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Finite Element Methods · Chapter 1 of 10 · Beginner

Introduction to the Finite Element Method

The finite element method turns an unsolvable continuous body into a large but solvable matrix problem. The whole subject is one move repeated: build a stiffness, assemble, and solve Ku = F.

Readiness check

This opening chapter needs mechanics and matrices. Tick only what you can do closed-notes.

Recall axial stiffness and Hooke's law.
Multiply matrices and solve a small linear system.
Recall a cantilever deflection formula.
Distinguish a node from an element.
Count degrees of freedom for a simple structure.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsRevisit axial loading in Mechanics of Materials, Chapter 2.

3 or more weak itemsReview linear systems in Math: Linear Systems.

The core idea

A continuous body is divided into simple elements joined at nodes. Each element gets a stiffness matrix; assembling them gives a global system Ku = F whose solution is the nodal displacements.

k = AE/L (element stiffness)Ku = Fstrains and stresses recovered from u

Exact solutions exist only for simple shapes, so the finite element method approximates a complex body by a mesh of elements with simple behaviour, connected at nodes. For each element a stiffness matrix relates the forces at its nodes to their displacements; the simplest, an axial bar, has stiffness k = AE/L, exactly the spring constant of mechanics of materials. These element matrices are assembled into one large global stiffness matrix K, the loads into a vector F, and the boundary conditions applied. Solving Ku = F gives every nodal displacement, from which strains, stresses, and reactions follow. The skill is not running the solver but judging whether the elements, mesh, and constraints make the answer trustworthy, which is why every result is checked against theory.

The skill works when: you build element stiffnesses, assemble, solve, and verify against a hand calculation.

The skill breaks down when: a contour plot is trusted without a convergence check or an equilibrium balance.

The concept. A bracket divided into triangular elements joined at nodes (blue). Each element contributes a stiffness; assembled, they form Ku = F, solved for the nodal displacements.

The skills, taught in order

Five skills frame what FEM is, the general steps, the element stiffness, degrees of freedom, and the hand-check discipline.

1.1 What FEM is and why

The finite element method finds an approximate solution to a field problem by dividing the domain into elements with simple assumed behaviour. It handles geometry, loads, and materials that defeat closed-form analysis, which is why it underlies almost all structural, thermal, and fluid simulation.

1.2 The general steps

Every linear FEM analysis follows the same path: discretize, choose the element, form each element stiffness, assemble the global system, apply boundary conditions, solve for displacements, and recover stresses and reactions.

Step	Action
1	discretize the body into elements and nodes
2	choose the element type and shape functions
3	form each element stiffness matrix
4	assemble the global K and load vector F
5	apply boundary conditions and solve Ku = F
6	recover strains, stresses, and reactions

1.3 The element stiffness

The building block is the element stiffness matrix relating nodal forces to nodal displacements. For an axial bar it is built from k = AE/L, the same stiffness as a spring; richer elements (beams, triangles) have larger matrices but the same role.

1.4 Degrees of freedom

Each node carries degrees of freedom, the independent displacements it can have: one per node for an axial bar, two for a plane truss, three (two translations and a rotation) for a plane frame. The total number of degrees of freedom sets the size of the global matrix.

1.5 The hand-check discipline

An FEM result is evidence only after validation. Before meshing, write the closed-form answer for a simplified version; after solving, confirm the mesh converges to it and that reactions balance the applied load. This habit runs through every chapter.

Engineering connection: the element stiffness and assembly introduced here are repeated, with richer elements, in every later chapter.

Worked example 1: the element stiffness of a bar

A steel bar has cross-sectional area A = 500 mm², length L = 1 m, and modulus E = 200 GPa. Find its axial stiffness as a single element, and the tip displacement under a 10 kN axial load.

Figure 1. A single bar element behaves like a spring of stiffness AE/L between its two nodes. The tip displacement is the load divided by that stiffness.

ProblemFind the bar's element stiffness and the tip displacement in Figure 1.
Given / findA = 500 mm², L = 1000 mm, E = 200 000 MPa, P = 10 kN. Find k and δ.
AssumptionsLinear elastic, uniform axial bar; one element captures it exactly.
ModelThe element stiffness is k = AE/L; the displacement is δ = P/k.
Equationsk = AE/Lδ = P/k
Solvek = (500 × 200 000)/1000 = 100 000 N/mm = 100 kN/mm. δ = 10 000/100 000 = 0.1 mm.
CheckThe same answer follows from δ = PL/AE = 10 000 × 1000/(500 × 200 000) = 0.1 mm, the mechanics-of-materials result. A single bar element is exact for uniform axial loading.
ConclusionThe element stiffness is just the spring constant of the bar. Every element in the course is this idea generalised to more degrees of freedom.

Result. k = 100 kN/mm, δ = 0.1 mm.

Worked example 2: counting degrees of freedom

A planar truss is modelled with 6 nodes, each free to move in x and y. Two of the nodes are pinned supports. Find the total number of degrees of freedom, the size of the global stiffness matrix, and the size of the system to solve after applying the supports.

Figure 2. Each node carries two degrees of freedom in a plane truss. The pinned supports remove four, so the unknown displacements form an 8-by-8 system.

ProblemFind the total DOF, the global matrix size, and the reduced system size for the truss in Figure 2.
Given / find6 nodes, 2 DOF each; 2 pinned supports (each removes 2 DOF). Find total DOF, K size, and reduced size.
AssumptionsPlane truss, so each node has two translational degrees of freedom and no rotation.
ModelTotal DOF = nodes × DOF per node; each pinned support fixes 2 DOF, shrinking the system.
Equationstotal DOF = 6 × 2 = 12constrained DOF = 2 × 2 = 4
SolveTotal DOF = 12, so the global K is 12 × 12. The two pins fix 4 DOF, leaving 8 unknowns, an 8 × 8 system to solve.
Check12 minus 4 equals 8, and the reduced matrix is square at the number of free DOF. The fixed DOF reappear when reactions are recovered.
ConclusionDegrees of freedom set the matrix size and the cost. Boundary conditions shrink the system and supply the supports that make it solvable.

Result. 12 DOF total; K is 12 × 12; the reduced system is 8 × 8.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
No hand check	Accepting a contour plot blindly	"What does theory predict here?"	Validate against a closed-form case before trusting FEM.
Miscounting DOF	Wrong matrix size or singular system	"How many DOF per node for this element?"	One for axial, two for a plane truss, three for a plane frame.
Forgetting boundary conditions	K is singular and will not solve	"Is the structure restrained against rigid motion?"	Apply enough supports to remove rigid-body modes.
Confusing nodes and elements	Stiffness assembled incorrectly	"Are these connecting points or pieces?"	Elements span between nodes; stiffness is per element.

Practice ladder

Level 1 · Direct skill

A bar has A = 300 mm², L = 600 mm, E = 70 GPa (aluminium). Find its element stiffness.

Show answer

k = AE/L = (300 × 70 000)/600 = 35 000 N/mm = 35 kN/mm.

Level 2 · Mixed concept

The Worked Example 1 bar carries 25 kN instead. What is the tip displacement?

Show answer

δ = P/k = 25 000/100 000 = 0.25 mm. Displacement scales linearly with load in a linear analysis.

Level 3 · Independent problem

A plane frame has 4 nodes (3 DOF each) with one fully fixed node. Find the total DOF and the reduced system size.

Show answer

Total DOF = 4 × 3 = 12. A fully fixed node removes 3, leaving 9, so the reduced system is 9 × 9.

Level 4 · Transfer to real engineering

For a real part you would analyse, list the FEM steps you would follow and the hand calculation you would use to validate the result.

What good work looks like

The general steps named in order and a specific closed-form benchmark (beam deflection, axial stress) chosen to check the FEM output.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that my element stiffness and DOF count are right."

"Give me four structures; I will count the degrees of freedom for each."

"Run the FEM for me." Building the stiffness and assembly yourself is the skill.

"Is this result correct?" Validating against theory is the point.

Portfolio task

Take a simple loaded part, count its degrees of freedom, identify the element type, and write the hand-calc benchmark you would validate an FEM result against.

Must include: an element type, a DOF count, and a closed-form benchmark.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. What does FEM approximate?

A continuous body, by a mesh of simple elements joined at nodes.

2. What is the central equation?

Ku = F, with K the assembled stiffness, u the nodal displacements, F the loads.

3. Write the bar element stiffness.

k = AE/L, the same as a spring constant.

4. How many DOF per node for a plane truss?

Two, the x and y translations.

5. Why hand-check an FEM result?

To confirm the mesh, elements, and constraints give a physically correct answer.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive the element stiffness and DOF count from a blank page.

+3 daysCount DOF for three new structures.

+7 daysCarry the stiffness idea into assembly, Chapter 2.

+30 daysReuse the hand-check discipline on every FEM result.

Textbook mapping

Item	Mapping
Primary source	Logan, A First Course in the Finite Element Method, Chapter 1 (Introduction)
Cross-reference	Hutton, Ch. 1 · Math: Linear Systems
Core topics	1.1 What FEM is · 1.2 General steps · 1.3 Element stiffness · 1.4 Degrees of freedom · 1.5 Hand-check discipline
Engineering connection	The element stiffness and assembly repeat with richer elements throughout.
Read next	Chapter 2: The Direct Stiffness Method: Spring Elements.