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Shearing Stresses in Reinforced Concrete Beams

The variation of shear in a rectangular reinforced concrete beam may be ascertained by considering a small portion of such a beam between any two sections a small distance, ds, apart, as shown in Fig. 26a, the breadth of the beam being taken as b inches. The forces acting on this bit of beam consist of the normal stresses (C and T)�� Fm 26 and the shear (V), it being assumed that the moment at BB' is larger than that at AA', and that the sections are so close together that the two shears May be considered equal. These forces are in equilibrium and applying the condition MM = 0. The tendency of the small portion of the, beam cdBA to be pulled to the right is resisted by the horizontal shear on the cd plane which may be expressed as the intensity of shear on that plane (v), assumed to be uniform, multiplied by the area bds. �As the concrete is assumed to take no tension, the shear intensity is constant between the neutral plane and the steel while above that plane it varies as in a homogeneous rectangular beam. Accordingly Equation (1) gives the maximum intensity of horizontal, and likewise of vertical, shear (as explained in Art. 50) at any section of a rectangular reinforced concrete beam.

This demonstration applies equally well in essential details to a rectangular concrete beam reinforced for both tension and compression and to a reinforced concrete tee beam. Tests confirm the conclusion that in the matter of shear a tee beam may be considered as equivalent to a rectangular beam of the same depth, with a width equal to that of the stem of the tee beam. The standard notation for this tee beam stem width is b and so for tee beams the formula is written. The value of j does not vary greatly for a wide range of conditions and an average value of 0.86 is usually taken for all shear computations. Since all computations in which the value of the shear is used are highly approximate greater precision than that obtained by the average value is unnecessary.

52. Diagonal Tension in Reinforced Concrete Beams. The concrete in a reinforced beam is no stronger in itself than when unreinforced and it cracks in any loaded beam when the tensile limit is exceeded, the line of cracking being indicated in a general way in Fig. 10, sloping more steeply toward the ends of the beam, tending to lie at right angles to the inclined web stress. The function of the reinforcement is not to prevent cracking, that being impossible, but to keep any one crack from opening up widely, thus compelling the formation of many minute cracks in place of a single large one which would cause failure. It is plain that so long as the cracks are vertical the horizontal bars are effective reinforcement, but where they are inclined horizontal bars are very ineffective, there being nothing but concrete to carry the vertical component of the inclined tension. When a beam is reinforced for normal stress only, failure occurs under small load somewhat as pictured in Fig. 27a, the part of the beam toward the center dropping below the end portion. To be accurate the sketch should show only gradual curves in the steel. There is insufficient strength in the concrete below the rods to the left of the rupture to resist the pressure brought upon it, and it spalls off in such a failure.

A beam is made secure against diagonal tension failure by supplying it with a sufficient amount of reinforcement, so placed as to cross a sufficient number of the inclined lines of potential failure. The more nearly perpendicular to the cracks the more effective are the rods. In practice use is made of stirrups (Fig. 27b) generally vertical, looped about the main steel, and of main longitudinal rods bent up at an angle across the region of diagonal tension stress in those portions of the beam where they are no longer needed to resist the normal tension. In order to proportion such reinforcement knowledge must be had of the amount of the diagonal tension. Unfortunately this cannot be computed accurately in a reinforced concrete beam since the concrete cracks irregularly and just how much tension is taken by the steel it is impossible to say. If there were no normal tension on any section below the neutral axis the maximum diagonal tension would act at 45 degrees and have intensity equal to that of the shear at the section. This is always the assumption made in design. In all discussions of diagonal tension these words from the 1916 Report of the. Joint Committee should be kept in mind: "In designing, resource is hard to the use of calculated vertical shearing stresses as a means of comparing or measuring the diagonal tension stresses developed, it being understood that the vertical shearing stress is not the numerical equivalent of the diagonal tensile stress, and that there is not even a constant ratio between them. It does not seem feasible to make a complete analysis of the action of web reinforcement and more or less empirical methods of calculation are therefore employed. Study of tests indicate that the concrete is effective in resisting small amounts of diagonal tension and may be counted on with safety to perform this duty unaided when the shearing stress is less than about 2 per cent of the ultimate compressive strength of the concrete, about 40 pounds per square inch for ordinary 1-24 mixes. When the shearing stress exceeds this limit, the concrete is ordinarily still counted on as carrying a portion of the diagonal tension.

The use of the shear as a measure of the diagonal tension accounts for the fact that diagonal tension failure and diagonal tension reinforcement are very commonly, and erroneously, spoken of as shear failure and shear reinforcement. It is hardly worth S while to quarrel with this usage so long as it is held clearly in exactly what the terms refer to.