Stress Transformation and Material Failure

Plane Stress Transformation

A point in a loaded body has a stress state (σx, σy, τxy). Rotate the element by angle θ and the stresses change:

σx' = (σx+σy)/2 + [(σx−σy)/2]cos2θ + τxy sin2θ τx'y' = −[(σx−σy)/2]sin2θ + τxy cos2θ

The quantity (σx+σy)/2 is the average normal stress and is invariant — it does not change with rotation. The goal is to find the orientation giving the largest normal stress (used for brittle failure) or the largest shear stress (used for ductile failure).

Principal Stresses and Maximum Shear

The principal stresses are the extreme normal stresses, occurring on planes where shear vanishes:

σ₁,₂ = (σx+σy)/2 ± √[((σx−σy)/2)² + τxy²]

The principal-plane angle is tan2θp = 2τxy/(σx − σy). The maximum in-plane shear stress equals the radical:

τ_max = √[((σx−σy)/2)² + τxy²] = (σ₁ − σ₂)/2

and acts on planes 45° from the principal planes. On those max-shear planes the normal stress equals the average (σx+σy)/2 — shear and principal planes are never the same plane.

Worked example — principal stresses and τ_max

Given σx = 80 MPa, σy = −40 MPa, τxy = 30 MPa:

1. Average = (80 + (−40))/2 = 20 MPa.
2. R = √[((80−(−40))/2)² + 30²] = √[(60)² + (30)²] = √(3,600 + 900) = √4,500 = 67.1 MPa.
3. σ₁ = 20 + 67.1 = 87.1 MPa; σ₂ = 20 − 67.1 = −47.1 MPa.
4. τ_max = R = 67.1 MPa.

The radius R is the maximum shear stress — a one-step shortcut once you build Mohr's circle.

Mohr's Circle

Mohr's circle turns the transformation equations into geometry on a σ–τ plot:

1. Center C = ((σx+σy)/2, 0).
2. Radius R = √[((σx−σy)/2)² + τxy²].
3. Principal stresses: σ₁ = C + R (right intercept), σ₂ = C − R (left intercept).
4. Maximum shear: τ_max = R (top/bottom of the circle).
5. Angles on the physical element double on the circle (a 90° physical rotation = 180° on the circle).

Mohr's circle is the fastest way to read off all three results — σ₁, σ₂, τ_max — at once.

Failure Theories for Ductile Materials

Maximum shear stress (Tresca)

Yielding begins when τ_max reaches σy/2, i.e. |σ₁ − σ₂| ≥ σy. Conservative and simple.

Distortion energy (von Mises)

Yielding begins when the equivalent stress reaches σy:

σ_VM = √(σ₁² − σ₁σ₂ + σ₂²) ≥ σy (plane stress)

Von Mises matches test data better than Tresca and is the standard choice in practice. For a pure shear state (σ₁ = −σ₂ = τ), von Mises predicts yield at τ = 0.577σy, whereas Tresca gives the more conservative 0.5σy.

Worked example — von Mises check

Using σ₁ = 87 MPa and σ₂ = −47 MPa from the earlier transformation, with σy = 250 MPa:

σ_VM = √(87² − (87)(−47) + (−47)²) = √(7,569 + 4,089 + 2,209) = √13,867 = 117.8 MPa.

Since 117.8 < 250 MPa, the material does not yield; the factor of safety is FS = σy/σ_VM = 250/117.8 = 2.1.

Maximum normal stress (brittle materials)

Brittle materials such as cast iron fail when σ₁ ≥ σ_ut (ultimate tensile) or |σ₂| ≥ σ_uc, not by yielding. For brittle parts the largest principal stress, not the shear, governs — which is why a chalk stick twisted in torsion fractures on a 45° helix (the plane of maximum tension).

Column Buckling — Euler's Formula

Long, slender columns fail by elastic instability long before the material yields:

P_cr = π²EI / (KL)²

or as a critical stress σ_cr = π²E/(KL/r)², where r = √(I/A) is the radius of gyration and KL/r is the slenderness ratio.

Worked example — Euler buckling load

A 3 m column, pinned–pinned (K = 1.0), E = 200 GPa, I = 5×10⁶ mm⁴:

1. KL = 1.0 × 3,000 = 3,000 mm.
2. P_cr = π²EI/(KL)² = π²(200,000)(5×10⁶)/(3,000²)
3. = (9.87)(1×10¹²)/(9×10⁶) = 1,097 kN.

Effective Length Factor K

Two non-negotiable rules: (1) use the smallest I — a column buckles about its weakest axis; (2) Euler's formula is valid only for slender columns (high KL/r); short columns fail by yielding instead, governed by σ = P/A.

Stress Concentrations

Near holes, notches, and fillets, local stress is amplified:

σ_max = K_t · σ_nominal

where K_t ≥ 1 is the stress concentration factor. For a small circular hole in a wide plate under tension, K_t ≈ 3 — the stress at the hole edge is triple the nominal. Ductile materials yield locally and redistribute, so K_t matters most for brittle materials and fatigue.

Fatigue and Creep

Quick reference

• σ₁,₂ = avg ± R; τ_max = R = (σ₁−σ₂)/2; avg = (σx+σy)/2.
• Von Mises: σ_VM = √(σ₁²−σ₁σ₂+σ₂²) ≥ σy (ductile). Brittle: σ₁ ≥ σ_ut.
• Buckling: P_cr = π²EI/(KL)²; smallest I; K = 0.5/0.7/1.0/2.0.