Centroids and Center of Gravity

Centroids and Center of Gravity

$Learning Objectives$

Distinguish geometric centroid, center of mass, and center of gravity.
Locate centroids of standard areas using symmetry and known formulas.
Determine the centroid of composite areas, including cutouts modeled as negative area.
Relate first moments of area to centroid coordinates.
Use centroid concepts to locate the resultant of a distributed load.
Apply the Pappus-Guldinus centroid theorems within their geometric limitations.

$Centroid$

The geometric center associated with a line, area, or volume. Its location depends on geometry rather than material density.

$Center of Mass$

The mass-weighted average position of a body. It depends on the distribution of mass.

$Center of Gravity$

The point through which the resultant gravitational force on a body acts. In a practically uniform gravitational field, it coincides with the center of mass.

$When the Three Centers Coincide$

For a homogeneous body in a uniform gravitational field, the geometric centroid of its volume, center of mass, and center of gravity coincide.

For a nonuniform-density body, the geometric centroid may differ from the center of mass. If the gravitational field varies appreciably over a body, the center of gravity can also differ from the center of mass, although that distinction is normally negligible for ordinary building-scale statics.

Center of Mass Coordinate

Mass-weighted coordinate of a continuous body along one axis.

\bar{x}=\frac{\int x\,dm}{\int dm}

Variables

Symbol	Description	Unit
$\bar{x}$	Center-of-mass x-coordinate	m
$dm$	Differential mass element	kg

Area Centroid Coordinates

Centroid coordinates of a continuous planar area.

\bar{x}=\frac{\int_A x\,dA}{A}, \qquad \bar{y}=\frac{\int_A y\,dA}{A}

Variables

Symbol	Description	Unit
$\bar{x}$	Area-centroid x-coordinate	m
$\bar{y}$	Area-centroid y-coordinate	m
$A$	Total area	$m^2$
$dA$	Differential area element	$m^2$

$Symmetry and Standard Shapes$

Symmetry is the fastest centroid check. If an area has one axis of symmetry, its centroid lies on that axis. If it has two intersecting symmetry axes, their intersection locates the centroid.

Frequently used results include:

rectangle: at its geometric center;
triangle: one-third of the altitude from the base toward the opposite vertex;
semicircular area: on the symmetry axis at $4r/(3\pi)$ from the diameter.

Coordinates must always be referenced to the same datum used in the calculation.

$First Moment of Area$

The area-weighted measure of distance from a reference axis. First moments locate centroids and later appear in beam-shear calculations.

First Moments and Centroid

Relates first moments of area to centroid coordinates.

Q_y=A\bar{x}, \qquad Q_x=A\bar{y}

Variables

Symbol	Description	Unit
$Q_y$	First moment of area about the y-axis	$m^3$
$Q_x$	First moment of area about the x-axis	$m^3$
$A$	Area	$m^2$
$\bar{x}$	Centroid x-coordinate	m
$\bar{y}$	Centroid y-coordinate	m

$Composite Areas$

A composite section is divided into standard subareas with known centroids. The overall centroid is an area-weighted average.

Holes and cutouts are represented as negative areas, with negative first moments. The same reference axes must be used for every component.

Composite-Area Centroid

Computes centroid coordinates from discrete component areas.

\bar{x}=\frac{\sum A_i x_i}{\sum A_i}, \qquad \bar{y}=\frac{\sum A_i y_i}{\sum A_i}

Variables

Symbol	Description	Unit
$A_i$	Signed area of component i; negative for a cutout	$m^2$
$x_i$	Component-centroid x-coordinate	m
$y_i$	Component-centroid y-coordinate	m

$Composite-Area Centroid$

STEP-BY-STEP

Choose common reference axes and dimensions.
Partition the section into nonoverlapping standard shapes.
Assign positive area to material and negative area to cutouts.
Locate each component centroid from the common axes.
Tabulate $A_i$ , $A_ix_i$ , and $A_iy_i$ .
Sum the signed areas and first moments.
Divide the summed first moments by the signed total area.
Check that the resulting centroid location is physically plausible.

$Interactive Exploration$

Change the flange and web dimensions of the T-shaped section and observe how the centroid shifts toward the region containing more area.

Simulation

Controls

160 mm

120 mm

40 mm

valid geometry

Composite area

12160.00 mm²

x centroid

80.000 mm

y centroid

102.105 mm

First-moment contributions

Flange

Aᵢ: 6400.00
xᵢ: 80.00
yᵢ: 140.00
Aᵢxᵢ: 512000.00
Aᵢyᵢ: 896000.00

Web

Aᵢ: 5760.00
xᵢ: 80.00
yᵢ: 60.00
Aᵢxᵢ: 460800.00
Aᵢyᵢ: 345600.00

\bar{x}=\frac{\sum m_i x_i}{\sum m_i},\qquad \bar{y}=\frac{\sum m_i y_i}{\sum m_i}

$Centroid and the Neutral Axis$

For a homogeneous, linearly elastic beam under elementary Euler-Bernoulli bending about a centroidal principal axis, the neutral axis passes through the cross-section centroid. More general unsymmetric, composite, nonlinear, or coupled bending cases require additional transformed-section or constitutive analysis.

The centroid is therefore a necessary geometric input to elementary flexure, but it should not be treated as a universal statement about every possible neutral axis.

$Distributed Loads$

The same centroid principle locates the line of action of a distributed-load resultant. If $w(x)$ is a load intensity, its resultant is the area under the load diagram and acts through the centroid of that load area. This is why triangular and trapezoidal loads are replaced at their load-diagram centroids.

Distributed-Load Resultant and Location

Uses the load-diagram area and first moment to determine the equivalent concentrated load.

$W=\int w(x)\,dx, \qquad \bar{x}=\frac{\int xw(x)\,dx}{W}$

Variables

Symbol	Description	Unit
$W$	Equivalent concentrated load	N
$w(x)$	Distributed-load intensity	N/m
$\bar{x}$	Location of the resultant from the chosen origin	m

$Pappus-Guldinus Theorems$

For an eligible plane curve revolved about a coplanar external axis that does not intersect the curve, the generated surface area equals the curve length times the distance traveled by its centroid.

For an eligible plane area revolved about a coplanar external axis that does not intersect the area, the generated volume equals the area times the distance traveled by its centroid.

Pappus Surface-Area Theorem

Surface area generated by a full revolution of a plane curve about an eligible external axis.

S=2\pi \bar{r}L

Variables

Symbol	Description	Unit
$S$	Generated surface area	$m^2$
$\bar{r}$	Perpendicular distance from the axis to the curve centroid	m
$L$	Length of the generating curve	m

Pappus Volume Theorem

Volume generated by a full revolution of a plane area about an eligible external axis.

V=2\pi \bar{r}A

Variables

Symbol	Description	Unit
$V$	Generated volume	$m^3$
$\bar{r}$	Perpendicular distance from the axis to the area centroid	m
$A$	Generating area	$m^2$

$Pappus Axis Restriction$

Do not apply the standard Pappus-Guldinus forms when the axis intersects the generating curve or area in a way that violates the theorem assumptions. Use direct integration or another valid geometric method instead.

Key Takeaways

A centroid is geometric, while centers of mass and gravity depend on physical distributions.
Symmetry provides an immediate constraint on centroid location.
First moments of area are the numerators of the centroid-coordinate equations.
Composite centroids are signed area-weighted averages; cutouts are negative areas.
The centroid of a load-intensity diagram locates the equivalent concentrated resultant.
The neutral-axis-through-centroid result belongs to specific elementary beam-theory assumptions, not every possible bending problem.
Pappus-Guldinus theorems connect centroid travel distance with eligible surfaces and volumes of revolution.