Why?   Consider re-installing the pipe at the design elevation.  If this can not be done, determine if there is a setting elevation that will allow for the minimum cover.

But, I hardly think 2 inches of cover is appropriate.

I agree with MPENN.  There's any number of methods to analyze this situation (Gaylord and Gaylord, "Structural Engineering Handbook" Chapter 25, for instance, but you'll have to stretch the assumptions because of your unconventional conditions and you'll almost certainly find a theoretical failure.  Then call the pipe manufacturer and he'll tell you the same thing.
This is not a problem to be engineered away.  This could fail at any time and somebody could get killed.  Make the contractor correct this.

Slapdog,

Don't bother wasting your time trying to determine whether 2" of cover will be sufficient.  It won't be, unless of course special installtion treatment and reinforcement is used, and then it would be cheaper to simply re-install the culvert at the design grade.  

I'm curious though that you have to consider the question of the cover now?  How much cover did the design originally allow for and just how tight are the drainage grade allowances for the culvert to be placed out of the contract allowable?

It doesn't quite sound like a contractor did this because generally the installaion would have failed the inspection and the contractor would be obligated or forced to install at the correct location.  Was this installation done incorrectly by another party or crew, whom, for whatever reason, is not willing to undertake the repair?  Or was the work not inspected propoerly and the invert elevation not checked against the gradesheet?

KRS Services
www.krs-services.com

The capability of the pipe to carry the load will depend on the size of the pipe. Generally larger pipes can have less cover because the load is spread out over a larger area. PipePac is a program distributed by the American Concrete Pipe association that will determine the required class of pipe under different soil, bedding and cover conditions. The software can be downloaded at http://www.concrete-pipe.org/software.htm.

As an example a 18" pipe installed using a Type B trench with HS-20 loading must be class III when there is 6" of cover and class IV when there is 3" of cover. A 24" pipe in the same conditions with 3" of cover need only be class III. I have heard that some places require a class V pipe any time the cover is less than 12". This seems overly conservative but it makes it easy for the designer and reviewer to just max out the strength of the pipe and move on without doing any further calculations.

I am new to the forum so I hope this is not too late to be of use.  Generally speaking minimum covers are not for load carrying capabilities, for RCP pipe the worst load is the three-edge load which is directly related to pipe class.  In the field, regardless of how shallow the cover, the backfill offers some lateral support and the bedding some foundation load distribution.  The pipe should withstand the design load without negative consequence.  

Minimum cover is necessary to prevent excessive settlement of the backfill around the pipe and subsequent exposure of the pipe.  There are two basic methods:

1) Since the soil will settle adjacent to the pipe more than over the pipe, there needs to be cover to dampen this effect.

2) There must be sufficient cover opposite a moving vehicle to resist the lateral component of the tire pressure.  Even minute movement of the pipe will lead to soil settlement an exposure of the pipe.

2" in nowhere near enough.  Generally speaking 9" of cover for non-traffic bearing situations is sufficient.  For traffic installations a minimum of 12" should be provided.

Hope this helps

regards,

KEW

Hope this is helpfull.