Machinery's Handbook, 31st Edition
240 Moment of Inertia 32.16 ft/sec 2 , and J is in ft 4 . If dimensions are in the inch-pound-second system, ρ is in lbs/in 3 , L is in inches, g is 386 in/sec 2 , and J is in inches 4 . Using metric SI units, the above formula becomes J M = ρ LJ , where ρ = den sity in kilograms/meter 3 , L = length in meters, and J = polar moment of inertia in meters 4 . The units of J M are kg·m 2 . Moment of Inertia of Built-Up Sections.— The usual method of calculating the moment of inertia of a built-up section involves the calculations of the moment of inertia for each element of the section about its own neutral axis and the transferring of this moment of inertia to the previously found neutral axis of the whole built-up section. A much simpler method, called the tabular method, can be used in the case of any section that can be divided into rectangular elements bounded by lines parallel and perpendicular to the neu- tral axis. It is based upon the formula: I = b ( h 1 3 - h 3 )/3 in which I = the moment of inertia about axis DE , Fig. 1, and b , h and h 1 are dimensions as given in the same illustration.
0.219
1.5
C
x
x
0.531
h 1
0.625
0.49
B A
b h
E
D
0.125
D
E
1.5
Fig. 1. Fig. 3. The method may be illustrated by applying it to the section shown in Fig. 2, and for simplicity of calculation shown “massed” in Fig. 3. The calculation may then be tabulated as shown in the accompanying table. The distance from the axis DE to the neutral axis xx (which will be designated as d ) is found by dividing the sum of the geometrical moments by the area. The moment of inertia about the neutral axis is then found in the usual way by subtracting the area multiplied by d 2 from the moment of inertia about the axis DE . Tabulated Calculation of Moment of Inertia Fig. 2.
I about axis DE b h h 3 1 3 3 − ^ h
Moment b h h 2
1 2 2 − ^ h
Area b ( h 1 - h )
Breadth b
Height h 1
Section
h 1 2
h 1 3
1.500 0.125 0.531 0.625 0.219 1.500
0.187 0.266 0.191
0.016 0.391 2.250
0.012 0.100 0.203
0.002 0.244 3.375
0.001 0.043 0.228
A B C
Σ A = 0.644 Σ I DE = 0.272 The distance d from DE , the axis at the base of the configuration, to the neutral axis xx is: Σ M = 0.315 . . 0644 0315 049 = = = The moment of inertia of the entire section with reference to the neutral axis xx is: . d A M
2
I I
. . Ad 0272 0644 049 0117 . . # −
= − =
N DE
2
=
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