Applied element method

The applied element method, or AEM is a method of analysis utilized in predicting the continuum and discrete behavior of structures. The modeling method in AEM adopts the concept of discrete cracking allowing it to automatically track structural collapse behavior passing through all stages of loading; elastic, crack initiation and propagation in tension-weak materials, reinforcement yield, element separation, element contact and collision, as well as collision with the ground and adjacent structures.

History

Research on the applied element method began in 1995 at the University of Tokyo as part of Professor 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.". Since then AEM has been the subject of research by a number of academic institutions and real-world application. Research has verified its accuracy for elastic analysis, crack initiation and propagation and estimation of failure loads at reinforced concrete structures, reinforced concrete structures under cyclic loading, buckling and post-buckling behavior, nonlinear dynamic analysis of structures under severe earthquakes, fault-rupture propagation, nonlinear behavior of brick structures, and analysis of glass reinforced polymers (GFRP) walls under blast loads .

Technical Discussion

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 and shear stresses from one element to the next.

Element Generation & Formulation

The modeling of objects in AEM is very similar to modeling objects in 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 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 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 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 stress or if the concrete reaches the separation strain.

Automatic Element 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.

Stiffness Matrix

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

$K_{n}={\frac {E\cdot T\cdot d}{a}}$

$K_{s}={\frac {G\cdot T\cdot d}{a}}$

Where $d={\text{distance between springs}}$, $T={\text{thickness of the element}}$, $a={\text{length of the representative area}}$, $E={\text{Youngs modulus}}$, and $G={\text{Shear modulus}}$ of the material. The above equation indicates that each spring represents the stiffness of an area $(T\cdot d)$ 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 $(T\cdot d)$ is replaced by the actual cross section area of the reinforcement bar. Similarly to model embedded steel sections, the area $(T\cdot d)$ may be replaced by the area of the steel section represented by the spring. Although the element motion moves as a rigid body, its internal 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 degree of freedom are determined by assuming a unit 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 are shown below:

${\begin{bmatrix}\sin ^{2}(\theta +\alpha )K_{n}&-K_{n}\sin(\theta +\alpha )\cos(\theta +\alpha )&\cos(\theta +\alpha )K_{s}L\sin(\alpha )\\+cos^{2}(\theta +\alpha )K_{s}&+K_{s}\sin(\theta +\alpha )cos(\theta +\alpha )&-\sin(\theta +\alpha )K_{n}L\cos(\alpha )\\-K_{n}\sin(\theta +\alpha )\cos(\theta +\alpha )&\sin ^{2}(\theta +\alpha )K_{s}&\cos(\theta +\alpha )K_{n}L\cos(\alpha )\\-K_{s}\sin(\theta +\alpha )\cos(\theta +\alpha )&+\cos ^{2}(\theta +\alpha )K_{n}&+\sin(\theta +\alpha )K_{s}L\sin(\alpha )\\\cos(\theta +\alpha )K_{s}L\sin(\alpha )&\cos(\theta +\alpha )K_{n}Lcos(\alpha )&L^{2}\cos ^{2}(\alpha )K_{n}\\-\sin(\theta +\alpha )K_{n}L\cos(\alpha )&+\sin(\theta +\alpha )K_{s}Lsin(\alpha )&+L^{2}\sin ^{2}(\alpha )K_{s}\end{bmatrix}}$

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 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
  • Progressive collapse
  • Blast analysis
  • Impact analysis
  • Seismic analysis
• Forensic engineering
• Performance based design
• Demolition analysis
• Glass performance analysis
• Visual effects

See also

• Structural engineering
• Failure analysis
• Earthquake engineering
• Progressive collapse
• Building implosion
• Multidisciplinary design optimization
• Young's modulus
• Shear modulus
• Physics engine

Further Reading

• Applied Element Method