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Not Quite Uniform Loadings Aug 2026

Not Quite Uniform Loadings Aug 2026

Intro

Structural engineers often refer to uniform loadings but such ideal loadings apply, at best, to dead loads (slabs for example). Actual live loads are unlikely to be uniform. I tabulate maximum moments for several variations in simple beam loading, in each case using a unit total load magnitude but different load geometries.

I found my results slightly surprising.

RESULTS

For the ideal uniform load: $Moment_{max} = M_{max} = \dfrac{w L^2}{8}$
Settng the total load $w L = 1$ , we have $M_{max} = \dfrac{L}{8} = 0.125 L$

For the single concentrated load: $M_{max} = \dfrac{L}{4} = 0.25 L$ (*)

For the double concentrated load: $M_{max} = \dfrac{L}{6} \approx 0.167 L$ (*)

For the triple concentrated load: $M_{max} = \dfrac{L}{6} \approx 0.167 L$ (*)

For the quadruple concentrated load: $M_{max} = \dfrac{3L}{20} \approx 0.15 L$ (*)

For the quintuple concentrated load: $M_{max} = \dfrac{3L}{20} \approx 0.15 L$ (*)

Yes, there’s a pattern.

For the N (N is odd) concentrated load: $M_{max} = \left(\dfrac{L}{8}\right) \left(\dfrac{N+1}{N}\right)$

For the N (N is even) concentrated load: $M_{max} = \left(\dfrac{L}{8}\right) \left(\dfrac{N+2}{N+1}\right)$

For the sine function load: $M_{max} = \dfrac{L}{2 \pi} \approx 0.159 L$ (*)

For the centered triangular load: $M_{max} = \dfrac{L}{6} \approx 0.167 L$ (*)

For the right side triangular load: $M_{max} = \dfrac{2 \sqrt{3} L}{27} \approx 0.128 L$ (*)

A recent post considered a different aspect of the triangular loading.

Conclusions

All of the loadings marked (*) produce a maximum moment that is larger than that of the unform load of equivalent magnitude.

Calculations

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