Fourier transform infrared spectroscopy (FT-IR) and thermogravimetric analysis (TGA). The wellbore stability performance of PMG was evaluated by linear swelling test, rolling recovery test, core immersion experiment and point load test and the wellbore stability mechanism was also investigated. The results reveal that the swelling height of sodium-based bentonite (Na-BT) in 2.0 % PMG solution at 150 °C is only 0.97 mm; the hot-rolling recovery rate in 2.0 % PMG solution at 200 °C is 72 %; the artificial core still maintains its original appearance after being immersed in 2.0 % PMG solution at 200 °C for 16 h, and the compressive strength increases by 12 %, effectively strengthening the wellbore. In addition, the EC50 value of PMG is > 106 mg/L, and the BOD5/CODcr value is 31 %, indicating that it is non-toxic and biodegradable. PMG has excellent wellbore stability performance, in addition to being able to adsorb on the surface of Na-BT by electrostatic action and hydrogen bonding, the PMG micelles that precipitated from the aqueous solution due to the breaking of hydrogen bonds in the molecular chain at high temperature also form a film on the clay surface, aggregating and cementing the clay particles together, further enhancing the wellbore stability.

Shell strength of Eucallista purpurata (Bivalvia) from life and death assemblages on sandy beaches of the Bonaerense coast, southwest Atlantic Argentina

ABSTRACT Mollusc shells are among the most abundant and well-preserved organisms in sediments worldwide, but their shells are usually subject to fragmentation due to exposure time in the TAZ, which compromises their preservation potential. The shell strength of valves from life and death assemblages from the sandy beaches of Santa Clara (37°50′49″S, 57°30′16″W) and Mar Chiquita (37°44′45″S, 57°25′1.5″W) on the Bonaerense coast, southwest Atlantic Argentina, was analysed to capture the effect of taphonomic processes over time and to determine how intrinsic properties and taphonomic variables influence their shell strength. To achieve this, the mechanical resistance profile of the bivalve Eucallista purpurata was assessed using point load and microindentation tests, and the mineralogical composition and organic matter content of the shell, as well as morphological variables, were measured. We concluded that, as expected, shells from life assemblages were more resistant to compression than those from death assemblages, but surprisingly, right valves were more resistant to compression than left valves in both Mar Chiquita and Santa Clara death assemblages. No differences were observed between broken and unbroken shells.

The Effect of Mineral Composition on the Correlation between Point Load Index with the Uniaxial Compressive Strength of Sulfate Rocks and their Point Loading Deformation

Point load test is a common, inexpensive and fast test for the indirect achievement of compressive or tensile strength of the rocks in the laboratory and the field. In this research, by conducting uniaxial compression test (UCT), axial (APLT), and diametral (DPLT) point load tests on gathered sulfate rock blocks from the Gachsaran Formation outcrops at the four under-construction reservoir dam sites in Iran, investigate the effect of mineral composition on the relationships between point load index, the uniaxial compressive strength (UCS) and their point load deformation. Regarding that, firstly by creating a correlation between axial (APLI) and diametral (DPLI) point load index and UCS, relationships for each specific mineral composition were provided. Secondly, by comparing the reliability of the APLT and DPLT results in predicting UCS, the conversion factors of the APLI and DPLI to the UCS were calculated. Thirdly, the effect of unique or multiple sampling locations in the analysis results was compared, and finally, for the first time, the deformation of loading points in point load tests was investigated. The results of this study confirmed that by variation of mineral composition of sulfate rocks, the relationships between APLI and DPLI, and UCS, conversion factors as well as loading points deformation patterns during point load tests are changed significantly, and the results of APLT and DPLT can be used to predict UCS with the same reliability in dry and saturated conditions.

Analisis Pengaruh Mineral terhadap Nilai Kuat Tekan Batuan

To determine the mineral content and composition of a rock, a petrographic test is carried out using a thin section method. Petrographic test is known as an efficient test of time and energy with accurate results in obtaining the composition and mineral content of rocks. The Point Load test is an index test that has been widely used to predict the UCS value of a rock indirectly in the field. This is due to the simple test procedure, easy sample preparation and can be done in the field, so that the strength of the rock can be quickly known in the field before testing in the laboratory.

Mechanical properties of reduced-sized limestone filler-cement pastes and their variability using an instrumented point load test

In spite of their relevance in the context of modelling of the mechanical properties of cementitious composites, the mechanical properties of cement pastes (without standard sand) and their variability are rarely determined, given the large experimental effort demanded. In the present work, an instrumented Point Load Test was used to measure the mechanical properties of reduced-sized limestone filler (LF)-cement pastes. Since limestone filler replacement can have different effects on the tensile strength of pastes, depending on how the mixture is proportioned, a Box–Behnken fractional factorial design, by varying the w/c and the w/s ratios, besides superplasticizer was employed for designing low-cement concretes. Indeed, limestone filler was used to reduce water to solids (w/s) ratio and the porosity of the pastes. For each w/s ratio, replacement of cement by LF increased the w/c ratio and the paste porosity (the dilution effect). The tensile strength of pastes depended essentially on the capillary porosity. For an equal capillary porosity, limestone filler contributed to increasing the strength due to the toughening effect, which was confirmed by the larger total displacements in load-displacement curves. The elastic modulus was mainly governed by total open porosity. Multiple regression equations were proposed to predict porosity and mechanical properties of limestone filler-cement pastes, whereas their variability was analyzed.

Evaluation of Ni-Cu Ore from Zapolyarnoe Based on Mineralogical and Physical Properties before and after Comminution

For the effective comminution and subsequent enrichment of mineral ores, comprehensive knowledge of their mineralogical and physical properties is required. Using an integrated methodology, this study evaluated samples of polymetallic Ni-Cu ore from Zapolyarnoe, Russia. Several analytical techniques were utilised, including optical microscopy, microindentation with Vickers geometry, the Point Load Test, and Mineral Liberation Analysis (MLA). The purpose of this study was to determine mineral associations, physical features, and enrichment during jaw crusher comminution. The acquired properties included the Point Load Strength Index, Vickers Hardness Number, and fracture toughness. The MLA method characterised seven fractions in terms of particle size distribution, degree of liberation, association, and modal mineralogy. Magnetite, pyrrhotite, pentlandite, and chalcopyrite were calculated in terms of wt% and their textural features. The enrichment of each ore phase in fractions with particle sizes smaller than 400 µm was determined. The influence of this enrichment was discovered to be correlated with various textural and structural parameters, such as intergrowth, grain size, and crack morphologies after indentations. In addition, the chromium content of magnetite contributed to an increase in the fracture toughness values. Despite the complexities involved, even limited samples of materials provide valuable insights into processing behaviour, emphasising the importance of considering mineralogical parameters in comminution studies.

In-site Investigation of Rock Mass Structure Effect on Tunnel Smooth Blasting Quality

Smooth blasting controls tunnel excavation contours and rock stability; however, it can lead to overbreaks and underbreaks due to rock mass anisotropy. Studying the Zigaojian tunnel of the Hangzhou-Huangshan Expressway, this research examined the effect of rock mass structure on smooth blasting quality. Four rock mass structure types–massive, layered, and cataclastic–were determined through manual measurements, three-dimensional scanning, point loading tests, and wave velocity measurements. The blasting overbreak and underbreak of the tunnel section were measured using a tunnel section laser distance measuring instrument, and the relationship between the blasting overbreak and rock mass structure type, surrounding rock grading, structural plane spacing, structural plane number, uniaxial compressive strength, and density was studied using multiple regression analysis. The results show that the overbreak area and maximum overbreak increase as the integrity of massive, massive, layered, and cataclastic rock masses decreases. The intersection of the rock mass structural plane and tunnel section was overbroken. The overbreak is positively correlated with the rock structure type, surrounding rock grading, and structural plane number and negatively correlated with the uniaxial compressive strength, density, and structural plane spacing. The results of this study can provide a reference for smooth blasting in similar projects.

Characterization of Rock Material Through Point Load Test Index, Granulometry, and Dimensions of Specimens

The characterization of rock material in mining projects is important, specifically with respect to compressive strength, since this parameter establishes safety factors for slopes in quarries and/or fortification systems for tunnels. This work proposes a simple and practical application methodology. It is proposed with accessible tests in situ, such as granulometric indices and point load test index, to more than a predetermined dimensioning of the specimens. The physical properties mentioned are established in a simple but methodical way; and in this work, 180 tests have been carried out on specimens of the same material, from the sector called Cojitambo, province of Cañar (Ecuador). The results obtained allow the characterization of the rock, establishing for this work that the point load test index has values between 3037 MPa and 4992 MPa, which categorizes rock material as a class of hard rock. It was concluded that the proposal avoids the permanent sending of samples to the laboratory, saving time and money, therefore, a calculation methodology has been structured for the stated objective. Keywords: granulometry, compression, rock, point load test index. Resumen Es importante la caracterización del material rocoso en proyectos mineros, específicamente respecto a la resistencia a compresión, ya que mediante este parámetro se establecen factores de seguridad para taludes en canteras y/o sistemas de fortificación para túneles. Este trabajo tuvo como objetivo proponer una metodología simple, sencilla y de aplicación práctica. Se la plantea con ensayos accesibles in situ, como los índices granulométricos e índice de carga puntual, a más de un predeterminado dimensionamiento de las probetas. Las propiedades físicas mencionadas se las establece de una forma sencilla, pero metódica y en este trabajo se han ejecutado 180 ensayos sobre probetas de un mismo material, proveniente del sector denominado Cojitambo, provincia del Cañar (Ecuador). Los resultados obtenidos permiten la caracterización de la roca, estableciendo para este trabajo que el índice de carga puntual tiene valores entre los 3.037 MPa y 4.992 MPa, lo que categoriza al material rocoso como una clase de roca dura. Se pudo concluir que la propuesta evita el envío permanente de muestras a laboratorio, ahorrando tiempo y dinero, por lo tanto, se ha estructurado una metodología de cálculo para el objetivo planteado. Palabras Clave: granulometría, compresión, roca, índice de carga puntual.

Recommendation of excavation method and support system based on rock mass classification of the Matenggeng Dam Tunnel, Cilacap, Central Java

The Matenggeng Dam is situated in Matenggeng village, Dayeuhluhur, Cilacap district, Central Java. It features a diversion channel equipped with a tunnel that has a diameter of five meters and is formed like a horseshoe. Therefore required engineering geological analysis for tunnel planning. The purpose of this study is to ascertain the tunneling technique and the corresponding support systems. This study will provide suggestions and input before and during the construction. This research uses rock mass classification and the Point Load Test for several samples. Tunneling method and support systems using empirical method Rock Mass Rating (RMR) and Geological Strength Index (GSI). The study location is composed of sandstone, conglomerate, andesite, and siltstone with alluvium deposits in Cijolang River area. Classification of rock masses is fair to good rock. The GSI and RMR analyses suggest that the excavation technique uses hammer and blasting methodologies and a max stand-up time without support of 20 to 144 hours. The tunneling method uses top heading and bench and support systems using rock bolts, wire mesh, shotcrete, and steel sets.

Investigation on Disaster Mechanism of Diversion Tunnel Induced by Gripper TBM in Hydrokarst Erosion Stratum and Engineering Measures

In gripper tunnel boring machine (TBM) tunneling through complex geological formations, the safe and efficient recovery from large-scale collapses remains a formidable challenge. In this study, we investigate the causes of a 1246 m3 collapse that occurred during the gripper TBM tunneling in the diversion tunnel in Xinjiang, China. Various techniques including TSP seismic waves, CFC advanced water exploration, laboratory point load tests and packer permeability tests were employed for thorough research. The examination discloses that the water softening in biotite-quartz schist in fractured zones contributes significantly to the loosening and dislocation of rock layers along joints. The gripper TBM’s cutterhead exacerbates this process through cutting action and vibrations, causing large-scale instability and eventual rock mass collapse. To tackle this engineering problem, we propose a three-step treatment scheme comprising “Reinforcement-Backfill-Re-excavation”. Furthermore, we propose a technique to handle TBM collapses by creating a “protective shell” within the cavity. The safety and feasibility of these proposed solutions were thoroughly validated through numerical simulations. Also, we utilized the Hoek-Brown theory and Rostami prediction formula to establish recommended values for the total thrust and total torque of the TBM during the collapsed section. The proposed treatment scheme and estimated parameters were successfully applied, resulting in a comprehensive solution from collapse handling to tunneling. This study offers valuable details on effectively managing large-scale collapses in gripper TBM tunneling, which can be useful for similar tunnel engineering and improve safety and efficiency.

APPLICATION OF SKELETAL BIOMECHANICS TO STRUCTURAL SYSTEMS

The concept of green construction enables a revolutionary change in construction sector in terms of design, production, and management. One such method is introducing the concept of biomimicry. Biomimicry is utilized in the field of design to solve problems. This paper mainly discusses the mimicking of human skeleton for structural design. The idea is mimicking humerus bone as a tension member and femur bone as a compression member. The optimized members of compression and tension (strut and tie) were put together to form the mimicked king post truss analytically with the conventional cross-section truss. Three cases were considered analytically with average diameter, maximum diameter, and equivalent self-weight to the members of mimicked truss, and experimentally testing with non-destructive test and point-load test. The result shows that the ultimate load-carrying capacity of critical compression member and tension member was 846.16 and 1952 kN, respectively, whereas the achieved load was 780.30 and 1729 kN. Also, the ratio of analytical stiffness to self-weight is 21.83 mm<sup>-1</sup> and the ratio of experimental stiffness to self-weight was 19.15 mm<sup>-1</sup>. Therefore, from the results it was observed that the equivalent results for mimic truss can be achieved in a truss which is modeled of equivalent self-weight. Hence, the development and use of structural elements using biomimicry is feasible and will lead to economic, green, and energy-efficient structures.

Assessment of Rock Slope Stability Along Degala-Koya Road, Erbil, NE Iraq

This article addresses the issue of instability in rock slopes along the main Degala-Koya road in Erbil northeastern Iraq, which poses a significant threat to human safety and property. the use of Slope Mass Rating Tool software and various techniques such as kinematic analysis and geomechanical classification to investigate cut slopes and assess slope stability. Geotechnical tests, including a point load test, were performed at eight stations in the study area to determine the unconfined compressive strength. comparison between discrete and continuous Slope Mass Rating values for rock slopes was carried out. The most unstable slope was identified as Station 4, while Stations 2, 5, 6, and 8 were deemed the most stable. The study found that rock fall accompanied by direct toppling was the most common type of failure, with plane sliding, rock fall, toppling, and wedge sliding as likely dominant failures. The Slope Mass Rating Tool-v205 software is more accurate When evaluating slope stability when strengthen with kinematic analysis by Dips-v6.008 software.

Reliability assessment of carbon fiber mortar: Combined pulse velocity, point load, and compressive strength tests

Cement mortar is used as an adhesive and plaster for brick walls, drainage channels or irrigation with river stones or mountain stones, and in the manufacturing of short revetments. The performance of cement mortar requires continuous improvement by construction materials researchers, scientists, and building contractors. Recently, carbon fiber sheet has gained popularity for use in retrofitting and improvement to the performance of building elements. This study aims to recycle small sheets of unused carbon fiber by cutting them into short carbon fiber (SCF) with a uniform length of 12 mm, which is used as an internal reinforcement in the cement mortar. All plain and fibrous mortar were prepared with a 50 mm cube mold. In addition, tests were conducted to provide reliable data regarding the application of recycled carbon fiber sheets as discrete fiber reinforcement in the improvement of cement mortar; these include ultrasonic pulse velocity (UPV), compressive strength, and point load strength index (PLSI) (known for performance rock testing in geotechnical and geological environments) tests. The test results for UPV, PLSI, and compressive strength showed that the addition of carbon fiber improves the performance of the mortar when compared to plain mortar. Furthermore, two simplified relationships are made: the first correlation was for PLSI with compressive strength, and the second correlation was for UPV with compressive strength. Finally, these correlations were compared with those of experimental tests obtained from published literature, thus demonstrating the correctness of the PLSI, compressive strength, and UPV values obtained in this study.

Study on the Numerical Relationship between Dynamic Compressive Strength and Crushing Energy Consumption in Dynamic Point Load Test under Different Corrosion Conditions

Compared with most dynamic tests, point load test is widely used in engineering practice because of its wide requirements for test conditions. In offshore mining engineering, the surrounding rock is affected by dynamic load impact and corrosion of various aqueous solutions for a long time. Therefore, it is of great significance to study the influence of hydrochemical corrosion on dynamic mechanical properties of rock to ensure the stability of surrounding rock. In this paper, a large number of dynamic point load and dynamic uniaxial compression tests were carried out on the granite mined from a large mine by using the Split Hopkinson pressure bar test device after the corrosion of pH values of 2, 5, and 7 solutions, respectively. Based on the energy theory, the crushing energy consumption in the test process is analyzed, and finally the fitting formula of the dynamic compressive strength and the crushing energy consumption in the dynamic point load test is obtained through fitting analysis. The results show that when the crushing energy consumption in the dynamic point load test is used to estimate the dynamic compressive strength, a certain crushing energy consumption (the final value of this test is 70 J in this paper) is more accurate, and it can estimate the dynamic compressive strength under various corrosion conditions under this crushing energy consumption. However, with the change of crushing energy consumption in the dynamic point load test, this transformation relationship is no longer accurate, and it is necessary to modify this transformation relationship with corrosion condition as the influencing factor. This conclusion can provide reference for obtaining the dynamic compressive strength of rocks under different corrosion conditions in engineering field in the future.

Probabilistic characterization of Brazilian tensile strength of rock from both direct and indirect measurements

The determination of Brazilian tensile strength (BTS) is an essential step in the analysis and design of mining structures at a particular site. The estimated design parameters are affected by the inherent variability of BTS, measurement errors arising from laboratory testing, and transformation uncertainty associated with the empirical model linking BTS to other rock properties when it is indirectly estimated. These uncertainties are usually lumped together as the total variability of BTS. However, it is the inherent variability resulting from natural geological factors, not the total variability, that directly affects the actual response of rock structures. Hence, there is a need for proper characterization of the inherent variability of BTS while the measurement errors and transformation uncertainty are explicitly incorporated. This paper develops a Bayesian approach which uses sequential updating, that is multi-input oriented for probabilistic characterization of the inherent variability of BTS of rock. The proposed approach systematically combines previous engineering experience and site information from both direct BTS data and data from indirect tests like point load test to inversely infer the inherent variability of BTS. The proposed approach quantitatively accounts for the effects of measurement errors and transformation uncertainty on the characterization of the inherent variability of BTS. The proposed approach is illustrated and validated using real-life data and simulated data. The result shows that the proposed approach provides a proper characterization of the inherent variability of BTS based on available information from multiple sources. Sensitivity studies are also performed to explore the effects of measurement errors on the performance of the proposed approach.

Stability Analysis of Strongly Weathered Muddy Slate Slopes Considering Softening Conditions of Water Immersion

To understand the stability of strongly weathered muddy slate slopes under water immersion effects, we obtained shear strength parameters of the weakly layered structures within this slate through direct shear tests. Point load tests were performed on in-site slate samples with varying water immersion durations to assess the water immersion’s softening impact on slate strength. Results highlight that muddy slate strength presents pronounced random variability, declining as water immersion duration increases. Drawing from shear strength parameters and the water immersion softening observed in laboratory and in-site tests, we formulated a numerical slope model that considers layered structures and water immersion conditions to evaluate slope stability. Numerical simulations suggest that the slate slope’s sliding surface, when layered, does not consistently form a basic circular arc or straight line. The slope safety factor (FOS) drops below 3, marking a notable decrease compared to a homogeneous slope (FOS = 3.22). In the model, multiple secondary sliding surfaces can emerge, leading to a sliding band with a specific thickness after introducing the random distribution of layer strength parameters. This further reduces the slope’s FOS to below 2.9. Water immersion makes slopes inclined to slide following the layered structure. If the dip angle of the slate’s layered structure is less steep than the slope’s dip angle, water immersion notably diminishes the FOS, which can dip to a minimum of 1.12.

Effects of Tire Crumb Rubber and Steel Fiber on Punching Shear Behavior of Two-way Alkali-activated Concrete Flat Slabs

As a result of the rapid increase in the demand for transportation vehicles recently, the accumulation of waste tires causes environmental problems. One of the methods that can contribute to the reduction of this environmental problem is the recycling of waste tires as a construction material in aggregate form. This research investigated the impacts of waste tire crumb rubber with/out steel fiber (SF) on the two-way punching-shear behavior alkali-activated concrete (AAC) flat slabs and performed center point load tests. In the study, one-type of SF and two kinds of scrap tire waste i.e., crumb rubber (CR) and tire particles, were used on the producing of rubberized AAC slabs obtained by 100% as binder made of ground-granulated blast furnace slag (GGBFS). Also, while both fine crumb rubber (FCR) and coarse crumb rubber (CCR) were used together in the AAC slabs, tire crumb rubber (TCR) was used alone at the same proportion. Additionally, the fine aggregate was substituted with 10% and 15% FCR and CCR, and coarse aggregate was substituted with TCR in the same proportions. Additionally, AAC slabs with recycled tire rubber (RTR) were produced as fibrous and non-fibrous using 1% by volume hook-end SF. In total, nine AAC slabs with sizes of 50 x 50 x 6 cm3 were manufactured for the study. Experimental data showed that the inclusion of RTR only slightly reduced the punching shear strength of AAC slabs and the punching shear strength of the slabs increased when SF was added to the mixtures containing RTR.

Theoretical model for the effect of porosity on correlations of point load tests of rocks

Point load test (PLT) is one of the most adopted indirect ways to estimate uniaxial compressive strength of rocks. Yet, the estimation can be interfered by rock porosity that has rarely been considered in model development. In this study, a theoretical model for the interpretation of results of PLTs considering the effect of porosity is proposed. The model was developed using the elastic stress field inside diametrically loaded sphere within the framework of linear elastic fracture mechanics and requires only basic physical parameters including porosity and Poisson's ratio. Apart from the point load index (PLI), an extra parameter, subtended angle, is included in the interpretation, which accounts partly for the wide range of the ratio between PLI and uniaxial compressive strength observed in experiments. The predictions of the proposed model compare reasonably well with data for a variety of rocks, which demonstrates the validity of the model.