Finite element method in structural mechanics

The finite element method (FEM) is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. In the FEM, the structural system is modeled by a set of appropriate finite elements interconnected at discrete points called nodes. Elements may have physical properties such as thickness, coefficient of thermal expansion, density, Young's modulus, shear modulus and Poisson's ratio. The origin of finite method can be traced to the matrix analysis of structures where the concept of a displacement or stiffness matrix approach was introduced. Finite element concepts were developed based on engineering methods in 1950s. The finite element method obtained its real impetus in the 1960s and 1970s by John Argyris, and co-workers; at the University of Stuttgart, by Ray W. Clough; at the University of California, Berkeley, by Olgierd Zienkiewicz, and co-workers Ernest Hinton, Bruce Irons; at the University of Swansea, by Philippe G. Ciarlet; at the University of Paris; at Cornell University, by Richard Gallagher and co-workers. The original works such as those by Argyris and Clough became the foundation for today’s finite element structural analysis methods. Straight or curved one-dimensional elements with physical properties such as axial, bending, and torsional stiffnesses. This type of element is suitable for modeling cables, braces, trusses, beams, stiffeners, grids and frames. Straight elements usually have two nodes, one at each end, while curved elements will need at least three nodes including the end-nodes. The elements are positioned at the centroidal axis of the actual members. Two-dimensional elements that resist only in-plane forces by membrane action (plane stress, plane strain), and plates that resist transverse loads by transverse shear and bending action (plates and shells). They may have a variety of shapes such as flat or curved triangles and quadrilaterals.

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Flexibility method

In structural engineering, the flexibility method, also called the method of consistent deformations, is the traditional method for computing member forces and displacements in structural systems. Its modern version formulated in terms of the members' flexibility matrices also has the name the matrix force method due to its use of member forces as the primary unknowns. Flexibility is the inverse of stiffness. For example, consider a spring that has Q and q as, respectively, its force and deformation: The spring stiffness relation is Q = k q where k is the spring stiffness.

Direct stiffness method

As one of the methods of structural analysis, the direct stiffness method, also known as the matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type. It is a matrix method that makes use of the members' stiffness relations for computing member forces and displacements in structures. The direct stiffness method is the most common implementation of the finite element method (FEM).

Finite element method in structural mechanics

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