Applied element method

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 "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".[1] Since then AEM has been the subject of research by a number of academic institutions and the driving factor in real-world applications. Research has verified its accuracy for: elastic analysis;[1] crack initiation and propagation; estimation of failure loads at reinforced concrete structures;[2] reinforced concrete structures under cyclic loading;[3] buckling and post-buckling behavior;[4] nonlinear dynamic analysis of structures subjected to severe earthquakes;[5] fault-rupture propagation;[6] nonlinear behavior of brick structures;[7] and the analysis of glass reinforced polymers (GFRP) walls under blast loads.[8]

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 and shear stresses from one element to the next.

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

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

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

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 indicate that each spring represents the stiffness of an area (T·d) within the length 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·d) is replaced by the actual cross section area of the reinforcement bar. Similar to modeling embedded steel sections, the area (T·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 × 6; the components of the upper left quarter of the stiffness matrix are shown below:

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

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 load and displacement control cases. The 3D stiffness matrix may be deduced similarly.

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

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

• Applied Element Method
• Extreme Loading for Structures - Applied Element Method