Beam Bending: Euler–Bernoulli Theory and 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.