“Computational Civil Engineering - "Intersections" International Journal

“Computational Civil Engineering 2005”, International Symposium 27· step 4) the transposition of the strains into the global system:with [ T σ ] = Cook’s matrix [5].T[ ε ] = [ Tσ ] ⋅ [ ε′](16)( 6x1)(6x6)(6x1)· step 5) the selection of an orthotropic constitutive matrix for the material:[ D '](6×6)⎡e11⎢⎢e12⎢e13= ⎢⎢ 0⎢ 0⎢⎢⎣0eee121113000eee131333000g0001300g00001300 ⎤0⎥⎥0 ⎥⎥0 ⎥0 ⎥⎥g33⎥⎦· step 6) the transposition of the constitutive matrix in global co-ordinates:(17)with [ T ε ] = Cook’s matrix [5].T[ D] [ T ] ⋅[ D '] ⋅[ T ](18)=εε(6×6) (6×6) (6×6) (6×6)· step 7) the application of the differential operator[ L ]( 6 × 1 )⎡ ∂⎢ ∂ x⎢⎢ ⋅⎢⎢⎢ ⋅= ⎢∂⎢⎢ ∂ y⎢⎢ ⋅⎢⎢ ∂⎢⎣∂ z⋅∂∂ y⋅∂∂ x∂∂ z⋅⎤⋅⎥⎥⋅ ⎥⎥∂ ⎥∂ z⎥⎥⋅ ⎥⎥∂ ⎥⎥∂ y⎥∂ ⎥∂ x ⎥⎦and the Gauss numerical integration of the following two matrices:(19)Tthe generalized stiffness matrix= [ K *] [ P] ⋅ [ D] ⋅ [ P] ⋅ dV (20)= ∫×(18× 18) (18×6) (6×6) (6 18)Vthe transformation matrix=⎛= ∫ ⎜⎝ ×T[ G] [ P] ⋅ [ D] ⋅ [ L] ⋅ [ N ] dV( 18× 24 ) (18×6) (6×6) (6×3) (3 24)V· step 8) the development of the stiffness matrix of the finite element:⎞⎟ ⋅⎠(21)T −1[ K ] = [ G] ⋅[ K *] ⋅ [ G](22)(24×24)(24×18)(18×18)(18×24)