Solving a system of equations using Symbolic Matlab

Hi, I am working with symbolic variables for the first time and I am struggling when it comes to solving systems of equations. I have tried the typical methods I would use to solve it if it were not symbolic, but in this case, I can't seem to make it work.

The system of equations, expressed matrix-wise, that I am trying to solve is the following one:

in which the 6x6 matrix is known, the loads vector (left hand side) is symbolic and the strains vector (remaining one) is the unknown. I created F as

F = @(x) [Nx; Ny; Nxy; Mx; My; Mxy] - [Aij, Bij; Bij, Dij]*[x(1); x(2); x(3); x(4); x(5); x(6)];

where Aij, Bij and Dij are the 3x3 matrices shown in the previous image, therefore, what I want to do is that I want to find the strains as a function of the loads, but I do not get the desired result using fsolve, neither the typical x=A\b.

If it helps, at the beggining of the code I declared:

syms Nx Ny Nxy Mx My Mxy t
Aij = [99.18 19.3 0; 19.3 27.50 0; 0 0 22.02]*t; 
Bij = [0 0 -2.24; 0 0 -2.24; -2.24 -2.24 0]*t^2;
Dij = [11.55 0.56 0; 0.56 1.10 0; 0 0 0.78]*t^3;

How would you solve the previous system of equations?

Thank you.

[Man Rsgo]

I have also tried

syms Nx Ny Nxy Mx My Mxy epsx epsy gamxy kx ky kxy t
F = [Nx; Ny; Nxy; Mx; My; Mxy] - [Aij, Bij; Bij, Dij]*[epsx; epsy; gamxy; kx; ky; kxy];
x = solve(F)

but it solves for epsx, gamxy, kx, kxy, ky and t, in that order, and I want to find epsx, epsy, gamxy, kx, ky, kxy plus the result is a structure with 2x1 fields for each variable, and it should be 1x1

Answer 1

[John D'Errico]

You CANNOT use fsolve, though you said you tried it. This is not a numerical problem, but a symbolic one. t is an unknown there.

syms t
Aij = [99.18 19.3 0; 19.3 27.50 0; 0 0 22.02]*t; 
Bij = [0 0 -2.24; 0 0 -2.24; -2.24 -2.24 0]*t^2;
Dij = [11.55 0.56 0; 0.56 1.10 0; 0 0 0.78]*t^3;

First, build the 6x6 matrix.

M = [Aij,Bij;Bij,Dij]

M = 

Next, what are the knowns? The unknowns?

syms Nx Ny Nxy Mx My Mxy  % these will be the knowns, though for now, we don't know them.
LHS = [Nx; Ny; Nxy; Mx; My; Mxy];

syms epsx epsy gamxy kx ky kxy % these are the unknowns. also unknown. ;-)
X = [epsx; epsy; gamxy; kx; ky; kxy];

Now the solve is almost trivial. But you need to tell solve what to solve for! The second argument to solve does that. As you can see, I give it a list of the unknowns to solve for.

sol = solve(LHS == M*X,[epsx epsy gamxy kx ky kxy])

sol = struct with fields:
     epsx: (25*(45920*Mxy + 41081*Nx*t - 25091*Ny*t))/(87182412*t^2)
     epsy: (25*(447328*Mxy - 25091*Nx*t + 180857*Ny*t))/(87182412*t^2)
    gamxy: (2*(864*Mx + 17584*My + 8851*Nxy*t))/(307151*t^2)
       kx: (4*(48011*Mx - 18284*My + 3024*Nxy*t))/(2150057*t^3)
       ky: (2*(178081*My - 5224*Mx + 17584*Nxy*t))/(307151*t^3)
      kxy: (50*(735925*Mxy + 5740*Nx*t + 55916*Ny*t))/(21795603*t^3)

Or, I could have used backslash at this point, just M\LHS, but solve does a nice job of returning the list of unknowns, and connecting them to their found values directly.