RESISTANCE TO IMPULSION.

THE forms of beams which are most likely to be exposed to impulsive forces in cases of practice are the rectangular, the grooved, and the open; therefore, in considering the nature of resistance to impulsion, we shall confine our inquiries to these three forms only.

Now, we have shewn in the solution of the first, fifth, and sixth problems, that the expressions for the resistance of those forms in the case of pressure are respectively as below, viz.

For the rectangular beam,

For the grooved beam,

lw = 850 b d2(1 − p3).

For the open beam, And in the solution of the tenth problem it is shewn that the deflexion for those forms is,

Hence, by taking the deflexion in feet, and introducing the number which expresses the power of gravity, viz. 32; we have, from the laws of the collision of bodies,

And, by squaring both sides of this equation, it becomes, after proper reduction,

17 m v2 = 772 bdl;

where v is the velocity of the moving body in feet per second, m its mass or weight in lbs., and b, d, and I, the dimensions of the beam as formerly employed: hence, when the moving body is urged by the power of gravity, we have,

1. When the beam is uniform in breadth and depth, 17 m v2 772 bdl.

2. When the beam is uniform in depth, and its transverse section in form of the letter I.

17 m v2 = 772 b d l ( 1 − q p3).

3. When the beam is uniform in breadth and depth, but has a part of the middle left out,

These three cases express the conditions of the beams when the depth is the same throughout the length; but, as we have remarked elsewhere, those forms are, for the sake of lightness and economy, frequently cast with the outline of the depth elliptical, and when this happens to be the case, the equations which express the conditions of resistance, are as under.

4. When the breadth is uniform, and the outline of the depth elliptical,

17 m v2 = 992 b d l.

5. When the outline of the depth is elliptical, and the transverse section in form of the letter I,

17 m v2 = 992 b d l ( 1 − q p3).

6. When the outline of the depth is elliptical, and part of the middle left out.

17 m v2 = 992 b d l (1 − p3).

The last three of these equations are deduced from the first three in the following manner. It has been shewn in Problem X. that the deflexion for a uniform beam, when the load is applied at the middle of the length, is

and at page 46 it is shewn, that the deflexion for a beam having the outline of the depth an ellipsis, is

Hence, 992 being substituted for 772 in each of the first three equations, gives the last three accordingly.

We may further observe, that in each of the preceding six cases, the beam is supposed to be supported at the ends, and the load to produce its effect at the middle point; but in the case of the three latter, this supposition is not necessary; for, as we

shall shew further on, the beams are alike, capable at every point throughout the length to resist the impulsion, and hence they are designated beams of equal strength.

We shall, in the next place, give a few examples to shew the method of reducing the formula.

Example 1. A uniform rectangular beam of cast iron, 1.53 inches broad, 6 inches deep, and 26 feet long, is exposed to a gravitating force of 8 feet per second; what is the weight of the gravitating body, the impulse being just within the elastic power of cast iron?

The formula belongs to Case 1, viz.

17 m v2 772 b d l.

Now b 1.53 inches, d = 6 inches, 726 feet, and v 8 feet: let these numbers be substituted in the equation for b, d, l, and v, and it becomes

17 x 82 x m = 772 x 1.53 × 6 × 26;

that is, m =

184260.96
1088

= 169.35 lbs. very nearly.

But by the first case of rectangular beams, when exposed to pressure, a beam of the same section, and 26 feet long, will bear, without permanent alteration, a load of not less than 1801 lbs. including its own weight, or 1418 lbs. excluding its own weight; hence it appears, that a load of 169 lbs. falling from a height of about one foot, will produce the same effect on a beam by impulsion, that a load of 1418lbs.

will produce by pressure, setting aside the consideration of the effect produced by the weight of the beam in both cases, but which would indeed have a very important influence in the comparison.

Example 2. A cast iron beam of uniform depth, having its transverse section in form of the letter I, is supported at the ends, and exposed to a blow from a weight of 100 lbs. falling freely through a height of 4 feet; what must be the depth of the beam to sustain the shock, supposing its length to be 30 feet, its greatest breadth 4 inches, the breadth in the middle 14 inches, and the depth in the middle .7 times the whole depth?

If h be the height in feet through which a body falls by the force of gravity, then, the writers on the laws of falling bodies have shewn that,

v2 = 644h.

Consequently, when the height of the fall is given instead of the velocity, we have only to employ 641 times that height in the calculation instead of v2, taking care, however, that the fall is reduced to feet, or otherwise, 64 must be reduced to inches, in order to render the terms homogeneous.

In the solution of the fifth problem it is shewn

D

that p = d. and q = b − B ; where B is the breadth

b

of the middle or grooved part of the beam, and D its