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.

Manner of Loading. | Description. | m and s at centre. | m and s at any point distant x from support p. | Location and amount of greatest bending-moment m. | Location and amount of greatest shearing-strain S. | Location and amount of greatest deflection δ. | ||

Uniform load on beam supported at both ends. | m = (u.l)/8 | When x greater than l/2 use (l-x) in place of x. | m = u/2. x. (1-x/l) | at centre | at support p or q. | at centre | ||

s = o | s = u. (1/2 - x/l) | m = (u.l)/8 | s = u/2 | δ = 5/384. (u.l3)e.i | ||||

Load at centre of beam supported at both ends. | m = (w.l)4 | m = w/2. x | at centre | at support p or q | at centre | |||

s = o | s = w/2 | m = (w.l)/4 | s = w/2 | δ = 1/48. (w.l3)/e.i | ||||

Load at any point on beam supported at both ends. | If y greater than l/2 | m = (w.z)/2 | At load use: | m = (w.y.z)/l | at load | at nearer support p If y smaller than 1/2 | near centre | |

s = o | ||||||||

s = (w.z)/l | ||||||||

For x use: | m =(w.z.x)/l | s= (w.z)/l | ||||||

If y smaller than l/2 | m = (w.y)/2 | s = (w.z)/l | m = (w.y.z)/l | at support q. | ||||

Form x use: | m = (w.y.x1)/l | If y greater than l/2 | ||||||

s = (w.y)/l | ||||||||

s = (w.y)/l | s = (w.y)/2 |

The comparative transverse strength of two or more rectangular beams or cantilevers is directly as the product of their breadth into the square of their depth, provided the span, material and manner of supporting and loading are the same, or x = bd2 (30)

Where x = a figure for comparing strength of beams of equal spans. Where b = the breadth of beam, in inches. Where d = the depth of beam, in inches.

Example. What is the comparative strength between a 3" x 12" beam, and a 6" x 12" beam? Also, between a 4" x 12" beam, and a 3" x 16" beam? All beams of same material and span, and similarly supported and loaded.

The strength of the 3" x 12" beam would be x1 = 3. 12. 12 = 432. The strength of the 6" x 12" beam would be x11 = 6. 12. 12=864, therefore, the latter beam would be just twice as strong as the former. Again, the strength of the 4" x 12" beam would be x11 = 4. 12. 12 = 576 and the strength of the 3" x 16" beam would be x1111 = 3. 16. 16=768.

The latter beam would therefore be 768/576 or just l 1/3 times as strong as the former, while the amount of material in each beam is the same, as 4. 12 = 3. 16 = 48 square inches in each. The reason the last beam is so much stronger is on account of its greater depth.

The comparative transverse strength of two or more beams or cantilevers of same cross-section and material, but of unequal spans, is inversely as their lengths, provided manner of supporting and loading are the same. That is, a beam of twenty-foot span is only half as strong as a beam of ten-foot span, a quarter as strong as one of five-foot span, etc.

Strength of beams of different cross-sections.

Strength of beams of different lengths.

All measurements in Table VII are in inches; all weights in pounds; c = modulus of elasticity in pounds inch; i = moment of inertia of cross-section of beam or cantilever around its neutral axis in inches; m =bending-moment in pounds inch; s= amount of shearing strain in pounds; 6 = total amount of deflection in inches.

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