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Area Moment of Inertia


The area moment of inertia is a property of a two-dimensional plane shape which characterizes its deflection under loading. It is also known as the second moment of area or second moment of inertia. The area moment of inertia has dimensions of length to the fourth power. Unfortunately, in engineering contexts, the area moment of inertia is often called simply "the" moment of inertia even though it is not equivalent to the usual moment of inertia (which has dimensions of mass times length squared and characterizes the angular acceleration undergone by a solids when subjected to a torque).

The second moment of area about the about the x-axis is defined by

 I_x=I_(xx)=inty^2dxdy,
(1)

while more generally, the "product" moment of area is defined by

 I_(xy)=intxydxdy.
(2)

Here, the positive sign convention is used (e.g., Pilkey 2002, p. 15).

More generally, the area moment of inertia tensor J_(ij) is given by

J_(ij)=int(r^2delta_(ij)-x_ix_j)dxdy
(3)
J=int[y^2 -xy; -xy x^2]dxdy
(4)

by analogy with the moment of inertia tensor, which has negative signs on the off-diagonal elements and, unlike the moment of inertia tensor, is not expressed in term of mass of the lamina.

For a closed lamina of uniform density with boundary specified by (x(t),y(t)) for t in [t_0,t_1] and the lamina on the left as the curve is traversed, Green's theorem can be used to compute the components of the area moment of inertia tensor as

I_(xx)=-1/3int_(t_0)^(t_1)y^3x^'dt
(5)
I_(xy)=-1/2int_(t_0)^(t_1)y^2xx^'dt
(6)
I_(yy)=1/3int_(t_0)^(t_1)x^3y^'dt.
(7)

The following table summarizes some area moments of inertia for some common shapes.

shapeaxisJ
annuluscentroid[1/4(a^4-b^4)pi 0; 0 1/4(a^4-b^4)pi]
diskcentroid[1/4pia^4 0; 0 1/4pia^4]
ellipsecentroid[1/4piab^3 0; 0 1/4pia^3b]
half-diskalong lower boundary[1/8pia^4 0; 0 1/8pia^4]
hexagondiameter[5/(16)sqrt(3)a^4 0; 0 5/(16)sqrt(3)a^4]
pentagonaxis through center and vertex[1/(96)sqrt(265+118sqrt(5))a^4 0; 0 1/(96)sqrt(265+118sqrt(5))a^4]
quarter-diskCartesian axis[1/(16)pia^4 -1/8a^4; -1/8a^4 1/(16)pia^4]
rectanglecentroid along Cartesian axis[1/(12)ab^3 0; 0 1/(12)a^3b]
squarecentroid along Cartesian axis[1/(12)a^4 0; 0 1/(12)a^4]

The area moments of inertia about axes along an inradius and a circumradius of a regular polygon with n sides (for n>=3) are given by

I_r=1/(24)A_n(6r_n^2-a^2)
(8)
=(a^4)/(192)n[cos((2pi)/n)+2]cos(pi/n)csc^2(pi/n)
(9)
I_R=1/(48)A_n(12R_n^2+a^2)
(10)
=(a^4)/(192)ncot(pi/n)[3cos^2(pi/n)+1]
(11)

(Roark 1954, p. 70).


See also

Geometric Centroid, Moment of Inertia, Radius of Gyration, Torsional Rigidity

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References

Dr. Drang. "Green's Theorem and Section Properties." Jan. 17, 2018. https://leancrew.com/all-this/2018/01/greens-theorem-and-section-properties/.Pilkey, W. D. Analysis and Design of Elastic Beams. New York: Wiley, 2002.Roark, R. J. Formulas for Stress and Strain, 3rd ed. New York: McGraw-Hill, 1954.

Referenced on Wolfram|Alpha

Area Moment of Inertia

Cite this as:

Weisstein, Eric W. "Area Moment of Inertia." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AreaMomentofInertia.html

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