The Solid Properties module calculates the mass properties of common homogeneous solids.  Calculated mass properties include centroid, volume, surface area, and moments of inertia.

The Solid Properties Module uses closed-form equations to calculate the mass properties of common solid types.  Choose a solid from the menu below to view the appropriate equations.

Thin Rectangular Plate	Rectangular Prism	Sphere	Cylinder	Cone
Thin Circular Disk	Tetrahedron	Hemisphere	Half Cylinder	Half Cone
Thin Annular Disk	Elliptical Cylinder	Spherical Shell	Cylindrical Shell	Conical Shell
Slender Rod	Quarter Slender Rod	Hemispherical Shell	Half Cylindrical Shell	Half Conical Shell

Mass Moment Of Inertia

The Mass Moment of Inertia of a solid body is a measure of that solid's ability to resist changes in its angular acceleration due to an applied torque.  Mass Moment of Inertia is defined with respect to a specific rotation axis.  Therefore a solid body does not have just one mass moment of inertia, but potentially an infinite number corresponding to all different pivot points.

The Mass Moment of Inertia of a solid body with density with respect to a given axis is defined by the volume integral:

where r is the perpendicular distance from the axis of rotation.

The Mass Moment of Inertia of a homogeneous solid body with respect to an arbitrary set of orthogonal axes is: 

Radius Of Gyration

The Radius Of Gyration represents the distance at which the entire mass of the solid must be concentrated if its moment of inertia about a specific axis is to remain unchanged.  The Radius of Gyration of a mass m about an axis x is defined by the relation:

Volume, Surface Area, Density

The Solid Properties module will calculate the volume, density, and surface area of common solid types, such as spheres, cylinders, and cones.  The module will not calculate this data for shell types, including slender rods and thin plates, because the area of the shell base is undefined.

For a solid of Volume V and mass m, the density is defined by the equation:

The calculated surface area includes the lateral area and base area of the specified solid, as shown in Figure 1.

Figure 1.  The calculated surface area includes lateral area and base area.

Figure 2. Solid Properties Module Input Form

The Solid Properties module calculates the following mass properties:

• Volume (Solid Types Only)

• Density (Solid Types Only)

• Surface Area (Solid Types Only)

• Centroid

• Moment of Inertia

• Radius of Gyration

Solid mass properties are output using the standard ETB result report shown in Figure 3.

Figure 3: Solid Properties Results Report