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The '''applied element method
== History ==
Research on the applied element method began in 1995 at the [[University of Tokyo]] as part of Dr.Hatem Tagel-Din's research studies. The term "Applied Element Method" was first coined in 2000 in a paper called "Applied Element Method for structural analysis: Theory and application for linear Materials."<ref name=AEMTheory>{{cite journal | last = | first = | authorlink = | coauthors = Meguro, K. and Tagel-Din, H. | title = Applied element method for structural analysis: Theory and application for linear materials | journal = Structural engineering/earthquake engineering. | volume = 17 | issue = 1 | pages = 21-35 | publisher = Japan Society of Civil Engineers(JSCE) | ___location = Japan | date = 2000 | url = http://sciencelinks.jp/j-east/article/200014/000020001400A0511912.php | issn = | doi = | id = F0028A | accessdate = 2009-8-10}}</ref>. Since then AEM has been the subject of research by a number of [[Academic institution |academic institutions]] and real-world application. 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 = | first = | authorlink = | coauthors = Tagel-Din, H. and Meguro, K | title = Applied Element Method for Simulation of Nonlinear Materials: Theory and Application for RC Structures | journal = Structural engineering/earthquake engineering. | volume = 17 | issue = 2 | pages = 137-148 | publisher = Japan Society of Civil Engineers(JSCE) | ___location = Japan | date = 2000 | url = http://www.jsce.or.jp/publication/e/book/book_seee.html#vol17 | issn = | doi = | id = | accessdate = 2009-8-10}}</ref>, [[Reinforced concrete |reinforced concrete]] structures under cyclic loading<ref>{{cite journal | last = | first = | authorlink = | coauthors = Tagel-Din, H. and Meguro, K | title = Applied Element Simulation of RC Structures under Cyclic Loading | journal = Journal of Structural Engineering | volume = 127 | issue = 11 | pages = 137-148 | publisher = ASCE | ___location = Japan | date = November 2001 | url = http://cedb.asce.org/cgi/WWWdisplay.cgi?0106179 | issn = 0733-9445 | doi = 10.1061 | id = | accessdate = 2009-8-10}}</ref>, [[Buckling |buckling]] and post-buckling behavior<ref>{{cite journal | last = | first = | authorlink = | coauthors = 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 | date = 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 under severe earthquakes<ref>{{cite conference | last = | first = | authorlink = | coauthors = Hatem Tagel-Din and Kimiro Meguro, 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 30th –February 4th, 2000 | url = | issn = | doi = | id = | accessdate = }}</ref>, fault-rupture propagation<ref>{{cite conference | last = | first = | authorlink = | coauthors = 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-6th, 2004 | url = | issn = | doi = | id = | accessdate = }}</ref>, nonlinear behavior of brick structures<ref>{{cite journal | last = | first = | authorlink = | coauthors = 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 = | accessdate = 2009-8-10}}</ref>, and analysis of [[Glass-reinforced plastic |glass reinforced polymers]] (GFRP) walls under blast loads <ref>{{citation | last = | first = | authorlink = | coauthors = 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 | date = 2005 | url = | issn = | doi = | id = | accessdate = 2009-8-10}}</ref>.
== Technical
In AEM, structures are modeled as an assembly of relatively small elements by dividing the structure virtually. The elements are connected together through a set of normal and shear springs located at contact points which are distributed along 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
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 together forming a mesh. However the main difference between AEM and FEM is how the elements are connected together. In AEM the elements are connected by a series of [[Nonlinear system |non-linear]] springs representing the material behavior.
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*'''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
When the average strain value at the element face reaches the separation strain, all springs at this face are removed and elements are not connected any more until they collide. If 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 contacted. For steel, the bars are cut if its stress reaches [[Ultimate tensile stress |ultimate stress]] or if the concrete reaches the [[Deformation (mechanics) |separation strain]].
===Automatic
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.
==Stiffness
The spring stiffness in a 2D model can be calculated from the following equations:
: <math>K_n=\frac{E\cdot T\cdot d}{a}</math
: <math>K_s=\frac{G\cdot T\cdot d}{a}</math
Where <math>d=\text {distance between springs}</math>, <math>T=\text {thickness of the element}</math>, <math>a=\text {length of the representative area}</math>, <math>E=\text {Youngs modulus}</math>, and <math>G=\text {Shear modulus}</math> of the material. The above equation indicates 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. 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.
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} ▼
▲<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) \\
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-\sin(\theta+\alpha)K_n L\cos(\alpha) & +\sin(\theta+\alpha)K_s Lsin(\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 ==
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{{reflist}}
== Further
* [http://www.appliedelementmethod.com/default.aspx Applied Element Method]
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