Finite Elements by Hand: Spring and Bar Elements, Stiffness Tables, Assembly, Boundary Conditions, Solving by Elimination, Truss Elements, Mesh Refinement & Result Checks
Open the black box behind simulation software. You will write element stiffness values for springs and bars, assemble them into a global table by node numbers, apply supports and loads, solve the system by Cramer's rule and Gaussian elimination step by step, recover stresses and reactions, build truss elements with direction cosines, refine a mesh until the answer converges and check every result for equilibrium, with every cell checked as you go.
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About this course
In this course you build the finite element method cell by cell instead of trusting a black box. You write the 2 × 2 table of spring elements, assemble a global stiffness table by node numbers with SUMIFS and check its symmetry and row sums. You remove supports, solve two unknowns with Cramer's rule and three unknowns with Gaussian elimination in step columns, then recover reactions, strains and stresses. Self-weight and temperature become equivalent nodal forces, truss bars get direction cosines with COS and SIN, a tapered bar is refined from 1 to 8 elements against an exact LN formula, and a capstone reviews a colleague's results for equilibrium, sensitivity and stress utilisation.
Lessons
- The Idea: Springs as Elements, Nodes and the 2 × 2 Element Table Pro
- Assembly: The Global Stiffness Table by Node Pairs with SUMIFS Pro
- Supports and Loads: The Reduced System, Cramer's Rule and Reactions Pro
- Bar Elements: k = E·A/L, Gaussian Elimination in Step Columns, Stresses Pro
- Distributed and Thermal Loads: Self-Weight and Temperature as Nodal Forces Pro
- Truss Elements: Direction Cosines, the Joint Table and Member Forces Pro
- Mesh Refinement: Tapered Bar with 1, 2, 4 and 8 Elements Pro
- Capstone: Checking a Colleague's Results – Equilibrium, Sensitivity, Stress Utilisation Pro