Learning topic
Geometric Properties of Plane Areas
Calculate centroids, first and second moments of area, polar moment, section modulus, principal axes, and properties of composite cross-sections.
Geometric properties of a plane area describe how area is distributed relative to selected axes and points. Unlike mechanical material properties, they depend only on the shape, dimensions, and orientation of the cross-section. These quantities are used in calculations of bending, torsion, stability, and combined loading.
Area and centroid
The area $A$ gives the overall size of a cross-section, while first moments of area are used to determine the position of its centroid. For composite sections, the centroid is obtained as an area-weighted average of the centroids of the simple component shapes.
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.
Second moments of area
The second moments of area $I_x$ and $I_y$ characterize the distribution of area about the corresponding axes. Area elements farther from an axis contribute more strongly because the distance enters quadratically.
The second moments of area about the x and y axes are:
$$I_x=\int_A y^2\,dA,\qquad I_y=\int_A x^2\,dA.$$
The product moment of area is:
$$I_{xy}=\int_A xy\,dA.$$
The polar second moment of area about the intersection of orthogonal axes is:
$$J_p=\int_A(x^2+y^2)dA=I_x+I_y.$$
Second and polar moments of area have dimensions of length to the fourth power, for example mm⁴.
Axis transfer and rotation
When the required axis is parallel to a known centroidal axis, the parallel-axis theorem is used. When the orientation of the axes changes, transformation formulas are applied to determine principal centroidal axes and principal second moments of area.
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.
Section moduli
In bending problems, the section modulus $W=I/y_{\max}$ is used, where $y_{\max}$ is the distance from the neutral axis to the corresponding extreme fiber. For an unsymmetrical cross-section, the section moduli for the upper and lower edges may differ.
Typical cross-section shapes
Simple and composite sections
| Shape | Area A | Centroidal second moment | Section modulus |
|---|---|---|---|
| Rectangle b × h, x-axis parallel to side b | $bh$ | $I_x=bh^3/12$ | $W_x=bh^2/6$ |
| Rectangle b × h, y-axis parallel to side h | $bh$ | $I_y=hb^3/12$ | $W_y=hb^2/6$ |
| Circle of diameter d | $\pi d^2/4$ | $I_x=I_y=\pi d^4/64$ | $W_x=W_y=\pi d^3/32$ |
| Circle of diameter d, polar property | $\pi d^2/4$ | $J_p=\pi d^4/32$ | $W_p=\pi d^3/16$ |
| Annulus D, d | $\pi(D^2-d^2)/4$ | $I_x=I_y=\pi(D^4-d^4)/64$ | $W_x=2I_x/D$ |
Complex profiles are divided into rectangles, triangles, circles, or other simple component areas. After the common centroid is found, the properties of each component are transferred to the common axes and summed.
- 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.
Section structure
The child topics separately cover first moments and centroids, second moments of area, the parallel-axis theorem, axis rotation, principal properties, and section moduli. This provides the geometric basis for the subsequent study of stresses and deformations in bending.
About this topic
Cross-sectional geometric properties dictate a member's resistance to various deformation modes independent of material composition. This section explores first moments of area, centroid coordinates, planar and polar moments of inertia, parallel axis theorem (Steiner's theorem), and methods for locating principal axes of inertia for complex built-up rolled steel shapes (channels, I-beams, angles).