This section is from the "Architectural Iron And Steel, And Its Application In The Construction Of Buildings" book, by WM. H. Birkmire.. Also see Amazon: Architectural Iron And Steel, And Its Application In The Construction Of Buildings.
In the following consideration of rolled struts of various shapes, the least radius of gyration of the cross-section taken around an axis through the centre of gravity is assumed as the effective radius of the strut. The resistance of any section per unit of area will in general terms vary directly as the square of the least radius of gyration, and inversely as the square of the length of the strut.* The shape of the section, and the distribution of the metal to resist local crippling strains, must also be considered. As a rule, that shape will be strongest which presents the least extent of flat unbraced surface. For instance, two I sections of unequal web widths may have the same web thickness, the same flange area, and the same least radius of gyration, but the wider-webbed section will be the weaker per unit of area, on account of the greater extent of unbraced web surface it contains. .For the same reason a hollow rectangular section composed of thin plates will be to some extent weaker than a circular section of the same length having the same area and radius of gyration.
As is well known, the method of securing the ends of the struts exercises an important influence on their resistance to bending, as the member is held more or less rigidly in the direct line of thrust.
In the table, page 46, struts are classified in four divisions, viz.: "Fixed-ended," "Flat-ended," "Hinged-ended," and "Round-ended".
* This applies only to long struts with free ends.
In the class of "fixed ends" the struts are supposed to be so rigidly attached at both ends to the contiguous parts of the structure that the attachment would not be severed if the member were subjected to the ultimate load. "Flat-ended" struts are supposed to have their ends flat and square with the axis of length but not rigidly attached to the adjoining parts. "Hinged ends" embrace the class which have both ends properly fitted with pins, or ball-and-socket joints, of substantial dimensions as compared with the section of the strut; the centres of these end joints being practically coincident with an axis passing through the centre of gravity of the section of the strut. "Round-ended" struts are those which have only central points of contact, such as balls or pins resting on flat plates, but still the centres of the balls or pins coincident with the proper axis of the strut.
If in hinged-ended struts the balls or pins are of comparatively insignificant diameter, it will be safest in such cases to consider the struts as round-ended.
When the pins of hinged-ended struts are of substantial diameter, well fitted and exactly centred, experiment shows that the hinged-ended will be equally as strong as flat-ended struts.
But a very slight inaccuracy of the centring rapidly reduces the resistance to lateral bending; and as it is almost impossible in practice to uniformly maintain the rigid accuracy required, it is considered best to allow for such inaccuracies to the extent given in the table.
In the table, the first column gives the effective length of the strut divided by the least radius of gyration of its cross-section, and the successive columns give the safe load per square inch of sectional area for each of the four classes aforesaid. By ultimate load is meant that pressure under which the strut fails. For hinged-ended struts the figures apply to those cases in which the axis of the pin is at right angles to the least radius of gyration, or in which the strut is free to rotate on the pin in its weakest direction.
It is good practice to increase the factor of safety as the length of the strut is increased, owing to the greater inability of the long struts to resist cross-strains, etc. For similar reasons it is advisable to increase the factor of safety for hinged and round ends in a greater ratio than for fixed or flat ends. Upon this consideration the table is arranged, with the factor of safety of 3 for the short to 6 for the long struts. The loads to be applied only under the most favorable circumstances, such as an invariable condition of the load, little or no vibration, etc.
Least Radius of Gyration.
If the strut is hinged by an uncertain method so that the centres of pins and axis of strut may not coincide, or the pins may be relatively small and loosely fitted, it is best in such cases to consider the strut as "round-ended".
In all cases the strut is supposed to stand vertical. In short struts this is immaterial; but when the length becomes considerable the deflection resulting from its own weight, if horizontal, would seriously affect the stability of the strict.
(a) An 8" I beam 65 pounds per yard, 18 feet long, is used as a strut having pins at both ends at right angles to the web. It would then be flat-ended in the direction of the flanges. By the table, page 50, for the radius of gyration of an 8" I beam, 65 lbs., r = .88. Then l/r = 216/.88=24.5
In the column - taking the greatest number nearest 245 (being 250), we find 1910 = greatest safe load in pounds per square inch of section.
(b) If braced in the direction of the flanges at two points 6 feet apart, it should then be considered as a series of flat ended struts 6 feet long, whose safe load would be l/r= 72/.88= say 80.
Opposite this number is 8420 = greatest safe load per square inch of section.
(c) When braced in the direction of its web it remains a hinged-ended strut 18 feet long; the radius of gyration is 3.25. Then l/r = 216/3.25 = 67.
Opposite this, taking the greatest number nearest 67 (i.e. 70), we have 9190 as the greatest safe load per square inch of section.