--- title: "Truss element stiffness matrices" subject: "Computational Techniques in Civil Engineering" chapter: 4 key: "semester_8/computational_techniques_in_civil_engineering:truss-element-stiffness-matrices" source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/truss-element-stiffness-matrices site: BCT Engineering --- # Truss element stiffness matrices Computational Techniques in Civil Engineering · Chapter 4: Finite element method - Asked in: 2070 Magh, 10 marks - Video: [truss stiffness matrix in 2D space](https://www.youtube.com/watch?v=UO7Jeq9vzAg) — Michael Sevier - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/truss-element-stiffness-matrices - Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-4 ## Question Determine the stiffness matrices for the elements as shown in Fig. 1. $A = 300$ mm$^2$ and $E = 2.1\times10^5$ MPa. [Figure: plane truss; node 1 at the bottom-left (supported), node 2 at 500 mm to the right of node 1, node 3 at 400 mm above node 2, node 4 at 400 mm above node 1 (supported); elements numbered 1 and 2 (diagonal from node 1 to node 3 and member 3-4 as drawn); a 50 kN horizontal load at node 2 and a 40 kN downward load at node 3] ## Short answer Use the standard plane-truss stiffness matrix. Element 1 (1-3) has AE/L=98389.47 N/mm, c=0.7809, s=0.6247. Element 2 (3-4) is horizontal with AE/L=126000 N/mm, so only the u-direction stiffness entries are nonzero; both matrices are symmetric and rows sum to zero. ## Answer The stiffness of a 2D truss element is $[k]=\dfrac{AE}{L}\begin{bmatrix}c^2&cs&-c^2&-cs\\cs&s^2&-cs&-s^2\\-c^2&-cs&c^2&cs\\-cs&-s^2&cs&s^2\end{bmatrix}$ with $c=\dfrac{x_j-x_i}{L}$, $s=\dfrac{y_j-y_i}{L}$. #### Geometry and assumptions Coordinates in mm: node 1 (0, 0), node 2 (500, 0), node 3 (500, 400), node 4 (0, 400). Element 1 is the diagonal 1-3; element 2 is the member 3-4, as drawn. $AE=300\times2.1\times10^5=6.3\times10^7$ N. Global dofs are $(u_n, v_n)$ at each node $n$. #### Element 1 (nodes 1 to 3) $$ L=\sqrt{500^2+400^2}=640.31\ \text{mm},\quad c=\frac{500}{L}=0.7809,\quad s=\frac{400}{L}=0.6247,\quad \frac{AE}{L}=98389.47\ \text{N/mm} $$ Rows and columns correspond to $(u_1,\ v_1,\ u_3,\ v_3)$: $$ [k]_1=\begin{bmatrix} 59993.58 & 47994.86 & -59993.58 & -47994.86 \\ 47994.86 & 38395.89 & -47994.86 & -38395.89 \\ -59993.58 & -47994.86 & 59993.58 & 47994.86 \\ -47994.86 & -38395.89 & 47994.86 & 38395.89 \end{bmatrix}\ \text{N/mm} $$ #### Element 2 (nodes 3 to 4) The member is horizontal: $L=500$ mm, $c=-1$ (from node 3 to node 4), $s=0$, $AE/L=126000$ N/mm. Only $c^2=1$ and $cs=0$ appear in the matrix, so the sign of $c$ does not matter. Rows and columns correspond to $(u_3,\ v_3,\ u_4,\ v_4)$: $$ [k]_2=\begin{bmatrix} 126000 & 0 & -126000 & 0 \\ 0 & 0 & 0 & 0 \\ -126000 & 0 & 126000 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\ \text{N/mm} $$ #### Checks - Both matrices are symmetric and every row sums to zero. - For element 1, $c^2+s^2=0.6098+0.3902=1$. - A horizontal bar has no stiffness in the $v$ direction, so the $v$ rows of $[k]_2$ are zero. **Answer:** $[k]_1$ and $[k]_2$ as above, with $AE/L=98389.47$ N/mm and $126000$ N/mm respectively. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How are c and s defined in the truss element stiffness matrix? A. c = (y_j - y_i)/L, s = (x_j - x_i)/L B. c = L/(x_j - x_i), s = L/(y_j - y_i) C. c = (x_j - x_i)/L, s = (y_j - y_i)/L D. c = x_i/L, s = y_i/L Answer: C. c and s are the direction cosines of the member from node i to node j with respect to the global x and y axes. (Key point: Truss element formula) 2. For element 1 from (0,0) to (500,400) mm, what are L, c and s? A. L = 640.31 mm, c = 0.7809, s = 0.6247 B. L = 900 mm, c = 0.5556, s = 0.4444 C. L = 640.31 mm, c = 0.6247, s = 0.7809 D. L = 500 mm, c = 1, s = 0 Answer: A. L = sqrt(500^2 + 400^2) = 640.31 mm, c = 500/L = 0.7809 and s = 400/L = 0.6247. (Key point: Element 1 length and cosines) 3. Why does the sign of c = -1 not matter for element 2 (node 3 to 4)? A. Because s is also negative B. Because the matrix contains only c^2 and cs, and cs = 0 as s = 0 C. Because the element is in compression D. Because AE/L becomes negative Answer: B. Only c^2 = 1 and cs = 0 appear, so the matrix is the same whether c is +1 or -1. (Key point: Horizontal member) 4. Which check applies to both element stiffness matrices? A. Each matrix is invertible B. The diagonal terms are zero C. The matrix is anti-symmetric D. Each is symmetric and every row sums to zero Answer: D. Symmetry and zero row sums (equilibrium of a free element) are properties of truss element matrices. 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