An unequal angle section of uniform thickness 0.5 cm has legs of lengths 6 cm and 4 cm. Estimate the positions of the principal axes, and the principal second moments of area.

Question

Accepted Answer

Firstly, find the position of the centroid of the cross-section :

[latex] \begin{aligned} ext{Total area, } A & = (0.06)(0.005) + (0.035)(0.005) \& = 0.475 imes 10^{-3} m^2 \ ext{Now,} \ A \bar{x} & =(0.055)(0.005)(0.0025)+(0.04)(0.005)(0.02) \ & =4.69 imes 10^{-6} m ^3 \ ext{Then} \ \bar{x} & =\frac{4.69 imes 10^{-6}}{0.475 imes 10^{-3}}=9.86 imes 10^{-3} m \ ext{Again} \ A \bar{y} & =(0.035)(0.005)(0.0025)+(0.06)(0.005)(0.03)=9.44 imes 10^{-6} m ^3 \ ext{Then} \ \bar{y} & =\frac{9.44 imes 10^{-6}}{0.475 imes 10^{-3}}=19.85 imes 10^{-3} m \ ext{Now,} \ I_x & =\frac{1}{3}(0.005)(0.06)^3+\frac{1}{3}(0.035)(0.005)^3-\left(0.475 imes 10^{-3} ight)(0.01985)^2 \ & =0.154 imes 10^{-6} m ^4 \ ext{and} \ I_y & =\frac{1}{3}(0.005)(0.04)^3+\frac{1}{3}(0.055)(0.005)^3-\left(0.475 imes 10^{-3} ight)(0.00986)^2 \ & =0.063 imes 10^{-6} m ^4 \end{aligned} [/latex]

With the axes Cx, Cy having the positive directions shown,

[latex] \begin{array}{l} I_{x y}=\iint_{-\bar{x}} x y d x d y \ =\int_{-\bar{x}}^{0.04-\bar{x}} x d x \int_{-\bar{y}}^{0.005-\bar{y}} y d y+\int_{-\bar{x}}^{0.005-\bar{x}} x d x \int_{-\bar{y}}^{0.06-\bar{y}} y d y \ =\frac{1}{4}\left\{\left[(0.04-\bar{x})^2-(-\bar{x})^2 ight]\left[(0.005-\bar{y})^2-(-\bar{y})^2 ight] ight. \ \left.\quad+\left[(0.005-\bar{x})-(-\bar{x})^2 ight]\left[(0.06-\bar{y})^2-(-\bar{y})^2 ight] ight\} \ =-0.0568 imes 10^{-6} m ^4 \end{array} [/latex]

From equation (9.29),

[latex] an 2 heta = \frac{2I_{xy}}{I_y - I_x} (9.29) \\ an 2 heta = \frac{2 imes 0.0568}{0.063 - 0.154} = 1.25[/latex]

Then

2θ = 51.3°

and

θ = 25.7°

From equations (9.36) the principal second moments of area are

[latex]I = \frac{1}{2}(I_x + I_y) \pm \sqrt{\frac{1}{4}(I_x - I_y)^2 + I_{xy}^2} (9.36) \\ \begin{aligned} \frac{1}{2}\left(I_x+I_y ight) \pm\left[\left\{\frac{1}{2}\left(I_x-I_y ight) ight\}^2+I_{x y}^2 ight]^{\frac{1}{2}} & =(0.1085 \pm 0.0728) 10^{-6} \ & =0.1813 ext { or } 0.0357 imes 10^{-6} m ^4 \end{aligned} [/latex]

Question 9.P.9: An unequal angle section of uniform thickness 0.5 cm has leg......

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