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 linear algebra, 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.
Researchers looked at various approaches for analysis of complex airplane frames.
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).
(1) can be integrated by making use of the following observations: where The system stiffness matrix K is square since the vectors R and r have the same size.
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.
are, respectively, the member-end displacements and forces matching in direction with r and R. In such case,
The first step when using the direct stiffness method is to identify the individual elements which make up the structure.
Each element is then analyzed individually to develop member stiffness equations.
This means that in two dimensions, each node has two degrees of freedom (DOF): horizontal and vertical displacement.
A frame element is able to withstand bending moments in addition to compression and tension.
Other elements such as plates and shells can also be incorporated into the direct stiffness method and similar equations must be developed.
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.
are the direction cosines of the truss element (i.e., they are components of a unit vector aligned with the member).
This form reveals how to generalize the element stiffness to 3-D space trusses by simply extending the pattern that is evident in this formulation.
When merging these matrices together there are two rules that must be followed: compatibility of displacements and force equilibrium at each node.
The global displacement and force vectors each contain one entry for each degree of freedom in the structure.
The element stiffness matrices are merged by augmenting or expanding each matrix in conformation to the global displacement and load vectors.
After inserting the known value for each degree of freedom, the master stiffness equation is complete and ready to be evaluated.
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 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.
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.