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MÖRSCH Of the Zurich Polytechnic Zurich Switzerland AUTHORIZED TRANSLATION FROM THE THIRD 1908 GERMAN EDITION REVISED AND ENLARGED BY E P GOODRICH Consulting Engineer NEW YORK THE ENGINEERING NEWS PUBLISHING COMPANY LONDON ARCHIBALD CONSTABLE AND COMPANY LTD 1909 Copyright 1909 BY THE ENGINEERING NEWS PUBLISHING COMPANY Entered at Stationers Hall London EC 1909 J F TAPLEY Co New York PREFACE TO SECOND EDITION In the absence of a uniform literature and in view of the number of profusely recommended systems the first edition of this work published by the firm of Wayss Freytag in 1902 effected the purpose of familiarizing those interested in the scientific principles of reinforced concrete with all the experimental researches available at that time The firm in question was impelled to publish it because systems based on wholly unscientific methods of calculation and offering no adequate security were being pushed into recognition by systematic advertisement so that the danger was imminent that reinforced concrete would forfeit a large proportion of the confidence it already enjoyed especially if a few failures should occur More than a year after the publication in connection with the first edition were published the Leitsätze Recommendations of the Verbands Deutscher Architekten und Ingenieure and of the Deutscher Beton Verein as well as the Regulations Bestimmungen of the Prussian government but they harmonized exactly with those of the first edition The publication of the second edition had another purpose The Leitsätze and the official Regulations had inspired widespread confidence in the new method of building but even the best of directions could not altogether obviate mistakes and failures where the proper knowledge of the coöperative effects of the two materialssteel and concretewas lacking In addition to this all directions presumed a knowledge of approved rules of construction as the Leitsätze could not possibly be amplified into a book of instructions on reinforced concrete This knowledge was however very difficult to obtain from the class journals and other literature because in these all sorts of systems were simultaneously described and conflicting opinions were also expressed The active part taken by the firm of Wayss Freytag as well as the undersigned Prof E Mörsch in the compilation of the preliminary Recommendations and the interest they manifested in making them final caused them to bring out the present second edition which represents a complete revision of the first edition and facilitates the application of the Leitsätze The general portion deals with examples chiefly relating to the practical reinforcement of Tbeams columns and arches under the most widely varied loads The succeeding and most comprehensive part treats of the theory of reinforced concrete covers exhaustively the properties of materials and then iii applies the theory in the closest possible manner to the results of the tests The author has avoided a repetition of useless theories on reinforced work of which there is no lack On the other hand he has succeeded in showing by means of tests that the methods of calculation given in the Leitsätze which are identical with those published in the first edition are well founded and useful At the same time the actual distribution of stress in reinforced sections was thoroughly studied The firm of Wayss Freytag placed the whole of their experimental data in great part hitherto unpublished at the disposal of the author in the preparation of the work In addition Bach gave the valuable results of the tests conducted for the reinforced concrete commission of the Jubiläumstiftung der Deutschen Industrie published in the course of the current year especially those relating to adhesion The third portion covering the uses of reinforced concrete reviews the most important fields of its utilization All the examples cited represent work done by the firm of Wayss Freytag and for the most part executed under the direction of the author in his capacity as director of the Technical Bureau of the beforementioned firm selected from their fifteen years experience in reinforced concrete work This limitation of the choice of examples is warranted inasmuch as all the reinforced construction work completed by the firm in question during the past five years has been calculated in accordance with the methods recommended in the Leitsätze and in accord with the rules given in the theoretical and general sections of the book regarding construction work The field of employment for reinforced concrete is constantly widening there can therefore be no claim raised that it has been completely covered only the most important features have been presented But the operations of this single firm give an excellent idea of the versatility of the employment of reinforced concrete The firm is well aware that the material herewith presented is of service to their competitors but believe that by a general deepening of knowledge of reinforced concrete they are rendering the most service to the subject Wayss Freytag Neustadt a d Haardt November 1905 Zurich November 1905 Professor E Mörsch OWING to the quick sale of the second edition at the request of the publishers and of the firm of Wayss Freytag the undersigned undertook the preparation of a third edition Of the new experiments conducted by the firm in the interim special attention must be called to those relating to shear in Tbeams and those made upon continuous beams These experiments in connection with the recently published results of the tests undertaken for the Reinforced Concrete Commission of the Jubiläumstiftung der Deutschen Industrie by the Testing Laboratory at Stuttgart made possible a detailed treatment of the subject in question Compared with the preceding editions it is here that the principal additions occur In addition the theoretical chapters relating to flexure and bending with axial stress were considerably extended In the applications the chapters on buildings columns and silos have likewise been enlarged In the preface to the second edition the grounds were given that led to the exclusive use of the work of the firm of Wayss Freytag These reasons still apply in regard to the new edition for most of the examples referred to in the applications were made under the authors direction and he also furnished the firm with the suggestions for the new tests The author has also collaborated as a member of the Commission in the program of tests conducted by the Testing Laboratory at Stuttgart In view of the present general development of reinforced concrete the standpoint of this work may possibly be designated as onesided It may be answered that the present advance in the art is in large part due to the efforts of the firm of Wayss Freytag and that on the other hand no complete presentation of all of the applications of reinforced concrete are contemplated because the scope of this work is much too limited Professor E Mörsch Zurich November 1907 PROFESSOR MÖRSCHs Eisenbetonbau is probably the clearest exposition of European methods of reinforced concrete construction that has yet been published It has for some years been a recognized standard in Europe and has also had a considerable demand in this country but the comparatively limited usefulness of the German edition to American engineers prompted us to make arrangements with Professor Mörsch for the rights of translation and publication of the book in the English language In the original German edition there is no division into chapters but for the sake of clearness and system and in conformity with American custom the translation has been divided into parts 1 The Theory of Reinforced Concrete and 2 The Applications of Reinforced Concrete which have been subdivided into Chapters and an Appendix On account of the impossibility of securing the original drawings and photographs from which to make reproductions for illustration it was necessary to import electros of the cuts used in the German book Wherever possible the wording of these has been translated into English and altered in the cut but in many cases such alterations were impossible and the German lettering has been left The measurements used in the German editions were in the metric system only in the translation the metric system has been retained but the English equivalents are given wherever measurements are quoted as well as in all tables Furthermore a table of metric and English equivalents has been included at the end of the book It is hoped that the efforts of the publishers to make available to Englishspeaking engineers the contents of this valuable work will merit their approval and appreciation The Engineering News Publishing Company Book Department New York November 1909 CONTENTS CHAPTER I Slabs 4 TBeams 8 CONCRETESTEEL CONSTRUCTION Der Eisenbetonbau CHAPTER I INTRODUCTION Reinforced concrete Eisenbeton is the name given to all varieties of construction in which are combined cementconcrete and steel in such manner that the two elements acting together statically resist all external forces In this connection it is to be understood that the concrete resists compressive stresses principally while the steel resists tensile ones in large measure that is gives the concrete a higher tensile strength In this type of construction many advantages and valuable properties result from the combination of these two quite dissimilar materials Buildings erected in this manner combine the massiveness of concrete with the lightness of steel construction and their wide distribution and daily growth in numbers is due to considerable economic advantages possessed by reinforced concrete over corresponding work in stone wood or iron Besides being cheaper in first cost than iron or wood practically all maintenance charges can be eliminated in reinforced concrete because of the rational manner in which use is made of the wearing qualities of the two elements Another excellent property of reinforcedconcrete work is its resistance to fire Because of this quality concrete has been employed for some time in building work in the shape of partitions and stairways and for the fireproofing of steel beams and columns Now columns and beams are built of the same materials which were formerly used simply for fireproofing purposes and in this way is secured a more uniform and cheaper fireproof construction These several advantages and the usefulness of reinforced concrete for the structural parts of beams columns and floor slabs arise from the following fundamental properties of concrete and steel in combination 1 Steel Covered with Concrete is most Perfectly Protected by it against Corrosion This is now a recognized fact but it should be added that only with relatively rich mixtures and with a plastic condition of the concrete not earthmoist can there be attained the intimate covering and adhesion necessary to give proper protection If a leaner and drier mixture is employed it 15 is necessary to wash the reinforcement with cement grout just before the deposit of the concrete to obtain the desired adhesion and security against rust As a proof of the existence of this property of protecting against rust there may be cited the numerous reinforcedconcrete reservoirs and sewers which have already stood for several decades and as yet show no signs of any corrosion of the reinforcement Some examinations of twentyyear old sewers showed the steel absolutely uninjured and of the same color as when it left the rolling mill Additional proofs are constantly being adduced by the repeated loading of structures and through the demolition of old reservoirs and floors in none of which has ever been disclosed any corrosion of properly covered reinforcement even when of considerable age Bauschinger gives the following report of some observations as to freedom from corrosion in several test specimens which had been broken in October 1887 and had lain in the open air till 1892 From several slabs the concrete covering the reinforcement was knocked away with a hammer The chips broke only in small pieces where the concrete was struck showing good adhesion between the steel and the concrete and the exposed reinforcement was entirely free from rust even close to fractured edges A tank was cracked and otherwise damaged through rough treatment during transportation so that the reinforcement was partially exposed Naturally the portion longest exposed showed corrosion and some rust was revealed when the concrete was removed adjacent to an old crack However when the metal was exposed under an unbroken hard surface no rust was revealed and the same adhesion was observed as in the slabs On July 23 1892 several fragments of floor slabs 6 to 8 cm 24 to 31 in thick were examined They had lain around the end of a sewer and the pieces next the entrance were most of the time covered with water which often contained sewage According to a statement of the owner the pieces had been in place about four years and had been purchased by him at the sale of the fragments of the tests made in 1887 They plainly showed the fractured ends from which the reinforcement stuck about 5 cm 2 in On one piece which lay somewhat lower than the others the reinforcement was scarcely 1 cm 04 in beneath the upper surface This upper layer was chiseled away the concrete proving very hard and adhering firmly to the steel The latter was absolutely rustless to within a distance of 1 cm 04 in from the fractured edge The coefficient for steel is usually assumed as 0000012 The several structural parts of reinforced concrete buildings are slabs Tbeams columns and archesthe characteristics of each of which will first be briefly described Their spacing should be from 5 to 15 cm 2 to 6 in and it is to be noted that light rods closely spaced carry more than larger rods with greater spacing Continuous reinforced concrete floors between Ibeams are usually constructed with slightly arched ceilings the arches being formed by constructing haunches down to the lower flanges of the beams The advantage of these haunches is that for the moments near the supports which exceed those at the centers the concrete has been increased in depth that no special increase in reinforcement is necessary An increase in the section of concrete at the supports is needed if the slab thickness at the center of the span is so thin as just to resist the compression at that point If this thickness were carried over the intermediate supports the concrete would be overstressed at those points According to the theory of continuous beams with variable section because of the arch form of the slab a slight reduction results in the moments at the centers of the spans with a corresponding increase of those over the supports Since ample reinforcement is generally provided at the latter points the exact and detailed computation of moments may be omitted in most practical cases In the same manner floor slabs which run continuously over reinforced concrete girders must be reinforced For want of accurate knowledge concerning the matter no account is taken in either case of the torsional resistance exerted by the rolled steel or reinforced concrete beams Thus a somewhat larger factor of safety is secured In thin slabs up to about 10 cm 39 in thickness the bending of the rods should be done with a slope of 13 In thicker and shorter slabs the slope can be steeper12 to 11½ It is evident in this connection that in all continuous slabs without regard to an arrangement to fit the distribution of moments so much reinforcement must be bent that the bent portion is able to carry the whole load of the central portion of the slab over into the ends which act as cantilevers even though the slab be cracked entirely through in the vicinity of the bends This rule is easy to follow and is the more important the less the amount of straight reinforcement and the more the concrete is exposed to outside stresses from shrinkage and temperature change Instead of finishing the ends of the straight and bent rods as hooks it is evident under such circumstances that the ends which lie next the centers of the slabs can remain straight and simply be anchored in the zone of compression of the concrete The number of systems of reinforced concrete floors is large and new systems are constantly being devised In most cases however their newness does not include any improvements As stated before many systems are at fault in that no reinforcement is provided near the upper surface over the beams as computations show necessary reinforcement being used only near the bottom while others employ a wrong distribution between the upper and lower systems of rods One improvement in such floor systems aims at a separation as far as possible of the zones of tension and compression without essentially increasing the total weight of the structure This is accomplished by employing numerous small ribs separated by hollow blocks or grooves filled with light pumice concrete The reinforcement is placed in the lower parts of the ribs If the hollow blocks above described or the other light filling material is omitted the floor construction consists of Tbeams of concrete with the steel enclosed by the stems of the Ts If the ribs are arranged further apart and built proportionately larger then what was formerly the compression zone must now be treated in accordance with established rules as a restrained reinforced concrete slab between beams In this way is developed a construction in which the slabs and beams combine to form a statically effective Tsection It is also possible to design slabs and independent beams of proper strength and of simple rectangular sections but it is clear that by making the slabs carry the compressive stresses a considerable economy is practised The stressing of the concrete slab in two directions at right angles to each other is not at all hazardous and occurs in numerous other types of construction From a theoretical standpoint a slab strengthened with ribs is more economical of material than a slab of uniform thickness At a certain span the greater cost of installing the ribs equals the saving in material so that Tbeams can first be built economically with spans of between 3 and 4 meters 10 to 13 ft Between the slabs and beams naturally occur shearing stresses for the transference of which most builders arrange special vertical reinforcing members called stirrups Bügel consisting of 6 to 10 mm ¼ to ⅜ in round rods or of thin flat iron These enclose the bottom reinforcing rods and thus prevent the formation in the concrete of the ribs of possible longitudinal cracks which might be caused by the hooked ends of the main reinforcing rods The stirrups thus increase the adhesive strength so that it equals at least that employed in calculations over several supports Similarly at points of negative moment steel must be introduced near the tops of the Tbeams or by carrying certain rods up and over the supports Under certain load conditions continuous top reinforcement may be necessary especially with unequal spans Furthermore at the simply supported ends of the slabs of heavilyloaded Tbeams some of the lower reinforcement should be bent upwards at an angle of about 45 so as to take up the shearing stresses or rather the diagonal tensile stresses in the slabs for which reinforcement must be provided Since the moments decrease toward the ends of the slabs not all of the rods are necessary close to the bottom in the vicinity of the supports so that a part can advantageously be bent upward At the intermediate supports where the greatest moments are found the compression occurs along the lower edge of the beam In order to lessen the unit stress the beam section is increased at such points by means of a bracket or knee producing a slightly arched effect In cases where the ribs are located above the slabs it is possible to do without these knees since the whole width of the slab between ribs serves as a zone of compression The knees or brackets at the intermediate supports have the added advantage of considerably reducing the unit shearing stresses partly because of the increased depth of beam but principally because the compressive stresses along the lower edges of the beam at such points act obliquely upward and thus equilibrate a part of the diagonal forces the lower rods straight over the supports The required number is to be determined by the necessary adhesion With large spans the standard lengths of rods will not suffice so that welding will be necessary The weld should be located where the rod is not fully loaded which in general is in a bend If a room of given dimensions is to be floored it is first divided into panels by main girders with intermediate supports if necessary These girders are then connected by simple slabs or beams may be introduced between the girders so as to diminish the slab spans In that case the slabs are supported on all four sides and require a correspondingly light reinforcement especially in a direction parallel with their greatest dimension The principal reinforcement is placed in the opposite direction or perpendicular to the beams A concrete column of any section contains a certain number of vertical rods which are placed close to the surface At certain points the rods are fastened together with horizontal wire ties The whole reinforcement thus forms a skeleton which encloses the concrete and prevents lateral bulging The result is that even in long columns ignoring the necessary safety against bending the strength of plain cubes will be attained The latter is higher than that of prisms The ties are placed from 20 to 40 cm 8 to 16 ins apart For a square column the reinforcement usually consists of four rods located in the corners with ties of 7 to 8 mm approximately ¼ to ⅜ in wire With large dimensions eight rods are used See Figs 15 and 16 The lower ends of the vertical reinforcing rods rest on a grid of flat bars so that the load carried by the rods may be distributed over a larger area of concrete This grid is usually placed in a separate concrete pedestal which distributes the column load over a larger surface of the foundation concrete proper corresponding with the lesser allowable unit stress of the latter In columns which extend through several stories of a building the sections diminish upward and the rods have to be offset at each change of diameter Further rods have to be spliced which can be done simply by slipping a short piece of pipe over the blunt ends Fig 17 Greater resistance against bending is afforded however by lapping the vertical rods from 50 to 80 cm 20 to 30 ins approximately and by having their ends hooked See Fig 18 Naturally the column section may be rectangular hexagonal octagonal circular etc and the number of reinforcing rods can be increased in proportion to the load With eccentric loading they should all be placed on one side The interiors of columns can also be made hollow by enclosing pipes in the concrete These can serve for rain leaders or may contain gas or water mains The diameter changes to correspond with the load to be carried and with the factor of safety desired It may run from 20 by 20 cm 8 by 8 ins to 70 by 70 cm and more 28 by 28 ins The diameter of the rods may vary from 14 to 40 mm ½ in to 1½ ins approximately The reinforcement for small arches can be determined in the same manner as for simple slabs Since no bending moments act on an arch with a parabolic profile and uniform loading a system of lightly interwoven reinforcement near the soffit is usually sufficient Usually however such simple reinforcement is not enough a second layer near the upper surface extending from the abutments over the haunches being needed In bridge arches which are subjected to variations of load reinforcement is introduced throughout near both the upper and lower arch surfaces Reinforced concrete arches have the advantage over arches of plain concrete that the reinforced arch can withstand tensile stresses as well as compressive ones For short spans it is thus possible to secure reinforced arches which make full use of the compressive strength of the concrete Under such conditions arches of much less thickness are secured than when nonreinforced concrete is used the thickness of which for short spans must be made so great as to prevent the appearance of appreciable tensile stresses PART I CHAPTER II THEORY OF REINFORCED CONCRETE STRENGTH AND ELASTICITY In the early stages of the development of reinforced concrete its builders had at hand no recognized methods of calculation and Monier and François Coignet erected their work solely by practical instinct and experience Of late a real rivalry has developed in the production of new theories concerning reinforced concrete and their authors have been anxious to explain the particular excellence inherent in a combination of steel and concrete with reference to their combined statical action Practice has here been far ahead of theory The principal question in controversy has been whether the tensile strength of the concrete in bending should be considered Among practical builders this question was really decided at the start and decided against its inclusion because absolutely no attention is paid to it and the steel is stressed to the maximum safe limit The tensile strength of the concrete is entirely ignored On this assumption was based the first method of theoretical computation of slabs devised by Koenen Government architect in Berlin in 1886 and his method has been used by the majority ever since Theoretical investigators unfamiliar with the practical side of concrete construction usually considered the tensile strength of the concrete and some even went so far in the older methods as to assume the elasticity in tension and compression as equal Later the modulus of elasticity in tension was accepted as smaller than that for compression and a parahola was assumed as the stressstrain curve Finally the stress curve for concrete in tension was found by Considerés investigations to be a straight line parallel with that of the steel It is evident that with such assumptions results are obtainable which appear extremely accurate to the several authors but long formulas are not attractive to practical builders and in this connection it is to be observed that the employment of a parabola for the stressstrain curve is actually less accurate than the use of a straight line because a certain amount of violence must be used if the stressstrain curve is forced into parabolic form But ignoring this point such methods of calculation do not provide the desired degree of safety and may even become actually dangerous if too small a percentage of reinforcement is used It is not the object of this book to give a review of all proposed methods of calculation This would be useless and furthermore the methods of checking designs contained in the Vorläufige Leitsätze für Eisenbetonbauten Tentative Recommendations concerning Reinforced Concrete Construction published in 1904 by the Verband Deutscher Architekten und Ingenieureverein and the Deutscher BetonVerein and in the Bestimmungen für die Ausführung von Konstruktionen aus Eisenbeton bei Hochbauten Regulations for the Execution of Constructions in Reinforced Concrete in General Building Work issued by the Prussian government are identical with those contained in the first edition of this book 1902 The new requirements of the French Ministry of Public Works of October 20 1906 for posts and telegraphs also contain the same assumptions and methods of computation Therefore here will be discussed only the theory above described which has been proved best by several years of trial and in a large number of constructions Since the first edition the results of numerous experiments have been secured which test the accuracy of these methods of calculation and especially explain the importance of shear in Tbeams Methods of calculation will therefore be found in close connection with the results of experiments In no other subject is it more important to rely as completely on the results of tests if disagreeable experiences are to be avoided since the present knowledge concerning reinforced concrete is at best imperfect and liable to surprises Before turning to the methods of calculation which are very simple a review will be made of the strength and elastic properties of steel and plain concrete so that the formulas may be more susceptible of daily use STEEL EISEN The properties of steel wrought iron or steel are well known today In calculations relative to steel construction the relation between stresses and strains is assumed and the limiting ratio will never be exceeded in actual loading Furthermore the tensile strength is the same as the compressive strength and the elastic behavior is the same under tensile and compressive stresses As to the modulus of elasticity and the safe working stress opinions do not differ materially Usually wrought iron in the form of rods is employed for reinforcement In it d represents the diameter of the machined test specimen not of the rod from which the specimen was prepared In special locations such as arch bridges the steel reinforcement can be the rod is anchored in a large mass of concrete but they will act in an opposite manner in the small stems of Tbeams especially at their bottoms where they will have a splitting effect and thus cause premature failure of bond It will be shown later that the adhesion in the case of ordinary round rods with hooked ends is ample to transfer all actual stresses and furthermore the arrangement of the principal reinforcement may be so designed with respect to the shearing stresses that no occasion should arise to make up any deficiency through the use of those costly special bars For reinforced concrete work only rich mixtures of finegrained materials should be employed Practically only with rich wet concrete will the necessary adhesion and rust prevention be secured because only then will the tamping force enough grout adheres to the concrete in spite of cracks and even rupture between the concrete and the steel and forms the real rust preventative as can be demonstrated When using drier and poorer concrete it is important to coat the reinforcement with cement grout immediately before depositing the concrete The sand aggregate exerts a great effect in determining the quality of the concrete The resistance which concrete offers to crushing is quite variable and changes with the proportions of the mixture and with the properties of the sand gravel and broken stone as well as with the tamping during making The form and size of the test specimen also influences the apparent strength The compressive strength per square centimeter decreases when the section of the specimen is enlarged The apparent strength is especially dependent upon the ratio of the height of the specimen to its base When this ratio is small as in mortar joints the strength is considerable But when the height is several times the diameter of the base failure will occur along a diagonal plane because the shearing strength has been exceeded and the compressive strength which is not involved appears small when the breaking load is divided by the area of section Elasticity Tests of Concrete Tensile Strength of Concrete Strength and Elasticity of Concrete When deformations are taken as ordinates and stresses as abscissas the curves of Figs 22 to 25 are obtained The deformation curves are quite regular in shape The tensile strength of large concrete specimens is always considerably less than of octagonal mortar ones since the latter can be compacted much better than can larger ones Stressstrain curves for concrete 3 months and 2 years old Table VIII Table X Table IX The bending strength of concrete is often used in connection with the compressive strength as a test of the quality of the material since tests of it are easier to make than tensile ones which latter depend largely on the degree of accuracy with which the load is applied at the exact center of the specimen So long as the fact is kept in mind that Naviers formula gives results good only for comparative purposes and that the actual tensile stresses are only about half those shown by it that method can conveniently be used SHEAR ADHESION ETC Shearing and Punching Strength of Concrete Schub und ScherfestigkeitThe great importance played by shearing forces in reinforced concrete construction and a study of the results of other tests led to the making of the following series of experiments partly by the writer and partly by the Testing Laboratory of the Royal Technical High School at Stuttgart The experiments disclosed a marked difference between the qualities of shear and punching resistance Schubfestigkeit and Scherfestigkeit As is known there exists in every section of a homogeneous beam loaded like those shown in Figs 29 and 30 normal stresses σ and shearing stresses τ which combine to form two inclined mutually perpendicular principal stresses socalled viz α₁σ2σ²4τ² and α₂σ2σ²4τ² the directions of which are found from tan 2α2τσ In distinction from the types of loading of Figs 29 and 30 is that of Fig 31 In the former only shearing stresses Schubspannungen were supposed to act that type being distinguished from other cases by the condition that the beam is subject only to flexure and consequently deflects The other variety is the case of pure shear This differs from the foregoing both spoken of as shear in that no bending takes place and the external force is here theoretically applied only on a single section while before it was constant through several adjoining sections or with a uniform load varied only slightly from one to another It is thus evident that pure shear is scarcely possible in practical work of small teeth Fig 33 along the infinitesimal faces of which compressive and tensile forces act in oblique but mutually perpendicular directions The horizontal components of these forces must balance among themselves and the vertical components must equal the total shearing force S Or in other words the shear c in the vertical section of a tooth Fig 34 is the resultant of the two normal forces bα and aσdα and must pass through their point of intersection which determines the perpendicularity of the faces of the teeth Because of the condition that a rupture of this series of teeth can occur only when the compressive stresses σd and the tensile stresses σs simultaneously reach their ultimate values a definite shape is imposed upon the right triangle abc and a definite relation must exist between the compressive tensile and shearing strengths In the triangle of forces c²ℓ²a²σ²b²σs² The equation of the horizontal components gives bαc b aσd αc or b²σs a²σd which in connection with the first equation gives c²ℓ²b²σsσda²σdσsσsσda²b² from which tσsσd The theoretical maximum pure shearing strength would therefore be the geometrical mean of the tensile and compressive strengths In an absolutely homogeneous material with equal tensile and compressive strengths t would equal σ or with regard to lateral dilation there is obtained t σ 11m In the case of actual tests of wrought iron and steel the strength in pure shear equals 07 to 08 of the tensile strength thus developing equally large shearing and torsional strengths compare Bach Elastizität und Festigkeit With concrete however of which the tensile strength is not as large as the compressive strength tests show that the shearing strength is considerably larger than the tensile one and close to the theoretical value tσsσd accurately with the width of the lower plate When the load was applied on the nonreinforced specimens a crack a first showed itself in the middle running from top to bottom This was doubtless caused by a bending of the specimens However the load on the machine could yet be considerably increased and only then did the load take full bearing on the edges of the plates as is necessary in order to obtain the real shearing strength 1 Test on three concrete specimens mixed 13 with 14 per cent of water 18 by 18 cm 7 by 7 in in section age 2 years Fig 35 The bending crack a appeared at a load P5 tonnes 11000 lbs but the load was increased to P40 t 88000 lbs when shearing along crack b took place In the second specimen the bending crack appeared at P10 t 22000 lbs and the shearing took place at P38 t 83600 lbs while the third specimen sheared at P50 t 110000 lbs On the assumption of an equal distribution of P between the two sections to be sheared the shearing strengths result as shown in Table XI broken at the Testing Laboratory of the Technical High School at Stuttgart gave the following average values Tensile strength σs 88 158 222 3 155 kgcm² 220 lbsin² Compressive strength σd350342233 3 308 kgcm² 5405 lbsin² In accordance with the theory described above the limit of shearing strength would be tσs σd15530869 kgcm² 981 lbsin² while the observed strength was 659 kgcm² 937 lbsin² 2 Test with 18 by 18 cm 7 by 7 in concrete prisms 1½ months old and 14 mixture with 14 per cent of water The aggregate consisted of 3 parts sand of 0 to 5 mm 0 to 02 in grains and 2 parts of gravel of 5 to 20 mm 02 to 078 in pebbles and was also of the same quality as the other specimens The arrangement is illustrated in Fig 36 Specimen 1 Bending crack in the middle at P15 t 33000 lbs sheared at P25 t 55000 lbs If a uniform distribution of stress is assumed the unit shearing strength will be t 125t² 815386 kgcm² 549 lbsin² Specimen 2 gave t417 kgcm² 593 lbsin² Specimen 3 t310 kgcm² 441 lbsin² Tension and compression tests were not made in connection with these specimens There exist however tests on concrete prisms like Fig 21 3 months old of similar composition of which the average of three strength tests were σs88 kgcm² 125 lbsin² and σd172 kgcm² 2446 lbsin² so that t88172388 kgcm² 439 lbsin² The average of the three shearing tests is t 386 47 317 3 371 kgcm² 528 lbsin² 3 Tests with reinforced concrete prisms a With straight rods only The experiments were performed on specimens of the same age size and mixture as the foregoing but each specimen was reinforced with four rods 10 mm 410 in in diameter near the upper and the lower surfaces as illustrated in Fig 37 The rods were not connected by ties They prevented a rupture of the specimen reduced the size of the cracks and allowed the load to be considerably increased after one shearing crack had appeared and until and even after the other crack had opened Specimen 1 At P12 t 26400 lbs a fine low horizontal crack showed itself At P15 t 33000 lbs a fine bending crack became visible in the center and shearing took place on the left at P20 t 44000 lbs t310 kgcm2 441 lbsin2 on the right at P30 t 66000 lbs t463 kgcm2 659 lbsin2 Average t386 kgcm2 550 lbsin2 In spite of these cracks the load was increased to P42 t 92400 lbs where the sole resistance against shear was the sixteen rod sections which then held Specimen 3 A bending crack appeared at P12 t 26400 lbs shearing occurred at the left at P15 t 33000 lbs t232 kgcm2 330 lbsin2 at the right at P28 t 61600 lbs t433 kgcm2 616 lbsin2 Average 333 kgcm2 473 lbsin2 Torsion Experiments with Concrete CylindersIn a cylinder undergoing a twist without any axial forces at play no normal stresses exist within any section only shearing stresses acting and at each point the latter are equal along directions parallel and perpendicular to the axis so that all elements in the body are stressed as is illustrated in Fig 31 page 32 It has been shown by the shearing experiments that the resistance offered by concrete to shear is somewhat greater than its tensile strength Consequently rupture of a cylinder subject to torsion must take place along a screw surface with a pitch of 45 at right angles to the major dilation or the oblique tensile stresses See Figs 3942 These torsion experiments were made at the Testing Laboratory of the Royal Technical High School of Stuttgart The mixture of the concrete was 14 and its age 2 to 3 months a Solid cylinder 26 cm 1024 in in diameter The length of the specimen under test was 34 cm 1338 in See Figs 39 and 40 The twisting moment was applied on the hexagonal heads See Table XIII The tensile strength of some hollow cylinders of similar section and equal age provided with the corresponding heads gave an average of τd80 kgcm² 1138 lbsin² while the similar above described tensile specimens like Fig 21 gave 77 kgcm² 1095 lbsin² The results found from the hollow cylinders agree quite satisfactorily with each other while the above described theory for solid cylinders has not been confirmed Aside from the greater age of the solid cylinders the greater value of τd is explained on the ground that since the modulus of elasticity diminishes with increase of stress the sections near the center carry a relatively large part of the load as shown by the formula τdMzπ 16 d³ so that the load is reduced on the outer portion The torsional strength of concrete therefore bears the same relation to its tensile strength as do the bending and tensile strengths In this manner can be explained the high value of 171 kgcm² 243 lbsin² when compared with the tension test specimens of the same material and mixture which gave about 9 kgcm² 128 lbsin² when 3 months old And with hollow cylinders in which the rupture takes place along a screw surface with a 45 pitch and at right angles to the maximum tensile strains the computed torsional stresses also correspond with the actual ones It must be mentioned however that only through the use of extremely plastic concrete will this agreement be obtained and only with wet concrete can the tamping be thoroughly effective as is especially necessary with hollow cylinders With regard to torsion investigations concerning spirally reinforced concrete hollow cylinders see page 53 The extensibility of concrete Shearing Experiments with Slotted Concrete BeamsThese tests were conducted on specimens with slits molded along the neutral axis so that with the method of loading shown in Fig 43 the failure would take place by a shearing of the connecting bridges at the ends The tests were made at the Testing Laboratory at Stuttgart At the ultimate load the shearing stresses existing in the sections aa are calculated as follows The unit shear at any point x along the neutral plane is τ P2 J b where S is the statical moment of the crosssection lying above the neutral axis in relation to it and J is the moment of inertia of the whole section Thus the total shear from 0 to l2 is Tπ t b l2 P SJ l2 and the shearing strength in aa is given by τ P S4 J lb d It must be explained that the side subject to tensile stresses had to be reinforced so that the weakest points in the body would be the bridges over the supports and so that the specimen would not fail prematurely through tension Example Specimen 85 wet mixture 13 age 105 days Under a load P1430 kg 3146 lbs the crack b₁ appeared conspicuously through the whole bridge At P1620 kg 3564 lbs a₁ showed itself through the whole bridge and m₁ started in the edge At P1170 kg 3894 lbs a₂ appeared At P2000 kg 4400 lbs m₂ appeared Under a load of P2410 kg 5320 lbs a wide crack formed at m₃ and m₂ widened considerably The load could not be further increased In Table XV the observed shearing strengths are given together with the tensile and compressive strengths of the specimens illustrated in Fig 21 page 21 The results are each averages of three specimens The shearing strength for 14 here observed of from 31 to 28 kgcm² 441 to 398 lbsin² is a little smaller than the one found by direct shear of 37 kgcm² 526 lbsin² The reason probably lies in the not entirely rigorous methods of calculation used in connection with the slotted prisms or else in that the solid end connections had an appreciable thickness so that partially inclined cracks could occur from diagonal tension In practice the case of pure shear is very rare Diagonal tensile stresses are always combined with shearing ones and the former become of critical importance long before the shear does as the torsion experiments plainly show This point will later be discussed more fully in connection with shearing tests on beams The percentage of water given is only nominal since the sand and gravel were moist The pressure of the testing machine was increased rather rapidly for the larger loads The results approach closely the shearing strength of similar concrete specimens The compressive stresses in the rods reached a maximum of 2140 kgcm² 30440 lbsin² and consequently were below their observed elastic limit of from 2600 to 3200 kgcm² 36980 to 45520 lbsin² Although the nonreinforced concrete cubes were not cracked by the pressing through of the rods their adhesive strength was smaller than was that of the ones containing spirals The experiments included tests for the determination of the influence of the amount of water used the quantity of sand the influence of jarring the specimen before the concrete had set and finally time tests of specimens up to three months old The following conclusions were deduced That percentage of water was best with which it was just possible to manufacture the specimens satisfactorily With the proportions above described this was 12 per cent Within certain limits the relative proportions of sand and gravel have no important influence on the resistance to sliding so long as the percentage of water is proportionately small when small amounts of sand are used The resistance to sliding will be increased by jarring the finished specimen before setting is completed at least when the specimen stands on a wooden bottom which gets jarred by being struck by other bodies This increase is more important when small percentages of water are used and is to be explained by the fact that through the jarring the grout which is necessary to a good bond will be enabled to collect around the reinforcement The sliding resistance is considerably greater in tests conducted at high rates of speed than at slower ones where the loads act for longer periods at each step Also tests in which rods are pushed through are somewhat higher than when they are pulled through In regard to the practical employment of these results it is to be noted primarily that it is impossible to obtain in actual work the exact percentage of water above mentioned on account of humidity of the various aggregates but that it is necessary to rely almost entirely on experience and good practice On the other hand an excess of water does not then have the harmful effect that it does on test specimens molded in solid castiron forms since the wooden molds absorb a part of the water and some more is lost through the cracks between the boards Furthermore in building construction the fresh concrete will receive plenty of jarring from the forms so that the highest value obtained from the experiments in which the specimens were shaken as well as tamped may be assumed as a proper working stress A very important point and one here brought out for the first time is that for steel stresses far below the elastic limit the unit adhesion diminishes with the length of rod embedded The explanation of this phenomena is as follows The tensile stress in the rod will decrease from the outside of the concrete to the inner end of the rod as the stress is transferred from its surface to the concrete Because of its elasticity the rod will stretch under the tension while the concrete will be thrown into compression and will shorten Consequently even under small tensile stress because of the changes of length in opposite directions in the two materials a sliding effect will be produced along the rod near its outer end so that the tensile stress in the steel will not be uniformly distributed over the whole length of the rod embedded in the concrete It will first be taken up by the adhesion at the outer end and only after that is exceeded and a slight displacement takes place will the distant parts of the concrete be stressed It follows from this unequal distribution of stress that the observed values of this stress are too small and that they should more properly be termed the frictional resistance as Bach has done The shorter is the embedded length of rod the smaller are the tension and elongation and the more nearly equally distributed will be the effect over the whole surface When the rods are pushed through there exist practically the same conditions but in less degree because then the steel and concrete are loaded in like kind Even then a slight sliding will occur very early along the outer portions of the rod This slight sliding explains the influence shown by the rate of application of the load It is easily seen that with a high rate the sliding does not have time to develop and that the adhesive stress is then more uniformly distributed over the embedded area of the rod In Fig 47 are given the principal results of Bachs tests They refer entirely to 14 concrete prisms with 15 per cent of water The earlier tests were conducted with applications of load for short periodseach step occupied onehalf a minute which is really long as compared with most experiments The embedded lengths of the rods are plotted as abscissas and the observed resistances to sliding as ordinates If the curves for adhesion on pushed and pulled rods under short load periods and also the curve showing the results for longer duration of load from nothing up to 110 minutes are extended to intersect the axis of ordinates all three meet at practically the same point which corresponds with an adhesive strength of 38 kgcm² 540 lbsin² At this value which corresponds to a length l0 the influences of the embedded length of rod of premature sliding of time and the difference between pulling and pushing all vanish This value of 38 kgcm² 540 lbsin² happens to correspond with that found by the author on specimens of the same mixture and age for shearing strength and also approaches closely that for the quickly operated adhesion experiments The low point of the middle curve at l200 mm may be explained by the fact that those specimens were first manufactured and the operator had not yet acquired proper experience In addition to the experiments on the slipping resistance of embedded round rods made at the Testing Laboratory in Stuttgart a series was also conducted with Thacher bars The specimens were again prepared of the same mixture of 1 part of Portland cement and 4 parts sand and gravel with 15 per cent water The height of the specimens was 20 cm 79 in while the length of the side of the square base was in some cases 22 cm 87 in some 16 cm 63 in and some 10 cm 39 in and the resistance to pulling out was found to vary with the diameter of the specimen since all split when the Thacher rods were withdrawn If the pull P is uniformly distributed over the embedded surface O the resistance to sliding for the several specimens was as given in Table XVIII UNIT ADHESION FOR DIFFERENT LENGTHS OF EMBEDMENT Metric English Metric English Metric English Length of side 22 cm 87 in 16 cm 63 in 10 cm 39 in Pmax 585 832 561 799 334 475 It is evident from the last figure that with a minimum thickness of specimen equal to 375 cm 15 in and with lesser values the splitting effect of the knots is so great that greater adhesion cannot be expected than that of common round rods as they come from the mills With greater thickness of concrete the splitting occurred when the elastic limit of the steel had been reached Only those adhesion experiments in which the steel stress remains under the elastic limit give a proper value of the adhesive strength to be used in the design of reinforced concrete structures and consequently all steel must be so arranged as to length shape and thickness that it will effect a safe transfer of stress to the concrete Actual tensile stresses are usually small however and an increase up to the elastic limit through overloading of beams is seldom to be feared In singly reinforced slabs the ends of the rods rest in large masses of concrete so that a diminishing of the adhesion because of premature cracking of the surrounding concrete is not to be feared In slabs the amount of the embedding is less but near the ends of beams stirrups are introduced which surround the concrete to some extent and so preserve its adhesive strength In this connection are here given the valuable results of the French Reinforced Concrete Commissions experiments Certain prisms with centrally located rods were manufactured in which only 2 to 25 cm 08 so 10 in of concrete existed between the rod and the outside surface Besides these some were made without stirrups as illustrated in Fig 49 A second series had three flat iron stirrups 30 by 2 mm 18 by 316 ins as are used in the Hennebique system and which enclosed the 30 mm 18 in diameter rods tightly In the third series open stirrups of the same flat iron were employed which enclosed a larger mass of concrete and were separated from the rods by a space of about 1 cm 38 in The concrete was composed of 300 kg 66 lbs of cement 400 l 14 cuft sand and 800 l 28 cuft gravel with 88 per cent by weight of water and were six months old at the time of the test The resistances shown in Table XIX were developed against pulling the rods out of the concrete TABLE IX ADHESION IN THE PRESENCE OF STIRRUPS Specimen Starting Resistance Average Sliding Resistance No of kgcm² of lbsin² kgcm² of lbsin² Specimens Surface Surface None 49a 2 72 102 81 115 Hennebique 49b 2 199 283 142 202 Open 49c 2 257 366 182 359 298 424 A repetition of these experiments with specimens three months old in which the stirrups of flat iron were replaced with 9 mm 38 in rods gave higher results see Table XX each being the average of three specimens TABLE XX ADHESION IN THE PRESENCE OF STIRRUPS Specimen Adhesion Sliding Resistance kgcm² lbsin² kgcm² lbsin² Figure 49a 247 351 88 125 Figure 49b 261 371 177 252 Figure 49c 312 457 300 284 Commission du ciment armé Expériences rapports etc relatives à lemploi du béton armé Paris 1907 then undergo considerable stretching Thus the total measured length may have increased only 20 per cent while in the neighborhood of the point of rupture the actual stretch has been 10 to 15 times this amount If it is supposed that the phenomenon known as reduction in area also applies to cement mortar then the total elongation measured between the ends will give only an average value and the mortar will in reality possess a very much greater ability to stretch than this value represents In reinforced construction the concrete is attached to the steel which latter possesses a much higher elastic limit than does the concrete When undergoing stress therefore the steel will tend to have the extension distributed uniformly over its whole length at a stress at which the concrete tends to contract locally But the adhesion makes it necessary for the concrete to follow the steel in its extensibility It will therefore endure throughout its whole length the maximum possible deformation and rupture will finally take place with an elongation measured over all which is considerably larger than if reinforcement were present This explanation given by Considère is obvious if the phenomenon of reduction of area really exists in concrete In computations concerning these bending tests Considère employed a method with reference to the relative distribution of stress between the concrete and steel which made the concrete show no greater tensile strength than that developed by plain concrete prisms This method was not entirely free from objections and therefore Considère subsequently made some true tension tests with reinforced concrete prisms Mortar prisms of square section 47 mm 185 ins on a side symmetrically reinforced with four wires 44 mm ⅝ in approx in diameter were subjected to tension and the stretch both in the reinforcement and the mortar was measured They were always found practically equal From the known modulus of elasticity of the reinforcement and the measured stretch of the steel could be computed the proportion of the total tensile stress P carried by the reinforcement The remainder divided by the section of concrete gave the unit tension in the mortar to which its measured elongations corresponded The observed law between stress and strain is shown in Fig 50 The ordinates represent the total tensile stress on the prisms while the abscissas give the corresponding stretch in the reinforcement As long as the load does not exceed a certain value Oa the strains increase uniformly and are very small They then increase suddenly but soon again become uniform and are represented by the flatter straight portion AB of the curve From the measured stretch and the known area of the reinforcement the part of the load carried by it can be calculated The curve for the steel is practically a straight line as long as the elastic limit is not exceeded In the figure this straight line is represented by OF which runs practically parallel with AB For any stretch OP is then PN equal to the part of the load PM carried by the steel NM concrete It therefore follows from the curve that the concrete in combination with the steel is able to stretch considerably but that after a certain elongation A the stress on the concrete does not materially increase The maximum stretch was 09 mm 0035 in which corresponds with a steel stress of 1800 kgcm² 25600 lbsin² This is less than the first value of 2 mm per meter found by Considère The lines CB CB CB represent repeated loadings and unloadings Considères tests were repeated by the French Government Commission with somewhat larger prisms of 124 concrete Similar results were obtained and it was further discovered that the extensibility of reinforced concrete which had set under water was greater than that which had set in air Considères tests very quickly became known and were at once used by theorists in the formation of new methods of calculation without waiting for confirmatory experiments or even considering the limitations placed by Considère himself on the practical value of his results In 1904 objections were raised to Considères theory by both American and German experimenters based on further tests made by them The experiments of A Kleinlogel conducted in the Testing Laboratory of the Royal Technical High School at Stuttgart were published in Beton und Eisen No II 1904 and also No 1 of the Forscherarbeiten aus dem Gebiete des Eisenbeton Vienna 1904 They comprised rectangular reinforced concrete beams 220 cm 866 ins long and 15 by 30 cm 59 by 118 ins in section The mixture was 1 cement 1 sand 2 crushed limestone For purposes of comparison some beams were made without reinforcement The beams were supported at the ends and loaded with two symmetrically placed loads 1 meter 394 ins apart The stretch of the lowest concrete layer was measured on a length of 80 cm 315 ins included within the central portion of a beam In order to make the cracks more evident the lower face and both sides of the beams were painted with a coat of whitewash The sixmonths old beams which had been kept in damp sand gave practically equal maximum extensions of the lower concrete layer for several different percentages of reinforcement This amounted to between 0148 and 0196 mm per meter 0000148 to 0000196 per foot Thus Considères law was not confirmed because the stretch of nonreinforced concrete was found about 0143 mm per meter 0000143 ft per foot According to Considère it was 01 to 02 mm per meter Kleinlogels tests also furnished important information about adhesion to which reference will be made later Because of the numerous objections raised concerning his hypothesis Considère repeated his experiments with larger specimens The concrete consisted of 400 kg 880 lbs of Portland cement 04 cubic meters 052 cuyds of sand and 08 cubic meters 104 cuyds of crushed limestone The beams were of rectangular section 3 meters 984 ft long 15 cm 6 ins wide and 20 cm 78 ins high and were reinforced on the lower side with two round rods 16 mm ⅝ in and three round rods 12 mm ½ in approx in diameter As in the beforementioned experiments they were tested with symmetrically placed loads 14 meters 55 ins apart within which distance the moment was uniform and no lateral forces acted Of two specimens one was kept under damp sand and one under water for six months at which age the specimens were tested It was shown that the first beam stood a stretch between the layers A and B of from 022 to 05 mm 000866 to 00196 in and the second which had been kept under water stood a similar stretch of from 056 to 107 mm 0022 to 004 in Fig 51 A crack could not be found even though the outer surface was coated with neat cement The concrete between the layers A and B was sawed out and still showed the same strength as untouched concrete Considère does not state and such is the case with all his tests whether he was able to cut away the section over the whole length of the beam in one piece or whether in several Of the experiments conducted by the Testing Laboratory of the Royal Technical High School at Stuttgart concerning the extensibility of reinforced concrete first will be discussed the Torsion Tests on Hollow Cylinders with Spiral ReinforcementHollow cylinders of the same dimensions as those described on page 39 were provided with spiral reinforcement having a pitch of 45 in the centers of the walls They were so arranged that torsion tests would produce tension in the spirals In Cylinder X which corresponded exactly with the other a fine crack appeared with Md 70000 cmkg 60600 inlbs The torque was however increased to 120000 cmkg 104000 inlbs when further parallel cracks appeared If there is subtracted from the value for Cylinder IX at which the first cracks appeared the torque Md 54560 cmkg 47200 inlbs carried by the nonreinforced hollow cylinders of equal age there remains in Specimen IX the moment Me 17940 cmkg 15600 inlbs This gives in the circle of 21 cm 827 in diameter in which the spirals lay a total horizontal circumferential strength S 17940 105 1710 kg 3762 lbs half of which must be taken up at the moment of cracking by the reinforcement which lies at an angle of 45 to this theoretical stress and half by the compressive resistance of the concrete acting at right angles to the direction of the reinforcement Consequently from Fig 53 Z D S 2 2 and the stress in the five spirals is σs 855 2 5 07²π 4 630 kgcm² 8960 lbsin² This stress may also be obtained from the torque Me 17940 by a proper distribution of the inclined tensile stresses τ over the section of the reinforcement For cylinder X the steel stress at the appearance of the first crack was found to be σe 540 kgcm² 7680 lbsin² With Cylinder XI with 10 spirals of 10 mm ⅜ in approx round rods otherwise like the foregoing the first crack α appeared at Md 125000 cmkg 108200 inlbs with other cracks running in the same direction and final rupture at Md 155000 cmkg 134200 inlbs With the same suppositions as before there is obtained for the steel stress at the appearance of the first crack in Cylinder XI σe 603 kgcm² 8580 lbsin² XI σe 560 7070 EXTENSIBILITY It is thus found that with the four hollow cylinders the first cracks in the concrete appeared at an extension which corresponded with an average steel stress of σe 630 540 603 560 4 583 kgcm² 8290 lbsin² The extension at this stress is 583 2160 027 mm per meter 000027 ft per foot If the shearing stresses at the appearance of the first crack and at rupture are computed from the formula τd Md π d⁴ d₀⁴ 16 d the results of Table XXI are obtained TABLE XXI SHEARING STRESSES AT FIRST CRACK AND AT RUPTURE Cylinder No At the First Crack τd kgcm² lbsin² At Rupture kgcm² lbsin² IX 252 358 302 430 X 244 347 420 597 XI 436 620 495 704 XII 418 595 540 768 It may be concluded from this that through a proper arrangement of the reinforcement that is by placing it in the direction of the maximum tensile stresses the shearing strength of reinforced concrete can be increased over that of plain concrete In specimens with weak reinforcement the stress at rupture rose to the ultimate stress in the steel while with heavier reinforcement such a stress could not be reached because the adhesion on the thicker rods was not sufficiently strong at the ends Bending Tests with Reinforced Beams of 15 by 80 cm Section These specimens had the same dimensions as those tested by Kleinogel but were made with 1 cement to 4 Rhine sand and gravel They were constructed in December 1902 and tested three months later at the Testing Laboratory at Stuttgart They were consequently older than Kleinogels specimens They were tested with two symmetrically placed loads so that a constant moment with no external forces acting was obtained throughout the central portion of 80 cm 315 ins between the loads Besides the stretch of the steel the shortening of the top concrete layer was also measured and the deflection within the measured length was also ascertained for different loads The stretch in the steel was measured between projecting lugs A4 which were clamped to the reinforcement In the ends of the beams the two reinforcing rods were arranged as shown in Fig 54 and several stirrups were provided to counteract the local effects of the forces P and the shearing and adhesive stresses These were such that no cracks appeared between the supports and the loads P The six specimens were severally reinforced with two 10 mm ⅜ in approx with two 16 mm ⅝ in and with two 22 mm ⅝ in rods Of these beams three were used for the determination of the steel stretch and three for the shortening of the top concrete layer because the apparatus was so designed that both observations could not be made simultaneously The tension face of each beam received a coat of whitewash to make the cracks easier of discovery The first cracks z were always noted next the lugs A probably because at those points the zone of tension in the concrete was weakened Afterward the cracks m m₁ and m₂ appeared within the central portion All indeed were so minute that they probably would not have been seen except for the coat of whitewash From the stretch in the plane of the reinforcement and the shortening of the top layer the extensibility of the lowest layer could be computed The tests gave the values shown in Table XXII at which the cracks appeared within the measured length TABLE XXII EXTENSIBILITY EXPERIMENTS Reinforcement Number of Round Rods Diameter Per Cent Stretch of the Steel Stretch of Lowest Concrete Layer mmm ftfoot mmm ftfoot 2 10 04 042 000042 050 000050 2 16 10 033 000033 040 000040 2 22 19 030 000030 038 000038 This was about treble that of nonreinforced concrete After the specimens were prepared they were kept moist for a considerable time but were tested in an airdry condition The difference between Considères tests and those of other experimenters can be partially explained since concrete which sets under water swells and therefore stands greater stretching than that which sets in air and decreases in volume It is also to be noted that with each repetition of his experiments Considère found smaller results From 2 mm they fell to o9 mm and finally to 05 mm per meter from 00020 to 00005 ft per foot The latter figure does not differ much from the results on pages 53 to 55 These bending tests will be discussed again later in connection with the subject of the exact location of the neutral axis and the distribution of stress in the section Also there will be given an independent explanation of the large extensibility observed by Considère and of the stress distribution between steel and concrete shown in Fig 50 A complete statement is impossible without having first discussed the theory of reinforced concrete Similar experiments were carried out for the Reinforced Concrete Commission of the Jubiläumstiftung der Deutschen Industrie in the Testing Laboratory at Stuttgart In them Bach thoroughly investigated the appearance of the first crack in beams of which the material proportions and load distribution were similar to those illustrated in Fig 54 and the outside of which was given a coat of whitewash With increasing load on the under side of the beams small damp spots first showed themselves These spots grew in size as the load was augmented With further increase cracks appeared always where a spot of water existed but not all such spots developed into cracks These phenomena which had been described by Turneaure Engineering News 1904 p 213 and also by R Feret Étude expérimentale du ciment armé 1906 developed in beams which had been kept under water and may be explained by their porosity in certain portions which were stretched by the tensile stresses and from which the moisture worked outward and so formed the spots of water on the surface The cracks appeared on the sides of the beams at somewhat higher loads than on the bottom It was further shown that the cracks usually commenced at the bottom corner furthest from the reinforcement In the section shown in Fig 55 a crack existed at a load of 6000 to 6500 kg 13200 to 14300 lbs at depths about as shown by the lines ab and cd and advanced under a load of 7000 kg 15400 lbs to the positions of ab1 cd1 In the beams with a single reinforcing rod the cracks appeared somewhat later in the narrower beams than in the wider ones The first corner crack was observed at a stretch of from 01270176 mm in a length of one meter 00001270000176 ft per foot for a beam 15 to 30 cm 59 to 118 ins wide with a single reinforcing rod The spots of moisture always appeared with a stretch of 008010 mm per meter 000008 to 000010 ft per foot depending on the distribution of the steel in the section This is however the ultimate stretch of plain concrete The formation of cracks will be delayed if the reinforcement in the vicinity of the porous spots in the stretched concrete receives additional assistance When the reinforcements were uniformly distributed over the whole width of the beam the cracks were actually found after greater stretching but were much smaller and correspondingly harder to discover In heavily reinforced beams the extension