 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 computerautomated 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.
Contents
History
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 (Felippa 2001).
Member stiffness relations
A typical member stiffness relation has the following general form:
where
 m = member number m.
 = vector of member's characteristic forces, which are unknown internal forces.
 = member stiffness matrix which characterises the member's resistance against deformations.
 = vector of member's characteristic displacements or deformations.
 = vector of member's characteristic forces caused by external effects (such as known forces and temperature changes) applied to the member while ).
If are member deformations rather than absolute displacements, then are independent member forces, and in such case (1) can be inverted to yield the socalled member flexibility matrix, which is used in the flexibility method.
System stiffness relation
 See also: Stiffness matrix
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 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 help to the keep the nodes in equilibrium under the nodal forces R. This implies that the righthandside of (1) will be integrated into the righthandside of the following nodal equilibrium equations for the entire system:
where
 = vector of nodal forces, representing external forces applied to the system's nodes.
 = system stiffness matrix, which is established by assembling the members' stiffness matrices .
 = vector of system's nodal displacements that can define all possible deformed configurations of the system subject to arbitrary nodal forces R.
 = 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' .
Solution
The system stiffness matrix K is square since the vectors R and r have the same size. In addition, it is symmetric because 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:
Subsequently, the members' characteristic forces may be found from Eq.(1) where can be found from r by compatibility consideration.
The direct stiffness method
It is common to have Eq.(1) in a form where and are, respectively, the memberend displacements and forces matching in direction with r and R. In such case, and can be obtained by direct summation of the members' matrices and . 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.
Example
Breakdown
The first step when using the direct stiffness method is to identify the individual elements which make up the structure.
Once the elements are identified, the structure is disconnected at the nodes, the points which connect the different elements together.
Each element is then analyzed individually to develop member stiffness equations. The forces and displacements are related through the element stiffness matrix which depends on the geometry and properties of the element.
A truss element can only transmit forces in compression or tension. This means that in two dimensions, each node has two degrees of freedom (DOF): horizontal and vertical displacement. The resulting equation contains a four by four stiffness matrix.
A frame element is able to withstand bending moments in addition to compression and tension. This results in three degrees of freedom: horizontal displacement, vertical displacement and inplane rotation. The stiffness matrix in this case is six by six.
Other elements such as plates and shells can also be incorporated into the direct stiffness method and similar equations must be developed.
Assembly
Once the individual element stiffness relations have been developed they must be assembled into the original structure. The first step in this process is to convert the stiffness relations for the individual elements into a global system for the entire structure. In the case of a truss element, the global form of the stiffness method depends on the angle of the element with respect to the global coordinate system (This system is usually the traditional Cartesian coordinate system).
(for a truss element at angle β)
After developing the element stiffness matrix in the global coordinate system, they must be merged into a single “master” or “global” stiffness matrix. When merging these matrices together there are two rules that must be followed: compatibility of displacements and force equilibrium at each node. These rules are upheld by relating the element nodal displacements to the global nodal displacements.
The global displacement and force vectors each contain one entry for each degree of freedom in the structure. The element stiffness matrices are merged together by augmenting or expanding each matrix in conformation to the global displacement and load vectors.
(for element (1) of the above structure)
Finally, the global stiffness matrix is constructed by adding the individual expanded element matrices together.
Solution
Once the global stiffness matrix, displacement vector and force vector have been constructed, the system can be expressed as a single matrix equation.
For each degree of freedom in the structure, either the displacement or the force is known.
After inserting the known value for each degree of freedom, the master stiffness equation is complete and ready to be evaluated. There are several different methods available for evaluating a matrix equation including but not limited to Cholesky decomposition and the brute force evaluation of systems of equations. If a structure isn’t properly restrained, the application of a force will cause it to move rigidly and additional support conditions must be added.
The method described in this section is meant as an overview of the direct stiffness method. Additional sources should be consulted for more details on the process as well as the assumptions about material properties inherent in the process.
Applications
The direct stiffness method was developed specifically to effectively and easily implement into computer software to evaluate complicated structures that contain a large number of elements. Today, nearly every finite element solver available is based on the direct stiffness method. While each program utilizes the same process, many have been streamlined to reduce computation time and reduce the required memory. In order to achieve this, shortcuts have been developed.
One of the largest areas to utilize the direct stiffness method is the field of structural analysis where this method has been incorporated into modeling software. The software allows users to model a structure and, after the user defines the material properties of the elements, the program automatically generates element and global stiffness relationships. When various loading conditions are applied the software evaluates the structure and generates the deflections for the user.
See also
 Finite element method
 Finite element method in structural mechanics
 Structural analysis
 Flexibility method
 List of finite element software packages
External links
 Application of direct stiffness method to a 1D Spring System
 Introduction to Stiffness Method
 Matrix Structural Analysis
 Animations of Stiffness Analysis Simulations
References
 Felippa, Carlos A. (2001), "A historical outline of matrix structural analysis: a play in three acts", Computers & Structures 79 (14): 1313–1324, doi:10.1016/S00457949(01)000256, ISSN 00457949, http://www.colorado.edu/engineering/CAS/Felippa.d/FelippaHome.d/Publications.d/Report.CUCAS0013.pdf
 Felippa, Carlos A. Introduction to Finite Element Method. Fall 2001. University of Colorado. 18 Sept. 2005 <http://www.devdept.com/fem/books.php>
 Robinson, John. Structural Matrix Analysis for the Engineer. New York: John Wiley & Sons, 1966
 Rubinstein, Moshe F. Matrix Computer Analysis of Structures. New Jersey: PrenticeHall, 1966
 McGuire, W., Gallagher, R. H., and Ziemian, R. D. Matrix Structural Analysis, 2nd Ed. New York: John Wiley & Sons, 2000.
Categories: Structural analysis
 Numerical differential equations
Wikimedia Foundation. 2010.
Look at other dictionaries:
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… … 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
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 — 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
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
Structural mechanics — Space frame used in a building structure Pipe frame used in a competi … Wikipedia
Truss — For other uses, see Truss (disambiguation). In architecture and structural engineering, a truss is a structure comprising one or more triangular units constructed with straight members whose ends are connected at joints referred to as nodes.… … Wikipedia
Beam (structure) — A statically determinate beam, bending (sagging) under an evenly distributed load. A beam is a horizontal structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the… … Wikipedia
List of mathematics articles (D) — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… … Wikipedia
List of structural engineering topics — This page aims to list all articles related to the specific discipline of structural engineering. For a broad overview of engineering, please see List of engineering topics. For biographies please see List of engineers.compactTOC NOTOC AA frame… … Wikipedia