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Architecture Engineering QUESTION #2361
Question 1
In the Finite Element Method (FEM) applied to structural analysis, the stiffness matrix \([K]\) of a structure is assembled from element stiffness matrices \([k_e]\). For a 2-node bar element of length \(L\), cross-sectional area \(A\), and modulus \(E\), the element stiffness matrix is:
  • \([k_e] = \dfrac{EA}{L} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}\)✔️
  • \([k_e] = \dfrac{EA}{L} \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\)
  • \([k_e] = \dfrac{EI}{L^3} \begin{bmatrix} 12 & -6L \\ -6L & 4L^2 \end{bmatrix}\)
  • \([k_e] = \dfrac{EA}{L^2} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}\)
Correct Answer Explanation
For a 2-node axial (bar) element with DOFs \(\{u_1, u_2\}\), the element stiffness matrix derived from the principle of virtual work is \([k_e] = \dfrac{EA}{L}\begin{bmatrix}1 & -1 \\ -1 & 1\end{bmatrix}\). This is symmetric and singular (as expected for an unrestrained element). Option C is part of the Euler-Bernoulli beam element stiffness matrix (for bending DOFs).