735 Riddle Road Cincinnati, Ohio 45220
The internal combustion generation hardly conceives that, not so long also, the topic of bursting pressure inspired raging debate. In the steam era, when boilers proliferated, tempers, like water, simmered on that subject. Knowledge of bursting pressure and the closely related concepts of working pressure and factors of safety mattered not only to designers of boilers and engines but also to investigators at explosion sites, insurance companies, implement dealers, and engineers. Equations for finding bursting pressure and working pressure abounded. Most formulas did not differ so much in substance as in degree of refinement; however, misunderstandings provoked controversy between one school of thought and another. In fact, to comprehend bursting pressure did require uncommon concentration. Descriptions in reputable mechanical engineering manuals and science textbooks from the late nineteenth and early twentieth centuries helped to unravel the mysteries of bursting pressure.
One such handbook, The Power Catechism, published by McGraw Hill in 1897, asked engineering students to imagine a ring like that depicted in Figure 1. If the ring were twelve inches in diameter and one inch wide (or, to be more exact, slong), and if it could contain an internal pressure of no more than one-hundred pounds to the square inch, the force tending to burst the ring would be 12 x 1 x 100, or 1,200 pounds. That force, thus, consists of the product found by multiplying the diameter, the length, and the pounds of pressure per square inch.
Now, a reasonable human being might think that the circumference, not the diameter, had better be used in the equation. After all, is not the internal pressure pushing against the whole ring, not merely an invisible diameter? Figure 2, however, depicts the physical law that pressure in any direction is effective on only those surfaces at right angles to that direction. Symbolized by the arrows, the pressure pushes against the ring and is resolved into the horizontal lines shown in the diagram. When placed end to end, these horizontal lines add up to a diameter.
A reasonable person might also consider it logical to assume that, if the pressure is, in effect, pulling a diameter in one direction, then an equal pressure must be pulling a diameter in the opposite direction. Figure 3 depicts such thinking. If this diagram represents the ring discussed earlier, then one arrow signifies a pressure of 1,200 pounds in one direction, and the other arrow indicates a pressure of 1,200 pounds in the opposite direction. A person might conclude that the bursting pressure is twice 1,200 pounds, or 2,400 pounds. The twin stresses of 1,200 pounds, though, were exerted against each other. As The Power Catechism points out, if two equally-strong people pull to the extent of their strength on opposite ends of a rope, the effect on the rope is the same as when one of the people pulls on the rope with its opposite end attached to a solid post. In the same way, one side of the ring pulls against the other side, and the tension in the ring is the force with which each side pulls not double the force. The bursting pressure for the ring, therefore, remains 1,200 pounds.
So far, the ring in these examples has been considered to have length (one inch) and diameter (twelve inches) but no thickness. Suppose it were one quarter of an inch thick. Further, assume it were made of boiler steel having a tensile strength of sixty thousand pounds per square inch. A bar of such steel one square inch in cross section breaks when a pull of sixty thousand pounds is applied in the direction of the bar's length. To tear the steel ring into two halves requires that it be torn apart at two points. Assume that, after the break, the area of each of these points will measure exactly one quarter of an inch by one inch. The area exposed by one of the breaks, then, will equal one-quarter of a square inch. The sum of the two areas will be one half of a square inch. To burst these two points apart will need a force equal to half of sixty thousand pounds per square inch, or thirty-thousand pounds. This force must be exerted on twelve square inches (the diameter of the ring multiplied by its length of one inch). The bursting pressure will be thirty thousand divided by twelve, or 2,500 pounds per square inch. The formula for this operation follows:
(Thickness x length x tensile strength x 2 breaks)/diameter x length = bursting pressure
Inserting the data makes the equation look like this:
(1/4 x 1 x 60,000 p.s.i. x 2)/I2x1 = 2,500 pounds per square inch
Because length appears in both the numerator and the denominator, it can be omitted. Also, dividing a diameter in two gives a radius; the 2 in the numerator no longer is needed when 'radius' is substituted for 'diameter.' The simplified equation now assumes this form:
(Thickness x tensile strength)/radius = bursting pressure
So long as the thickness and the radius are expressed in inches, this formula will yield the bursting pressure in pounds per square inch of a cylinder. A careful examination of this formula shows that, with each increase in the radius, a cylinder becomes weaker. If the thickness of a cylinder were one inch and the tensile strength were sixty-thousand pounds per square inch, a cylinder of a two inch radius would burst at thirty-thousand p.s.i., whereas a cylinder with a radius of four inches would burst at fifteen-thousand p.s.i.
Now enters the factor of safety, which is the number by which the bursting pressure is divided to calculate the allowable working pressure. In the steam era, factors of safety differed from year to year, from manufacturer to manufacturer, and from model to model of boiler and engine. Like bursting pressure the concept of a factor of safety led to confusion and arguments.
If a builder were to allow a factor of safety of six, that would not mean that the bursting pressure of a cylinder made of boilerplate could be divided by six to get the safe pressure at which the boiler and engine might be worked. A boiler takes a shape far more complicated than that of a simple cylinder and has seams which subtract from the strength of the plate itself. A glance at Figure 4 will suffice to show how complex is the shape of the space occupied by water and steam in a locomotive styled boiler. (The schematic diagram of a boiler in Figure 4 is not intended to resemble that of a particular builder and does not include such parts as the tubes, the fire door, the grates, the smoke box, and the smokestack.) The gray areas reveal how water flows beyond the cylindrical form to cover the top, sides, and in this case bottom of the firebox. Above the level of the water, steam fills the space including the dome. Not shown are the riveted seams, which hold together all of these shapes, the braces, which support flat surfaces at various angles, and the numerous stay bolts, which lend rigidity to the overall structure. The equation for the bursting pressure of a simple cylinder cannot serve to indicate what would be an appropriate factor of safety or the best working pressure for a locomotive boiler. Nor does the cylinder formula apply to a return flue boiler, wherein the large central flue effectively changes the shape to that of a cylinder within a cylinder.
Engineers in the design department of a boiler manufacturer had to wield numerous theories and equations. In their efforts to find the most accurate means of determining bursting pressure, authorities disagreed. After all, their task was complicated. They had to calculate precisely the relative strengths of various forms of lap seam boilers and of butt strap boilers. Knowing that the strongest joint results when the tensile strength of the sheet between the rivet holes equals the shearing strength of the rivets, engineers invented several configurations of seams, from lap seams with one, two, or three rows of rivets to but strap joints with one or two straps and with as many as six rows of rivets (see Figure 5 for front and side views of a butt strap joint). The Power Catechism cautions that any seam constitutes a delicate balance: 'If you give the rivets more pitch in order to increase the sheet section, you lessen the number of rivets, increase the stress on each, and the joint will fail by shearing. If you put in more rivets, you reduce the section of plate between them, and the plate will pull apart. You can only gain strength in one factor by sacrificing strength in the other, and the greatest strength will obviously be when both are equal' (41). To determine the proper pitch, or distance between the centers of the rivets, designers of boilers often used a five step formula: (A) multiply the number of rows of rivets by the cross sectional area of the rivet hole; (B) multiply this product by the shearing strength of the rivet; (C) multiply the thickness of the plate by its tensile strength; (D) divide the product obtained in step B by the products obtained in step C, and (E) add the diameter of the rivet to the quotient obtained after dividing. Engineers understood that even this detailed method of deriving the pitch would not apply in every situation.
Designers commonly used averages to simplify the mathematics involved in building boilers. Such shortcuts included the application of standard percentages of efficiency assigned to certain configurations of seam used with specified thicknesses of boilerplate and with particular riveting methods. For example, a butt-joint with double straps, as compared to a seamless sheet of boilerplate, might be assigned an efficiency of 87.5 percent. This seam-efficiency percentage, then, could reduce the complexity of calculating the bursting pressure and, hence, the factor of safety and the working pressure. In his Steam Engine Guide (published in 1910 by the American Thresher-man Company in Madison, Wisconsin), Phillip S. Rose uses such percentages in a rule for determining the safe working pressure of a cylindrical boiler: 'Multiply twice the thickness of the boiler plate expressed in inches by the ultimate tensile strength of the plate and this product by the efficiency of the joint in percent. Divide this product by five hundred times the diameter of the boiler, and the result is the safe working pressure for a new boiler' (29). Again, such an equation applies only to a cylinder, not to the complicated shapes of most boilers.
While figuring bursting pressures, mechanical engineers also had to recognize that the stress on a longitudinal seam exceeds that on roundabout seams. For example, if a tube sheet were solid, its entire circumference would resist the boiler's pressure per square inch multiplied by the area, or the number of square inches in that tube sheet. The area is found by this rule:
Diameter2 x .7854
The circumference is found by this method:
Diameter x 3.1416
For each inch of the roundabout seam, the resistance is found to be in this proportion:
(Diameter2 x .7854)/Diameter Diameter x 3.1416 = Diameter/4
The stress on each inch of the boiler's length is in this ratio: Diameter/2
The stress on a longitudinal seam, therefore, is twice that on a roundabout seam, or, to put the same idea another way, a roundabout seam is twice as strong as a longitudinal seam (all other matters being equal). This observation, coupled with the fact that cylinders having larger diameters will burst sooner than those with smaller diameters, helps to explain why, with no increase in the thickness of the boilerplate, larger and larger diameters of boilers are weaker and weaker. The comparative weakness of the longitudinal seam partly accounts for the need to thicken the plate in accordance with each increase in the diameter of the boiler.
As if these calculations were not sufficiently daunting, Figure 4 plainly exhibits the difficulty of determining the forces at work in the water and steam spaces surrounding the firebox. Designers asked themselves what is the effect of the tube sheet holes which accept the tubes, does this factor change once the tubes are installed, what kind of supports and how many are needed to hold the crown sheet, how should the firebox walls be strengthened, what can be done to add durability to the upper portion of the front tube sheet, how can the dome and waist be made strong, and similar questions.
One rule of thumb for figuring the amount of stress permitted on diagonal braces was to set the limit at six thousand pounds per square inch of sectional area of a given brace. The allowance on such a brace of a given diameter could be determined by multiplying six-thousand by the square of the diameter and by .7854.
Next, to find the approximate stress which each stay bolt could sustain, the engineer multiplied the pressure per square inch by the area in square inches bounded by lines drawn between a given stay bolt and its neighbors. Figure 6 shows such lines between stay bolts, A, B, C, and D in the diagram at the left. To the right, a side view of the subject reveals how the stay bolts support parallel surfaces. If the distance between A and B equals eight inches, and if the distance between B and D equals eight inches, then the area is sixty-four square inches. Multiplying sixty four by the pressure per square inch gives the approximate stress on any of these stay bolts. For designers to determine the exact stress on a given stay bolt demanded detailed knowledge of engineering and of mathematics. To derive the diameter of a stay bolt, the designer could begin by dividing the approximate stress by six-thousand (the allowable working pressure on stay bolts). For example, if a stay bolt were supporting sixty four square inches at 150 pounds per square inch, the pressure would equal 9,600 pounds. Dividing this figure by six thousand yields 1.6 of an inch. Dividing 1.6 by .7854 yields 2. The square root of 2 is 1.4. The diameter of the required stay bolt, thus, is 1.4 of an inch. The best size of stay bolt to use would round up to a diameter of one and a half inches.
Calculating the dynamic effects of pressure on a flat surface required sophisticated computations. Rose hints at this complexity. 'The tendency of pressure on the inside of any structure is to form that structure into the shape of a sphere. If the walls of the structure are flat they are subjected to a cross bending strain as well as a tensile strain. A flat plate in a boiler must be treated as a beam under a uniform load, and supported at the ends' (30). Rose adds that a complete treatment of the problem is beyond the scope of his book. He gives a rule for approximating the allowable working pressure for plates seven sixteenths of an inch and under. ' 'Multiply the square of the thickness of the plate in sixteenths of an inch, by 112 and divide this product by the square of the distance from center to center of the stay bolts.' For thicker plates, 120 can be used in place of 112.
Regardless of the level of complication, figuring how to support flat, rectangular surfaces was a snap, compared to determining how to brace flat surfaces having curved perimeters, such as the segment of the circle formed in the front tube sheet (or head) above the tubes. Figure 7 depicts this area. Tables giving the area of circular segments for a wide range of heights enabled designers to simplify their work, but they still confronted a challenge in knowing exactly where to place stay bolts for maximum effect in a surface bounded by a curve. An even greater problem arose in determining how to brace curving surfaces.
Figure 8 offers a schematic illustration of a Case 110 horsepower boiler. (Measurements are approximate.) Double thickness of boilerplate in key areas, numerous stay bolts, and braces reveal the genius of the designers. A comparison of Figure 4 and Figure 6 suggests the amount of stay s bolts and braces which mechanical engineers deemed necessary for ensuring strength in a locomotive styled boiler.
No wonder the topic of bursting pressure invoked controversy in the steam era! The calculation of bursting pressure, appropriate factors of safety, and working pressure challenged designing engineers. In his Encyclopedia of American Steam Traction Engines, Jack Norbeck writes, 'One weakness of the Scheidler engine was the support of the crown sheet. A lifetime engine man once remarked that no one in his right mind would buy a Scheidler, since there were always around the factory several old Scheidler boilers every one of which had blown down in the crown sheet' (247). Norbeck explains that Reinhardt Scheidler did not support his crown sheets in the customary way. Norbeck continues, 'On the afternoon of April 29, 1903, an engine under test in the factory blew down her crown sheet, killing Scheidler instantly and injuring several workmen near the engine.' Mechanical engineers needed to determine bursting pressure as accurately as possible. Then as now, the difference between a good design and a great design can spell the difference between death and life.