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An introduction to the concept of area moment of inertia, including calculations for pressure, bending stress, and torsion. It covers definitions of rectangular and polar moments of inertia, as well as the polar moment of inertia and radius of gyration. The document also explains the transfer of axes and the parallel axis theorem.
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Calculation the moment of distributed forces.
Examples of distributed forces: Pressure, stress
Pressure dMAB = py dA = ky^2 dA
ᠹ 㐄 ᡣ 㔅 ᡷ䙦ᡷᡖᠧ䙧
Bending Stress
= ky^2 dA
ᠹ 㐄 ᡣ 㔅 ᡷ⡰ᡖᠧ
Torsion
The total moment involves an integral of the form:
This integral is called moment of inertia of an area
or more fitting: The second moment of area, since the first moment ydA is multiplied by the moment arm y to obtain the second moment for the element dA.
(Centroid; First moment of area)
The moment of inertia of an area is a purely mathematical property of the area and in itself has no physical significance.
The polar moment of inertia
The polar moment of inertia of dA about z-axis:
ᡖᠵこ 㐄 ᡰ⡰ᡖᠧ
The polar moment of inertia of the entire area about the z-axis:
ᠵこ 㐄 㔅 ᡰ⡰^ ᡖᠧ
Because of ᡰ⡰^ 㐄 ᡶ⡰^ ㎗ ᡷ⡰ We get:
ᠵこ 㐄 ᠵけ ㎗ ᠵげ
Other symbols: J, IP, Ir
The second moment of area is always a positive quantity. (x^2 , y^2 , r^2 , square of a distance)
Radius of Gyration:
The radius of gyration k is a measure of the distribution of the area from the axis of rotation. It is defined as:
ᡣ 㐄 㒓ᠵ/ᠧ Furthermore:
ᡣけ 㐄 㒓ᠵけ/ᠧ ᡣげ 㐄 㒓ᠵげ/ᠧ ᡣこ 㐄 㒓ᠵこ/ᠧ
For the moments of inertia we get then:
