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Concept# Direct stiffness method

Summary

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). In applying the method, the system must be modeled as a set of simpler, idealized elements interconnected at the nodes. The material stiffness properties of these elements are then, through matrix mathematics, compiled into a single matrix equation which governs the behaviour of the entire idealized structure. The structure’s unknown displacements and forces can then be determined by solving this equation. The direct stiffness method forms the basis for most commercial and free source finite element software.
The direct stiffness method originated in the field of aerospace. Researchers looked at various approaches for analysis of complex airplane frames. These included elasticity theory, energy principles in structural mechanics, flexibility method and matrix stiffness method. It was through analysis of these methods that the direct stiffness method emerged as an efficient method ideally suited for computer implementation.
Between 1934 and 1938 A. R. Collar and W. J. Duncan published the first papers with the representation and terminology for matrix systems that are used today. Aeroelastic research continued through World War II but publication restrictions from 1938 to 1947 make this work difficult to trace. The second major breakthrough in matrix structural analysis occurred through 1954 and 1955 when professor John H. Argyris systemized the concept of assembling elemental components of a structure into a system of equations. Finally, on Nov. 6 1959, M. J. Turner, head of Boeing’s Structural Dynamics Unit, published a paper outlining the direct stiffness method as an efficient model for computer implementation .

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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

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.

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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.

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