§mull Wlmvmity ptaeg BOUGHT WITH THE INCOME PROM THE SAGE ENDOWMENT FUND THE GIFT OP Hcnrtj m. Sage 1891 &J?.6J?<>3. ?/3./.Jsi0..y... 5474 Cornell University Library TA 445.C75 1903 Exper mental researches on re nlorced co 3 1924 021 408 863 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924021408863 EXPERIMENTAL RESEARCHES REINFORCED CONCRETE ARMAND CONSIDERE Ingenieur en Chef des Ponts et Chausstes, TRANSLATED AND ARRANGED LEON S. MOISSEIFF, C. E. Assoc. Mem. Am. Soc. C. E. WITH AN INTRODUCTION THE TRANSLATOR. AUTHORIZED EDITION. NEW YORK McGRAW PUBLISHING COMPANY 114 Liberty Street 1903 COPYHIGHTKD, 1903, BY THB McGRAW PUBLISHING COMPANY, New Yobk. INTRODUCTION. Concrete as a material for structures, or parts of them, for the many and various needs of modern industrial life, had, as is well known, an extensive and growing application before iron or steel rods were embedded in it for reinforc- ing. The great advantages of concrete, viz., stability of characteristic properties, small effects caused by changes in temperature, protection against rust and heat, fireproof qual- ities, and, finally, the facility of adaptation to different forms and shapes, combined with a low cost of manufacture, made its still more extensive application desirable. But the re- sistance of concrete to the stresses and strains caused in it by external forces is low compared to that of the materials generally used by engineers, such as steel and wrought iron. Especially is the resistance of concrete to tensile and shearing stresses so small that structures or parts thereof which are subjected to such stresses to a considerable extent become un- economical and impractical. When it, therefore, became known from the applications made by Monier, Wayss, and others that iron embedded in concrete would act together with the latter and thus virtually strengthen it, engineers all over the world were eager to take advantage of this method of reinforcing concrete. Many and multiform applications of this principle were made and numerous letters-patent taken, each claiming superiority over the other. It is due especially to the initiative and boldness of French and German engineers and their imtiring energy in overcoming difficulties and objections, both engineering and legal, that reinforced concrete has had such a rapid and successful development. Practically within the last decade' reinforced concrete structures began to be universally used in [iii] iv INTRODUCTION. all civilized countries and to compete, in many instances successfully, with steel structures. The importance of the new material-has become such that, no civil engineer can well afford to be without a thorough knowledge of its properties. As is generally the case with new materials, the practical advantages of reinforced concrete were demonstrated a long time before the theoretical analyses of its properties were attempted. Kather to explain than to study the increased resistance of concrete and steel combined, various assump- tions were made as to the behavior of the new material. The most rational of these asstimptions, were based on the analogy of composite structures, assuming that steel embedded and well distributed in concrete will act the same as does a rod laid parallel with a piece of timber. Formulas founded on these . various assumptions have been- deduced, and, while they have a rational appearance, they are empirical only. As such they answer very well their purpose of giving quick and safe rules for the computation of certain reinforced con- crete constructions, such as beams, especially, provided the limits of their range of application are observed. They are useful and convenient to the busy engineer in estimating and preliminary work. But it should be observed that these empirical formulas give satisfactory results only because in beams the practical proportions of depth to length vary within narrow limits and that it is comparatively easy to fit em- pirical constants into the chosen formulas. This is not stated to depreciate the value of the empirical formulas, but to point out that all these coefficients of elas- ticity of concrete, the ratios of the latter to that of steel, the allowed unit stresses, etc., are not what they claim to be in name, but are merely numerical constants to be applied to given empirical formulas. It therefore follows that the nu- merical values given to the above constants cannot furnish us a true insight into the behavior of reinforced concrete. To fully understand the action and utilize the properties, of a material, its stress and strain story must be thoroughly INTRODUCTION. v known and not merely results of tests of given structures. The great value of Considered researches consists in the fact that they represent practically the first systematic attempt to study the properties of reinforced concrete in a scientific manner, by one of the world's foremost experimenters. How fruitful in results such researches can be, is proved by Con- sidered discovery of " hooped " concrete for compression members, and the Italian engineer Maciachini's adaptation of this principle to hooped beams. It is in the continuation of such researches and their full discussion that the future development of reinforced con- crete lies. Prom such investigations the designing engineer will be able to judge for himself the use he can make of this material in the vast field of modern engineering construc- tion. This is well understood by engineers the world around, and very recently the Swiss Commission on Reinforced Con- crete has decided to undertake the investigation of the sub- ject, at a cost of about $8,000, on the lines practically laid down by Considered researches. The researches, the results of which are given in this book, were undertaken in the year 1898 and cover the period of time from that date to the end of 1902. The first published report of the results obtained is found in a paper by the ex- perimenter before the Trench Academy of Sciences at the end of 1898. As the work proceeded M. Considere pub- lished a number of papers containing the results of his la- bors, the list of which is given below.* The book presented "The following are the publications of the author on the subject: Academie des Sciences. Influence des armatures mgtalliques sur les proprigtgs des mortiers et bgtons, 12 dgcembre 1898 et 2 Janvier, 1899. Academie des Sciences. Variations de volume, 18 sept., 1899. Academie des Sciences. Resistance a la traction, 18 aout, 1902. Academie des Sciences. Etude thSorique du beton frettg,25 aout, 1902. " Influence des armatures mgtalliques," Genie Civil, 1899. " Resistance a la compression du bgton arme et du beton frettg," Genie Civil, 1903. " Methode d'eupreuve des constructions en beton arme," congres in- ternational desmethodes d'essai a Paris, 1900. " Contribution a l'gtude des proprigtgs du bgton armg," congrgs in- ternational des mgthodes d'gssai a Budapest, 1901. vi INTRODUCTION. here consists of a compilation of these publications arranged and classified so as to make as far as possible one coherent treatise. It has been the intent to adhere to the author's wording and treatment, avoiding at the same time unneces- sary repetitions. The chapters of the book follow in gen- eral, as will be seen, the titles of the several papers as they were published and also their chronological order. Of all arrangements this appeared to be the best, containing, as each paper does, the further development of the author's views. It is hoped that this book will be found to present adequately before American engineers the famous researches of the author; and if it should, as it is hoped it will, be of any value to the engineering profession, the objects of this trans- lation will be attained. THE TRANSLATOR. CONTENTS. CHAPTER I. page. Reinforced Concrete in Bending 1-57 1. The Influence of Reinforcing on Concrete 1 2. Results of Known Experiments 13 3. The Resistance and Elasticity of the Materials Generally Used 15 4. The Graphic Solution of the Problem 15 5. The Approximate Solution , 18 6. The Displacements of the Neutral Axis 21 7. The Influence of the Proportion of Iron or Steel 23 8. The Influence of the Quality of the Concrete and the Metal. . 25 9. The Cost of the Different Types of Beams 26 10. The Determination of the Most Economic Proportion of Metal. 28 11. The Computation of Reinforced Concrete Beams 32 12. The Deformation of Concrete-Steel Under Repeated Loads . . 35 13. The Assumption of an Increase in the Coefficient of Elasticity of the Steel '. 45 14. The Possible Results of Poor Workmanship 47 15. The Influence of the Variations of the Coefficient of Elastic- ity of Concrete on the Resistance of Reinforced Beams 51 16. Symmetrical Reinforcing 52 17. Conclusions 54 CHAPTER II. The Deformation and Testing of Reinforced Concrete Beams. 58-86 1. The Necessity for Direct Tests of Reinforced Concrete Struc- tures 58 2. The Insufficiency of the Assumptions Made for the Computa- tion of Deformations 59 3. The Law of Deformation of Concrete in Compression 60 4. The Law of Deformation of Concrete Under a First Tension Load 61 5. The Effect of Repealed Stresses 62 [vii] viii CONTENTS. PAGE. 6. The Effect of Very Small Cracks on Stress and Deformation. 63 7. The Exact Method for the Determination of the Deformations. 64 8. The Approximate Method of Computation of the Deformation. 64 9. The Complete Curves of Deformation 69 10. The Comparison of the Actual Deformations with the Com- puted 70 11. Results of Assumptions Made by Different Authors 7l 12. Effect of Percentage of Iron and Quality of Concrete . . 73 13. Effect of Longitudinal Sliding of the Reinforcing Metal 73 14. Effect of Surroundings in which Concrete is Placed 74 15. Informations Obtained from Tests 77 16. The Deformations of Beams Symmetrically Reinforced 80 17. The Computation of Deflections. , 82 18. Limits of Test Loads 85 19. Conclusions. 85 CHAPTER III. Effects of Changes in Volume of Concrete 87-94 1. Shrinking of Concrete in Air , 87 2. Swelling of Concrete in Water 89 3. Effects of the Proportion of Moisture , 91 CHAPTER IV. Tensile and Compressive Resistance of Reinforced Concrete. 95-103 1. The Tensile Resistance of Reinforced Concrete 95 2. Effect of Shrinking and Swelling on Deformations and Stresses 99 3. Compressive Resistance of Concrete 100 CHAPTER V. Resistance of Concrete to Shearing and Sliding 104-109 1. Shearing Stresses. 104 2. Sliding Deformation and Resistance , 105 CHAPTER VI. Effect of Cracks on Stresses and Deformations 110-118 1. Effect of Cracks 110 2. Effects of the Convexity of the Curves of Deformation of Mortar and Concrete , 115 3. Computation of the Dimensions and Deformations of Rein- forced Concrete Beams 117 CONTENTS. ix CHAPTER VII. page. The Compressive Resistance of Reinforced and Hooped Con- crete 119-188 1. Concrete Reinforced by Longitudinal Rods 119 2. Concrete Reinforced by Transverse Rods 122 3. Theoretical Considerations on the Resistance of Hooped Con- crete 124 4. The Resistance of Hooped Sand 125 5. Experimental Researches 128 6. General Properties of Reinforced Concrete and Hooped Con- crete 131 7. The Spacing of the Hoops 135 8. The Ductility of Hooped Concrete 138 9. The Elastic Behavior of Hooped Concrete. Experimental Data 141 A. The Elastic Behavior Under a First Load 141 B. The Elastic Behavior Under Repeated Loads 143 10. The Elastic Behavior of Hooped Concrete. Analysis of Facts. 149 11. The Elasticity and Resistance of the Concrete in Hooped Members 154 12. Practical Rules for the Computation of Hooped Concrete Members 157 13. The Column Resistance of Hooped Concrete 159 14. Factor of Safety 1L3 15. Types of Hooped Concrete 168 16. The Method of Manufacture of Compression Members 172 17. The Influence of the Character of the Materials 177 18. First Cost 180 19. Conclusions 183 20. Additional Experiments 186 REINFORCED CONCRETE. CHAPTER I. Reinforced Concrete in Bending. 1. The Influence of Reinforcing on Concrete. The first attempts to reinforce concrete by embedding iron rods in it were made by practical men, not theorists, to whom much honor is due, for it is probable that theo- reticians would never have advised a priori, a combination of materials, the heterogeneous character of which did not inspire them with confidence. With increasing bold- ness reinforced concrete structures have been successfully built for shops, magazines, grain elevators, water reser- voirs, bridges, floors, and other structures or parts of structures. But engineers in charge of great public and private works, with some few exceptions, have hesitated to adopt the new material on a larger scale. They thought that a clearer and fuller knowledge must first be obtained of the phenomena resulting from the combination of con- crete and steel and of its probable durability and resist- ance to atmospheric influences. One of the most serious objections which naturally pre- sented itself was the following: Numerous experiments made throughout the civilized world have shown that ce- ment mortars of usual proportions cannot, when in ten- sion, endure, without breaking, elongations exceeding on the average one hundredth of 1 per cent, of the total length of the test specimen. But when sustaining an elongation up to this limit the stress in the embedded steel does not exceed about 2,900 pounds per square inch, [1] 2 REINFORCED CONCRETE. that is, about a fourth or a fifth of the stress which could be allowed for the same steel in order to realize the full advantages of its use. It seemed, therefore, that the mortar or concrete of concrete-steel members, where the steel, is stressed from 9,000 to 15,000 pounds per square inch, must crack, and would thus be in a very poor condi- tion as to durability. . ^To this objection the partisans of concrete-steel, reply that the strength and solidity of their structures can be seen and that only very seldom are al- most imperceptible cracks produced. But to this again their opponents answer that very small cracks escape ob- servation, and that mortar, even without being cracked,. may lose its resistance, owing to some unknown form of disintegration; and again, that while the momentary sta- bility of a member the mortar of which is disintegrated in the portion subjected to great elongations can be ex- plained, if the steel remains solidly embedded in the unbroken cement at the ends, no reliance can be put upon the permanent stability of such a structure if it is ex- posed to repeated loads. In fact, the steel near the limits of the broken portions will have to be very highly stressed, causing thereby excessive elongations, the transmission of which to the surrounding unbroken mortar must by and by extend the disintegration to the complete failure of the member. Expressions of these uncertainties are some- times heard even from some of the partisans of the new method of construction. The known observations and experiments do not seem to be of a character to throw sufficient light on the sub- ject. Observations have been generally limited to the measurements of the deflections of beams subjected to loads; but it is wholly impossible to deduce from them the elongations of the " fibres " in tension and the shorten- ings of the compressed " fibres " produced after the elastic limit has been exceeded at any point. Sometimes, it is true, these deformations have been measured, but. mostly REINFORCED CONCRETE IN BENDING. 3 the intention was to determine the elongation of the steel itself. To reach the steel with the measuring apparatus cuts were made in the concrete which destroyed its con- tinuity and prevented getting from the observations a clear picture of the phenomena which really take place in beams having no cracks. In some cases the accuracy of the measurements of the loads employed is also, of a doubtful character. When the loads have been made up of numerous bags of sand piled on the beams to a height of 5 to 1 feet, arch action was produced which must have relieved the middle por- tion of the beams to a noticeable degree. With loads produced by means of a hydraulic press errors from manometer readings could occur, and it seems that such was the case with some experiments where, contrary to what has been found elsewhere, observations seemed to indicate that the deformations increase more and more slowly as the stresses increase. For these reasons, and also because no sure deductions can be based on experiments whose details were not fol- lowed carefully, the author decided to make new tests. Since it was clearly necessary to have numerous tests, small prisms only were used. In doing so the author was guided by the consideration that the laws governing the materials would be shown in small prisms, and could then be verified by a small number of tests on large prisms so arranged as to furnish directly the necessary data. It appeared to be proper to abandon the measuring of deflections which to the other complications of the action of bodies in flexure adds one more, because the deflection is the resultant of different deformations produced in all the sections of the member in flexure. The author, there- fore, adhered exclusively to the measuring of the elonga- tions of the surfaces in tension and the shortenings of the compressed faces. In order that these deformations be uniform for the whole length between observation 4 REINFORCED CONCRETE. marks it was necessary that the prism 'be subjected to a constant bending movement. These several conditions were fulfilled by adopting the following arrangement: Prisms 23.6 inches long by 2.36 inches square were moulded, some without reinforcing for the determination of the properties of the mortar, independent of the steel, and others reinforced on the tension side only by wires or rods of steel or iron varying from 0.075 to 0.303 inches diameter. The greatest number *had no transverse rein- forcing in order to simplify the observed phenomena. Each prism was placed vertically with its lower end fixed in a solid testing block, Eig. 1, and the upper end Fig. 1. provided with a cap which also formed a fixed attach- ment and had a horizontal lever 27.5 inches long suspend- ing a scale. In the fixed portions shearing stresses of varying intensity were produced by changing their lengths and their distances from the ends of the prisms. In the portion between the fixed ends, there was no shearing stress and the simple phenomenon of flexure could be observed. These experiments were not completed at the date of writing, but the results already obtained will be described and their practical consequences indicated. Among the numerous tests, all showing concurring results, the follow- REINFORCED CONCRETE IN BENDING. 5 ing gave particularly precise data and is chosen for a complete statement. From mortar mixed 730 pounds of Portland cement and 4.3 cubic feet of water per cubic yard of good quartzy sea sand, were made one prism, Eb. 31, not reinforced, and five prisms reinforced by 17 wires of 0.075 inch diameter, by three wires of 0.167 inch diameter, or by iron rods 0.303 inch diameter. The author him- self tamped the prisms with exceptional care. Prism No*. 31, with a section 2.36 inches deep and 2.4 inches wide, broke after having resisted for several minutes a bending moment of about 83 foot-pounds, which pro- duced a shortening of the extreme compressed " fibres " Pig. 2. of 0.0131 per cent, and elongation of the " fibres " in ten- sion of 0.0201 per cent, of the length. The mirror index used to show elongations by its displacements could alone be followed by the eye until the rupture, which took place after an elongation of 0.0266 per cent. Prism E"o. 34 had the section shown in Fig. 2. The wires had a diameter of 0.167 inch. Only the value's for this prism will be given in detail, but the results of the other prisms reinforced by wires 0.167 and 0.075 inch diameter were almost absolutely identical. The results of the prism reinforced by a rolled rod 0.303 inch diame- ter differed only in the smaller elasticity of rolled iron and the greater distance from the centers of the rods to the surface of the prism, made necessary by the larger 6 REINFORCED CONCRETE. diameter. Prism No. 34 was kept in water 210 days and tested 3 days after it was taken out. Its bending moment was increased to about 569 foot-pounds without causing failure. Afterward, to study the effect of re- peated deformations, this prism was subjected to 139,052 repetitions of the bending moment, varying from 250 to 402 foot-pounds, alternating with the same number of .returns to the position of equilibrium. After this double test the prism appeared intact for the whole length be- tween the fixed points, although the mortar had sustained, during the first bending, an elongation of 0.198 per cent., that is, almost twenty times as much as the 0.01 per cent, which similar mortars cannot sustain without breaking, and afterward withstood 139,052 elongations varying be- tween 0.0545 and 0.127 per cent. To determine whether the elongated portions of the mortar were cracked the author detached this mortar, by means of a sand saw, from the rods and the body of the prism, and found that it remained perfectly intact except for two superficial cracks 0.08 to 0.16 inch long. Notwithstanding the strain due to sawing, pieces 0.59 by 0.47 inch in section and 3.15 to 7.9 inches long, that is, more than half the length between supports, were cut off, and, tested by bend- ing, gave resistances up to 313 pounds per square inch, thus proving that, almost throughout, the surface layers, which had sustained an elongation twenty times that con- sidered dangerous, not only were not disintegrated, but retained nearly the same strength as new mortar. The total resistance of a reinforced prism cannot be anything but the sum of the resistances of its two ele- ments. The elongation of the steel can be computed from the deformations as measured on the two opposite faces of the prism, making the classical assumption of the theory of flexure that the cross-sectional planes remain planes after bending; and the tension in the reinforcing bars can be easily deduced when the coefficient of elas- REINFORCED CONCRETE IN BENDING. 7 ticity of the steel is known. In this case the coefficient was established by a direct tension test on an identical test piece. Multiplying the . tension in the reinforcing members by its lever arm, that is, by its distance from the resultant of the mortar in compression, a portion of which forms a couple with the tension in the bars, the moment of the couple is obtained. (See .Table I, column 10.) Subtracting this moment from the total bending moment gives the moment of the mortar in tension and of that fraction of the resultant of the mortar in compression, which forms a couple with it. Table I shows the results obtained. In column 11 the fourth and the last values are unex- pectedly high, and the author first attributed this to errors of observation. But the same anomaly was observed on similar prisms and is, probably, satisfactorily explained by the longitudinal sliding of the rods in the mortar. The moment of resistance of the mortar tension increased with the deformation, first rapidly and uniformly, then more and more slowly up to 1,400 inch-pounds, and did not fall below this amount afterward, though the elonga- tion at the tension-face reached 0.198 per cent. Column 11 suffices to prove that mortar reinforced by metal can sustain elongations very much greater than those allowed for it up to the present and still contribute efficiently to the resistance of the member. At first sight this result appears inadmissible, but the objection vanishes when one remembers the phenomena of the deformation of metals and admits that identical phenomena can take place in mortars, though the extreme smallness of the deformations has precluded their observa- tion. "When a round rod of soft steel, for instance, is subjected to simple tension, it first undergoes a uniform elongation, increasing to 18 to 22 per cent., and then it suddenly strangles itself at the " point of stricture " until rupture ensues after a local elongation of 200 to 300 per 9 2 Moments Caused by Concrete; Difference Between Cols. 1-10. 11 O SIOQOOOO 5 Moments Caused by Steel. 10 Inch-lbs. 110 296 522 1,200 2.145 2,880 4,110 5,170 Lever Arm on Tension Side. 9 ggo-t-fc-fc-r-fc-t-t- s |™ En 1 °° ^ rt OX «D C* cular property of mortar. This prism was consequently made with exceptional care, which cannot be expected in practice. To obtain information on which rules for de* signing can be based, observations must be made on the actual construction and behavior of structures. As stated the majority of recorded observations were made under conditions which do not allow very precise conclusions. However, it has been clearly established that generally the first cracks have not appeared before the bending moment acting on the highest stressed section caused tensile stresses in the iron from 23,000 to 28,000 pounds per square inch and, therefore, elongations of about 0.1 per cent, of the length of the test piece. The author has attempted to prove that if concrete-steel can endure much greater elongations than unreinforced concrete, it is be- cause the steel efficiently aids the sections which show a tendency to deform more than the others and thus com- pels the concrete to take all the molecular elongation of which it is capable over the whole length of its extreme " fibres." The aid given to the concrete by the steel is evidently proportional to the coefficient of elasticity of the latter ; but this coefficient drops suddenly to one-tenth and sometimes even to one-fiftieth of its initial value when the elastic limit is exceeded. From the point of view of the protection given the concrete against premature cracks, the passing of the elas- tic limit produces the same result as if the sectional area of the steel were reduced to one-tenth or one-fiftieth, which then becomes negligeable. The concrete must nec- essarily crack in the weakest sections as soon as the steel is stressed above its elastic limit. It is impossible for concrete to sustain without cracking an elongation ex- 14 REINFORCED CONCRETE. ceeding about 0.1 per cent, in prisms reinforced by wrought iron or extra soft steel. If in the prism So. 34 an elongation of nearly 0.2 per cent, was obtained, it was because the iron wire had an elastic limit exceeding 54,000 pounds. It was seen in section 1 that the study of the curve of the bending moments added a new proof to the results of the direct examination of the mortar cut from prism No, 34. It confirmed that the mortar was not broken before the elongation of 0.2 per cent, was reached, since it maintained until then a resisting moment of great value. It is a matter of research to discover if well-known experiments will permit a verification of this phenome- non. This can be easily done even when no information is at hand as to the quality and resistance of the concrete and steel used in the beams whose successive loadings and corresponding deformations are known. If the result of the displacement of the neutral axis, which is of small importance, be neglected, it may be said that the bending moments induced by the tension of the reinforcing metal are very closely proportional to the deformations. If, therefore, a curve be plotted with deforma- tions for abscissas and bending moments for ordinates, the moments caused by the metal will be represented by a straight line, O F beginning at the origin of co-ordinates O, as long as the elastic limit of the metal is not exceeded. (See- Figure 3.) The total moments, being the sum of the moments of the steel and the concrete, would give a discontinu- ous curve, partly concave, such as O A B C F if the concrete break under a small elongation such as REINFORCED CONCRETE IN BENDING. 15 0.01 per cent., causing the rupture of unreinforced con- crete. If, on the contrary, the concrete has neither dis- integrated nor noticeably cracked, the elastic limit not having been reached, and has produced a bending mo- ment which increases rapidly up to a value below which it does not fall, the curve of moments must take a form like A D, first convex, and then practically straight. But this is exactly what is shown by all known experi- ments, and especially those made at Lausanne, Switzer- land, by M. Ferrari, in 1893 and 1894. 3. The Resistance and Elasticity of the Materials Generally Used. Examined in line with the above ideas the results of the experiments known to us seem to confirm the conclu- sions drawn from the tests on prism ~No. 34 and from all the rest of the author's experiments. A commonly used concrete is made of 550 pounds of Portland cement to one cubic yard of a mixture of equal parts of sand and small gravel. This concrete often gives a tensile resist- ance of 215 pounds, and more than 2,600 pounds per square inch in compression, with an elastic coefficient of 2,800,000 to 3,700,000 pounds; but the author would adopt, for the sake of prudence, as the least and safe values, 170, 2,140, and 2,700,000. These values give for the curve of deformation the following figures: TABLE III. Elongations or shortenings in per cents of length . . 0.004 0.010 0.025 0.050 0.100 0.150 Corresponding tension? in pounds per square inch. 107 156 170 170 170 170 Compressions in pounds per square inch 107 256 568 925 1495 2140 4. The Graphic Solution of the Problem. Plotting the elongations or shortenings as abscissas and tensions and compressions as ordinatesthe curve of de- 16 REINFORCED CONCRETE. formation in Fig. 4 is obtained/- from which by trial computations the stresses at all points of the concrete and steel of a section of a concrete-steel beam can be determined, if a known deformation is given to any of. the fibres. Suppose, for instance, that the bending mo- ment produced in a prism made of the above concrete Fig. 4. is to be determined when the reinforcing steel is stressed somewhat below the elastic limit, say at 23,000 pounds per square inch, and will take an elongation of 0.09 per cent. In Fig. 5, let A B represent the depth of the cross- section of the prism and F the center of gravity of the reinforcing metal. Let an abscissa F f be drawn at F rep- REINFORCED CONCRETE IN BENDING. 17 resenting an elongation of 0.09 per cent. If the usual hypothesis of the conservation of the planes of the sec- tions after bending is accepted as practically accurate, section A B will after bending take the position A' B', which must pass through the point f for steel stressed at 23,000 pounds per square inch. The position A' B', as yet unknown, must satisfy the condition that the sum of the tensions in the metal and concrete in the portion O B must be equal to the sum of Fig. 5. the compressive stresses of the concrete in the portion O A. To test this let us try a direction A' f B'. In.each " fibre " of section A B the elongation or the shortening will be equal to the corresponding ordinate of the straight line A' f B'. By scaling from the curve' of deformation, Fig. 4, the ordinate corresponding to each elongation or shortening, the value of the tension or compression which it will cause will be found. It will be easy to com- pute graphically the sums of the tensions and compres- sions in all the " fibres " of the concrete and metal and 2 18 REINFORCED CONCRETE. to see whether the two sums are equal, and thus deter-, mine whether the trial direction of A' B' satisfies the con- dition of equilibrium. When the correct position of A' B' is found, the intersection of A B and A' f B' will give the position of the neutral axis O, and it will be easy to de- termine graphically the bending moment caused by the tensions and compressions in the metal, and concrete of the prism. 5. The Approximate Solution. The preceding method gives an exact solution; but if the irregularity of the properties of concrete be taken into account, it seems a waste of energy to try to obtain absolute accuracy in the methods of computation of the dimensions of the members, and so an approximate and rapid method will be given. By substituting for the curve of elongation of the concrete, O M5in Fig. 4, the horizontal line H N, which coincicdes with the greatest portion of this curve, only a very small error will be committed as far as the computation of the moments is concerned, since the triangular space OHM has a small area and a very much reduced lever arm. Similarly the straight line O B, can be substituted for the curve O P Q. Equations can be now established and solved directly, but before proceeding, the object to be attained must be pre- cisely stated. The danger which threatens the structure if the" stress in the metal exceeds the elastic limit has been stated. The greater or less elongation to which concrete is sub- jected within the limits of practice does not appear to be of great importance, because the tension does not vary correspondingly and because rupture depends much less upon the intrinsic value of the elongation than upon the passing of the elastic limit of its reinforcing members. It thus appears that, except the elastic limit of the metal, nothing but the crushing resistance of the concrete in REINFORCED CONCRETE IN BENDING. 19 compression needs to be considered as far as the resist- ance to bending is concerned. A third important factor to be considered is the sliding of the reinforcing metal in the concrete,, but this factor has no direct relation to the value of the bending moment and depends solely on the shearing force. Hence it can be left aside while considering the bending effect and be investigated separately. To develop the method for computing algebraically the compressive stress induced in the concrete of a given prism when the reinforcing metal has reached the elastic limit, and determining the moment of bending resistance which is then produced by the prism proceed as follows: Denote by In, the depth and by e the width of a ^cross-section of the prism in inches; by p the percentage or ratio of the area of the metal reinforcing to the area of the concrete ; by hu the dis- tance from the center of gravity of the reinforcing bars to the extreme "fibres" in tension; by I the elastic limit of the metal; by t the stress caused in concrete in tension when its elongation reaches 0.015 to 0.020 per cent., and which then remains constant with an increase of de- formation; by c the compression in the most compressed " fibre " of the concrete ; by K the ratio of the angular coefficient of the line O E., Fig. 4, which represents the average coefficient of elasticity of the concrete in com- pression, to the angular coefficient of the line O W, which represents the coefficient of elasticity of the metal. If we assume the average coefficient of elasticity at 28,'000,000 for steel and at 2,800,000 for concrete, K will have the value of 0.1 during the perfectly elastic state and decreasing values below 0.1 as increased deformations of the concrete in compression cause a greater change in the elastic behavior. The position of the neutral axis must be determined first. If hx denote the distance from the neutral axis of the prism to the extreme " fibres " in tension, the value of x 20 REINFORCED CONCRETE. will be obtained by equating the tensions in the steel and concrete to the compressions in the concrete. The value of the greatest compression of the concrete is given by the formula, in which e==Klx l^. (1) x — u The value of x which gives the ratio of the distance from the neutral axis to the extreme " fibres " in tension to the total depth h of the prism then follows from the above condition : 2 x — u The moment of resistance to bending is given by the formula : -*, ,,/ 4 — a? , a!-3i*+2\ ,g\ M = e hHt x—y, 1- Ijp g — z — I ( 3 ) When the quality of the concrete and metal to be used are selected the values to be given to I, t and K will be' known. For h u the most suitable value will be chosen, and it will generally be practically the same in all cases, namely about 0.12 h. For the ratio of steel to concrete sections different values such as 0.01, 0.02, 0.03, 0.04 will' be assumed. The equations will then be solved for each value of p and, by interpolation, there will be found for any value of p the position of the neutral axis, the amount of the greatest compression in the concrete and the moment of resistance to bending when the stress in- creases until the elastic limit of the metal is reached. In a very short time a table can be computed which will furnish in two or three minutes the bending moment which a prism of any dimensions and percentage of metal will be able to sustain. Table IV has been computed for a beam of one inch square section. For the first three cases a concrete poor in cement and for the rest a rich concrete was assumed. To find the moment which a REINFORCED CONCRETE IN BENDING. 21 prism of height h and width e will sustain the figures in column 11 of table IV have to be multiplied by e h? ex- pressed in inches. The quality of concretes is much too variable to ascribe to the figures of columns 5 and 6 any general applicability. They are simply illustrative of a method of computation the results of which will be interpreted in each case according to the quality of the materials to be used. Intentionally the author has taken values below the average. 6. The Displacements of the Netttbal Axis. Column 9 of Table IV shows a series of displacements of the neutral axis under eight assumed different con- ditions. It gives the variations of the distance of the neutral axis from the extreme' " fibre " of the tension side expressed in parts of the depth of the cross-section. The values which are computed for the tension and compres- sion in the concrete and metal depend upon this distance to the neutral axis, no matter which method of computa- tion may be used. If the first three lines of Table IV be examined it will be seen that the distance from the neutral axis to the tension face is reduced from 0.57 to 0.42 of the depth when the percentage of the metal to the concrete increases from 1 to 3 per cent. The area of the concrete in compression increases with the area of the metal, the tensile stress of which it must balance. Not only does the neutral axis occupy different positions in beams of different character, but in the same beam it very frequently shifts in a very noticeable manner during the application of load. While the elastic behavior of the concrete is still perfect, the reinforced portion has neces- sarily a much higher coefficient of, elasticity than the remaining portion, if the concrete is homogeneous. The neutral axis will hence be found between the reinforcing members and the middle of the section. But with the increase of the loads the elasticity of the concrete in ten- sion changes more and more, while that of the concrete in •panoj-^ooa „; J8a jsop ' " «5 N M •pjBi oiqno« j9d}soo " jo ^aeuioji •" ■e^sjonoQ jo noissejd 2 ■moo I«n}OT ■sob j nois ■ne,LO}S!xy [BJ^naiiniojj GOUB}ST(I imora a ■oaoo a oiipsg; =2« — < o co SI § s •jaquiriK- [22] REINFORCED CONCRETE IN BENDING. 23 compression changes at first tfnly slightly. The neutral axis must thus shift,' and it really does shift, away from the reinforcing members. Thus in tests made on nu- merous beams reinforcel by 1.1 per cent, of metal, the neutral axis, which was at first very near the middle of the section, has shifted up to 0.58 or 0.60 of the depth of the section with the increase in load. These figures are very near the value of 0.57 which is given in Table IV for 1 per cent, of metal. A precisely' opposite effect will take place at the end ■of the loading if the reinforcing members have too much .area as compared to the concrete. This is because in order to balance the tension of the reinforcing members the concrete in compression must be very highly stressed, and the change in its elasticity will finally be the deciding element The author has never experimented on beams so highly reinforced, and he is, therefore, not in a position "to verify the accuracy of his conclusions as far as these beams are concerned." 7. The Influence of the Proportion of Iron or Steel. A study of columns 10 and 11 of Table IV leads to the following conclusions : For the beams made of a con- crete poor in cement, reinforced by ordinary iron, the moment of resistance increases from 18.6 to 42.6 foot- pounds when the percentage of the iron is increased from 1 to 3. The cost due to the increase in volume of metal grows much less rapidly than the increase in resistance. It would thus appear that there is a considerable advan- tage in increasing indefinitely the percentage of the iron if the figures of column 10, which give the stress of the concrete in compression, are ignored. To fully appreciate the meaning of column 10 some preliminary remarks are necessary. A concrete containing 500^- pounds of cement to the 24 REINFORCED CONCRETE. cubic yard of sand and stone sustains a pressure of 2,140 pounds per square inch, "without alteration, and even much more when it is well made, but if the stress be repeated the concrete will crush under a much smaller load. There is very little information on the effect produced on mortars by repeated loads. M. de Joly published in. the "Annales des Ponts et Chaussees " a study which tends to prove that mortars in tension finally break under re- peated stress' at about half the tension which they can re- sist at a single application. Since concrete resists com- pression much better than tension, it is to be expected that this superiority in compression would show its good effects on the resistance under repeated compressive stresses, but in the absence of conclusive experiments it is prudent to admit that the repeating stress must in all cases remain below two-thirds of the crushing strength, which has been indicated in column 6 of Table IV. This is in accordance with the proportion determined by Woehler for metals. ~No similar progressive change in the nature of the reinforcing metal need be feared, for metal stressed within the elastic limit, because the experi- ments of Woehler have also proved that iron and steel can sustain an indefinite number of repetitions as long as the stress remains within the elastic limit. If, therefore, it be assumed that the concrete will, crush under the pressures given in column 6, or urder two-thirds of their value, according to whether the load is permanent or repeating, it will be seen from an inspec- tion of the table that a certain percentage of the metal must not be exceeded in order that the concrete shall not fail by crushing before the reinforcing metal has reached its elastic limit. By interpolation this limiting- percentage is found at 2.17 per cent, for beams subjected to a permanent load, and at 0.8 per cent, for beams sub- jected to repeated application of loads. Among existing structures a great number can be found where the pro- REINFORCED CONCRETE IN BENDING. 25 portion of iron reaches 2 and even 3 per cent. This is mostly due to the fact that in the experiments on which the design of the beams were based one or only a few applications of load have been made. The author be- lieves that experiments with repeated application of loads will lead to somewhat different rules. 8. The Influence oe the Quality op the Concrete and the Metal. The knowledge that the danger of the crushing of the concrete under repeated application of load is nearer than generally supposed leads to the question whether larger proportions of cement could not be employed to advan- tage. The fourth to sixth lines of Table IV are based on a concrete supposed to contain 1,340 pounds of cement ■to the cubic yard of gravel and sand, mixed so as to give the highest resistance. By interpolation in column 10 it is found that such a concrete can sustain a proportion of metal of 5.6 per cent., if the load be constant, and about 3.3 per cent, with a recurring load. It thus may be con- cluded that the percentage of reinforcing metal can and should increase with the increase in strength of the concrete. The above table also throws some light on the effects of a substitution of high steel, like rail steel, for the ordi- nary structural variety. For good and sufficient reasons such high steel is excluded from ordinary structures, and especially from riveted construction. It becomes very brittle after being punched or after having undergone deformation in the cold state, but it maintains a more than sufficient ductility when used in thin bars which re- ceive no cold hammering. Its elongation, measured on a length of about 4 inches, is in this case at least 16 per cent. It thus seems to be advantageous to utilize its great resistance in concrete-steel construction. i Using this high steel with a concrete rich in cement. 26 REINFORCED CONCRETE. and allowing > for .the- steel- unit- stresses^ exceeding -those generally allowed for iron and soft steel within the elastic limits indicated in column 4 of Table IV, moments of resistance are obtained as high as 38.8 and 61. 7 foot- pounds, according to whether the per cent, of metal is 1 or 2 (seventh and eighth line). The choice between an increase in the proportion of the steel and the sub- stitution of a steel of higher resistance, stressed almost twice the amount allowed for iron, depends on numerous • -considerations. An increase in the proportion of the metal results in increased rigidity, and the use of a steel work- ing under the conditions indicated in the above, gives, on the contrary, to the reinforced members, a higher elasticity and ability to sustain without cracking twice as great de- formations. It will, therefore, be advantageous to use iron or soft steel for structures where the vibrations might prove dan- gerous, and high steel where spreading or unequal set- tling of supports are to be feared. Such, for instance, is the case when rigid abutments connecting various parts ■of a complex structure may, by unequal yielding, cause dangerous reactions among the connected members of the structure. The advantage of increased elasticity is mani- fest in this case. So it is also for structures liable to I' shocks. In fact, it may be stated that for an equal moment (s. of resistance, a beam will absorb twice the kinetic energy when reinforced by a high steel as when reinforced by iron or soft steel because it will be in a condition to sus- tain double the deformation before breaking. 9. The Cost of the Different Types of Beams. When the choice of reinforcing metal and of the pro- portion of concrete is not determined by purely engineer- ing considerations it will be made in accordance with con- siderations of the greatest economy. To throw some light ■on the economic side of the subject, the cost of each kind of beam per cubic yard has been given in column 12 of REINFORCED CONCRETE IN BENDING. 27 Table IV, and in column -13 .-the ratio of the cost to the corresponding moment of resistance has been indicated.^ A concrete containing 500 pounds of cement per cubic i yard of sand and gravel can be easily had 'at $7 per J cubic yard, all work of making the concrete-steel beams included. This is a good average price. An addition of 840 pounds of cement, per cubic yard of sand and gravel, will increase the cost by about another $7, but, allow- ing for the increased volume, the cost of the rich concrete used in Table IV will not exceed $12.50'. The cost of wrought; iron and soft steel has been estimated at 2.5 cents per pound, and that of the high steel at 3 cents. ^ Column 13 shows that the use of a rich concrete and a high steel will result in considerable economy. But. to be more definite on the advantages offered by the latter variety of reinforced concrete, the ultimate crushing re- sistance of the concrete, which must not be exceeded, needs also to be considered. This consideration fixes the limits for the increase in the proportion of the metal. If only the figures of the last column of Table IV are considered it must be concluded that the use of a high steel and a concrete rich in cement will result in economy in almost every case, and especially for beams subjected to repeated applications of load. But such an absolute statement cannot be made safely. Reasons for limitations exist which must be taken into account and which will change somewhat the aspect of the question. When using selected materials the least dimensions must be given to the beams, and if this should lead to a reduction in depth as well as in width a decrease in economy will be the re- sult, because the moment of resistance is proportional to the square of the depth, while the cost is proportional to the first power. It will be well, therefore, to limit the conclusion to the statement that a concrete rich in cement and reinforced by a high steel seems to be advantageous in certain cases, and that it should be studied, instead of being excluded 28 REINFORCED CONCRETE. from concrete-steel structures, as has been the tendency heretofore. In any case, and engineers have already come to the understanding of it, a concrete rich in cement must be tfsed for maritime works where impermeability is one of the main requirements for masonry. It also seems that liability to shocks and repetitions of load should, in general, lead to the use of good materials. In preparing Table IV the author has taken the richest concrete and the strongest steel in order to emphasize the differences in the possible results, but it is probable that in many cases it will be better to use propo.rtions of cement between those indicated in the table, and, it may be, steel of average strength, not so brittle as rail steel. There seems to be no objection to the substitution of medium and high steel for soft steel, reducing its volume only enough to make the cost the same; that is, about one-tenth. In fact, the elastic limit of medium steel exceeds that of soft steel by about 10 per cent., and if, therefore, the steel be used with this reduction in weight conditions will remain the same as if the soft steel were used, as long as the stresses in the reinforcing steel remain within the elastic limit. But when the elastic limit is exceeded, the beams reinforced by wrought iron or soft steel will crack and completely break to pieces, while the beams reinforced by the higher steel will not show any damage so long as there is no crushing on the compression side. This manifests an evident advantage which does not appear to be offset by any serious incon- venience. 10. The Detebmination of the Most Economic Pro- portion of Metal. For a certain proportion of metal to concrete the com- pressive stress induced in the concrete reaches its limit, which cannot be exceeded without danger. If the pro- portion of metal be still increased, the stress caused in the concrete in compression cannot be kept within the REINFORCED CONCRETE IN BENDING. 29 desired limits otherwise than by reducing the working tensile stress allowed on the metal. It is probable that the proportion for which the steel and the concrete in compression simultaneously reach their greatest allowable stresses will give the greatest possible economy. To. de- termine the value of this critical percentage introduce into equation (1) the values of c and I, which will give the greatest allowable unit stresses, compute the value of x and introduce it into equation (2), which will furnish the desired proportion p of metal to concrete. There will then remain unknown the moments of beams in which the proportion of steel is higher than found by the preceding method of computation. For the determi- nation of these moments it is sufficient to establish formulas similar to (1), (2), and (3) by the introduction of a constant value for the compression c of the concrete, instead of the tensile stress I in the metal, which need not be considered, as it is known that the reinforcing has an excess of strength. _ C_ X — U /£> K 1-x V ' <«, + «£ £Z_« = J(i - ») (5) K \ — x 2 These formulas give the values of the breaking moments due to the crushing of the concrete for higher propor- tions of metal than the so-called critical percentage, while equations (1) to (3) give the breaking moments due to exceeding the elastic limit of the steel for the lower pro- portions of metal. In Table V are given the results as computed for beams made of the poorer concrete, 500 pounds of cement to the cubic yard of sand and gravel, and having iron for reinforcing. Both constant and re- peated loads have been computed. 30 REINFORCED CONCRETE. TABLE V. Concrete with 500 - Pounds . Cement to the Cubic Yard, and Iron Revnforcvng. Constant Loads. Repeated Loads. Percentage of 0.83 1 a 2.17 S3. 3 0.82 16.5 1.5 19.3 3 23.4 Breaking mo- ments, f t-lbs.. . . 16.5 18,6 31.1 42.6 Cost per foot- pound, cents. 59 55 ' 44 1,2 40 69 62 72 Tension in metal, lbs. per sq. in.. 22.800 £2,800 32,800 23,800 18,350 22,800 16,900 11,709 Compression in concrete, lbs. per sq. inch.... 1,485 1,525 2,040 2,U0 2,140 2,11,0 1,425 1,425 While the meaning of the figures of Table V is obvious enough, a diagram, Fig. 6, has been drawn for the con- fig. 6. stant loads, with proportions of metal to concrete as abscissas. As ordinates for the curve ABC, the break- ing moments were taken, for D E F the cost of a cubic yard of beam per foot-pound sustained, for g H I, the ten- sions per square inch in the reinforcing iron, and for REINFORCED CONCRETE IN BENDING. 31 j K L, the greatest compressive stresses in the concrete. The balancing proportion is 2.17 per cent, and evidently it is also quite advantageous as to cost per foot-pound, pro- vided that beams of the same depth of cross-section are compared. It can be seen from the diagram that the cost increases less rapidly for an excess than for a deficiency of metal. The increase in cost in the first case is very slight if the excess of steel be kept within certain limits; and it seems that there is no great objection to an increase in the percentage of the metal, but its possible dangerous action should be kept in mind. In fact, with the rein- forcing members relatively too strong, the concrete will fail first by crushing without preliminary signs of failure, while, if the reinforcing is too weak it will before failing first exceed the elastic limit and herald the approaching danger to the structure by the cracks which will develop in the concrete. Serious accidents could thus be pre- vented. Tables VI and VII give the costs per cubic yard of beam per foot-pound sustained, which correspond to the different proportions of metal for beams rich in cement and reinforced by iron or steel. It is unnecessary to in- dicate here the tensile and compressive stresses since they vary according to the same laws as those given in Table V. TABLE VI. Concrete with 1,340 Pounds Cement to the Cubic Yard and Iron Reinforcing. Percentage of metal Breaking moments, ft. -lbs. Cost per ft. -lb., cents Constant Loads. 45 4 61.0 5.6 8L.5 S7 6.5 91.0 38 Repeated Loads. 1 25.6 3 49.4 45 S.S BS.S 5.6 59.4 52 32 REINFORCED CONCRETE. TABLE VII. Concrete with l,3Jfi Pounds Cement to the Cubic Yard and Steel Reinforcing. Percentage of metal Breaking moments, f t.-lbs. Cost per ft.-Ib, cent Constant Loads. 61 7 33 76.9 3 5 E6.0 31 Repeated Xoads. 1 38.8 43 1.2 U2.6 48.9 From Table VI it is seen that the theoretically most advantageous percentage of iron for beams, of a concrete as rich in cement as 1,340 pounds, and subjected to a con- stant load is 5.6. Such high percentages are practically out of question. It thus appears that a concrete contain- ing about 850 pounds of Portland cement to the cubic yard will in most cases be sufficient for beams, reinforced by ordinary wrought iron or soft steel if they are to sus- tain largely constant loads. A richer concrete, on the con- trary, will be advantageous for beams liable to repeated loads and reinforced by a steel of high resistance, the full capacity of which has to be utilized. 11. The Computation of Reinforced Concrete Beams. Tables V, VI, and VII give for the various kinds of material and percentages of metal the bending moments which will cause either the cracking of the concrete in tension or the crushing of the concrete in compression, the limits to exceed which is dangerous. To compute the di- mensions to be given the beams for practical purposes a suitable factor of safety must be adopted. It is well to consider first what has been done for iron and steel struc- tures. For structures of some magnitude, where shocks are REINFORCED .CONCRETE IN RENDING. 33 little to be feared, ordinary wrought iron may be stressed up to 12,000 pounds per square inch, and soft steel to 15,000 pounds, or about one-quarter of the ultimate strength, and less than one-half of the least elastic limit. If it be considered that there is no metal structure in existence which would not collapse after exceeding the elastic" limit, especially in its compression members, it must be admitted that the real factor of safety does not exceed two in the large structures. For small structures the greatest stresses are reduced to 9,000 and 12,000, re- spectively, and are sometimes still less; but this decrease is hardly sufficient to cover the increased effect of vibra- tions, shocks, and rust. These destructive causes need little to be feared for concrete-steel structures, and one would thus be tempted to adopt a factor of safety of 2 with respect to the breaking moments as calculated for the above tables and below which there is no danger of failure. Before reaching a conclusion, however, the results ob- tained to the present time from reinforced concrete struc- tures should be considered. These results are certainly too recent to furnish conclusive information as to the dura- tion of such structures; many are, however, several years old, and the increasing application of concrete-steel struct- ures for many purposes goes to show that they have so far proved satisfactory. It is, therefore,- of interest to com- pare the breaking moments as computed in the tables to the bending moments which would be really applied to the beams according to the formula used by M. Henne- bique. Let M be the bending moment applied to a beam and 2H the depth' of that portion of the section which is in compression. Assume that the compressive stress in the concrete is throughout the section equal to 350 pounds per square inch, and that its moment taken for the whole com- pressed portion must be in equilibrium with one-half of 3 34 REINFORCED CONCRETE. M. If e denote the width of the beam, this moment of resistance due to compression in the concrete will be: 2 .26X350x^=3500 IP=~, whence, 2 2 JET=0 .038 /— . v e The other half of the moment M mast be sustained by the metal, according.: to the Hennebique assumption. If then the distance of the metal from the line limiting the compressed portion be denoted by H, and if it be desired to stress its area A at 14,000 pounds per square inch, , M 28,000 II ' It is unnecessary to prove that the Hennebique formula is based on two theoretical errors: First, the uniform distribution of the compressive stress over the whole com- pressed area, and, second, on the equality of the tensile and compressive moments with reference to the neutral axis. The bending moments as computed by the above Henne- bique formula are given in Table VIII in comparison with the breaking moments computed according to the author's formulas. TABLE VIII. Concrete with 500 Pounds Cement to the Cubic Yard and Iron Reinforcing. Percentage of metal Breaking moments in ft. -lbs. . . . Working moments(Hennebique) Factor of safety. Constant Loads. 0.82 16.5 7.3 2.3 1 18.6 8.8 31.1 11.8 2.6 8. 17 33.2 12.3 2.7 3 43.6 13.5 3.1 Repeated Loads. 0.82 16.5 7.2 2.3 1 5 19.3 10.0 1.9 3 23.4 13.5 1.7 It can be seen that for structures subjected to constant loads, the factor of safety ranges from 2.3 to 3.1 and varies REINFORCED CONCRETE IN BENDING. 35 little from an average value of 2.5. For an empirical formula this is a remarkable result and it shows the good practical sense of its originator. For structures subjected to frequently repeated loads, the factor of safety still re- mains satisfactory for small percentages in the neighbor- hood of 1 per cent., but it drops to 1.7 when .the. propor- tion of metal increases to 3 per cent., a somewhat too small a degree of safety. From all the preceding considerations, we are led to the conclusion that as long as the condition of the oldest structures has not been fully investigated, it will be wise to adopt a factor of safety of 2.5 with respect to the break- ing loads. These breaking loads will be determined for each type of construction after having made tests and hav- ing collected definite and reliable information on the ma- terials to be used. 12. The Deformation oe CWckete-Steee uktdee Re- peated Loads. Until tests extending over a great length of time fur- nish exact information on the effects of repeated loads on concrete-steel, it will prove of interest to study the defor- mations which take place in beams immediately after the load applied to them has been removed; in other words, after subjecting the beam to a load, a repetition of which seems dangerous. For an example, test prism No. 35 has been selected, which had exactly the same composition as prism No. 34. This prism gave results almost identi- cal with those of No. 34 during. the period of the appli : cation of the load. In Table IX are the elongations and shortenings caused in the opposite faces of the prism when subjected to the bending moments indicated in the first line. 36 REINFORCED CONCRETE. TABLE IX. Bending moments in foot-pouids. Elongations of concrete in thou sandths of length . . Shortenings of concrete in thou- sandths of length Bending moments in foot-pounds. Elongations of concrete Shortenings of concrete Bending moments in foot-pounds. Elongations of concrete Shortenings of concrete ....... Bending moments in foot-pounds. Elongations of concrete Shortenings of concrete 4.12 S3.0 0.143 0.146 98.1 0.313 0.231 219.7 0.751 0.580 37.5 0.035 0.008 4.12 022 0.026 4.12 084 0.052 174.0 0.575 0.377 83.0 0.089 0.057 219.7 0.408 0.322 174.0 432 0.296 98.1 0.S86 0.276 128.9 0.149 0.125 280.3 0.694 0.489 280.3 0.729 0.506 4.12 0.180 0.073 174.0 0.254 0.213 219.7 0.597 0.447 372.0 1.187 0.741 128.9 0.208 0.200 174.0 0.478 0.246 280.3 0.929 0.647 The load was applied and gradually removed three times after having caused bending moments of 174.0, 280.3, and 372.0 foot-pounds. We will discuss simply the results of the deformations when the bending moment after having been increased to 372 foot-pounds was reduced to 4.1 foot-pounds, the value required to keep in place the levers of the loading apparatus. By means of computations similar to those made in the first section of this chapter, Table X, has been prepared. The depth of the prism, 2.362 inches, was divided, in columns 2 and 3, proportionately to the observed elongation and shortening, into two parts, one in tension and the other in compression. In column 5 the elongations of the metal have been tabulated as com- puted from the elongation of the mortar, admitting the common assumption of the conservation of plane sections. By multiplying the figures in column 5 by the coefficients in column 6, column 7 of unit tensile stresses in the iron has been obtained. Multiplying the total tensile stresses by the distances from the axis of the reinforcing metal to the resultants of the compressive stresses, the moments of resistance of the metal were obtained. Subtracting these partial moments from the total bend- ing moments in column 1, the resisting moments of the X ■H P"« tt -sioo 98BJ0AV •81 P™ 81 - »IOO ■« jo stonpojj-' •noisnsi Him „ jo rajy J8A9T* ■nojsnsi ni N jtj;jo^ jo ua-iy** S !S M in a o aaio CVS0C«O»r-ll- " cSSSincraoo -ipi aim mesas ■ ffloooooc jo uuy J8A3T t- rid op *« lis — COMNI-- •ogs csotsvt-i gssrais -^CO COO ^< l- CD j aVooo'o 2 « « eo m og eg ot5 3 too© oj t- to -v 55 th oooooo a Sohwoo ct in t— coco -h a> t- in 55 -h n'ddo'o'd CI . Is® eg 8° «•_■■■■ "OOOOOO [37] 38 REINFORCED CONCRETE. concrete in tension and the corresponding portions of the Concrete in compression are found. These figures are tabulated in column 11, and cannot be far from the true amounts, because they are the logical outcome of the ob- served deformations, if the assumption of the conserva- tion of plane sections is admitted. This common assump- tion is very nearly true for beams subjected to a simple bending moment without shear, and such was the case of prism No. 35, which was tested in the same manner as prism ~No. 34. To deduce with the least possible error from column 11, the values taken successively by the tension in the different elongated " fibres " of the concrete during the unloading a delicate and somewhat uncertain analysis would have to be made, which may well be omitted here. However, some useful indications, though not exactly pre- cise, will be obtained by the aid of the following method: In column 12 are the areas of the portions of the section in tension, the products of the width of the prism, 2.44 inches, by column 3. In column 13, the lever arms of the tension in the concrete are given, computed on the as- sumption that during unloading all the elongated " fibres " had at the same instant the same tension, as actually oc- curred in almost all of them during the loading, from the time when the elastic limit for tension was exceeded. This assumption has not only not been proved, but it is probable that it is not at all true. Columns 13 to 15 must, therefore, be regarded as much less exact than those of column 11. They have, however, been computed because they are the only ones which afford even an approximate idea of the changes in the coefficient of elasticity, without which the true character of the phenomena caused by the repeated application of loads could not be clearly under- stood. Column 14 gives the products of columns 12 and 13, and by dividing the resisting moments of the concrete in REINFORCED CONCRETE IN BENDING. 39 tension in column 11 by these products, the average value of the tension in the elongated " fibres " have been ob- tained, in accordance with the assumption the uncertainty of which has already been stated. These average tensions are given in column 15. If now the variations of the tensions, At, in column 15, and the corresponding varia- tions of the elongations A a, obtained by taking the differ- ences of the figures in column 4, are divided out, the successive values of the ratio — . are obtained. This ratio Aa may be called the" instantaneous coefficient of elasticity, because it characterizes the elasticity of, the concrete at a given state of its deformation and free of the effects of the preceding deformations. Table -XI contains these in- stantaneous coefficients of the concrete in tension as com- puted for prism No. 35. It has been stated above that the tensile stresses in the concrete as given in column 15 of Table X, cannot be con- sidered as very exact. The same is true of the instan- taneous coefficients of elasticity, derived from them, column 5, Table XI ; and while they should not be credited TABLE XI. Elongations of Mortar, a. 1 Values of a a. 2 Average Ten- sions of Mor- tar t. 3 Values of a t. i Instant. Coeft. of Elasticity: ^L. P AU O In thousandths. 1.137 Lbs. per Sq. In. 326 Lbs. per Sq. In. 0.208 137 659,000 0.929 189 6. ire 60 337,000 0.751 129 0.176 33 188,000 0.575 96 0.169 78 413,000 0.386 18 6.256 88 344,000 130 -70 with an accuracy not possessed, their variations show a wide enough range to make with certainty some conclu- 4Q REINFORCED CONCRETE. sions and to give others a high degree of probability., From the figures it can be stated with assurance that wheyi a beam is gradually unloaded after having been subjected to a high bending moment, the instantaneous coefficient of elasticity of the portion in tension has at the beginning a high value; and it is probable that it may reach,. at least, one-fourth or one-fifth of the perfect coefficient of elas- ticity of the concrete from which this beam is made. If the unloading be continued, the instantaneous coefficient of elasticity then decreases very rapidly, and it is probable that it becomes about one-tenth of the initial coefficient of elasticity of the concrete. The experiment has not been continued far enough to show what would take place if after the bending moment has changed -its sign, the part at first in tension should gradually become compressed. It is probable that the coefficient would finally assume a value very near the initial value of the coefficient of elasticity of concrete in compression. It would then follow that between the high value at the beginning of the unloading and the very high value under high compressive stress, the coefficient passes through a minimum which appears to be less than one- tenth of the normal coefficient. This phenomenon should not be surprising. To show its meaning, if not its full extent, it may be stated that as shown in all the experi- ments, concrete-steel beams after unloading have only a small fraction of their deformation left. To be more definite, take prism Ed. 35 as an exam- ple. Under a bending moment of 4,460 inch-pounds the elongation of the concrete went up to 1.137 of the length; when the moment was reduced to 50 inch- pounds the elongation dropped to 0.130, a decrease of 1.007. If, during this return to equilibrum, the concrete had maintained its initial coefficient of elasticity of 3,470,000, its tensile stress, first positive then negative, would have undergone in its extreme portion an algebraic REINFORCED CONCRETE IN BENDING. 41 variation of 3,470,000x0.001007 = 3,494 pounds per square inch, giving some 6,350 pounds for the total stress in the- tension portion of the initial section, which is erroneous. Since this portion of the section showed, under the action of the greatest moment, a total tension of about 1,050 pounds (col. 12 x col. 15 of Table X), it must after unloading dispose of a compression of 6,350 — 1,050 = 5,300 pounds. But this is evidently impossible, because this pressure will not be in equilibrum either with the external forces, which have practically no value for the least moment of 50 inch-pounds, or with the tensile stress in the reinforcing metal which is only 3,270 pounds per square inch or 215 pounds total, as given in Table X. The shortened " fibres " are much too far away from this group of opposite stresses to effect their impossible equili- brium. The considerable decrease in the value of the coefficient of elasticity of the concrete in tension during the unloading of reinforced beams thus appears to be incontestible. Another fact, though of much less importance, should also be stated here. The reinforcing metal remains in tension in reinforced members which have been unloaded after having been subjected to great bending, but this is in quite a small degree. In prism No. 35, for instance, the reinforcing bars kept an enlongation of 0.013 of 1 per cent, of the length and a tension of 3,270 pounds per square inch after an elongation of 0.1137 per cent., and a stress of 26,970 pounds. It is of interest to determine the effect of a test load of greater amount than the repeated working load. "When at the beginning of the experiment prism ISTo. 35 was subjected to a bending moment of from 50 to 2,090 inch-pounds, without having been previously subjected to a higher load, the elongation of the extreme portions of the concrete varied from 0.0022 to 0.0254 per cent., the resisting moment of the iron from 43 to 695 inch-pounds 42 • REINFORCED CONCRETE. and that of the mortar from 6 to 1,391 inch-pounds. The results were very different when the bending moment was varied between the same limits of 50 to 2,0.90' inch- pounds after the prism had been subjected to a moment of 4,460 inch-pounds. The elongation of the concrete then varied from 0.013 to 0.0575 per cent., the moment of the iron from 390 to 1,650 inch-pounds, and the mo- ment of the concrete from — 340 to 440 inch-pounds. It is thus seen that the phenomenon is quite different in the two cases, and that the application of a preliminary test load has the effect of considerably increasing the absolute value and the variations of the elongations of the conierate,,of„almQst,doubling .the variations of the stress in the iron and of reducing those in the concrete by one- half. It is impossible for the present to say whether this modification is advantageous and whether the concrete will more or less change its state by undergoing great deformations, producing, however, small stresses because of the considerable decrease of its coefficient of elasticity; or whether it will undergo small deformations enduring, nevertheless, a more considerable stress with a smaller reduction of the coefficient of elasticity. In any case, the reinforcing metal produces, as far as the resistance of the concrete to repeated loads is concerned, an effect not less important than that produced on the elongation under a single application of load. Indeed, the reinforced con- crete of prism No. 34 has resisted without breaking 139,052 repetitions of an elongation exceeding 0.127 per cent., while concrete not reinforced breaks under a repeti- tion of very slight tensile stresses, causing very small deformations. From experiments, the interesting results of which M. Joly has published in the "Annales des Ponts et Chaussees," it is seen that prisms of neat cement broke after a certain number of repetitions of stress which, according to the given coefficients of elasticity, correspond REINFORCED CONCRETE IN BENDING. 43 to elongations much, below 0.005 per cent., that is, at least twenty-five times smaller than the elongation which prism 34 sustained for a great many repetitions without apparent alteration. In conclusion, it may be stated that the resistance of reinforced concrete to repeated loads is due to a small extent to the permanent tension which the deformations cause in the reinforcing metal and for the greatest part to a considerable decrease in the coefficient of elasticity of the concrete without a corresponding decrease in the tensile resistance. Deformations of concrete in reinforced members give it new properties very desirable for its resistance. If bending moments be successively applied and re- moved alternately a certain number of times, the elon- gation of the portion of a reinforced beam in tension ^in- creases with each repetition, but by smaller and smaller amounts. Thus, on applying and removing four times the moment of 4,460 inch-pounds, prism No. v 35 showed successive increases in elongation of 0.0023, 0.0019, 0.0015, and 0.0014 per cent. After these results it is almost certain that the successive deformations of rein- forced beams decrease without interruption and tend tcwrard zero when the repeated stress is kept within the limits in which an indefinite repetition of stress can be sustained. Experience and experiment will determine these limits. The progressive increase in the elongation of the concrete in tension necessarily causes an increase in the elongation of the metal embedded in it and, conse- quently, an increase in the stress of the latter. Obser- vations which have been made and the preceding con- siderations seem to justify the following explanation. "When a reinforced concrete beam is subjected to repe- titions of stress which do not exceed its resistance, its tension side elongates until, owing to the progressively increasing aid of the metal embedded in it, the tension 44 REINFORCED. CONCRETE.. in the concrete is so reduced as to be within the limits of the stress, of which it can sustain an indefinite number of repetitions. After having worked under these condi- tions for some time the concrete, which has, so to say, economized its resistance, will doubtless regain its strength to its full extent if it be afterward subjected to higher, stresses and deformations. Such, at least, was the case with prism l^o. 34, because the small pieces cut out of its tension side after 139,052 repetitions of a con- siderable elongation showed the same bending resistance as the identical concrete not subjected to fatigue. Only numerous and prolonged experiments can furnish data for the complete knowledge of the laws of resistance of reinforced concrete under repeated loads; .but it- seems quite probable that these laws are similar to the laws established by Woehler for metals, that is, that the limits of stress within which reinforced concrete can sustain an indefinite repetition are the higher the smaller the range of the stresses. If this hold true, the stresses must be reduced to a minimum when they change signs, and if, on the contrary, they are of the same character they can be raised to so much higher limits, the less the range of change below their maximum. For the sake of greater accuracy, the limits of the elongation of the concrete rather than the limits of stress should be spoken of, be- cause the concrete can evade an excess of stress put on it by yielding sufficiently to throw the excess of stress on to the reinforcing metal. It is hardly necessary to remark that a higher proportion of cement while increasing the quality of the concrete must also raise the limits of stress or elongation within which the concrete can sustain any number of repetitions. In the preceding, only one of the constituent elements of concrete-steel construction has been considered, namely, the concrete in tension. As to the reinforcing metal it is only necessary to remember that when embedded in REINFORCED CONCRETE IN BENDING. 45 concrete it must never be stressed above its elastic limit, and the well-known investigations of Woehler have proved that under these conditions it does not undergo any alterations when subjected to an infinite number of repetitions of stress of the same character. As to the concrete in compression, the little that is known about it has been stated, and it is probable that the limiting stress for repeated applications of load is below two-thirds of the crushing resistance. It is of a great interest to determine the elongations of the concrete, as commonly made, in beams of large dimen- sions, used in experiments or tested in existing structures. The author could not obtain any information from re- cently-made tests. The attention" of the experimenters was not directed to this point, and to his inquiries it was replied that no arrangements were made for the observa- tion of the beginning of the cracks and the corresponding deformations. But it is the current opinion among build- ers of concrete-steel structures that, as has already been stated, the concrete does not generally crack until the reinforcing metal is stressed nearly to its elastic limit, which requires an elongation in the neighborhood of 0.08 to 0.1 per cent. 13. The Assumption- or an Incbease in the Coeffi- cient of Elasticity of the Steel. All engineers who have studied concrete-steel construc- tion agree, we believe, that the deformations of rein- forced beams are so ■ small, within their working limits, that they cannot be sufficiently explained by the assump- tion that the reinforcing metal alone takes care of the tension in the beam without any aid from the concrete. Convinced that reinforced concrete breaks under small elongations, as had been shown by tension tests of con- crete not reinforced, several authors have assumed that 46 REINFORCED CONCRETE. the properties of the iron or steel itself are modified by the adhesion of the concrete which surrounds it, and that its coefficient of elasticity is thus increased. This assump- tion should have been examined' as to its correctness, but, after having consulted several mathematicians who are most conversant with molecular mechanics, the author considers it to be hardly admissible that of two combined materials the stronger should -have its -properties thor- oughly modified by the relatively weak adhesion of the other on its surface. He believes the case is decided in the negative by the fact that the amounts of the bending moments sustained by the reinforced prism are explained by admitting that the concrete maintains its resistance well above the elongations which cause its rupture in members not reinforced, and also by the other fact that when cut off from the reinforcing the concrete, which un- derwent such an elongation, had maintained the necessary resistance to produce the bending resistance attributed tait. It is hardly necessary to state that the resistance of the concrete in tension in reinforced concrete beams varies considerably with the proportions of the mortar, the pro- portion of the water used and the efficiency of the tamp- ing. The tension reached about 300 pounds per square inch in all the prisms of series ~No. 34, which were made with exceptional care, but this is a maximum for mortars of ordinary proportions. In a series of ten experiments on concretes made without special care and of the pro- portions employed by M. Hennebique, the tensile resist- ance of the concrete fell to 170. pounds, and even to the minimum of. 115 pounds. It should, however, -be stated that an examination, after rupture, of the prism which gave such low resistance showed that it had been made very poorly and that between the reinforcing bars the concrete had received altogether insufficient tamping and showed grains- of sand hardly in contact. REINFORCED CONCRETE IN BENDING. 47 14. The Possible Results of PoOb Workmanship. The question naturally arises whether, even recognizing the property of concrete to sustain without breaking the elongations caused in reinforced beams, its tensile resist' ance should not be assumed as zero in computations for dimensions of proposed structures. It is well known that workmen are often careless- and that poor workmanship of" all -grinds; may occur during construction. It is -especi- ally certain that a lack of adhesion between different lay- ers must always be expected, and that transverse cracks may result in reinforced members when the injured sur- faces are perpendicular to the reinforcing; metal, as is the case in vertical members. In horizontal beams, on the contrary, the layers are parallel to the direction of the reinforcing bars and it is more difficult to see how dangerous cracks may be developed because of faulty construction. Observations on structures have proved that cracks per- ceptible to the naked eye are very rarely met. Prom the practical point of view it must be admitted that if there exist cracks, and that if, in spite of the effects of time, they remain such as never to be noticed, they are inoffensive as far as resistance proper is concerned. Like observations on existing structures, experiments prove that cracks do not indicate a near danger when they be- come visible; because the load must be much increased after their first appearance to cause noticeable deforma- tions and final rupture. These facts are reassuring as to the strength of con- cretfe-steel constructions, and -allay much of- the fear of results of possible poor workmanship. But it is import- ant to discover the causes of poor workmanship and to determine whether the factor of safety is the same for all types of construction. For this purpose compute the breaking moments which the different types of beams could sustain if the portion of the concrete in tension had 48 REINFORCED CONCRETE. transverse cracks for the full width, and compare them to the breaking moments of uncracked beams as given in the preceding tables. To obtain values for the former, the expressions for the tension of the concrete, t, must be eliminated from both groups of formulas, (1) to (3) and (4) to (6). Results of such computations are given in Table XII, the first five columns referring to beams of a concrete not rich in cement and reinforced by wrought iron. TABLE XII. Concrete with 500 Pounds Concrete with 1,340 Pounds Cement to the Cubic Cement to the Cubic Yard, and Iron. Yard, and Steel. Percentage of metal 1 2 2.17 2.4 3 1 2 2.5 3.3 3.5 Moment for beam i without cracks. >- 18.6 31.1 33.2 34.4 42.6 38.8 61.7 76.9 83.0 F6.0 in f t.-lbs ) Homentforbeams 1 with cracks, in / 14. S 27.8 29.6 32.4 42.6 27.2 51.9 63.8 81.2 86.0 ft.-lbs ) Loss of resistance i due to cracks, > 23 13 11 6 30 16 17 2" A study of this table, together with Tables VI, VII and VIII, will show that the most economical propor- tion of metal is higher for beams with cracks. This was to be expected, because in these beams the reinforcing metal gets no aid from the concrete in tension to make up the moment required for the equilibrium of the con- crete in compression, and it, consequently, requires more metal. Eor the beams with cracks 2.4 per cent, of metal is the most advantageous, giving a breaking moment which differs by not more than 6 per cent, from that of the beams without cracks, made of the same concrete and iron. If the percentage of metal be increased to 3 per cent., the breaking moments of the cracked and uncracked beams become the same. The appearance of cracks, as far as the resistance of the beam under a single appHca- REINFORCED CONCRETE IN BENDING. 49 tion of load is concerned, can, therefore, be considered as being of no importance for beams well reinforced. This is easily explained. It has been shown that for the percentages of metal exceeding the most economical percentage the breaking moment must be computed in such a manner that the extreme portion in compression should not be stressed beyond the limiting compressive stress, thus getting the elongation corresponding to that indicated in Fig. 7 by the ordinate a A. The depth of Fig. 7. the cross-section of the beam is represented by a b. By solving equation (5) it is found that in beams without cracks the neutral axis cuts the line ab in a point, 0, b representing 0.42 of a b. Of course, the sum of the com- pressive stresses in the concrete in compression a O, on one side, and the sum of the tensile stresses 1 in the concrete in tension O b and in the reinforcing metal E, on the other side, are equal. The computation also shows that if the beam is cracked the neutral axis occupies a position 0' so that 0' b is equal to 0.44 of ab. The sum of the compressive stresses cor- responding to the area of the new triangle O' a A is some- what smaller than the sum of the stresses corresponding to the initial triangle O a A ; if the distance between the re- sultant of the compressive and of the tensile stresses is 4 50 REINFORCED CONCRETE. increased sufficiently, the bending moment remains the same in spite of the cracks. In prisms with cracks the lever arm is the distance between the center of the metal and the center of gravity of the triangle O' a A. Tn prisms without cracks it is only the tensile stress taken up by the metal which acts at F, the remaining stress being taken up by the concrete in tension with resultant acting at C, near the middle of O b. The average lever arm of the components of the bending moment is, there- •fore, appreciably greater in a beam with cracks than in one without. Quite different is the result if the percentage of rein- forcing metal is below the balancing or best percentage. As the percentage falls from 2 to 1 the loss in resistance caused by the cracks increases from 12 to 23 per cent. Similar results are found for beams made of a concrete very rich in cement and reinforced by high steel, to which the last five columns of Table XII refer. If, in- stead of employing the algebraic formulas which are ap- proximate only, the exact curve of deformation were used, the figures obtained would not be perfectly identi- cal; but the above conclusions would not be modified. The conclusion is thus reached that the cracks in the concrete in tension materially decrease the resistance of beams reinforced by a low percentage of metal, but exert almost no influence on the resistance of beams in which the percentage of metal is appreciably higher than that giving the most economic beam. The decrease in resist- ance caused by the cracks does not exceed 11 to 17 per cent, if the most economic percentage is chosen. This offers a considerable guarantee for safety, at least so far as structures subjected to permanent loads are concerned. Experience only will teach whether cracks are also of so little danger for beams subjected to repeated loads, and whether the disintegration will extend more and more. Another consequence of the preceding must be noted REINFORCED CONCRETE IN BENDING. 51 here. When -in reinforced beams the total breaking of the concrete in tension .cannot cause a sensible decrease in their resistance,- it seems! that* light. cracks-produced by any cause whatever cannot extend as long as the elastic limit of the metal has not been reached. The author has a small prism cut from prism No. 34 at the place where a slight crack was observed after the application of a moment of 6,830 inch-pounds. This crack did not extend any farther after 139,052 repetitions of stress. This re- sult may serve as a good indication as to the effect of repeated loads. 15. The Influence of the Variations of the Co- efficient of Elasticity of ;.Concrete on the Resistance of Reinforced Beams. It is well known that the properties, and especially the coefficient of elasticity, of mortar and concrete vary with the quantity of water used and the efficiency and dura- tion of the tamping. It is important to determine the consequences which result from this in concrete-steel con- struction. For beams of a concrete containing a small proportion of cement, reinforced by 1 per cent, of metal, for example, the ratio of the coefficient of elasticity of the concrete to that of the steel, K, can in practice reach the value of 0.11 instead of 0.07, as was assumed for it in Table IV. By adopting this value it is found that the breaking moment will then increase to 22.4 instead of 18.6 foot-pounds. The variation will thus not be more than 17 per cent, of the resistance, while the value of the coefficient of elasticity has increased 60 per cent. Similar results are obtained for beams strongly rein- forced. Thus, for beams having 3 per cent, of metal, an increase in the value of the coefficient of elasticity of 60 per cent, causes an increase in the breaking moment of less than 17 per cent. Decreases in the value of K, 52 REINFORCED CONCRETE. of course, cause similar effects in the opposite direction. The influence of the variations of the coefficient of elas- ticity of the concrete on the resistance of reinforced beams is thus quite slight, and this fact is explained by the displacement of the neutral axis. Reinforced concrete possesses thus the two following properties of great practical importance. When the pro- portion of reinforcing metal is sufficient, cracks exert a slight influence on its resistance to bending and they show no tendency to grow. Again, the coefficient of elasticity of the concrete can vary within wide limits with- out causing a proportional change in the resistance of the reinforced beams. This change is somewhat below one- third of the variation of the coefficient of elasticity. Thus, as far as the bending resistance is concerned, which is the only resistance so far considered, the consequences of poor workmanship are less dangerous than might have been expected. As has been indicated before, a factor of safety of 2.5 will be sufficient. 16. Symmetrical Reinforcing. The idea of reinforcing concrete with iron was inspired by its low tensile resistance, which is, on the average, not more than 1/10 to 1/12 of its compressive resistance. Recently it has occurred to many engineers that it can be shown that the widely-extended practice of reinforcing beams assymmetrically, that is, placing the reinforcing metal in the tension side only, is not rational and that the highest resistance and economy will be obtained by symmetrical reinforcing. The reasoning is strictly cor- rect in the logic of its deductions, but it assumes at the start that the coefficient of elasticity of concrete is the same for tension as for compression; and that this holds true not only for the very small deformations, for which this assumption is correct, but also for the much greater REINFORCED CONCRETE IN BENDING. 53 elongations which take place in reinforced beams, and even up to the ultimate strength, which is of very great importance, because of the elimination of a given quantity expressed by the factor of -safety. Not only does the coefficient of elasticity of concrete decrease considerably as soon as the elongation exceeds 0.01 to 0.015 of 1 per cent., but it soon becomes almost zero; and the tensile resistance remains constant, while that of the concrete in compression continues to increase rapidly. It is, there- fore, quite difficult to admit that it is of advantage to reinforce concrete beams symmetrically where they are not subjected to reversals of bending moments, which is the only kind of beams treated here. However, if it is not rational to reinforce symmetrically a material the properties of which are essentially unsymmetrical, it does not follow that there is never any advantage in reinforc- ing the compressed side of concrete beams by a less sec- tion of metal than that used for the tension side. For example, let us find the result which will be ob- tained in reinforcing in this manner the prism reinforced by 2 per cent, of metal, to which the second line of Table IV refers. If two symmetrical reinforcing sets, each having a section equal to one-hundredth of the cross- sectional area of the beam, be added to its previous re- inforcing, the breaking moment will be increased by 14.4 foot-pounds, and will thus become 45.5 foot-pounds, an increase of about 46 per cent. The cost per cubic yard will be increased from $13.60 to $22.20, or by about 48 per cent. This would give a beam reinforced in its ten- sion side by 3 per cent, and in its compression side by 1 per cent, of metal. Referring to the different tables, it will be seen that better results can be obtained at less cost by increasing the proportion of cement at the same time with the percentage of the reinforcing metal. 54 REINFORCED CONCRETE. 17. Conclusions. We have observed how the quality of concrete can. vary with the care given to its fabrication, and, therefore, understand that even in experiments made at laboratories for mortars considered identical, values of the coefficient, of elasticity have been found varying from 1,400,000 to 5,700,000 pounds, and even more. In the author's first experiments, where the making of the test specimens was not sufficiently watched, he found coefficients which ap- proached the one or the other of these extreme values. It seems that such variations must also be met in practice. With such widely differing results it is impossible to give figures which should hold true for all cases, and the author, therefore, does not attribute an absolute value to any of the figures given in this treatise, except, of course, those resulting directly from each . experiment, in so far as they apply to the prisms for which they were obtained. On the other side, the figures given in the different tables are deduced from algebraic formulas, which are not absolutely exact because in their deduction the two parts of the curve of deformation were replaced by straight lines. The error so committed is not appreciable except for the value of the compressive stress in the con- crete, which is actually noticeably lower than the one given by the computation. The figures obtained in the above have no value if they are considered separately; but since they have all been deduced by the same method from identical or concurring data, their relations are of importance, and the laws which appear to result from these relations well deserve attention. They are not pre- sented here, however, as being finally established; and it is desirable that numerous experiments be made to enable engineers to accurately establish the laws of be- havior of reinforced concrete. It is with the above re- REINFORCED CONCRETE IN BENDING. 55 strictions in mind that the ^following conclusions are drawn. Concrete sufficiently reinforced by iron or steel can, ■without cracking or disintegrating, sustain elongations much higher than those observed in the ordinary tension tests. The reinforcing metal cannot insure the uniform elongation of the concrete when the elastic limit of the metal has been reached. When the elongation of the reinforced concrete ex- ceeds the ordinary elongation due to tension, the tensile stress increases more and more slowly and the coefficient of elasticity consequently decreases more and more rap- idly. The tensile stress becomes almost constant, and the coefficient of elasticity is very low from the point where the elongation reaches the greatest value due to bending in unreinforced concrete, which is 2 to 2.5 times as much as the elongation due to direct tension. When a beam which has sustained a high bending mo- ment is gradually unloaded the coefficient of elasticity of the concrete in tension has at the beginning quite a high value, though much below the initial coefficient of elasticity; then it decreases to a very low value. If concrete be subjected to the repetition of stresses below the greatest stress which it has once resisted, it sustains so much the greater deformations and undergoes so much the smaller changes, the more the greatest stress has ex- ceeded the present stress. The curves of deformation of tests, the results of which are known to the author, agree with these facts and the explanations given above. It thus appears to be logical to eliminate the hypothesis of an increase in the coeffi- cient of the metal embedded in the concrete. The curve of deformation of a concrete, plotted from the results of experiments, enables one to determine graphically all the stresses which are developed in rein- forced beam made of the same concrete, when it is sub- 56 . REINFORCED CONCRETE. jected to simple bending moment. Algebraic equations easily furnish a sufficiently accurate solution. For a given quality of concrete and metal it is thus easy to compute the most economical percentage of metal ■which is characterized by the fact that the metal and the concrete in compressing both reach their allowable limits at the same time. The allowable limit of stress is doubtless for the con- crete in compression, as well as for all materials which have been investigated, so much lower, the more fre- quently the stresses vary and the wider their range. The typical percentage of reinforcement for a beam is the higher the higher the resistance of the concrete and the lower the coefficient of elasticity, and especially the elastic limit. For beams subjected to repeated loads, the percent- age of the metal must be reduced or, what is better, the proportion of cement increased. The substitution of steel of the quality of rail-steel for wrought iron, with a section reduced in inverse propor- tion to the cost, seems to have advantages only when the reinforcing metal will not undergo deformation before being put in place. The use of rail-steel stressed in pro- portion to its higher elastic limit is dangerous where stiffness is required, and of advantage where ductility and resistance to impact is required. When the breaking moments have been computed the factor of safety must be determined. It appears that in accordance with the results obtained from existing struc- tures the factor of safety can be fixed at 2.5. The transverse rupture of the portion in tension de- creases materially the resistance of weakly-reinforced beams, but it has little effect upon beams which have a reinforcing as high as the typical percentage. It has no influence at all on beams reinforced by a still higher percentage. The cracks thus have no tendency to extend when the reinforcing is sufficiently strong. REINFORCED CONCRETE IN BENDING. , 57 The variations of the coefficient of elasticity of the con- crete exert only a relatively small influence on the resist- ance of reinforced beams. Because of these two facts, had ■workmanship in the construction of concrete-steel beams is s less to be feared than might have been' supposed, as far as their resistance to simple bending moments, with- out shearing stresses, is concerned. Symmetrical reinforcing is not to be recommended for beams which will surely always bend in one direction, be- cause the coefficient of elasticity and the resistance of the concrete are essentially unsymmetrical in bending. Some advantages may be obtained by adding symmetrical rein- forcing to the unsymmetrical reinforcing required to obtain equilibrium by balancing the difference in the re- sistance of concrete in tension and in bending; but it seems that the same result will be obtained, at less cost, by increasing the proportion of cement and the section of the reinforced metal in tension. CHAPTER II. The Deformation and Testing of Reinforced Concrete Beams. 1. The Necessity foe Dibect Tests of Reinfobced CoNCBETE StBTJCTUBES. The engineer who is designing a steel structure speci- fies that tests shall be made at the shops which will give a clear indication of the character of the materials used. These tests refer to the ultimate strength, elastic limit, ultimate elongation and reduction of area. He also in- spects the construction of his structure, and bad workman- ship is getting more and more rare. If poor work is sometimes done it can be discovered by careful inspection, and when the structure is tested on completion nothing unexpected will take place, if its type and design are based on practical experience. Thus the deflections which steel structures show under their test loads are found to be almost identical with those computed for them and their determination is, therefore, not of great value. The measuring and observation of the local deforma- tions, on the contrary, furnish valuable information on the distribution of stresses, and enable the engineer to appreciate the advantages and disadvantages of the vari- ous types of construction ; but it is very seldom that they disclose faulty construction or bad material. It can thus be said that for steel or iron structures the preliminary tests of the materials used and the inspection of construc- tion and erection furnish all the necessary assurances. Quite different is the case with concrete-steel structures, because laboratory tests tell us only of the quality of the materials employed, and the most active inspection will not be able to prevent positively poor workmanship and [58] DEFORMATION AND TESTING OF BEAMS. 59 faulty construction -which can , destroy^the .' streirgth of structures made of the best materials. In fact, the proportions of the concrete may, in spite of careful watching, not be in all parts in accordance with the specifications; the quantity of water used in mixing must, in order to produce identical results, vary within a wide range, according to the condition of moisture in the materials and the atmosphere, and it is quite sure that it will be sometimes badly proportioned. If too much water be added the strength of the concrete, and especially its coefficient of elasticity, will be decreased to a degree which may be considerable; if too little water be added the adhesion of the concrete to the reinforcing metal will not be sufficient. The thoroughness of the tamping has a still greater influence on the strength of the work. To the faults of execution, faults of design may be added. The latter must especially be guarded against in a new type of construction, the theory of which is not yet fully established. 2. The Insufficiency of the Assumptions Made foe the Computation of Deformations. Whatever the results of the tests of the materials may be, very little information on the strength of a concrete- steel structure can be obtained without direct tests of the structure itself. But observed deformations will furnish really useful indications only when compared to the nor- mal deformations which, according to the computations, should have been caused in accordance with the quality and disposition of the materials. Hitherto no method has been established to enable the engineer to compute these normal deformations. However, at one time it was as- sumed that reinforced concrete, whatever its deformation, preserves the coefficient of elasticity as determined in or- dinary tension tests, and at another that the resistance of (50 REINFORCED CONCRETE. the concrete in tension can be neglected in reinforced members. Sometimes it has also been assumed that the resistance of concrete in tension can be neglected only when its deformations exceed the elongation which causes rupture in common tension tests. None of these assump- tions has given results agreeing with the actual behavior of reinforced concrete. We may conclude that in rein- forced concrete construction certainly some particular phenomena occur, a knowledge of which is necessary to predict their resistance and deformations. ■ 3. The Law of Deformation of Concrete in Com- pression. Concrete in compression is generally not reinforced and it cannot be expected that the phenomena mentioned in the preceding section will be found to be caused by it. It suffices to recall the well-known law of deformation of concrete in compression and to make it more precise by naming the ratio of an infinitely small variation of com- pression to the variation of length caused by it, the " in- stantaneous coefficient of elasticity." When the compression increases, but remains within an amount, which, in general, is nearly a third of the ulti- mate strength, the instantaneous coefficient of elasticity decreases, but in a very small degree. When the stress increases still more the change in the elasticity gradually increases; it is appreciable for a compressive stress near one-half the value of the ultimate, and it then increases so rapidly before failure that the instantaneous coefficient may fall below one-twentieth of its value under a light stress. This aspect of the wdef formation is similar to the one generally observed on all materials. The simple law which determines the deformation of concrete in com- pression thus cannot furnish the explanation of observed irregularities, and it must be looked for in the phenomena which are produced in tension. DEFORMATION AND TESTING OF BEAMS. CI 4. The Law of Defoemation of Coetceete tjndee a Fiest Tension Load. Engineers who have studied the subject have encoun- tered the two apparently contradictory facts that mortar and concrete of the usual proportions break in tension with elongations below one-ten-thousandth of the length, and that in reinforced concrete beams which have sus- tained on their tension sides elongations up to one- thousandth of the measured length, no cracks are ob- served. It was, however, not known whether the portions which apparently are not injured do not contain invisible cracks which are prevented from becoming visible by the reinforcing metal, and whether they have not been altered : in some other way, so as to reduce their resistance. Of the experiments which the author undertook to in- vestigate this subject, those on reinforced concrete prisms in bending have been described and discussed in Chap- ter I. Experiments on reinforced concrete prisms in simple tension were made afterward, and their results are given in Chapter III. It would require a long and intricate analysis to show how the laws of simple tension can be deduced from the complex phenomena of flexure. A few words will, on the contrary, suffice to explain the method followed to establish these laws by means of the results of simple tension tests. IVIortar prisms reinforced by symmetrical reinforcing members were subjected to tension acting in the line and direction of their gravity axis. For each load were meas- ured, first, the elongations of the reinforcing members, and, second, those of the sides of the prism. The latter were found to be practically equal to the elongations of the reinforcing members, except at their ends. To 1 an elongation X, of the reinforcing members, having a sec- tion a and a coefficient of elasticity E R , corresponds a. stress in the metal equal to A aE B . If the total load on 6 2 REINFORCED CONCRETE. the prism which has caused these deformations is denoted by Pythe total -stress in the. mortar was P — X a E^ and its ■ J p X n W tensile stress per unit of area will be j- ?' A be- ing the cross-sectional area of the mortar. 5. The Effect of Bepeated Stbesseb. Knowledge of the curve of deformation correspond- ing to a first load will be sufficient for computing the de- formations for the usual tests of reinforced concrete structures, which include a single loading only; but it is important also to know how these structures will be de- formed under repeated test loads. In Fig. 8 the curves Fig. 8. represent the law of deformation of the mortar in a rein- forced mortar prism subjected to repeated and increasing loads, alternating with returns to equilibrium. If the load is gradually decreased after first having continued the test to the point B, the curve B C, which represents the deformation during the period of unloading, is almost a straight line except at its ends. The permanent de- formation, which remains after complete unloading, is indicated by O C. If the prism is loaded again to E and again unloaded, the curves CDE and E E G will be suc- cessively obtained. These curves, like B C, are almost straight and their inclinations to the horizontal are so much smaller as their deformation has been extended further. DEFORMATION AND TESTING OF BEAMS. 63 The precise notion of the instantaneous coefficient of elasticity; that is, of the inclination to the horizontal of the tangent to the curve of deformation, enables us to summarize briefly the laws of the deformation of mortar or concrete in tension in reinforced concrete members. These laws are as follows: 1. During the first loading the instantaneous coefficient of elasticity is equal to that of unreinforced mortar, as long as the elongation does not exceed the amount which will cause rupture in the same mortar when not rein- forced, and it decreases very rapidly as soon as this amount is exceeded, approaching zero. 2. When a reinforced concrete member is subjected to a successive series of unloadings and reloadings the in- stantaneous coefficient of elasticity has, during any of these operations, an almost constant value, and this value is the smaller the greater the maximum deformation has been. 3. The repetition of the same stresses causes a decrease in the coefficient of elasticity which is at first appreciable, but approaches zero. 6. The Effect of Veby Small Graces on Steess and Deformation. These laws inform us what a perfect mortar or con- crete can do and not what will be certainly obtained from materials as they are used practically. Reinforced con- crete members sometimes contain cracks, and this must be taken into account in computing their proper dimensions. It is an established fact that in well-built reinforced con- crete structures, which are not exposed to an excessive drying out, only very seldom are cracks visible under a magnifying glass produced before or during the usual tests. Furthermore, • laboratory experiments prove that the cracks, caused by sufficiently increasing the load above 64 REINFORCED CONCRETE. the generally accepted test loads, have no appreciable in- fluence on the deformation as long as they cannot be seen with the naked eye. This becomes evident when it is considered that a crack influences only a very small frac- tion of the length of the tested member. It must, there- fore, be concluded that the deformations caused during the test loads generally used for structures of this type are not influenced to any appreciable degree by the in- visible cracks which may have existed before the test or have been produced during the same. 7. The Exact Method for the Determination of the Deformations . The law of deformation of the reinforcing metal and the concrete of a concrete-steel member gives all the elements necessary for computing the deformations under any load sufficiently moderate not to cause wide cracks, and especially under the test load. The comparison of the actual and computed deformations under the test loads will furnish valuable information not only on the resist- ance of the member tested, but also on the nature of the danger most likely to threaten it. This problem of com- puting the deformation for a given stress can be solved only by successive trials to find the position of the neutral axis, and such an inclination of the line A' B', of Fig. 4, that the resulting stresses will satisfy the double condition that the sums of the tensile and compressive stresses shall be equal, and that the moments produced by the stresses shall be equal, during the test. 8. The Approximate Method of Computation of the Deformation. It is a waste of precision to attempt absolute accuracy in computations based on the physical properties of con- crete. * DEFORMATION AND TESTING OF BEAMS. * C5 It is, therefore, proper to substitute for the deforma- tion curve Q P O M, in Fig. 4, the polygon. E.OII, -which is very little different, but allows the introduction of ■simple equations in the usual case of concrete beams, re- inforced on the tension side only. In Chapter I, section 5, the polygon EOHM was taken, including the small tri- Fig. 4. angle H L, which was sufficiently accurate for the pur- poses there intended, but for the computation of the de- formations the more accurate polygon EOLM will be used. Using the same notation as before and denoting by s the tensile stress in the reinforcing metal in pounds per .square inch, the following equations are obtained: 5 CO REINFORCED CONCRETE. From the assumption of the conservation of planes after flexure, it follows that the tensile stress s in- the reinforcing metal is to the compressive stress in the ex- treme portions of the concrete c as the products of the distances from the neutral axis by the coefficients of elasticity of the two materials. Hence: c = Ksx^ r (7> os — u The condition that the sum of the tensile stresses in the metal and concrete must equal the sum of the compressive stresses in the concrete furnishes the following equation: to - £- (1 - 0) + sp= J 1 -x ( 8 > It G £ K The moment of resistance of the reinforced beam which is. produced by the tensile stresses in the reinforcing metal and concrete in tension and by the compressive stresses* of the concrete in compression is: , M = e W V-f- ^ {l-xf+sp (x-u)- °- (1-x)*] ■ • . ■ (9> The tests which will generally be made on the metal and concrete give the values of t and K; only three un- knowns thus remain to be evaluated x, s, and c. The three equations (7), (8), and (9) furnish the means for determining their values corresponding to each value of the test moment M. The form of the equations is such that their direct solution will be difficult, and the same practical result can be obtained in the following manner. From equations (7) and (8) eliminate s, then: '-& 1 -»+5rT^i-s a -o (10> This equation contains three variables c, K, and x. If the' value of the compression c in the concrete is taken arbi- trarily, the. curve of deformation of the concrete used in the structure will give the value K, which corresponds to this compression. Introducing the values of c and K in. equation (10), the corresponding value of x will be found- DEFORMATION AND TESTING OF BEAMS. 67 Substituting the numerical values of c, K, x thus found in equations (7) and (9) the corresponding values of s and M are obtained. By giving c a certain -number of successive values, con- veniently chosen between the limits which may be caused under the test load, a series of values of c, s, and M will be obtained, each of which will satisfy the equations and will consequently characterize one of the possible posi- tions of equilibrium of the beam in question.* If the ulti- mate strength of the beam is to be determined, the values obtained are just the elements required; however, it is not the deformations which will be caused by s and c which are looked for, but the deformations in the beam under normal conditions when tested by the usual trial load. The elongation of the metal X is equal to -nr and the shortening of the most compressed portion of the con- Si crete r is equal to -pr where E s and E s are the coefficients of elasticity of the metal and concrete. The coefficient of elasticity of steel is known; that of the concrete for each value of c is equal to the E s multiplied by the value of K corresponding to that of c, previously determined for equation (10). To apply the results to all beams hav- ing the same proportion of metal p, whatever their width e, and their depth h, the total moments M must be reduced M to the unit moments m= -^ tliat is > tbe resisting moments for a beam of unit width and depth. The values M, s, c which will be obtained will enable the designer to compute m, X, r, each of which will be realized in the tested beam when the test load produces the absolute moment M, and the moment m referred to a section a unit deep and a unit wide. By numerical in- terpolation the deformations ;, r, which will be caused under the action of any unit moment m, whose value lies 68 REINFORCED CONCRETE. between the extreme values, can be easily determined. Still more easily will the same result be obtained by con- structing two curves having for abscissas values of m "and for ordinates the corresponding values of X in one of them and r in the other. Such curves are shown in Pigs. 9 and 10. IK. 9. — CURVE OF ELONGATION OF REINFORCING METAL. FIG. 10. — CURVE OF GREATEST SHORTENINGS OF CONCRETE. The curves A C represent the elongations computed by formulas (7) to (10), and the lines O E, O D, O H I, the results of the three assumptions made by different writers, which will be considered below. The full line O B repre- sents the actual deformations of prism No. 34. deformation and testing of beams. c? 9. The Complete Cukves of Deformation. If the assumption for the deduction of equations (7) to (10) be remembered, it is seen that these are not appli- cable before the elasticity in the extreme portion of the concrete in tension has changed its character. For loads- causing stresses below the above assumption the coeffi- cient of elasticity preserves its normal value, and it is very easy to compute the deformations by the usual formulas. But care must be taken in computing the moments of inertia to account for the difference in the initial coeffi- cients of elasticity of the metal and concrete. Complet- ing thus, for light loads, the curves given by equations ( 7 ) to (10) for the period when the elasticity of the concrete in tension is changed, the dot-and-dash lines A C and O' A' C, Figs. 9 and 10, are obtained. These lines will necessarily show up any irregularity, as E L ¥, in Fig. 4, which has served as a basis for the deduction of the- formulas. It is evident that the real condition will be still more nearly approached in joining by curves the two sides of the angle formed by O A C, as the curve of de- formation of the concrete in tension joins both lines E L and L M. Finally the two complete curves O A C and 'A' C will be obtained. The former represents the elongations of the reinforcing members, and the latter the greatest short- enings of the concrete which will be caused in a beam reinforced by the proportion of metal p when subjected to a unit bending moment; that is, to a total bending moment M — m e h 2 , e being its width and h its depth. Similar curves may be constructed for a number of values of the proportion of metal p, and all necessary elements, will then be at hand to determine at once, by numerical or graphical interpolation, the deformations of any beam of reinforced concrete which has the characteristic quali- ties E s , K, and t for which the computations have been made. 70 REINFORCED CONCRETE. 10. The Comparison of the Actual Deformations with the Computed. The method just given should be checked experi- mentally to determine its reliability. Prism No. 34 will again be taken for comparison. The deformations ob- served during the test are shown in full lines in Figs. 9 and 10, and the deformations computed by formulas (7) to (10) in dot-and-dash lines. In each diagram the two curves are almost identical, and it is natural to question whether this remarkable agreement does not result from the fact that the values of K and t introduced in the formulas have been determined by a method which should lead to this agreement. Such is not the case. E s was determined by a tension test on a wire identical with the reinforcing wires. The experiments gave the position of the neutral axis, and, consequently, for each load the area of the compressed portion of the prism, thus permitting the computation of the compressive stresses in the mortar within limits of accuracy which would still be very narrow if even appreciable errors were committed in the evaluation of the tensile stresses t in the concrete. These values of the compressive stresses once being known, the values of K were determined by the corre- sponding values of the shortenings of the concrete, which were measured directly. The tension t, maintained al- most without variation from the time the elastic limit was exceeded, was determined by direct experiments made on small prisms detached from the reinforcing, as described. Hence for prism 34 formulas (7) to (10) have served to compute the deformations with a very satisfactory ac- curacy by introducing in them the real characteristics of the mortar and iron employed in its construction. DEFORMATION AND TESTING OF BEAMS. 71 11. Results of Assumptions Made by Diffeeent Authoes. To the author's knowledge only three assumptions have been made in attempting to explain the resistance and de- formation of reinforced concrete beams. The first as- sumption, that the concrete preserves its coefficient of elasticity throughout the test, no matter how improbable it is, forms the necessary basis for certain methods adopted by some engineers, who assume that in the computations the reinforcing metal can be replaced by an equivalent area of concrete in the ratio of the initial coefficients of elasticity of the two materials. This assumption gives for the computed- deformations the ordinates of the two straight lines E, O' E' of Figs. 9 and 10. They differ much from the actual deformations, except under light loads. The second and directly opposite assumption, that the concrete in tension has no effect on the resistance, has been made up to the present time by the majority of prac- titioners, and the author has explained in Chapter I, how it happened that this assumption, though inaccurate, has given acceptable results, in the computation of dimensions of reinforced concrete beams. Quite different is the case of the deformations. The values on this assumption are obtained by means of equations (7) to (10), by making £ = 0. They are represented by the ordinates of the lines O D and O' D', Eigs. 9 and 10. The third assumption, apparently the most reasonable before the experiments described in Chapter I were made, that the concrete remains perfectly elastic while its elonga- tion does not exceed the value causing rupture in simple tension tests of unreinf orced concrete — ■ which is true — and — which is not true — that it breaks as soon as this limit is exceeded, requires that in a beam subjected to an increasing bending moment cracks start in the extreme portions when the elongations become equal to breaking elongations, and as the load increases, that the cracks grow 72 REINFORCED CONCRETE. progressively until approaching very near to the neutral, axis. If the beam would actually behave so, the defor- mations would be represented during the elastic period by the ordinates of the lines O E and O' E' ; they would then rapidly increase, owing to the rupture of the portion in tension, the resistance of which would approach zero, and they would then be represented by the lines OHID and. 0' H' I' D' forming asymptotes to the lines O D and O' D*. Though it is interesting from a theoretical point of view to see how the different assumptions agree with the facts observed imtil rupture, it is especially useful to study the differences shown within the limits of loads which may actually be used for testing purposes. These limits, are drawn on the diagrams and the differences shown, within these limits, between the computed deformations and those observed are given in the following table. These figures lead to several conclusions : TABLE XIII. § Difference between actual deformations and results of formulas (7) to (10), per cent Difference between actual deformations and results, first assumption, per cent Difference between actual deformations and results, second assumption, per cent Difference between actual, deformations and results, third assumption, per cent 5 45 110 i 100 o 50 25 20 It is seen at a glance that all the. assumptions previously made have predicted deformations very different from the DEFORMATION AND TESTING OF BEAMS. 73 actual, while a very satisfactory approximation is obtained by the use of formulas (7) to (10). Within practical limits, the third assumption gives almost as poor results as the second, owing to the fact that it is true only during the period of perfect elastic behavior, or tfor loads not ex- ceeding the same, while practical tests require much heavier loads. 12. Effect of Percentage of Iron and Quality of Concrete. The errors resulting from the second assumption are due to the behavior of the concrete in tension, and must, therefore, increase with the relative importance of the concrete to the reinforcing. Prism Wo. 34 was in con- ditions favorable to this. It should be noted that the per- centage of reinforcing metal of 1.12 per cent, is smaller than the percentages generally adopted, and that, there- fore, the relative importance of the mortar is greater, its importance being increased also by its quality. It was made very carefully and showed a tensile resistance of about 300 pounds per square inch, at least 30 per cent, higher than sustained by concretes generally employed in building construction. For ordinary structures which set in air, the second assumption must, for these reasons, lead to smaller errors than are shown in the comparison with the results on prism 34. On the other hand, if for hydraulic works rich concrete is used, a still higher re- sistance will be obtained for the concrete than that of the mortar of prism 34, and the difference between the re- sults of the different formulas will be increased. 13. Effect of Longitudinal Sliding of the Kein- forcing Metal. The experiments on prism !No. 34 differ from most practical tests, also in that the force causing the bending 74 REINFORCED CONCRETE. moment was applied near the ends, and, hence, there was no shearing stress in the central portion where the deforma- tions were observed. The shearing stress* causes the sliding of the reinforcing bars in the concrete and increases the deformations, but in what degree cannot as yet be stated. New experiments will be required to determine the neces- sary corrections for sections subjected to shearing stresses in order to apply the formulas established on the assump- tion of conservation of planes and, hence, on that of the absence of shearing stress. 14. Effect of Sueeoundings in which Concrete is Placed. Prism ISTo. 34 was kept in water 210 days and was taken out only three days before the test, while structures built in the open air are tested after an almost complete drying out. To explain the important differences for the deformations which must result from the different sur- roundings, it is necessary to state briefly the effect of the hardening of concrete in air and water. It is well known that according as mortar or concrete has set in the open air or in water they shrink or swell. These changes of volume reach 0.2 per cent, for blocks made of pure Port- land cement and decrease with the proportion of cement to sand and gravel, but without falling appreciably below 0.04 per cent, for the concrete poorest in cement generally employed on construction. The metal reinforcing embedded in mortar or concrete tends to preserve its initial length, but the surrounding material adheres to it. A complex state of equilibrium is established in which the longitudinal sliding exerts a great influence, and compression results in the reinforc- ing bars and tension in the concrete, if it has been kept in the air and, on the contrary, tension takes place in the reinforcing bars and compression in the concrete, if it DEFORMATION AND TESTING OF BEAMS. 75 has been kept under water. It is in order to determine the effect which these stresses have on the deformation of re- inforced concrete beams. Let us consider two otherwise identical prisms one of which was kept in air and the other in water. Let us apply the same load to both prisms. As long as the elastic limit of the concrete be not exceeded, the deformations will be the same for the two prisms, be- cause, if even the absolute values of the stresses produced in the metal and concrete be different, their variations will be identical, and in the elastic period the deformations de- pend only on the variations of the stresses and not on their absolute values. When the elasticity of the concrete changes, the defor- mations of the two prisms will, on the contrary, be very different. To illustrate this more easily, we shall neglect the intermediate period and consider the period when the elongation of the concrete in tension exceeds at all points the elastic elongation, except in a practically negligable portion nearest the neutral axis. If, for simplicity's sake, the slight difference in the position of the neutral axis in the two prisms be temporarily neglected, with the intention of returning to it, the following reasoning will be true. Let M be the moment to which each of the two prisms is subjected, and p. the moment of resistance pro- " duced by the tensile stresses in the elongated portion of the concrete and an equal compressed portion, which com- pletes the resisting couple. Then p. will be of the same value in both prisms, because the elastic elongation being, by assumption, exceeded, the portion in tension will cause in one as in the other prism the same tensile stress t. The reinforcing metal in the two prisms must, for equilibrium, produce the same moment M — //, and consequently it must take the same absolute elongation ;. But in the prism kept in the air the reinforcing bars were compressed by the shrinking of the concrete and shortened by the amount r before the load was applied, while in the prism kept in 76 REINFORCED CONCRETE. water they had elongated by the amount d. Consequently to obtain the same absolute elongation k, the reinforcing bars of the prism kept in air must elongate by the amount k _j_ r, while those in the prism kept in water must elonr gate by k — d only. Generally the values of r and d are from 0.01 to 0.02 per cent, of the length of the member. Such is not exactly the case becaiise of the displacement of the neutral axis which shifts farther away from the re- inforcing bars the more these elongate. In prisms kept in air, an increase in the lever arm of the couple resisting bending is the result, and hence a decrease in the tensile stress required to prodtice the moment M and the cor- responding elongation of the reinforcing bars. The op- posite evidently takes place in prisms kept in water, so that really the variations of length which the test load causes in the reinforcing members are somewhat less than k -j- r for the former and somewhat more than k — d for the latter. But without injuring the beams by holes in the concrete the variations in length of the reinforcing bars cannot be measured, and the concrete only is measured. The tendency for longitudinal sliding, which the shear- ing force produces in the reinforcing bars, has a direction opposite to the one caused by the shrinking in beams kept in air, while in the prisms kept in water it has the same direction as that due to the swelling. This is a new cause of the relative increase in the deformations of concrete, and hence, of deflections of beams kept in air. It is easily understood from the preceding that transverse reinforcing bars which decrease the sliding tendency must hence reduce the deformations of structures of rein- forced concrete. It has been seen how many elements have an effect on the deformation of reinforced beams, and how numerous investigations must be made to determine exactly the role and importance of each. However, the conceptions ac- DEFORMATION AND TESTING OF BEAMS. 77 quired seem to be sufficient to obtain practical results, with the condition of making on each structure some tests of normally built members, under the supervision of the engineer, with the materials and form of reinforcing used in the structure. From the results of these tests the modifications will be deduced which must be applied to the figures given by the formulas, based on the assump- tions of the conservation of planes and the absence of initial internal stresses, in order that their results should agree with the actual deformations of the beams built of the same materials and with the same care as for the structures. 15. Informations Obtained from Tests. The chances for poor workmanship in reinforced con- crete construction are numerous, and poor sand or a low grade cement may reduce all the qualities of the concrete. The proportion of the water used exerts a considerable in- fluence on the coefficient of elasticity, which may fall to 2,100,000 pounds for materials from which 3,500,000 pounds can be obtained by using a proper proportion of water. The resistance also decreases but in a less pro- portion than the coefficient of elasticity, if there be an excess of water in the concrete. The tamping is of no less importance, and insufficient tamping seems, to reduce the resistance and the coefficient of elasticity' in pretty much the same degree. It may also happen that the proportion of cement pre- • scribed by the specifications is not observed. Should this occur the tensile and compressive resistances will be de- creased as well as the coefficient of elasticity, but from numerous experiments it is known that the coefficient of elasticity will be decreased in a smaller proportion. The study of the curves of deformation permits us to dis- tinguish the different modifications which characterize REINFORCED CONCRETE. the properties of concrete. In fact, during the perfectly- elastic period under light loads the deformations depend exclusively on the coefficient of elasticity. By comparing the inclinations to the horizontal of the computed and actually observed lines O H and O' H', in Figs. 9 and •n Fig. 11. Prism No. 2, 500 pounds of cement, one cubic yard sand and gravel equal parts. Excess of water. Seven months in air. Prism No. 6, 500 pounds of cement, one cubic yard sand and gravel equal parts. Insufficient water. Seven months in air. Prism No. 12, 500 pounds of cement, one cubic yard sand and gravel equal parts. Normal amount of water. Seven months in air. Prism No. 34, 720 pounds of cement, one cubic yard sand. Normal amount of water. Seven months in water. 10, which represent the elastic deformations, indications will be obtained as to the value of the coefficient of elasticity. As to the elastic limit, it is proportional to the abscissas of the points H and Ii', where the inclination of the curves changes almost suddenly. DEFORMATION AND TESTING OF BEAMS. 79 Tlie abscissas of the lines D and 0' D' represent, as has been shown, the deformations which would have taken place if the concrete could resist no tension. The portions of the abscissas between these lines and the curves of -observed deformations B and O' B' are thus almost pro- portional to the actual tensions, which are nearly identical with the breaking resistances in tension. From the curves of deformation and the variation of the position of the neutral axis, information can also be deduced as to the sliding of the reinforcing metal in the concrete, but the study of this question wo'uld lead too far from the present subject. It would involve still more time and space to investigate the precise effect which the causes modifying the deformation of reinforced concrete beams have on the stress and fatigue of each of the constituent elements. It will be sufficient for the present purposes to illus- trate the differences caused in the deformation of rein- forced concrete beams by reproducing the diagrams of the elongations of the concrete or mortar of the tension •side of four prisms. These prisms were identical in di- mensions and reinforcing with prism ISTo. 34, which was one of the four, that is, they had a square section of 2.36 inches on a side and were reinforced by three wires 0.17 inch in diameter. Three prisms, JSTos. 2, 6, and 12, con- tained 500 pounds of cement to the cubic yard of a mix- ture of sand and fine gravel, the proportions employed by M. Hennebique. ' The fourth prism, ~No. 34, contained 720 pounds of cement to the cubic yard of sand, which the author believed to be equivalent; but these proportions may possibly be somewhat too rich. The first three prisms differed only in the intentional excess or insuffi- ciency of water. The deformations were the smallest in the prisms made under the most normal conditions. The difference in deformation between the prism ~So. 34 and the Others is very great, and is in a measure due to the difference in the coefficients of elasticity of the mortar 80 REINFORCED CONCRETE. or concrete, but still more to the fact that the first three prisms were kept in the air and the last in water. These examples will demonstrate that the differences which practical load tests will show between reinforced concrete beams will be great enough not to require meas- uring instruments of ^reat precision. 16. The Deformations op Beams Symmetrically Ke- iktfoeced. In the preceding sections of this chapter beams rein- forced on the tension side only have been discussed, but beams reinforced also on the compression side are some- times used. Their deformations can be easily deter- mined by means of the diagrams which have been estab- lished for beams reinforced on one side only. Let AR in Fig. 12 be the cross-section of a beam reinforced by rods F on one side only. If this section be subjected to a unit bending moment m, the neutral axis will be in the position 0, which can be determined by means of equa- tions (7) to (10). K _-.., — S! i * «! »: t i A. i s S' FIG. 12. If to this beam a supplementary reinforcing area of steel S in tension, and one S' in compression 'be added, and if the distances from the neutral axis be denoted by u and u', the total stresses of opposite signs which will be produced in these new reinforcing rods will be equal to DEFORMATION AND TESTING OF BEAMS. 81 each other, provided the areas of the additional rein- forcing satisfy the conditions 8 u = S' u', because the stresses which the reinforcing will sustain per square inch will be proportional to the deformations, and, therefore, to the distances from the neutral axis, u and u'. The equi- librium of the beam with reinforcing on one side only under the moment m will thus not be disturbed, if at the same time with the addition of the new reinforcing rods, the moment of the external forces also be increased by an amount equal to the moment of resistance furnished by the tension in S and the compression in S'. The total value of these forces is easily determined, because the additional reinforcing metal in tension, being placed at the same distance from the neutral axis as the reinforcing steel of the initial beam, will sustain the same elongation and, consequently, also the same stress per square inch s as the steel in this initial beam. This stress s is given by the formulas (7) to (10) or by the tables or diagrams es- tablished by means of these formulas. The additional moment of resistance is hence : . S s (u + u')=S s d, where d is the distance between the reinforcing members. Consequently nothing is changed in the unit stresses in the deformations caused in any part of the steel or con- crete of a beam reinforced on one side only if to both sides reinforcing rods be added the sectional areas of which are to each other as u : u', and if the bending moment S s d be increased at the same time. The ratio u : v! is deduced directly from the position of the neutral axis, that is, from the value of x, which is given by the formulas (7) to (10). Thus, in order that the diagrams or tables prepared for beams reinforced on one side only should also furnish the necessary information for beams reinforced on both sides, with any reinforcing members, it suffices to add to them a concurring table. 6 82 REINFORCED CONCRETE. "Let r represent the value which the ratio u : u' assumes under the test load for a beam reinforced on one side only by the proportion of metal to concrete p. The deforma- tions computed for this beam will be exactly the same as for all the beams reinforced on both sides by additional re- inforcing members, the sectional areas of which will be in the ratio y, provided that they be subjected to the bend- ing test increased by 8 s d. Let us, for an example, assume that for beams rein- forced on one side by 2 per cent, of metal, the value of y for the test load will be 1.10. The deformations of these beams will then be the same as those of all beams in which the reinforcing rods in tension have an area of 0.02 +y, with a reinforcing in compression of 1.10 y, where y has any value whatever. Thus the same table will be ap- plicable for all beams, the double reinforcing of which will have cross-sectional areas given by the following fig- ures or interpolations between them: Keinforcing in tension, per cent., 2 2 5 3 3.5 4. Reinforcing in compression, percent., 0.55 1.1 1.65 2.2 It is needless to explain here in more detail the very simple methods of interpolation by means of which, once these concurring tables have been established for the dif- ferent percentages of single reinforcing, the corresponding tables for double reinforcing of any sectional area can be deduced. It should, however, be understood that the values of the unit moment of resistance will be replaced by m + S s d. It is hardly necessary to state that by means similar to those just indicated for the computation of the deformations of beams reinforced on both sides, the strength of these beams can also be computed. 17. The Computation of Deflections. It has been shown that tests based on the measuring of the elongations and shortenings which take place in cer- DEFORMATION AND TESTING OF BEAMS. 83 tain sections of beams in bending readily furnish iiseful information on the quality of the materials of which they are made, and it is probable that engineers who are pro- vided with the necessary instruments for the observations will make use of this method. The deflections of prisms loaded as beams are due to the deformations of an indefinite number of sections, and it is, from the beginning, evident that by observing this complicated phenomenon it will not be possible to obtain as easily such accurate results as by the study of the de- formation of the sections. However, since the measuring of the deflections is of almost general use and does not re- quire any special instruments, it is probable that it will continue to be used frequently, and it should here be men- tioned, though somewhat hastily. To compute the deflections yield to a large extent to the stresses which the reinforc- ing rods produce in it. Thus -reinforced members which have set in the open air remain of a greater length than that which they would have taken freely without restraint from the reinforcing. The crystallizations which take' place during the setting are between the artificially sep- arated molecules, and a decrease in density, elasticity, and resistance is the result. The opposite should be caused in reinforced members which have hardened in .water, but the author has not had the occasion to verify whether an improvement in the quality of the concrete actually takes place. The effects of the variations in the hygrometric condition on com- pletely hardened mortars are very different from those resulting from gradual hardening. There is a struggle be- CHANGES IN VOLUME OF CONCRETE. 93 tween the two associated materials, the coefficients of elas- ticity of which have arrived at their final values, and the difference between the length which each material tries to assume and the one it is compelled to take by the combi- nation is inversely proportional to its coefficient of elas- ticity. From these considerations it follows that account should be taken of the fact that the difference in the variations of length which take place in a mortar accord- ing to whether it is reinforced or not will be considerably greater during the beginning of the hardening than dur- ing the following hygrometric changes. Experience has confirmed this fact. .... It, also follows -that the variations in volume which the hygrometric changes tend to produce in hardened mortars are smaller than those produced dur- ing gradual hardening, since internal stresses, and espe- cially tensile stresses in the reinforcing rods, may be •caused, which attain 5,500 to 8,500 pounds per square inch. Contrary to what takes place during the slow hard- ening of mortar, the hygrometric variations appear to be the more dangerous the less rich in cement the mortar is, because its resistance is smaller while the internal stresses are, at least, as great. It should be added that these disadvantages are practi- cally of no account for masonry always exposed to the air because the changes in the proportions of moisture in the air have a very small effect. The question is only of im- portance for members made in the open air which begin to harden before being put in water where they finally remain submerged, as in the case of piles, caissons, etc. By keeping them moistened until put in place, not only will the cracks which are often caused, as has been shown by experience, be prevented, but also the changes in the internal stresses, which cannot be of any advantage. It would be both interesting and useful to determine experi- mentally the results to be obtained by keeping reinforced 94 REINFORCED CONCRETE. concrete members, to be permanently exposed to the air, as moist as possible by abundant and repeated sprinkling for several weeks. It is obvious that the final shrinking, as well as the disadvantages resulting therefrom, would be decreased. In concluding attention should be called to a fact worthy of research. Reinforced concrete members pre- viously subjected to test loads have shown very little ef- fect due to changes of the moisture in them. This will seem probable when it is remembered that the test load reduces the coefficient of elasticity of the concrete very considerably. CHAPTER IV. Tensile and Compressive Resistance of Reinforced Concrete. 1. The Tensile Resistance oe Reinforced Conceete. In. Chapter I the bending experiments made by the author on reinforced concrete prisms were fully discussed. Some objections were raised against them which had the semblance of being justified. In these experiments the deformations of the mortar were measured, and in order to determine from them those of the reinforcing metal the generally accepted assumption of the conservation of plane sections had to be made. The greatest care, how- ever, was taken to justify this assumption by limiting the observations to the central portion of the prisms where the shearing stresses, which are the ones to distort the plane sections, vanish. To avoid any uncertainty on this point — the only one open to dispute — new tension and bending experiments were subsequently made, in which the elongations of the reinforcing rods were, measured di- rectly and simultaneously with the deformations of the mortar. The tension experiments especially are simple and seem to be free of error. Prisms made of mortar 1.85 by 1.85 inches in cross-section, were symmetrically rein- forced by four wires of 0.17 inch diameter and subjected to a direct pull. For each load the elongation of the rein- forcing wires X and that of the mortar A i3 were measured. As shown in Chapter II, the stress per square . inch in both the wires and the mortar can be easily de- duced from these observations. In a similar manner the stresses in the bending experiments were determined. Except measuring the compressed fibres of the mortar, [95] 96 REINFORCED CONCRETE. the computations required for the determination of the position of the neutral axis were of a more complicated character and admitted only very small inaccuracies, which were, however, too small to affect in any way the conclusions resulting from the experiments. The laws established on the basis of the first series of experiments, Chapter I, have been fully confirmed, supplemented, and also extended to the effects of repeated loads. To illustrate easily and precisely the phenomena ob- served in the numerous and different tests made, the diagram shown in Fig. 15 was plotted for the simplest case, that of direct tension. The total loads applied suc- Fig. 15. cessively on the prism were plotted as ordinates and the corresponding elongations of the reinforcing rods as ab- scissas. The elongations of the mortar, also measured di- rectly, were found to be sensibly equal to those of the iron in the middle portion of the prisms subjected to simple tension, as well as in the portions of prisms in flexure which are sufficiently distant from the fixed ends and, therefore, free of shearing stresses. As long as the load did not exceed a certain value a, the elongations in- creased regularly and were very small, then the increase TENSILE RESISTANCE OF REINFORCED CON0RETE. 97 became almost suddenly much greater, but soon assumed a regular course, represented by the straight line AB. The load was limited to b, then gradually reduced to zero. The resulting shortenings are represented by the dotted line B C which is almost a straight line except at the ends. A series of new applications of load to b, and corre- sponding unloadings caused the deformations represented by C B', B' C, C B", B" C", etc., that is by curves more and more distant from the vertical axis b and less in- clined to the horizontal. The alternate loading and un- loading was continued until the final state was estab- lishedj and it was observed that the points B and 0' ap- proached more and more slowly the limits C n B n . It was also observed that in prisms, where there were no pre- liminary stresses caused by the gradual hardening in air or water, the final value O C n of the permanent elongation was always less than one-fifth and often less than one- tenth of the total elongation under the load b. To illustrate more precisely the other characteristics of the results obtained, it is necessary to plot the curve which represents the tensions taken by the reinforcing rods un- der different loads. This curve is evidently the straight line O F, the inclination of which to the horizontal is the product of the cross-section of the reinforcing rods by their •coefficient of elasticity. If M be any point on the curve of deformation, W P represents the tension' taken by the reinforcing rods when the reinforced prism sustained the total load M P, and consequently the tension resisted by the mortar was M !N~. The observations to which the sttidy of the results leads are the following: During the first period O A, where the elongation a A has not much exceeded the elongation which a prism of the same mortar, but not reinforced, will sustain without breaking, the tensile resistance of the reinforced mortar is the same as if not reinforced. The 7 98 , REINFORCED CONCRETE. two combined materials act as if independent and divide the stress proportionately to their coefficients of elasticity. Above this limit the elongations of the prism become much more rapid, the tension MNof the mortar increases only very slowly and hence its coefficient of elasticity becomes very small. It is especially important for practical purposes to de- termine clearly, for the case of repeated loads, the final value of the resistance B n Q which the mortar will sustain when subjected to an infinite number of repetitions of load without exceeding the limits O b and Q E. This is the dif- ference of the stress existing at the same time between the mortar and the reinforcing metal. In prisms which were kept in the air and which were subjected during the test to a maximum elongation b B n of 0.09 per cent, of the length the final value of the tension B n Q resisted by the mortar was equal to 70 per cent. . of the greatest tension M !N~ and remained somewhat higher than the tensile resistance of the. same concrete not reinforced. In prisms which were kept under water and then in the air for several days be- fore the test so as to eliminate the internal stresses due to* gradual hardening, the final tensile stress was higher than 70 per cent, of the greatest tensile stress. Considering the elastic behavior of the prisms, the phenomena which take place during the tension of rein- forced concrete lead to the conclusions which have already been formulated in Chapter I, for the tension portions of reinforced concrete in flexure. They hold true for both cases. After these conclusions it is certainly of import- ance to determine what would take place if, due to an aecident, a reinforced concrete member be subjected to a load greater than that of the repeated test. The law of the phenomenon which would take place is represented by the curve B n D E. It shows that in this case the concrete regains, after a certain increase in load, the greatest re- sistance which it had before the repeated applications of TENSILE RESISTANCE OF REINFORCED CONCRETE. 99 load. This is an important property which would prove valuable in some cases, as, for instance, in' that of an accident. These new and original tests throw some light on the behavior of reinforced -concrete when subjected to a single application of a high load or to a repetition of loads. It seems that these properties could not be estab- lished either by theoretical reasoning or by the study of the results of previous experiments. 2. Effect of Shrinking and Swelling on Defor- mations and Stresses. \ The laws established in Chapter I and in the preceding section are for reinforced concrete members which have been exposed to the air and water in such a manner as to reduce the internal stresses in them sufficiently to be neg- lected in the tests. In which direction the shrinking and swelling of the members will effect the stresses can be readily seen. It has been shown that as soon as the con- crete exceeds a certain elongation, which is easily ex- ceeded in all highly stressed reinforced structures, it of- fers almost a constant resistance to an increase in elonga- tion. It follows, therefrom, that if an external force is applied to a reinforced concrete member, whether it be exposed to the air or water, the reinforcing bars will in any case offer the same resistance to deformation as the concrete and will thus have the same absolute elongation. The relative elongation will, of course, be greater in struc- tures exposed to the air, as they were in initial compres- sion before the test, and hence were shortened below their true length, while in structures under water elongations already exist. Hence the deformations must, under equal conditions, be greater in structures exposed to the air than in structures kept under water before being tested. The effect of shrinking and "swelling on the deforma- tion of reinforced concrete structures is the greater the 100 REINFORCED CONCRETE. greater the forces acting in them. In fact, the shrinking in the air causes the portions of beams, in tension to elon- gate more than the portions in compression, and thus shifts the neutral axis toward the compression side, reducing its cross-sectional area. An increase in the unit stresses necessarily follows, since the concrete in compression tends to maintain equilibrium with the stresses in the reinforcing bars and in the cohci'ete in tension. Besides the shrinking of the concrete causes tensile stresses in the concrete surrounding the reinforcing metal •and compressive- stresses in- the- opposite portions. The shrinking thus has a double action to cause compressive stresses in portions opposite the reinforcing metal before the test and then to increase the compressive stresses caused by the test load. The effect of swelling is, of course, exactly the opposite. Experiments prove these conclusions. 3. COMPEESSIVE EeSISTAETCE OF CONCRETE. Generally the compressive resistance of the concrete is not increased by the imbedding of iron rods in the portions of reinforced concrete structures subjected to compressive stresses. It thus follows well-known laws to which the author's experiments have added nothing. However, the curve of deformation has been plotted for a prism in com- pression made of a mortar containing 1,000 pounds of cement to the cubic yard of sand, which was tested by the director of the laboratory of l'Ecole des Ponts et Chaussees after hardening 30 days in air. The curve, which is given in Fig. 16, shows the following important practical characteristics: The shortenings are at the beginning proportional to the pressures and the coefficient of elasticity, hence, re- mains almost constant up to a load of about two-thirds of the ultimate resistance. The deformations increase "with increased loads and attain relatively very high values. TENSILE RESISTANCE OF REINFORCED CONCRETE. 101 It will further be seen that this property has an im- portant effect on the ultimate resistance of concrete masonry in general and on reinforced concrete especially.. It is well known that mortar in thin joints can resist^ without crushing, much higher pressures than can be re- sisted by prisms of considerable length. M. Harel de la. Comprosaion. Fig. 16. Noe has correctly concluded from these facts and theoreti- cal investigations that the compressive resistance of con- crete will be increased by reinforcing it with bars or rods in a direction perpendicular to the direction of the com- pressive force. Experiments confirm this deduction, but they also bring out a fact which should not be overlooked. (See Chapter VII.) 102 REINFORCED CONCRETE. It has been seen that mortar and concrete have a ten- dency to reduce their volume when kept in the air. The reinforcing members prevent this contraction or shrink- ing the more effectively, the stronger and the more nu- merous they are and the more directions in which they spread. Hence it is probable that the stresses thus pro- duced during the gradual hardening of the concrete will cause stresses in it which will act in a sense opposite to the compressive stresses and have a favorable effect, as has been pointed out by M. Candlot. Comparative tests on two prisms, one of which was re- inforced as usual by longitudinal rods in the tension side and the other also by additional cross rods in the com- pressed portion, gave the following results. The coeffi- cient of elasticity of the portions in compression of the first prism was found to be twice as high as in the second. The volume of the reinforcing metal in the compressed portion was 1 per cent, of that of the mortar. These prisms, which suffered by the sliding of the longitudinal reinforcing rods, did not furnish any information as to the crushing resistance, but it is probable that they vary proportionally to- the coefficient of elasticity, as generally occurs in mortar of the same proportion which is subjected to shocks or various compressive forces. It would be of interest to carry out more numerous, systematic experiments on the subject, and to determine how much the reduction in quality, which the transverse reinforcing rods produce in the compressed concrete, is counteracted by the co-operating action of the resistance* of the reinforcing rods themselves. This co-operating ac- tion is only of importance when the elastic limit has been exceeded. It follows that a similar but less pronounced reduction in quality also takes place in the concrete in tension which is reinforced by longitudinal rods. In all of the author's experiments, more than fifty, this was proved. The neutral axis was, in spite of the metal re- TENSILE RESISTANCE OF REINFORCED CONCRETE. 103 inf orcing, always found to be at the middle of the rec- tangular cross-section of reinforced members kept in the open air. It thus appeared as if the concrete had the same coefficient of elasticity throughout its whole mass, while the neutral axis, which generally shifts away from the less resistant parts, really ought to have moved nearer to the reinforcing, the coefficient of elasticity of which is, on the average, ten times as great as that of the concrete. CHAPTER V. Resistance of Concrete to Shearing and Sliding. Though shearing and sliding show much similarity there are, however, essential differences between them.. In shearing rupture is caused by sliding of the sections- in directions which make an angle of 45° with the direction of the force, as has been clearly demonstrated by M. Mesnager. The reinforcing bars, on the contrary, which adhere strongly to the concrete, can be displaced only by sliding longitudinally, parallel to the longitudinal forces. 1. Shearing Stresses. The experiments smade by M. Mesnager tend to show that the resistance of mortar to shearing exceeds its ten- sile resistance as it is determined by the usual tests. Tests, which may be too few to allow of general conclu- sions, have shown a difference of 20 to 30 per cent, be- tween these two resistances. A fact recently observed has thrown some light on the ductibility of cement subjected to enormous shearing stresses at the same time with other complicated stresses, the compressive stresses much exceeding the tensile stresses. On a rock was placed a buoy made of a hollow iron pipe 1% inches in diameter and filled with neat cement. The waves bent this buoy to a radius of curva- ture of about 22 inches, measured on the axis of the buoy, and it was expected to find the cement all pulverized. Cutting the buoy along its bent axis it was found that the cement showed some sliding surface's only betw.een which solid pieces were found, the deformation of which indi- cated a sliding of the "fibres" on each other of at least 20 per cent, of their length. [104] RESISTANCE OF CONCRETE TO SLIDING. 105 It must be concluded that, the same as reinforced con- crete in tension, mortar or concrete can sustain enormous sliding when it is compressed, as it was in the interior of. the above-described buoy. This fact, which is of import- ance for the resistance of masonry in deep foundations- subjected to high pressures, does not seem to be so for the study of reinforced concrete construction. However, compared to the observations made on the sliding of the reinforcing members, to be described later, it will make them" more complete and lead to conclusions which are not without importance. 2. Sliding Deformation and Resistance. The author made numerous experiments on the sliding of the reinforcing metal by different methods, but all having as the object the measurement of the longitudinal displacements of certain points of the bars, relative to the adjacent surfaces of the prism which were in the same transverse section before the application of the load. The -I to 3 Perxxnt of Lenqrth. *l : t Sliding- Fig. 17. results obtained from different prisms were very different . in their absolute values, but when they were represented graphically by diagrams, with the loads as ordinates and the corresponding sliding as abscissas, the uniform aspect of the deformations was seen clearly, Fig. 17. The relative sliding or displacement of points of the reinforcing rods and the surrounding concrete, which were 10G REINFORCED CONCRETE. distant from each other by the small distance of 0.2 to 0.3 inch, were extremely small as long as the sliding stresses did not exceed certain values, which evidently correspond to the elastic limit. Then, almost suddenly, the sliding increased rapidly, and for an increase in load of about one-tenth to one-fifth, at the most, they attained relatively considerable values, corresponding to a sliding from 1 to 3 per cent. The similarity of the form of the curves, represented by Fig. 17, to that of the curve of deformation of concrete in tension, which differs so much from the results on concrete not reinforced, is evident. But the values of the deformations are much higher for sliding than for tension, possibly ten or fifteen times as much. This result would seem to be doubtful if the examination of the above-mentioned buoy had not revealed still greater sliding. It should be observed that these sliding curves, which haA^e a very definite practical value, do not have a precise scientific explanation. In fact, for reasons which need not be given here, the stresses per square inch of surface of contact which tend to produce sliding of the reinforcing rods are not at all proportional to the applied loads taken for the ordinates of the curve in Fig. 17. On the other hand, these unit stresses vary with the distance from the reinforcing rods, since cylindrical rings of mortar are con- sidered on the circumferences of which the stresses are distributed. The observed sliding is thus the complex re- sultant of different displacements which occur in the variously stressed portions. This lack of scientific pre- cision does not diminish the importance of the observa- tions on the considerable ductility which mortar or con- crete possess as to sliding, and which increases with the increase in pressure on the concrete. The observations of the author cover not only the de- formations, but also the resistance of the reinforcing rods to sliding. For reasons similar to those for the deforma- RESISTANCE OF CONCRETE TO SLIDING. 107 tions, their useful" practical results cannot establish the elementary laws of the sliding phenomenon. It is, how- ever, of no less interest for practical purposes to know the results. Prisms were loaded the same as the prism described in Chapter I, until rupture by the sliding of the reinforcing members. In the absence of a better method the usual but, as seen, inaccurate assumption was made that sliding stresses are proportional to the shear- ing stresses, and it was found that sliding resistance varies from 70 to 170 pounds per square inch of surface of con- tact for prisms kept in the air and made of concrete con- taining 500 pounds of cement per cubic yard of a mixture ■of equal parts of- good, sand and small, gravel. The re- inforcing consisted of drawn iron wires of 0.17 inch diameter, the surface of which was perfectly clean, shin- ing, and possibly somewhat greasy. The resistance to sliding increased to 256 pounds for prisms of the same concrete, reinforced by rolled iron wires 0.24 inch in diameter, the surface of which was similar to that of the rods usually employed in practice. In another series of prisms made of mortar containing 730 pounds of cement to the cubic yard of sand and kept in water, reinforced by iron wires 0.17 inch in diameter and slightly rusted, the sliding resistance, computed the same as above, varied from 330 to 500 pounds per square inch of surface of contact. It was determined, for mortars kept in the air, that the proportion of water used in mixing exerts a con- siderable influence on the adhesion to the reinforcing metal. In three prisms, otherwise identical, concrete mixed with an excess of water, normal concrete and too dry concrete were used, and the resistances to sliding were respectively 155, 170, and 70 pounds per square inch. These results agree with the opinion of practical men, and the weak adhesion to the reinforcing rods of concrete which is too dry is easily understood to be due to the 108 REINFORCED CONCRETE. smaller number of points in contact. On the contrary, in concrete made by experienced workmen an excess of water gives to the cement the necessary fluidity to circu-' late between the grains of the sand and to fill all voids around the reinforcing metal. These advantages of wet concrete for adhesion are, however, counteracted by an appreciable decrease in the tensile and compressive re- sistances of the concrete, as has been shown by M. Candlot. The results stated above should be verified by tests on masonry of usual dimensions, because concrete fails under altogether special conditions when the dimensions of the prisms are as small as the test prisms, which did not ex- ceed 2.4 inches on a side. It will be noticed at once how much the sliding .resistance as determined is below the values of 570 to 710 pounds per square inch, generally ac- cepted according to the experiments of Bauschinger and Jolly. Part of the difference at least must be attributed to the fact that the reinforcing rods of a prism in bend- ing are surrounded by concrete, which sustains at the same time very high tensile stresses, generally much above its elastic limit. The experiments of Bauschinger and Jolly, on the contrary, were made on metal bars well embedded in concrete blocks where all other stresses but the sliding, could be neglected. The results of the author's experiments on sliding seem to agree with the experimental results obtained by Hart- mann, and the theory developed by Mesnager and others. Sliding, which has a deciding role in the deformation of the metals, and even in that of all other bodies, is resisted by two kinds of stresses: One is due to the strength of the material, the other to the f rictional resistances, which are proportional to the normal pressures on the sliding surfaces. It is thus natural that the sliding and shearing resistances are influenced by all kinds of stresses which are acting in the reinforced concrete, and that they are higher in the parts closely adhering to the metal than in RESISTANCE OF CONCRETE TO SLIDING. 109 reinforced concrete beams; the concrete of which has spent part of its resistance on the elongations due to bend- ing, which have exceeded its elastic limit. If it be also noticed that the sliding of the reinforcing members and the deformations of the plane sections in- crease the deflection, the conclusion is reached, which is contrary to the elementary and accepted notions on the resistance of materials, that the bending moments can have much effect on the resistance of concrete to shearing and sliding, and that the effects of the shearing stresses on the deformations cannot be absolutely neglected. CHAPTER VI. Effect of Cracks on Stresses and Deformations. 1. Effect of Oeacks. It has been assumed above .that the reinforced concrete was in good condition and that the elongations due to the applied loads did not cause rupture. For reasons pointed out in Chapter III, this is frequently otherwise, especially in structures exposed to the air, and it is of im- portance to investigate the effects of the cracks which interrupt the continuity. Generally practical builders to avoid any errors as to the tensile resistance of the con- crete assume this resistance to be zero. They deter- mine in an arbitrary way the position of the neutral axis and, hence, also the resistance of the portion in compres- sion, which are made a function of the assumed greatest stress. It is easy to determine exactly the position which will be taken by the nexitral axis since the concrete in ten- sion has no. effect. In Chapter I the formulas were re- printed, which led to this assumption. The following considerations will demonstrate that they do not hold true: In the first critical period, which is the period we have to look out for, only a small number of widely separated cracks are generally formed. The curves the ordinates of which represent the tensile stresses of the concrete and reinforcing members adjacent to one of these cracks show clearly the forms given in Fig. 18. The tensile stress in the concrete which has the value a A in the uninjured sections decreases at the cracked section to zero. The tensile stress of the reinforcing steel, on the contrary, rises from b B to c C, since it tends to compensate for the gradually decreasing resist- ance of the concrete. The difference in the stresses in adjacent sections necessarily causes longitudinal action [110] EFFECTS OF CRACKS ON DEFORMATIONS. Ill due to the adhesion of the steel to the concrete, which thus causes sliding stresses, represented by the ordinates of the curve D E F in Fig. 18. Each force causes a relative displacement, the crack thus opens, the bending of the -member at the cracked section shifts, to the ad- Tensile Stress in Concrete. . ! © z Tensile Stress in Sieel. SUqJing Stresses., Fig. 18. jacent sections and the neutral axis approaches the com- pressed side, the sectional area of which is thus decreased, while the stress per square inch is increased. This phe- nomenon differs according to whether the concrete is un- der water or in the air and, consequently, whether the 112 REINFORCED CONCRETE. concrete adjacent to the reinforcing steel has an initial tension or compression. It is doubtless true that the effect of the cracks will be the more prominent the greater the cross-sectional area of the reinforcing rods is to their surface of contact. Actually the differences in stress C c — Bb which have to counteract the sliding stresses are proportional to the areas of the reinforcing members, while the adhesion which equalizes them is proportional to their area of contact. The greatest value of the sliding stress which causes a crack and the length of the displaced reinforc- ing are to each other as the cross-sectional area to the circumference of the reinforcing members. To verify these assumptions the following experiments were made: Of a mortar containing 730 pounds of cement per cubic yard of sand twelve prismatic test pieces were made of 35.4 inches length and of a square cross-section of 2.36 inches. Three of these had no reinforcing and were in- tended to characterize the properties of the mortar alone. Each of the remaining prisms was reinforced with metal of a total area of 0.07 square inch arranged so that three prisms were reinforced by a single rod, three others by three rods, and the remaining three by five -rods, which all had the same cross-sectional area. A series consisting of one of each of these four types was kept all the time in the air and was tested after two months. A second series was successively and alternately kept in the air and under water so as to reduce the shrink- ing and swelling to the least possible and thus to avoid all initial stress before the tests. This second series was also tested after two months' time. The third series was kept under water, and it was the intention to keep it there until the swelling had such an effect as to show elonga- tions in the reinforcing rods. In each of the reinforced prisms a slit was provided in the tension side by inserting a small and thin metal piece smeared with wax which EFFECTS OF CRACKS ON DEFORMATIONS. 113 did not adhere to the concrete. The prisms of the first two series, as well as those of the third series, were sub- jected to bending moments which increased to much higher limits than would be allowable for practical load tests. The elongations of the mortar in tension were measured as well as those of the reinforcing rods and the shortenings of the concrete in compression. These measurements were made on lengths of 2.36 inches, equal to the depth of the prisms, on parts having the above- mentioned slits in their middle and also on parts of com- pletely continuous and uninjured mortar. The results were completely in accordance with the as- sumptions in the prisms of the second series where the internal stresses due to gradual hardening were almost of zero value. An increase in the bending of the re- inforcing rods was found in them near the cracks, also a sliding of the iron on its surrounding concrete, and finally an increase in the shortenings of the mortar in compression. The intensity of all of these phenomena increased with the diameter of the reinforcing rods. In the prisms kept in the air the reinforcing rods were short- ened during the gradual hardening process by 0.021 to 0.03 per cent. The effects of the slits in them were the same as for the second series, but with more intensity and have by far exceeded the author's assumptions. In the prism reinforced by a single iron rod of 0.3 inch diameter the sliding of the rod relative to the sur- face of the mortar reached the value of 0.015 per cent., which is very near the ultimate stress. It should, how- ever, be added that the diameter of the reinforcing rod of this prism was intentionally made much larger than it would be made in practice, to illustrate more clearly the effect of the dimensions of the reinforcing metal. But the most surprising fact was the amount of the short- enings of the compressed portions. In the prism rein- forced by three iron wires of 0.17 inch diameter, where 8 114; REINFORCED CONCRETE. the dimensions of concrete and iron agreed with tried' practical cases, the shortenings near the slit were found to he ten times as great as in the uninjured portion of. the prism, though the load was a light one. It was 5- per cent, below a similar load admissible for load tests. ■ If the forces were always proportional to the shorten-: ings it would have to be concluded that the crack or slit increased the compressive stresses in the portion opposite • the reinforcing members five to ten times, which would give absurd values. The careful separation of the re-: suits of these experiments furnishes the explanation of' this evident irregularity. These results prove that the coefficient' of elasticity of the mortar decreased consid- erably at the cracked sections. Though the forces are proportional to the elastic deformations the compressive stresses at the cracks could not increase to a very high degree, in spite of the very great shortenings. However it be, the change in the' elastic- behavior at the cracked section is above doubt, and the following re- marks will throw some light on the cause of this change: The shortenings increased very much with the increase in diameter of the reinforcing rods though the cross-sec- tional area of the iron was the same in all prisms., But it seems that for equal areas the diameter can haVe an effect on the contraction only by the rigidity of the iron wires, and the rigidity increases rapidly with the diame- ter. It must, therefore, be concluded that the rigidity of the reinforcing rods, which offers a resistance to the contraction very different from that caused by the longi- tudinal stresses, had most of its effect on the sections weakened by cracks. And, contrary to what might have been supposed, the portions opposite the reinforcing rods maintained their stress during the contraction, and also underwent a change in their elasticity when the contrac- tion was prevented, as was actually observed in the ex- periments. This is especially true of prisms reinforced ' by transverse reinforcing bars. EFFECTS OF CRACKS ON DEFORMATIONS. 115 . It should also be remarked that the above test prisms sustained no load during their hardening. It is easily pos- sible that prisms sustaining permanent loads, as beams and floors, for instance, do not, after the removal of the falsework, undergo the same deformations as the test prisms, because the loads cause sufficiently high com- pressive stresses to equilibrate the rigidity of the rein- forcing members. In view of so complicated phenomena it would be pre- sumptuous to predict the effect of cracks on the failure of a beam by basing these predictions on experiments on the elasticity of prisms which were not so conducted as to insure their crushing before the sliding of the' rein- forcing rods. Crushing experiments are also required- to make the subject clearer. It seems that it is always possible to make with all due care certain assumptions which are based on the following considerations, the gen- eral value of which for the explanation of the facts ob- served on reinforced concrete structures has been- established. 2. Effects of the Convexity of the Curves of Deformation" of Mortar and Concrete. It has been seen that the curves of deformation are convex toward the horizontal axis and that this convexity is quite marked for concrete in compression, but it is much stronger for the concrete in tension of reinforced concrete structures and for the sliding resistances of the reinforcing members. It follows that when a portion of the concrete has reached the elastic limit for any one of the deformations, the stresses which are caused thereby increase from then on only very slowly. The adjacent portions, on the contrary, which have been less de- formed, sustain stresses which grow rapidly so as to com- pensate for the overstressing and to delay rupture. This is the same with all elastic bodies. If a masonry 110 REINFORCED CONCRETE. structure contains in addition bending stresses, the por- tions in tension and compression do not possess the cor- responding resistances. The weakest portion undergoes a greater deformation than the rest. The neutral axis, therefore, shifts away from this portion and there results an increase in the cross-sectional area of the portions which must resist the greatest stresses. If the elasticity in a portion of the structure has been initially reduced because of contractions or any other cause, the deforma- tions increase, the neutral axis shifts still farther away, and new portions come to the assistance of those having too little resistance. It is thus always correct to consider the favorable ef- fects of the mutual aid of the parts of concrete-steel structures as a very important fact. It is known that rupture may be caused by the repetition of smaller stresses, perhaps by two-thirds or one-half of the resist- ance of a single application of load which causes rupture. Especially is this true for loads the single application of which is up to the ultimate resistance. It is, therefore, so much more permissible to make assumptions, and it seems thus to be probable, that cracks reduce the resistance of concrete-steel structures much more with repeated loads than with a single application of the greatest load. Ex- periments will settle this question. Finally it should be stated that the above experiments refer only to the slits or cracks provided during the mak- ing of the test prisms. The cracks which are formed during the loading cause no alteration in the concrete. It is thus probable that the effects which result from them are also, for structures built' in the air, similar to the effects which have been observed in prisms kept partly in the air and partly in water. The latter was done in order to reduce as near to zero as possible the internal stresses in the prisms and the reductions in elasticity cor- responding to them. These effects will consequently be modified and much smaller than those observed on prisms EFFECTS OF CRACKS ON DEFORMATIONS. 117 kept in the air only. The diagrams of Fig. 18 repre- sent the results of these tests. They were presented be- fore the members of the Testing Commission who found them to demonstrate the correctness of most of the laws formulated in this and in the preceding chapters. 3. Computation oar the Dimensions and Deforma- tions of Reinforced Concrete Beams. From the facts observed in all of the above experiments a great complication of the different conditions follows. But these conditions must be known to understand the phenomena observed on structures erected and during test loads; hence, also the effects of the properties and proportions of the elements of which the concrete-steel structures- consist. For the sake of simplicity in compu- tation, the complications must be reduced; and to estab- lish rules for computations, it is quite logical not to pro- ceed with absolute rigor, since the properties of concrete are as variable as its materials and depend to a great extent on its careful mixing. It is only important to know exactly the errors made voluntarily in order to simplify the subject; and to determine the limits of these errors the laws governing the behavior of concrete-steel structures must be known. In Chapters I and II it was shown that the knowledge of the curves of deformation of the concrete in tension and compression is sufficient to compute the stresses and deformations caused by the bending moments, if the fol- lowing conditions are fulfilled: The concrete in tension must be unbroken and continuous; there must be no in- itial internal stresses before the loading; there must be no noticeable disturbances by the shearing stresses. The rapid algebraic method of computation shown in Chapters I and II as a substitute for the exact method, which suffers from graphical inaccuracies, lacks elegance; and it will certainly be improved by other engineers* 118 REINFORCED CONCRETE. -but its results appear to be safe. To take the internal stresses into account, especially those caused by slow hardening in the air and under water, it will suffice to correct the results by certain coefficients. These coeffi- cients will give somewhat higher values for the computed deformations and unit stresses of the concrete in com- ..pression and somewhat lower values, for the stresses in the reinforcing metal of structures kept in the air. The opposite will be the result for structures under water. Finally it would be very proper to assume cracks for all structures kept in the air and it would then be possible to judge when the same assumption should also be made ■for structures under water. The discontinuity of the con- crete will then be taken into account by means of coeffi- cients, which will actually increase the computed values of the tensile stresses in the reinforcing metal and the com- pressive in the concrete. As to the compressive stress of the concrete, an increase in the same would be the more important the larger the cross-sectional areas of the rein- forcing members are to their circumferences. Thus the shifting of the neutral axis will be injurious to the in- crease in the tensile stresses of the reinforcing, but in so slight a degree that it is not worth the while to take ac- count of it. One of the results of these researches is that the en- gineer who knows the heterogeneous nature of concrete- steel structures will be able to judge and estimate the dan- gerous conditions. The experiments on the sliding of the reinforcing members and on the effects of cracks raise it above doubt that concrete-steel construction the more in- 'sures the strength and durability of the structure the greater the number and the smaller the cross-sectional areas of the reinforcing members relatively to their circumfer- ence. But it is easily seen that the great multiplication of reinforcing units has practical objections. It is very diffi- cult to establish in each case the correct effect of the oppos- ing conditions. CHAPTER VII. The Compressive Resistance of Reinforced Concrete and Hooped Concrete. 1. Concrete Reinforced by Longitudinal Rods. The first idea which presented itself to engineers to in- crease the resistance of concrete in compression was to re- inforce it, similarly to tension pieces, by rods laid longi- tudinally in the direction of the stress. For purposes of construction, to keep the rods better in place, the rein- forcing rods were tied together by a network or a belt of smaller rods. Some engineers understood that these belts perform another important role, that they protect the lon- gitudinal rods from premature flexure and retard the swelling of the concrete and, hence, its ultimate failure. It will be seen below that by hooping or completely sur- rounding the concrete by steel rods a considerably higher resistance can be obtained, and it is. evident that between this method, supplemented by the addition of longitudinal rods, on one side, and the method of reinforcing by longi- tudinal main rods tied together by belts of lighter ma- terial on the other side, there is an intermediate continuous series of methods of reinforcing. The conclusions reached by this study will enable us to foresee the effects of these complex combinations, but before making the synthesis the influence of each element should be investigated separately. Concrete reinforced by longitudinal rods tied together by netting or belts of dimensions too small or spaced too far apart to exert noticeable influence on the resistance of con- crete will, therefore, be treated first. It was admitted, up to the present time, that the dif- ferent varieties of stone, mortars, and concrete, when un- der compression, always fail by shearing along planes [119] 120 REINFORCED CONCRETE. which are inclined to the direction of the stress. The recent experiments made in Germany by Eoeppel and re- peated by Mesnager at the laboratory of l'Ecole des Ponts et Chaussees have proved that this mode of failure is due to the friction exerted on the lower.planes of the test. speci- mens by the plates transmitting the pressure. And it has further been proved that by sufficiently reducing this fric- tion by the introduction of a greased surface, the failure^ will take place along surfaces which will be parallel to the direction of the pressure. It is not clear how longitudinal reinforcing bars, which are parallel to the lines of rupture, could prevent the sepa- ration of the molecules and increase the resistance of the- concrete, and it seems that the only effect of longitudinal reinforcing in compression members consists in adding the- resistance of the steel to that of the concrete without, strengthening the latter. Experience has shown that such is the case. The effects of the reinforcing bars are, how- ever, complicated, for the reasons which follow. As has been shown in Chapter III, the tendency to- shrink which concrete shows when hardening in air causes in reinforced concrete internal stresses of great intensity; tension in the concrete and compression in the metal. Ex- periments made in 1902 at the laboratory of l'Ecole des Ponts et Chaussees, according to the program laid out by the French Commission on Concrete-Steel, have deter- mined the effect due to the shrinking of large concrete-steel, specimens of the most commonly employed mixture, 420 pounds of Portland cement to the cubic yard of sand and 1-inch gravel in the proportion of 1:2. Measurement of the variations in length of the reinforcing bars has shown that after three months the shrinking of the con- crete had compressed the metal, 6,540 pounds per square inch, in prisms 6.5 feet long of a section about 4x4 inches, and reinforced near the edges by 4 iron wires \ inch in diameter. The compressive stress in the metal has reached. 10,800 to 14,200 pounds per square inch in beams 13.1 COMPRESSIVE RESISTANCE. 121 feet long having a cross-section about 8x16 inches and re- inforced near one of the smaller sides by 4 metal rods of f-inch diameter placed 1.3 inches from the face. The latter specimens were prepared to be tested for bending. It is superfluous to point out the importance of the above statement as to the magnitude of the interior stresses in members of the usual mixtures and of dimensions sim- ilar to those met in practice. Neglecting this kind of stresses, some engineers have made grave mistakes in the interpretation of bending experiments and have established incorrect formulas and rules, especially on the subject of stresses in compression members. They have assumed that if a certain specimen has undergone a shortening, %, its reinforcing bars, which had a coefficient of elasticity E, were compressed to a stress E i, neglecting the addition which has to be made to the latter stress for the shrinking of the concrete, if it has hardened in air, and which usually exceeds it in amount. The above considerations are suffi- cient to compute the stresses in compression members as long as the elastic limits have not been surpassed, neither in the concrete nor in the metal; but this is only one side of the question. Without entering into a discussion of the unit stresses which may be allowed for the various elements of con- crete-steel structures, it is evident that the basis of any computation must be the knowledge of the stresses which are induced in these elements at the instant at which, for the first time, there appears any danger for the one or the other of them. It is, therefore, important to know the stress caused by the reinforcing steel in a member in com- pression at the instant where it begins to fail by the crush- ing of the concrete, which takes place a long time before that of the steel. A concrete of common quality can stand without crush- ing a reduction in length of 0.07 to 0.10' per cent, and sometimes more. Such a compression will cause a stress in the metal of 20,000 to 29,000 pounds per square inch, if 122 . . REINFORCED CONCRETE. •the coefficient of elasticity be 29,000,000 pounds. This stress added to the previous stress of 7,000 to 14,000 pounds, gives a total of 27,000 to 43,000 pounds per square inch of the metal, which is equal and even superior to the elastic limit of the iron and mild steel which is usually employed. Therefore, before the crushing of the concrete, -the reinforcing bars are almost always stressed up to their elastic limit, unless the elastic limit of the bars be excep- tionally high or the concrete exceptionally poor. This stress cannot be appreciably surpassed because a very great decrease takes place in the value of the coeffi- cient of elasticity of the metal as soon as the elastic limit has been exceeded, and the stresses increase, therefore, with an extreme slowness which is limited by the small deformations which the concrete can still undergo without crushing. Hence, it may be stated that in concrete members rem- forced by longitudinal rods connected by cross-pieces or ties too weak or too far apart to bind the concrete suffi- ciently together crosswise, the total resistance to crushing ■varies little from the sum of the resistances offered by the crushing strength of the concrete and the longitudinal bars when stressed up to their elastic limit. During the elastic period the metal, which has been compressed before by the shrinking tendency of the concrete, causes important stresses. 2. CoNCKETE ReINTOKCED BY TrANSVEBSE E.ODS. Whatever the mode of rupture of concrete in compres- sion, the crushing of the same must be retarded by the use of reinforcing rods put in planes perpendicular to the direction of the external pressure and sufficiently near to each other. The tendency to slide along oblique planes is, indeed, resisted by reinforcing bars which cut these planes, whether parallel or perpendicular to the direction of the pressure. Rupturing along surfaces parallel to the pres- sure is directly opposed by transverse reinforcing. COMPRESSIVE RESISTANCE. 123 The idea of using transverse reinforcing is not new, and, while it may be still older, it is sufficient to mention that it was experimented upon in 1892 by Koenen and Wayss. Since then Harel de la Noe has theoretically explained the advantages of transverse reinforcing and has made and inspired some very interesting applications. The trans- verse reinforcing may consist of a series of rods placed on diameters, all passing through the center of the section, or of a net with rectangular openings, or of circumferential rods which constitute hoops embedded in the concrete to a depth required to protect the metal from the action of at- mospheric influences. The author has not made any experiments on the first system which concentrates the metal around the center where it is the least useful. He has limited his prelimi- nary experiments to reinforcing consisting either of cir- cumferential hoops or of netting wires at right angles and parallel to the sides of the section. For equal weights of metal the resistance to crushing was appreciably more than twice as great for the circumferential reinforcing as for the wire netting. Without entering into a theoretical discussion, the above result can be explained by a simple observation. If the , external layers of a prism reinforced by rectangular wire nets are considered the lateral thrust outwards to which they are subjected by the pressure at their base will in nowise be resisted by the rods parallel to these layers or faces, and nothing prevents them from separating from the central mass simultaneously with the concrete in which they are embedded. Of course, the bars at right angles to the faces considered offer a resistance to the outward thrust, but only to such extent as they adhere to the con- crete. This adhesion is proportional to the area of con- tact, and is zero at the ends and only increases in intensity as the distance along the bars increases from the faces, but these faces are just the layers most exposed to crushing. 124 REINFORCED CONCRETE. To remedy this fault the author has first employed iron rods so connected as to support each other, and then nets, of wires interwoven in a manner which promised the best results. After all these arrangements the crushing be- ginning at the face has gradually spread toward the center and it became apparent why, for equal weights of steel, not more than one-half of the resistance shown by the hooped concrete was obtained. It was as a result of the above experiments that all further investigations were di- rected to concrete reinforced by hoop-like rods. 3. Theoeetical Oonsidebations oit the Kesistawce of Hooped Cojstckete. The inner forces acting in solid bodies are often placed in two different classes. The name, cohesion, is generally given to the inter-molecular action, and it is known that it varies in proportion to the distances of the molecules from each other up to a certain point which is called the "elastic limit." As a premise nothing is supposed to be known of the effects produced by cohesion above the elastic limit; but at the same time it is generally admitted that friction exerts an action in the interior. of bodies similar to that exerted on their surface. However, this division, which may appear arbitrary, is not generally accepted and the deductions made from it may be disputed. We will leave the purely theoretical considerations and will attempt to attain the practical aim of the engineer, which is to formulate rules which will enable him to predict the mechanical properties of the ma- terials. The following method was adopted for investiga- tion: A certain number of prisms of concrete of different qualities and surrounded by hoops of various arrange- ments and sizes was prepared. Some had, also, longi- tudinal reinforcing rods. These prisms were submitted to increasing pressures and the shortenings produced were COMPRESSIVE RESISTANCE. 125 measured together with the loads. By a well-known for- mula for the thrust of a granular mass, the resistance was computed which would be offered by a prism of the same dimensions, reinforced in the same way if sand without cohesion were put in place of the concrete. The same co- efficient of friction was assumed and the same percentage ■of swelling of cross-section to decrease of length. This was computed for each observed deformation. It is evident that the excess of the observed resistance of a concrete prism over the similar resistance of sand corresponding to the same deformation can only be attributed to the direct or indirect effects of the cohesion of the concrete. Without entering into a. discussion on the character of this difference in resistance and without attributing to the name a precise scientific meaning, we shall call this excess the " specific resistance of the concrete." From this definition it follows that to determine the total compressive resistance of a hooped concrete prism it will suffice to add the specific resistance of the concrete to the resistance of a prism of sand having the same hoop- ing and the same coefficients of friction and transverse swelling. The latter resistance can be computed. To make use of this arbitrary distinction it must be possible to predict the specific resistance of the concrete in hooped members from the resistance of concrete of the same qual- ity not hooped. It will be seen that this can be done as far as is required. 4. The Resistance of Hooped Saito. Of course, sand without cohesion cannot be actually hooped otherwise than by a continuous shell of the same weight, as the hooping rings of concrete. This difference is of no importance as far as the following considerations are concerned. The crushing resistance which the hoop- ing will give to sand without cohesion is easily computed fcy any formula for the thrust of -a granular mass. 120 REINFORCED CONCRETE. If p represents the pressure per square inch which is exerted on the upper end of a vertical cylinder of a non- cohesive material, the angle of friction of which is /, and the weight of which can be neglected relatively to the external pressure, it is known that to prevent crushing P a pressure must be applied on the sides equal to —p? per square inch, where K=tan 2 f . By the aid of this formula the effect of hooping can be easily computed for a cylinder. In fact, if A be the area of each of the two symmetrical sections which a meridianal plane cuts in the hooping; if, further, r be the radius of the base of the cylinder and h its height, the pressure per unit area of surface of contact which the hooping exerts on the sand A will be equal to 7- , for each unit of tensile stress in ^ r h the metal. From the formula for K it follows that the upper base of the cylinder will sustain — =- per unit of t.KAt area and -. for the whole area of the base, r.r 2 . h The volume of the hooping metal being 2 - r A, the" ratio u of the resistance given by the hooping to the sand to the volume of the metal employed will be u== ■ — . It is evident that the corresponding ratio U],for the longi- tudinal reinforcing members which sustain the pressure directly as they are generally employed in reinforced con- crete construction, will be 11!,= ■ . , ■= —= — . We thus J-7- Jt h A get — == ~k-> an d experiments have given iT=:4.ft for sand and appear to lead to the same value for con- crete. It thus follows. that the resistance given to sand by the hooping is 2.4. times greater than the direct resist- COMPRESSIVE RESISTANCE. 127 arice of the longitudinal reinforcing members of same weight when the tensile stress in the former is equal to the compressive stress in the latter. Thus 2.4 is also the ratio of the crushing resistances of the two types of reinforcing for equal weights of rein- forcing metal. This is so because crushing takes place in hooped members as well as in members longitudinally reinforced when the elastic limit of the metal has been reached, which is the same for tension as for compression. In fact, the hoops yield then too rapidly to prevent swelling. Crushing is not the only danger which has to be taken care of in compressed members; they can also fail by flexure, as columns, and then their resistance is propor- tional to their coefficient of elasticity, that is, to the ratio of the unit pressure they sustain to the shortening which, they undergo. If m represents the ratio of the transverse swelling to the longitudinal shortening the hoops will elongate by m i only when the longitudinal members will be shortened by i. The tensile stresses in the former and the compressive in the latter are thus proportional to these deformations, mi and i, and, hence, the corresponding increases in the coefficients of elasticity due to the hoop- ing or longitudinal reinforcing are to each other as — — — to K \ . . , . to i, or as — - — :J-j which gives the numerical value of 2.4 m : 1. 2 It is, of course, out of the question to use hooped sand, and the preceding considerations are of interest only for the indications which can be deduced from them as to the resistance of solids. The value m, which has been mentioned in the above, refers to hooped concrete and no conclusive experiments have been made for its determina- tion. The only experiment which can be mentioned re- ferred to unreinforced concrete and gave for m the value of 0.4. It is evident that smaller values will be found 128 REINFORCED CONCRETE. for hooped concrete, and they will be the smaller the greater the percentage of metal, because the hoops resist the swelling, increase the density, and make the mole- cules approach each other, which is the cause of the in- crease in the coefficient of elasticity. The author has adopted 0.375 for the value of m be- cause this value gives resistances which, on the average, agree sufficiently with the results of experiments made on prisms of hooped concrete. If noticeable difference occur, they are on the safe side. There is evidently no reason why this value of m should also be taken for sand. The following conclusion may thus be made without rais- ing objection. If there existed a noncohesive material for which K and in had the values 4.8 and 0.375, which appear to agree with concrete, the friction caused by the hooping would give it a coefficient of elasticity which would be 0.375 x 2.4 = 0.90 of that of longitudinal reinforcing members of the same weight as the hooping. ••>- The modifications which cohesion will cause in the ef- fects of friction cannot be easily predicted, and the con- clusions drawn as to noncohesive bodies are of interest only in so far as they permit, in the absence of an exact theory of hooped concrete, the deduction of sufficient rules for the computation of its resistance and coefficient of elasticity. 5. Experimental Keseaeches. The above considerations referred to bodies without co- hesion; and it is important to determine whether in con- crete the effects of cohesion act in addition to friction and in what way. Experiments were made to verify the agree- ment of the resistance of a sand cylinder in a shell with the formula for earth thrust. It was thus that the value 4.8 for K Avas obtained, which agrees with the angle of friction. A series of experiments was then made at Quimper, in 1901, on small prisms of mortar 1.575 COMPRESSIVE RESISTANCE. 129 inches in diameter hooped by a fine iron wire. The deformations were not measured and the results can, therefore, only be used to verify the resistance to crushing and to compare the same to the assumptions made above. The accompanying table shows the results obtained from some of the prisms. The figures given in the table are significant. The iron wire employed for the hooping was drawn cold and did not have a definite elastic limit. From its curve of de- formation, 78,200 pounds per square inch appeared to be the value, after passing which the stress in the metal increased too slowly for its elongation to be able to main- tain the real efficiency of the reinforcing. This value of 78,200 pounds per square inch has been multiplied in the table by the ratio of iron to concrete, giving the values in the next to last line, representing the compressive re- sistance which would have been offered by the metal if it were used as longitudinal reinforcing bars instead of hooping wires. The ratios obtained in the last line of the table thus give the coefficient of efficiency from the point of view of the compressive resistance of metal employed in longitudinal reinforcing or in hooping. For sand, as seen in section 4, this ratio is 2.4 and the figures of the last line show that for mortar this ratio has not deviated far in these experiments. The compressive values obtained should be noted. Of the prisms one was of mortar which had had time enough to set. With a volume of metal equal to 0.034 of the total volume and without any longitudinal reinforcing it showed a resistance of 10,500 pounds per square inch of total section. It will be useful to compare the resistance of this prism with the resistance of an iron prism of the same weight. The hooped concrete had a density of 2.4, and that of iron is 7.8. To compute the pressure which an iron bar of the same weight will get per square inch of section, 10,500 must be multiplied by 3.2, the ratio of 9 130 REINFORCED CONCRETE. the densities of the two materials, giving 33,600 pounds per square inch. Since not more than 36,000 to 39,000 pounds per square inch can be expected of the total area of a riveted iron section weakened by numerous holes, it may be said that a prism of an ordinary mixture, rein- forced by an average percentage of hoops, has shown a compressive resistance in the neighborhood of that of ordi- nary iron. TABLE XIII. Weight of cement per cubic yard of „ , „„ „ gan( j r J . 675 Pounds , 730 Pounds Age of mortar tested, days 8 14 22 23 100 Ratio of volume of iron to volume of concrete 0.02 0.03 0.04 0.02 0.034 Resistance to crushing in pounds per square inch of total section. . 4,870 6,540 7,360 4,930 10,500 Resistance to crushing of concrete not reinforced 569 711 853 853 2,420 Increase of resistance due "to hoop- ing 4,301 5,829 6,507 4,077 8,080 Product of ratio of iron to concrete by 78,200 pounds 1,564 2,346 3,128 1,564 2,658 Ratio of the values of the last two lines 2.7 2.5 2.1 2.6 3.0 It will be seen in what follows that the results of hoop- ing are less advantageous for the coefficient of elasticity, and, therefore, for the resistance to flexure as a column than for that of the crushing resistance. To study this question experiments had to be made on long members and their deformations measured. With the aid of M. Hennebique, 38 prisms of octagonal section of 5.9 inches diameter were made. Different reinforcing was em- bedded, and the concrete consisted of the usual mixture of 420 pounds of Portland cement to the cubic yard of sand and gravel in the proportion of 1 : 2 in some prisms and double the amount of cement in others. Some prisms had a length of 1.64 feet and were especially intended to test the crushing resistance; the others had a length of 4.25 feet and were used to study the elasticity and the COMPRESSIVE RESISTANCE. 131 ductility of hooped concrete, which is one of its character- istic properties and one of the most important for safety. Each specimen teaches its lesson, but it is plainly im- possible to describe all the results obtained, comprising about 1,200 observations of deformations. They have, therefore, been grouped so as to throw most light on the important points. The first group consists of six prisms. Prism A was not reinforced. It crushed under a load of 1,050 pounds per square inch. Prism B was reinforced with helicoidal spirals of 5.5 inches average diameter made of cold-drawn i-inch iron wire spaced 1.18 inches centers to centers. It crushed under a pressure of 5,120 pounds per square inch of total section. Prism C was reinforced by helicoidal spirals of 5.5 inches average diameter made of cold-drawr> iron wire of 0.17 inch diameter and spaced 0.59 inch cen- ters to centers. Without crushing it stood a pressure of 5,400 pounds per square inch, which was the highest press- ure supplied by the testing machine employed. Prisms D and E were reinforced respectively the same as B and C, and in addition by 8 longitudinal wires J inch in diame- ter leaning against the inside of the spirals. They failed as columns before crushing, under pressures of 4,550 to 4,700 pounds per square inch. Prism F had 8 longitu- dinal reinforcing wires 0.35 inch diameter tied together by belts of iron wire 0.17 inch diameter spaced 3.15 inches apart, that is, closer than they are usually in concrete steel constructions. It failed under a pressure of 2,420 pounds per square inch. The following phenomena were noticed without the aid of measuring devices. 6. Geneeal Peopebties of Beinfobced Concrete and Hooped Ooetcbete. The unreinforced prism A broke suddenly, without any signs of approaching danger. The failure of F was almost 132 REINFORCED CONCRETE. as sudden. Its breaking load did not exceed by more than 7 per cent, the load producing the first cracks. The rein- forcing rods bent outwards between their cross-connecting ties and the concrete crushed. Concrete in compression when not reinforced or when reinforced by longitudinal Fig. 19. rods must be classified among the materials which break suddenly without much deformation. Quite different is the behavior of hooped concrete, as will be seen. The hooped prisms behaved at the beginning the same as the others and showed only very small deformations, un- COMPRESSIVE RESISTANCE. 133 der small loads; but this so-called elastic period did not end with, the failure of the specimen. The shortening was observed to increase rapidly and cracks appeared in the concrete covering the hoops, first fine, then more and more pronounced. The curves of deformation of these prisms are represented in Fig. 19. The cross-marks in them indi- cate the appearance of cracks in the prisms B, D, E. These superficial injuries appeared in prism only after the considerable shortening of 0.355 per cent. The apparatus used did not measure exactly the short- enings above the figures given in Table XIV, but rough measurements showed that failure took place after de- formations greater than 3 per cent, of the lengths. Table XIV gives the shortenings observed under the pressures indicated in the first line. > M X I oo ■«* SS 00 £- •TfltD in S3 rii J? o CM :g8 H esf •COCO 1 00 e to :SS8 *e *§ 88 ■ rtH CO 5* •t-W rf£ tx • oo S §8 :feS c w • op s m 60 Tl a 00 T eg sss &h 1 ooo CO o oo© ^3 »o o ooo ff «3 o 1 in T-lOO ooo! - e>> o ooo ooo CO p ° fen i^ sea to to to o< PhPh^ m Hi o 1 • g 00 CO • CO •00 CO [3 o ■5 • o 1 CO CO So CO CO CO CO* COCO cxcS CO en CO of torn in CO CO CO WHO en om-^t OS 00 f- ooo O 5 OOO s CO CO CO OOO 1 oii-iw ooo h S a to a,g go P-li— < to a 13 ~ a c 1 to CO Pi * p £ T p. + ■H £ — ► [134] COMPRESSIVE RESISTANCE. 135 The prisms A, B, 0, were 35.4 inches long and the prisms D, E, F, 51.2 inches. On each of the curves of deformation shown in Fig. 19 are inscribed two figures and their sum. The first of these figures gives the pro- portion of the area of the longitudinal reinforcing mem- bers to that of the concrete, and the second that of the hoops, or, in their absence, of the iron ties. The third figure gives the total proportion of metal. By compar- ing the percentages of the iron to the concrete, with the corresponding crushing resistances, the considerable su- periority of hooped concrete, from this point of view, be- comes evident. The table giving the results of the Quim- per experiments shows this very clearly. In concluding it may be added that the first examina- ' tion of the results of this group of experiments has shown at once that while concrete not reinforced or reinforced by longitudinal rods, even when tied together by ties ap- preciably nearer to each other than is usually the case, breaks in compression under small deformations and with cut any notice, hooped concrete sustains, without crush- ing, considerably heavier loads and only fails a long time after cracks in the faces and exaggerated deformations have called attention to the approaching danger. 1. The Spacing op the Hoops. Examining the curves of Fig. 19 more closely, an ap- parent anomaly is noticed. The ordinates which corre- spond to a given abscissa, or, in other words, the pressures sustained by the differently hooped prisms for the same shortening, have no relation to the proportion of metal in the hoops or in the longitudinal reinforcing members. The explanation of this fact has been furnished by the observation of the conditions . of the tests. While testing prism B, cracks appeared under the light load of 1,730 pounds per square inch, and soon after the 136 REINFORCED CONCRETE. concrete began to chip off and finally failed between the spirals which served as hooping. These spirals were. 1.18" inches apart. The failure of the prisms, which took place" under a pressure of 5,120 pounds per square inch was- due to the failure of the concrete, and there was nothing' to indicate that the metal had reached its elastic limit. In prism C, whose spirals were only 0.59 inch apart, the 1 cracks did not show before a pressure of 2,480 pounds per square inch was exerted. Chipping occurred also later- than in B, and under a pressure of 5,400 pounds no failure of the concrete was observed, and the prism did not fail. In D and E cracks appeared under pressure of 2,900 to- 1 3,360 pounds per square inch. These prisms contained,; besides the same spirals as B and O, longitudinal rods, which were in contact with the interior surfaces of the spirals, and formed with them a network which efficiently resisted the lateral failure of the concrete. Computations, prove that the superiority of D and E, in this respect, con- siderably surpasses the resistance which could have been offered by the longitudinal rods alone. Connecting these facts to the figures given in Table XIII, and to the obser- vations made on other series of experiments, certain con- clusions are arrived at as to the results obtained on the subject of the spacing of hoops and longitudinal reinforc- ing rods. When the spacing of the spirals did not exceed one-fifth, the diameter of the prisms, resistances were obtained al- most independent of this spacing. On the contrary as far as it concerned the appearance of cracks in the external layer, the elasticity, and, hence, also the column resist- ance, the results were the more successful the nearer the hoops were to each other. The resistance of these prisms was still much more increased by the addition of longitudi- nal rods leaning against the interior surface of the hoops. The facts mentioned and others which have been observed lead to the adoption of a spacing of the spirals of one-sev- COMPRESSIVE RESISTANCE. 137 enth to one-tenth the diameter when longitudinal reinforc- ing rods are added. The question may be raised whether it is proper to ad- mit that the effects of hooping are the same, whatever the absolute values of the dimensions, as long as the rela- tive values of these dimensions to the diameter of the post remain the same. Experiments on widely different dimensions have proved it to be so, and the following con- sideration will explain it: If, taking the simplest case, the spirals are replaced by continuous hoops extending between planes perpendicular to the axis, the deformation which the continuity of hooping will indirectly prevent will consist of the swelling of the little cylinders inclosed between the tangent planes of the spirals. If the end pressure which is exerted on two similar prisms of dif- ferent diameters has the same unit value, the lateral pres- sure required to prevent swelling will, in all probability,, also have the same 1 unit value. Such is certainly the case with sand. The total value of the lateral pressure will, for the small cylinders considered, be proportional to their sections along a diametral plan© which is equal to the product of their height by the diameter; that is, to the square of the diameters since the prisms are similar. The hoops will offer this resistance to swelling, but it can only be transmitted from the metal to the small cylinders by the friction and cohesion acting on their bases whose areas are also proportional to the square of the diameter. If two cylinders of different diameters are considered, all dimensions of which are proportional to each other, the shearing stress which will be produced in the planes tan- gent to the hoops will have the same value when the unit pressure on the prisms are the same. In other words, prisms of proportional dimensions behave the same way under the same unit pressures, and it seems to be justifi- able to express the rules as to the dimensions,, of the hoop- ing element as a function of the ratio of these dimen- sions to the diameter, as was done above. 138 reinforced concrete. 8. The Ductility of Hooped Concrete. A metal tube 7£ inches in diameter was filled with Port- land cement and bent to a radius of curvature of the neutral line of 21.6 inches. The metal shell was then removed and a piece of the cement was cut out. The cement which underwent such great deformations did not break, and on the compressed side only infrequent cracks were observed. Tests proved that it still had a great, re- sistance. This observation on concrete inclosed in a metal shell led to the belief that similar results would be ob- tained for hooped concrete. Numerous experiments have indeed proved that in prisms which had bent under heavy pressures the concrete did not break and it maintained its cohesion. A hooped prism of the proportion of 840 pounds of cement per cubic yard of gravel and sand, which was subjected to a pressure of 7,940 pounds per square inch of original section showed great deformation, bending in the shape of the letter S with a greatest versed sine of 0.4 inch in a length of 13 inches. The curvature was much sharper at the middle so that the least radius of curvature was about 2 feet. The portion of concrete nearest to the outside did not show any transverse cracks, and hence could not have elongated much. The flexure of the prism was, therefore, produced almost wholly by the shortening of the opposite fibers, for which computa- tion gave the enormous figure of 17 per cent. I o •« hi bo a 10 a i* ^ s «S to El 583 ;S (Ofl a: CO oaoioo CO CO CD *3* -OOO ^3&S OCDNO coin coo -000 co *h com 'OOO w;c»t-o XOMW cot— coco OO — 11 •OOO • OOO ■ bo be bo C.9.S-9 gggg K - - U ©000 Phcgcogq ® © 0 oao bo m .g .9 -3 ~ — d ra- ta — CO 10 bi) if**. t-o* TOCO •■o ^, 0(0 r*^ cat- •d ws .9 a a U3CO ti CD to COT* fl 60 a Ti O oco 1-1 coo TJ 1(3 £" c 0) r/i =s TH ■ B.CQ O [139] 140 REINFORCED CONCRETE. It is not certain -whether the concrete could have stood this deformation if the sharpest curvature had not ex- tended over only a short length ; the obliquity and warp- ing of the cross-sections could have had an important in- fluence on the curvature. But the fact remains that the above prism stood considerable deformation. The hoop- ing spirals and longitudinals were afterward removed from this prism and the remaining concrete was coherent enough for its whole length of 4.25 feet to be handled without breaking. It was put on two blocks 3.61 feet apart and it required 55 pounds to break it by bending. One of the halves of the prism which was less bent by the warping, but which stood the same average pressure of 7,940 pounds was put on two supports 20.5 inches apart and a load of 428 pounds was required to break it. The tensile resistance indicated by this bending te:t figures, by the formula B = ■, 205 pounds per square inch, in which M represents the bending moment and d the diameter, which the removal of the reinforcing had reduced to 4f inches. This tensile resistance of 205 pounds does not much differ from the initial tensile re- sistance of the concrete. In the compression tests with- out column flexure not more than 3 per cent, of shorten- ing was observed before failure. The difference of these results is due to the considerable swelling which, in this case, is required by such shortening as would lead to fail- ure, outside the elastic limit. In two other prisms the compressive resistance instead of the resistance to bending was tested after the removal of the reinforcing hoops and longitudinals. The first, of the same proportions as the one above, bent under a pressure of 6,970 pounds per square inch, which caused a shortening of 0.6 per cent. The average compressive resistance of the plain concrete was 1,420 pounds, and its real resistance must have much exceeded this figure, as the pressure was not well applied at the ends. A similar COMPRESSIVE RESISTANCE. 141 test was more carefully made on a prism mixed in the proportion of 630 pounds cement per cubic yard, and which stood a pressure of 10,270 pounds per square inch with a shortening of 2.4 per cent, on the average and 2.8 per cent, on the most stressed side. After removal of the reinforcing spirals, the inside cylinder, which had ahout 10.5 square inches section, sustained a pressure of 9,700 pounds. From the above tests, selected as they are from many similar tests, it must be concluded that hooped concrete sustains without disintegrating considerable shortening and conserves a great part of its original resistance; and that with the small deformations which occur in structures, the resistance of hooped concrete can he considered to be constant from the instant when it reaches its maximum. The analogy between this phenomenon and the behavior of concrete in tension is evident. 9. The Elastic Behavior of Hooped Conceete. Ex- peeimental data. The amount of deformations of a structure, which is, in certain cases, of importance, as well as the column resist- ance, which is of prime importance for all columns, both depend on the coefficient of elasticity. It is, therefore, absolutely necessary to investigate carefully the elastic behavior of hooped concrete. A. The Elastic Behavior under a Eirst Load. Erom among a great number of results obtained from tests of prisms those given by four octagonal prisms 5.9 inches in diameter and 4.27 feet long have been chosen for the interpretation of the results. These prisms were reinforced by helicoidal spirals and longitudinal rods, as follows : Prisms G and H, 840 pounds cement per cubic yard, spirals ^-inch wire spaced 0.79 inch, 8 rods, 5/16 inch 142 REINFORCED CONCRETE. diameter; Prism I, concrete and spirals same as above, 20 rods, 0.276 inch diameter; Prism J, 420 pounds cement per cubic yard, spirals same as above, 8 rods, 0.276 inch diameter. Table XV shows the shortening of these prisms under the given loads, the loading being re- peated as indicated. In all of these specimens the spirals had the same dimen- sions, and in the first three the concrete was of the same mixture. Similar results might have been expected of them, but the facts proved otherwise. While for the first prism the pressures below 2,845 pounds per square inch gave a coefficient of elasticity of 7,111,000 pounds, the sec- ond of identical composition gave only 2,845,000. This great difference was due to the quantity of water used for the mixing of the concrete ; it was correct for the first and excessive for the second, and gave a soft concrete which crumbled under the hammer and could not acquire the required compactness for a high coefficient of elasticity. The first lesson taught by these experiments is the great irregularity of concrete as usually fabricated and not care- fully inspected, and the doubtful value of results based on the coefficient of elasticity. If the general behavior of the deformation curves of the prisms be studied, it is noticed that they show an im- portant change of inclination after a certain pressure has been exceeded, the same as is shown for ductile metals. Tbis point may be called the elastic limit, but without attributing to it the sense usually given to this limit. An elastic limit, in the proper meaning of the words, does not exist for concrete. Prisms of the same proportions do not show, so much irregularity in their elastic limit as in their coefficient. The first three specimens, G, II, I, thus had an elastic limit varying between 4,830 and 5,400 pounds per square inch, while the coefficient of elasticity varied between 2,850,000' and 7,100,000 pounds. The elastic limit, moreover, varies with the proportion of COMPRESSIVE RESISTANCE. 143 cement, while the coefficient of elasticity, on the con- trary, is little influenced by it. For prism J it reached only 2,845 pounds per square inch. These results are in accord with what is known of concrete that is not reinforced. B. The Elastic Behavior under Repeated Loads. The observation of the deformations during the un- loading and reloading of the specimens, remaining always below the first load, has given results of great practical interest. The first thing shown, which could have be*en predicted, was a permanent shortening which increases if the same load is repeated, but more and more slowly, and tends rapidly toward a final' limit. A reduction of the final deformations is thus obtained and with it an appreciable increase in the coefficient of elasticity for the succeeding unloadings and reloadings of the speci- mens. The second result, which is more important, could not have been foreseen. It is the form of the deforma- tion curves and especially the direction of their curvature, which is concave to the pressure axis, while during the first application of load it curves in the inverse sense. It follows that the coefficient of elasticity which is repre- sented by the inclination of the tangent to the curve of de- formation increases with the pressure in the unloading and reloading instead of decreasing with inereasing press- ure as under the first application of load. The import- ance of this fact will be more fully discussed under " Column Resistance," in section 13 ; but it is well to indi- cate here its bearing. Evidently flexure is to be feared in a column under high pressures and it is, therefore, unfortunate that the coefficient of elasticity, which is directly proportional to the column resistance, decreases with the increase of pressure. Such is the case under the first application of the load for hooped or otherwise reinforced concrete and Mi REINFORCED CONCRETE. also for structural iron and steel. On the other hand, it must be considered especially fortunate that hooped concrete which has been subjected to a first load has a coefficient of elasticity which is the greater the higher "the pressure becomes, provided it does not exceed the first load. This fact has, to the author's knowledge, never before been observed on other materials. To bet- ter discuss this point the results of a very exact experi- ment, made at the laboratory of' the Ponts et Ohaussees, are given in Table XVI. The prism tested had an oc- tagonal section of 4.3 inches diameter and was made of concrete of the proportion of 1,000 pounds of cement to 32 cubic feet of gravel 0.2 to 1 inch in diameter and 10.7 •cubic feet of sand passed through a screen of 0.2-inch holes. The helicoidal spirals were of iron wire 0.17 inch in diameter and 0.82 inch centers to centers. The average diameter of their generating cylinder was 3.76 inches. In addition the prism was reinforced by 8 lon- gitudinal wires of the same size and material. The length of the prism was 51.18 inches. £ tq 60 a J*] ■Hi 5 T3 *- 9 Si, (3 ■i ^ s S> f=i 6$ §9 §°! ' THeO CO YH S3 g OCO (NO IS onto 03 iH 1-1 _ to • COTjt r-t • c -T-t 1-t O 1H C 3 = o-+ 2» CO O 1 oo OCO OiH ih ©" 04 CO 1* ■ QO i> COCO ' OS"* ^°° •■tH CO <*:« o es ©« Ofc- ' S 1 "" 1 QO w lH CO AS o • T-ltO CSt* -Ci •.-'■■ , •■cm t-H iO t- OS 1 ^ >oao Si 50 o • en • CO «3 • lO • C» ; iH *" 60 -OS lO -to bb o" • B ■ A ~ " O «= 5? OQ0 oca d '• °*°. CD CC £*! Td rlN CO o o 01 ^ o S» »— 1 •lO -CD CI a •■# t-TH ■ fl » p [a p O CC -d OCO 1= in o « ■ ^SS OS "" § OS ■ ID CO bo OCO 60 Ot-< be • 2 >oo O CO .9 CI t- CO -a 0-c- COO "S 83^ eg '« ^ —1-1 C5 «-T-4 w o Hi CO c3 O to o t- t-5 o -; ,_7 o 8 «~ h -OS US i O Ft-. O CO GQ OCO O** oca i-H CO ana CM CO '^ , ^: GO ■iH OCO COCO T-H 0* OCO OCO 3! **■ t-ao % °. 'I s ? OCO OIO ss " -rH 2>> "s -„ :-,-" pound rich . . . in one parts. . B£ 5 ■"fflM'S . to W) bo «a-i cs A □ C-S CD«eJ q.f- b s° « £ a (- el £. d 3 to © h 3 4) e) of the equality for the values of the coefficient of elasticity and the elastic limit at different points of the same section is easily understood. The lack- of sym- metry causes curving and invites flexure. But it is prac- tically impossible to obtain elastic symmetry in members concreted while lying longitudinally, as it is seen at once that there must exist differences between the tipper and lower layers as to tamping and distribution of water, cement, sand, and stone. By examining and sounding with a hammer different parts of a member appreciable dif- ferences can almost always be noticed in the compactness. This consideration tends to the adoption of vertical moulding for members compressed endwise ; but it should be noted that this does not apply to beams whose hori- zontal layers are subjected to different stresses. For them COMPRESSIVE RESISTANCE. 173 the sum of their moments only 13 of importance and not the distribution over the section. Vertical moulding has also other advantages. It tends to place the stones with their flat sides perpendicular to the direction of the pressure, the same as in well-built masonry. Horizontal moulding, on the contrary, places them in the most unfavorable position. The concrete being better confined in vertical moulds than in horizon- tal, tamps more qxiickly and much better. In members reinforced by hoops and longitudinals pressing against the inner surface of the hoops the whole central portion is free for vertical ramming, and the tamping tool can strike the concrete without shaking the reinforcing and disturbing the previously tamped portions. If members which are to be built in a different position are moulded vertically they get the advantage of being made in im- proved moulds provided with stops for the spacing of the spirals and in a covered shop protected from the rain, Avhich is dangerous during moulding, and the sun, which is not less dangerous during the early time of setting. Thorough inspection in a shop is easy and only a small uumber of selected workmen are required. For com- plete safety all members should be carefully examined and their quality judged by sounding with a hammer, which gives valuable information to the experienced man, and by weighing from which the density of the member could he computed and which would add additional infor- mation on the character of the concrete. Indeed, for a concrete of a given proportion the elastic limit and the resistance vary in the same sense as the density and are in proportion to the intensity of the sound. Members uot up to the standard could then be rejected. Vertical concrete members can be taken out of the moulds at the end of six days. The same moulds can then be used over again many times. Beginning suffi- ciently early with the making of the members, they will 174 REINFORCED CONCRETE. acquire all the desired resistance before being put in place. The members will be put during their hardening- in conditions which are best fitted for them, in moist air or even in water. On the other hand, there are some disadvantages attached to the shop process and the ver- tical moulding. It may be feared that the tamping of the central portion does not compact the concrete in the annular space between the hoops and the external surface though experience shows that the cement and the finer sand and gravel flow into this space and fill it to a satis- factory .degree. But it has been shown by the experi- ments reported in this chapter that the central portion only is of real importance as to resistance. The external layer only protects the metal from atmospheric influences. After the removal of the moulds it is easy to see whether the metal will be well protected and, if required, the surface may get a finishing touch. Visible and easily repaired faults have only a relatively small value and no others can exist in hooped-concrete members moulded ver- tically. The second disadvantage of shop moulding con- sists in the required handling and transportation to the work. The increased cost due to this may in whole or 1 in part be compensated by the saving of the forms used on the work. The third and, apparently, most serious objection of shop moulding is the inconvenience of as- sembling. Careful examination shows that this objection is not serious for hooped concrete. It is impossible to foresee with certainty the ways and means which practice will develop and adopt for the as- sembling of the members, but some indications may be given which, while open to improvement, will assure safe results. In consideration of local stresses of high in- tensity which may be induced at the joints, the ends of the members will be reinforced by additional hoops of finer wire than that used for the regular hoops. The joints will thus be strengthened to any desired degree in- COMPRESSIVE RESISTANCE. 175 dependent of the other parts of the structure. To give an idea of the efficiency of hooping after ramming it is sufficient to state that the prism tested at Quimper, whose crushing resistance, as given in section 5, had the high value of 10,500 pounds per square inch, consisted of a mortar cylinder three months old which was then hooped and covered by cement and tested ten days later. It will generally be useless to provide for the continuity of the longitudinal rods in hooped members since their duty is mainly to form with the hoops a network which will prevent the swelling of the concrete and they fulfill it perfectly by simply stopping at the joints. Cases may, however, occur in which it will be prudent to insure trans- mission of tensile stresses in the longitudinals which may be caused by bending moments. In such cases it will be easy to embed in the concrete at the ends of the rods, while moulding, iron tubes which will come opposite each other and in which metal rods of the same section as the longi- tudinals will be placed and grouted by cement. Or the longitudinal rods on the end of one member may be left projecting and some tubes be placed in the corresponding end of the other member. Joints made in this way will not, as in other structures, be weak joints. If, due to settling of centers, irregular filling of joints or any other cause, unequal distribution of stress occurs, the great ductility of hooped concrete and the enormous excess of resistance at the ends of members will exclude all danger. Concrete reinforced by longitudinal rods without hoops and made in single members will offer difficulties similar to masonry, at least when no hoops are used near the ends. But even when laid in place without joints the longitudi- nally reinforced concrete in long compression members has a defect which has not as yet attracted sufficient at- tention. In this type of construction the longitudinal rods make up a very important part of the resistance and the transmission of pressure through them must be per- 17G REINFORCED CONCRETE. fectly assured. Their connection is generally made by placing the adjacent ends of the rods in very short tubes. These ends are cut very irregularly and are not generally in bearing contact except by some projection which yields under the pressure and allows the abutting ends to ap- proach, producing thereby local deformations in the con- crete whose magnitude and effect cannot be foreseen. To avoid this objection reliance is placed on the cement which penetrates into the tubes during the ramming, but this is quite uncertain and it is also to be noted that badly- rammed cement will not be able to resist pressures of 11,000 to 17,000 pounds per square inch which the rods transmit to it. Sometimes another method is used. The rods are straddled over each other and reliance is placed on the adhesion of the concrete for the transmission of the pressure. This method is good for hooped concrete be- cause the rods have here a secondary function only and are embedded in a concrete which is protected from all danger by the hoops. But the same method is objection- able in structures in which the longitudinals perform a main function, and in which there is nothing to prevent breaking up and crushing the concrete which is highly stressed by the transmission of the pressure. Numerous reasons thus render advisable the making of hooped-concrete members under cover, protected from weather and by selected workmen under a continuous in- spection of responsible foremen. Referring to section 9 it will be seen that to obtain the greatest efficiency from this type the members must, before being put in place, be subjected to preliminary pressures of higher intensities than they will Be called upon to resist in the structure. Independently of the important guaranty which the pre- liminary testing of all members of the structure gives and the convenience of rejecting doubtful members, the pre- liminary testing results in a great increase of elasticity and column resistance. COMPRESSIVE RESISTANCE. 177 Responsible contractors, to whom important structures are let, will not find any difficulty in providing the neces- sary testing apparatus and accessories. This will be cheaper than may be expected because the shortenings which will have to be produced in the tested members are quite small, less than 0.04 per cent., and the stress required is so small that a cylinder and piston -of a hy- draulic press with a hand pump worked by one or two men will be sufficient to test very strong members. Neither motors nor accumulators will be required. The methods of manufacture and erection which have been discussed in the above are similar to those used on metal structures and guarantee the good execution required for works of some magnitude. Of course, small structures will not re- quire these methods, as has been proved by many existing structures. 17. The Influence of the 'Character of the Materials. Most builders use for the concrete the same proportions, 500 pounds of cement to 1.2 cubic yards of sand and gravel, which yield one cubic yard of concrete put in place . and tamped. In floors and beams and other members sub- jected to bending, the tensile resistance of the concrete is not taken into account. There is, therefore, little advan- tage in improving the character of the concrete in the parts in tension. It seems that such is not the case in the compressed portions where the concrete supplies the most, if not the whole, of the resistance. But the two fol- lowing points should be considered. If the concrete. for the space between the main beams is made richer, this will be done in order to decrease their thickness, but the mo- ment of resistance decreases as the square of the thick- ness, while the volume of the concrete decreases as its first power only. The cost decreases still more slowly than the volume because certain expenses remain the same, as, 178 REINFORCED CONCRETE. for instance, that of the moulds. The conditions are simi- lar if not identical for the main beams. Finally, it is known, and the author has proved it by exact data, that the cracks which appear in tension members exposed to dry air and especially to the sun while under the partial load which they carry during their hardening period, are the more important the richer the concrete is. Neither, of these reasons holds true for compression members, except the first in the quite rare case where the members are so long as to require provision against flexure. On the contrary, for compression members which will not be tested in advance, there is a decided reason for richer concrete. For the first load imposed on concrete the elastic limit, which has a considerable influence on the column resistance, increases in a high degree with the proportion of cement when the same is increased from 500 to 1,000 pounds. This fact is shown by all of the author's expert ments, and an example can be seen by comparing the de- formations reported in section 9 for three prisms of 1,000 pounds and one of 500 pounds of cement. No doubt the importance of the initial elastic limit de- creases after the testing of the members before putting them in place but it does not vanish altogether. This method of construction will only be used when high re- sistances with all possible guaranty will be desired, and the small increase in cost of a richer concrete will be will- ingly paid. The whole difference is not more than 10 to 20 per cent, of the cost of concrete. The author believes that in any case for members subjected to great stresses 750 pounds and frequently 1,000 pounds of cement per cubic yard of concrete laid in place should be used for a mixture of sand and gravel so balanced as to have the least amount of voids. The choice of the metal to be used also deserves atten- tion, and it should be here remembered that the longi- tudinal rods are compressed beforehand by the shrinking COMPRESSIVE RESISTANCE. 179 of the concrete and reach rapidly their elastic limit, white the hQops must be compressed before taking tension, and that their stresses are small when the longitudinal rein- forcing is already at the end of its resistance. The elonga- tions of the hoops increase, therefore, about three times as. slowly as the shortenings of the longitudinals. The experi- ments show that the greatest shortenings which will have to be -considered, not for working stresses but for dan- gerous limits, do not exceed 0.2 to 0.3 per cent., and it, is certain that the corresponding elongations of the hooping metal do not reach 0.07 to 0.1 per cent, and thus remain below the elastic elongation of common wrought iron and soft steel. The elastic limit and the ultimate resistance of the hoops is, therefore, not import- ant and their effect depends solely on the coefficient of elasticity. But it is known that the coefficient of elasticity of soft steel is on the average 10 per cent, higher than that of wrought iron, and that for the high carbon steels it does not have any higher value. Drawn wire will thus for hooping be inferior rather than superior to bars. These considerations lead to the use of high-carbon steel for the longitudinal reinforcing and of soft steel or wrought iron for the hoops, according to the cost of the metal. The rolling into spirals is facilitated by the use of malleable metals. It is now understood why the author has not attached much importance to the fact that the hooping of the tested prisms was made of drawn iron wire. ISTo doubt, with bars, the ultimate resistance would not have shown such an enormous and useless excess of crushing strength, but the interesting period of the deformation would have been exactly the same, and the conclusions, which have been based solely on facts observed during this period, would not have been different. It may be concluded that for hooped concrete intended to resist high pressures it is proper to use concrete con- ISO REINFORCED CONCRETE. taming 750 to 1,000 pounds of cement per cubic yard of concrete laid in place, and to use high-carbon steel for the longitudinal reinforcing and wrought iron or soft steel for the helicoidal spirals which form the hooping. Gen- erally it will suffice to make the metal between 1 and 2 per cent, of the volume for the longitudinal reinforcing, and between 2 and 3 per cent, for the hoops. This will give a high ductility and a resistance which will exceed all requirements. 18. Piest C'CST. The question of first cost is a very delicate one, and to find a complete solution for it long discussions are re- quired. The author will only attempt here to furnish some first thoughts on the subject. The study of first cost must lead to a comparison of the cost of the different types of construction, among which the engineer must choose. Hooped concrete must evidently be compared to concrete-steel as ordinarily reinforced and to riveted steel. If the structure considered is to be such as can be built c-f concrete-steel of any of the old types the advantages Which the substitution of hooped concrete under the most economic conditions will offer will have to be considered first. Then investigation should determine the desirability of adopting the more or less expensive methods which can be applied in the making of hooped concrete. For con- crete of the same proportions and same grade and percent- age of metal, hooping will give much more resistance than longitudinal reinforcing, and will at the same time modify the fragility to a very high ductility, which will allow of a reduction in the factor of safety. It does not seem that the cost per cubic yard containing the same materials should appreciably differ for the two types of construc- tion. There are, however, two reasons which might outweigh the general advantages of hooped concrete and lead to the COMPRESSIVE RESISTANCE. 181 adoption of the other type. The first of these reasons is that it is somewhat inconvenient to introduce for the col- umns a method of construction different from that used fcr beams, and to which the workmen are not accustomed. The second reason is that for columns or supports carry- ing light loads the least section is' generally fixed by practii- cal considerations and is frequently sufficiently strong with ■any kind of reinforcing. It is the shop manufacture of hooped-concrete members that must be mainly considered. It may be made in big shops with the special care and economy resulting from, the employment of skilled labor and mechanical means. Columns and other members of hooped concrete with longitudinal reinforcing rods project- ing out at the ends, so as to assure the connection between them and the beams or other parts of the structure, of high efficiency could then be manufactured. More difficult is the comparison of hooped concrete to riveted construction. It will be limited to some indica- tions of the cost of the materials of which the hooped con- crete is made and the additional cost of its making. The cost of the hooped concrete in place so determined will then be compared to the total cost of the steel construction of the same strength. It will then be seen whether the- difference in cost is sufficient to cover the handling, erec- tion and falsework required for hooped concrete as well as the contractor's profit. The fact must also be consid- ered that the longer the spans the heavier will be the weight of the hooped concrete as compared to steel, and that the fixed load of the structure will, therefore, be pro- portionately increased and its required resistance will have to be proportionately higher. But it will be found on closer examination that the difference in weights does not have the importance which it may seem to have at first sight. On the other hand, it should be considered whether the hooped-concrete structures do not have the advantage over the steel structures as to maintenance, 182 REINFORCED CONCRETE. length of service, solidarity of parts, and, for short dis- tances, rapid absorption of vibrations and resistance to impact. To make a rough comparison riveted construction of ordinary soft steel will be assumed. It can be bought for 3 cents a pound, which is a low value for present prices. An allowable working stress of 10,000 pounds per square inch will be assumed on the steel. The gross sec- tion of the members will exceed the net section by 10 to 20 per cent., and, if in addition the weight of rivets and details will be considered, the allowable unit stress of 10,000 pounds is reduced to 7,500 pounds per square inch of gross section as far as weight is concerned. For hooped-concrete, as given in section 14, a unit stress of 950 to 2,150 pounds may be allowed, according whether the concrete has been molded in place without extra care and reinforced with a low percentage of metal or whether it has been made in the shop with extra care and strict supervision of a good concrete and a higher percentage of metal, and has been previously tested above its working load. Omitting the cost of the moulds, staging, etc., the cost of the concrete may be taken at $10 per cubic yard, which is a high figure and is taken to be on the safe side. For the same reason the. higher unit pressures mentioned in the above as being allowable for hooped concrete will not be used and the following will be assumed : Allowable pressure in pounds per square inch 900 1,350 1,800 A block or short column one square foot in section will carry a load, in pounds, of. . . 129,600 194,400 259,200 A steel member of same resistance at 7,500 pounds per square inch will have a sec- tion, in square inches, of 17.3 26.0 34.6 It will weigh, per linear foot, in pounds. . . 59 88.5 116 At three cents per pound, it will cost, per < linear foot $1.77 $2.0© $3.54 COMPRESSIVE RESISTANCE. 183 Adding to the cost of the concrete the cost of the rein- forcing steel, the cost of the hooped-concrete members will be found. The percentage of the metal varies between 3 and 6 per cent, of the concrete, which gives 400 to 800 pounds of steel per cubic yard at a cost of $12 to $24. The total cost of the hooped concrete will then be $22 to $34 per cubic yard. Per cubic foot it will be $0.82 to $1.26. Comparing this to the above table it is seen that more than a sufficient margin is left for the cost of moulds, scaffolding, etc. 19. Conclusions. Hooped concrete has a high ductility, its crushing re- sistance is very high and exceeds the sum of the resistance of the concrete used for it, the resistance of longitudinal reinforcing rods stressed up to their elastic limit, and the resistance at the same rate of imaginary longitudinals representing a volume of 2.4 times that of the hooping metal. If the concrete were very poor and had only the cohesion strictly necessary to prevent crumbling away be- tween the spirals, its resistance consisting of the two latter elements would still be very great. The danger due to poor work in concreting which forms an objection to concrete- steel constructions is hence almost altogether avoided by the use of hooping. Under the first load the hooping has much less influence on the coefficient of elasticity than on the crushing resistance, but even for the most ordinary concretes the value of this coefficient is sufficient to pre- vent flexure under high loads in structures having mem- bers of usual sizes. When a hooped-concrete member is subjected to a test load its elastic limit increases up to the value of this load, and likewise the coefficient of elas- ticity under a heavy load is increased in a high degree, and the more so the less satisfactory its value has been ^previously. The concrete is thus improved to any required degree 184 REINFORCED CONCRETE. by the hoops which, due to their mode of action and co- efficient of efficiency, 2.4, can produce a high resistance without being much stressed themselves. The adhesion of the concrete to the metal which has such an important effect on members in bending is almost of no importance in hooped concrete. in compression. It is called into ac- tion at the ends of the spirals only, which are bent in toward the center of section where its resistance is much assisted. by the considerable friction developed in the core by the strong pressures exerted on all sides. To obtain this result it is necessary to space the hoops closely enough to each other to prevent the crumbling of the concrete be- tween them. This object is attained, without spacing the hoops excessively close, by placing against the inside but-. face of the hoops longitudinal bars which in their turn also increase the tensile resistance. The necessity of having the hoops sufficiently close to- each other leads to making them of wire or bars bent to spirals and embedded in the concrete near the surface. The reduced number of joints is made by simple over- lapping of the adjacent spirals and bending their ends, into the hooped core. The strength of the joints is thus made equal to. the strength of the members. The spirals are made beforehand, and it is, therefore, easy to check their dimensions before concreting and to make sure o£ the high character of the reinforcing which, as has been shown in the above, will suffice to secure the safety of the member even if the concrete should turn out of a lower grade than could have been foreseen. By decreasing the spacing between the spirals at the ends of the members. a local excess of crushing strength can be obtained of any desired amount and' also a considerable ductility. It is also easy to assure the transmission of tensile stresses from one member to another, 'and practical methods have been pointed out showing how to attain this result in a simple manner. The serious . objections which are attached • to COMPRESSIVE RESISTANCE. 185 joints in masonry and usual concrete-steel constructions can thus be entirely eliminated. It appears that for structures for which it is desired to obtain the highest resistance and safety together with the least weight the method of building by members made In the shop will have the following advantages : Elastic symmetry; execution of the work under supervision in well-prepared moulds by a few experienced workmen ; con- tinuous inspection and protection from the sun and the Tain; possibility of keeping the members in water or in moist air to prevent cracks ; opportunity to sound with the hammer, to weigh and even to test all the members, and to reject those below the standard; beginning the making of the members at the starting time of preliminary opera- tions; putting the members in place after they have been hardened. Where the members will be tested they will greatly improve in elasticity and column resistance. The deformations after the removal of the falsework will be much reduced, even with considerable working pressures. The necessary rules to compute the resistance and elas- ticity of a hooped member of any proportions have been given. The high ductility of hooped concrete, its resistance to impact and atmospheric influences and the general solidar- ity of the parts in concrete-steel structures logically lead to the adoption for hooped concrete of factors of safety in regard to the elastic limit and column flexure below 2 or 2.5 as are generally used for steel structures. But because of the novelty of this type of construction 3 to 3.5 has been proposed, even indicating the probability of begin- ning with factors of safety still higher. The reasons which sometimes prompt the use of smaller proportions of cement for concrete in tension or bending do not hold for com- pression members, and it is proper to use richer concretes for these members. It is advantageous to make longitudi- nal reinforcing rods of the highest steel that can be used 186 REINFORCED CONCRETE. without trouble. For the hooping, on the contrary, wrought iron or soft steel are perfectly suitable. Of two concrete members made of the same quantities of the same materials and costing the same, the one which is hooped will have more strength and a very much higher ductility than the one made according to any of the usual types. The approaching danger will in the first be, heralded by the scaling 'off of the surface which has no influence on the strength. In the second nothing will an* nounee the coming failure or, at best, heavy injuries which will endanger the structure will indicate it. The cost of members made of hooped concrete is ap- preciably less than one-half of the cost of steel members of same resistance. In each case the fixed load of the structure must be considered, the -cost of erecting the hooped-concrete members must be calculated, together with the general expenses and profits, and the different methods of construction then compared. Finally, the properties of hooped concrete appear to insure for it a place as a mate* rial for compression members between ordinary concrete and longitudinally or transversely reinforced concrete, on one side, and riveted steel members, on the other. 20. Additional Expekiments. Since the publication of the above chapter the author has made some additional experiments on hooped-concrete prisms. These experiments have all confirmed the results previously obtained, and some of the numerical values of these additional tests are given. 1. A prism 19.7 inches long of octagonal section, 4.3 inches in diameter, was made of concrete containing 840 pounds of Portland cement per cubic yard of gravel and sand mixed in the proportion of 1 to 3. The gravel had a greatest diameter of • 1 inch and the sand of 0,2 inch. The spirals were made of iron wire 0.17 inch in diameter wound around a cylinder of 3f inches diameter with a COMPRESSIVE RESISTANCE. 187 maximum pitch of 0.7 inch. Eight longitudinal reinforc- ing rods of the same wire were added. The total sectional area of the prism was 15.5 square inches. The sectional area of the core inclosed by the spirals was 11.16 square inches. The prism failed by bending as a column, with- out rupture, under a load of 142,000 pounds, correspond- ing to a pressure of 9,150 pounds per square inch of initial section and 12,700 pounds per square inch of the section of the core, which alone remained intact under the high pressures. The first cracks in the external layer of con- crete surrounding the spirals appeared under a pressure of 3,420 pounds per square inch. The shortening of the prism was not measured at this stage, but was 0.12 per cent, under a pressure of 286 pounds and increased to 2.4 per cent, of its length when the last measurement was taken before failure. An identical prism made of the same concrete, but without reinforcing, gave a resistance of 2,240 pounds per square inch. 2. A prism of the same length, 19.7 inches of octagonal section, 12.6 inches in diameter, was made of the same concrete as No. 1. The spirals were of iron wire 0.39 inch in diameter and wound around a cylinder of 10.6 inches diameter with a maximum pitch of 1.46 inches. Eight longitudinal rods of rolled iron 0.59 inch in diameter were added. The total section of the prism was 131 square inches. The sectional area of the core inclosed by the spirals was 88.5 square inches. The prism crushed under a total load of 820,600 pounds by the rupture of one of the spirals. This load corresponds to a pressure of 6,260 pounds per square inch of initial section and 9,280 pounds per square inch of core section. The first cracks appeared under a pressure of 285 pounds per square inch. The shortening of the prism before rupture amounted to 4.2 per cent, of its length. No comparative identical concrete prism was made with this one. Examination of the con- crete of this prism showed that it was not of as good a 188 REINFORCED CONCRETE. quality as prism No. 1. The iron wire of 0.39 inch diameter, which formed the spirals, had, of course, a less resistance per square inch of section than the wire of 0.17 inch diameter. It may be stated that the least results obtained exceeded the values as computed according to the rules 'given. 3. Though the fact that hollow prisms of hooped con- crete will not resist pressures well and will crush on the in- ner side could have been foretold, it was found to be neces- sary to verify it by experiment. Two cylinders of 23.6 inches length and an external diameter of 7.1 inches were made. The diameter of the hollow space was 4.9 inches for one cylinder and 4.5 inches for the other. The resist- ance was found to be 2,580 pounds per square inch of con- crete section for the first cylinder and 2,290 pounds for the second. The resistance of an identical concrete prism without reinforcing was 2,404 pounds per square inch. It was thus proved that hooped members must be of solid formation or have small openings only. CATALOGUE . OF. Engineering Books . . . ON . . . POWER TRANSMISSION ELECTRIC LIGHTING ELECTRIC RAILWAYS THE TELEPHONE THE TELEGRAPH DYNAMOS AND MOTORS ELECTRICAL MEASUREMENTS TR-ANSFORMERS STORAGE BATTERIES WATER. WORKS ■ BR.IDGES STEAM POWER. PLANTS VENTILATION AND HEATING STEAM AND HOT WATER. HEATING PLUMBING. Etc. PUBLISHED BY THE McGraw Publishing Co., Publishers, Importers and Booksellers, 114 Liberty Street. *» V New York City. McGRAW PUBLISHING COMPANY, BOOKS AND PAMPHLETS. American Street Railway Investments Financial data of over 1,300 American city, suburban and inter- urban electric railways, statistics of operation, details of plant and of equipment, and names of officers. Published annually. Vol. X, 1903 edition 5 00 We can still supply Vols. I (1894) to IX (190a) at $5.00 each. 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