The stress induced in tin' sustaining part, whether tensile, compressive, or torsional, is caused by the application of forces, either acting directly without leverage, or with leverage'in the production of moments.
The forces applied from external sources are at constant war with the resisting forces due to the strength of the fibres of the material composing the machine members. The moments of the external forces are constantly exerted against and balanced by the momeuts of the internal resistance of the material. Hence, design, from a strength standpoint, is merely a balancing of internal strength against external force. In other words, we may in all cases write a sign of equality, place the applied effort on one side, the effective resistance on the other, and we shall have an equation, which, if capable of solution, will give the proper proportions of the parts considered.
External Force = Internal Resistance. External Moment - Internal Moment of Resistance. Expressed in terms of the "Mechanics:"
P = AS (1)
B or T = SI/C (2)
In these formulas, which are perfectly general.
P = direct loud in pounds.
A = area of effective material, in square inches.
S = workig fibre stress of the material (tensile, compressive, or shearings, in pounds per square inch. external moment (bendings or torsional), in inch- pounds., direct or polar), of the resisting section moto fibre of the resisting section from the P may produce direct tensile, compressive, or shearing stress.
B may produce tensile or compressive stress, and requires use of direct moment of inertia in either case.
T produces shearing stress, and requires use of polar moment of inertia.
The origin of formula (1) is obvious, the assumption being that the fibre stress So equally distributed to every particle in the area "A." .
The development of formula (2) is given in any text-book in Mechanics. It requires the aid of the Calculus, however. Any good handbook gives values for both the direct moment of inertia and the polar moment of inertia for quite a large variety of sections, bo that further reference is an easy matter for the student. These values are also obtained through the methods of the Calculus.
The reason for introducing these formulas at this time is to call the attention of the student especially to the fact of their universal and fundamental use in all problems concerning the strength of machine parts. Nearly every computation may be reduced to or expanded from these two simple equations. Many complex combinations occur, of course, which will not permit simple and direct application of these formulas, but the student will do well to place himself in perfect command of these two. Assuming that he is able to analyze forces, and compute the simple moment at the point where he wishes to find the strength of section, the rest is the mere insertion of the assumed working fibre stress of the material in the formula (2) above, and solution for the quantity desired.
When the case is one of combined stress, the relation becomes more complicated and difficult of analysis and solution. The most common case is where bending is combined with torsion, as in the case of a shaft transmitting power, and at the same time loaded transversely between bearings. In fact there are very few cases of shafts in machines, which, at some part of their length, do not have this combined stress. In this case the method of procdure is to find the simple bending moment and the si moment separtely, in the. ordinary elasticity furninished us with a formulas or an epuivalentt torsional moment the same effect upon the fibre of the action of the two simple moments acting together. In other words, the separate moments combined in action, being impossible of solution in that form, are reduced to an equivalent simple moment and the solution then becomes the same as for the previous case.
MACHINE FOR PUNCHING OR SHEARING. Showing Motor and Genring.
The Long and Allstatter Co.
These equivalent equations are given below, the subscript "e" being added to express separation from the simple moment:
Be = B/2 +½ √ B2+T2 (3) Te= B +√ B2 + T2 (4)
Bc and Te, found from these equations, are the external moments, and are to be equated to the internal moments of resistance of the section precisely as if they were simple bending or torsional moments. Either may be used. For shafts (4) is generally used, being the simpler of the two in form.