Formulas for Bending of Reinforoed Oonoriste Beams

Position" of Neutral Axis. The first step is to determine the position of the neutral axis. Since cross sections that were plane before bending remain plane after bending, the unit deforma tions of the fibers vary as their distances from the neutral axis; and hence, in Fig. 27, For simple bending the total tension is equal to the total compression and hence Combining equations 1 and 2, substituting p = A _ bd, and solving, gives: Equation 3 shows that the position of the neutral axis depends only upon the proportion of the steel and the ratio of the modulii of elasticity. For val ues of n = 15 and of p be tween 0.75 and 1 per cent, k varies from 0.38 to 0.42; but is ordinarily taken as j.

449.

Arm of Resisting Couple. To find the arm of the resisting couple, jd, Fig. 27, notice that the distance of the centroid of the compressive stress from the top of the beam is kd; and therefore the arm of the couple, jd = d — i kd or It should be noticed that j does not vary much with p, and that for n = 15 and p between 0.75 and 1.0 per cent—common values, the average value of j is about {.

Resisting Moment.

If the amount of reinforcement is insufficient to utilize the full compressive resistance of the concrete, then the resisting moment of the beam depends upon the steel, and is On the other hand, if the beam is over-reinforced, its resisting moment depends upon the concrete and is To find the actual resisting moment in any particular case, the two values of M must be computed, and the smaller one taken.

Application of Preceding Equations. The preceding equa tions are all that are really necessary in solving problems involving the bending moment of reinforced concrete beams.

To determine the unit fiber stress on the steel

for a given bending moment, M, solve equation 5 thus: To find the fiber stress on the concrete, solve equation 6 thus: To find the value of in terms of eliminate M between equations 7 and 8, and find To determine the amount of steel required, solve equations 1 and 9, and then Equation 10 shows that for the same values of of the area of the steel to that of the concrete above the center of the steel is the same for all sizes of beams; that is, the amount of steel depends only upon the two ratios above.

To find the area of the beam: If the value of p addpted is less than that given by equation 10, then the area of the beam should be deter mined from equation 5, that is, from the relation but if the value of p selected is more than that determined by equa tion 10, then the area of the beam should be determined from equa tion 6, that is, from the relation Additional Information. For a description and discussion of three series of carefully conducted and comprehensive experi ments, giving much interesting and instructive information con cerning the strength and theory of flexure of reinforced concrete beams, see Bulletins No. 1, 4, and 14 of the University of Illinois Engineering Experiment Station.

T-BasMs. Sometimes in the construction of concrete floors for buildings and bridges, the slab and the reinforced beam supporting it are constructed as a monolith; and consequently a portion of the slab on each side of the beam acts as compression area to balance the tension in the steel in the lower part of the beam. Such a member is usually called a T-beam, but sometimes a ribbed slab. In a T-beam the slab is called the flange, and the beam proper the stem.

The formulas for T-beams are more complicated than those for simple beams, because the neutral axis may be in either the flange or the stem, and it is not possible to determine which except by trial. The computations for T-beams are further complicated by the fact that such beams are often made continuous over the sup ports, which produces tension on the upper side of the beam at the support and requires the introduction of reinforcement at this point.

For formulas for T-beams, see Turneaure and Maurer's Principles of Reinforced Concrete Construction, pages 78-84; and for a de scription and discussion of a series of carefully conducted experiments on T-lieams, see Bulletin No. 12 University of Illinois Engineering Experiment Station.