Matrix stiffness method

Matrix stiffness method

In structural engineering, the matrix stiffness method (or simply "stiffness method", also known as Direct stiffness method) is a "matrix" method that makes use of the members' stiffness relations for computing member forces and displacements in structures. For example, if "k" is the stiffness of a spring that is subject to a force "Q", the spring's stiffness relation is:

:Q = kq

where "q" is the spring deformation. This relation gives "q = Q/k" as the resulting spring deformation.

Member stiffness relations

A typical member stiffness relation has the following general form:

:mathbf{Q}^m = mathbf{k}^m mathbf{q}^m + mathbf{Q}^{om} qquad qquad qquad mathrm{(1)}where:"m" = member number "m".:mathbf{Q}^m = vector of member's characteristic forces, which are unknown internal forces.:mathbf{k}^m = member stiffness matrix which characterises the member's resistance against deformations.:mathbf{q}^m = vector of member's characteristic displacements or deformations.:mathbf{Q}^{om} = vector of member's characteristic forces caused by external effects (such as known forces and temperature changes) applied to the member while mathbf{q}^m = 0 ).

If mathbf{q}^m are member deformations rather than absolute displacements, then mathbf{Q}^m are independent member forces, and in such case (1) can be inverted to yield the so-called "member flexibility matrix", which is used in the flexibility method.

ystem stiffness relation

For a system with many members interconnected at points called nodes, the members' stiffness relations such as Eq.(1) can be integrated by making use of the following observations:
* The member deformations mathbf{q}^m can be expressed in terms of system nodal displacements r in order to ensure compatibility between members. This implies that r will be the primary unknowns.
* The member forces mathbf{Q}^m help to the keep the nodes in equilibrium under the nodal forces R. This implies that the right-hand-side of (1) will be integrated into the right-hand-side of the following nodal equilibrium equations for the entire system:

:mathbf{R} = mathbf{Kr} + mathbf{R}^o qquad qquad qquad mathrm{(2)}where:mathbf{R} = vector of nodal forces, representing external forces applied to the system's nodes.:mathbf{K} = system stiffness matrix, which is established by "assembling" the members' stiffness matrices mathbf{k}^m .:mathbf{r} = vector of system's nodal displacements that can define all possible deformed configurations of the system subject to arbitrary nodal forces R.:mathbf{R}^o = vector of equivalent nodal forces, representing all external effects other than the nodal forces which are already included in the preceding nodal force vector R. This vector is established by assembling the members' mathbf{Q}^{om} .


The system stiffness matrix K is square since the vectors R and r have the same size. In addition, it is symmetric because mathbf{k}^m is symmetric. Once the supports' constraints are accounted for in (2), the nodal displacements are found by solving the system of linear equations (2), symbolically::mathbf{r} = mathbf{K}^{-1} (mathbf{R}-mathbf{R}^o ) qquad qquad qquad mathrm{(3)}

Subsequently, the members' characteristic forces may be found from Eq.(1) where mathbf{q}^m can be found from r by compatibility consideration.

The direct stiffness method

It is common to have Eq.(1) in a form where mathbf{q}^m and mathbf{Q}^{om} are, respectively, the member-end displacements and forces matching in direction with r and R. In such case, mathbf{K} and mathbf{R}^o can be obtained by direct summation of the members' matrices mathbf{k}^m and mathbf{Q}^{om} . The method is then known as the direct stiffness method.

The advantages and disadvantages of the matrix stiffness method are compared and discussed in the flexibility method article.

ee also

*Finite element method
*Finite element method in structural mechanics
*Structural analysis
*Flexibility method

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Matrix - получить на Академике действующий промокод Pharmacosmetica или выгодно matrix купить со скидкой на распродаже в Pharmacosmetica

  • Direct stiffness method — As one of the methods of structural analysis, the direct stiffness method (DSM), also known as the displacement method or matrix stiffness method, is particularly suited for computer automated analysis of complex structures including the… …   Wikipedia

  • Matrix (mathematics) — Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. In mathematics, a matrix (plural matrices, or less commonly matrixes)… …   Wikipedia

  • Matrix method — The matrix method is a structural analysis method used as a fundamental principle in many applications in civil engineering. The method is carried out, using either a stiffness matrix or a flexibility matrix. The flexibility method is not… …   Wikipedia

  • Flexibility method — In structural engineering, the flexibility method is the classical consistent deformation method for computing member forces and displacements in structural systems. Its modern version formulated in terms of the members flexibility matrices also… …   Wikipedia

  • Finite element method in structural mechanics — 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… …   Wikipedia

  • Finite element method — The finite element method (FEM) (sometimes referred to as finite element analysis) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based …   Wikipedia

  • Engineering treatment of the finite element method — This is a draft of a new explanation as suggested on . The finite element method (FEM) is a technique for finding approximate solutions to differential equations that is particularly useful in engineering. As of 2005, FEM is the primary analysis… …   Wikipedia

  • Moment distribution method — The moment distribution method (not to be confused with moment redistribution) is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal.[1] The method only …   Wikipedia

  • Trefftz method — In mathematics, the Trefftz method is a method for the numerical solution of partial differential equations named after the German mathematician Erich Trefftz (1888 1937). It falls within the class of finite element methods. Introduction The… …   Wikipedia

  • Tangent stiffness matrix — In computational mechanics, a tangent stiffness matrix is a matrix that describes the stiffness of a system in response to small changes in configuration. It represents tangent in that the energy of the system can be thought of as a high… …   Wikipedia