# Equation for the spring constant of an orthotropic, rectangular wire spring

I hope I can get some help with this problem as I've been struggling with it for a long time. I have a helical tension spring with radius $$R$$ and a square wire cross section and made of orthotropic material. I'm trying to derive an equation for the spring constant $$k$$ by equating the energy stored by the spring $$\frac{1}{2}kx^2$$, where $$x$$ is the distance stretched, with the energy stored $$U_t$$ by a shaft of equivalent length so $$\frac{1}{2}kx^2=U_t$$

As explained in the title, I have a particular situation because the material is orthotropic and the cross-section of the shaft is a rectangle as shown in the image above. I already have an expression for the torsional rigidity $$GJ$$ of an orthotropic rectangular shaft with shear moduli $$G_{zx}$$ and $$G_{zy}$$: $$GJ=a b^3 G_{zx} \beta$$ where $$\beta=\sum_{n=1,3,5...}^{\infty}\frac{32 c^2}{\pi ^4 n^4} \left(1-\frac{2 c }{\pi n}\tanh \left(\frac{\pi n}{2 c}\right) \right)$$ and $$c=\frac{a}{b}\sqrt{\frac{G_{zy}}{G_{zy}}}$$. When I equate $$\frac{1}{2}kx^2=U_t=\frac{GJ}{2 L}\theta^2$$ I can plug in $$\theta=\frac{x}{R}$$ (the angle the shaft has been twisted by is roughly equal to the angle each coil of the spring gets stretched by) so $$k=\frac{GJ}{L R^2}$$ where $$L=2\pi R N_a$$ and $$N_a$$ are the number of active coils in the spring. To me the equation looks complete but the problem is that when I enter values like $$a=0.005$$m, $$b=0.0025$$m, $$G_{zx}=1159.1$$MPa, $$G_{zy}=974.3$$MPa, $$N_a=5$$, and $$R=0.01$$ I get $$k=632.642$$ instead of $$k=2780$$, which is the right answer.

Since I have that the torque on the shaft is $$T=ab^3G_{zx}\theta\beta$$ I tried other equations for $$U_t$$ where $$U_t=\frac{1}{2}T\theta$$ or $$U_t=\frac{1}{2}\frac{T^2 L}{GJ}$$ but none of the answers come even close, $$198.75$$ and $$62.4393$$. Am I doing something wrong when equating the two energies?