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.
The writer has so often been asked for more information as to the meaning of the term Moment of Inertia that a few more words on this subject may not be out of place.
All matter, if once set in motion, will continue in motion unless stopped by gravity, resistance of the atmosphere, friction or some other force; similarly, matter, if once at rest, will so remain unless started into motion by some external force. Formerly it was believed, however, that all matter had a certain repugnance to being moved, which had to be first overcome, before a body could be moved. Probably in connection with some such theory the term arose.
In reality matter is perfectly indifferent whether it be in motion or in a state of rest, and this indifference is termed "Inertia." As used to-day, however, the term Moment of Inertia is simply a symbol or name for a certain part of the formula by which is calculated the force necessary to move a body around a certain axis with a given velocity in a certain space of time; or, what amounts to the same thing, the resistance necessary to stop a body so moving.
In making the above calculation the "sum of the product of the weight of each particle of the body into the square of its distance from the axis" has to be taken into consideration, and is part of the formula; and, as this sum will, of course, vary as the size of the body varies, or as the location or direction of the axis varies, it would be difficult to express it so as to cover every case, and therefore it is called the "Moment of Inertia." Hence the general law or formula given covers every case, as it contains the Moment of Inertia, which varies, and has to be calculated for each case from the known size and weight of the body and the location and direction of the axis.
In plane figures, which, of course, have no thickness or weight, the area of each particle is taken in place of its weight; hence in all plane figures the Moment of Inertia is equal to the "sum of the products of the area of each particle of the figure multiplied by the square of its distance from the axis."
Thus if we had a rectangular figure (119) b inches wide and d inches deep revolving around an axis M-N, we would divide it into many thin slices of equal height, say n slices each of a height = 2. X.
The distance of the centre of gravity of the first slice from the axis M-N will, of course be= 1/2. 2. X. = 1. X
The distance of the centre of gravity of the second slice will be = 3.X, that of the third slice will be= 5. X, that of the fourth slice will be = 7. X, that of the last slice but one will be= (2 n - 3). X. and that of the last slice will be = (2 n - 1). X
The area of each slice will, of course, be = 2. X. b; therefore the Moment of Inertia of the whole section around the axis M-N will be (see p. 8), i = 2. X. b. (1. X)2 + 2. X. 6. (3.X)2+ 2. X. b. (5. X)2+
2. X. b. (7. X)+ etc......+ 2. X. b. [(2n - 3).X]2
+ 2. X. b.[(2n - l).X]2or, i=2.X.3b,. [l2+32 + 52+72 + etc......+ (2n - 3)2
+ (2n-l)2] now the larger n is, that is the thinner we make our slices, the nearer will the above approximate:
Calculation of Moment of Inertia.
i = 2. X3.b. [4/3. n8]
= 8. X3. n3. b/3
Therefore, as: 2. X. n = d we have, by cubing,
8. X3. n3 = d3; inserting this in above, we have: i = d3.b/3 or b.d3/3
The same value as given for i in Table I, section No. 29. Of course it would be very tedious to calculate the Moment of Inertia in every case; besides, unless the slices were assumed to be very thin, the result would be inaccurate; the writer has therefore given in Table I, the exact Moments of Inertia of every section likely to arise in practice.
The Moment of Inertia applies to the whole section, the " Moment of Resistance," however, applies only to each individual fibre, and varies for each; it being equal to the Moment of Inertia of the whole section divided by the distance of the fibre from the axis.
Now to show the connection of the Moments of In-ertia and Resistance with transverse strains, let us consider the effect of a weight on a beam (sup ported at both ends).
If we consider the beam as cut in two and hinged at the point A (where the weight is applied), Fig. 120; further, if we consider a piece of rubber nailed to the bottom of each side of the beam, it is evident that the effect of the weight will be, as per Fig. 121.
Examining this closer we find that the cor ners of the beams above A (or their fibres) will crush each other, while those below A, are separated farther from each other, and the piece of rubber at B greatly stretched. It is evident, therefore, that the fibres nearest A experience the least change, and that the amount of change of all the fibres is directly proportionate to their distance from A (as the length of all lines drawn parallel to the base of a triangle, are proportionate to their distance from the axis); further, that the fibres at A experience no change whatever. Now, if instead of considering the effect of a load on a hinged beam we took an unbroken beam, the effect would be similar, but, instead of being concentrated at one point, it would be distributed along the entire beam; thus the beam A B D C (Fig. 122) which is not loaded, becomes when loaded, the slightly curved beam (A BDC) Fig. 123.
Moment of Resistance.
Effect of load on beam.
It is evident that the fibres along the upper edge are compressed or A B is shorter than before; on the other hand the fibres along C D are elongated, or in tension, and C D is longer than before; if we now take any other layer of fibres as E F, they - being below the neutral (and central)1 axis X-Y - are evidently elongated; but not so much so, as C D: and a little thought will clearly show that their elongation is proportioned to the elongation of the fibres C D, directly as their respective distances from the neutral axis X-Y. It is further evident that the neutral axis X-Y is the same length as before, or its fibres are not strained; it is, therefore, at this point that the strain changes from one of tension to one of compression.
In Fig. 124 we have an isometrieal view of a loaded beam.
1 As a rule the neutral axis can be safely assumed to be central, but it is not necessarily so. In materials, such as cast-iron, stone, etc., where the resistance of the fibres to compression and tension varies greatly, the axis will be far from the centre, nearer the weaker fibres.
Let us now consider an infinitesimally thin (cross) section of fibres A B C D in reference to their own neutral axis M-N. It is evident that if we were to double the load on the beam, so as to bend it still more, that the fibres along A B would be compressed towards or would move towards the centre of the beam; the fibres a-long D C on the contrary would be elongated or would move away from the centre of the beam.
The fibres along M-N, being neither stretched nor compressed, would remain stationary.
The fibres between M-N and A B would all move towards the centre of the beam, the amount of motion being proportionate to their distance from M-N; the fibres between M-N and D C on the contrary would move away from the centre of the beam the amount of motion being proportionate to their distance from M-N; a little thought, therefore, shows clearly that the section ABCD turns or rotates on its neutral axis M-N, whenever additional weight is imposed on the beam.
This is why we consider in the calculations the moment of Inertia or the moment of resistance of a cross-section as rotating on its neutral axis.
Now let us take the additional weight off the beam and it will spring back to its former shape, and, of course, the fibres of the in-finitesimally thin section ABCD will resume their normal shape; that is, those that were compressed will stretch themselves again, while those that were stretched will compress themselves back to their former shape and position, and those along the neutral axis will remain constant; or, in other words, this thin layer of fibres A BCD can be considered as a double wedge - shaped figure A B A1 B1MNDC
Rotation around neutral axis.