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<title>Applied element method</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Applied element method</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>The <b>applied element method</b> (<b>AEM</b>) is a numerical analysis used in predicting the <a href="Linear_continuum" title="Linear continuum">continuum</a> and <a href="Discrete_mathematics" title="Discrete mathematics">discrete</a> behavior of structures. The modeling method in AEM adopts the concept of discrete cracking allowing it to automatically track <a href="Structural_failure" class="mw-redirect" title="Structural failure">structural collapse</a> behavior passing through all stages of loading: elastic, <a href="Crack_propagation" class="mw-redirect" title="Crack propagation">crack initiation and propagation</a> in tension-weak materials, reinforcement <a href="Yield_(engineering)" title="Yield (engineering)">yield</a>, element separation, element contact and <a href="Collision" title="Collision">collision</a>, as well as collision with the ground and adjacent structures.

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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>Exploration of the approach employed in the applied element method began in 1995 at the <a href="University_of_Tokyo" title="University of Tokyo">University of Tokyo</a> as part of Dr. Hatem Tagel-Din's research studies. The term "applied element method" itself, however, was first coined in 2000 in a paper called "Applied element method for structural analysis: Theory and application for linear materials".<sup id="cite_ref-AEMTheory_1-0" class="reference"><a href="#cite_note-AEMTheory-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Since then AEM has been the subject of research by a number of <a href="Academic_institution" title="Academic institution">academic institutions</a> and the driving factor in real-world applications. Research has verified its accuracy for: elastic analysis;<sup id="cite_ref-AEMTheory_1-1" class="reference"><a href="#cite_note-AEMTheory-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> crack initiation and propagation; estimation of <a href="Structural_failure" class="mw-redirect" title="Structural failure">failure loads</a> at reinforced concrete structures;<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="Reinforced_concrete" title="Reinforced concrete">reinforced concrete</a> structures under cyclic loading;<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="Buckling" title="Buckling">buckling</a> and post-buckling behavior;<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> nonlinear dynamic analysis of structures subjected to severe earthquakes;<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> fault-rupture propagation;<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> nonlinear behavior of brick structures;<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and the analysis of <a href="Glass-reinforced_plastic" class="mw-redirect" title="Glass-reinforced plastic">glass reinforced polymers</a> (GFRP) walls under blast loads.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Technical_discussion">Technical discussion</h2></div>

<p>In AEM, the structure is divided virtually and modeled as an assemblage of relatively small elements. The elements are then connected through a set of normal and shear springs located at contact points distributed along with the element faces. Normal and shear springs are responsible for the transfer of <a href="Normal_stress" class="mw-redirect" title="Normal stress">normal</a> and <a href="Shear_stress" title="Shear stress">shear</a> stresses from one element to the next.

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<div class="mw-heading mw-heading3"><h3 id="Element_generation_and_formulation">Element generation and formulation</h3></div>

<p>The modeling of objects in AEM is very similar to modeling objects in <a href="Finite_element_method" title="Finite element method">FEM</a>. Each object is divided into a series of elements connected and forming a mesh. The main difference between AEM and FEM, however, is how the elements are joined together. In AEM the elements are connected by a series of <a href="Nonlinear_system" title="Nonlinear system">non-linear</a> springs representing the material behavior.

</p><p>There are three types of springs used in AEM:

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<ul><li><b>Matrix Springs</b>: Matrix springs connect two elements together representing the main <a href="Material_properties" class="mw-redirect" title="Material properties">material properties</a> of the object.</li>

<li><b>Reinforcing Bar Springs</b>: Reinforcement springs are used to implicitly represent additional reinforcement bars running through the object without adding additional elements to the analysis.</li>

<li><b>Contact Springs</b>: Contact Springs are generated when two elements collide with each other or the ground. When this occurs three springs are generated (Shear Y, Shear X and Normal).</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Automatic_element_separation">Automatic element separation</h3></div>

<p>When the average strain value at the element face reaches the separation strain, all springs at this face are removed and elements are no longer connected until a collision occurs, at which point they collide together as rigid bodies.

</p><p>Separation strain represents the strain at which adjacent elements are totally separated at the connecting face. This parameter is not available in the elastic material model. For concrete, all springs between the adjacent faces including reinforcement bar springs are cut. If the elements meet again, they will behave as two different rigid bodies that have now contacted each other. For steel, the bars are cut if the stress point reaches <a href="Ultimate_tensile_stress" class="mw-redirect" title="Ultimate tensile stress">ultimate stress</a> or if the concrete reaches the <a href="Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">separation strain</a>.

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<div class="mw-heading mw-heading3"><h3 id="Automatic_element_contact/collision">Automatic element contact/collision</h3></div>

<p>Contact or collision is detected without any user intervention. Elements are able to separate, contract and/or make contact with other elements. In AEM three contact methods include Corner-to-Face, Edge-to-Edge, and Corner-to-Ground.

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<div class="mw-heading mw-heading2"><h2 id="Stiffness_matrix">Stiffness matrix</h2></div>

<p>The spring stiffness in a 2D model can be calculated from the following equations:

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<annotation encoding="application/x-tex">{\displaystyle K_{n}={\frac {E\cdot T\cdot d}{a}}}</annotation>

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</math></span><img src="./1ef5b7f21d3c74c3f528a8c7254e8fbf8d2e27d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.112ex; height:5.343ex;" alt="{\displaystyle K_{n}={\frac {E\cdot T\cdot d}{a}}}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle K_{s}={\frac {G\cdot T\cdot d}{a}}}</annotation>

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</math></span><img src="./eeffd4fe19a3aea22a4ab66afc2eb966a6e00311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.948ex; height:5.343ex;" alt="{\displaystyle K_{s}={\frac {G\cdot T\cdot d}{a}}}" loading="lazy"></span></dd></dl>

<p>Where <i>d</i> is the distance between springs, <i>T</i> is the thickness of the element, <i>a</i> is the length of the representative area, <i>E</i> is the <a href="Young's_modulus" title="Young's modulus">Young's modulus</a>, and <i>G</i> is the <a href="Shear_modulus" title="Shear modulus">shear modulus</a> of the material. The above equation's indicate that each spring represents the stiffness of an area (<i>T</i>·<i>d</i>) within the length of the studied material.

</p><p>To model reinforcement bars embedded in concrete, a spring is placed inside the element at the location of the bar; the area (<i>T</i>·<i>d</i>) is replaced by the actual cross section area of the reinforcement bar. Similar to modeling embedded <a href="Steel_sections" class="mw-redirect" title="Steel sections">steel sections</a>, the area (<i>T</i>·<i>d</i>) may be replaced by the area of the steel section represented by the spring.

</p><p>Although the element motion moves as a <a href="Rigid_body" title="Rigid body">rigid body</a>, its internal <a href="Deformation_(engineering)" title="Deformation (engineering)">deformations</a> are represented by the spring deformation around each element. This means the element shape does not change during analysis, but the behavior of assembly of elements is deformable.

The two elements are assumed to be connected by only one pair of normal and shear springs. To have a general stiffness matrix, the locations of element and contact springs are assumed in a general position. The stiffness matrix components corresponding to each <a href="Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degree of freedom</a> are determined by assuming a unit <a href="Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a> in the studied direction and by determining forces at the <a href="Centroid" title="Centroid">centroid</a> of each element. The 2D element stiffness matrix size is 6 × 6; the components of the upper left quarter of the <a href="Stiffness_matrix" title="Stiffness matrix">stiffness matrix</a> are shown below:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\sin ^{2}(\theta +\alpha )K_{n}&-K_{n}\sin(\theta +\alpha )\cos(\theta +\alpha )&\cos(\theta +\alpha )K_{s}L\sin(\alpha )\\+\cos ^{2}(\theta +\alpha )K_{s}&+K_{s}\sin(\theta +\alpha )\cos(\theta +\alpha )&-\sin(\theta +\alpha )K_{n}L\cos(\alpha )\\\\-K_{n}\sin(\theta +\alpha )\cos(\theta +\alpha )&\sin ^{2}(\theta +\alpha )K_{s}&\cos(\theta +\alpha )K_{n}L\cos(\alpha )\\+K_{s}\sin(\theta +\alpha )\cos(\theta +\alpha )&+\cos ^{2}(\theta +\alpha )K_{n}&+\sin(\theta +\alpha )K_{s}L\sin(\alpha )\\\\\cos(\theta +\alpha )K_{s}L\sin(\alpha )&\cos(\theta +\alpha )K_{n}L\cos(\alpha )&L^{2}\cos ^{2}(\alpha )K_{n}\\-\sin(\theta +\alpha )K_{n}L\cos(\alpha )&+\sin(\theta +\alpha )K_{s}L\sin(\alpha )&+L^{2}\sin ^{2}(\alpha )K_{s}\end{bmatrix}}}</annotation>

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</math></span><img src="./ec32303516cfe5495a476641bda8d76e93fd4421.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.676ex; margin-bottom: -0.329ex; width:84.738ex; height:27.176ex;" alt="{\displaystyle {\begin{bmatrix}\sin ^{2}(\theta +\alpha )K_{n}&-K_{n}\sin(\theta +\alpha )\cos(\theta +\alpha )&\cos(\theta +\alpha )K_{s}L\sin(\alpha )\\+\cos ^{2}(\theta +\alpha )K_{s}&+K_{s}\sin(\theta +\alpha )\cos(\theta +\alpha )&-\sin(\theta +\alpha )K_{n}L\cos(\alpha )\\\\-K_{n}\sin(\theta +\alpha )\cos(\theta +\alpha )&\sin ^{2}(\theta +\alpha )K_{s}&\cos(\theta +\alpha )K_{n}L\cos(\alpha )\\+K_{s}\sin(\theta +\alpha )\cos(\theta +\alpha )&+\cos ^{2}(\theta +\alpha )K_{n}&+\sin(\theta +\alpha )K_{s}L\sin(\alpha )\\\\\cos(\theta +\alpha )K_{s}L\sin(\alpha )&\cos(\theta +\alpha )K_{n}L\cos(\alpha )&L^{2}\cos ^{2}(\alpha )K_{n}\\-\sin(\theta +\alpha )K_{n}L\cos(\alpha )&+\sin(\theta +\alpha )K_{s}L\sin(\alpha )&+L^{2}\sin ^{2}(\alpha )K_{s}\end{bmatrix}}}" loading="lazy"></span></dd></dl>

<p>The stiffness matrix depends on the contact spring stiffness and the spring location. The stiffness matrix is for only one pair of contact springs. However, the global stiffness matrix is determined by summing up the stiffness matrices of individual pairs of springs around each element. Consequently, the developed stiffness matrix has total effects from all pairs of springs, according to the stress situation around the element. This technique can be used in both <a href="Structural_load" title="Structural load">load</a> and displacement control cases. The 3D stiffness matrix may be deduced similarly.

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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>The applied element method is currently being used in the following applications:

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<ul><li>Structural vulnerability assessment

<ul><li><a href="Progressive_collapse" title="Progressive collapse">Progressive collapse</a></li>

<li>Blast analysis</li>

<li>Impact analysis</li>

<li><a href="Seismic_analysis" title="Seismic analysis">Seismic analysis</a></li></ul></li>

<li><a href="Forensic_engineering" title="Forensic engineering">Forensic engineering</a></li>

<li>Performance based design</li>

<li>Demolition analysis</li>

<li>Glass performance analysis</li>

<li><a href="Visual_Effects" class="mw-redirect" title="Visual Effects">Visual effects</a></li></ul>

<ul><li><a href="Building_implosion" title="Building implosion">Building implosion</a></li>

<li><a href="Earthquake_engineering" title="Earthquake engineering">Earthquake engineering</a></li>

<li><a href="Extreme_Loading_for_Structures" title="Extreme Loading for Structures">Extreme Loading for Structures</a></li>

<li><a href="Failure_analysis" title="Failure analysis">Failure analysis</a></li>

<li><a href="Multidisciplinary_design_optimization" title="Multidisciplinary design optimization">Multidisciplinary design optimization</a></li>

<li><a href="Physics_engine" title="Physics engine">Physics engine</a></li>

<li><a href="Progressive_collapse" title="Progressive collapse">Progressive collapse</a></li>

<li><a href="Shear_modulus" title="Shear modulus">Shear modulus</a></li>

<li><a href="Structural_engineering" title="Structural engineering">Structural engineering</a></li>

<li><a href="Young's_modulus" title="Young's modulus">Young's modulus</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFMeguroTagel-Din2000" class="citation journal cs1">Meguro, K.; Tagel-Din, H. (2000). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120229032846/http://sciencelinks.jp/j-east/article/200014/000020001400A0511912.php">"Applied element method for structural analysis: Theory and application for linear materials"</a>. <i>Structural Engineering/Earthquake Engineering</i>. <b>17</b> (1). Japan: Japan Society of Civil Engineers: <span class="nowrap">21–</span>35. F0028A. Archived from <a rel="nofollow" class="external text" href="http://sciencelinks.jp/j-east/article/200014/000020001400A0511912.php">the original</a> on 2012-02-29<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-08-10</span></span>.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFTagel-DinMeguro2000" class="citation journal cs1">Tagel-Din, H.; Meguro, K (2000). <a rel="nofollow" class="external text" href="https://www.jsce.or.jp/publication/e/book/book_seee.html#vol17">"Applied Element Method for Simulation of Nonlinear Materials: Theory and Application for RC Structures"</a>. <i>Structural Engineering/Earthquake Engineering</i>. <b>17</b> (2). Japan: Japan Society of Civil Engineers: <span class="nowrap">137–</span>148<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-08-10</span></span>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFTagel-DinMeguro2001" class="citation journal cs1">Tagel-Din, H.; Meguro, Kimiro (November 2001). <a rel="nofollow" class="external text" href="https://cedb.asce.org/cgi/WWWdisplay.cgi?0106179">"Applied Element Simulation of RC Structures under Cyclic Loading"</a>. <i>Journal of Structural Engineering</i>. <b>127</b> (11). Japan: ASCE: <span class="nowrap">137–</span>148. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1061%2F%28ASCE%290733-9445%282001%29127%3A11%281295%29">10.1061/(ASCE)0733-9445(2001)127:11(1295)</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0733-9445">0733-9445</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2009-08-10</span></span>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFTagel-DinMeguro2002" class="citation journal cs1">Tagel-Din, H.; Meguro, K (2002). "AEM Used for Large Displacement Structure Analysis". <i>Journal of Natural Disaster Science</i>. <b>24</b> (1). Japan: <span class="nowrap">25–</span>34.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFTagel-DinKimiro_Meguro2000" class="citation conference cs1">Tagel-Din, Hatem; Kimiro Meguro, K (January 30 – February 4, 2000). <i>Analysis of a Small Scale RC Building Subjected to Shaking Table Tests using Applied Element Method</i>. New Zealand: Proceedings of the 12th World Conference on Earthquake Engineering. pp. <span class="nowrap">25–</span>34.</cite></span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFHATEMKimiro_MEGURO2004" class="citation conference cs1">HATEM, Tagel-Din; Kimiro MEGURO, K (August 1–6, 2004). <i>Dynamic Modeling of Dip-Slip Faults for Studying Ground Surface Deformation Using Applied Element Method</i>. Vancouver, Canada: Proceedings of the 13th World Conference on Earthquake Engineering.</cite></span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFMayorkaKimiro_Meguro2003" class="citation journal cs1">Mayorka, Paola; Kimiro Meguro, K (October 2003). <a rel="nofollow" class="external text" href="http://www.jstage.jst.go.jp/article/seisankenkyu/55/6/581/_pdf">"Modeling Masonry Structures using the Applied Element Method"</a>. <i>Seisan Kenkyu</i>. <b>55</b> (6). Japan: Institute of Industrial Science, The University of Tokyo: <span class="nowrap">123–</span>126. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1881-2058">1881-2058</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2009-08-10</span></span>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFMayorkaKimiro_Meguro2005" class="citation book cs1">Mayorka, Paola; Kimiro Meguro, K (2005). <i>Blast Testing and Research Bridge at the Tenza Viaduct</i>. Japan: University of Missouri-Rolla, TSWG Contract Number N4175-05-R-4828, Final Report of Task 1.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.appliedelementmethod.com/">Applied Element Method</a></li>

<li><a rel="nofollow" class="external text" href="https://www.extremeloading.com/extreme-loading-technology/">Extreme Loading for Structures - Applied Element Method</a></li></ul></div><div><div class="zim-footer">

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