Probabilistic Finite Element Analysis Research Articles

Probabilistic Finite Element Analysis of Human Rib Biomechanics: A Framework for Improved Generalizability.

In dynamic impact events, thoracic injuries often involve rib fractures, which are closely related to injury severity. Previous studies have investigated the behavior of isolated ribs under impact loading conditions, but often neglected the variability in anatomical shape and tissue material properties. In this study, we used probabilistic finite element analysis and statistical shape modeling to investigate the effect of population-wide variability in rib cortical bone tissue mechanical properties and rib shape on the biomechanical response of the rib to impact loading. Using the probabilistic finite element analysis results, a response surface model was generated to rapidly investigate the biomechanical response of an isolated rib under dynamic anterior-posterior load given the variability in rib morphometry and tissue material properties. The response surface was used to generate pre-fracture force-displacement computational corridors for the overall population and a population sub-group of older mid-sized males. When compared to the experimental data, the computational mean response had a RMSE of 4.28N (peak force 94N) and 6.11N (peak force 116N) for the overall population and sub-group respectively, whereas the normalized area metric when comparing the experimental and computational corridors ranged from 3.32% to 22.65% for the population and 10.90% to 32.81% for the sub-group. Furthermore, probabilistic sensitivities were computed in which the contribution of uncertainty and variability of the parameters of interest was quantified. The study found that rib cortical bone elastic modulus, rib morphometry and cortical thickness are the random variables that produce the largest variability in the predicted force-displacement response. The proposed framework offers a novel approach for accounting biological variability in a representative population and has the potential to improve the generalizability of findings in biomechanical studies.

Probabilistic three-dimensional finite element analysis of geosynthetic-reinforced pile-supported embankments

The current study focuses on a probabilistic three-dimensional finite element analysis of geosynthetic-reinforced pile-supported embankments. The probabilistic analysis utilizes a finite element technique and was executed using Python scripting in Abaqus. A deterministic model is established for both unreinforced and geosynthetic-reinforced pile-supported embankments, employing a basic linear elastic perfectly plastic Mohr-Coulomb constitutive model. This numerical model adopted a simplified methodology that eliminates the need to simulate actual piles and subsoil. The results are validated across different embankment heights and mesh sizes. Based on results from Monte-Carlo simulations, we aimed to show how results can vary even by considering only one random variable. Probability density functions for various parameters, including pile efficacy, maximum differential settlement, and maximum strain in the reinforcement, are shown. The findings illustrate the ability of probabilistic approaches to predict the most likely occurrences for different governing parameters of geosynthetic-reinforced pile-supported embankments. They also highlight the potential of probabilistic approaches to expand further into a full-scale comprehensive study which can offer deeper insights into the stability of these embankments considering various sources of uncertainties.

Metamodel-assisted hybrid optimization strategy for model updating using vibration response data

In this study, an effective and novel method, termed Metamodel Assisted Hybrid of Particle Swarm Optimization with Genetic Algorithm (MA-HPSOGA), is developed to identify unknown structural dynamic parameters. The method first constructs four popular metamodels to substitute the computationally expensive numerical analysis based on the Latin hypercube sampling method and probabilistic finite element analysis, and their accuracy is assessed by R-squared. Subsequently, a suitable and low-cost metamodel is selected in combination with a hybrid optimization strategy by incorporating Genetic Algorithm (GA) into Particle Swarm Optimization (PSO). Two examples with measured vibration response data and different levels of complexity are used to verify the effectiveness and practicality of the presented method. The results showed that polynomial chaos expansion assisted HPSOGA has the highest computational efficiency and accuracy in the four coupled methods. Besides, compared to the conventional iteration-based dynamic parameter identification methods, the presented method shows an overwhelming advantage in terms of computational efficiency. Furthermore, the performance of HPSOGA is compared with its sub-algorithms, showing that the hybrid strategy offers faster convergence and stronger robustness. Our findings reveal that the MA-HPSOGA may be used as a promising method for achieving high-efficiency model updating in large-scale complex structures.

Probabilistic finite element analysis of failures in concrete dams with large asperities in the rock–concrete interface

Common analytical assessment methods for concrete dams are unlikely to predict material fracture in the dam body because of the assumption of rigid body behavior and uniform- or linear stress distribution along a predetermined failure surface. Hence, probabilistic non-linear finite element analysis, calibrated from scale model tests, was implemented in this study to investigate the impact of concrete material parameters (modulus of elasticity, tensile strength, compressive strength, fracture energy) on the ultimate capacity of scaled model dams. The investigated dam section has two types of large asperities, located near the downstream and/or upstream end of the rock–concrete interface. These large-scale asperities significantly increased the interface roughness. Post-processing of the numerical simulations showed interlocking between the buttress and the downstream asperity leading to fracture of the buttress with the capacity being determined mainly by the tensile strength of the buttress material. The capacity of a model with an asperity near the upstream side, with lower inclination, was less dependent on the material parameters of the buttress as failure occurred by sliding along the interface, even with inferior material parameters. Results of this study show that material parameters of the concrete in a dam body can govern the load capacity of the dam granted that significant geometrical variations in the rock–concrete interface exists. The material parameters of the dam body and their impact on the capacity with respect to the failure mechanism that developed for some of the studied models are not commonly considered to be decisive for the load capacity. Also, no analytical assessment method for this type of failure exists. This implies that common assessment methods may misjudge the capacity and important parameters for certain failure types that may develop in dams.

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.

Probabilistic finite element analysis of fatigue life of additively manufactured clasp.

The present study was aimed to develop a probabilistic finite element method (FEM) that predicts the variability in the fatigue life of additively manufactured clasp so that it can be used as a virtual test in the design phase before manufacturing. Titanium alloy (Ti-6Al-4V) clasp with integrated chucking part, which was designed for experimental fatigue test to validate the computational method, was investigated. To predict the lower bound, an initial spherical defect was assumed in the region where stress concentration was predicted. The Smith-Watson-Topper (SWT) method, Bäumel & Seeger rule, elasto-plastic FEM, and zooming FEM were used. The influence of assumed initial defect on the fatigue life was significant, and the large variability in the fatigue life was predicted. This study demonstrated that the proposed practical computational method can simulate the large variability in the fatigue life of titanium alloy clasp, which is useful in its design before manufacturing.

An efficient method for probabilistic finite element analysis with application to contact assessment of CANDU fuel channels

In CANDU reactors, irradiation induced creep and growth of the fuel channel (FC) assembly leads to Pressure Tube (PT) sag and to a reduction in the gap between the PT and the Calandria Tube (CT) (PT–CT gap). Excessive PT sag can lead to contact between the PT and the CT. Uncertainties in the parameters that influence PT–CT contact require probabilistic assessment methods to assess risk of PT-CT contact and demonstrate adherence to CSA Standard N285.8 provisions. Improved models in the prediction of PT–CT contact require computationally expensive 3D finite element analysis (3D FEA), a tool that is prohibitively difficult to use in probabilistic assessments. Current probabilistic analyses rely on 1D FEA models for the prediction of PT-CT contact estimates that are generally less accurate. Therefore, an improved and computationally more efficient approach for conducting probabilistic finite element analyses (PFEA) of CANDU FCs is highly desirable. In this paper, such a PFEA approach is proposed by coupling the multiplicative dimensional reduction method (M-DRM) and the polynomial chaos expansion (PCE) method. The analysis that follows studies the PT–CT gap predictions using deterministic 3D FEA by considering two different PT configurations present within the population of the FCs in a CANDU reactor core. The PFEA is then carried out using 1D and 3D FEA by employing the ABAQUS software suite. The accuracy of the probabilistic analysis results using the proposed approach is initially checked against Monte Carlo Simulation (MCS) results using 1D FEA. Following a good agreement, the proposed approach was applied for PFEA based on 3D FEA. The results indicate that the two different PT orientations significantly influence the probabilistic contact results. They also show that the probabilistic results based on 1D FEA, which are currently used by the nuclear industry, are not reliable. Moreover, it is also shown that a sensitivity analysis can be performed without any additional effort. The low computational cost, predictive capability and the generality of the proposed method are important attributes for adopting it in the future in probabilistic assessments of an entire reactor core.

Modelling and strength prediction of pre-tensioned hybrid bonded joints for structural steel applications

ABSTRACT Strength prediction of bonded joints remains a challenging task made even further complicated for hybrid joints in which adhesive is supplemented by mechanical fasteners. The present paper complements a series of experimental investigations on hybrid joints combining adhesive bonding with pre-tensioned bolts on galvanised and coated steel substrates, and considering two adhesives. The paper starts with modelling the strength of the bonded connection under the simultaneous action of tensile and compressive normal stresses, and shear. It then presents how these data are moulded into appropriate failure criteria and used the latter as input data for a subsequent full probabilistic finite element analysis. Following the paths of Weibull and subsequent researchers, joint capacity is predicted with reasonable accuracy for a dozen different combinations of substrates, adhesive, and pre-tension level – for which, on average, an accuracy slightly below 10% has been achieved. The results deliver meaningful insights into the subtle relationship between adhesive bonding, mechanical fastening, failure criteria, and resulting predictions methods.

Blast Vulnerability of Multi-Storey Masonry Infill Reinforced Concrete Frames

The prototype or experimental study of blast effect on structures is not always possible due to high risk of life and environmental wellbeing. However, advancement in computational science had made it possible to solve such complex and risky problems. Researchers in the past few decades have presented ample computational and numerical studies on blast effect on masonry structure using deterministic approach. The studies have neglected the effect of various uncertainties that may perhaps occur during an actual blast scenario. Therefore, in present study probabilistic Finite Element (FE) analysis was performed on three statistically stable masonry infill RC (Reinforced Concrete) frames having single (G+0), two (G+1) and three (G+2) storey levels respectively to arrive at the blast fragility curves. The probability of failure was computed based on the inelastic tension damage curves developed for the masonry. The effect of scaled distance and different storey levels on blast vulnerability of masonry infill RC frame walls are investigated through these fragility curves. The results indicate a noteworthy effect of scaled distance and storey levels on the blast resistance of the structure. Frame G+0 is found to be comparatively more vulnerable against the blast load effect than other multi-storey frames. The critical value of scaled distance (Z_critical) corresponding to PF 90% in panel P1 was found as 6.67m/kg^1/3, 4.81m/kg^1/3 and 4.66m/kg^1/3 respectively of frames G+0, G+1 and G+2; in panel P2 was found as 4.3m/kg^1/3 and 4.6m/kg^1/3 respectively of frames G+2 and G+3 and; in panel P3 was found as 4.32 m/kg^1/3 of frame G+3. It is also observed that RC frame is negligibly affected with respect to the masonry panel in terms of blast load mitigation.

Mandibular reconstruction system reliability analysis using probabilistic finite element method

The aim of this study was to design for mandibular reconstruction of large lateral defect with minimum target reliability with designated confidence interval under bite force range of 300 ± 102 N. The performance of the models has been evaluated by numerical analysis considering the uncertainty of input parameters. Computer-Aided design was used to develop the models of three designs according to the patient's anatomy and to achieve to near symmetry of the mandible. Stress-strength modeling was utilized for the probabilistic physics of failure analysis under assumption of a quasi-static load. Monte-Carlo simulation was also applied for probabilistic finite element analysis and reliability assessment. The sensitivity analysis of the models was developed to reflect the significance of the variables in the models. The deterministic stress analysis shows that the highest stress and the second maximum stress are 110 MPa and 85 MPa for cortical bone around the screws, respectively. Also, it is determined that the maximum plate stress of the titanium conventional plate model is 580 MPa. The reconstruction system success rate was improved in all models by observing the anatomy of the patient's mandible in the plate designs by computer-aided design and additive manufacturing techniques. Based on the results, the reliability of plate strength and pull-out screws strength are 99.99% and 96.71% for the fibula free flap model, respectively, and 99.99% and 94.17%, respectively, for the customized prosthesis model. Probability sensitivity factors showed that uncertainty in the elastic modulus of the cortical bone has the greatest effect on the probability of screws loosening.

Impact of structural uncertainties on the buckling strength of cylindrical GLARE panels subjected to uniform compression

The objective of this article is the investigation of the elastic buckling strength of cylindrical simply supported GLARE (GLAss REinforced) panels subjected to axial compression using probabilistic analysis methods, so that the effect of uncertainties associated with material properties and dimensions of the panels on their elastic buckling load can be evaluated. The mechanical properties of aluminum along with the dimensions of aluminum and unidirectional (UD) glass-epoxy layers are considered to be random input variables whereas the critical buckling load is defined as a random output parameter. The employed eigenvalue buckling analysis and the probabilistic finite element analysis were carried out with ANSYS software. The Probabilistic Design System (PDS), along with the Monte Carlo Simulation and the Latin Hypercube Sampling method were used for the calculations. It is found that the thickness of aluminum layers has the strongest effect on the buckling strength, among the considered random input variables. It is also demonstrated that there is a considerable probability for the buckling load of GLARE panels to be overestimated when a deterministic analysis is conducted.

Probabilistic Finite Element Analysis of Cooled High-Pressure Turbine Blades—Part A: Holistic Description of Manufacturing Variability

Abstract Modern high-pressure turbine (HPT) blade design stands out due to its high complexity comprising three-dimensional blade features, multipassage cooling system (MPCS), and film cooling to allow for progressive thermodynamic process parameters. During the last decade, probabilistic design approaches have become increasingly important in turbomachinery to incorporate uncertainties such as geometric variations caused by manufacturing scatter and deterioration. Within this scope, the first part of this two-part article introduces parametric models for cooled turbine blades that enable probabilistic finite element (FE) analysis taking geometric variability into account to aim at sensitivity and robustness evaluation. The statistical database is represented by a population of more than 400 blades whose external geometry is captured by optical measurement techniques and 34 blades that are digitized by computed tomography (CT) to record the internal geometry and the associated variability, respectively. Based on these data, parametric models for airfoil, profiled endwall (PEW), wedge surface (WSF), and MPCS are presented. The parametric airfoil model that is based on the traditional profile theory is briefly described. In this regard, a methodology is presented that enables to adapt this airfoil model to a given population of blades by means of Monte Carlo-based optimization. The endwall variability of hub and shroud are parametrized by radial offsets that are applied to the respective median endwall geometry. WSFs are analytically represented by planes. Variations of the MPCS are quantified based on the radial distribution of cooling passage centroids. Thus, an individual MPCS can be replicated by applying adapted displacement functions to the core passage centroids. For each feature that is considered within this study, the accuracy of the parametric model is discussed with respect to the variability that is present in the investigated blade population and the measurement uncertainty. Within the scope of the second part of this article, the parametric models are used for a comprehensive statistical analysis to reveal the parameter correlation structure and probability density functions (PDFs). This is required for the subsequent probabilistic finite element analysis involving real geometry effects.

Probabilistic Finite Element Analysis of Cooled High-Pressure Turbine Blades—Part B: Probabilistic Analysis

Abstract Modern high-pressure turbine (HPT) blade design stands out due to high complexity comprising three-dimensional blade features, multipassage cooling system (MPCS), and film cooling to allow for progressive thermodynamic process parameters. During the last decade, probabilistic design approaches have become increasingly important in turbomachinery to incorporate uncertainties such as geometric variations caused by manufacturing scatter. In Part B of this two-part article, real geometry effects are considered within a probabilistic finite element (FE) analysis that aims at sensitivity evaluation. The knowledge about the geometric variability is derived based on a blade population of more than 400 individuals by means of parametric models that are introduced in Part A. The HPT blade population is statistically assessed, which allows for reliable sensitivity analysis and robustness evaluation taking the variability of the airfoil, profiled endwalls (PEWs) at hub and shroud, wedge surfaces (WSFs), and the MPCS into account. The probabilistic method—Monte Carlo simulation (MCS) using an extended Latin hypercube sampling (eLHS) technique—is presented subsequently. Afterward, the FE model that involves thermal, linear-elastic stress, and creep analysis is described briefly. Based on this, the fully automated process chain involving computer-aided design (CAD) model creation, FE mesh morphing, FE analysis, and postprocessing is executed. Here, the mesh morphing process is presented involving a discussion of the mesh quality. The process robustness is assessed and quantified referring to the impact on input parameter correlation. Finally, the result quantities of the probabilistic FE simulation are evaluated in terms of sensitivities. For this purpose, regions of interest are determined, wherein the statistical analysis is conducted to achieve the sensitivity ranking. A significant influence of the considered geometric uncertainties onto mechanical output quantities is observed, which motivates to incorporate these in modern design strategies or robust optimization.

Probabilistic finite element analysis of the deflection on a beam

The reliability of complex systems can be quantified using probabilistic finite element analysis. A widely accepted approach is interlacing of probabilistic methods with commercial finite element solvers for evaluating the fatigue life. In this paper, the deflection of a beam is calculated using finite element method (ABAQUS® commercial software), then the results of the simulated model are employed in MATLAB®, which is used to determine the probability, reliability and the reliability index. It was found that the reliability index is around 1.2428, while the reliability value is calculated to be around 0.8930. From the calculations the values of the probability of failure (POF) was found to be around 0.1070. Finally, the results of the calculated maximum deflection have been compared and a good agreement is found.

Probabilistic thermo-mechanical finite element analysis for the fire resistance of reinforced concrete structures

This paper presents a probabilistic methodology based on the thermo-mechanical finite elements analysis to investigate the impact of the variability of the thermal properties of the concrete in the fire safety of structures. This is meant to evaluate if characteristic values or safety factors for the conductivity and specific heat are required during the semi-probabilistic structural fire safety assessment. To illustrate the use of the proposed methodology, this work includes a case-study with a tunnel lining which considers the uncertainties related to the thermal and mechanical properties of the concrete, the soil load, and the temperatures described by the standard temperature-time curve. Two failure criteria are considered: one was the maximum temperature of 300 °C at the reinforcement and the other based on the temperature-dependent strength as provided in the Eurocode EN 1992-1-2. Several finite element analyses are performed. The design of experiments is executed by a Correlation Latin Hypercube Sampling. The calculated probability of failure has different values depending on the adopted failure criteria. A sensitivity analysis using the Spearman's rank correlation coefficient was carried out and demonstrates that the uncertainty related to the specific heat has the greatest impact on the results.

Influence of morphology and material properties on the range of motion of the costovertebral joint – a probabilistic finite element analysis

The motion of the costovertebral joint (CVJ) is governed by the material properties and its morphology. The goal of this numerical study was to identify the material and morphology parameters with the greatest influence on the motion of the CVJ. A fully parametric finite element model of the anatomy and material properties of the CVJ was developed. The impact of five morphology and thirteen material parameters was investigated and compared to in vitro data. The motion was influenced in particular by the rotational stiffness of the articulatio capitis costae and the lateral position of the fovea costalis transveralis.

How the height and the area of the annulus fibrosus affect the range of motion behavior: A stochastic analysis

The annulus fibrosus has substantial variations in its geometrical properties (among individuals and between levels), and plays an important role in the biomechanics of the spine. Few works have studied the influence of the geometrical properties including annulus area, anterior / posterior disc height, and over the range of motion, but in general these properties have not been reported in the finite element models. This paper presents a probabilistic finite element analyses (Abaqus 6.14.2) intended to assess the effects of the average disc height ( hp) and the area ( A) of the annulus fibrosus on the biomechanics of the lumbar spine. The annulus model was loaded under flexion, extension, lateral bending, and axial rotation and analyzed for different combinations of hpand A in order to obtain their effects over the range of motion. A set of 50 combinations of hp(mean = 18.1 mm, SD = 3.5 mm) and A (mean = 49.8%, SD = 4.6%) were determined randomly according to a normal distribution. A Yeoh energy function was used for the matrix and an exponential function for the fibers. The range of motion was more sensitive to hpthan to A. With regard to the range of motion the segment was more sensitive in the following order: flexion, axial rotation, extension, and lateral bending. An increase of the hpproduces an increase of the range of motion, but this decreases when A increases. Comparing the range of motion with the experimental data, on average, 56.0% and 73.0% of the total of data were within the experimental range for the L4–L5 and L5–S1 segments, respectively. Further, an analytic equation was derived to obtain the range of motion as a function of the hpand A. This equation can be used to calibrate a finite element model of the spine segment, and also to understand the influence of each geometrical parameter on the range of motion.

Probabilistic Slope Stability Analysis: A Case Study

Probabilistic slope stability analyses are becoming more and more popular to evaluate the safety level of slopes and the associated risk and reliability, especially in the recent years. Probabilistic approach can take into account the uncertainties and natural variability in material properties, as well as in environmental factors, by using various statistical distribution fuctions (such as normal, lognormal etc.) for random variables. It is already noted by various researchers that, a slope with a deterministic factor of safety larger than 1.00 using average values of soil parameters may have a significant level of probability of failure, if the material properties are unknown, or contain significant uncertainty/variability. In this study, a well-documented landslide case study is used to demonstrate the importance of probabilistic approach in slope stability; to investigate the effects of considering variabiliy in material properties; and to compare deterministic and probabilistic slope stability analyses results. Deterministic limit eqilibrium, probabilistic limit equilibrium and probabilistic finite element analyses are conducted for Lodalen landslide in Oslo, Norway and the results are compared with each other. The factor of safety, the probability of failure and the most critical failure surface are investigated with and without statistical cross-correlation of soil’s shear strength parameters. The results of this study can provide further insights into the comparison of deterministic and probabilistic approaches in slope safety.

Forensic Investigation on Cracking in Hanger-to-Girder Connections of Long-Span Suspension Bridges

The hanger-to-girder connection is a vital load-bearing component of suspension bridges. However, two types of premature cracking (types A and B) in hanger-to-girder connections of the Runyang Suspension Bridge have been observed only several years after the bridge was open to traffic. This paper presents the field inspections on the pattern, location and propagation of cracks. To further investigate the mechanism of cracking, a multi-scale finite element (FE) model of the bridge is developed. It is observed from FE analysis results that under vehicle loads, large stress range and significant stress concentration exist. The type A cracks are mainly due to the discontinuity of hanger-to-girder connections and in-plane shear/bending deformation, and out-of-plane bending deformation of the welded connections accelerated type B cracking. To this end, probabilistic FE analyses are performed, and the predicted fatigue lives of the hanger-to-girder connections are obtained and compared with those in the field, where good agreement is observed.

Structural failure investigations through probabilistic nonlinear finite element analysis: Methodology and application

Abstract Structural failure investigations can be strongly influenced by high levels of uncertainty in modelling parameters, particularly in the case of historical constructions. This suggests forensic analysts to perform probabilistic simulations, allowing a risk-informed diagnosis and prognosis of structural failures. In this study, a failure investigation methodology including uncertainty characterisation, modelling and propagation is presented and applied to a historic piperno stone balcony, the collapse of which caused four casualties. High uncertainty in physical and mechanical properties of piperno stone, which has been widely used for a long time in the architectural heritage of Naples and Southern Italy, motivated stochastic finite element (SFE) simulations to account for spatial variability of material properties throughout the balcony. Based on field inspections, laboratory surveys and experimental testing, a three-dimensional finite element (FE) model with four alternative restraint conditions was developed and material properties were statistically characterised. Experimental data were found to be in agreement with those available in the literature for similar piperno stones. Deterministic nonlinear FE simulations with mean material properties showed a major influence of restraint conditions, providing an initial identification of the most realistic model that was able to reproduce the observed damage. Then, SFE simulations were performed on structural models having random fields of material properties. It is shown that the selected SFE model of the balcony had a mean load capacity very close to the total load expected at the time of collapse, allowing the lowest uncertainty level in the output of forensic analysis.