Mechanics of Materials

123/3337–11FE Reference Handbook 10.4 · Mechanics of Materials · Failure Theories

Handbook formula

von Mises (Distortional Energy) Stress

σ_{v} = σ_{x}^{2} - σ_{x} σ_{y} + σ_{y}^{2} + 3 τ_{x y}^{2}

Yield when σ_v ≥ Sy (or σ1,σ2,σ3 form). Ductile metals on FE Civil (steel pins, shafts) use von Mises or max-shear (Tresca). Tresca is more conservative.

$σ_{v}$: Equivalent von Mises stress
$S y$: Yield strength

Step-by-step solved example

σx=80 MPa, σy=0, τ=30 MPa. σ_v?

σ_v from the in-plane principals / Mohr radius plus the third principal.

1. Plug in
σ_v=√(80²−0+0+3×900)=√(6400+2700)=√9100=95.4 MPa.
$σ_{v} = 95.4 MPa$
2. Compare
Yield if 95.4 ≥ Sy.

Answer: σ_v ≈ 95 MPa

10 practice questions

0/10 correct

1.Pure shear τ (σx=σy=0): σ_v =

2.Uniaxial σ: σ_v =

3.Ductile yield (von Mises) vs brittle (max-normal) — FE uses

4.σx=σy=σ, τ=0: σ_v =

5.FS = Sy / σ_v for

6.Plane stress σz=0 is already built into

7.Hydrostatic stress (σ,σ,σ) has σ_v

8.If σ_v=180 and Sy=360, FS=

9.Principal-stress form is

10.Compared with Tresca at the same Sy, von Mises