Module 13 · Lesson 13.1
Deriving the beam deflection equation
Euler–Bernoulli theory: from exact curvature to EI d²v/dx² = M, and where it fails.
Why this matters
Every deflection formula you will ever use comes from one differential equation. Deriving it once shows you why span appears to the third or fourth power, and why the boundary conditions matter as much as the loading.
By the end of this lesson you should be able to
- Write the exact curvature of a deflected line and justify the small-slope approximation
- Combine curvature with moment–curvature to get the beam equation
- Produce the shear and load forms by differentiating
- Apply the right boundary conditions for each support type
From first principles
The Euler–Bernoulli beam equation
We want to show: that EI d²v/dx² = M(x), and hence EI d⁴v/dx⁴ = w(x).
We already know from the bending derivation that a bending moment bends the beam to a curvature 1/R = M/EI. So if we can express the curvature of the deflected shape in terms of the deflection v(x), we can turn that physical statement into a differential equation and solve it. The only mathematical subtlety is that the exact curvature of a curve is awkward. Structural beams deflect very little — a span/300 limit means the slope is tiny — and that lets us replace the awkward exact expression with a very simple one.
Practice
A simply supported beam of span 4.0 m carries a central point load of 25 kN. Using δ = PL³/48EI with E = 205 000 N/mm² and I = 9.0 × 10⁷ mm⁴, what is the mid-span deflection?
Practice
A simply supported beam of span 8.0 m carries a uniform load of 20 kN/m. With E = 205 000 N/mm² and I = 3.0 × 10⁸ mm⁴, what is the mid-span deflection, in mm?
Practice
How many times must the deflection v be differentiated with respect to x to obtain the distributed load w?
Summary
- Exact curvature simplifies to d²v/dx² when slopes are small
- Combining it with 1/R = M/EI gives EI v″ = M
- Differentiating gives the shear and load forms
- Boundary conditions, not the differential equation, decide the answer
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint