Shape change of buckled sheet under change of load

I need to solve the following problem: a sheet of flexible but inextensible material (can be modelled as cable or chain in 2D) is fixed in endpoints and buckles up. Then a variable force is applied in one or more points. Under this force shape of the buckle changes because the sheet/chain/cable needs to preserve its length, it cannot compress or stretch. I need to compute numerically the shape of the buckle with and without force.

In my model I represent it as a chain, then I compute forces at both ends of each link. Force difference gives a rotating moment, so I rotate each link proportionally. However I cannot figure out how to constrain individual rotations so that the length of the chain is preserved. If each link has angle $phi_i$, I need to fulfill $sum(cos(phi_i))=A$ and $sum(sin(phi_i))=0$, but that seems not trivial how to translate this into constraints on all $phi_i$. Is there a method?
Or is there a whole different approach for solving this type of problems?