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);

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);

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

See similar textbooks

Related questions

Question

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);

У
7 (1.2, 3.0)
8 (0.8, 2.0)
6 (2.3, 3.2)
1 (1.0, 1.0) 2 (2.0, 1.0)
5 (4.0, 3.3)
4 (3.6, 2.3)
3 (3.5, 1.3)
X

Expert Solution

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Types of Phase Diagrams and Its Applications in Metallurgy

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.