Unimodal Weibull Research Articles

New insights into the interpretation of the results of four point bending tests on float glass

The four point bending test is one of the most commonly used and standardised tests to determine the mechanical properties of materials. For its use on float glass there are both the ASTM C158-02R17 and European EN 1288-3:2001 standards. However when testing float glass the results tend be a statistical muddle. This impacts the reliability of the design strength of float glass determined using four point bending tests. In an attempt to resolve this problem a series of four point bending tests were conducted which were designed to be more systematic than those previously reported in the literature and which use newly developed digital microscopy techniques for pre- and post-test analysis. By systematically testing, the test results can be divided into different groups based on air side, Sn side and source of failure, allowing the data to be divided into clear and separate statistical groups. Secondly the results can potentially be used to validate the lower bound glass strength theory proposed by Ballarini et al. (J Eng Mech 142(12):04016100, 2016). The glass specimens were industrially cut, ground, chamfered and flat polished on the long sides. Specimens were checked using advanced digital microscopy before and after testing. The results suggest that uni-modal Weibull behaviour only applies above a critical failure stress. Failures at stresses below this critical failure stress as a group have a separate and steeper Weibull slope. This supports Ballarini’s theory for a lower bound failure strength although there are important differences between the air side and Sn side of the specimens which this theory does not currently allow for. These differences seem to be inherent to the differences between the glass and the Sn sides. The results also show that the strength of cut, ground, chamfered and flat polished glass can be high but that inconsistency in process control and the irregular occurrence of surface failures are the main causes for the statistical spread. Digital microscopy can reliably measure the quality of the various surfaces and intersections of surfaces of a glass specimen but there is no absolute relation between the size of a detected defect and the probability that this defect actually leads to failure.

SIR dynamics with infection age in complex heterogeneous networks

This paper models the spread of an SIR-epidemic with infection age in a heterogeneous complex network and the basic reproduction number is obtained. Next, the accuracy of the main results is confirmed in simulations. Finally, we focus on the impact of infection age on the spread of the disease and show that the recovery time distribution has a significant effect on the disease dynamics. When the recovery time conforms to Uniform distribution or unimodal Gamma distribution, keeping the mean unchanged and increasing the variance will reduce the peak point, the final cumulative epidemic incidence and the basic reproduction number, and will increase the epidemic duration. Furthermore, we show the different effects of different distributions. Compared with the unimodal Gamma distribution with an expectation of 1 and a variance less than 1, the unimodal Weibull distributions with the same expectation and variance bring the greater peaks and the greater cumulative epidemic incidences. Beyond the above facts, we try to explain the reasons behind them.

Probabilistic finite element analysis of sheet molding compound composites with an extended strength distribution model

Sheet Molding Compound (SMC) composites exhibit large scatters in mechanical properties and a strong correlation between the strength and stress distribution. For example, SMC demonstrates a much higher strength in the 3-pt bending test compared to that demonstrated in the uniaxial tension test. As a result, finite element (FE) simulations with the mean mechanical properties underpredict the flexural response of the SMC structures. Also, probabilistic finite element (PFE) analysis with a unimodal statistical strength model built from the uniaxial tensile data did not solve this problem. To account for the dependency of strength on stress distribution, an extended strength distribution model (ESDM), in the form of a trimodal Weibull distribution consisting of unimodal Weibull distribution models of tension and flexural strengths, has been proposed. The ESDM was subsequently tested in PFE simulations of SMC tensile and flexural experiments following a randomization algorithm which considers the tensile strength as a random variable and assigns it according to ESDM across finite element models. Simulations with this approach successfully reproduced both the tensile and flexural responses with the predicted mean peak load, post-peak behavior, and energy absorption similar to experimental results. • A computational efficient method to include the scatter and size effect in FEA. • A trimodal Weibull strength model based on both tension and flexure strength data. • Simulations reproduced the tensile and flexural responses in experiments.

Experimental determination of bimodal strength distribution of S-glass fibers

Tensile strength distribution of S-Glass fibers at small gage lengths on the order of the ineffective length is required for both analytical and numerical models of composite unidirectional tensile strength. Historically unimodal Weibull strength distributions measured at large gage lengths are extrapolated to smaller length scales (5–200 μm). The accuracy of this approach is assessed by using 2 different experiments – the filament tensile test (gage lengths > 4 mm) and a continuously-monitored Single Fiber Fragmentation Test (SFFT, gage length = 544 μm). Extrapolation of the unimodal Weibull distribution to the 544 μm gage length leads to significant overprediction of fiber strengths (by ∼60%). The Weibull distribution obtained from SFFT also shows a significantly higher Weibull Modulus (12.9) as opposed to the tension dataset (4.5). This indicates that a different population of defects dominate at the shorter gage lengths. A bimodal Weibull equation has been fit to the combined data to account for the 2 different defect populations to provide accurate strength predictions over fiber lengths from ∼500 μm to 30 mm. An independent validation experiment was performed at 2 mm gage length to validate the Bimodal Equation.

From small forest samples to generalised uni‐ and bimodal stand descriptions

Abstract Forests provide multiple services, and in the face of global change adaptive management strategies are needed, which inevitably must be based on models. However, most locally accurate forest models are tied to the stand scale and cannot readily be applied across large areas. Empirical data for model initialisation are often not available at large spatial scales. National Forest Inventories (NFIs) provide spatially representative tree and stand samples, but their samples are typically small, that is, only a few trees are measured per plot, and they are truncated, that is, not each tree has the same probability of being observed. To overcome these issues, we develop and apply a methodology to derive stand descriptions from small sample data, taking the Swiss NFI as a case study. We extended the traditional Weibull function to (multi‐)truncated unimodal and bimodal forms that are suitable for the representation of samples from survey designs with multiple callipering thresholds. Subsequently, we applied these functions in an extended parameter prediction method to derive stand diameter distributions from representative samples. Additionally, we predicted species compositions using a multinomial logistic regression model and assigned them to the diameter distributions of the stands. The diameter distribution of 9.1% of the Swiss NFI samples was better described by a bimodal than a unimodal Weibull function. The uni‐ and bimodal diameter model in combination with the model to determine species composition can be used to predict stand descriptions from single small samples or entire forest types in the target area. Thereby, the bimodal form is suitable for capturing stand structures with distinct under‐ and overstorey. In Switzerland, the diameter distributions of stands are typically positively skewed. Our method can be applied to any large‐scale dataset (e.g. NFI) and allows to generate initial conditions in terms of spatially representative stands. These, in turn, are suitable for forest stand simulators, which allows for developing adaptive forest management strategies at large scales, by simulating realistic and site‐specific stand development while still reflecting detailed management measures. Furthermore, stand descriptions can be used to assess tree species diversity, regeneration and harvest potentials.

Characterizing Distributions of Tensile Strength and Crack Precursor Size to Evaluate Filler Dispersion Effects and Reliability of Rubber.

Undispersed filler agglomerates or other substantial inclusions/contaminants in rubber can act as large crack precursors that reduce the strength and fatigue lifetime of the material. To demonstrate this, we use tensile strength (stress at break, σb) data from 50 specimens to characterize the failure distribution behavior of carbon black (CB) reinforced styrene-butadiene rubber (SBR) compounds. Poor mixing was simulated by adding a portion of the CB late in the mixing process, and glass beads (microspheres) with 517 μm average diameter were introduced during milling to reproduce the effects of large inclusions. The σb distribution was well described with a simple unimodal Weibull distribution for the control compound, but the tensile strengths of the poor CB dispersion material and the compounds with the glass beads required bimodal Weibull distributions. For the material with the lowest level of glass beads—corresponding to less than one microsphere per test specimen—the bimodal failure distribution spanned a very large range of σb from 13.7 to 22.7 MPa in contrast to the relatively narrow σb distribution for the control from 18.4 to 23.8 MPa. Crack precursor size (c0) distributions were also inferred from the data, and the glass beads introduced c0 values in the 400 μm range compared to about 180 μm for the control. In contrast to σb, critical tearing energy (tear strength) was unaffected by the presence of the CB agglomerates and glass beads, because the strain energy focuses on the pre-cut macroscopic crack in the sample during tear testing rather than on the microscopic crack precursors within the rubber. The glass beads were not detected by conventional filler dispersion measurements using interferometric microscopy, indicating that tensile strength distribution characterization is an important complementary approach for identifying the presence of minor amounts of large inclusions in rubber.

Prediction of Diameter Distributions with Multimodal Models Using LiDAR Data in Subtropical Planted Forests

Tree diameter distributions are essential for the calculation of stem volume and biomass, as well as simulation of growth and yield and to understand timber assortments. Accurate and reliable prediction of tree diameter distributions is critical for optimizing forest structure compositions, scheduling silvicultural operations and promoting sustainable management. In this study, we investigated the potential of airborne Light Detection and Ranging (LiDAR) data for predicting tree diameter distributions using a bimodal finite mixture model (FMM) and a multimodal k-nearest neighbor (KNN) model (compared to the unimodal Weibull model (UWM)) over a subtropical planted forest in southern China. To do so, we first evaluated the capability of various LiDAR predictions (i.e., the bimodality coefficient (BC) and Lorenz-based indicators) to stratify forest structural types into unimodal and multimodal stands. Once the best LiDAR prediction for the differentiation was determined, the parameters of UWM (in non-specific and species-specific models) and FMM (in structure-specific models) were estimated by LiDAR-derived metrics and the tree diameter distributions of stands were generated by the estimated LiDAR parameters. When KNN was applied for constructing diameter distributions, optimal KNN strategies, including number of neighbors k, response configurations and imputation methods (i.e., Most Similar Neighbor (MSN) and Random Forest (RF)) for different species were heuristically determined. Finally, the predictive performance of estimated LiDAR the parameters of UWM, FMM and KNN for predicting diameter distributions were assessed. The results showed that LiDAR-predicted Lorenz-based indicators performed best for differentiation. Parameters of UWM and FMM were predicted well and the species-specific models had higher accuracies than the non-specific models. Overall, RF imputation from KNN with an optimal response set (i.e., DBH) were was stable than MSN imputation when k = 5 neighbors. In addition, the inclusion of bimodal FMM for differentiated all plots generally produced a more accurate result (Mean eR = 40.85, Mean eP = 0.20) than multimodal KNN (Mean eR = 52.19, Mean eP = 0.26), whereas the UWM produced the lowest performance (Mean eR = 52.31, Mean eP = 0.26). This study demonstrated the benefits of multimodal models with LiDAR for estimating diameter distributions for supporting forest inventory and sustainable forest management in subtropical planted forests.

Uni- and bimodal Weibull distribution for analyzing the tensile strength of Diss fibers

ABSTRACTThe present work considered several statistical distribution functions with the aim to critically analyze the strength data of natural Diss fibers. Besides the Weibull two-parameter distributions, three-parameter and bimodal Weibull were used. Kolmogorov–Smirnov, Anderson–Darling and chi-square goodness-of-fit test were used to judge how accurate the theoretical Weibull probability density function (PDF) fits with tensile strength data.Tensile test results indicated that the fibers have good tensile strength and Young’s modulus. However, their fracture strength exhibits high scatters due to the presence of defects within the fiber and/or on the fiber surface. Microscopic observation revealed that the rupture of the fibers is governed by two types of defects, the rough surface of the fiber formed by several spines and the internal defects such as the defects of connection between fibrils.The statistical analysis of variability in strength showed that the results of the shape and scale parameters calculated using bimodal Weibull distribution give larger values compared to those using the single Weibull distribution. Also, the strength data points fit well to the bimodal Weibull PDF curve. All three goodness-of-fit tests indicated also that the bimodal Weibull model gives a more accurate prediction of data strength compared to the unimodal Weibull model.

A Comparison of Airborne Laser Scanning and Image Point Cloud Derived Tree Size Class Distribution Models in Boreal Ontario

Airborne Laser Scanning (ALS) metrics have been used to develop area-based forest inventories; these metrics generally include estimates of stand-level, per hectare values and mean tree attributes. Tree-based ALS inventories contain desirable information on individual tree dimensions and how much they vary within a stand. Adding size class distribution information to area-based inventories helps to bridge the gap between area- and tree-based inventories. This study examines the potential of ALS and stereo-imagery point clouds to predict size class distributions in a boreal forest. With an accurate digital terrain model, both ALS and imagery point clouds can be used to estimate size class distributions with comparable accuracy. Nonparametric imputations were generally superior to parametric imputations; this may be related to the limitation of using a unimodal Weibull function on a relatively small prediction unit (e.g., 400 m2).

Characterization of surface winds over the Iberian Peninsula

ABSTRACTThis work presents a characterization of the surface wind climatology over the Iberian Peninsula (IP). For this objective, an unprecedented observational database has been developed. The database covers a period of 6 years (2002–2007) and consists of hourly wind speed and wind direction data recorded at 514 automatic weather stations. The original observations underwent a quality control process to remove rough errors from the data set. In the first step, the annual and seasonal mean behaviour of the wind field are presented. This analysis shows the high spatial variability of the wind as a result of its interaction with the main orographic features of the IP. In order to simplify the characterization of the wind, a clustering procedure was applied to group the observational sites with similar temporal wind variability. A total of 20 regions are identified. These regions are strongly related to the main landforms of the IP. The wind behaviour of each region, characterized by the wind rose (WR), annual cycle (AC) and wind speed histogram, is explained as the response of each region to the main circulation types (CTs) affecting the IP. Results indicate that the seasonal variability of the synoptic scale is related with intra‐annual variability and modulated by local features in the WRs variability. The wind speed distribution not always fit to a unimodal Weibull distribution consequence of interactions at different atmospheric scales. This work contributes to a deeper understanding of the temporal and spatial variability of surface winds. Taken together, the wind database created, the methodology used and the conclusion extracted are a benchmark for future works based on the wind behaviour.

Tensile strength distribution of carbon fibers at short gauge lengths

In this study, we determined the strength distribution of high-strength polyacrylonitrile-based carbon fibers of different short gauge lengths (~1 mm) using both the single-fiber-composite four-point bending test and the single-fiber tensile test. We employed the bimodal Weibull model to explain the experimental data. We found that the value of the Weibull shape parameter for short gauge lengths was higher than that for long gauge lengths. This implies that the tensile strength distribution of carbon fibers is governed by two different flaw populations. The tensile strength of resin-impregnated fiber bundles predicted on the basis of the bimodal Weibull distribution was in better agreement with the experimental result than the tensile strength of those predicted on the basis of the unimodal Weibull distribution.

Effect of silane coupling agent on mechanical performance of glass fibre

Mechanical performance of commercially manufactured unsized and γ-aminopropylsilane sized boron-free E-glass fibres has been characterised using single-fibre tensile test. Both apparent fibre modulus and fibre strength were found to strongly depend on fibre gauge length. The average strength of sized fibres was found 40–80 % higher than unsized fibres at different gauge lengths. Weibull analysis suggested that the failure mode of unsized fibres could be described by unimodal Weibull distribution, whereas the strength distribution of sized fibres appeared to be controlled by two exclusive types of flaw population, types A and B. Comparison of the Weibull plots between unsized and sized fibres revealed that the strength of unsized fibres was likely to be dominated by type A flaws existing on the bare glass surface and type B flaws may be related to the defects on the glass surface coated with silane. This was partially supported by the observation of fractured cross-sectional area using SEM. It was, therefore, proposed that the strength difference between unsized and sized glass fibres may be more reasonably interpreted from the surface protection standpoint as opposed to the flaw healing effect. The results obtained from this study showed that silane coupling agent plays a critical role in the strength retention of commercially manufactured E-glass fibres and the silane effect on the fibre strength is also affected by the change in gauge length of the sample.

On the surface wind speed probability density function over complex terrain

[1] The physical mechanisms determining the shape of the surface wind speed probability density function over complex terrain are investigated using observations from a dense mesoscale network and a high spatial resolution mesoscale simulation for the 1992 to 2005 period. Results indicate that the atmospheric stability plays a major role in controlling the shape of the wind speed distribution but its effects are strongly modulated by the contribution of different atmospheric scales of motion such as the mesoscale or synoptic scale. The local topographic features further modulate the relative contribution of each mechanism. As a consequence of the complicated interaction of these atmospheric processes the surface wind speed distribution can present a complicated shape that is not always expected to fit a unimodal Weibull type distribution over complex terrain regions.

Glass Strengthening Via High-Intensity Plasma-Arc Heating

A high-intensity plasma-arc lamp was used to irradiate the surface of soda–lime silicate glass tiles to determine whether an increase in strength could be achieved. The lamp has a maximum power density of 3500 W/cm2, a processing area of 2 cm × 10 cm, a broad-spectrum energy distribution between 0.2 and 1.4 μm, and was controlled to unidirectionally sweep across 50 mm2 tiles at a constant speed of 8 mm/s. Ring-on-ring (RoR) equibiaxial flexure and four-point unidirectional flexure testings of entire tiles were used to measure and compare uncensored and unimodal Weibull failure stress distributions of treated and untreated glass. Even under nonoptimized processing conditions, the RoR Weibull characteristic failure stress increased by approximately 25% and the four-point-bend Weibull characteristic failure stress increased by approximately 65%. Additionally, the failure stresses of the weakest samples of the heat-treated glass were significantly higher than those of the untreated glass. Strengthening was due to a fire-polishing-like mechanism. The arc-lamp heat treatment caused the location of the strength-limiting flaws in the four-point-bend tiles to change; namely, failure initiation occurred on the gauge section surface for the treated glass, whereas it occurred at a gauge section edge for the untreated glass. Arc-lamp heat treatment is attractive not only because it provides strengthening, but because it can (noncontact) process large amounts of glass quickly and inexpensively, and is a process that either a glass manufacturer or an end-user can readily use.

An experimental study on the tensile behavior of Kevlar fiber reinforced aluminum laminates at high strain rates

Abstract In this paper, tensile behavior of Kevlar fiber reinforced aluminum laminates (KRALL) has been determined at strain rates 100, 500 and 1300 s −1 . Yield strength, tensile strength and failure strain of the composite all increased with increasing strain rate. The application of pre-stress on KRALL increased the yield strength, decreased the failure strain, but had no effect on the tensile strength. A linear damage model has been combined with Weibull distribution function to establish a constitutive equation for KRALL. The results show that strength of KRALL obeyed the bimodal Weibull distribution, and the strength of pre-stressed KRALL obeyed unimodal Weibull distribution.

Bimodal versus Weibull Wind Speed Distributions: An Analysis of Wind Energy Potential in La Venta, Mexico

The International Standard IEC 61400-12 and other international recommendations suggest the use of the two-parameter Weibull probability distribution function (PDF) to estimate the Annual Energy Production (AEP) of a wind turbine. Most of the commercial software uses the unimodal Weibull PDF as the default option to carry out estimations of AEP, which in turn, are used to optimise wind farm layouts. Furthermore, AEP is essential data to assess the economic feasibility of a wind power project. However, in some regions of the world, the use of these widely adopted and recommended methods lead to incorrect results. This is the case for the region of La Ventosa in Mexico, where the frequency of the wind speed shows a bimodal distribution. In this work, mathematical formulations by using a Weibull PDF and a bimodal distribution are established to compare the AEP, the capacity factor and the levelised production cost for a specific wind turbine. By combining one year of wind speed data with the hypothetic power performance of the Vestas V27-225 kW wind turbine, it was found that using the Weibull PDF underestimates AEP (and thus the Capacity Factor) by about 12%.

A bimodal lognormal model of the distribution of strength of carbon fibres: effects of electrodeposition of titanium di (dioctyl pyrophosphate) oxyacetate

The tensile strength distribution of Fortafil-3 carbon fibres of circular cross-section has been investigated at different gauge lengths. The unimodal Weibull, unimodal lognormal and bimodal lognormal models were tested. Estimating the model parameters by the method of maximum likelihood, and testing each model by the Kolmogorov-Smirnov goodness-of-fit statistic at prescribed levels of significance, it is found that the data for untreated unsized fibres fit a bimodal lognormal model best. The proportions of the low and high strength populations,p andq, respectively, did not show any well-defined trend with gauge length and had average values close to 0.5. But the lognormal mean for each population showed an increasing trend with decreasing gauge length. It is inferred thatp andq are, respectively, related to the presence of surface flaws and internal defects, both of which probably have the same structural origin. After electrodeposition of titanium di (dioctyl pyrophosphate) oxyacetate (TDPO), the fibre strength was still best approximated by a bimodal lognormal distribution. But the weighting factorp for the weak population was reduced markedly, with a corresponding increase ofq, indicating the healing of surface flaws during electrodeposition of a protective layer of TDPO. Furthermore, in contrast to the observations with untreated fibres, the lognormal means for both the low and high strength populations of the electrocoated fibres were essentially unchanged with gauge length. Changes were also indicated in the number and severity of surface flaws, caused by concurrent electrochemical processes.

Weibull analysis of strengths of coir fibres

The large scatter observed in the fracture strengths of coir fibres ranging in length from 0·001 m to 0·065 m and in diameter from 0·15 × 10 −3 m to 0·35 × 10 −3 m, is analysed statistically using a two-parameter Weibull treatment. It was found that the strength distribution at any length (and for any diameter of fibre) can be represented satisfactorily by unimodal Weibull distributions. The plot on logarithmic coordinates of mean fracture strength ( σ ƒ ) against test length for a 0·25 × 10 −3 m diameter coir fibre, shows a linear relationship between 0·065 m and 0·006 m; below 0·006 m the strength is relatively independent of length. Apparently below a length of 0·006 m the flaws are evenly distributed and the mean distance between gross flaws is of the order of 0·006 m. The plot of percentage coefficient of variation against mean fracture strength shows that the experimental results fall in two parallel lines showing a discontinuity at a given strength corresponding to a length of about 0·006 m. The increase in Weibull parameter, m, with increase in diameter shows that the flaw distribution is more uniform in fibres of large diameter. This conclusion is in agreement with the fibre strength histograms plotted for various diameters. The origin of flaws in coir fibres may be from two sources: (a) growth defects and (b) processing defects. These flaws and flaw distributions in coir fibres are responsible for the observed scatter in fracture strength values.