Statics

Centroid

Aka. centre of area, geometric centre. The point where the area of a figure is assumed to be concentrated. Defined for a 2D shape.

Located on the axes of symmetry, if one exists.

Center of gravity

The point where the mass of an object is concentrated. Defined for a 3D shape.

First moment of area

Measure of spatial distribution of a shape in relation to an axis.

$$\text{About x-axis} = \int_A {y\; \text{d}A} = A\bar{x}$$ $$\text{About y-axis} = \int_A {x\;\text{d}A} = A\bar{y}$$

Here:

• $\bar{x}$ - Centroid’s $x$ coordinate
• $\bar{y}$ - Centroid’s $y$ coordinate
• $A$ - Total area

About an axis of symmetry, first moment of area is $0$. $$

Second moment of area

Aka. area moment of inertia. Describes how the cross-sectional area of a shape is distributed relative to an axis.

Not to be confused with moment of inertia.

$$\text{About x-axis} = I_{xx} = I_x = \int_A {y^2\;\text{d}A}$$ $$\text{About y-axis} = I_{yy} = I_y = \int_A {x^2\;\text{d}A}$$

Always positive.

For common shapes

Shape	Description	$I_{xx}$
Rectangle or Parallelogram	Base $b$. Height $h$. About centroidal axis parallel to base.	$\cfrac{bh^3}{12}$
Triangle	Base $b$. Height $h$. About base.	$\cfrac{bh^3}{12}$
Triangle	Base $b$. Height $h$. About centroidal axis parallel to base.	$\cfrac{bh^3}{36}$
Circle	Diameter $d$. About centroidal axis.	$\cfrac{\pi d^4}{64}$

The product of moment of area about x,y axes

$$I_{xy} = \int_A {xy\; \text{d}A}$$

The polar moment of area about z axis

$$I_{zz} = J_0 = \int_A {r^2\; \text{d}A} = I_{xx} + I_{yy}$$

Radius of gyration

$$\text{About x-axis} = r_x^2 = \frac{I_{xx}}{A}$$ $$\text{About y-axis} = r_y^2 = \frac{I_{yy}}{A}$$ $$\text{About z-axis} = r_z^2 = \frac{I_{zz}}{A}$$

Moment of inertia

Describes the resistance of a body to changes in its rotational motion.