# Elastic Section Properties - Area, Centroid and Second Moment of Area

Units given for each property are examples showing the order of the unit in question.
SectionArea, A (mm²)Centroid in x, Cxx (mm)Centroid in y, Cy (mm)Second Moment of Area in x, Ixx (mm⁴) Second Moment of Area in y, Iyy (mm⁴)Torsional Constant, J (mm⁴) Section Moduli in x, Zx (mm³)Section Moduli in y, Zy (mm³)Comment
$$A = bd$$$$C_{x} = \frac{b}{2}$$$$C_{y} = \frac{d}{2}$$$$I_{xx} = \frac{bd^3}{12}$$$$I_{yy} = \frac{db^3}{12}$$$$J=\frac{d^3}{3}\bigg[b-0.63d$$ $$\qquad\left(1-\frac{d^4}{12b^4}\right)\bigg]$$ $$\text{ for } d < b$$ $$Z_{x,max} = Z_{x,min}$$ $$= \frac{bd^2}{6}$$$$Z_{y,max} = Z_{y,min}$$ $$= \frac{db^2}{6}$$Can be used for a square section, setting $$d=b$$
$$A = BD-bd$$$$C_{x} = \frac{B}{2}$$$$C_{y} = \frac{D}{2}$$$$I_{xx} = \frac{BD^3}{12} - \frac{bd^3}{12}$$$$I_{yy} = \frac{DB^3}{12} - \frac{db^3}{12}$$$$\frac{2t^2(D-t)^2(B-t)^2}{Bt+Dt-2t^2}$$ $$\text{where } t = \frac{B-b}{2}$$ $$= \frac{D-d}{2}$$ $$Z_{x,max} = Z_{x,min}$$ $$= \frac{1}{6D}\left(BD^3 - bd^3\right)$$$$Z_{y,max} = Z_{y,min}$$ $$= \frac{1}{6B}\left(DB^3 - db^3\right)$$Can be used for a square section, setting $$d=b$$
$$A = \frac{\pi d^2}{4}$$$$C_{x} = \frac{d}{2}$$$$C_{y} = \frac{d}{2}$$$$I_{xx} = \frac{\pi d^4}{64}$$$$I_{yy} = \frac{\pi d^4}{64}$$$$J = \frac{\pi d^4}{32}$$ $$Z_{x,max} = Z_{x,min}$$ $$= \frac{\pi d^3}{32}$$$$Z_{y,max} = Z_{y,min}$$ $$= \frac{\pi d^3}{32}$$-
$$A = \frac{\pi D^2}{4} -\frac{\pi d^2}{4}$$$$C_{x} = \frac{D}{2}$$$$C_{y} = \frac{D}{2}$$$$I_{xx} = \frac{\pi D^4}{64} - \frac{\pi d^4}{64}$$$$I_{yy} = \frac{\pi D^4}{64} - \frac{\pi d^4}{64}$$$$J = \frac{\pi}{4}(D-t)^3t$$ $$\text{where } t = \frac{D-d}{2}$$ $$Z_{x,max} = Z_{x,min}$$ $$= \frac{1}{32D}\left(D^3 - d^3\right)$$$$Z_{y,max} = Z_{y,min}$$ $$= \frac{1}{32D}\left(D^3 - d^3\right)$$-
$$A = \frac{bd}{2}$$$$C_{x} = \frac{b}{2}$$$$C_{y} = \frac{d}{3}$$$$I_{xx} = \frac{bd^3}{36}$$$$I_{yy} = \frac{db^3}{48}$$$$J = \frac{b^3 d^3}{(15b^2 + 20d^2)}$$ $$\text{for } \frac{2}{3} < \frac{b}{d} < \sqrt{3}$$ $$Z_{x,max} = \frac{bd^2}{12}$$ $$Z_{x,min} = \frac{bd^2}{24}$$$$Z_{y,max} = Z_{y,min}$$ $$= \frac{db^2}{24}$$-
$$A = \frac{bd}{2}$$$$C_{x} = \frac{b}{3}$$$$C_{y} = \frac{d}{3}$$$$I_{xx} = \frac{bd^3}{36}$$$$I_{yy} = \frac{db^3}{36}$$- $$Z_{x,max} = \frac{bd^2}{12}$$ $$Z_{x,min} = \frac{bd^2}{24}$$$$Z_{y,max} = \frac{db^2}{12}$$ $$Z_{y,min} = \frac{db^2}{24}$$-
$$A=2(bt_{f})+$$ $$(d-2t_{f})t_{w}$$$$C_{x} = \frac{b}{2}$$$$C_{y} = \frac{d}{2}$$$$I_{xx} = \frac{1}{12}(bd^3-$$ $$[b-t_{w}][d-2t_{f}]^3)$$$$I_{yy} = \frac{1}{12}(b^3d-$$ $$[b-t_{w}]^3[d-2t_{f}])$$$$\frac{2t_{f}^3b + t_{w}^3d}{3}$$ $$Z_{x,max} = Z_{x,min}$$ $$= \frac{1}{6d}(bd^3-$$ $$[b-t_{w}][d-2t_{f}]^3)$$$$Z_{y,max} = Z_{y,min}$$ $$= \frac{1}{6b}(b^3d-$$ $$[b-t_{w}]^3[d-2t_{f}])$$-
$$A=2(bt_{f})+$$ $$(d-2t_{f})t_{w}$$$$C_{x} = \frac{1}{A}\bigg(bt_{f}^2+$$ $$\qquad\frac{{t_f}^2}{2}\bigg[d-2t_{f}\bigg]\bigg)$$$$C_{y} = \frac{d}{2}$$$$I_{xx} = \frac{1}{12}(bd^3-$$ $$[b-t_{w}][d-2t_{f}]^3)$$$$\frac{1}{3}(t_{f}b^3+2dt_{w}^3$$ $$-2t_{w}^3t_{f})-AC_{y}^2$$$$\frac{2t_{f}^3b + t_{w}^3d}{3}$$ $$Z_{x,max} = Z_{x,min}$$ $$= \frac{1}{6d}(bd^3-$$ $$[b-t_{w}][d-2t_{f}]^3)$$$$Z_{y,max} = \frac{I_{yy}}{C_{x}}$$ $$Z_{y,min} = \frac{I_{yy}}{b-C_{x}}$$ -
$$A=(bt_{f})+$$ $$(d-t_{f})t_{w}$$$$C_{x} = \frac{b}{2}$$$$C_{y} = \frac{t(bt+d^2-t^2)}{2A}$$ $$\text{when } t_{f}=t_{w}=t$$$$\frac{t}{3}(bt^2+d^3$$ $$-t^3)-AC_{y}^2$$ $$\text{when } t_{f}=t_{w}=t$$$$\frac{1}{12}(b^3t+t^3(d-t))$$ $$\text{when } t_{f}=t_{w}=t$$$$\frac{t_{f}^3b + t_{w}^3d}{3}$$ $$Z_{x,max} = \frac{I_{xx}}{C_{y}}$$ $$Z_{x,min} = \frac{I_{xx}}{d-C_{y}}$$$$Z_{y,max} = \frac{I_{yy}}{C_{x}}$$ $$Z_{y,min} = \frac{I_{yy}}{b-C_{x}}$$Assumes constant thickness, i.e. $$t_{f} = t_{w}$$
$$A=t[b+(d-t)]$$$$C_{x} = \frac{t(dt+b^2-t^2)}{2A}$$$$C_{y} = \frac{t(bt+d^2-t^2)}{2A}$$$$\frac{t}{3}(bt^2+d^3$$ $$-t^3)-AC_{y}^2$$$$\frac{t}{3}(dt^2+b^3-t^3)-AC_{x}^2$$$$\frac{t_{f}^3b + t_{w}^3d}{3}$$ $$Z_{x,max} = \frac{I_{xx}}{C_{y}}$$ $$Z_{x,min} = \frac{I_{xx}}{d-C_{y}}$$$$Z_{y,max} = \frac{I_{yy}}{C_{x}}$$ $$Z_{y,min} = \frac{I_{yy}}{b-C_{x}}$$Assumes constant thickness. Also has u-u and v-v principal axes.