NOTE K.

ON THE STRENGTH OF COLUMNS.

MR. HODGKINSON has obligingly communicated the following observations on Art. 430. :

1. The reader must be made to understand that the rounding of the ends of the pillars is to make them moveable there, as if they turned by means of a universal joint; and the flat-ended pillars are conceived to be supported in every part of the ends by means of flat surfaces, or otherwise rendering the ends perfectly immoveable.

2. The coefficient (13) for hollow columns with rounded ends is deduced from the whole of the experiments first made, including some which were very defective on account of the difficulty experienced in the earlier attempts to cast good hollow columns so small as were wanted. The first castings were made lying on their side; and this, notwithstanding every effort, prevented the core being in the middle: some of the columns were reduced, too, in thickness, half way between the middle and the ends, and near to the ends, and this slightly reduced the strength. These causes of weakness existed much more among the pillars with rounded ends than those with flat ones; they are alluded to in the paper (Art. 47.). Had it not been for them, the coefficient (13) would, I conceive, have been equal to that for solid pillars (or 14.9).

3. The fact of long pillars with flat ends being about three times as strong as those of the same dimensions with rounded ends is, I conceive, well made out, in cast iron, wrought iron, and timber; you have, however, omitted it, being perhaps led to do it through the low value of the coefficient (13) above mentioned.

The same may be mentioned with respect to the near approach in strength of long pillars with flat ends, and those of half the length with rounded ends. It may be said that the law of the 17 power of the length would nearly indicate the latter; but this last, and the other powers 3.76 and 3·55, are only approximations, and not exactly constant, though nearly so, and I do not know whether the other equal quantities are not, with some slight modifications, physical facts.

4. The strength of pillars of similar form and of the same materials varies as the 1-865 power, or near as the square of their like linear dimensions, or as the area of their cross section.

If we conceive the bar now to return to its former temperature, contracting by the same amount (A) per linear unit; then the point B (fig. 2.) will by this contraction be made to ascend through the space BX.λ=x\

Total descent of B by elongation and contraction is therefore determined by the equation

B

To determine the pressure upon a nail driven through the rod at any point P fastening it to the plane.

It is evident, that in the act of extension the part BP of the rod will descend the plane and the part AP ascend; and conversely in the act of contraction; and that in the former case the nail B will sustain a pressure upwards equal to that necessary to cause BP to descend, and a pressure downwards equal to that necessary to cause PA to ascend; so that, assuming the pressure to be downwards, and adopting the same notation as before, except that AP is represented by p, AB by a, and the pressure upon the nail (assumed to be downwards) by P, we have in the case of extension

EXAMPLE OF THE DESCENT OF THE LEAD ON THE ROOF OF BRISTOL

CATHEDRAL.

My attention was first drawn to the influence of variations in temperature to cause the descent of a lamina of metal resting on an inclined plane

THE TABLES OF M. GARIDEL.

TABLE II.

Showing the Angle of Rupture ¥ of an Arch whose Loading is of the same Material with its Voussoirs, and whose Extrados is inclined at a given Angle to the Horizon. (See Art. 344.)

a = ratio of lengths of voussoirs to radius of intrados.

cratio of depth of load over crown to radius of intrados, so that c=ẞ(1+a). (Art. 338.)

Y is a minimum when cos.0= cos. ☺1, in which case

There will therefore be a jump of the pendulum upon its bearings at each oscillation, if the amplitude 0, of the oscillation be such, that

The theory of the falsely-balanced carriage-wheel differs from that of the rolling cylinder,-1st, in that the inertia of the carriage applied at its axle influences the acceleration produced by the weight of the wheel, as its centre of gravity descends or ascends in rolling; and, 2ndly, in that the wheel is retained in contact with the plane by the weight of the carriage. The first cause may be neglected, because the displacement of the centre of gravity is always in the carriage-wheel very small, and because the angular velocity is, compared with it, very great.

If W, represent that portion of the weight of the carriage which must be overcome in order that the wheel may jump (which weight is supposed to be borne by the plane), and if Y, be taken to represent the pressure upon the plane, then (equation 52.)

In order that there may be a jump, this expression must be negative.

by observing, in the autumn of 1853, that a portion of the lead which covers the south side of the choir of Bristol Cathedral, which had been renewed in the year 1851, but had not been properly fastened to the ridge beam, had descended bodily eighteen inches into the gutter; so that if plates of lead had not been inserted at the top, a strip of the roof of that length would have been left exposed to the weather. The sheet of lead which had so descended measured, from the ridge to the gutter, 19ft. 4in., and along the ridge 60ft. The descent had been continually going on from the time the lead had been laid down. An attempt made to stop it by driving nails through it into the rafters had failed. The force by which the lead had been made to descend, whatever it was, had been found sufficient to draw the nails. * As the pitch of the roof was only 161° it was sufficiently evident that the weight of the lead alone could not have caused it to descend. Sheet lead, whose surface is in the state of that used in roofing, will stand firmly upon a surface of planed deal when inclined at an angle of 30° †, if no other force than its weight tends to cause it to descend. The considerations which I have stated in the preceding articles led me to the conclusion that the daily variations in the temperature of the lead, exposed as it was to the action of the sun by its southern aspect, could not but cause it to descend considerably, and the only question which remained on my mind was, whether this descent could be so great as was observed. To determine this I took the following data:—

Mean daily variation of temperature at Bristol in the month of August; assumed to be the same as at Leith (Komtz Meteorology, by Walker, p. 18.)

8° 21' Cent.

⚫0028436.

232 inches.

Linear expansion of lead through 100° Cent.
Length of sheets of lead forming the roof from the ridge
to the gutter

Inclination of roof

16° 32'.

Limiting angle of resistance between sheet lead and deal 30°
Whence the mean daily descent of the lead, in inches, in the month of
August, is determined by equation (2.) to be

* The evil was remedied by placing a beam across the rafters, near the ridge, and doubling the sheets round it, and fixing their ends with spike-nails.

This may easily be verified. I give it as the result of a rough experiment of my own. I am not acquainted with any experiments on the friction of lead made with sufficient care to be received as authority in this matter. The friction of copper on oak has, however, been determined by General MORIN (see a table in the preceding part of this work) to be 0.62, and its limiting angle of resistance 31° 48'; so that if the roof of Bristol Cathedral had been inclined at 31° instead of 16°, and had been covered with sheets of copper resting on oak boards, instead of sheets of lead resting on deal, the sheeting would not have slipped by its weight only.