Centroids and moments

Where does a plate balance? Where does the resultant of a distributed load act? Where is the neutral axis of a beam, and how stiff is the beam in bending? These questions all come down to moments: integrals that weight each small piece of an object by its distance from an axis.

This lesson develops first moments, which locate centroids and centres of mass, and second moments of area, which measure how a cross-section resists bending. Both are among the most-used integrals in structural and mechanical engineering, and the methods you learn here are exactly the ones in your mechanics of solids and design courses.

Moments and balance

Picture a light rod with masses \(m_1, m_2, \ldots\) at positions \(x_1, x_2, \ldots\). Each mass has a moment \(m_i x_i\) about the origin: its tendency to turn the rod. The rod balances at the point \(\bar{x}\) where the total moment equals the total mass placed there:

This is a weighted average of the positions. A heavy mass pulls the balance point toward itself.

For a continuous object, replace the sum with an integral. If a rod lying along \([a, b]\) has density \(\rho(x)\) (mass per unit length), a short piece of length \(dx\) has mass \(\rho(x)\,dx\) and moment \(x\,\rho(x)\,dx\):

Centroids of plane regions

For a flat plate of uniform thickness and material, the centre of mass depends only on the shape. This geometric balance point is the centroid. Engineers need centroids constantly, because:

• a uniformly distributed load acts as a single force through the centroid of the load diagram;
• the neutral axis of a beam in bending passes through the centroid of its cross-section;
• the resultant of water pressure on a surface is found from moments of the pressure distribution.

For the region under \(y = f(x)\) from \(a\) to \(b\), a vertical strip has area \(f(x)\,dx\), its own centroid at \(\big(x, \tfrac12 f(x)\big)\), and therefore moments \(x f(x)\,dx\) about the y-axis and \(\tfrac12 f(x)\cdot f(x)\,dx\) about the x-axis:

The integrals \(\int x\,dA\) and \(\int y\,dA\) are called the first moments of area, written \(Q_y\) and \(Q_x\). The first moment \(Q\) reappears in beam theory, where it determines the shear stress at each level of a cross-section.

Standard results

A few centroids are used so often that it is worth remembering them:

• Rectangle: at the centre.
• Triangle: one-third of the height up from the base (two-thirds down from the apex).
• Semicircle: \(4r/(3\pi)\) from the flat edge.
• Quarter circle: \(4r/(3\pi)\) from each straight edge.
• Region under \(y = kx^n\) from 0 to \(b\): \(\bar{x} = \dfrac{n+1}{n+2}b\).

Composite shapes

Real cross-sections, such as T-beams, angles, channels and built-up girders, are made of simple pieces. Their centroid is the area-weighted average of the pieces' centroids:

A hole counts as a negative area. In practice, engineers set this up as a table: one row per piece, with columns for area, centroid position and area × position.

The theorem of Pappus

Centroids give a shortcut for volumes of revolution. When a region of area \(A\) is rotated about an axis it does not cross, the volume swept out is the area times the distance travelled by its centroid:

Second moment of area

The first moment, \(\int y\,dA\), finds where an area balances. The second moment of area, often called the area moment of inertia,

measures how far the area is spread from an axis. Squaring the distance means area far from the axis counts heavily. This is what makes a beam stiff in bending. The bending stress in a beam is \(\sigma = My/I\) and its deflection is inversely proportional to \(EI\), so a larger \(I\) means lower stresses and less sag.

The parallel axis theorem

To find \(I\) for a composite section about its centroid, you need each piece's \(I\) about that same axis. The parallel axis theorem shifts a second moment from a piece's own centroid to any parallel axis a distance \(d\) away:

This is the reason I-beams look the way they do. Most of the material sits in the flanges, far from the neutral axis, where it contributes the most to \(I\). The thin web holds the flanges apart and carries shear.

Mass moment of inertia

The second moment of area has a twin for rotating bodies. The mass moment of inertia,

measures how hard a body is to spin up or slow down, just as mass measures resistance to linear acceleration. You met it in Mechanics as \(I = \tfrac12 MR^2\) for a solid disk. Now you can derive it.

For a thin ring, where all the mass is at radius \(R\), the same idea gives \(I = MR^2\), twice as much. This is why flywheels are often built with heavy rims and light spokes: mass far from the axis stores more rotational energy for the same weight.

The parallel axis theorem works for mass moments too: \(I = I_c + Md^2\). Engineers use it for connecting rods, robot arms and any part that rotates about a point other than its centre of mass.

Centre of pressure

The same ideas locate where a distributed force acts. On a vertical rectangular gate holding water of depth \(H\), the pressure grows linearly with depth. Its resultant acts at the centroid of the triangular pressure diagram: two-thirds of the way down, at depth \(\tfrac23H\). In general, the depth of the centre of pressure is \(\bar{h} + \dfrac{I_c}{A\bar{h}}\), which uses both a centroid and a second moment of area. Gates, dams and retaining walls are all designed using this result.