The Parallel Axis Theorem

The parallel axis theorem is important for both stability and structural analysis.  Area moments of inertia are representative of the stiffness of an area to tipping (stability) or flexure (structures).  The parallel axis theorem calculates the moment of inertia with respect to any axis selected.  This theorem makes moment of inertia calculations convenient and easier to handle.

The parallel axis theorem states that the moment of inertia about an arbitrarily selected axis is equal to the moment of inertia about an area’s axis plus the square of the distance between these axes multiplied by the area.  Written in a formula as:

I_A = I_i + (R_i)² x A_i

Where I_A is the moment of inertia about the arbitrary axis, I_i is the moment of inertia of the area about it’s own axis, R_i is the distance between the arbitrary axis and the area axis, and A_i is the area under consideration.

This theorem has several important practical applications in naval architecture and in boat and yacht design.  Among the primary are calculating complex cross sectional area properties for structures and calculating the water plane area properties for stability.

Structures and the Parallel Axis Theorem

For structures the amidships cross sectional area properties are an important calculation.  This theorem is used in two forms for this calculation. 

The first form is used in the initial summing up of component area properties within the cross section.  It is as follows:

I_A = I_i + R_i² x A_i

Where I_A is the moment of inertia about the arbitrarily selected axis, I_i is the moment of inertia about the component’s own centroidal axis,
R_i is the distance between the arbitrarily selected axis and the component’s own centroidal axis, and A_i is the cross sectional area of the component.

The second form is in determining the area properties of all the components combined, that is the area properties for the entire section.  This calculation is needed to represent physical reality.  A structure flexes about its neutroidal axis.  For the case at hand the neutroidal axis is for the entire section and not the arbitrary axis selected for convenience of calculation.  This second formula is expressed mathematically as follows:

I_NA = S I_A + R_NA² x A_T

Where S I_A equals the summation of the individual components moments of inertia about the arbitrarily selected axis, R_NA equals the distance between the arbitrarily selected axis and the neutral axis of the entire section.  This value is found by summing first moments of each component area, SR_i x A_i and then dividing it by the summation of all the component areas, A_{T =} SA_i. 

Stability and the Parallel Axis Theorem

For stability, the parallel axis theorem is used for moment of inertia calculations for water plane area properties.  One important example is the calculation of longitudinal area properties.  Longitudinal area properties are often computed about amidships (the arbitrary axis in this case).  But the boat trims (tips or pitches) longitudinally about the center of area of the water plane.  So the area properties need to be recalculated about the area center (centroid) of the water plane.  The area center is located at the LCF, the longitudinal center of floatation.  The moment of inertia about the LCF is calculated as follows:

I_LCF = I_A + (R_A-LCF)² x A_WP

Where I_LCF is the moment of inertia about the LCF for the water plane in the longitudinal direction, I_A is the longitudinal moment of inertia about amidships, R_A-LCF is the distance between amidships and the LCF, and A_WP is the area of the vessel’s waterplane.

Catamaran Transverse Stability

Catamarans present an especially interesting application of this theorem for transverse moment of inertia calculations.  Catamarans are excellent example for utilizing the parallel axis theorem.

I_CL = 2 x [I_H + (R_H-CL)² x A_H]

This is for both hulls so the amount above is multiplied by 2.  I_CL is the moment of inertia about the centerline.  The boat tips (lists or heels) about the centerline for hulls that symmetrical on both sides of the centerline.  I_H is the transverse moment of inertia of a hull about a hulls axis, R_H-CL is the distance between the centerline and the hull axis, A_H is the water plane area for the hull.

Conclusion

A deep understanding and application of the parallel axis theorem, benefits ship design engineers, naval architects, yacht designers, boat designers and designers of any type of floating structure.  It also benefits ocean, structural and civil engineers as well. 

Coast guards, classification societies and other regulatory agencies may require a computer model of a vessel.  The software that works with the computer model usually automatically applies this theorem to give the values needed for analysis.  It is good to have a handle on parallel axis theorem in order to check the hull software for accuracy.