86. When a beam is supported at the ends in a horizontal position, and a weight rests upon the middle, if the deflection be of small amount, the laws regulating it may be determined as follows: - The extension of any part of the beam is directly as the force by which it is produced, Art. 79; and it is known by experiment that the deflection is in proportion to the weight, all other things being constant; therefore the deflection is as the extension. Now, the effect of a load to produce extension in a beam is as the leverage, and as the load itself; or, in other words, as the bending moment M described in Art. 44; but the leverage is proportional to the length, therefore the force producing the extension is as the length and weight directly, and the resistance to extension is inversely as the breadth, and square of the depth, as shown by writers on the Strength of Materials. But the extension of each fibre will be directly as the number of its parts extended, that is also as the length; and as the quantity of angular motion, which will again be as the length directly, and depth inversely. Uniting these equivalents of the extending and resisting forces, we find the deflection to be as the weight, and cube of the length directly; and as the breadth, and cube of the depth inversely - that is making L = the length of bearing in feet, W = the weight in pounds, B = the breadth in inches, D = the depth in inches, a = the constant number which depends upon the nature of the material, and A = the deflection in inches.

We have L3xWxa/BxD3 = [2]

Before the deflection A can be found, the value of a. must be obtained (it = 1/40 of a in Tables VI. to XI.).

87. In order that the beam may be sufficiently stiff for the carpenter's purpose, the deflection should not exceed 1/40 of an inch for each foot of the total length. If we substitute a/40 for a. in the above formula, it becomes

L3xWxa/40 xBxD3= [3]

And to find a we have

40 x B x D3 x /L3xW=a [4]

From this formula the values of a in Tables VI. to XI. have been calculated.

88. The quantity of timber being the same, the stiffness of a beam will vary according to its depth; but there is a certain proportion between the depth and breadth, which if exceeded will render the beam liable to overturn and break sideways. To avoid which, the breadth should never be less than that given by the following rule, unless the beam be supported laterally.

Rule III. - Divide the length in feet by the square root of the depth in inches, and the quotient multiplied by the decimal 0.6 will give the least breadth that should be given to the beam.

When the depth is not determined by other circumstances, the nearer its form is made to approach that determined by the rule the stronger it will be; and, from the same rule, another is easily derived which will show the advantage of making beams deep in proportion to the thickness.

89. To find the strongest form for a beam so as to use only a given quantity of timber.

Rule IV. - Multiply the length in feet by the decimal 0.6, and divide the given area in inches by the product; and the square of the quotient will give the depth in inches.

Example. - If the bearing be 20 feet, and the given area of section be 48 inches; then 48/0.6x20 = 48/12 = 4, and the square of 4 is 16 inches, the depth required; and the breadth will be 3 inches. A beam 16 inches by 3 would support more than twice as much as a square beam of the same area of section; which shows how important it is to make beams deep and thin. In many old buildings, and even in new ones, in country places, the very reverse of this has been practised; the principal beams being more frequently laid on their broad than on their narrow side.

90. The stiffest beam that can be cut out of a round tree is found by laying off the thickness which is equal to half of the diameter of the circle, from the extremities of the diameter, and completing the triangles as in Fig. 41. The figure thus formed gives the breadth to the depth as 1 is to the square root of 3,* or as 1 is to 1.732, nearly; or as .58 is to 1; this being in general a good proportion for beams that have to sustain a considerable load, and where it would be impossible to get them deeper on account of the size of the tree, we may substitute it for the breadth in equation [3] which then gives for the depth: -

Fig. 41.

[5]

* Young's 'Nat. Phil.'

And for inclined beams, c being the angle of inclination,

[6]

91. The deflection of square beams is the same, whether strained in direction of the side or diagonal.