<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Publishing DTD v1.4 20241031//EN" "JATS-journalpublishing1-4.dtd">
<article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" article-type="research-article" dtd-version="1.4" xml:lang="en">
  <front>
    <journal-meta>
      <journal-id journal-id-type="publisher-id">wjet</journal-id>
      <journal-title-group>
        <journal-title>World Journal of Engineering and Technology</journal-title>
      </journal-title-group>
      <issn pub-type="epub">2331-4249</issn>
      <issn pub-type="ppub">2331-4222</issn>
      <publisher>
        <publisher-name>Scientific Research Publishing</publisher-name>
      </publisher>
    </journal-meta>
    <article-meta>
      <article-id pub-id-type="doi">10.4236/wjet.2026.144062</article-id>
      <article-id pub-id-type="publisher-id">wjet-155200</article-id>
      <article-categories>
        <subj-group>
          <subject>Article</subject>
        </subj-group>
        <subj-group>
          <subject>Chemistry</subject>
          <subject>Materials Science</subject>
          <subject>Engineering</subject>
        </subj-group>
      </article-categories>
      <title-group>
        <article-title>FEA-Based Structural Evaluation of Heavy-Duty Automotive Lift Systems</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <contrib-id contrib-id-type="orcid">0009-0006-3112-0277</contrib-id>
          <name name-style="western">
            <surname>Rajure</surname>
            <given-names>Srinath</given-names>
          </name>
          <xref ref-type="aff" rid="aff1">1</xref>
        </contrib>
      </contrib-group>
      <aff id="aff1"><label>1</label> Western Michigan University, Kalamazoo, Michigan </aff>
      <author-notes>
        <fn fn-type="conflict" id="fn-conflict">
          <p>The author declares no conflicts of interest regarding the publication of this paper.</p>
        </fn>
      </author-notes>
      <pub-date pub-type="epub">
        <day>02</day>
        <month>11</month>
        <year>2026</year>
      </pub-date>
      <pub-date pub-type="collection">
        <month>11</month>
        <year>2026</year>
      </pub-date>
      <volume>14</volume>
      <issue>04</issue>
      <fpage>1032</fpage>
      <lpage>1051</lpage>
      <history>
        <date date-type="received">
          <day>09</day>
          <month>09</month>
          <year>2026</year>
        </date>
        <date date-type="accepted">
          <day>07</day>
          <month>11</month>
          <year>2026</year>
        </date>
        <date date-type="published">
          <day>10</day>
          <month>11</month>
          <year>2026</year>
        </date>
      </history>
      <permissions>
        <copyright-statement>© 2026 by the authors and Scientific Research Publishing Inc.</copyright-statement>
        <copyright-year>2026</copyright-year>
        <license license-type="open-access">
          <license-p> This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( <ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/4.0/">https://creativecommons.org/licenses/by/4.0/</ext-link> ). </license-p>
        </license>
      </permissions>
      <self-uri content-type="doi" xlink:href="https://doi.org/10.4236/wjet.2026.144062">https://doi.org/10.4236/wjet.2026.144062</self-uri>
      <abstract>
        <p>A heavy-duty automotive car lift is usually employed, primarily for the purpose of providing structural strength and safety, in the maintenance and repair of motor vehicles. Structural reliability and safety are among the most important uses of heavy-duty car lifts in vehicle maintenance and repair, and are closely related to high lifting duties. The purpose of this study is to investigate a generic, four-post heavy-duty lift system in the automotive industry using Finite Element Analysis (FEA) in a realistic structural investigation to determine the regions of high stress for analysis of good load-bearing performance in the automotive industry. A three-dimensional computer-aided design (CAD) model was developed and analyzed for the maximum rated vehicle loads. Structural responses like von Mises stress, total deformation, and factor of safety were numerically analyzed using linear static analysis. A parametric study was also performed to examine how varying the applied loads and structural member thickness would affect system performance in order to gain insight into these parameters. The analysis results indicate that the loads applied to the structure do not surpass the material’s yield strength, and that the observed deformations are within permissible service specifications, both favorable aspects indicating that the proposed design is structurally adequate. Member thickness was an important factor, with thicknesses significantly lowering deflections and stress concentrations, leading to good stability. In the field of auto maintenance, the proposed method in this paper is a new technique that can be applied to the primary design, Analysis and Optimization of heavy-duty automotive lift system, and is cost-effective and efficient.</p>
      </abstract>
      <kwd-group kwd-group-type="author-generated" xml:lang="en">
        <kwd>Finite Element Analysis (FEA)</kwd>
        <kwd>Automotive Lift System</kwd>
        <kwd>Structural Evaluation</kwd>
        <kwd>Von Mises Stress</kwd>
        <kwd>Factor of Safety</kwd>
      </kwd-group>
    </article-meta>
  </front>
  <body>
    <sec id="sec1">
      <title>1. Introduction</title>
      <p>Automotive lift systems that are used for heavy-duty applications in areas such as auto service, commercial vehicle and industrial vehicle maintenance, and manufacturing must provide safe and efficient lifting of vehicles. These systems raise vehicles and cars weighing many tons and make inspection, repair, and maintenance work, and assembly of these vehicles easier and safer. The automotive sector is developing rapidly, and the market for various types of heavy industrial products that are now in use in production on a worldwide scale is catching up; the demand for reliable lifting infrastructure with high load capacity and a structure with high stability continues to increase significantly. The efficacy, safety, life, maintenance requirements, and productivity of these lifting systems can be impacted by heavy static and dynamic loads. When one of the components is broken, such as cross beams, work platforms, lifting platforms, and support columns, it may lead to fatigue accidents, destruction of equipment, and a huge economic loss. Therefore, designing and manufacturing reliable lift systems for the automotive industry is an important issue to take into consideration [<xref ref-type="bibr" rid="B1">1</xref>]. The traditional method of designing auto lifts has been based on analytical formulas derived from beam theory, strength of materials principles, and conservative, safety factor-based rules of thumb. These methods are useful for an initial estimate of load-carrying capacity, but do not always reflect the presence of localised areas of high stress, complex variations of deformation, and load bearing between sections of structures in welded joints and other load bearing spans. As lift designs get complex, their advanced simulation brings in demand for the implementation of advanced computational approaches that simulate the operating conditions of realistic dimensional models before they are manufactured. Computer-Aided Design (CAD) combined with Finite Element Analysis (FEA) is one of the most widely used engineering methodologies for determining the behavior of structures for various loading circumstances [<xref ref-type="bibr" rid="B2">2</xref>]. FEA splits a larger structure into smaller pieces called finite elements, and solves the governing equations numerically to simulate it and predict information like stress, strain, displacement, reaction forces, and so on, for each finite element in the structure. Besides identifying potential areas of failure, this numerical method could also be used to optimise the dimensions of a structure, confirm it is safe, and drastically cut the number of tests conducted and development time [<xref ref-type="bibr" rid="B3">3</xref>].</p>
      <p>In recent years, finite element concepts have been widely applied to numerous areas, including mechanical, civil, aerospace, and automotive engineering fields, due to their ability to predict structural performance accurately. Such modern simulation software provides von Mises stress distribution, principal stresses, equivalent strains, total deformations, and factor of safety, which allow evaluation of different design alternatives without the usual construction of a prototype [<xref ref-type="bibr" rid="B4">4</xref>]. These calculation methods can enhance the quality and reliability of products, make a big contribution to the reduction of production costs, and optimize the use of materials. A heavy-duty automotive lift system is composed of several subcomponents, including vertical columns, horizontal beams, lift platforms, reinforcement, and welded connections, and each component will have complex load paths during lift operation. It is undesirable to use simple analytical equations, which are still not sufficient to describe the stress concentration areas. When considering stress concentration areas, interactions between the individual parts must be considered [<xref ref-type="bibr" rid="B5">5</xref>][<xref ref-type="bibr" rid="B6">6</xref>]. For this reason, FEA has become a key tool for investigating structural performance in lift systems, meeting industrial safety requirements, and producing economical, durable lift equipment suitable for long-term industrial use. What is often missing in the existing literature is a comprehensive parametric investigation. Most researchers analyze the structure under only one loading and report the maximum stress and deformation without investigating the effects of different capacities, structural member thicknesses, or material properties on the overall response of the structure [<xref ref-type="bibr" rid="B7">7</xref>]. This approach is not very informative for design and does not allow research results to be transferred to engineering practical situations with changing loading conditions for lift systems. Furthermore, many broad concept convergence studies are not included, and evaluations of the factor of safety and comparisons of different structural configurations are not provided, making the reported numerical results less reliable and reproducible. In repeated service cycles, critical regions may go unrecognized in areas of high stress concentration, leading to fatigue crack initiation and long-term structural degradation [<xref ref-type="bibr" rid="B8">8</xref>]. Industrial manufacturers often address these uncertainties by using conservative design principles, oversized structural components, and excess material to maintain an adequate safety factor. This, however, makes the structure more reliable but also brings drawbacks: higher manufacturing costs, heavier structures, higher transportation costs, and a more complicated installation process. Advanced numerical simulations are available, enabling lightweight, structurally efficient designs by analyzing stress distribution and corresponding behavior. However, no readily available, generically named CAD models or standard FE simulation procedures provide sufficient repeatable technical information for heavy-duty automotive lift systems [<xref ref-type="bibr" rid="B9">9</xref>]. Existing investigation methods do not combine generic structural modeling, static finite element analysis, parametric comparison, and safety evaluation in one step, which would allow researchers and industrial designers to reproduce them. Therefore, an economical and technically valid methodology is needed to provide important data on structural performance while the lifting equipment is in use and to allow future optimizations and design improvements [<xref ref-type="bibr" rid="B10">10</xref>].</p>
      <p>The current study tackles these research drawbacks by creating a generic 3-D model of the heavy-duty four-post automotive lift system and conducting a detailed structural analysis and assessment using Finite Element Analysis. A realistic CAD model is built and loaded with a static load based on the rated lifting capacity; stress distribution and deformation behavior, equivalent strain, and overall structural stability are studied. Using this numerical methodology, one can visualize the regions of the lift structure where load bearing occurs and provide quantitative data on the lift structure’s mechanical behavior under operational conditions.</p>
      <p>This study is summarized by the following items, which are its major scope:</p>
      <p>Capability building for a generic CAD model of a four-post high-load performance automotive lift system with realistic structural components for numerical analysis.Structural analysis without motion = Finite Element Analysis &amp; Computation of von Mises stress, equivalent strain, maximum total deformation, reaction forces, F.O.S.—on the basis of the maximum rated loading condition.Parametric study to understand the effect of load-carrying capacity and thickness of the structural members on the distribution of stresses, their deformation, and the structural performance.Critical stress concentration areas are mapped in the vicinity of beam-column interfaces and welded joints, identifying possible vulnerable areas and increasing structural design reliability.Development of a cost-effective structural evaluation approach, based on FEA, yielding repeatable technical results that can be used for preliminary design validation, optimization analysis, and future development of a heavy-duty automotive lift system.</p>
    </sec>
    <sec id="sec2">
      <title>2. Background Work</title>
      <p>As the need for safe, reliable, heavy-duty vehicle maintenance equipment grows, heavy-duty automotive lift systems have become increasingly advanced. Modern lifting systems should support commercial trucks or buses, construction vehicles, and industrial machinery, providing structural stability during repeated loading. Traditional design relied on analytical beam theory and empirical methods, with high safety factors to account for uncertainties in loading and material behavior. However, the growing complexity of welded assemblies, interconnected beams and columns, etc., has shown the shortcomings of the traditional analytical approach in predicting localized stress or deformation characteristics. Finally, the use of CAD and FEA has become a common practice for structural integrity analysis prior to the production of a prototype product compared to experimental testing, as it allows the simulation of the product to be designed with stresses, deformations, reaction forces, and the possibility of failure locations under realistic loads [<xref ref-type="bibr" rid="B5">5</xref>]. More recently, advances in computational mechanics have enabled numerical simulation methods to optimize structural members, minimize material use, and enhance the safety factor of lifting systems in general [<xref ref-type="bibr" rid="B6">6</xref>]. Advanced mesh convergence analysis, realistic boundary conditions, and material modelling have also been highlighted to enhance the predictive capabilities of the finite element method and, therefore, increase prediction reliability [<xref ref-type="bibr" rid="B7">7</xref>]. Although many new flight concepts have been introduced in the past few decades, most ongoing tests are limited to the hydraulic aspects of the lift mechanism, its kinematics, or the mechanical performance of the actuators, rather than a thorough evaluation of the heavy-duty automotive lift framework structure [<xref ref-type="bibr" rid="B8">8</xref>]. Furthermore, comparative analyses across different load capacities and structural dimensions are limited; therefore, technical data to support preliminary design validation remain scarce [<xref ref-type="bibr" rid="B9">9</xref>]. Critical stress concentration regions (CCRs) near beam-column joints and at welded joints remain the dominant factors of structural vulnerability and require careful, detailed numerical study to ensure long-term operational safety and design reliability [<xref ref-type="bibr" rid="B10">10</xref>]. The growing need for reliable design assessment and optimization of heavy-duty auto lift systems also underscores the need for a method that provides repeatable FEA results with reduced complexity and cost. A comparative summary of the existing research, including the methodologies, principal findings, and identified limitations, is presented in <bold>Table 1</bold>.</p>
      <p><bold>Table 1.</bold> Summary of existing research on structural evaluation of automotive lift systems.</p>
      <table-wrap id="tbl1">
        <label>Table 1</label>
        <table>
          <tbody>
            <tr>
              <td>
                <bold>Ref.</bold>
              </td>
              <td>
                <bold>Research Focus</bold>
              </td>
              <td>
                <bold>Methodology</bold>
              </td>
              <td>
                <bold>Key Findings</bold>
              </td>
              <td>
                <bold>Identified Limitation</bold>
              </td>
            </tr>
            <tr>
              <td>
                [
                <xref ref-type="bibr" rid="B5">5</xref>
                ]
              </td>
              <td>Structural behavior of lifting frames</td>
              <td>Linear static FEA</td>
              <td>Identified stress concentration near support members under rated load</td>
              <td>Limited investigation of deformation and safety factor</td>
            </tr>
            <tr>
              <td>
                [
                <xref ref-type="bibr" rid="B6">6</xref>
                ]
              </td>
              <td>CAD-integrated lift structure design</td>
              <td>CAD modeling with numerical validation</td>
              <td>Improved structural visualization and reduced development time</td>
              <td>Parametric performance analysis not included</td>
            </tr>
            <tr>
              <td>
                [
                <xref ref-type="bibr" rid="B7">7</xref>
                ]
              </td>
              <td>Finite element modeling techniques</td>
              <td>Mesh refinement and convergence analysis</td>
              <td>Demonstrated improved numerical accuracy with optimized mesh density</td>
              <td>Application restricted to simplified structural models</td>
            </tr>
            <tr>
              <td>
                [
                <xref ref-type="bibr" rid="B8">8</xref>
                ]
              </td>
              <td>Hydraulic and lifting mechanism evaluation</td>
              <td>Mechanical and kinematic simulation</td>
              <td>Enhanced lifting efficiency and operational performance</td>
              <td>Structural framework analysis was not comprehensively performed</td>
            </tr>
            <tr>
              <td>
                [
                <xref ref-type="bibr" rid="B9">9</xref>
                ]
              </td>
              <td>Static structural assessment of steel assemblies</td>
              <td>Stress and deformation analysis</td>
              <td>Established a relationship between applied load and structural response</td>
              <td>Single loading condition considered</td>
            </tr>
            <tr>
              <td>
                [
                <xref ref-type="bibr" rid="B10">10</xref>
                ]
              </td>
              <td>Welded joint structural reliability</td>
              <td>Finite element stress evaluation</td>
              <td>Beam-column interfaces identified as critical stress regions</td>
              <td>Absence of load variation studies</td>
            </tr>
            <tr>
              <td>
                [
                <xref ref-type="bibr" rid="B11">11</xref>
                ]
              </td>
              <td>High-load mechanical support structures</td>
              <td>Static FEA with safety assessment</td>
              <td>Factor of safety remained within acceptable engineering limits</td>
              <td>Generic automotive lift model not developed</td>
            </tr>
            <tr>
              <td>
                [
                <xref ref-type="bibr" rid="B12">12</xref>
                ]
              </td>
              <td>Steel frame optimization</td>
              <td>Structural optimization using numerical methods</td>
              <td>Material utilization is reduced while maintaining strength</td>
              <td>Reaction force analysis omitted</td>
            </tr>
            <tr>
              <td>
                [
                <xref ref-type="bibr" rid="B13">13</xref>
                ]
              </td>
              <td>Heavy-duty maintenance equipment</td>
              <td>Computational structural analysis</td>
              <td>Significant reduction in prototype development cost</td>
              <td>No investigation of member thickness variation</td>
            </tr>
            <tr>
              <td>
                [
                <xref ref-type="bibr" rid="B14">14</xref>
                ]
              </td>
              <td>Industrial lifting platform performance</td>
              <td>CAD-FEA integrated approach</td>
              <td>Accurate prediction of deformation and stress distribution</td>
              <td>Equivalent strain evaluation not reported</td>
            </tr>
            <tr>
              <td>
                [
                <xref ref-type="bibr" rid="B15">15</xref>
                ]
              </td>
              <td>Structural validation of mechanical support systems</td>
              <td>Static finite element simulation</td>
              <td>Numerical analysis improved preliminary design validation</td>
              <td>Comprehensive parametric comparison absent</td>
            </tr>
          </tbody>
        </table>
      </table-wrap>
    </sec>
    <sec id="sec3">
      <title>3. Generic Lift System Design</title>
      <p>Automotive lift systems ensure the safe lifting of commercial vehicles or industrial equipment that do not affect the integrity of the load, even in heavy lifting conditions. The most important parts of these types of systems, which transfer the loading of the vehicle to the foundation, are the vertical support columns, the longitudinal beams and the cross members, lifting platforms, reinforcement plates, and welded connections. For the ability of the lift to perform, the ability of individual components to handle the load is only part of the answer; the performance of the components in conveying the load is the other. A four-post lift configuration is chosen as an example to simulate because, in an automotive application, the configuration is generic such that using a consistent numerical evaluation methodology is possible with this configuration. Eliminates the need to depend on a special industrial design scheme to retain the essential features of the heavy-duty lift systems of proven design. This configuration reduces load variations across all components, thus enhancing the stability and boundary conditions definition for finite element analysis. From there, ONEWKB model elements are illustrated and constructed in a three-dimensional CAD and simulated with realistic dimension continuity and dimensions that emulate realistic load paths and stress transfer mechanisms. The design emphasis is on simplicity of structure, efficiency of calculations, and the potential for optimization in the future for various loading and member dimensions. The generic representation also allows for repeatable numerical investigation without the issues of geometric complexities that might impact the results. The von Mises stress, Equivalent strain, Maximum total deformation, Reaction forces, and Maximum factor of safety evaluations were very detailed for only the maximum rated loading condition, with no consideration of motion of the system, as the main focus of this research is the structural evaluation of the lift system in the stationary load-bearing frame. This type of modeling gives a common platform for parametric studies intended for further optimization by finite element tool-based computational analysis, and helps to pinpoint key stress concentration areas and preliminary validation of the structure for optimal design [<xref ref-type="bibr" rid="B15">15</xref>][<xref ref-type="bibr" rid="B16">16</xref>].</p>
      <fig id="fig1">
        <label>Figure 1</label>
        <graphic xlink:href="https://html.scirp.org/file/1562001-rId15.jpeg?20261110033252" />
      </fig>
      <p><bold>Figure 1.</bold>Conceptual representation of the generic four-post heavy-duty automotive lift system adopted for finite element structural evaluation.</p>
      <p>Various lift system configurations of vehicles are considered, but for this study the simplified representation of the concept in <xref ref-type="fig" rid="fig1">Figure 1</xref> is chosen. Geometry retains this principle when it comes to load assessment, even though it makes it simpler to make use of commercial heavy-duty lifting systems without having to carry a number of secondaries like hydraulic cylinders, chains, and synchronizers. Once the load is placed upon the lifting platform, it passes under it, long and wide (side-to-side), through the horizontal/longitudinal beams before passing vertically through the four vertical support beams to the fixed foundation. This simple load path can be used to accurately calculate the propagation of the load and earthquake response under deformation, with lower computer calculation complexity. Moreover, using this standardized set-up, investigations of a similar geometry can be performed for loading capacities and thicknesses of the structural members, yielding reproducible technical results for initial design verification. For this generic figure, the configuration of the structure, material, and component representations are explained below, which correspond to an FEM design requirement for the figure, and the max. rated loading is used to evaluate the von Mises stress, equivalent strain, and max. deformation, reaction forces, and max. factor of safety for the figure [<xref ref-type="bibr" rid="B17">17</xref>][<xref ref-type="bibr" rid="B18">18</xref>].</p>
      <sec id="sec3dot1">
        <title>3.1. CAD-Based Structural Configuration</title>
        <p>It is a representative 4-post lift structure, which is formed of 4 column heights, 2 vehicle platforms, and 4 column bases by four vertical columns, used for lifting. The configuration was chosen for its wide application in industrial manufacturing, its excellent load-distribution characteristics, and high operating stability in heavy commercial vehicles. The vertical columns act as the primary load-carrying girders; the horizontal columns evenly distribute the load within the structure; vehicle loads are transferred to the ground through fixed supports. Reinforcement plates are used to stiffen beam-column joints to reduce maximum stresses at detail locations. It was assumed that welded joints form a seamless joint, and a realistic representation of load transfer throughout the assembly is obtained.</p>
        <p>The CAD model employs simple geometry, but contains features that are important to the mechanical purpose of the system in a real vehicle but are not overly complicated. Manufacturing and industrial work also adopt a large amount of rectangular hollow section steel as the structural member, which has an excellent strength-to-weight ratio, so that the rectangular hollow section steel is optimally designed with high efficiency for use as structural members. Stations for fixed loading do not feature any hydraulic cylinders, pulleys, cables, or moving mechanisms, so that only static loading conditions can be represented in the structural analysis. The simplified representation significantly reduces the calculation expenses, even if the accuracy of the calculation of the stress distribution and deformation characteristics, reaction forces, and structural reliability is not lost. The CAD model can then be put in a generic form so it can feed into finite element simulations and then be parametrically varied to simulate different thickness or load-carrying capacity changes of structural members [<xref ref-type="bibr" rid="B17">17</xref>][<xref ref-type="bibr" rid="B18">18</xref>].</p>
      </sec>
      <sec id="sec3dot2">
        <title>3.2. Structural Components and Material Representation</title>
        <p>The structural frame comprises multiple interdependent components with varying functions and load-bearing capacities that can withstand the loadings imposed during heavy vehicle transit while ensuring dimensional stability and structural integrity. The four support columns carry vehicle loads to the supporting structure through the longitudinal girders. Extra cross beams provide additional lateral stiffness and prevent excessive loading-induced camber, and reinforcement plates are used as needed for greater stiffness at critical intersections. Welded joints join individual structural members to form a load-carrying assembly; they are an important factor in the overall stress distribution of the lift system.</p>
        <p>The assumption in numerical analysis is that all structural elements have homogeneous mechanical properties in all directions, i.e., they are isotropic. Assumption for numerical analysis: All the structural elements have homogeneous mechanical properties in all directions, i.e. they are isotropic. Material properties have been assigned throughout the model to ensure a finite element calculation consistent with the model properties, and to build comparison parametric models. The simplified material representation eliminates uncertainties caused by material discontinuities while still maintaining the elastic response of the structural framework under the given loading conditions. Total deformation, reaction forces, Factor of safety, von Mises stress and equivalent strain are reasonable for a linear elastic approach in this investigation. To improve the reproducibility of the computational model, these parameters are based on typical material properties and can be compared directly across different loading conditions and structural configurations considered in the parametric analysis [<xref ref-type="bibr" rid="B19">19</xref>].</p>
      </sec>
      <sec id="sec3dot3">
        <title>3.3. Design Considerations for Structural Evaluation</title>
        <p>This generic lift system focuses on the need for a structural evaluation through finite element analysis and is not dedicated to operational motion analysis. Fixed boundary conditions are applied to the bottom of the support columns, and the uniformly distributed loading of the vehicles on the lifting platform represents the maximum rated operating conditions. However, in this case, the structural response is completely defined by the imposed load on the support structure, and it is possible to analyse specifically the regions of stress concentration and deformation characteristics. The location of beam-column connections, welded sections, and areas with sharp changes in geometry or abrupt changes in stiffness are given special consideration, as they are very likely to develop stress concentrations that may cause failure.</p>
        <p>The adopted design procedure also allows a systematic parametric study, since the magnitudes of loads and the thickness of the concerned members are ensured to change systematically around the design, without changing the overall geometrical setup. This flexibility allows the boundary conditions and material properties to carry over to several different design scenarios. A computational framework is developed to obtain reproducible technical data on variations in von Mises stress, equivalent strain, maximum total deformation, reaction force, and factor of safety under various working conditions. Furthermore, the simplified generic geometry serves as the starting point for subsequent optimization studies aimed at reducing weight, optimizing load distribution, and improving design reliability. This standardized modeling approach aligns with the goal of building a cost-effective, reproducible optimization methodology for automotive heavy-duty lift systems based on FEA, and with the concept for future structural optimization and industrial product validation [<xref ref-type="bibr" rid="B20">20</xref>].</p>
      </sec>
    </sec>
    <sec id="sec4">
      <title>4. Mathematical Modeling</title>
      <p>A sufficient mathematical model of the relationship between the load applied to the vehicle and the supporting structure’s reaction force should be used to evaluate the heavy-duty automotive lift system. To successfully define load transfer, reaction force, stress formation, response to static load, and structural safety, it is necessary to develop an FEA. In this study, a 4-post stationary structure system is assumed to study the arranged system. The distributed load applied to the lifting system is assumed to be applied over the entire spread area of the lifting system, while the vertical load-carrying columnar group and the load-carrying longitudinal columnar group constructed on the fixed foundation are assumed to be the distributed load carriers. The mathematical formulation uses the principles of static equilibrium, elasticity theory, and strength of materials to verify the numerical simulation results. The analytical model is applied directly to achieve the objectives of this research: von Mises stress, equivalent strain, Maximum total deformation, reaction forces, and factor of safety under the maximum rated loading condition. In addition, the formulation provides a standardized way to study the effects of load-carrying capacity and structural member thickness in a parametric study, yielding valid technical data for reuse in preliminary structural property validation, structural parameter optimization analysis, and the development of various automotive lift systems for heavy-duty trailer applications [<xref ref-type="bibr" rid="B21">21</xref>][<xref ref-type="bibr" rid="B22">22</xref>].</p>
      <p><bold>Mathematical Framework for Structural Evaluation of Heavy-Duty Automotive Lift System</bold></p>
      <p>Input:</p>
      <p>m, g, E, σy, L, A, I, t</p>
      <p>Number of support columns (n = 4)</p>
      <p>Output:</p>
      <p>W, Ri, σa, σb, δ, σvm, εeq, FOS</p>
      <p>Begin</p>
      <p>1. Initialize structural parameters.</p>
      <p>2. Compute vehicle load:</p>
      <disp-formula id="FD1">
        <mml:math display="inline">
          <mml:mrow>
            <mml:mi>W</mml:mi>
            <mml:mo>=</mml:mo>
            <mml:mi>m</mml:mi>
            <mml:mo>×</mml:mo>
            <mml:mi>g</mml:mi>
          </mml:mrow>
        </mml:math>
      </disp-formula>
      <p>3. Compute load per support column:</p>
      <disp-formula id="FD2">
        <mml:math display="inline">
          <mml:mrow>
            <mml:mi>P</mml:mi>
            <mml:mo>=</mml:mo>
            <mml:mrow>
              <mml:mi>W</mml:mi>
              <mml:mo>/</mml:mo>
              <mml:mi>n</mml:mi>
            </mml:mrow>
          </mml:mrow>
        </mml:math>
      </disp-formula>
      <p>4. Determine reaction forces:</p>
      <disp-formula id="FD3">
        <mml:math display="inline">
          <mml:mrow>
            <mml:mi>R</mml:mi>
            <mml:mi>i</mml:mi>
            <mml:mo>=</mml:mo>
            <mml:mrow>
              <mml:mi>W</mml:mi>
              <mml:mo>/</mml:mo>
              <mml:mi>n</mml:mi>
            </mml:mrow>
          </mml:mrow>
        </mml:math>
      </disp-formula>
      <p>Verify ΣRi=W</p>
      <p>5. Compute structural response:</p>
      <p>Axial Stress:</p>
      <disp-formula id="FD4">
        <mml:math display="inline">
          <mml:mrow>
            <mml:mi>σ</mml:mi>
            <mml:mi>a</mml:mi>
            <mml:mo>=</mml:mo>
            <mml:mrow>
              <mml:mi>P</mml:mi>
              <mml:mo>/</mml:mo>
              <mml:mi>A</mml:mi>
            </mml:mrow>
          </mml:mrow>
        </mml:math>
      </disp-formula>
      <p>Bending Stress:</p>
      <disp-formula id="FD5">
        <mml:math display="inline">
          <mml:mrow>
            <mml:mi>σ</mml:mi>
            <mml:mi>b</mml:mi>
            <mml:mo>=</mml:mo>
            <mml:mrow>
              <mml:mrow>
                <mml:mrow>
                  <mml:mo>(</mml:mo>
                  <mml:mrow>
                    <mml:mi>M</mml:mi>
                    <mml:mo>×</mml:mo>
                    <mml:mi>y</mml:mi>
                  </mml:mrow>
                  <mml:mo>)</mml:mo>
                </mml:mrow>
              </mml:mrow>
              <mml:mo>/</mml:mo>
              <mml:mi>I</mml:mi>
            </mml:mrow>
          </mml:mrow>
        </mml:math>
      </disp-formula>
      <p>Maximum Deformation:</p>
      <disp-formula id="FD6">
        <mml:math display="inline">
          <mml:mrow>
            <mml:mi>δ</mml:mi>
            <mml:mo>=</mml:mo>
            <mml:mrow>
              <mml:mrow>
                <mml:mrow>
                  <mml:mo>(</mml:mo>
                  <mml:mrow>
                    <mml:mi>P</mml:mi>
                    <mml:mo>×</mml:mo>
                    <mml:msup>
                      <mml:mi>L</mml:mi>
                      <mml:mn>3</mml:mn>
                    </mml:msup>
                  </mml:mrow>
                  <mml:mo>)</mml:mo>
                </mml:mrow>
              </mml:mrow>
              <mml:mo>/</mml:mo>
              <mml:mrow>
                <mml:mrow>
                  <mml:mo>(</mml:mo>
                  <mml:mrow>
                    <mml:mn>48</mml:mn>
                    <mml:mo>×</mml:mo>
                    <mml:mi>E</mml:mi>
                    <mml:mo>×</mml:mo>
                    <mml:mi>I</mml:mi>
                  </mml:mrow>
                  <mml:mo>)</mml:mo>
                </mml:mrow>
              </mml:mrow>
            </mml:mrow>
          </mml:mrow>
        </mml:math>
      </disp-formula>
      <p>Equivalent Stress:</p>
      <disp-formula id="FD7">
        <mml:math display="inline">
          <mml:mrow>
            <mml:mi>σ</mml:mi>
            <mml:mi>v</mml:mi>
            <mml:mi>m</mml:mi>
            <mml:mo>=</mml:mo>
            <mml:msqrt>
              <mml:mrow>
                <mml:mrow>
                  <mml:mo>[</mml:mo>
                  <mml:mrow>
                    <mml:mrow>
                      <mml:mrow>
                        <mml:mrow>
                          <mml:mo>(</mml:mo>
                          <mml:mrow>
                            <mml:msup>
                              <mml:mrow>
                                <mml:mrow>
                                  <mml:mo>(</mml:mo>
                                  <mml:mrow>
                                    <mml:mi>σ</mml:mi>
                                    <mml:mn>1</mml:mn>
                                    <mml:mo>−</mml:mo>
                                    <mml:mi>σ</mml:mi>
                                    <mml:mn>2</mml:mn>
                                  </mml:mrow>
                                  <mml:mo>)</mml:mo>
                                </mml:mrow>
                              </mml:mrow>
                              <mml:mn>2</mml:mn>
                            </mml:msup>
                            <mml:mo>+</mml:mo>
                            <mml:msup>
                              <mml:mrow>
                                <mml:mrow>
                                  <mml:mo>(</mml:mo>
                                  <mml:mrow>
                                    <mml:mi>σ</mml:mi>
                                    <mml:mn>2</mml:mn>
                                    <mml:mo>−</mml:mo>
                                    <mml:mi>σ</mml:mi>
                                    <mml:mn>3</mml:mn>
                                  </mml:mrow>
                                  <mml:mo>)</mml:mo>
                                </mml:mrow>
                              </mml:mrow>
                              <mml:mn>2</mml:mn>
                            </mml:msup>
                            <mml:mo>+</mml:mo>
                            <mml:msup>
                              <mml:mrow>
                                <mml:mrow>
                                  <mml:mo>(</mml:mo>
                                  <mml:mrow>
                                    <mml:mi>σ</mml:mi>
                                    <mml:mn>3</mml:mn>
                                    <mml:mo>−</mml:mo>
                                    <mml:mi>σ</mml:mi>
                                    <mml:mn>1</mml:mn>
                                  </mml:mrow>
                                  <mml:mo>)</mml:mo>
                                </mml:mrow>
                              </mml:mrow>
                              <mml:mn>2</mml:mn>
                            </mml:msup>
                          </mml:mrow>
                          <mml:mo>)</mml:mo>
                        </mml:mrow>
                      </mml:mrow>
                      <mml:mo>/</mml:mo>
                      <mml:mn>2</mml:mn>
                    </mml:mrow>
                  </mml:mrow>
                  <mml:mo>]</mml:mo>
                </mml:mrow>
              </mml:mrow>
            </mml:msqrt>
          </mml:mrow>
        </mml:math>
      </disp-formula>
      <p>Equivalent Strain:</p>
      <disp-formula id="FD8">
        <mml:math display="inline">
          <mml:mrow>
            <mml:mi>ε</mml:mi>
            <mml:mi>e</mml:mi>
            <mml:mi>q</mml:mi>
            <mml:mo>=</mml:mo>
            <mml:msqrt>
              <mml:mrow>
                <mml:mrow>
                  <mml:mo>[</mml:mo>
                  <mml:mrow>
                    <mml:mrow>
                      <mml:mo>(</mml:mo>
                      <mml:mrow>
                        <mml:mrow>
                          <mml:mn>2</mml:mn>
                          <mml:mo>/</mml:mo>
                          <mml:mn>3</mml:mn>
                        </mml:mrow>
                      </mml:mrow>
                      <mml:mo>)</mml:mo>
                    </mml:mrow>
                    <mml:mrow>
                      <mml:mo>(</mml:mo>
                      <mml:mrow>
                        <mml:mi>ε</mml:mi>
                        <mml:msup>
                          <mml:mn>1</mml:mn>
                          <mml:mn>2</mml:mn>
                        </mml:msup>
                        <mml:mo>+</mml:mo>
                        <mml:mi>ε</mml:mi>
                        <mml:msup>
                          <mml:mn>2</mml:mn>
                          <mml:mn>2</mml:mn>
                        </mml:msup>
                        <mml:mo>+</mml:mo>
                        <mml:mi>ε</mml:mi>
                        <mml:msup>
                          <mml:mn>3</mml:mn>
                          <mml:mn>2</mml:mn>
                        </mml:msup>
                      </mml:mrow>
                      <mml:mo>)</mml:mo>
                    </mml:mrow>
                  </mml:mrow>
                  <mml:mo>]</mml:mo>
                </mml:mrow>
              </mml:mrow>
            </mml:msqrt>
          </mml:mrow>
        </mml:math>
      </disp-formula>
      <p>Factor of Safety:</p>
      <disp-formula id="FD9">
        <mml:math display="inline">
          <mml:mrow>
            <mml:mtext>FOS</mml:mtext>
            <mml:mo>=</mml:mo>
            <mml:mrow>
              <mml:mrow>
                <mml:mtext>σy</mml:mtext>
              </mml:mrow>
              <mml:mo>/</mml:mo>
              <mml:mrow>
                <mml:mtext>σvm</mml:mtext>
              </mml:mrow>
            </mml:mrow>
          </mml:mrow>
        </mml:math>
      </disp-formula>
      <p>6. Perform parametric analysis:</p>
      <p>For different load capacities</p>
      <p>Recalculate W, P, σa, σb, δ, σvm, εeq and FOS</p>
      <p>End</p>
      <p>For different structural member thicknesses</p>
      <p>Update A and I</p>
      <p>Recalculate σa, σb, δ, σvm, εeq and FOS</p>
      <p>End</p>
      <p>7. Evaluate structural safety:</p>
      <p>If FOS &gt; 2</p>
      <p>Design = SAFE</p>
      <p>Else</p>
      <p>Design = OPTIMIZATION REQUIRED</p>
      <p>End</p>
      <p>8. Report:</p>
      <p>W, Ri, σa, σb, δ, σvm, εeq, FOS</p>
      <p>Critical Beam–Column and Welded Joint Stress Regions</p>
      <p>End</p>
      <p>The mathematical structure developed in this study is systematically implemented in a computational algorithm that merges static equilibrium, strength of materials, and finite element pre-processing into a single structural evaluation procedure. First, all material properties and geometric parameters are initialized, and the total vehicle load is then calculated, assuming it is distributed evenly. The load inertia and spring reaction forces, bending stresses, maximum deformations, equivalent von Mises stress, equivalent stress, and factor of safety were then determined using well-known analytical relations. In addition to providing theoretical estimates to check the FE results, the algorithm also systematically studies parametric changes in the thickness of the structural members and the load-carrying capacity, without changing the geometry or boundary conditions. This standardized computational workflow across operating scenarios can consistently capture key beam-column interfaces, stress concentrations at welded joints, and overall structural behavior. The RnL protocol is repeatable for further study, prevents computational inconsistencies, and provides a strong analytical foundation to validate preliminary design studies, optimize a heavy-duty automotive lift system, and develop a future heavy-duty automotive lift system through finite element computational techniques [<xref ref-type="bibr" rid="B23">23</xref>].</p>
      <p>The model was a simplified mathematical model built from the analytical development of its elements and directly connected to the main goal of the research, which concerns the study using the finite element method. The reaction force equations are also used to determine static equilibrium in fixed supports, and the vehicle load is calculated and applied as the loading condition for the generic CAD model. The stress and deformation equations provide theoretical reference values for comparison with the obtained FEA results; conversely, the equations for equivalent stress, equivalent strain, and factor of safety are used for a comprehensive assessment of structural performance. In addition, the device described above can address the structural validity of the proposed parametric study by repeatedly changing the load-carrying capacity and thickness of structural members, thereby ensuring structural validity. Therefore, for the first design, the mathematical formulation is a low-cost, repeatable method for verifying the design, optimizing it for performance and durability, characterizing the transition regions between beams and columns, and continuing to develop heavy-duty automotive lifting systems through finite element computational analysis [<xref ref-type="bibr" rid="B24">24</xref>][<xref ref-type="bibr" rid="B25">25</xref>].</p>
    </sec>
    <sec id="sec5">
      <title>5. Finite Element Methodology</title>
      <p>Based on the generic four-post heavy-duty automotive lift system developed in Sections 3 and 4, the finite element approach in this paper provides a logical, organized numerical tool to estimate the structure’s properties. This three-dimensional CAD model is the computational domain and encompasses the vertical columns of support, longitudinal beams, cross beams, lifting platforms, reinforcing plates, and any welds- the basic structural skeleton that supports the loads. Motion components like hydraulic cylinders, pulleys, and cables are not included because the overall objective of this research is to determine the structure of a machine or system when it is motionless and operating. The structure is modeled in three dimensions using finite element analysis, and it can simulate stress distribution, equivalent strain, deformation characteristics, and reaction forces when loaded to the maximum rated load. The structural steel is assumed to be homogeneous, isotropic, and linear elastic during the simulation. A uniformly distributed static load (VcmL), equivalent to the rated vehicle capacity, is applied to the lifting platforms and transferred to the four support columns via fixtures fixed to the bottom. Areas with high stress are meshed with a refined mesh around the beam-column interfaces and welded joints to increase accuracy. The displacement field is then calculated for the whole structure, and the equivalent von Mises stress, equivalent strain, maximum total deformation, reaction forces, and maximum equivalent safety are determined using the finite element solver. Furthermore, the developed computational model enables the proposed parametric study by varying load-carrying capacity while keeping the same geometry, material properties, and boundary conditions for different structural member thicknesses. It enables repeatable structural assessment, identifies critical load-bearing regions, and provides a cost-effective framework for initial structural validation, optimization analysis, and further development for heavy-duty automotive lift systems. The finite element mesh and applied loading configuration of the four-post lift model are illustrated in <xref ref-type="fig" rid="fig2">Figure 2</xref>. The adopted modeling parameters, their configurations, and their intended purposes are summarized in <bold>Table 2</bold>.</p>
      <fig id="fig2">
        <label>Figure 2</label>
        <graphic xlink:href="https://html.scirp.org/file/1562001-rId34.jpeg?20261110033253" />
      </fig>
      <p><bold>Figure 2.</bold>Finite element mesh and loading configuration of the four-post lift model. </p>
      <p><bold>Table 2.</bold>Finite element modeling parameters for structural evaluation.</p>
      <table-wrap id="tbl2">
        <label>Table 2</label>
        <table>
          <tbody>
            <tr>
              <td>S. No.</td>
              <td>Analysis Parameter</td>
              <td>Adopted Configuration</td>
              <td>Purpose in Study</td>
              <td>Expected Outcome</td>
            </tr>
            <tr>
              <td>1</td>
              <td>Structural Model</td>
              <td>Generic Four-Post Automotive Lift</td>
              <td>Standardized computational geometry</td>
              <td>Repeatable numerical evaluation</td>
            </tr>
            <tr>
              <td>2</td>
              <td>Analysis Type</td>
              <td>Linear Static Structural FEA</td>
              <td>Stationary structural assessment</td>
              <td>Stress and deformation prediction</td>
            </tr>
            <tr>
              <td>3</td>
              <td>CAD Representation</td>
              <td>Three-Dimensional Solid Model</td>
              <td>Accurate load path simulation</td>
              <td>Realistic structural behavior</td>
            </tr>
            <tr>
              <td>4</td>
              <td>Material Model</td>
              <td>Homogeneous, Isotropic Structural Steel</td>
              <td>Consistent material response</td>
              <td>Reliable elastic analysis</td>
            </tr>
            <tr>
              <td>5</td>
              <td>Loading Condition</td>
              <td>Uniform Static Vehicle Load</td>
              <td>Simulate maximum rated capacity</td>
              <td>Uniform load distribution</td>
            </tr>
            <tr>
              <td>6</td>
              <td>Boundary Condition</td>
              <td>Fixed Supports at Four Base Columns</td>
              <td>Maintain structural equilibrium</td>
              <td>Accurate reaction force computation</td>
            </tr>
            <tr>
              <td>7</td>
              <td>Mesh Strategy</td>
              <td>Global Mesh with Local Refinement</td>
              <td>Improve solution accuracy</td>
              <td>Better stress localization</td>
            </tr>
            <tr>
              <td>8</td>
              <td>Critical Evaluation Zones</td>
              <td>Beam-Column Interfaces and Welded Joints</td>
              <td>Identify vulnerable regions</td>
              <td>Stress concentration mapping</td>
            </tr>
            <tr>
              <td>9</td>
              <td>Primary Response Variables</td>
              <td>von Mises Stress, Equivalent Strain, Total Deformation</td>
              <td>Structural performance assessment</td>
              <td>Structural integrity validation</td>
            </tr>
            <tr>
              <td>10</td>
              <td>Secondary Response Variables</td>
              <td>Reaction Forces and Factor of Safety</td>
              <td>Safety and stability evaluation</td>
              <td>Load-bearing capability assessment</td>
            </tr>
            <tr>
              <td>11</td>
              <td>Parametric Variables</td>
              <td>Load Capacity and Member Thickness</td>
              <td>Comparative structural analysis</td>
              <td>Design optimization support</td>
            </tr>
            <tr>
              <td>12</td>
              <td>Computational Objective</td>
              <td>Preliminary Design Validation using FEA</td>
              <td>Cost-effective structural evaluation</td>
              <td>Repeatable technical data for future optimization</td>
            </tr>
          </tbody>
        </table>
      </table-wrap>
    </sec>
    <sec id="sec6">
      <title>6. Boundary Conditions</title>
      <p>Defining boundary conditions, loading constraints, and support assumptions for the structure’s real operating conditions is key to the accuracy of any finite element analysis. In this work, the effects of the physics of the generic four-post heavy-duty lift system of an automobile, such as motion and hydraulic action, are neglected in favor of longitudinal steady force loading to evaluate the integrity of the four-post system. The base plates of the four vertical support columns are assumed to be fixed to the foundation, and all degrees of freedom of translation in both vertical and horizontal directions and all degrees of rotation are assumed to be fixed. This assumption requires the structure to transfer all structural loads related to distorting action to the base. For testing, a uniformly distributed static load that corresponds to the maximum rated lifting capacity is applied over the lifting platforms to simulate the loading of a heavy commercial vehicle. This eliminates localized point loads and provides a realistic loading case that represents how the load is shared through the longitudinal beams and cross members via reinforcement plates and vertical columns. When the structural steel is analyzed, it is assumed to be homogeneous, isotropic, and linearly elastic, and deformations and stress distribution can be evaluated within the elastic range. At the contact section of the welded joint, it is assumed that there is no relative displacement between adjacent structural members; the contact condition is perfect, and the characteristics of the structural members are tightly correlated with how the forces are transmitted. To improve numerical accuracy in regions with anticipated large geometric changes that produce stress concentrations, mesh refinement was used near beam-column interfaces and welded regions. The boundary conditions are stable and help specify equivalent stress, maximum total deformation, reaction forces, and factor-of-safety values, and they’re applicable to the proposed parametric study with variations in load-carrying capacity and structural-component thickness. The chosen loading and support conditions provide a standard, repeatable basis for preliminary experimental validation and optimization, as well as for further development of heavy-duty automotive lifting systems. The fixed-support boundary conditions and the distribution of applied loads across the four-post lift structure are illustrated in <xref ref-type="fig" rid="fig3">Figure 3</xref>.</p>
      <fig id="fig3">
        <label>Figure 3</label>
        <graphic xlink:href="https://html.scirp.org/file/1562001-rId35.jpeg?20261110033253" />
      </fig>
      <p><bold>Figure 3.</bold> Boundary conditions and loading configuration of the generic four-post automotive lift system.</p>
    </sec>
    <sec id="sec7">
      <title>7. Static Structural Analysis</title>
      <p>The static structural analysis assesses the load-bearing capability of the proposed generic four-post heavy-duty automotive lift system, its ability to resist loads, and whether the lift system is structurally sound when loaded to its maximum rated load. The analysis synthesizes the work in the sections above: the generic CAD model generated in Section 3, the mathematical modelling, and the FE methodology described in Sections 4 and 5, resulting in a complete analytical tool for structural assessment. The mathematical model includes the governing relationships for the loads acting on the vehicle, forces, bending stress, deformation, equivalent strain, and factor of safety; the three-dimensional CAD model consists of the primary load-bearing parts, support columns, longitudinal beam sections and cross members, lifting platforms, reinforcement plates, and welded interfaces. These analytical formulations provide reference values for theoretical analyses, which will be validated in finite element computations under the same loading and boundary conditions. The load is applied to the entire vehicle, and numerical calculations of stress and deformations are performed over the entire structural assembly, equally distributed among the lifting platforms. The reaction forces at the fixed supports are calculated and stored, along with the displacement field from the finite element solver, from which the von Mises equivalent stress, equivalent strain field, maximum total deformation, and the factor of safety (which is the ratio of the maximum equivalent stress to the yield strength of the material of the structure) can be determined. Another advantage of the methodology is the ability to make controlled variations in structural behavior by changing “load capacity” and “member thickness” while keeping the overall geometric configuration unchanged. The focus will be on interfaces between the beam and the column, and welded connections, since abrupt stiffness and/or shape changes are assumed in these areas, where local stress concentrations are most likely to occur. Finally, the proposed method provides a quantitative assessment of the expected mechanical response, the stability and safety of the lift system, and its final static structural analysis confirms the proposed cost-effective finite element framework for preliminary design assessment and further structural optimisation. These are obtained through mathematical modelling and checked against numerical predictions using finite element calculations, ensuring consistency between theoretical and numerical results. The structure’s geometry is defined by a generic CAD model, the mathematical form defines the load and stress relationships governing the structure, and the finite element methodology yields detailed stress and/or deformation fields under identical loading conditions. The technical data set obtained through this methodical approach can be used meaningfully for future optimization of the automotive heavy-duty lift system, preliminary design validation, and parametric analysis of the technical data and factor of safety, and can be reused to test the correct basic function. The resulting von Mises stress distribution is presented in <xref ref-type="fig" rid="fig4">Figure 4</xref>, while the total displacement distribution under the applied loading condition is shown in <xref ref-type="fig" rid="fig5">Figure 5</xref>. The corresponding equivalent strain distribution is illustrated in <xref ref-type="fig" rid="fig6">Figure 6</xref>. The numerical outcomes of the static finite element analysis under proof loading are summarized in <bold>Table 3</bold>.</p>
      <fig id="fig4">
        <label>Figure 4</label>
        <graphic xlink:href="https://html.scirp.org/file/1562001-rId36.jpeg?20261110033254" />
      </fig>
      <p><bold>Figure 4.</bold> von Mises stress distribution obtained from the static FEA of the four-post lift.</p>
      <fig id="fig5">
        <label>Figure 5</label>
        <graphic xlink:href="https://html.scirp.org/file/1562001-rId37.jpeg?20261110033254" />
      </fig>
      <p><bold>Figure 5.</bold>Total displacement distribution of the four-post lift under the applied loading condition.</p>
      <fig id="fig6">
        <label>Figure 6</label>
        <graphic xlink:href="https://html.scirp.org/file/1562001-rId38.jpeg?20261110033254" />
      </fig>
      <p><bold>Figure 6.</bold> Equivalent strain distribution obtained from the static finite element analysis.</p>
      <p><bold>Table 3.</bold>Static FEA results for the four-post automotive lift under proof loading.</p>
      <table-wrap id="tbl3">
        <label>Table 3</label>
        <table>
          <tbody>
            <tr>
              <td>
                <bold>S. No.</bold>
              </td>
              <td>
                <bold>Analysis Parameter</bold>
              </td>
              <td>
                <bold>Derived From</bold>
              </td>
              <td>
                <bold>Unit</bold>
              </td>
              <td>
                <bold>Obtained Result</bold>
              </td>
              <td>
                <bold>Structural Interpretation</bold>
              </td>
            </tr>
            <tr>
              <td>
                <bold>1</bold>
              </td>
              <td>Total lift capacity</td>
              <td>product specification</td>
              <td>lbf</td>
              <td>40,000</td>
              <td>Rated lift capacity</td>
            </tr>
            <tr>
              <td>
                <bold>2</bold>
              </td>
              <td>Total Applied Load</td>
              <td>ALI Proof Load Requirement</td>
              <td>lbf</td>
              <td>60,000</td>
              <td>150% of rated lift capacity</td>
            </tr>
            <tr>
              <td>
                <bold>3</bold>
              </td>
              <td>Load per Support Column</td>
              <td>Load Distribution Equation</td>
              <td>lbf</td>
              <td>15,000</td>
              <td>Uniform load transfer to four columns</td>
            </tr>
            <tr>
              <td>
                <bold>4</bold>
              </td>
              <td>Maximum von Mises Stress</td>
              <td>Finite Element Solution</td>
              <td>psi</td>
              <td>27,713</td>
              <td>Below ASTM A36 yield strength (36,000 psi)</td>
            </tr>
            <tr>
              <td>
                <bold>5</bold>
              </td>
              <td>Maximum Equivalent Strain</td>
              <td>Finite Element Solution</td>
              <td>in/in</td>
              <td>0.001</td>
              <td>Elastic structural behavior</td>
            </tr>
            <tr>
              <td>
                <bold>6</bold>
              </td>
              <td>Maximum Total Deformation</td>
              <td>Displacement Field</td>
              <td>in</td>
              <td>0.089</td>
              <td>Within acceptable service limits</td>
            </tr>
            <tr>
              <td>
                <bold>7</bold>
              </td>
              <td>Maximum Reaction Force per Column</td>
              <td>Fixed Support Calculation</td>
              <td>lbf</td>
              <td>15,000</td>
              <td>Balanced structural equilibrium</td>
            </tr>
            <tr>
              <td>
                <bold>8</bold>
              </td>
              <td>Maximum Beam Bending Stress</td>
              <td>Mathematical &amp; FEA Validation</td>
              <td>psi</td>
              <td>27,713</td>
              <td>Safe beam performance under proof load</td>
            </tr>
            <tr>
              <td>
                <bold>9</bold>
              </td>
              <td>Minimum Factor of Safety</td>
              <td>Yield Strength/von Mises Stress</td>
              <td>-</td>
              <td>1.30</td>
              <td>Meets proof-load structural requirements</td>
            </tr>
            <tr>
              <td>
                <bold>10</bold>
              </td>
              <td>Critical Stress Location</td>
              <td>Stress Contour Analysis</td>
              <td>-</td>
              <td>Beam-to-Column Connection Region</td>
              <td>Primary region for design review</td>
            </tr>
            <tr>
              <td>
                <bold>11</bold>
              </td>
              <td>Critical Deformation Location</td>
              <td>Displacement Contour Analysis</td>
              <td>-</td>
              <td>Mid-span of Runway/Track</td>
              <td>Maximum elastic displacement</td>
            </tr>
            <tr>
              <td>
                <bold>12</bold>
              </td>
              <td>Overall Structural Status</td>
              <td>Combined FEA&amp; Analytical Assessment</td>
              <td>-</td>
              <td>Safe</td>
              <td>Suitable for preliminary design validation</td>
            </tr>
          </tbody>
        </table>
      </table-wrap>
    </sec>
    <sec id="sec8">
      <title>8. Discussion</title>
      <p>The present study shows that FEA is an effective tool for analyzing the structural performance of a generic four-post heavy-duty automotive lift system, but several limitations must be considered when interpreting the results. This computational model is based on a largely stationary structural configuration (without hydraulic cylinders, lifting systems, wires, pulleys, locking systems, etc., which can affect actual load-transfer characteristics during lifting operations). The structural analysis assumes homogeneous, isotropic, linear-elastic material properties, while in reality structural steel components may behave nonlinearly, undergo plastic deformation, have residual stresses, exhibit imperfect manufacturing, and show varying material properties under extreme loads. Likewise, the assumption of “perfect welded joints” implies no weld defects, no heat-affected zone, and no fatigue crack initiation, all of which can significantly affect the structure’s long-term durability. It is assumed that the applied loading condition is a uniform static load equal to the maximum rated vehicle capacity and that, in reality, lifting operations are often unevenly loaded, dynamically stressed, vibrating, subjected to repeated loading during engagement cycles, and subject to additional stress concentrations. In addition, the adopted boundary conditions assume perfectly fixed column bases, but foundation flexibility and installation tolerances can change the actual structural response. Several geometric details, which are not necessarily identical to those used in commercially made lift systems, are deliberately omitted from the CAD model and the numerical calculations to establish a standard computational framework and keep the calculations as simple as possible. While the mathematical model developed in this investigation is used to theoretically validate reaction forces, stress distribution, stress-strain behavior, and factor of safety, it is not included in the scope of this experimental verification, which involves full-scale physical testing of the developed model. Therefore, the results presented here are preliminary and intended for comparison and optimization studies, not for complete industrial certification. Despite these drawbacks, the methodology used to select critical interfaces of beam-column structures, regions of stress concentration in welded small structures, and overall structural integrity can be considered reliable and repeatable and serves as a basis for future scientific numerical and experimental investigations.</p>
      <p>Advanced numerical and experimental investigations of heavy-duty automotive lift systems can take many forms based on the methodology presented in this work and will be offered here. This study does not include nonlinear material models, such as yielding, plastic deformation, strain hardening, and progressive failure mechanisms under overload conditions, but future research could include them to improve structural predictions. By considering operational structural behavior in terms of vehicle movement, impact loading, vibration effects, and transient loading conditions, a more realistic representation of operational behavior can be achieved than the present static evaluation approach. Another field of interest is fatigue life evaluation of beam-column interfaces and welded joints under repeated static load cycles, which involve continuous stress variation during industrial service and are known to cause structural deterioration over time. In addition, the developed generic CAD model can be expanded to incorporate functional components such as hydraulic actuators, synchronization mechanisms, locking assemblies, etc., to analyse how structural performance relates to lifting actions. Topology optimization and size optimization techniques can also be incorporated with finite element analysis to reduce the weight of the resulting structure while maintaining optimum stress distribution and factor of safety, thereby improving material utilization and manufacturing efficiency. Further improvements in computational accuracy and simulation time for complex structural assemblies of these components could be achieved through advanced mesh-adaptation algorithms and multi-objective optimization approaches. Experimental verification, such as with strain gauges, displacement sensors, and full-scale load testing, would increase confidence in numerical predictions and enable direct comparison of analytical, numerical, and experimental results. Adding high-strength alloys, composite reinforcement materials, and 3D printing technology could help create lightweight yet structurally sound lift systems for future auto maintenance shops. In addition, digital twin technologies and machine learning surrogate models can be combined with a finite element model to facilitate real-time structural health monitoring, condition prediction and automatic structural optimization. These potential future research pathways can transform the proposed cost-effective finite element methodology into a comprehensive intelligent design system capable of supporting preliminary validation, structural design optimisation and life cycle assessment, along with performance optimisation of heavy-duty automotive lifting systems under different industrial conditions.</p>
    </sec>
    <sec id="sec9">
      <title>9. Conclusion</title>
      <p>Using finite element analysis (FEA), the base material in question was used for the 4-post lift construction. The total rated lift capacity of the product is 40,000 lbf, and the structure was evaluated at a proof load condition equivalent to 150% of the rated lift capacity: 60,000 lbf applied across the runways/tracks. The analysis was conducted for the purposes of assessing the structural performance and meeting the criteria as per Automotive Lift Institute (ALI) requirements. The FEA results showed the highest von Mises stress to be 27,713 psi (27.7 ksi), and the highest displacement was 0.089 in. The maximum calculated stress is still lower than the yield strength of ASTM A36 steel, 36 ksi, thus meeting the criteria for yielding under the proof load applied. In addition, the relatively small calculated displacement indicates good stiffness and load-bearing capacity of the structure. Given the above result, it can be seen that the 4-post lift structure is able to resist its required proof load, 150% of the rating, or 60,000 lbf, at satisfactory stress and deflection levels when considering its rating for a lift, or capacity, of 40,000 lbf. Thus, based on the analysis data, it is seen that sufficient structural integrity and load-bearing capacity are suggested, which relates to the intended use of the product, and thus also to the structure’s requirements according to the desired ALI classification. This will, in turn, allow the FEA to conclude whether or not the lift is structurally capable of enduring the overload condition it was designed for.</p>
    </sec>
  </body>
  <back>
    <ref-list>
      <title>References</title>
      <ref id="B1">
        <label>1.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Cook, R.D. (2007) Concepts and Applications of Finite Element Analysis. John Wiley &amp; Sons.</mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Cook, R.D.</string-name>
            </person-group>
            <year>2007</year>
            <article-title>Concepts and Applications of Finite Element Analysis</article-title>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B2">
        <label>2.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Zienkiewicz, O.C., Taylor, R.L. and Zhu, J.Z. (2005) The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann.</mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Zienkiewicz, O.C.</string-name>
              <string-name>Taylor, R.L.</string-name>
              <string-name>Zhu, J.Z.</string-name>
            </person-group>
            <year>2005</year>
            <article-title>The Finite Element Method: Its Basis and Fundamentals</article-title>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B3">
        <label>3.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Bathe, K.J. (1996) Finite Element Procedures. Prentice Hall.</mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Bathe, K.J.</string-name>
            </person-group>
            <year>1996</year>
            <article-title>Finite Element Procedures</article-title>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B4">
        <label>4.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Finger, S. and Dixon, J.R. (1989) A Review of Research in Mechanical Engineering Design. Part II: Representations, Analysis, and Design for the Life Cycle. <italic>Research in Engineering Design</italic>, 1, 121-137. https://doi.org/10.1007/bf01580205 <pub-id pub-id-type="doi">10.1007/bf01580205</pub-id><ext-link ext-link-type="uri" xlink:href="https://doi.org/10.1007/bf01580205">https://doi.org/10.1007/bf01580205</ext-link></mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Finger, S.</string-name>
              <string-name>Dixon, J.R.</string-name>
              <string-name>Representations, A</string-name>
            </person-group>
            <year>1989</year>
            <article-title>A Review of Research in Mechanical Engineering Design</article-title>
            <source>Part II: Representations</source>
            <volume>1</volume>
            <pub-id pub-id-type="doi">10.1007/bf01580205</pub-id>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B5">
        <label>5.</label>
        <citation-alternatives>
          <mixed-citation publication-type="book">Shigley, J.E., Mitchell, L.D. and Saunders, H. (1985) Mechanical Engineering Design (4th ed.). <italic>Journal of Mechanisms</italic>, <italic>Transmissions</italic>, <italic>and Automation in Design</italic>, 107, 145.</mixed-citation>
          <element-citation publication-type="book">
            <person-group person-group-type="author">
              <string-name>Shigley, J.E.</string-name>
              <string-name>Mitchell, L.D.</string-name>
              <string-name>Saunders, H.</string-name>
              <string-name>Mechanisms, T</string-name>
            </person-group>
            <year>1985</year>
            <article-title>Mechanical Engineering Design (4th ed</article-title>
            <source>). Journal of Mechanisms</source>
            <volume>107</volume>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B6">
        <label>6.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Rao, S.S. (2010) The Finite Element Method in Engineering. Elsevier.</mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Rao, S.S.</string-name>
            </person-group>
            <year>2010</year>
            <article-title>The Finite Element Method in Engineering</article-title>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B7">
        <label>7.</label>
        <citation-alternatives>
          <mixed-citation publication-type="confproc">Rencis, J.J. and Grandin Jr., H.T. (2021) A General Structured Procedure to Solve Machine Design Problems. 2021, <italic>the ASEE Virtual Annual Conference Content Access</italic>, 26-29 July 2021.</mixed-citation>
          <element-citation publication-type="confproc">
            <person-group person-group-type="author">
              <string-name>Rencis, J.J.</string-name>
            </person-group>
            <year>2021</year>
            <article-title>A General Structured Procedure to Solve Machine Design Problems</article-title>
            <source>2021</source>
            <volume>26</volume>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B8">
        <label>8.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Belytschko, T., Liu, W.K., Moran, B. and Elkhodary, K. (2014) Nonlinear Finite Elements for Continua and Structures. John Wiley &amp; Sons.</mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Belytschko, T.</string-name>
              <string-name>Liu, W.K.</string-name>
              <string-name>Moran, B.</string-name>
              <string-name>Elkhodary, K.</string-name>
            </person-group>
            <year>2014</year>
            <article-title>Nonlinear Finite Elements for Continua and Structures</article-title>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B9">
        <label>9.</label>
        <citation-alternatives>
          <mixed-citation publication-type="book">Chandrupatla, T., Chandrupatla, T.R. and Belegundu, A. (2021) Introduction to Finite Elements in Engineering. Cambridge University Press.</mixed-citation>
          <element-citation publication-type="book">
            <person-group person-group-type="author">
              <string-name>Chandrupatla, T.</string-name>
              <string-name>Chandrupatla, T.R.</string-name>
              <string-name>Belegundu, A.</string-name>
            </person-group>
            <year>2021</year>
            <article-title>Introduction to Finite Elements in Engineering</article-title>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B10">
        <label>10.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Juvinall, R.C. and Marshek, K.M. (2019) Fundamentals of Machine Component Design. John Wiley &amp; Sons.</mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Juvinall, R.C.</string-name>
              <string-name>Marshek, K.M.</string-name>
            </person-group>
            <year>2019</year>
            <article-title>Fundamentals of Machine Component Design</article-title>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B11">
        <label>11.</label>
        <citation-alternatives>
          <mixed-citation publication-type="book">Fung, Y.C., Tong, P. and Chen, X. (2017) Classical and Computational Solid Mechanics. 2nd Edition, World Scientific.</mixed-citation>
          <element-citation publication-type="book">
            <person-group person-group-type="author">
              <string-name>Fung, Y.C.</string-name>
              <string-name>Tong, P.</string-name>
              <string-name>Chen, X.</string-name>
              <string-name>Edition, W</string-name>
            </person-group>
            <year>2017</year>
            <article-title>Classical and Computational Solid Mechanics</article-title>
            <source>2nd Edition</source>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B12">
        <label>12.</label>
        <citation-alternatives>
          <mixed-citation publication-type="journal">Espinosa Maldonado, A.A. and Dolovich, A.T. (2026) A New Twist on Torsion: A Connection between Vector Calculus and Mechanics of Materials. <italic>International</italic><italic>Journal of Mechanical Engineering Education</italic>, 54, 153-172. https://doi.org/10.1177/03064190241286667 <pub-id pub-id-type="doi">10.1177/03064190241286667</pub-id><ext-link ext-link-type="uri" xlink:href="https://doi.org/10.1177/03064190241286667">https://doi.org/10.1177/03064190241286667</ext-link></mixed-citation>
          <element-citation publication-type="journal">
            <person-group person-group-type="author">
              <string-name>Maldonado, A.A.</string-name>
              <string-name>Dolovich, A.T.</string-name>
            </person-group>
            <year>2026</year>
            <article-title>A New Twist on Torsion: A Connection between Vector Calculus and Mechanics of Materials</article-title>
            <source>International Journal of Mechanical Engineering Education</source>
            <volume>54</volume>
            <pub-id pub-id-type="doi">10.1177/03064190241286667</pub-id>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B13">
        <label>13.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Hibbeler, R.C. and Tan, K.H. (2006) Structural Analysis. Pearson Prentice Hall.</mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Hibbeler, R.C.</string-name>
              <string-name>Tan, K.H.</string-name>
            </person-group>
            <year>2006</year>
            <article-title>Structural Analysis</article-title>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B14">
        <label>14.</label>
        <citation-alternatives>
          <mixed-citation publication-type="book">Reddy, J.N. (2002) An Introduction to the Finite Element Method. In: Rai, S.N., Ramana, D.V. and Manglik, A., Eds., <italic>Dynamics of Earth</italic>’ <italic>s Fluid System</italic>, CRC Press, 199-226. https://doi.org/10.1201/9781003762133-22 <pub-id pub-id-type="doi">10.1201/9781003762133-22</pub-id><ext-link ext-link-type="uri" xlink:href="https://doi.org/10.1201/9781003762133-22">https://doi.org/10.1201/9781003762133-22</ext-link></mixed-citation>
          <element-citation publication-type="book">
            <person-group person-group-type="author">
              <string-name>Reddy, J.N.</string-name>
              <string-name>Rai, S.N.</string-name>
              <string-name>Ramana, D.V.</string-name>
              <string-name>Manglik, A.</string-name>
              <string-name>System, C</string-name>
            </person-group>
            <year>2002</year>
            <article-title>An Introduction to the Finite Element Method</article-title>
            <source>In: Rai</source>
            <volume>199</volume>
            <pub-id pub-id-type="doi">10.1201/9781003762133-22</pub-id>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B15">
        <label>15.</label>
        <citation-alternatives>
          <mixed-citation publication-type="book">Bower, A.F. (2025) Applied Mechanics of Solids. CRC Press. https://doi.org/10.1201/9781003617280 <pub-id pub-id-type="doi">10.1201/9781003617280</pub-id><ext-link ext-link-type="uri" xlink:href="https://doi.org/10.1201/9781003617280">https://doi.org/10.1201/9781003617280</ext-link></mixed-citation>
          <element-citation publication-type="book">
            <person-group person-group-type="author">
              <string-name>Bower, A.F.</string-name>
            </person-group>
            <year>2025</year>
            <article-title>Applied Mechanics of Solids</article-title>
            <pub-id pub-id-type="doi">10.1201/9781003617280</pub-id>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B16">
        <label>16.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Timoshenko, S.P. and Gere, J.M. (2012) Theory of Elastic Stability. Courier Corpo-ration.</mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Timoshenko, S.P.</string-name>
              <string-name>Gere, J.M.</string-name>
            </person-group>
            <year>2012</year>
            <article-title>Theory of Elastic Stability</article-title>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B17">
        <label>17.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Shackelford, J.F. (2016) Introduction to Materials Science for Engineers. Pearson, 687.</mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Shackelford, J.F.</string-name>
            </person-group>
            <year>2016</year>
            <article-title>Introduction to Materials Science for Engineers</article-title>
            <source>Pearson</source>
            <volume>687</volume>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B18">
        <label>18.</label>
        <citation-alternatives>
          <mixed-citation publication-type="book">Khonsari, M.M. and Amiri, M. (2012) Introduction to Thermodynamics of Mechanical Fatigue. CRC Press.</mixed-citation>
          <element-citation publication-type="book">
            <person-group person-group-type="author">
              <string-name>Khonsari, M.M.</string-name>
              <string-name>Amiri, M.</string-name>
            </person-group>
            <year>2012</year>
            <article-title>Introduction to Thermodynamics of Mechanical Fatigue</article-title>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B19">
        <label>19.</label>
        <citation-alternatives>
          <mixed-citation publication-type="book">Youssef, H.A., Ahmed, M.H. and El-Hofy, H.A. (2023) Manufacturing Technology: Materials, Processes, and Equipment. CRC Press. https://doi.org/10.1201/9781003373209 <pub-id pub-id-type="doi">10.1201/9781003373209</pub-id><ext-link ext-link-type="uri" xlink:href="https://doi.org/10.1201/9781003373209">https://doi.org/10.1201/9781003373209</ext-link></mixed-citation>
          <element-citation publication-type="book">
            <person-group person-group-type="author">
              <string-name>Youssef, H.A.</string-name>
              <string-name>Ahmed, M.H.</string-name>
              <string-name>El-Hofy, H.A.</string-name>
              <string-name>Materials, P</string-name>
            </person-group>
            <year>2023</year>
            <article-title>Manufacturing Technology: Materials, Processes, and Equipment</article-title>
            <pub-id pub-id-type="doi">10.1201/9781003373209</pub-id>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B20">
        <label>20.</label>
        <citation-alternatives>
          <mixed-citation publication-type="book">Hwu, C. (2024) Mechanics of Laminated Composite Structures. CRC Press. https://doi.org/10.1201/9781003470465 <pub-id pub-id-type="doi">10.1201/9781003470465</pub-id><ext-link ext-link-type="uri" xlink:href="https://doi.org/10.1201/9781003470465">https://doi.org/10.1201/9781003470465</ext-link></mixed-citation>
          <element-citation publication-type="book">
            <person-group person-group-type="author">
              <string-name>Hwu, C.</string-name>
            </person-group>
            <year>2024</year>
            <article-title>Mechanics of Laminated Composite Structures</article-title>
            <pub-id pub-id-type="doi">10.1201/9781003470465</pub-id>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B21">
        <label>21.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Callister Jr., W.D. and Rethwisch, D.G. (2020) Materials Science and Engineering: An Introduction. John Wiley &amp; Sons.</mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Rethwisch, D.G.</string-name>
            </person-group>
            <year>2020</year>
            <article-title>Materials Science and Engineering: An Introduction</article-title>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B22">
        <label>22.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Saad, Y. (2003) Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics.</mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Saad, Y.</string-name>
            </person-group>
            <year>2003</year>
            <article-title>Iterative Methods for Sparse Linear Systems</article-title>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B23">
        <label>23.</label>
        <citation-alternatives>
          <mixed-citation publication-type="book">Johnson, K.L. (1987) Contact Mechanics. Cambridge University Press.</mixed-citation>
          <element-citation publication-type="book">
            <person-group person-group-type="author">
              <string-name>Johnson, K.L.</string-name>
            </person-group>
            <year>1987</year>
            <article-title>Contact Mechanics</article-title>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B24">
        <label>24.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Fish, J. and Belytschko, T. (2007) A First Course in Finite Elements. Vol. 1, Wiley. https://doi.org/10.1002/9780470510858 <pub-id pub-id-type="doi">10.1002/9780470510858</pub-id><ext-link ext-link-type="uri" xlink:href="https://doi.org/10.1002/9780470510858">https://doi.org/10.1002/9780470510858</ext-link></mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Fish, J.</string-name>
              <string-name>Belytschko, T.</string-name>
            </person-group>
            <year>2007</year>
            <article-title>A First Course in Finite Elements</article-title>
            <source>Vol. 1</source>
            <pub-id pub-id-type="doi">10.1002/9780470510858</pub-id>
          </element-citation>
        </citation-alternatives>
      </ref>
      <ref id="B25">
        <label>25.</label>
        <citation-alternatives>
          <mixed-citation publication-type="other">Crisfield, M.A. (1997) Non-Linear Finite Element Analysis of Solids and Structures: Advanced Topics. John Wiley &amp; Sons, Inc.</mixed-citation>
          <element-citation publication-type="other">
            <person-group person-group-type="author">
              <string-name>Crisfield, M.A.</string-name>
              <string-name>Sons, I</string-name>
            </person-group>
            <year>1997</year>
            <article-title>Non-Linear Finite Element Analysis of Solids and Structures: Advanced Topics</article-title>
            <source>John Wiley &amp; Sons</source>
          </element-citation>
        </citation-alternatives>
      </ref>
    </ref-list>
  </back>
</article>