Stress Transformation and Material Failure

Plane Stress Transformation

For a 2D stress state (σx, σy, τxy), the stresses on a plane at angle θ:

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

$\tau_{x'y'} = -\frac{\sigma_x - \sigma_y}{2}\sin 2\theta + \tau_{xy}\cos 2\theta$

Principal Stresses

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

Principal plane angle: $\tan 2\theta_p = \frac{2\tau_{xy}}{\sigma_x - \sigma_y}$

On principal planes, shear stress is zero.

Maximum Shear Stress

$\tau_{max} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2} = \frac{\sigma_1 - \sigma_2}{2}$

Maximum shear stress planes are at 45° to the principal planes.

Mohr's Circle

Mohr's circle is a graphical method for stress transformation:

Construction:

1. Plot point A: (σx, τxy) — note: some conventions use (σx, -τxy)
2. Plot point B: (σy, -τxy)
3. Center C is at ((σx + σy)/2, 0)
4. Radius R = √((σx - σy)/2)² + τxy²)
5. Principal stresses: σ₁ = C + R, σ₂ = C - R
6. Maximum shear stress: τmax = R

Failure Theories

Maximum Shear Stress (Tresca) — Ductile Materials

Failure when: τmax ≥ σy/2

Or equivalently: |σ₁ - σ₂| ≥ σy

Distortion Energy (von Mises) — Ductile Materials

Failure when: $\sigma_{VM} = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2} \geq \sigma_y$

For general 3D stress: $\sigma_{VM} = \frac{1}{\sqrt{2}}\sqrt{(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2}$

Von Mises is more accurate than Tresca for ductile materials and is used more often in practice.

Maximum Normal Stress — Brittle Materials

Failure when: σ₁ ≥ σut (ultimate tensile) or |σ₂| ≥ σuc (ultimate compressive)

Column Buckling (Euler's Formula)

Critical buckling load for long, slender columns:

$P_{cr} = \frac{\pi^2 EI}{(KL)^2}$

Or in terms of critical stress: $\sigma_{cr} = \frac{\pi^2 E}{(KL/r)^2}$

where KL/r is the slenderness ratio and r = √(I/A) is the radius of gyration.

Effective Length Factor K

Always use the smallest I (about the weakest axis) for buckling calculations — the column buckles about its weakest axis.

Stress Concentrations

Near geometric discontinuities (holes, notches, fillets), stress is amplified:

$\sigma_{max} = K_t \cdot \sigma_{nominal}$

where Kt is the stress concentration factor (always ≥ 1).

For a small circular hole in a wide plate under tension: Kt ≈ 3.

Fatigue and Creep