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Appendix C Properties of Shapes
Table C.0.1. Centroids of Common Shapes
|
\(A = b h\) |
\(b/2\) |
\(h/2\) |
|
\(\dfrac{bh}{2}\) |
\(b/3\) |
\(h/3\) |
|
\(\dfrac{(a+b) h}{2}\) |
\(\dfrac{a^2 +ab + b^2}{3 (a+b)}\) |
\(\dfrac{h(2a+b)}{3(a+b)}\) |
|
\(\pi r^2\) |
\(r\) |
\(r \) |
|
\(\dfrac{\pi r^2}{2}\) |
\(r\) |
|
|
\(\dfrac{\pi r^2}{4}\) |
\(\dfrac{4 r}{3 \pi}\) |
\(\dfrac{4 r}{3 \pi}\) |
Table C.0.2. Moments of Inertia of Common Shapes
|
\((b/2, h/2)\) |
\(\bar{I}_{x'} = \dfrac{1}{12} b h^3\)
\(\bar{I}_{y'} = \dfrac{1}{12} b^3 h\)
|
\(I_{x} = \dfrac{1}{3} b h^3\) \(I_{y} = \dfrac{1}{3} b^3 h\)
|
|
\((b/3, h/3)\) |
\(\bar{I}_{x'} = \dfrac{1}{36} b h^3\) \(\bar{I}_{y'} = \dfrac{1}{36} b^3 h\)
|
\(I_x = \dfrac{1}{12} b h^3\) \(I_y = \dfrac{1}{12} b^3 h\)
|
|
\((r,r)\) |
\(\bar{I}_{x'}=\bar{I}_{y'}= \dfrac{\pi}{4} r^4\) |
\(I_{x}=I_{y}= \dfrac{5 \pi}{4} r^4\) |
|
\(\left (r, \dfrac{4r}{3\pi} \right)\) |
\(\bar{I}_{x'} = \left(\dfrac{\pi}{8} - \dfrac{8}{9\pi}\right) r^4\)
\(\bar{I}_{x'} \approx 0.1098\ r^4\)
\(\bar{I}_{y'} = \dfrac{\pi}{8} r^4 \)
|
\(I_x =\dfrac{\pi}{8} r^4\)
\(I_y =\dfrac{5 \pi}{8} r^4\)
|
|
\(\left (\dfrac{4r}{3\pi}, \dfrac{4r}{3\pi} \right)\) |
\(\bar{I}_{x'} = \dfrac{1}{2}\left(\dfrac{\pi}{8} - \dfrac{8}{9\pi}\right) r^4 \)
\(\bar{I}_{x'} \approx 0.0549\ r^4\)
\(\bar{I}_{y'} = \dfrac{\pi}{8} r^4 \)
|
\(I_x = I_y = \dfrac{\pi }{16}r^4\) |
