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Queensferry

Module 13 · Lesson 13.2

Curvature and deflected shapes

Picture the shape before you calculate the number.

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

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
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This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint