I am currently trying to isolate the variable A from the following list of equations so I can solve for it in Excel using only the input variables. I have used a list of known formulae to describe it and isolate it, but I am getting different answers to the textbook which makes me think I have made a mistake in my rearranging. I was hoping that I could get some advice on where I have gone wrong and what I can simplify/rearrange better.
INPUT VARIABLES
Max axial force: $$F_{max}$$ Min axial force: $$F_{min}$$ Max shear force: $$V_{max}$$ Min shear force: $$V_{min}$$ Ultimate tensile strength: $$S_{ut}$$ Ideal endurance limit from RR More Test: $$S_n\prime$$ Endurance limit of material: $$S_n$$ Factor of safety: $$FS$$
FORMULAE
Max normal stress:
$$\sigma_{max}=\frac{F_{max}}{A}$$
Min normal stress:
$$\sigma_{min}=\frac{F_{min}}{A}$$
Max shear stress:
$$\tau_{max}=\frac{V_{max}}{A}$$
Min shear stress:
$$\tau_{min}=\frac{V_{min}}{A}$$
Max Von Mises stress:
$${\sigma\prime}_{max}=K_f\sqrt{{\sigma_{max}}^2+3{\tau_{max}}^2}$$
Min Von Mises stress (signed):
$${\sigma\prime}_{min}=-K_f\sqrt{{\sigma_{min}}^2+3{\tau_{min}}^2}$$
Mean stress:
$$\sigma_m=\frac{{\sigma\prime}_{max}+{\sigma\prime}_{min}}{2}$$
$$=\frac{K_f}{2A}\left(\sqrt{{F_{max}}^2+3{V_{max}}^2}-\sqrt{{F_{min}}^2+3{V_{min}}^2}\right)$$
Alternating stress:
$$\sigma_m=\frac{{\sigma\prime}_{max}-{\sigma\prime}_{min}}{2}$$
$$=\frac{K_f}{2A}\left(\sqrt{{F_{max}}^2+3{V_{max}}^2}+\sqrt{{F_{min}}^2+3{V_{min}}^2}\right)$$
Alternating stress to Mean stress ratio:
$$r=\frac{\sqrt{{F_{max}}^2+3{V_{max}}^2}+\sqrt{{F_{min}}^2+3{V_{min}}^2}}{\sqrt{{F_{max}}^2+3{V_{max}}^2}-\sqrt{{F_{min}}^2+3{V_{min}}^2}}$$
$\therefore$ Mean stress:
$$\sigma_m=\frac{\sigma_a}{r}$$
Goodman relation:
$$\frac{1}{FS}=\frac{\sigma_a}{S_n}+\frac{\sigma_m}{S_{ut}}$$
$$=\frac{\sigma_a}{S_n}+\frac{\sigma_a}{rS_{ut}}$$
$$=\frac{K_f}{2A}\left(\frac{\sqrt{{F_{max}}^2+3{V_{max}}^2}+\sqrt{{F_{min}}^2+3{V_{min}}^2}}{S_n}-\frac{\sqrt{{F_{max}}^2+3{V_{max}}^2}+\sqrt{{F_{min}}^2+3{V_{min}}^2}}{rS_{ut}}\right)$$
$$=\frac{1}{A}\left(\frac{K_f\left(rS_{ut}-S_n\right)\left(\sqrt{{F_{max}}^2+3{V_{max}}^2}+\sqrt{{F_{min}}^2+3{V_{min}}^2}\right)}{2S_nrS_{ut}}\right)$$
Cross sectional area:
$$A=\frac{1}{FS}\left(\frac{K_f\left(rS_{ut}-S_n\right)\left(\sqrt{{F_{max}}^2+3{V_{max}}^2}+\sqrt{{F_{min}}^2+3{V_{min}}^2}\right)}{2S_nrS_{ut}}\right)$$