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. 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

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 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.

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 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

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 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

......

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 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

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.