Let V represent this value in a Beam, liar, or Cylinder, one foot in length, and one inch square, side, or in diameter: W the weight; l the length in feet; b the breadth, and d the depth in inches; m the distance of the weight from one end; and n the distance of it from the other in feet.

Note

In cylinders, for b d2 put d3.

l W

1. Fixed at one End, weight suspended from the other,-----=V.

6 d2 l W

2. Fixed at both Ends, weight suspended from the middle.------=V.

6bd2

8. Supported at both Ends, weight suspended from the middle lW

------=V.

4bd2

4. Supported at both Ends, weight suspended at any other point than mn W the middle,--------=V.

lb d2

5. Fixed at both Ends, weight suspended at any other point than 2mn W the middle,-----------=V.

3lb d2

From which formulae, the weight that may be borne, or any of the dimensions, may be computed by the following:

1. Vd b² / l =W: Vd b² / W =1; lw / Vd² =b: √ lw / bv = d. In rectangular beams, eta b and d= V³ lw / V.

2. 6 b d² V / l =W:6 b d³ V / W =l; lw / 6 d² V =b; √ lw / 6b V = d. In rectangular beams, etc., b and d=V³ lw / 6 V.

3. 4b d² V / l =W: 4b d² V / W =l; lw / 4 d² V = b: √ lw / 4 b V =d. In rectangular beams.

etc, 6 and d= V³ lw / 4V.

4V

4. lb d² V / mn = W, mn W / 'b d² V = 1; mn W / l d³ V = b; √ mn W / lb V = d. In rectangular beams, etc., b and d= V³ mnw / lV.

5. 3 l b d² V / 2 mn =W; 2 mnW / 3 b d² V = 1 ; 2mn W / 3 l d² V = b; √ 2 mn W / 3 l b V =d In rectangular beams, etc, b and d= V³ 2 mn W / 3 lV.

When the weight is uniformly distributed, the same formulae will apply, W representing only half the required or given weight.