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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

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.

- 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)

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 dependant. 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,

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# Stiffness matrix of bending plate

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