# How can I triple integral the matrix k between -h/2 to h/2, -b/2 to b/2, -a/2 to a/2 wrt. z,y,x respectively?

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Ahmed on 28 Dec 2022
Answered: Paul on 28 Dec 2022
clc
clear all
E=70*10^6;
v=0.3;
p=50*10^3;
h=0.01;
a=0.5;
b=0.5;
syms x y z
B=-h*[6*x*(b-2*y)/(a^(3)*b) 0 (a-6*x)*(b-2*y)/(2*a^2*b) -6*x*(b-2*y)/(a^3*b) 0 -(a+6*x)*(b-2*y)/(2*a^2*b) -6*x*(b+2*y)/(a^3*b) 0 -(a+6*x)*(b+2*y)/(2*a^2*b) 6*x*(b+2*y)/(a^3*b) 0 (a-6*x)*(b+2*y)/(2*a^2*b);6*(a-2*x)*y/(a*b^3) -(a-2*x)*(b-6*y)/(2*a*b^2) 0 6*(a+2*x)*y/(a*b^3) -(a+2*x)*(b-2*y)/(2*a*b^2) 0 -6*(a+2*x)*y/(a*b^3) (a+2*x)*(b+6*y)/(2*a*b^2) 0 -6*(a-2*x)*y/(a*b^3) (a-2*x)*(b+6*y)/(2*a*b^2) 0;4*(-3*b^2*x^2+a^2*(b^2-3*y^2))/(a^3*b^3) (b-2*y)*(b+6*y)/(2*a*b^2) -(a-2*x)*(a+6*x)/(2*a^2*b) (12*b^2*x^2-4*a^2*(b^2-3*y^2))/(a^3*b^3) -(b-2*y)*(b+6*y)/(2*a*b^2) -(a-6*x)*(a+2*x)/(2*a^2*b) 4*(-3*b^2*x^2+a^2*(b^2-3*y^2))/(a^3*b^3) -(b-6*y)*(b+2*y)/(2*a*b^2) (a-6*x)*(a+2*x)/(2*a^2*b) (12*b^2*x^2-4*a^2*(b^2-3*y^2))/(a^3*b^3) (b-6*y)*(b+2*y)/(2*a*b^2) (a-2*x)*(a+6*x)/(2*a^2*b)];
Bt=transpose(B);
D=(E/(1-v^2))*[1 v 0;v 1 0;0 0 (1-v)/2];
k=Bt*D*B;

Paul on 28 Dec 2022
Hi Ahmed
Try int
clc
clear all
E=70*10^6;
v=0.3;
p=50*10^3;
h=0.01;
a=0.5;
b=0.5;
syms x y z
B=-h*[6*x*(b-2*y)/(a^(3)*b) 0 (a-6*x)*(b-2*y)/(2*a^2*b) -6*x*(b-2*y)/(a^3*b) 0 -(a+6*x)*(b-2*y)/(2*a^2*b) -6*x*(b+2*y)/(a^3*b) 0 -(a+6*x)*(b+2*y)/(2*a^2*b) 6*x*(b+2*y)/(a^3*b) 0 (a-6*x)*(b+2*y)/(2*a^2*b);6*(a-2*x)*y/(a*b^3) -(a-2*x)*(b-6*y)/(2*a*b^2) 0 6*(a+2*x)*y/(a*b^3) -(a+2*x)*(b-2*y)/(2*a*b^2) 0 -6*(a+2*x)*y/(a*b^3) (a+2*x)*(b+6*y)/(2*a*b^2) 0 -6*(a-2*x)*y/(a*b^3) (a-2*x)*(b+6*y)/(2*a*b^2) 0;4*(-3*b^2*x^2+a^2*(b^2-3*y^2))/(a^3*b^3) (b-2*y)*(b+6*y)/(2*a*b^2) -(a-2*x)*(a+6*x)/(2*a^2*b) (12*b^2*x^2-4*a^2*(b^2-3*y^2))/(a^3*b^3) -(b-2*y)*(b+6*y)/(2*a*b^2) -(a-6*x)*(a+2*x)/(2*a^2*b) 4*(-3*b^2*x^2+a^2*(b^2-3*y^2))/(a^3*b^3) -(b-6*y)*(b+2*y)/(2*a*b^2) (a-6*x)*(a+2*x)/(2*a^2*b) (12*b^2*x^2-4*a^2*(b^2-3*y^2))/(a^3*b^3) (b-6*y)*(b+2*y)/(2*a*b^2) (a-2*x)*(a+6*x)/(2*a^2*b)];
Bt=transpose(B);
D=(E/(1-v^2))*[1 v 0;v 1 0;0 0 (1-v)/2];
k=Bt*D*B;
K = int(int(int(k,x,-a/2,a/2),y,-b/2,b/2),z,-h/2,h/2)
K =
vpa(K,10)
ans =

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