Extracting the square root of the first member of the equation, and indicating the operation for the second member,

k + pn = √2pn + (pn)2

and

k = √2pn + (pn)2 — pn

(7)

This is the BASIC FORMULA that determines the LOCATION OF THE NEUTRAL AXIS of the cross-section of a beam, when p and n are known. From Formula (4), by transposition, and by substituting M for Mc and M.,

These last two formulas may be used to CHECK THE STRESSES IN THE STEEL AND CONCRETE, respectively, for members already designed.

Formula (5), gives the value of the RESISTING MOMENT, which a member with given dimensions is capable of resisting without exceeding a certain working strength in the concrete. Therefore,

This is the formula for determining the DEPTH OF BEAM to resist a given bending moment without exceeding a specified unit stress in the concrete.

These are the formulas for determining the CROSS-SECTIONAL AREA of the main tensile reinforcement to resist a given bending moment.

In practice the values of the working stresses, fe and fs, are known, and the ratio, n, of the assumed moduli of elasticity of the steel and concrete. In order to adapt the formulas to practical use it is necessary to determine the values of p and j for a BALANCED BEAM. A BALANCED BEAM is one in which the sectional areas of the concrete and the steel are such that each develops its full working stress simultaneously with the other.

6. Determination of the Balancing Percentage of Steel. From Formula (6),

This equation gives the percentage of steel required to develop simultaneously the stresses f, and fe when n is known.

For example, let the assumed working unit stresses be f

sq in, and f. = 650 lb per sq in, and let n = 15. Then, substituting in Formula (13),

Ρ

=

1

[2 X 16 000/650] × [{(16 000/(15 x 650)} + 1]

is the balancing percentage for the data assumed.

=

16 000 lb per

In this manner may be found the BALANCING PERCENTAGES of STEEL REINFORCEMENT for all rectangular members. Table I, Values for FormulaFactors, page 20, computed by this formula, gives the value of p for the working stresses and for the two values of n ordinarily used.

7. Determination of the Value of j. Since the compression-area of the cross-section of a beam can be represented as a triangle, by reason of the assumption of STRAIGHT-LINE VARIATION OF STRESSES (Fig. 1), and since the distance of the center of gravity of the triangle from its base is one-third its altitude, measured from its base,

In order then to find the value of j for any given values of fs, fe, n, and p,

Table I. Values for Formula-Factors for Rectangular Beams and Slabs

*The value of k applies to all types of beams, including T beams. † In members with compression-reinforcement, p becomes pi

it is necessary to first solve for k directly, from Formulas (6) or (7), using the working stresses assumed, and the corresponding value of p as computed above. From Formula (6), using the assumed stresses and the value of p just found, 2 × 0.0077 × (16 000/650) = 0.38

k

=

2pfs/fc

=

8. Working Formulas. As the stresses in the steel and concrete, and the the value of n, usually remain unchanged for large portions of the work, the fundamental Formulas (10), (11), and (12), should be adapted for practical use by substituting the value of fe, k, and j corresponding to any given values, and the economical value of p. For fe 650 lb per sq in, f. 16 000 lb per sq in,

=

M

√325 × 0.874 × 0.38 × b

0.0077 bd

16 000 X 0.874 X d 14 000d

=

M is in inch-pounds, b and d in inches, and A, in square inches.

These are the WORKING FORMULAS for depth of concrete and cross-sectional area of longitudinal steel in rectangular beams and slabs, for the stresses and the value of p as noted. For different working stresses, or a different value of p, a similar procedure is followed to determine corresponding coefficients. In Table I, page 20 are given values of p, j and k for balanced reinforcement. The table also contains a moment factor K

inforcement, K = 1⁄2f.jk or pf.j.

M

=

bd2

in which, for balanced re

It is seen that both Formulas (16) and (17), give A,, the STEEL-AREA required. Formula (16) is used in the design of slabs and rectangular beams whenever it is feasible to balance the steel and concrete; otherwise it is necessary to use Formula (17). It should be remembered that, as this formula contains a value of j which was determined for the particular value of p required in a BALANCED BEAM, there is a slight inaccuracy when it is used with any other percentage of reinforcement. The exact calculation should be made in the case of typical members which are repeated many times, and for which a corrected value of k, and its dependent j, can be determined by trial, after the approximate percentage of steel has been computed by using 0.874 as the value of j.

Having thus reduced the fundamental formulas to the final forms expressed in Formulas (15), (16), and (17), the depth d of the beam, as governed by the compressive strength of the concrete, and the cross-sectional area, A,, of the longitudinal steel reinforcement, can be computed, as soon as the bending moments have been determined.

CHAPTER III

BENDING MOMENTS

1. Bending Moments for Uniform Loads. The bending moments, positive and negative, depend upon the DEGREE OF RESTRAINT at the supports. For instance, a simple beam, that is, one merely resting on two supports, has a maximum positive moment at the middle, equal to WL/8 in-lb, in which W is the total uniformly-distributed load in pounds and L the span in inches; and there is a zero bending moment at each support. If this beam is fixed at one end and simply supported at the other, the maximum positive bending moment, at 3 L in from the free end, is reduced to 9 WL/128 in-lb, or approximately WL/14.2 in-lb; and a negative moment, equal to WL/8 in-lb, is developed at the fixed end. Likewise, a beam fixed at both ends, has a positive moment at the middle, WL/24 in-lb, and a negative moment at each support twice as great, WL/12 in-lb.

A REINFORCED-CONCRETE BEAM CONTINUOUS OVER A SUPPORT ALWAYS DEVELOPS A NEGATIVE BENDING MOMENT OVER THE SUPPORT, and this is usually greater than the maximum positive bending moment near the mid-span. The design-moments used in practice should never be less than the theoretical moments, and may be taken somewhat greater, in order to provide for variations in live-load distribution, as in the case of equal, continuous spans, where WL/12 in-lb or WL/16 in-lb is used as the value of the positive design-moment adjacent to the middle of the span. Here, theoretically, the moment is only WL/24 in-lb. Slabs, beams, or girders assumed to be simply supported at one end and continuous at the other end, are often referred to as SEMICONTINUOUS. This condition exists, in monolithic construction in the end-spans of a series of slabs, beams, or girders. When both ends are continuous, the member is said to be FULLY-CONTINUOUS. This last condition ordinarily exists in all except the end-spans in a series of slabs, beams, or girders. The following formulas are abstracted from the report of the Joint Committee on Standard Specifications for Concrete and Reinforced Concrete, 1924. In order to conform with the notation of the text, W has been substituted for ul in the equations:

M = bending moment in inch-pounds;

W = total uniformly-distributed load on the member in pounds;
L = span, in inches;

moment of inertia of a cross-section of a beam, girder, or column about the neutral axis for bending in biquadratic inches;

unsupported length of a column in inches.