Direct stiffness method

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.

${\displaystyle {\begin{bmatrix}f_{x1}\\f_{y1}\\f_{x2}\\f_{y2}\\\end{bmatrix}}={\begin{bmatrix}k_{11}&k_{12}&k_{13}&k_{14}\\k_{21}&k_{22}&k_{23}&k_{24}\\k_{31}&k_{32}&k_{33}&k_{34}\\k_{41}&k_{42}&k_{43}&k_{44}\\\end{bmatrix}}{\begin{bmatrix}u_{x1}\\u_{y1}\\u_{x2}\\u_{y2}\\\end{bmatrix}}}$

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 in-plane rotation. The stiffness matrix in this case is six by six.

${\displaystyle {\begin{bmatrix}f_{x1}\\f_{y1}\\m_{z1}\\f_{x2}\\f_{y2}\\m_{z2}\\\end{bmatrix}}={\begin{bmatrix}k_{11}&k_{12}&k_{13}&k_{14}&k_{15}&k_{16}\\k_{21}&k_{22}&k_{23}&k_{24}&k_{25}&k_{26}\\k_{31}&k_{32}&k_{33}&k_{34}&k_{35}&k_{36}\\k_{41}&k_{42}&k_{43}&k_{44}&k_{45}&k_{46}\\k_{51}&k_{52}&k_{53}&k_{54}&k_{55}&k_{56}\\k_{61}&k_{62}&k_{63}&k_{64}&k_{65}&k_{66}\\\end{bmatrix}}{\begin{bmatrix}u_{x1}\\u_{y1}\\\theta _{z1}\\u_{x2}\\u_{y2}\\\theta _{z2}\\\end{bmatrix}}}$

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. 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).

${\displaystyle {\begin{bmatrix}f_{x1}\\f_{y1}\\f_{x2}\\f_{y2}\\\end{bmatrix}}={\frac {EA}{L}}{\begin{bmatrix}c^{2}&sc&-c^{2}&-sc\\sc&s^{2}&-sc&-s^{2}\\-c^{2}&-sc&c^{2}&sc\\-sc&-s^{2}&sc&s^{2}\\\end{bmatrix}}{\begin{bmatrix}u_{x1}\\u_{y1}\\u_{x2}\\u_{y2}\\\end{bmatrix}}{\begin{array}{r }s=\sin \beta \\c=\cos \beta \\\end{array}}}$ (for a truss element at angle β) Equivalently, ${\displaystyle {\begin{bmatrix}f_{x1}\\f_{y1}\\\hline f_{x2}\\f_{y2}\end{bmatrix}}={\frac {EA}{L}}\left[{\begin{array}{c c|c c}c_{x}c_{x}&c_{x}c_{y}&-c_{x}c_{x}&-c_{x}c_{y}\\c_{y}c_{x}&c_{y}c_{y}&-c_{y}c_{x}&-c_{y}c_{y}\\\hline -c_{x}c_{x}&-c_{x}c_{y}&c_{x}c_{x}&c_{x}c_{y}\\-c_{y}c_{x}&-c_{y}c_{y}&c_{y}c_{x}&c_{y}c_{y}\\\end{array}}\right]{\begin{bmatrix}u_{x1}\\u_{y1}\\\hline u_{x2}\\u_{y2}\end{bmatrix}}}$

where ${\displaystyle c_{x}}$ and ${\displaystyle c_{y}}$ 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.

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 by augmenting or expanding each matrix in conformation to the global displacement and load vectors.

${\displaystyle k^{(1)}={\frac {EA}{L}}{\begin{bmatrix}1&0&-1&0\\0&0&0&0\\-1&0&1&0\\0&0&0&0\\\end{bmatrix}}\rightarrow K^{(1)}={\frac {EA}{L}}{\begin{bmatrix}1&0&-1&0&0&0\\0&0&0&0&0&0\\-1&0&1&0&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\\\end{bmatrix}}}$ (for element (1) of the above structure)

Finally, the global stiffness matrix is constructed by adding the individual expanded element matrices together.

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.