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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:

IA = Ii + (Ri)2 x Ai

Where IA is the moment of inertia about the arbitrary axis, Ii is the moment of inertia of the area about it’s own axis, Ri is the distance between the arbitrary axis and the area axis, and Ai 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.

In this write up, the terms neutroidal axis and centroidal axis are used interchangeably.  They refer to an axis that goes through the center of an area.   

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:

IA = Ii + Ri2 x Ai

Where IA is the moment of inertia about the arbitrarily selected axis, Ii is the moment of inertia about the component’s own centroidal axis,
Ri is the distance between the arbitrarily selected axis and the component’s own centroidal axis, and Ai 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:

INA = S IA + RNA2 x AT

Where S IA equals the summation of the individual components moments of inertia about the arbitrarily selected axis, RNA 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, SRi x Ai and then dividing it by the summation of all the component areas, AT = SAi

This is all conveniently summarized and neatly presented on popular spreadsheets that are available on this website.  The product codes are SMs-e, SMs-m, SMf-e and SMf-m.  Composite cross sections are similar but more complex.  For composites see the article and example on composite sections and the product codes SMc-e and SMc-m. 

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:

ILCF = IA + (RA-LCF)2 x AWP

Where ILCF is the moment of inertia about the LCF for the water plane in the longitudinal direction, IA is the longitudinal moment of inertia about amidships, RA-LCF is the distance between amidships and the LCF, and AWP is the area of the vessel’s waterplane.

This longitudinal moment of inertia calculation is generally applied to both monohulls and multihulls.  For catamarans the results must be multiplied by two, if only one hull was used in the initial calculations. 

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.

ICL = 2 x [IH + (RH-CL)2 x AH]

This is for both hulls so the amount above is multiplied by 2.  ICL 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.  IH is the transverse moment of inertia of a hull about a hulls axis, RH-CL is the distance between the centerline and the hull axis, AH 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.