It is also necessary to provide expansion-joints when sections of concrete are poured between rigid terminations. For example, a concrete bridge between two sections of a warehouse should be jointed with the main structure; concrete roof-fills should have joints adjacent to all parapets, or penthouse walls (see page 339); concrete sidewalks, especially those poured in cold weather, should have expansion-joints at frequent intervals to avoid the possibility of buckling under summer heat. The angles of retaining walls, the junctions between horizontal slabs and vertical walls, like those formed by the changes in level of basement-floors, and similar sections, should be designed to permit freedom of movement. It should be constantly borne in mind that the FACTOR

OF EXPANSION OF CONCRETE UNDER THE INFLUENCE OF HEAT IS PRACTICALLY

THAT OF STEEL and when this temperature change is combined with the VOLUME CHANGE DUE TO MOISTURE, it becomes a most important element of the design.

11. Protection of Reinforcing-Steel. In order to protect the steel reinforcement of concrete buildings from rusting, IT IS OF PRIME IMPORTANCE TO

ENCASE ALL BARS BY A SUFFICIENT MASS OF CONCRETE TO PREVENT ATMOSPHERIC MOISTURE, OR GROUND-WATER, FROM COMING IN CONTACT WITH THE STEEL. There is probably no other single cause that has contributed to so many partial failures, and unsightly exteriors, as the neglect of this most vital requirement. As the atmospheric conditions vary greatly in different parts of the country, it is extremely difficult to lay down any definite rules for the thickness of insulation which will be adequate under all conditions. In general, the reinforcement on the exteriors of buildings should be protected by a minimum of 2 in of wellmade concrete, and this dimension should apply to hoops and stirrups as well as to the main reinforcement. Footings should have a minimum of 3 in of nonporous concrete to protect the steel. The requirements for concrete in alkali soils are given on page 589, and for concrete exposed to the action of sea water, on page 590. The insulation required for the purpose of FIRE-RESISTANCE depends, to a large extent, upon the fire-hazard to which the building will be exposed, and the character of the aggregate employed. Most building codes require 2 in for columns and girders, 11⁄2 in for beams and walls, and 1 in for slabs. The Joint Committee, 1924, recommends, for fire-resisting construction, 2 in for beams, girders and columns, and 1 in for slabs and walls, provided aggregate showing an expansion not materially greater than that of limestone or trap-rock is used. Where the fire-hazard is limited, a protection of in for slabs and walls and 14 in for beams, girders, and columns is permitted, without reservation relative to the aggregate.

CHAPTER II

FUNDAMENTAL PRINCIPLES AND FORMULAS

1. Concrete and Steel in Combination. As concrete is very much stronger in COMPRESSION than it is in TENSION, it has become the generally accepted practice to provide STEEL REINFORCEMENT in the concrete to resist all of the principal tensile stresses developed by any system of loading. Steel is also used to resist compressive stresses, as in columns, in rectangular beams with limited cross-sectional areas, where sufficient compressive resistance would not otherwise be developed, and in the webs of T beams to increase the resistance to negative bending moments over supports; but it is not as economical as concrete for this purpose and should be specially introduced only when it is impracticable to increase the sectional area of the member.

In a typical reinforced-concrete beam, of either rectangular or T-shaped cross-section, all compression through the middle part of the span is resisted by the concrete and all tension assumed to be resisted by the steel. The compressive stress is a maximum in the extreme fibers, f. (see Fig. 1, Diagrams of Deformations and Stresses), and diminishes to zero at the neutral surface or axis. The RESULTANT OF THE COMPRESSIVE STRESSES in the concrete and the RESULTANT OF THE TENSILE STRESSES in the longitudinal reinforcement, are the forces of a RESISTING MECHANICAL COUPLE. Accordingly, as the FORCES OF A COUPLE must be equal, the most economical beam is one in which the respective areas of steel and concrete are so adjusted that each will develop, simultaneously with the other, its full unit working stress. Rectangular beams and slabs are usually designed according to this principle. In the case of T beams, however, the slab-thickness usually results, at the mid-span, in an excess of concrete, and the required amount of steel in that portion of the beam is computed from the tensile strength required, irrespective of the compressive strength of the member.

2. Modulus of Elasticity. The ratio of the unit stress to the corresponding unit deformation is called the MODULUS OF ELASTICITY, which is constant within the ELASTIC LIMITS of the materials. Since in a reinforced-concrete member, such as a column, the UNIT DEFORMATIONS under the load are the same for both materials, and equal in each case to the UNIT STRESS divided by the MODULUS OF ELASTICITY, it follows that

This RATIO OF THE TWO MODULI is usually designated by the letter n, and the following are the generally accepted values:

For concrete assumed to have a compressive strength at 28 days of

1 500 lb per sq in to 2 200 lb per sq in, n = 15

2 200 to 2 900 lb per sq in, n = 12

More than 2 900 lb per sq in, n = 10

For example, in a reinforced-concrete column, the concrete of which is capable of developing, by test, 2 000 lb per sq in, in compression, and has a working stress of 500 lb per sq in, the vertical steel is considered capable of carrying only

notwithstanding the fact that the steel in compression may safely resist a much higher stress.

3. Fundamental Assumptions. The following method of analysis of reinforced-concrete beams is based on FIVE ASSUMPTIONS:

(1) A PLANE CROSS-SECTION of a beam, before bending, remains a plane section after bending.

(2) The MODULUS OF ELASTICITY of the concrete in compression remains constant within the assumed working stresses.

(3) PERFECT ADHESION exists between concrete and steel within the range of working stresses.

(4) ALL TENSION is borne by the steel.

(5) THE INITIAL STRESS in the steel caused by contraction or expansion of the concrete is negligible, except in some column-formulas. (See Report of Joint Committee, 1924.)

4. Notation. The following NOTATION is used in this discussion (see Fig. 1).

d

=

b =

k

l:d

j

=

=

effective depth of beam, distance from extreme fibers in compression to center of gravity of tensile reinforcement in inches;

width of beam or of section of slab in inches;

ratio of distance of neutral axis of cross-section from extreme fibers in compression to effective depth of beam;

distance of neutral axis from extreme fibers in compression in inches; ratio of distance between the center of compression of concrete and center of tension of steel to effective depth of beam or ratio of arm of resisting couple to d;

jd = distance between center of compression in concrete and center of tension in steel or arm of resisting couple in inches;

fe = compressive unit stress in extreme fibers of concrete in pounds per square inch;

tensile unit stress in extreme fibers of concrete in pounds per square inch;

tensile unit stress in steel in pounds per square inch;

f's

=

compressive unit stress in steel in pounds per square inch; A, = area of cross-section of main tensile reinforcement in square inches; modulus of elasticity of steel, in pounds per square inch;

E.

=

Ec modulus of elasticity of concrete, in pounds per square inch;

=

n = modulus of elasticity of steel divided by modulus of elasticity of

=

concrete, or Es/Ec;

M, resisting moment of reinforcement in inch-pounds; when computed on the basis of the full area of steel, As, times an assumed stress, fs; resisting moment of concrete in inch-pounds when computed on the basis of an assumed extreme fiber stress in the concrete, fc;

Mc

M

=

=

bending moment in inch-pounds;

p = percentage of reinforcement, or A,/bd.

5. Flexure-Formulas.

tions.

These formulas are based on the above assump

By assumption (1) the DEFORMATION in any fiber of the beam is proportional to its distance from the neutral surface, or from the neutral axis of the cross

section of the beam. And since, with a constant modulus of elasticity, by assumption (2), the stresses are proportional to their deformations, these stresses, also, vary in proportion to their distances from the neutral axis, and their varying magnitudes or intensities can be graphically represented by straight lines. At the neutral surface of the beam, or neutral axis of the cross-section of the beam, the stress is zero, and the stresses increase to a maximum, fe, in the extreme compressive fibers. Thus, they form, for the concrete, a triangle of compressive stresses, with the CENTER OF COMPRESSION at one-third the altitude of the triangle, measured from its base. The CENTER OF TENSION is at the center of gravity of the cross-section of the steel. In Fig. 1, A. and ▲, represent the MAXIMUM DEFORMATIONS in the concrete and steel, respectively, when the beam is in a state of flexure under any system of loading. The RESULTANT COMPRESSIVE STRESS in the concrete is the sum of the varying compressive stresses, and its line of action is through the center of gravity of the triangle of stresses.

Since the modulus of elasticity of each material equals the unit stress in the material divided by the corresponding unit deformation, the deformation. Ac in the concrete is equal to fc/Ec and the deformation As in the steel equals fs/Es.

Since the RESISTING MOMENT computed from the resistance of the steel must equal the unit tensile stress in the steel, f., times the area of the steel, pbd, times its lever arm, jd,

or

8

M1 = f. × pbd × jd

This is the RESISTING MOMENT at the section of the beam considered, as determined by the strength of the steel. And the resisting moment computed from the resistance of the concrete equals the average compressive unit stress in the concrete, 1⁄2fe times the area of the cross-section of the concrete in compression, times its lever-arm, jd.

M. (1⁄2)fc kdb × jd, or, M.

=

=

(2)fcjkbd2

(5)

This is the RESISTING MOMENT at the section of the beam considered, as determined by the strength of the concrete.

BUT SINCE EITHER OF THESE EXPRESSIONS IS THE MEASURE OF THE RESISTING MOMENT, THEY MAY BE PLACED EQUAL TO EACH OTHER, AND HENCE

fs X pbd × jd = 1⁄2fc × kdb × jd

Canceling common factors,

(2f./fc) X (p/k) = 1

or

30

or

and

fe/f. = 2p/k

Equating the values of fe/f, in Formulas (3) and (6),

k/[n(1 − k)] = 2p/k

p = k2/[2n(1k)]

(6)

Completing the square of the first member of this last equation by adding (pn)2 to both members,

k2 + 2pnk + (pn)2 = 2pn + (pn)2