TABLE XXVI.—THE 3·5TH POWER OF THE DIAMETERS, OR d35.

1.035 1.00

3.735 97·43 5.635 415:58 7.535 1155 35

TABLE XXVII.—THE 1.63RD POWER OF THE LENGTHS, OR 7163.

Example 1.-What is the breaking weight of a solid cylindrical column of cast iron whose length is 20 feet and diameter 6 inches; the ends being flat and well bedded? From equation (3) we get, by substitution,

Example 2.-A hollow cylindrical cast-iron pillar is 24 feet long, and its external and internal diameters are 9 inches and 7 inches respectively. Calculate its breaking weight, its ends being flat and well bedded.

From equation (5)

Hodgkinson's formulæ are not well adapted for determining the strengths of pillars unless we have a table of 35th and 1.63rd powers. In case such a table is not at hand it will be necessary to have recourse to a table of logarithms.

In equation (1), if W and be known d may be found, or if W and d be known 7 may be found.

By transposing the members of the equation we get

By means of this equation the diameter of a solid cylindrical column may be found when its breaking weight and length are known. In the same way it may be shown that

which gives an expression for the length of a solid cylindrical pillar whose breaking weight and diameter are known.

Example 3.-What will be the diameter of a solid cylindrical cast-iron pillar 10 feet long whose breaking weight is 40 tons, the ends being flat?

Substituting these values in equation (7) we obtain—

log dlog 40+ 1.63 log 10

3.5

log 44.16

= 0.4534

Example 4.-A solid cast-iron cylindrical pillar with both ends rounded, and whose diameter is 3 inches, fails with a load of 5 tons. What is its length?

By substitution in equation (8) we get

log = log 14.9 +3.5 log 3 – log 5·5

1.63

.719.5 feet.

The external and internal diameters and also the lengths of

hollow cylindrical columns may be found by transposing the members of equation (2). Thus

The values of d, da, and I may be found from equations (9), (10), and (11) with the aid of a table of logarithms.

155. Strength of Columns of Medium Length.-A column of medium length, as has been explained, is one whose length varies between 5 and 30 times its diameter when applied to those with flat ends. Columns of this class fail partly by crushing and partly by bending, and the formulæ given for long columns do not apply to them. Mr. Hodgkinson has, from his experiments, deduced the following formula for the strength of medium pillars:

Where W1 = breaking weight of the medium column in tons,

1

W = breaking weight in tons as calculated from equa

tions (1) or (2),

C= sectional area of the columns multiplied by the crushing weight of the material.

Example 5.-What is the breaking weight of a solid column of cast iron 10 feet long and 7 inches in diameter, the ends being flat and well bedded, and the crushing strength of the iron being 40 tons per square inch.

From equation (3) we get

W = 44.16

(7.5)3:5
10163

= 44.16 ×

1155.35
42.66

= 1,196 tons.

This would be the breaking weight on the assumption that the column failed by flexure alone.. Sectional area of column = 44·17 square inches, hence

156. Safe Working Load on Cast-iron Columns.-The factor of safety to be used for columns cannot be fixed on any hard and fast lines as a number of considerations have to be taken into account. Generally speaking, for those made of cast iron and exposed to steady loads, 4th of the breaking weight may be considered as a safe working load, provided that proper precautions be taken to make the column bed properly, but even then it is not often advisable to load them to a greater extent than of their breaking load.

th

If the columns be exposed to loads of a vibratory character the margin of safety should be still greater, varying from th toth of the breaking weight. We have, therefore, the following rules for the factor of safety for cast-iron columns:

1st. For steady loads the factor of safety should vary from 6 to 10.

2nd. For vibratory loads it should vary from 12 to 20.

Example 6.-What is the safe stationary load which may be applied in practice to a hollow cast-iron column (of the same quality of iron as in example 5) 12 feet long, the external diameter being 8 inches and the thickness of metal being 1 inch, and the ends well bedded?

The safe load in ordinary practice may be taken as th of this or 45.6 tons.

157. Proportions of Hollow Cast-iron Columns. There is no rule to guide us in determining the best proportions between the lengths and diameters of columns; several considerations have to be taken into account in fixing this. The relative pro