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Course: ME 441, Fall 2009
School: Rochester
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Element 3 (7.78, Example 2.73) (7.95, 2.7) 2 1 (8.22, 2.46) 4 (7.79, 2.47) Coord(Node_1, 1)=8.22 Coord(Node_1, 2)=2.46 Coord(Node_2, 1)=7.95 Coord(Node_2, 2)=2.7 Coord(Node_3, 1)=7.78 Coord(Node_3, 2)=2.73 Coord(Node_4, 1)=7.79 Coord(Node_4, 2)=2.47 E=200000 T=0.1 v=0.3 Elemental Stiffness Subroutine for QUAD4 Sub_stiffness( coordinates, material, thickness) Calculate SE End Sub_stiffness( coordinates,...

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Element 3 (7.78, Example 2.73) (7.95, 2.7) 2 1 (8.22, 2.46) 4 (7.79, 2.47) Coord(Node_1, 1)=8.22 Coord(Node_1, 2)=2.46 Coord(Node_2, 1)=7.95 Coord(Node_2, 2)=2.7 Coord(Node_3, 1)=7.78 Coord(Node_3, 2)=2.73 Coord(Node_4, 1)=7.79 Coord(Node_4, 2)=2.47 E=200000 T=0.1 v=0.3 Elemental Stiffness Subroutine for QUAD4 Sub_stiffness( coordinates, material, thickness) Calculate SE End Sub_stiffness( coordinates, material, thickness, elm_num) T=elm_num_thickness E=elm_num_material v=elm_num_poissons For i=1 to 4 x(i)= X_coord of elm_num y(i)=Y_coord of elm_num End Calculate SE (x3, y3) (x2, y2) (x1, y1) (x4, y4) - 1 1 T B E B t J d d - 1 1 2 point Gauss integration 2 2 i=1 j=1 B E B t J -0.57735,0.57735 0.57735,0.57735 3 2 T For i=1 to 2 xi=intpoint(i) For j=1 to 2 eta=intpoint(j) 1 End End 0.57735,-0.57735 4 -0.57735,-0.57735 Constitutive Relationships If Plane Stress E stress 0 1 E 1 0 2 1- 1- 0 0 2 If Plane Strain E strain 0 1 - E 1- 0 ( 1 + ) ( 1 - 2 ) 1-2 0 0 2 Jacobian Derivatives of the shape functions d N1 d d N1 d d d N2 d d N3 d d N4 DN d d d N2 N3 N4 d d d DN 1 -( 1 - ) ( 1 - ) 4 -( 1 - ) - ( 1 + ) ( 1 + ) -( 1 + ) (1 + ) (1 - ) -0.57735,0.57735 0.57735,0.57735 3 2 J= DN x Coordinates For i=1 to NumIntPoints For j=1 to NumIntPoints 4 1 0.57735,-0.57735 eta=eta(i) xi=xi(i) -0.57735,-0.57735 J sub x 1 1 -( 1 - ) ( 1 - ) ( 1 + ) - ( 1 + ) x 2 4 -( 1 - ) -( 1 + ) ( 1 + ) ( 1 - ) x 3 x 4 y1 y2 y3 y4 J = J + Jsub Next Next J detJ B Matrix 1 0 0 0 0 0 0 1 0 1 1 0 Never changes Inverse of Jacobian J- 1 0 0 J- 1 J -1 J -1 0 0 12 11 -1 -1 J 22 0 0 J 21 -1 -1 0 0 J 11 J 12 -1 -1 0 0 J 21 J 22 Derivatives of shape functions. Evaluate them at integration points d N1 d d N1 d 0 0 d d d d d d 0 N2 0 N3 0 N4 0 d d d d d d d N1 0 N2 0 N3 0 N4 d d d d d d d d N1 0 N2 0 N3 0 N4 d d d d 0 d N2 0 d N3 0 d N4 0 Sub_quad_4 (coord, material, thickness, elm_num) % Initialize the stiffness matrix !!!!!!!!!!!!!!!!!!!!!! SE=0.0 Int_point(1)=-0.5773502692 For i=1 to 4 x(i)= X_coord of y(i)=Y_coord elm_num of elm_num Next For i=1 to 2 xi=int_point(i) For j=1 to 2 eta=int_point(i) % Calculate the derivatives of the shape functions 4 1 Int_point(2)=-0.5773502692 -0.57735,0.57735 0.57735,0.57735 3 2 0.57735,-0.57735 DN 1 -( 1 - ) ( 1 - ) 4 -( 1 - ) - ( 1 + ) ( 1 + ) -( 1 + ) (1 + ) (1 - ) -0.57735,-0.57735 % Calculate the Jacobian and its determinant. Also, check the Jacobian!!!!!!!!!! J=DN x Coords Jdet=Det(J) % Populate alfa, beta and gamma matrices 1 0 0 0 0 0 0 1 0 1 1 0 J -1 J -1 0 0 12 11 -1 -1 J 22 0 0 J 21 -1 -1 0 0 J 11 J 12 -1 -1 0 0 J 21 J 22 d N1 d d N1 d 0 0 d d d 0 N2 0 N3 0 N4 0 d d d d d d d N1 0 N2 0 N3 0 N4 d d d d d d d d N1 0 N2 0 N3 0 N4 d d d d 0 d d N2 0 d d N3 0 d d N4 0 % Calculate the stiffness matrix contribution of the current integration point K=B'.E.B.t.Jdet where B = alfa x beta x gamma % Add it to the elemental stiffness matrix SE=SE+K Next Next Example Element SE Homework #5 (1.29, 1.4) 4 3 (1.57, 1.21) 1 (1.25, 1.07) 2 (1.59, 0.97) E=200000 T=0.1 v=0.3 Calculate the stiffness matrix for the above element for Plane Strain and Plane Stress cases. For Plane Strain, use t=1 Homework #5 Solutions SE Plane Stress 1 1 2 3 4 5 6 7 8 14656 5420.6 -3860.4 1168.6 -10646 -459...

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