Substituting 0.675 for the quantity P0.0982 S in the formula D3 = P0.0982 S X gives the formula D3 = 0.675 X, from which the diameters at all cross sections of a tapered pin above its shoulder, that will give it a strength just equal to that of a section of 1.5 inches diameter and 5 inches from the line wire, may be found. To use the formula for this purpose it is only necessary to substitute any desired values of X therein and then solve in each case for the corresponding values of D. Let it be required, for instance, to determine what diameter a pin should have at a cross section one inch below the line wire in order that the maximum strain at that cross section may equal the corresponding strain at a cross section five inches below the line wire and of 1.5 inch diameter. Substituting one as the value of X, the last-named formula becomes D3 = 0.675, and from this, D = 0.877, which shows that the diameter of the pin one inch below the line wire should be 0.877-inch. A similar calculation will show that if a pin is long enough so that a cross section above the cross-arm is 12 inches below the line wire, the diameter of this cross section should be equal to the cube root of 0.675 × 12 = 8.1, which is 2.008, or practically two inches. It should be observed that the calculations just made have nothing to do with the ability of a pin to resist any particular pull of its line wire. These calculations simply show what diameters a pin should have at different distances below its line wire in order that the maximum stress at each of its cross sections may equal that at a cross section 5 inches below the wire where the diameter is 1.5 inches. In Vol. xx., A. I. E. E., pp. 415 to 419, specifications are proposed for standard insulator pins based on calculations like those just made. As a result of such calculations, the following table for the corresponding values of X and D, as used in the above formula, are there presented, each expressed in inches.

XD
10.877
21.106
31.263
41.395
51.500
61.592
71.678
81.754
91.825
101.888
111.95
132.06
152.17
172.25
192.34
212.42

A pin twenty-one inches long between the line wire and the cross-arm will have a uniform strength to resist the pull of the wire if it has the diameter given in this table at the corresponding distances below the line wire. From this it follows that a pin of any length between wire and cross-arm corresponding to X in the table will be equally strong to resist a pull of the line wire as a standard 1.5-inch diameter pin with its wire five inches above the cross-arm. In other words, if a pin that is twenty-one inches long between the line wire and the cross-arm has the diameters given in the table at the corresponding distances below the wire, then a pin of equal strength to resist bending, and of any shorter length, would correspond in the part above the cross-arm to an equal length cut from the top end of the longer pin. Designating that part of a pin that is above the cross-arm as the “stem,” and that part in the cross-arm as the “shank,” each pin in the specifications under consideration is named by the length of its stem, as a 5-, 7- or 11-inch pin. It is proposed that each pin of whatever length be threaded for a distance of 2.5 inches at the top of its stem with four threads per inch, the sides of each thread being at an angle of ninety degrees with each other. Each thread is to cut into the pin about 3⁄32 inch, come to a sharp angle at the bottom, and be about 1⁄16 inch wide on top. At the end of the pin the proposed diameter over the thread is one inch in all cases, and at the lower end of the threaded portion the outside diameter is 1.25 inches. Near the end of the pin the diameter at the bottom of the thread is thus only 13⁄16 inch, and the corresponding diameter at the lower end of the threaded portion is about 11⁄16 inches on all pins. Each pin is to have a square shoulder to rest on the cross-arm, and the diameter of this shoulder is to be 3⁄8 inch greater than the nominal diameter of the shank of the pin. The proposed length of this shoulder on all pins is 1⁄4 inch before the taper begins. The actual diameter of the shank of each pin just below its shoulder is to be 1⁄32 inch less than the nominal diameter, and the actual diameter of the lower end of each shank is to be 1⁄16 inch less than the nominal diameter. With these explanations the proposed sizes of pins have dimensions as follows in inches:

Length
of
Stem.		Length
of
Shank.		Nominal
Diameter
of Shank.
541⁄411⁄2
741⁄413⁄4
941⁄417⁄8
1143⁄42
1343⁄421⁄8
1543⁄421⁄4
1753⁄423⁄8
1953⁄421⁄2

In order rightly to appreciate the utility of this table of proposed standard pins, it is necessary to have in mind the fact that all the dimensions are based on the assumption that a wooden pin with a shank of one and one-half inches diameter, and with its line wire attached five inches above the cross-arm, is strong enough for general use on transmission lines. Such an assumption covers a wide range of practice, but its truth may well be doubted for many cases. That this assumption does form the basis of the entire table is clearly shown by the fact that the calculated diameter at the shank of each pin is made to depend on a uniform pull, P, of the line wire, giving a uniform maximum stress, S, in the outer fibres of the wood just where the shank joins the stem. In other words, every pin in the table is designed to break with a uniform pull of the line wire, provided that the point on the insulator where the wire is attached is just on a level with the top of its pin in each case. It will at once occur to practical men that while a five-inch pin with one and one-half inch shank, or a larger pin of equal ability to resist the pull of a line wire, may be strong enough for the conductors of some transmission lines, this same pin may be entirely too weak for the longer spans, sharper angles, and heavier conductors of other lines.

Thus, on the sixty-five-mile line between Cañon Ferry and Butte, Mont., each conductor is of copper and has a cross section of 106,500 cm., while on the older line between Niagara Falls and Buffalo each copper conductor has a cross section of 350,000 cm. Evidently with equal conditions as to length of span, amount of sag, and sharpness of angles on these two lines, pins ample in strength for the smaller wire might be much too weak for the larger wire.

A little consideration will show that it is neither rational nor desirable to adopt pins of uniform strength for all transmission lines, but that several degrees of strength are necessary to correspond with the range in sizes of conductors in regular use. The size of pins for use on any transmission line, when the maximum bending strain exerted by the conductors has been determined, should be found by calculation and experiment, or by experiment alone. According to Trautwine, the average compressive strength of yellow locust is 9,800 pounds, of hickory 8,000 pounds, and of white oak 7,000 pounds per square inch in the direction of the grain. These compressive strengths are less than the tensile strengths of the same woods, and should therefore be employed in calculation, since the fibres on one side of a bending pin are compressed while the fibres on the other side are elongated. Substituting 1,000 for the value of S in the formula, S = P X.0982 D3, and also 5 for the value of X, and 11⁄2 for the value of D, the resulting value of P is found to be 736.5 pounds. This result shows that with a locust pin of 11⁄2 inches diameter at the shank, and with its line wire attached five inches above the shoulder, the unbalanced side pull of the wire that will break the pin by bending is 736 pounds, provided that the wood of the pin has a strength of 1,000 pounds per square inch in compression. As all of the proposed standard pins in the above table are designed for uniform strength to resist the same pull of a line wire attached on a level with the top of the pin in each case, it follows that the pull of 736 pounds by the wire will break any one of these pins under the conditions stated.

The calculation just made takes no account of the fact that the actual diameter of the shank of each pin just below the shoulder is 1⁄32 inch less than the nominal diameter, but this of course reduces the strength somewhat. Trautwine states that the figures above given for the compressive strengths of wood are only averages and are subject to much variation. Of course no pin should be knowingly loaded in regular practice to the breaking point, and to provide against variations in the strength of wood, and for unexpected strains, a liberal factor of safety, say four, should be adopted in fixing the maximum strains on insulator pins. Applying this factor to the calculations just made, it appears that the maximum pull of the line wire at the top of any one of the above proposed standard pins should not exceed 736 ÷ 4 = 184 pounds in regular work. A little calculation will readily show that the side pull of some of the larger conductors now in use on transmission lines will greatly exceed 184 pounds under conditions, as to sag, angles and wind pressure, that are frequently met in practice.

On page 448, Vol. xx., A. I. E. E., some tests are reported on six locust wood pins with shank diameters of 17⁄16 to 11⁄2 inches. Each of these pins was tested by inserting its shank in a hole of 11⁄2 inches diameter in a block of hard wood, and then applying a strain at about right angles to the pin and about 41⁄2 inches from the block by means of a Seller’s machine. The pull on each pin was applied gradually, and in most of the pins the fibres of the wood began to part when the side pull reached 700 to 750 pounds, though the maximum loads sustained were about ten per cent above these figures. The average calculated value of S, the compressive strength of the wood in these pins, was 11,130 pounds per square inch on the basis of the loads at which the fibres of the wood began to break, and 13,623 pounds per square inch for the loads at which the pins gave way. On pages 650 to 653 of the volume last cited, results are reported of tests on twenty-two pins of eucalyptus wood, which is generally used for this purpose in California. Twelve of these pins were of a size much used in California on lines where the voltage is not above 30,000. Each of the twelve pins was 67⁄8 inches long in the stem, 45⁄8 inches long in the shank, 11⁄2 inches in diameter at the shank, 2 inches in diameter at the square shoulder where the shank joins the stem, and 13⁄8 inches in diameter at the top of the thread. The pins were tested by mounting each of them in a cross-arm, securing the cross-arm in a testing machine so that the pin was horizontal, placing an insulator on the pin, and exerting the strain on a cable wrapped around the side groove of the insulator. This cable varied a little from right angles to the axis of each pin, but the component of the strain at right angles to this axis was calculated and the breaking load here mentioned is that component. Nearly all of these twelve pins broke square off at the cross-arm.

For a single pin, the lowest breaking strain was 705 pounds, the largest 1,360 pounds, and the average for the twelve pins was 1,085 pounds. Unfortunately, the exact distance of the cable from the cross-arm is not stated, but as the cable was wound about the side groove of the insulator it was probably either in line with or a little below the top of the pin. It seems probable also that the diameter of these pins at the shoulder—that is, two inches—may have increased the breaking strain somewhat by giving the shoulder a good bearing on the cross-arm. The ten other pins were of the size in use on the 60,000-volt line between Colgate power-house and Oakland, Cal. Each of these pins had a length of 53⁄8 inches and a maximum diameter of 21⁄8 inches in the shank, and a length of 103⁄8 inches in the stem, with a diameter of 21⁄2 inches at the shoulder. This shoulder was not square, but its surface formed an angle of forty-five degrees with the axis of the pin, and this bevel shoulder took up 1⁄4 inch of the length just given for the stem of the pin. At 21⁄2 inches from its threaded end the stem of the pin had a diameter of 115⁄16 inches, and the diameter slopes to 13⁄8 inches at two inches from the end. The two inches of length at the top of the stem has the uniform diameter of 13⁄8 inches, and is threaded with four threads per inch for the insulator. Each of these ten pins was tested, as already described, until it broke, but the break in this case started as a split at the lower end of the threaded portion and ran down the stem to the shoulder in a line nearly parallel with the axis of the pin. The pull on the cable at right angles to the axis of each pin had a maximum value of 1,475 pounds in one case, and a corresponding value of 3,190 pounds in another, while the average breaking strain for the ten pins was 2,310 pounds. Unfortunately, the report of this test above named does not distinctly state just how far the testing cable was attached above the shank of each of these large pins; but it seems probable that the same insulator was used with the larger as with the smaller pins, and if this was so the testing cable was attached near the end of each pin, as this cable was wound about the side groove of the insulator used on the smaller pins. With the types of insulator in actual use on the Colgate and Oakland line the wire is carried at the top groove and its centre is about two and a half inches above the top of the pin. It is therefore probable that these pins would not withstand as great strains on the lines as they did in these tests. The bevel shoulder on each of these larger pins no doubt increases its ability to resist a bending strain, because the bevel surface fits tightly down into a counterbore in the cross-arm. Where the pin has a shoulder at right angles with the axis, as is more usually the case, and the top of the cross-arm is a little rounding, the square shoulder does not have a firm seat and is of slight importance as far as the strength of the pin to resist a bending strain is concerned. Evidently the weakest point in the ten larger pins of this test was at the lower end of the threaded portion, since in each case the break was in the form of a long split starting where the thread ended. There seems to be no sufficient reason for the reduction of the diameter of a pin intended for a heavy line wire to a diameter as small as one inch at the threaded end, or for limiting the length of the threaded portion to 2.5 inches, as proposed in the specifications for standard pins. It is certain that the cost of the pin would be no more if its diameter at the threaded end were 11⁄4 or 13⁄8 inches with a uniform taper from the end of the pin down to the shoulder and with the thread cut down the stem for three or four inches. Furthermore, any increase in the cost of insulators for these larger threaded ends of pins would no doubt be a small matter. Some excess of strength in the stem of a pin over that of its shank is to be desired, for the stem is more exposed to the weather and to charring by leakage currents over the surface of the insulator. On high-voltage lines, this charring is usually worse at that part of each pin just below its thread, and the commonest breaks of pins on these lines leave the insulators with the threaded portions of their pins hanging on the wire, while the remainder of each pin remains on the cross-arm. From the tests just noted it is evidently poor design to give the threaded portion of a pin a short length of uniform diameter, and then to increase the diameter at once by a shoulder, as was done with the pins on the Colgate and Oakland line. This design evidently leads to failure of pins by splitting from the lower end of the threads. The better design is the more common one which gives the stem of the pin a uniform taper from the shoulder to the top. Where the line wire is secured to the top of its insulator, anywhere from one to three inches above the top of the pin, there is a strong tendency for the insulator to tip on its pin, and this tendency is more effectively met the longer the joint between the pin and insulator.