This section is from the book "Cement And Concrete", by Louis Carlton Sabin. Also available from Amazon: Cement and Concrete.

Mod.Elas. Concrete | 1,000,000. | 2,000,000. | 3,000,000. | 4,000,000. | ||||||||

Working Stress of Concrete in Compression. | Moment Equals d^2 Times. | Moment Equals d2 Times. | Steel. | Moment Equals d^2 Times. | Steel. | Moment Equals d^2 Times. | Steel. | |||||

Sq. Ins. Area Equals d Times. | PerCent. of Area Section. | Sq. Ins. Area Equals d Times. | PerCent. of Area Section. | Sq. Ins. Area Equals d Times. | PerCent. of Area Section. | Sq. Ins. Area Equals d Times. | Per Cent, of Area Section. | |||||

Lbs. per Sq. In. | a | b | c | a | b | c | a | b | c | a | b | c |

100 200 300 400 500 600 | 14.1 43.1 78.1 115.0 156.0 195.0 | .018 .060 .114 .174 .240 .309 | .15 .50 .95 1.45 2.00 2.58 | 8.2 28.2 54.7 86.2 120.0 156.0 | .013 .046 .090 .142 .200 .262 | .11 .38 .75 1.18 1.67 2.18 | 5.9 20.9 42.2 68.1 97.1 129.0 | .007 .027 .055 .092 .133 .180 | .06 .23 .46 .77 1.11 1.50 | 4.5 16.4 34.4 56.4 81.7 109.4 | .006 .021 .044 .074 .109 .149 | .05 .18 .37 .62 .91 1.24 |

Note: — Above table derived from equations (4), (5) and (6) by making the following assumptions: E. = 30,000,000, fs = 10,000 lbs. per sq. in.

595. For example, suppose we wish to know the strength of a beam ten inches deep (d = h — i = 10 in.) and the amount of steel required to develop a stress in the concrete of 400 lbs. per square inch when the stress in steel is 10,000 lbs. per sq. in., and the modulus of elasticity of the concrete is assumed at 3,000,000. In column a under 3,000,000 modulus, and opposite 400 lbs. stress, we find 68.1; then the moment of resistance of a beam one inch wide is 68.1 inch-lbs. X 10 X 10 = 6,810 inch-lbs., and the resistance of a beam 12 inches wide is 6,810 foot-lbs. The area of steel required in 12 inches width of beam is .092 d or 0.92 sq. in. This beam is reinforced with .77 of one per cent, steel. Similar tables may be prepared for other values of Es and fs if desired.

596. In Table 161 the equations have been completely solved for certain typical values of Ec and fc, assuming the values for Es and fs of thirty million and ten thousand respectively, as in Table 160. Having computed the bending moment, and fixed upon the probable safe working stress and modulus of elasticity of the concrete which it is proposed to use, it is only necessary to take from the table the required depth of beam and the amount of steel reinforcement required.

For example, a girder 10 feet long supported at the ends carries two loads of 5,000 pounds, each load being 2.5 feet from a support.

If the width of girder is 15 inches, working stress of concrete 300 lbs. per sq. in. and modulus of elasticity of concrete 1,500,000, what is the required depth of girder and area of steel in tension side?

The maximum bending moment (neglecting weight of beam) is 12,500 ft.-lbs. throughout the central five feet. The required moment of resistance for twelve inches in width is 12/15 of 12,500 = 10,000 ft.-lbs. Looking in the table for this bending moment under 300 lbs. stress and 1,500,000 modulus, we find it is between d = 12 and d = 14, or at about d = 13 inches. If we allow 2 inches below center of steel reinforcement, we have total depth of beam, h = 13 + 2 = 15 inches. In the same lines we find area of steel for 12 inch width between 1.08 and 1.26, or, say, 1.17; then for 15 in. width the required area is 15/12 X 1.17 = 1.46 sq. in. The bars should not be more than 3 to 6 inches apart. We may use, then, 5 bars 9/16 inch square or 5/8 inch diameter, spaced three inches apart. In large beams it is necessary to consider the bending moment occasioned by the weight of the beam after making a first approximation to the size required.

d = depth in inches from top of beam or slab to center of steel reinforcement. Moments in ft.-lbs. per foot width of beam (or inch-pounds per inch width). Area steel in square inches per foot width of beam.

Note:—Above table derived from equations (4), (5) and (6) by making the following assumptions: Mod. Elast. of Steel, Es = 30,000,000, Stress in Steel, fs = 10,000 lbs. per sq. in.

(Tensile strength of concrete is neglected).

Working Stress of Concrete in Compr. | 200 Lbs. per Sq. Inch. | 300 Lbs. per Square Inch. | 400 Lbs. per Square Inch. | 500 Lbs. per Square Inch. | d | ||||||||||

Mod. Elast. Concrete. | 1,500,000. | 2,500,000. | 1,500,000. | 2,500,000. | 1,500,000. | 3,500,000. | 2,500,000. | ||||||||

d = Depth. | Moment. | Steel. | Moment. | Steel. | Moment. | Steel. | Moment. | Steel. | Moment. | Steel. | Moment. | Steel. | Moment. | Steel. | |

Inches. | Ft.-Lbs. | Sq. In. | Ft.-Lbs. | Sq. In. | Ft.-Lbs. | Sq. In. | Ft.-Lbs. | Sq. In. | Ft.-Lbs. | Sq. In. | Ft.-Lbs. | Sq. In. | Ft.-Lbs. | Sq. In. | In. |

1 2 3 4 6 6 7 8 9 10 12 14 16 18 21 24 27 30 33 36 | 34 136 307 546 852 1228 1671 2182 2762 3410 4910 6680 8730 11050 15040 19640 24860 30690 37140 44200 | .05 .09 .14 .18 .23 .27 .32 .37 .41 .46 .55 .64 .73 .82 .96 1.10 1.24 1.37 1.51 1.65 | 48 191 429 763 1192 1717 2337 3053 3864 4770 6870 9350 12210 15450 21040 27480 34770 42930 51940 61820 | .06 .13 .19 .26 .32 .39 .45 .51 .58 .64 .77 .90 1.03 1.15 1.35 1.54 1.73 1.92 2.11 2.31 | 65 258 581 1034 1615 2326 3165 4134 5233 6460 9300 12660 16540 20930 28490 37210 47090 58140 70350 83720 | .09 .18 .27 .36 .45 .54 .63 .72 .81 .90 1.08 1.26 1.44 1.62 1.89 2.16 2.43 2.70 2.97 3.24 | 76 303 682 1213 1895 2729 3714 4850 6140 7580 10910 14860 19400 24560 33430 43660 55260 68220 82550 98240 | .11 .21 .31 .42 .52 .63 .73 .83 .94 1.04 1.25 1.46 1.67 1.87 2.19 2.50 2.81 3.12 3.43 3.75 | 99 396 891 1584 2475 3564 4851 6336 8020 9900 14260 19400 25340 32080 43660 57020 72170 89100 107800 128300 | i .14 .29 .43 .57 .71 .85 1.00 1.14 1.28 1.42 1.71 1.99 2.27 2.56 2.98 3.41 3.84 4.26 4.69 5.11 | 89 355 800 1420 2220 3190 4350 5680 7190 8875 12780 17400 22700 28740 39120 51100 64700 79900 96600 115000 | .12 .24 .36 .48 .60 .72 .84 .96 1.08 1.20 1.32 1.68 1.92 2.16 2.52 2.88 3.24 3.60 3.96 4.32 | 108 433 975 1733 2707 3900 5310 6930 8770 10830 15600 21230 27720 35090 47770 62400 79000 97300 117900 140300 | .15 .30 .45 .60 .75 .90 1.05 1.20 1.35 1.50 1.80 2.10 2.40 2.70 3.15 3.60 4.05 4.50 4.95 5.40 | 1 2 3 4 5 6 7 8 9 10 12 14 16 18 21 24 27 30 33 36 |

597. The above tables are prepared on the assumption that the stress in concrete shall be equal to the value selected when the stress in the steel reinforcement reaches 10,000 lbs. per sq. in. From the equations, other tables may be prepared if desired, in which the working stress in steel shall be 12,500, 16,000 or any other assumed value. The tables are not suited to the computation of beams in which excessive reinforcement is used.

As to actual tests of the performance of concrete and steel in combination, the possible variations in material are so diverse and the Cost of experiments so great that the results thus far obtained appear somewhat fragmentary, but each investigator has selected a small branch of the subject for experiment. Among the more valuable tests in this line may be mentioned the following : —

Tests at Massachusetts Institute Technology, Prof. Gaetano Lanza, Trans. Amer. Soc. C. E., vol. 50, p. 486. Tests at Purdue University, Prof. W. K. Hatt, Jour. Western Soc. Engrs., June, 1904. Tests at Rose Polytechnic Institute, Prof. Malvard A. Howe, Jour. Western Soc. Engrs., June, 1904. Tests at University of Illinois, Prof. A. N. Talbot, Proc. Amer.

Soc. for Testing Materials, 1904. Tests at University of Wisconsin, Prof. F. E. Turneaure, Proc. Amer. Soc. for Testing Materials, 1904.

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