Problems in strength of materials

For the average student to obtain a working knowledge of any scientific subject it is necessary that he solve numerous problems. This is especially true in the study of mechanics. In teaching Strength of Materials the author has found that the text-books do not give a sufficient number of examples to completely familiarize the student with the application of the theory.

The aim of this book is to furnish a large variety of problems on each part of the subject, and thus relieve the instructor of tedious dictation in the class room.

A discussion of riveted joints is given for use in the computation and design of such joints as are often found in boiler construction.

No definite notation is adopted in order that the book may be used in connection with a course of lectures or with any text-book on the subject.

Tables at the back of the book give all the data necessary for solving the problems, but answers have been omitted in order to emphasize that the goal is a proper solution and not a mere numerical answer.

I wish to thank Professor C. B. Eichards for suggesting numerous examples and for other valuable assistance in compiling this book.

In structural work, as in girders, trusses, etc., and in many forms of receptacles, such as tanks, the shells of steam boilers, etc., composed of plates, the plates are joined together by riveted joints.

When the plates are in tension the rivets transfer the tension from one plate to another. This brings a stress upon each rivet, which tends to shear it across in the plane of the surfaces of contact of the plates. A compressive stress is also brought upon the rounded surface of the rivet, where it bears upon the plate, which tends to crush it against the metal of the plate in front of the rivet. This is called a bearing stress, and the exact manner in which this stress acts between the cylindrical surface of the rivet and the hole in the plate through which the rivet passes is not known. Experiment and experience, however, show that for our computations we may suppose this stress to be uniformly distributed over an area which is the projection of the curved surface of the rivet hole up on a plane through its axis. We then compute for this projected area a working unit-stress whose safe value has been determined by experiment.

The general discussion of riveted joints covers their use in all kinds of structures, but we shall limit our attention to their use in uniting plates of pipes and shells which are subjected to internal fluid pressure, and have to be designed for tightness as well as strength. The special case is that of cylindrical boiler shells.

In connecting the plates, the rivets may be arranged in many different ways, but in general they are distributed in rows extending parallel to the edges of the plates that are joined, as is shown in the diagrams of a few forms of joints (see Figs. 3-9).

In each single row the rivets are spaced uniformly, although the uniform spacing in one row may be different from that in another row. The uniform spacing, measured from the center of one rivet to the center of the next one in the same row, parallel to the edge of the plate, and in the row in which the rivets are most widely spaced, is called the pitch.

By examining the diagrams it can be seen that there is in every case a repeating uniformity in the grouping of the rivets along the joint, so that the joint may be divided by lines perpendicular to the edge, into sections which are in every respect alike. These are called repeating sections, and in computing the strength of the joint we may compute the strength of one repeating section and assume that the strength of the whole joint is that of the aggregate of all such sections.

The width of a repeating section will be denoted by p, the thickness of the plate by t, and the diameter of the rivet holes by d.

The diameter of the rivet hole is taken instead of the original diameter of the cold rivet, because the rivet, when properly driven and headed, completely fills the hole, the size of which therefore determines the effective diameter of the driven rivet. The cold rivet is usually about 1/2 of an inch smaller than the hole, so that when heated red hot it may be easily and quickly inserted.

A riveted joint may fail in one of several ways.

1. The rivets may be sheared, as shown in Pig. 10.

2. The plate in front of the rivet may be sheared out, as in a of Fig. 11.

3. The plate may crush in front of the rivet, as in I or c of Fig. 11.
 
4 The plate may break along the rivet holes, as in d, or along lines from the center of a rivet in one row to the center of the next rivet in the adjacent row, as in e of Fig. 11. Experiments have shown that unless the bearing stress be excessive there is no danger of the joint failing in the manner of 2 or 3, if the " margin," that is, the distance between the edge of the rivet hole and the edge of the plate, be made sufficiently great. It should be made at least as great as d.

In a butt joint the main plates do not overlap, but cover plates are used to connect them. When tension is applied to the main plates of a butt joint having two cover plates, one half of this applied tension is transferred to each cover plate. Hence, theoretically, the thickness of each cover plate should be one half that of the main plate; but the cover plates, or straps, must be thick enough to remain tight against leakage arising from their flexure between the rivets, and so thick that their edges will admit of effective calking. It is customary, therefore, to make the thickness of each cover plate about five eighths that of the main plates. In the case of a single-strap joint, in which the strap is subject to a bending stress as well as to stress from calking, the strap is made 1J times the thickness of the main plates. Single-strap joints ought not to be used for the seams of boiler shells.

In a butt joint with single or double riveting, there are twice as many rivet sections to be sheared in a repeating section as in the corresponding case for a lap joint. Hence the strength of the joint against shearing the rivets is twice as great. The effective rivet-bearing surfaces in a butt joint are those surfaces only which are in front of the rivets where they pass through the main plate, and their number, therefore, is equal to the number of rivets in one of the main plates in the repeating section, one surface for each rivet. It must be recognized that in butt joints the number of rivets to be considered in a repeating section is the number on one side only of the line of separation of the mam plates. Thus, in Figs. 6 and 7, only one rivet can be considered, in Fig. 8 two rivets are taken, and in Fig. 9 five rivets.