# Derivation of an Equivalent Spring Constant

 P: 9 Derivation of an Equivalent Spring Constant This problem difficult because you can't determine the reaction moments through equilibrium alone (statically indeterminate). Therefore you need to use information about the deflection to solve the problem. With the symmetry of the problem you know: $$y(0) = y(L) = 0$$ $$\frac{dy}{dx}_{x = 0} = \frac{dy}{dx}_{x = L/2} = \frac{dy}{dx}_{x = L} = 0$$ Split the problem into two parts. Integrate the equation to get y' and y. Use the 3 relevant boundary conditions to solve for the two integration constants and the reaction moment Ma. $$M(x) = EI \frac{d^2y}{dx^2} = -M_{A} + \frac{P}{2}(x)$$ Good luck Attached Thumbnails