Machinery's Handbook, 31st Edition
Curved Beams 275 Table 4. Values of Stress Correction Factor K for Various Curved Beam Sections
Factor K
Factor K
Inside Fiber
Outside Fiber
Inside Fiber
Outside Fiber
y 0 a
y 0 a
Section
Section
R ∕ c
R ∕ c
1.2 3.41 .54 .224 R 1.4 2.40 .60 .151 R 1.6 1.96 .65 .108 R 1.8 1.75 .68 .084 R 2.0 1.62 .71 .069 R 3.0 1.33 .79 .030 R 4.0 1.23 .84 .016 R 6.0 1.14 .89 .0070 R 8.0 1.10 .91 .0039 R 10.0 1.08 .93 .0025 R 1.2 2.89 .57 .305 R 1.4 2.13 .63 .204 R 1.6 1.79 .67 .149 R 1.8 1.63 .70 .112 R 2.0 1.52 .73 .090 R 3.0 1.30 .81 .041 R 4.0 1.20 .85 .021 R 6.0 1.12 .90 .0093 R 8.0 1.09 .92 .0052 R 10.0 1.07 .94 .0033 R 1.2 3.01 .54 .336 R 1.4 2.18 .60 .229 R 1.6 1.87 .65 .168 R 1.8 1.69 .68 .128 R 2.0 1.58 .71 .102 R 3.0 1.33 .80 .046 R 4.0 1.23 .84 .024 R 6.0 1.13 .88 .011 R 8.0 1.10 .91 .0060 R 10.0 1.08 .93 .0039 R 1.2 3.09 .56 .336 R 1.4 2.25 .62 .229 R 1.6 1.91 .66 .168 R 1.8 1.73 .70 .128 R 2.0 1.61 .73 .102 R 3.0 1.37 .81 .046 R 4.0 1.26 .86 .024 R 6.0 1.17 .91 .011 R 8.0 1.13 .94 .0060 R 10.0 1.11 .95 .0039 R 1.2 3.14 .52 .352 R 1.4 2.29 .54 .243 R 1.6 1.93 .62 .179 R 1.8 1.74 .65 .138 R 2.0 1.61 .68 .110 R 3.0 1.34 .76 .050 R 4.0 1.24 .82 .028 R 6.0 1.15 .87 .012 R 8.0 1.12 .91 .0060 R 10.0 1.10 .93 .0039 R 1.2 3.26 .44 .361 R 1.4 2.39 .50 .251 R 1.6 1.99 .54 .186 R 1.8 1.78 .57 .144 R 2.0 1.66 .60 .116 R 3.0 1.37 .70 .052 R 4.0 1.27 .75 .029 R 6.0 1.16 .82 .013 R 8.0 1.12 .86 .0060 R 10.0 1.09 .88 .0039 R
1.2 3.63 .58 .418 R 1.4 2.54 .63 .299 R 1.6 2.14 .67 .229 R 1.8 1.89 .70 .183 R 2.0 1.73 .72 .149 R 3.0 1.41 .79 .069 R 4.0 1.29 .83 .040 R 6.0 1.18 .88 .018 R 8.0 1.13 .91 .010 R 10.0 1.10 .92 .0065 R 1.2 3.55 .67 .409 R 1.4 2.48 .72 .292 R 1.6 2.07 .76 .224 R 1.8 1.83 .78 .178 R 2.0 1.69 .80 .144 R 3.0 1.38 .86 .067 R 4.0 1.26 .89 .038 R 6.0 1.15 .92 .018 R 8.0 1.10 .94 .010 R 10.0 1.08 .95 .0065 R 1.2 2.52 .67 .408 R 1.4 1.90 .71 .285 R 1.6 1.63 .75 .208 R 1.8 1.50 .77 .160 R 2.0 1.41 .79 .127 R 3.0 1.23 .86 .058 R 4.0 1.16 .89 .030 R 6.0 1.10 .92 .013 R 8.0 1.07 .94 .0076 R 10.0 1.05 .95 .0048 R 1.2 3.28 .58 .269 R 1.4 2.31 .64 .182 R 1.6 1.89 .68 .134 R 1.8 1.70 .71 .104 R 2.0 1.57 .73 .083 R 3.0 1.31 .81 .038 R 4.0 1.21 .85 .020 R 6.0 1.13 .90 .0087 R 8.0 1.10 .92 .0049 R 10.0 1.07 .93 .0031 R 1.2 2.63 .68 .399 R 1.4 1.97 .73 .280 R 1.6 1.66 .76 .205 R 1.8 1.51 .78 .159 R 2.0 1.43 .80 .127 R 3.0 1.23 .86 .058 R 4.0 1.15 .89 .031 R 6.0 1.09 .92 .014 R 8.0 1.07 .94 .0076 R 10.0 1.06 .95 .0048 R
R
t
3 2
4 t 1 2
4 t
t
c
h
c
R
3 t 2 t
t
t
c
6 t
4 t
R
c
R
b
c
t 4 t t
3 t
3 t
2 b
b
c
t
R
R
2 d
3 b
c
2 b
b
d
R
c
R
5 b
t 2
4 t
c
4 t
b
4 b
2 t
t
t
R
t 2
c
R
3 5
Example: The fiber stresses of a curved rectangular beam are calculated as 5000 psi using the straight beam formula, S = Mc / I . If the beam is 8 inches deep and its radius of curvature is 12 inches, what are the true stresses? Solution: R / c = 12 ∕ 4 = 3. The factors in the table corresponding to R / c = 3 are 0.81 and 1.30. Outside fiber stress = 5000 3 0.81 = 4050 psi; inside fiber stress = 5000 3 1.30 = 6500 psi.
b
c
b
R
a y 0 is the distance from the centroidal axis to the neutral axis of curved beams subjected to pure bending and is measured from the centroidal axis toward the center of curvature.
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