ON

THE DEFLEXION AND STIFFNESS

OF

CAST IRON BEAMS

EXPOSED TO TRANSVERSE STRAINS.

PROBLEM X. To determine an expression for the deflexion of a uniform rectangular beam of cast iron, when the strain is equal to the elastic force of the material, or, according to TREDGOLD, 15300 lbs. on a square inch.

It is sufficiently proved by experiment, (see TREDGOLD on Cast Iron, arts. 45 and 50), that when a beam is bent or deflected by means of a certain strain, the quantity of deflexion is directly proportional to the force that produces it, while the elastic force of the material remains perfect; hence, by an inference similar to that employed in the solution of the first problem, we find that the

square of the length in feet, divided by the depth in inches drawn into the deflexion, is a constant quantity; therefore, in a beam whose dimensions are known, if the deflexion produced by a given force be correctly ascertained, the deflexion for other beams similarly circumstanced can easily be calculated.

For, let D, L and ▲ be the depth, length, and deflexion used in the experiment, and d, I and d the depth, length, and deflexion for which the calculation is made; then, from the foregoing inference, we have

L'
ᎠᎪ

=

And this, by exterminating the fractions, becomes dd L2=DA 1o.

Now, by Mr. Tredgold's experiment, (see TREDGOLD on Cast Iron, art. 45), it appears, that a load of 300 lbs. on the middle of a beam one inch square, and thirty-four inches between the supports, produced a deflexion of 16 of an inch, while the elastic force of the material remained unimpaired. Let these dimensions be substituted for D, L and ▲ in the preceding equation, and taking the length in feet, we obtain for the deflexion as follows:

1. When the beam is supported at the ends, and loaded in the middle.

dd 0212.

,,

2. When the beam is supported at the ends, but the load not in the middle.

It has been shewn, in the second case of the first problem, that the strain at the point where the load is applied, is proportional to the rectangle of the segments, into which the length of the beam is divided at that point; and we have stated above, that the deflexion is proportional to the force that produces it, while the strain is within the elastic power of the material; hence, by equality of ratios, the deflexion is proportional to the rectangle of the segments into which the length of the beam is divided at the point of strain. Let m and n represent these segments, or the respective distances of the straining force from the points of support; then, since the deflexion produced by the load in the middle of the beam, where the rectangle of the segments is 12, has been shewn by the last case to be as 02/2, we have

12 mn: 02 1o 08 m n.

Hence, when the beam is supported at the ends, but the load not in the middle, the equation for the deflexion is

dd = '08 mn.

3. When the beam is supported at the ends, and loaded uniformly over the length.

It is demonstrated by writers on the resistance

of solids, that a load uniformly distributed over the length of a beam, produces the same deflexion as if five-eighths of that load were collected at the middle point. (See TREDGOLD on Cast Iron, arts. 18 and 217). Now, in the third case of the first problem it is shewn, that the load which a rectangular beam will support, when uniformly distributed over the length, is and by the first case of the same problem, the load supported in the middle is

850 b d2

1700 b d2

; and we have shewn, in the first case preceding, that the deflexion produced by the load

Hence the equation for the deflexion of a beam, when loaded uniformly over the length, is

dd025 12.

4. When the beam is fixed at one end, and loaded at the other. It has been stated, in the fourth case of the first problem, that the strain on a beam fixed at one end and loaded at the other, is quadruple the strain on a beam supported at the ends and loaded in the middle, and the deflexion is proportional to the strain; therefore, if be the angle of deflexion,

and R the quotient that arises from the division of the fixed part by the projecting part of the beam (the length of which is ), then the general expression for the deflexion is

dò = •08 (1+Rcos.).

But, since, the angle of deflexion, is always very small in practical cases, its cosine will not differ sensibly from unity; in which case, the preceding expression becomes

If the fixed part of the beam be greater than the projecting part, or such, that its flexure is very small, we have

do 0812.

If the fixed part be equal to the projecting part, or R = 1, we have

dd = 1612.

It seems unnecessary to calculate the deflexion for the other two cases of the problem, viz.

When the beam is fixed at one end and loaded uniformly over the length; and when the beam is supported at the ends, and the load increases as the distance from one of the supports; and moreover, we may here remark, that the equations for the deflexion which we have just investigated, are equally applicable to the same cases of beams, of the several forms of section given in the table, at page 18, when