Cement Sand Stone 7 Da. 1 Mo. 3 Mo. 6 Mo. 7 Da. 1 Mo. 3 Mo. 6 Mo. 7 Da. 1 Mo. 3 Mo. 6 Mo.

Strength and Weight of Plain Concrete.-The average ultimate strength of concrete in tension, compression, and shear is given in the accompanying table for different proportions of mixture, the aggregate of which is broken stone. Concrete made of gravel is 75% as strong and concrete made with cinders is about 65% as strong.

As the strength of concrete increases with age, it is necessary for the engineer to know when the concrete will be loaded. It is customary to assume a factor of safety based on the strength of the concrete after 6 mo. The engineer must be careful that the concrete, in the first few months after being laid, is not subjected to too great stresses. For general work,

a factor of safety of 5 on concrete 6 mo. old is recommended. This will give the required strength for the first few months, and yet will not be wasteful of material at any time. A factor of safety of 4 on concrete 6 mo. old may be used for steady loads, such as earth fills, water pressure, etc.

The weight of concrete depends mainly on the kind of aggregate used. It averages about 140 to 150 lb. per cu. ft., for broken-stone and gravel concrete, and 110 to 115 for cinder concrete.

REINFORCED CONCRETE

FORMULAS FOR RECTANGULAR BEAMS Reinforced concrete is concrete in which steel or iron is .embedded in order to increase the strength of the former.

Fundamental Principles.-Many theories have been advanced as a basis for the design of reinforced-concrete beams, and it is not yet known which is most nearly correct. The formulas that follow are based on the so-called straight-line theory, which has been almost universally adopted in the United States and has been recommended by a Joint Committee composed of members of the leading engineering societies of this country. This theory is based on the following assumptions, and principles derived from these assumptions:

1. A plane section of a beam remains plane after it has been subjected to bending.

2. For one and the same material, the unit stresses at different points of a beam subjected to bending are proportional to their distances from the neutral axis.

3. The unit stresses in steel and concrete at points equidistant from the neutral axis are proportional to their respective moduli of elasticity.

4. The concrete is assumed to take only compressional stresses, all the tensional stresses being carried by the steel.

5. The internal stresses in the section of a reinforced-concrete beam subjected to bending form a couple consisting of the resultant of all compressional stresses taken by the concrete, on one hand, and the tensional stresses taken by the steel, on the other hand.

It is also assumed that the value of the ratio of the moduli of elasticity of steel and concrete (usually denoted by n) is constant within the limits of the working stresses of the mate rials. This value of n greatly varies with the qualities of the material and labor employed in the manufacture of the concrete, and is usually specified by city ordinances.

The reinforced-concrete tables given later are computed for n = 12 and n = 15, which are prevalent in the present engineering practice.

Definitions.-The economic steel ratio is that ratio of the area of steel to the area of concrete at which both the steel and concrete can be stressed to their maximum allowable limit at the same time, and is denoted by Pe. If a lower ratio is used, the stress in the concrete will not reach its limit without overstressing the steel, and if a higher ratio is employed the full strength of the steel cannot be utilized without overstressing the concrete. The economic steel ratio, or as it is also called the critical value of steel, is not a fixed quantity; it depends on the ratio of the allowable maximum unit stresses of steel and concrete.

The stress ratio is the ratio of the stresses actually produced in the steel and concrete by a given external moment. When n is constant, the value of the stress ratio depends only on the amount of steel used. For the critical value of steel, the stress ratio equals the ratio of the allowable maximum unit stress in steel and that in concrete.

Notation. The accompanying illustration shows a section of The following notation is used

a reinforced-concrete beam.

fs and fc stresses in steel and concrete, respectively,

Fs and

actually produced by the bending moment M;

F= maximum allowable unit stresses in steel and concrete, respectively;

Ms and Mc= working moment of resistance of steel and concrete for the unit stresses Fs and Fc, respectively;

r = stress ratio=fs: fc;

re=stress ratio when economic percentage of steel

is used, which is equal to Fs: Fc;

Es and E=moduli of elasticity of steel and concrete,

Cs-section-modulus coefficient for steel=jp;

Cc-section-modulus coefficient for concrete =

kj 2

R= coefficient whose values are given later in tabu

lar form.

Formulas. Following are the formulas for rectangular reinforced-concrete beams:

When the economic steel ratio is used, Ms = Mc; also,

Formulas 7 and 8 furnish the fundamental equations for designing and investigating rectangular reinforced-concrete beams. Formula 7, expresses the resistance of the beam for steel and is to be used when the steel ratio is below its critical value, while formula 8 gives expression to the moment of resistance of concrete and governs the design in cases when the amount of steel is above the economic ratio. Cs and Cc can be determined by formulas 4 and 5, and for finding the economic steel ratio formula 11 is available. For n = 12 and n = 15, C, and Ce can be taken directly from the accompanying table; also, the economic ratio of steel may be ascertained from this table, as will be explained presently.

Reinforced-Concrete Tables and Their Application.-For n = 12 or 15 the accompanying table of properties of reinforced.