Learning topic
Parallel-Axis Theorem and Composite Sections
Transfer second moments to parallel axes and calculate composite-section centroids and moments of inertia, including holes and simple components.
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For a composite section, tabulated properties of individual simple shapes are not sufficient because their second moments must first be referred to a common axis. The parallel-axis theorem provides this transfer.
The second moment of area about an axis parallel to a centroidal axis is found from the parallel-axis theorem:
$$I_x=I_{x_c}+Aa^2,$$
where Ixc is the second moment about the parallel centroidal axis, A is the area, and a is the distance between the axes.
Similarly:
$$I_y=I_{y_c}+Ab^2.$$
For a composite section, the transferred contributions of all parts are summed algebraically; holes are treated with negative area.
Why the centroid comes first
Centroidal second moments of a composite section are calculated about axes through the centroid of the complete area. Therefore, the first step is to determine that centroid.
The first moments of area about the x and y axes are:
$$S_x=\int_A y\,dA,\qquad S_y=\int_A x\,dA.$$
The centroid coordinates are:
$$x_c=\frac{S_y}{A},\qquad y_c=\frac{S_x}{A}.$$
For a composite section divided into simple parts:
$$x_c=\frac{\sum_i A_i x_i}{\sum_i A_i},\qquad y_c=\frac{\sum_i A_i y_i}{\sum_i A_i}.$$
Holes can conveniently be treated as negative areas in algebraic summation.
Procedure
- Divide the composite section into simple shapes with known geometric properties. Treat holes as negative areas.
- Choose a convenient reference coordinate system and determine the areas Ai and centroid coordinates xi, yi of all parts.
- Find the centroid of the complete section from the algebraic sums Ai xi and Ai yi.
- Draw centroidal axes through the calculated centroid.
- Determine the centroidal second moments of each part and transfer them to the common centroidal axes using the parallel-axis theorem.
- Sum the contributions algebraically to obtain Ix, Iy, and, if required, Ixy.
- If principal axes are required, determine their rotation angle and the principal second moments of area.
- For bending calculations, determine the section moduli for the required extreme fibers and verify units and geometric symmetry.
Holes
A hole can be treated as a negative component: its area, first moments, and transferred second moment are subtracted from the corresponding sums for solid parts.
Example structure
For a T-section, represent the flange and web as two rectangles. First determine the centroid coordinate of the complete section. Then calculate the centroidal $I_x$ of each rectangle and add $A_i a_i^2$, where $a_i$ is the distance between the component centroid and the common centroidal axis.
Check
Verify that all distances are measured to the same reference axis, units are consistent, and the $Aa^2$ term has not been added to a second moment that is already taken about the required common axis.