# Stiffness matrix of bending plate

by Ronankeating
Tags: bending, matrix, plate, stiffness
 P: 64 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 ke= t∫∫BT*Db*B*det(J)dζdη+∫∫BT*Ds*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. Db is the bending coefficient matrix (5x5) (scalar) Ds 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(Ni = 1/4(1 + ξ*ξi )(1 + η*ηi)) is bilinear of ζ and η. B matrix is 1st order bilinear, so BT*Db*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,
 Engineering Sci Advisor Thanks P: 6,064 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.
 P: 64 Thanks in advance, So as a result Gauss integration can be performed?
Engineering