Optimization of truss with regards to mass, with cross sectional areas and node position as design variables

This is a homework assignment so I only need a push in the right direction.

I have a 4 bar truss with 4 nodes, bar 1 and 2 are fixed in one end so there are 4 DOFs. Picture of truss

The objective function to be minimized is the mass of the truss. The design variables are the cross sectional areas of the bars and the location of node 3.

The constraints are displacement on node 4 and the stresses in the bars.

My question is, how do I start? I am somewhat familiar with optimizing functions of multivariable equations, but since the constraints are stresses and displacements I am having some troubles figuring out how to go about the problem, since I cant very well include the equations for the stresses in one constraint function?

So my idea was to use penalty functions for the stress constraints, but so far I have only used exterior penalty functions and that seem to be a very bothersome way to go about the problem.

I have the hint from our professor to use the optimization toolbox, but the fmincon that I thought of first does not seem applicable here, thankful for any suggestions!

Best regards

/T

Answer 1

[John D'Errico]

Well, you did ask for only a push.

There are 4 degrees of freedom, so the 4 cross-sectional areas of the bars.

You can compute the truss mass trivially given the bar lengths, the cross sectional areas, and the density of the material.

As far as the constraints go, you apparently have a constraint on displacement at node 4. So IF you knew the areas of each bar, and the force applied, then you could compute the displacement. That is a simple function of the unknowns.

Similarly, you apparently have a constraint on the stress in any of the bars, so a maximum stress allowed. Again, if you knew the cross-sections of each bar, and the force applied, then you can compute the displacement of each node due to that force. Therefore the strain in each bar is a trivial computation, and so also the stress. So you will have 4 constraints. In fact, those constraints should be linear, if I recall my grad school Mech E days from 30+ years ago. (Ok, linear as long as F is not sufficiently high that the truss exceeds the limits of linear elasticity and undergoes buckling. That gets nasty to deal with.)

And, yes, you WILL use fmincon. In fact, as I said, if you write down the equations carefully, I think the constraints will be linear inequality constraints, although you could write them as nonlinear inequalities. The latter would be the lazy way out, but still viable. :)

[Tobias Rattfelt]

Yes, thank you for your answer!

Below is how I have written the constraints so far, but what I'm having trouble with really is how to do include the constraints in the optimization, I get that I need to recalculate the stiffness matrix each time, but how do I do this with regards to the constraints?

Maybe I asked my first question poorly.

Best regards

/T

%%

% Constraints

% sigmax=2;

% umax=0.5;

% C(1)=Sig(1)-sigmax; % stress constraint bar 1

% C(2)=Sig(2)-sigmax; % stress constraint bar 2

% C(3)=Sig(3)-sigmax; % stress constraint bar 3

% C(4)=Sig(4)-sigmax; % stress constraint bar 4

% C(5)=sqrt(u(7)^2+u(8)^2)-umax; % displacement constraint

[Tobias Rattfelt]

How can I use the fmincon here? Or are you suggesting that I write the constraints so that they calculate the displacement as a constraint? I thought it could only be done using the steps I have in my script but I guess I can try to come up with another formula for the constraints. Thank you for your help!

[aws ali]

Hi friend, Have you came throug the book of (Applied Optimization with MATLAB Programming) by P. Venkataraman? If not you can go to the website of the book, they allow to download the examples of optimization (Matlab codes), which I assume they very good to start with. You need chapter 10 from that book where the examples associted with the matlab codes stated there (i.e. ch.10).