#089 - Engineering Mathematics | Part 1: Linear Algebra – The Unsung Hero of Structural Integrity

Connecting the Dots with Real-World Examples.

I know what you’re thinking. He wants to talk about algebra, disgusting. Just hold on a second.

For many of us, school mathematics felt transactional: memorize the formula, solve the problem, pass the exam. I moved through it just fine, but I was executing steps, not building understanding. The principles remained abstract, their implications didn’t really mean anything to me.

The education system, in my experience, as a student, seems designed for compliance, not for understanding.

Then I started working as an engineer and began to see the practical application of this stuff in unexpected and fascinating ways.

This series isn't about rediscovering a love for tedious derivations or slamming your head off a textbook, it’s about trying to understand the engineering landscape as it relates to mathematics.

We use sophisticated software, powerful black boxes that churn out results. But relying solely on these tools is like taking an Uber. You might get where you're going, but you have no idea how you got there. While you can still reach your objective, you may miss out on deeper understanding, the ability to diagnose problems, to innovate, to engineer.

Does this even matter? That’s up to you and your personal philosophy.

My frustration, and I suspect many peoples, stems from the fact that this connection, the practical relevance, is often buried under layers of academic bureaucracy/entrenchment, the massive scope of ‘engineering fundamentals’ and the tight timelines of the academic calendar. Teachers and students simply don’t have the time to fit it all in and even if you do, it’s impossible to retain because your brain is constantly saturated with new, complex information. It’s hard to parse the core details from the peripheral details.

This series is an attempt, admittedly imperfect, to re-open the door to understanding some of these fundamentals.

We'll focus on the why and the how, connecting some theory to real-world applications, and maybe rekindle a spark of appreciation or interest for the mathematical foundations of our profession.

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Part 1: Linear Algebra

In this first installment, we'll review linear algebra, a foundational tool that underpins much of structural analysis and design.

Not glamorous, but essential.

Engineering is often about managing forces and ensuring equilibrium. Structures must resist loads, distribute stresses, and remain stable. This leads to systems of linear equations.

Linear algebra provides the framework to represent and solve these systems, allowing us to move from qualitative concepts to quantitative predictions.

We are all familiar with this but it helps to start at the beginning.

Matrices and Systems of Equations: The Stiffness Method

Consider the following problem: determining the forces and deflections in a structure. The stiffness method, the basis of most structural analysis, relies entirely on matrix algebra.

• The Core Principle: The stiffness method relates nodal displacements (how much points on the structure move) to nodal forces through a stiffness matrix. This matrix encapsulates the structure's resistance to deformation.

Bear with me on this, it’s a concept that is really worth revisiting.

The Fundamental Equation:

F = K * d

Where:

• F = Vector of external forces applied to the structure's nodes.

• K = Stiffness matrix (a square matrix).

• d = Vector of nodal displacements.

Simple Example: A Spring System Imagine two springs connected in series, with spring constants:

• k1​=100 N/m

• k2​=200 N/m.

A force of 50 N is applied to the connection point, and the other end is fixed.

We can represent this as a system of equations:

[ k1+k2   -k2 ] [ d1 ]   [ 50 ]
[ -k2      k2 ] [ d2 ] = [  0 ]

To dust off your matrix algebra, remember that they are really just shorthand expressions of regular algebra equations. When you multiply a matrix (the stiffness) by a vector (the displacement), you're creating a system of equations you can solve.

SOLUTION

Starting equation:
[ 300  -200 ] [ d1 ]   [ 50 ]
[ -200  200 ] [ d2 ] = [  0 ]

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STEP 1: Convert matrix equation to regular equations

Matrix multiplication rule: Each row × column gives one equation

Row 1: (300)(d1) + (-200)(d2) = 50
Row 2: (-200)(d1) + (200)(d2) = 0

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STEP 2: Use boundary condition

Given: d2 = 0 (fixed support - this point cannot move)

Substitute d2 = 0 into both equations:
Equation 1:  300d1 - 200(0) = 50  →  300d1 = 50
Equation 2: -200d1 + 200(0) = 0  →  -200d1 = 0

═══════════════════════════════════════════════════════════════

STEP 3: Solve for d1

From Equation 1:
300d1 = 50
d1 = 50 ÷ 300 = 0.167 m

Check: Does this satisfy Equation 2?
-200(0.167) = -33.4 ≈ 0 ✓ 

═══════════════════════════════════════════════════════════════

ANSWER: Node 1 moves 0.167 m, Node 2 stays fixed at 0 m

This simple example extends to complex structures with thousands of degrees of freedom. The principle remains the same: construct the stiffness matrix, apply the load vector, and solve for the displacement vector.

The stiffness matrix works because it's a systematic way to enforce force equilibrium at every node simultaneously. The method systematically enforces the three fundamental requirements of structural mechanics:

1. Compatibility - displacements must be continuous

2. Equilibrium - forces must balance at every node

3. Constitutive relations - element behavior follows material laws

The mathematical elegance lies in how this local element-by-element assembly automatically produces a global system that satisfies all physical requirements simultaneously. Each element "knows" only about its two connected nodes, but when assembled, the global matrix captures the entire structural behavior.

This is why the same method works for everything from simple springs to complex 3D frames, shells, and solids - the assembly process is identical, only the element stiffness relationships change.

Statically Indeterminate Structures: The stiffness method is particularly crucial for statically indeterminate structures, where the equations of static equilibrium alone are insufficient. The stiffness matrix provides the additional relationships needed to solve for the unknowns.

This is why FEA tools are so helpful, you can imagine the existential dread of trying to figure out multiple nodal displacements by hand, like the good old days.

Conclusion

You made it. Today was a purposefully basic refresher and I’m cutting it short even though what I really wanted to talk about are Eigenvalues and Eigenvectors. I will cover that in the next algebra installment.

If you find this stuff as interesting as I do, there are a few useful resources that I have found helpful.

I find math skills are like any other skill, you can’t learn skateboarding or the guitar by reading an article. You need to step on the board or try holding a chord.

• Khan Academy is an incredible resource for learning Math and everything else.

• Brilliant is expensive but another great app for learning and brain training games.

• Youtube Channels that I like:

• 3Blue1Brown - Animated Maths and Physics Content - this guy is the man, 5 stars.

• Veritasium - Science, Physics and Math

If you have any other good suggestions, post them below in the comments.

See you in the next one.

James 🌊