Finite Elements by Hand: Spring and Bar Elements, Stiffness Tables, Assembly, Boundary Conditions, Solving by Elimination, Truss Elements, Mesh Refinement & Result Checks

What you'll build

What you will be able to do

Each lab ends with a concrete result that you build yourself and the checker confirms, cell by cell.

• Describe a chain of three springs as elements and nodes, write out the four stiffness entries of every element, and turn measured node displacements into elongations, spring forces and the force one element puts on a node.
• List every element entry with its row node and column node, assemble the global 4 × 4 stiffness table of a spring network with SUMIFS, and check it for symmetry and zero row sums.
• Remove the clamped nodes of a three-spring chain, solve the reduced 2 × 2 system for the two free displacements with Cramer's rule written out, and recover the support reactions.
• Turn a stepped bar into three bar elements with k = E·A/L, solve the 3 × 3 reduced system by forward elimination and back substitution in step columns, and recover the stress in each element.
• Turn the self-weight of a hanging rod into equivalent nodal forces and a temperature rise in a clamped bar into an initial-strain load, then find the displacements and the thermal stress.
• Rotate two bar elements into the x–y plane with c = cos θ and s = sin θ, assemble the 2 × 2 table of one free joint, solve its displacements with Cramer's rule and check the member forces against a statics hand result.
• Model a tapered bar with 1, 2, 4 and 8 constant-area elements, add up the element elongations to the tip displacement, compare each mesh with the exact formula and chart how the error shrinks.
• Review a colleague's hand model of a bar between two walls: recompute the displacement, test the reported reactions for equilibrium, measure the sensitivity to a 10 % smaller area and compare the stresses with a fictional allowable value.