Explicit Finite Element Analysis Research Articles

Effects of Dental Implant Diameter and Tapered Body Design on Stress Distribution in Rigid Polyurethane Foam During Insertion

Anchorage, evaluated by the maximum insertion torque (IT), refers to mechanical engagement between dental implant and host bone at the time of insertion without external loads. Sufficient anchorage has been highly recommended in the clinic. In several studies, the effects of implant diameter and taper body design under external loading have been evaluated after insertion; however, there are few studies, in which their effects on stress distribution during insertion have been investigated to understand establishment of anchorage. Therefore, the objective of this study was to investigate the effects of dental implant diameter and tapered body design on anchorage combining experiments, analytical modelling, and finite element analysis (FEA). Two implant designs (parallel-walled and tapered) with two implant diameters were inserted into rigid polyurethane (PU) foam with corresponding straight drill protocols. The IT was fit to the analytical model (R2 = 0.88-1.0). The insertion process was modelled using explicit FEA. For parallel-walled implants, normalized IT and final FEA contact ratio were not related to the implant diameter while the implant diameter affected normalized IT (R2 = 0.90, p<0.05, β1 = 0.20 and β2 = 0.93, standardized regression coefficients for implant diameter and taper body design) and final FEA contact ratio of tapered implants. The taper design distributed the PU foam stress further away from the thread compared to parallel-walled implants, which demonstrated compression in PU foam established by the tapered body during insertion.

Higher‐order inverse mass matrices for the explicit transient analysis of heterogeneous solids

AbstractNew methods are presented for the direct computation of higher‐order inverse mass matrices (also called reciprocal mass matrices) that are used for explicit transient finite element analysis. The motivation of this work lies in the need of having appropriate sparse inverse mass matrices, which present the same structure as the consistent mass matrix, preserve the total mass, predict suitable frequency spectrum and dictate sufficiently large critical time step sizes. For an efficient evaluation of the reciprocal mass matrix, the projection matrix should be diagonal. This condition can be satisfied by adopting dual shape functions for the momentum field, generated from the same shape functions used for the displacement field. A theoretically consistent derivation of the inverse mass matrix is based on the three‐field Hamilton principle and requires the projection matrix to be evaluated from the integral of these shape functions. Unfortunately, for higher‐order FE shape functions and serendipity FE elements, the projection matrix is not positive definitive and can not be employed. Therefore, we study several lumping procedures for higher order reciprocal mass matrices considering their effect on total‐mass preserving, frequency spectra and accuracy in explicit transient simulations. The article closes with several numerical examples showing suitability of the direct inverse mass matrix in dynamics.

Finite Element Modeling and Analysis of Perforated Steel Members under Blast Loading

Perforated steel members (PSMs) are now frequently used in building construction due to their beneficial features, including their proven blast-resistance abilities. To safeguard against structural failures from explosions and terrorist threats, perforated steel beams (PSBs) and perforated steel columns (PSCs) offer a viable alternative to traditional steel members. This is attributed to their impressive energy absorption potential, a result of their combined high strength and ductile behavior. In this study, numerical examinations of damage assessment under the combined effects of gravity and blast loads are carried out to mimic real-world scenarios of external explosions close to steel structures. The damage assessment for PSBs and PSCs considers not just the initial deformation from the blast, but also takes into account the residual capacities to formulate dependable damage metrics post-explosion. Comprehensive explicit finite element (FE) analyses are performed with the LSDYNA software. The FE model, when compared against test results, aligns well across all resistance phases, from bending and softening to tension membrane regions. This validated numerical model offers a viable alternative to laboratory experiments for predicting the dynamic resistance of PSBs and PSCs. Moreover, it is advisable to use fully integrated solid elements, featuring eight integration points on the element surface, in the FE models for accurate predictions of PSBs’ and PSCs’ behavior under blast loading. A parametric study is presented to investigate the effect of web-opening shapes, retrofitting, and different blast scenarios. The results obtained from the analytical FE approaches are used to obtain the ductile responses of PSMs, and are considered an important key in comparisons among the studied cases.

Evaluation of roof cutting by directionally single cracking technique in automatic roadway formation for thick coal seam mining

Automatic roadway formation by roof cutting is a sustainable nonpillar mining method that has the potential to increase coal recovery, reduce roadway excavation and improve mining safety. In this method, roof cutting is the key process for stress relief, which significantly affects the stability of the formed roadway. This paper presents a directionally single cracking (DSC) technique for roof cutting with considerations of rock properties. The mechanism of the DSC technique was investigated by explicit finite element analyses. The DSC technique and roof cutting parameters were evaluated by discrete element simulation and field experiment. On this basis, the optimized DSC technique was tested in the field. The results indicate that the DSC technique could effectively control the blast-induced stress distribution and crack propagation in the roof rock, thus, achieve directionally single cracking on the roadway roof. The DSC technique for roof cutting with optimized parameters could effectively reduce the deformation and improve the stability of the formed roadway. Field engineering application verified the feasibility and effectiveness of the evaluated DSC technique for roof cutting.

Predictive modeling of jute‐polyester composite tubes for impact performance: A comprehensive finite element analysis approach

AbstractThis paper examines the feasibility of utilizing jute‐polyester composites as a substitute for glass fiber composites in demanding automotive impact protection. Further, the study focuses on developing accurate modeling procedures using LS‐DYNA explicit finite element analysis to predict the load–displacement responses and failures of jute‐polyester (JP) composite tubes under static and dynamic loading. Experimental tests validate the model, comparing numerical and experimental results using the Gross Correlation Index (GCI). The GCI ranged from a minimum of 0.83 to a maximum of 0.97. Strong agreement is observed, confirming the reliability of the developed finite element (FE) model. The validated modeling procedures offer insights for optimizing the design and performance of jute‐polyester composites, promoting sustainable materials in the automotive sector, particularly for applications related to vehicle interior trims.Highlights JP composite tubes studied under axial and transverse impact loads Developed accurate modeling procedure using LS‐DYNA FE model validated through experimental test results GCI used to compare numerical and experimental results Successful prediction of failure modes and impact parameters

Development of Numerical Simulation Techniques for the Analysis of a Rollover Tractor

AbstractOne of the leading causes of death in the word is the injury in the field of agriculture. Agricultural machinery, in fact, is very dangerous for workers and rollover accidents are due to several reasons. First, it must be considered that these machines operate on nonlinear land and on extremely steep slopes, so the causes are due to operational, tractor, and soil characteristics. Furthermore, recent studies have shown that in 33% of the rollovers that are analyzed, passive safety devices, such as seat belt, are not used by workers (who got thrown out of the safety zone). Another cause of fatal accidents is that, until now, the mechanical design of tractors has not provided for occupant protection. In fact, there are no standard tests that assess the safety of the occupant. This study aims to conduct an explicit finite element (FE) analysis of a rollover test of a generic T5 category tractor to assess the structural characteristics and the biomechanical performance of the occupant.

Investigating the impact of roller position on pressure tube-end fitting rolled joint performance using 3D explicit finite element analysis

Pressurized Heavy Water Reactors (PHWRs) utilize Zr-2.5 Nb pressure tubes that are connected to SS-403 end fittings through rolled joints. These rolled joints are made using a rolling tool equipped with five rollers and a tapered mandrel. The action of the rollers induces plastic deformation on the inner surface of the tube. The tapered mandrel promotes rotation and axial advancement of the rollers into the pressure tube through friction. However, prolonged usage of this rolling tool can lead to the wearing down of both the mandrel and the rollers. Consequently, this wear and tear can cause the rollers to deviate from their intended position within the rolled joint. Such misalignment can lead to excessive rolling of the pressure tube into the entry taper section of the end fitting. This, in turn, generates higher residual stresses than initially anticipated. This issue may also arise from accumulated tolerances within the rolling tool or positioning errors made by operators during the rolling operation. The overextension of the rollers can significantly increase tensile residual stresses within the pressure tube, and management of these stresses is critical to prevent failure of the pressure tube due to Delayed Hydrogen Cracking.The primary objective of the present investigation is to explore the effects of the overextension of rollers on the residual tensile stresses in the transition region of the rolled joint. This objective is achieved by employing a three-dimensional explicit finite element analysis methodology. In order to ensure the reliability and authenticity of the finite element model, the computational results are cross-validated with the experimental findings documented in the existing literature. The current investigation involves the examination of two distinct configurations of pressure tube to end fitting rolled joints, which are typically found in PHWRs. The results of the present inquiry reveal an optimal roller position that minimizes residual tensile stresses within the transition region of a rolled joint. Furthermore, it is observed that any deviations in the roller position within a range of 5 mm, beyond the optimal position, result in a negligible increase in residual tensile stress.

In-silico modelling of multi-strike insertion and torsional resistance of tapered revision hip stems: Insight into spline design philosophy.

Despite the clinical success of tapered splined titanium stems, a knowledge gap still exists between spline design and its primary mechanical stability, which is critical to the long-term success of revision hip arthroplasty. Additionally, almost all published pre-clinical studies relied on resource-intensive benchtop and cadaveric testing. Hence, the present study developed a novel computational model to investigate effects of spline geometry and configuration on axial and torsional stability of tapered stem. Dynamic explicit Finite Element Analysis coupled with a state-of-the-art adaptive meshing technique was used to simulate the highly non-linear contacts and large bony material deformations. Hybridising primary straight splines with secondary angled splines results in 41% and 10% increases of peak insertion force and post-seating moment than the predicate device for the same seating position. The primary straight splines cut at multiple circumferential bony locations, enhancing torsional stability; while the alternatively placed secondary angled splines form wedges with the bone, providing reliable seating and additional torsional resistance. To the best knowledge of the author, this is the first in-silico investigation of its kind to simulate multi-strike seating and torsional resistance of revision hip stems, offering an effective and efficient platform for future multi-factorial parametric study and uncertainty quantification.

A gradient-based non-local GTN model: Explicit finite element simulation of ductile damage and fracture

Non-local models have over the years been established as an effective approach to solve the pathological mesh dependency problem observed in finite element simulation of strain-softening materials. This paper presents the formulation, implementation, and application of a gradient-based non-local Gurson–Tvergaard–Needleman (GTN) model for explicit finite element analysis. The porosity is taken as the non-local variable where the increment in porosity is averaged over the volume using an implicit gradient model. The gradient model is implemented in Abaqus/Explicit by utilising the coupled thermal–mechanical solver, which proves to be both a simple and computationally efficient approach. Due to the use of an explicit integration scheme, a transient term is introduced to the partial differential equation of the gradient formulation. The non-local GTN model is compared to the local counterpart for increasing mesh refinements using a plane strain shear band specimen, a plane strain tension specimen, and a plane strain compact tension specimen. The proposed approach can remedy the pathological mesh dependency problem. For the plane strain tension specimen, it is shown that the non-local GTN model will preserve the fracture mode (slant versus cup–cup fracture mode) when the mesh is refined. The non-local GTN model is also able to predict the same fracture mode as observed in ductile tearing experiments. However, non-local averaging can over-smooth the fields and exclude the slant fracture mode from occurring if the material length Lc is too large.

Hull Girder Ultimate Strength Analysis of 17500 LTDW Oil Tanker under the Combined Global and Local Load using Explicit Finite Element Analysis

This study investigates the hull girder’s ultimate strength of 17500 LTDW Oil Tanker. The explicit finite element analysis is carried out to estimate the hull girder longitudinal strength of the Oil Tanker. The numerical analysis starts with the longitudinal strength analyses that examine the ship’s structural response behavior due to sagging and hogging bending moment load. The structural strength analyses were conducted to determine the maximum structural response behavior for withstanding the operational load. Afterward, the hull girder’s ultimate strength is undertaken to reflect the progressive collapse behavior of the oil tanker structure under the combined global bending moment and local hydrostatic pressure load. The explicit finite element analysis is performed to estimate the ultimate strength behavior, considering the effect of the hydrostatic pressure load and the material nonlinearities. The typical 17500 LTDW oil tanker represents Indonesia’s common oil vessel that supports national fuel oil distribution. Furthermore, this study can recognize the oil tanker’s ultimate hull girder strength characteristics.

Comprehensive computational investigations on various aerospace materials under complicated loading conditions through conventional and advanced analyses: a verified examination

Most failures develop as a result of a lack of resistivity information at the internal structure level during typical loading situations such as shock load and impact load. Impact loads have a significant impact on a component’s structural performance. A careful, organized examination of impact load settings and their side effects can reveal how well something can withstand peak loads. First, this study investigated the impact analyses on nine varied lightweight composite materials through a conventional experimental setup and computational tools. So, the best three lightweight materials are shortlisted for further investigation under complicated explicit analysis. Second, the study investigated the behavior of composite materials subjected to rapid loading circumstances in several real-time applications. The applications chosen include bullet crash analysis, unmanned aerial vehicle (UAV) propellers, and car bumpers. The three different principal composites, carbon fiber-reinforced polymer (CFRP), glass fiber-reinforced polymer (GFRP), and Kevlar fiber-reinforced polymer (KFRP), are selected and applied in crash analysis using ANSYS Workbench’s explicit technique-based finite element analysis (FEA). The comparison assessments are conducted using stumpy structural characteristics such as impact stress and equivalent strain. Two distinct grid convergence tests were performed to check whether the computational processes and discretization were correct. The standard methodologies were used on all three selected real-time applications, resulting in error percentages that were within acceptable bounds, ensuring the generation of dependable structural outputs. The ideal composite material is a Kevlar fiber-based composite with minimal defect affectability for all types of crash applications. Furthermore, multidisciplinary optimizations are performed, and the KFRP is verified to give good crash load resistance with reduced dense contribution.

An experimental and numerical study on the behaviors of adhesively-bonded steel double-hat section components under axial impact loading

A significant consideration in the deployment of adhesively-bonded joints in applications such as automotive body structures is their performance under impact loading. Due to the distinct behaviors of an epoxy-based structural adhesive under normal and shear loads combined with its propensity to undergo brittle fracture in cohesion and adhesion, special care needs to be taken in the constitutive modeling of such an adhesive as traditional Von Mises type yielding and elastoplastic material behavior do not seem to be applicable. In the current study, to start with, experimental load-displacement behaviors till failure of single lap shear and T-peel joints with steel substrates are predicted with the aid of a cohesive zone material model using LS-DYNA, an explicit nonlinear finite element analysis solver. With confidence established on cohesive zone modeling of the joints mentioned, the procedure is extended to robust prediction of the load-displacement responses and deformed shapes of steel double-hat section components subjected to axial impact loading. Three variants of steel hat section components have been considered from the viewpoint of joining of flanges with conventional discrete spot-welds, only continuous adhesive bonding, and a hybrid configuration with a combination of adhesive bonding and sparse spot-welds. The study reported here provides insights into mechanical behaviors of adhesively-bonded steel double-hat sections under impact loading, and demonstrates an efficient finite element modeling approach for adhesively-bonded steel hat sections using cohesive zone elements with fracture mechanics-based criteria for failure.

Finite Element Simulation of Jet Fracture and Distribution of Grooving Charge Cover on Inner Surface

According to the analysis of the application of grooves on the inner surface of the conical charge cover in JET warhead, when the charging structure of the conical charge cover and the caliber of the charge cover are determined, Ansys LS-DYNA explicit finite element analysis software was used to study the JET fracture time, JET granulation and JET distribution caused by the groove depth and groove spacing of the inside surface of the charge cover of five groups. On this basis, parallel group experiments were conducted to analyze the data structure of the five groups. The influence law of groove structure on jet after effect on the inner surface of conical drug cover was obtained. It was found that when groove depth was 0.1mm, the effect on jet after effect was good, while the effect on jet after effect was poor when groove depth was 0.2mm. Reducing groove spacing on the inner surface would optimize jet fracture.

Dynamic finite element analysis of hip replacement edge loading: Balancing precision and run time in a challenging model

An important aspect in evaluating the resilience of hip replacement designs is testing their performance under adverse conditions that cause edge loading of the acetabular liner. The representation of edge loading conditions in finite element models is computationally challenging due to the changing contact locations, need for fine meshes, and dynamic nature of the system.In this study, a combined mesh and mass-scaling sensitivity study was performed to identify an appropriate compromise between convergence and solution time of explicit finite element analysis in investigating edge loading in hip replacement devices. The optimised model was then used to conduct a sensitivity test investigating the effect of different hip simulator features (the mass of the translating fixture and mediolateral spring damping) on the plastic strain in the acetabular liner. Finally, the effect of multiple loading cycles on the progressive accumulation of plastic strain was then also examined using the optimised model.A modelling approach was developed which provides an effective compromise between mass-scaling effects and mesh refinement for a solution time per cycle of less than 1 h. This ‘Recommended Mesh’ model underestimated the plastic strains by less than 10%, compared to a ‘Best Estimate’ model with a run time of ∼190 h. Starting with this model setup would therefore significantly reduce any new model development time while also allowing the flexibility to incorporate additional complexities as required. The polyethylene liner plastic strain was found to be sensitive to the simulator mass and damping (doubling the mass or damping had a similar magnitude effect to doubling the swing phase load) and these should ideally be described in future experimental studies. The majority of the plastic strain (99%) accumulated within the first three load cycles.

Manual of GUI Program Governing ABAQUS Simulations of Bar Impact Test for Calibrating Bar Properties, Measured Strain, and Impact Velocity

Bar impact instruments, such as the (split) Hopkinson bars and direct impact Hopkinson bars, measure blast/impact waves or mechanical properties of materials at high strain rates. To effectively use such instruments, it is essential to know (i) the elastic properties of the bar, (ii) the correction factor of the measured strain, and (iii) information on impact velocity. This paper presents a graphic-user-interface (GUI) program prepared for solving these fundamental issues. We describe the directory structure of the program, roles and relations of associated files, GUI panels, algorithm, and execution procedure of the program. This program employs a separately measured bar density value and governs the ABAQUS simulations (explicit finite element analyses) of the bar impact test at a given impact velocity for a range of bar properties (elastic modulus and Poisson’s ratio) and two correction factors (in compression and tension) of the measured strain. The simulation is repeated until the predicted elastic wave profile in the bar is reasonably consistent with the experimental counterpart. The bar properties and correction factors are determined as the calibrated values when the two wave profiles are reasonably consistent. The program is also capable of impact velocity calibration with reference to a reliably measured bar strain wave. The quantities of a 19.1 mm diameter bar (maraging steel) were successfully calibrated using the presented GUI program. The GUI program, auxiliary programs, pre-processing files, and an example ABAQUS input file are available in a publicly accessible data repository.

Fracture mechanism and failure strain of TA31 titanium alloy for deep-sea pressure hulls based on continuum damage mechanics

Titanium alloys has high fatigue resistance, high corrosion resistance, high temperature resistance, and other excellent properties, and have been widely used in deep-sea equipment and aviation industries. In this paper, the fracture mechanism and failure strain of TA31 titanium alloy, which has been widely used in deep-sea equipment, were studied experimentally and numerically in different stress states. Considering the pressure sensitivity, the Modified Johnson-Cook (MJC) model and the Bonora damage model were used to study the fracture behavior. In order to obtain the parameters of models, four types of specimens under different stress triaxiality were conducted, and a hybrid experimental-numerical approach was employed in this paper. Then, the coupled constitutive elastic–plastic-damage model was developed and implemented in ABAQUS explicit finite element analysis (FEA) code. Finally, to validate the suggested model, FEA simulation was carried out and compared with the experimental results. The comparison revealed that the Bonora model with constant parameters was not enough to predict the failure strain. The damage parameters were sensitive to the stress triaxiality. In addition, the fracture morphology was observed by scanning electron microscope (SEM), which revealed the micro-mechanism of failure for TA31 titanium alloy. It is concluded that a higher stress triaxiality and shear mechanism lead to lower plastic deformation, and will inhibit the void growth on the damage evolution.

Experimental–numerical studies of failure behavior of PLC optical splitters under uniaxial tensile loading

This work presents an experimental and numerical study of the failure behavior of planar lightwave circuit (PLC) optical splitters under uniaxial tensile loading. Based on the experimental results, the specific fracture position, ultimate failure load (Pm), failure mode and load-displacement behavior of the PLC optical splitter under uniaxial tensile loading were determined. In order to numerically simulate the real fracture behavior and stress distribution of adhesively bonded joints, a three-dimensional (3-D) explicit finite element analysis method based on material constitutive and tiebreak contact algorithm was proposed, which treats cohesive failure and adhesive/interfacial failure separately. The Pm predicted by the numerical simulation agrees well with the experimentally obtained Pm with a relative error of ∼4.3%, and the predicted failure mode is also similar to the experimentally observed mixed failure mode, which proves the accuracy and rationality of the adopted finite element method and model. This study can provide useful references for the reliability design and optimization of PLC optical splitters.

Finite element analysis based design of biomimetic functionally graded Ti-6Al-4V alloy scaffolds for human cortical bone applications

Bone defects may impede the functionality of a bone as an organ. The bone defects can be repaired using osteogenic cells, growth factors, and bone scaffolds. Functionally graded scaffolds ideally meet the purpose of repair of these damaged bones as they have mechanical and biological properties similar to the surrounding tissues. The properties of these biomaterial structures can be mimicked to that of the human bone for improved osseointegration and bone regeneration without causing any stress shielding effect. Triply Periodic Minimal Surface (TPMS) lattices have geometric properties similar to human bones, making them suitable for orthopaedic implant applications. In this study, the implicit modelling approach was adopted to create a TPMS gyroid unit cell using nTopology software. Thereafter, the gyroid TPMS lattice was used to parametrically design a functionally graded scaffold with radial grading, mimicking the topology of human bone. Here, the scaffold porosity and pore size were varied from 62 to 82% and 236–320 µm, respectively, from the outer to the inner layer. The mechanical responses and compression behaviour of the Ti-6Al-4V alloy scaffold were studied by employing Abaqus the commercial Finite Element Analysis (FEA) software. Capitalizing on the cyclic symmetry, only 1/24th of the geometric model was considered for Dynamic Explicit FEA, yielding a computationally efficient solution. The obtained Young's modulus of 22.1 GPa and compressive strength of 205 MPa of the designed scaffold closely matches with those of human cortical bone, confirming their ability to meet its structural requirements. The objective of this work is to develop a robust numerical framework to parametrically design functionally graded metallic bone scaffolds.

A Novel Performance-Based Evaluation of Built-Up Wide-Flange Steel Columns Subjected to Close-Range Detonations

Transportation facilities are crucial civil engineering structures, which are vulnerable to increasing risks of potential explosive loadings. The structural members of older facilities were typically made of steel and built up from angles and plates joined together with rivets, rather than directly rolled in steel mills. Most existing blast mitigation design approaches were made for military or petrochemical facilities in rural areas with significant stand-off distances. However, transportation facilities, which are located in congested major cities, with structural members exposed to pedestrians and passengers, are subject to the risks of potential close-range or near-contact detonations. Recently, new approaches were developed to address the potential progressive collapse of high-rise buildings in urban environment, but the approaches do not address the localized deformations or failures on column flange or web elements that are common in close-range or near-contact detonations. For this reason, the existing approaches are not appropriate to the blast mitigation design of the urban transportation structures. Instead, a high-fidelity, nonlinear, explicit finite element analysis is suggested. Typical practicing structural engineers do not have opportunities for education or training for such analysis due to technical challenges, cost, and time constraints. The main goal of the present research is to provide practical guides of the nonlinear, explicit finite element analyses for close-in or near-contact detonation problems that can be followed by practicing structural engineers and to provide simple design charts that can be used as a preliminary tool without performing the analyses. To accomplish the goal, more than 200 parametric study models were built for nonlinear, explicit finite element analysis of built-up steel columns subjected to close-in detonations. Characteristics of the steel materials for the built-up columns were investigated and incorporated in the numerical models. The models were determined so that they can cover practical ranges of columns, stand-off distances, and charge weights. The analysis results were presented on contour graphs using novel performance-based damage criteria. The graphs and design methodology presented in this study can be used as an efficient and practical tool to quickly assess blast resistance of built-up steel columns in transportation structures.

An investigation on the size-effects on thickness-distribution of directionally rolled multi-stage deep-drawn copper cups through finite element analysis, and its validation through experimentation employed scaling law

Most sectors are moving towards downsizing products that retain high standards of quality and satisfy customer needs. Miniaturisation of components typically makes use of scaling rules. In this work, we use the idea of similarity theory to investigate the micro-and nano-scale thinning behaviours of multi-stage deep-drawn cups. Deep drawing with a limiting draw ratio (LDR) of 1.69 was used to shape ETP copper sheets into cylindrical cups via annealing and directionally rolling. Each rolling pass consists of nine incremental passes that result in a 50% reduction in thickness. The influence of scaling laws for uni and bi-directional rolling was studied, and scaled dies and punch sets were employed to create scaled cups of five different thicknesses, each 50% of the original 6 mm. For this simulation, we used Altair INSPIRE, a sort of explicit finite element analysis that uses real-world material test data. To investigate the impact of thickness variation across all scaled cups, measurements of the thickness were taken at three points on multi-stage deep-drawn cups. Bi-directional rolling produced cups with less thickness variance than uni-directional rolling. All the scaled cups had a consistency greater than the mean at the lip point and less at the cup base for both ways of rolling, it was also noticed. Physical evaluations of the drawn cups for extreme values verified that the inferred variations lie within ±15% of their original thickness for all cups, and the mean thickness likewise has 50% of its multi-stage-drawing, proving the applicability of the scaling rules.