Area Section Properties By X, Y Coordinates

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The properties of a polygon can be calculated using the equations below. The shape can be solid or have one or more “holes.” The coordinates for the solid section are input clockwise and holes or voids are entered anti-clockwise.

Input

Solid Section

Solid Section

Section With Holes

Section With Holes

Equations

A=-\sum_{i=0}^{n-1}{\frac{(y_{i+1}-y_i)(x_{i+1}+x_i)}{2} }

\bar{x}=\frac{-1}{A}\sum_{i=0}^{n-1} ( \frac{y_{i+1}-y_i}{8})[ (x_{i+1}+x_i)^2+\frac{(x_{i+1}-x_i)^2}{3}]

\bar{y}=\frac{1}{A}\sum_{i=0}^{n-1} ( \frac{x_{i+1}-x_i}{8})[ (y_{i+1}+y_i)^2+\frac{(y_{i+1}-y_i)^2}{3}]

I_x=\sum_{i=0}^{n-1} [\frac{{(x_{i+1}-x_i)}{(y_{i+1}+y_i)}}{24}][{(y_{i+1}+y_i)^2}+{(y_{i+1}-y_i)^2}]

I_y=-\sum_{i=0}^{n-1} [\frac{{(y_{i+1}-y_i)}{(x_{i+1}+x_i)}}{24}][{(x_{i+1}+x_i)^2}+{(x_{i+1}-x_i)^2}]

I_{\bar{x}}=I_x-A\bar{y}^2

I_{\bar{y}}=I_y-A\bar{x}^2

Definitions

A – is the area of the shape

\bar{x} – is the x coordinate of the centroid

\bar{y} – is the y coordinate of the centroid

I_x – is the moment of inertia about the x axis

I_y – is the moment of inertia about the y axis

I_{\bar{x}} – is the moment of inertia about the x axis translated to the centroid

I_{\bar{y}} – is the moment of inertia about the y axis translated to the centroid

Web Application

Bridge Automation’s Section Property Calculator uses the above equations and can be found at this link.

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