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+PRACTICAL
+REINFORCED CONCRETE
+STANDARDS
+
+FOR THE DESIGN OF REINFORCED
+CONCRETE BUILDINGS
+
+BY
+H. B. ANDREWS, M. A.M. SOC. C. E.
+
+FIRST EDITION
+FIRST THOUSAND
+
+PUBLISHED BY
+SIMPSON BROS. CORPORATION
+BOSTON
+
+COPYRIGHT 1938 BY H. B. ANDREWS
+
+
+
+ | 123504 |
+ |
+
+
+ | OCT 22 1908 |
+ |
+
+
+ | SDKC |
+ |
+
+
+ | ·AN2 |
+ |
+
+
+
+# PREFACE
+
+Among the many publications relating to reinforced concrete, there seem to be few that answer the requirements of the architectural designer.
+
+The majority of architects are men who studied for their profession before any thorough investigation of reinforced concrete construction had been made. They are now called upon by their clients to design in a material requiring special and intimate study of its characteristics, and the solution of intricate formulae to determine the proper composition, arrangement, and dimensions of its component parts.
+
+Many theories have been advanced by many authorities during the progress of scientific research and experiment. Revisions in these theories have been made from time to time until they now tend to converge toward a common focus, but as yet these theories have not been put in practical working shape, and as a result many structures for which reinforced concrete is specially adapted are built of other material.
+
+It is therefore the purpose of the author to publish information of reinforced concrete in the shape of standard sections, tables, and specifications, that will enable designs to be made in this material as rapidly and as intelligently as in wood or steel.
+
+The tables contained herein have been in practical use in designing and constructing several large buildings in the City of Boston and elsewhere, and the work designed has been approved by conservative concrete specialists.
+
+The need of a standard form of construction with standard specifications is often brought forcibly to mind by the failure of reinforced concrete structures through no fault of the materials entering therein, but through the lack of knowledge by the architect, inspector, or contractor of the design or handling of the materials which enter into the work.
+
+This work is divided into five chapters:
+
+1. A Brief Theory of Reinforced Concrete Construction, including original formulae by the author for moments of resistance of T-beams and tables of standard sections.
+2. Miscellaneous Tables.
+3. A Reinforced Concrete Code.
+4. Standard Specifications.
+5. Foundations.
+
+It is the author's purpose to make this book as valuable as possible for practical designing and building, and to that end will invite suggestions from all interested so that use can be made of them, if found practical, for future editions.
+
+H. B. ANDREWS.
+
+Boston, January, 1908.
+
+I
+
+CONTENTS
+
+CHAPTER I
+STANDARD SECTIONS
+
+Design of Reinforced Concrete Beams .................................................. 1
+Illustrations of Use of Diagrams and Tables ........................................... 4
+Standard Sections - Typical Section of Floor ......................................... 6
+Bending Moments - Formulae ............................................................. 7
+Diagram for Designing Reinforced Concrete Slabs .................................... 9
+Diagram of Bending Moments .............................................................. 10
+Elements of Reinforced Concrete Beams .................................................. 11
+Working Loads for Reinforced Concrete Columns ....................................... 17
+
+CHAPTER II
+TABLES
+
+Weights and Areas of Square and Round Steel Rods. Welded and Expanded Metal .................................................................................................................. 18
+Proportions of Concrete Aggregates .......................................................... 19
+Crushing Strength of Portland Cement Concrete .......................................... 20
+Material for 100 Sq. Ft. Concrete Sidewalk or Floor .................................... 21
+Safe Loads for Wooden Beams ..................................................................... 22
+
+CHAPTER III
+A REINFORCED CONCRETE CODE .................................................................. 23
+
+CHAPTER IV
+REINFORCED CONCRETE SPECIFICATIONS .................................................. 28
+
+CHAPTER V
+FOUNDATIONS
+
+Loading ........................................................................................................... 41
+Classes of Foundations .............................................................................. 42
+Foundations directly upon the Soil ............................................................ 42
+Pile Foundations ....................................................................................... 45
+
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+
+PRACTICAL REINFORCED CONCRETE
+STANDARDS
+
+CHAPTER I
+STANDARD SECTIONS
+
+DESIGN OF REINFORCED CONCRETE BEAMS
+
+
A diagram showing the dimensions of a reinforced concrete beam. The diagram includes labels for various parts such as the web (b), flange (b'), effective depth of beam (d), distance from top to neutral axis (d/2), full depth of beam (h), depth of flange (h'), distance from lower side of flange to neutral axis (m/d'h'), distance between centroid of compression and neutral axis (n), distance between centroid of compression and centroid of tension (c), unit compression in concrete at top of flange (c'), unit compression in concrete at bottom of flange (e'), unit stress in steel (s), total compression in concrete (S), and total stress in steel (C). The diagram also shows the centroids of compression and tension.
+
+NOTATION
+
+\begin{align*}
+b & - \text{breadth of web.} \\
+b' & - b + 4h' - \text{breadth of flange.} \\
+d & - \text{effective depth of beam.} \\
+\frac{d}{2} & - \text{distance from top of beam to neutral axis.} \\
+h & - \text{full depth of beam.} \\
+h' & - \text{depth of flange.} \\
+m & - \frac{d}{q} h' - \text{distance from lower side of flange to neutral axis.} \\
+n & - \text{distance between centroid of compression and neutral axis.} \\
+a & - \text{distance between centroid of compression and centroid of tension} - \text{Moment arm.} \\
+c & - \text{unit compression in concrete at top of flange.} \\
+c' & - \text{unit compression in concrete at bottom of flange.} \\
+s & - \text{unit stress in steel.} \\
+C & - \text{Total compression in concrete} = S, \text{total stress in steel.}
+\end{align*}
+
+Pn. 1
+
+2
+PRACTICAL REINFORCED CONCRETE STANDARDS
+
+FORMULAE
+
+The following assumptions are made in obtaining formulae:
+
+I. A uniform horizontal compression in flange for a distance of twice the depth of flange each side of web. Making this the maximum width of flange tends to avoid the danger of shear along the web.
+
+II. That the compression in the concrete varies uniformly from the neutral axis to the top of the flange.
+
+III. That under working loads, and until the steel is stressed beyond its elastic limit, the neutral axis will lie approximately midway between the top of the beam and the centre of tension. This assumption has been corroborated by tests made to destruction of several T-beams, reinforced with different percentages of steel, at the Massachusetts Institute of Technology, the location of the neutral axis being carefully determined at each increment of load.
+
+IV. That the total compression in the concrete will be balanced by an equal tension in the steel.
+
+V. No allowance is made for tensile strength of concrete.
+
+In Fig. 1 consider first a rectangular section of width $b'$ and of depth $d$, then the theoretical compression due to any load
+
+$$\frac{b'd}{2} \times \frac{c'd'}{2} = \frac{cb'd'}{4}$$ (1)
+
+This can be represented by a triangle as shown by Fig. No. 2, where it is shown that the centre of compression will be at the centre of gravity of the triangle or two thirds of the distance $\frac{d}{3}$ above the neutral axis.
+
+The resistance moment about the neutral axis
+
+$$\frac{cb'd}{4} \times \frac{d}{3} = \frac{cb'd'}{12}$$ (2)
+
+Considering next the two areas bounded by $m$ and $2h'$, the theoretical compression
+
+$$\frac{4ch'm}{2} = 2ch'm,$$ (3)
+
+and its moment about the neutral axis
+
+$$-2ch'm \cdot \frac{2m}{8} = -\frac{4ch'm^2}{5},$$ (4)
+
+The actual compression in the T-section equals the difference between formulae (1) and (3)
+
+$$-\frac{cb'd}{4} - 2ch'm = C - S.$$ (5)
+
+STANDARD SECTIONS s
+
+and the resultant resistance moment about the neutral axis equals the difference between formulae (8) and (4)
+
+$$\frac{c'b^3}{4} - \frac{4c'h'm^3}{3}.$$ (6)
+
+The quotient of the resultant resistance moment divided by the total compression is the resultant moment arm, or
+
+$$n = \frac{\frac{12}{3}}{\frac{c'b^3}{4} - 2c'h'm} = \frac{c'b^3 - 16c'h'm^3}{3(c'b^3 - 24c'h'm)},$$ (7)
+
+$$c' = \frac{2cm}{d};$$ eliminating $c'$ by substitution in equation (7) and dividing both members by $c$ it becomes
+
+$$n = \frac{b'd^3 - 92h'm^3}{8b'd^3 - 48h'm^3}. $$ (8)
+
+To determine the moment of resistance of a beam when the tension in the steel is known, take moments around the centre of compression in the concrete with a moment arm
+
+$$a = \frac{d}{2} + n,$$ then $Mr = aS.$ (9)
+
+$S$ must never exceed the value of $C$ obtained by assuming the maximum unit stress $c$; it may, however, be less than this value, and the moment of resistance obtained by using the maximum value of $c$ will be decreased in proportion to the decreased value of $S$.
+
+**SHEAR**
+
+The diagonal tension existing in the web of a concrete beam may be resolved into vertical and horizontal components, each of which equals the vertical shear due to load at the section considered. The horizontal component will be taken care of by the horizontal beam rods. The vertical component will be taken care of by the concrete, provided it is not stressed over 60 lbs. per sq. inch of effective cross-section, i.e. the area included in the web between the centroid of compression in the concrete and the centroid of tension in the steel, or distance "a" in the tables. If stressed beyond this, the full vertical component must be taken care of by stirrups of steel, in a horizontal distance equal to "a".
+
+The stirrups should not be farther apart than $\frac{1}{2}$ "a," as with any wider spacing they would lose part of their value.
+
+Let $V$ = total external vertical shear at cross section considered,
+
+$$v = \text{shear per sq. in. cross-section},$$
+
+$$a = \text{effective depth between centroid of compression in concrete and centroid of tension in steel};$$ then
+
+$$v = \frac{V}{ab}. $$ (10)
+
+1 This assumption is made on the basis of using a 1-8-4 mixture of concrete.
+
+PRACTICAL REINFORCED CONCRETE STANDARDS
+
+If $v$ exceeds 50, provide stirrups spaced so that their tensile strength in a length of beam not exceeding $a$ is equal to $V$.
+
+If beam rods are trussed, the value of the vertical component of the trussed rods may be utilized.
+
+This value we will call $W$.
+
+$$W = A \times s \times \frac{a}{3.1}$$ (11)
+
+Where $A$ - sectional area of steel in trussed rods,
+$s$ - working stress of steel,
+$a$ - effective depth,
+$3.1$ - horizontal length trussed portion; all dimensions being used as inches.
+
+If $T$ represents tensile strength of stirrups in length of beam = "a," then
+
+$$T = V - W.$$ (18)
+
+To locate the section where the vertical shear is just 60 lbs per sq. inch, let $x$ - distance of this section from point of support, $l$ - span in feet, $d$ and $b$ as already used in previous formula, and $W$ the load per linear foot of beam or girdor; then
+
+$$x = \frac{l}{2} \cdot \frac{30db}{W}$$ (19)
+
+ILLUSTRATIONS OF USE OF DIAGRAMS AND TABLES
+
+The diagram shown on page 9 is for use in obtaining graphically the thickness and reinforcement of floor slabs. For illustration, assume a superimposed load of 125 lbs. per sq. ft. and a dead load which includes the weight of the slab, approximated, of 75 lbs. per sq. ft., making a total load of 200 lbs. per sq. ft. to be carried on a span of, say 12 feet.
+
+The horizontal lines measure the span in feet and the curved diagonal lines the bending moment in foot pounds. Follow the vertical line from the figure 12 to the point where it intersects the diagonal line marked 200 lbs. per sq. ft., and thence horizontally left to the columns marked "Thickness of slab in inches." The thickness of slab may be selected from one of these columns, and the amount of reinforcement to be used with it is shown by the figures in the column, remembering, as a general rule, that the minimum thickness of slab and the maximum amount of reinforcement is the most economical. For the example given the thickness of slab would be 6" and the reinforcement about .58 sq. inch in sectional area for one foot in width of slab. Interpolation can be made in both the diagram and figures for any of the factors entering into the problem.
+
+The diagram shown on page 10 is used similarly for obtaining the bending moments due to combined live and dead loads for beams. After the bending moment due to the load is obtained, a section of beam whose moment of resistance is equal to this bending moment may be selected from the tables marked "Elements of reinforced concrete beams."
+
+For illustration, if the combined live and dead loads on a beam with a span of
+
+STANDARD SECTIONS 5
+
+20 feet is 3000 lbs. per lin. ft. of beam, then the bending moment of 150,000 foot lbs. is obtained at the left hand side of the diagram. Referring to the tables, it is found that beams $E-10-26$, $F-10-26$, $G-10-26$, $D-10-80$, $E-12-44$, $F-18-44$, $G-18-24$, and $E-18-26$ with moments of resistance varying from 140,730 to 157,000 foot lbs. will practically fill the requirements. Take, for example, the beam $E-18-26$. The letters from $C$ to $G$ represent thickness of the slab on flange of the T-beam of from 8" to 7". The letter $E$, therefore, represents a thickness of 8". The first figure, 12, is the thickness of the stem, and the last figure, 26, the total depth of beam including slab. The first column in the table shows the maximum unit compressive stress in the concrete; the second column, the total compressive stress in concrete or tensile stress in steel; the third, the moment arm or distance between centroids of compression and tension; the fourth, the moment of resistance of beam in foot pounds; the fifth and sixth, the size of straight and trussed round rods used for reinforcement; the seventh, the sectional area of reinforcement; the eighth, the weight of reinforcement per lin. ft., and the ninth, the sectional area of concrete under the slab.
+
+
+
+ |
+ |
+ |
+ |
+ |
+ |
+ |
+ |
+ |
+
+
+ | $E$ |
+ $F$ |
+ $G$ |
+ $D$ |
+ $E$ |
+ $F$ |
+ $G$ |
+ $D$ |
+ $E$ |
+
+
+ | 12 |
+ 12 |
+ 12 |
+ 12 |
+ 12 |
+ 12 |
+ 12 |
+ 12 |
+ 12 |
+
+
+ | 8" |
+ 8" |
+ 8" |
+ 8" |
+ 8" |
+ 8" |
+ 8" |
+ 8" |
+ 8" |
+
+
+ | 9" |
+ 9" |
+ 9" |
+ 9" |
+ 9" |
+ 9" |
+ 9" |
+ 9" |
+ 9" |
+
+
+ | 10" |
+ 10" |
+ 10" |
+ 10" |
+ 10" |
+ 10" |
+ 10" |
+ 10" |
+ 10" |
+
+
+ | Total Depth of Beam Including Slab (in.) |
+
+
+ | Maximum Unit Compressive Stress in Concrete (psi) |
+
+
+ | Total Compressive Stress in Concrete or Tensile Stress in Steel (psi) |
+
+
+ | Moment Arm or Distance Between Centroids of Compression and Tension (in.) |
+
+
+ | Moment of Resistance of Beam in Foot Pounds |
+
+
+ | Size of Straight and Trussed Round Rods Used for Reinforcement (in.) |
+
+
+ | Sectional Area of Reinforcement (sq. in.) |
+
+
+ | Weight of Reinforcement Per Lin. Ft. |
+
+
+ | Sectional Area of Concrete Under Slab (sq. ft.) |
+
+
+
+
A page from a technical manual or textbook discussing standard sections for beams and slabs.
+
+The first figure, 12, is the thickness of the stem, and the last figure, 26, the total depth of beam including slab. The first column in the table shows the maximum unit compressive stress in the concrete; the second column, the total compressive stress in concrete or tensile stress in steel; the third, the moment arm or distance between centroids of compression and tension; the fourth, the moment of resistance of beam in foot pounds; the fifth and sixth, the size of straight and trussed round rods used for reinforcement; the seventh, the sectional area of reinforcement; the eighth, the weight of reinforcement per lin. ft., and the ninth, the sectional area of concrete under the slab.
+
+6 PRACTICAL REINFORCED CONCRETE STANDARDS
+
+STANDARD SECTIONS.
+
+TYPICAL SECTION OF FLOOR.
+
+
A diagram showing various standard sections of reinforced concrete structures.
+
+1
+
+STANDARD SECTIONS
+BENDING MOMENTS
+
+FORMULAE FOR BENDING MOMENTS
+
+(1)
+Beam fixed at one end, with concentrated load.
+$$B.M. = WL$$
+
+(2)
+Beam fixed at one end, with uniformly distributed load.
+$$B.M. = \frac{WL}{2}$$
+
+(3)
+Beam fixed at one end, with combination of uniformly distributed and concentrated loads.
+$$B.M. = PL + \frac{WL}{2}$$
+
+(4.)
+Beam supported at both ends, with concentrated load in middle.
+$$B.M. = \frac{WL}{4}$$
+
+(5.)
+Beam supported at both ends, with uniformly distributed load.
+$$B.M. = \frac{WL}{8}$$
+
+(6.)
+Beam supported at both ends, with concentrated load not at centre.
+$$B.M. = \frac{WMN}{L}$$
+
+8
+PRACTICAL REINFORCED CONCRETE STANDARDS
+
+(7)
+Beam supported at both ends, with equal and symmetrical concentrated loads.
+B. M. = WM.
+
+GRAPHICAL METHOD OF DETERMINING BENDING MOMENTS.
+
+(1) Beam supported at both ends, with one concentrated load, to find the bending moment at any part of the beam.
+
+Let $W$ be the weight as shown; then, as previously given, the bending moment at $W = \frac{WMN}{L}$. Plot the beam and the location of $W$ to some convenient scale, then to this, or some other scale, measure the line $WB$ equal to the bending moment already found. Connect $B$ with each end of the beam. Then if we wish to find the bending moment at some point, as $E$, draw $DE$ vertically to line $CB$. Measure $DE$ with same scale used in measuring $WB$. The result will be the bending moment at $E$.
+
+(8) Beam with two concentrated loads.
+
+Let $W$ and $P$ be the two concentrated loads as shown. Plot the bending moments $WB$ and $PC$ due to each of these loads by formula already given. Complete the diagram for each load by drawing $ABD$ and $ACD$. Now the total bending moment at $W$ would be $WE$, due to load $W$, plus $WE$, due to load $P$, or $WB$; and the total bending moment at $P$ would be $PC$, due to load $P$, plus $PF$, due to load $W$, or $PC$.
+
+Draw the outline $ABCD$, and this will represent the bending moment due to both loads, and will be the greatest where the vertical height scales the most.
+
+This method can be employed to find the bending moment due to any number of concentrated loads.
+
+(8) Beam with uniformly distributed load.
+
+At the middle of the beam draw the line $AB = \frac{WL}{8}$.
+
+Connect the points $C B D$ by a parabola and it will give the outline of the bending moments.
+
+(4) Beam loaded with both distributed and concentrated loads,
+
+Plot the outline of the bending moments due to the concentrated loads as per Case No. 2, and for the distributed load as per Case No. 3. The vertical distance between the upper and the lower outline at any point will be the bending moment at that point.
+
+STANDARD SECTIONS
+
+
+
+ | Diagram for Designing Reinforced Concrete Slabs |
+
+
+ | Thickness of Slab in inches |
+ B.M. |
+ W.L. |
+
+
+ | 3/4 5/6 7/8 9/10 |
+ 10 Piles |
+ 9,000 |
+
+
+ |
+ 8 Piles |
+ 8,500 |
+
+
+ |
+ 6 Piles |
+ 8,000 |
+
+
+ |
+ 5 Piles |
+ 7,500 |
+
+
+ |
+ 4 Piles |
+ 7,000 |
+
+
+ |
+ 3 Piles |
+ 6,500 |
+
+
+ |
+ 2 Piles |
+ 6,000 |
+
+
+ |
+ 1 Pile |
+ 5,500 |
+
+
+ |
+ 1/2 Pile |
+ 5,000 |
+
+
+ |
+ 1/4 Pile |
+ 4,500 |
+
+
+ |
+ 1/8 Pile |
+ 4,000 |
+
+
+ |
+ 1/16 Pile |
+ 3,500 |
+
+
+ |
+ 1/32 Pile |
+ 3,000 |
+
+
+ |
+ 1/64 Pile |
+ 2,500 |
+
+
+ |
+ 1/128 Pile |
+ 2,000 |
+
+
+ |
+ 1/256 Pile |
+ 1,500 |
+
+
+ | Reinforcement, Sectional Area Per Ft. Width: |
+
+ | | |
|---|
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|---|
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|---|
| Span in Feet: |
|---|
+
+
A graph showing the relationship between span length (in feet) and reinforcement requirements for reinforced concrete slabs. The x-axis represents span lengths from 3 to 12 feet. The y-axis represents reinforcement requirements in pounds per linear foot. The graph shows different curves for different thicknesses of slab (3/4", 5/6", etc.) and pile counts (8 piles, 6 piles, etc.). The curves are labeled with reinforcement values in pounds per linear foot.
+
+9
+
+10
+PRACTICAL REINFORCED CONCRETE STANDARDS
+DIAGRAM OF BENDING MOMENTS
+B. M. = W L² + 8
+
+
A graph showing bending moment in foot pounds versus span in feet. The x-axis ranges from 10 to 30 feet, and the y-axis ranges from 50,000 to 600,000 foot pounds. The lines on the graph represent different values of bending moment.
+
+BENDING MOMENT IN FOOT POUNDS
+
+SPAN IN FEET
+
+50,000
+100,000
+150,000
+200,000
+250,000
+300,000
+350,000
+400,000
+450,000
+500,000
+550,000
+600,000
+
+STANDARD SECTIONS
+11
+
+ELEMENTS OF REINFORCED CONCRETE BEAMS
+
+
+
+
+ | No. of Beams |
+ s |
+ 0=8 |
+ a |
+ Mr In ft. lbs. |
+ Size of Rods |
+ Sec. Area of Steel per ft. ft. lbs. |
+ Wt. of Steel per ft. ft. lbs. |
+ Cu. ft. of Concrete under slab |
+
+
+ |
+ |
+ |
+ |
+ |
+ Bent |
+ Straight |
+ |
+ |
+
+
+
+
+ | 4-8 |
+ 663 |
+ 3976 |
+ 5 |
+ 1657 |
+ One 9-10" 2485 .845 |
+ 1.043 .222 |
+ |
+
+
+ | C-4-8 |
+ 401 |
+ 9621 |
+ 5 |
+ 4009 |
+ " 7-8" 6013 2.044 .138 |
+ |
+ |
+
+
+ | D-4-8 |
+ 321 |
+ 9621 |
+ 5 |
+ 4009 |
+ " 7-8" 6013 2.044 .111 |
+ |
+ |
+
+
+ | E-4-8 |
+ 267 |
+ 9621 |
+ 5 |
+ 4009 |
+ " 7-8" 6013 2.044 .083 |
+ |
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+ | No. of Beams s | 0=8 | a | Mr In ft. lbs. | Size of Rods | Sec. Area of Steel per ft. ft. lbs. | Wt. of Steel per ft. ft. lbs. | Cu. ft. of Concrete under slab |
+
+
+| Elements of Reinforced Concrete Beams: |
+
+No. of Beams s C-4-8 D-4-8 E-4-8 F-4-8 G-4-8 H-4-8 I-4-8 J-4-8 K-4-8 L-4-8 M-4-8 N-4-8 O-4-8 P-4-8 Q-4-8 R-4-8 S-4-8 T-4-8 U-4-8 V-4-8 W-4-8 X-4-8 Y-4-8 Z-4-8 A-5-8 B-5-8 C-5-8 D-5-8 E-5-8 F-5-8 G-5-8 H-5-8 I-5-8 J-5-8 K-5-8 L- |
|---|
+
+
+ | No. of Beams s | C=5, D=5, E=5, F=5, G=5, H=5, I=5, J=5, K=5, L=5, M=5, N=5, O=5, P=5, Q=5, R=5, S=5, T=5, U=5, V=5, W=5, X=5, Y=5, Z=5, A=6, B=6, C=6, D=6, E=6, F=6, G=6, H=6, I=6, J=6, K=6, L=6, M=6, N=6, O=6, P=6, Q=6, R=6, S=6, T=6, U=6, V=6, W=6, X=6, Y=6, Z=6, |
|---|
+
+
+ | No. of Beams s | C=7, D=7, E=7, F=7, G=7, H=7, I=7, J=7, K=7, |
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+
+