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:

Diameter^{2} 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:

(Diameter^{2} 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.