Propped Cantilever with Uniform Load

Reference

Built in at one end, seated at the other: the propped cantilever is the first genuinely indeterminate beam most people meet, and it behaves accordingly.

The equation

Rprop = 3wL/8 and Rwall = 5wL/8
Mwall = wL2/8 hogging
Msag = 9wL2/128 at 3L/8 from the prop
δmax = wL4/185EI at 0.4215L from the prop
  • w is the load per unit length, in kN per meter
  • L is the span from wall to prop, in meters
  • E and I are the modulus and second moment of area
  • δ is the vertical deflection
w = 10 kN/mL = 6 mBending moment, sagging plotted up9wL2 /128 = 25.3 kNmwL2 /8 = 45 kNm hogging at the wallcalculatgeek.com
One clamped end changes everything: the wall takes five eighths of the load and a moment equal to what the whole span would carry at midspan if both ends were pinned.

Worked example

A 6 m span carrying 10 kN/m, built in at the left and propped at the right, EI = 16000 kNm2. The prop takes 3 × 10 × 6/8 = 22.5 kN and the wall 37.5 kN. The wall moment is 10 × 62/8 = 45 kNm hogging, the sagging peak is 9 × 10 × 62/128 = 25.3 kNm at 2.25 m from the prop, and the maximum deflection is 10 × 64/(185 × 16000) = 4.4 mm at about 2.53 m from the prop. All of it checks against an independent numerical solve; the 185 is itself a rounded coefficient, 184.6 exactly.

When to use it

The classic model for a beam cast into a wall at one end and seated at the other, and the first real lesson in indeterminacy: the split between wall and prop depends on stiffness, so a prop that settles even a few millimetres sheds load and moment back to the wall. When the support conditions are in doubt, bracket the answer with the pinned and fixed cases, or model the beam properly.

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