FORMULAS FOR BENDING OF REINFOROED OONORISTE BEAMS. Reasons for Differences. Numerous formulas have been proposed for the strength of reinforced concrete beams; but they may be grouped into two classes, viz.: (1) emperical formulas, those that express the results of experiments; and (2) rational formulas, those that are based upon the principles of mechanics and the laws of the strength of materials involved. Only the latter will be considered here.
The numerous rational formulas differ among themselves for three reasons, viz.: (1) according to the amount of the tensile re sistance of the concrete considered; (2) according to the distribution of the compressive fiber stresses assumed; and (3) according to the method employed of applying the factor of safety.
1. When the load is first applied to a reinforced concrete beam, the tensile resistance of the beam is the sum of the tension in the steel and that in the concrete; but as the load increases, the con crete cracks on the lower side of the beam, and thus decreases the amount of tension taken by the concrete. When the beam fails, these cracks extend nearly to the neutral axis; and hence the un broken tensile area is quite small, and as it is quite near the neutral axis the moment of its resistance is practically negligible. It is the almost universal custom to neglect in formulas for practice, the effect of the tensile resistance of the concrete.
2. As shown in Fig. 25, the stress-deformation curve for concrete in compression is nearly a straight line up to and even beyond ordinary working stresses, and the most common working formulas are based upon a straight-line stress-deformation relation. In ex perimental work it is usually necessary to use the curved stress deformation relation; but some engineers have added useless com plication by taking account of the curvature of the stress-deformation diagram in deducing working formulas. When the curvature is taken into account, the stress-deformation curve is usually assumed to be the arc of a parabola with its vertex at A' (Fig. 25) or above.
3. There are employed two methods of applying the factor of safety. One is to apply the factor to the ultimate strength of the concrete and of the steel, and employ the safe working strength in the formula for the safe strength of the beam; and the other method is to deduce a formula for the ultimate strength of the beam, and then apply a factor of safety to this result to determine the safe load for the beam. The second method was formerly the more common;
but the first is the more simple and the more logical, and has now become the more common. One objection to the use of formulas for the ultimate strength is that most of them do not take account of the curvature of the stress-deformation line; and the few that do, thereby add complication without compensating advantage.
Fig. 26 shows the distribution of fiber stress in the concrete, assumed in different formulas for reinforced concrete beams.* No. 9 represents the distribution usually assumed and the one employed in this volume.
2. The stress diagram for compression is a straight line up to the safe compressive strength of the concrete.
3. There are no temperature or shrinkage stresses in either the steel or the concrete.
The following nomenclature will be used: * f, = unit fiber stress in the steel; • = unit fiber stress in the concrete at its compressive face; e, = unit elongation of the steel due to the stress h; = unit shortening of the concrete due to the stress f,; E. = modulus of elasticity of the steel; E. = modulus of elasticity of the concrete in compression; n= ratio E,=E.; T = total tension in the steel at any section of the beam; C = total compression in the concrete at any section of the beam; M. = resisting moment as determined by the steel; = resisting moment as determined by the concrete; M = bending moment or resisting moment in general; b = breadth of a rectangular beam; d = distance from the compressive face to the plane of the steel; k = ratio of the depth of the neutral axis of a section below the top to the distance d; j = ratio of the arm of the resisting couple to the distance d; A = area of cross section of the steel; • p = A _ b d, and is called the steel ratio.