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Geometric Section Properties

The section properties are first calculated with respect to the section definition coordinate axes. From these the centroid of the section is calculated and then the properties with respect to the centroid.

For a perimeter section the section properties are calculated with respect to the coordinate axes using the following formulae.

A=12∑(xy′−x′y)Ax=16∑(xy′−x′y)(y−y′)Ay=16∑(xy′−x′y)(x−x′)Iox=112∑(xy′−x′y)(y2−yy′+y′2)Ioy=112∑(xy′−x′y)(x2−xx′+x′2)Ioxy=124∑(xy′−x′y)(2xy+x′y+xy′+2x′y′)\begin{aligned} A &= \frac{1}{2} \sum (xy'-x'y)\\ A_x &= \frac{1}{6} \sum (xy'-x'y)(y-y')\\ A_y &= \frac{1}{6} \sum (xy'-x'y)(x-x')\\ I_{ox} &= \frac{1}{12} \sum (xy'-x'y)(y^2-yy' +y'^2)\\ I_{oy} &= \frac{1}{12} \sum (xy'-x'y)(x^2-xx' +x'^2)\\ I_{oxy} &= \frac{1}{24} \sum (xy'-x'y)(2xy + x'y + xy' + 2x'y') \end{aligned}

For a line segment section the section properties relative to the coordinate axes are calculated using the following formulae.

A=12∑(lt)Ax=12∑lt(x+x′)Ay=12∑lt(y+y′)Iox=∑lt[l212−l2−t212l2(x−x′)2+(y+y′)2)4]Ioy=∑lt[l212−l2−t212l2(y−y′)2+(x+x′)2)4]Ioxy=∑lt[l2−t212l2(x−x′)(y−y′)+(x+x′)(y+y′)4]J={4a2∑ltwhen it is a single closed section (a - area enclosed by the line segments)∑lt33when it is not a single closed section\begin{aligned} A &= \frac{1}{2}\sum(lt)\\ A_x &= \frac{1}{2}\sum lt(x+x') \\ A_y &= \frac{1}{2}\sum lt(y+y')\\ I_{ox} &= \sum lt \left[ \frac{l^2}{12} - \frac{l^2-t^2}{12l^2}(x-x')^2 + \frac{(y+y')^2)}{4}\right]\\ I_{oy} &= \sum lt \left[ \frac{l^2}{12} - \frac{l^2-t^2}{12l^2}(y-y')^2 + \frac{(x+x')^2)}{4}\right]\\ I_{oxy} &= \sum lt \left[ \frac{l^2-t^2}{12l^2}(x-x')(y-y') + \frac{(x+x')(y+y')}{4}\right]\\ J &= \begin{cases} \frac{4a^2}{\sum\frac{l}{t}} & \text{when it is a single closed section (a - area enclosed by the line segments)}\\ \frac{\sum lt^3}{3} & \text{when it is not a single closed section} \end{cases} \end{aligned}

Then, for both perimeter and line segment sections, the centroid of the section and the section properties with respect to the centroid are calculated as follows.

xc=AyAyc=AxAIxx=Iox−Ayc2Iyy=Ioy−Axc2Ixy=Ioxy−Axcyc\begin{aligned} x_c &= \frac{A_y}{A}\\ y_c &= \frac{A_x}{A}\\ I_{xx} &= I_{ox} - Ay_c^2\\ I_{yy} &= I_{oy} - Ax_c^2\\ I_{xy} &= I_{oxy} - Ax_cy_c \end{aligned}

where (x,y)(x, y) are the coordinates of a point and (xc,yc)(x_c, y_c) are the coordinates of the previous point. The summation is carried out for each point round the section if it is a perimeter section or for each segment if it is a line segment section.

Note that the torsion constant, JJ of Perimeter sections, and shear factors, KyK_y and KzK_z of both perimeter and line segment sections are not calculated and are set to zero.