I am trying to optimize the angles alpha and theta, for which members of truss would carry the less loads. Given is location of force and value of force.

Problem is, that I get contradictory equations, where decrease of one angle is favorable for one member but unfavorable for other one (as more vertical member gets, more vertical loads it will carry). So I came up with the equation, but I am unable to solve it for angles to get minimum force values in the members.

On one hand, I know that it has something to do with derivatives, because if we set derivative of force with respect to angle to zero, I think one should be able to get the most efficient angle. But, there are 2 equations and I am not sure if this approach is correct or how to set up derivative equation.

Any ideas you may suggest?


I am attaching the drawing of the truss structure below

enter image description here

  • $\begingroup$ I assume $L_1$ and $L_2$ are constants? or is the length of the rods constant? $\endgroup$
    – NMech
    Commented Jan 15, 2023 at 15:22
  • $\begingroup$ I changed the task and refined it a little bit. What is given is $ L_{3} $ and $ F $ , idea is to find optimal angles with absolute value of $ F_{AB} + $$ F_{BC} $ is minimum. $\endgroup$ Commented Jan 15, 2023 at 17:07
  • $\begingroup$ Is also H constant? $\endgroup$
    – NMech
    Commented Jan 15, 2023 at 17:17
  • $\begingroup$ No, $H$ is the height, which is defined by the angles and $L_{3}$ as shown in equation 3 in the Geometry part of the Figure attached. Only known quantities (constants) are $F$ and $L_{3}$, angles are to be found and all other geometries, as well as internal forces in members are defined by angles. $\endgroup$ Commented Jan 15, 2023 at 17:50
  • 1
    $\begingroup$ It's an underconstrained solution, you'll minimise the axial forces by maximising H, ad infinitum. Both will tend to F/2. $\endgroup$ Commented Jan 15, 2023 at 21:59


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