Max Principal Stress⚠ unverified

Understanding the Parameters

• The average stress $\tfrac12(\sigma_x+\sigma_y)$ — the first term is the mean of the two normal stresses, which is also the centre of Mohr's circle (Mohr Circle Center). Principal stresses sit symmetrically about this centre.

• The radius term $\sqrt{(\tfrac{\sigma_x-\sigma_y}{2})^2 + \tau_{xy}^2}$ — this is the radius of Mohr's circle (Mohr's Circle Radius), the maximum in-plane shear stress $\tau_{max}$. The principal stresses are the centre plus and minus this radius, so $\sigma_1 = \text{centre} + \tau_{max}$. A large shear $\tau_{xy}$ or a large difference $\sigma_x-\sigma_y$ swells the circle and raises the peak tension.

• The output $\sigma_1$ — the algebraically largest normal stress. On its plane the shear is zero. For a brittle material (glass, ceramics, a composite matrix, cast iron) this is the quantity that governs failure — the maximum-normal-stress theory — because such materials part along the plane of greatest tension.

• Sign matters. $\sigma_1$ can be compressive if all stresses are compressive; it is simply the most tensile (least compressive) of the principal values.

Derivation (Approaching a Proof)

Consider the stress on a plane rotated by angle $\theta$ from the x-axis. The transformation equations give the normal stress on that plane,

$$\sigma_\theta = \frac{\sigma_x+\sigma_y}{2} + \frac{\sigma_x-\sigma_y}{2}\cos 2\theta + \tau_{xy}\sin 2\theta.$$

The principal planes are where $\sigma_\theta$ is stationary (and, equivalently, where shear vanishes). Setting $d\sigma_\theta/d\theta = 0$:

$$-(\sigma_x-\sigma_y)\sin 2\theta + 2\tau_{xy}\cos 2\theta = 0 \quad\Longrightarrow\quad \tan 2\theta_p = \frac{2\tau_{xy}}{\sigma_x-\sigma_y}.$$

Substituting this angle back (using $\cos 2\theta_p$ and $\sin 2\theta_p$ from the right triangle with opposite $2\tau_{xy}$ and adjacent $\sigma_x-\sigma_y$) yields the extreme normal stresses as the average plus or minus the hypotenuse-derived radius:

$$\sigma_{1,2} = \frac{\sigma_x+\sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2}.$$

Taking the plus sign gives the maximum, $\sigma_1$. $\blacksquare$ This is exactly the eigenvalue problem for the symmetric 2-D stress tensor $\left(\begin{smallmatrix}\sigma_x & \tau_{xy}\\ \tau_{xy} & \sigma_y\end{smallmatrix}\right)$ — principal stresses are its eigenvalues, principal directions its eigenvectors.

Dimensional check. The average is a stress; the root is $\sqrt{\text{Pa}^2}=\text{Pa}$; both terms are $\text{Pa}$, so $\sigma_1$ is $\text{Pa}$. $\checkmark$

History and Development

• Cauchy and the stress tensor. The idea that the state of stress at a point is a tensor with real principal values on mutually perpendicular planes is due to Augustin-Louis Cauchy (1820s), one of the foundational results of continuum mechanics.

• Mohr's circle. Otto Mohr (1882) gave the graphical construction that makes principal stresses and maximum shear immediately visible — the centre and radius that appear directly in this formula (Mohr Circle Center, Mohr's Circle Radius). It remains a standard teaching and checking tool.

• Which theory for which material. Maximum principal stress governs brittle failure; ductile metals instead follow distortion-energy (Von Mises Stress 2D) or maximum-shear (Tresca Stress). Knowing which applies is the crux of choosing a failure criterion.

Related Concepts: Min Principal Stress, Mohr Circle Center, Mohr's Circle Radius, Principal Stresses, Von Mises Stress 2D, Tresca Stress, Wing Bending Stress

Notes: Registry calculator max-principal-stress (unverified). Correct as shipped. $\sigma_1 = $ centre $+$ radius of Mohr's circle; governs brittle fracture and fatigue-crack initiation (peak-tension driven). Paired with Min Principal Stress ($\sigma_2 = $ centre $-$ radius). All defaults $1.0$.