## Using Hudson's Equation for Riprap

written by: Jayant R Row • edited by: Lamar Stonecypher • updated: 10/27/2010

Riprap using rock or concrete materials help to protect structures built on coastal lands. Calculating the size of the riprap is important if it has to be effective and Hudson’s Equation is used for this.

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### Riprap and Its Need to Protect Shorelines

Riprap is rock material used to strengthen shorelines and save them from erosion. The most common rocks used are granite or limestone and this has more to do with their abundant availability in areas near the shore. Occasionally concrete elements or rubble can also serve this function of protecting shorelines on seas, rivers and other water bodies from wind and wave action that can affect the natural shores. This need to protect shorelines arises more from mankind’s need to protect coastal properties and the structures that come up on it.

Image source: Wikimedia: Riprap

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Riprap absorbs and deflects the impact of waves before they can reach the structure or shoreline being defended. Riprap has a size and mass that make it impossible for water from the waves to move or shift it and this enables it to absorb the energy from the wave and cause the water to trickle into the crevices between individual pieces of the riprap, further reducing any chance of erosion or damage.

You will find riprap laid around bridge foundations and other piled structures. This ensures that the foundations are not undercut by the erosion caused by water and waves thus protecting the structure of the bridge or other buildings to remain intact. Quite often riprap which consists of large angular and loose stones is laid over a geotextile or granular layer that gives further protection to the surface being protected. This layer helps to separate the underlying soils from the water and also helps to prevent any movement of the soils into the cavities of the riprap. This ensures that there is no settlement or undercutting of the base on which the riprap rests.

Riprap works best if it is placed by hand and large voids are filled with smaller stones. This allows the riprap to interlock and present a more stable surface to the action of water. If riprap is dumped by cranes or other mechanical devices, the rip rap stones would shift and adjust themselves over time because of the wave action, but during this process the underlying soil remains at a greater danger. Dumped riprap will require greater thicknesses to remain stable, whereas riprap placed by hand can do with much less material. Labor for manual placing, however, can be quite a cost and the engineer would have to balance the two to arrive at the correct decision. The time required for hand placing is much greater than for dumped rubble and this may make a difference, especially in cases where there is a struggle against elements and inclement weather.

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### Hudson’s Equation and Riprap

Hudson’s Equation is the formula used to obtain a preliminary design of rock size to be used for riprap. The equation itself is:

W= wrH3 / KD (Sr – 1)3 cot Ɵ

Where wr is the unit weight of the rock being used and H is the height of waves in the region. The specific gravity of the material is indicated by Sr. Ɵ is the angle that the riprap slope forms with the horizontal, while KD is a stability coefficient for the riprap which is worked out in the laboratory.

This formula deals only with regular waves and makes no other allowances that can affect the calculations of riprap size like duration of storm, the wave period which is the time between waves and permeability of the structure. These factors can affect riprap sizes calculated but Hudson’s Equation does allow an initial estimate to be made which can be used for further refinements and laboratory tests needed to get more accurate data. Hudson’s Equation is used for most standard sea or water body shore protection, while more sophisticated formulas may be used in the case of expensive canals or water entrances that require a lot of work to be carried out.

Image source: Wikimedia: Riprap along shoreline

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### References

Revetment Design

The purpose of using riprap