This section is from the book "Cyclopedia Of Architecture, Carpentry, And Building", by James C. et al. Also available from Amazon: Cyclopedia Of Architecture, Carpentry And Building.

If the fibre stress in the beam due to the loading was less than 16,000, the deflection would be obtained by multiplying the result given in the table by the ratio of given fibre stress to 16,000.

The formula (2) applies directly to beams and channels only. If, therefore, a table based on this formula is made, and it is desired to use it for determining the deflection of unsymmetrical shapes, such as angles, tees, etc., the coefficients given must be divided by twice the distance of the neutral axis from extreme fibre, since both numerator and denominator of (1) has been multiplied by 2.

If a beam had a center load, its deflection could be obtained from this table by multiplying by 8/5, this being the ratio of the deflection of a beam supported at the ends and loaded with a center load, to that of a similar beam with the same total load uniformly distributed.

In the table of safe loads it will be noted that a heavy black line divides the capacities specified. This is to denote the limit of span beyond which the deflection of the beam, if loaded to its full capacity, would be likely to cause the ceiling to crack. This limit of span can be determined from the formulae given above, as follows:

A deflection of 1/360 of the span can be safely allowed without causing cracks. Substituting 1/360 for d, therefore, we have l = 5l2 X 8 X 16,000 360 384 E y and l = 48.3 y

Making the substitutions of the value of y for different sized beams, gives limits agreeing with those in the Cambria Hand Book. The limits given in the Carnegie book are fixed arbitrarily at 20 times the depth of beam and some less than these.

Expressing the above formula in a different form, we have fl = 384 E y

360x5x8 and f l/y = 384 E = C (a constant).

360x5x8 f l/y = 773,333.

Distance between Supports in feet. | Coefficient for Fibre Stress of 16 000 lbs. per Square Inch. | Coefficient for Fibre Stress of 12 500 lbs. per Square Inch. | Distance between Supports in Feet. | Coefficient for fibre Stress of 16 000 lbs. per Square Inch. | Coefficient for Fibre Stress of 12500 lbs. per Square Inch. |

L | H | H | L | H | H |

4 | .265 | .207 | 23 | 8.756 | 6.841 |

5 | .414 | .323 | 24 | 9.534 | 7.448 |

6 | .596 | .466 | 25 | 10.345 | 8.082 |

7 | .811 | .634 | 26 | 11.189 | 8.741 |

8 | 1.059 | .828 | 27 | 12.066 | 9.427 |

9 | 1.341 | 1.047 | 28 | 12.977 | 10.138 |

10 | 1.655 | 1.293 | 29 | 13.920 | 10.875 |

11 | 2.003 | 1.565 | 30 | 14.897 | 11.638 |

12 | 2.383 | 1.862 | 31 | 15.906 | 12.427 |

13 | 2.797 | 2.185 | 32 | 16.949 | 13.241 |

14 | 3.244 | 2.534 | 33 | 18.025 | 14.082 |

15 | 3.724 | 2.909 | 34 | 19.134 | 14.948 |

16 | 4.237 | 3.310 | 35 | 20.276 | 15.841 |

17 | 4.783 | 3.737 | 36 | 21.451 | 16.759 |

18 | 5.363 | 4.190 | 37 | 22.659 | 17.703 |

19 | 5.975 | 4.668 | 38 | 23.901 | 18.672 |

20 | 6.621 | 5.172 | 39 | 25.175 | 19.668 |

21 | 7.299 | 5.703 | 40 | 26.483 | 20.690 |

22 | 8.011 | 6.259 |

This equation shows that if the table of properties is used to determine the capacity of a beam for a certain span which will be within the plaster limits of deflection, the product of the fibre strain and the span must be kept constant for a given depth of beam.

For example, if it is desired to know the fibre strain allowable for a 12-inch beam on an effective span of 30'-0" (30 feet 0 inches) such that the plaster deflection will not be exceeded, we have f = 773,333 X 6 = 12,888. 30 X 12

The formula can be more quickly used by comparison with the limiting span given by the table of safe loads. In the above case the limit of span for a 12-inch beam and a fibre strain of 16,000 lbs. is 24 feet; therefore the required f = 24/30 X 16,000

= 12,800 Lateral Deflection of Beams. When beams are used for long spans, and the construction is such that no support against side deflection is given, the beam will not safely carry the full load

Ratio of Span or Distance between Lateral Supports to Flange Width. | Allowable Unit Stress for Direct Flexure in Extreme Fibre. | Proportion of Tabular Safe Load to be Used. | Ratio of Span or Distance between Lateral Supports to Flange Width. | Allowable Unit Stress for Direct Flexure in Extreme Fibre. | Proportion of Tabular Safe Load to be Used |

l/b | P | l/b | P | ||

19.37 | 16000 | 1.0 | 65 | 7474 | .47 |

20 | 15882 | .97 | 70 | 6835 | .43 |

25 | 14897 | .93 | 75 | 6261 | .39 |

30 | 13846 | .87 | 80 | 5745 | .36 |

35 | 12781 | .80 | 85 | 5281 | .33 |

40 | 11739 | .73 | 90 | 4865 | .30 |

45 | 10746 | .67 | 95 | 4595 | .29 |

50 | 9818 | .61 | 100 | 4154 | .26 |

55 | 8963 | .56 | 105 | 3850 | .24 |

60 | 8182 | .51 | 110 | 3576 | .22 |

indicated by the table, and the allowable fibre stress in top flange must be reduced. If such a beam were to carry a load giving a fibre stress of 16,000 lbs. per square inch, the actual fibre stress in top flange would be greater than this, as the deflection sideways would tend to distort the top flange and thus cause the additional stresses.

The length of beam which it is customary to consider capable of safely carrying the full calculated load without support against lateral deflection, is twenty times the flange width. The reason for thus fixing upon twenty times the flange width may be seen from the following:

In any consideration of a reduction of stress in a compression member due to bending caused by its unsupported length, it is customary to use Gordon's formula for the safe stress in columns. This formula is: fc = f/1+ l2/ar2

For columns with fixed ends, a =36,000. Now if we consider a

5-inch 9.75-lb. I, the moment of inertia about the neutral axis coincident with center line of web is I' = 1.23.

Since the moment of inertia of the web alone about this axis is inappreciable, the moment of inertia of each flange about this axis is I' f = .62.

The area of the whole section = 2.87 sq. in.

Web = .86 Area of flanges = 2.01 sq. in. Area of one flange = 1.00 ,, Therefore r't2= .62 r'f= .79 The width of flange for 5-in. beam = b = 3.00 in.

Therefore r1f, = b/3.80

Tests on full-sized columns show that columns of length less than ninety times the radius of gyration bend little if any under their load. It is, therefore, generally customary to disregard the effect of bending for lengths less than 90 radii. If in the above we multiply, we have:

90 r'f = 23.7 b

The assumption that with full fibre stress of 16,000 lbs. beams should be supported at distances not greater than twenty times the flange width, brings the limit under that of 90 radii.

Approximately the same result will be obtained if we assume the flange a rectangle and substitute 18,000 for f in Gordan's formula.

Then r2 = b2/12 and fc = 18,000/1 + l2/3,000b2 and for l = 20b fc = 15,900.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Section Index | Depth of Beam Inches | Weight per Foot Pounds | Area of Section Square Inches | Thickness of Web Inches | Width of Flange Inches | Mom, of Inertia Neutral Axis Perpendicular to Web at Center | Mom. of Inertia Neutral Axis Coincident with Center Line of Web | Radius of Gyration Neutral Axis Perpendicular to Web at Center |

I | I | r | ||||||

100.00 | 29.41 | 0.754 | 7.254 | 2380.3 | 48.56 | 9.00 | ||

95.00 | 27.94 | 0.692 | 7.192 | 2309.6 | 47.10 | 9.09 | ||

B 1 | 24 | 90.00 | 2G.47 | 0.631 | 7.131 | 2239.1 | 45.70 | 9.20 |

85.00 | 25.00 | 0.570 | 7.070 | 2168.6 | 44.35 | 9.31 | ||

80.00 | 23.32 | 0.500 | 7.000 | 2087.9 | 42.86 | 9.46 | ||

100.00 | 29.41 | 0.884 | 7.284 | 1655.8 | 52.65 | 7.50 | ||

95.00 | 27.94 | 0.810 | 7.210 | 1606.8 | 50.78 | 7.58 | ||

B 2 | 20 | 90.00 | 26.47 | 0.737 | 7.137 | 1557.8 | 48.98 | 7.67 |

85.00 | 25.00 | 0.663 | 7.063 | 1508.7 | 47.25 | 7.77 | ||

80.00 | 23.73 | 0.600 | 7.000 | 1466.5 | 45.81 | 7.86 | ||

75.00 | 22.06 | 0.649 | 6.399 | 1268.9 | 30.25 | 7.58 | ||

B 3 | 20 | 70.00 | 20.59 | 0.575 | 6.325 | 1219.9 | 29.04 | 7.70 |

65.00 | 19.08 | 0.500 | 6.250 | 1169.6 | 27.86 | 7.83 | ||

70.00 | 20.59 | 0.719 | 6.259 | 921.3 | 24.62 | 6.69 | ||

B80 | 18 | 65.00 | 19.12 | 0.637 | 6.177 | 881.5 | 23.47 | 6.79 |

60.00 | 17.65 | 0.555 | 6.095 | 841.8 | 22.38 | 6.91 | ||

55.00 | 15.93 | 0.460 | 6.000 | 795.6 | 21.19 | 7.07 | ||

100.00 | 29.41 | 1.184 | 6.774 | 900.5 | 50.98 | 5.53 | ||

B 4 | 15 | 95.00 | 27.94 | 1.085 | 6.675 | 872.9 | 48.37 | 5.59 |

90.00 | 26.47 | 0.987 | 6.577 | 845.4 | 45.91 | 5.65 | ||

85.00 | 25.00 | 0.889 | 6.479 | 817 8 | 43.57 | 5.72 | ||

80.00 | 23.81 | 0.810 | 6.400 | 795.5 | 41.76 | 5.78 | ||

75.00 | 22.06 | 0.882 | 6.292 | 691.2 | 80.68 | 5.60 | ||

B 5 | 15 | 70.00 | 20.59 | 0.784 | 6.194 | 663.6 | 29.00 | 5.68 |

65 00 | 19.12 | 0 686 | 6 096 | 636.0 | 27.42 | 5.77 | ||

60.00 | 17.67 | 0.590 | 6.000 | 609.0 | 25.96 | 5.87 | ||

55.00 | 16.18 | 0.656 | 5.746 | 511.0 | 17.06 | 5.62 | ||

B 7 | 15 | 50.00 | 11.71 | 0.558 | 5.648 | 483.4 | 16.04 | 5.73 |

45.00 | 13.24 | 0.460 | 5.550 | 455.8 | 15.00 | 5.87 | ||

42.00 | 12.48 | 0.410 | 5.500 | 441.7 | 14.62 | 5.95 | ||

55.00 | 16.18 | 0.822 | 5.612 | 321.0 | 17.46 | 4.45 | ||

B 8 | 12 | 50.00 | 14.71 | 0.699 | 5.489 | 303.3 | 16.12 | 4.54 |

45.00 | 13.24 | 0.576 | 5.366 | 285.7 | 14.89 | 4.65 | ||

40.00 | 11.84 | 0.460 | 5.250 | 268.9 | 13.81 | 4.77 | ||

B 9 | 12 | 35.00 | 10.29 | 0.436 | 5.086 | 228.3 | 10.07 | 4.71 |

31.50 | 9.26 | 0.350 | 5.000 | 215.8 | 9.50 | 4.83 | ||

40.00 | 11.76 | 0.749 | 5.099 | 158.7 | 9.50 | 3.67 | ||

Bll | 10 | 35.00 | 10.29 | 0.602 | 4.952 | 146.4 | 8.52 | 3.77 |

30.00 | 8.8S | 0.455 | 4.805 | 134.2 | 7.65 | 3 90 | ||

25.00 | 7.37 | 0.310 | 4.660 | 122.1 | 6.89 | 4.07 | ||

85.00 | 10.29 | 0.732 | 4 772 | 111.8 | 7.31 | 3.29 | ||

B13 | 6 | 80.00 | 8.82 | 0.5G9 | 4.609 | 101.9 | 6.42 | 3.40 |

25.00 | 7.3c | 0.406 | 4.446 | 91.9 | 5.65 | 3.54 | ||

21.00 | 6.31 | 0.290 | 4.330 | 84.9 | 5.16 | 3.67 | ||

25.50 | 7.50 | 0.541 | 4.271 | 68.4 | 4.75 | 3.02 | ||

B15 | 8 | 23.00 | 6.76 | 0.449 | 4.179 | 64.5 | 4.39 | 3.09 |

20.50 | 6.03 | 0.357 | 4.087 | 60.6 | 4.07 | 3.17 | ||

18.00 | 5.33 | 0.270 | 4.000 | 56.9 | 3.78 | 3.27 | ||

20.00 | 5.88 | 0.458 | 3.868 | 42.2 | 3.24 | 2.68 | ||

B17 | 7 | 17.50 | 5.15 | 0.353 | 3.763 | 39.2 | 2.94 | 2 70 |

15.00 | 4.42 | 0.250 | 3.660 | 36.2 | 2.67 | 2.86 | ||

17.25 | 5.07 | 0.475 | 3.575 | 26.2 | 2.36 | 2 27 | ||

B 19 | 6 | 14.75 | 4.34 | 0.352 | 3.452 | 24.9 | 2.09 | 2.35 |

12.25 | 3.61 | 0.230 | 3.330 | 21.8 | 1.85 | 2.46 | ||

14.75 | 4.34 | 0.504 | 3.294 | 15.2 | 1.70 | 1.87 | ||

B21 | 5 | 12.25 | 8.60 | 0.357 | 3.147 | 13 6 | 1 45 | 1 91 |

9.75 | 2.87 | 0.210 | 3.000 | 12.1 | 1.23 | 2.05 | ||

10.50 | 3.09 | 0.410 | 2.880 | 7.1 | 1.01 | 1.52 | ||

B23 | 4 | 9.50 | 2.79 | 0.337 | 2.807 | 6.7 | 0.93 | 1.55 |

8 50 | 2.50 | 0 263 | 2.733 | 6 4 | 0 85 | 1 59 | ||

7.50 | 2.21 | 0.190 | 2.660 | 6.0 | 0.77 | 1.64 | ||

7.50 | 2.21 | 0.361 | 2.521 | 2.9 | 0.60 | 1.15 | ||

B77 | 3 | 6.50 | 1.91 | 0.263 | 2.423 | 2.7 | 0.53 | 1.19 |

5.50 | 1.63 | 0.170 | 2.330 | 2.5 | 0.46 | 1.23 |

10 | 11 | 12 | 13 | 14 | 15 |

Radius of Gyration Neutral Axis Coincident with Center Lino of Web | Seetion Modulus Neutral Axis Perpendicular to Web at Center | Coefficient of Strength for Fiber Stress of 16,000 lbs. per sq. in. Used for Buildings | Coefficient of Strength for Fibor Stress of 12,500 lbs. per 6q.in. Used for Bridges | Distance Center to Center Required to make Radii of Gyration equal | Section Index |

r' | S | c | C | ||

1.28 | 103.4 | 21l5800 | 1653000 | 17.82 | |

1.30 | 192.5 | 2052900 | 1603900 | 17.99 | |

1.31 | 186.6 | 1990300 | 1554900 | 18.21 | B 1 |

1.33 | 180.7 | 1927600 | 1505900 | 18.43 | |

1.36 | 174.0 | 1855900 | 1449900 | 18.72 | |

1.34 | 165.6 | 1766100 | 1379800 | 14.76 | |

1.35 | 160.7 | 1713900 | 1339000 | 14.92 | |

1.36 | 155.8 | 1661600 | 1298100 | 15.10 | B 2 |

1.37 | 150.9 | 1609300 | 1257200 | 15.30 | |

1.39 | 146.7 | 1564300 | 1222100 | 15:47 | |

1.17 | 126.9 | 1353500 | 1057400 | 14.98 | |

1.19 | 122.0 | 1301200 | 1016600 | 15.21 | B 3 |

1.21 | 117.0 | 1247600 | 974700 | 15.47 | |

1.09 | 102.4 | 1091900 | 853000 | 13.20 | |

1.11 | 97.9 | 1044800 | 816200 | 13.40 | B 80 |

1.13 | 93.5 | 997700 | 779500 | 13.63 | |

1.16 | 88.4 | 943000 | 736700 | 13.95 | |

1.31 | 120.1 | 1280700 | 1000600 | 10.75 | |

1.32 | 116.4 | 1241500 | 969900 | 10.86 | |

1.32 | 112.7 | 1202300 | 939300 | 10.99 | B 4 |

1.32 | 109.0 | 1163000 | 908600 | 11.13 | |

1.32 | 106.1 | 1131300 | 883900 | 11.25 | |

1.18 | 92.2 | 983000 | 768000 | 10.95 | |

1.19 | 88.5 | 943800 | 737400 | 11.11 | B 6 |

1.20 | 84.8 | 904600 | 706700 | 11.29 | |

1.21 | 81.2 | 866100 | 676600 | 11.49 | |

1.02 | 68.1 | 726800 | 567800 | 11.05 | |

1.04 | 64.5 | 687500 | 537100 | 11.27 | B 7 |

1.07 | 60.8 | 648200 | 506400 | 11.54 | |

1.08 | 58.9 | 628300 | 490800 | 11.70 | |

1.04 | 53.5 | 570600 | 415800 | 8.65 | |

1.05 | 50.6 | 539200 | 421300 | 8.83 | B 8 |

1.C3 | 47.6 | 507900 | 396800 | 9.06 | |

1.08 | 44.8 | 478100 | 373500 | 9.29 | |

0.99 | 38.0 | 405800 | 317000 | 9.21 | B 9 |

1.01 | 36.0 | 383700 | 299700 | 9.45 | |

0.90 | 31.7 | 338500 | 264500 | 7.12 | |

0.91 | 29.3 | 312400 | 244100 | 7.32 | B 11 |

0.93 | 26.8 | 286300 | 223600 | 7.57 | |

0.97 | 24.4 | 260500 | 203500 | 7.91 | |

0.84 | 24.8 | 265000 | 207000 | 6.36 | |

0.85 | 22.6 | 241500 | 7.58 | B 13 | |

0.88 | 20.4 | 217900 | 170300 | 6.86 | |

0.90 | 18.9 | 201300 | 157300 | 7.12 | |

0.80 | 17.1 | 182500 | 142600 | 5.82 | |

0.81 | 10.1 | 172000 | 134400 | 5.96 | B 15 |

0.82 | 15.1 | 161600 | 126200 | 6.12 | |

0.84 | 14.2 | 151700 | 118500 | 6.32 | |

0.74 | 12.1 | 128600 | 100400 | 5.15 | |

0.70 | 11.2 | 119400 | 93300 | 5.31 | B 17 |

0.78 | 10.4 | 110400 | 86300 | 5.50 | |

0.68 | 8.7 | 93100 | 72800 | 4.33 | |

0.69 | 8.0 | 85300 | 66600 | 4.49 | B 19 |

0.72 | 7.3 | 77500 | 60500 | 4.70 | |

0.63 | 6.1 | 64600 | 50500 | ....... | |

0.63 | 5.4 | 58100 | 45100 | ....... | B 21 |

0.65 | 4.8 | 51600 | 40300 | ....... | |

0.57 | 3.6 | 38100 | 29800 | ...... | |

0.58 | 3.4 | 36000 | 28100 | ...... | B 23 |

0.58 | 3.2 | 33900 | 26500 | ...... | |

0.59 | 3.0 | 31800 | 2490O | ....... | |

0.S2 | 1.9 | 20700 | 1620 | ....... | |

0.52 | 1.8 | 19100 | 15000 | ....... | B 77 |

0.53 | 1.7 | 17600 | 13800 | ...... |

Section Index | Size Inches | Thickness Inches | Weight per Foot Pounds | Area of Section Square Inches | Moment of Inertia Neutral Axis Parallel to Length | Section Modulus Axis as Before | Radius of Gyration Axis as Before |

I | s | r | |||||

M 10 | 9 1/2x3 3/4 | 1/2 | 16.3 | 4.8 | 3.08 | 1.38 | 0.91 |

M 11 | 9 1/2x3 3/4 | 9/16 | 18.0 | 5.3 | 4.13 | 1.57 | 0.91 |

M 12 | 9 1/2x3 3/4 | 5/8 | 19.7 | 5.8 | 4.57 | 1.77 | 0.90 |

M 13 | 9 1/2x3 3/4 | 11/16 | 21.4 | 6.3 | 5.02 | 1.90 | 0.90 |

M 14 | 9 1/2x3 3/4 | 3/4 | 23.2 | 6.8 | 5.46 | 2.15 | 0.90 |

Section Index | Size Inches | Thickness Inches | Weight per Foot Pounds | Area of Section Square Inch | Moment of Inertia Neutral Axis Parallel to Length | Section Modulus Axis as before | Radius of Gyration Axis as before |

I. | S | r | |||||

M 30 | 8 3/4x1 1/2 | 1/4 | 8.1 | 2.4 | 0.64 | 0.80 | 0.52 |

M 31 | 8 3/4xl 9/16 | 5/16 | 10.1 | 3.0 | 0.95 | 1.13 | 0.57 |

M 32 | 8 3/4xl 5/8 | 3/8 | 12.0 | 3.5 | 1.25 | 1.42 | 0.62 |

M 33 | 12 3/16x2 3/4 | 3/8 | 17.75 | 5.2 | 4.79 | 3.33 | 0,96 |

M 34 | 12 3/16x2 13/16 | 7/16 | 20.71 | 6.1 | 5.81 | 3.90 | 0.98 |

M 35 | 12 3/16x2 7/8 | 1/2 | 23.67 | 7.0 | 6.82 | 4.46 | 0.99 |

Table IV gives values to use for fibre stress, and proportions of full tabular load to use for different ratios of length and width of flange.

Tables V, VI, VII, and VIII give the properties of the minimum and maximum sizes of the different shapes. These tables are for use in choosing sections to meet the requirements of design, and will be explained in detail in the pages that treat of design of members in which these shapes are used.

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