Machine design - Fastenings
MACHINE DESIGN - FASTENINGS
BY WILLIAM LEDYARD CATHCART
NEW YORK, D. VAN NOSTRAND COMPANY, 1903
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Machine design - Fastenings
PREFACE
The main purpose of this book is to present, in compact form for the use of the student and designer, modern American data from the best practice in the branch of Machine Design to which the work refers. The theoretical treatment of the subject has also been given fully; but this has been done for completeness only, since that field has been covered exhaustively by able writers. Scientific analysis and the records of practice are both essential to success in the design of machine members, but neither alone is trustworthy. The former predicts only those stresses which prevail under normal conditions arid ignores the overload, the rough handling, or the slight accident which the machine may meet and against which it should not fail. Practical data, on the other hand, show only the proportions which constructors have given in specific cases of stress and service and empirical formulae founded upon them may give results wide of the mark, if the inherent limitations of these formulae be exceeded. The problem of design is one whose many elements vary continually in number, character, and magnitude, and, for its solution, theoretical analysis, precedent, and the ripened judgment of the designer are required.
CONTENTS
- SHRINKAGE AND PRESSURE JOINTS
- SCREW FASTENINGS
- RIVETED JOINTS: THEORY AND FORMULAE
- RIVETED JOINTS: TESTS AND DATA FROM PRACTICE
- KEYED JOINTS: PIN-JOINTS
SCREW FASTENINGS.
A screw-surface or helicoid is described by a right line revolving about and advancing along an axis, as directrix, one extremity, of the line remaining upon the axis and the angle of advance, between the latter and the line or generatrix being constant.
The nominal, or outer diameter, D, of a screw is that of the outside or top of the thread. The effective diameter, d, is that of the base or root of the thread and of the cylinder or core upon which the latter is described. The depth of the thread is the radial distance between its base and top. The pitch, p, is the axial distance between adjacent convolutions of the same thread, i. e., the axial distance which the nut traverses during one revolution. The pitch-angle, d, of any helix of the thread, is the inclination between that helix and a plane perpendicular to the axis of the cylinder. While, in a normal screw, the pitch of all helices is the same, the pitch-angle of each depends upon its diameter. Calculations with regard to stresses within the thread are referred to the mean thread-diameter, d , at which all forces are assumed to be concentrated. This diameter may be taken also, with sufficient accuracy, as that of the mean helix, equally distant from the helices at base and top of thread. The projected area of the thread is used in computations for bearing pressure.
In addition to differences in the forms of the threads, screws are distinguished further as right or left-handed and single or multiple threaded. In a right-handed screw, the thread ascends contra-clockwise from left to right. Screw-fastenings have usually a single, right-handed, approximately triangular thread. A multiple (double, triple) threaded screw is one in which the cylinder is traversed by two or more threads, parallel and similar in all respects. Such screws, having ample bearing surface, are used for the transmission of power.
The screw and its nut form, kinematically, a pair; the relative motion of whose two elements consists of rotation about an axis and translation along the latter. If the material of the nut be relatively inelastic, as metal, the requirement for motion as above, is that the ratio between translation and rotation shall be constant, i. e., that there shall be uniform pitch. When, however, the screw revolves in a mobile medium or nut, as water, its surface may have a varying pitch throughout. The marine propeller is a transverse section of a multiple-threaded screw, the pitch of whose blade-surface may be either constant or expand in either or both of two ways radially outward or from the leading to the following edge of the blade.
The form of the thread is determined by the character of its service. The more important differences between the square thread and the full or modified triangular type lie in the relative strengths of these forms and the friction of operation. The load in a bolt is usually axial. It is transmitted to the bolt-thread and supported by the reaction of the nut-thread. The load-action and the nut-reaction must be, for equilibrium, equal. These mutual actions are, disregarding friction, normal to the contact surfaces, i. e., to the threads. Considering friction, the reactions are diverted from the normal by the angle of friction.
1. FRICTION. This is directly proportional to the normal pressure upon the contact-surfaces.
2. STRENGTH. In the triangular thread, the section at the root is the full length of the nut, while, in the square form, the section is but one half this length. Against shearing and flexure at the root, the latter thread is, therefore, proportionately the weaker.
3. NUT. As noted, the triangular type has a bursting action upon the nut, which action, disregarding friction, does not exist with the square thread.
In general, the triangular form is more suitable for screw-fastenings, owing to its greater strength, its increased frictional holding power which prevents backing off under load, and the finer pitches permissible by the full section at the base of the thread. On the other hand, the square thread is better adapted for power-transmission, since it has less friction and its bursting effect upon the nut is so small as to be negligible.
Requirements of the Screw-Thread.
The screw is used as a detachable fastening in joining the members of a structure or machine; in producing pressure or tension, as in the screw-jack and testing-machine ; and for the transmission of power and conversion of motion, as in the worm-gear and screw propeller. Its requirements for these uses are :
1. POWER. This depends upon the pitch and form. The effect of the latter upon the strength and power of thread has been discussed. With a given applied force, the less the pitch, the greater the axial load may be, since the pitch fixes the angle of the inclined plane upon which the load virtually moves.
2. STRENGTH. This is governed by the pitch, form and depth of the thread. With constant load, the steeper the pitch, the greater must be the applied power and the consequent normal pressure upon the thread. For the same load and nominal diameter, the deeper the thread, the less its mean bearing-pressure will be; but the moment of the load upon the root will be larger and the effective diameter of the bolt to resist tension, will be reduced.
3. DURABILITY. The most durable thread is one whose form produces the least friction, whose depth gives minimum bearing pressure, and which is most accurately fitted.
The requirements of the screw-thread make its elements inter-
dependent. Consider:
1. EFFECTIVE DIAMETER. This depends upon the axial load and the torsional stress produced by friction between the threads in setting up the nut. The magnitude of the latter stress is governed by the applied power, and that of the power by the axial load and pitch.
2. PITCH. The relations between pitch and diameter in the prevailing systems of screw-threads are the outcome less of logical analysis than of long experience. For screw-fastenings, the limit in one direction lies in the fact that, with an excessively coarse pitch, the depth will be too great and the effective diameter will be reduced unduly. Again, that component of the pressure which is parallel to the thread -surfaces will exceed the force of friction between the latter, and, owing to this excess, the nut will back off. On the other hand, with an unduly small pitch-angle, the surface-friction will form too large a proportion of the total work of setting up the nut, the torsional action upon the bolt will be excessive, and the latter may be sheared. In general, fine pitches are unsuitable for soft metals and coarse pitches for shallow holes.
3. FORM. As stated, the square thread is the form best adapted for power-transmission. For large fastenings requiring to be read- ily and frequently removed and which are strained heavily, but in one direction only, as the breech-block of a gun, the trapezoidal thread is most suitable. This thread has the acting face normal to the axis, the rear face at an angle thereto, and combines the greatest strength and least friction attainable.
For screw-fastenings in general, the triangular thread, with blunt top, straight sides, and filled-in base-angle, was adopted through various considerations with regard to strength, friction, durability, ease of manufacture, and conformity with general practice. Thus, in strength and frictional holding power, this form is superior; its straight sides give even wear and maximum bearing surface; the angle between them is fixed, in the various systems, by compromises between the conditions as to strength, friction, bursting action upon the nut, and facility of verification and production; the flat or rounded top reduces the liability to injury; and the filling in of the reentrant base angle increases the effective diameter of the bolt and, in the Seller's system, the resilience of the latter also.
4. NUT. The nut may yield either by the shearing or rupture of its threads or by bursting from the action of the outward component of the pressure upon the thread. The latter, both on bolt and nut, acts as a cantilever beam, fixed at the root and loaded uniformly over the bearing surface. When worn, the area of the latter is reduced, the bearing becomes irregular, the load is practically concentrated, and the bending moment at the root may be increased. If the nut is of a metal materially weaker than that of the bolt, its depth should be greater than the normal. In any event, this depth should be sufficient to give ample strength against flexure and shear at the root of the thread, to provide sufficient bearing surface to prevent abrasion, and to afford a good hold for the wrench.
5. MULTIPLE THREADS. In power-transmission screws of large pitch, a single thread will provide adequate bearing surface only by having a depth so great as to give an unduly small effective diameter of bolt. When the pitch is sufficient to permit it, the use of two or more parallel threads of usual proportions will secure the required surface with a normal effective diameter. Such threads are usually of square or trapezoidal form.
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The nominal, or outer diameter, D, of a screw is that of the outside or top of the thread. The effective diameter, d, is that of the base or root of the thread and of the cylinder or core upon which the latter is described. The depth of the thread is the radial distance between its base and top. The pitch, p, is the axial distance between adjacent convolutions of the same thread, i. e., the axial distance which the nut traverses during one revolution. The pitch-angle, d, of any helix of the thread, is the inclination between that helix and a plane perpendicular to the axis of the cylinder. While, in a normal screw, the pitch of all helices is the same, the pitch-angle of each depends upon its diameter. Calculations with regard to stresses within the thread are referred to the mean thread-diameter, d , at which all forces are assumed to be concentrated. This diameter may be taken also, with sufficient accuracy, as that of the mean helix, equally distant from the helices at base and top of thread. The projected area of the thread is used in computations for bearing pressure.
In addition to differences in the forms of the threads, screws are distinguished further as right or left-handed and single or multiple threaded. In a right-handed screw, the thread ascends contra-clockwise from left to right. Screw-fastenings have usually a single, right-handed, approximately triangular thread. A multiple (double, triple) threaded screw is one in which the cylinder is traversed by two or more threads, parallel and similar in all respects. Such screws, having ample bearing surface, are used for the transmission of power.
The screw and its nut form, kinematically, a pair; the relative motion of whose two elements consists of rotation about an axis and translation along the latter. If the material of the nut be relatively inelastic, as metal, the requirement for motion as above, is that the ratio between translation and rotation shall be constant, i. e., that there shall be uniform pitch. When, however, the screw revolves in a mobile medium or nut, as water, its surface may have a varying pitch throughout. The marine propeller is a transverse section of a multiple-threaded screw, the pitch of whose blade-surface may be either constant or expand in either or both of two ways radially outward or from the leading to the following edge of the blade.
The form of the thread is determined by the character of its service. The more important differences between the square thread and the full or modified triangular type lie in the relative strengths of these forms and the friction of operation. The load in a bolt is usually axial. It is transmitted to the bolt-thread and supported by the reaction of the nut-thread. The load-action and the nut-reaction must be, for equilibrium, equal. These mutual actions are, disregarding friction, normal to the contact surfaces, i. e., to the threads. Considering friction, the reactions are diverted from the normal by the angle of friction.
1. FRICTION. This is directly proportional to the normal pressure upon the contact-surfaces.
2. STRENGTH. In the triangular thread, the section at the root is the full length of the nut, while, in the square form, the section is but one half this length. Against shearing and flexure at the root, the latter thread is, therefore, proportionately the weaker.
3. NUT. As noted, the triangular type has a bursting action upon the nut, which action, disregarding friction, does not exist with the square thread.
In general, the triangular form is more suitable for screw-fastenings, owing to its greater strength, its increased frictional holding power which prevents backing off under load, and the finer pitches permissible by the full section at the base of the thread. On the other hand, the square thread is better adapted for power-transmission, since it has less friction and its bursting effect upon the nut is so small as to be negligible.
Requirements of the Screw-Thread.
The screw is used as a detachable fastening in joining the members of a structure or machine; in producing pressure or tension, as in the screw-jack and testing-machine ; and for the transmission of power and conversion of motion, as in the worm-gear and screw propeller. Its requirements for these uses are :
1. POWER. This depends upon the pitch and form. The effect of the latter upon the strength and power of thread has been discussed. With a given applied force, the less the pitch, the greater the axial load may be, since the pitch fixes the angle of the inclined plane upon which the load virtually moves.
2. STRENGTH. This is governed by the pitch, form and depth of the thread. With constant load, the steeper the pitch, the greater must be the applied power and the consequent normal pressure upon the thread. For the same load and nominal diameter, the deeper the thread, the less its mean bearing-pressure will be; but the moment of the load upon the root will be larger and the effective diameter of the bolt to resist tension, will be reduced.
3. DURABILITY. The most durable thread is one whose form produces the least friction, whose depth gives minimum bearing pressure, and which is most accurately fitted.
The requirements of the screw-thread make its elements inter-
dependent. Consider:
1. EFFECTIVE DIAMETER. This depends upon the axial load and the torsional stress produced by friction between the threads in setting up the nut. The magnitude of the latter stress is governed by the applied power, and that of the power by the axial load and pitch.
2. PITCH. The relations between pitch and diameter in the prevailing systems of screw-threads are the outcome less of logical analysis than of long experience. For screw-fastenings, the limit in one direction lies in the fact that, with an excessively coarse pitch, the depth will be too great and the effective diameter will be reduced unduly. Again, that component of the pressure which is parallel to the thread -surfaces will exceed the force of friction between the latter, and, owing to this excess, the nut will back off. On the other hand, with an unduly small pitch-angle, the surface-friction will form too large a proportion of the total work of setting up the nut, the torsional action upon the bolt will be excessive, and the latter may be sheared. In general, fine pitches are unsuitable for soft metals and coarse pitches for shallow holes.
3. FORM. As stated, the square thread is the form best adapted for power-transmission. For large fastenings requiring to be read- ily and frequently removed and which are strained heavily, but in one direction only, as the breech-block of a gun, the trapezoidal thread is most suitable. This thread has the acting face normal to the axis, the rear face at an angle thereto, and combines the greatest strength and least friction attainable.
For screw-fastenings in general, the triangular thread, with blunt top, straight sides, and filled-in base-angle, was adopted through various considerations with regard to strength, friction, durability, ease of manufacture, and conformity with general practice. Thus, in strength and frictional holding power, this form is superior; its straight sides give even wear and maximum bearing surface; the angle between them is fixed, in the various systems, by compromises between the conditions as to strength, friction, bursting action upon the nut, and facility of verification and production; the flat or rounded top reduces the liability to injury; and the filling in of the reentrant base angle increases the effective diameter of the bolt and, in the Seller's system, the resilience of the latter also.
4. NUT. The nut may yield either by the shearing or rupture of its threads or by bursting from the action of the outward component of the pressure upon the thread. The latter, both on bolt and nut, acts as a cantilever beam, fixed at the root and loaded uniformly over the bearing surface. When worn, the area of the latter is reduced, the bearing becomes irregular, the load is practically concentrated, and the bending moment at the root may be increased. If the nut is of a metal materially weaker than that of the bolt, its depth should be greater than the normal. In any event, this depth should be sufficient to give ample strength against flexure and shear at the root of the thread, to provide sufficient bearing surface to prevent abrasion, and to afford a good hold for the wrench.
5. MULTIPLE THREADS. In power-transmission screws of large pitch, a single thread will provide adequate bearing surface only by having a depth so great as to give an unduly small effective diameter of bolt. When the pitch is sufficient to permit it, the use of two or more parallel threads of usual proportions will secure the required surface with a normal effective diameter. Such threads are usually of square or trapezoidal form.
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