Skip to content
Queensferry

Module 13 · Lesson 13.5

Macaulay's method and moment–area

Two ways to get a deflection when the loading is not in any table.

Why this matters

Standard deflection formulae cover a handful of arrangements. Real beams have loads at awkward positions, several loads at once, and overhangs. Macaulay's method integrates the beam equation once for the whole span however many loads there are, and the moment–area theorems get an answer straight off the bending moment diagram — often in two lines.

By the end of this lesson you should be able to

  • Write a bending moment expression using Macaulay brackets
  • Integrate it and apply boundary conditions to find deflection
  • State the two moment–area theorems
  • Use moment–area to find a cantilever deflection from the M/EI diagram

What you should already know

  • Bending moment diagrams (Module 3)
  • Moment–curvature and EI v″ = M (Module 13)
  • Standard deflection formulae (Module 13)

The obstacle to integrating EI v″ = M directly is that M(x) changes its algebraic form at every load. A beam with three point loads needs four separate expressions, four pairs of integration constants, and continuity conditions matching them at each junction. That is eight constants for one beam.

Macaulay's method removes all of it with one piece of notation. Write terms that begin at a particular position inside angled brackets:

⟨x − a⟩ means: zero when x < a, and equal to (x − a) when x ≥ a

So the term simply switches itself on as you pass the load. One expression then covers the whole beam, and the rules that make it work are these:

The procedure is then exactly the one you already know, applied once:

  1. 1.Find the reactions.
  2. 2.Write M(x) with Macaulay brackets, measuring x from the left-hand end throughout.
  3. 3.Integrate once for EI·(slope), introducing a constant A.
  4. 4.Integrate again for EI·(deflection), introducing a constant B.
  5. 5.Apply two boundary conditions to find A and B.
  6. 6.Substitute the x you care about, discarding negative brackets.

Measuring from one end throughout is not optional. The switching behaviour only works if every bracket turns on in the same direction.

Worked example

Central point load by Macaulay's method

Given

  • Simply supported beam, span L = 6.00 m
  • Point load P = 40.0 kN at mid-span
  • E = 205 000 N/mm², I = 2.00 × 10⁸ mm⁴

Find

The mid-span deflection, derived rather than looked up.

    The moment–area theorems take a different route. Instead of integrating symbols, they read areas off the M/EI diagram.

    First theorem. The change in slope between two points on a beam equals the area of the M/EI diagram between them.

    θB − θA = area of M/EI from A to B

    Second theorem. The deviation of B from the tangent drawn at A equals the first moment of that area, taken about B.

    t_(B/A) = (area of M/EI from A to B) × (distance from its centroid to B)

    Both follow directly from EI v″ = M: the first is one integration, the second is two.

    They are at their best on cantilevers, where the tangent at the fixed end is horizontal and known, so the deviation is the deflection. On simply supported beams a little more care is needed, because neither tangent is horizontal.

    Worked example

    Cantilever tip deflection by moment–area

    Given

    • Cantilever, length L = 3.00 m, fixed at the left
    • Point load P = 15.0 kN at the free end
    • E = 205 000 N/mm², I = 1.20 × 10⁸ mm⁴

    Find

    The tip deflection, using the moment–area theorems.

      Practice

      A simply supported beam of span 6.00 m carries a 40.0 kN point load at mid-span, with E = 205 000 N/mm² and I = 2.00 × 10⁸ mm⁴. Macaulay's method gives EIv = −180 kN·m³ at mid-span. What is the deflection, in mm?

      Practice

      A cantilever 3.00 m long carries a 15.0 kN point load at its free end, with E = 205 000 N/mm² and I = 1.20 × 10⁸ mm⁴. Using moment–area, what is the tip rotation, in radians?

      Practice

      In the Macaulay expression M(x) = 20x − 40⟨x − 3⟩ for a 6 m beam, what is the value of the bracket term when x = 2.0 m?

      Summary

      • A Macaulay bracket ⟨x − a⟩ is zero until x reaches a, then behaves normally
      • Never expand it; integrate it as a unit; discard it when negative
      • One expression and one pair of constants covers a beam with any number of loads
      • Always measure x from the same end
      • Moment–area theorem 1: change in slope = area of the M/EI diagram
      • Moment–area theorem 2: deviation from a tangent = first moment of that area
      • Moment–area is quickest when one tangent is already known, as at a fixed end
      Progress is kept in this browser only.

      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