Calculate the stiffness matrix for the Q8 element shown below. Make use of the shape functions in natural coordinates and the Jacobian calculated in previous problems. Use 2x2 Gaussian numerical integration. Use plane strain conditions (for unit thickness) with E=200E9 (MPa) and nu=0.25. I have attached my matlab code below. Please look thru it and correct it. Thanks. u=sym('u') %%zta v=sym('v') %%eta x1=1 x2=2 x3=3.5 x4=3.6 x5=4.0 x6=2.3 x7=1.2 x8=0.8 y1=1 y2=1 y3=1.3 y4=2.3 y5=3.3 y6=3.2 y7=3.0 y8=2.0 X=[x1;x2;x3;x4;x5;x6;x7;x8] Y=[y1;y2;y3;y4;y5;y6;y7;y8] %% diff of zta DN1Du=1/4(1-v)(2u+v); DN2Du=-u(1-v); DN3Du=1/4(1-v)(2u-v); DN4Du=1/2(1-v^2); DN5Du=1/4(1+v)(2u+v); DN6Du=-u(1+v); DN7Du=1/4(1+v)(2u-v); DN8Du=-1/2(1-v^2); %% diff of eta DN1Dv=1/4(1-u)(2v+u); DN2Dv=-1/2(1-u^2); DN3Dv=1/4(1+u)(2v-u); DN4Dv=-v(1+u); DN5Dv=1/4(1+u)(2v+u); DN6Dv=1/2(1-u^2); DN7Dv=1/4(1-u)(2v+u); DN8Dv=-v(1-u); %% find dx/du dx/dv, dy/du and dy/dv a=[DN1Du, DN2Du, DN3Du, DN4Du, DN5Du, DN6Du,DN7Du, DN8Du] dxdu=aX b=[DN1Dv, DN2Dv, DN3Dv, DN4Dv, DN5Dv, DN6Dv, DN7Dv,DN8Dv] dxdv=bX dydu=aY dydv=bY %% Jacobian J=dxdudydv-dxdvdydu %% define DNiDx DN1Dx=1/J(dydvDN1Du-dyduDN1Dv) DN2Dx=1/J(dydvDN2Du-dyduDN2Dv) DN3Dx=1/J(dydvDN3Du-dyduDN3Dv) DN4Dx=1/J(dydvDN4Du-dyduDN4Dv) DN5Dx=1/J(dydvDN5Du-dyduDN5Dv) DN6Dx=1/J(dydvDN6Du-dyduDN6Dv) DN7Dx=1/J(dydvDN7Du-dyduDN7Dv) DN8Dx=1/J(dydvDN8Du-dyduDN8Dv) %% defrine DniDx DN1Dy=1/J(-dxdvDN1Du+dxduDN1Dv) DN2Dy=1/J(-dxdvDN2Du+dxduDN2Dv) DN3Dy=1/J(-dxdvDN3Du+dxduDN3Dv) DN4Dy=1/J(-dxdvDN4Du+dxduDN4Dv) DN5Dy=1/J(-dxdvDN5Du+dxduDN5Dv) DN6Dy=1/J(-dxdvDN6Du+dxduDN6Dv) DN7Dy=1/J(-dxdvDN7Du+dxduDN7Dv) DN8Dy=1/J(-dxdvDN8Du+dxduDN8Dv) %% B matrix B=[DN1Dx, 0, DN2Dx, 0, DN3Dx, 0, DN4Dx, 0, DN5Dx, 0, DN6Dx, 0, DN7Dx, 0, DN8Dx, 0; 0, DN1Dy, 0, DN2Dy, 0, DN3Dy, 0, DN4Dy, 0, DN5Dy, 0, DN6Dy, 0, DN7Dy, 0, DN8Dy; DN1Dy, DN1Dx, DN2Dy, DN2Dx, DN3Dy, DN3Dx,DN4Dy,DN4Dx,DN5Dy,DN5Dx,DN6Dy,DN6Dx, DN7Dy,DN7Dx,DN8Dy,DN8Dx;] BT=transpose(B) %% define D strain E=20010^9 nu=0.25 Q=20010^9/((1+0.25)(1-20.25)) R=[1-0.25, 0.25, 0; 0.25, 1-0.25, 0; 0, 0, (1-20.25)/2;] D=QR %% u1=-1/sqrt(3); v1=-1/sqrt(3); u2=1/sqrt(3); v2=1/sqrt(3); w1=1; w2=1; % Substitute u and v with u1 and v1 J_sub = subs(J, [u, v], [u1, v1]); B_sub = subs(B, [u, v], [u1, v1]); BT_sub = subs(BT, [u, v], [u1, v1]); F1=BT_subDB_subJ_sub % Substitute u and v with u1 and v2 J_sub2 = subs(J, [u, v], [u1, v2]); B_sub2 = subs(B, [u, v], [u1, v2]); BT_sub2 = subs(BT, [u, v], [u1, v2]); F2=BT_sub2DB_sub2J_sub2 % Substitute u and v with u2 and v1 J_sub3 = subs(J, [u, v], [u2, v1]); B_sub3 = subs(B, [u, v], [u2, v1]); BT_sub3 = subs(BT, [u, v], [u2, v1]); F3=BT_sub3DB_sub3J_sub3 % Substitute u and v with u2 and v2 J_sub4 = subs(J, [u, v], [u2, v2]); B_sub4 = subs(B, [u, v], [u2, v2]); BT_sub4 = subs(BT, [u, v], [u2, v2]); F4=BT_sub4DB_sub4J_sub4 F=(F1w1w2)+(F2w1w2)+(F3w1w2)+(F4w1w2);

Question

Calculate the stiffness matrix for the Q8 element shown below. Make use of the shape functions in natural coordinates and the Jacobian calculated in previous problems. Use 2x2 Gaussian numerical integration. Use plane strain conditions (for unit thickness) with E=200E9 (MPa) and nu=0.25.

I have attached my matlab code below. Please look thru it and correct it. Thanks.

u=sym('u') %%zta
v=sym('v') %%eta

x1=1
x2=2
x3=3.5
x4=3.6
x5=4.0
x6=2.3
x7=1.2
x8=0.8

y1=1
y2=1
y3=1.3
y4=2.3
y5=3.3
y6=3.2
y7=3.0
y8=2.0
X=[x1;x2;x3;x4;x5;x6;x7;x8]
Y=[y1;y2;y3;y4;y5;y6;y7;y8]
%% diff of zta
DN1Du=1/4*(1-v)*(2*u+v);
DN2Du=-u*(1-v);
DN3Du=1/4*(1-v)*(2*u-v);
DN4Du=1/2*(1-v^2);
DN5Du=1/4*(1+v)*(2*u+v);
DN6Du=-u*(1+v);
DN7Du=1/4*(1+v)*(2*u-v);
DN8Du=-1/2*(1-v^2);

%% diff of eta
DN1Dv=1/4*(1-u)*(2*v+u);
DN2Dv=-1/2*(1-u^2);
DN3Dv=1/4*(1+u)*(2*v-u);
DN4Dv=-v*(1+u);
DN5Dv=1/4*(1+u)*(2*v+u);
DN6Dv=1/2*(1-u^2);
DN7Dv=1/4*(1-u)*(2*v+u);
DN8Dv=-v*(1-u);
%% find dx/du dx/dv, dy/du and dy/dv
a=[DN1Du, DN2Du, DN3Du, DN4Du, DN5Du, DN6Du,DN7Du, DN8Du]
dxdu=a*X
b=[DN1Dv, DN2Dv, DN3Dv, DN4Dv, DN5Dv, DN6Dv, DN7Dv,DN8Dv]
dxdv=b*X
dydu=a*Y
dydv=b*Y

%% Jacobian
J=dxdu*dydv-dxdv*dydu

%% define DNiDx

DN1Dx=1/J*(dydv*DN1Du-dydu*DN1Dv)
DN2Dx=1/J*(dydv*DN2Du-dydu*DN2Dv)
DN3Dx=1/J*(dydv*DN3Du-dydu*DN3Dv)
DN4Dx=1/J*(dydv*DN4Du-dydu*DN4Dv)
DN5Dx=1/J*(dydv*DN5Du-dydu*DN5Dv)
DN6Dx=1/J*(dydv*DN6Du-dydu*DN6Dv)
DN7Dx=1/J*(dydv*DN7Du-dydu*DN7Dv)
DN8Dx=1/J*(dydv*DN8Du-dydu*DN8Dv)

%% defrine DniDx

DN1Dy=1/J*(-dxdv*DN1Du+dxdu*DN1Dv)
DN2Dy=1/J*(-dxdv*DN2Du+dxdu*DN2Dv)
DN3Dy=1/J*(-dxdv*DN3Du+dxdu*DN3Dv)
DN4Dy=1/J*(-dxdv*DN4Du+dxdu*DN4Dv)
DN5Dy=1/J*(-dxdv*DN5Du+dxdu*DN5Dv)
DN6Dy=1/J*(-dxdv*DN6Du+dxdu*DN6Dv)
DN7Dy=1/J*(-dxdv*DN7Du+dxdu*DN7Dv)
DN8Dy=1/J*(-dxdv*DN8Du+dxdu*DN8Dv)

%% B matrix

B=[DN1Dx, 0, DN2Dx, 0, DN3Dx, 0, DN4Dx, 0, DN5Dx, 0, DN6Dx, 0, DN7Dx, 0, DN8Dx, 0;
0, DN1Dy, 0, DN2Dy, 0, DN3Dy, 0, DN4Dy, 0, DN5Dy, 0, DN6Dy, 0, DN7Dy, 0, DN8Dy;
DN1Dy, DN1Dx, DN2Dy, DN2Dx, DN3Dy, DN3Dx,DN4Dy,DN4Dx,DN5Dy,DN5Dx,DN6Dy,DN6Dx, DN7Dy,DN7Dx,DN8Dy,DN8Dx;]

BT=transpose(B)

%% define D strain E=200*10^9 nu=0.25
Q=200*10^9/((1+0.25)*(1-2*0.25))
R=[1-0.25, 0.25, 0;
0.25, 1-0.25, 0;
0, 0, (1-2*0.25)/2;]
D=Q*R
%%
u1=-1/sqrt(3);
v1=-1/sqrt(3);
u2=1/sqrt(3);
v2=1/sqrt(3);

w1=1;
w2=1;

% Substitute u and v with u1 and v1
J_sub = subs(J, [u, v], [u1, v1]);
B_sub = subs(B, [u, v], [u1, v1]);
BT_sub = subs(BT, [u, v], [u1, v1]);

F1=BT_sub*D*B_sub*J_sub

% Substitute u and v with u1 and v2
J_sub2 = subs(J, [u, v], [u1, v2]);
B_sub2 = subs(B, [u, v], [u1, v2]);
BT_sub2 = subs(BT, [u, v], [u1, v2]);

F2=BT_sub2*D*B_sub2*J_sub2

% Substitute u and v with u2 and v1
J_sub3 = subs(J, [u, v], [u2, v1]);
B_sub3 = subs(B, [u, v], [u2, v1]);
BT_sub3 = subs(BT, [u, v], [u2, v1]);

F3=BT_sub3*D*B_sub3*J_sub3

% Substitute u and v with u2 and v2
J_sub4 = subs(J, [u, v], [u2, v2]);
B_sub4 = subs(B, [u, v], [u2, v2]);
BT_sub4 = subs(BT, [u, v], [u2, v2]);

F4=BT_sub4*D*B_sub4*J_sub4

F=(F1*w1*w2)+(F2*w1*w2)+(F3*w1*w2)+(F4*w1*w2);