Beam Bending Explained — Euler–Bernoulli Theory & Roark's Formulas

What beam bending theory gives you

Load a beam and you want to know four things along its length: the reactions at the supports, the shear force, the bending moment, and the deflection. The bending moment tells you the stress (and whether it yields); the deflection tells you whether it's stiff enough. Euler–Bernoulli beam theory is the standard framework that produces all four, and it's what the tabulated formulas in Roark's Formulas for Stress and Strain are built on.

The Euler–Bernoulli assumptions

The theory is simple because it assumes a lot — and knowing the assumptions is knowing when it applies:

Prismatic — uniform cross-section along the length.
Linear-elastic — stress proportional to strain, no yielding.
Small deflections — the beam bends a little, not into a big curve.
Plane sections remain plane — a flat cross-section stays flat and perpendicular to the beam axis as it bends (this is what neglects shear deformation — fine for slender beams, less so for short deep ones).

The shear → moment → deflection chain

The four quantities are linked by integration. Start from the distributed load, integrate to get shear, integrate again for bending moment, then the deflection comes from double-integrating M(x)/EI (moment over the bending stiffness — E is the material's modulus, I the section's second moment of area). So a stiffer material or a deeper section (bigger EI) bends less for the same moment. This chain is exactly why the bending stress depends on the section shape through I, and the deflection depends on it again.

Determinate vs indeterminate beams

The solution method depends on whether statics alone can find the reactions:

Statically determinate (simply supported, cantilever, overhanging) — the support reactions come straight from force and moment equilibrium. Solvable by hand.
Statically indeterminate (fixed-fixed, propped cantilever) — there are more supports than equilibrium equations, so you need an extra condition: the deflection or slope at a redundant support must be zero. That compatibility requirement, solved by the force (flexibility) method using virtual work, closes the problem.

Worked reasoning, section choice & checklist

Why the same load deflects a beam 16× more as a cantilever than simply supported, how to pick a section for stiffness vs strength, and a checklist for a beam check.

Support conditions matter enormously

The same point load at midspan produces very different results depending on the supports. A cantilever with an end load deflects far more than the same beam simply supported — deflection scales with the cube of the effective length, so how the ends are held dominates the answer. This is why "will it be stiff enough?" almost always comes down to the support condition and the span before the section even matters.

Choosing a section: strength vs stiffness

Bending stress is M·c/I and deflection scales with 1/(E·I). Both improve with a larger second moment of area I — which rewards putting material far from the neutral axis (why I-beams and tubes are efficient). But strength cares about I/c (the section modulus) while stiffness cares about I alone, so a section optimised for one isn't automatically best for the other. Decide which governs — a floor beam is usually deflection-limited, a highly-stressed bracket strength-limited.

Checklist

Confirm the beam is slender and elastic — Euler–Bernoulli understates deflection for short, deep beams (shear matters there).
Get the support condition right first; it dominates both moment and deflection.
Find the maximum bending moment and check the stress (M·c/I) against the allowable with a safety factor.
Check the maximum deflection against the serviceability limit (often a span fraction like L/360).
For indeterminate beams, trust a compatibility-based solution (or a validated tool), not statics alone.

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Beam Bending: Euler–Bernoulli Theory and Roark's Formulas