JOINT COMMITTEE ON REINFORCED CONCRETE SECOND REPORT . LONDON THE ROYAL INSTITUTE OP BRITISH ARCHITECTS 9 CONDUIT STREET, REGENT STREET, W. 1911 Price One Shilling (Lop. a. p I B R.A RY OF THE U N I VLRS ITY or ILLINOIS From the Library of Prof. Arthur Newell Talbot Mun. and San. Engineering Theo. and App. Mechanics Faculty 1885-1942 Presented by his Family G93.S T(8Is Digitized by the Internet Archive in 2016 with funding from University of Illinois Urbana-Champaign Alternates https://archive.org/details/jointcommitteeonOOroya JOINT COMMITTEE ON KEINFOKC E 1 ) CONC RE F E SECOND REPORT LONDON THE ROYAL INSTITUTE OF BRITISH ARCHITECTS 9 CONDUIT STREET, REGENT STREET, W. 1911 THE JOINT COMMITTEE ON EEINEOECED CONCEETE : Chairman : Sir Henry Tanner, I.S.O., E.V.O., F.K.I.B.A., President Concrete Institute. Vice-Chairman : Professor W. C. Unwin, F.E.S., Hon A.E.I.B.A., President Institution Civil Engineers. Max Clarke [i^.] WiLLiABi Dunn [F.], Assoc. Inst. C.E. A. T. Walmisley, M. Inst. C.E. [H.A.] H. D. Searles-Wood [A^.J, Hon. Sec. E. Dru Drury [A’.] Thomas Henry Watson [Ai] Benjamin I. Greenwood Frank May, J.P. J. W. CocKRiLL, M.Inst.C.E. [A.] A. E. Collins, M.Inst.C.E. Captain J. Gibson Fleming, E.E., Assoc.Inst.C.E. C. H. Colson, M.Inst.C.E. Frank Capon E. Fiander Etchells, F.Phys.Soc., A.M.I.Mech.E. Charles F. Marsh, M.Inst.C.E., M.Am.Soc.C.E. William G. Kirkaldy, A.M.Inst.C.E. [Rep'esenting the Royal Institute of j British Architects ] Representing the District Surveyors' j Association I Representing the Institute of Builders ] Representing the Municipal and County I . Engineers' Association Representing the War Office Representing the Admiralty I Representmg the London County j Council I Representing the Concrete Institute. tWlV£Bs/TY OF Illinois URBANM KEY TO THE NUTATION. The notation is built up on the principle of an index. The signiticant words in any term are abbreviated down to their initial letter, and there are no exce])tions. Capital letters indicate moments, areas, volumes, total forces, total loads, ratios, and constants, &c. Small letters indicate intensity of forces, intensity of loads, and in- tensity of stresses, lineal dimensions (lengths, distances, &c.), ratios, and constants, &c. Dashed letters indicate ratios, such as a, c, n, &c., where the u, c, and n indicate the numerators in the respective ratios. The dash itself is mnemonic and is an abbreviation of that longer dash which indicates division or ratio. Subscript letters are only used where one letter is insufficient ; and the subscript letters themselves are the initial or distinctive letters of the qualify- ing words. Greek letters indicate ratios and constants. They are sparingl}^ used and are subject to the “initial letter” principle. The symbols below are arranged in ali)habetical order for facility of reference. STANDARD NOTATION. (In pillars) A = the effective area of the pillar (see definition on page 15). .^K = Area equivalent to some given area or area of an equivalent section or equivalent area. A, = cross-sectional area of a vertical or diagonal shear member, or group of shear members, in the length p, where p = pitch of stirrups. A, = Area of tensile reinforcement (in square inches). (In pillars) Ay = Area of vertical or longitudinal reinforcement in square inches. a = arm of the resisting moment or lever arm (in inches). a, = arm ratio = a jd . a, d = a. B = Bending moment ot the external loads and reactions (in pomid inches). B, B.^ = Bending moments at consecutive cross sections. Generally b = breadth. (In tee beams) b = breadth of flange of beam (in inches). h,. = breadth of rib of T beam (in inches). 0, O3 = a series of constants. (In beams) c = compressive stress on the compressed edge of the concrete (in pounds per square inch). (In pillars) c = wnrking compressive stress on the concrete of the hooped core. Cp = the working compressive stress on a prism of con- crete (not hooped) or the working compressive stress of plain concrete. c„ = compressive stress on concrete at the underside of the slab (in tee beams). c^= c jt = the ratio of c to t. In circular sections generurlly d — diameter. In rectangular sections generally d = depth. (In pillars) d = the diameter of the hooped core in inches. (In beams) d = effective depth of the beam (in inches). (In beams) d^ = depth or distance of the centre of compression from ' the compressed edge. d„ = defection. d, = total depth of the slab (in inches). (In pillars) c?„ = distance between the centres of vertical bars measured perpendicular to the neutral axis. Ec = Elastic modulus of concrete (in pounds per square inch). E,, = Elastic modulus of steel (in pounds per square inch). e = ecccntricitu of the load measured from the centre of the pillar (in inches). (Ill beams) / = extreme fbre stress, i.e. stress at the extreme “ fibre ” of any member under transverse load. (In pillars) / = a form factor or constant which will vary accord- ing to whether the hooping is curvilinear or rectilinear, &c. I = Inertia moment of a member. Ic = Inertia moment of concrete only. Ig = Inertia moment of steel only. - Inertia moment on axis xx when necessary. lyY = Inertia moment on axis yy when necessary. I = length of a pillar or effective length of span of beam or slab. m = modular ratio = E.,/E,.. N = a numerical coefficient. n = neutral axis depth, i.e. depth of neutral axis from the extreme compressed edge (in inches). w,= njd = the neutral axis ratio n, d = n. N., N., N, = a series of numerical coefficients. P = total safe pressure. (In pillars) p = the pitch of the laterals in inches {i.e. the axial spacing of the laterals). (In shear formula;) p = pitch of distance apart (centre to centre) of the shear members or groups of shear members (measured horizontally). 7T = peripheral ratio or the ratio of the circumference of a circle to its diameter. R^ = Compressive Hesistance moment = Besistunce moment of the beam in terms of the compres- .sive stress (in pound inch units). R, = Tensile Besistance moment or Resistance moment in terms of the tensile stress (in pound inches). (In beams) r = A.,/bd = ratio of area of tensile reinforcement to the area bd. (In pillars) r = V/, /V = the ratio of volumes, i.e. the ratio of the volume of helical or horizontal reinforcement to the volume of hooped core. (In beams) H = the total shear in pounds at a vortical section. = the section modulus. (In pillars) s — Spacing factor or constant which will vary with the pitch of the laterals. (In beams) s = intensity of the shearing stress on concrete in pounds per square inch. Sg = shearing stress on the steel (in units of force per unit of area). (Ill beams) s,= ds/d = the slab depth ratio. T = Total tension in the steel (in pounds). T| T., = Total tensile forces at consecutive cross sections. t = tensile stress on the steel (in pounds per square inch). U = Total ultimate breaking load on any member. [Compare W = Working load.) u = intensity of ultimate crushing resistance of plain concrete per unit of area or ultimate compres- sive stress on prisms of concrete not hooped. (In pillars) V = Volume of hooped core in cubic inches. (In pillars) V/, = Volume of hooping reinforcement in cubic inches. W = total working load or weight on any member. (In pillars) Wf = the working factor = c^fu = the reciprocal of the safety factor. w = weight or load per unit of length of span. INTEODUCTOEY NOTE. The First .Report of this Committee appeared in 1907. Since that date the use and knowledge of Reinforced Concrete for Archi- tectural and Engineering Constructions have steadily increased. It therefore appeared desirable that it should be reconsidered in the yj light of further experience, and this Second Report is the result of the Committee’s labours. The section on Materials has been modified in certain details. The section on Methods of Calculation has been re-cast in form, and the standard notation (proposed by the Concrete Institute) has been adopted. The sub-section on Columns has been revised and the formulge i ''' proposed have been re-cast so as to include the cases in which the n lateral or helical binding is a material factor in the strength. The suggestions which have been made from time to time by w Institutes and individuals, for which the Committee’s thanks are ^ due, have been of much value and have been fully considered. V d 'TO r X r\ C A A A 1201091 CONTENTS. PAGE Members of the Joint Committee .......... 2 Key to the Notation ....... Inset between ]jp. ^ and Z Introductory Note 3 Prefatory Note to the Keport 5 Materials 6 Methods of Calculation — Data . 9 Beams 10 Shear Keinforcement 14 Pillars and Pieces under Direct Thrust 15 Pillars eccentrically loaded 19 APPENDICES. I. Calculations for Singly Eeinforced Beams [Captain Fleming] . . .22 II. Shear Stresses in Eeinforced Concrete Beams [W. Dunn] . . . . 29 III. The Strength of Eectangular Slabs [E. F. Etchells] 32 IV. Strength of Pillars [E. F. Etchells] 36 V. The Moment of Inertia of Sections of Eeinforced Concrete [W. C. Unwin] . 38 VI. Tensile and Shearing Stresses in Web Eeinforcement [E. F. Etchells] . . 41 VII. Bach’s Theory of the Eesistance of Flat Slabs supported on all Edges and uniformly loaded [W. C. Unwin] 44 VIII. Comparison of the Eesults given by various Eules for the Strength of Flat Eectangular Slabs supported on all Edges and uniformly loaded [Wm. Dunn] 46 Tabular Statement of the Equations for solving Singly Eeinforced Beams . Inset at end SECOND EEPOET. PEEFATOEY NOTE. 1. Keinforced concrete is used so much in building and engineering construction that a general agreement on the essential requirements of good work is desirable. The proposals which follow are intended to embody these essentials, and to apply generally to all systems of reinforcement. Good workmanship and materials are essential in reinforced concrete. With these and good design structures of this kind appear to be trustworthy. It is essential that the workmen employed should be skilled in this class of construction. Very careful superintendence is required during the execution of the work in regard to — {a) The quality, testing, and mixing of the materials. {b) The sizes and positions of the reinforcements. (c) The construction and removal of centering. {d) The laying of the material in place and the thorough punning of the concrete to ensure solidity and freedom from voids. If the metal skeleton be properly coated with cement, and the concrete be solid and free from voids, there is no reason to fear decay of the rein- forcement in concrete of suitable aggregate and made with clean fresh water. 2. The By-laws regulating building in this country require external walls to be in brick, or stone, or concrete of certain specified thicknesses. In some places it is in the power of the local authorities to permit a reduced thickness of concrete when it is strengthened by metal ; in other districts no such power has been retained. We are of opinion that all By-laws should be so altered as to expressly include reinforced concrete amongst the recog- nised forms of construction. A section should be added to the By-laws declaring that when it is desired to erect buildings in reinforced concrete complete drawings showing all details of construction and the sizes and positions of reinforcing bars, a specification of the materials to be used and proportions of the concrete, and the necessary calculations of strength based on the rules contained in this Eeport, signed by the person or persons responsible for the design and execution of the work, shall be lodged with the local authority. 3. Fire Eesistance. — {a) Floors, walls, and other constructions in steel and concrete formed of incombustible materials prevent the spread of fire in varying degrees according to the composition of the concrete, the thick- ness of the parts, and the amount of cover given to the metal. (/;) Experiment and actual experience of fires show that concrete in which limestone is used for the aggregate is disintegrated, crumbles and loses coherence when subjected to very fierce fires, and that concretes of gravel or sandstones also suffer, but in a rather less degree."^' The metal reinforcement in such cases generally retains the mass in position, but the strength of the part is so much diminished that it must be renewed. Con- crete in which coke-breeze, cinders, or slag forms the aggregate is only superficially injured, does not lose its strength, and in general may be repaired. Concrete of broken brick suffers more than cinder concrete and less than gravel or stone concrete. (c) The material to be used in any given case should be governed by the amount of fire resistance required as well as by the cheapness of, or the facility of procuring, the aggregate. (fZ) Eigidly attached web members, loose stirrups, bent-up rods, or similar means of connecting the metal in the lower or tension sides of beams or floor slabs (which sides suffer most injury in case of fire) with the upper or compression sides of beams or slabs not usually injured are very desirable. (e) In all ordinary cases a cover of i inch on slabs and 1 inch on beams is sufficient. It is undesirable to make the covering thicker. All angles should be rounded or splayed to prevent spalling off under heat. (/) More perfect protection to the structure is required under very high temperature, and in the most severe conditions it is desirable to cover the concrete structure with fire-resisting plastering which may be easily renewed. Columns may be covered with coke-breeze concrete, terra-cotta, or other fire-resisting facing. MATERIALS. 4. Cement. —Only Portland cement complying with the requirements of the specification adopted by the British Engineering Standards Committee should be employed ; in general the slow-setting quality should be used. Every lot of cement delivered should be tested, and in addition the tests for soundness and time of setting, which can be made without expensive appa- ratus, should be applied frequently during construction. The cement should be delivered on the work in bags or barrels bearing the maker’s name and the weight of the cement contained. 5. Sand . — The sand should be composed of hard grains of various sizes up to particles which will pass a quarter-inch square mesh, but of which at least 75 per cent, should pass ^-inch square mesh. Fine sand alone is not so suitable, but the finer the sand the greater is the quantity of cement required for equal strength of mortar. It should be clean and free from ligneous, organic, or earthy matter. The value of a sand cannot always be judged from its appearance, and tests of the mortar prepared with the cement and the sand proposed should always be made. Washing sand does not always improve it, as the finer particles which may be of value to the com- pactness and solidity of the mortar are carried away in the process. * The smaller the aggregate the less the injury. 7 G. Aggregate . — The aggregate, consisting of gravel, hard stone, or other suitable material,'" should be clean and preferably angular, varied in size as much as possible between the limits of size allowed for the work. In all cases material which passes a sieve of a quarter-inch square mesh should be reckoned as sand. The maximum allowable size is usually J inch. The maximum limit must always be such that the aggregate can pass between the reinforcing bars and between these and the centering. The sand should be separated from the gravel or broken stone by screening before the materials are measured. 7. Proportions of the Concrete. — Jn all cases the proportions of the cement, sand, and aggregate should be separately specified in volumes. The amount of cement added to the aggregate should be determined on the work by weight. The weight of a cubic foot of cement for the purpose of proportioning the amount of cement to be added may be taken at 90 lbs. As the strength and durability of reinforced concrete structures depend mostly on the concrete being properly proportioned, it is desirable that in all important cases tests should be made as described herein with the actual materials that will be used in the work before the detailed designs for the work are prepared. In no case should less dry cement be added to the sand when dry than will suffice to fill its interstices, but subject to that the proportions of the sand and cement should be settled with reference to the strength required, and the volume of mortar produced by the admixture of sand and cement in the proportions arranged should be ascertained.! The interstices in the aggregate should be measured and at least suffi- cient mortar allowed to each volume of aggregate to fill the interstices and leave at least 10 per cent, surplus. For ordinary work a proportion of one part cement to two parts sand will be found to give a strong, practically watertight mortar, but where special watertightness or strength is required the proportion of cement must be increased. 8. Metal . — The metal used should be steel having the following qualities : {a) An ultimate strength of not less than 60,000 lbs. per square inch. if) A yield point of not less than 32,000 lbs. per square inch. (c) It must stand bending cold 180° to a diameter of the thickness of pieces tested without fracture on outside of bent portion. {d) In the case of round bars the elongation should not be less than 22 per cent., measured on a gauge-length of eight diameters. In the case of bars over one inch in diameter the elongation may be measured on a gauge-length of four diameters, and should then be not less than 27 per cent. * Coke breeze, pan breeze or boiler-ashes ought not to be used for reinforced concrete. It is advisable not to use clinker or slag, unless the material is selected with great care. t For convenience on small works the following figures may be taken as a guide, and are probably approximately correct for medium silicious sand : — Parts Cemeut Parts Sami Parts Mortar 1 + 1 2 = 1-20 1 + 1 = 1-50 1 + = 1-90 Parts Ccmeirt Parts Sand Parts Mortar 1 + 2 = 2-35 1 + 2| = 2-70 I + 3 = 300 8 For other sectional material the tensile and elongation tests should be those prescribed in the British Standard Specification for Structural Steel. If hard or special steel is used, it must be on the architect’s or engineer’s responsibility and to his specification. Before use in the work the metal must be clean and free from scale or loose rust. It should not be oiled, tarred, or painted. Welding should in general be forbidden ; if it is found necessary, it should be at points where the metal is least stressed, and it should never be allowed without the special sanction of the architect or engineer responsible for the design. The reinforcement ought to be placed and kept exactly in the positions marked on the drawings, and, apart from any consideration of fire resistance, ought not to be nearer the surface of the concrete at any point than 1 inch in beams and pillars and J inch in floor slabs or other thin structures. 9. Mixing : General . — In all cases the concrete should be mixed in small batches and in accurate proportions, and should be laid as rapidly as possible. No concrete which has begun to set should be used. Hand-mixmg . — When the materials are mixed by hand they are to be turned over dry and thoroughly mixed on a clean platform until the colour of the cement is uniformly distributed over the aggregate. Machine-mixing . — Whenever practicable the concrete should be mixed by machinery. 10. Laying . — The thickness of loose concrete that is to be punned should not exceed three inches before punning, especially in the vicinity of the reinforcing metal. Special care is to be taken to ensure perfect contact between the concrete and the reinforcement, and the punning to be con- tinued till the concrete is thoroughly consolidated. Each section of con- creting should be as far as possible completed in one operation * ; when this is impracticable, and work has to be recommenced on a recently laid surface, it is necessary to wet the surface ; and where it has hardened it must be hacked off, swept clean, and covered with a layer of cement mortar \ inch thick, composed of equal parts of cement and sand. Work should not be carried on when the temperature is below 34° Fahr. The concrete when laid should be protected from the action of frost, and shielded against too rapid drying from exposure to the sun’s rays or winds, and kept well wetted. All shaking and jarring must be avoided after setting has begun. The efficiency of the structure depends chiefly on the care with which the laying is done. Water . — The amount of water to be added depends on the temperature at the time of mixing, the materials, and the state of these, and other factors, and no recommendation has therefore been made. Sea-water should not be used. 11. Centering or Casing . — The centering must be of such dimensions, and so constructed, as to remain rigid and unyielding during the laying and punning of the concrete. It must be so arranged as to permit of easing and removal without jarring the concrete. Provision should be made wherever * In particular the full thickness of floor slab should be laid in one operation. 9 practicable for splaying or rounding the angles of the concrete. Timber when used for centering may be advantageously limewashed before the con- crete is deposited. 12. Striking of Centres . — The time during which the centres should remain up depends on various circumstances, such as the dimensions or thickness of the parts of the work, the amount of water used in mixing, the state of the weather during laying and setting, &c., and must be left to the judgment of the person responsible for the work. The casing for columns, for the sides of beams, and for the soffits of floor slabs not more than 4 feet span must not be removed under eight days ; soffits of beams and of floors of greater span should remain up for at least fourteen days, and for large span arches for at least twenty-eight days. The centering of floors in build- ings which are not loaded for some time after the removal of same may be removed in a short time ; the centering for structures which are to be used as soon as completed must remain in place much longer. If frost occurs during setting, the time should be increased by the duration of the frost. 13. Testing . — Before the detailed designs for an important work are prepared, and during the execution of such a work, test pieces of concrete should be made from the cement, sand, and aggregate to be used in the work, mixed in the proportions specified. These pieces should be either cubes of not less than 4 inches each way, or cylinders not less than 6 inches diameter, and of a length not less than the diameter. They should be pre- pared in moulds, and punned as described for the work. Not less than four cubes or cylinders should be used for each test, which should be made twenty-eight days after moulding. The pieces should be tested by compres- sion, the load being slowly and uniformly applied. The average of the results should be taken as the strength of the concrete for the purposes of calculation, and in the case of concrete made in proportions of 1 cement, 2 sand, 4 hard stone the strength should not be less than 1,800 lbs. per square inch. Such a concrete should develop a strength of 2,400 lbs. at 90 days. Loading tests on the structure itself should not be made until at least two months have elapsed since the laying of the concrete. The test load should not exceed one and a half times the accidental load. Consideration must also be given to the action of the adjoining parts of the Structure in cases of partial loading. In no case should any test load be allowed which would cause the stress in any part of the reinforcement to exceed two-thirds of that at which the steel reaches its elastic limit. METHODS OF CALCULATION. Data. 1. Loads . — In designing any structure there must be taken into account : — {a) The weight of the structure. {h) Any other permanent load, such as flooring, plaster, &c. (c) The accidental or superimposed load. {d) In some cases also an allowance for vibration and shock. ]0 Of all probable distributions of the load, that is to be assumed in calcu- lation which will cause the greatest straining action. (i.) The weight of the concrete and steel structure may be taken at 150 lbs. per cubic foot. (ii.) In structures subjected to very varying loads and more or less vibration and shock, as, for instance, the floors of public halls, factories, or workshops, the allowance for shock may be taken equal to half the accidental load. In structures subjected to considerable vibration and shock, such as floors carrying machinery, the roofs of vaults under passage ways and court- yards, the allowance for shock may be taken equal to the accidental load. (iii.) In the case of columns or piers in buildings, which support three or more floors, the load at different levels may be estimated in this way. For the part of the roof or top floor supported, the full accidental load assumed for the floor and roof is to be taken. For the next floor below the top floor 10 per cent, less than the accidental load assumed for that floor. For the next floor 20 per cent, less, and so on to the floor at which the reduction amounts to 50 per cent, of the assumed load on the floor. For all lower floors the accidental load on the columns may be taken at 50 per cent, of the loads assumed in calculating those floors.* Beams. 2. Spans . — These may be taken as follows : — For beams the distance from centre to centre of bearings ; for slabs supported at the ends, the clear span + the thickness of slab ; for slabs continuous over more than one span the distance from centre to centre of beams. 3. Bending moments . — The bending moments must be calculated on ordinary statical principles, and the beams or slabs designed and reinforced to resist these moments. In the case of beams or slabs continuous over several spans or fixed at the ends, it is in general sufficiently accurate to assume that the moment of inertia of the section has a constant value. Where the maximum bending moments in beams or floor slabs continuous over three or more equal spans and under uniformly distributed loads, are not determined by exact calculation, the bending moments should not be taken 1 /^ less than -j- at the centre of the span and . at the intermediate 12 12 supports. When the spans are of unequal lengths, when the beam or slab is continuous over two spans only, or when the loads are not uniformly distributed, more exact calculations should be made. If the bending moments are calculated by the ordinary theory of con- tinuous beams it should be remembered that the supports are usually assumed le^^el, and if this is not the case, or the supports sink out of level, the bending moments are altered. 4. Stresses . — The internal stresses are determined, as in the case of a homogeneous beam, on these approximate assumptions : — * In the case of many warehouses and buildings containing heavy machines it is desirable not to make any reduction of the actual loads. 11 (a) The coefficient of elasticity in compression of stone or gravel con- crete, not weaker than 1:2:4, is treated as constant and taken at one- lifteenth of the coefficient of elasticity of steel. lb«. per sq. in. Coefficient for concrete „ „ steel E Ee = 2,000,000 E, = 30,000,000 = 15. It follows that at any given distance from the neutral axis, the stress per square inch on steel will be fifteen times as great as on concrete. (5) The resistance of concrete to tension is neglected, and the steel rein- forcement is assumed to carry all the tension. (c) The stress on the steel reinforcement is taken as uniform on a cross- section, and that on the concrete as uniformly varying. In the case of steel of large section it may be necessary to consider the stress as varying across the section. 5. Workmg stresses . — If the concrete is of such a quality that its crushing strength is 1,800 lbs. per square inch after twenty-eight days as determined from the test cubes made in accordance with Clause 13, and if the steel has a tenacity of not less than 60,000 lbs. per square inch, the following stresses may be allowed : — lbs. Ber so. in. Concrete, in compression in beams subjected to bending 600 Concrete in columns under simple compression . . 600 Concrete in shear in beams 60 Adhesion * or grip of concrete to metal .... 100 Steel in tension 16,000 Steel in compression . . . fifteen times the stress in the surrounding concrete Steel in shear 12,000 When the proportions of the concrete differ from those stated above, the stress allowed in compression on the concrete may be taken at one-tliird the crushing stress of the cubes at twenty-eight days as determined above. If stronger steel is used, the allowable tensile stress may be taken at one-half the stress at the yield point of the steel, but in no case should it exceed 20,000 lbs. per square inch. Beams with Single Eeinfokcement. Beams with single reinforcement can be divided into three classes : {a) Beams of T form in which the neutral axis falls outside the slab. {h) Beams of T form in which the neutral axis falls within the slab. (c) Eectangular beams. The equations found for (a) are general equations, from which the equations for (h) and (c) may be deduced. In the calculation of all beams, the area upon which the ratio of tensile * It is desirable that the reinforcing rods should be so designed that the adhesion is sufficient to resist the shear between the metal and concrete. Precautions should in every case be taken by splitting or bending the rod ends or otherwise to provide additional security against the sliding of the rods in the concrete. 12 reinforcement is taken is considered as a rectangle of breadth equal to the greatest breadth of the beam and of depth equal to the greatest effective depth of the beam. In designing beams where the rib is monolithic with a slab, the beam may be considered to be of T form. The slab must first be calculated and designed having its own reinforcing bars transverse to the rib. The whole of the slab cannot in general be considered to form part of the upper flange of the T beams. The width h of the upper flange may be assumed to be not greater than one-third the span of the beams, or more than three-fourths of the distance from centre to centre of the reinforcing ribs or more than fifteen 4- <. C ; Td ^.s n 1 7^ i 1 1 1 * * • . - - N / Fig. 1. times the thickness of slab. The width h, of the ri Fig. 2. one-sixth of the width h of the flange. (a) Beams of T section where the neutral axis falls outside the slab. In this case the small compression in the rib between the underside of the slab and the neutral axis may be neglected. In a homogeneous beam the stresses are proportional to the distances from the neutral axis. In a discrete beam, such as a beam of concrete and steel, on account of the greater rigidity of steel, at a given distance from the neutral axis the stress in the steel will be m times as great as in concrete. Hence me _ n,d _ n, t d{l — n,) 1— n, c n, t m(l —n,) The mean compressive stress in the flange is 1 I i\c + c - [ n and the total compression is ds^ _ c 2n — ds ) 2 ‘ n -i) 7 T c 2?!' dff hds ^ . 2 n The area of reinforcement At = rbd and the total tension is irhd 18 Equating total compression and total tension M,- . = trbd ■ 2 n c ^rUf t (2n, - s,)s, Equating these two values for n, (2n^— s^)s, . 2(s,+mr)* The value of the lever arm is d- ds 3n — ^ds 3 2n — ds s, /3n, The compressive resistance moment of the beam is (5/+4mr5/ — 12mr5,+ 12mr) 6(5/+2mr) The tensile resistance moment is T> —ih ^2 (5/H-4mr5,^ — 12mr5,+ 12wr) 6m(z — To obtain stresses in the concrete and steel equal to c and t respectively r must have a value 2mcs,— mcs/ — 2mt When r exceeds the value given by this equation the equation to must be used in determining the moment of resistance. When r is less than the above value the equation to E^ must be used. The following equation gives the value for r which causes the neutral axis to be at the underside of the slab 2m(l — 5,) {h) When the neutral axis falls within the slab, or is at the bottom edge of the slab, the equations for values of Eg and E^ can be simplified and become n,- = \/(mV- E. cbd‘^n^ 1 R* = trhd'^ f \ To obtain stresses in the concrete and steel equal to c and t respectively r must equal me" 14 (c) For rectangular beams, not of T form, the equations given for T beams under (b) apply. The ratio of reinforcement may be taken on any other suitable sectional area if the formulae are modified in accordance. Slabs supported or fixed on more than tioo sides. It does not appear that there is either a satisfactory theory or trust- worthy experiments from which the strength of rectangular slabs supported or fixed on all four edges can be determined. \_See Appendices for a statement of some rules'which have been used in determining the strength of slabs.'] SHEAK EEINFOECEMENT. It is always desirable to provide reinforcement to resist the shearing and diagonal tension stresses in reinforced concrete beams. The diagonal tension stresses depend on the vertical and horizontal shear and also on the longitudinal tension at the point considered. As the longitudinal tension in the concrete at any given point is very uncertain, the amount and direc- tion of the diagonal tension cannot be exactly determined. It is the general practice to determine the necessary reinforcement by taking the vertical and horizontal shearing only intc consideration. The following equations may be used to determine the necessary resist- ance to shearing. When S, the total shear in lbs. at a vertical section, does not exceed 605a, no shear reinforcement is required.* When S exceeds 605a, vertical shea,r members may be provided to take the excess and proportioned by the following rule : — A''“^^"" = S- 60 to V or _(S-605a)p. > a where Ss is the unit resistance of the steel to shearing, and p is the pitch, or distance apart of the vertical shear members or groups of shear members, of area kg. In the case of T beams, b,. should be substituted for 5. In important cases, when extra security is required, the resistance of the concrete to shear, represented by 605a, should be disregarded. When the shear members are inclined at an angle of about 45° to the horizontal, the area kg may be decreased in the proportion of ^ . 2 These equations, though based on somewhat uncertain assumptions, give reasonable results. But experience shows that : (a) In general, floor slabs require no special reinforcement against shear- ing, and that the bending up of alternate bars near the end is sufficient. * The value of Sp is shown in the appendix to be - a 15 (h) In beams, especially in T beams, shearing reinforcement should be provided at distances apart not exceeding the depth of the beam. (c) It is desirable to bend up one or more of the bars of the tension re- inforcement near the supports. When bent at an angle of about 45° the effect of this may be taken into account in the manner set out above ; when bent at a small angle to the horizontal the effect is very indeterminate. (d) As the resistance of the shear members to the pull depends on the adhesion, and the anchorage at the ends, it is desirable to use bars of small diameter, and to anchor the stirrups at both their ends. In all cases the stirrups must be taken well beyond the centre of compression. PILLAKS AND PIECES UNDEE DIEECT THEUST. Definitions. The length is to be measured between lateral supports (neglecting ordinary bracketing). The effective diameter of a pillar means the least width and should be measured to the outside of the outermost vertical reinforcement. The effective area of a pillar means the area contained by the outermost lateral reinforcement, and should be measured to the outside of the outermost vertical reinforcement. Loading and Length of Pillars. If the load is strictly axial the stress is uniform on all cross-sections. Lateral bending of the pillar as a whole is not to be feared provided : (a) That the ratio of length to least outside diameter does not exceed 18. (b) That the stress on the concrete does not exceed the permissible working stress for the given pillar. (c) That the load be central. (d) That the pillar be laterally supported at the top and base. Construction. Lateral reinforcement properly disposed raises the ultimate strength and increases the security against sudden failure, by preventing the lateral expansion of the concrete and the sudden disruption of the pillar. Practical considerations lead to the addition of longitudinal bars, and the formation of an enveloping network of steel. The total cross-sectional area of the vertical reinforcement should never be less than 0.8 per cent, of the area of the hooped core. There should be at least six vertical bars when curvilinear laterals are used, and four for square pillars having rectilinear laterals. In the case of rectangular pillars in which the ratio between the greater and the lesser width (measured to the outside of the vertical bars) exceeds one and a half, the cross-section of the pillars should be subdivided by cross ties ; and the number of vertical bars should be such that the distance between the vertical bars along the longer side of the rectangle should not exceed the distance between the bars along the shorter side of the rectangle. 16 The most efficient disposition of the lateral reinforcement wo aid appear to be in the form of a cylindrical helix, the pitch or distance between the coils being small enough to resist the lateral expansion of the concrete. Jointed circular hoops as ordinarily made are apparently not quite so efficient. Rectilinear ties are still less adapted to resist the lateral or radial expan- sion of a highly stressed core. The volume of curvilinear laterals should never be less than 0*5 per cent, of the volume of hooped core. The diameter of rectilinear laterals should not be less than of an inch. Steength. The amount of the increase of strength in hooped pillars depends upon 1. The form of hooping (whether curvilinear or rectilinear, &c.). 2. The spacing or distance between the hoops. 3. The quantity of hooping relative to the quantity of concrete in the core of the pillar. 4. The quality of the concrete. Consequently the increase of strength may be shown to be equal to the product of the four factors (w./.s.r). u — the ultimate compressive stress on concrete not hooped (per unit of area). / = a form factor or constant which will vary according to whether the hooping is curvilinear or rectilinear, &c. s = Spacing factor or constant wffiich will vary with the pitch of the laterals. V/i = Volume of hooped reinforcement in cubic inches. V = Volume of hooped core in cubic inches. r = V^/ V = the ratio of volumes, i.e. the ratio of the volume of helical or horizontal reinforcement to the volume of hooped core. The ultimate compressive stress on concrete not hooped being = u, and the increase of strength due to hooping being u.f.s.r, the total resistance of the hooped material per unit of area wall then be = U + U.f.S.T = u[l-\- f.s.r]. Let Cp = the working compressive stress on a prism of concrete (not hooped) = Wpw Wp = the working factor = Cpju, Then the safe compressive stress on the hooped core = c, where c=Wpw[l+/,s.r] = C2>[l+/.s.r] 17 The values of /, s and their product may bo obtained fiom the following table : — Form of Ijateral Reinforcement Form Factor =f Spacing of Laterals in Terms of Diameter of Hooped Core 1 Spacing Factor = s Value of / s Helical . 1 0'2d 32 32 J) • • • ‘ 1 0-3d 1 24 24 • • • i 1 0-icl 1 16 16 Circular Hoops 0-75 i 0-U 32 24 • i 0-75 ! 0-3d 24 18 0-75 0-id 16 12 Rectilinear 0-5 0'2d 32 16 0-5 0-3d 1 24 - 12 ? J 0-5 1 OAd 16 8 • • 0*5 ' 0'5d 1 8 4 n • • 0*5 0'6f^ 1 1 0 0 Let p = the pitch of the laterals in inches [i.e. the axial spacing of the laterals]. d = the effective diameter of the hooped core in inches. The spacing factor should not be taken at more than 32, even if p is less than '2d, but intermediate values of the spacing factor may be obtained from the equation S =48 - 80^ d It will be seen from the above table that the advantage of hooping disappears with an increase in the spacing of the laterals, irrespective of the volume of hooping or the value of r. Before the safe stress on the hooped core can bo obtained it will be necessary to give values to Wp and A table for this purpose will be found below. The value of the working compressive stress on the concrete of the hooped core having been obtained, the maximum permissible pressure or load may be obtained from the equation P = c [A + (m — 1) Ay], where A = the effective area of the pillar. = modular ratio. Ay = Area of vertical reinforcement. P = total safe pressure on pillar. 18 Working Btrbsses. A safety factor of 4 at 90 days is recommended for all pillars. The following table of working stresses is suitable if good materials are used, and is liased on the assumption that test cubes have at least the strength given at the periods stated : — Table showing the Value of u and Cp for Pillars. Proportions of Concrete Mea- sured by Volume ! Pounds’ of Cement to 13^ cubic feet of Sand and 27 cubic feet of Shingle or Broken Stones Value of u at 28 days in pounds per square inch Value of u at 90 days in pounds per square inch Value of C), at 90 days in pounds per square inch (Safety factor = 4 1 (Working factor = 1/4) 1:2 : 4 610 1,800 2,400 600 1 : 1^ : 3 810 1 2,100 2,800 700 1:1:2 1 i 1,220 2,700 3,600 900 It is assumed that the tests of the strength of the concrete are made on unrammed cubes and of the same consistency as the concrete used on the work.* Limitations of Stress on Pillars. The following limits of stress should be observed in pillars : (a) The stress on the metal reinforcement (i.e., the value of m.c) should not exceed 0*5 of the yield point of the metal ; {b) Whatever the percentage of lateral reinforcement the working stress on the concrete of pillars should not exceed (0’34 + 0'32 /) u where f = form factor. u = ultimate crushing resistance of the concrete. Form of Laterals Kectilinear . Independent Circular Hoops Helical Value of Form Factor (0*34 + 0*32/) 0’5 0’5 20 0- 75 0*58 20 1 - 00 0-66 If these limits are adopted, the working stress on hooped concrete will always fall within the limit of continued endurance ” for plain concrete. * The limit of 2,400 pounds per square inch given in the previous report of the Committee was adopted on the assumption that the cubes would be rammed witli iron rammers under laboratory conditions, 19 /= W riLLAUH ECOKNITJCALLY LOA])EJ). Tf a pillar initially straight is loaded eccentrically, as when a beam rests on a bracket attached to the pillar, it may be regarded as fixed at the base and free at the loaded end. Then it must Lend in the plane passing through the load, the deflection at the top being Let e be the eccentricity of the load measured from the centre of the pillar when straight. Then the bending moment at the base of the pillar is W But it is known that will be small compared with e, provided that W is small compared with 2'EIIP, and this will be the case in such conditions as are likely to occur in designing concrete pillars. Then the bending moment may be taken as W^, and the extreme “ fibre ” stress at the edge of the base of the pillar, treating it as homogeneous, will be ^ 1 A j very nearly, where A is the whole section of the pillar and the section modulus relatively to an axis through the centre of gravity and at right angles to the plane of bending. In dealing with reinforced pillars which are not homogeneous, it is convenient to substitute for the actual section of the pillar what may be termed the equivalent section, or section of concrete equivalent in resistance to the actual pillar. If A is the effective area of section of the pillar (including the area of reinforcement), and Ay is the area of vertical reinforcement, then the equivalent section is Ae = A + (m — l)Ay. If d is the depth of the section in the i)lane of bending, the Inertia moment relatively to the neutral axis can be expressed in the form I = nkd‘\ and the section modulus in the form S„, = 2tiAd. {See Appendix V.) It is desirable in pillars that there should be no tension, and generally when the vertical load is considerable there is none. Cases in which the eccentricity is so great that there is tension must be treated by the methods applicable to beams if it is made a condition that the steel carries all the tension. In the following cases it is assumed that there is no tension. Case I.— Pillar of Circular Section, Beinforceonents Symmetrical and Equidistant from the Neutral Axis. Let m be the modular ratio = E.,/E, A the effective cross-section of the column in square inches. Ay the area of vertical reinforcement in square inches, d the diameter of r> 2 20 the pillar, the distance between the vertical reinforcing bars perpendi- cular to the neutral axis. Then the equivalent section is Ae = a + — l)Ay, and the section modulus is {Appendix V.) = -^A<^ -f- ^{'iii — l)Av ^ . d The stress at the edges of the section can then be calculated by the general equation /= W J f 1 where e is the eccentricity of the load in inches and W the weight or load in pounds. The greater value of stress must not exceed the safe stress stated above. Case II. — Bectangtdar Section ivitli Beinf or cement Sijmmetrical and Equidistant from the Neutral Axis. Using the same notation as in the last case, d being now the depth of the section in the plane of bending, the section modulus is {Appendix V.) Sm = gA^^ + — l)Ay ^ , d and the stresses are given by the same equation as in the previous case. Case III . — Column of Circular Section with Beinforcing Bars arranged in a Circle. — Using the same notation as in Case I., h^ being the diameter of the circle of reinforcing bars, the section modulus is {Appen- dix V.) + \(m - V)kA-i, d and the stresses are given by the same equation as in Case I. (c) Long Pillars axially loaded. For pillars more than 18 diameters in length there is risk of lateral buckling of the pillar as a whole. The strength of such pillars would be best calculated by Gordon’s formula, but there are no experiments on long pillars by which to test the values of the constants for a concrete or concrete and steel pillar. There does not seem, however, to be any probability of serious error if the total load is reduced in a proportion inferred from Gordon’s formula to allow for the risk of buckling. Let, as before, A = the area of the column in inches ; Ay = the area of vertical reinforcement. Then Ae = A + — l)Ay is the equivalent section. Let N be the numerical constant in the equation, I = NA'?“ {Ap- pendix F.), and d the least diameter of the pillar. Then for a pillar fixed in direction at both ends Gordon’s formula is W _ 1 _ 1 ku ^ p 1 + c; CfUP so that the pillar will carry less than a short column of the same dimen- 21 sions in the ratio of 1 + C.^ to 1, or, in other words, the column will be safe if calculated as a short column, not for the actual weight or pressure P, but for a weight or pressure = (1 + C2)W. The constant Cj has not been determined experimentally for reinforced long columns. But its probable value is ^ ^ 47T‘E(j ^ u where it is the ultimate crushing stress. Butting = 2,000,000 and tt = 2,500, then = 82,000. Looking at the well-understood uncertainty of the rules for long columns, very exact calculation is useless. Some values of N for ordinary types of column are given in Appendix V. Taking these values, the following are the values of 1 + C.^ : — 1 d Case I. N = 0-098 Values of 1 + C^. Case II. N = 0-075 Case 111. N = 00646 20 1-18 1-17 1-19 25 1-20 1-26 1-80 80 1-29 1-88 1-44 The differences of 1 + Cg for considerable differences of N are not vei'y great. In any case N can be found by the method in the Appendix with little trouble. In the case of columns fixed at one end and rounded or unfixed at the other, 2C2 must be substituted for C2. If the column is rounded at both ends, 4C2 must be substituted for C. 22 APPENDIX L CALCULATIONS POP SINGLY REINPOKCED BEAMS. By Capt. J. G. Pleming, R.E. To find the moment of resistance of a T beam in which the neutral axis falls outside the slab A- c The total compression is equal to the area ABDEx width of flange. The area ABDE ^ AB=c AB.OE _ c . {n—d^) AE= ds. area ABDE= f c V 71 J 2 ^ (2n-4) ds ~ n 2‘ I'heu tlie total compression = ^ .h. The area of tensile reinforcement At = rhd. The total tension = T = trhd. 28 Equating total compression to total tension e . . b=trbd. n 2 Let c,= ratio of c to t, then _ rhd 2n rd 2n {2n—dfi)ds * Substituting n,d for n and s,d for dg. r d ^n.d n ' {2n,d—s,d ) . s,d' _ 2rn,d^. s,d‘^' _ 2rn, ~{2n,-s,)s,' It has already been shown on page 12 of the report that another value for cjt can be obtained, deduced from the supposition that the stress on any layer is jointly proportional to its distance from the neutral axis and to the rigidity of the material. This value for cl t is m(l— n,) Equating these two values for cjt, n, _ ^rn, m(l — n,) (2n,— 5,)6-/ whence _ s/+2mr 2(5,+mr)‘ It has been found that clt = . (2n— 5,)s To express this in terms of m, r and s, ^ ^ s/+2mr ' 2(5,+mr) substituting for in the equation for cjt cjt = ^ 2(5,+mr) 2 ( 5 /+ 2mr) ] 2(s,-|-mr) 2r(s/+2mr) (2s/^-j-4mr— 2s/— 27nrs,)s, 2r(s/+2mr) 2mr{2~-s,)s. s/+2mr m(2— s,)s 24 arm. To find the moment of resistance it is first necessary to find the lever The centre of pressure is at a distance 4 /2c„+c\ B V Ca-\-C ) from the extreme fibre in compression. __ c{n — ds) On d,= 4- n 2c{n — di n B y c{n — ds) _|_c n __ dj^ V^cn — 2cc?^ B L 2cn — cds - _ ds fBn — 2df 1 V 2n s The lever arm is a = d — dc ^ dg f o?z — B * \ — ds _ 6nd — ddds — Snds + 2ds^ expressing n and ds as n,d and s^d 8(2n — ds) ^ _ 671, d^ — Ss,d'^ — Qn,s,d^ + 2s, ‘^d' 6d(2n, — s,), _ d(6rt, — 3s,(l + n,) + 2^/) 3(2n, — s,) The total compression _ c{2n — ds) ds ^ n 2 cbds f2n — d, 2 r 2n-ds l L n J __ cbds d{2n, — s,) 2 * d . n, _ cbds (2n, — s,) 2 n. Then the resistance moment in terms of the compressive stress is found by the equation * 2 n ' cbds [2n, — s,) . {671, — B5,(l + ^0 + ^ 5 /’) 2 ‘ n, ‘ B(2n, - s,) cbdds . (6n,~ Bg,(l -|- 'fh) + 6n, 25 Expressing n, in terms of s„ m and r f 6 . (s/ + 2mr ) __ o / . _j_ + 2mr\ , ^ , U 2(s. 4- mr) ' V ‘ 2(s,-i-mr)/ ' II =: 2(5, + 5,^ + 2mr 2(5, + mr) _ cbdds f65/^ + 12mr — 65,“ — 12wr5,— 85 ,'’ + 45,'^ + 4?/ir5,'^‘ 6 [ 5,^ + 2mr _ cbdds\s/ + 4c.mrs/ — 12mrs, + 127?iri . 6 |_ 5,^ + 2mr J The resistance moment in terms of the te7isile stress is found by the equation Ki = trhda trhd\^n, — 35,(1 + n,) + 25,*) "" pn, Expressing n, in terms of 5, m and r ^^_2mr _ o , 5,^ + 2mr\ . , ’ 2(5,-f-wr) ' ' \ 2(5, 4-mr)/ ' f 2 . (5,^ -h 2mr) r* 6 N = irhd‘ . thd [ 2 ( 5 , + '^nr) ^/-\-4mrSf^ — 12mr5,4- 12mr — 5 , 6m(2 — 5 ,) To find the ratio of reinforcement to give stresses of c and t in the concrete and steel. Ec = Ef. cbdds (s/ + ^mrs,- — 12mr5, + 12mr) 6 ( 5 ,^ + 2mr) _ tbd4{Sf^ + 4mr5,‘^ — 12mr5,-l- 12mr) 6m(2 — 5 ,) cds _ id 5 ,^ + 2mr m(2— 5 ,) ’ C5,m(2— 5 ,) = t{s,‘^ + 2mr). 2mcs,— mcSf- — is/ + 2mtr. _ 2nics,— mcs,- — i5,‘ 2mt To find the value of r such that the neutral axis is at the underside of the slab. d^ — n s/ + 2?/tr 5 , = n,= r. 2 ( 5 , -|- mr) 2Sf^ + 2mrs, — 5 ,^ + 2mr. 2mr{\ — 5 ,) = 5 ,^ r = 2m (1 — 5 ,) 26 When the neutral axis falls at the underside of the slab or within the slab n,= 5,. The equation _ 5 / + 2mr mr) becomes n‘^ + 2mr 2(n,+ mr) 2n/“ + ^mrn,= + 2mr rif — \/mV^ + 2mr — mr. The equation R<.= chdds{^n, — ds,{l -f n^) + 2g,^) becomes B,. = cbdn,d ^^'^' ~ ~ 6n, cbd /-I % \ 2 - 3 )• The equation becomes Tj _ 35,(1 +n,)+ 2s/) T) _ trhd'\6n^ — 3n,(l + n,) + 2n,^) * “ 3n, To find the value of r to give values of c and t — c _ s/ + 2mr t m(2 — s,)s. _ 2r(s/^ + 2mr) (4mr — ^mrs^Sf _ 2r(s,‘^ + 2mr) (4mr + 2s/^ — 2s, ^ — 2mrs,)s, + ^mr) [2(s,^ 4 - 2mr) — 2s,(s, + mr)]s. 2 ^ (s,^ + 2w r) 2(s, 4 - (s ,2 4 - 2mr) _ 2 s ' 2(s, 4 - 'mr) ' 2(s,4- mr). s, ^rn, (2rt,-s/)s, 27 and if s,= n, t r = n, cn, Tt Also ^ ^ t m{\-n,) me— men f = n,t 71, {tne-^t) = me me me + 1 Substituting in the equation eu/ '2t r — me^ 2t{me + t)' The equations found for a T beam where the neutral axis does not fall outside the slab apply equally to rectangular beams. These equations for rectangular beams may be equally easily deduced from first principles as outlined below. < d c _ n, t m(l — n^) applies equally to this type of beam as to a T beam, depending entirely on the stress in any fibre varying directly with its distance from the neutral axis and the rigidity of the material. The total compression — \ebdn,. The total tension = trhd. Equating total compression and total tension e 2r ~t n, ‘ c Equating these two values for-, t n, _ ^ m(l — n, * 28 n, = 2mr — 2mrn, “1- -f = 2mr + n, + mr = V + 2mr. n, = V mV' + 2mr — mr. I’lie resistance moment with regard to the concrete under compression is found by the equation K,. = total compression X lever arm n 3* 71, d The lever arm a = d — = d- cbd Be =2-”/ ebd^ = - ^ n (>-?)■ _ 3 J d 1 The resistance moment with regard to the tensile stress on the steel K« = total tension X lever arm To find the value for r to give values of c and t — c _ n, t m(l — n,y me — men, = tn, n,(me 1) = me me n, = - — me-\-t Also by equating total compression and total tension c _ 2r t n, 2rt = en. r = 2t substituting for n, its value 7UC me t r = me' 2«(mc + 0’ 29 APPENDIX IT. BHEAK STKESSES IN EEINFOECED CONCEETE BEAMS. By W. Dunn. In a reinforced concrete beam the steel reinforcement is assumed to take all the tension. Accordingly, if Bg is the bending moment at a section AAj, and B^ is the bending moment at a section DD„ distant p from AA , the total stress on the steel at the former section is B, a and at the latter section is where a is a the lever arm. The difference T, -To Bi-B, a is communicated from the steel to the 30 concrete l)y adhesion, and produces a shearing stress in the concrete above it, which is distril)uted over an area of fh. If s is the unit shearing re- sistance of the concrete, we have T, -T.,= = phs. a If the concrete cannot safely resist this unit shearing stress s, we must add steel shear members. If we add vertical steel shear members, let the sectional area of steel in the length p equal A.s and the safe unit shearing resistance of the steel be s,s. This area A,s- may be provided in one member or in a group of members. Then we must have vhs + A„Ss = ?i— ~ a The ratio in which the shear stress is distributed between the steel and concrete has not yet been determined, for which reason and where greater safety is desired, the shearing resistance of the concrete is often neglected and the vertical shear members determined by the formula A ^1 — ^2 a Now Bj — B .2 is the difference between the bending moments in the length p. From the principles of statics, this difference equals the average shearing force on the length p multiplied by the length p ; that is, from the ordinary equation ^ fix S, we get (since dB = B^ B^— B2=Sp. B, B^ and dx = p) We may then put yhs + AsSs= If the resistance of the concrete is neglected, we have a a * A^Sg Sp a These rules fix the sectional area and spacing of the vertical shear members. If there is a shear stress (and a shear stress only) on one face of a body, there must (by Eankine’s theory of internal stress) be a shear stress of the same intensity (and a shear stress only) on any plane at right angles to the first plane. Also, on the plane inclined at 45° in one direction to the first plane there must be a normal tension only, of the same intensity, and on the plane inclined at 45° in the other direction a normal compression only, of the same intensity. Whenp is the horizontal distance between shear members inclined at 45°, the extent of the planes, on which these normal tensions and compressions act, is less than the planes on which the shears act, in the ratio of 1 to \/2 and the amount of the normal tension or compression to bo ta.kon l)y the shear member is also lesf^ in the ratio ^ . Accordingly, when shear members are inclined at 45° to the horizontal. 31 and the tensile resistance of the concrete is neglected, the diagonal shear inenibers are proportioned by the rule a \/'Z when the resistance of the concrete is neglected, and aV2 when the resistance of the concrete is taken into account. 82 APri^.NDTX ITT. THE STRENGTH OF RECTANGULAR SLABS. By E. P. Etchells. Grashof’s formula for the maximum bending moment on flat plates is perfectly general in form. It is based inter alia upon the identity of the deflection (at the point of intersection) of any two cross-sections at right angles to each other. Applying this principle to a rectangular slab, we see that a point at the centre of the area must have the same deflection along the major as along the minor axis. The curvature will, however, be greater in the direction of the minor axis. As the bending moment and the stress are both proportional to the curvature, it follows that the bending moment and the stress will both be greater in the direction of the minor axis. Rankine’s formula for the strength of rectangular plates is in form identical with Grashof’s. The Progress Report of the American Joint Committee on Concrete presented to the American Society of Civil Engineers on January 20, 1909, gives the Grashof-Rankine rule for slabs. The Austrian Government Report, 1908, adopts the Grashof-Rankine rule. The French Government adopt an empirical formula by which the bend- ing moment at the centre of a square slab fixed in direction on four sides and uniformly loaded = ^ as against obtainable by the Grashof-Rankine rule. The references in the French Government Reports are as follows “ S’il s’agit d’un hourdis porte par deux cours de nervures orthogonales, d’ecartements respectifs I V , pour calculer le moment de flexion dans le sens de la portee I, on pourra, faiite de mieux, le calculer comme si les nervures de portee I existaient seules, en multipliant le chiffre obtenu par le coefficient de reduction : 1 7 ‘ 1+2 I, “ On fera do memo en permutant les lottres I et V pour obtenir le moment de flexion dans le sens de la portee 33 Another form of this equation for the bending moment reduction factor is also given in the French report : — “ Hourdis portes par deux cours de poutres orthogonaux. On represente par I et V les espacements, d’axe en axe, des poutres des deux cours ortho- gonaux. “ Pour calculer le moment de flexion dans le sens de la portee I, on deter- mine la valeur qu’il aurait si le second cours de poutres n’existait pas et on la multiplie par le coefflcient de reduction : V ^ V 4 “ On procede de meme pour calculer le moment de flexion dans le sens de la portee V en rempla 9 ant dans la formule, I par I et ?'par L” No other reason or proof of the French rule is given, and it would appear to be based on a limited range of tests and gives practically the average bend- ing moment for the whole of the cross-section of a slab. It is considered that it would be inadvisable to form conclusions as to the behaviour of slabs outside the limited range of a few experiments. Moreover, it is considered injudicious to use formulae for the average bending moment in lieu of the maximum bending moment on slabs which may measure as much as 30 feet by 60 feet and have a thickness of about 9 inches. The Grashof-Kankine rule gives the position and maximum bending moment at any part of the cross-section of the two central strips of the slab. When the span of the slab is small in relation to the thickness of the slab, it will be seen that the central strip of the slab will get a certain amount of support by the sides of the slab parallel to the central strip. It will, of course, be obvious that the strips next to the side and parallel to the side cannot possibly have the deflection which will occur on the central strip. As the ratio of the span to the depth of the slab increases, the effect of the supports which are parallel to the central Strip decreases. The tendency is to make slabs larger and larger, and a slab 9 inches thick may measure 30 feet by 60 feet. Let us take a strip across the centre of the 60-foot edge having a span of 30 feet ; it is submitted in this case that the supports along the 30-foot edge will not do much to decrease the deflection of the central strip, which is 30 feet away from the shorter supporting edges. It is therefore urged that the use of the Grashof-Kankine formula for the bending moment of central strips is justifiable and reasonable. It should be noted that the Grashof-Kankine rule purports to deal with the point of maximum stress in a homogeneous material, such as cast iron or plain concrete. In the case of reinforced concrete, however, we are able to vary the spacing of the tensile reinforcement according to the bending moment at any part of the cross-section, c B4 Narrow BtripB wliicli aro parallel to the sides and are near the sides are sustained for their whole length. They cannot deilect, and cannot have a camber in the direction of their length so long as the side supports remain rigid. If we take a cross-section through the centre of the slab, we will find that the deflection at the midspan of the central strip will be a maximum, and the deflection at the midspan of the two outermost strips will be reduced to zero. Moreover, the maximum bending moment will occur at some section of he central strip, while the bending moment on the cross-section of the two outermost strips (which are supported along one edge for the whole of their length) will be reduced and will approach zero. Any exact calculation of the variation of the bending moment in a cross- section of all the adjacent strips will be nullified by any deflection of the side supports. If we assume that the bending moments in a cross-section of all the adjacent strips will vary as the ordinates of a parabola, the average bending moment over the whole of the cross-section of the strips will be two-thirds of the maximum bending moment (on the central strip). On this assumption it will be found that in the case of the square slab, at any rate, the Grashof-Kankine rule and the French rule give the same results. This may be shown as follows : The bending moment reduction factor (for the shorter span) in the Grashof-Eankine rule ^ V Dividing the numerator and the denominator by will give 1 _ 1 7;V1- 1 + 1 + In the case of the square slab : and the bending moment at the centre of the square slab i 1 ^ wz 24 ^ 2 24 ^ 2 48' If the bending moments across the section of the slab are taken as varying as the ordinates of a parabola, the average bending moment across the slab will be two-thirds of the maximum bending moment, viz. the average bend- ing moment across the whole of the given cross-section will be 2 m 3 48 72 * This is, of course, the result which is given by the French rule. The advantage of the Grashof-Eankine rule is. however, that it ])vovides B5 for the maximum l)cri(liug moment at the centre of a Blal>, whereas th(? hrencli rule seems more ai)])lical)le to the average bending moment across the width of the slab without any consideration as to the manner and dis- tribution of the bending moment and stresses, and without any regard to the fact that the maximum bending moment at the central strip of the section is probably one and a-half times the average bending moment over the whole width of the cross-section. It might, therefore, he deemed desirable to consider the advisability of retaining unchanged the Grashof-Kankine rule for the maximum bending moment for the central strips of slabs, and taking a decreased bending moment for the parallel strips between the central strip and the sides ; the bending moment for the side strips being reduced according to some such law as here suggested. This procedure is in strict accordance with the letter and the spirit or the Grashof-Kankine rule. As a precautionary measure, it might furthermore he deemed advisable not to reduce the bending moment of the side strips to zero, but to adopt some working rule, which could, in a way, make some allowance for the fact that, if the side supports are the ribs of Tee-Beams, they must themselves suffer some deflection under load. The tensile reinforcement parallel to and near the sides of the slabs will, by their intersection in the corners of the slabs, tend to reduce the extent of those diagonal tensions which must inevitably occur in the extreme corners of rectangular slabs when the central stiips assume their permissible and intended deflection. The varying bending moments across a section of the slab can be pro* vided for, either by having uniform spacing of bars and varying the diameter of the bars ; or by varying the spacing and keeping the diameter of the bars uniform throughout the full width of the cross-section of the slab. It will be found much more convenient, both in the design and in the actual execution of the work, to keep the diameter of the bars uniform and to vary the spacing. c 2 36 APPENDIX IV. ' STEENGTH OF PILLARS. By E. F. Etchells. The precise amount of the increase of strength in hooped pillars depends, inter alia, upon 1. The form of hooping (whether curvilinear or rectilinear, &c.). 2. The closeness of the spacing of the hoops in relation to the diameter of the hooped core. 3. The quantity of the hooping relative to the quantity of concrete in the core of the pillar. 4. The quality of the concrete. Consequently the increase of strength may he shown to be proportional to the product of the four factors ufs'r where u — the ultimate compressive stress on concrete not hooped (in pounds per square inch). /' = a form coefficient which will vary according to whether the hooping is curvilinear or rectilinear, and which should, to a certain extent, represent the relative value of the form of hooping to resist deformation due to the expansion of the stressed core. s' = spacing ratio = p/d, the ratio of the pitch of the laterals to the diameter of the hooped core. V/, volume of helical reinforcement in cubic inches. V = volume of hooped core in cubic inches. r == V/,/V = the ratio of volumes, i.e., the ratio of the volume of helical or horizo7ital reinforcement to the volume of hooped core. (Note. — If V is for the whole pillar, then Vy^ must be for the whole pillar. If V is for the volume of unit length of pillar, then Vy, must be the volume of helical reinforcement in unit length of pillar.) p — the pitch or axial spacing of the laterals in inches. d = the diameter of the hooped core in inches. By the use of suitable constants we may change a relationship of pro- portionality into a relationship of equality. Let 71 ^, n- 2 , n.^, n^ be such a series of numerical coefficients. Thus instead of saying that “ the increase of strength is 'proportio7ial to the product of the four factors ” ufs'r we may say that “ the increase of strength is equal to ” {n,u){7i,f){n./){n,r). 87 Let n^{nj'){n./)n^ = m', then we may say that “ the increase of strength due to the lateral hooping is equal to ” mn'r. The value of m' is fairly well known from experiments, and although each of its component factors has not yet been isolated, yet the separate effects of the “ form ” and “ spacing ” factor are traceable, so that instead of m' we can use the equation m' = fs, where / and s are constants, depending, inter alia, upon the closeness of the spacing and the form of the laterals. The ultmiate compressive stress on a prism of concrete 7iot hoofed = u. The working stress on a prism of concrete (not hooped) = The increase of strength due to hooping being um'r the total resistance of the hooped material will then be -Increase of streiigth due to hoofing of concrete = u-\- um'r = u\\ -f mV]. Let Wp = the worhmg factor — Cj,/u ^ the reciprocal of the safety factor s; The safe compressive stress on the concrete within the hooped core = c where inmai sirengm oj non- hoo'ped concrete + or since m' = fs _ u[l -|- mV] = Wpw[l + m'r] c = Cj,[l 4- fsr]. 38 APPENDIX V. THE MOMENT OF INEETIA OF SECTIONS OF EEINFOECED CONCEETE. By W. C. Unwin. If m is the modular = ratio E^/Ec of the coefficients of elasticity of steel and concrete, then an area A^of steel is equivalent in resistance to mkt of concrete. If A is the area of a section (including the area of reinforcing bars), and A„ the area of the vertical reinforcing bars, then the section is equivalent to a section of area A = A^ + {Hi — l)k^, of concrete only. h This will be called the equivalent section.' The moment of inertia of a section about its neutral axis can always be put in the form I = where d is the depth at right angles to the neutral axis and N is a numerical coefficient depending on the form of the section. Thus for a rectangular section Afp. I = 1 and for a circular section I = In dealing with reinforced sections it is convenient in many cases to express the moment of inertia in terms of the equivalent area. The equivalent area is found by adding to the actual area of the section portions of a total area {m • • 1) A,, at the same distance from the neutral axis as the reinforcing bars. The figure shows a section for which p b is the plane of bending, and n a the neutral axis passing througli the centre of gravity of the section. The reinforcing bars are supposed symmetrical to the neutral axis. The projecting parts of total area (m — 1)A„ are the concrete areas equivalent to B9 the steel. If I,, is the moment of inertia of the section without reinforcing bars, the moment of inertia with reinforcing l)ars is I = Ig + Thus for a rectangular section I = + i{m - and the modulus of the section is — l)Ay . (A/ For a circular section I = - l)AvC/^^ and the section modulus is s,„ = ^kd + him - l)Av-*-. d Example 1. — Eectangular Section. Let m = 15, Ay = 0*01 A, and = O'^d. The equivalent area is A]^ = A -f- — l)Av — A -|- 14Ay. I = TT 4 X 100 = 0*1117A^Z2^ But A = 0-877 Ae I = 0-098Acl2 S„* = 0*196Afi. In this case in the general expression I = NAf?^, N = 0*098 when A^ is the equivalent section. Example 2. — Circular Section. Let ni =15, Ay = 0*01 A, and d^ = 0*8c/. The area of the equivalent section is 1*14A, as before. I = yVAcl^ + i X 14 X *01A X *646^^ = 0*0849A b c -*■-6 --*■ i 1^5 T ; ■ -f- • IN — ;■ — — -;-A 1 M _-|_ --- 1 ^ 2 1 i d 1 1 “i ”A if (cl) (b) (C) T beam T beon^ Rect- (neutral (neutral angular axis OXIS beam outside within slab) slab) Value of n, j"/ + 2 mr 2 (j-, + TTxr) ■ynfr-^ 2 m'r- 7 nr Lev Mean comt:>ression stress Value of Value of Rf VLIues of T to giv-e Stresses of c and t Value of r atwFicK d{'S 2t4'rn + /e7n7^) 6 m-r(!?-xS,) cmr ( 2 -s,) cbdd Value of A, 6 i.s,^t 27 nr) ^■Ljs ^^/t-Amr^^-\27n,rsA t2mr S^C2- ^ zmcs,- mcs/- tj, ^ 2mt r= 2 m (i ~ S/) Tbd z ^ s ' (rSd^(i-f) T‘ 7 nc‘ 2t(m,ct-t) Tbd UNIVEB9ITY OF ILUN04S-URBANA ^ 3 0 12 050060737