Since you have not provided a complete AMPL model, I cannot try solving it on my computer to see the trouble that Ipopt is having. However, I can make a good guess. When beta = 1e-8, beta^2 = 1e-16. That is such a small number that it is likely being ignored by Ipopt.
Of course, the model is mathematically correct for any value of beta > 0. But Ipopt’s algorithms must work with the limited precision that is available in computer arithmetic. As a result, when beta is very small, Ipopt is unable to compute correct results.
Do you get an acceptable solution when beta = 1e-7 or even 1e-6? Then maybe you could use that solution for your application. Alternatively you could try modeling F_f in a different way. To get advice about that, you would have to say which of v, v_s, F_c, F_v, F_s, F_a, and F_r are variables and which are parameters. (In AMPL terminology, if the value is given as part of the data, it’s a parameter; if the value is determined by the solver, it’s a variable.)