The problem is ο¬rst solve for scalar ο¬eld with the interpolating polynomial . Writing
| (1) |
Evaluating the ο¬eld at each node gives
Hence
Where is the determinant
Substituting (2) into (1) gives
Where
For constant stress triangle, the ο¬eld is a vector ο¬eld. Hence replacing with equation (3) becomes
From the above
Hence
From (4) all of the terms are evaluated. Substituting the result into (5) gives the BΒ matrix
And
Hence becomes
| (6) |
Letting and , the above becomes
But the area of triangle is give by
And the determinant was found above to be , hence
Substituting the above into found above in equation (6) gives