Length of semi major axis for the ellipse in "Total Strain Energy theory"

Question

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?

Answer 1

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.