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Structural Analysis Fundamentals / Beam deflection

Curvature and deflected shapes

Picture the shape before you calculate the number.

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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.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

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Why this matters

Sketching the deflected shape takes ten seconds and catches errors that a page of algebra will not. It is also how you decide whether a computer model has been supported the way you intended.

By the end of this lesson you should be able to

  • Tell deflection from rotation
  • Relate moment to curvature
  • Sketch shapes from boundary conditions

Deflection is how far a point moves. Rotation is how much the beam has tilted there. They are different, and a point can easily have one without the other. At a simple support the deflection is zero but the beam is free to rotate. At a fixed support both are zero — the beam leaves the support horizontally.

Moment and curvature
1R=MEIEId2ydx2=M\frac{1}{R} = \frac{M}{EI} \qquad EI\frac{d^2 y}{dx^2} = M

What it calculates: How sharply the beam bends at a section, given the bending moment there.

1/R
Curvature — how sharply the beam bends (1/mm)
M
Bending moment at the section (N·mm)
E
Young's modulus (N/mm²)
I
Second moment of area (mm⁴)
y
Deflection (mm)

This assumes

  • Elastic material
  • Small deflections
  • Bending about a principal axis

In plain terms: The product EI is the beam's flexural stiffness. Where the moment is large the beam curves sharply; where the moment is zero the beam is momentarily straight. Increasing EI reduces curvature everywhere.

Predict first

Where does the deflected shape of a cantilever have zero slope?

A cantilever fixed at the left, curving downwards to its greatest deflection at the free end, leaving the support horizontally.cantilever with a tip load
The beam leaves the fixed support horizontally — that is the boundary condition doing its work.

Practice

A beam carries a bending moment of 80 kN·m at a section, with E = 205 000 N/mm² and I = 1.5 × 10⁸ mm⁴. What is the curvature 1/R, in units of 1/mm?

Practice

If the second moment of area of a beam is doubled while the bending moment stays the same, what happens to the curvature? Enter the factor it is multiplied by.

Practice

At a fixed (built-in) support, how many of the following are zero: the deflection, and the slope? Enter the number.

Summary

  • Deflection and rotation are different
  • Curvature = M/EI, so EI is flexural stiffness
  • Boundary conditions set the shape; sketch before you calculate