The table below gives properties of common cross sections. More extensive tables can be found in the listed references.

The properties calculated in the table include area, centroidal moment of inertia, section modulus, and radius of gyration. For information on cross section properties, see our on cross section properties reference.

Shape Representation Properties
Rectangle

Area [in2]:

 $$A = bh$$

Moment of Inertia [in4]:

 $$I_{cx} = {b h^3 \over 12}$$ $$I_{cy} = {h b^3 \over 12}$$ $$J_c = {bh \over 12} (b^2 + h^2)$$

Section Modulus [in3]:

 $$S = {b h^2 \over 6}$$

Radius of Gyration [in]:

 $$r_{cx} = {h \over 2 \sqrt{3}}$$ $$r_{cy} = {b \over 2 \sqrt{3}}$$
Circle

Area [in2]:

 $$\begin{eqnarray} A &=& \pi r^2 \nonumber \\ &=& {\pi d^2 \over 4} \nonumber \end{eqnarray}$$

Moment of Inertia [in4]:

 $$\begin{eqnarray} I_{cx} &=& I_{cy} \nonumber \\ &=& {\pi r^4 \over 4} \nonumber \\ &=& {\pi d^4 \over 64} \nonumber \end{eqnarray}$$ $$\begin{eqnarray} J_c &=& {\pi r^4 \over 2} \nonumber \\ &=& {\pi d^4 \over 32} \nonumber \end{eqnarray}$$

Section Modulus [in3]:

 $$S = {\pi d^3 \over 32}$$

Radius of Gyration [in]:

 $$\begin{eqnarray} r_{cx} &=& r_{cy} \nonumber \\ &=& {r \over 2} \nonumber \\ &=& {d \over 4} \nonumber \end{eqnarray}$$
Circular Tube

Area [in2]:

 $$\begin{eqnarray} A &=& \pi (r_o^2 - r_i^2) \nonumber \\ &=& {\pi \over 4} (d_o^2 - d_i^2) \nonumber \end{eqnarray}$$

Moment of Inertia [in4]:

 $$\begin{eqnarray} I_{cx} &=& I_{cy} \nonumber \\ &=& {\pi \over 4} (r_o^4 - r_i^4) \nonumber \\ &=& {\pi \over 64} (d_o^4 - d_i^4) \nonumber \end{eqnarray}$$ $$\begin{eqnarray} J_c &=& {\pi \over 2} (r_o^4 - r_i^4) \nonumber \\ &=& {\pi \over 32} (d_o^4 - d_i^4) \nonumber \end{eqnarray}$$

Section Modulus [in3]:

 $$S = {\pi (d_o^4 - d_i^4) \over 32 d_o}$$

Radius of Gyration [in]:

 $$\begin{eqnarray} r_{cx} &=& r_{cy} \nonumber \\ &=& \sqrt{ r_o^2 + r_i^2 \over 4 } \nonumber \\ &=& \sqrt{ d_o^2 + d_i^2 \over 16 } \nonumber \end{eqnarray}$$
I-Beam

Area [in2]:

 $$\begin{eqnarray} A &=& bh - (b - t_w) h_w \nonumber \\ &=& 2 b t_f + t_w h_w \nonumber \end{eqnarray}$$

Moment of Inertia [in4]:

 $$\begin{eqnarray} I_{cx} &=& {bh^3 \over 12} - {(b - t_w) h_w^3 \over 12} \nonumber \\ &=& {1 \over 12} (bh^3 - b h_w^3 + t_w h_w^3) \nonumber \end{eqnarray}$$

Section Modulus [in3]:

 $$S = {1 \over 6} \left[ b h^2 - {h_w^3 \over h} (b - t_w) \right]$$

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