Tuesday, May 22, 2012

Direct Stiffness Method - An Overview

A brief overview of the direct stiffness method as applicable to skeletal structures

The direct stiffness method is a matrix method of structural analysis. This post pertains only to the analysis of skeletal structures, that is, structures that can be modelled using only 1D truss/beam elements in either 2D or 3D space. Further, element stiffness is explicitly derived from first principles.

In general, a direct stiffness method such as the Finite Element Method, is quite general and can model structures using 1D, 2D (Plate, Plane Stress, Plane Strain elements) and 3D (solid elements) in 2D or 3D space.

Subject to the above mentioned limitations, the direct stiffness method is a procedure to solve the structure stiffness equation, namely
Eq. 1                                                    

where [K] is the structure stiffness matrix, {x} is the displacement vector and {P} is the load vector. Depending on the specific problem being solved, the stiffness matrix [K] and the displacement and load vectors {x} and {P} can be partitioned into sub-matrices, as follows:

Eq. 2                        

The rows and columns of [K] are:
  1. First row and first column corresponding to the unknown displacement components at the nodes, namely {x1}.  Elements of {P1} are known and can be assembled based on known loads applied at the nodes and directly on elements.
  2. Second row and second column corresponding to the known non-zero displacements at the nodes, namely{x2}.  Elements of {P2} are unknown and correspond to the reaction components corresponding to displacements that are known and non-zero.
  3. Third row and third column corresponding to the known zero displacements at the nodes,namely {x3} = {0}.  Elements of {P3} are unknown and correspond to the reaction components corresponding to displacements that are known to be zero.
Rewriting the above equation by indicating unknows with a question mark (?) and known zeros with zero (0), the rest are known non-zeros:

 Since {x3} = {0}, all elements of {x3} are zeros. The stiffness equation can be written as three separate equations, the first of which is:
Eq. 3                      

and can be rearranged by moving all known values to the right as follows:
Eq. 4                        

Solving the above equation yields the unknown displacements {x1}, which are no longer unknown. The second and third equations are:
Eq. 5                        

with the unknowns on the right side and all knowns on the left side (now that {x1} is known), it is possible to determine the reactions corresponding to the known (non-zero and zero) displacements.

Since the sub-matrices in the third column are multiplied with {x3}={0}, they do not appear in any of the three equations and hence not required in the analysis by the direct stiffness method.

If the structure has only known zero displacements at supports and no known non-zero displacements (support settlements), the entire second row and second column of [K] as well as second sub-matrix {P2} and {x2} are null, and the equation simplifies to the following form:
Since {x3}={0}, it is only necessary to assemble [K11] in order to determine the unknown displacements {x1}. In order to determine the reactions corresponding to known zero displacements, it is sufficient to assemble [K31] so that [K31]{x1}={P3}.

According to this scheme of solving the stiffness equation, the degrees of freedom are numbered in the sequence adopted above, namely, unknown displacements first, known non-zero displacements (if any) next and known zero displacements last.

Accordingly, while assembling the stiffness matrix, the following stiffness matrices are assembled:
  1. [K11] is required always
  2. [K21] and  [K22] are required only if the the structure has one or more known non-zero displacements. Both are not required if the structure has no known non-zero displacements.
  3. [K31] and  [K32] are required if reactions are to be computed.  [K32] is not required if   {x2}={0}.   [K31] is not required if reactions are computed from member end forces.
Since it is possible to compute the reactions from the known forces at the ends of members (which in turn are obtained from known displacements at nodes), it is not necessary to use the second and third equations above (expanded from the stiffness equation expanded in terms of sub-matrices) to calculate the reactions. However, if there are known non-zero displacements in a structure, it is necessary to assemble [K12] in order to calculate {x1}.

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