This section is from the book "Safe Building", by Louis De Coppet Berg. Also available from Amazon: Code Check: An Illustrated Guide to Building a Safe House.

The stiffness of beams or cantilevers of same cross-section and material (and similarly loaded and supported), however, diminishes very rapidly, as the length of span increases, or what is the same thing, the deflection increases much more rapidly in proportion than the length; the comparative stiffness or deflection being directly as the cube of their respective lengths or L8.

That is if a beam 10 feet long deflects under a certain load one-third of an inch, the same beam with same load, but 20 feet long will deflect an amount x as follows: x: 1/3=203:103, or x=(203.1/3)/103 = 8000/3000 = 2 2/3"

Stiffness of beams of different lengths.

The comparative stiffness, that is amount of deflection of two or more beams or cantilevers, similarly supported and loaded, and of same material and span, but of different cross-sections, is inversely as the product of their respective breadths into the cubes of their respective depths or

Where x = a figure for comparing the deflection of beams of same material, span and load.

Where b = the breadth of beam, in inches. Where d = the depth of beam, in inches.

Example.

If a beam 3" x 8" deflects 1/2" under a certain load, what will a beam 4" x 12" deflect, if of same material and span, similarly supported and with same load?

For the first beam we should have x1 = 1/3.83 = 1/1536 = 0,00065

For the second beam we should have x11 = 1/4.123 = 1/6912 = 0,00014

The deflection of the latter beam will be as

δ: 0",5 =0,00014: 0,00065, or δ = 0",108

Stiffness of beams of different cross-sections.

The comparative strength of rectangular beams or cantilevers of different cross-sections and spans, but of same materials and similarly loaded and supported, is, of course, directly as the product of their breadth into the cross-sections.

Squares of their depths, divided by their length of span, or x = b.d2/L (32)

Where x = a figure for comparing the strength of different beams of same material, but of different cross-sections and spans. Where b = the breadth, in inches. Where d=the depth, in inches. Where L = the length of span, in feet.

The comparative stiffness or amount of deflection of different rectangular beams or cantilevers of same material, and similarly loaded and supported, but of different cross-sections and spans, would be directly as the cubes of their respective lengths, divided by the product of their respective breadths into the cubes of their depths or

Where x = a figure for comparing the amount of deflections of beams of same material and load, but of different spans and cross-sections.

Where L = the length of span, in feet.

Where b = the breadth, in inches.

Where d = the depth in inches.

Stiffness of beams of different lengths& cross-sections.

If it is desired to calculate a wooden girder supported at both ends and to carry its full safe uniform load, and yet not to deflect enough to crack plaster, the following will simplify the calculation:

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