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Again, Mr. Godfrey seems to misunderstand the influence of Poisson's ratio in multiple-way reinforcement. If Mr. Godfrey's ideas are correct, it will be found that a slab supported on two sides, and reinforced with rods running directly from support to support, is stronger than a similar slab reinforced with similar rods crossing it diagonally in pairs. Tests of these two kinds of slabs show that those with the diagonal reinforcement develop much greater strength than those reinforced directly from support to support. Records of small test slabs of this kind will be found in the library of the Society.
Mr. Godfrey makes the good point that the accuracy of an elastic theory must be determined by the elastic deportment of the construction under load, and it seems to the writer that if authors of textbooks would pay some attention to this question and show by calculation that the elastic deportment of slabs is in keeping with their method of figuring, the gross errors in the theoretical treatment of slabs in the majority of works on reinforced concrete would be remedied.
Although he makes the excellent point noted, Mr. Godfrey very inconsistently fails to do this in connection with his theory of slabs, otherwise he would have perceived the absurdity of any method of calculating a multiple-way reinforcement by endeavoring to separate the construction into elementary beam strips. This old-fashioned method was discarded by the practical constructor many years ago, because he was forced to guarantee deflections of actual construction under severe tests. Almost every building department contains some regulation limiting the deflection of concrete floors under test, and yet no commissioner of buildings seems to know anything about calculating deflections.
In the course of his practice the writer has been required to give surety bonds of from $50,000 to $100,000 at a time, to guarantee under test both the strength and the deflection of large slabs reinforced in multiple directions, and has been able to do so with accuracy by methods which are equivalent to considering Poisson's ratio, and which are given in his book on concrete steel construction.
Until the engineer pays more attention to checking his complicated theories with facts as determined by tests of actual construction, the view, now quite general among the workers in reinforced concrete regarding him will continue to grow stronger, and their respect for him correspondingly less, until such time as he demonstrates the applicability of his theories to ordinary every-day problems.
PAUL CHAPMAN, ASSOC. M. AM. SOC. C. E. (by letter).—Mr. Godfrey has pointed out, in a forcible manner, several bad features of text-book design of reinforced concrete beams and retaining walls. The practical engineer, however, has never used such methods of construction. Mr. Godfrey proposes certain rules for the calculation of stresses, but there are no data of experiments, or theoretical demonstrations, to justify their use.
It is also of the utmost importance to consider the elastic behavior of structures, whether of steel or concrete. To illustrate this, the writer will cite a case which recently came to his attention. A roof was supported by a horizontal 18-in. I-beam, 33 ft. long, the flanges of which were coped at both ends, and two 6 by 4-in. angles, 15 ft. long, supporting the same, were securely riveted to the web, thereby forming a frame to resist lateral wind pressure. Although the 18-in. I-beam was not loaded to its full capacity, its deflection caused an outward flexure of 3/4 in. and consequent dangerous stresses in the 6 by 4-in. angle struts. The frame should have been designed as a structure fixed at the base of the struts. The importance of the elastic behavior of a structure is forcibly illustrated by comparing the contract drawings for a great cantilever bridge which spans the East River with the expert reports on the same. Due to the neglect of the elastic behavior of the structure in the contract drawings, and another cause, the average error in the stresses of 290 members was 18-1/2%, with a maximum of 94 per cent.
Mr. Godfrey calls attention to the fact that stringers in railroad bridges are considered as simple beams; this is theoretically proper because the angle knees at their ends can transfer practically no flange stress. It is also to be noted that when stringers are in the plane of a tension chord, they are milled to exact lengths, and when in the plane of a compression chord, they are given a slight clearance in order to prevent arch action.
The action of shearing stresses in concrete beams may be illustrated by reference to the diagrams in Fig. 3, where the beams are loaded with a weight, W. The portion of W traveling to the left support, moves in diagonal lines, varying from many sets of almost vertical lines to a single diagonal. The maximum intensity of stress probably would be in planes inclined about 45 deg., since, considered independently, they produce the least deflection. While the load, W, remains relatively small, producing but moderate stresses in the steel in the bottom flange, the concrete will carry a considerable portion of the bottom flange tension; when the load W is largely increased, the coefficient of elasticity of the concrete in tension becomes small, or zero, if small fissures appear, and the concrete is unable to transfer the tension in diagonal planes, and failure results. For a beam loaded with a single load, W, the failure would probably be in a diagonal line near the point of application, while in a uniformly loaded beam, it would probably occur in a diagonal line near the support, where the shear is greatest.
It is evident that the introduction of vertical stirrups, as at b, or the more rational inclined stirrups, as at c, influences the action of the shearing forces as indicated, the intensity of stress at the point of connection of the stirrups being high. It is advisable to space the stirrups moderately close, in order to reduce this intensity to reasonable limits. If the assumption is made that the diagonal compression in the concrete acts in a plane inclined at 45 deg., then the tension in the vertical stirrups will be the vertical shear times the horizontal spacing of the stirrups divided by the distance, center to center, of the top and bottom flanges of the beam. If the stirrups are inclined at 45 deg., the stress in them would be 0.7 the stress in vertical stirrups with the same spacing. Bending up bottom rods sharply, in order to dispense with suspenders, is bad practice; the writer has observed diagonal cracks in the beams of a well-known building in New York City, which are due to this cause.
In several structures which the writer has recently designed, he has been able to dispense with stirrups, and, at the same time, effect a saving in concrete, by bending some of the bottom reinforcing rods and placing a bar between them and those which remain horizontal. A typical detail is shown in Fig. 4. The bend occurs at a point where the vertical component of the stress in the bent bars equals the vertical shear, and sufficient bearing is provided by the short cross-bar. The bars which remain horizontal throughout the beam, are deflected at the center of the beam in order to obtain the maximum effective depth. There being no shear at the center, the bars are spaced as closely as possible, and still provide sufficient room for the concrete to flow to the soffit of the beam. Two or more adjacent beams are readily made continuous by extending the bars bent up from each span, a distance along the top flanges. By this system of construction one avoids stopping a bar where the live load unit stress in adjoining bars is high, as their continual lengthening and shortening under stress would cause severe shearing stresses in the concrete surrounding the end of the short bar.
The beam shown in Fig. 5 illustrates the principles stated in the foregoing, as applied to a heavier beam. The duty of the short cross-bars in this case is performed by wires wrapped around the longitudinal rods and then continued up in order to support the bars during erection. This beam, which supports a roof and partitions, etc., has supported about 80% of the load for which it was calculated, and no hair cracks or noticeable deflection have appeared. If the method of calculation suggested by Mr. Godfrey were a correct criterion of the actual stresses, this particular beam (and many others) would have shown many cracks and noticeable deflection. The writer maintains that where the concrete is poured continuously, or proper bond is provided, the influence of the slab as a compression flange is an actual condition, and the stresses should be calculated accordingly.
In the calculation of continuous T-beams, it is necessary to consider the fact that the moment of inertia for negative moments is small because of the lack of sufficient compressive area in the stem or web. If Mr. Godfrey will make proper provision for this point, in studying the designs of practical engineers, he will find due provision made for negative moments. It is very easy to obtain the proper amount of steel for the negative moment in a slab by bending up the bars and letting them project into adjoining spans, as shown in Figs. 4 and 5 (taken from actual construction). The practical engineer does not find, as Mr. Godfrey states, that the negative moment is double the positive moment, because he considers the live load either on one span only, or on alternate spans.
In Fig. 6 a beam is shown which has many rods in the bottom flange, a practice which Mr. Godfrey condemns. As the structure, which has about twenty similar beams, is now being built, the writer would be thankful for his criticism. Mr. Godfrey states that longitudinal steel in columns is worthless, but until definite tests are made, with the same ingredients, proportions, and age, on both plain concrete and reinforced concrete columns properly designed, the writer will accept the data of other experiments, and proportion steel in accordance with recognized formulas.
Mr. Godfrey states that the "elastic theory" is worthless for the design of reinforced concrete arches, basing his objections on the shrinkage of concrete in setting, the unreliability of deflection formulas for beams, and the lack of rigidity of the abutments. The writer, noting that concrete setting in air shrinks, whereas concrete setting in water expands, believes that if the arch be properly wetted until the setting up of the concrete has progressed sufficiently, the effect of shrinkage, on drying out, may be minimized. If the settlement of the forms themselves be guarded against during the construction of an arch, the settlement of the arch ring, on removing the forms, far from being an uncertain element, should be a check on the accuracy of the calculations and the workmanship, since the weight of the arch ring should produce theoretically a certain deflection. The unreliability of deflection formulas for beams is due mainly to the fact that the neutral axis of the beam does not lie in a horizontal plane throughout, and that the shearing stresses are neglected therein. While there is necessarily bending in an arch ring due to temperature, loads, etc., the extreme flanges sometimes being in tension, even in a properly designed arch, the compression exceeds the tension to such an extent that comparison to a beam does not hold true. An arch should not be used where the abutments are unstable, any more than a suspension bridge should be built where a suitable anchorage cannot be obtained.
The proper design of concrete slabs supported on four sides is a complex and interesting study. The writer has recently designed a floor construction, slabs, and beams, supported on four corners, which is simple and economical. In Fig. 7 is shown a portion of a proposed twelve-story building, 90 by 100 ft., having floors with a live-load capacity of 250 lb. per sq. ft. For the maximum positive bending in any panel the full load on that panel was considered, there being no live load on adjoining panels. For the maximum negative bending moment all panels were considered as loaded, and in a single line. "Checker-board" loading was considered too improbable for consideration. The flexure curves for beams at right angles to each other were similar (except in length), the tension rods in the longer beams being placed underneath those in the shorter beams. Under full load, therefore, approximately one-half of the load went to the long-span girder and the other half to the short-span girder. The girders were the same depth as the beams. For its depth the writer found this system to be the strongest and most economical of those investigated.
E.P. GOODRICH, M. AM. SOC. C. E.—The speaker heartily concurs with the author as to the large number of makeshifts constantly used by a majority of engineers and other practitioners who design and construct work in reinforced concrete. It is exceedingly difficult for the human mind to grasp new ideas without associating them with others in past experience, but this association is apt to clothe the new idea (as the author suggests) in garments which are often worse than "swaddling-bands," and often go far toward strangling proper growth.
While the speaker cannot concur with equal ardor with regard to all the author's points, still in many, he is believed to be well grounded in his criticism. Such is the case with regard to the first point mentioned—that of the use of bends of large radius where the main tension rods are bent so as to assist in the resistance of diagonal tensile stresses.
As to the second point, provided proper anchorage is secured in the top concrete for the rod marked 3 in Fig. 1, the speaker cannot see why the concrete beneath such anchorage over the support does not act exactly like the end post of a queen-post truss. Nor can he understand the author's statement that:
"A reinforcing rod in a concrete beam receives its stress by increments imparted by the grip of the concrete; but these increments can only be imparted where the tendency of the concrete is to stretch."
The latter part of this quotation has reference to the point questioned by the speaker. In fact, the remainder of the paragraph from which this quotation is taken seems to be open to grave question, no reason being evident for not carrying out the analogy of the queen-post truss to the extreme. Along this line, it is a well-known fact that the bottom chords in queen-post trusses are useless, as far as resistance to tension is concerned. The speaker concurs, however, in the author's criticism as to the lack of anchorage usually found in most reinforcing rods, particularly those of the type mentioned in the author's second point.
This matter of end anchorage is also referred to in the third point, and is fully concurred in by the speaker, who also concurs in the criticism of the arrangement of the reinforcing rods in the counterforts found in many retaining walls. The statement that "there is absolutely no analogy between this triangle [the counterfort] and a beam" is very strong language, and it seems risky, even for the best engineer, to make such a statement as does the author when he characterizes his own design (Diagram b of Fig. 2) as "the only rational and the only efficient design possible." Several assumptions can be made on which to base the arrangement of reinforcement in the counterfort of a retaining wall, each of which can be worked out with equal logic and with results which will prevent failure, as has been amply demonstrated by actual experience.
The speaker heartily concurs in the author's fourth point, with regard to the impossibility of developing anything like actual shear in the steel reinforcing rods of a concrete beam; but he demurs when the author affirms, as to the possibility of so-called shear bars being stressed in "shear or tension," that "either would be absurd and impossible without greatly overstressing some other part."
As to the fifth point, reference can be given to more than one place in concrete literature where explanations of the action of vertical stirrups may be found, all of which must have been overlooked by the author. However, the speaker heartily concurs with the author's criticism as to the lack of proper connection which almost invariably exists between vertical "web" members and the top and bottom chords of the imaginary Howe truss, which holds the nearest analogy to the conditions existing in a reinforced concrete beam with vertical "web" reinforcement.
The author's reasoning as to the sixth point must be considered as almost wholly facetious. He seems to be unaware of the fact that concrete is relatively very strong in pure shear. Large numbers of tests seem to demonstrate that, where it is possible to arrange the reinforcing members so as to carry largely all tensile stresses developed through shearing action, at points where such tensile stresses cannot be carried by the concrete, reinforced concrete beams can be designed of ample strength and be quite within the logical processes developed by the author, as the speaker interprets them.
The author's characterization of the results secured at the University of Illinois Experiment Station, and described in its Bulletin No. 29, is somewhat misleading. It is true that the wording of the original reference states in two places that "stirrups do not come into action, at least not to any great extent, until a diagonal crack has formed," but, in connection with this statement, the following quotations must be read:
"The tests were planned with a view of determining the amount of stress (tension and bond) developed in the stirrups. However, for various reasons, the results are of less value than was expected. The beams were not all made according to the plans. In the 1907 tests, the stirrups in a few of the beams were poorly placed and even left exposed at the face of the beam, and a variation in the temperature conditions of the laboratory also affected the results. It is evident from the results that the stresses developed in the stirrups are less than they were calculated to be, and hence the layout was not well planned to settle the points at issue. The tests, however, give considerable information on the effectiveness of stirrups in providing web resistance."
"A feature of the tests of beams with stirrups is slow failure, the load holding well up to the maximum under increased deflection and giving warning of its condition."
"Not enough information was obtained to determine the actual final occasion of failure in these tests. In a number of cases the stirrups slipped, in others it seemed that the steel in the stirrups was stretched beyond its elastic limit, and in some cases the stirrups broke."
"As already stated, slip of stirrups and insufficient bond resistance were in many cases the immediate cause of diagonal tension failures, and therefore bond resistance of stirrups may be considered a critical stress."
These quotations seem to indicate much more effectiveness in the action of vertical stirrups than the author would lead one to infer from his criticisms. It is rather surprising that he advocates so strongly the use of a suspension system of reinforcement. That variety has been used abroad for many years, and numerous German experiments have proved with practical conclusiveness that the suspension system is not as efficient as the one in which vertical stirrups are used with a proper arrangement. An example is the conclusion arrived at by Moersch, in "Eisenbetonbau," from a series of tests carried out by him near the end of 1906:
"It follows that with uniform loads, the suspended system of reinforcement does not give any increase of safety against the appearance of diagonal tension cracks, or the final failure produced by them, as compared with straight rods without stirrups, and that stirrups are so much the more necessary."
Again, with regard to tests made with two concentrated loads, he writes:
"The stirrups, supplied on one end, through their tensile strength, hindered the formation of diagonal cracks and showed themselves essential and indispensable elements in the * * * [suspension] system. The limit of their effect is, however, not disclosed by these experiments. * * * In any case, from the results of the second group of experiments can be deduced the facts that the bending of the reinforcement according to the theory concerning the diagonal tensile stress * * * is much more effective than according to the suspension theory, in this case the ultimate loads being in the proportion of 34: 23.4: 25.6."
It is the speaker's opinion that the majority of the failures described in Bulletin No. 29 of the University of Illinois Experiment Station, which are ascribed to diagonal tension, were actually due to deficient anchorage of the upper ends of the stirrups.
Some years ago the speaker demonstrated to his own satisfaction, the practical value of vertical stirrups. Several beams were built identical in every respect except in the size of wire used for web reinforcement. The latter varied from nothing to 3/8-in. round by five steps. The beams were similarly tested to destruction, and the ultimate load and type of failure varied in a very definite ratio to the area of vertical steel.
With regard to the author's seventh point, the speaker concurs heartily as far as it has to do with a criticism of the usual design of continuous beams, but his experience with beams designed as suggested by the author is that failure will take place eventually by vertical cracks starting from the top of the beams close to the supports and working downward so as to endanger very seriously the strength of the structures involved. This type of failure was prophesied by the speaker a number of years ago, and almost every examination which he has lately made of concrete buildings, erected for five years or longer and designed practically in accord with the author's suggestion, have disclosed such dangerous features, traceable directly to the ideas described in the paper. These ideas are held by many other engineers, as well as being advocated by the author. The only conditions under which the speaker would permit of the design of a continuous series of beams as simple members would be when they are entirely separated from each other over the supports, as by the introduction of artificial joints produced by a double thickness of sheet metal or building paper. Even under these conditions, the speaker's experience with separately moulded members, manufactured in a shop and subsequently erected, has shown that similar top cracking may take place under certain circumstances, due to the vertical pressures caused by the reactions at the supports. It is very doubtful whether the action described by the author, as to the type of failure which would probably take place with his method of design, would be as described by him, but the beams would be likely to crack as described above, in accordance with the speaker's experience, so that the whole load supported by the beam would be carried by the reinforcing rods which extend from the beam into the supports and are almost invariably entirely horizontal at such points. The load would thus be carried more nearly by the shearing strength of the steel than is otherwise possible to develop that type of stress. In every instance the latter is a dangerous element.
This effect of vertical abutment action on a reinforced beam was very marked in the beam built of bricks and tested by the speaker, as described in the discussion[J] of the paper by John S. Sewell, M. Am, Soc. S. E., on "The Economical Design of Reinforced Concrete Floor Systems for Fire-Resisting Structures." That experiment also went far toward showing the efficacy of vertical stirrups.
The same discussion also contains a description of a pair of beams tested for comparative purposes, in one of which adhesion between the concrete and the main reinforcing rods was possible only on the upper half of the exterior surfaces of the latter rods except for short distances near the ends. Stirrups were used, however. The fact that the beam, which was theoretically very deficient in adhesion, failed in compression, while the similar beam without stirrups, but with the most perfect adhesion, and anchorage obtainable through the use of large end hooks, failed in bond, has led the speaker to believe that, in affording adhesive resistance, the upper half of a bar is much more effective than the lower half. This seems to be demonstrated further by comparisons between simple adhesion experiments and those obtained with beams.
The speaker heartily concurs with the author's criticism of the amount of time usually given by designing engineers to the determination of the adhesive stresses developed in concrete beams, but, according to the speaker's recollection, these matters are not so poorly treated in some books as might be inferred by the author's language. For example, both Bulletin No. 29, of the University of Illinois, and Moersch, in "Eisenbetonbau," give them considerable attention.
The ninth point raised by the author is well taken. Too great emphasis cannot be laid on the inadequacy of design disclosed by an examination of many T-beams.
Such ready concurrence, however, is not lent to the author's tenth point. While it is true that, under all usual assumptions, except those made by the author, an extremely simple formula for the resisting moment of a reinforced concrete beam cannot be obtained, still his formula falls so far short of fitting even with approximate correctness the large number of well-known experiments which have been published, that a little more mathematical gymnastic ability on the part of the author and of other advocates of extreme simplicity would seem very necessary, and will produce structures which are far more economical and amply safe structurally, compared with those which would be produced in accordance with his recommendations.
As to the eleventh point, in regard to the complex nature of the formulas for chimneys and other structures of a more or less complex beam nature, the graphical methods developed by numerous German and Italian writers are recommended, as they are fully as simple as the rather crude method advocated by the author, and are in almost identical accord with the most exacting analytical methods.
With regard to the author's twelfth point, concerning deflection calculations, it would seem that they play such a small part in reinforced concrete design, and are required so rarely, that any engineer who finds it necessary to make analytical investigations of possible deflections would better use the most precise analysis at his command, rather than fall back on simpler but much more approximate devices such as the one advocated by the author.
Much of the criticism contained in the author's thirteenth point, concerning the application of the elastic theory to the design of concrete arches, is justified, because designing engineers do not carry the theory to its logical conclusion nor take into account the actual stresses which may be expected from slight changes of span, settlements of abutments, and unexpected amounts of shrinkage in the arch ring or ribs. Where conditions indicate that such changes are likely to take place, as is almost invariably the case unless the foundations are upon good rock and the arch ring has been concreted in relatively short sections, with ample time and device to allow for initial shrinkage; or unless the design is arranged and the structure erected so that hinges are provided at the abutments to act during the striking of the falsework, which hinges are afterward wedged or grouted so as to produce fixation of the arch ends—unless all these points are carefully considered in the design and erection, it is the speaker's opinion that the elastic theory is rarely properly applicable, and the use of the equilibrium polygon recommended by the author is much preferable and actually more accurate. But there must be consistency in its use, as well, that is, consistency between methods of design and erection.
The author's fourteenth point—the determination of temperature stresses in a reinforced concrete arch—is to be considered in the same light as that described under the foregoing points, but it seems a little amusing that the author should finally advocate a design of concrete arch which actually has no hinges, namely, one consisting of practically rigid blocks, after he has condemned so heartily the use of the elastic theory.
A careful analysis of the data already available with regard to the heat conductivity of concrete, applied to reinforced concrete structures like arches, dams, retaining walls, etc., in accordance with the well-known but somewhat intricate mathematical formulas covering the laws of heat conductivity and radiation so clearly enunciated by Fourier, has convinced the speaker that it is well within the bounds of engineering practice to predict and care for the stresses which will be produced in structures of the simplest forms, at least as far as they are affected by temperature changes.
The speaker concurs with the author in his criticism, contained in the fifteenth point, with regard to the design of the steel reinforcement in columns and other compression members. While there may be some question as to the falsity or truth of the theory underlying certain types of design, it is unquestioned that some schemes of arrangement undoubtedly produce designs with dangerous properties. The speaker has several times called attention to this point, in papers and discussions, and invariably in his own practice requires that the spacing of spirals, hoops, or ties be many times less than that usually required by building regulations and found in almost every concrete structure. Moersch, in his "Eisenbetonbau," calls attention to the fact that very definite limits should be placed on the maximum size of longitudinal rods as well as on their minimum diameters, and on the maximum spacing of ties, where columns are reinforced largely by longitudinal members. He goes so far as to state that:
"It is seen from * * * [the results obtained] that an increase in the area of longitudinal reinforcement does not produce an increase in the breaking strength to the extent which would be indicated by the formula. * * * In inexperienced hands this formula may give rise to constructions which are not sufficiently safe."
Again, with regard to the spacing of spirals and the combination with them of longitudinal rods, in connection with some tests carried out by Moersch, the conclusion is as follows:
"On the whole, the tests seem to prove that when the spirals are increased in strength, their pitch must be decreased, and the cross-section or number of the longitudinal rods must be increased."
In the majority of cases, the spiral or band spacing is altogether too large, and, from conversations with Considere, the speaker understands that to be the inventor's view as well.
The speaker makes use of the scheme mentioned by the author in regard to the design of flat slabs supported on more than two sides (noted in the sixteenth point), namely, that of dividing the area into strips, the moments of which are determined so as to produce computed deflections which are equal in the two strips running at right angles at each point of intersection. This method, however, requires a large amount of analytical work for any special case, and the speaker is mildly surprised that the author cannot recommend some simpler method so as to carry out his general scheme of extreme simplification of methods and design.
If use is to be made at all of deflection observations, theories, and formulas, account should certainly be taken of the actual settlements and other deflections which invariably occur in Nature at points of support. These changes of level, or slope, or both, actually alter very considerably the stresses as usually computed, and, in all rigorous design work, should be considered.
On the whole, the speaker believes that the author has put himself in the class with most iconoclasts, in that he has overshot his mark. There seems to be a very important point, however, on which he has touched, namely, the lack of care exercised by most designers with regard to those items which most nearly correspond with the so-called "details" of structural steel work, and are fully as important in reinforced concrete as in steel. It is comparatively a small matter to proportion a simple reinforced concrete beam at its intersection to resist a given moment, but the carrying out of that item of the work is only a start on the long road which should lead through the consideration of every detail, not the least important of which are such items as most of the sixteen points raised by the author.
The author has done the profession a great service by raising these questions, and, while full concurrence is not had with him in all points, still the speaker desires to express his hearty thanks for starting what is hoped will be a complete discussion of the really vital matter of detailing reinforced concrete design work.
ALBIN H. BEYER, ESQ.—Mr. Goodrich has brought out very clearly the efficiency of vertical stirrups. As Mr. Godfrey states that explanations of how stirrups act are conspicuous in the literature of reinforced concrete by their absence, the speaker will try to explain their action in a reinforced concrete beam.
It is well known that the internal static conditions in reinforced concrete beams change to some extent with the intensity of the direct or normal stresses in the steel and concrete. In order to bring out his point, the speaker will trace, in such a beam, the changes in the internal static conditions due to increasing vertical loads.
Let Fig. 8 represent a beam reinforced by horizontal steel rods of such diameter that there is no possibility of failure from lack of adhesion of the concrete to the steel. The beam is subjected to the vertical loads, [Sigma] P. For low unit stresses in the concrete, the neutral surface, n n, is approximately in the middle of the beam. Gradually increase the loads, [Sigma] P, until the steel reaches an elongation of from 0.01 to 0.02 of 1%, corresponding to tensile stresses in the steel of from 3,000 to 6,000 lb. per sq. in. At this stage plain concrete would have reached its ultimate elongation. It is known, however, that reinforced concrete, when well made, can sustain without rupture much greater elongations; tests have shown that its ultimate elongation may be as high as 0.1 of 1%, corresponding to tensions in steel of 30,000 lb. per sq. in.
Reinforced concrete structures ordinarily show tensile cracks at very much lower unit stresses in the steel. The main cause of these cracks is as follows: Reinforced concrete setting in dry air undergoes considerable shrinkage during the first few days, when it has very little resistance. This tendency to shrink being opposed by the reinforcement at a time when the concrete does not possess the necessary strength or ductility, causes invisible cracks or planes of weakness in the concrete. These cracks open and become visible at very low unit stresses in the steel.
Increase the vertical loads, [Sigma] P, and the neutral surface will rise and small tensile cracks will appear in the concrete below the neutral surface (Fig. 8). These cracks are most numerous in the central part of the span, where they are nearly vertical. They decrease in number at the ends of the span, where they curve slightly away from the perpendicular toward the center of the span. The formation of these tensile cracks in the concrete relieves it at once of its highly stressed condition.
It is impossible to predict the unit tension in the steel at which these cracks begin to form. They can be detected, though not often visible, when the unit tensions in the steel are as low as from 10,000 to 16,000 lb. per sq. in. As soon as the tensile cracks form, though invisible, the neutral surface approaches the position in the beam assigned to it by the common theory of flexure, with the tension in the concrete neglected. The internal static conditions in the beam are now modified to the extent that the concrete below the neutral surface is no longer continuous. The common theory of flexure can no longer be used to calculate the web stresses.
To analyze the internal static conditions developed, the speaker will treat as a free body the shaded portion of the beam shown in Fig. 8, which lies between two tensile cracks.
In Fig. 9 are shown all the forces which act on this free body, C b b' C'.
At any section, let
C or C' represent the total concrete compression; T or T' represent the total steel tension; J or J' represent the total vertical shear; P represent the total vertical load for the length, b - b';
and let [Delta] T = T' - T = C' - C represent the total transverse shear for the length, b - b'.
Assuming that the tension cracks extend to the neutral surface, n n, that portion of the beam C b b' C', acts as a cantilever fixed at a b and a' b', and subjected to the unbalanced steel tension, [Delta] T. The vertical shear, J, is carried mainly by the concrete above the neutral surface, very little of it being carried by the steel reinforcement. In the case of plain webs, the tension cracks are the forerunners of the sudden so-called diagonal tension failures produced by the snapping off, below the neutral surface, of the concrete cantilevers. The logical method of reinforcing these cantilevers is by inserting vertical steel in the tension side. The vertical reinforcement, to be efficient, must be well anchored, both in the top and in the bottom of the beam. Experience has solved the problem of doing this by the use of vertical steel in the form of stirrups, that is, U-shaped rods. The horizontal reinforcement rests in the bottom of the U.
Sufficient attention has not been paid to the proper anchorage of the upper ends of the stirrups. They should extend well into the compression area of the beam, where they should be properly anchored. They should not be too near the surface of the beam. They must not be too far apart, and they must be of sufficient cross-section to develop the necessary tensile forces at not excessive unit stresses. A working tension in the stirrups which is too high, will produce a local disintegration of the cantilevers, and give the beam the appearance of failure due to diagonal tension. Their distribution should follow closely that of the vertical or horizontal shear in the beam. Practice must rely on experiment for data as to the size and distribution of stirrups for maximum efficiency.
The maximum shearing stress in a concrete beam is commonly computed by the equation:
V v = ——————- (1) 7 —- b d 8
Where d is the distance from the center of the reinforcing bars to the surface of the beam in compression:
b = the width of the flange, and V = the total vertical shear at the section.
This equation gives very erratic results, because it is based on a continuous web. For a non-continuous web, it should be modified to
V v = ——————- (2) K b d
In this equation K b d represents the concrete area in compression. The value of K is approximately equal to 0.4.
Three large concrete beams with web reinforcement, tested at the University of Illinois[K], developed an average maximum shearing resistance of 215 lb. per sq. in., computed by Equation 1. Equation 2 would give 470 lb. per sq. in.
Three T-beams, having 32 by 3-1/4-in. flanges and 8-in. webs, tested at the University of Illinois, had maximum shearing resistances of 585, 605, and 370 lb. per. sq. in., respectively.[L] They did not fail in shear, although they appeared to develop maximum shearing stresses which were almost three times as high as those in the rectangular beams mentioned. The concrete and web reinforcement being identical, the discrepancy must be somewhere else. Based on a non-continuous concrete web, the shearing resistances become 385, 400, and 244 lb. per sq. in., respectively. As none of these failed in shear, the ultimate shearing resistance of concrete must be considerably higher than any of the values given.
About thirteen years ago, Professor A. Vierendeel[M] developed the theory of open-web girder construction. By an open-web girder, the speaker means a girder which has a lower and upper chord connected by verticals. Several girders of this type, far exceeding solid girders in length, have been built. The theory of the open-web girder, assuming the verticals to be hinged at their lower ends, applies to the concrete beam reinforced with stirrups. Assuming that the spaces between the verticals of the girder become continually narrower, they become the tension cracks of the concrete beam.[N]
JOHN C. OSTRUP, M. AM. SOC. C. E.—The author has rendered a great service to the Profession in presenting this paper. In his first point he mentions two designs of reinforced concrete beams and, inferentially, he condemns a third design to which the speaker will refer later. The designs mentioned are, first, that of a reinforced concrete beam arranged in the shape of a rod, with separate concrete blocks placed on top of it without being connected—such a beam has its strength only in the rod. It is purely a suspension, or "hog-chain" affair, and the blocks serve no purpose, but simply increase the load on the rod and its stresses.
The author's second design is an invention of his own, which the Profession at large is invited to adopt. This is really the same system as the first, except that the blocks are continuous and, presumably, fixed at the ends. When they are so fixed, the concrete will take compressive stresses and a certain portion of the shear, the remaining shear being transmitted to the rod from the concrete above it, but only through friction. Now, the frictional resistance between a steel rod and a concrete beam is not such as should be depended on in modern engineering designs.
The third method is that which is used by nearly all competent designers, and it seems to the speaker that, in condemning the general practice of current reinforced designs in sixteen points, the author could have saved himself some time and labor by condemning them all in one point.
What appears to be the underlying principle of reinforced concrete design is the adhesion, or bond, between the steel and the concrete, and it is that which tends to make the two materials act in unison. This is a point which has not been touched on sufficiently, and one which it was expected that Mr. Beyer would have brought out, when he illustrated certain internal static conditions. This principle, in the main, will cover the author's fifth point, wherein stirrups are mentioned, and again in the first point, wherein he asks: "Will some advocate of this type of design please state where this area can be found?"
To understand clearly how concrete acts in conjunction with steel, it is necessary to analyze the following question: When a steel rod is embedded in a solid block of concrete, and that rod is put in tension, what will be the stresses in the rod and the surrounding concrete?
The answer will be illustrated by reference to Fig. 10. It must be understood that the unit stresses should be selected so that both the concrete and the steel may be stressed in the same relative ratio. Assuming the tensile stress in the steel to be 16,000 lb. per sq. in., and the bonding value 80 lb., a simple formula will show that the length of embedment, or that part of the rod which will act, must be equal to 50 diameters of the rod.
When the rod is put in tension, as indicated in Fig. 10, what will be the stresses in the surrounding concrete? The greatest stress will come on the rod at the point where it leaves the concrete, where it is a maximum, and it will decrease from that point inward until the total stress in the steel has been distributed to the surrounding concrete. At that point the rod will only be stressed back for a distance equal in length to 50 diameters, no matter how far beyond that length the rod may extend.
The distribution of the stress from the steel rod to the concrete can be represented by a cone, the base of which is at the outer face of the block, as the stresses will be zero at a point 50 diameters back, and will increase in a certain ratio out toward the face of the block, and will also, at all intermediate points, decrease radially outward from the rod.
The intensity of the maximum stress exerted on the concrete is represented by the shaded area in Fig. 10, the ordinates, measured perpendicularly to the rod, indicating the maximum resistance offered by the concrete at any point.
If the concrete had a constant modulus of elasticity under varying stress, and if the two materials had the same modulus, the stress triangle would be bounded by straight lines (shown as dotted lines in Fig. 10); but as this is not true, the variable moduli will modify the stress triangle in a manner which will tend to make the boundary lines resemble parabolic curves.
A triangle thus constructed will represent by scale the intensity of the stress in the concrete, and if the ordinates indicate stresses greater than that which the concrete will stand, a portion will be destroyed, broken off, and nothing more serious will happen than that this stress triangle will adjust itself, and grip the rod farther back. This process keeps on until the end of the rod has been reached, when the triangle will assume a much greater maximum depth as it shortens; or, in other words, the disintegration of the concrete will take place here very rapidly, and the rod will be pulled out.
In the author's fourth point he belittles the use of shear rods, and states: "No hint is given as to whether these bars are in shear or in tension." As a matter of fact, they are neither in shear nor wholly in tension, they are simply in bending between the centers of the compressive resultants, as indicated in Fig. 12, and are, besides, stressed slightly in tension between these two points.
In Fig. 10 the stress triangle indicates the distribution and the intensity of the resistance in the concrete to a force acting parallel to the rod. A similar triangle may be drawn, Fig. 11, showing the resistance of the rod and the resultant distribution in the concrete to a force perpendicular to the rod. Here the original force would cause plain shear in the rod, were the latter fixed in position. Since this cannot be the case, the force will be resolved into two components, one of which will cause a tensile stress in the rod and the other will pass through the centroid of the compressive stress area. This is indicated in Fig. 11, which, otherwise, is self-explanatory.
Rods are not very often placed in such a position, but where it is unavoidable, as in construction joints in the middle of slabs or beams, they serve a very good purpose; but, to obtain the best effect from them, they should be placed near the center of the slab, as in Fig. 12, and not near the top, as advocated by some writers.
If the concrete be overstressed at the points where the rod tends to bend, that is, if the rods are spaced too far apart, disintegration will follow; and, for this reason, they should be long enough to have more than 50 diameters gripped by the concrete.
This leads up to the author's seventh point, as to the overstressing of the concrete at the junction of the diagonal tension rods, or stirrups, and the bottom reinforcement.
Analogous with the foregoing, it is easy to lay off the stress triangles and to find the intensity of stress at the maximum points, in fact at any point, along the tension rods and the bottom chord. This is indicated in Fig. 13. These stress triangles will start on the rod 50 diameters back from the point in question and, although the author has indicated in Fig. 1 that only two of the three rods are stressed, there must of necessity also be some stress in the bottom rod to the left of the junction, on account of the deformation which takes place in any beam due to bending. Therefore, all three rods at the point where they are joined, are under stress, and the triangles can be laid off accordingly.
It will be noticed that the concrete will resist the compressive components, not at any specific point, but all along the various rods, and with the intensities shown by the stress triangles; also, that some of these triangles will overlap, and, hence, a certain readjustment, or superimposition, of stresses takes place.
The portion which is laid off below the bottom rods will probably not act unless there is sufficient concrete below the reinforcing bars and on the sides, and, as that is not the case in ordinary construction, it is very probable, as Mr. Goodrich has pointed out, that the concrete below the rods plays an unimportant part, and that the triangle which is now shown below the rod should be partially omitted.
The triangles in Fig. 13 show the intensity of stress in the concrete at any point, or at any section where it is wanted. They show conclusively where the components are located in the concrete, their relation to the tensile stresses in the rods, and, furthermore, that they act only in a general way at right angles to one another. This is in accordance with the theory of beams, that at any point in the web there are tensile and compressive stresses of equal intensity, and at right angles to one another, although in a non-homogeneous web the distribution is somewhat different.
After having found at the point of junction the intensity of stress, it is possible to tell whether or not a bond between the stirrups and the bottom rods is necessary, and it would not seem to be where the stirrups are vertical.
It would also seem possible to tell in what direction, if any, the bend in the inclined stirrups should be made. It is to be assumed, although not expressly stated, that the bends should curve from the center up toward the end of the beam, but an inspection of the stress triangles, Fig. 13, will show that the intensity of stress is just as great on the opposite side, and it is probable that, if any bends were required to reduce the maximum stress in the concrete, they should as likely be made on the side nearest the abutment.
From the stress triangles it may also be shown that, if the stirrups were vertical instead of inclined, the stress in the concrete on both sides would be practically equal, and that, in consequence, vertical stirrups are preferable.
The next issue raised by the author is covered in his seventh point, and relates to bending moments. He states: "* * * bending moments in so-called continuous beams are juggled to reduce them to what the designer would like to have them. This has come to be almost a matter of taste, * * *."
The author seems to imply that such juggling is wrong. As a matter of fact, it is perfectly allowable and legitimate in every instance of beam or truss design, that is, from the standpoint of stress distribution, although this "juggling" is limited in practice by economical considerations.
In a series of beams supported at the ends, bending moments range from (w l^{2})/8 at the center of each span to zero at the supports, and, in a series of cantilevers, from zero at the center of the span to (w l^{2})/8 at the supports. Between these two extremes, the designer can divide, adjust, or juggle them to his heart's content, provided that in his design he makes the proper provision for the corresponding shifting of the points of contra-flexure. If that were not the case, how could ordinary bridge trusses, which have their maximum bending at the center, compare with those which, like arches, are assumed to have no bending at that point?
In his tenth point, the author proposes a method of simple designing by doing away with the complicated formulas which take account of the actual co-operation of the two materials. He states that an ideal design can be obtained in the same manner, that is, with the same formulas, as for ordinary rectangular beams; but, when he does so, he evidently fails to remember that the neutral axis is not near the center of a reinforced concrete beam under stress; in fact, with the percentage of reinforcement ordinarily used in designing—varying between three-fourths of 1% to 1-1/2%—the neutral axis, when the beam is loaded, is shifted from 26 to 10% of the beam depth above the center. Hence, a low percentage of steel reinforcement will produce a great shifting of the neutral axis, so that a design based on the formulas advocated by the author would contain either a waste of materials, an overstress of the concrete, or an understress of the steel; in fact, an error in the design of from 10 to 26 per cent. Such errors, indeed, are not to be recommended by good engineers.
The last point which the speaker will discuss is that of the elastic arch. The theory of the elastic arch is now so well understood, and it offers such a simple and, it might be said, elegant and self-checking solution of the arch design, that it has a great many advantages, and practically none of the disadvantages of other methods.
The author's statement that the segments of an arch could be made up of loose blocks and afterward cemented together, cannot be endorsed by the speaker, for, upon such cementing together, a shifting of the lines of resistance will take place when the load is applied. The speaker does not claim that arches are maintained by the cement or mortar joining the voussoirs together, but that the lines of pressure will be materially changed, and the same calculations are not applicable to both the unloaded and the loaded arch.
It is quite true, as the author states, that a few cubic yards of concrete placed in the ring will strengthen the arch more than a like amount added to the abutments, provided, however, that this material be placed properly. No good can result from an attempt to strengthen a structure by placing the reinforcing material promiscuously. This has been tried by amateurs in bridge construction, and, in such cases, the material either increased the distance from the neutral axis to the extreme fibers, thereby reducing the original section modulus, or caused a shifting of the neutral axis followed by a large bending moment; either method weakening the members it had tried to reinforce. In other words, the mere addition of material does not always strengthen a structure, unless it is placed in the proper position, and, if so placed, it should be placed all over commensurately with the stresses, that is, the unit stresses should be reduced.
The author has criticized reinforced concrete construction on the ground that the formulas and theories concerning it are not as yet fully developed. This is quite true, for the simple reason that there are so many uncertain elements which form their basis: First, the variable quantity of the modulus of elasticity, which, in the concrete, varies inversely as the stress; and, second, the fact that the neutral axis in a reinforced concrete beam under changing stress is migratory. There are also many other elements of evaluation, which, though of importance, are uncertain.
Because the formulas are established on certain assumptions is no reason for condemning them. There are, the speaker might add, few formulas in the subject of theoretical mechanics which are not based on some assumption, and as long as the variations are such that their range is known, perfectly reliable formulas can be deduced and perfectly safe structures can be built from them.
There are a great many theorists who have recently complained about the design of reinforced concrete. It seems to the speaker that such complaints can serve no useful purpose. Reinforced concrete structures are being built in steadily increasing numbers; they are filling a long needed place; they are at present rendering great service to mankind; and they are destined to cover a field of still greater usefulness. Reinforced concrete will undoubtedly show in the future that the confidence which most engineers and others now place in it is fully merited.
HARRY F. PORTER, JUN. AM. SOC. C. E. (by letter).—Mr. Godfrey has brought forward some interesting and pertinent points, which, in the main, are well taken; but, in his zealousness, he has fallen into the error of overpersuading himself of the gravity of some of the points he would make; on the other hand, he fails to go deeply enough into others, and some fallacies he leaves untouched. Incidentally, he seems somewhat unfair to the Profession in general, in which many earnest, able men are at work on this problem, men who are not mere theorists, but have been reared in the hard school of practical experience, where refinements of theory count for little, but common sense in design counts for much—not to mention those self-sacrificing devotees to the advancement of the art, the collegiate and laboratory investigators.
Engineers will all agree with Mr. Godfrey that there is much in the average current practice that is erroneous, much in textbooks that is misleading if not fallacious, and that there are still many designers who are unable to think in terms of the new material apart from the vestures of timber and structural steel, and whose designs, therefore, are cumbersome and impractical. The writer, however, cannot agree with the author that the practice is as radically wrong as he seems to think. Nor is he entirely in accord with Mr. Godfrey in his "constructive criticism" of those practices in which he concurs, that they are erroneous.
That Mr. Godfrey can see no use in vertical stirrups or U-bars is surprising in a practical engineer. One is prompted to ask: "Can the holder of this opinion ever have gone through the experience of placing steel in a job, or at least have watched the operation?" If so, he must have found some use for those little members which he professes to ignore utterly.
As a matter of fact, U-bars perform the following very useful and indispensable services:
(1).—If properly made and placed, they serve as a saddle in which to rest the horizontal steel, thereby insuring the correct placing of the latter during the operation of concreting, not a mean function in a type of construction so essentially practical. To serve this purpose, stirrups should be made as shown in Plate III. They should be restrained in some manner from moving when the concrete strikes them. A very good way of accomplishing this is to string them on a longitudinal rod, nested in the bend at the upper end. Mr. Godfrey, in his advocacy of bowstring bars anchored with washers and nuts at the ends, fails to indicate how they shall be placed. The writer, from experience in placing steel, thinks that it would be very difficult, if not impractical, to place them in this manner; but let a saddle of U-bars be provided, and the problem is easy.
(2).—Stirrups serve also as a tie, to knit the stem of the beam to its flange—the superimposed slab. The latter, at best, is not too well attached to the stem by the adhesion of the concrete alone, unassisted by the steel. T-beams are used very generally, because their construction has the sanction of common sense, it being impossible to cast stem and slab so that there will be the same strength in the plane at the junction of the two as elsewhere, on account of the certainty of unevenness in settlement, due to the disproportion in their depth. There is also the likelihood that, in spite of specifications to the contrary, there will be a time interval between the pouring of the two parts, and thus a plane of weakness, where, unfortunately, the forces tending to produce sliding of the upper part of the beam on the lower (horizontal shear) are a maximum. To offset this tendency, therefore, it is necessary to have a certain amount of vertical steel, disposed so as to pass around and under the main reinforcing members and reach well up into the flange (the slab), thus getting a grip therein of no mean security. The hooking of the U-bars, as shown in Plate III, affords a very effective grip in the concrete of the slab, and this is still further enhanced by the distributing or anchoring effect of the longitudinal stringing rods. Thus these longitudinals, besides serving to hold the U-bars in position, also increase their effectiveness. They serve a still further purpose as a most convenient support for the slab bars, compelling them to take the correct position over the supports, thus automatically ensuring full and proper provision for reversed stresses. More than that, they act in compression within the middle half, and assist in tension toward the ends of the span.
Thus, by using U-bars of the type indicated, in combination with longitudinal bars as described, tying together thoroughly the component parts of the beam in a vertical plane, a marked increase in stiffness, if not strength, is secured. This being the case, who can gainsay the utility of the U-bar?
Of course, near the ends, in case continuity of action is realized, whereupon the stresses are reversed, the U-bars need to be inverted, although frequently inversion is not imperative with the type of U-bar described, the simple hooking of the upper ends over the upper horizontal steel being sufficient.
As to whether or not the U-bars act with the horizontal and diagonal steel to form truss systems is relatively unessential; in all probability there is some such action, which contributes somewhat to the total strength, but at most it is of minor importance. Mr. Godfrey's points as to fallacy of truss action seem to be well taken, but his conclusions in consequence—that U-bars serve no purpose—are impractical.
The number of U-bars needed is also largely a matter of practice, although subject to calculation. Practice indicates that they should be spaced no farther apart than the effective depth of the member, and spaced closer or made heavier toward the ends, in order to keep pace with cumulating shear. They need this close spacing in order to serve as an adequate saddle for the main bars, as well as to furnish, with the lighter "stringing" rods, an adequate support to the slab bars. They should have the requisite stiffness in the bends to carry their burden without appreciable sagging; it will be found that 5/16 in. is about the minimum practical size, and that 1/2 in. is as large as will be necessary, even for very deep beams with heavy reinforcement.
If the size and number of U-bars were to be assigned by theory, there should be enough of them to care for fully 75% of the horizontal shear, the adhesion of the concrete being assumed as adequate for the remainder.
Near the ends, of course, the inclined steel, resulting from bending up some of the horizontal bars, if it is carried well across the support to secure an adequate anchorage, or other equivalent anchorage is provided, assists in taking the horizontal shear.
The embedment, too, of large stone in the body of the beam, straddling, as it were, the neutral plane, and thus forming a lock between the flange and the stem, may be considered as assisting materially in taking horizontal shear, thus relieving the U-bars. This is a factor in the strength of actual work which theory does not take into account, and by the author, no doubt, it would be regarded as insignificant; nevertheless it is being done every day, with excellent results.
The action of these various agencies—the U-bars, diagonal steel, and embedded stone—in a concrete beam, is analogous to that of bolts or keys in the case of deepened timber beams. A concrete beam may be assumed, for the purposes of illustration, to be composed of a series of superimposed layers; in this case the function of the rigid material crossing these several layers normally, and being well anchored above and below, as a unifier of the member, is obvious—it acts as so many bolts joining superimposed planks forming a beam. Of course, no such lamination actually exists, although there are always incipient forces tending to produce it; these may and do manifest themselves on occasion as an actual separation in a horizontal plane at the junction of slab and stem, ordinarily the plane of greatest weakness—owing to the method of casting—as well as of maximum horizontal shear. Beams tested to destruction almost invariably develop cracks in this region. The question then naturally arises: If U-bars serve no purpose, what will counteract these horizontal cleaving forces? On the contrary, T-beams, adequately reinforced with U-bars, seem to be safeguarded in this respect; consequently, the U-bars, while perhaps adding little to the strength, as estimated by the ultimate carrying capacity, actually must be of considerable assistance, within the limit of working loads, by enhancing the stiffness and ensuring against incipient cracking along the plane of weakness, such as impact or vibratory loads might induce. Therefore, U-bars, far from being superfluous or fallacious, are, practically, if not theoretically, indispensable.
At present there seems to be considerable diversity of opinion as to the exact nature of the stress action in a reinforced concrete beam. Unquestionably, the action in the monolithic members of a concrete structure is different from that in the simple-acting, unrestrained parts of timber or structural steel construction; because in monolithic members, by the law of continuity, reverse stresses must come into play. To offset these stresses reinforcement must be provided, or cracking will ensue where they occur, to the detriment of the structure in appearance, if not in utility. Monolithic concrete construction should be tied together so well across the supports as to make cracking under working loads impossible, and, when tested to destruction, failure should occur by the gradual sagging of the member, like the sagging of an old basket. Then, and then only, can the structure be said to be adequately reinforced.
In his advocacy of placing steel to simulate a catenary curve, with end anchorage, the author is more nearly correct than in other issues he makes. Undoubtedly, an attempt should be made in every concrete structure to approximate this alignment. In slabs it may be secured simply by elevating the bars over the supports, when, if pliable enough, they will assume a natural droop which is practically ideal; or, if too stiff, they may be bent to conform approximately to this position. In slabs, too, the reinforcement may be made practically continuous, by using lengths covering several spans, and, where ends occur, by generous lapping. In beams the problem is somewhat more complicated, as it is impossible, except rarely, to bow the steel and to extend it continuously over several supports; but all or part of the horizontal steel can be bent up at about the quarter point, carried across the supports into the adjacent spans, and anchored there by bending it down at about the same angle as it is bent up on the approach, and then hooking the ends.
It is seldom necessary to adopt the scheme proposed by the author, namely, a threaded end with a bearing washer and a nut to hold the washer in place, although it is sometimes expedient, but not absolutely necessary, in end spans, where prolongation into an adjacent span is out of the question. In end spans it is ordinarily sufficient to give the bars a double reverse bend, as shown in Plate III, and possibly to clasp hooks with the horizontal steel. If steel be placed in this manner, the catenary curve will be practically approximated, the steel will be fairly developed throughout its length of embedment, and the structure will be proof against cracking. In this case, also, there is much less dependence on the integrity of the bond; in fact, if there were no bond, the structure would still develop most of its strength, although the deflection under heavy loading might be relatively greater.
The writer once had an experience which sustains this point. On peeling off the forms from a beam reinforced according to the method indicated, it was found that, because of the crowding together of the bars in the bottom, coupled with a little too stiff a mixture, the beam had hardly any concrete on the underside to grip the steel in the portion between the points of bending up, or for about the middle half of the member; consequently, it was decided to test this beam. The actual working load was first applied and no deflection, cracking, or slippage of the bars was apparent; but, as the loading was continued, deflection set in and increased rapidly for small increments of loading, a number of fine cracks opened up near the mid-section, which extended to the neutral plane, and the steel slipped just enough, when drawn taut, to destroy what bond there was originally, owing to the contact of the concrete above. At three times the live load, or 450 lb. per sq. ft., the deflection apparently reached a maximum, being about 5/16 in. for a clear distance, between the supports, of 20 ft.; and, as the load was increased to 600 lb. per sq. ft., there was no appreciable increase either in deflection or cracking; whereupon, the owner being satisfied, the loading was discontinued. The load was reduced in amount to three times the working load (450 lb.) and left on over night; the next morning, there being no detectable change, the beam was declared to be sound. When the load was removed the beam recovered all but about 1/8 in. of its deflection, and then repairs were made by attaching light expanded metal to the exposed bars and plastering up to form. Although nearly three years have elapsed, there have been no unfavorable indications, and the owner, no doubt, has eased his mind entirely in regard to the matter. This truly remarkable showing can only be explained by the catenary action of the main steel, and some truss action by the steel which was horizontal, in conjunction with the U-bars, of which there were plenty. As before noted, the clear span was 20 ft., the width of the bay, 8 ft., and the size under the slab (which was 5 in. thick) 8 by 18 in. The reinforcement consisted of three 1-1/8-in. round medium-steel bars, with 3/8-in. U-bars placed the effective depth of the member apart and closer toward the supports, the first two or three being 6 in. apart, the next two or three, 9 in., the next, 12 in., etc., up to a maximum, throughout the mid-section, of 15 in. Each U-bar was provided with a hook at its upper end, as shown in Plate III, and engaged the slab reinforcement, which in this case was expanded metal. Two of the 1-1/8-in. bars were bent up and carried across the support. At the point of bending up, where they passed the single horizontal bar, which was superimposed, a lock-bar was inserted, by which the pressure of the bent-up steel against the concrete, in the region of the bend, was taken up and distributed along the horizontal bar. This feature is also shown in Fig. 14. The bars, after being carried across the support, were inclined into the adjacent span and provided with a liberal, well-rounded hook, furnishing efficient anchorage and provision for reverse stresses. This was at one end only, for—to make matters worse—the other end was a wall bearing; consequently, the benefit of continuity was denied. The bent-up bars were given a double reverse bend, as already described, carrying them around, down, in, and up, and ending finally by clasping them in the hook of the horizontal bar. This apparently stiffened up the free end, for, under the test load, its action was similar to that of the completely restrained end, thus attesting the value of this method of end-fixing.
The writer has consistently followed this method of reinforcement, with unvaryingly good results, and believes that, in some measure, it approximates the truth of the situation. Moreover, it is economical, for with the bars bent up over the supports in this manner, and positively anchored, plenty of U-bars being provided, it is possible to remove the forms with entire safety much sooner than with the ordinary methods which are not as well stirruped and only partially tied across the supports. It is also possible to put the structure into use at an earlier date. Failure, too, by the premature removal of the centers, is almost impossible with this method. These considerations more than compensate for the trouble and expense involved in connection with such reinforcement. The writer will not attempt here a theoretical analysis of the stresses incurred in the different parts of this beam, although it might be interesting and instructive.
The concrete, with the reinforcement disposed as described, may be regarded as reposing on the steel as a saddle, furnishing it with a rigid jacket in which to work, and itself acting only as a stiff floor and a protecting envelope. Bond, in this case, while, of course, an adjunct, is by no means vitally important, as is generally the case with beams unrestrained in any way and in which the reinforcement is not provided with adequate end anchorage, in which case a continuous bond is apparently—at any rate, theoretically—indispensable.
An example of the opposite extreme in reinforced concrete design, where provision for reverse stresses was almost wholly lacking, is shown in the Bridgeman Brothers' Building, in Philadelphia, which collapsed while the operation of casting the roof was in progress, in the summer of 1907. The engineering world is fairly familiar with the details of this disaster, as they were noted both in the lay and technical press. In this structure, not only were U-bars almost entirely absent, but the few main bars which were bent up, were stopped short over the support. The result was that the ties between the rib and the slab, and also across the support, being lacking, some of the beams, the forms of which had been removed prematurely, cracked of their own dead weight, and, later, when the roof collapsed, owing to the deficient bracing of the centers, it carried with it each of the four floors to the basement, the beams giving way abruptly over the supports. Had an adequate tie of steel been provided across the supports, the collapse, undoubtedly, would have stopped at the fourth floor. So many faults were apparent in this structure, that, although only half of it had fallen, it was ordered to be entirely demolished and reconstructed.
The cracks in the beams, due to the action of the dead weight alone, were most interesting, and illuminative of the action which takes place in a concrete beam. They were in every case on the diagonal, at an angle of approximately 45 deg., and extended upward and outward from the edge of the support to the bottom side of the slab. Never was the necessity for diagonal steel, crossing this plane of weakness, more emphatically demonstrated. To the writer—an eye-witness—the following line of thought was suggested:
Should not the concrete in the region above the supports and for a distance on either side, as encompassed by the opposed 45 deg. lines (Fig. 14), be regarded as abundantly able, of and by itself, and without reinforcing, to convey all its load into the column, leaving only the bending to be considered in the truncated portion intersected? Not even the bending should be considered, except in the case of relatively shallow members, but simply the tendency on the part of the wedge-shaped section to slip out on the 45 deg. planes, thereby requiring sufficient reinforcement at the crossing of these planes of principal weakness to take the component of the load on this portion, tending to shove it out. This reinforcement, of course, should be anchored securely both ways; in mid-span by extending it clear through, forming a suspensory, and, in the other direction, by prolonging it past the supports, the concrete, in this case, along these planes, being assumed to assist partly or not at all.
This would seem to be a fair assumption. In all events, beams designed in this manner and checked by comparison with the usual methods of calculation, allowing continuity of action, are found to agree fairly well. Hence, the following statement seems to be warranted: If enough steel is provided, crossing normally or nearly so the 45 deg. planes from the edge of the support upward and outward, to care for the component of the load on the portion included within a pair of these planes, tending to produce sliding along the same, and this steel is adequately anchored both ways, there will be enough reinforcement for every other purpose. In addition, U-bars should be provided for practical reasons.
The weak point of beams, and slabs also, fully reinforced for continuity of action, is on the under side adjacent to the edge of the support, where the concrete is in compression. Here, too, the amount of concrete available is small, having no slab to assist it, as is the case within the middle section, where the compression is in the top. Over the supports, for the width of the column, there is abundant strength, for here the steel has a leverage equal to the depth of the column; but at the very edge and for at least one-tenth of the span out, conditions are serious. The usual method of strengthening this region is to subpose brackets, suitably proportioned, to increase the available compressive area to a safe figure, as well as the leverage of the steel, at the same time diminishing the intensity of compression. Brackets, however, are frequently objectionable, and are therefore very generally omitted by careless or ignorant designers, no especial compensation being made for their absence. In Europe, especially in Germany, engineers are much more careful in this respect, brackets being nearly always included. True, if brackets are omitted, some compensation is provided by the strengthening which horizontal bars may give by extending through this region, but sufficient additional compressive resistance is rarely afforded thereby. Perhaps the best way to overcome the difficulty, without resorting to brackets, is to increase the compressive resistance of the concrete, in addition to extending horizontal steel through it. This may be done by hooping or by intermingling scraps of iron or bits of expanded metal with the concrete, thereby greatly increasing its resistance. The experiments made by the Department of Bridges of the City of New York, on the value of nails in concrete, in which results as high as 18,000 lb. per sq. in. were obtained, indicate the availability of this device; the writer has not used it, nor does he know that it has been used, but it seems to be entirely rational, and to offer possibilities.
Another practical test, which indicates the value of proper reinforcement, may be mentioned. In a storage warehouse in Canada, the floor was designed, according to the building laws of the town, for a live load of 150 lb. per sq. ft., but the restrictions being more severe than the standard American practice, limiting the lever arm of the steel to 75% of the effective depth, this was about equivalent to a 200-lb. load in the United States. The structure was to be loaded up to 400 or 500 lb. per sq. ft. steadily, but the writer felt so confident of the excess strength provided by his method of reinforcing that he was willing to guarantee the structure, designed for 150 lb., according to the Canadian laws, to be good for the actual working load. Plain, round, medium-steel bars were used. A 10-ft. panel, with a beam of 14-ft. span, and a slab 6 in. thick (not including the top coat), with 1/2-in. round bars, 4 in. on centers, was loaded to 900 lb. per sq. ft., at which load no measurable deflection was apparent. The writer wished to test it still further, but there was not enough cement—the material used for loading. The load, however, was left on for 48 hours, after which, no sign of deflection appearing, not even an incipient crack, it was removed. The total area of loading was 14 by 20 ft. The beam was continuous at one end only, and the slab only on one side. In other parts of the structure conditions were better, square panels being possible, with reinforcement both ways, and with continuity, both of beams and slabs, virtually in every direction, end spans being compensated by shortening. The method of reinforcing was as before indicated. The enormous strength of the structure, as proved by this test, and as further demonstrated by its use for nearly two years, can only be explained on the basis of the continuity of action developed and the great stiffness secured by liberal stirruping. Steel was provided in the middle section according to the rule, (w l)/8, the span being taken as the clear distance between the supports; two-thirds of the steel was bent up and carried across the supports, in the case of the beams, and three-fourths of the slab steel was elevated; this, with the lap, really gave, on the average, four-thirds as much steel over the supports as in the center, which, of course, was excessive, but usually an excess has to be tolerated in order to allow for adequate anchorage. Brackets were not used, but extra horizontal reinforcement, in addition to the regular horizontal steel, was laid in the bottom across the supports, which, seemingly, was satisfactory. The columns, it should be added, were calculated for a very low value, something like 350 lb. per sq. in., in order to compensate for the excess of actual live load over and above the calculated load.
This piece of work was done during the winter, with the temperature almost constantly at +10 deg. and dropping below zero over night. The precautions observed were to heat the sand and water, thaw out the concrete with live steam, if it froze in transporting or before it was settled in place, and as soon as it was placed, it was decked over and salamanders were started underneath. Thus, a job equal in every respect to warm-weather installation was obtained, it being possible to remove the forms in a fortnight.
In another part of this job (the factory annex) where, owing to the open nature of the structure, it was impossible to house it in as well as the warehouse which had bearing walls to curtain off the sides, less fortunate results were obtained. A temperature drop over night of nearly 50 deg., followed by a spell of alternate freezing and thawing, effected the ruin of at least the upper 2 in. of a 6-in. slab spanning 12 ft. (which was reinforced with 1/2-in. round bars, 4 in. on centers), and the remaining 4 in. was by no means of the best quality. It was thought that this particular bay would have to be replaced. Before deciding, however, a test was arranged, supports being provided underneath to prevent absolute failure. But as the load was piled up, to the extent of nearly 400 lb. per sq. ft., there was no sign of giving (over this span) other than an insignificant deflection of less than 1/4 in., which disappeared on removing the load. This slab still performs its share of the duty, without visible defect, hence it must be safe. The question naturally arises: if 4 in. of inferior concrete could make this showing, what must have been the value of the 6 in. of good concrete in the other slabs? The reinforcing in the slab, it should be stated, was continuous over several supports, was proportioned for (w l)/8 for the clear span (about 11 ft.), and three-fourths of it was raised over the supports. This shows the value of the continuous method of reinforcing, and the enormous excess of strength in concrete structures, as proportioned by existing methods, when the reverse stresses are provided for fully and properly, though building codes may make no concession therefor.
Another point may be raised, although the author has not mentioned it, namely, the absurdity of the stresses commonly considered as occurring in tensile steel, 16,000 lb. per sq. in. for medium steel being used almost everywhere, while some zealots, using steel with a high elastic limit, are advocating stresses up to 22,000 lb. and more; even the National Association of Cement Users has adopted a report of the Committee on Reinforced Concrete, which includes a clause recommending the use of 20,000 lb. on high steel. As theory indicates, and as F.E. Turneaure, Assoc. M. Am. Soc. C. E., of the University of Wisconsin, has proven by experiment, failure of the concrete encircling the steel under tension occurs when the stress in the steel is about 5,000 lb. per sq. in. It is evident, therefore, that if a stress of even 16,000 lb. were actually developed, not to speak of 20,000 lb. or more, the concrete would be so replete with minute cracks on the tension side as to expose the embedded metal in innumerable places. Such cracks do not occur in work because, under ordinary working loads, the concrete is able to carry the load so well, by arch and dome action, as to require very little assistance from the steel, which, consequently, is never stressed to a point where cracking of the concrete will be induced. This being the case, why not recognize it, modify methods of design, and not go on assuming stresses which have no real existence?
The point made by Mr. Godfrey in regard to the fallacy of sharp bends is patent, and must meet with the agreement of all who pause to think of the action really occurring. This is also true of his points as to the width of the stem of T-beams, and the spacing of bars in the same. As to elastic arches, the writer is not sufficiently versed in designs of this class to express an opinion, but he agrees entirely with the author in his criticism of retaining-wall design. What the author proposes is rational, and it is hard to see how the problem could logically be analyzed otherwise. His point about chimneys, however, is not as clear.
As to columns, the writer agrees with Mr. Godfrey in many, but not in all, of his points. Certainly, the fallacy of counting on vertical steel to carry load, in addition to the concrete, has been abundantly shown. The writer believes that the sole legitimate function of vertical steel, as ordinarily used, is to reinforce the member against flexure, and that its very presence in the column, unless well tied across by loops of steel at frequent intervals, so far from increasing the direct carrying capacity, is a source of weakness. However, the case is different when a large amount of rigid vertical steel is used; then the steel may be assumed to carry all the load, at the value customary in structural steel practice, the concrete being considered only in the light of fire-proofing and as affording lateral support to the steel, increasing its effective radius of gyration and thus its safe carrying capacity. In any event the load should be assumed to be carried either by the concrete or by the steel, and, if by the former, the longitudinal and transverse steel which is introduced should be regarded as auxiliary only. Vertical steel, if not counted in the strength, however, may on occasion serve a very useful practical purpose; for instance, the writer once had a job where, owing to the collection of ice and snow on a floor, which melted when the salamanders were started, the lower ends of several of the superimposed columns were eaten away, with the result that when the forms were withdrawn, these columns were found to be standing on stilts. Only four 1-in. bars were present, looped at intervals of about 1 ft., in a column 12 ft. in length and having a girth of 14 in., yet they were adequate to carry both the load of the floor above and the load incidental to construction. If no such reinforcement had been provided, however, failure would have been inevitable. Thus, again, it is shown that, where theory and experiment may fail to justify certain practices, actual experience does, and emphatically. |
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