(Smntll Winivmii^ pibat;g BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF Henrg M. Sage | SM&:^^^. siMii. 972* camall University Library The elements of reinforced concrete bull 3 1924 015 344 025 THE ELEMENTS OF REINFORCED CONCRETE BUILDING. By the same Author. ENGLISH CHURCH ARCHITECTURE From the Earliest Times to the Reformation. Profusely Illustrated by Sketches and Measured Drawings. Crown 8vo., 2/6 net. LONDON: FRANCIS GRIFFITHS. THE ELEMENTS OF REINFORCED CONCRETE BUILDING BY G. A. T. MIDDLETON, A.R.I.B.A. (Past Vice-President of the Society of Architects) m New York SPON & CHAMBERLAIN, 123-125 Liberty Street, 1910 CONTENTS. PAGE Preliminary Considerations . . . . . . i Chapter I. — Rectangular Beams with Supported Ends ; Bending Moments — Resistance Moments — Designing from given data . . . . . . 5 Chapter II. — Vertical Reinforcements . . ■ • 25 Chapter III. — T- and L-Beams with Supported Ends 31 Chapter IV. — Continuous Beams. . .. •■42 Chapter V. — Beams with Double Reinforcement 51 Chapter VI. — Floor Slabs . . . . . . 62 Chapter VII. — Foundations . . . . • • 69 Chapter VIII. — Column's . . . . . . ■ • 75 Chapter IX. — Walls and Retaining Walls . . 8r Chapter X. — Materials ; Mixing and Laying the Concrete — Falsework, or Centering — Connec- tions . . . . . . . . . . . . 87 Chapter XI. — Some Patent Systems . . • • 99 PREFACE. Though many books have been published on the subject of reinforced concrete, none of them have hitherto dealt with the elementary underlying principles thereof in an introductory spirit. It was to meet the requirements of the rudimentary student that the articles which have now been put together in volume form were first prepared and published in the " Building News." The hook must, therefore, be considered as introductory rather than exhaustive, it being intended to clear the ground for further investigation by such readers as may feel inclined thereto. G. A. T. MIDDLETON March, 1909. The Elements of Reinforced Concrete Buildinor. PRELIMINARY CONSIDERATIONS. THROUGHOUT this Httle book it will be assumed that readers are acquainted with the elementary- principles of statics as apphed to the determination of stresses in girders, cantilevers, and roofs. These principles remain the same, whatever materials are employed for the purpose of resisting the stresses set up ; it is only the application which differs. Hitherto — that is, until the last ten years or so — every member in a structure has consisted •of a single material, whose resistance to stresses, whether compressional, tensional, or shear, has been well known. Even when compound structures of steel and concrete were erected, each material had its particular function to per- form, steel girders being designed as such to carry the ^^'hole weight brought on them by concrete slabs which served as floors, and the concrete itself being treated independently, so far as the small floor spans were concerned. In some comparatively rare and recent instances, the fact that a steel girder was encased in concrete for the purpose of fire resistance was considered as adding stiffness to the girder, with the result that its section was reduced, and the strength was still found to be sufficient for its purpose. This was particularly the case with small steel joisl.s em- bedded in floors. But this system of construction is some- thing essentially different upon that of reinforced concrete, in which steel rods are embedded in the more bulk}^ material, not so much in order to protect them from the effect of 2 ELEMENTS OF REINFORCED CONCRETE. fire as to add the power of resisting tension where tensional stress is set up. Elementarily, the idea seems simple enough; but it is soon found to be complicated in practice by many considerations about which it is necessary to have some knowledge before the subject can be properly investigated, even from the theoretical point of view. To begin with, the combination of steel and concrete in this way does not produce a single homogeneous material, but a compound one. Before theoretical considerations can be met at all, it is necessary that there should be adhesion between the steel and the concrete. This, by means of careful work- manship, it may generally be assumed that there is. TheR the concrete is by no means necessarily uniform, either in composition or strength. It is impossible to test it in the same way that steel is tested ; it is itself a compound, dependent for all its essential qualities, not upon one factor, but upon many. It is certainly possible to test the cement, but even moderate uniformity of the mass can only be obtained by using a standard aggregate as well as a standard cement, and by extremely careful mixing in order to produce an approximately standard result, an absolute standard being unobtainable. It is now generally assumed that concrete in compression will safely resist 6oolb. per square inch. It is, however, more convenient, as a rule, to calculate in tons than in pounds, and as this is not much more than a quarter of a ton, that value will be assumed herein, any error being upon the side of safety. Unfortunately, its resistance to- shear is so extremely slight, only 6olb. per square inch being safely allowable, that it is infinitely best to neglect it altogether, and to trust entirely to steel. This introduces a new problem, for, although the shears are the same in reinforced beams as in any other, they have to be resisted PRELIMINARY CONSIDERATIONS. 3 in quite a new manner. The adhesion of concrete to metal may be safely taken as loolb. per square inch. This is sufficient for no special calculations to be necessary, as, for all practical purposes, it may be considered as perfect. The resistance of concrete to tension may be absolutely neglected. The whole object of reinforcement is to introduce steel for this purpose. Steel may be safely loaded in tension up to 7J tons per square inch, if it complies, as it should do, with the specification of the Engineering Standards Committee. The modulus of elasticity (E) of steel is approximately ten times that of concrete ; and this ratio is, as will be seen later, of great importance. The question of adhesion between cement and concrete is one to which a very great deal of attention has been given, and many patent forms of bar have been introduced in order to secure it. This matter, however, will be deali" with at a later stage. In one respect only are steel and concrete practically equal, and that is in their expansion under heat. This is fortunate, as it enables the materials to be used in combina- tion with the assurance that they will not separate from one another when a fire occurs. Another important point which has a bearing upon the theory of slabs and beams is the fact that these are fre- quently used in continuous connection, and that the inter- mediate supports are themselves beams which may very well be wanting in the absolute rigidity which is given by a wall or some other solid bearing. This introduces an element of uncertainty which goes far to upset all accepted theory. Allowance has to be made for it, just as for possible imperfections in the concrete. This can only be done by 4 ELEMENTS OF REINFORCED CONCRETE. " rule of thumb," or, to put it into more polite language, by the use of experienced judgment. Yet another consideration is introduced when economy has in any degree to be considered, and that is the cost of erecting the necessary staging to carry the concrete until it has fully set. Theory may even dictate that, in the same building, different sizes of beams may be used in different positions, when, on the other hand, practical experience will say that it is more economical to use the timbering over and over again than to make up, say, different girder boxes for each girder. The two things have to be put against one another, and it is generally found that it is economical in the long run to make the girders larger in some cases than they need be, in order to standardise them and utilise the same boxings. Although externally alike, however, there is no reason why the reinforcement should not vary according to the respective loads, and the calculations then are based upon a known size of girder ; that is, the per- centage of steel is altered, while the girder remains the same externally, instead of everything being designed afresh for each particular load and span, to meet the theoretic needs of every case. CHAPTER I. RECTANGULAR BEAMS WITH SUPPORTED ENDS ; BENDING MOMENTS— RESISTANCE MOMENTS- DESIGNING FROM GIVEN DATA. SIMPLE rectangular beams of reinforced concrete, having the ends merely supported, and not rigidly fixed, are designed, like those of any other substance, so that the Moment of Resistance shall equate with or exceed the Bending Moment. As a rule, it is the maximum B.M. only which is considered, as the reinforcement, once decided upon, is continued uniformly throughout the length of the beam, it being cheaper and safer to do this than to run the risk of short rods being embedded in the wrong places, while greater homogeneity is secured also. It is consequently of httle importance where the maximum B.M. occurs, so long as its value is known. In the case (see Fig. i) of a fixed load, \^', carried by a girder having a span, I, at a point at distance x from the left and y from the right-hand abutment, the maximum B.M., immediately under the point of application of the 6 ELEMENTS OF REINFORCED CONCRETE. load, is R^ X X or Rj X y, where Rj and R2 represent the reactions of the abutments. Now, Wy W a; R, = and R, = I I W yx W xy . 13 A/T ^- , , r> ivi — or I I which come to the same thing. When W acts at the centre of the span, then x = y I and, consequently- — 1 I W X — X — 2 2 W/ RM 2 I 4 A httle examination of Fig. i will show that the further W is located from the centre of the span, the less becomes the maximum B.M. due to it, until the B.M. disappears, when the load is placed directly over either support. It follows that, if the beam be loaded throughout its length by a series of infinitely small loads infinitely close together, the total maximum B.M., which occurs in the centre, is not the same as if a single load of equal value to the total were employed. If, again, the total load — that is, the sum of all the infinitely small loads — be called W,the maximum w; BM = — 8 or just half what it would have been had the same load been single and located centrally. RECTANGULAR BEAMS. 7 WTien a girder carries se\'eral loads of different \-alues at different points, it is easier to determine the B.il. graphically than by calculation. An example is shown in Fig. 2, the girder being loaded with 5, 8, and 7 tons (or ■cw-ts., or lbs., according to the unit of weight to be used) at the several points indicated on the sketch. It is first necessary' to ascertain the reactions. To do this, lines are first drawn vertically downwards through the p>oints of ■appUcation of the loads, and the spaces between the \'arious forces are lettered. Thus, beginning at the left-hand side, the space between R^ and the hne below the load 5 is lettered a, the space between the lines below the loads 5 and S is lettered h, the next space c, and the next d. A \ertical line is dropped, and this is divided oft to scale, to represent the loads, in sequence downwards, as they occur from left to right on the girder. Thus, a distance of 5 units to scale is set vertically downwards from a to h, to represent the load of 5 tons lying between the spaces a and h ; a distance ■of 8 units from h to c, to represent the load of 8 tons between b and c ; and a distance of 7 units from c to d, to represent the load of 7 tons between c and d. Any point, o (known £is the pole), not being in the vertical line a — d, is now selected, and a line — a is drawn •(numbered i on the diagram), joining and a. A line parallel to this, also numbered i, is then drawn on the -sketch, between the hne of R^ and the Hne of the load of 5 tons, separating space a from a new space to be lettered •0, to agree with the lettering of the pole. Similarly, line 2 is drawn from to 6 on the diagram, and Une 2 is drawn parallel to it on the sketch, separating ■space h from space o. Lines 3 and 4 are similarlj- drawn on both diagram and sketch. 8 ELEMENTS OF REINFORCED CONCRETE. The extremities of lines i and 4, where they meet the reaction lines on the sketch, are now joined by the dot-and- j stroke line 5, and a similar and parallel line 5 is drawn , on the diagram. On the sketch this new line 5 separates,' space o from an outer space x, while on the polar diagram/ it fixes the point, x, which determines the value of the tw(^ reactions, R2 from d, to x, and R^ from x to a, by scale. This- completes the Polar Diagram for Reactions. Another vertical load-line is now set down as the basis of a second Polar Diagram, the points a-^, h^, x^, c^, and d^ being projected horizontally to it from their positions on the Polar Diagram for Reactions, and locating points a^, bi, «j, Cj, and d-^, respectively ; and a new pole, P, is located in a horizontal direction from x^, and at a distance there- from equal to one unit of the length-scale to which the girder-span is drawn on the sketch, and in which the B.M.'s are required. Thus, if the distance component of the B.M. is desired in feet, the length x^ — P must equal ift. to the scale to which the girder is drawn on the sketch ; if it be desired in inches, the length must equal lin., and so on. Lines are now drawn vertically upwards on the sketch through the points of application of the loads, and the space P, representing the pole, P, occurs above the hori- zontal girder line, just as the space x-^ occurs below it. Thus the horizontal girder line separates space P from space A'l, and the similarly lettered points P and x^ are joined by the horizontal line 6 on the diagram. P is now joined to a^ b^, q, and d^, by lines 7, 8, 9, and 10 ; and space P is separated by parallel lines to these, similarly numbered, from spaces a^, b-^, c-y, and di, in regular sequence. If the draughtsmanship has been accurate the line 10 will close at the extremity of line 6, where it joins the right-hand abutment. 8 O, / i \ / MAX. B.M. I \ I -a- |.>t ■^. +p. SKETCH . \^ f P, It \ I r ,1 / ^^^ 5 \ \ \ \ \ .1 Fblar DiaqrAm fop Reactions. \ 5 8 c 10 i « — I,- p, <- 17. ^f polar Diagram -fcr ' rolar uiagram tor Polar diaqram Tor D.Mi Equivalenl" Ceatral Load Kig 2. [face I'AGK 7. RECTANGULAR BEAMS. ' 9 The outline made by the lines 7, 8, 9, and 10 on the sketch now denotes the values of the B.M.'s at all points along the girder, by the length of the various ordinates to it from the girder line — to the scale of loads employed. Thus if the vertical scale of loads employed on the Polar Diagrams be 8 tons per inch, the B.M.'s, as indicated by the outline above the girder line on sketch, will be 8 per' inch in inch-tons, or foot-tons, or any other length-unit x tons, according, to the unit represented by x^^ P on the- Polar Diagram for B.M.'s. It consequently follows that if the Hrte x-^ P be set out. to a length equal to 2ft. on the scale of sketch, the loads being in tons, then the B.M.'s are ascertained in values, of 2-foot-tons, and so on. This is often a great convenience, as if the length x^ P be too short, the radial lines on the Polar Diagram become so acute that it is difficult to draw accurate parallels, and to make the eventual B.M. outline- close properly. In such a case, x^^ P may be set out as 2ft., 3ft., or more, and the B.M.'s thus ascertained multiphed. by 2, 3, etc., correspondingly, in order to reduce them to- foot-tons. It will be noticed that the greatest ordinate above the girder line represents the maximum B.M. There are many occasions when it is useful to discover what single load, centrally placed, will produce the same maximum B.M. as many loads, such a single load being commonly known as the " Equivalent central load." To ascertain this, the max. B.M. is set up over the centre of the span, as shown in the supplementary sketch to the right of the principal sketch, and lines 11 and 12 are drawn from its top to the extremities of the girder, enclosing a new space, Pj, represented by Pj on a new Polar Diagram, X2 Pi exactly corresponding with x^ P. Line 11 is now drawn rxo ELEMENTS OF REINFORCED CONCRETE. irom Pj on this diagram parallel to line ii on the new ■sketch, till it reaches the load line at e, the line P^ e thus corresponding with the line between spaces T-y and e on the new sketch. Line 12 is similarly drawn parallel to hne 12, and similarly determines / on the load line. Then e f on the load line represents, to the scale of loads, a single •central load, such as would give the same max. B.M. as has already been found to be that due to many loads previously applied. In other words, e / is their " Equivalent central load." The " Equivalent distributed load " is twice this, or 2 e f. In practice, it is customary to super-impose the three polar diagrams and the two sketches which have been ■severed for the sake of clearness, preferably working each in a different colour. Beam or girder designing depends to a large extent upon the satisfactory equating of the maximum Bending Moment with the Moment of Resistance of a vertical section, taking the safe values of resistance of the material employed both to compression and tension. In beams of reinforced concrete the calculations are complicated, owing to the neutral axis occurring, as a rule, elsewhere than through the centre of gravity of the section, its position \-arying according to the proportion of reinforcement employed. Theoretically, the greatest advantage of reinforcement is obtained when the extreme layer of concrete in the com- pression, or upper, side of the neutral axis is placed under the extreme safe compression of J ton per square inch, the steel on the tension, or lower side of the neutral axis being simultaneously placed under the extreme safe tension of 7 J tons per square inch. All concrete below the neutral axis is neglected, its tensional resistance being slight ; but the centre of reinforcement is generally taken instead of RECTANGULAR BEAMS. ir its outer layer, roughly compensating for the neglect of the •concrete, and even then, as experience shows, leaving a margin on the safe side. The depth, d, may, therefore, be taken as from top of beam to centre of reinforcement, as shown in Fig. 3. If the distance of the neutral axis from top of beam be denoted by x, then its distance from centre of reinforcement will he. d- x. Then X : d - X Extreme safe resistance of concrete Modulus of elasticity of concrete Extreme safe resistance of steel ' Modulus of elasticity of steel J ton 7^ tons .'. X : d- X : : - — : E. E, But the modulus of elasticity (E,) of steel is ten times that of concrete (EJ. Therefore E,= 10 E„. .' . X . d - X . : - : — I 10 •• : i : f d .'. a; = — 4 This may be determined graphically, as shown in Fig. 3, by setting the value - — to scale along the top of the beam fjX tons io the right of a vertical line, and the value - — to the E« ■same scale along the centre hne of reinforcement to the left of the same line, and joining the extremities of these 12 ELEMENTS OF REINFORCED CONCRETE. FiR. 3- scale values. The joining line will intersect the vertical hne where the neutral axis occurs. The beam may now be assumed to have the section shown in Fig. 4, with a width b, and an effective sectional area b d, everything below the centre of the reinforcement being neglected. The extreme topmost layer of concrete being under compression to the utmost permissible extent. RECTANGULAR BEAMS. 13 each layer which Ues nearer to the neutral axis is under a less compression, until this becomes nil when the neutral axis is reached. The result is approximately the same as if there were a mass of concrete lying above the neutral axis, contained in a parabohc outline, as indicated in Fig. 4, with every layer equally under compression to the utmost o. h % permissible extent. The sectional area of this mass = 3 T arm of momeai" of res/'s/Sace . Fig. 4- 14 ELEMENTS OF REINFORCED CONCRETE. The resistance of this mass to compression may be con- sidered to be located at its centre of gravity, which occurs. at a distance of — from the topmost layer, and of — from the neutral axis. The distance between this and the centre of the reinforcement (where the tensional resistance- of the steel may be considered to be located) is, conse- quently, d-- This is the lever-arm of the Moment of 5 Resistance. When X = — (that is, as already determined, when the 4 extreme layers of concrete and steel are under the extreme permissible stresses of J ton and 7^ tons per square inch^ respectively), then this lever-arm becomes 2 d 4 A <^d 5 10 10 Similarly, the parabola-bounded sectional area of con- ^ 2&.1;, zh d h d „ Crete, becomes = For convenience, the sec- 3 12 6 tional area of the concrete h d may be designated a„, and that of the steel, a^. Then the area of concrete — = — - 6 6 The Moment of Resistance of the concrete is then the product of this area in square inches, the resistance per square inch, and the lever-arm. In other words, in the case under consideration M.R. = - X J ton x^— = --- = ^—^ — inch tons. 6 10 240 80 RECTANGULAR BEAMS. 15 Similarly, the Moment of Resistance of the steel is the- product of the area of steel in square inches, the resistance per square inch, and the lever-arm. In other words, in the same case, M.R. = a,X7itonsx^'=^l^ 10 4 Consequently, as the M.R. of the concrete = the M.R. of the steel, 3 a^ d 27 a^ d 80 4 .'. fl,. = 180 ffg. Thus, to meet the theoretical advantage of concrete resisting safely {--ton per square inch in compression, and steel resisting 7 J tons per square inch in tension, being- both stressed to their utmost safe limit in their extreme fibres, a beam is produced whose steel has only — per cent, of the area of the concrete, as measured from top of beam to centre of reinforcement. Practically, so small a proportion is rarely used. It would not be economical. It may, however, be considered as an extreme case — that in which the neutral axis is si tuated at a distance — from 4 the top of the beam. If the proportion of steel be increased, the position of the neutral axis is lowered, and the stress per square inch, on the steel is decreased, the understanding being that the stress per square inch on the concrete may not be increased beyond J ton per square inch. It is as rare to bring the 36 ELEMENTS OF REINFORCED CONCRETE. meutral axis below - from the top of the beam as it is to 2 place it nearer thereto than — This may then be con- 4 ^idered as the other extreme case, and, as such, is worthy ■of consideration. As is shown graphically in Fig. 5 : — "^Toa Ibp of be&m. ReiniorcemeaK Fis- 5- RECTANGULAR BEAMS. 17 J-ton y tons and as E, = 10 E„- E» E, \-\.or\ y tons , 10 E„ X i-ton , ^ .•. y tons = = = 2| tons This shows that when a sufficient proportion of reinforce- ment is used to bring down the neutral axis to halfway between its centre and top of the beam, a much weaker material' than steel, which is capable of safely resisting 7I tons per square inch, may be used, provided that it has the same Modulus of Elasticity, namely, 10 E^,. This con- dition is met by wrought iron, though it is rarely employed, as it is not commercially procurable now in rod or bar form much, if at all, cheaper than steel. When the neutral axis is thus placed, so that x = — then the parabola-bounded 2 area of concrete (see Fig. 4) becomes — 2 bd 2 b d a„ 3 3 3 and the lever-arm of the Moment of Resistance becomes — 2 d 2 4^ 5 5 i8 ELEMENTS OF REINFORCED CONCRETE. Then the M.R. of the concrete is — a,. ^d - X i-ton X — 3 5 and that of steel is — flg X 2j tons X — 5 Equating these — «„ ^d ^d - X J-ton X — = a^ X 2\ tons X — 3 5 5 .-. — = 2i a, .'. fl„ = 30 a, 12 In such a case, the proportion of steel used is 3.3 per cent, of the area of concrete, as measured from top of beam, to centre of reinforcement. It is a high proportion, but not uneconomical, and by no means infrequently employed. So far, at any rate, as the concrete and the main reinforce- ment — the tensional rods — are concerned, enough is now known to enable a beam to be designed, under given con- ditions of loading and depth, if its ends be merely supported. A simple case will serve as well for illustration as any other — say, that of a beam having a span of 20ft. from centre to centre of bearing, carrying an evenly-distributed load of 10 tons {\ ton per ft. run), or its equivalent central load of 5 tons, or any number of other loads placed indis- criminately, but equivalent thereto. It may also be assumed that this is the most heavily-loaded of several beams over similar spans which it will be economical and convenient — and sightly — to make of the same section. It will, consequently, be the most heavily reinforced, and RECTANGULAR BEAMS. 19 may very well carry 3^ per cent, of steel in its section, calculated above the C. of reinforcement. As in the case of a steel girder, it is first necessary to determine the depth, d, and also, as in such a case, this should not be less than ^ of the span, /, if excessive deflection is to be avoided, while it is rareh' more than — 12 This does not determine the absolute total depth, but that from top of beam to C. of reinforcement — that is, the theoretically total depth, any concrete added below this level being merely placed there in order to protect the steel against fire, its tensile resistance being negligible. Taking d = ~ , then in this case ^= 2oin., I being 24oin. 12 Fig 6. 2 ELEMENTS OF REINFORCED CONCRETE. Then the lever arm of the M.R. Ad 4x20 ,. 5 5 for, as aheady explained, a reinforcement of 3J per cent. involves a centrally-placed neutral axis. bd Then the area of concrete above the neutral axis = — 2 of which two-thirds is operative within the parabolic outline (Fig. 6), and has a safe compressional resistance of J ton per square inch. Therefore the M.R. of the concrete — bd = — X i ton X i6in. 3 6 X 20 = X i ton X i6in. This equates with the B.M., which is — W / 10 tons X 24oin. or — ~ - X 20 , ^ ^ 10 tons X 240in. X i ton X 16 = 3 .". Sob = goo b = iijin. The area of concrete above C. of reinforcement is, there- fore, ii| X 20 = 225 sq. in. ; and the area of steel to be 22^ provided is 3^ per cent, of this, or — =■■ yl sq in 30 ■"' ^ This can be provided either by three square bars, ijin. by i^in., as shown blacked in on Fig. 6 ; or by four circular RECTANGULAR BEAMS. 21 rods, two of liin. and two of ifin. diameter, as shown in open circles ; or by Kahn or any other patent bars holding the same total sectional area. Of course, in either case, the steel must be entirely embedded in concrete, in order to secure perfect adherence, and to a sufficient thickness to protect it against the effect of fire. A thickness of i^in. is generally considered to be enough, but careful men prefer to use 2in. — at any rate wherever a high temperature may be anticipated. So far, all has been plane sailing, though it will be noticed that more steel is needed in the reinforcing rods than would have been required in the whloe of a steel joist of similar depth to carry the same load, this being due to the fact that the steel is only effective to resist 2^ tons per sq.in. in tension (as explained in relation to Fig. 5), whereas it is capable of resisting 7^ tons safely, and can be employed to do so in a steel joist, or built-up girder. It consequently would appea-r at first sight that it would be simpler and cheaper to use a steel joist, and dispense with the concrete, until it is remembered that the concrete is needed for fire resistance in many cases, and that simple rectangular girders, such as this, are rarely met with, a concrete flange, consisting of part of the floor slab which the girder carries, materially altering the case. This is a more com- plicated problem, however, not ripe for discussion at this stage. The next matter for present consideration is a beam of the same span (20ft.) and of the same external dimensions {b d = 11^ X 20), carrying a lesser load — say, an evenly distributed load of yi tons, or its equivalent. The B.M. is then— 7J X 240 _. , = 225m.-tons. 3^-P ^ 1 d o :^ i__j 22 RECTANGULAR BEAMS. 23 This equates with the M.R. of the concrete, which is (see Fig. 7)— 2b X X iton X 3 ^5 2 X Tl\x (-7) X i X 3 ^5 15 X 100 -2X = X 8 5 By solving the resulting quadratic equation of B.M. "with M.R., it is ascertained that x = 7in. almost exactly ; and d- X is consequently 20 - 7 = I3in. (See Fig. 7.). Then if y = the stress in the steel per square inch ■equivalent to these conditions, then — y i ton : 13 : : _^ : 7 E« E, Then, as E^ = 10 E^, by multiplying extremes and means — y i X 7 = — X 13 10 E, E, y i X 13 10 7 .-. y — 4^»^ tons. B.M Now, the total tensional stress to be resisted is , 2x a - — 5 according to the usual law of the equation ot stress and 24 ELEMENTS OF REINFORCED CONCRETE. bending moments. As d = 20 and a; = 7, while B.M. = 225in.-tons, this amounts to 13J tons. The amount of steel necessary is consequently —^ sq. in., 4tt or a trifle under 3 sq. in., which can easily be provided by two ijin. X ijin. square bars ; or by four circular rods — each of fin. diameter — as shown in black squares and open circles respectively in Fig. 7. The reinforcement, it will be noticed, is still subject to so slight a tensional stress, less than 5 tons per sq. in., that wrought iron might safely be used instead of steel, while the proportion of metal to concrete is as 3 : 225 sq. in., or nearly ij per cent. CHAPTER 11. \^ERTICAL REINFORCEMENTS. EXPERIENCE and experiments alike have demon- strated that reinforced concrete beams and slabs, when they fail, do so almost invariably by curved cracks near the abutments, as indicated in Fig. 8, following- very closely the theoretical lines of maximum compressional stresses as given by Rankin, and shown in Fig. 9. These are at right angles to the theoretical lines of max. tensional stresses — stresses compounded of the direct horizontal tension and the tensional component of the shear — as alsa shown in Fig. 9. With regard to this, Mr. A. W. Buel, in the well-known American book on the subject, " Reinforced :^\ Fig. 8. Concrete Construction," says : " The hues of compressive stress cross the Hues of tensile stress at the neutral axis, making angles of 45deg. with the latter. The intensity of stress on these lines is maximum where they are horizontal — at the top of the beam for the compressive-, and at the bottom for the tensile-stress lines — and are zero when they become vertical at the bottom or top." From this Mr. Buel argues that any cracks near the centre of span are due primarily to the tension of flexure, and are extended by the tensile stress of shear acting normally to the crack,, and tending to pull the concrete apart, while any cracks: near the abutments would most hkely be primarily due to 26 ELEMENTS OF REINFORCED CONCRETE. longitudinal shearing between the reinforcing bars and the concrete, which is maximum towards the ends. I Cenfre Lime rig. 9. Whether Mr. Buel is entirely right in this contention it is almost impossible to determine ; but, at any rate, it is obviously necessary to, as it were, lace the concrete with reinforcement approximately at right angles to the probable ■direction of the cracks — in other words, as nearly as may be along the lines of max. tensional stresses, as shown in Fig. 9. As there are obvious practical difficulties in the way of introducing curved reinforcing-rods which shall follow the direction of these hues, it is usual to employ vertical " stirrups,'' as they are called, or else diagonal ties made by turning up, at the required angle, flanged portions of the main reinforcing-rods, of patented sections, specially •designed for this purpose. Vertical stirrups are more .generally used — they are not necessarily of any patent form — one being placed close to each abutment, propor- tioned so as to resist the shear, which at this point is equal "to the reaction of that support. In almost all cases where reinforced concrete girders are used, the load may be considered to be uniformly distributed. In the rare instances when this is not the case, it simplifies matters so greatly to use the equivalent dis- tributed load, that this is almost invariably done. The VERTICAL REINFORCEMENTS. 27 stirrup next the support is therefore nearly always propor- tioned to resist half the calculated distributed load — or else half the equivalent distributed load — acting as a shear. Thus, for a beam having a span of 2oft., and carrying an evenly-distributed load of 10 tons, as assumed in connec- tion with Figs. 6 and 7, the end sets of stirrups would each have to be capable of safely resisting 5 tons in tension. If made of steel, safelj' resisting 7J tons per square inch, they would therefore have to contain f sq. in. in each set ; and as there would, in the same cross-vertical section, be a stirrup attached to each reinforcing rod, and each stirrup is frequently double, this would be provided by either two stirrups of two parallel lin. by f\in. pieces of hoop steel, or possibly by enough vertically-placed wires having the same total sectional area ; wires having the advantage over' hoop-metal that they can be easily twisted round, and so somewhat rigidly attached to, the main reinforce- ment. To determine the number of sets of such stirrups neces- sary on each side of the centre of the span, it is necessary to divide the max. tensional stress by the safe resistance of a set of stirrups to tension. Practically, however, the divisor used should be the shear at the abutment, thus leaving any balance between this and the resistance of the stirrups which are provided as an additional margin of safety. Now, the max. tensional stress — . _ B .M. Lever arm of the M.R. Reverting to the examples given in Figs. 6, and 7, it will be remembered that the max. B.M. was 225in.-tons, while the lever-arm in the one case was i6in., and in the other I3in. Thus, with a stirrup-resistance of 5 tons, the number 28 ELEMENTS OF REINFORCED CONCRETE. of stirrups each side of the centre (counting the centre each time) will be, taking the nearest unit in excess in each instance — In case Fig. 6. ¥/ - 5 In case Fig. 7. 225 . „ TF — 5 = 3 = 4 or 5 stirrups in all. or 7 stirrups in all. The latter of these cases will be considered here for exemplification. — i = cfoc'of abutrrxnf Fig. How these stirrups are theoretically located longitudinally is shown in Fig. 10. The max. tensional stress is set up to any scale vertically over the centre of the span, and a VERTICAL REINFORCEMENTS. 29 parabola is constructed as shown, indicating the stresses at other points along the beam by ordinates from the horizontal line. Then vertical distances are set up above the beam, each equal to the shear at abutment (in this case 5 tons), and horizontals are drawn, cutting the parabolic ■outline where shown. Verticals are then dropped, meeting the horizontal Une of the beam at corresponding points. These are the spots where the stirrups should be introduced, in addition to one at the centre. Alternatively, the spaces between the stirrups may all be ■equal, and the sectional area thereof determined propor- tionally to the stress as then found (see Fig. 11). 30 ELEMENTS OF REINFORCED CONCRETE. Thus, points aj and b^ are found as before, but the distance ^1 — Cg is made equal to a^ — b^. A vertical is then raised from C2 to the parabolic outline, which is met at Cg. The vertical distance between the horizontal lines through b and Cg then represents the tension in the set of stirrups at Cj, which may be proportioned accordingly. There are often practical difficulties in the way of insur- ing that workmen put the stirrups exactly where determined by Fig. 10 ; and just as great in varying the sizes of the stirrups and insuring that the right ones are always used in the right places. Consequently, it is a common practice, and a safe one, to adopt the method of Fig. 11, so far as the spacing is concerned, but to make all the stirrups of uniform section, as determined for the positions a^ and b^. CHAPTER III. T- AND L-BEAMS WITH SUPPORTED ENDS. FROM what has been already said, it is evident that somewhat large rectangular beams, heavily rein- forced, are required in order to carry even compara- tively hght loads ; and if beams of this form were commonly employed it might be reasonably questioned whether they could not be more economically produced as steel girders- embedded in concrete than in true reinforced concrete. As a matter of fact, however, beams of rectangular section are extremely rare. It has only been necessary to consider them first because the problems involved in designing them are simple, and need to be mastered before those of the more common T-beam are investigated. In general, beams are required to carry floors, and even when they carry walls there are usually floors resting on them also. These floors, like the beams, are of reinforced concrete ; and, moreover, they do not merely rest on the beams, but are homogeneously connected therewith, con- crete to concrete, if. not reinforcement to reinforcement. Almost invariably the reinforcement of the floors runs from beam to beam, so that the stresses in the floors are at right angles to those in the beam. As a result, the concrete in the floor-slabs is available for use to resist compression in the upper part of the beam. In other words, the beam is. no longer of rectangular section, consisting only of the portion which projects below the floor - slabs, but is of T-section, the portion below the floor being merely the vertical flange, introduced to contain and form connection with the reinforcement, while half of the concrete in the floor-slabs to right and left theoretically forms the table 32 ELEMENTS OF REINFORCED CONCRETE. of the T, as indicated in Fig. 12. (Practically, this state- ment needs modification, but consideration of this point must be deferred for some considerable time yet) . ^ 10-0' ^ — M Li X — \6-o" 5*f- /o'o"- - ^ Occasionally, though much more rarely, the reinforce- ment of the floor-slabs lies parallel to the beams — this ■occurring, for instance, in relation to large main beams, ■which carry the smaller ones, which primarily support the floors. Even then, however, the concrete in the floor-slabs jnay, to a certain, and even a considerable, extent, be con- sidered as offering resistance to the compression of flexure in the beam, it being only a small proportion of the upper part of the floor-slab section which is being utilised to resist the compression of flexure in itself. In such a case it is usually safe to assume that one-fourth of the concrete in the floor-slabs to right and left of the beam forms the table of the T, as shown in Fig. 13. It is abundantly clear that T-beams, in which the resistance of the flooring-slab to compression is taken advantage of, as in Figs. 12 and 13, are infinitely more economical than simple rectangular beams. T- AND L-BEA.MS. 33 To take an illustration, let it be supposed that a long room, 2oft. wde, \\ith another similar room over it, has the floor of the upper room constructed as shown in Fig. 12, with reinforced floor-slabs 4in. thick, resting on and con- nected to reinforced beams spaced at loft. centres. The reinforcement of the floor-slabs would be placed across from beam to beam, and would comply with the conditions of Fig. 12 ; while the beams may be supposed to rest on their supporting walls, and not to be rigidly attached thereto. If the floor be expected to carr\-, as a ma.ximum, 2cwt. per square foot (which is a fair load on a light warehouse floor, but insufficient allowance for heav\- goods), this ^^•ould gi\e i ton per foot run on each girder. The beam would probably have 6in. or gin. bearing on the supporting waUs. Taking it as gin., the theoretic span becomes 20ft. gin., and the theoretic load 2o| tons. This is even!}- distributed throughout its length, so that the max. B.M.= = — — t ft. -tons = 647in. -tons. 8 8 -^ The depth of the beam, from top of concrete to C. of reinforcement, may be taken as clear span I = — = I2in. 20 20 the thickness of the floor-slab being included in this. «- lo'o'' ^ ±. Fig. 14 34 ELEMENTS OF REINFORCED CONCRETE. If it be assumed that the neutral axis lies within the floor-slab — which should be arranged for, if possible — the problem, as shown in Fig. 14, is an easy one, corre- sponding very closely with that already considered in Fig. 7 ; for, taking x as the distance of neutral axis from top of concrete, then the area of concrete within the para- 2 & jv bolic outline is & being i2oin. ; and the lever-arm 3 2 % of the Moment of Resistance is 12 - — . Then, taking the safe resistance of concrete to compression to be J-ton per square inch, and equating M.R. and B.M., we have G-t) 2,h X X i X I 12 J = 647 Solving this quadratic equation, we have A; = 3in. As it happens, this exactly corresponds with the position of the neutral axis, - , from the top of the concrete, as 4 already ascertained, in accordance with Fig. 3, to be that which is theoretically most advantageous ; but this will not often occur. As it is, we know that enough steel has to be introduced, at 7J tons per square inch, to meet the tensional stress, and this (the stress) is equal to B.M. 647 2 X 2X3 d 12 5 5 647 ,. = = 60 tons nearly. 10.8 T- AND L-BEAMS. 35 In order to resist this, no less than 8J sq. in. of steel would be needed, This would obviously be uneconomical, requiring no less than four 2^in. diameter rods to make it up. The easiest way to remedy this is to deepen the beam, the amount of metal needed varying inversely as the depth 2 X — that is, the theoretic depth oi d-^^ ; for the limit safe stress for steel, 7I tons per square inch, has already been reached, and may not be exceeded. This means that X remains — , so that the area of concrete within the para- 4 bolic outline increases rapidly as the depth d increases. In other words, the concrete is then called upon to resist less than J-ton per square inch in compression, which is ■on the side of safety. This is done, it will be noticed, without thickening the floor-slab, the additional cost being only that of a little extra concrete in the flange of the beam as it is deepened, to set against the saving of metal achieved by reducing the area of the reinforcing-rods. It is not often that d is taken as more than — , but there 12 is no theoretic reason for this limitation. <- - 5 >, IZ Fig. IS. It frequently happens that the last of a series of T-beams is a L-beam, or inverted L-Beam, where it forms, say, the trimmer of a staircase-opening. It then carries only half 36 ELEMENTS OF REINFORCED CONCRETE. the load which is, borne by any other similar beam of its series ; but, so far as the concrete is concerned, this automatically rights itself, the shaded portion in Fig. 15, within half-parabolas above the neutral axis, obviously bearing the same relation to the load as does the larger shaded area in Fig. 14. Only half the steel, however, will be needed, if the depth of the beam be uniform with that of the rest of the series, as would probably be the case. Small beams, such as have just been considered, are frequently only secondary to main supporting beams, on which they bear, and to which they are homogeneously attached — concrete to concrete, and steel to steel — just as the floor-blabs arij attached to themselves. Suppose that the beams of 20ft. span rest on such main beams, with a span of 40ft. between centre and centre of bearings, instead of walls, the main beams, A (see Fig. 16), being 2oft. gin. between centres, and the secondary beams, B, loft. from c to c of each other, and of the wall-bearings of the main beams A. <. . 100 *-— ZO'9"— 20- 9- T 1 I, - ro'o I, ' 10- I _1_ -E>- •&■ •5 ■B- B- ■B 4oa Fig. 16. T- AND L-BEAMS. 37 Then each beam A carries a definite load of 2o| tons at €ach of the three points where girders B rest on it, and the Max. B.M. (at centre) 2a| X 40 , , 20| . , = -^ — + (2—^ X 20) 4 4 = 2oyi + 207J = 415ft. -tons = 4,98oin.-tohs. The depth, d, of the beam may, for the purposes of calcu- lation, be assumed to be 3oin. from top of concrete to c of reinforcement, this being more than — and less than — . 20 12 As the floor-slabs are only 4in. thick, it is tolerably ■obvious that the neutral axis will lie below their bottom ■surface. In such a case the area of concrete, which may safely be assumed to be resisting compression at J ton per square inch, is that which is diagrammatically shown, shaded, in Fig. 17. This consists of so much of the beam and floor slab as is contained within the combination of two parabolic outlines, each with its vertex at the centre ■of the neutral axis, K, one having the breadth of the active part of the floor-slab, a c, as its base, while the base of the other is the total breadth of the beam e /. In other words, it is the shaded portion, a, c, m, h, k, g, I, made up of the truncated parabola, a, c, m, I, in the floor-slab, with the addition of the parabola h, k, g, in the beam. This area is difficult to determine so long as x is unknown, -especially as the length of the lever-arm of the Moment of Resistance is also unknown until x is ascertained. This -difficulty may be got over, for preliminary calculations, at any rate, by assuming that the whole rectangle of the floor- slab, a, c, p, n, is included within the shaded area, thus neglecting to notice that the small unshaded triangles ' 38 ELEMENTS OF REINFORCED CONCRETE. a, I, n, and c, m, f lie outside the parabolic area of extreme- fibre stress. This appears to be a material error, as shown in Fig. 17 ; but it is not so in reality, when, as is usual,, the floor-slab is much shallower in comparison than it is. shown therein — merely for the sake of clearness of explana- tion. It may also, as a rule, be assumed without material error that the centre of gravity of the shaded portio-n lies- in a line with the lower surface of the floor-slab, so that the length of the lever-arm is d- 4in. in the case now under discussion. Practical considerations, to be explained later, modify this considerably ; but for the present this will suffice, especially as any error is on the side of safety. As one-fourth only of the floor-slab lying between A and A may, on each side thereof, be taken as forming the table of the T-beam, instead of one-half (on account of the reinforcement of the slabs running from B to B, and consequently parallel to the compression in A) , the sectional area of floor-slab available for compressional resistance is « c X 4in. = loft. 4jin. x 4in. = I24^in. X 4in. = 598 sq. in. The additional area of concrete within the area h k g = g hx(x- 4) square inches. The lever-arm = d - 4= 30-4 = 26in. •. the M.R. = [598+f gh {x- 4)] I ton x 26 ; and the B.M., as already ascertained, is 4,98oin.-tons. Equating the M.R. and B.M., on the assumption that X — -. it is found that the breadth, g h, needs to be just 2 over 23in., which would be excessive. As the steel might then only be trusted to resist 2-^ tons per square in., as already discovered for a neutral axis thus placed (Fig. 6), it would be absurdly extravagant in steel, as well as in. concrete. 1 J ^- H - ^ 40 ELEMENTS OF REINFORCED CONCRETE. lix = -, then gh = I2in. (which can be more than obtained 4 by making e f= I5in.), the steel may be calculated to resist 7i tons per square inch in tension. Even then, as the tension in the steel is -^ — = iq2 tons, this is an absurd result, 26 ^ requiring no less than 25J sq. in. of armouring to resist it ; and it would not put this right to take the lever-arm of the M.R. as 27in., instead of 26in., more nearly approxi- mating with the true centre of gravity of the shaded outline a, c, m, h, k, g. I. The only thing to do, therefore, is to increase the depth d of the beam, and try again, taking d this time to be equal to — = 40in., and the lever-arm consequently to be 12 40 - 4 = 36ins. Then [598 + f g A (;v - 4 )] J ton x 36 = 4,980 and, a X = — = loin. 4 then [598 + f g A (10 - 4) i ton 4- 36 = 4,980 ■ • (i49i 4- g /*) X 36 = 4,980 •■• 149J 4- g ^ = 140 .'. gh = just under i in. One effect of raising the neutral axis and increasing the depth is, therefore, to reduce the width g h to almost a vanishing point ; while the amount of steel needed has also been reduced, the tension therein being now only —^ = 140 tons. Even this, at 7^ tons per square inch, would demand i8fsq.in. of reinforcement, or just over eight T- AND L-BEAMS. 41 square bars, each i^in. by i|in. ; or just over six square bars, each ifin. by if in. ; or ]\^si^- under six circular rods of 2in. diameter each, or eight of i|in, diameter each. By still further increasing the depth, still taking x= — 4 it is obvious that the amount of reinforcement could be further decreased. Already the breadth g h — and conse- quently the breadth of the beam — has ceased to enter seriousty into the calculations, as very much more than lin. would be needed to cover the steel. Whatever further calculations may be made, therefore, will be based on (a) the practicability of increasing the depth under the special circumstances, and (b) the proportionate extra cost of concrete, as this was done, as compared with the lessened cost of steel. CHAPTER IV. CONTINUOUS BEAMS. IT will have been obvious to all readers that the introduction of the main beams A A in Fig. i6, in place of walls, to support the ends of the secondary beams B B, converted the latter into what are known as " continuous beams," homogeneously connected to one another end to end. Such beams -are essentially different from those which are merely supported at their ends. They tend to bend as shown diagrammatically in Fig. i8. Fig. i8. the portions h c and d e, which are carried on the inter- mediate supports B and C, being cantilevers whose ends support the ends of the true beams ah, c d, and e f. Follow- ing the usual laws, the cantilever portions have so much as is above the neutral axis in tension and what is below it in compression ; while the reverse is the case with the portions which are true beams. The points b, c. d, and e, where the change takes place, are known as " Points of Contraflexure." Their positions, under varying positions of load, are exceedingly difficult to determine. It is customary to calculate the B.M.'s by the " Equivalent distributed load " rather than by actual irregular loads, if such exist, and to take the points b and e as or distant from B or C in the case of end spans. CONTINUOUS BEAMS. BC 43 and the points c and d as distant from B and C 4 respectively in the case of central spans. This is not exactly correct, though it is quite near enough for all practical pur- poses — theoretically B c and C d would each equal .211 X B C ; and in order that B b may equal B c, and C d equal C e, A B and C D (the end spans) should each be .789 B C. Occasionalh' — very occasionally — the end spans, such as A B and C D, are made three-fourths of each central span, or .75 B C ; but more often all the spans are equal, unless they are controlled by the exigencies of plan, when any sort of inequahty may exist. All such inequahties affect the positions of the points of contraflexure, just as inequalities of loading do ; while the uncertainty is added to by the possibihty that the main supporting beams A A (Fig. 16) may themselves deflect somewhat. Assuming equal spacing and evenly distributed loading, the max. B.M.'s of continuous beams may be found by multiplying W / by the following — No. of Spans Co-efficients to give B.il. in Centre of Span. t .0703 .0703 .S .08 .025 .08 4 .0772 .0364 .0364 .0772 5 ■0775 .0329 .0461 .0329 -0775 44 ELEMENTS OF REINFORCED CONCRETE. In no case does the co-efficient for intermediate spans exceed .0461, which is that for the central of five spans ; while that for end spans does not exceed .08. If these maxima be replaced by .05 (or aV) for intermediate spans, and .1 (or tV) for end spans, sufficient allowance should be made for the various possible inequalities already men- tioned, provided that certain precautions be taken with the reinforcement, which will be referred to presently. No. of Spans. Co-efficients to give B.M. over Supports. 2 ■125 3 .1 .1 4 .1071 .0714 .1071 5 .105 .079 .079 .105 In no case does the co-efficient over the supports next to the walls exceed .125, nor that over the more nearly central supports exceed .1. If these maxima be replaced by .16 (or i) and .125 (or |) respectively, sufficient allow- ance is again made for possible irregularities. Thus the max. B.M.'s to be provided against, taking W to represent the " Equivalent distributed load " in each case, are as follows : CONTINUOUS BEAMS. 45 In Centre of Continuous Beams. Over Supports of Con- tinuous Beams. Intermediate Spans. End Spans. ' Central Supports. Outer Supports. 20 10 6 8 -Applying these data to the secondary beams B B in Fig. i6, quite different results are arrived at from what were obtained when they were considered, in Fig. 14, as being mereh- supported at the ends. The B.M.'s are greatest over the central supports, where they rise to '- with upper 6 fibres in tension and the lower ones in compression. This means that the floor slab cannot be utihsed to resist compression, and that enough concrete for the purpose must be included in the section of the beam itself adjoining the supports. Taking the load W as before at 20| tons, the span / as 2oJft. (or 249in.), and the depth d as 2iin. ( rather more than axis, then — ^} and assuming a central neutral B.M. over support = W I 2o| X 249 86iin.-tons. . . h d Ad The M.R. of the concrete requured is x \ ton x , 3 5 as explained in connection with Fig. 4. Thus, substituting its assumed value for d — 21 b X i X 4 X 21 861. ■ . & = 30 inches (nearly) . 46 ELEMENTS OF REINFORCED CONCRETE. This would be eminently unsatisfactory — wasteful and clumsy. But the main beams A A are to be 4oin. deep, and the secondaries B B may consequently be also of the same depth where they join. They would look better, perhaps, if a little less, say, 36in. Taking this as the value ■of d, then — 36 6 , 4 X 36 .^ X i X ^ -- = 861. 3 5 .•. h = 10 inches (nearly). With a central neutral axis, as already explained, the reinforcement may only be calculated to resist 2J tons per square inch ; and the M.R. of the steel, hke that of the ■concrete, must equate with the B.M. Therefore — 4 X 36 a, X 2* X • = 861. = 12 sq. I 36" I 36: Tkutral Axis. | -A. Central Supports. Outer Supports. END SECTIONS. Fig 19. CONTINUOUS BEAMS. 47 This can be made up, and a little to spare, by four 2in. circular rods, and the eventual section of the beam, near the central supports, will be as shown in Fig. 19. Externally it would be of the same dimensions, for the sake of sym- metry, at the outer supports, but need not be so heavily reinforced, as the B.M. would then be — W I 20J X 2o| X 12 ^ . ^ -g- = = 647m.-tons. This would be more than met if a, = gsq.in., which could be made up by three 2in. circular rods instead of four. {See Fig. 19). As will be explained later, it is rarely that in practice •such large rods are used ; but the dimensions given are sufficient for purposes of illustration. What is essential is to provide sufficient area by the use of rods which are ■commercially procurable and easy to handle. The great depth d may be gradually reduced, theoretically to nil, practically to that of the central portion of the beam, at the point of contraflexure. It would perfectly well suffice for the depth d of the -central portion to be I2in., as assumed in the calculations for the beam when supported at its ends, when it was shown (Fig. 14) that there was ample concrete in the floor- slab to resist the compression occasioned by a B.M. of '- with a neutral axis situated at — from the top of the 8 4 beam. So there is obviously more than enough to resist wz w ; that due to a B.M. of , or even of , with asimilarly- 20 10 placed neutral axis, which will now be assumed. 48 ELEMENTS OF REINFORCED CONCRETE. It does not necessarily follow, it may be said, that the neutral axis of a beam will be central throughout if it be central at its support, in the case of a continuous beam,, especially when it is not of uniform depth throughout. Its. position depends upon considerations already explained, which may not be identical on either side of the point of contraflexure. I I I Z,nd Spans. Intefmedi&ie Spans. CENTRAL SECTIONS. Fig. 20. If the neutral axis be at — from the top of the central 4 portion, and the depth d be I2in., then — a, X 7i X 9 X 12 10 for intermediate spans 20 20 for end spans ■'■ ^s = 3i' sq. in. for intermediate spans = 7 sq. in. for end spans These areas cannot quite be provided by one (and two)' 2in. circular rods respectively. Yet, for the sake of con- tinuity, it is better to employ them than to introduce a 50 ELEMENTS OF REINFORCED CONCRETE. different section, as 2in. rods have been already determined upon for the portions of the girder over the supports. The only thing to do is slightly to increase the value of d again, making it I2jin. instead of I2in. The central sections will then be as shown in Fig. 20, while the theoretical arrangement of the reinforcing rods of the secondary beams (Fig. 16) will be as shown in Fig. 21. It must be understood, however, that they are only shown diagrammatically. In practice means are taken to connect the rods, while vertical stirrups, as already explained, have also to be introduced. CHAPTER V. BEAMS WITH DOUBLE REINFORCEMENT. UP to the present it has been considered that the M^concrete can alone be depended upon to resist the ^compressional stresses in beams, using the steel to resist tension only ; but there are many cases when it is advantageous to employ steel both in tension and com- pression. For one thing, it is theoretically better to intro- duce the metal rather as numerous wires of small diameter than as a few rods of large size, and to connect these wires by means of the stirrups or vertical reinforcement ; and, for another thing, it is often actually economical to use metal both in the upper and lower portions of a beam section, raising the Umiting stress in each case to 7i- tons per square inch. By way of an example, the case alreadj^ dealt with in connection with Fig. 6 may well be reconsidered. It will be remembered that it was a beam, supported at both ends, carrying an evenly-distributed load of lo tons over a span of 20ft., and with a depth of 2oin. from top of con- crete to centre of reinforcement-. Under these coriditions, with a central neutral axis, its breadth worked out at iijin,, while the sectional area of the steel required to resist tension was 7I sq. in. But the tensional stress wasonly — ■ B.M. 10 tons X 24oin. Effective depth 8 x i6in. = i8| tons Thus the steel was^'only employed to resist 2\ tons per* square inch, while it might safely have been subjected to' a tension of 7^ tons per square inch. 52 ELEMENTS OF REINFORCED CONCRETE. If the neutral axis had been raised to a point distant — 4 from the top of the concrete, in accordance with Figs. 3 and 4, then, as already shown, the steel might have been employed to do its full work ; but the breadth of the beam would have had to be greatly widened if the concrete alone were to be trusted to resist compression. I< h' ll\" Jl i I I I I d=2o: -_i Fig. 22. The same breadth of iijin., as determined in relation to Fig. 6, may, however, be retained if steel be introduced in compression as well as tension, and quite possibly with economy, when the neutral axis is thus raised. The metal in compression (see Fig. 22) would necessarily be located on the compressional axis of resistance — that is, horizontally centring with the centre of gravity of the parabolic outline. BEAMS WITH DOUBLE REINFORCEMENT. 53 ■which is located at — from the top of the concrete. As 10 d = 2o'm., this provides ample cover from a practical standpoint. Also, the lever arm of the Moment of Resist- ance = ^— = i8m. 10 Then the stress to be resisted — B.M 30oin.-tons Effective depth i8in. = 16.6 tons. This requires less than 2^ sq.in. of steel to resist it safely, where the stress is tensile, at 7J tons per square inch, and this can be supplied by three rods of lin. diam. each. Three similar rods would also suffice to resist the com- pression, without taking the concrete into consideration at all ; and if such were used for the sake of uniformity, the total amount of steel needed would only come to six lin. rods, or 4.70 sq. in., as compared with 7J sq. in., shown in Fig. 6 in tension alone. But the concrete within the parabolic outline in Fig. 22 may be used up to |- ton per square inch. The area thus available being f x iij X 5in. = 37J sq. in. Its resistance is consequently 9I tons. Thus there is onlj' 16.6 - 9.375 = 7.2916 tons to be resisted by compressional reinforcement at 7J tons per square inch, requiring less than i sq. in. of metal, which can be easily made up by two |in. diam. rods, or three of fin. diam. The latter are shown in Fig. 22, for they are somewhat the more ■economical. The total amount of steel used in the section now amounts to three lin. and three fin. rods, or 3.67 sq. in. only. 54 ELEMENTS OF REINFORCED CONCRETE. It might at first sight appear risky to subject small com- pressional rods to the full working stress of 7J tons per square inch ; and so it would be if it were not for the sur- rounding concrete preventing buckling, aided also by careful connections with the tensional rods by means of stirrups. While it is thus possible to economise some simple beams by using double reinforcement, at the same time pro- viding for more secure vertical lacing, and using rods of small diameters which adhere well to the concrete, no such advantages are to be gained in other cases by similar means. For example, the T and L beams in Figs. 14 and 15 are already as economically designed as possible, both steel and concrete doing their full work. So, too, is the larger T-beam (Fig. 17) so far as its central section is concerned ; though in this case the amount of metal in the reinforcement is so great that it would pay to reduce it when the B.M. diminishes sufficiently, as the abutments are approached, for some of the rods to be safely omitted. When, however, the short beams are continuous, the advantages of double reinforcement are great and obvious, particularly over the supports. So far as the central sections are concerned, the investigations are much like those just made, remembering that the max. central B.M. is for intermediate and for end spans, while T- 20 10 beams rarely need secondary reinforcement (in the upper or compressional portion) at all in their central portions. What occurs over and near the abutments, on the other hand, is worth inquiring in. Taking the cases already investigated in connection with Fig. 19, it will be remembered that the B.M. over the central supports was discovered to be 861 inch-tons. BEA:\IS with double reinforcement. 55 The depth d of the central portion was eventually taken as I2|in. ; but although this resulted in the use of large rods of 2in. diameter, it would be convenient, for the purposes of exemphfication, to adopt this depth again, using it through- out, and not in the centre only. Assuming that the neutral axis lies at — above the 4 bottom surface of the end section {d being the depth from this bottom surface to the C. of the upper reinforcement), Q X 12— then the effective depth would be ? = iijin. The 10 centre of the bottom reinforcement \\-ould then be located at only 12^- 11 J = ijin. from the bottom of the concrete, which would not provide a sufficient covering to resist fire. Retaining the same total depth, it is then clear that the effecti\e depth, between the centres of upper and lower reinforcement, must be decreased, with the result of enhanced stresses and increased area of metal : otherwise the total depth must be increased. It is better to choose the latter alternative. For the present, then, assume an effective depth of i5in., with a total depth of 2oin. Then, the stress in the reinforcement (making no allow- ance for the resistance of the concrete) — B.M. 861 = — = 57.4 tons. 15 This requires 7|sq.in. of steel to resist it, at 7J tons per square inch, whether in tension or compression ; and, with the steel so far from the outer edge of the concrete propor- tionally, enough should be provided to meet the whole 56 ELEMENTS OF REINFORCED CONCRETE. stress ; else some risk is run of the outer fibres of the con- crete being called on to do too much work. This area would be almost provided by four i^in. diam. circular rods ; which would quite suffice if the effective depth between c and c of reinforcement were increased to iSjin. The breadth might be gin., leaving 2in. cover of concrete outside the reinforcement everywhere ; which would make the extreme depth 24^in., as shown in Fig. 23. Using a beam of the same dimensions (24jin. hx gin.), the section at the outer supports, as indicated in Fig. 23, would admit of much less steel being used, for the B.M. would only be , or 647 in. -tons, while there would be an effective depth of iqin., with the same covering of concrete as before. „, B.M. 647 Then stress = = -^ d ig = 34TTf tons. This requires rather more than 4I sq. in. of steel to resist it, which can be supplied by two if in. diam. rods. Probably, however, three i|in. rods would be used in preference, so as to employ rods of the same diameter as those used over the central supports, the additional ones being introduced as shown by open circles in Fig. 23. This would just suffice, the effective depth being then ly^m., and the stress 37 tons. The external section, ha\'ing now been determined, would be retained throughout the whole length of the beam, one of the principal objects of the double reinforcement being to do away with the deep shoulders introduced at the abutments in Fig. 21. It has already been shown that no reinforcement in the compressional portion is needed in the central section of the --H «0 o Oj L I 09 TS I 1 ^ 11 I*- I M — k — effccti' depth = > # • ' # 1^ ,1 • • 9 1 1 a 3 10 u z o F u o z (O h C£ O Q. Q. D U) < h z u U 58 ELEMENTS OF REINFORCED CONCRETE. beams (Fig. 20), with a much less total depth than 24^in., and the neutral axis distant — from the top of concrete. 4 In the recent calculations a central neutral axis has been assumed, no stress being thrown upon the concrete at all, but the steel being calculated to resist the whole of it, both in tension and compression. Under these conditions — assuming that the neutral axis is central from end to end — the safest assumption that can be made — the concrete is again obviously more than ample to meet the compres- sion. All that is needed, consequently, is to calculate the reinforcement for tension. Taking the resistance of the concrete to be concentrated at the level of the bottom of the floor slab, or 4in. from the top, and the centre of reinforcement to be 2fin. from the bottom, the effective depth is 244 - 6f = lyf in. Then the B.M. for intermediate spans — _ W^ 20f X 249 20 20 = 2 5 Sin. -tons and the B.M. for end spans — W / = — = 517 m.-tons. ID Dividing these by the effective depth, it is found that the stress for intermediate spans = 14^ tons, and the stress for end spans = 29 tons. Thus 6 sq. in. of steel is the area required in the centre portions of the intermediate spans, and 12 sq.in. in the end spans ; for, if the concrete is depended upon in compres- sion and the neutral axis be central, the steel may only be subjected to 2-| tons per square inch. This would not do at all. !<- I T + si? t 0) (0 I I - ^ 1 I + T 0^ 5 a z • u CO z o I- u in < H Z UJ i h < q: bl §- z 6o ELEMENTS OF REINFORCED CONCRETE. It would be obviously more economical to again introduce double reinforcement, with a total effecti\-e depth of igin. (for it is not likely that more than two rods will be needed). Under these conditions the stress for intermediate spans = iStV tons, the stress for end spans = 26TTr tons. So that with the full allowance of yl tons per square inch, now safely admissible, nearly 1.8 sq.in. of steel is required in the central portion of intermediate spans and nearly 3.6 sq.in. in end spans, which can be almost exactly pro- vided by one and two rods of ijin, diameter respecti\-ely. The central sections are thus as shown in Fig. 24, while the longitudinal distribution of the rods, as determined by all these calculations and sections, is as shown in Fig. 25, it being noticeable that the extra rods required over the supports need not run right out to the points of contra- flexure, being only required o\'er such distances as corre- spond with greater stresses than can be safely met by the continuous rods -d o p^ p CHAPTER VL FLOOR SLABS. FIGS. 26 and 27 illustrate how two floors of unrein- forced concrete slabs supported (not fixed) on all sides fractured when tested to breaking point by Col. Seddon, R.E., in 1874. The results are instructive, showing that the nearty square slab failed along its diagonals while the oblong slab cracked along its major axis, as well as along Hnes drawn at angles of 45'' from the corners. In other words, the cracks have approximately the appearance of the plan of a hipped roof. Nearly thirty j-ears later, in 1900, a floor of reinforced concrete, having a network of reinforcing rods parallel to the sides and ends, which was tested to breaking h\ the French Ministerial Commission on Reinforced Concrete, failed, as shown in Fig. 28 ; which, like Figs. 26 and 27, ias been extracted from Marsh and Dunn's " I\Ianual of FLOOR SLABS. 63 Reinforced Concrete." This floor was more than merely supported — it was partially fixed on all sides ; yet the Fi^. 27. •directions taken by the cracks confirm the results arrived at by Col. Seddon so long previously, while suggesting by their multiplicit}? that the method of reinforcement adopted was practically correct, no line of especial weak- ness being displayed. 64 ELEMENTS OF REINFORCED CONCRETE. This is the method now generally employed, the principal reinforcing rods, spaced about 6in. apart, crossing the floor in the shorter direction, while secondary rods lie above them, their object being principally to distribute loads on to the main rods. They are generally spaced about ift. Unfortunately, there have not been enough experiments made yet for a reliable theory of stresses to be evolved. It is only possible to calculate for each strip of flooring as if it were independent, knowing well that any error will be on the safe side. Taking a floor carrying 2cwt. per foot super., with a span of loft. and reinforcing rods 6in. apart, and considering a single strip of this, 6in. wide, as a separate beam carrying an evenly distributed load of locwt. (or J ton), it is easily seen that the max. B.M., Wl T — = - ft. -tons = 7J in. -tons. 8 8 ^" If the total depth be 4in., and the rods be lin. above the bottom of the slab, then d = 2m., and, with a neutral axis distant — from the top, the effective depth will be - — - 4 10 = 2.7in., and the area of concrete (to be taken at J ton per square inch) will be — • Therefore the M.R. of the concrete 6 will be — 1 bd , 6x3 2.7 X I X — = 2.7 X i X 6 ' ■» ■ 6 = 2.7 X| = 2in.-tons. This, being less than the B.M., would not suffice. FLOOR SLABS. 65 If a central neutral axis be assumed, the effective depth would be — - = 2.4in., while the area of concrete would be — . The M.R. of the concrete would then be — 3 2.4 X J X — =.6x6 = 3.6in.-tons. 3 This would still be insufficient ; but not to so great an extent as would appear at first sight. The B.M. must be multiplied by a reducing co-efficient, found by experiment, which varies according to the pro- portion of the length to the span (1) of the slab. According to the French Government rule, which alone seems to take the effect of reinforcement fully into account, this co-efficient — I* length* Adopting this, it is found that — When length = I, then co-eff. = .33 „ =1-25 A „ „ =.55 = 1-50^, ,, ,, =71 = 2 1, ,, ,, =.89 It will thus be seen that, multiplying the B.M. of 7-2in.- tons by each of these co-efficients in sequence, the result is less than 3.6in.-tons in the case of a square slab, rather more than equates with it when the length of the slab is x\ I, and is in considerable excess in the case of longer slabs, when it would be necessary to increase the depth of the slab. 66 ELEMENTS OF REINFORCED CONCRETE. Taking the case of equality, when the reduced. B.M. (after multiplying by the co-efficient) = 3.6in.-tons (merely for the sake of argument, and not because the actual case presents itself), then with an effective depth or lever-arm oi 2.4in., the stress in the reinforcement is ' — = 15 tons. As, with a central neutral axis, the steel may only be subjected to 2J tons per square inch, this would call for one |in. diam. rod. This is larger than it is usual to use in floors, so that the inevitable conclusion is come to that the depth is insuffi- cient ; or else smaller rods may be used nearer together. Two -fin. rods just exceed in sectional area one |in. rod, so 1hat fin. rods may be used 3in. apart, instead of -^in. lods 6in. apart. As a matter of practice it is much better to use tiiicker floors, as they resist the passage of sound to a greatly enhanced extent. One that is 5in. thick is in this respect infinitely superior to one that is 4in. thick. It is also more usual to calculate for an entire slab at once than for narrow sections one at a time. On this basis, taking each btiy of the floor to be 20ft. gin. In? loft., and the. total depth to be 6in., it is found that the load is 2o| X 10 X 2 cwts., or 2o| tons, and the B.M. is consequenlly — 20|XI20 ^3iiin.-tons in centre if supported all round, and over outer supports if continuous. 20|XI20 = 415111. -tons ditto over central suj^ports ditto FLOOR SLABS. 67 ~ =249in.-tons ditto in centre of end spans ditto, and = i24^in.-tons ditto intermediate spans ditto. ^0 and 20|XI20 20 Allowing lin. cover for the reinforcing-rods of approxi- dia matelv Jin. diam., the value of ^ = Gin. - lin. - — ' = 4iin. 2 The effecti\-e depth, with central neutral axis = — = 4.2in. ; and with neutral axis distant — from top = - — = 4.275in. 4 10 The area of concrete within the parabohc outline would be, for central neutral axis — ^ 19 , . 2o| X 12 X — = 249 X -- = 394i sq. m. 3 12 Taking the safe resistance of concrete per square inch to be 1 ton, the M.R. would be— 394t X i X 4.2 = 4i2in.-tons. Without utihsing the co-efficient for slabs, this more than suffices to meet the B.iL in all cases, except over central supports of continuous slabs, and even then is but httle on the wrong side. Using it, ho\^e\-er, the M.R. is in every case in excess of the B.M., which becomes — Inch-tons. 311 X. 89 = 277 in centre if supported all round and over outer supports if continuous. 415 X .89 = 369 ditto over central supports ditto. 249 X .89 = 202 ditto in centre of end spans ditto. 124^ X .89 = loi ditto intermediate spans ditto. 68 ELEMENTS OF REINFORCED CONCRETE. Dividing these by the effective depth of 4.2in., the stress in the reinforcement becomes, in the various cases, 67, 88, 48, and 24 tons respectively, requiring 27, 36, 19J, and 9I square inches of metal, allowing 2^ tons per square inch, as is right for a central neutral axis. If fin. dia. rods are used, each with a sectional area of .44 sq.in., then in a length of 20ft. gin. the following rods are necessitated, those in the centre occurring at lin., from the bottom of the slab, and those over the supports at lin. from the top — 27 -^-.44 = 62 in centre if supported all round and over outer supports if continuous, or one every 4in. 36 -;-.44 = 82 ditto over central supports ditto, or one every 3in. i9J-i-.44 = 45 ditto in centre of end spans ditto, or one every 5 Jin. 91^.44 = 23 ditto intermediate spans ditto, or one every lofin. CHATTER VII. FOUNDATIONS. IN few parts of a building can greater economies be effected by using reinforced concrete than in foundations, where it is combined with a degree ■of security against cracks due to unequal settlement which the old method of construction was powerless to provide. The great object of all foundation work is to spread the weight of walls, piers, and columns over a large surface, so that the pressure on the earth below may be practically uniform, and to bridge over weak places. The weight is ^generally distributed by means of footings and a broad concrete bed, any risk of disruption on account of a weak spot being roughty provided against by thickening the •concrete. By introducing reinforcement it is easy to secure dis tribution without using footings — always unreliable, owing to their construction — and to provide a theoretically sound " bridge " without any excessive thickness of "Concrete. ok- ^- 14 — ^ J — irods y^ Saport. K — i=2n. — ^ 15 ferods Bapari. Fig. 29. Take, for example, the I4in. wall shown in Fig. 29, ■which may be assumed to carry the unusually heavy load 70 ELEilENTS OF REINFORCED CONCRETE. (including its o\ra weight) of 2 tons per foot run, and to rest upon very weak soil which may only be trusted to bear i ton per square foot safely. It is thus obvious that the foundation must be 4ft. wide. It is also obvious that for e\'ery foot run of the wall there will be an evenly dis- tributed upward pressure of 2 tons exercised by the earth upon the foundation. Half of this acts on each side of the centre of the wall, which constitutes the fulcrum of an inverted double csmtilever. ilaking the bed of concrete 7in. thick, and placing the main transverse reinforcing-rods 5in. from top of concrete, the lever-arm is ihen-^ — = 4iin., and the parabola of 10 concrete is — X 12 x -^^ =10 sq. in. The max. B.M. is i ton x I2in. = i2in.-tons, and the Moment of Resistance of the concrete is 10 x 4^ X J ton = iijin.-tons. This assumes the neutral axis at — from top 4 of concrete ; with a central neutral axis the lever arm would be = 4in., and the parabola of concrete would be 5 2 =) — X 12 X — = 20 sq. m., so that the M.R. of the concrete 3 2 would be 20 X 4 X i ton = 2oin.-tons. Thus, if the reinforcement be calculated at 7I tons per square inch, there is some sHght theoretical risk that the concrete may, at its outer layer, be in compression to more than J ton per square inch. On the other hand, with more FOUNDATIONS. 71 steel and a lowered neutral axis in consequence — the steel only taking 2| tons per square inch — the margin of safety in the concrete would be large. \\'ithout going into exactitudes, it is tolerably obvious that safety will be secured with steel stressed to 7 tons per square inch. The maximum stress per foot run is only B M 12 = 2| tons, which can be safely met by le\-er arm 4^ two iin. circular rods for every foot run of the wall. So far as the transverse distribution of the load is con- cerned, it is only therefore necessary to introduce Un. rods at 6in. centres. The longitudinal rodding to secure bridging over possible weak pockets in the earth cannot be calculated, as their extent and location cannot even be guessed at. All that can be done is to introduce longitudinal rods at about gin. centres below the transverse rods, as shown in Fig. 29. making them of any suitable diameter. It is not often that such weak soils as this are met with ; and it may be remarked in parenthesis here that, owing to various circumstances which are to be explained later on, elaborate calculations are not often needed in any ordinary reinforced concrete work. However, it is essential to under- stand the theory, even if it need not always be applied. Under the building regulations which are in force almost all over the country it is necessary to cover the site of every building with a 6in. layer of cement concrete. This layer might very well be continuous with such a foundation 72 ELEMENTS OF REINFORCED CONCRETE. as that which has just been considered. It is generally not reinforced at all ; but in the case of a weak and water- bearing soil there is just the possibihty that the weight of the walls may carry them down uniformly, while the upward pressure on the floor may suffice to burst it. In such a case the floor needs reinforcing, as a continuous slab Fig 30. with the weight applied from below, as shown in Fig. 30> the points of contraflexure being fixed at the terminations of the cantilevers, as determined in Fig. 29. Only small rods need be used ; but neither their number nor their size can be theoretically determined without knowing what is the extent of the upward pressure. Most people would be contented to use a network of Jin. rods running both ways and spaced at gin. centres ; and in practice the rods shown in Fig. 30 would be connected by bending them, and not dissociated. Properly calculated foundations are obviously most urgently needed under heavy piers and weight-bearing columns, whether the latter be of reinforced concrete them- selves or of steel or iron. The method of calculation adopted is much the same as in the case of foundations under a wall. Say that a pier 3ft. square carries a load of 100 tons, and rests on ordinarily good soil (chalk, clay, or gravel), which FOUNDATIONS. 73 Avill take 2 tons per square foot. This calls for 50ft. super, of bearing surface, or rather more than 7ft. square. Con- servative and careful workers would make it 7ft. 6in. j^ -2-3".- ^^- 3'-0"- -^« -2'3''--^ d-12 SECTION. I — .- ^^^^^% " ^^^^^^P h ---- - "" " """ ^^f^^/^^ - '-'-"- : /y . ----._----- PLAN. Fig. 31. The cantilever thus only extends 2ft. 3in. on either side of the pier, as shown in Fig. 31, and with an upward pressure of 2 tons per square foot, the B.M. per foot run of it is^ 4| tons X -^' = 6o|in.-tons. 2 74 ELEMENTS OF REINFORCED CONCRETE. ^^'ith concrete which resists J ton per square inch, this requires — 6o| ^ 243 i X lever arm lever arm inches of concrete within the parabolic area for every I2in. run. This is secured, if the neutral axis be — from 4 top of concrete, with a depth of I2in. from top of concrete to c. of reinforcement, and an effective lever arm of — = lo.Sm. 60J The max. stress in the steel per foot run = — -^ = 5I tons. 10.8 This can easily be met by Mn. circular rods at 3in. centres, made of steel which will safely resist 7^ tons per square inch ; and similar rods would be similarly placed in the opposite direction also, as shown on plan in Fig. 31. Thus the total depth of the concrete need be not more than I4in., and footings may be dispensed with. CHAPTER J 'III. COLUMNS. CONCRETE is itself so reliable in compression that it could be used in columns and piers without reinforcement, except for the economy of bulk which can be effected. This is so great that, except in massive engineering structures, homogeneous concrete piers have become almost unknown. On the other hand, steel stanchions and cast-iron columns of many sections, encased in concrete as a protection against fire, have been commonly employed, the metal being calculated to carrj^ the whole of the load, and the concrete being merely added as a protective skin. Protected metal work of this description is now sometimes used both for columns and girders in combination with reinforced concrete flooring, and probably \\-ill continue to be thus used for many years to come. It is not, of course, reinforced concrete, but pro- tected steel construction. In unskilled hands it is at present, perhaps, easier of construction : but it is not more economi- cal, and has the disadvantage that homogeneous connection cannot be made with the properly reinforced work in the floors. It is usual to build reinforced columns and piers of a series of vertical steel rods, bound together by stout wire, either in rings (see Fig. 32), or in the form of a spiral, and attached to the verticals by wire. The rods must be kept 2in. within the concrete, as protection against fire, and also to pre\-ent bending, in which the wire bands also assist, and between them they do this so well that almost the same load per square inch may be safely put upon such steel rods as upon the steelwork in stanchions whose greatest diam.eter bears the same proportion to the length as does 76 ELEMENTS OF REINFORCED CONCRETE. the whole diameter of the concrete column. Neither in steel stanchions nor in reinforced concrete columns is it usual to make the height much more than twenty times the ■diameter. Fig. 32. In the case of reinforced concrete, however, there is no fixed rule to this effect, nor much reason for one. The section depends almost entirely upon the weight to be borne — at least, it appears so, according to present knowledge. Theoretically the steel, supported throughout its length by its concrete wrappings, could, under any reasonable COLUMNS. ir proportions of length to diameter, be loaded up to its safe total per square inch for short cubes, if it were not for the fact that when the concrete sets the cement which binds it contracts, and, adhering to the steel, puts the metal into compression to an extent ^^'hich cannot be accurately ascertained. All that experiments have so far shown is that steel which complies with the specification of the Engineering Standards Committee may, when employed as- reinforcing rods in columns, be safely subjected to a compression of three tons per square inch, while the concrete may be taken as resisting \ ton per square inch. This would be equivalent to allowing 8cwt. (or f ton) per square inch o\"er the entire sectional area, of which. 95 per cent, should consist of concrete and 5 per cent, of steel. Experience shows that this would be in excess of the necessity of the case, if the concrete be properly made of proper materials, and that it is quite safe to introduce only 4I per cent, of steel, with the same allowance of 8c wt. per square inch over the entire sectional area ; and this- agrees with the generally accepted dictum that concrete is stronger in columns, when subjected to direct compres- sion, than in beams, when subjected to compression due to bending. It is almost exactly correct for steel taking 3 tons- per square inch safely, and concrete 5^-cwt. per square inch. Take, for example, a column i8ft. high \\'hich has to- carry 100 tons. Its total sectional area, at I ton per square inch, would need to be ^^— = 250 sq. in. ; and of this the U amount of steel in the vertical rods could be 4^ per cent.,, or 10 sq. in. 78 ELEMENTS OF REINFORCED CONCRETE. The column would be i6in. X i6in. if square, or i8in. -diam. if circular, with a very small margin on the side of safety in either case, while the necessary steel would be made up of either ten lin. or eighteen fin. square bars, or thirteen lin. or twenty-three fin. diam. rods. Larger sizes (and fewer of them) might be used, but would be more likely to be wasteful, except that in this case six liin. ■diam. rods would only just exceed the required area. rig. 33- Fig. 33 shows a i6in. square column with 24 rods of ^in. •diam. (23 only being required), and Fig. 34 shows a circular •column, iSin. diam., with six rods of ijin. diam. each. These sections indicate tolerably clearly how difficult it would be to properly pack the concrete round numerous -wired rods, as in Fig. 33, by pouring it into a casing from above, without displacement of the rods either during the COLUMNS. 79 pouring or the subsequent tamping to secure homogeneity ; and this difficulty would be rather accentuated than ■diminished by distributing the rods amongst the mass. 'Consequently the use of comparatively few rods of large ■diameter, as shown in Fig. 34, is more common. It is not only easier to fill the mould and properly pack the concrete, but the larger and stiffer rods are much more readily kept in position than smaller ones. When used in warehouses, circular columns have the advantage over square ones that they have no arrises to be knocked off. On the other hand, the moulds for them are more difficult to make and support. In practice a com- promise is generally come to, square piers being used with 8o ELEMENTS OF REINFORCED CONCRETE. the corners chamfered off ; and even then the surface should be protected by cement rendering or granohthic facing, or even by steel plates, if rough usage is to be anticipated. \A'hen reinforced concrete is used for piles, it is almost essential to employ one of the patent forms, which will be described later on. They may, whate^'er their length, be generally assumed to be capable of carrying the same load at least as columns of the same section, for although they are often ^-ery long in comparison with their sectional area, they are partially supported throughout their length by friction, and are also prevented from moving laterally. The principal risk in their use is that the heads might be crushed during driving. This is generally pro\'ided against by binding the top of the pile extemalh* with wire, and by interposing a wooden " dolty," and sometimes a sack of sand also, between the head of the pile and the ram, which is often of considerable weight, up to 2 tons, but allowed only a short drop, not exceeding 6ft. CHAMER IX. WAlAJi AND RETAINING WALLS. IT frequently happens that reinforced concrete walls, whether external or internal, are no more than fillings between piers and girders which carry the loads. When this is the case they need only be thick enough and sufficiently reinforced to resist dri\'ing rain and any occasional accidental blow to which they may happen to be subjected. L'nder these circumstances walls need not be thicker than floors, and internal ones, carrying no weight, may be as thin as it is practicable to make them — that is, about 3in., with light fin. vertical rods about 6in. apart, laced horizontally by similar rods spaced I2in. apart; placed in the middle of the wall, as shown in Fig. 35, the rods being wired together where they cross, and in all cases having fish-tailed ends. Fi:,-. 35. 82 ELEMENTS OF REINFORCED CONCRETE. External walls would be similar, but would generally be 6in. thick, and have two layers of reinforcement— one being set about an inch from inside and outside respectively, as shown in Fig. 36. Weight-carrying walls would almost invariably be of similar construction, the only difference being that the thickness and the amount of reinforcement might be increased to carry the load, the usual allowance being made, as for columns, of J ton per square inch on the concrete and 3 tons per square inch on the steel. Fie. Retaining walls are entirely different things. They are of the nature of continuous vertical cantilevers acted upon by earth or water pressures, as the case may be, whose amount must be calculated in the usual way. WALLS AND RETAINING WALLS. 7^- 83 A-^^ Shrrups. E^^c Fig. A typical case is shown in Fig. 37, where a retaining wall to resist earth-pressure is shown, whose height is h above C, the centre of gravity of the adjoining concrete floor. The earth-pressure is concentrated at A, distant x above h C, when % = — ; giving a max. B.M. at C of A x. This must 3 equate with B y to prevent overturning ; and it is not usually necessary to carry the toe of the wall down far to obtain a sufficient resistance at B, even though the lever arm y is short, with the usual allowance for the kind of soil met with ; for such a toe would certainly be carried into something solid, like good clay or gravel, capable of resisting at least two, and probably three, tons per square foot safely. 84 ELEMENTS OF REINFORCED CONCRETE. The fulcrum, C, should then be made strong enough to resist the sum of the pressures A and B, as a horizontall}-- laid column ; for although B appears to be met by the earth beneath C, yet its pressure is transferred to C as soon as it is utilised to counterbalance A. The calculation of the concrete and the reinforcement is now quite a simple matter. Say that the value of A has been ascertained to be three tons per foot ran of the wall, and that r = 4ft., or 48in. Then the max. B.M. at C = 3 X 4S = i44in.-tons per foot run. 12 A The tensional reinforcement will be located near the earth side of the wall, as indicated in the foot run of the plan shown to a large scale in Fig. 38, and the thickness from centre of reinforcement to outside of concrete may safely be assumed as = ift. or I2in. Then, assuming a central neutral axis, the effective lever arm = 4 Fig. 55- SOME PATENT SYSTEMS. 107- The Hennebique patents include a stout w're clip- (Fig. 55), which is used for tying together the circular rods- used as the reinforcements of columns. Grasping them in pairs, as indicated in the diagram, the clips connect them in a rigid manner ; and, of course, they are introduced at. regular intervals in the height. SECTION A B. SECTION C D CONSlDERE SPIRAL REINf ORCEMENT OF COLUMM AND CONTINUOUS • GIRDER Fig. 56. In point of efficiency it is, however, apparently difficult to surpass the Considere spiral system of reinforcmg. To8 ELEMENTS OF REINFORCED CONCRETE. columns and other corapressional portions of a structure, as illustrated in Fig. 56. The effect of the spiral wire or rod is, it seems, not so much to take up some of the thrust itself as to bind the enclosed concrete, preventing it from bursting under load, and giving it considerable additional compressional resistance. According to the French Ministerial Regulations, which the Considere Construction Co. adopt as the basis of their own calculations, and have found by experiment to produce results which are remarkably close to actual fact, the permissible compres- sional stress (SJ per square inch on the concrete enclosed within the spiral may be determined by the formula — S3 = S, (I + 32 />. + 15 />.) Where — S; = admissible stress per square inch on plain concrete. p^ = sectional proportion of metal in spiral as compared with concrete. p^ = sectional proportion of metal in verticals as compared with concrete. Suppose, for instance, that it has been determined to use 4 per cent, of metal, of which 2 per cent, (or t;V) is to be in the vertical rods and 2 per cent, (or tV) in the spirals, the concrete being of such quality that it may be taken as safely resisting 70olb. per square inch. Then — S, = 700 (i + 32 X TO + 15 X aV) ^- 700 + 448 + 210 = 1,3581b. per square inch. If the concrete be of i : i : 2 mixture, capable of with- standing goolb. per square inch safely in compression, then S, = 1,7461b., or more than | ton per square inch. SOME PATENT SYSTEMS. 109 Taking the lower figure of 1,3581b. per square inch, and assuming that it is needed to carry 100 tons, then the sec- tional area within the spiral must be — L = i65sq.in.,. 1,358 which is contained in a circle whose diameter is 14^-in. This represents the concrete within the spiral, the external envelope of concrete being entirely neglected. It peels off long before crushing takes place, and thus acts as a most \'aluable sign of overloading ; but it cannot be trusted to< carry any of the weight. If the effective sectional area of the concrete be 165 sq.in.,. then that of the steel, both in vertical rods and spiral, will- be W, or 2 per cent, of this, which is 3.3 sq.in., or rather- more than four circular inches. Eight vertical rods of fin. diameter each may, therefore, be used, or a correspondingly larger number of smaller rods, or smaller number of larger rods. f Each spiral, when unwrapped, is 14^ X 3.1416 = 45 Jin., long. This, divided by the pitch of the spiral, gives the corre- sponding number of short rods, each the length of the spiral, whose total section must equal 3.3 sq.in., or just over four circular inches. If \m. diameter wire be used, containing .25 of a circular- inch in section, then it needs 4-^^, or just over 16 such rods, to contain just over four circular inches. This can be obtained with 2|in. pitch ; for 16x21 = 46,. whereas the unwrapped spiral is 45jin. long. If fin. diameter wire were used, ifin. pitch would be- ample ; and this would be better, as it would bind the concrete more continuoush' 4 r^i i" Lv 4^ f ? f ^ ;.0 . o to < I- -^X) \ — f XOCNtT REINFORCED CONCRETE PILE ^''8- 57- SOME PATENT SYSTEMS. iii The illustration (Fig. 56) shows the same principle of •spiralling applied to the compressional portion of a con- tinuous girder. If the same system of reinforcing be applied to piles, they may be driven direct without the intervention of any dolly or pad between the head of the pile and the ram ; but the head must be spirally reinforced externally as well as within the envelope. The Coignet pile (Fig. 57) has a special shoe, into which the vertical rods are bent. These rods are connected at short intervals by circular wire bands, which act somewhat ■similarly to the Considere spiral so far as the prevention of bursting of the contained concrete is concerned ; but a dolly has to be used to prevent disruption during driving. It will be noticed that the pile is externally circular in section except for two fiat faces which act as guides during -driving, for which purpose they are useful. The Williams pile (Fig. 58), made by Messrs. S. Williams & Sons, Ltd., is made of concrete with a rolled steel joist as a core, whose flanges are bent and cut to form the point, while holes are provided at intervals throughout the length for lifting purposes, and near the head to enable further lengths of joist to be fished on when lengthening is neces- sary, as often occurs in pier and jetty work, after the pile lias been driven. It is for such purposes that this form of pile is most suitable. THE END. ELECTRICAL Circuits and Diagrams Electrical CircuitsudDiagrams PMeaaacHU*. Electric Gas Igrnlting APPERATUS. ELEBTRie Bells. aMMu/veiaroRS. AND Alarms. c Alternating Currents SIMPLY eXPUUNED f&S^ 1 i i^ Study of Elegtrigity rOD DCGINNU!6 .THE.. Model Library OF Practical Books M».25co. SMALL ACCUMULATORS. I Size ol Books, 5 In. x 71 in. Over SO Books, 25ets. each MODERN PDIMAIiY BATTERJES. lia Woodwork Joints hMT ro Hut viD micsr 10 iME mot INVENTIONS pROTEa SCLl AND BUV THEM FREDEBIC B WHIOMT wm\ [E^ Mechanical Drawing SOfPLY EXPLAINED nODEL STEAM ENGINE DESIGN 25c. BOOKS. ELECTRICITY. The study of, and its laws for beginners, com- prising the laws of electric current generation and flow. Ohm's law, galvanism, magnetism, induction, principles of dynamos and motors, wiring, with explanations of simple mathematics as applied to elec- trical calculations. By N. H. Schneider. With 55 original illustra- tions and 6 tables. DRY BATTERIES. A practical handbook on the designing, fil- ling and finishing of dry batteries, with tables, for automobiles, gas engine, medical and coil work, electric bells, alarms, telephones, ex- periments and all purposes requiring a first-rate battery. Fully il- lustrated with 30 original drawings. ELECTRICAL CIRCUITS AND DIAGRAMS. Being a selec- tion of original up-to-date and practical diagrams for installing an- nunciators, alarJns, bells, electric gas lighting, telephones, electric power light and wirmg circuits, induction coils, gas engine igniters, dynamos and motors, armature windings. By N. H. Schneider. ELECTRIC BELLS AND ALARMS. How to install them. By N. H. Schneider. Including batteries, wire and wiring, circuits, pushfes, bells, burglar alarms, high and low water alarms, fire alarms, therinostats, annnuciators, and the locating and remedying of faults. With 56 original diagrams. MODERN PRIMARY BATTERIES. Their construction, use and maintenance, including batteries for telephones, telegraphs, motors, electric lights, induction coils, and for all experimental work. By N. H. Schneider. 94 pages, 55 illustrations. The best and latest American book on the subject. EXPERIMENTING WITH INDUCTION COILS. H. S. Norrie, author of "Induction Coils and Coil Making." A most instructive little book, full of practical and interesting experiments, fully ex- plained in plain language with numerous hints and suggestions for evening entertainments. Arranged under the following headings: Introduction; The Handling of Ruhmkorffi Coil; Experiments with Sparks; Effects in the Vacuum; Induction and Wireless Telegraphy. With 36 original illustrations. SMALL ACCUMULATORS. How made and used, by P. Mar- shall. Giving full descriptions how to make all the parts, assemble them, charge the cells and run them, with examples of their practi- cal application. Useful receipts and memoranda and a glossary of technical terms. 80 pages, 40 illustrations, paper. ^ ELECTRIC GAS LIGHTING. How to install Electric gas ignit- ing apparatus including the jump spark and multiple systems for all purposes. Also the care and selection of suitable batteries, wiring and repairs, by H. S. Norrie. ioi pages, 57 illustrations, paper. 25c. BOOKS. MAKING WIRELESS OUTFITS. By Newton Harrison, E.E. A concise and simple explanation on the construction and vise of simisle and inexpensive wireless equipments, for sending and re- ceiving up to 100 miles, giving full details and drawings of apparatus; diagrams of circuits and tables. Including the Morse and Con- tinental Codes. 61 pages, 27 illustrations. CIRCUITS AND DIAGRAMS. Part 2. By Norman H. Schneider. Alternating Current Generators and Motors: Single Phase and Polyphase Transformers : Alternating Current and Direct Current Motor Starters and Reversers: Arc Generators and Cir- cuits: Switch- Wiring: Storage Battery : Meter Connections : etc. etc. 69 original drawings, with full explanations. ALTERNATING CURRENTS SIMPLY EXPLAINED. An Ele- mentary Handbook on Alternating Current Generators, Trans- formers, and Motors. By Alfred W. Marshall. This book is written for those who desire elementary information about Alter- nating electric currents, simply written and yet intensely interest- ing. Contents of Chapters: — 1. What an Alternating Current is. 2. How Alternating Currents are Produced. 3. How Alternating Currents are Measured. 4. Transformers and Choking Coils. 5. Alternating Current Motors. 6. Rotary Converters. 7. Rectifiers. 82 pages, 32 illustrations. INDUCTION COILS. How to Make and Use Them, by P. Marshall. New edition revised and enlarged by K. Stoye. A practical handbook on the construction and use of medical and sparking coils for wireless telegraphy, gas engines, automobiles, gas lighting. X-rays, and all other purposes. With complete tables of windings for coils giving } in. spark up to 12 in. sparks. With full description for the construction of mercury interrupters. 76 pages, 35 illustrations. SIMPLE EXPERIMENTS IN STATIC ELECTRICITY. By P. C. Bull, M.A. Contents of Chapters: — 1. Production of electricity by various means. Viz.: friction, heat, pressure, chemical action, etc. 2. Electrical attraction, repulsion, and distribution. 3. Induction. 4. Leyden jars and other condensers. 5. Mechanical, chemical and heating effects. 6. Luminous effects. 7. Miscel- laneous experiments. Being a series of instructive and entertaining electrical experiments. 72 pages, 51 illustrations. THE MAGNETO TELEPHONE. Its construction, fitting up and use, by Norman Hughes. Giving full particulars for planning out a short line, putting up the insulators, stringing wires, con- necting instruments, suitable batteries. 80 pages, 23 illustrations. PRACTICAL ELECTRICS. A universal handy book on everyday electrical matters, including connections, alarms, batteries, bells, carbons, induction and resistance coils, dynamos, measuring, micro- phones, motors, telephones, phonographs, photophones, etc. 135. pages, 126 illustrations. 25c. BOOKS. THE SLIDE VALVE SIMPLY EXPLAINED. A practical treatise for. locomotive engineers by W. J. Tennant, revised and enlarged by J. H. KiNEALY. Contents of Chapters: — Introduction. 1. The simple slide. 2. The eccentric, a crank. Special model to give quantitative results. 3. Advance of the eccentric. 4. Dead center. Order of cranks. Cushioning and lead, 5. Expansion — lap and lead; advance, compression. 6. Doiible ported and piston valves. 7. The effect of alterations to valve and eccentric. 8. Notes on link: motions. 9. Cut-offs, reversing gears, etc. 89 pages, 41 illustrations. MANAGEMENT OF BOILERS. The Fireman's Guide. A Handbook on the Care of boilers. By K. P. Dahlstrom. Espec- ially written in plain English for the use of beginners and firemen. Contents of Chapters: — Introduction. 1. Firing and Economy of fuel. 2. Feed and Water-line. 3. Low water and foaming or priming. 4. Steam-pressure. 5. Cleaning and Blowing out. 6. General directions. Summary of rules. INJECTORS. THEIR CONSTRUCTION, CARE AND MANAGE- MENT. By Frederick Keppy. Second edition. The best and most practical treatise on this subject as it is written by a practical engineer for the instruction of engineers. 69 pages, 45 illustrations, price 25c. REFRIGERATION AND ICE-MAKING. By W. H. Wakeman. Fourth edition. Consisting of practical notes and information for engineers. 43 pages, tables and illustrations, price 25c. STEAM TURBINES. How to design and build them. By H. H. Harrison. A practical handbook for model makers. Con- tents of Chapters. 1. General Consideration. 2. Pressure De- veloped by an Impinging Jet; Velocity and Flow of Steam Through Orifices. 3. Method of Designing a Steam Turbine. 4. Com- plete Designs for DeLaval Steam Turbines; Method of Making Vanes; Shrouding. 5. The Theory of Multiple Stage Turbines With detail drawings and tables. 85 pages, 74 illustrations. MODEL BOILER MAKING. A practical handbook on the de- signing. Making and Testing of small Steam Boilers. By E. L. Pearce. Contents of Chapters: — 1. General principles of boiler design, materials, shape, proportions, strength, capacity, heating surface: 2. Stationery boilers. 3. Launch boilers. 4. Locomo- tive boilers. 5. Setting out plates, spacing tubes, etc. 6. Boiler fittings. 7. Fuel, lamps, fire-grates. 86 pages, 35 illustrations. GAS AND OIL ENGINES SIMPLY EXPLAINED. A practical handbook for Engine attendants. By W. C. Runciman. Contents of Chapters: — Preface. 1. Introductory. 2. The component parts of an engine. 3. How a gas engine works. 4. Ignition devices. 5. Magneto ignition. 6. Governing. 7. Cams and valve settings. 8. Oil Engines. 88 pages, 51 illustrations. 25c. BOOKS. THE BEGINNERS GUIDE TO THE LATHE. An elementary instruction book on turning in wood and metal. By Percival Marshall. Specially written in plain language for the beginner and as an elementary text-book for manual training schools. Con- tents of Chapters: — 1. The lathe and its parts. 2. Method of holding and driving work. 3. Turning in wood. 4. ' Turning in metal. 5. The Slide-Rest 6. Drilling and boring in the lathe. 76 pages, 75 illustrations. MECHANICAL DRAWING SIMPLY EXPLAINED. By F. E. Powell. The threefold object of this book is to show how draw- ings are made, how to read other peoples' drawings, and how to make practical working drawings. Contents of Chapters: — 1. Introduction. 2. The use and care of drawing instruments. 3. On " reading " and setting out drawings. 4. Inking-in and finish- ing drawings. 5. On drawing for reproduction. 78 pages, 44 illustrations. MODEL STEAMER BUILDING. By Percival Marshall. A practical handbook on the design and construction of model steamer hulls, deck-fittings, and other details, including a model torpedo boat destroyer, and a -side-wheel passenger steamer. With laying- ofi tables. 64 pages, 39 illustrations. MACHINERY FOR MODEL STEAMERS. By Percival Mar- shall. A practical handbook on the design and construction of engines and boilers for model steamers, single and double cylinder engines, side wheel engines. The use of liquid, fuel, and the pro- portions of machinery for model boats. 64 pages, 44 illustrations. SIMPLE MECHANICAL WORKING MODELS. How to make and use them. By Percival Marshall. How to make the following: Water and wind motors; hot-air engines; steam en- gine and pump; slide valve launch engine; model steam boats; working locomotive in cardboard, model gravttation railway, etc. 64 pages, 34 illustrations. MODEL STEAM ENGINES. How to Understand Them and How to Run Them. By H. Greenly. Including examples of stationary locomotive, portable and marine engines. With different kinds of boilers and methods of getting up steam, as well as engine de- tails and valve mechanisms, etc. 87 pages and 55 illustrations. MODEL STEAM ENGINE DESIGN. A handbook for the De- signer of small Model Steam Engines, including original tables and calculations for speed, power, proportions of pumps, compound engines, and valve diagrams. By Robert M. de Vignier. Con- tents of Chapters: 1. Various Types. Speed of Model Engines. 2. Power Calculations. Materials. 3. Feed Pumps. 4. Com- pound Engines. 5. The Valve Diagram. 6. Engine Layout. Pat- terns. 102 pages, 34 illustrations. 25c. BOOKS. WIRELESS TELEPHONE CONSTRUCTION. By Newton Harrison. A comprehensive explanation oi the making of a Wireless Telephone Equipment. Both the transmitting and re- ceiving stations fully explained with details of construction suffi- cient to give an intelligent reader a good start in building a Wireless Telephone system and in operating it. 74 pages and 43 illustrations . THE WIMSHURST MACHINE. HOW TO MAKE AND USE IT. A practical handbook on the construction and working of Wimshurst machines, including radiography and wireless telegraphy and other static electrical apparatus. By A. W. Marshall. Second edition, revised and enlarged. Containing a number of sectional drawings and details to scale. 112 pages, fully illustrated. SMALL ELECTRICAL MEASURING INSTRUMENTS. How to Make and Use Them. By Percival Marshall. Contents of Chapters: — 1. Instruments for testing the presecne of an electric current, detectors, galvanometers. 2. Instruments for measuring the pressure or quantity of an electric current, amperemeters; voltmeters. 3. Instruments for measuring electrical resistance, wheatstone bridge. 4. Instruments for measuring static elec- tricity. 5. Practical details for construction. 6. The principles upon which electrical measuring instruments work. 7. How to use electrical measuring instruments. 8. How to choose electrical measuring instruments. 90 pages, 59 illustrations. INVENTIONS. How to Protect, Sell and Buy Them. By Frederic B. Wright. Counsellor in Patent Causes This book is especially written for the use of Inventors, instructing them how to place their inventions before an Attorney clearly; the rights given them under the Law, Patent specifications. Legal forms, and the many points necessary for an Inventor to know to protect himself under the American'Laws. The most practical and clearly written American book on this subject, especially intended for the un- initiated. 114 pages, and 1 sample pattern drawing. UNIVERSAL TIME CARD MODEL. By setting to the desired hour at any one place the movable model will show at a glance the actual time of all the other places in the world. Printed on stiff card in two colors, size 7 in. by 9 in. HOW TO BUILD A 20 FT. BIPLANE GLIDING MACHINE, that will carry a man. By A. P. Morgan. A practical handbook on its construction and management. Enabling an intelligent reader to make his first step in the field of aviation with a compre- hensive understanding of some of the principles involved. Fully illustrated with detailed drawings. 25c. BOOKS. SMALL DYKAMOS AND MOTORS. How to make and use them. A practical handbook, by P. E. Powell. Contents of Chapters: — 1. General Considerations. 2. Pield Magnets. 3. Armatures. 4. Commutators and Other. Details. 5. Tables of Windings. 6. How to Build a Small Machine. 7. Useful Data. 8. Testing and Repairing. 76 pages, fully illustrated with detail drawings. SMALL ELECTRIC MOTORS. How to make and use them. By P. E. Powell. Contents of Chapters: — 1. Some points in the design of electric motors. 2r Examples of small motors to be worked by battery power. 3. A Model four-pole electro motor. 4. Motors for use on electric lighting circuits. 5. Applications of small motors and the power required for certain work. 6. Start- ing and speed controlling switches; fuses. 7. Reversing switches for Model motor; gearing, with tables of windings. 75 pages, 48 illustrations. ELECTRIC BELLS AND ALARMS. A practical handbook on their construction, installation and repair. By P. E. Powell. 77 pages, 51 illustrations. ELECTRIC BATTERIES. How to make and use them. Prac- tically describing the common forms of primary batteries. By Percival Marshall. 63 pages, 34 illustrations. TELEPHONES AND MICROPHONES. A practical handbook on their construction and use. By Percival Marshall. In- cluding testing, faults and their remedies. 80 pages, 33 illustrations. SIMPLE ELECTRICAL WORKING MODELS. By Percival Marshall. Showing the construction of electrical toys and novelties, easily constructed with a few tools from simple materials. 69 pages, 43 illustrations. X-RAYS SIMPLY EXPLAINED. A handbook on the theory and practice of Radio-telegraphy. By R. P. Howgrave-Graham. A most instructive and interesting work. 93 pages, profusely illustrated. ELECTRIC LIGHTING FOR AMATEURS. A Practical Guide to the installation of light on a small scale, describing the construc- l^ion of lamps, lamp-holders, .switches, batteries, etc., etc. By Percival Marsha il. 80 pages, 46 illustrations. ELECTRICAL APPARATUS SIMPLY EXPLAINED. A first-rate little book describing the principles and working of some of the electrical appliances in general use. 80 pages, 35 illustrations. SIMPLE SCIENTIFIC EXPERIMENTS. How to perform en- tertaining and instructive experiments with simple home-made apparatus with 59 illustrations. 25c. BOOKS. WOODWORK JOINTS. How to make and where to use them. A new revised and enlarged edition. Contents of Chapters: 1. Mortise and tenon joints. 2. Lap joints. 3. Dove-tail joints. 4. " Glue " joints. 5'. Scarfing joints and joints for lengthening timbers. 6. Circular work, showing how to make joined woodwork frames in the form of ovals and circles. The work describes clearly the construction of the principle joints used in carpentry and joinery, and shows not only how to set them out, but indicates for what purpose they are best suited. 101 pages, 178 illustrations. THE LOCOMOTIVE, simply explained. By Chas. S. Lake. A first introduction to the study of the locomotive engine, their designs, construction and erection, with a short catechism in the form of questions and answers. 72 pages, 26 illustrations STANDARD SCREW THREADS AND TWIST DRILLS. A Guide to. By George Gentry. The tables given are for small sizes of the following makers: — Whitworth Standard; British a'ssociation standard; bicycle screw threads; cycle standard; V Standard and U. S. Standard forms; International standard thread, metric system; " Progress " metric system of screws for watches; 77 pages, 6 illustrations. SIMPLE CHEMICAL EXPERIMENTS. A series of instructive experiments in inorganic chemistry. By T. T. Baker. Contents of Chapters: — -1. How to fit up and equip a small chemical labora- tory. ■ 2. How to fit up apparatus. 3. Elements and compounds. 4. Preparation of Chlorine. Ammonia, hydrochloric acid, etc. 5. Combustion. 6. How to make oxygen; hydrogen; ozone; etc. 7. Preparation of metallic salts. 8. Sulphur. 9. The atmosphere. 10. Making Chemicals. 72 pages, 19 illustrations. THE MODEL VAUDEVILLE THEATRE. By Norman H. Schneider. Describing the construction of a model theatre and the making of numerous devices to be used with it. With suggestions for various novelties for an evenings' entertainment, including chapters on shadowgraphs, the use of a polyopticon, lighting effects, wave effects, storms, etc. etc. One of these small theatres can be made very easily and at small expense and will afford many hours of amusement not only to the young but also to the grown-up, as there is no limit to the scope of the entertainments that can be produced thereon, fully illustrated. SIMPLE SOLDERING BOTH HARD AND SOFT. Together witK. a description of inexpensive home-made apparatus. By Edward Thatcher, Instructor of Manual Training, Teachers' College, Col- umbia University, (in the press). FOR SALE BY SPON ® CHAMBERLAIN Publishers of Engineering, Electrical and Industrial Booics 123-125 Liberty Street. New YorK, U.S.A.