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*WORKS OF PROFESSOR MERRIMAN
PUBLISHED BY
JOHN WILEY & SONS
Treatise on Hydraulics. 8vo, 593 pages,	$5.00
Mechanics of Materials. 8vo, 518 pages,	$5.00
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By Professors MERRIMAN and JACOBY TEXT-BOOK ON ROOFS AND BRIDGES;
Part I. Stresses. 8vo, 326 pages,	$2.50
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By Professors MERRIMAN and BROOKS
Handbook for Surveyors. 121110, 246 pages.	$2.00
Edited by Professors MERRIMAN and WOODWARD
Series of Mathematical Monographs.
Eleven Volumes, 8vo.
Each, $1.00STRENGTH OF MATERIALS.
A TEXT-BOOK
FOR
MANUAL TRAINING SCHOOLS
BY
MANSFIELD MERRIMAN,
Membïii of International Association for Testing Materials.
FIFTH EDITION, ENLARGED. TOTAL ISSUE, ELEVEN THOUSAND.
NEW YORK:
JOHN WILEY & SONS
London : CHAPMAN & HALL, Limited.Copyright, 1897, 1906,
BY
MANSFIELD MERRIMAN.
First Edition, November, 1897. Reprinted, i8y8. Second Edition, December, 1898. Reprinted, 1899, Third Edition. November, iyoo. Reprinted, iqoi, 1903, Fourth Edition, Revised, October, 1905.
Fifth Edition, Enlarged, September, 1906. Reprinted 1907, 1908.
ROBERT DRUMMOND. PRINTER, NEW YORK.PREFACE.
In the following pages the attempt is made to give a presentation of the subject of the strength of materia^, beams, columns, and shafts, which may be understood by those not acquainted with the calculus. The degree of mathematical preparation required is merely that now given in high schools, and includes only arithmetic, algebra, geometry, and such a course in mechanics as is found in elementary works on physics. In particular the author has had in mind the students in the higher classes of manual training schools, and it has been his aim to present the subject in such an elementary manner that it may be readily comprehended by them and at the same time cover all the essential principles and methods.
As the title implies the book deals mainly with questions of strength, the subject of elastic deformations occupying a subordinate place. As the deductions of the deflections of beams are best made by the calculus they are not here attempted, but the results are stated so that the student may learn their uses; later, if he continues the study of engineering, his appreciation of the proofs that he will then read will be accompanied with true scientific interest.
All the rules for the investigation and design of common beams, including the subject of moment of inertia, are here presented by simple algebraic and geometric
34
PREFACE.
methods. No Greek letters are used, and algebraic operations are made âs simple as possible. As the mechanical ideas involved are by far the most difficult paft flfl the subject, a Special effort has been made to clearly present them, and to illustrate them by numerous pradnpS^ numerical examples.
A chapter dtilrthe manufacture and general properties of matô£|&ÎS 0' given, as also one on resilience and ■impact. Problems for students to solve are ptCSShled, Ifld ifcèbbùldIbe strongly insisted upon that these should pe thoroughly and completely worked out. It is indeed only by the Solution of many numerical exercises that % good knowledge of the theory of the subject can be acquired.
NOTE TO THE FIFTH EDITION.
To this edition a new chapter has been added which deals with reinforced concrete, especially with columns and beams ; in this the attempt has been made to clearly set forth the laws of distribution of the stresses between the concrete and the steel and td explain the fundamental principies for investigation and design. Minor Changes have been made in other chapters, and many tiew praBms for students have been added. Compared with the fourth edition, the number of problems has been increased from 84 to 140, and the number of pages from 128 to 156.
M. M.CONTENTS
Chapter I.
ELASTIC AND ULTIMATE STRENGTH.
PAGB
Art. i. Direct Stresses. 2. The Elastic Limit. 3. Ul-
timate Strength. 4. Tension. 5. Compression. 6. Shear. 7. Working Unit-stresses............... 7
Chapter II.
GENERAL PROPERTIES.
Art. 8. Average Weights. 9. Testing Machines. 10. Timher. 11. Brick. 12. Stone. 13. Cast Iron. 14. Wrought Iron. 15. Steel. 16. Other Materials..	22
Chapter III.
MOMENTS FOR BEAMS.
Art. 17. The Principle of Moments. 18. Reactions for Supports. 19. Bending Moment. 20. Resisting Moments.
It, Centers of Gravity. 22. Moments of Inertia....	40
Chapter IV.
CANTILEVER AND SIMPLE BEAMS.
Art. 23. Definitions and Principles. 24. Resistance to Shearing. 25. Resistance to Bending. 26. Safe Loads for Beams. 27. Investigation of Beams. 28. Design of Beams. 29. Comparative Strengths. 30. Steel I Beams. 31. Beams of Uniform Strength...... ....... 56
Chapter V.
COLUMNS OR STRUTS.
Art. 32. General Principles. 33. Radius of Gyration.
34.	Formula for Columns. 35. Safe Loads for Columns.
35.	Investigation of Columns. 37. Design of Columns.. 75
56
CONTENTS.
Chapter VI.
THE TORSION OF SHAFTS.
PAGB
Art. 38. Phenomena of Torsion. 39. Polar Moment of Inertia. 40. Formula for Torsion. 41. Shafts to Transmit Power. 42. Solid Shafts. 43. Hollow Shafts. 85
Chapter VII.
ELASTIC DEFORMATIONS.
Art. 44. The Coefficient of Elasticity. 45. Elongation under Tension. 46. Shortening under Compression.
47. Deflection of Cantilever Beams. 48. Deflection of Simple Beams. 49. Restrained Beams. 50. Twist in Shafts............................................ 95
Chapter VIII.
RESILIENCE OF MATERIALS.
Art. 51. Fundamental Ideas. 52. Elastic Resilience of Bars. 53. Elastic Resilience of Beams. 54. Ultimate Resilience. 55. Sudden Loads. 56. Stresses Due to Impact.................................... 105
Chapter IX.
MISCELLANEOUS APPLICATIONS.
Art. 57. Water and Steam Pipes. 58 Riveted Lap Joints. 59. Riveted Butt Joints. 60. Stresses Due to Temperature. 61. Shrinkage of Hoops. 62. Shaft Couplings. 63. Rupture of Beams and Shafts........ 114
Chapter X.
REINFORCED CONCRETE.
Art. 64. Concrete and Steel. 65. Compound Bars.
66. Short Columns. 67. Beams with Symmetric Reinforce-
ment. 68. Unsymmetric Reinforcement. 69. Design of Beams. 70. General Discussions.........'............ 125
Appendix.
Art. 71. Miscellaneous Problems. 72. Answers to Problems............................................ 145
155
IndexSTRENGTH OF MATERIALS,
Chapter I.
ELASTIC AND ULTIMATE STRENGTH.
Art. 1. Direct Stresses.
A ‘stress’ is an internal resistance which balances an exterior force. If a weight of 500 pounds be suspended by a rope a stress of 500 pounds exists in every cross-section of the rope ; or if the rope be cut anywhere and the ends be connected by a spring balance this will register 500 pounds. Stresses are measured in pounds» tons, or kilograms.
A ‘ unit-stress ’ is the stress on a unit of area; this is expressed in pounds per square inch or in kilograms per square centimeter. Thus, when a bar of three square inches in cross-section is subject to a pull of 12 000 pounds, the unit-stress is 4000 pounds per square inch, if the total stress be uniformly distributed over the cross-section.
Three kinds of direct stress are produced by exterior forces which act on a body and tend to change its shape; these are,
Tension, tending to pull apart, as in a rope.
78
ELASTIC AND ULTIMATE STRENGTH.
Ch. I.
Compression, tending to push together, as in a wall or column.
Shear, tending to cut across, as in punching a plate.
The forces which produce these kinds of stress may be Called tensile, compressive, and shearing forces, while the Stresses themselves are frequently called tensile, compressive, and shearing stresses.
A Stresses always accompanied by a ‘ deformation jjj or change of shape of the body. As the applied force ifrCfeases the deformation and the stress likewise in-cr£3j&S*.?and if the force be large enough it finally overcomes the stress and the rupture of the body follows.
Tension and compression differ only in regard to direction* A tensile stress in a bar occurs when two forces of equal intensity act upon its ends, each acting
Fig. I.
away from the end of the bar. In compression the direction of the forces is reversed, each acting toward pWiHmd of the bar. The tensile force produces a defor-ihatron called * elongation * and the compressive force prpddfees a deformation called ‘ shortening.* If P be th^Torce in pounds then the total stress in every section *of the bar is equal to P.
Shear implies the action of tw$ forces in parallel pl£ffi#&nd very near together, like the forces in a pair of shears, from which analogy the name is derived. Tlius, if a bar "be laid upon two supports and two loads, each P pounds,';be ap|died to it near the supports, theM are hence produced near each support two parallelArt. 1.
DIRECT STRESSES.
9
forces which tend to cut the bar across vertically. In each of these sections the shearing stress is equal
to P. The deformation caused by the shearing force P is a vertical sliding between the upward and downward forces; and the bar will be cut across if the external shear overcomes the internal stress.
In all cases of direct stress the total stress Pis supposed to be uniformly distributed over the area of the cross-section ; this area will be called the ‘ section area.* Thus if A be the section area and 5 be the unit-stress, then
from which one of the quantities may be computed when the other two are given. For example, it is known that a wrought-iron bar will rupture under tension when the unit-stress 5 becomes 50000 pounds per square inch ; if the area A is 4^- square inches, then the tensile force required to rupture the bar is P — 4^ X 50000 = 225 000 pounds.
Problem x.' A cast-iron bar which is to be subjected to a tension of 34 000 pounds is to be designed so that the unit-stress shall be 2 500 pounds per square inch. What should be the section area in square inches ? If the bar is round what should be its diameter ? r
P
P
Fig. 2.
p=AS,	5 = ^
m
2
/ hIO
ELASTIC AND ULTIMATE STRENGTH.
Ch. I.
Art. 2. The Elastic Limit.
When a tensile force is gradually applied to a bar it elongates, and up to a certain limit the elongation is proportional to the force. Thus, if a bar of wrought-iron one square inch in section area and ioo inches long be subjected to a tension of 5 000 pounds it will be found to elongate 0.02 inches; if 10000 pounds be applied the elongation will be 0.04 inches ; if 15 000 pounds it, will be 0.06 inches, for 20000 pounds 0.08 inches, for 25 000 pounds o. 10 inches. Thus far each addition of 5 000 pounds has produced an elongation of 0.02 inches. But when the next 5 000 pounds is added, mak'ing a total stress of 30000 pounds, it will be found that the total elongation is about 0.13 or 0.14 inches, and hence the elongations are increasing more rapidly than the stresses.
The £ elastic limit ’ is defined to be that unit-stress at which the deformations begin to increase in a faster ratio than the stresses. In the above illustration this limit is about 25 000 pounds per square inch, and this indeed is the average value of the elastic limit for wrought iron. The term ‘ elastic strength ’ is perhaps more expressive than elastic limit, but the latter is the one in general use.
When the unit-stress in a bar is less than the elastic limit the bar returns, when the stress is removed, to its original length. When the unit-stress is greater than the elastic limit the bar does not fully spring back, but there remains a so-called permanent set. In other words, the elastic properties of a bar are injured if it is stressed beyond the elastic limit. Hence it is aArt. 3.
ULTIMATE STRENGTH.
II
fundamental rule in designing engineering constructions that the unit-stresses should never exceed the elastic limit of the material.
The following are average values of the elastic limits of the four materials most used in engineering construction under tensile and compressive stresses.
Table I. Elastic Limits.
• Material.	Pounds per Square Inch.		Kilos per Sq. Centimeter.	
	Tension.	Compression.	Tension.	Compression
Timber	3 OOO	3 OOO	210	210
Cast Iron	6 000	20 OOO	420	I 4OO
Wrought Iron	25 OOO	25 OOO	1750	1750
Steel, structural	35 OOO	35 000	2 450	2 450
Those in English units must be carefully kept in mind by the student ; and it may be noted that pounds per square inch multiplied by 0.07 will give kilos per square centimeter. But little is known concerning the elastic limit in shear ; it is probably about three-fourths of the values above given for tension.
Prob. 2. A square stick of timber is to carry a compressive load of 81 000 pounds. What should be its size in order that the unit-stress may be one-third of the elastic limit ?	H I
Art. 3. Ultimate Strength.
When a bar is under stress exceeding its elastic limit it is usually in an unsafe condition. As the stress is increased by the application of exterior forces the deformation rapidly increases, until finally the rupture of the bar occurs. By the term ‘ ultimate12
ELASTIC AND ULTIMATE STRENGTH.
Ch. I.
strength* is meant that unit-stress which occurs just before ruptufe, it being the highest unit-stress that the bar will bear.
The ultimate strengths of materials are from two to four times theiirelastic limits, but for some materials they are much greater in compression than in tension. The average valwKS will be given in subsequent ar-ticles.
The ‘ factdiilof safety’ is a number which results by diRhng the ultimate strength by the actual unit-stress that	in a bar. For example, a stick of timber,
6x6 inches in section area, whose- ultimate strength in teHon isHHooo pounds per square ™:h,Hunder a te^^Hstress of 32 400 pounds. The unit-stress then is 32 40o/3^H§6^R)oun(Hper square inch, and the factor of safety is 10000/900 — 11. The factor of safety was foiHfcrly much used in dBBgmng, but it is now considered tHWbcttcr plan to judge of the security of a body under sReS» by inference to its elastic limit. Thus in tjHabovMrise, as the unit-stress is only one-third the elastic limit for timber, the degree of security may be regarded a$ sufficient.
Prob. 3. A	wrought iron 2$ inches in diameter
ruptures under a tension of 271 000 pounds. What is its ultimate strengthH| pcfflBIBper square inch ?
■■ I ■
Art. 4. Tension.	L U'Cw
When a barHteffld under tension, it is done by loads which Bl^^^^Hlly applied. The elongations increase proportiojgdly to the stresses until the elastic limit is reached. After' the unit-stress has exceededArt. 4.
TENSION.
r3
the elastic limit the elongations increase more rapidly than the stresses, and this is often accompanied by a reduction in area of the cross-section of the bar. Finally the ultimate strength of the material is reached, and the bar tears apart.
A graphical illustration of these phenomena may be made by laying off the unit-stresses as ordinates and the elongations per unit of length as abscissas. At various intervals, as the test progresses, the applied loads are observed and the resulting elongations are
measured. The loads divided by the section area give the unit-stresses, while the total elongations divided by the length of the bar give the unit-elongations. On the plot a point is made for the intersection of each unit-stress with its corresponding unit-elongation, and a curve is drawn connecting the several points for each material. In this way curves are plotted showing the properties of each material. It is seen that each curve is a straight line from the origin o until the14
ELASTIC AND ULTIMATE STRENGTH.
Ch. I.
elastic limit is reached, showing that the elongations increase proportionally to the unit-stresses. At the elastic limit a sudden change in the curve is seen, and afterwards the elongation increases more rapidly than the stress. The end of the curve indicates the point of rupture. The curve for steel in the diagram is for a quality much stronger than structural steel, this being the kind mostly used in bridges and buildings.
The ultimate elongation is an index of the ductility of the material, and is hence generally recorded for wrought iron and steel; this is usually expressed as a percentage of the total length of the bar, or it is ioo times the unit-elongation. The following table gives mean values of the ultimate strengths and ultimate elongations for the principal materials used in tension.
Table II. Tensile Strengths.
Material.	Ultimate Strength.		Ultimate Elongation. Per Cent.
	Pounds per Square Inch.	Kilos per Sq. Centimeter.	
Timber	IO OOO	700	1-5
Cast Iron	20 OOO	1400	0.5
Wrought Tron	50 000	3 500	30
Steel, structural	65 000	4 500	25
All these values should be regarded as rough averages, since they are subject to much variation with different qualities of the material; for instance poor timber may be as low as 6000 while strong timber may be as high as 20 000 pounds per square inch in ultimate tensile strength. The ultimate strengths given in the table should, however, be memorized by the stu-Art. 5.
COMPRESSION.
IS
dent as a basis for future knowledge, and they will be used for all the examples and problems in this book, unless otherwise stated.
Prob. 4. What should be the diameter of a wrought-iron bar so as to carry a tension of 200 000 pounds with a factor of safety of 5 ? If the bar is cast iron, what should be its diameter ?	,
Art. 5. Compression.
The phenomena of compression are similar to those of tension provided that the elastic limit is not exceeded, the shortening of the bar being proportional to the applied force. After the elastic limit is passed the shortening increases more rapidly than the stress. If the length of the specimen is less than about ten times its least thickness, failure usually occurs by an oblique splitting or shearing, as seen in the diagram.
Fig. 4.
If the length be large compared with the thickness, failure usually occurs under a sidewise bending, so that this is not a case of simple compression. All the values given in the following table refer to theió
ELASTIC AND ULTIMATE STRENGTH.
Ch. I.
short specimens; longer pieces are called ‘columns’ or ‘struts,’ and these will be discussed in Chapter V.
The mean values of the ultimate compressive strengths of the principal materials are tabulated below. These are subject to much variation in different qualities of the materials, but it is necessary for the student to fix them in his mind as a preliminary basis for more extended knowledge. It is seen that
Table III. Compressive Strengths.
Material.	Ultimate Strength.	
	Pounds per Square Inch.	Kilos per Sq. Centimeter.
Timber	8 OOO	56°
Brick	3 OOO	210
Stone	6 000	420
Cast Iron	90 OOO	6 300
Wrought Iron	50 OOO	3 5°°
Steel, structural	65 OOO	4 500
timber is not quite as strong in compression as in tension, that cast iron is 4] times as strong, that wrought iron and structural steel have the same ultimate strength in tension and compression.
The investigation of a body under compression is made by formula (1) of Art. 1. For example if a stone block 8 X 12 inches in cross-section is subjected to a compression of 230000 pounds, the unit-stress produced is 230000/96^=2400 pounds per square inch, and the factor of safety is 6 000/2 400= 2-J-; this is not sufficiently high for stone, as will be seen later.
Prob. 5. A brick 2X4X8 inches weighs about 4.1 pounds. What will be the height of a pile of bricks soArt. 6.
SHEAR.
17
that the unit-stress on the lowest brick
its ultimate strength ?	/ 1	~~t
h	y 1 / j
Art. G. Shear.
shall be one-half of
c * w#
I y y
A4*
Shearing stresses exist when two forces acting like a pair of shears tend to cut a body between them. When a hole is punched in a plate, the ultimate shearing strength of the material must be overcome. If two thin bars be connected by a rivet and these be subjected to tension, the cross-section of the rivet between the plates is brought into shear. If a bolt is in tension, the forces acting on the head tend to shear or strip it off.
The following table gives the average ultimate shearing strength of different materials as determined
Tabi.k IV. Shearing Strengths.
Material.	Ultimate Strength.	
	Pounds per Square Inch.	Kilos per Sq. Centimeter.
(longitudinal Timber ■<	600	42
(transverse	3 OOO	210
Cast Iron	20 OOO	I 400
Wrought Iron	40 OOO	2 800
Steel, structural	50 OOO	3 500
by expei iment. For timber this is much smaller along the grain than across the grain ; in the first direction it is called the longitudinal shearing strength, and in the second the transverse shearing strength. Rolled plates of wrought iron and steel, where the process of manufacture induces a fibrous structure, arei8
ELASTIC AND ULTIMATE STRENGTH.
Ch. I.
also sheared more eS^ly in the longitudinal than in the transverse direction*
Wooden specimens,-for tensile tests like that shown in tire figure will fairly sheafing off the ends if the lerigth ab is not SUfRefeotly great. For instance, sup-ab to fee 6 ifiche^and the .diameter of the central part tO bd 2 inches. T|p$, ends afe grasped tightly by
a_I b
H 8.
the machine and the cross-section of the central part Chtite brought under tensile stress. The forcO retired to ‘Cause rupture by tension is
/*j-s /43 = 3.14 X i4 X 10000 = 31 400 pounds.
rH$$t the ends also tend to shear off on the surface of |tyhnd£&whose diameter is 2 inches and whose length thè force required to cause this rupture by re
fwSP AS%=< 3*14 X 2 X GtX 600 5=^22 600 ]3<fflnd^M
and heltce the specimen: Will fail by shearing off the C11E® * “0 prevent this the distance ab must be made B9p:d&han/6 inches.
wrough t-ir^Bbol#’ 15 f^ies in deleter has a head 1^ inches Iona,. When a tension of 15 000 pounds ^HSpUed to the bolt, find ■thjfctgglile ur||BlmP§and the for tei^Hjns. Also find the unit-stress to shear off the head of the bolt, and the fac^H of safety sjfiejxr.Art. 7.
WORKING UNIT-STRESSES.
*9
Art. 7. Working Unit-Stresses.
When a body of cross-section A is under a stress P, the unit-stress if produced is found by dividing P by A. By comparing this value of if with the ultimate strengths and elastic limits given in the preceding articles, the degree of security may be inferred. This process is called investigation. The student may not at first be able to form a good judgment with regard to the degree of security, this being a matter which involves some experience as well as acquaintance with engineering precedents and practice. As his knowledge increases, however, his ability to judge whether unit-stresses are or are not too great will constantly improve.
When a body is to be designed to stand a total stress P, the unit-stress a is first assumed in accordance with the rules of practice, and then the section area A is computed. Such assumed unit-stresses are often called working unit-stresses, meaning that these are the unit-stresses under which the material is to act or work. In selecting them, two fundamental rules are to be kept in mind :
1.	They should be considerably less than the elastic
limits.
2.	They should be smaller for sudden stresses than
for steady stresses.
The reason for the first requirement is given in Art. 2. The reason for the second requirement is that experience teaches that suddenly applied loads and shocks are more injurious and produce higher unit-stresses than steady loads. Thus a bridge subject20
ELASTIC AND ULTIMATE STRENGTH.
Ch. I.
to the traffic of heavy trains must be designed with lower unit-stresses than a roof where the variable load consists only of snow and wind.
It will be best for the student to begin to form his engineering judgment by fixing in mind the following average values of the factors of safety to be used for different materials under different circumstances. The working unit-stress will then be found for any special
Table V. Factors of Safety.
Material.	For Steady Stress. (Buildings.)	For Varying Stress. (Bridges.)	For Shocks. (Machines.)
Timber	8	10	15
Brick and Stone	15	25	35
Cast Iron	6	15	20
Wrought Iron	4	6	IO
Steel, structural	4	6	IO
case by dividing the ultimate strengths by these factors of safety. For instance, a short timber strut in a bridge should have a working unit-stress of about 8 ooo/io = 800 pounds per square inch.
It frequently happens that a designer works under specifications in which the unit-stresses to be used are definitely stated. The writer of the specifications must necessarily be an engineer of much experience and with a thorough knowledge of the best practice. It may be noted also that the particular qualities of timber or steel to be used will influence the selection of working unit-stresses, and in fact different members of a bridge truss are often designed with different unit-Art.
WORKING UNIT-STRESSES.
21
stresses. The two fundamental rules above stated are, however, the guiding ones in all cases.
Modern engineering is the art of economic construction. In numerous instances this will be secured by making all parts of a structure of equal strength, for if one part is stronger than another it has an excess of material which might have been saved.
Prob. 7. A wrought-iron rod is to be under a stress of S2 000 pounds. Find its diameter when it is to be used in a building, and also when it is to be used in a bridge. #«
IN22
GENERAL PROPERTIES.
Ch. II.
Chapter II.
GENERAL PROPERTIES.
Art. 8. Average Weights.
The average weights of the six principal materials used in engineering constructions are given in the following table, together with their specific gravities. These are subject to more or less variation, according to the quality of the material. For instance, brick
Taisle VI. Weight.
Material.	Pounds per Cubic Foot.	Kilos per Cubic Meter.	Specific Gravity.
Timber	40	600	0.6
Brick	125	2 OOO	2.0
Stone	160	2 600	2.6
Cast Iron	450	7 200	7.2
Wrought Iron	480	7 700	7-7
Steel	490	7 800	7.8
may weigh as low as ioo or as high as 150 pounds per cubic foot, depending upon whether it is soft common quality or fine hard-pressed. Unless otherwise stated, the above values will be used in all the examples and problems in the following pages, and hence those in pounds per cubic foot must be carefully kept in the memory.
For computing the weights of bars, beams, andArt. 9.
TESTING MACHINES.
23
pieces of uniform section area, the following approximate simple rules are much used by engineers :
A wrought-iron bar one square inch in section area and one yard long weighs ten pounds.
Timber is one-twelfth the weight of wrought iron.
Brick is one-fourth the weight of wrought iron.
Stone is one-third the weight of wrought iron.
Cast iron is six per cent lighter than wrought iron.
Steel is two per cent heavier than wrought iron.
For example, if a bar of wrought iron be l}X3 inches in section and 22 feet long, its section area is 4^ square inches and its weight is 45 X 7^"—'33° pounds. A steel bar of the same dimensions will weigh 330 -f- 0.02 X 330 « 337 pounds, and a cast-iron bar will weigh 330 — 0.06 X 330« 310 pounds.
By reversing the above rules the section areas are readily found when the weights per linear yard are given. Thus, if a stick of timber 15 feet long weighs 120 pounds, its weight per yard is 24 pounds and its section area is 2.4 X 12 — 28.8 square inches.
Prob. 8. What is the weight of a stone block 12 X 18 inches and 43 feet long ? How many square inches in the cross-section of a steel railroad rail which weighs 95 pounds per yard ? If a cast-iron water pipe 12 feet long weighs 10 000 pounds, what is its section area ?
Art. 9. Testing Machines.
The simplest method of testing is by tension, a specimen being used like that shown in Art. 6. The heads are either gripped in jaws, or they are provided with threads so that they may be screwed into nuts to which the forces are applied. The power may be fur-24
GENERAL PROPERTIES.
Ch. II.
nished by a lever, a screw, or by hydraulic pressure, the last being the method in machines of high capacity. In these tests the elastic limit, ultimate strength, and the ultimate elongation are generally recorded, the latter being expressed as a percentage of the original length. For ductile materials the contraction of area of the fractured specimen is also noted, as this does not vary with the length of the specimen to the same extent as the ultimate elongation. In such tensile tests the load is applied gradually, and not suddenly or with impact.
The elastic limit is detected by taking a number of measurements of the elongation for different loads, and then noting when these begin to vary more rapidly than the stresses. For ductile materials the change is a sudden one, and it may be often noted by the drop of the scale beam of the machine.
Compressive tests are confined mainly to brick and stone, and are but little used for commercial tests of metals on account of the difficulty of securing a uniform distribution of pressure over the surfaces. Cement, which is always used in compression, is indeed usually tested by tension, this being found to be the cheaper and more satisfactory method.
The capacity of a testing machine is the number of pounds it can exert as tension or compression. A small machine for testing wire or cement need not have a capacity greater than I ooo or 2 ooo pounds. Machines of 50000, IOOOOO, and 150000 pounds for testing metals are common. The Watertown machine has a capacity of 1 000000 pounds, and can test a small hair or a steel bar of 10 square inches section area withArt. 10.
TIMBER.
25
equal precision. The heaviest machine is that at Phoenixville, Pa., for testing eye-bars, which has a capacity of 2 160000 pounds.
Tests are also made by loading beams transversely and measuring the deflections, as well as finding the load required to produce rupture. The loads which produce rupture give comparative measures of the ultimate strengths of the beams.
Prob. 9. A steel eye-bar tested at Phoenixville was io X 2§ inches in size and 47 feet long. The length after rupture was 57.6 feet, and the area of the fractured cross-section was 13.0 square inches. Compute the percentage of ultimate elongation and the percentage of reduction of area.
Art. 10. T imber.
Good timber is of uniform color and texture, free from knots, sap wood, wind shakes, and decay. It should be well seasoned, which is best done by exposing it to the sun and wind for two or three years to dry out the sap. The heaviest timber is usually the strongest ; also the darker the color and the closer the annular rings the stronger and better it is, other things being equal. The strength of timber is always greatest in the direction of the grain, the sidewise resistance to tension or compression being scarcely one-fourth of the longitudinal.
The following table which gives average values of the ultimate strength cf a few of the common kinds of timber will be useful for reference. These values have been determined from tests of small specimens carefully selected and dried. Large pieces of timber such as2Ô
GENERAL PROPERTIES.
Ch. U
Table VII. Strength of Timber.
Kind.	Pounds per Cubic Foot.	Pounds per Tensile Strength.	Square Inch. Compressive Strength.
Hemlock	25	8 OOO	5 OOO
White Pine	27	8 OOO	5 500
Chestnut	40	12 OOO	5 000
Red Oak	42	9 OOO	6 000
Yellow Pine	45	15 OOO	q 000
White Oak	48	12 OOO	8 000
are actually used in engineering structures will probably have an ultimate strength of from fifty to eighty per cent of these values. Moreover the figures are liable to a range of 25 per cent on account of variations in quality and condition arising from place of growth, time when cut, and method of seasoning. To cover these variations the factor of safety of 10 is not too high, even for stready stresses.
The shearing strength of timber is still more variable than the tensile or compressive resistance. White pine across the grain may be put at 2 500 pounds per square inch, and along the grain at 500. Chestnut has 1 500 and 600 respectively, yellow pine and oak perhaps 4 000 and 600 respectively.
The elastic limit of timber is poorly defined. In precise tests on good specimens it is sometimes observed at about one-half the ultimate strength, but under ordinary conditions it is safer to put it at one-third. The ultimate elongation is small, usually being between I and 2 per cent.
Prob. 10. If a piece of white cedar 2X2 inches inArt. 11.
BRICK.
27
cross-section ruptures under a compression of 20 800 pounds, what is the size of a square section that will stand 25 000 pounds with a factor of safety of 10?
Art. 11. Erick.
Brick is made of clay which consists mainly of silicate of alumina with compounds of lime, magnesia, and iron. The clay is prepared by cleaning it carefully from pebbles and sand, mixing it with about one-half its volume of water, and tempering it by hand stirring or in a pug mill. It is then moulded in rectangular boxes by hand or by special machines, and the green bricks are placed under open sheds to dry. These are piled in a kiln and heated for nearly two weeks until those nearest to the fuel assume a partially vitrified appearance.
Three qualities of brick are taken from the kiln ; ‘arch-brick’ are those from around the arches where the fuel is burned, these are hard and often brittle ; * body-brick,’ from the interior of the kiln, are of the best quality; ‘soft brick,’ from the exterior of the pile, are weak and only suitable for filling. Paving brick are burned in special kilns, often by natural gas or by oil, the rate of heating being such as to insure toughness and hardness.
The common size is 2 X 4 X 8J inches, and the average weight 4J- pounds. A pressed brick, however, may weigh nearly 5^ pounds. Good bricks should be of regular shape, have parallel and plane faces, with sharp angles and edges. They should be of uniform texture, and when struck a quick blow should give a sharp metallic ring. The heavier the brick, other things being equal, the stronger and better it is.GEN K E A L 1‘ R O P5 R 1 IES.
Ch. II,
Poor brick will absorb when dry from 20 to 30 per cent of its weight of- water, ordinary qualities absorb from 10 to 20 per cent, while hard paving brick should not absorb more than 2 or 3 per cent. An absorption test is valuable in measuring the capacity of brick to resist the disintegrating action of frost, and as a rough general rule the greater the amount of water absorbed the less is the strength and durability.
The crushing strength of brick is variable; while a mean value may be 3 000 pounds per square inch, soft brick will scarcely stand 500, pressed brick may run to 10000, and the best qualities of paving brick have given 15 000 pounds per square inch, or even more. Crushing tests are difficult and expensive to make on account of the labor of preparing the specimen cubes so as to secure surfaces truly parallel. Tensile and shearing tests of bricks are rarely made and but little is known of their behavior under such stresses; the ultimate tensile strength may perhaps range from 50 to 500 pounds per square inch.
Prob. 11. Compute the unit-stress at the base of a brick wall 17 inches thick and 55 feet high. What is the factor of safety ?
Art. 12. Stone.
Sandstone, as its name implies, is sand, usually quartzite, which has been consolidated under heat and pressure. It varies much in color, strength, and durability, but many varieties form most valuable building material. In general it is easy to cut and dress, but the variety known as Potsdam sandstone is very hard in some localities.STONE.
29
Art. 12.
Limestone is formed by consolidated marine shells, and is of diverse quality. Marble is limestone which has been reworked by the forces of nature so as to expel the impurities, and leave a nearly pure carbonate of lime; it takes a high polish, is easily cut, and makes one of the most beautiful building stones.
Granite is a rock of aqueous origin metamorphosed under heat and pressure; its composition is quartz, feldspar, and mica, but in the variety called gneiss the mica is replaced by hornblende. It is fairly easy to work, usually strong and durable, and some varieties will take a high polish.
Trap, or basalt, is.an igneous rock without cleavage. It is hard and tough, and less suitable for building constructions than other rocks, as large blocks cannot be readily obtained and cut to size. It has, however, a high strength, and is remarkable for durability.
The average weight of sandstone is about 150, of limestone 160, of granite 165, and of trap 175 pounds per cubic foot. The ultimate compressive strength of sandstone is about 5000, of limestone 7000, of granite 12000, and of trap 16000 pounds per square inch; these figures refer to small blocks, but the ultimate strength of large blocks is materially smaller.
The quality of a building stone cannot be safely inferred from tests of strength, as its durability depends largely upon its capacity to resist the action of the weather. Hence corrosion and freezing tests, impact tests, and observations of the behavior of stone under conditions of actual use are more important Ilian the determination of crushing strength in a com-pression machine.3o
GENERAL PROPERTIES.
Ch. II.
Prob. 12. A stone pier 12 X 30 feet at the base, 8 IK 24 feet at the top, and 16% feet high is to be built at $6.37 per cubic yard. What is this total cost ?
Art. 1$. Cast iRoJfc-j^
Cast iron is a modern product, having been first made in England about the beginning of the fifteenth century. Ores of iron are melted in a blast furnace, producing pig iron. The pig iron is remelted in a cupola furnace &fld poured into moulds, thus forming tastings. Beams, columns, pipes, braceSy and blocks of every shape required in engineering structures are thus produced.
Pig iron is divided into two classes* founcfry pig and forge pig, the ft^Hter being used for castings and the latter for making wrought iron. Foundry has a dark-gray fracture, with large crystals and a metallic luster; forge pig has a light-gray or silver-white fracture, with small crystals. Foundry pig has a specific gravity of from 7.1 to 7-2> and it contains from 6 to 4 per cent of carbon ; forge pig has a specific gravity of from 7.2 to 7.4, and it contains from 4 to 2 per CCnt of carbon. The hfflter the, percentage of carbon the is thflpecific gravity, and the easier it is to. 'ihelt the pig. Besi^R the carbon there are present from 1 to per (Ht of other impurities, such as silicon, mangas! nese, and phosphorus.
The properties and strength of castingapldepend upon the	and the method of their
mlfacture in both the kHt and the cupola fmn^Re. Cold blast) pig p«ucH stronger itot> than theBBoM blast, but it is more expensive. Long continued'Art. 13.
CAST IRON.
3r
fusion improves the quality of the product, as also do repeated meltings. '1 he darkest grades of foundry pig make the smoothest castings, but they are apt to be brittle ; the light-gray grades make tough castings, but they are apt to contain blow holes or imperfections.
The percentage of carbon in cast iron is a controlling factor which governs its strength, particularly that percentage which is chemically combined with the iron. As average values for the ultimate strength of cast iron, 20000 and 90000 pounds per square inch in tension and compression respectively are good figures. In any particular case, however, a variation of from 10 to 20 per cent from these values may be expected, owing to the great variation in quality. The elastic limit is poorly defined, there being no sudden increase in deformation, as in ductile materials.
The high compressive strength and cheapness of cast iron render it a valuable material for many purposes ; blit its brittleness, low tensile strength and ductility forbid its use in structures subject to variations of load or to shocks. Its ultimate elongation being scarcely one per cent, the work required to cause rupture in tension is small compared to that for wrought iron and steel, and hence as a structural material the use of cast iron must be confined entirely to cases of compression.
I’rob. 13. A cast-iron bar weighing 31 pounds per linear yard is to be subjected to tension. How many pounds are required to rupture it ?32
GENERAL PROPERTIES.
Ch. IL
Art. 14. Wrought Iron.
The ancient peoples of Europe and Asia were acquainted with wrought iron and steel to a limited extent. It is mentioned in Genesis, iv. 22, and in one of the oldest pyramids of Egypt a piece of iron has been found. It was produced, probably, by the action of a hot fire on very pure ore. The ancient Britons built bloomaries on the tops of high hills, a tunnel opening toward the north furnishing a draft for the fire, which caused the carbon and other impurities to be expelled from the ore, leaving behind nearly pure metallic iron.
Modern methods of manufacturing wrought iron are mainly by the use of forge pig (Art. 13), the one most extensively used being the puddling process. Here the forge pig is subjected to the oxidizing flame of a blast in a reverberatory furnace, where it is formed into pasty balls by the puddler. A ball taken from the furnace is run through a squeezer to expel the cinder and then rolled into a muck bar. The muck bars are cut, laid in piles, heated, and rolled, forming what is called merchant bar. If this is cut, piled, and rolled again, a better product, called best iron, is produced. A third rolling gives‘best best’ iron, a superior quality, but high in price.
The product of the rolling-mill is bar iron, plate iron, shape iron, beams, and rails. Bar iron is round, square, and rectangular in section ; plate iron is from \ to 1 inch thick, and of varying widths and lengths ; shape iron includes angles, tees, channels, and other forms used in structural work; beams are I-shaped,Art. 14.
WROUGHT IRON.
33
and of the deck or rail form. Structural shapes and beams are, however, now more extensively rolled in mild steel than in wrought iron.
Wrought iron is metallic iron containing less than 0.25 per cent of carbon, and which has been manufactured without fusion. Its tensile and compressive strengths are closely equal, and range from 50 000 to 60000 pounds per square inch. The elastic limit is well defined at about 25 000 pounds per square inch, and within that limit the law of proportionality of stress to deformation is strictly observed. It is tough and ductile, having an ultimate elongation of from 20 to 30 per cent. It is malleable, can be forged and welded, and has a high capacity to withstand the action of shocks. It cannot, however, be tempered, nor can it be melted, except by the highest temperatures.
The cold-bend test for wrought iron is an important one for judging of general quality. A bar perhaps Ixf inches and 15 inches long is bent when cold either by pressure or by blows of a hammer. Bridge iron should bend through an angle of 90 degrees to a curve whose radius is twice the thickness of the bar, without cracking. Rivet iron should bend through 180 degrees until the sides of the bar are in contact, without showing signs of fracture. Wrought iron that breaks under this test is lacking in both strength and ductility.
The process of manufacture, as well as the quality of the pig iron, influences the strength of wrought iron. The higher the percentage of carbon the greater is the strength. Best iron is 10 per cent stronger than or-34
GENERAL PROPERTIES.
Ch. II.
dinary merchant iron owing to the influence of the second rolling. Cold rolling causes a marked increase in elastic limit and ultimate strength, but a decrease in ductility or ultimate elongation. Annealing lowers the ultimate strength, but increases the elongation. Iron wire, owing to the process of drawing, has a high tensile strength, sometimes greater than iooooo pounds per square inch.
Good wrought iron when broken by tension shows a fibrous structure. If, however, it be subject to shocks or to repeated stresses which exceed the clastic limit, the molecular structure becomes changed so that the fracture is more or less crystalline. The effect of a stress slightly exceeding the elastic limit is to cause a small permanent set, but the elastic limit will be found to be higher than before. This is decidedly injurious to the quality of the material on account of the accompanying change in structure, and hence it is a fundamental principle that the working unit-stresses should not exceed the elastic limit. For proper security indeed the allowable unit-stress should seldom be greater than one-half the elastic limit.
In a rough general way the quality of wrought iron may be estimated by the product of its tensile strength and ultimate elongation, this product being an approximate measure of the work required to produce rupture. Thus high tensile strength is not usually a good quality when accompanied by a low elongation.
Prob. 14. What should be the length of a wronghf-iron bar, so that when hung at its .upper end it will rupture there under the stress produced by its own weight ?Art. 15.
STEEL.
35
Art. 15. Steel.
Steel was originally produced directly from pure iron ore by the action of a hot fire, which did not remove the carbon to a sufficient extent to form wrought iron. The modern processes, however, involve the fusion of the ore, and the definition of the United States law is that “steel is iron produced by fusion by any process, and which is malleable.” Chemically, steel is a compound of iron and carbon generally intermediate in composition between cast and wrought iron, but having a higher specific gravity than either. The following comparison points out the distinctive differences between the three kinds of iron :
Per cent of Carbon. Spec Grav.	Properties.
Cast Iron,	5 to 2	7.2	Fusible, not malleable.
Steel,	i 50100.10	7.S	Fusible and malleable.
Wrought Iron,	0.30 to 0.05	7.7	Malleable, not fusible.
It should be observed that the percentage of carbon alone is not sufficient to distinguish steel from wrought iron; also, that the mean values of specific gravity stated are in each case subject to considerable variation.
The three principal methods of manufacture are the crucible process, the open-hearth process, and the Bessemer process. In the crucible process impure wrought iron or blister steel, with carbon and a flux, are fused in a sealed vessel to which air cannot obtain access; the best tool-steels are thus made. In the open-hearth process pig iron is melted in a Siemens furnace, wrought-iron scrap being added until the proper degree of carbonization is secured. In the Bessemer process pig iron is completely decarbonized in a converter byGENERAI, PROPERTIES.
Ch. II.
36
an air blast and then recarbonized to the proper degree by the addition of spiegeleisen. The metal from the open hearth furnace or from the Bessemer converter is cast into ingots, which are rolled in mills to the required iorms. The open-hearth process produces steel for guns, armor plates, and for some structural purposes : the Bessemer process produces steel for railroad rails anti also for structural shapes.
The physical properties of steel depend both upon the method of manufacture and upon the chemical composition, the carbon having the controlling influence upon strength. Manganese promotes malleability and silicon increases the hardness, while phosphorus and sulphur tend to cause brittleness. The higher the percentage of carbon within reasonable limits the greater is the ultimate strength and the less the elongation.
A classification of steel according to the percentage of carbon and its physical properties of tempering and welding is as follows :
Extra hard, 1.00 to 0.60^ C., takes high temper, but not weldable. Hard,	0.70 to c.jo% C., lemperable, but welded with difficulty.
Medium, 0.50 to 0.20% C , poor temper, but weldable.
Mild,	0.40 to 0.05% C., not temperable, but easily welded.
It is seen that these classes overlap so that there are no distinct lines of demarcation. The extra-hard steels are used for tools, the hard steels for piston-rods and other parts of machines, the medium steels for rails, tires, and beams, and the mild or soft steels for rivets, plates, and other purposes.
The structural steel used in bridges and buildings has an ultimate tensile strength of from 60 OOO toA.RT. 15.
STEEL.
37
^OOOO pounds per square inch, with an elastic limit from 30 000 to 40 000 pounds per square inch. The hard and extra-hard steels are much higher in strength. By the use of nickel as an alloy steel has been made with an ultimate tensile strength of 277 000 and an elastic limit of over 100000 pounds per square inch.
The compressive strength of steel is always higher than the tensile strength. The maximum value recorded for hardened steel is 392 000 pounds per square inch. The expense of commercial tests of compression is, however, so great that they are seldom made. The shearing strength is about three-fourths of the tensile strength.
Steel castings are extensively used for axle-boxes, cross-heads, and joints in structural work. They contain from 0.25 to 0.50 per cent of carbon, ranging in tensile strength from 60000 to 100000 pounds per square inch.
Steel has entirely supplanted wrought iron for railroad rails, and largely so for structural purposes. Its price being the same, its strength greater, its structure more homogeneous, the low and medium varieties are coming more and more into use as a satisfactory and reliable material for large classes of engineering constructions.
Prob. 15. A short steel piston-rod is to be designed to be used with a piston which is 20 inches in diameter and subject to a steam-pressure of 150 pounds per square inch. If the ultimate strength of the steel is go 000 pounds per square inch, what should be its diameter, allowing the a factor of safety of 15 ?general properties.
Ch. II.
Art. lG. Other Materials.
Common mortar is composed of one part of lime to five parts of sand by measure. When six months old its tensile strength is from 15 to 30, and its compressive strength from 150 to 300 pounds per square inch. Its strength slowly increases with age, and it mciy be considerably increased by using a smaller proportion of sand.
Hydraulic mortar is composed of hydraulic cement and sand in varying proportions. The less the proportion of sand the greater is its strength. A common proportion is 3 parts sand to 1 of cement, the strength of this being about one-fourth of the neat cement. The natural cements are of lighter color, lower weight, and lesser strength than the Portland cement, but they arc quicker in setting and cheaper in price. When one week old, neat natural cement has a tensile strength of about 125 and Portland cement about 300 pounds per square inch ; when one year old the tensile strengths are about 300 and 500 pounds per square inch respectively. The compressive strength is from 6 to 8 times the tensile strength, and it increases more rapidly with age.
Concrete, composed of hydraulic mortar and broken stone, is an ancient material, having been extensively used by the Romans. It is mainly employed for foundations and monolithic structures, but in some cases large blocks have been made which are laid together like masonry. Like mortar, its strength increases with age. When six months old its mean compressive strength ranges from 1 000 to 3000 pounds per squareArt. 1G.
OTHER MATERIALS.
39
inch, and when one year old it is probably about fifty per cent greater.
Ropes are made of hemp, of manilla, and of iron or steel wire with a hemp center. A hemp rope one inch in diameter has an ultimate strength of about 6000 pounds, and its safe working strength is about 800 pounds. A manilla rope is slightly stronger. Iron and steel ropes one inch in diameter have ultimate strengths of about 36000 and 50 ooo pounds respectively, the safe working strengths being 6 000 and 8 000 pounds. As a fair rough rule, the strength of ropes may be said to increase as the squares of their diameters.
Aluminum is a silver-gray metal which is malleable and ductile and not liable to corrode. Its specific gravity is about 2.65, so that it weighs only 168 pounds per cubic foot. Its ultimate tensile strength is about 25 600 pounds per square inch, and its ultimate elongation is also low. Alloys of aluminum and copper have been made with a tensile strength and elongation exceeding those of wrought iron, but have not come into use as structural materials.
Prob. 16. Ascertain the weight of lead and brass pe» cubic foot, and their ultimate tensile strengths.MOMKNTS FOR REAMS.
Ch. III.
46
Chapter III.
MOMENTS FOR BEAMS.
Art. 17. The Principle of Moments.
The moment of a force with respect to a point is a quantity which measures the tendency of the force to cause rotation about that point. The moment is the product of the force by the length of its lever-arm, the lever-arm being a line drawn from the point perpendicular to the direction of the force. Thus if P be any
lorce and p the length of a perpendicular drawn to it from any point, the product Pp is the moment of the force with respect to that point. As P is in pounds and p is in feet or inches, the moment is a compound quantity which is called pound-feet or pound-inches.
The most important principle in mechanics is the principle of moments. This asserts that if any number of forces in the same plane be in equilibrium, the algebraic sum of their moments about any point in that plane is equal to zero. This principle results from theArt. 17.
THE PRINCIPLE OF MOMENTS.
41
meaning of the word equilibrium, which implies that the body on which the forces act is at rest ; and since it is at rest the forces taken collectively have no tendency to turn it around any point. All experience teaches that the principle of moments is, indeed, a law of nature whose truth is universal.
The point from which the lever-arms are measured is often called the ‘ center of moments.’ Forces which tend to turn around this center in the direction of motion of the hands of a watch have positive moments, and those which tend to turn in the opposite direction have negative moments. Thus, in the above figure, the numerical values of Pp and Z3,/’, are negative, while that of P% p% is positive. If the forces be in equilibrium, the sum Pp + A,/’, has the same numerical value as P„p„, or the algebraic sum of the three moments is zero; this will be the case wherever the center of moments be taken.
In all investigations regarding the strength of beams, the principle of moments is of constant application. A beam is a body held in equilibrium by the downward loads and the upward pressures of the supports. As the beam is at rest these forces are in equilibrium, and the algebraic sum of their moments is zero about any point in the plane. Moreover by further use of the principle of moments the stresses in all parts of the beam due to the given loads may be determined.
Prob. 17. In the above figure the three forces are in equilibrium, P% being 500 pounds, P% being 866 pounds, and the lever-arms being /> — r.5 feet, />, — 3.5 feet, p2 — 5.1 feet. Show that the force P is 1 777 pounds.42
MOMENTS FOR BEAMS.
Ch. Ill
-L
Art. 18. Reactions of Supports.
Let a simple beam resting on two supports near its ends be subject to a load P situated at 6 feet from the left support, and let the span be 24 feet. Taking the
P

Fig. 7.
center of moments at the right support, the lever arm of A*, is 24 feet, that of /’is 18 feet, and that of is o; then by the application of the principle of moments A?, X 24 —P X 18 — o, or R} = \P. Again, taking the center of moments at the left support the lever-arm of R} is c, that of P is 6 feet, and that of A\ is 24 feet; then likewise from the principle of moments
—	R^ X 24 -(- P X 6 — o, or A’2 = \P. The sum of these two reactions is equal to Pt as should of course be the case.
The reactions caused by the weight of the beam itself may be found in a similar manner, the uniform load being supposed concentrated at its center of gravity in stating the equations of moments. Thus, if the weight of the beam be Hi the two equations of moments are found to be A*, X 24 — WX 12 = 0, and
— R^ X 24 -f- W X 12 — 0, from which A5, = ^ JV and R, = \W.
The reactions due to both uniform and concentrated loads on a simple beam may also be computed in one operation. As an example, let there be a simple beam 12 feet long between the supports and weighing 35Art. 13.
REACTIONS OR SUPPORTS
43
pounds per linear foot, its total weight being 420 pounds. Let there be three loads of 300, 60, and 150 pounds, placed 3, 5, and 8 feet respectively trom the left support. To find the left reaction A,, the center
Fig. 8.
of moments is taken at the right support and the weight of the beam regarded as concentrated at its middle: then the equation of moments is
A5, X 12 — 420 X 6 — 300 X 9 — 60 X 7 — 150 X 4. from which R — 520 pounds. In like manner, to find R.t the center of moments is taken at the left support, anel then
— A’, x 12 -f 420 x6-f 300 X 3 + 60 X 5 + x 50 X 8, from which A, = 410 pounds. As a check the sum of A, and Af, is found to be 930 pounds, which equals the weight of the beam and the three loads.
By means of the principle of moments other problems relating to reactions of beams may also be solved. For instance, if a simple beam 12 feet long weighs 30 pounds per linear foot and carries a load of 600 pounds, where should this load be put so that the left reaction may be twice as great as the right reaction ? Here let x be the distance from the left support to the load ; let A, be the left reaction and Aa the right reaction. Then taking the centers of moments at the right and left support in succession there are found
A, = 180	50 (12 — x), Aa — 180 -j- 50.19
and placing A, equal to 2Aa there results x = 2.8 feet.44
MOMENTS FOR REAMS.
Ch. III.
Prob. 18. A beam weighing 30 pounds per linear foot rests upon two supports 16 feet apart. A weight of 400 pounds is placed at 5 feet from the left end, and one of 600 pounds is placed at 8 feet from the right end. Find the reactions due to the total load.
Art. 19. Bending Moments.
The ‘bending moment ’ at any section of a beam is the algebraic sum of the moments of all the vertical forces on the left of that section. It is a measure of the tendency of those forces to cause rotation around that point. At the ends of a simple beam there are no bending moments, but at all other sections they exist, and the greater the bending moment the greater are the horizontal stresses in the beam, these stresses in fact bemg produced by the bending moment.
For example, let a beam 30 feet long have three loads of 100 pounds each, situated at distances of 8, 12, and 22 feet from the left support. By the method of
the previous article the left reaction Rt is 160 pounds and the right reaction A2 is impounds. For a section 4 feet from the left support the bending moment is 160X4—640 pound-feet, and for a section at 8 feet from the left support the bending moment is 160 X 8 = 1 280 pound-feet. For a section 10 feet from the left support there are two vertical forces on the left of theArt. 19.
BENDING MOMENTS.
45
section, 160 acting up and ioo acting down, so that the bending moment is 160 X 10 — ioo X 2 ~ 1 400pound-feet. For a section at the middle of the beam the bending moment is 160X 15 — 100 X 7 — 100 X 3 = I 400 pound-feet. For a section under the third load the bending moment is, in like manner, 1 120 pound-feet, and for a section at 3 feet from the right support it is 420 pound-feet. The vertical ordinates underneath the beam represent the values of these bending moments, , and the diagram thus formed shows how the bending moments vary throughout the length of the beam.
For a simple beam of span l and uniformly loaded with zv pounds per linear unit, each reaction is zvl. For any section distant x from the left support, the
bending moment is $zv/X x — wx X \x, where the lever-arm of the reaction is x and the lever-arm of the load zvx is ix. If za be 80 pounds per linear foot and / be 30 feet, the bending moment at any section is then 1 200 x — 40jr» For x — 10 feet, the bending moment is S 000 pound-feet ; for ,r = 15 feet it is 9 OOO pound-feet; for .r ~ 20 feet it is 8000 pound-feet, and so on. The diagram shows the distributions of moments throughout the beam, and it can be demonstrated that the curve joining the ends of the ordinates is the common parabola.46
MOMENTS FOR BEAM?.
Ch. III.
When a beam is loaded both uniformly and with concentrated loads, the bending moments for all sections may be found in a similar manner. The maximum bending moment indicates the point where the beam is under the greatest horizontal stresses; this will usually be found near the middle of the beam and often under one of the concentrated loads. For simple beams resting on two supports at their ends all the bending moments are positive. It may further be noted that if the vertical forces on the right of the section be used the same numerical values will be found for the bending moments.
Prob. 19. A simple beam of 12 feet span weighs 60 pounds per linear foot, and has a load of 150 pounds at 8 feet from the left end. Compute the bending moments for sections distant 2, 4, 6, 8, 10 feet from the left support, and construct the diagram of bending moments.
Art. 20. Resisting Moments.
Suppose a simple beam to be cut by an imaginary vertical plane MN and the portion on the right of that plane to be removed. In order that the remaining
P
C
%
Fig. it.
M
net			
Jz	c	D	
		£ c	
f -		_J		
part may be in equilibrium, forces must be applied to the section ; in the figure horizontal forces are shown, and these represent the horizontal stresses in the sec-Art. 20.
RESISTING MOMENTS.
47
tion. It is found by experiment that there is a certain line CD on the side of the beam which does not change in length under the bending, and hence there is no horizontal stress upon it. Below this neutral line the fibers in a simple beam are found to be elongated and above it they are shortened ; thus the stresses below the neutral line are tension and those above it are compression.
The reaction and loads on the left of the section MN together with the stresses acting on that section constitute a system of forces in equilibrium. The algebraic sum of the moments of the reaction and loads with respect to the point D is the bending moment for the section MN, the value of which may be found by the methods of the last article. This bending moment causes a bending which is resisted by the horizontal stresses in the section.
A neutral line like CD\s also found in all longitudinal vertical sections of the beam. There is in fact a ‘neutral surface’ extending throughout the entire width of the beam, and the intersection of this neutral surface with any section area gives a line CC which is called the ‘ neutral axis ’ of that section. It is found by experiment that the horizontal stresses increase uni-iormly from the neutral axis to the top and bottom of the beam, provided that the elastic limit of the mate* rial be not exceeded. Thus, if 6' be the unit-stress at the upper or lower side of the beam, the unit-stress half-way between that side and the neutral axis is ^ Also if c be the distance from the neutral axis to the upper or lower side of the beam, and s be any distance
less than c, then the unit-stress at the distance s is S-.
c48
MOMENTS FOR BEAMS.
Cus hi.
‘ Resisting moment ’ is the term used to denote the algebraic sum of the moments of all the horizontal stresses in a section with respect to its neutral axis. Let a be any small elementary area of the cross-section at a distance z from the neutral axis. The unit-stress
on this small area is S~, and hence the total stress on
c
this small area is SB
az
The moment of this stress with
respect to the point D is S— multiplied by its iever-
511 _.	. .	.....
arm z, or ——. The resisting moment is the algebraic c
sum of all the values of--for all possible values of z.
Or, letting 2 denote this process of summation,
„ . .	5
Resisting moment = -2az7. h	c
Now the quantity 2az* is what is called the ‘moment of inertia ’ of the cross-section, and it will be shown in Art. 22 how its value is found. The moment of inertia is designated by /, and hence
„	.	Sf
Resisting moment = —. b	c
This is a general expression applicable to all kinds of cross-sections, and in the next chapter it will be constantly used in the investigation and design of beams.
The term ‘moment of inertia’has no reference to inertia when it is applied to plane surfaces, as is here the case. It is merely a name for the quantity and this quantity is found by multiplying each elemen-Art. HI.
CENTERS OF GRAVITY.
49
tar}’ area by the square of its distance from the given axis and taking the sum of the products. As s' is always positive whether s be positive or negative, or the moment of inertia / is always positive.
Prob. 20. In the above figure let Ml) and DN be each 6 inches and let the width of the beam CC be 8 inches. If the tensile unit-stress .S on the bottom of the beam is 6oo pounds per square inch, the compressive unit-stress on the top of the beam is also 6oo pounds per square inch. Show that the total tensile stress is 14400 pounds, and that the total compressive stress is also 14 400 pounds.
Art. 21. Centers ok Gravity.
The center of gravity of a plane surface is that point upon which a thin sheet of cardboard, having the same shape as the given surface, can be balanced when held in a horizontal position. In the investigation of beams its section area is the given surface, and it is required to know the distances from the top or bottom of the section to the center of gravity. The letter c will be used to denote these distances when they are equal, and the longest of these distances when they are unequal.
For a square, rectangle, or circle, whose depth is d, it is evident that c = id. Also for a section of I shape, where the upper and lower flanges are equal in size, it is plain that c — \d.
For a T section c is greater than \d, and its value is to be found by using the principle of moments. If the width of the horizontal flange be 4 inches and its thickness ij inches, the area of the flange is 5 square inches; if the height of the vertical web be 65°
MOMENTS FOR BEAMS.
Ch. III.
inches and its thickness I inch, the area of the web is 6 square inches. The total area of the cross-section is then ii square inches. Now if this section be a thin
1
c
KlG. 13.
sheet held in a horizontal plane the weights of the two parts and the whole are represented by 5, 6, and ij. With respect to an axis at the end of the web the lever-arms of these weights are 6|- inches, 3 inches, and c inches ; the equation of moments then is
5 X 6f -f 6 X 3 - 11 X c = o, from which the value of c is found to be 4.65 inches.
The method of moments may thus be applied to areas as well as to forces. If a be any area and z the distance of its center of gravity from an axis, the product as is called the static moment of the area. The sum of the static moments of all parts of the figure is represented by 2az, and if A be the total section area, then
2as C = ~A
is a general expression of the method of finding the distance c. If the axis be taken within the section, some of the jj's are negative, and if the axis passes through the center of gravity of the section then the quantity 2az is zero.
For the channel section m the same method is to be followed as for the T._ When the cross-section isArt. 22.
MOMENTS OF INERTIA.
5*
bounded by curved lines, as in a railroad rail, it is to be divided up into small rectangles and the value of a be found for each; the sum of all the a's is A, and then by the above method the value of t is computed. For the various rolled shapes found in the market the values of c are thus determined by the manufacturers and published for the information of engineers.
Triangular beams are never used, but it is often convenient to remember that for any triangle whose depth is d the value of c is § d.
Prob. 2i. A deck-beam used in buildings has a rectangular flange 4 X § inches, a rectangular web 5 X 1 inches, and an elliptical head which is 1 inch in depth and whose area is 1.6 square inches. Find the distance of the center of gravity from the top of the head. ^;
Art. 22. Moments of Inertia.
The moment of inertia of a plane surface with respect to an axis is the sum of the products obtained by multiplying each elementary area by the square of its distance from that axis. In the discussion of beams the axis is always taken as passing through the center of gravity of the cross-section and parallel to the top and bottom lines of the cross-section. Let I be this moment of inertia, as in Art. 20; its value is to be found by determining the quantity 2az'\
To find / for a rectangle of breadth b and depth d, let CC be the axis through the center of gravity and parallel to b. Let the elementary area d be a small strip HR parallel to CC and at a distance z from it. Let a line gh be drawn parallel to the depth d of the5*
MOMEKTS FOR BEAMS.
Ch. Ill,
rectangle, artd normal to gh let lines be drawn equal to the squares of s\ thus ee is the square of CE, and gg is the square of CG. Now the‘:"elementary product is the elementary area EE multiplied by the ordr nate ee; hence Sae? is represented by a solid standing on bd ^'hose Variable height is shown by the shaded area ghhcg. But the volume of this solid,is the product of its length b and this shaded area. The curve ceg is» a parabola because each line ee is the square of the
corresponding altitude ce; accordingly the shaded area is onBHhird of ghkg. But gh is equal to d, and gg is (^)*; thus thi^ffiiaded area is represented by £. d. \d\ or j\d\ Hence
b X ^d3 =
is the mo^jnt of inertia of a rectangle about an axis of gravity and parallel to ItSt'base. The moment of inertia is a compound quantit^ffiSttlt1 ing by multiplying an area by the square of a distance ; it thHconta^R the linear unit-Tour tim^T If b = 3 inches ;Hl d = 4 tn^^Hthen^^Hid inched or the nu^^^^Bun^Rf I is biquadratic inches.,
Mc^Bsnts of inertia when referred to the same axis ■1 be	subtracted like any other qualitiesArt. 22
MOMENTS OF INERTIA.
S3
which are of the same kind. Thus, let there be a hollow rectangular section whose outside depth and
Fig. 14.
breadth are b and dand whose inside depth and breadth are //, and d,, the thickness of the metal being the same throughout. Then the moment of inertia of this section is found by subtracting the moment of inertia of the inner rectangle from that of the outer one, or
I ss febd* — xV^i3
is the moment of inertia for the section whose area is bd — bxdx.
For the common I beams whose flanges are equal the same method applies. Let b be the width of the flanges and d the total depth ; also let t be the thickness of the web and t1 the thickness of the flanges. The moment of inertia of the area (b — t){d — 2t,) is then to be subtracted from the moment of inertia of the area bd, or
/ =	— -^{b — t){d — 2 t,y
is the moment of inertia for the I section.
For the T section the distance c from the end of the web to the axis through the center of gravity must first be computed by the method of the last article. Then the distance from the outside of the flange to54
MOMENTS KOR REAMS.
Ch. III.
the axis, is also known. Let b be the breadth of the flange and tx its thickness, and let t be the thickness of the web. Then the moment of inertia of the area tc is one-half of that of a rectangle of depth 2c, or£ X tV(2^)S>
which is \tc%; also the moment of inertia of the area bcx is one-half of that of a rectangle of depth 2cx. Adding these together and subtracting the moment of inertia of the area (b — t)(cx — tx), there results
/ = W + iK - Ub - t)(cx - /,)*,
which is the moment of inertia for the T section. The same formula applies to the n section if t be the thick ness of the two webs.
The above formulas for I and T sections are correct for cast-iron beams where the corners are but little rounded. For wrought iron and steel beams, however, the flanges are not usually of uniform thickness, and all the corners are rounded off by curves, so that the formulas are not strictly correct; for such shapes the numerical values of the moments of inertia for all the sections in the market are published by the manufacturers, so that it is not necessary for engineers to compute them. (See Art. 30.)
In this chapter the fundamental applications of moments to be used in discussing beams have been presented, and it is now possible to take up the subject and give the theory of equilibrium of beams clearlyArt. 22.
MOMENTS OF INERTIA.
55
and logically, so that the student may undertake practical problems in the most satisfactory manner.
Prob. 22. A steel I beam weighing 8o pounds per linear foot is 24 inches deep, its flanges being 7 inches wide and inches mean thickness, while the web is 0.5 inches thick. The moment of inertia stated by the manufacturer is 2088 inches*. Compute it by the formula here given. Q56
CANTILEVER AND SIMPLE BEAMS
Ch. IV.
Chapter IV.
CANTILEVER BEAMS AND SIMPLE BEAMS.
Art. 23. Definitions and Principles.
A simple beam is a bar resting upon supports at its ends, and is the kind most commonly in use. A cantilever beam is a bar resting on one support at the middle, or if a part of a beam projects out from a wall or beyond a support this part is called a cantilever beam. In a simple beam the lower part is under tension and the upper part under compression ; in a cantilever beam the reverse is the case. Unless other wise stated, all beams will be regarded as having the section area uniform throughout the entire length.
Since a beam is at rest the internal stresses in any section hold in equilibrium the external forces on each
Fig. 16.
side of that section. Thus, if a beam be imagined to be cut apart and the two parts separated, as in the figure, forces must necessarily be required to prevent the parts from falling. These internal forces or stresses may be resolved into horizontal and vertical com.Art. 23.
DEFINITIONS AND PRINCIPLES.
57
ponents. The horizontal components are stresses of tension and compression, while the vertical components add together and form a stress known as the resisting shear.
Each side of the beam is held in equilibrium by the vertical forces and stresses that act upon it. The vertical forces are the reaction and the loads, the stresses are the horizontal ones of tension and compression, and the vertical one of shear. The sum of all the horizontal tensile stresses must be equal to the sum of all the horizontal compressive stresses, or otherwise there would be longitudinal motion. The sum V of the vertical stresses must equal the algebraic sum of the reaction and loads, or otherwise there would be motion in an upward or dowmvard direction. Lastly, the sum of the moments of the stresses in the section must equal the sum of the moments of the vertical forces, or otherwise there would be rotation. These statical principles apply to each part into which the beam is supposed to be divided.
It is found by experiment that the fibers on one side of the beam are elongated and on the other shortened, while between is a neutral surface, which is
c
T*------
Z
1 da
I
c
I
I
- _______________
	*	
Neutral Surface	■«- *■
		
Fig. 17.
unchanged in length. It is also found that the amount of elongation or shortening of any fiber is directly proportional to its distance from the neutral surface. Hence, if the elastic limit be not surpassed, the stresses58
CANTILEVER AND SIMPLE BEAMS.
Ch. IV.
are also proportional to their distances from the neutral surface.
From the above it can be shown that the neutral axis passes through the center of gravity of the cross-section. For if 6' be the unit-stress on the remotest fiber and c its distance from the neutral axis, then the
unit-stress at the distance z is S-, and the total stress
c
on an elementary area a is A'—. The algebraic sum of
all the horizontal stresses in the section then is -2az.
c
From the above statical principles this sum must be zero, and it hence follows that 2az must be zero ; that is, the sum of the moments of the elementary areas is zero with respect to the neutral axis. Hence the neutral axis passes through the center of gravity, for the center of gravity is that point upon which the surface can be balanced or for which 2az — o.
Prob. 23. An I beam which is 20 feet long weighs 700 pounds and the area of its cross-section is 10.29 S(luare inches. What is the kind of material ?
Art. 24. Resistance to Shearing.
When a beam is short it sometimes fails by shearing in a vertical section near one of the supports. The force that produces this shearing is the resultant of all the vertical forces on one side of the section. Thus, in the simple beam of the first diagram this resultant is the reaction minus the weight of the beam between the reaction and the section ; in the cantilever beamArt. 24.
RESISTANCE TO SHEARING.
59
of tlie second diagram it is the loads and the weight of the beam on the left of the section.
4 Vertical shear* is the name given to the algebraic sum of all the vertical forces on the left of the section under consideration. Thus in the first diagram, if the
Fig i8
reaction be 6 ooo pounds, the vertical shear F just at the right ot the support is 6 ooo pounds, if the beam weigh ioo pounds per linear foot, the vertical shear at a section one foot from the support and on the left of the single load is 5 900 pounds. Again in the second diagram, if the beam weigh 100 pounds per linear foot and if each concentrated load be 800 pounds, and the distance from the end to the section shown be 4 feet, the vertical shear in that section is 2000 pounds.
It is seen from these illustrations that in a simple beam tlie greatest vertical shear is at the supports, and that in a cantilever beam it is at the wall. Only these sections, then, need be investigated in a solid beam. For a simple beam of length l and carrying w pounds per linear unit, the greatest vertical shear is the reaction \wl. For a cantilever beam of length /,to
CANTILEVER AN# SIMPLE BEAMS.
Ch. IV.
the greatest vertical sheaf due to uniform load is the total weight tvl.
The Vertical shear V produces in the cross-section an equsffijShearing stress. If A be the section area and S thé shearing unit-stress acting over that area, then
V	V
V=AS, S=L	(3)
are the equations similar to (l) of Art. i : these are used for the practical computations regarding ilhear in solid beams.
For example, consider a steel I beam weighing 2$o pounds per yard and 12 feet long, over which roll th^^Bcomotive wheels 4 feet apart and each bearing" 14 000 pounds. The greatest shear will occur when
..'	1	""^"TSr
Pig 19.
one wheel is almost at the support as shown in the figure. By Art. 18 the reaction is found to be 28 500 pounds. aiHthis is the greatest vertical shear V. By Art. 8 the area of the cross-section is found to be 24.5 square inches. Then the shearing unit-stress in the se^Sm is
.S ~ "24 5° ~ 1 I^° Pounc^s Per S(luare inch.
which is a low working unit-stress for steel.
As a secon^Rxample, consider a wooden cantilever beam w^fth projects out from a bridge floor and sup-Art. 25.
RESISTANCE TO BENDING.	6l
ports a sidewalk. Let it be 6 inches wide, 8 inches deep, and 7 feet long, and let the maximum load that comes upon it be 7 500 pounds. The vertical shear at the section where it begins to project is then 7 590 pounds, or the load that it carries plus its own weight. As the section area is 48 square inches, the shearing unit-stress is a little less than 160 pounds per square inch. The factor of safety against shearing is hence about 19 (Art. 6), so that the security is ample.
It is indeed only in rare instances that solid beams of uniform cross-section are subject to dangerous stresses from shearing. Beams almost universally fail by tearing apart under the horizontal tensile stresses, and hence the following articles will be devoted entirely to the consideration of these bending stresses.
Prob. 24. A simple beam of cast iron is 3 X 3 inches in section and 5 } feet long between supports. Besides its own weight it is to carry a load of 4 000 pounds at the middle and a Iqad of 1 000 pounds at 2I feet from the left end. Find the factor of safety against shearing.
Art. 25. Resistance to Bending.
In Art. 24 it was shown that the resisting moment of the internal stresses in any section is equal to the bending moment of the external forces on each side of the section. Art. 19 explains how to find the bending moment, which hereafter will be designated by the letter J/. In Art. 20 an expression for the resisting moment is derived. Therefore
SI
T = M	^
is the fundamental formula for the discussion of theÔ2
CANTILEVER AND SIMPLE BEAMS.
Ch. IV.
flexure of beams, provided that the elastic limit is not exceeded. Here S is the unit stress of tension or compression on the top or bottom of the beam, c is the vertical distance of .S' from the center of gravity of the cross-section, and / is the moment of inertia of the cross-section. Art. 21 explains how to find c, and Art. 22 shows how I is determined.
This formula shows that 5 varies directly with M, that is, the greatest tensile or compressive stress in the beam occurs at the section where M has its maximum value. For a simple beam under uniform load the bending moment M at any section distant * from the left support is, as shown in Art. 19,
M — \wl. x — wx .\x — \iv(lx — x2),
and if x — il, this gives M — \rvl% as tlm maximum bending moment; or if IV be the total load ul, this may be written as M — ill7. When concentrated loads are on a simple beam the maximum bending mo-
Fig; 20.
ment must usually be found by trial; it will generally be under one of those loads.
For a cantilever beam of length / the maximum bending moment always occurs at the wall. For aArt. 25.
RESISTANCE TO BENDING.
63
uniform load of tv per linear unit the bending moment at a section distant x from the end is the load wx into its lever arm t}.r, and this is negative as it tends to produce rotation in a direction opposite to that of the hands of a watch (Art. 17). Thus, for any section M — — 2~vx*> an^ when x becomes equal to / the maximum value is M = — ift'/*. The negative sign shows merely the direction in which rotation tends to occur, and when using formula (4) the value of J\I is to be inserted without sign. The diagram of bending moments for this case is a parabola, since J\I increases as the square of x.
For concentrated loads on a cantilever beam the bending moment M is — I\x until ur passes beyond the second load ; then M — — Pvx — Pt(x — a) where a is the distance between the two loads. Thus the dia-
gram of bending moments is composed of straight lines, and the maximum value of M occurs when x becomes equal to /.
It may be noted that the only difference in stating moment equations for a cantilever beam and for a simple beam lies in the fact that for the former there is no reaction at the left end. A cantilever beam is hence really simpler than a simple beam, as no reactions need be computed.M
CANTILEVER AND SIMPLE REAMS.
Ch. IV.
Prob. 25. A cantilever beam has a load of 800 pounds at it$ end and is also uniformly loaded with 125 pounds per linear foot ; its length is 5 feet. Compute the bending moments for five sections, one foot apart, and construct the diagram of bending moments.
Art. 26. Safe Loads for Beams.
A safe load for a beam is one that produces a tensile or compressive unit-stress which is safe according to the principles set forth in Chapter I. To find such a safe load for a given beam the safe value of S is to be assumed from those principles. Then in formula (4) the values of / and c are known. The maximum bending moment M is to be expressed in terms of the unknown load, and thus an equation is derived from which the load is found.
For example let a wooden cantilever beam be 2 inches wide, 3 inches deep, and 72 inches long, and let it be required to find what load P at the end will produce a unit-stress of 800 pounds per square inch. Here the maximum value of M is P X 72. From Art. 21 the value of c is inches and from Art. 22 the value of /is 4I inches4. Then by (4) of Art. (25),
SI I 800x4,5 I p
c	1.5	'
from which A* is found to be 33-J- pound.s.
Again, let a simple beam of cast iron be 3 inches wide, 4 inches deep, and 36 inches long, and let it be required to find what uniform load will produce a unit, stress of 2 000 pounds per square inch. Here let w be the uniform load per linear inch ; the total load is wl,Art. 27.
INVESTIGATION OF REAMS.
6S
each reaction is \wl, and the maximum bending moment M is	The value of c is 2 inches, and that
of 1 is 16 inches4. Then since / is 36 inches,
SI _ 2 000 X 16 c	2
162 ft',
from which w = 98.8 pounds per linear inch, and hence the total uniform safe load that can be put on the beam is about 3 560 pounds.
The student should notice that in using formula (4) all lengths must be expressed in the same unit. If the length of a beam is given In feet it must be reduced to inches for use in the formula, because S, /, and c are expressed in terms of inches. Formula (4) cannot be used to find the load that will rupture a beam, except in the manner indicated in Art. 63.
Prob. 26. A steel I beam 7 inches deep and weighing 22 pounds per foot has for the moment of inertia of its cross-section 52.5 inches4, and it is to be used as a simple beam with a span of 18 feet. What load P can it carry when the greatest unit-stress S is required to be 12000 pounds per square inch.
Art. 27. Investigation of Beams.
To investigate a beam acted upon by given loads the greatest unit-stress 5 produced by those loads is to be found from formula (4). From the given dimensions of the beam / and c are known, from the given loads the maximum value of M is to be found ; hence
- Me66
CANTILEVER AND SIMPLE BEAMS.
Ch. IV.
is the equation for computing the value of S. Then by the rules of Chapter I the degree of security of the beam is to be inferred. As formula (4) is deduced under the laws of elasticity, it fails to give reliable values of .S when the elastic limit is exceeded.
For example, consider a cast-iron u section which is used as a simple beam with a span of 6 feet, and upon which there is a total uniform load of 80 000 pounds. Let the total depth be 16 inches, the total width 12 inches, the thickness of the flange 2 inches, and the thickneW^f the webs 1 inch. By Art. 25 the greatest value of M is at the middle of the beam, this being
X 80000 X 6 X 12 — 720OOO pound-inches, By Art. 21 the value of c is found to be 10.7 incites. By Art. 22 the value of / is found to be 1 292 inches*. Then,
S «	5 960 pounds per square inch.
This, is the compressive unit-stress in the end of the Web when the beam is placed in the l_j position, as ill usually the case in buildings. On the base of the beam the tensile unit-strcsjBH about half this value, since ct is abHt one-half of e. Thus under the compressive stress the beam has a factor of safety of about 15, and truin' the tensile ^stress it has a factor of safety of abHt 7. As the least factor of safety for cast iron 10, the beam has not the full degree of security required by the best practice.
Prob. 27. A piece of wooden scantling 2 inches square atifls feet long is hung horizontally by a rope at each end and a student weighing 175 pounds stands upon it. Is is safe ?Art. 28.
DESIGN OF BEAMS.
67
Art. 28. Design of Beams.
The design of a beam consists in determining its size when the loads and its length are given. The allowable working unit-stress S is first assumed according to the requirements of practice. From the given loads the maximum bending moment M is then computed. Thus in formula (4) everything is known except / and c, and
c ~ S
is an equation which must be satisfied by the dimensions to be selected.
For a rectangular beam of breadth b and depth d the value of c is %d, and the value of / is bd*. Thus the equation becomes
and if either b or dbe assumed the other can be computed. For example, let it be required to design a rectangular wooden beam fora total uniform load of 80 pounds, the beam to be used as a cantilever with a length of 6 feet, and the working value of .S' to be 800 pounds per square inch. Here the maximum value of J/ is 80 X 3 = 240 pound-feet = 2 880 pound-inches. Thus bd'1 — 21.6 inches8. If b be taken as 1 inch, d = 4.65 inches ; if b be 2 inches, d — 3.29 inches ; if b be 3 inches, d — 2.68 inches. With due regard to sizes readily found in the market 3X3 inches are perhaps good proportions to adopt.
Prob. 28. A simple cast-iron beam of 14 feet span car-68
CANTILEVER AND SIMPLE BEAMS.
CH. IV.
ries a load of ioooo pounds at the middle. If its width is 4 inches find its depth for a factor of safety of 10 ; also find its width for a depth of 12 inches.
Art. 29. Comparative Strengths.
The strength of a beam is measured by the load it can carry with a given unit-stress S. Let it be required to investigate the relative strengths of the four following cases :
1st. A cantilever loaded at the end with W.
2d. A cantilever loaded uniformly with W.
3d. A simple beam loaded at the middle with W.
4th. A simple beam loaded uniformly with W.
Let / be the length in each case, and the cross-section be of breadth b and depth d. Then c = and 1 — ^bd*. Then from Art. 25, and formula (4),
For 1st,	K II s# &	and	W =	Sbd' 61 ’
For 2d,	M = % Wl,	and	W =	Sbda 2~6T
For 3d,	M= ÌWI,	and	W =	Sbd5 4 61
For 4th,		and	W =	„SU' Sr6T
Hence the comparative strengths of the four cases are as the numbers 1, 2, 4, 8; that is, if four such beams are of equal size and length and of the same material, the second is twice as strong as the first, the third is four times as strong, and the last is eight times as strong as the first.Art. 30.
STEEL I BEAMS.
69
From these equations the following important laws regarding rectangular beams are derived :
The strength varies directly as the breadth and directly as the square of the depth.
The strength varies inversely as the length.
A beam is twice as strong under a distributed load as under an equal concentrated load.
The second and third of these laws apply also to beams having cross-sections of any shape.
The reason why rectangular beams are placed with the longest dimension vertical is now seen to be that the strength increases in a faster ratio with the depth than with the breadth. If the breadth be doubled the strength is doubled ; if the depth be doubled the strength is four times as great as before.
Prob. 29. Show that a beam 3 inches wide, 6 inches deep, and 4 feet long, is nine times as strong as a beam 7 inches wide, 4 inches deep, and io| feet long.
Art. 30. Steel I Beams.
Wrought-iron rolled beams have been much used in bridge and building construction, but now medium-steel beams are almost exclusively employed. The ultimate tensile strength of such steel will be taken as 65 000 pounds per square inch, and its elastic limit as 35000 pounds per square inch, in the solution of examples and problems hereafter given. These beams are manufactured in about thirteen different depths, and of each depth there are several different sizes or weights, so that designers have a large variety from which to select. In the following table only the heaviest and lightest sections of each depth are given. The7o
CANTILEVER AND SIMPLE BEAMS,
Ch. IV.
Table Vili. Steel I Beams.
Depth.	Weight per Foot.	Section Area. A	Moment of Inertia. /	Section Modulus. /	Moment of Inertia. /'
Inches.	Pounds.	Sq. Inches.	Inches4.	Inches3.	Inches4.
24	IOO	29.4	2 380	198	48.6
24	80	23-5	2 088	174	42.9
20	75	22.1	I 269	127	30 2
20	65	I9.I	I 170	II7	27.9
l8	70	20.6	92I	102	24.6
l8	55	15-9	796	88.4	21.2
15	55	15.9	SU	68.1	I7.I
15	42	12.5	442	58.9	14.6
12	35	10.3	228	38.0	IO.I
12	3i£	9-3	216	36.0	9.5O
IO	40	ii.8	159	317	9 50
IO	25	7-4	122	24.4	6.89
9	35	10.3	112	24.8	7.31
9	21	6.3	85.0	18.9	5.16
8	25i	7.50	68.4	17.1	4-75
8	18	5 33	569	14.2	3-73
7	20	5 88	42.2	12.i	3-24
7	15	4.42	36-2	10.4	2 67
6	I7i	5-07	2Ó.2	8.73	2.36
6	12i	3.61	21.8	7-27	1.85
5	I4J	4*34	15-2	6.08	1.70
5	9*	2.87	12.1	4.84	1.23
4	10 i	3-09	7-1	3-55	i.or
4	7i	221	6.0	3-co	0.77
3	7}	2.21	2.9	1.93	0.60
3 !	k	1.63	2.5	1.71	0.46
proportions of one of the sizes of the two 6-inch beams are shown in the accompanying figure, “the outer lineArt. 30,
STEEL I BEAMS.
71
on the right-hand side indicating the heavier section and the other one the lighter section.
In this table the moments of inertia / in the fourth column are those about an axis through the centers of gravity and perpendicular to the web, and are those to be used in all beam computations. The values 1' given in the last column are with respect to an axis through the center of gravity but parallel to the web; these are for use in the next chapter in the discussion of struts.
The quantity — is often called the ‘section modulus’ c
as it contains all the dimensions of the cross-section. The process of selecting an I section depends merely
on finding a value of - which corresponds to the value
of as shown in Art. 28; hence for convenience these
values are tabulated in the fifth column of the table.
For example, an I beam in a floor is to have 20 feet span and to carry a uniform load of 13 500 pounds; what size is to be selected ? The bending moment is72
CANTILEVER AND SIMPLE BEAMS.
Ch. IV.
M = ■§• X 13 500 X 20 X 12 * 405 000 pound-inches, and the working unit-stress 5 should be -Jt X 65 OOO pounds per square inch. Then from formula (4),
/	405 000	. .	,
- sas ------= 31.2 inches ,
c 13 000
and hence, from the table, the heavy 10-inch beam should be »std.
Prob. 30. A heavy i5*inch steel I beam of 12 feet span carries a Uniform load of 42 net tons. Find its factor of safety if the span be 6 feet; also if the span be 9 feet.
Art. 31, Beams of Uniform Strength.
The beams thus far dfecussed have been of uniform section throughout their entire length. As the bending moments are small near the ends of the beam the unit-stress <S is there also small, and hence more material is used than is really needed. A beam of uniform strength is one so shaped that the unit-stress S is the same at all part^of the length. Such beams are sometimes made in cast iron, but rarely in other materials.
Fig. 23.
For a cantilever beam loaded with P at the end, the bending moment at any distance x from the end is Px. If the section is rectangular, formula (4) reducesArt. 31.
BEAMS OF UNIFORM STRENGTH.
73
to	— Px, in which /’and S are constant. If b is
made the same throughout, then
6Px
~SP
d'
and therefore dP must vary directly as x. If x — /, the value of d is the depth dx at the wall, and accordingly 6P/Sb =ss d*/l\ hence the equation becomes
d= d,\'
Thus, \{ x — \I, d
K
if x — \l, d — \dx, and so
on. As the squares of the depths vary with the distances from the end, the curve of the beam should be the common parabola.
For a rectangular cantilever beam uniformly loaded with to per linear unit the bending moment M is ^wx5
and formula (4) becomes ±Sbd‘ = %wx*. If the breadth be uniform throughout, then
d2
ywx_
Sb
Here if x — l, the value of d is the depth d, at the wall, and thus y^u/Sb — d*/P. Accordingly,
d74
CANTILEVER AND SIMPLE BEAMS.
Ch. IV.
gives the depth for any value of x, and it shows that the elevation of the beam should be a triangle.
The vertical shear near the end of the beam modifies slightly the form near the end. Thus for the first case above, if S' be the working shearing unit-stress there must be a section at the end whose area A is equal at least to P/S'.
Prob. 31. A simple beam of uniform strength is to be designed to carry a heavy load P at the middle. If be the depth at the middle, show that the depths at distances 0.1/, 0.2/, 0.3/, and 0.4/ should be 0.45^/,, 0.63^,, 0.77*/,, and 0.89 dvArt. 32.
GENERAI. PRINCIPLES.
75
Chapter V.
COLUMNS OR STRUTS.
Art. 32. General Principles.
A bar under compression whose length is greater than about ten times its thickness is called a column or a strut. For shorter lengths the case is one of direct compression where the rules of Art. 5 apply. For the short specimen failure occurs by the shearing or splintering of the material. P'or the strut or column, however, failure generally occurs by a sidewise bending ; this induces bending stresses, so that the phenomena of stress are more complex than in a beam.
Wooden and cast-iron columns are usually square or round, and are sometimes built hollow. Wrought-iron columns are made by riveting together channels, plates, and angle-irons. It is clear that a square or round section is preferable to a rectangular one, since then the tendency to bend is the same in all directions. For a rectangular section the bending will evidently occur in a plane parallel to the shorter side of the rectangle ; thus in investigating such a column the depth d is this shorter side instead of the longer one, as in beams. When a single I beam is used as a column it tends to bend in a plane parallel to the flanges, and hence the moment of inertia to be used in its discussion is F, which is given in the last column of the table in Art. 30, the axis for this coinciding with the middle line of the web.76
COLUMNS OR STRUTS.
Ch. V.
If a short prism whose section area is A be loaded with the weight P, the unit-stress is PJA, and this is uniformly distributed over the area A. For a column, however, this is not the case ; while the mean unit-stress is still PJA the unit-stress on the concave side, if bending occurs, may be very much greater than PJA. The longer the column the greater is this unit-stress on the concave side liable to become, and hence a long column cannot carry so large a load as a short one.
There are three ways of arranging the ends of columns. Class (a) includes those with ‘round ends’ or
those having their ends hinged on pins. Class (5) includes those with one end round and the other fixed ; the piston-rod of a steam-engine is of this type. Class (c) includes those having fixed ends ; these are used in bridge and building constructions. The figure here given is a symbolical representation, and is not intended to imply that the ends of the columns are necessarily enlarged in practice. It is found by experiment that class (r) is stronger than (b), and that (b) is stronger than (a).
Prob. 32. In a certain test wrought-iron tubes 2.37 inches in outer diameter and having a section area of 1.08Art. 33.
RADIUS OF GYRATION.
77
square inches were used. A tube 8 feet long failed under 24800 pounds and a tube 3^ feet long failed under 38200 pounds. What load would be required to cause failure for a tube only 6 or 8 inches long ?
Art. 33. Radius of Gyration.
In the discussion of columns a quantity r, called ‘radius of gyration of the cross-section,’ is frequently used. It is defined to be that quantity whose square is equal to the least moment of inertia of the cross-section divided by the area of that cross-section, or
Thus, for a heavy 10-inch jeam it is found by the table in Art. 30 that r'2 — 9.50/11.8 = 0.805 inches’.
The values of r* for rectangles are readily obtained from the moments of inertia given in Art. 22 :
For a solid rectangle, r1 = tV2^»
^	„	,	, bd -b,d?
for a hollow rectangle, r = —---------=—
&	\2{bd—bxd$
in which d is the least side of the rectangle.
The reason why the least moment of inertia is used for columns is that the bending tends to occur in a plane perpendicular to the axis about which the moment of inertia is the least. Thus, a rectangular strut bends in a plane parallel to the least side of its cross-section.*8
COLUMNS OR STRUTS.
Ch. V.
As circular cross-sections are frequency used for columns, the values of the moment of inertia for these will here be stated without proof. Let d be the outer diameter and the inner diameter. Then
For a hollow circle, /as -fen(d*— d*), pz^fe^d*-^d?).
Here the values of P are found by dividing the first / by \nd2 and the second 1 by \nici'1—d*), these being the areas of the cross-sections.
Prob. 33. Prove that the square of the radius of gyration for a hollow square section is	T </,*)«
Columns and struts generally fail under the stresses produced by combined compression and bending. The phenomena are so complex that no purely theoretical formula will fully represent all cases. The formula of Rankine is^that which has the best rational basis, but this cannot here be fully developed, as the laws of deflection have not yet been discussed.
Let P be the load on the vertical column, and let a horizontal plane ab cut it at the middle. If A is the section area, the average compressive unit-stress P/A may be represented by the line cd. But in consequence of the bending this is increased to aq on the concave side and decreased to bq on the convex side. The triangles pdq and qdp represent the longitudinal bending stresses, as in beams. Let the maximum unit-stress aq be denoted by .S'. The part ap is equal
For a solid circle, I — fend*,
Art. 34. Formula for Columns.Art. 34.
FORMULA FOR COLUMNS.
1
to cd or to P/A. The part fq is due to the bending and will be denoted by Sr Hence the maximum unit-stress aq is given by
Now from the formula (4) established for cases of bending in Art. 25, the value of S] is Mt'/f, where ill is the bending moment of the external forces. Here
the only external force is /\ and its lever arm is the lateral deflection of the central line of the column. Let this lateral deflection be called f; then Jlf = P/, and accordingly,
.S'
P , Pcf I ’
where c represents tlie distance ac in the figure.
Now let I be replaced by At“*, where r is the radius of gyration of the cross-section. Then the preceding equation becomes
s =
which shows how the unit-stress S on the concave side8o
COLUMNS OR STRUTS.
Ch. V.
increases with the lateral deflection f By a discussion of the subject of deflection, such as is given in ‘ Mechanics of Materials,’ it is shown that the value of
p
p
Fig. 27.
/"which is liable to occur increases as the square of the length of the beam, or cf may be made equal to ql‘, where q depends upon the kind of material and the arrangement of the ends (see Art. 48). The last equation may now be written,
P
A
c
iJ
(5)
which is Rankine’s formula for columns.
The values of q to be used in problems and examples in this book are given in the following table. These mean values have been derived by the consideration of numerous experiments on the rupture ofArt. 35.
SAFE LOADS FOR COLUMNS.
81
Table IX. Column Constants g.
r~ Material.	Both Ends Fixed.	One Fixed End and One Round End.	Both Ends Round.
Timber	I	1.78	4
	3 OOO	3 OOO	3 OOO
Cast Iron	I	1.7s	4
	5 OOO	5 000	5 OOO
Wrought Iron	I	1.78	4
	35 000	35 000	35 000
Steel ..	1	i.7s	4
	25 OOO	25 OOO	25 OOO
columns and	struts. It is	seen that in	all cases q is
four times as large for round ends as for fixed ends,
o	7
which results from the fact that a very loner column with round ends has only one-fourth the strength of one with fixed ends.
Prob. 34. If r/A — 500 pounds per square inch for a timber column with fixed ends, find from (5) the values of .S when //r = o, l/r — 50, and 1/r — 100.
Art. 35. Safe Loads for Columns.
To find a safe load for a column of given size and material the working value of .S' is to be assumed by Art. 7. The value of r is determined by Art. 33, and q by the table in Art. 34. Then, from the formula (5),
P-
I q ~‘t
1 i r
vrhlth gives the safe load for the column.82
COLUMNS OR STRUTS.
Ch. V.
For example, let it be required to find the safe load for a timber strut 3X4 inches and 5 feet long, so that the greatest compressive unit-stress A may be 800 pounds per square inch. Here b — 4 inches, d = 3 inches, r3 = -fed' s* £ inches3, /a = 3 600 inches3, P/P = 4 800, q = 1/3000, ql'/r* — 1.6. Then
which is the safe load for the strut. If the length be only about one foot, the safe load will be simply P — 12 X 800 = 9600 pounds. If the length be 12 feet, P will be found by the formula to be only 940 pounds. The influence of the length on the safe load is hence very great.
Prob. 35= A hollow cast-iron column to be used in a building is 6 X 6 inches outside dimensions and 5X5 inches inside dimensions, the length being 18 feet, and the ends fixed. Find its safe load.
The investigation of a column under a given load consists in computing the unit-stress A' from formula (5) and then comparing this with the ultimate strength and elastic limit of the material, having due regard to
p _ 1 2^X/8oo
“ 1 —1.6
— 3 690 pounds,
Art. 36. Investigation of Columns.
whether the stresses are steady, variable, or sudden (Art. 7). The value of S is
and the given data will include all the quantities in the second member.Art. 3G.
INVESTIGATION OE COLUMNS.
»3
For example, a wrought-iron tube used as a column with fixed ends carries a load of 38 000 pounds. Its outside diameter is 6.36 inches, its inside diameter 6.02 inches, and its length 18 feet. It is required to find the unit-stress 6' and the factor of safety. Here P = 38000 pounds, A = ^7r(6-36J — 6.023) m 3.31 square inches, q — 1/35 000, /= 18 X 12 = 216 inches, r* = ^^6.36* -|- 6.023) = 4.79 inches3. Then by the formula
or .S' = 14 700 pounds per square inch. The factor of safety is thus about 4, which is a safe value if the column is used under steady stress, but too small if sudden stresses or shocks are liable to occur. If the length of this column be 36 feet, the unit-stress .S will become about 25 000 pounds per square inch, so that its factor of safety will be only 2.2, a value far too low for proper security.
As a second example let a heavy 10-inch steel I beam which is 25 feet long be used as a strut in a bridge truss, the ends being hinged on pins. Let the compression on it be 5 900 pounds. Here from the table in Art. 30 A — 11.8 square inches and F — 9.50 inches4, whence r3 = 0.80 inches5; also q — 4/25 000, / — 300 inches, P — 5 900 pounds. Then from the formula 5 is found to be 9 500 pounds per square inch, which is about one-third of the elastic limit of the material, and hence a safe value.
Prob. 36. A pine stick 3X3 inches and 12 feet long is used as a column with fixed ends. Find its factor of safety under a load of 3 000 pounds. If its length be only one foot, what is the factor of safety ?84
COLUMNS OR STRUTS.
Cn. V.
Art. 37. Design of Columns.
When the length of a column is given and the load to be carried by it, the design consists in selecting the proper material and then finding the dimensions so that .S' may have the proper working value. This in general must be done by trial, dimensions being assumed and inserted in formula (5), and if these do not fit changes are to be made in them until a satisfactory agreement is found.
For example, let it be required to find the size of a square wooden column with fixed ends and 24 feet long to carry a load of 100000 pounds with a factor of safety of 10. Here as the ultimate strength of timber is 8 000 pounds per square inch the working value of .S' is 800 pounds per square inch. If the column be very short the area A will be 125 square inches and the side of the square about 11 inches, hence for the actual case the side of the square is more than 11 inches. Assume it 18 inches, and then from the formula of the last article find the value of S; this being less than 800, shows that 18 inches is too great. Again, trying 16 inches, this will be found to give a value of .S' a little larger than 800, while 17 inches makes 5 a little smaller than 80c. Thus, i6| inches is a satisfactory solution of the problem.
Prob. 37. Find what steel I beam 12 feet long may Le used as a column to carry a load of 100000 pounds, taking the working value of ,S at 12 000 pounds per square inch.Art. 38
PHENOMENA OF TORSION.
85
Chapter VI.
THE TORSION OF SHAFTS.
Art. 38. Phenomena of Torsion.
Torsion is that kind of stress which occurs when external forces tend to twist a body round an axis. A shaft which transmits power is twisted by the forces applied to the pulleys, and thus all its cross-sections are brought into stress. This stress is a kind of shearing, but the forces acting in different parts of a section are not parallel.
Let one end of a horizontal bar be rigidly fixed, and to the free end let a lever be attached at right angles
to its axis. A weight P hung at the end of this lever will twist the shaft so that a line ab which originally was horizontal will assume a spiral form ad, while the radial fine cb will move to the position cd. It has been shown by experiments that, if the material is not86
THE TORSION OF SHAFTS.
Ch. VI.
stressed beyond its elastic limit, the angles bed and bad are proportional to the applied weight P, and that on the removal of this weight the lines cd and ad will return to their original position. If the elastic limit be exceeded this proportionality does not hold, and if the stress be made great enough the bar will be ruptured.
Let / be the lever-arm of P with respect to the axis c. Then experience also shows that the amount of twist is proportional to p. The product Pp is the moment of Pwith respect to the axis, and it is called the ‘twisting moment.’ If there be several forces Pl, P3, etc., acting on the shaft with lever-arms /,,/,, etc., the total twisting moment Pp is the algebraic sum of the separate moments /*,/,, P^Piy etc., those being positive which tend to turn in the direction of the hands of a watch, and those negative which turn in the opposite direction.
For example, let the three lever-arms be applied to a bar at the points B, C, and D, whose distances from A
are 5, 8, and 12 feet. Let the forces in the figure be Py — 30 pounds, P3 = 60 pounds, and P3 = 100 pounds, their lever-arms being/, — 2.5 feet,/, = 2.0 feet, and p3 = 3.5 feet. Then for all sections between D and C the twisting moment is -f- 30 X 2.5 = -f- 75 pound-feet ; for all sections between C and B the twisting mo-Art. 39.
POLAR MOMENTS OF INERTIA.
87
meat is -f- 30 X 2.5 — 60 X 2.0 = — 45 pound-feet; and for all sections between B and A the twisting moment is —|- 30 X 2.5 — 60 X 2.0 -J- 100 X 3-5 = —1~ 305 pound feet. ' Thus the tendency to twist between B and C is in the opposite direction to that in the other parts of the bar.
Prob. 38. If a force of 600 pounds acting at 5 inches from the axis twists the end of a shaft 30 degrees, what force acting at 12 inches from the axis will twist it 60 degrees.
Art. 39. Polar Moments of Inertia.
In the discussion of shafts the moments of inertia of cross-sections are required with respect to a point at the center of the shaft and not with respect to an axis in the same plane, as in beams and columns. The ‘polar moment of inertia ’ of a surface is defined as the sum of the products obtained by Multiplying each elementary area by the square of its distance from the center of the surface. Thus if a be any elementary area and x its distance from the center the quantity 'Sax* is the polar moment of inertia.
In the figure let a be any elementary area and z its distance from an axis AB passing through the center
n
		1—v—zzoa
		
		
		C
Fig		3°-
of gravity of the section ; then Saz* is the moment of inertia with respect to the axis AB (Art. 22). Also, if88
THE TORSION OF SHAFTS.
Cu. VI.
y is the distance from a to an axis CD which is normal to AB, then 'ZZaf' is the moment of inertia with respect to the axis CD. But since z* y* = x2, the product 2ax5 is equal to 'Zaz1 -f- 2ay'; that is, the polar moment of inertia is the sum of the moments of inertia taken with respect to any two rectangular axes.
The polar moment of inertia is represented by J. By the aid of the above principle its value is readily found from the values of / given in Arts. 22 and 33. Let d be the diameter of a circle ; then,
For a solid circle, J — -fand*.
Also, in the case of a hollow section let d be the outer and dx be the inner diameter; then,
For a hollow circle, J = xzTt{d* — d j4).
The circular sections are most frequently used for shafts, and the discussions of this chapter apply only to such shafts. The theory of the torsion of square and rectangular bars is very complicated and cannot be given here.
Prob. 39. Show that the polar moment of inertia for the hollow circular section is -&A {d* -J- dx), where A is the section area.
Art. 40. Formula for Torsion.
If two cross-sections are taken in a shaft very near together, each section tends to twist with respect to the other, and shearing stresses are found to exist in all parts of the section. These stresses are zero at the center and greatest at the boundary of a circular sec-Art. 40.
FORMULA FOR TORSION.
89
tion, and they act everywhere perpendicular to the lever-arms drawn to them from the center. If the elastic limit be not exceeded it is found that the stresses are proportional to their lever-arms.
Let /’be the force acting with the lever-arm p which produces the twisting moment Pp. This must be equal
to the resisting moment of the internal stresses. Let a be the shearing unit-stress at the remotest part of the section whose distance from the center is c. Then the stress at a distance 'f from the center is 4A, and the
stress at a distance x from the center is S-. The total
c
stress on an elementary area a at a distance x from the
center is then .S -, and the moment of this stress with c
respect to the center is S----. The resisting moment is
c
ci x’1
the sum of all the values of A'— , or, since 5 and c are
c
constants, this sum is ' Sax*. But, as seen in the last
c
article, the quantity 2ax2 is the polar moment of iner-9°
THE TORSION OF SHAFTS.
Ch. VI.
tia J. Accordingly the resisting moment of the internal shearing stresses is —, and, equating this to the
c
twisting moment Pp, there results
^ = Pp,	(«)
which is the fundamental formula for the torsion of shafts with circular cross-sections.
This formula is analogous with formula (4) for beams, and is used in a similar manner to investigate and design shafts. The unit-stress 5 is here always a shearing stress, and its working values are to be determined by applying factors of safety to the ultimate shearing strengths given in Art. 6. Shafts which transmit power are subject to variable loads and often to shocks, and hence their values of S' should be taken low. Formula (6) is subject to the same limitation as formula (4), namely, it is only true when the unit-stress S is less than the elastic limit of the material (see Art. 63).
For example, if Pp =. 20000 pound-inches, let it be required to find the shearing stress produced by it in a circular shaft 4 inches in diameter. Here c — 2 inches, J = 25.13 inches1, and then by (6) the value of S is found to be 1 590 pounds per square inch. If the shaft be wood this is too low a value of S, it being about one-half of the ultimate shearing strength ; if it be wrought iron or steel there is a high factor of safety.
Prob. 40. A round steel shaft is subject to a twisting moment of 2 500 pound-inches. What should be its diameter so that the greatest shear S may be 6 000 pounds per square inch ?Art. 41.
SHAFTS TO TRANSMIT POWER.
91
Art. 41. Shafts to Transmit Power.
* Work ’ is the product of a force by the distance through which it is exerted. Thus, if a weight of 10 pounds be lifted vertically a distance of 5 feet there are performed 50 foot-pounds of work. If this weight be moved horizontally, however, the force required depends only on frictional and other resistances ; if these require a force of 3 pounds and this be exerted through a distance of 5 feet, then 15 foot-pounds of work are performed.
‘Power’ is work performed in a given time. The unit of power is the ‘ horse-power ’ which is defined as 33 000 foot-pounds of work performed in one minute. Thus, if 99000 foot-pounds of work are performed in one minute, the power exerted is 3 horse-powers ; if 99000 foot-pounds of work be performed in two minutes, the power exerted is 11 horse-powers.
Power from a motor is usually transmitted to a shaft by belts, and the shaft then transmits the power to the places where the work is to be performed. In doing this the shaft is brought under stress. Let H be the power transmitted through a belt to a pulley. Let P be the tangential force in pounds brought by the belt on the circumference of the pulley, and let p be the radius of the pulley in inches. Let n be the number of revolutions made by the shaft and pulley in one minute. In one revolution P pounds is exerted through 2np inches, and the work of P X 27xp pound-inches, or \nPp pound-feet, is performed. In one minute the work performed is \nnPp pound-feet. The number of92
THE TORSION OF SHAFTS.
Ch. VI.
horse powers exerted is found by dividing this work by 33 ooo, or
H f«||.
198 OOO
The twisting moment Pp may now be replaced by the resisting moment S//c, and hence
II 108000H	, I
— = s---------!	(7)
c	mt
which is the formula for the discussion of round shafts that are used to transmit power.
For such a shaft c is equal to one-half of the outer diameter, whether its section be solid or hollow. The unit-stress .S' is here always that for shearing, and in selecting its safe value a high factor of safety is to be used, as the shaft is subject to variable stresses. It is noticed that .S' varies inversely with n, that is, for a given power transmitted the slower the speed the greater is the stress in the shaft.
Prob. 41. If a round shaft one inch in diameter transmits one horse-power at 100 revolutions per minute, show that the shearing stress produced is about 3 200 pounds per square inch.
Art. 42. Solid Shafts.
For round solid shafts of diameter d, the polar moment of inertia is	the value of c is \d, and
formula (7) then reduces to
321000—,
J	n
in which d must be taken in inches and A in pounds per square inch. From this formula A may be found for a given shaft which transmits power, or d may beArt. 42.
SOLID SHAFTS.
93
computed when it is required to design a shaft for that purpose.
For example, let it be required to find the factor of safety of a round solid shaft of wrought iron, 2^ inches in diameter, when transmitting 25 horse-power at 100 revolutions per minute. Here d — 2.5 inches, H — 25, n sa 100, and the formula gives
„	321 000 X 25	.	.	,
o =	100 ~	I4° Poun“s Per s<4uare mch,
so that the factor of safety is about 10; this is a high value for a shaft not subject to shocks.
As an example of design, let it be required to find the diameter of a wrought-iron shaft when transmitting 90 horse-power at 250 revolutions per minute. Here the factor of safety will be taken at 8, or the allowable unit-stress 5 at 7 000 pounds per square inch. Then, from the formula,
rf" = —°°° X.9g I 7000 X 250
and hence the diameter d should be 2f inches.
The above formulas do not apply to square shafts ; in these the greatest stress is not at the corners but along the middle of the sides. It is shown in Mechanics of Materials (tenth edition) that the formula for a solid square shaft of side d is
II
Sd — 283600—.
n
For example, let it be required to find how many horsepowers are transmitted by a wooden shaft 12 inches square when it makes 25 revolutions per minute and X is 200 pounds per square inch. Here all quantities in94
THE TORSION OF SHAFTS.
Ch. VI.
the formula arc known except H} and the solution gives H — 305 horse-powers.
Prob. 42. Find the horse-power that can be transmitted by a solid steel shaft of inches diameter when making 150 revolutions per minute, S being 7 500 pounds per square inch.
Art. 43. Hollow Shafts.
Hollow forged steel shafts are now much used for ocean steamers, as their strength is greater than solid shafts of the same area of cross-section. If d be the outside and dt the inside diameter, the value of J is
n
32
O
c/,4) and c is \d. These inserted in (7) give
5
di - d*	H
-----:-- = 321 OOO 4
d J n'
which is the formula for investigation and discussion.
For example, a nickel steel shaft of 17 inches outside diameter is to transmit 16000 horse-powers at 50 revolutions per minute; what should be the inside diameter so that the unit-stress 6" may be 25 000 pounds per square inch? Here everything is given except dlt and from the equation its value is found to be 11 inches nearly. The area of the cross-section of this shaft will be about 132 square inches, and its weight per linear foot about 449 pounds.
Prob. 43. If a hollow shaft have the same area of cross-section as a solid one, and if the inside diameter of the hollow shaft be one-half of the outside diameter, prove that the hollow shaft is 44 per cent stronger than the solid
one.Art. 44.
THE MODULUS OF ELASTICITY,
95
Chapter VII.
ELASTIC DEFORMATIONS.
Art. 44. The Modulus of Elasticity.
It was explained in Chapter I that, when a bar is subject to stresses produced by gradually applied forces, the elongations increase proportionately with the stresses, if the elastic limit is not exceeded. This .law of elasticity enables the elongations of bars and the deflections of beams to be computed, provided that none of the stresses exceeds the elastic limit of the material.
The ‘modulus of elasticity’ in tension is the ratio of the unit-stress to the unit-elongation. Thus, if a bar one inch long and one square inch in cross-section be under the stress S an elongation s is produced, and
(8)
s
is the modulus of elasticity. If the bar have a section area A which is acted on by the pull P, then the unit-stress .S' is P/A ; if the bar have the length l an elongation e is produced and the unit-elongation s is given by e/I.
For compression E is the ratio of the unit-stress to the unit-shortening accompanying that stress, and in general h is the ratio of the unit-stress to the unit-deformation. As s is an abstract number, E is ex-96
ELASTIC DEFORMATIONS.
Ch. VII.
pressed in the same unit as S, that is, in pounds per square inch or kilos per square centimeter.
Within the elastic limit 5 increases at the same rate as s, and thus E is a constant; beyond the classic limit there is no proper modulus of elasticity. For different materials under the same unit-stress 5, the value of E increases as s decreases; thus if is a measure of the stiffness of materials. Formula (8) also gives
•y -
which is the change of a unit of length of a bar under a given unit-stress
The values of the moduluses of elasticity for tension and compression are practically the same, and their mean values for the different materials are given in the following table. For shear the moduluses of
Table X. Moduluses of Elasticity.
Material.	Pounds per Square Inch.	Kilos per Sq. Centimeter.
Timber	I 500 OOO	105 OOO
Cast Iron	15 OOO OOO	I 050 COO
Wrought Iron	25 OOO OOO	I 750 <00
Steel	30 OOO OOO	2 IOO OOO
elasticity are about one-third of those stated in the table. These values show that, within the elastic limit, steel is the stiffest of the four materials, it being .i j per cent stiffer than wrought iron, twice as stiff as cast iron, and twenty times as stiff as timber. In other words, a given stress less than the elastic limit will elongate a timber bar twenty times as much as aArt. 45.
ELONGATION UNDER TENSION.
97
steel bar, a cast-iron bar twice as much, and a wrought-iron bar 20 per cent more.
Prob. 44. A bar one inch square and 2 inches long elongates 0.0004 inches under a tension of 5 000 pounds. Compute the modulus of elasticity.
Art. 45. Elongation under Tension.
Let a bar whose section area is A and whose length is l be under the tension P, and let e be the elongation produced. The unit-stress ,S is P/A and the unit-elongation s is c/l. Then the modulus of elasticity E is
s Ac
and hence, if P/A be less than the elastic limit,
PI
e —
All
is the elastic elongation of the bar due to the applied tension P.
For example, let it be required to find the elongation of a wrought-iron bar 30 feet long when stressed up to its elastic limit. Here 1P/A = 25 OOO pounds per square inch, E s= 25 ooo000 pounds per square inch, and / = 360 inches. Then from the formula c ~ 0.36 inches. This is the elastic elongation ; the ultimate elongation will be about 72 inches. In all cases, as seen from the figure in Art. 4, the elastic elongations are very small compared with the ultimate elongations.
Prob. 45. A steel eye-bar 30 feet long is if X 6 inches in size. How much does it elongate under a pull of 90 ooo pounds ?98
ELASTIC DEFORMATIONS.
Ch. VII.
Art. 46. Shortening under Compression.
If a bar of cross-section A and length l be under the compression P it shortens the amount e. For a short bar where no lateral deflection can occur the unit-stress .S' is uniform over the cross section, and the shortening follows the same law as does the elongation in tension, and hence
PI
e ~ AE
Here, as before, the unit-stress P/A must not exceed the elastic limit of the material.
For example, let a cast-iron bar one inch in diameter and 5 inches long be under a compression of ioooo pounds. Here P — ioooo pounds, A — 0.785 square inches, l — 5 inches, and E — 15000000 pounds per square inch. Then from the formula e — 0.0042 inches; but this result is of no value, because the unit-stress P/A is nearly double the elastic limit of cast iron. If, however, P be given as 3 000 pounds, then the formula properly applies, and e is found to be 0.0013 inches.
Prob. 46. A wrought-iron bar 18 inches long weighs 24 pounds. How much will it shorten under a compression of 7 250 pounds ?
Art. 47. Deflection of Cantilever Beams.
The best method of deriving formulas for the deflections of beams is by the help of the calculus. These methods are given in higher works on the subject; see for instance Merriman’s Mechanics of Materials. The formulas will be stated here without proof, butArt 4r.
DEFLECTION OF CANTILEVER BEAMS.
99
be accompanied by illustrations showing their value and importance.
When a load P is at the end of a cantilever beam whose length is /, a deflection of that end results, which will be designated by f. This deflection will evidently
Fig. 32.
be the greater the greater the load and the longer the length of the beam. The formula for it is
in which P. is the modulus of elasticity of the material (Art. 44) and / is the moment of inertia of the cross-section (Art. 22).
When a uniform load is on the beam let this be called IV. Then the deflection is
/ =
ivr
8EÏ
It :s thus seen that the deflection varies as the cube of the length of the beam, so that if the length be doubled
Fig. 33.
the deflection wili be eight times as great. It is also seen that a uniform load produces only three-eighths as much deflection as a single load at the end.
For example, let it be required to compute the deflection of a cast-iron cantilever 2X2 inches and 6IOO
ELASTIC DEFORMATIONS.
Ch. VII.
feet long, due to a load of ioo pounds at the end. Here P = ioo pounds, / — y2 inches, E — 15000000 pounds per square inch, and / = -^2*= i-J- inches4. Then from the formula f — 0.622 inches, which is the deflection at the end.
For a rectangular cross-section of breadth b and depth d the value of / is -jVA/3. Thus the deflections of rectangular beams vary inversely as b and d\ As stiffness is the reverse of deflection, it is seen that the stiffness of a beam is directly as its breadth, directly as the cube of its depth, and inversely as the cube of its length. The laws of stiffness are hence quite different from those of strength.
Prob. 47. A steel I beam 8 inches deep and 6 feet long is used as a cantilever to cany a uniform load of 240000 pounds. What will be its deflection ?
Art. 48. Deflection of Simple Beams.
When a simple beam of span l has a load P at the middle, each reaction is \P. If this beam be imag-
ined to be inverted it will be seen to be equivalent to two cantilevers of length \l, each having the load \P at the end. Hence in the first formula for the deflection of a cantilever, given in Art. 47, if / be replaced by \l and P by ^P, it becomes
Fig. 34.Art. 48.
IIP'.FLECTION OF SIMPLE BEAMS.
IOI
which gives the deflection of a simple beam due to a load at the middle.
When a simple beam is loaded with w per linear unit the total load zvl is represented by W. The de-flection at the middle due to this load is
. 384^/’
which is only five-eighths of the deflection caused by the same load at the middle.
The formulas of this and the preceding article are only valid when the greatest horizontal stress S produced by the load is less than the elastic limit. These formulas can be expressed in terms of 5 by substituting the values of P and IF from the formula (4) of Art. 25. Thus for the simple beam with load at the middle \Pl .= SI/c, and for the uniform load \Wl — SI/c. Hence,
SP
For the single load P, f	;
12 Ec
KSP
For the uniform load IV, f — ■■■ - - ;
48 be
which show that the deflections of beams under the same unit-stresses increase directly as the squares of their lengths.
Prob. 48. In order to find the modulus of elasticity of oak, a bar 2X2 inches, and 6 feet long, was loaded at the middle with 50 pounds, and then with 100 pounds, the corresponding deflections being found to be o. 16 and 0.31 inches. Compute the modulus of elasticity E.102
ELASTIC DEFORMATIONS.
Ch. VII.
Art. 49. Restrained Beams.
A beam is said to be restrained at one end when that end is horizontally fixed in a wall and the other end rests on a support. In this case the reaction of
the support is less than for a simple beam. For a uniform load of w per linear unit over the span / it is proved in ‘ Mechanics of Materials ’ that the reaction at the support is \%vl, provided the elastic limit be not exceeded. The bending moment at any section distant x from the support then is \wlx — £wx2, and this shows that when x = \l there is no bending moment; when x — •§/ the greatest positive bending moment is •$r§wl2, and when x =■ l the greatest negative bending moment is •§zvl2; the distribution of bending moments being as shown in the figure. Also, the maximum deflection is
which occurs when x has the value 0.4215/.
For a beam fixed at both ends and uniformly loaded there is a negative bending moment	at each
wall and a positive bending moment -rfawP at the middle; also the deflection at the middle is
Fig. 35.
f- Wl'	- Wl%
' 7 185EI~ 185El'
wr
in which W is the total uniform load wl.Art. 50.
TWIST IN SHAFTS.
I03
1't is seen that in these restrained beams the lower side is partly in tension and partly in compression, since a positive bending moment indicates the former and a negative one the latter (Art. 25). For a simple beam the greatest bending moment is 117, for a beam fixed at both ends the greatest bending moment is ih 1/7; hence if botli be the same size the restrained
beam will carry the greater load, or if both carry the same load the restrained beam may be of smaller size than the simple one. Thus if beams can be fixed horizontally at their ends the construction may be more economical.
Prob. 49. When a beam is fixed at one end and supported at the other, the reaction of the supported end due to a load P at the middle is ,5eP. Show that there is a positive bending moment PI under the load, and a negative bending moment ^PI at the wall. Also draw the diagram of bending moments.
Art. 50. Twist in Shafts.
When a shaft of length / transmits//horse-powers at a speed of n revolutions per minute, one end of the shaft is twisted with respect to the other through an angle of J) degrees. If the elastic limit be not exceeded, this angle is, for a round shaft,
. B
= 3 5C0 000
DELASTIC DEFORMATIONS.
Cn. VII.
104
in which Jis the polar moment of inertia of the section (Art. 39) and / is the modulus of elasticity for shear (Art. 44). Here / must be taken in inches, J in inches4, and Ain pounds per square inch.
For example, let a steel shaft 125 feet long, 17 inches outside diameter, and 11 inches inside diameter transmit 16000 horse-powers at 50 revolutions per minute. Here//— 16 000 horse-powers, / = 1 500 inches, ^ = 50, F= 10000000 pounds per square inch, J = 6765 inches1. Then from the formula D — 25.3 degrees, which is the angle through which a point on one end is twisted relative to the corresponding point on the other end.
If this shaft revolve with a speed of only 25 revolutions per minute while doing the same work, its angle of twist will be twice as great and the stresses in it also twice as great as before. The formula also shows that the angle of twist varies directly as the length of the shaft.
Prob. 50. A solid steel shaft 125 feet long and 16 inches in diameter transmits 8 000 horse-powers at a speed of 25 revolutions per minute. Compute the angle of twist.Art. 51.
FUNDAMENTAL IDEAS.
1°5
ClIAFTER VIII.
RESILIENCE OF MATERIALS.
Art. 51. Fundamental Ideas.
When a force of uniform intensity P is exerted through the distance c the work performed is measured by the product Pc. When a bar is tested in a machine, however, the force gradually and uniformly increases from o up to the value P and produces the elongation e ; here the work performed is \Pe, because the average value of the uniformly increasing force is In the first place the work may be represented by a rectangle of height Pand base c; in the second case the work may be represented by a triangle of height P and base e\ and the area of the triangle is one-half that of the rectangle.
As the external force increases from o up to P the internal stress in the bar increases gradually and uniformly from o up to A. The internal work of these stresses is called the * resilience ’ of the bar. As the internal work equals the external work \Pe, this quantity is a measure of the resilience.
Strength is the capacity of a body to resist force; stiffness is the capacity of a body to resist deformation; resilience is the capacity of a body to resist work. The higher the resilience of a material the greater is its capacity to resist the work of external forces.
Elastic resilience is that internal work which has beenio6
RESILIENCE OK MATERIALS.
Ch. VITI.
performed when the internal stress reaches the elastic limit. Ultimate resilience is that internal work which has been performed when the body is ruptured. Ultimate strength is usually, from two to three times the elastic strength ; ultimate elongation is always much greater than elastic elongation ; and ultimate resilience is very much larger than elastic resilience.
Resilience, like work, is expressed in foot-pounds, or inch-pounds, usually in the latter unit. Thus, if a bar be subject to a stress which gradually and uniformly increases from oup to 5 000 pounds and is accompanied by an elongation of 0.5 inches, the resilience is 1 250 pound-inches.
Prob. 51. If a wrought-iron bar weighing 30 pounds per linear foot be subject to a stress of 5 000 pounds per square inch which is accompanied by an elongation of 0.5 inches, what is the resilience ?
Art. 52. Elastic Resilience of Pars.
Let a bar of length / and section area A be under a tension P, which produces a unit-stress .S' equal to the elastic limit of the material and an elongation e. The elastic resilience of the bar is then equal to \Pc. Now P = SA, and by Art. 45 the elastic elongation is e — Pl/AE — Sl/E; hence letting K represent the elastic resilience, the product \Pe becomes
(9)
or the elastic resilience of a bar is proportional to its section area and to its length', that is, to its volume.Art 52.
ELASTIC RESILIENCE OF EARS.
107
If the bar have a section of one square inch and a length of one inch, then Al is one cubic inch, and the elastic resilience is
in which 5" is the elastic limit of the material.
This quantity is called the modulus of resilience, since for any given material it is a constant. Tor bars under tension the average values of .S are given in Art. 2, and those of /: in Art. 44. Using these constants, the ‘ modulus of resilience ’ k has the following values for tension :
For timber,	k — 3 inch-pounds;
For cast iron,	k — 1 inch-pound ;
For wrought iron, k = 12 inch-pounds;
For steel,	k = 42 inch-pounds.
These figures show that the capacity of steel to resist work within the elastic limit is the greatest of the four materials, and that of cast iron the least.
For a bar of any size the elastic resilience is found by multiplying its volume by the modulus of resilience k. Thus, a bar of timber whose volume is 50 cubic inches has an elastic resilience of about 150 inch-pounds, that is, the external work required to stress it up to the elastic limit is 1 50 inch-pounds. The particular shape of the bar is unimportant ; it maybe 5 inches in section area and 10 inches long, or 2 inches in section area and 25 inches long, or any other dimensions which give a volume of 50 cubic inches.
The above formula (9) also gives the work required to produce any unit-stress A' which is less than theRKSII.IKNCK OK MATERIALS.
Ch. VIII.
elastic limit. For example, let it be required to find the work needed to stress a bar of wrought iron up to 12 500 pounds per square inch, the diameter of the bar being 2 inches and its length 18 feet. Here 6' = 12 500 pounds per square inch, li = 25000000 pounds per square inch, A = 3.14 square inches, and / = 216 inches. Then
If the bar is required to undergo this stress 250 times per minute, the work required in one minute is 250 X 2 120 s* 530000 inch-pounds = 44200 footpounds. The power expended in stressing the bar is hence 44 200/33 000 — 1 -34 horse-powers.
When a bar is under a unit-stress .S', and this is in. creased by additional exterior loads to the resilience is
provided that 5, be not greater than the elastic limit.
Prob. 52. A bar of steel 10 feet long and weighing 490 pounds is stressed in one second from 4 000 up to 9 oco pounds per square inch. What work and what horse-power are expended in doing this ?
Art. 53. Elastic Resilience of Beams.
When a simple beam of span / is brought into stress by a load P applied gradually and uniformly at the middle, the deflection f results and the work ^Pf is performed. This work equals the resilience of the beam. The value of f in terms of the horizontal unit-stress is given in Art. 48, and the value of P in terms
„ _ /2_5oo’ X 3-H X 216 2x25 000 000
= 2 120 inch-pounds.Art. 54.
U Lf IM AT E RESILIENCE.
IO9
of the unit-stress A' is given by (4) of Art. 25. Accordingly the [product ^Pf has the value
K
4L
S' P zE ‘ 3?
in which /, the moment of inertia of the section, has been replaced by its equivalent Ar\ where A is the section area and r its least radius of gyration (Art. 33).
For a simple beam under a full uniform load the elastic resilience is given by
$*	8P_
2 A 15 c2
K
AL
which is if times that of the simple beam with a single load at the middle.
These expressions show that the elastic resilience of beams of similar cross-sections are proportional to their volumes. For rectangular sections where the depth is d, the value of c is \d and that of P is	thus P/P
is f. Hence a rectangular bar under tensile stress has nine times the resilience of a rectangular beam loaded at the middle and 5f times that of the same beam under a full uniform load.
Prob. 53. What horse power is required to stress in one second a heavy 20-inch steel I beam of 24 feet span from 500 up to 8 000 pounds per square inch, this being done by a load at the middle ?
Art. 54. Ultimate Resilience.
The ultimate resilience of a body is equal to the external work required to produce rupture. The ultimate resilience greatly7 surpasses the elastic resilience, it being for wrought iron and steel sometimes five hundred times as large. It is not possible, however, to establishI IO
RESILIENCE OF MATERIALS.
Ch. VIII.
a formula by which the ultimate resilience can be computed, because the law of increase of the deformations beyond the elastic limit is unknown.
If a diagram be made showing the increase of elongation with stress, as in the figure of Art. 4, the abscissas indicating the elongations and the ordinates the stresses, then the area included between the curve and the axis of elongations represents the ultimate resilience for one cubic inch of the material. The total ultimate resilience is then found by multiplying this by the volume in cubic inches.
In Art. 14 it was remarked that the product of the ultimate strength and ultimate elongation is an index of the quality of wrought iron and steel. This is so because it is a rough measure of the ultimate resilience. A measure which more closely fits the area given by a stress-diagram is
k = ~h 2St)>
in which Se is the elastic limit, St the ultimate tensile strength, and s the ultimate unit-elongation. For example, take a wrought iron specimen where Se — 25 000 and St = 50000 pounds per square inch, while s = 30 per cent = 0.30; then k — J2 500 inch-pounds is the ultimate resilience for one cubic inch of the material.
Prob. 54. Show from the values given in Arts. 2 and 4 that the average ultimate resilience of timber in tension is about 50 per cent greater than that of cast iron.
Art. 55. Sudden Loads.
When a tension is gradually applied to a bar it increases from o up to its final value, while the elongation increases from o to e and the unit-stress increasesArt. 55.
SUDDEN LOADS.
Ill
from o to 5. A ‘sudden load’ is one which has the same intensity from the beginning to the end of the elongation ; this elongation being produced, the bar springs back, carrying the load with it, and a series of oscillations results, until finally the bar comes to rest with the elongation c. The temporary elongation produced is greater than c, and hence also the temporary stresses produced are greater than .S.
Let P be a suddenly applied load and y the temporary elongation produced by it; the external work performed during its application is Py. Now let Q be the internal stress corresponding to the elongation y; this increases gradually and uniformly from o up to Q, and hence its resilience or internal work is %Qy. But as internal work must equal external work,
| Oy s* Py, or Q — 2P;
that is, the sudden load A produces a temporary inter, nal stress equal to 2P.
Now after the oscillations have ceased, the bar comes to rest under the steady load A and has the elongation e. If the elastic limit of the material be not exceeded, corresponding elongations are proportional to their stresses; thus
y Q
~ — p~ 2, or y — 2e\
that is, the sudden load produces a temporary elongation double that caused by the same load when gradually applied.
If A be the section area of the bar the unit-stress S under the gradual load is P/A, and the temporary unit-stress produced under the sudden load is 2PJA or 2S.112
RESILIENCE OF MATERIALS.
Ch. VIII.
The unit-stresses temporarily produced by sudden loads are hence double those caused by steady loads. It is for this reason that factors of safety are taken higher for variable loads than for steady ones.
Prob. 55. A simple beam of wrought iron, 2X2 inches and 18 inches long, is to be loaded with 3000 pounds at the middle. Show that the beam will be unsafe if this be applied suddenly.
Art. 56. Stresses due to Impact.
Impact is said to be produced in a bar or beam when a load falls upon it from a certain height. The temporary stresses and deformations in such a case are greater than for sudden loads, and may often prove very injurious to the material. If the elastic limit be not exceeded, it is possible to deduce an expression showing the laws that govern the stresses produced by the impact. This will here be done only for the case of impact on the end of a bar.
If the load P falls from the height h upon the end of a bar and produces the momentary elongation y, the work performed is P(h -f-y). The stress in the bar increases gradually and uniformly from o up to the value Q, so that the resilience or internal work is \Qy. Hence there results
\Qy — P(]i + y)-
Also, if e be the elongation due to the static load P, the law of proportionality of elongation to stress gives
y Q
~e=P'Art. 56.
STRESSES DUE TO IMPACT.
i*3
By solving these equations the values of Q and y are
which give the temporary stress and elongation produced by the impact.
If h — o these formulas reduce to Q — 2/?andjy — 2e, as found in the last article for sudden loads. If h = 4^ they become Q = 4P and y = 4<?; if h — \2e they give Q st 6P and y — 6e. Since e is a small quantity for any bar it follows that a load P dropping from a moderate height upon the end of a bar may produce great temporary stresses and elongations. If these stresses exceed the elastic limit they cause molecular changes which result in brittleness and render the material unsafe.
The above expressions for Q and_y are not exact, as the resistance against motion due to the inertia of the material has not been taken into account. In Chapters XIII and XIV of Mechanics of Materials (tenth edition) the subject is discussed far more completely than has been possible here.
Prob. 56. In an experiment upon a spring a steady weight of 15 ounces on the end produced an elongation of 0.4 inches. Wlaat temporary elongation would be produced when the same weight is dropped upon the end of the spring from a height of 7 inches ?Ii4
MISCELLANEOUS APPLICATIONS.
Ch. IX.
Chapter IX,
MISCELLANEOUS APPLICATIONS.
Art. 57. Water and Steam Pipes.
The pressure of water or steam in a pipe is exerted in every direction and tends to tear the pipe apart longitudinally. This external force is resisted by the internal tensile stresses which act in the walls of the pipe normal to the radii. If p be the pressure per square inch exerted by the water or steam, d the diameter and l the length of the pipe, the total pressure P exerted on any diametral plane is p .Id. If t be the thickness of the pipe and X the tensile unit-stress, the total resisting stress will be S. 2It, if the thickness is not large compared with the diameter. Hence
pld — 2Slt, or pd — 2St, is the formula for discussing water or steam pipes.
Water-pipes are made of cast iron or wrought iron, the former being more common, while for steam the latter is preferable. A water-pipe liable to the shock of water-ram should have a high factor of safety, and in steam-pipes the factors should also be high. The formula above deduced shows that the thickness of a pipe must increase with its diameter, as also with the internal pressure to which it is to be subjected.
For example, let it be required to find the proper thickness for a wrought-iron steam-pipe of 18 inches diameter to resist a steam-pressure of 250 pounds perArt. 58.
RIVETED LAP JOINTS.
“5
square inch. With a factor of safety of io the working unit-stress is 5 S°° pounds per square inch. Then from the formula
t
250 X 18 j . .
-----= 0.45 inches,
2 X 5 500
so that a thickness of $ inch would probably be selected.
Prob. 57. Find the factor of safety of a cast-iron water-pipe 12 inches in diameter and § inches thick under a pressure of 130 pounds per square inch.
Art. 58. Riveted Lap Joints.
When two plates are joined together by rivets and the plates then subject to tension, there is brought a shear upon the rivets which tends to cut them off. A riveted lap joint is one where one plate simply laps over the other.
For a lap joint with a single row of rivets, let P be the tensile stress which is transmitted from one plate
to another by means of one rivet ; let a be the pitch of the rivets, or the distance from the center of oneMISCELLANEOUS APPLICATIONS.
Ch. IX.
116
rivet to the center of the next; let d be the diameter of a rivet and t the thickness of the plate. The plate tends to tear apart on the section area (a — d)t, while the rivet tends to shear off on the section area ^nd*. Accordingly, if St and be the unit-stresses for tension and shear,
P=zt(a-d)St, P=\7td'Ss,
are formulas for the discussion of this case.
For example, a steel water-pipe 30 inches in diameter has a longitudinal rivet seam with one row of rivets, the diameter of the rivets being £ inches, theii pitch 2 inches, and the thickness of the plate \ inches. If the interior water-pressure be 130 pounds per square inch, what are the unit-stresses in tension and shear ? Here the total pressure on a diametral plane of length equal to the pitch is P = 130 X 2 X 30 = 7 800 pounds. Then for tension on the plate
St B 7— 7 N 6 240 pounds per square inch,
2 (2 iv
and for shear on the rivet
S, = —q-^~° „ ■ = 8 860 pounds per square inch, °-7°5 X -je
Here the factors of safety are about 10 for the plates and about 6 for the rivets, so that the joint may be regarded as a satisfactory one.
When two rows of rivets are used, these are staggered, so that the rivets in one row come opposite the middle of the pitches in the other row. Here the tension Pis distributed over two rivet sections instead of one, and
P = t{a — d)St, P ~ ^Ttd'S"Art. 58.
RIVETED LAP JOINTS.
II7
are the formulas for investigation. Thus for the data of the above example, if there are two rows of rivets,
St is 12500 pounds per square inch, and ^ = 8900 pounds per square inch, which are good working values for mild steel if the pipe is not subjected to the shock of water-ram.
The working unit-stress for shear should be about three-fourths of that for tension, or Ss = $St. Equating the above values of P under this condition gives a joint whose security in tension is the same as that in shear; thus
a ~ d
d'
°.59y
a — d A
the first for single lap riveting and the second for double lap riveting. These are approximate rules for finding the pitch when the thickness of plates and diameter of rivets are given.
Prob. 58. A steel water-pipe 30 inches in diameter has118
MISCELLANEOUS APPLICATIONS.
Ch. IX.
rivets § inches in diameter and plates £ inches thick. If double riveting is used, what should be the pitch of the rivets ?
Art. 59. Riveted Butt Joints.
When two plates butt together cover plates are used on one or both sides; if on both sides the covers are one-half the thickness of the main-plate. The shear on each rivet is here divided between the upper and the
Fig. 39.
lower cross-section, this being called a case of double shear. Thus if P be the tension which is transmitted through one rivet, d the diameter of a rivet, a the pitch, and t the thickness of the main plate,
P — t{a — d)Sf\ P — 2 . \ird'S., which are the same as for two rows of lap riveting.
The ‘ efficiency ' of a riveted joint is the ratio of the strength of the joint to that of the solid plate. If the joint be designed so as to be of equal strength in tension and shear this efficiency is
t(a — d)St 	d
taSt	a
Th us if the rivets be f inches in diameter and the pitch be 2 inches the efficiency is I —	— 0.625, that
is, the riveted joint has only 62.5 per cent of the strength of the solid plate. Single lap riveting has usually an efficiency of about 60 per cent, while double lap riveting and common butt riveting has from 70 toArt. GO.
STRESSES DUE TO TEMPERATURE.
ll9
75 per cent. By using three or more rows of rivets efficiencies of over 80 per cent can be secured.
When a joint is not of equal strength in tension and shear there are two efficiencies, one being the ratio of the tensile strength of the joint to that of the solid plate, and the other the ratio of the shearing strength to that of the solid plate. The least of these is the true efficiency.
Prob. 59. A butt joint with two cover-plates has the main plate \ inches thick, the rivets l inches in diameter, and the pitch of the rivets 2J inches. Compute the efficiency.
Art. GO. Stresses due to Temperature.
A bar which is free to move elongates when the temperature rises and shortens when it falls. But if the bar be under stress so that it cannot elongate or contract, the change in temperature produces a certain unit-stress. This unit-stress is that which would cause a change of length equal to that produced in the free bar by the change in temperature.
The coefficient of linear expansion is the elongation of a bar of length unity under a rise of temperature of one degree. For the Fahrenheit degree the average values of the coefficients of expansion are as follows : For brick and stone, C — 0.000 00 50;
For cast iron,	C — 0.000 OO 62 ;
For wrought iron, C = 0.0000067;
For steel,	C = 0.000 00 65.
Thus a free bar of cast iron 1000 inches long will elongate 0.0062 inches for a rise of one degree, and 0.62 inches for a rise of 100 degrees.120
MISCELLANEOUS APPLICATIONS.
Ch. IX.
The elongation of a bar of length unity for a change of t degrees is hence s = Ct. But (Art. 44) the unit-stress due to the unit-elongation s is A = sE, where E is the modulus of elasticity. Therefore A = CtE
is the unit-stress produced by a change of t degrees on a bar which is fixed. If the temperature rises A is compression, if the temperature falls 5 is tension.
For example, consider a wrought-iron rod which is used to tie together two walls of a building and which is screwed up to a stress of 10 000 pounds per square inch. If the temperature falls 50 degrees there is produced a tensile unit-stress,
A la 0.0000067 X 50 X 25 000000 = 8 400,
and hence the total stress in the rod is 18400 pounds per square inch. If the temperature rises 50 degrees the stress in the bar is reduced to 1 600 pounds per square inch. In all cases the unit-stresses due to temperature are independent of the length and section area of the bar.
Prob. 60. A cast-iron bar 6 feet long and 4X4 inches in section is confined between two immovable walls. What pressure is brought on the walls by a rise of 40 degrees in temperature ?
Art. 61. Shrinkage of Hoops.
A hoop or tire is frequently turned with the inner diameter slightly less than that of the cylinder or wheel upon which it is to be placed. The hoop is then expanded by heat and placed upon the cylinder, and upon cooling it is held firmly in position by the radialArt. 62,
SHAFT COUPLINGS.
I 2 I
stress produced. This radial stress causes tension in the hoop.
Let D be the diameter of the cylinder upon which the hoop is to be shrunk and d the interior diameter to which the hoop is turned. If the thickness of the hoop is small, D will be unchanged by the shrinkage and
d will be increased to D. The unit-elongation of the hoop is then s = (D — d)/d, and hence the unit-stress produced is
S=sE =
D-d
E,
where E is the modulus of elasticity of the material.
A common rule in steel hoop shrinkage is to make D — d equal to	that is, the cylinder is turned
so that its diameter is y^Vuth greater than the inner diameter of the hoop. Accordingly the tangential unit-stress which occurs in the hoop after shrinkage is 30000000/1 500 as 20 000 pounds per square inch.
When the hoop is thick the above rule is not correct, for a part of the stress produced by the shrinkage causes the diameter of the cylinder to be decreased. The rules for this case are complex ones, and cannot be developed in an elementary text-book ; they will be found in Chapter XVI of Mechanics of Materials.
Prob. 61. Upon a cylinder 18 inches in diameter a thin wrought-iron hoop is to be placed. The hoop is turned to an inner diameter of 17.98 inches and then shrunk on. Compute the tensile unit-stress in the hoop.
Art. 62. Shaft Couplings.
Let a shaft be in two parts which are connected by a flange coupling. In the figure A shows the end view and B the side view of the coupling. The flanges122
MISCELLANEOUS APPLICATIONS.
Ch. IX.
of the coupling are connected by bolts which are brought into shearing stress in transmitting the torsion from one part of the shaft to the other.
Fig. 40.
Let the shaft be solid and of diameter D, let there be n bolts of diameter^, and let h be the distance from the center of the shaft to the center of a bolt. If D and d be assumed, as also the distance h, then, as shown in Art. 124 of Mechanics of Materials, the formula
_ (d-\-- 2h)D'
n ~ WT WyA
gives the number of bolts required in order that the strength of the bolts may be the same as the strength of the shaft.
For example, let D= 8 inches, d— 1 inch, and h = 12 inches, then the formula gives 11 = ii.i,so that 12 bolts should be used. If D — 8 inches, d—\\ inches, and h — 12 inches, the formula gives n = 5.01, so that five or six bolts should be used.
The case shown at CD in the above figure is one that should never occur in practice, because here the bolts are subject to a bending stress as well as to the shearing stresses due to the torsion. It is clear that this bending stress will increase with the length between the flanges, and that the bolts should be greater in diameter than for the case of pure shearing.
Prob. 62. A solid steel shaft 16 inches in diameterArt. 63.
RUPTURE OF REAMS AND SHAFTS.
I23
transmits 16 000 horse-powers at 50 revolutions per minute* Design a flange coupling for this shaft.
Art. 63. Rupture of Beams and Shafts.
The formulas (4) and (6) deduced in Arts. 25 and 40 for the discussion of beams and shafts are only valid when the unit-stress .S is less than the elastic limit of the material. Formula (4) can, however, be used for the rupture of beams, if 5 be taken as a certain quantity intermediate between the ultimate compressive and tensile strengths of the material. This quantity, which is called the ‘ modulus of rupture,’ has been determined by breaking beams and then computing S from the formula. In like manner formula (6) can be used for the rupture of round shafts if the modulus of rupture, as found by experiment, be used instead of the ultimate shearing strength.
The following table gives average values of the mod-ulus of rupture as determined by testing beams and
Table XI. Moduluses of Rupture.
Material.	For Beams.		For Round Shafts.	
	Pounds per Square Inch.	Kilos per Square-Centimeter.	Pounds per Square Inch.	Kilos per Square Centimeter.
Timber	9 000	630	2 OOO	140
Stone	2 OOO	140		
Cast Iron	35 000	2450	25 OOO	I 750
Wrought Iron			40 OOO	2 800
Steel, hard	80 OOO	5 600	70 OOO	4 90°
columns to destruction. Wrought iron and mild steel have no proper modulus of rupture when used asr 2 A
MISCELLANEOUS APPLICATIONS.
Ch. IX.
beams, since they continually bend and do not break however great the load may be. By the use of these values formulas (4) and (6) may be applied to the solution of numerous problems relating to the rupture of beams and shafts.
For example, let it be required to find the size of a square cast-iron simple beam of 6 feet span that will rupture under its own weight. Let x be the side of the square. The weight of a cast-iron bar X square inches in section area and one yard long is 9.4a' pounds; thus the weight of the beam is 18.8a' pounds. The bending moment is \ Wl or 169.2a' pound-inches. The value of c is and that of / is j\x\ Then, by formula (4),
35000 X a-4
” 6,-------- =
and the solution of this gives x — 0.17 inches.
It is to be noted, when formulas (4) and (6) are used for cases of rupture, that they are entirely empirical and have no rational basis.
Prob. 63. What force J\ acting at 24 inches from the axis of a steel shaft 1.4 inches in diameter, will cause failure by torsion ?Art. 64.
CONCRETE AND STEEL.
I25
Chapter X.
REINFORCED CONCRETE.
Art. 64. Concrete and Steel.
Columns and beams are made by ramming concrete into wooden forms or boxes which surround their sides, these being removed after the concrete has hardened. Steel rods are often placed in the forms and the concrete rammed around them, and this combination is often called ‘reinforced concrete.’ The object of inserting the steel is to make a safer and stronger construction than is possible with concrete alone, and to do this at a lower cost than is possible when only steel is used. Since 1895 there has been a great development in this kind of construction, steel rods being now extensively used for the reinforcement not only of columns and beams but also in walls, sewers, and arches.
The kind of concrete generally used for this purpose is made of Portland cement, sand, and broken stone, an excellent grade being of the proportions 1 cement, 2 sand, 4 stone by measure, and a lower grade being of the proportions 1 cement, 3 sand, 6 stone; these two grades are frecpiently called ‘ 1:2:4 concrete ’ and ‘1:3:6 concrete ’ ; the former is stronger and better than the latter but its cost is higher. The average weight of concrete is about 150 pounds per cubic foot, or about the same as that of sandstone.REINFORCED CONCRETE.
CH X.
126
The strength of concrete increases with its age, reaching nearly the highest value by the end of the first year. Its ultimate compressive strength is much higher than the ultimate tensile strength, and-the following are average values of these in pounds per square inch for concrete one year old:
Compressive Strength. Tensile Strength.
For 1:2:4 concrete	3500	300
For 1:3:6 concrete	2500	200
These figures, like all those given in this chapter, refer only to concrete made with Portland cement. The ultimate shearing strength of concrete is from 800 to 1000 pounds per square inch.
It is seen from these values of the ultimate stiengths that concrete is not well adapted to resist tension, and under tensile stresses it is almost impracticable to use concrete, unless it be strengthened by reinforcing rods of metal.
Concrete suffers a greater change of shape under a given applied unit-stress than steel, or the stiffness of steel is much greater than that of concrete. Mean values of the modulus of elasticity for the two grades of concrete are in pounds per square inch;
For 1:2:4 concrete,	E = 3 000 000.
For 1 : 3 : 6 concrete,	E — 2 000 000.
For steel the mean value of if is 30000000 pounds per square inch (Art. 44) and hence it is seen that concrete suffers an elastic deformation ten or fifteen times as great as that of steel when subjected to the same stress per square inch.Art. G4.
CONCRETE AND STEEL.
127
The elastic limit oi concrete is not well defined, but as a rough average it may be taken in compression at about one-sixth and in tension at about one-fifth 6f the ultimate strength. The allowable working unit-stress for concrete under compression is generally taken as about one-seventh of the ultimate strength that is, about 500 pounds per square inch for 1:2:4 concrete and about 350 pounds per square inch for 1:3:6 concrete.
The rods or bars used for reinforcement are generally of structural steel having an ultimate strength of about 60000 and an elastic limit of about 35 oco pounds per square inch. These may be the round, square, and rectangular shapes such as are everywhere found in the market, and various special patented forms are also widely used. Tire Ransome rods are square bars which have been twisted so that the comer lines are spirals; the Thacher and Johnson bars are rolled so as to have protuberances or swellings at intervals along the length; these forms are claimed to possess special advantages in preventing the rods from slipping in the concrete. The Kahn bar has projecting fins which are intended to prevent beams from shearing, and there are also several other forms which are advertised and used. Some of these are claimed to be advantage us in having an elastic limit much higher than that of structural steel.
Prob. 64. Consult the advertising columns of the engineering journals and obtain pictures of several kinds of reinforcing bars for beams. Compute the amount of shortening of a 1:2:4 concrete column, 12 inches square and 9 feet long, under a load of 60 000 pounds.128
REINFORCED CONCRETE.
Ch. X.
Art. 65. Compound Bars.
A bar which is under tension is usually called a rod, while one under compression is called a strut. The rods and struts considered in the previous chapters may be called simple ones, as each has been only of one material; a compound rod or strut, however, is one formed of two or more kinds of materials. For example, a steel rope with a hempen center, or a concrete column with a steel bar within it parallel to the axis, are instances of compound tension and compression members.
When a rod of two materials is subject to a long!“ tudinal tension P, part of this is resisted by orte material and part by the other. Let Ax be the section area of one material and A2 that of the other, and let be ture unitgflress over the area A]t and S2 the unit-str®S®ver the area A2. Then vljSj and A^Sa are the totalfflresses on the two sections, and hence, since the resisting stresses must equal the applied tension,
P = A^-t-A2S2	(io)
is a necessary equation of equilibrium. When P, Alt and A2 are siren, the values of and S* cannot be determ^gd from this equation, and hence a second cc^^^^Hbe^wKHi them must be derived.
This se^^p condition is established from the fact that the elongation ® the two parts of the bar is the same.	be the modulus of elasticity of the first
material and E2 that of the second. Then, if the elastic limit of neither material® exceeded, the elongationArt. G5.
COMPOUND BARS.
129
of a unit of length of the first material is Sl/El (Art. 44) and that of the second material is SjEr Since these must be equal,
E,
.S'.,
e:
or
s.
£
e:
<*0
which shows that the unit-stresses in the two materials are proportional to their moduluses of elasticity. If E2 is 10 times as great as E{, then S„ must be 10 times as great as Sl.
By the help of formulas (10) and (11) the values of 5, and S'., due to a load P on a compound rod of two materials may be easily found. The above reasoning applies also to compression if the length of the strut is not more than about 10 times its thickness, so that lateral flexure does not change the uniform distribution of the stresses. For example, consider a wooden bar having wrought-iron straps fastened along two opposite sides; here /4 for the timber is 1 500 000 and A, for the wrought iron is 25 000 ooo pounds per square inch, so that Ex iE2 is 0.06, and accordingly the unit-stress St in the timber is equal to0.o65f, so that, if Sn is 5 OOO pounds per square inch, .S'1 will be 300 pounds per square inch. Formula (1 1) cannot, however, be used if 5, exceeds the elastic limit of the timber or if 51 exceeds the elastic limit of the wrought iron.
The determination of the values of 5. and St for this compound rod is now easily made when .4., A~, and P are given. Let /ll for the timber be 36 square inches and A for the wrought iron be 4 square inches. Let the load P be Co ooo pounds. Since 5 = 0.06S.; formula (10) gives
60 ooo = 36 X o.o6S2 -J- 4.S'2,130
REINFORCED CONCRETE.
Ch. X.
from which S.2 — 9740 pounds per square inch for the wrought iron, and hence	580 pounds per square
inch for the timber. It is here seen that the total stress which comes on the wrought iron is 4 X 9740 or about 39000 pounds, while that on the timber is about 21 000 pounds. The metal hence carries the greater part of the load, and it does this largely by virtue of its greater stiffness. In general, the higher the value of E for a material in a compound bar, the greater is the part of the load carried by it.
Prob. 65. A short timber strut, 8X8 inches in section, has four steel plates fastened to its sides, each being 6 X f inches in size, and it carries a load of 180000 pounds. Compute the compressive unit-stresses in the two materials.
Art. 66. Reinforced Columns.
Concrete columns are generally square or round. Fig. 41 shows a square form having four steel rods
Fig. 41.	Fig. 42,	Fig. 43.
imbedded in it near the corners. Fig. 42 shows a round column having a single rod through its axis. Fig. 43 shows a round column formed 1 y a hollowArt. G6.
REINFORCED COLUMNS.
metal cylinder which is filled with concrete. The length of these columns will be considered short compared with the thickness, so that no tendency to lateral flexure exists.
The investigation of a reinforced column consists in determining the compressive unit-stresses due to the given load P, and comparing them with the allowable unit-stresses. Let At be the section area of the concrete and A., that of the metal, and let A, and E2 be the corresponding moduluses of elasticity. Let n represent the known ratio /'(//f,; then, according to (i i) in Art. 65, the value of S„/St must also be n, provided the elastic limit of neither material is exceeded. Inserting in (10) for S2 its value nS1, and solving, there results
51 =
P
Ax + uA2’	(l2)
from which the unit-stress in the concrete may be computed, and then the unit-stress in the metal is found from S, = fiSt.
As an example of investigation, take a reinforced column of 1 : 3 : 6 concrete which is 14 inches square and has four steel rods, each f inches in diameter parallel with its length near the corners, as in Fig. 41, while the load P is 71 000 pounds. Here the section area of the four rods is A2^s o 442 square inches and that of the concrete is = 196 — 0.44— 195.56 square inches. Then, since n is 15 (Art. 64), formula (12) gives 5, = 71 000/202.2 =351 pounds per square inch for the concrete, and hence S2~ 5270 pounds per square inch for the steel. These are safe working stresses, that for the steel being very low.*32	REINFORCED CONCRETE.	Ch. X.
When a reinforced column is to be designed, the load P and the unit-stress 5J for the concrete are known or assumed, while n is to be taken as 10 or 15, depending on the kind of concrete (Art. 64). Then the above investigation shows that the section areas At and A2 of the two materials are to be determined so as to satisfy the equation
P
»	(!3)
In order to do this, one section area is usually assumed and the other can then be computed Evidently many different sets of values of Ai and A 2 can be found, and the one to be used will generally be determined by convenience and local conditions. For example, let the column in Fig. 42 be of 1:2:4 concrete for which is to be 500 pounds per square inch, and it be required that the diameter shall be 6 inches, while the load/5 is 20000 pounds. Let it be required to find the diameter d of the single steel rod. Mere A2 = jirf1! and .4X = ^(36 — d2), and then
1^(36 — d- -f- ic)d2) = 20 000/500,
from which d is found to be 1.28 inches, so that a rod iA inches in diameter should be used.
For a column like Fig. 43, cast iron is often used for the outer casing, and since A is 15 000 000 pounds per square inch for cast iron, the value to be used for n will be about /h for 1:3:6 concrete, which is the grade that would be most likely to be used for a column of this kind. The safe load P which such a column can carry may then be found by formula (10), in which Si = 350 and S2= yb X 350 = 2 620 poundsArt. 67. beams with symmetric reinforcement.
!33
per square inch, while the investigation or design of such a column may be made by the methods above explained.
Prob. 66. A column like Fig. 43 is to be made as described in the last paragraph. The diameter of the concrete is 2 feet 6 inches, and the load to be carried is 500000 pounds. Compute the thickness of the cast-iron casing.
Art. 67. Beams with Symmetric Reinforcement.
Although beams of timber are sometimes reinforced by fastening metal plates upon the sides, the most common example of reinforcement is that of a concrete beam in which steel plates are imbedded. Concrete beams are usually rectangular and Fig. 44 represents
IT c2
Fig. 44.	Fig. 45.	Fig. 46.
OOO
OOO
a section of one with no reinforcing rods, this being called a plain concrete beam; such a beam is not well adapted for carrying heavy loads on account of the low tensile strength of the concrete (Art. 64). Fig. 45 gives a section of a concrete beam in which a steel I beam is imbedded; although this form is sometimes used, the steel part is generally made sufficiently large to carry the given loads, the office of the concrete being to protect the steel from fire. Fig„ 46 is a form in which steel rods are imbedded near both theT34
REINFORCED CONCRETE.
Ch. X.
top and the bottom of the beam, and symmetrically arranged with respect to the neutral axis of the concrete section.
The discussion of a plain concrete beam is made by the help of formula (4) and the methods explained in Arts. 25-28. For example, let the beam be 8 inches wide, 10 inches deep, 6 feet in span, and let it be required to compute the total uniform load which it can carry when the concrete is stressed to 100 pounds per inch on the tensile side. The maximum bending moment for the total load JV is |JFx 72 in inch-pounds (Art. 19). The section modulus I/c is, from Arts. 21 and 22, found to be | X 8 X io2 = 133.3 inches3. Accordingly, the flexure formula (4) gives
M=S(I/c), or gJV = 100 X 133-3*
from which W = 1480 pounds is the total uniform load which the beam can carry. Since this beam weighs about 500 pounds, a uniform load of about 980 pounds will stress the concrete on the tensile side to 100 pounds per square inch.
For cases where the imbedded steel is placed symmetrically with respect to the concrete section as in Figs. 45 and 46, the flexure formula (4) may be modified so as to take the two ‘different materials into account. Let be the unit-stress in the concrete on the remotest fiber the distance from the neutral axis; the unit-stress in the steel at the distance c2, and /, and I2 the moments of inertia of the concrete and steel sections. Jll being the maximum bending moment carried by the beam, this must be equal toAkt. GÏ. beams with symmetric reinforcement.
*35
the sum of the resisting moments of the two sections (Art. 23). Therefore, for compound beams,
x-M+SA.
C'l
Now, for the beam as for the column, the change of length of any line drawn on the sides parallel with the length must be the same for both concrete and steel. At the distance unity from the neutral axis the unit-stress in the concrete is Sx/cx and the change in a unit of length is then Sr/i\J^y, if Ey is the modulus of elasticity of the concrete. Similarly for the steel the change in a unit of length at the distance unity from the neutral axis is S2/c2E2. Hence
Oj	c,
yi;— 7 A- whence A m -tiSu	(14)
in which n represents the ratio E /Eu the value of which is 10 or 15 (Art. C4). Also inserting this value of S in the above expression for m, there results
AT^p{Jx + nIt),
5, =
«t _
h + nr
V
which are formulas for the design and investigation of reinforced-concrete beams.
As an example, let it be required to find the total uniform load W which a beam like Fig. 46 can carry, Nj being IOO pounds per square inch and n being 10. Let the width be 8 inches, the depth IO inches, the span 6 feet, and each of the six steel rods' be ^ inch in diameter and have its center 3 inches from the neu-136
REINFORCED CONCRETE.
Ch. X.
tral axis. The value of Min pound-inches is^Wx 72, or 9IV. The section area of each rod is 0.196 square inches, so that the moment of inertia /2 is 6 X 0.196 X 32 = 10.6 inches4. The moment of inertia of the 8X 10-inch rectangle is TV X 8 X io3 = 666.7 inches4, and hence Ix — 666.7 — 10.6 = 656 inches4. Then, since Sl/cl is 100/5,the first formula in (15) gives9fF = 20(656-!- 106), from which W — 1 720 pounds, which is 13 per cent more than a plain concrete beam of the same dimensions can carry.
For the above case the unit-stress S2 in the steel rod is quite small. If the rods are at the upper and lower surfaces of the beam S2 is nSu but since they are only 3 inches from the neutral axis formula (14) gives S2 — f X 10 X 100 p: 600 pounds per square inch, while the steel may safely bear twenty-five times as great a unit-stress. The full safe strength of the steel rods cannot indeed be developed unless the tensile strength of the concrete is entirely overcome. A symmetric arrangement of the rods, like that in Fig. 46, is not economical and is rarely used for beams. The above discussion indicates, however, the principles involved, and the formulas will be modified in the next article so as to apply to the usual cases of un-symmetric reinforcement.
Prob. 67. A beam like Fig. 46 is 10 inches wide, 12 inches deep, 14 feet in span, and has eight steel rods each f inches in diameter, four being 1 inch below the top and four being 1 inch above the bottom. Compute the unit-stresses for the 1:2:4 concrete and for the steel when the beam is loaded with 200 pounds per linear foot besides its own weight.Art. 68.
UNSYMMETRIC REINFORCEMENT.
137
Art 68. Unsymmetric Reinforcement.
For the reasons stated in the last article, reinforced-concrete beams are generally built with imbedded rods only on the tensile side, as shown in Fig. 47.
Fig.
47-
	<	b—H
t d	
	1___	' A, '
	
	© © ©
STS
When the loads on the beam are light, so that the unit-stress on the tensile side does not exceed about one-half of the tensile strength of the concrete, the distribution of stresses in the vertical section is that shown in the diagram. Let be the compressive unit-stress on the upper surface of the beam, Tx the tensile unit-stress on the lower surface, and T2 the tensile unit-stress in the steel. Let 11, as before, represent the ratio Et/Ex found by dividing the modulus of elasticity of the steel by that of the concrete. Let b be the breadth and d the depth of the rectangular section, a the section area of the steel rods, the centers of which are at the distance h below the middle of the beam. Let M be the bending moment of the loads for the given section, and let it be required to find the values of .Sj, Tx, and T„ due to M.
Space cannot be taken in this elementary book to develop the theory applicable to this case, and hence138
REINFORCED CONCRETE.
Ch. X.
the following formulas, demonstrated in Mechanics of Materials (tenth edition), will be merely stated and exemplified. The neutral surface in this case lies at a certain distance k below the middle of the beam, and the value of k may be computed from
I
rHE
na
Then the unit-stresses for the concrete are
Sl — (d -j- ^k)
6M
Id3'
Tx = (d -
2k)
6M
Id3'
while that for the steel is
T2 m 2n(/i — k)
6M
Id-I
and from these formulas the beam may be investigated, provided the load does not produce a value of Tx greater than about 150 pounds per square inch.
For example, let b = 10 and d = 12 inches, h ■=. 4 inches, a = 2.4 square inches, and n = 1 5 for 1:3:6 concrete (Art. 64). Then k — 0.923 inches is the distance of the neutral surface below the middle of the beam. Now let the span of the beam be 8 feet and the uniform load upon it be 2400 pounds; the bending moment at the middle is M= \wl =28 800 inch-pounds. The compressive unit-stress on the upper surface of the concrete is then found to be S. =s 138 pounds per square inch, and the tensile unit-stress on the lower surface is 7\ = 102 pounds per square inch, while the tensile unit-stress in theArt. 68.
UNSYMMETRIC REINFORCEMENT.
J3 9
steel is T2 = 923 pounds per square inch. Both Sx and T2 are quite small, but Tx is about one-half of the ultimate tensile strength.
When a heavy load is applied to a beam of this kind the tensile resistance of the concrete is first overcome, vertical cracks extending upward from the lower side, and thus a greater stress is thrown upon the steel. The theoretic analysis for this case is a difficult one unless it be assumed that the concrete below the neutral surface exerts no material resistance, and this assumption is the one usually made. Tig. 48
Fig. 48.
shows the distribution of stresses, the neutral surface being usually above the middle of the beam. Let b be the breadth of the beam, d the distance of the centers of the reinforcing rods below the top, and g the distance of the neutral surface below the top. The method of investigating this case is as follows: the value of g is to be computed from the formula
pr
&
na
T
nat
~b v
na
*>•
and then t.he unit-stresses are
C =
6M
(3^
cr
&
)bg
r—
M
(3^ - g)ai4o
REINFORCED CONCRETE.
Ch. X.
the first being for compression on the concrete and the second for tension on the steel. As an example, let b — 12 and d = 4.5 inches, a = 0.6 square inches,
/ = 60 inches, the uniform load be 1800 pounds, and n — 10. Then g = 1.68 inches, which is the depth of the neutral surface below the top of the beam. From the given load and span, M is found to be 13 500 inch-pounds, and then the compressive stress on the upper surface of the concrete is C = 340 pounds per square inch, and the tensile stress in the steel rods is T = $710 pounds per square inch. Both of these are lower than maximum allowable values.
Prob. 68. Let a reinforced beam of 1: 2:4 concrete be 24 inches wide, 5 inches deep, and 6 feet span, with 1.2 square inches of steel at 2 inches below the middle; compute the unit-stresses Sit T,, T3 due to the light uniform load of 1125 pounds. Also compute the unit-stresses C and T due to the heavy uniform load of 4500 pounds.
Art. 69. Design of Beams.
When a reinforced-concrete beam is to be designed the allowable unit-stresses for concrete and steel are given or assumed, as also the span and width, and it is then required to compute the depth of the beam and the section area of the steel. When this is done according to the formulas applicable to Fig. 47, the steel is stressed but slightly; if Tx be taken as 100 pounds per square inch for the concrete, T2 will be less than 1000 pounds per square inch for 1:2:4 concrete and less than 1500 for 1:3:6 concrete. It is found impossible to economically design a beam on this theory and have unit-stresses prevail that areArt. 69.
DESIGN OF BEAMS.
141
satisfactory, this being due to the low tensile strength of the concrete. Nothing remains to be done, therefore, but to allow the concrete to crack on the tensile side and thus to stress the steel higher in tension than is otherwise possible. The formulas above given for the distribution of stresses shown in Fig. 48 may be transferred so as to be applicable to cases of design.
The quantities usually given in designing are the allowable compressive unit-stress C on the concrete, the allowable tensile unit-stress T on the steel, the ratio E.JE1 = n, the bending moment M, and the breadth b of the rectangular beam. Let the ratio TIC be designated by t, then for the case of Fig. 48,
d =	/6jl
1/2+ 3«A/ bC' ~ 2ttn + 0
are the formulas for computing d and a. The unit-stress C should be taken as high as allowable by the specifications; T should not be higher than the highest allowable value, but it may be taken lower than this value if economy in cost is promoted.
For example, a rectangular beam of 1:3:6 concrete is to have a span of 14 feet, a breadth of 20 inches, and is to carry a uniform load of 300 pounds per square foot, including its own weight. It is required to find the depth of the beam and the section area of the reinforcing rods so that the unit-stresses C and T shall be 350 and 14000 pounds per square inch respectively. Here n = 15,/ = T/C = 40, and b — 20 inches. The total load on the beam is 300 X 14 X 20/12 = 7000 pounds, and the bending moment is142
REINFORCED CONCRETE.
Ch. X
M = l X 7ooo x 14 X 12 = 147000 inch-pounds. Inserting these values in the first of the above formulas, there is found d — 13.o inches.	Then
bd = 20 X 13 = 260 square inches, and from the second formula a — 0.90 square inches.
For the above case the section area of the steel is about one-third of one per cent of the section area bd of the concrete. Higher percentages of steel are frequently used, from 0.60 to 1.25 per cent being common values, but the author considers that these are not economical except for high-class concrete and low-priced steel. The depth d computed by the above formula is that from the top of the beam to the centers of the reinforcing rods. The actual depth of the beam is, however, greater than d by 1 or inches, the extra thickness of concrete serving to protect the steel from corrosion and from the effects of fire.
Prob. 69. For the above numerical data except that for a compute the depth d and the section a, taking the value of T as 12 000 pounds per square inch. Also taking the value of T as 9000 pounds per square inch. If steel costs 50 times as much as concrete, per cubic unit, which of the three beams is the cheapest ?
Art. 70. General Discussions.
The formulas and methods above presented for reinforced-concrete beams are valid when the unit-stresses in the concrete are proportional to their distances from the neutral surface, and this is the case only when the changes of length are proportional to the unit-stresses. Concrete is a material in which this proportionality does not exist for compressiveArt. 70.
GENERAL DISCUSSIONS.
143
stresses higher than 500 pounds per square inch, so that it cannot be expected that the formulas will apply to cases of rupture. Formulas for rupture have been deduced by Hatt and others in which the unit-stresses are taken as varying according to a parabolic law with their distances from the neutral surface, and such formulas are sometimes used for designing beams by applying proper factors of safety.
The phenomena of failure of reinforced concrete beams have been fully ascertained by the experiments made by Talbot in 1905. The beams were tested by applying two concentrated loads at the third points of the span, and the deflections at the middle were measured for several increments of loading, as also horizontal changes of length near the top and bottom. Under light loads the tensile resistance of the concrete was plainly apparent; when the tensile unit-stress in the concrete approached the ultimate strength, the neutral surface rose and the stress in the steel increased. A little later fine vertical cracks appeared on the tensile side, while the tensile stresses in the steel and the compressive stresses in the concrete increased faster than the increments of the load. The last stage was a rapid increase in the deformations, and rupture generally occurred by the crushing of the concrete on the upper surface, the steel being then stressed beyond its elastic limit.
In some cases, especially in short beams, failure is observed to occur by an oblique shearing near the supports, and this may be prevented by inclined or vertical reinforcing rods. In designing a beam, however,144
REINFORCED CONCRETE.
CH. X.
it is usually not necessary to make computations for this shearing stress because the dimensions obtained by the flexure formulas are for a beam which is to carry only about one-fifth or one-sixth of the load that causes rupture.
The above discussions give only an introduction to the subject of reinforced concrete. Many questions in regard to beams remain yet to be settled, but it is believed that the methods above given are fundamental and not liable to essential change. It seems likely that the use of combined concrete and steel, not only for beams and columns, but for arches, walls, piers, dams, and aqueducts, is to increase rapidly and become a most important feature in engineering construction.
Prob. 70. A plain concrete beam, 12 inches wide, 13^ inches deep, and 14 feet span, broke under two single loads, each of 1300 pounds and placed at the third points of the span. Compute the modulus of rupture by the common flexure formula. If this beam has one per cent of steel placed 11 inches above the lower surface, compute the values of C and T.Art. 71.
REVIEW PROBLEMS.
M5
APPENDIX.
Art. 71. Review Problems.
The following problems may be used in connection with those in the text, or may serve for review exercises. For each of the preceding articles there is given one additional problem.
CHAPTER I.
Prob. 71. If a cast-iron bar, iF X 2J inches in section area, breaks under a tension of 66 000 pounds, what tension will probably break a bar 1^ inches in diameter?
Prob. 72. What should be the size of a round bar of structural steel to carry a tension of 200 000 pounds with a unit-stress of one-half the elastic limit?
Prob. 73. What' should be the size of a round bar of structural steel to carry a tension of 125 000 pounds with a factor of safety of 5 ?
Prob. 74. What force is required to rupture in tension a cast-iron bar 8 inches in diameter ?
Prob. 75. A bar of wrought iron one square inch in section area and one yard long weighs 10 pounds. Find the length of a vertical bar which ruptures under its own weight when hung at its upper end.146
APPENDIX.
Art. 71.
Prob. 76. The beam in Fig. 2 is 3 X 4 inches in section area, and -P is 13 000 pounds. Compute the shearing unit-stress.
Prob. 77. A short cast-iron column is 12 inches in outside diameter and 10 inches in inside diameter. Compute its factor of safety when carrying a load of 165 000 pounds.
CHAPTER II.
Prob. 78. Find the diameter of a wrought-iron bar which is 24 feet long and weighs 1344 pounds.
Prob. 79. What is the diameter of the largest bar of structural steel which can be tested in a machine of 100 000 pounds capacity ?
Prob. 80. What should be the size of a short piece of yellow pine which is to carry a steady load of 80 000 pounds?
Prob. 81. A brick weighs 4.42 pounds when dry and 4.75 pounds after immersion for one day in water. What percentage of water has it absorbed ?
Prob. 82. Find the weight of a granite column 18 feet high and 18 inches in diameter.
Prob. 83. Compute the weight of a cast-iron water-pipe 12 feet long, and 20 inches inside diameter, the thickness of the metal being 1 inch.
Prob. 84. A wrought-iron bar, 1 inch in diameter and 30 feet long, is hung at its upper end. What load applied at the lower end will stress it to the elastic limit ?
Prob. 85. If steel costs 3 cents per pound and nickel costs 35 cents per pound, what is the minimum cost of a pound of nickel steel which contains 3.25 per cent of nickel?
Prob. 86. A bar of aluminum copper, i; X if inches inArt. 71.
REVIEW PROBLEMS.
147
section, breaks under a tension of 42 800 pounds. What tension will probably break a bar of the same material which is 1J X 2$ inches in section ?
CHAPTER III.
Prob. 87. Three men carry a stick of timber, two taking hold at a common point and one at one of the ends. Where should be the common point so that each man may carry one-third of the weight?
Prob. 8S. Two locomotive wheels, six feet apart and each weighing 20000 pounds, roll over a beam of 27 feet span. Find the greatest reaction which can be caused by these wheels.
Prob. 89. Compute the bending moments under each concentrated load for Fig. 8, taking the weight of the beam into account.
Prob. 90. For Fig. 12 let c= 6 and cx = 3 inches. If the unit-stress S at the top of the web is 6 400 pounds per square inch, what is the unit-stress Sx on the lower side of the flange?
Prob. 91. The two bases of a trapezoid are 8 and 5 inches and its height is 4 inches. Find its center of gravity.
Prob. 92. For a solid circular section the moment of inertia with respect to an axis through the center is -^tzv/4. Find the moment of inertia for a hollow circular section with outside diameter dx and inside diameter cir
CHAPTER IV.
Prob. 93. Locate the neutral axis for a T section which is 3 X 3 inches and f inches thick.
Prob. 94. A timber 4X6 inches in section projects 6 feet out of a wall. What load must be put upon it so that148
APPENDIX.
Art. 71.
the greatest shearing stress shall be 120 pounds per square, inch ?
Prob. 95. Draw the moment diagrams for Fig. 7 and Fig. 8.
Prob. 96. A simple wooden beam, 8 inches wide, 9 inches deep, and 14 feet in span, carries two equal loads, one being 2.5 feet at the left and the other 2.5 feet at the right of the middle. Find these loads so that the factor of safety of the beam shall be 8.
Prob. 97. A simple wooden beam, 3 inches wide, 4 inches deep, and 16 feet span, has a load of 150 pounds at the middle. Compute its factor of safety.
Prob. 98. A simple beam of structural steel, § inches deep and 16 feet span, is subject to a rolling load of 500 pounds. What must be its width in order that the factor of safety may be 6 ?
Prob. 99. Compare the strength of a joist, 3X8 inches, when laid with long side vertical with that when it is laid with short side vertical.
Prob. 100. Compare the strength of a light 9-inch steel I beam with that of a wooden beam 8 inches wide and 12 inches deep, the span being the same for both.
Prob. 101. A cast-iron cantilever beam is to be 4 feet long, 3 inches wide, and to carry a load of 15 000 pounds at the end. Find the proper depths for every foot of length, using 3000 pounds per square inch for the horizontal unit-stress and 4000 for the vertical shearing unit-stress.
CHAPTER V.
Prob. X02. Find the safe steady load for a hollow short cast-iron column which is 12 inches in outside and 9 inches in inside diameter.Art. 71.
REVIEW PROBLEMS.
149
Prob. 103. Compute the radius of gyration for the column section in the last problem.
Prob. 104. Given <j =	and ■S' = 5000 pounds per
square inch for a cast-iron column. Plot a curve for formula (5), taking values of l/r as abscissas and values e£ P/A as ordinates.
Prob. 105. Determine the safe load for a fixed-ended timber column, 3X4 inches in section and 10 feet long, so that the greatest compressive unit-stress may be 800 pounds per square inch.
Prob. 106. A cylindrical wrought-iron column with fixed ends is 12 feet long, 6.36 inches in outside diameter, 6.02 inches in inside diameter, and carries a load of 49 000 pounds. Find its factor of safety.
Prob. 107. Compute the size of a square timber column with fixed ends to carry a load of 100 000 pounds with a factor of safety of 10.
CHAPTER VI.
Prob. xo8. If a force of 80 pounds, acting at 18 inches from the axis, twists the end of a shaft through 15 degrees, what force will produce the same result when acting at 4 feet from the axis ?
Prob. 109. Compute the polar moment of inertia for a hollow shaft with outside diameter r8 inches and inside diameter 10 inches.
Prob. no. Compute the shearing unit-stress for the shaft of the last problem when it is subject to a twisting moment of 2500 pound-inches.
Prob. hi. Find the horse-power that can be transmitted by a wrought-iron shaft 3 inches in diameter when making*5°
APPENDIX.
Art. 71
50 revolutions per minute, the value of S being 6000 pounds per square inch.
Prob. 112. Find the diameter of a solid wrought-iron shaft to transmit 90 horse-power at 250 revolutions per minute, the value of S being 7000 pounds per square inch.
Prob. 113. Find the ratio of the strength of a hollow shaft to that of a solid one, the section areas being equal, and the outside diameter of the hollow section being double that of the inside diameter.
CHAPTER VII.
Prob. 114. Show, for timber and wrought-iron bars strained to their elastic limits, that the change of length of the former is double that of the latter.
Prob. 115. Compute the tensile force required to stretch a bar of structural steel, if X 9f inches in section area and 23 feet 3^ inches long, so that its length may become 23 feet 3! inches.
Prob. 116. What unit-stress will shorten a block of cast-iron 0.4 per cent of its length ?
Prob. 117. A cast-iron bar, 2 inches wide, 4 inches deep and 6 feet long, was balanced upon a support and a weight of 4000 pounds hung at each end, when the deflection of each end was found to be 0.401 inches. Compute the modulus of elasticity.
Prob. 118. Compute the elastic deflection of a light steel 10-inch beam of 30 feet span, due to its own weight, when resting on supports at the ends.
Prob. 119. Compute for the beam of the last problem the deflection when the beam is fixed at both ends.
Prob. 120. A wrought-iron shaft, 5 feet long and 2 inches in diameter, is twisted through an angle of 7 degrees whenArt. 71.
REVIEW PROBLEMS.
*5*
transmitting 4 horse-power at 120 revolutions per minute. Compute the shearing modulus of electricity.
CHAPTER VIII.
Prob. 121. How many foot-pounds of work are required to stress a steel piston-rod, 3 inches in diameter and 4 feet long, from o up to 16000 pounds per square inch ?
Prob. 122. What horse-power is required to stress the rod of the last problem 120 times in one minute ?
Prob. 123. Compute the horse-power required to deflect, 59 times per second, a wrought-iron cantilever beam, 2X3X72 inches, so that at each deflection the unit stress S shall range from o to 9 000 pounds per square inch.
Prob. 124. What work is required to rupture by tension a wrought-iron bar which weighs 325 pounds ?
Prob. 125. Discuss the case where a sudden load P is applied to a bar will'd* is already under the static load P.. What is the maximum stress?
Prob. 126. What is the height from which a weight must fall upon the end of a bar in order to produce a deformation equal to three times the static deformation ?
CHAPTER IX.
Prob. 127. What steam pressure is required to burst a pipe of structural steel which is 18 inches in diameter and f inches thick?
Prob. 128. Discuss a lap joint with two rows of rivets, for which a — 2I, d — -J, and t = J inches, P being 8 000 pounds.
Prob. 129. Compute the efficiency of the above joint.*5«
APPENDIX.
Art. 71.
Prob. 130. A railroad rail is laid in place at a temperature of 40°. If it cannot expand, what unit-stress is produced when the temperature rises to 90°.?
Prob. 131. Find the radial unit-pressure between the rim and tire of a locomotive driving-wheel when the shrinkage is y-jinr, the diameter of the tire being 60 inches and its thickness f- inches.
Prob. 132. A solid shaft 6 inches in diameter is coupled by bolts 1^ inches in diameter with their centers 5 inches from the axis. How many bolts are necessary ?
Prob. 133. What center load will rupture a cast-iron beam 1 inch square and 12 inches in span ?
CHAPTER X.
Prob 134. Find the safe load for a short column of 1:2:4 concrete which is 2 X 3 feet in section area.
Prob. 135. A short timber column, 6X8 inches in section, has two steel plates, each § X 8 inches, bolted to the 8-inch sides. Compute the load P when the timber is stressed to 750 pounds per square inch.
Prob. 136. A pier of 1:3:6 concrete, 6 feet in diameter, is surrounded by a cast-iron casing 1.15 inches thick. What part of the total load is carried by the concrete ?
Prob. 137. For a 1:2:4 concrete beam like Fig. 46 let b = 10 and d — 12 inches, and the six steel rods be 5 inches from the neutral surface. Find the size of the rods when the beam is 13 feet in span and carries a total uniform load of 5 000 pounds.
Prob. 138. For the dimensions in Prob. 68, what uniform load will probably cause the concrete to begin to fail in tension ?Art.
ANSWERS TO PROBLEMS.
r53
Prob. 139. Design a beam of 1:2:4 concrete with three reinforcing rods, the width to be 8 inches and the span 6 feet, so that it will safely carry a total uniform load of 2 500 pounds.
Prob. 140. Consult a paper by Sewall and the accompanying discussions in Vol. 56 of Transactions of American Society of Civil Engineers, and ascertain different opinions as to what should be the comparative cost of concrete and steel in order to produce the most economical reinforced beam.
Art. 72. Answers to Problems.
The following are the answers to some of the problems stated in the preceding pages, the numbers of the problems being in parentheses. In solving problems it is very desirable that the student should follow a neat and systematic method. The practice of making computations with a pencil on loose scraps of paper should be discontinued by every student who has followed it, and he should hereafter solve his problems in a special book, using pen and ink. Before beginning the solution, a diagram should be drawn whenever possible, for a diagram helps the student to understand the problem, and a problem thoroughly understood is really half solved. Before beginning the computation of a numerical problem, it is best to make a mental estimate of the answer, for thus the engineering judgement of the student will be developed. In no event should a student look in this article for an answer to a problem until he has completed its solution.*54
APPENDIX.
Art. 72.
r	i/
(i) 4-j^- inches:	(4) 4J inches for the vvrought-iron bar.
(5) 1780 feet. (6) Factor of safety = 20, nearly. (7) 31-inches for second case. (9) Percent of reduction of area = 505.	(12) $1058.84. '(13) 65 700 pounds. (15) 3^ inches.
(21) 4.04 inches. (22) 2097 inches4. (26) 3136 pounds. (28) 18 inches. (30) 5.9 and 3.9.	(32) 59000 pounds.
(35) 58000 pound^. (38)500 pounds. (40) 1.3 inches (45) 0.12 inches. (46) 0.001 x inches. (51) 11250 inch-pounds. (56) 2.8 inches. (60) 59 500 pounds. (63) 1680 pounds. (68) T.} = 735 pounds per square inch. (74) About 1000000 pounds. (78) 4§ inches. (83)	140 pounds.
(85) 4.04 cents. (86) 105 000 pounds. (94) 2840 pounds. (97) Nearly 8.	(98) 2.0 inches. (99) 2§ to #•	(103) 3-75
nches. (106) 3J.	(107) 132 inches. (108) 30 pounds.
(112) 2§ inches. (113) 144 to 100.	(117) 14500000
per square inch. (126) h — i.5<r.	(131) 500 pounds per
square inch. (135) 126 000 pounds.INDEX.
*55
INDEX.
Absorption test, 28 Aluminium, 39 Answers to problems, 152
Bars, ii, 13, 75, 106, 127 Beams, 40-74. 106. 123 cantilever, 56 reinforced, 133, 137, 140 restrained, 102 simple, 56
uniform strength, 72 Bending moments, 44, 62 Bending of beams, 61
columns, 77 Brick. 16, 27 Building stone, 28 Butt joints, 118
Cantilever beams, 56, 9S Cast iron, II, 14, 16, 17, 30 Cement, 38 Center of gravity, 49 moments, 41 Cold-bend test, 33 Columns, 75, 140 Compound bars, 128 Compression, 8, 15, 75 Compressive strength, 16 tests, 24
Concrete, 38, 125
and steel, 125 beams, 133, 137 columns, 130
Deflection of beams, 99, ioo Deformations, 8, 95 Design of beams, 67, 140 columns, 84, 132 Direct stresses, 7
Economic construction, 21 Elastic deformations, 95 limit, 10, It resilience, 106 108 strength, 7,11 Elongation, 8, 97 Ends of columns, 76
Factor of safety, 12, 20
Granite, 29
Gyration, radius of, 77
lloops, 120 Hollow shafts, 94 Horse-power, 91 Hydraulic cement, 38
I beams, 53, 69 Impact, 112
Investigation, 65, 82, 90
Lap joints, 115 Limestone, 29
Modulus of elasticity, 96 resilience, 107INDEX.
156
Modulus of rupture, 123 Moments for beams, 40, 44, 62 of inertia, 48, 51, 87 Mortar, 38
Neutral axis, 47, 87, 137, 139 surface, 47 Nickel steel, 37, 92
Pig iron, 30 Pipes, 114
Polar moments of inertia, 87
Radius of gyration, 77 Rankine’s formula, 80 Railroad rails, 37 Reactions of beams, 42 Reinforced concrete, 125 Resilience, 105
of bars, 106 of beams, 107 Resisting moment, 46 Restrained beams, 102 Riveted joints, 115, 118 Round shafts, 90, 92 Ropes, 39
Sandstone, 28 Safe loads for beams, 64 columns, 81 Shaft couplings, 121 Shafts, 85, 91, 103, 123 Shear, 8, 17, 58 Shortening, 8, 46 Simple beams, 58 Square shafts, 93 Steam pipes, 114 Steel, 11, 14, 16, 17, 35, 127 Steel castings, 37 Steel I beams, 69, 70, 133 Stone, 16, 28
Stress, 7, 13, 19, 112, 120 Struts, 75
Sudden loads, 19, no
T sections, 54 Tables:
Compressive strengths, 16 Constant for columns, 81 Elastic limits, 11 Factors of safety, 20 Modulus of elasticity, 96 Modulus of rupture, 123 Shearing strengths, 17 Steel I beams, 70 Strength of timber, 26 Tensile strengths, 14 Weights of materials, 22 Temperature, 119 Tensile tests, 24 Tension, 7, 8, 12, 14 Testing machines, 23 Test specimens, 18 Timber, It, 14, 16, 17, 25 Torsion, 85, 124 Trap rock, 29 Twist in shafts, 103
Ultimate elongation, 14 resilience, 109 strength, 7, II Unit-stress, 7, 19
Vertical shear, 59
Water pipes, 114 Weights of bars, 23
materials, 22 Work, 91, 105
Wrought iron, 11, 14, 16, 17, 32



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i 50
ASTRONOMY.
Comstock’s Field Astronomy for Engineers...............................8vo,	2	50
Craig’s Azimuth........................................................4to,	3	50
Crandall’s Text-book on Geodesy and Least Squares......................8vo,	3	00
Doolittle’s Treatise on Practical Astronomy............................8vo,	4	00
Gore's Elements of Geodesy.............................................8vo,	2	50
Hayford’s Text-book of Geodetic Astronomy..............................8vo,	3	00
Merriman's Elements of Precise Surveying and Geodesy...................8vo,	2	50
*	Michie and Harlow's Practical Astronomy.............................8vo,	3	o°
Rust’s Ex-meridian Altitude, Azimuth and Star-Finding Tables. (In Press.)
*	White's Elements of Theoretical and Descriptive Astronomy...........nmo,	2	00
3CHEMISTRY.
Abderhalden’s Physiological Chemistry in Thirty Lectures. (Hall and Defren).
(In Press.)
*	Abegg’s Theory of Electrolytic Dissociation, (von Ende.)...........i2mo,	i	25
Adriance’s Laboratory Calculations and Specific Gravity Tables........i2mo,	1	25
Alexeyeff’s General Principles of Organic Syntheses. (Matthews.).......8vo,	3	00
Allen’s Tables for Iron Analysis.......................................8vo,	3	00
Arnold’s Compendium of Chemistry. (Mandel.).................... .Large i2mo, 3 50
Association of State and National Food and Dairy Departments, Hartford
Meeting, 1906.................................................8vo,	3	00
Jamestown Meeting, 1907.........................................8vo,	3	00
Austen’s Notes for Chemical Students.................................12010,	1	50
Baskerville’s Chemical Elements. (In Preparation).
Bernadou’s Smokeless Powder.—Nitro-cellulose, and Theory of the Cellulose
Molecule.....................................................i2mo,	2	50
*	Blanchard’s Synthetic Inorganic Chemistry..........................i2mo,	1	00
*	Browning's Introduction to the Rarer Elements.......................8vo,	1	50
Brush and Penfield’s Manual of Determinative Mineralogy................8vo,	4	00
*	Claassen’s Beet-sugar Manufacture. (Hall and Rolfe.)................8vo,	3	00
Classen’s Quantitative Chemical Analysis by Electrolysis. (Boltwood.). .8vo,	3	00
Cohn’s Indicators and Test-papers.....................................i2mo,	2	00
Tests and Reagents................................................8vo,	3	00
*	Danneel’s Electrochemistry. (Merriam.).............................i2mo,	1	25
Duhem's Thermodynamics and Chemistry.	(Burgess.)..................8vo,	400
Eakle’s Mineral Tables for the Determination of Minerals by their Physical
Properties....................................................8vo,	1	25
Eissler’s Modern High Explosives.......................................8vo,	4	00
Effront's Enzymes and their Applications.	(Prescott.)..................8vo,	3	00
Erdmann’s Introduction to Chemical Preparations. (Dunlap.).............nmo,	1	25
*	Fischer’s Physiology of Alimentation........................Large l2mo, 2 00
Fletcher’s Practical Instructions in Quantitative Assaying with the Blowpipe.
i2mo, mor.	1	50
Fowler’s Sewage Works Analyses........................................i2mo,	2	00
Fresenius’s Manual of Qualitative Chemical Analysis. (Wells.)..........8vo,	5	00
Manual of Qualitative Chemical Analysis. Part I. Descriptive. (Wells.) 8vo,	3	00
Quantitative Chemical Analysis. (Cohn.) 2 vols....................8vo,	12	50
When Sold Separately, Vol. I, $6. Vol. II, $8.
Fuertes’s Water and Public Health.....................................i2mo,	1	5°
Furman’s Manual of Practical Assaying..................................8vo,	3	00
*	Getman’s Exercises in Physical Chemistry...........................i2mo,	2	00
Gill’s Gas and Fuel Analysis for Engineers............................i2mo,	1	25
*	Gooch and Browning’s Outlines of Qualitative Chemical Analysis.
Large nmo,	1	25
Grotenfelt’s Principles of Modern Dairy Practice. (Woll.)..............nmo,	2	00
Groth's Introduction to Chemical Crystallography (Marshall)............nmo,	1	25
Hammarsten's Text-book of Physiological Chemistry. (Mandel.)...........8vo,	4	00
Hanausek’s Microscopy of Technical Products. (Winton.).................8vo,	5	00
*	Haskins and Macleod’s Organic Chemistry............................i2mo,	2	00
Helm’s Principles of Mathematical Chemistry. (Morgan.)................i2mo,	1	50
Hering’s Ready Reference Tables (Conversion Factors)...........i6mo, mor. 2 50
*	Herrick’s Denatured or Industrial Alcohol...........................8vo,	4	00
Hinds's Inorganic Chemistry............................................8vo,	3	00
*	Laboratory Manual for Students................................   i2mo,	1	00
*	Holleman’s Laboratory Manual of Organic Chemistry for Beginners.
(Walker.)....................................................i2mo,	1	00
Text-book of Inorganic Chemistry. (Cooper.).......................8vo,	2	50
Text-book of Organic Chemistry. (Walker and Mott.)................8vo,	2	50
Holley and Ladd’s Analysis of Mixed Paints, Color Pigments, and Varnishes.
Large i2mo 2 50
4Hopkins’s Oil-chemists’ Handbook.
Johannsen’s Dete Keep’s Cast Iron.
Landauer’s Spectrum Analysis. (Tingle.)...................
* .Langworthy and Austen’s Occurrence of Aluminium in
ucts, Animal Products, and Natural Waters........
Lassai-Cohn’s Application of Some General Reactions to ]
Organic Chemistry. (Tingle.).....................
Leach’s Inspection and Analysis of Food with Special R
Lob’s Electrochemistry of Organic Compounds. (Lorenz.)..........
Lodge’s Notes on Assaying and Metallurgical Laboratory Experiments.
Lunge’s Techno-chemical Analysis. (Cohn.).................
* McKay and Larsen’s Principles and Practice of Butter-making Maire’s Modem Pigments and their Vehicles.................
	3	00
...8vo,	5	00
y..8vo,	I	25
;.. .8vo,	4	00
	2	50
.. i2mo,	I	00
	3	oo
le Prod-		
...8vo,	2	00
ions in		
. i2mo,	I	00
to State		
	7	50
...8vo,	3	00
... .8vo,	3	00
...8vo,	3	00
	I	00
...8vo,	I	50
. i2mo,	2	00
. I2IT10,	I	50
*	Martin’s Laboratory Guide to Qualitative Analysis with the Blowpipe. . nmo, Mason’s Examination of Water. (Chemical and Bacteriological.). . . .nmo,
Water-supply. (Considered Principally from a Sanitary Standpoint.)
8vo,
Matthews’s The Textile Fibres. 2d Edition, Rewritten..................8vo,
Meyer’s Determination of Radicles in Carbon Compounds. (Tingle.). .i2mo,
Miller’s Cyanide Process...............................................izmo,
Manual of Assaying................................................i2mo,
Minet’s Production of Aluminum and its Industrial Use. (Waldo.). . . .nmo,
Mixter’s Elementary Text-book of Chemistry..............................nmo,
Morgan’s Elements of Physical Chemistry.................................nmo,
Outline of the Theory of Solutions and its Results...............121110,
*	Physical Chemistry for Electrical Engineers........................nmo,
Morse’s Calculations used in Cane-sugar Factories................i6mo, mor.
*	Muir’s History of Chemical Theories and Laws.......................8vo,
Mulliktn’s General Method for the Identification of Pure Organic Compounds.
Vol. I.................................................Large 8vo,
O’Driscoll’s Notes on the Treatment of Gold Ores......................8vo,
Ostwald’s Conversations on Chemistry. Part One. (Ramsey.).............nmo,
“	%	**	“	Part Two. (Turnbull.).........nmo,
*	Palmer’s Practical Test Book of Chemistry..........................nmo,
*	Pauli’s Physical Chemistry in the Service of Medicine. (Fischer.).... nmo,
*	Penfield’s Notes on Determinative Mineralogy and Record of Mineral Tests.
8vo, paper,
60 I 25
4 00 4 00 I 00 I CO
1	00
2	50 1 50
3	co I 00 I 50
1	50
4	00
5	00
2	00
1	50
2	00 I 00 I 25
50
Tables of Minerals, Including the Use of Minerals and Statistics of
Domestic Production...........................................8vo,	1	00
Pictet’s Alkaloids and their Chemical Constitution. (Biddle.)...........8vo,	5	00
Poole’s Calorific Power of Fuels........................................8vo,	3	00
Prescott and Winslow’s Elements of Water Bacteriology, with Special Reference to Sanitary Water Analysis.........................................nmo,	1	50
* Reisig’s Guide to Piece-dyeing........................................8vo,	25	00
Richards and Woodman’s Air, Water, and Food from a Sanitary Standpoint..8vo,	2	00
Ricketts and Miller’s Notes on Assaying.................................8vo,	3	00
Rideal’s Disinfection and the Preservation of Food......................8vo,	4	00
Sewage and the Bacterial Purification of Sewage....................8vo,	4	00
Riggs’s Elementary Manual for the Chemical Laboratory...................8vo,	1	25
Robine and Lenglen’s Cyanide Industry. (Le Clerc.)......................8vo,	4	00
Ruddiman’s Incompatibilities in Prescriptions...........................8vo,	2	00
Whys in Pharmacy ................................................ i2mo,	1	00
5Ruer’s Elements of Metallography. (Mathewson). (In Preparation.)
Sabin’s Industrial and Artistic Technology of Paints and Varnish......8vo,	3	00
Salkowski’s Physiological and Pathological Chemistry. (Orndorff.).....8vo,	2	50
Schimpf’s Essentials of Volumetric Analysis...........................nmo,	1	25
*	Qualitative Chemical Analysis...................................  8vo,	1	25
Text-book of Volumetric Analysis................................i2mo,	2	50
Smith’s Lecture Notes on Chemistry for Dental Students................8vo,	2	50
Spencer’s Handbook for Cane Sugar Manufacturers................i6mo, mor. 3 00
Handbook for Chemists of Beet-sugar Houses................i6mo, mor. 3 00
Stockbridge’s Rocks and Soils.........................................8vo,	2	50
*	Tillman’s Descriptive General Chemistry........................... 8vo,	3	00
*	Elementary Lessons in Heat......................................  8vo,	1	50
Treadwell’s Qualitative Analysis. (Hall.).............................8vo,	3	00
Quantitative Analysis. (Hall.)...................................8vo,	4	00
Turneaure and Russell’s Public Water-supplies.........................8vo,	5	00
Van Deventer’s Physical Chemistry for Beginners. (Boltwood.)..........nmo,	1	50
Venable’s Methods and Devices for Bacterial Treatment of Sewage......8vo, 3 00
Ward and Whipple’s Freshwater Biology. (In Press,)
Ware’s Beet-sugar Manufacture and Refining. Vol. I.............Small 8vo, 4 00
“	“	“	“	“	Vol. II..........Small 8vo, 5 co
Washington’s Manual of the Chemical Analysis of Rocks, ...............8vo,	2	00
*	Weaver’s Military Explosives.......................................8vo,	3	00
Wells’s Laboratory Guide in Qualitative Chemical Analysis.............8vo,	1	50
Short Course in Inorganic Qualitative Chemical Analysis for Engineering
Students....................................................nmo,	1	50
Text-book of Chemical Arithmetic.................................nmo,	1	25
Whipple’s Microscopy of Drinking-water................................8vo,	3	50
Wilson’s Chlorination Process......................................... nmo	1	50
Cyanide Processes..............................................   nmo	1	50
Winton’s Microscopy of Vegetable Foods.................................8vo	7	50
CIVIL ENGINEERING.
BRIDGES AND ROOFS. HYDRAULICS. MATERIALS OF ENGINEERING. RAILWAY ENGINEERING.
Baker’s Engineers’ Surveying Instruments............................12mo,
Bixby’s Graphical Computing Table..................Paper 19^X24! inches.
Breed and Hosmer’s Principles and Practice of Surveying.............8vo,
*	Burr’s Ancient and Modern Engineering and the Isthmian Canal.....8vo,
Comstock’s Field Astronomy for Engineers.............................8vo,
*	Corthell’s Allowable Pressures on Deep Foundations...............l2mo,
Crandall’s Text-book on Geodesy and Least Squares...................8vo,
Davis’s Elevation and Stadia Tables..................................8vo,
Elliott’s Engineering for Land Drainage.............................i2mo,
Practical Farm Drainage........................................i2mo,
*Fiebeger’s Treatise on Civil Engineering............................8vo,
Flemer’s Phototopographic Methods and Instruments....................8vo,
Folwell’s Sewerage. (Designing and Maintenance.).....................8vo,
Freitag’s Architectural Engineering..................................8vo,
French and Ives’s Stereotomy.........................................8vo,
Goodhue’s Municipal Improvements....................................i2mo,
Gore’s Elements of Geodesy...........................................8vo,
*	Hauch and Rice’s Tables of Quantities for Preliminary Estimates,.l2mo,
Hayford’s Text-book of Geodetic Astronomy..........................  8vo,
Bering’s Ready Reference Tables (Conversion Factors)..........i6mo, mor.
3	00
	25
3	00
3	50
2	50
i	25
3	00
i	00
i	50
i	00
5	00
5	00
3	00
3	50
2	50
i	50
2	50
i	25
3	00
2	50
i	25
6*	Ives’s Adjustments of the Engineer’s Transit and Level.....i6mo, Bds. 25
Ives and Hilts’s Problems in Surveying........................i6mo, mor. 1 50
Johnson’s (J. B.) Theory and Practice of Surveying............Small 8vo, 4 00
Johnson’s (L. J.) Statics by Algebraic and Graphic Methods............8vo,	2	00
Kinnicutt, Winslow and Pratt’s Purification of Sewage. (In Preparation).
Laplace’s Philosophical Essay on Probabilities. (Truscott and Emory.)
i2mo, 2 00
Mahan’s Descriptive Geometry..........................................8vo,	1	50
Treatise on Civil Engineering. (1873.) (Wood.)...................8vo,	5	00
Merriman’s Elements of Precise Surveying and Geodesy..................8vo,	2	50
Merriman and Brooks’s Handbook for Surveyors..................i6mo, mor. 2 00
Morrison’s Elements of Highway Engineering. (In Press.)
Nugent’s Plane Surveying..............................................8vo,	3	50
Ogden’s Sewer Design..................................................nmo,	2	00
Parsons’s Disposal of Municipal Refuse................................8vo,	2	00
Patton’s Treatise on Civil Engineering...................8vo, half leather, 7 50
Reed’s Topographical Drawing and Sketching............................4to,	5	00
Rideal’s Sewage and the Bacterial Purification of Sewage..............8vo,	4	00
Riemer’s Shaft-sinking under Difficult Conditions. (Corning and Peele.). .8vo,	3 00
Siebert and Biggin’s Modern Stone-cutting and Masonry.................8vo,	1	50
Smith’s Manual of Topographical Drawing. (McMillan.)..................8vo,	2	50
Soper’s Air and Ventilation of Subways. (In Press.)
Tracy’s Plane Surveying.......................................l6mo, mor. 3 00
*	Trautwine’s Civil Engineer’s Pocket-book...................i6mo, mor. 5 00
Venable’s Garbage Crematories in America..............................8vo,	2	00
Methods and Devices for Bacterial Treatment of Sewage..........8vo,	3	00
Wait’s Engineering and Architectural Jurisprudence....................8vo,	6	00
Sheep, 6 50
Law of Contracts.................................................8vo,	3	00
Law of Operations Preliminary to Construction in Engineering and Architecture..........................................................8vo,	5	00
Sheep, 5 5°
Warren’s Stereotomy—Problems in Stone-cutting.........................8vo,	2	50
*	Waterbury’s Vest-Pocket Hand-book of Mathematics for Engineers.
21X 5)! inches, mor. 1 00
Webb’s Problems in the Use and Adjustment of Engineering Instruments.
i6mo, mor.	1	25
Wilson’s Topographic Surveying........................................8vo,	3	50
BRIDGES AND ROOFS.
Boiler’s Practical Treatise on the Construction of Iron Highway Bridges. .8vo,	2	00
Burr and Falk’s Design and Construction of Metallic Bridges...........8vo,	5	00
Influence Lines for Bridge and Roof Computations.................8vo,	3	00
Du Bois’s Mechanics of Engineering. Vol. II...................Small 4to, 10 00
Foster’s Treatise on Wooden Trestle Bridges...........................4*0»	5	00
Fowler’s Ordinary Foundations.........................................8vo,	3	5°
French and Ives’s Stereotomy..........................................8vo,	2	50
Greene’s Arches in Wood, Iron, and Stone..............................8vo,	2	50
Bridge Trusses...................................................8vo,	2	50
Roof Trusses.....................................................8vo,	1	25
Grimm’s Secondary Stresses in Bridge Trusses..........................8vo,	2	50
Heller’s Stresses in Structures and the Accompanyin Deformations....8vo,
Howe’s Design of Simple Roof-trusses in Wood and Steel................8vo,	2	00
Symmetrical Masonry Arches.......................................8vo,	2	50
Treatise on Arches............................................... 8vo,	4	00
Johnson, Bryan, and Turneaure’s Theory and Practice in the Designing of
Modern Framed Structures.............................Small 4to, 10 00
7Merriman and Jacoby’s Text-book on Roofs and Bridges:
Part I.	Stresses in Simple Trusses...............................8vo,	2	50
Part II.	Graphic Statics.........................................8vo,	2	50
Part III.	Bridge Design...........................................8vo,	2	50
Part IV.	Higher Structures.......................................8vo,	2	50
Morison’s Memphis Bridge.......................................Oblong 4to, 10 00
Sondericker’s Graphic Statics, with Applications to Trusses, Beams, and Arches.
8vo, 2 00
Waddell’s De Pontibus, Pocket-book for Bridge Engineers....... i6mo, mor, 2 00
* Specifications for Steel Bridges...................................i2mo, 50
Waddell and Harrington’s Bridge Engineering. (In Preparation.)
Wright’s Designing of Draw-spans. Two parts in one volume.............8vo, 3 50
HYDRAULICS.
Barnes’s Ice Formation..............................................8vo,
Bazin’s Experiments upon the Contraction	of	the	Liquid	Vein	Issuing from
an Orifice. (Trautwine.)...................................8vo,
Bovey’s Treatise on Hydraulics......................................8vo,
Church’s Diagrams of Mean Velocity of Water in Open Channels.
Oblong	4to,	paper,
Hydraulic Motors. .............................................8vo,
Mechanics of Engineering.......................................8vo,
Coffin’s Graphical Solution of Hydraulic Problems.......i6mo, morocco,
Flather’s Dynamometers, and the Measurement of	Power.........i2mo,
Folwell’s Water-supply Engineering..................................8vo,
Frizell’s Water-power...............................................8vo,
Fuertes’s Water and Public Health.............................i2mo,
Water-filtration Works...................................i2mo,
Ganguillet and Kutter’s General Formula for	the	Uniform	Flow of Water in
Rivers and Other Channels. (Hering and Trautv
Filtration of Public Water-supplies...........
Hazlehurst’s Towers and Tanks for Water-works..........
Herschel’s 115 Experiments on the Carrying Capacity of Large, Riveted, Metal
Conduits...................................................8vo,
Hoyt and Grover’s River Discharge...................................8vo,
Hubbard and Kiersted’s Water-works Management and Maintenance......8vo,
*	Lyndon’s Development and Electrical Distribution of Water Power. .. .8vo, Mason’.s Water-supply. (Considered Principally from a Sanitary Standpoint.)
8vo,
Merriman’s Treatise on Hydraulics...................................8vo,
*	Michie’s Elements of Analytical Mechanics.........................8vo,
Molitor’s Hydraulics of Rivers, Weirs and Sluices. (In Press.)
Schuyler’s Reservoirs for Irrigation, Water-power, and Domestic Water-
supply...............................................Large 8vo,
*	Thoma> and Watt’s Improvement of Rivers...........................4to,
Turneaure and Russell’s Public Water-supplies.......................8vo,
Wegmann’s Design and Construction of Dams. 5th Ed., enlarged........4*0,
Water-supply of the City of New York from 1658 to 1895.........4*°»
Whipple’s Value of Pure Water...............................Large i2mo,
Williams and Hazen’s Hydraulic Tables...............................8vo,
Wilson’s Irrigation Engineering..............................Small 8vo,
Wolff’s Windmill as a Prime Mover...................................8vo,
Wood’s Elements of Analytical Mechanics.............................8vo,
Turbines......................................................  8vo,
3 00
2 00 5 00
50
00
00
50
00
00
00
1	50
2	50
)	8vo,	4	00
. . Large i2mo,	1	5o
	8 vo,	3	00
	8vo,	2	50
2 00
2	00 4 00
3	00
4	00
5	00 4 00
5	00
6	00
5	00
6	00 10 00
1 00
1	50 4 00 3 00 3 00
2	50
SMATERIALS OF ENGINEERING.
Baker’s Roads and Pavements..........................................8vo,
Treatise on Masonry Construction................................8vo,
Birkmire’s Architectural Iron and Steel..............................8vo,
Compound Riveted Girders as Applied in Buildings..............8vo,
Black’s United States Public Works............................Oblong 4to,
Bleininger’s Manufacture of Hydraulic Cement. (In Preparation.)
*	Bovey’s Strength of Materials and Theory of Structures............8vo,
Burr’s Elasticity and Resistance of the Materials of Engineering.....8vo,
Byrne’s Highway Construction..........................................8vo,
Inspection of the Materials and Workmanship Employed in Construction.
i6mo,
Church’s Mechanics of Engineering.....................................8vo,
Du Bois’s Mechanics of Engineering.
Vol. I. Kinematics, Statics, Kinetics.....................Small 4to,
Vol. II. The Stresses in Framed Structures, Strength of Materials and
Theory of Flexures......................................Small 4to,
♦Eckel’s Cements, Limes, and Plasters.. ..............................8vo,
Stone and Clay Products used in Engineering. (In Preparation.)
Fowler’s Ordinary Foundations.........................................8vo,
Graves’s Forest Mensuration...... ....................................8vo,
Green’s Principles of American Forestry..............................i2mo,
*	Greene’s Structural Mechanics......................................8vo,
Holly and Ladd’s Analysis of Mixed Paints, Color Pigments and Varnishes
Large i2mo,
Johnson’s Materials of Construction...........................Large 8vo,
Keep's Cast Iron......................................................8vo,
Kidder’s Architects and Builders’ Pocket-book........................i6mo,
Lanza’s Applied Mechanics.............................................8vo,
Maire’s Modern Pigments and their Vehicles ......................... i2mo,
Martens’s Handbook on Testing Materials. (Henning.) 2 vols...........8vo,
Maurer’s Technical Mechanics..........................................8vo,
Merrill’s Stones for Building and Decoration......................... 8vo,
Merriman’s Mechanics of Materials.....................................8vo,
*	Strength of Materials..........................................i2mo,
Metcalf’s Steel. A Manual for Steel-users.............................nmo,
Patton’s Practical Treatise on Foundations............................8vo,
Rice’s Concrete Block Manufacture.....................................8vo,
Richardson’s Modern Asphalt Pavements...............................  8vo,
Richey’s Handbook for Superintendents of Construction.. .	.... i6mo, mor.,
*	Ries’s Clays: Their Occurrence, Properties, and Uses..............8vo,
Sabin’s Industrial and Artistic Technology of Paints and Varnish.....8vo,
♦Schwarz’s Longleaf Pine in Virgin Forest..,..........................nmo,
Snow’s Principal Species of Wood......................................8vo,
Spalding’s Hydraulic Cement..........................................i2mo,
Text-book on Roads and Pavements.................................nmo,
Taylor and Thompson’s Treatise on Concrete, Plain and Reinforced.....8vo,
Thurston’s Materials of Engineering. In Three Parts...................8vo,
Part I. Non-metallic Materials of Engineering and Metallurgy....8vo,
Part II. Iron and Steel..........................................8vo,
Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents.................................................8vo,
Tillson’s Street Pavements and Paving Materials.......................8vo,
Tumeaure and Maurer’s Principles of Reinforced Concrete Construction.. . 8vo, Wood’s (De V.) Treatise on the Resistance of Materials, and an Appendix on
the Preservation of Timber.................................. 8vo,
Wood’s (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and Steel...........................................................8vo,
10
6
3
4
1
2
2
6
2
5 7 2
7
4
5 5
1
2 5 2
3
4
5 3
1
3
2 2
5
8
2
3
2
4
3
00
00
50
00
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50
50
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50
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5°
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50
50
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50
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50
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50
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25
50
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50
5«
00
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00
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9RAILWAY ENGINEERING.
Andrews’s Handbook for Street Railway Engineers..........3x5 inches, mor. i 25
Berg’s Buildings and Structures of American Railroads..............4to, 5 00
Brooks’s Handbook of Street Railroad Location..................i6mo, mor.	1	50
Butt’s Civil Engineer’s Field-book.............................i6mo, mor.	2	50
Crandall’s Railway and Other Earthwork Tables..................... • 8V0>	1	5°
Transition Curve........................................  i6mo,	mor.	1	50
♦Crockett’s Methods for Earthwork Computations.....................&vo9 1 50
Dawson’s “Engineering” and Electric Traction Pocket-book.....i6mo, mor.	5	00
Dredge’s History of the Pennsylvania Railroad: (1879)..............Paper, 5 00
Fisher’s Table of Cubic Yards................................Cardboard, 25
Godwin’s Railroad Engineers’ Field-book and Explorers’ Guide. . . i6mo, mor. 2 50 Hudson’s Tables for Calculating the Cubic Contents of Excavations and Embankments..........................................................8vo, i 00
Ives and Hilts’s Problems in Surveying, Railroad Surveying and Geodesy
i6mo, mor. 1 50
Molitor and Beard’s Manual for Resident Engineers..................i6mo, 1 00
Nagle’s Field Manual for Railroad Engineers....................i6mo,	mor.	3	00
Philbrick’s Field Manual for Engineers.........................i6mo,	mor.	3	00
Raymond’s Railroad Engineering. 3 volumes.
Vol. I. Railroad Field Geometry. (In Preparation.)
Vol. II. Elements of Railroad Engineering...................8vo, 3 50
Voi III. Railroad Engineer’s Field Book. (In Preparation.)
Searles’s Field Engineering....................................i6mo,	mor.	3	00
Railroad Spiral...........................................i6mo,	mor.	1	50
Taylor’s Prismoidal Formulas and Earthwork.........................8vo, 1 50
♦Trautwine’s Field Practice of Laying Out Circular Curves for Railroads.
i2mo. mor, 2 50
*	Method of Calculating the Cubic Contents of Excavations and Embank-
ments by the Aid of Diagrams................................8vo,	2	00
Webb’s Economics of Railroad Construction....................Large i2mo, 2 50
Railroad Construction...................................iómo, mor. 5 00
Wellington’s Economic Theory of the Location of Railways.......Small	8vo,	5	0<>
DRAWING.
Barr’s Kinematics of Machinery.......................................8vo,	2	50
*	Bartlett’s Mechanical Drawing.....................................8vo,	3	0(>
*	“	“	“	Abridged Ed.......................8vo, 1 5°
Coolidge’s Manual of Drawing.................................8vo, paper, 1 00
Coolidge and Freeman’s Elements of General Drafting for Mechanical Engineers........................................................Oblong 4to, 2 5<>
Durley’s Kinematics of Machines......................................8vo,	4	00
Emch’s Introduction to Projective Geometry and its Applications......Svo,	2	50
Hill’s Text-book on Shades and Shadows, and Perspective..............8vo,	2	00
Jamison’s Advanced Mechanical Drawing................................8vo,	2	00
Elements of Mechanical Drawing..................................8vo,	2	50
Jones’s Machine Design:
Parti. Kinematics of Machinery..................................8vo,	1	5 o
Part II. Form, Strength, and Proportions of	Parts...............8vo,	3	00
•MacCord’s Elements of Descriptive Geometry..........................8vo,	3	oc
Kinematics; or, Practical Mechanism.............................8vo,	5	00
Mechanical Drawing..............................................4to,	4	oa
Velocity Diagrams...............................................8vo,	1	5°
McLeod’s Descriptive Geometry................................Large i2mo, 1 5C
*	Mahan’s Descriptive Geometry and Stone-cutting....................8vo,	1	50,
Industrial Drawing. (Thompson.).................................8vo,	3	50
10Moyer’s Descriptive Geometry..........................................8vo,
Reed’s Topographical Drawing and Sketching............................4to,
Reid’s Course in Mechanical Drawing...................................8vo,
Text-book of Mechanical Drawing and Elementary Machine Design.8vo,
Robinson’s Principles of Mechanism................................... 8vo,
Schwamb and Merrill’s Elements of Mechanism...........................8vo,
Smith’s (R. S.) Manual of Topographical Drawing. (McMillan.).........8vo,
Smith (A. W.) and Marx’s Machine Design..............................8vo,
* Titsworth’s Elements of Mechanical Drawing...................Oblong 8vof
barren’s Drafting Instruments and Operations.........................i2mo,
Elements of Descriptive Geometry, Shadows, and Perspective.....8vo,
Elements of Machine Construction and Drawing....................8vo,
Elements of Plane and Solid Free-hand Geometrical Drawing. . . i .2mo,
General Problems of Shades and Shadows..........................8vo,
Manual of Elementary Problems in the Linear Perspective of Form and
Shadow...................................................121110,
Manual of Elementary Projection Drawing........................i2mo,
Plane Problems in Elementary Geometry..........................i2mo,
Problems, Theorems, and Examples in Descriptive Geometry........8vo,
Weisbach’s Kinematics and Power of Transmission. (Hermann and
Klein.).....................................................8vo,
Wilson’s (H. M.) Topographic Surveying................................8vo,
Wilson’s (V. T.) Free-hand Lettering..................................8vo,
Free-hand Perspective...........................................8vo,
Woolf’s Elementary Course in Descriptive Geometry..............Large 8vo,
ELECTRICITY AND PHYSICS.
*	Abegg’s Theory of Electrolytic Dissociation, (von Ende.).......i2mo,
Andrews’s Hand-Book for Street Railway Engineering.....3X5 inches, mor.,
Anthony and Brackett’s Text-book of Physics. (Magie.)........Large i2mo,
Anthony’s Lecture-notes on the Theory of Electrical Measurements. . . .i2mo, Benjamin’s History of Electricity....................................8vo,
Voltaic Cell....................................................8vo,
Betts’s Lead Refining and Electrolysis..............................8vo,
Classen’s Quantitative Chemical Analysis by Electrolysis. (Boltwood.). 8vo,
*	Collins’s Manual of Wireless Telegraphy........................i2mo,
Mor.
Crehore and Squier’s Polarizing Photo-chronograph....................8vo,
*	Danneel’s Electrochemistry. (Merriam.)...........................i2mo,
Dawson’s “Engineering” and Electric Traction Pockei-book.....i6mo, mor
Dolezalek’s Theory of the Lead Accumulator (Storage Battery), (von Ende.)
i2mo,
Duhem’s Thermodynamics and Chemistry. (Burgess.).....................8vo,
Flather’s Dynamometers, and the Measurement of Power..............nmo,
Gilbert’s De Magnete. (Mottelay.)....................................8vo,
*	Hanchett’s Alternating Currents..................................i2mo,
Hering’s Ready Reference Tables (Conversion Factors')........i6mo, mor.
Hobart and Ellis’s High-speed Dynamo Electric Machinery. (In Press.) Holman’s Precision of Measurements...................................8vo,
Telescopic Mirror-scale Method, Adjustments, and Tests. . . .Large 8vo,
*	Karapetoff’s Experimental Electrical Engineering..................8vo,
Kinzbrunner’s Testing of Continuous-current Machines.................8vo,
Landauer’s Spectrum Analysis. • (Tingle.)............................8vo,
Le Chatelier’s High-temperature Measurements. (Boudouard—Burgess.) i2mo, Lob’s Electrochemistry of Organic Compounds. (Lorenz.)...............8vo,
*	Lyndon’s Development and Electrical Distribntion of Water Power . . . .8vo,
*	Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II. 8vo, each,
*	Michie’s Elements of Wave Motion Relating to Sound and Light.....8vo,
11
2 oa 5 00
2	00
3	00 3 oa 3 00
2	50
3	00 1 25 1 25. 3 50 7 50 1 00 3 OO
1 00 I 50
1	25
2	50
5 oa
3	5^
1	oa
2	5a
3	oa
1 25 1 25 3 00 1 00 3 00
3	00
4	00 3 oa
1	50^
2	00
3	00
1	25
5	00
2	50
4	oo-
3	00 2 50
1	00
2	50
2 00 75
6	00
2	00
3	00 3 00 3 oa
3	00 6 oa
4	oaMorgan’s Outline of the Theory of Solution and its Results............nmo,	i	oo
*	Physical Chemistry for Electrical Engineers.......................i2mo,	i	50
Niaudet’s Elementary Treatise on Electric Batteries. (Fishback).... nmo.	2	50
*	Norris’s Introduction to the Study of Electrical Engineering.......8vo,	2	50
*	Parshall and Hobart’s Electric Machine Design.......4to, half morocco, 12 50
Reagan’s Locomotives: Simple, Compound, and Electric. New Edition.
Large 12mo, 3 50
*	Rosenberg’s Electrical Engineering. (Haldane Gee—Kinzbrunner.). . .8vo,	2	00
Ryan, Norris, and Hoxie’s Electrical Machinery. Vol. 1................8vo,	2	50
Schapper’s Laboratory Guide for Students in Physical Chemistry.......i2mo,	1	00
Thurston’s Stationary Steam-engines...................................8vo,	2	50
*	Tillman’s Elementary Lessons in Heat...............................8vo,	1	50
Tory and Pitcher’s Manual of Laboratory Physics...............Large nmo, 2 00
Ulke’s Modern Electrolytic Copper Refining............................Svo,	3	00
LAW.
*	Davis’s Elements of Law............................................8vo,
*	Treatise on the Military	Law	of	United States....................8vo,
*	Sheep,
*	Dudley’s Military Law and the Procedure of Courts-martial . .. .Large 12mo,
Manual for Courts-martial......................................i6mo, mor.
Wait’s Engineering and Architectural Jurisprudence....................8vo,
Sheep,
Law of Contracts.................................................8vo,
Law of Operations Preliminary to Construction in Engineering and Architecture..........................................................8vo
Sheep,
2 50 7 00 7 50
2	50 1 50 6 00 6 50
3	00
5 00 5 50
MATHEMATICS.
Baker’s Elliptic Functions.........................................8vo,	1	50
Briggs’s Elements of Plane Analytic Geometry. (Bocher).............nmo,	1	00
*	Buchanan’s Plane and Spherical Trigonometry.....................8vo,	1	00
Byerley’s Harmonic Functions.......................................8vo,	1	00
Chandler’s Elements of the Infinitesimal Calculus.*............. i2mo, . 2 00
Compton’s Manual of Logarithmic Computations.......................nmo,	1	50
Davis’s Introduction to the Logic of Algebra.......................8vo,	1	50
* Dickson’s College Algebra.................................Large nmo, 1 50
* s Introduction to the Theory of Algebraic Equations.......Large nmo, 1 25
Emch’s Introduction to Projective Geometry and its Applications....8vo,	2	50
Fiske’s Functions of a Complex Variable..........................  8vo,	1	00
Halsted’s Elementary Synthetic Geometry............................8vo,	1	50
Elements of Geometry.............................................8vo,	1	75
*	Rational Geometry...............................................nmo,	1	50
Hyde’s Grassmann’s Space Analysis..................................8vo,	1	00
*	Jonnson’s (J B.) Three-place Logarithmic Tables: Vest-pocket size, paper, 15
100 copies, 5 00
*	Mounted on heavy cardboard, 8X10 inches, 25
10 copies, 2 00
Johnson’s (W. W.) Abridged Editions of Differential and Integral Calculus
Large nmo, 1 vol. 2 50
Curve Tracing in Cartesian Co-ordinates............................nmo,	1	00
Differential Equations.............................................8vo,	1	00
Elementary Treatise on Differential Calculus. (In Press.)
Elementary Treatise on the Integral Calculus...............Large nmo> 1 50
* Theoretical Mechanics.............................*....................nmo, 3 00
Theory of Errors and the Method of Least Squares...................nmo,	1	50
Treatise on Differential Calculus..........................Large nmo, 3 00
Treatise on the Integral Calculus..........................Large 12mo, 3 00
Treatise on Ordinary and Partial Differential Equations.. Large i2mo, 3 50
12Laplace’s Philosophical Essay on Probabilities. (Truscott and Emory.).i2mo,	2 00
*	Ludlow and Bass’s Elements of Trigonometry and Logarithmic and Other
Tables.....................................................8vo, 3 00
Trigonometry and Tables published separately....................Each, 2 00
* Ludlow’s Logarithmic and Trigonometric Tables......................8vo, 1 00
Macfarlane’s Vector Analysis and Quaternions.........................8vo, 1 00
McMahon's Hyperbolic Functions.......................................8vo, 1 00
Manning’s IrrationalNumbers and their Representation bySequences and Series
12010,	1 25
Mathematical Monographs. Edited by Mansfield Merriman and Robert
S. Woodward........................................Octavo, each 1 00
No. 1. History of Modern Mathematics, by David Eugene Smith.
No. 2. Synthetic Projective Geometry, by George Bruce Halsted.
No. 3. Determinants, by Laenas Gifford Weld. No. 4. Hyperbolic Functions, by James McMahon. No, 5. Harmonic Functions, by William E. Byerly. No. 6. Grassmann’s Space Analysis, by Edward W. Hyde. No. 7. Probability and Theory of Errors, by Robert S. Woodward. No. 8. Vector Analysis and Quaternions, by Alexander Macfarlane. No. g. Differential Equations, by William Woolsey Johnson. No. 10. The Solution of Equations, by Mansfield Merriman. No. 11. Functions of a Complex Variable, by Thomas S. Fiske.
Maurer’s Technical Mechanics.........................................8vo,	4	00
Menfman’s Method of Least Squares....................................8vo,	2	00
Solution of Equations.........................................8vo,	1	00
Rice and Johnson’s Differential and Integral Calculus. 2 vols. in one.
Large 12 mo, 1 50
Elementary Treatise on the Differential Calculus........Large i2mo, 3 00
Smith’s History of Modern Mathematics................................8vo,	1	00
*	Veblen and Lennes’s Introduct;on to the Real Infinitesimal Analysis of One
Variable...................................................8vo, 2 00
*	Waterbury’s Vest Pocket Hand-Book of Mathematics for Enginecrs.
2IX 5s inches, mor.,	1	00
Weld’s Determinations................................................8vo,	1	00
Wood's Elements of Co-ordinate Geometry..............................8vo,	2	00
Woodward's Probability and Theory of Errors..........................8vo,	1	00
MECHANICAL ENGINEERING.
MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS.
Bacon’s Forge Practice...............................................i2mo,
Baldwin's Steam Heating for Buildings................................i2mo,
Bair's Kinematics of Machinery........................................8vo,
*	Bartlett’s Mechanical Drawing..................................... 8vo,
*	“	*	** Abridged Ed.............................8vo,
Benjamin’s Wrinkles and Recipes......................................i2mo,
*	Burr’s Ancient and Modern Engineering and the Isthmian Canal......8vo,
Carpenter’s Experimental Engineering..................................8vo,
Heating and Ventilating Buildings................................8vo,
Clerk’s Gas and Oil Engine....................................Large i2mo,
Compton’s First Lessons in Metal Working.............................i2mo,
Compton and De Groodt’s Speed Lathe..................................12mo,
Coolidge’s Manual of Drawing..................................8vo, paper,
Ccolidge and Freeman’s Elements of General Drafting for Mechanical Engineers......................................................Oblong 4to,
Cromwell's Treatise on Belts and Pulleys........................... .i2mo,
Treatise on Toothed Gearing.....................................i2mo,
Durley’s Kinematics of Machines.....................................  8vo,
50
50
50
00
50
00
50
00
00
00
i 50 i 50 i 00
2 50 i 50 i 50 4 00
13Flather’s Dynamometers and the Measurement of Power,.................i2mo,	3	00
Rope Driving....................................................i2mo,	2	00
Gill’s Gas and Fuel Analysis for Engineers...........................i2mo,	1	25
Goss':? Locomotive Sparks.............................................8vo,	2	00
Hall’s Car Lubrication...............................................i2mo,	1	00
Hering's Ready Reference Tables (Conversion Factors)........i6mo, mor., 2 50
Hobart and Ehis’s High Speed Dynamo Electric Machinery. (In Press.)
Hutton’s Gas Engine...................................................8vo,	5	00
Jamison’s Advanced Mechanical Drawing.................................8vo,	2	00
Elements of Mechanical Drawing.................................8vo,	2	5a
Jones’s Machine Design:
Parti. Kinematics of Machinery...................................8vo,	1	50
Part II. Form, Strength, and Proportions of Parts................8vo,	3	00
Kent’s Mechanical Engineers* Pocket-book....................i6mo, mor , 5 00
Kerr’s Power and Power Transmission...................................8vo,	2	00
Leonard’s Machine Shop Tools and Methods .............................8vo,	4	00
*	Lorenz’s Modern Refrigerating Machinery. (Pope, Haven, and Dean.) . . 8vo,	4	00
MacCord’s Kinematics; or, Practical Mechanism.........................8vo,	5	00
Mechanical Drawing...............................................4to,	4	00
Velocity Diagrams................................................8vo,	1	50
MacFarland’s Standard Reduction Factors for Gases.....................8vo,	1	50
Mahan’s Industrial Drawing. (Thompson.)...............................8vo,	3	50
*	Parshall and Hobart’s Electric Machine Design..Small 4to, half leather, 12 50
Peele’s Compressed Air Plant for Mines. (In Press.)
Poole’s Calorific Power of Fuels......................................8vo,	3	00
*	Porter’s Engineering Reminiscences, 1855 to 1882...................8vo,	3	00
Reid’s Course in Mechanical Drawing...................................8vo,	2	00
Text-book of Mechanical Drawing and Elementary Machine Design.8vo,	3	00
Richard’s Compressed Air.............................................i2mo,	1	50
Robinson’s Principles of Mechanism....................................8vo,	3	00
Schwamb and Merrill’s Elements of Mechanism...........................8vo,	3	00
Smith’s (O.) Press-working of Metals..................................8vo,	3	00
Smith (A. W.) and Marx’s Machine Design...............................8vo,	3	00
Sorel ’ s Carbureting and Combustion in Alcohol Engines. (Woodward and Preston).
Large i2mo, 3 00
Thurston’s Animal as a Machine and Prime Motor, and the Laws of Energetics.
i2mo, 1 00
Treatise on Friction and Lost Work in Machinery and Mill	Work... 8vo,	3	00
Tillson’s Complete Automobile Instructor..........................i6mo,	1	50
mor., 2 00
*	Titsworth’s Elements of Mechanical Drawing...............Oblong 8vo, 1 25
Warren’s Elements of Machine Construction and Drawing.................8vo,	7	50
*	Waterbury’s Vest Pocket Hand Book of Mathematics for Engineers.
2gX 5£ inches, mor., 1 00
Weisbach’s Kinematics and the Power of Transmission. (Herrmann—
Klein.)......................................................8vo,	5	00
Machinery of Transmission and Governor«. (Herrmann—Klein.). .8vo,	5	00
Wolff’s Windmill as a Prime Mover.....................................8vo,	3	00
Wood’s Turbines.......................................................8vo,	2	50
MATERIALS OF ENGINEERING.
*	Bóvey’s Strength of Materials and Theory of Structures.............8vo,	7	50
Burr’s Elasticity and Resistance of the Materials of Engineering.	 .8vo,	7	50
Church’s Mechanics of Engineering.....................................8vo,	6	00
*	Greene’s Structural Mechanics...................................   8vo,	2	50
Holley and Ladd’s Analysis of Mixed Paints, Color Pigments, and Varnishes.
Large	i2mo,	2	50
Johnson’s Materials of Construction...................................8vo,	6	00
Keep’s Cast Iron......................................................8vo,	2	50
Lanza’s Applied Mechanics.............................................8vo,	7	50
14Maire’s Modern Pigments and their Vehicles.........................nmo, 2 00
Martens’s Handbook on Testing Materials. (Henning.)................8vo, 7 50
Maurer’s Technical Mechanics.......................................8vo, 4 00
Merriman’s Mechanics of Materials..................................8vo, 5 00
* Strength of Materials............................................nmo, 1 00
Metcalf’s Steel. A Manual for Steel-users............................nmo,	2	00
Sabin’s Industrial and Artistic Technology of Paints and Varnish.....8vo,	3	00
Smith’s Materials of Machines........................................nmo,	1	00
Thurston’s Materials of Engineering..........................3 vols., 8vo, 8 00
Part I. Non-metallic Materials of Engineering, see Civil Engineering, page 9.
Part II. Iron and Steel................................................8vo,	3	50
Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents.......................................................8vo,	2	50
"Wood’s (De V.) Elements of Analytical Mechanics.....................8vo,	3	00
Treatise on the Resistance of Materials and an Appendix on the
Preservation of Timber.............................................8vo,	2	00
Wood’s (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and
Steel..............................................................8vo,	4	00
STEAM-ENGINES AND BOIIERS.
Berry’s Temperature-entropy Diagram.................................i2mo,	1	25
Carnot’s Reflections on the Motive Power of Heat. (Thurston.).......i2mo,	1	50
Chase’s Art of Pattern Making.......................................i2mo,	2	50
Creighton’s Steam-engine and other Heat-motors. . .................. 8vo,	5	00
Dawson’s “Engineering” and Electric Traction Pocket-book. .	.	.i6mo, mor.,	5	00
Pord’s Boiler Making for Boiler Makers..............................i8mo,	1	00
Goss’s Locomotive Performance........................................8vo,	5	00
Hemenway’s Indicator Practice and Steam-engine Economy..............i2mo,	2	00
Hutton’s Heat and Heat-engines.......................................8vo.	5	00
Mechanical Engineering of Power Plants..........................8vo,	5	00
Kent’s Steam boiler Economy..........................................8vo,	4	00
Kneass’s Practice and Theory of the Injector.........................8vo,	1	50
MacCord’s Slide-valves...............................................8vo,	2	00
Meyer’s Modern Locomotive Construction...............................4*°»	10	00
Moyer’s Steam Turbines. (In Press.)
Peabody’s Manual of the Steam-engine Indicator.......................nmo,	1	50
Tables of the Properties of Saturated Steam and Other Vapors....8vo,	1	00
Thermodynamics of the Steam-engine and Other Heat-engines.......8vo,	5	00
Valve-gears for Steam-engines...................................8vo,	2	50
Peabody and Miller’s Steam-boilers...................................8vo,	4	00
Pray’s Twenty Years with the Indicator.......................Large 8vo, 2 50
Pupin’s Thermodynamics of Reversible Cycles in Gases and Saturated Vapors.
(Osterberg.)...............................................i2mo,	1	25
Reagan’s Locomotives: Simple, Compound, and Electric. New Edition.
Large 12mo,	3	50
Sinclair’s Locomotive Engine Running and Management.................i2mo,	2	00
Smart’s Handbook of Engineering Laboratory Practice.................i2mo,	2	50
Snow’s Steam-boiler Practice.........................................8vo,	3	00
Spangler’s Notes on Thermodynamics..................................i2mo,	1	00
Valve-gears.....................................................8vo,	2	50
Spangler, Greene, and Marshall’s Elements of Steam-engineering.......8vo,	3	00
Thomas’s Steam-turbines..............................................8vo,	4	00
Thurston’s Handbook of Engine and Eoiler Trials, and the Use of the Indicator and the Prony Brake............................................8vo,	5	00
Handy Tables....................................................8vo,	1	50
Manual of Steam-boilers, their Designs, Construction, and Operation..8vo,	5	00
15Thurston’s Manual of the Steam-engine........................2 vols., 8vo, 10 on
Parti. History, Structure, and Theory.......................8vo,	6	OO'
Part II. Design, Construction, and Operation................8vo,	6	00
Stationary Steam-engines......................................8vo,	2	50
Steam-boiler Explosions in Theory and in Practice............12mo,	1	50
Wehrenfenning’s Analysis and Softening of Boiler Feed-water (Patterson) 8vo,	4 00
Weisbach’s Heat, Steam, and Steam-engines. (Du Bois.)................8vo,	5	00
Whitham’s Steam-engine Design........................................8vo,	5	00
Wood’s Thermodynamics, Heat Motors, and Refrigerating Machines.. .8vo,	4 00
MECHANICS PURE AND APPLIED.
Church’s Mechanics of Engineering...................................8vo, 6 00
Notes and Examples in Mechanics................................8vo, 2 00
Dana’s Text-book of Elementary Mechanics for Colleges and Schools. .i2mo, 1 50 Du Bois’s Elementary Principles of Mechanics:
Vol. I. Kinematics............................................8vo,	3	50
Vol. II. Statics..............................................8vo,	4	00
Mechanics of Engineering. Vol. I........................Small 4to, 7 5°
Vol. II......................Small 4to, 10 00
*	Greene’s Structural Mechanics.....................................8vo,	2	50
James’s Kinematics of a Point and the Rational Mechanics of a Particle.
, Large 12mo,	2 00
*	Johnson’s (W. W.) Theoretical Mechanics..........................12mo,	3	00
Lanza’s Applied Mechanics............................................8vo,	7	50
*	Martin’s Text Book on Mechanics, Vol. I, Statics.................12mo,	1	25
*	Vol. 2, Kinematics and Kinetics I .i2mo, 1 50
Maurer’s Technical Mechanics.........................................8vo,	4	00
*	Merriman’s Elements of Mechanics.................................12mo,	1	00
Mechanics of Materials........................................8vo,	5	00
*	Michie’s Elements of Analytical Mechanics.........................8vo,	4	00
Robinson’s Principles of Mechanism...................................8vo,	3	00
Sanborn’s Mechanics Problems.................................Large 12mo, 1 50
Schwamb and Merrill’s Elements of Mechanism..........................8vo,	3	00
Wood’s Elements of Analytical Mechanics..............................8vo,	3	00
Principles of Elementary Mechanics.............................12mo,	1	25
MEDICAL.
Abderhalden’s Physiological Chemistry in Thirty Lectures. (Hall and Defren).
(in Press).
von Behring’s Suppression of Tuberculosis. (Bolduan.)...............i2mo,	1	00
*	Bolduan’s Immune Sera............................................i2mo,	1	50
Davenport’s Statistical Methods with Special Reference to Biological Variations........................................................ibmo, mor., 1 50
Ehrlich’s Collected Studies on Immunity. (Bolduan.).................8vo, 6 00
*	Fischer’s Physiology of Alimentation...............Large i2mo, cloth, 2 00
de Fursac’s Manual of Psychiatry. (Rosanoff and Collins.)....Large i2mo, 2 50
Hammarsten’s Text-book on Physiological Chemistry. (Mandel.).........8vo,	4	00
Jackson’s Directions for Laboratory Work in Physiological Chemistry. ,.8vo,	1	25
Lassar-Cohn’s Practical Urinary Analysis. (Lorenz.).................i2mo,	1	00
Mandel’s Hand Book for the Bio-Chemical Laboratory..................i2mo,	1	50
*	Pauli’s Physical Chemistry in the Service of Medicine. (Fischer.).... i2mo,	1	25
*	Pozzi-Escot’s Toxins and Venoms and their Antibodies. (Cohn.)....i2mo,	1	00
Rostoski’s Serum Diagnosis. (Bolduan.)..............................i2mo,	1	00
Ruddiman’s Incompatibilities in Prescriptions......................  8vo,	2	00
Whys in Pharmacy.............................................i2mo,	r	00
Salkowski’s Physiological and Pathological Chemistry. (Orndorff.)....8vo,	2	50
*	Satterlee’s Outlines of Human Embryology..........................nmo,	1	25
Smith’s Lecture Notes on Chemistry for Dental Students...............8vo,	2	50
16Steel’s Treatise on the Diseases of the Dog.........................8vo, 3 50
*	Whipple’s Typhoid Fever...................................Large i2mo, 3 00
Woodhull’s Notes on Military Hygiene................................i6mo, 1 50
*	Personal Hygiene................................................i2mo, i 00
Worcester and Atkinson’s Small Hospitals Establishment and Maintenance,
and S uggestions for Hospital Architecture, with Plans for a Small Hospital...................................................i2mo,	1	25
METALLURGY.
Betts’s Lead Refining by Electrolysis...............................8vo.	4	00
Bolland’s Encyclopedia of Founding and Dictionary of Foundry Terms Used
in the Practice of Moulding................................12mo,	3	00
Iron Founder................................................12mo.	2	50
“ Supplement..........................................i2mo,	2	50
Douglas’s Untechnical Addresses on Technical Subjects...............i2mo,	1	00
Goesel’s Minerals and Metals:	A Reference Book.............. .. . i6mo, mor. 3 00
*	Iles’s Lead-smelting.............................................12mo,	2	50
Keep’s Cast Iron....................................................8vo,	2	50
Le Chatelier’s High-temperature Measurements. (Boudouard—Burgess.) 12mo,	3	00
Metcalf’s Steel. A Manual for Steel-users...........................12mo,	2	00
Miller’s Cyanide Process............................................12mo	1	00
Minet’s Production of Aluminum and its Industrial Use. (Waldo.)... . 12mo,	2	50
Robine and Lenglen’s Cyanide Industry. (Le Clerc.)..................8vo, 4 00
Ruer’s Elements of Metallography. (Mathewson). (In Press.)
Smith’s Materials of Machines.......................................12mo,	1	00
Thurston’s Materials of Engineering. In Three Parts.................8vo, 8 00
part I. Non-metallic Materials of Engineering, see Civil Engineering, page 9.
Part II. Iron and Steel.......................................8vo, 3 50
Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents........................................8va,	2	50
Ulke’s Modern Electrolytic Copper Refining..........................8vo,	3	00
West’s American Foundry Practice....................................12mo,	2	50
Moulders Text Book..........................................12mo,	2	50
Wilson’s Chlorination Process.......................................l2mo,	1	50
Cyanide Processes...........................................12mo,	1	50
MINERALOGY.
Barringer’s Description of Minerals of Commercial Value. Oblong, morocco, 2 50
Boyd’s Resources of Southwest Virginia...............................8vo	3	00
Boyd’s Map of Southwest Virginia.......................Pocket-book form. 2 00
*	Browning’s Introduction to the Rarer Elements....................8vo, 1 50
Brush’s Manual of Determinative Mineralogy. (Penfield.).. ..........8vo, 4 00
Butler’s Pocket Hand-Book of Minerals........................16mo, mor. 3 00
Chester’s Catalogue of Minerals..............................8vo, paper, 1 00
Cloth, 1 25
Crane’s Gold and Silver. (In Press.)
Dana’s First Appendix to Dana’s New u System of Mineralogy. .**. .Large 8vo, 1 00
Manual of Mineralogy and Petrography..........................i2mo 2 00
Minerals and How to Study Them................................i2mo, 1 50
System of Mineralogy..........................Large 8vo, half leather, 12 50
Text-book of Mineralogy.......................................8vo, 4 00
Douglas’s Untechnical Addresses on Technical Subjects...............i2mo, 1 00
Eakle’s Mineral Tables..............................................8vo, 1 25
Stone and Clay Froducts Used in Engineering. (In Preparation).
Egleston’s Catalogue of Minerals and Synonyms. . ...................8vo, 2 50
Goesel’s Minerals and Metals:	A Reference Book...............i6mo, mor. 3 00
Groth’s Introduction to Chemical Crystallography (Marshall)...... 12mo, 1 25
17*	Iddings's Rock Minerals............................................8vo, 5 00
Johannsen’s Determination of Rock-forming Minerals in Thin Sections...8vo, 4 00
*	Martin’s Laboratory Guide to Qualitative Analysis with the Blowpipe. i2mo, 60
Merrill's Non-metallic Minerals: Their Occurrence and Uses............8vo, 4 00
Stones for Building and Decoration......... .....................8vo, 5 00
*	Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests.
8vo, paper, 50
Tables of Minerals, Including the Use of Minerals and Statistics of
Domestic Production............................................8vo, 1 00
Pirsson’s Rocks and Rock Minerals. (In Press.)
*	Richards's Synopsis of Mineral Characters...................i2mo,mor. 1 25
*	Ries’s Clays: Their Occurrence, Properties, and Uses...............8vo, 5 00
*	Tillman's Text-book of Important Minerals and Rocks.................8vo,	2	00
MINING.
*	Beard's Mine Gases and Explosions...........................Large i2mo, 3 00
Boyd's Map of Southwest Virginia.........................Pocket-book form, 2 00
Resources of Southwest Virginia.................................8vo,	3	00
Crane’s Gold and Silver. (In Press.)
Douglas's Untechnical Addresses on Technical Subjects.................i2mo,	I	00
Eissler’s Modern High Explosives.	 8vo.	4	00
Goesel's Minerals and Metals :	A Reference Book.. ,,...........i6mo, mor. 3 00
Ihlseng's Manual of Mining.............................................8vo,	5	00
*	Iles's Lead-smelting.............................................  i2mo,	2	50
Miller's Cyanide Process..............................................i2mo,	1	00
O'Driscoll's Notes on the Treatment of Gold Ores.......................Svo,	2	00
Peele's Compressed Air Plant for Mines. (In Press.)
Riemer’s Shaft Sinking Under Difficult Conditions. (Corning and Peele).. .8vo,	3 00
Robine and Lenglen's Cyanide Industry. (Le Clerc.).....................8vo,	4	00
*	Weaver's Military Explosives........................................8vo,	3	00
Wilson’s Chlorination Process.........................................limo,	1	50
Cyanide Processes................................................12mo,	1	50
Hydraulic and Placer Mining. 2d edition, rewritten...............i2mo,	2	50
Treatise on Practical and Theoretical Mine Ventilation...........i2mo,	1	23
SANITARY SCIENCE.
Association of State and National Food and Dairy Departments, Hartfora Meeting,
1906.........................................................8vo,	3	00
Jamestown Meeting, 1907........................................8vo,	3	00
*Bashore's Outlines of Practical Sanitation...........................12mo,	1	25
Sanitation of a Country House.................................12mo,	1	00
Sanitation of Recreation Camps and Parks......................12mo,	1	00
Folwell's Sewerage. (Designing, Construction,	and Maintenance.;........8vo,	3	00
Water-supply Engineering..........................................8vo,	4	00
Fowler's Sewage Works Analyses........................................12mo,	2	00
Fuertes's Water-filtration W’orks.....................................12mo,	2	50
Water and Public Health.......................................12mo,	1	50
Gerhard's Guide to Sanitary House-inspection..........................16mo,	1	00
* Modern Baths and Bath Houses ........................................8vo, 3 00
Sanitation of Public Buildings................................. 12mo, 1 50
Hazen’s Clean Water and How to Get It........................ Large l2mo, 1 50
Filtration of Public Water-supplies............................8vo,	3	00
Kinnicut, Winslow and Pratt’s Purification of Sewage. (In Press.)
Leach's Inspection and Analysis of Food with Special Reference to State
Control......................................................8vo,	7	00
Mason’s Examination of Water. (Chemical and	Bacteriological)..........12mo,	1	25
Water-supply. (Considered principally from a Sanitary Standpoint;.. 8vo, 4 00
18*	Merriman’s Elements of Sanitary Engineering......................8vo,
Ogden’s Sewer Design.................................................12mo,
Parsons’s Disposal of Municipal Refuse............................... .8vo,
Prescott and Winslow’s Elements of Water Bacteriology, with Special Reference to Sanitary Water Analysis.............................12mo,
*	Price's Handbook on Sanitation...................................12mo,
Richards’s Cost of Food. A Study in Dietaries.......................12mo,
Cost of Living as Modified by Sanitary Science.................12mo,
Cost of Shelter.................................................12mo,
*	Richards and Williams’s Dietary Computer.........................8vo,
Richards and Woodman’s Air, Water, and Food from a Sanitary Standpoint.................................................................8vo,
Rideal’s Disinfection and the Preservation of Food..................8vo,
Sewage and Bacterial Purification of	Sewage....................8vo,
Soper’s Air and Ventilation of Subways. (In Press.)
Turneaure and Russell’s Public Water-supplies.......................8vo,
Venable’s Garbage Crematories in America....... ....................8vo,
Method and Devices for Bacterial Treatment of Sewage...........8vo,
Ward and Whipple’s Freshwater Biology. (In Press.)
Whipple’s Microscopy of Drinking-water................................8vo,
*	Typhod Fever............................................Large 12mo,
Value of Pure Water.....................................Large l2mo,
Winton’s Microscopy of Vegetable Foods................................8vo,
MISCELLANEOUS.
Emmons’s Geological Guide-book of the Rocky Mountain Excursion of the
International Congress of Geologists...................Large 8vo,
Ferrel’s Popular Treatise on the Winds...............................8vo,
Fitzgerald’s Boston Machinist.........................................i8mo,
Gannett’s Statistical Abstract of the World...........................24mo,
Haines’s American Railway Management..................................12mo,
* Hanusek’s The Microscopy of Technical Products. (Winton^...........8vo,
Ricketts’s History of Rensselaer Polytechnic Institute? 1824-1894.
Large i2mo,
Rotherham’s Emphasized New Testament............................Large 8vo,
Standage’s Decoration of Wood, Glass, Metal, etc.....................12mo,
Thome ’s Structural and Physiological Botany. (Bennett)..............16mo,
Westermaier’s Compendium of General Botany. (Schneider)..............8vo,
Winslow’s Elements of Applied Microscopy..............................12mo,
HEBREW AND CHALDEE TEXT-BOOKS.
Green’s Elementary Hebrew Grammar.............................i2mo,
Gesenius’s Hebrew and Chaldee Lexicon to the Old Testament Scriptures.
(Tregelles.)..........................Small 4to, halfrinorocco,
19
2 oo> 2 00 2 00
i 50 i 50 i 00 i 00 i 00
1	50
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4	00
5	00
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3 60 3 00 i 00 7 50
I 50
4	00
1	00 75
2	50
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3	00 2 00 2 00 2 25 2 00 i 50
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