Triangular loadHalf of the rectangle, at a third from the high side

Why at a third? (30 seconds)

• Most of the load sits on the high side → the resultant is pulled there too.
• The centre of gravity of a triangle — at a third from the high base. In a beam of 6 m: 6 / 3 = 2 m from the high side.
• The load grows along the beam → S does not fall at a constant rate → parabola.

A beam of 6 m — a triangle from 0 to 3

• Resultant
  3 · 6 / 2 = 9 t, 2 m from B → x = 4
• ΣMA
  9·4 − BY·6 = 0 → BY = 6 t
• ΣMB
  AY·6 − 9·2 = 0 → AY = 3 t ✓ 3 + 6 − 9 = 0
• Z
  S is a parabola → Z = √(2·S·L / Q) = √(2·3·6 / 3) → Z = 3.46 m from the side where the load goes to zero
• Mmax
  Up to Z a load of 3 t has built up — exactly AY, so S = 0. Its resultant sits Z / 3 before Z:
  3·3.46 − 3·1.15 → Mmax = 6.93 t·m

What every letter in the Z formula means: S — the shear on the side where the load goes to zero (here AY = 3) · L — the length of the triangular load · Q — the intensity at the high end. Z is measured from the side where the load goes to zero.

The load is high at A instead? Everything is mirrored: Z is measured from B, and at the cut you take the forces to the right of the point — BY with lever arm Z, and the resultant of the load between B and Z with lever arm Z/3.

Why not by areas? The area under the parabola of S is not a rectangle, a triangle or a trapezoid. So M at Z is calculated with a cut: cut the beam at Z, and take about the cut point the moments of all the forces to its left — AY with lever arm Z, and the resultant of the load up to Z with lever arm Z/3.

That's it. Resultant at a third, Z from the formula, and Mmax with one cut.

4 pitfalls everyone falls into

• Resultant in the middle, as in a rectangle. You get BY = 4.5 instead of 6 — and everything after it is wrong.
• Forgetting the half. Q · L without / 2 → double the load.
• Z = S / Q. This formula works only when S falls along a diagonal line. Here S is a parabola.
• Measuring Z from the high side. Z is measured from the side where the load goes to zero.