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The '''applied element method''' ('''AEM)''') is a method ofnumerical analysis utilizedused in predicting the [[Continuum (mathematics)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 |collision]], as well as collision with the ground and adjacent structures.
== History ==
ResearchExploration onof 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 "Appliedapplied Elementelement Methodmethod" itself, however, was first coined in 2000 in a paper called "Applied Elementelement Methodmethod for structural analysis: Theory and application for linear Materialsmaterials"."<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 = 21-3521–35 | publisher = Japan Society of Civil Engineers(JSCE) | ___location = Japan | date year= 2000 | url = http://sciencelinks.jp/j-east/article/200014/000020001400A0511912.php | issn id=F0028A | doi access-date= 2009-08-10 | id url-status= F0028Adead | accessdate archive-url=https://web.archive.org/web/20120229032846/http://sciencelinks.jp/j-east/article/200014/000020001400A0511912.php 2009|archive-8date=2012-1002-29 }}</ref>. Since then AEM has been the subject of research by a number of [[Academicacademic institution |academic institutions]]s and the driving factor in real-world applicationapplications. Research has verified its accuracy for: elastic analysis;<ref name="AEMTheory"/>, crack initiation and propagation and; 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 engineering.Engineering | volume = 17 | issue = 2 | pages = 137-148137–148 | publisher = Japan Society of Civil Engineers(JSCE) | ___location = Japan | dateyear = 2000 | url = httphttps://www.jsce.or.jp/publication/e/book/book_seee.html#vol17 | issn = | doi = | id = | accessdateaccess-date = 2009-808-10}}</ref>, [[Reinforced concrete |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 = 137-148137–148 |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 | doi = 10.1061 | id = | accessdateaccess-date = 2009-808-10}}</ref>, [[Buckling |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 = 25-3425–34 | publisher = | ___location = Japan | date year= 2002 | url = http://www.drs.dpri.kyoto-u.ac.jp/jsnds/download.cgi?jsdn_24_1-3.pdf | issn = | doi = | id = | accessdate = 2009-8-10}}</ref>, nonlinear dynamic analysis of structures undersubjected 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 = 25-3425–34 | ___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 = SEISANSeisan KENKYUKenkyu | volume = 55 | issue = 6 | publisher = Institute of Industrial Science, The University of Tokyo | pages = 123-126123–126 | ___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 = 2009-808-10}}</ref>, and the analysis of [[Glass-reinforced plastic |glass reinforced polymers]] (GFRP) walls under blast loads .<ref>{{citationCite | last =book | first first1= Paola| authorlink last1= Mayorka| coauthors last2= Paola Mayorka and Kimiro Meguro,|first2= 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 | dateyear = 2005 | url = | issn = | doi = | id = <!-- | accessdate = 2009-808-10 -->}}</ref>.
== Technical discussion ==
In AEM, structuresthe arestructure is divided virtually and modeled as an assemblyassemblage of relatively small elements by dividing the structure virtually. The elements are connectedthen togetherconnected through a set of normal and shear springs located at contact points which are 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===
The modeling of objects in AEM is very similar to modeling objects in [[Finite element method |FEM]]. Each object is divided into a series of elements that are connected togetherand forming a mesh. However theThe main difference between AEM and FEM, however, is how the elements are connectedjoined together. In AEM the elements are connected by a series of [[Nonlinear system |non-linear]] springs representing the material behavior.
There are three types of springs used in AEM:
*'''Matrix Springs''': Matrix springs connect two elements together representing the main [[Material properties |material properties]] of the object.
*'''Reinforcing Bar Springs''': Reinforcement springs are used to implicitly represent additional reinforcement bars running through the object without adding additional elements to the analysis.
*'''Contact Springs''': 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).
===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 notno longer connected anyuntil morea untilcollision theyoccurs, collide.at Ifwhich point they collide together they collide 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 itsthe stress point reaches [[Ultimate tensile stress |ultimate stress]] or if the concrete reaches the [[Deformation (mechanics) |separation strain]].
===Automatic dlementelement contact/collision===
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.
: <math>K_n=\frac{E\cdot T\cdot d}{a}</math>
: <math>K_s=\frac{G\cdot T\cdot d}{a}</math>
Where ''d'' is the distance between springs, ''T'' is the thickness of the element, ''a'' is the length of the representative area, ''E'' is the [[Young's modulus]], and ''G'' is the [[shear modulus]] of the material. The above equation's indicatesindicate that each spring represents the stiffness of an area <math>(''T\cdot ''·''d'')</math> within the length a of the studied material.
To model reinforcement bars embedded in concrete, a spring is placed inside the element at the ___location of the bar; the area <math>( ''T \cdot ''·''d '') </math> is replaced by the actual cross section area of the reinforcement bar. SimilarlySimilar to modelmodeling embedded [[ Steel sections |steel sections]], the area <math>( ''T \cdot ''·''d '') </math> may be replaced by the area of the steel section represented by the spring. ▼
Although the element motion moves as a [[ Rigid body |rigid body]], its internal [[Deformation (engineering) |deformations]] 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. ▼▲To model reinforcement bars embedded in concrete, a spring is placed inside the element at the ___location of the bar; the area <math>(T\cdot d)</math> is replaced by the actual cross section area of the reinforcement bar. Similarly to model embedded [[Steel sections |steel sections]], the area <math>(T\cdot d)</math> may be replaced by the area of the steel section represented by the spring.
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 x× 6; the components of the upper left quarter of the [[ Stiffness matrix |stiffness matrix]] are shown below: ▼▲Although the element motion moves as a [[Rigid body |rigid body]], its internal [[Deformation (engineering) |deformations]] 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 [[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 x 6; the components of the upper left quarter of the [[Stiffness matrix |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
\end{bmatrix}</math>
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 the 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
*Demolition analysis
*Glass performance analysis
*[[Visual Effects |Visual effects]]
==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]]
==References==
{{reflist}}
== Further reading ==
* [http://www.appliedelementmethod.com/default.aspx Applied Element Method]
*[https://www.extremeloading.com/extreme-loading-technology/ Extreme Loading for Structures - Applied Element Method]
{{DEFAULTSORT:Applied Element Method}}
[[Category:Structural analysis]]
[[Category:Structural engineering]]
[[Category:Building engineering]]
[[Category:Glass engineering and science]]
[[Category: Numerical analysis ]]
[[Category: Scientific simulation software]]
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