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The '''Appliedapplied Elementelement Methodmethod''' ('''AEM)''') is a numerical analysis used in predicting the [[Linear continuum|continuum]] and [[Discrete mathematics|discrete]] behavior of structures. The modeling method in AEM adopts the concept of discrete cracking allowing it to automatically track [[Structural failure|structural collapse]] behavior passing through all stages of loading: elastic, [[Crack propagation|crack initiation and propagation]] in tension-weak materials, reinforcement [[Yield (engineering)|yield]], element separation, element contact and [[collision]], as well as collision with the ground and adjacent structures.
== History ==
Exploration of the approach employed in the applied element method began in 1995 at the [[University of Tokyo]] as part of Dr. Hatem Tagel-Din's research studies. The term itself "Appliedapplied Elementelement Method,method" itself, however, was first coined in 2000 in a paper called "Applied element method for structural analysis: Theory and application for linear Materials.materials".<ref name=AEMTheory>{{cite journal | last last1=Meguro | first first1=K. | authorlink last2= | coauthors = Meguro, K. and Tagel-Din, |first2=H. | title = Applied element method for structural analysis: Theory and application for linear materials | journal = Structural engineeringEngineering/earthquake engineering.Earthquake Engineering | volume = 17 | issue = 1 | pages = 21–35 | publisher = Japan Society of Civil Engineers(JSCE) | ___location = Japan | year = 2000 | url = http://sciencelinks.jp/j-east/article/200014/000020001400A0511912.php| doi = | id = F0028A | accessdate access-date= 2009-08-10 |url-status=dead |archive-url=https://web.archive.org/web/20120229032846/http://sciencelinks.jp/j-east/article/200014/000020001400A0511912.php |archive-date=2012-02-29 }}</ref>. Since then AEM has been the subject of research by a number of [[academic institution]]s and the driving factor in real-world applications. Research has verified its accuracy for: elastic analysis;<ref name="AEMTheory"/>; crack initiation and propagation; estimation of [[Structural failure|failure loads]] at reinforced concrete structures;<ref>{{cite journal | last last1= Tagel-Din| first first1= H.| authorlink last2= Meguro| coauthors first2= Tagel-Din, H. and Meguro, K | title = Applied Element Method for Simulation of Nonlinear Materials: Theory and Application for RC Structures | journal = Structural engineeringEngineering/earthquakeEarthquake engineeringEngineering | volume = 17 | issue = 2 | pages = 137–148 | publisher = Japan Society of Civil Engineers(JSCE) | ___location = Japan | year = 2000 | url = httphttps://www.jsce.or.jp/publication/e/book/book_seee.html#vol17| doi = | id = | accessdateaccess-date = 2009-08-10}}</ref>; [[reinforced concrete]] structures under cyclic loading;<ref>{{cite journal | last last1= Tagel-Din| first first1= H.| authorlink last2= Meguro| coauthors first2= Tagel-Din, H. and Meguro, K Kimiro| title = Applied Element Simulation of RC Structures under Cyclic Loading | journal = Journal of Structural Engineering | volume = 127 | issue = 11 | pages = 137–148 |doi=10.1061/(ASCE)0733-9445(2001)127:11(1295)| publisher = ASCE | ___location = Japan | date = November 2001 | url = httphttps://cedb.asce.org/cgi/WWWdisplay.cgi?0106179 | issn = 0733-9445 | id = | accessdateaccess-date = 2009-08-10 | last1 = Meguro | first1 = Kimiro}}</ref>; [[buckling]] and post-buckling behavior;<ref>{{cite journal | last last1=Tagel-Din | first first1=H. | authorlink last2=Meguro | coauthors first2= Tagel-Din, H. and Meguro, K | title = AEM Used for Large Displacement Structure Analysis | journal = Journal of Natural Disaster Science | volume = 24 | issue = 1 | pages = 25–34 | publisher = | ___location = Japan | year = 2002 | url = http://www.drs.dpri.kyoto-u.ac.jp/jsnds/download.cgi?jsdn_24_1-3.pdf | issn = | doi = | id = | accessdate = 2009-08-10}}</ref>; nonlinear dynamic analysis of structures subjected to severe earthquakes;<ref>{{cite conference | last first1= Hatem| first last1= | authorlink = | coauthors = Hatem Tagel-Din and |last2=Kimiro Meguro,|first2= K | title = Analysis of a Small Scale RC Building Subjected to Shaking Table Tests using Applied Element Method | publisher = Proceedings of the 12th World Conference on Earthquake Engineering | pages = 25–34 | ___location = New Zealand | date = January 30th30 –February– 4thFebruary 4, 2000 | url = | issn = | doi = | id = | accessdate = }}</ref>; fault-rupture propagation;<ref>{{cite conference | last first1= Tagel-Din| first last1= HATEM| authorlink last2=Kimiro MEGURO| coauthors first2= Tagel-Din HATEM and Kimiro MEGURO, K | title = Dynamic Modeling of Dip-Slip Faults for Studying Ground Surface Deformation Using Applied Element Method | publisher = Proceedings of the 13th World Conference on Earthquake Engineering | pages = | ___location = Vancouver, Canada | date = August 1st-6th1–6, 2004 | url = | issn = | doi = | id = | accessdate = }}</ref>; nonlinear behavior of brick structures;<ref>{{cite journal | last first1= Paola| first last1= Mayorka| authorlink last2=Kimiro Meguro| coauthors first2= Paola Mayorka and Kimiro Meguro, K | title = Modeling Masonry Structures using the Applied Element Method | journal = Seisan Kenkyu | volume = 55 | issue = 6 | publisher = Institute of Industrial Science, The University of Tokyo | pages = 123–126 | ___location = Japan | date = October 2003 | url = http://www.jstage.jst.go.jp/article/seisankenkyu/55/6/581/_pdf | issn = 1881-2058 | doi = | id = | accessdateaccess-date = 2009-08-10}}</ref>; and the analysis of [[Glass-reinforced plastic|glass reinforced polymers]] (GFRP) walls under blast loads .<ref>{{Cite book | last first1= Paola| first last1= Mayorka| authorlink last2=Kimiro Meguro| coauthors first2= Paola Mayorka and Kimiro Meguro, K | title = Blast Testing and Research Bridge at the Tenza Viaduct | publisher = University of Missouri-Rolla, TSWG Contract Number N4175-05-R-4828, Final Report of Task 1| ___location = Japan | year = 2005 | url = | issn = | doi = | id = <!-- | accessdate = 2009-08-10 -->}}</ref>.
== Technical discussion ==
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 [[Normal stress|normal]] and [[Shear stress|shear]] stresses from one element to the next.
===Element generation and formulation===
===Automatic element separation===
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.
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 [[Ultimate tensile stress|ultimate stress]] or if the concrete reaches the [[Deformation (mechanics)|separation strain]].
: <math>K_n=\frac{E\cdot T\cdot d}{a}</math>
</br>
: <math>K_s=\frac{G\cdot T\cdot d}{a}</math>
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 [[Degrees of freedom (physics and chemistry)|degree of freedom]] are determined by assuming a unit [[Displacement (vector)|displacement]] in the studied direction and by determining forces at the [[centroid]] of each element. The 2D element stiffness matrix size is 6 × 6; the components of the upper left quarter of the [[stiffness matrix]] are shown below:
: <math>\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
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 [[Structural load|load]] and displacement control cases. The 3D stiffness matrix may be deduced similarly.
== Applications ==
The applied element method is currently being used in the following applications:
*Structural vulnerability assessment
==See also==
* [[StructuralBuilding engineeringimplosion]]
* [[FailureEarthquake analysisengineering]]
* [[Extreme Loading for Structures]] ▼*[[Earthquake engineering]]
* [[ProgressiveFailure collapseanalysis]]
* [[Multidisciplinary design optimization]] ▼*[[Building implosion]]
* [[Physics engine]]
▲*[[Multidisciplinary design optimization]]
* [[Young'sProgressive moduluscollapse]]
* [[Shear modulus]]
* [[PhysicsStructural engineengineering]]
* [[Young's modulus]]
▲*[[Extreme Loading for Structures]]
==References==
{{reflist}}
== Further reading ==
*[http://www.appliedelementmethod.com/default.aspx Applied Element Method]
*[httphttps://www.extremeloading.com/ExtremeLoadingTechnology.aspxextreme-loading-technology/ Extreme Loading for Structures - Applied Element Method]
{{DEFAULTSORT:Applied Element Method}}
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