SME
Advanced Solid Mechanics (Solid Mechanics II) · Problem type
Von Mises and Tresca from three principal stresses
Use this for “The principal stresses are 80, 20 and −40 MPa in steel that yields at 300 MPa. Find σᵥₘ, σ₁ − σ₃ and both safety factors.”
Diagram
Sorted: σ₁ = 80, σ₂ = 20, σ₃ = −40 MPa.
The
biggest
circle,
from
σ₃
to
σ₁,
gives
τₘₐₓ
=
σ₁ − σ₃
2
=
60
MPa.
Tresca:
yield
when
τₘₐₓ
reaches
σY
2
=
150
MPa
(the
dashed
line),
n
=
2.5.
Von Mises: σᵥₘ = 104 MPa, n = 2.89.
Von Mises stress σᵥₘ = 103.923 MPa
Safety factor, von Mises nᵥₘ = 2.8868
Safety factor, Tresca nT = 2.5
This is an example. Type your own values to start a new problem.
2
×
(
σᵥₘ (MPa)
)
2
=
(
σ₁ (MPa)
−
σ₂ (MPa)
)
2
+
(
σ₂ (MPa)
−
(
σ₃ (MPa)
)
)
2
+
(
σ₃ (MPa)
−
σ₁ (MPa)
)
2
nᵥₘ
=
σY (MPa)
σᵥₘ (MPa)
nT
=
σY (MPa)
σ₁ (MPa)
−
(
σ₃ (MPa)
)
The first line is σᵥₘ = √(((σ₁ − σ₂)² + (σ₂ − σ₃)² + (σ₃ − σ₁)²) ÷ 2), squared and multiplied by 2.
σTresca
Tresca stress
calculated · MPa
Formulas
σᵥₘ
=
√((
(
σ₁
−
σ₂
)
2
+
(
σ₂
−
σ₃
)
2
+
(
σ₃
−
σ₁
)
2
)
÷
2)
103.923
=
√((
(80 − 20)
2
+
(20
−
(−40))²
+
(−40 − 80)
2
)
÷
2)
σTresca
=
σ₁
−
σ₃
120 = 80 − (−40)
nᵥₘ
=
σY
σᵥₘ
2.8868
=
300
103.923
nT
=
σY
σTresca
2.5
=
300
120
Assumptions
•
σ₁ ≥ σ₂ ≥ σ₃, numbered from the largest; tension is positive.
•
A ductile metal yields when σᵥₘ, or σ₁ − σ₃ for Tresca, reaches σY from a tensile test.
•
Tresca never reads lower than von Mises, so it is the safer of the two.
Step-by-step
WE KNOW
σ₁ = 80 MPa, σ₂ = 20 MPa, σ₃ = −40 MPa, σY = 300 MPa
FIND
von Mises stress (σᵥₘ), Tresca stress (σTresca), safety factor, von Mises (nᵥₘ), safety factor, Tresca (nT)
Step 1 · Find von Mises stress (σᵥₘ)
Use σᵥₘ = √(((σ₁ − σ₂)² + (σ₂ − σ₃)² + (σ₃ − σ₁)²) ÷ 2)
Square the three differences, add them, halve, then take the square root.
σᵥₘ
=
√((
(80 − 20)
2
+
(20
−
(−40))²
+
(−40 − 80)
2
)
÷
2)
σᵥₘ
=
√((
60
2
+
60
2
+
(−120)
2
)
÷
2)
σᵥₘ
=
√(
3,600 + 3,600 + 14,400
2
)
σᵥₘ
=
√(
21,600
2
)
σᵥₘ
=
√
10
,800
σᵥₘ = 103.923 MPa
Step 2 · Find Tresca stress (σTresca)
Use σTresca = σ₁ − σ₃
The diameter of the biggest Mohr circle, twice the largest shear stress.
σTresca = 80 − (−40)
σTresca = 120 MPa
Step 3 · Find safety factor, von Mises (nᵥₘ)
Use nᵥₘ = σY ÷ σᵥₘ
How many times σᵥₘ fits under σY; below 1 the part yields.
nᵥₘ
=
300
103.923
nᵥₘ = 2.8868
Step 4 · Find safety factor, Tresca (nT)
Use nT = σY ÷ σTresca
How
many
times
σ₁
−
σ₃
fits
under
σY;
below
1
the
part
yields.
nT
=
300
120
nT = 2.5
Check
Put the numbers back in. Both sides should match.
103.923
=
√((
(80 − 20)
2
+
(20
−
(−40))²
+
(−40 − 80)
2
)
÷
2)
✓
120 = 80 − (−40) ✓
2.8868
=
300
103.923
✓
2.5
=
300
120
✓
Topic check
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