Stiffness matrix of bending plate

[Ronankeating]

hi,

I'm studying the book for elasticity theory and I stuck at one of the equation in that book, the stiffness matrix for elements(thin or thick plate) in bending is given in that form

k_e= t∫∫B^T*D_b*B*det(J)dζdη+∫∫B^T*D_s*B*det(J)dζdη

• B is the strain matrix(5x12) where the inner products include the shape function and derivation of shape function regarding the ζ and η.
• D_b is the bending coefficient matrix (5x5) (scalar)
• D_s is the shear coefficient matrix (5x5) (scalar)
• det(J) is the determinant of Jacobian matrix (2x2)

and book suggests that the equation can be solved by carrying out the Gauss integration, with 2 Gauss points for 1st term and 1 Gauss point for 2nd term, in order to avoid the shear locking phenomenon.

The question is:
Since I know that shape function(N_i = 1/4(1 + ξ*ξ_i )(1 + η*η_i)) is bilinear of ζ and η. B matrix is 1st order bilinear, so B^T*D_b*B yieleds 2nd order. Form of det(J) is also ζ and η dependent. Multiplication of all of this terms will result in at least 3 order form of equation , is that really can be solved with Gauss integration?

Regards,

 

[AlephZero]

Gaussian integration with n points (in one dimension) is exact for polynomials of order 2n-1.

In any case, there are often reasons NOT to want to do "exact" integration. (For example you mentioned one reason, to avoid shear locking). Gaussian integration is similar to fitting an approximate polynomial to the "exact" function by least squares, and integrating the approximate polynomial.

 

[Ronankeating]

Thanks in advance,

So as a result Gauss integration can be performed?

 

[AlephZero]

Well, obviously it CAN always be performed. The question is whether it is a good or bad way to evaluate the integrals. Usually, it's a good way, or at least "good enough", considering the output from the FE model is only an approximation to the "exact" solution of a continuum mechanics problem, and the continuum mechanics problem is only an approximation to the real-world situation.

There are other numerical integration methods which you might find out about later in your course, so don't get the idea that it's the ONLY way to do numerical integration for finite elements.

 

[Ronankeating]

Thanks a lot