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.

## Table III. Coefficients For Deflection In Inches For Cambria Shapes Used As Beams Subjected To Safe Loads Uniformly Distributed

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

## Table IV. Reduction In Values Of Allowable Fibre Stress And Safe Loads For Shapes Used As Beams Due To Lateral Flexure

 Ratio of Span or Distance betweenLateralSupports toFlange Width. Allowable Unit Stress for DirectFlexure in Extreme Fibre. Tabular SafeLoad to beUsed. Ratio of Span or Distance betweenLateralSupports toFlange Width. Allowable Unit Stress for DirectFlexure in Extreme Fibre. Proportion of Tabular Safe Load to beUsed 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.

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.

## Table V. Properties Of I-Beams

 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 InertiaNeutral AxisPerpendicular to Web atCenter Mom. of Inertia Neutral AxisCoincident withCenter 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

## Table V. - (Continued.) Properties Of I-Beams

 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 ofStrength forFibor Stress of12,500 lbs. per6q.in. Used for Bridges Distance Center to CenterRequired to makeRadii ofGyration 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 ......

## Table V. - (Continued.) Properties Of Carnegie Trough Plates

 Section Index Size Inches Thickness Inches Weight per Foot Pounds Area of Section Square Inches Moment of InertiaNeutral 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

## Table V. - (Concluded.) Properties Of Carnegie Corrugated Plates

 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.