Applied element method
Encyclopedia
The Applied Element Method (AEM) is a numerical analysis used 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.
as part of Dr.Hatem Tagel-Din's research studies. The term itself "Applied Element Method," however 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 institution
s and the driving factor in real-world applications. Research has verified its accuracy for: elastic analysis; crack initiation and propagation; estimation of failure loads
at reinforced concrete structures; reinforced concrete
structures under cyclic loading; buckling
and post-buckling behavior; nonlinear dynamic analysis of structures subjected to severe earthquakes; fault-rupture propagation; nonlinear behavior of brick structures; and the analysis of glass reinforced polymers
(GFRP) walls under blast loads .
stresses from one element to the next.
. Each object is divided into a series of elements connected together and forming a mesh. The main difference between AEM and FEM, however, is how the elements are joined 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:
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 stress or if the concrete reaches the separation strain
.
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 x 6; the components of the upper left quarter of the stiffness matrix are shown below:
Linear continuum
In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line.Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two members there is another, and which "lacks gaps" in the...
and discrete
Discrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...
behavior of structures. The modeling method in AEM adopts the concept of discrete cracking allowing it to automatically track structural collapse
Structural failure
Structural failure refers to loss of the load-carrying capacity of a component or member within a structure or of the structure itself. Structural failure is initiated when the material is stressed to its strength limit, thus causing fracture or excessive deformations...
behavior passing through all stages of loading: elastic, crack initiation and propagation in tension-weak materials, reinforcement yield
Yield (engineering)
The yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed...
, element separation, element contact and collision
Collision
A collision is an isolated event which two or more moving bodies exert forces on each other for a relatively short time.Although the most common colloquial use of the word "collision" refers to accidents in which two or more objects collide, the scientific use of the word "collision" implies...
, 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 TokyoUniversity of Tokyo
, abbreviated as , is a major research university located in Tokyo, Japan. The University has 10 faculties with a total of around 30,000 students, 2,100 of whom are foreign. Its five campuses are in Hongō, Komaba, Kashiwa, Shirokane and Nakano. It is considered to be the most prestigious university...
as part of Dr.Hatem Tagel-Din's research studies. The term itself "Applied Element Method," however 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 institution
Academic institution
Academic institution is an educational institution dedicated to education and research, which grants academic degrees. See also academy and university.- Types of academic institutions include :...
s and the driving factor in real-world applications. Research has verified its accuracy for: elastic analysis; crack initiation and propagation; estimation of failure loads
Structural failure
Structural failure refers to loss of the load-carrying capacity of a component or member within a structure or of the structure itself. Structural failure is initiated when the material is stressed to its strength limit, thus causing fracture or excessive deformations...
at reinforced concrete structures; reinforced concrete
Reinforced concrete
Reinforced concrete is concrete in which reinforcement bars , reinforcement grids, plates or fibers have been incorporated to strengthen the concrete in tension. It was invented by French gardener Joseph Monier in 1849 and patented in 1867. The term Ferro Concrete refers only to concrete that is...
structures under cyclic loading; buckling
Buckling
In science, buckling is a mathematical instability, leading to a failure mode.Theoretically, buckling is caused by a bifurcation in the solution to the equations of static equilibrium...
and post-buckling behavior; nonlinear dynamic analysis of structures subjected to severe earthquakes; fault-rupture propagation; nonlinear behavior of brick structures; and the analysis of glass reinforced polymers
Glass-reinforced plastic
Fiberglass , is a fiber reinforced polymer made of a plastic matrix reinforced by fine fibers of glass. It is also known as GFK ....
(GFRP) walls under blast loads .
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 the element faces. Normal and shear springs are responsible for the transfer of normal and shearShear stress
A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...
stresses from one element to the next.
Element generation and formulation
The modeling of objects in AEM is very similar to modeling objects in FEMFinite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
. Each object is divided into a series of elements connected together and forming a mesh. The main difference between AEM and FEM, however, is how the elements are joined 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 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 stress or if the concrete reaches the separation strain
Deformation (mechanics)
Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body...
.
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:
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
Young's modulus
Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...
, 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
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
, 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
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
are determined by assuming a unit displacement
Displacement (vector)
A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...
in the studied direction and by determining forces at the centroid
Centroid
In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...
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:
-
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 loadStructural loadStructural loads or actions are forces, deformations or accelerations applied to a structure or its components.Loads cause stresses, deformations and displacements in structures. Assessment of their effects is carried out by the methods of structural analysis...
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 collapseProgressive collapseA building undergoes progressive collapse when a primary structural element fails, resulting in the failure of adjoining structural elements, which in turn causes further structural failure, similar to a house of cards....
- Blast analysis
- Impact analysis
- Seismic analysisSeismic analysisSeismic Analysis is a subset of structural analysis and is the calculation of the response of a building structure to earthquakes...
- Progressive collapse
- Forensic engineeringForensic engineeringForensic engineering is the investigation of materials, products, structures or components that fail or do not operate or function as intended, causing personal injury or damage to property. The consequences of failure are dealt with by the law of product liability. The field also deals with...
- Performance based design
- Demolition analysis
- Glass performance analysis
- Visual effectsVisual effectsVisual effects are the various processes by which imagery is created and/or manipulated outside the context of a live action shoot. Visual effects involve the integration of live-action footage and generated imagery to create environments which look realistic, but would be dangerous, costly, or...
See also
- Structural engineeringStructural engineeringStructural engineering is a field of engineering dealing with the analysis and design of structures that support or resist loads. Structural engineering is usually considered a specialty within civil engineering, but it can also be studied in its own right....
- Failure analysisFailure analysisFailure analysis is the process of collecting and analyzing data to determine the cause of a failure. It is an important discipline in many branches of manufacturing industry, such as the electronics industry, where it is a vital tool used in the development of new products and for the improvement...
- Earthquake engineeringEarthquake engineeringEarthquake engineering is the scientific field concerned with protecting society, the natural and the man-made environment from earthquakes by limiting the seismic risk to socio-economically acceptable levels...
- Progressive collapseProgressive collapseA building undergoes progressive collapse when a primary structural element fails, resulting in the failure of adjoining structural elements, which in turn causes further structural failure, similar to a house of cards....
- Building implosionBuilding implosionIn the controlled demolition industry, building implosion is the strategic placing of explosive material and timing of its detonation so that a structure collapses on itself in a matter of seconds, minimizing the physical damage to its immediate surroundings...
- Multidisciplinary design optimizationMultidisciplinary design optimizationMulti-disciplinary design optimization is a field of engineering that uses optimization methods to solve design problems incorporating a number of disciplines. As defined by Prof. Carlo Poloni, MDO is "the art of finding the best compromise"...
- Young's modulusYoung's modulusYoung's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...
- Shear modulus
- Physics enginePhysics engineA physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics , soft body dynamics, and fluid dynamics, of use in the domains of computer graphics, video games and film. Their main uses are in video games , in which case the...
- Extreme Loading for StructuresExtreme Loading for StructuresExtreme Loading for Structures - a commercial structural analysis software program based on the applied element method for the automatic tracking and propagation of cracks, separation of elements, element collision, and collapse of structures under extreme loads...
Further reading
- Structural vulnerability assessment