Math 2.0

Mechanics of deformable solids Definition

Direct stiffness method

Builds on Stiffness matrixTruss

Definition

Consider a linear elastic framed structure, for example a truss. It is made of members e=1,…,Me = 1, \ldots, M joined at nodes. Let d∈ℝmd \in \R^m be the vector of all nodal degrees of freedom in global coordinates: displacements and, for frames, rotations. Let F∈ℝmF \in \R^m be the vector of external nodal loads. The direct stiffness method consists of the following steps.

  1. Member stiffness: for each member, a matrix kek_e maps its local end displacements to its end forces in local coordinates.
  2. Transformation: a matrix TeT_e maps the global degrees of freedom of the member's nodes to its local end displacements.
  3. Assembly: the global stiffness matrix is K=∑eLeTTeTkeTeLeK = \sum_e L_e^T T_e^T k_e T_e L_e. Here LeL_e selects from dd the degrees of freedom of the nodes of member ee.
  4. Solution: the rows and columns of the supported degrees of freedom are removed, and the reduced system Kd=FK d = F is solved. Member forces are then keTeLedk_e T_e L_e d, and support reactions follow from the removed rows.

Remarks

Example: two springs of stiffness k1k_1 and k2k_2 in series along a line, with nodes 0,1,20, 1, 2, node 00 fixed and a load PP at node 22. Each spring has ke=k(1−1−11)k_e = k\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}, and assembly gives K=(k1−k10−k1k1+k2−k20−k2k2).K = \begin{pmatrix} k_1 & -k_1 & 0 \\ -k_1 & k_1 + k_2 & -k_2 \\ 0 & -k_2 & k_2 \end{pmatrix}. Removing node 00 and solving gives d1=P/k1d_1 = P/k_1 and d2=P/k1+P/k2d_2 = P/k_1 + P/k_2.

Non-example: if no node is supported, step 4 removes nothing and KK is singular, since a rigid translation d=(1,1,1)d = (1, 1, 1) gives Kd=0Kd = 0; the method then has no unique solution.

For a pin-jointed bar of axial stiffness EAEA and length LL, ke=EAL(1−1−11)k_e = \frac{EA}{L}\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} in the axial direction.

Reading path

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Further reading

  1. Web https://en.wikipedia.org/wiki/Direct_stiffness_method
About this page
Tag3ZGBB22YWW
URIm2://3ZGBB22YWW
Kinddefinition
WritingW0 stub
MSC74S05, 74K10
DefinesDirect stiffness method
UsesStiffness matrix, Truss
Changed3 November 2026

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