Structural Analysis Fundamentals / Beam deflection
Deriving the beam deflection equation
Euler–Bernoulli theory: from exact curvature to EI d²v/dx² = M, and where it fails.
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This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Checks needed for project use
- An independent structural reviewer must check remaining derivations, diagrams, sign conventions and the interactive models across their full input ranges.
- Independently check deflection-lab implementations, principal-axis direction conventions and shear coefficients for non-rectangular sections; serviceability limits are project-specific.
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
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Read the lesson register and outstanding review notesWhy 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), for upward-positive deflection and downward-positive loading.
Elastic bending relates moment to curvature through M = EI/R. For small slopes, curvature can be approximated by the second derivative of deflection with the stated sign convention. This gives a differential equation to solve with the support and end conditions. Check slopes as well as deflection magnitude before adopting the approximation.
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