Hodgkinson's Formulæ for Columns.-The principal practical use of Euler's formula was to furnish a general form of expression for the breaking load to the experiments of Eaton Hodgkinson.

Applying Euler's formula to solid cylindrical columns with

For a square column, in which the length of side is denoted by b, Euler's formula would give

Inserting the constants determined by Hodgkinson, his formulæ become, when the length exceeds 30 times the diameter

44d3-55

זיין

for cast-iron columns with flat ends, where I is in feet and d in inches, and P, the breaking load in tons 134d3'55

12

10.95b*

for wrought-iron columns, ditto, ditto

for dry oak, ditto, ditto

12

7.8164

12

for dry fir, ditto, ditto

For hollow columns of cast iron we have, where D and d denote the external and internal diameters respectively in inches

Hodgkinson found that the strength of columns with flat ends, one end flat and the other rounded, both ends rounded, are as 3:2:1, approximately.

Rankine's Formula. This formula is based on the assumption that the load acts along the axis of the column, and that there is no eccentricity. It may be investigated in the following

1

manner :-1

Let the eccentricity.

Pthe load.

the central deflection. Rthe radius of curvature.

A

f

the area of the cross-section.

r = the radius of gyration.

f

y

=

=

the intensity of stress due to bending within the elastic limit of the material.

the intensity of stress at rupture.

the distance from the neutral axis to the extreme fibre.

7 = the length of the column.

It can be proved that, in a column fixed at one end and free at the other—

or P(2EI P2 + 2E0yA) = ƒA (2EI - P2)

Since P neglect it.

Р

A

is small compared with 2EI + 2AE0y, we may

If 0 = 0

C is a constant depending on the material and the conditions of fixing, or otherwise, at the ends. This is Rankine's formula, and it is clear that it is only true when there is no eccentricity, and when the stress produced is within the elastic limit of the material. In order not to exceed the elastic limit, the intensity

Р
A'

of buckling stress, should be divided by a suitable factor of

safety. If the factor of safety is expressed thus,

which is convenient for variable loads, where d is the least dimension of the cross-section, then, if we denote the working stress by W, we have

The usual factor of safety for dead loads is four.

Gordon's Formula. This formula is of the same form as Rankine's, but, instead of using the least radius of gyration, r, the least dimension of the cross-section is substituted, and the value of c modified accordingly. This will be true for circular and rectangular sections, sincer is a function of d, but for all other sections r is not a simple function of d, hence Rankine's formula is more general than Gordon's.

1

In Rankine's formula ƒ is taken as 36,000 for wrought iron, and C = 600 for flat ends, 18000 for round ends, = 24000 for one end flat and the other end round.

In hollow cylindrical struts, if d is substituted for r

3

for cast iron f = 72,000, c = 0; 400; 1600

800

for wrought iron f = 36,000, c = 4500; 2250; 3000

Professor Bauschinger recommends the following formulæ for columns in fire-proof buildings, as the result of a long series of experiments in which the columns were heated and sprinkled with water

where

the length, and r = the least radius of gyration. The following tests of full-sized columns were made at the Watertown Arsenal in 1881.

The columns were placed in the testing-machine on accurate bearings formed by inserting thin pieces of rolled brass between the ends of the column and the face of the holder when there was not a perfect fit.

Counterpoise weights were attached to the long columns at two points of their length, to counteract the tendency to bend from the weight of the column.

(Tested with cap at end A, and pin of hard cast steel 3 inches in diameter at end B.

Plate on pin end cracked at middle rivet, also channel beam under this crack.

(Cap and pin not used; both ends take square bearings on compression plate.

Tested in a similar manner to No. 2.

24,120 Failed suddenly, springing upwards, breaking every brace
on each side, excepting in the middle of the column.
(Cap not used. Pin end tested with pin used in No. 1
column.

Cracked upper channel beam through middle rivet, then
lower channel on side of hole near pin end.

Tested in a similar manner to No. 4.

25,050 Failed at lower channel beam, a crack extending from
pin 5 inches long through centre rivet-hole.
(Tested as in Nos. 4 and 5.

24,600 Maximum deflection 1.15 inch.

End braces gave out at rivet-holes.