# Length of semi major axis for the ellipse in “Total Strain Energy theory”

The equation describing the "Total Strain Energy Theory" (Haigh & Beltrami's Theory of failure) is given as follows:

$$σ_1 + σ_2 - 2μ σ_1 σ_2 ≤ (Syt)^2$$ where $$σ_1$$ and $$σ_2$$ are Principal stresses.

Since this is an equation of ellipse symmetric about the line $$x=y$$, the semi major axis is along this line. So to obtain the length of semi major axis, we substitute $$σ_2 = σ_1$$. And it gives:

$$2 σ_1^2 - 2μ σ_1^2 = (Syt)^2$$

Therefore,

$$σ_1 = \frac{Syt}{\sqrt{2(1-μ)}}$$

But various textbooks suggest that the value is:

My derivation is having the term "2" which the textbooks are missing. What is the error/mistake here?

• For one thing, you introduced an extra "^2" when you combined the first two terms of the first equation. – Dave Tweed Jun 22 at 10:12

You calculated the $$\sigma_1$$ = $$\sigma_2$$ at the end of the major axis. Calculate the semi-major axis as distance to origin, $$a ={(2 \sigma_1}^2)^{1/2}$$.
In your first equation, you left off square in the $$\sigma$$ terms.