--- title: "Displacements of a spring assembly" subject: "Finite Element Method" chapter: 3 key: "semester_8/finite_element_method:spring-system-displacements" source: https://www.bctengineer.com/q/semester_8/finite_element_method/spring-system-displacements site: BCT Engineering --- # Displacements of a spring assembly Finite Element Method · Chapter 3: Direct Stiffness Method: Discrete Finite Elements - Asked in: year not given, 8 marks - Video: [FEM Spring Problems | Finite Element Methods on Spring Elements Problem | Spring Problems in Physics](https://www.youtube.com/watch?v=b2Asq-gbizQ) — Mahesh Gadwantikar - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/finite_element_method/spring-system-displacements - Whole chapter: https://www.bctengineer.com/md/semester_8/finite_element_method/chapter-3 ## Question Three springs of stiffness $k_1 = 200$ N/mm, $k_2 = 300$ N/mm and $k_3 = 100$ N/mm are connected in series between nodes 1-2, 2-3 and 3-4. Node 1 is fixed to a wall. Axial loads of 400 N and 600 N act on nodes 3 and 4 respectively (no load on node 2). Using the direct stiffness method, find (a) the global stiffness matrix, (b) the nodal displacements, (c) the reaction at node 1 and (d) the force in each spring. ## Short answer Assemble global stiffness: [K] = [[200,-200,0,0],[-200,500,-300,0],[0,-300,400,-100],[0,0,-100,100]] N/mm. Apply u1=0, solve: u2=5.00 mm, u3=8.33 mm, u4=14.33 mm. Reaction at node 1 is 1000 N opposing loads. Spring forces: 1000 N (springs 1 and 2), 600 N (spring 3). ## Answer #### Element stiffness matrices For a spring element, $[k^{(e)}] = k\begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$. ```text wall |--k1--o--k2--o--k3--o--> 600 N 1 2 3 4 --> 400 N (axial, at node 3) ``` #### (a) Global stiffness matrix (N/mm), degrees of freedom $u_1, \dots, u_4$ $$ [K] = \begin{bmatrix} 200 & -200 & 0 & 0 \\ -200 & 500 & -300 & 0 \\ 0 & -300 & 400 & -100 \\ 0 & 0 & -100 & 100 \end{bmatrix} $$ #### (b) Apply boundary condition $u_1 = 0$ Delete row and column 1: $$ \begin{bmatrix} 500 & -300 & 0 \\ -300 & 400 & -100 \\ 0 & -100 & 100 \end{bmatrix}\begin{Bmatrix} u_2 \\ u_3 \\ u_4 \end{Bmatrix} = \begin{Bmatrix} 0 \\ 400 \\ 600 \end{Bmatrix} $$ From row 3: $-100u_3 + 100u_4 = 600$, so $u_4 = u_3 + 6$. Adding rows 2 and 3 gives $-300u_2 + 300u_3 = 1000$, so $u_3 = u_2 + 3.333$. Row 1: $500u_2 - 300(u_2 + 3.333) = 0$ gives $200u_2 = 1000$, so $u_2 = 5.000$ mm. Then $u_3 = 8.333$ mm and $u_4 = 14.333$ mm. #### (c) Reaction From row 1 of the full system: $R_1 = 200u_1 - 200u_2 = -200(5) = -1000$ N. The wall pulls back with 1000 N, which equals the total applied load $400 + 600$. #### (d) Spring forces | Spring | Elongation (mm) | Force = $k\delta$ (N) | |---|---|---| | 1 (nodes 1-2) | 5.000 | 1000 (tension) | | 2 (nodes 2-3) | 3.333 | 1000 (tension) | | 3 (nodes 3-4) | 6.000 | 600 (tension) | Check at node 3: $1000 - 600 = 400$ N, equal to the applied load. **Answer:** $u_2 = 5.00$ mm, $u_3 = 8.33$ mm, $u_4 = 14.33$ mm; reaction at node 1 = 1000 N (acting towards the wall, opposite to the loads); spring forces 1000 N, 1000 N and 600 N. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the diagonal term K22 of the global matrix? A. 500 B. 200 C. 300 D. 100 Answer: A. Node 2 is shared by springs 1 and 2, so K22 = k1 + k2 = 200 + 300 = 500 N/mm. (Key point: Global stiffness) 2. How is the fixed wall at node 1 applied? A. Set F1 = 0 B. Set u1 = 0 and delete row and column 1 C. Delete row and column 4 D. Set k1 = 0 Answer: B. Since u1 = 0, row and column 1 are removed, leaving a 3 x 3 system for u2, u3, u4. (Key point: Boundary condition) 3. What are u2, u3 and u4 in mm? A. 5.00, 3.33, 6.00 B. 3.33, 8.33, 14.33 C. 5.00, 8.33, 14.33 D. 2.5, 5.0, 8.0 Answer: C. u2 = 5.000, u3 = u2 + 3.333 = 8.333 and u4 = u3 + 6 = 14.333 mm. (Key point: Displacements) 4. What is the reaction at the wall, and why? A. 600 N B. 400 N C. 200 N D. 1000 N, equal to the total applied load 400 + 600 Answer: D. R1 = -200 u2 = -1000 N; equilibrium requires the wall to resist the total applied load. 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