Learning topic

Centroid and First Moments of Area

First moments of area Sx and Sy, centroid coordinates for simple and composite plane sections, symmetry rules, and treatment of holes.

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This topic explains first moments of area and the determination of centroid coordinates for plane sections. It covers simple and composite shapes, algebraic summation of areas, treatment of holes as negative areas, and symmetry as a useful geometric check.

The centroid of an area is the geometric point through which the centroidal axes of a plane section pass. Its location is determined using first moments of area.

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.

Property of centroidal axes

The first moment of the complete area about an axis passing through its centroid is zero. If a shape has one axis of symmetry, its centroid lies on that axis; with two symmetry axes, the centroid lies at their intersection.

Composite section

Divide a composite shape into simple parts. Determine each area $A_i$ and the coordinates of its own centroid, then use the algebraic sums $\sum A_i x_i$ and $\sum A_i y_i$. Holes are treated as negative areas.

Example

Suppose two rectangles have areas $A_1=2000\ \text{mm}^2$ and $A_2=1000\ \text{mm}^2$, with centroid $y$-coordinates of $20$ and $80\ \text{mm}$. Then:

$$y_c=\frac{2000\cdot20+1000\cdot80}{3000}=40\ \text{mm}.$$

Common errors

Do not average component centroid coordinates without weighting them by area. All coordinates must be measured from the same reference axis, and the sign of a hole area must be handled consistently.