General Section Properties & Stresses

This spreadsheet calculates the section properties of a general shape defined by nodal co-ordinates together with the stress distribution due to a direct axial load and bending moments about the principal axes.

Stresses are calculated at all nodal positions and illustrated by line contours over the outline.

Calculation Reference
Section Properties
Steel Design
Steel Designers Manual

To calculate the section properties of a general shape defined by nodal coordinates along with the stress distribution due to a direct axial load and bending moments about the principal axes, follow these steps:

1. Define the shape using nodal coordinates: Create a list of nodal coordinates (x, y) that define the shape's geometry. Ensure that the coordinates follow a consistent order, either clockwise or counterclockwise.

2. Calculate the area (A) and centroid (C) of the shape: Use the following formulae to calculate the area and centroid:

A = (1/2) * Σ[(xi * yi+1) - (xi+1 * yi)]

Cx = (1/(6*A)) * Σ[(xi + xi+1) * ((xi * yi+1) - (xi+1 * yi))]

Cy = (1/(6*A)) * Σ[(yi + yi+1) * ((xi * yi+1) - (xi+1 * yi))]

where (xi, yi) and (xi+1, yi+1) represent adjacent nodal coordinates.

3. Calculate the moments of inertia (Ix, Iy) and product of inertia (Ixy) of the shape:

Ix = (1/12) * Σ[((yi^2 + yi * yi+1 + yi+1^2) * ((xi * yi+1) - (xi+1 * yi))])

Iy = (1/12) * Σ[((xi^2 + xi * xi+1 + xi+1^2) * ((xi * yi+1) - (xi+1 * yi))])

Ixy = (1/24) * Σ[((xi * yi+1 + 2 * xi * yi + 2 * xi+1 * yi+1 + xi+1 * yi) * ((xi * yi+1) - (xi+1 * yi))])

4. Calculate the principal moments of inertia (I1, I2) and the orientation of the principal axes:

The principal moments of inertia can be calculated using the following equations:

I1 = (Ix + Iy) / 2 + sqrt(((Ix - Iy) / 2)^2 + Ixy^2)

I2 = (Ix + Iy) / 2 - sqrt(((Ix - Iy) / 2)^2 + Ixy^2)

The orientation angle (θ) of the principal axes can be calculated using:

θ = (1/2) * atan(2 * Ixy / (Ix - Iy))

5. Calculate the stress distribution due to the direct axial load (P) and bending moments (Mx, My) about the principal axes:

Stress due to axial load: σ_axial = P / A

Stress due to bending moments: σ_bending = (Mx * y - My * x) / I1

Total stress: σ_total = σ_axial + σ_bending

By following these steps, you can calculate the section properties of a general shape defined by nodal coordinates and analyze the stress distribution due to a direct axial load and bending moments about the principal axes.