Basic equations of beam deflection

In the preface to his book Strength of Materials, J. P. Den Hartog says

After the alphabet and the tables of multiplication, nothing has proved quite so useful in my professional life as these six little expressions.

The six expressions he refers to are nicknamed the vergeet-me-nietjes in Dutch, which translates to forget-me-nots in English. They are also known as Dr. Myosotis’s equations because myosotis is the genus for forget-me-nots. The equations give the angular and linear deflections of a cantilever beam.

Imagine a beam anchored at one end and free on the other, subject to one of the kinds of load: a bending moment M at the opposite end, a point force P at the opposite end, or a force w distributed over the length of the beam. The equations below give the rotation (angular deflection) and displacement (linear deflection) of the free end of the beam.

Rotation Displacement
Bending moment  ML/EI  ML2/2EI
Point load  PL2/2EI  PL3/3EI
Distributed load  wL3/6EI  wL4/8EI

Here E is the modulus of elasticity, L is the length of the beam, and I is the area moment of inertia.

One thought on “Basic equations of beam deflection

  1. Another excellent bite-sized post. Thanks, John!
    The symmetry of these is one of the “hidden gems of STEM”.

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