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 $$\frac{1}{2}kx^2=U_t$$

Cross section of the wire 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?


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