SME
Advanced Solid Mechanics (Solid Mechanics II) · Problem type
Octahedral stresses
Use this for “The principal stresses are 65, 15 and −30 MPa. Find the octahedral normal and shear stresses.”
Diagram
Sorted:
σ₁
=
65,
σ₂
=
15,
σ₃
=
−30
MPa.
σoct
=
σ₁ + σ₂ + σ₃
3
=
16.7
MPa,
the
mean
stress;
τoct
=
(
1
3
)√(
(σ₁ − σ₂)
2
+
(σ₂ − σ₃)
2
+
(σ₃ − σ₁)
2
)
=
38.8
MPa.
The octahedral planes’ point lies in the shaded band between the circles.
This is an example. Type your own values to start a new problem.
σoct (MPa)
=
σ₁ (MPa)
+
σ₂ (MPa)
+
(
σ₃ (MPa)
)
3
9
×
(
τoct (MPa)
)
2
=
(
σ₁ (MPa)
−
σ₂ (MPa)
)
2
+
(
σ₂ (MPa)
−
(
σ₃ (MPa)
)
)
2
+
(
σ₃ (MPa)
−
σ₁ (MPa)
)
2
The second line is τoct = (1 ÷ 3) × √((σ₁ − σ₂)² + (σ₂ − σ₃)² + (σ₃ − σ₁)²), squared and multiplied by 9.
Formulas
σoct
=
σ₁
+
σ₂
+
σ₃
3
+16.6667
=
65 + 15 + (−30)
3
τoct
=
(
1
3
)
×
√(
(
σ₁
−
σ₂
)
2
+
(
σ₂
−
σ₃
)
2
+
(
σ₃
−
σ₁
)
2
)
38.8015
=
(
1
3
)
×
√(
(65 − 15)
2
+
(15
−
(−30))²
+
(−30 − 65)
2
)
Assumptions
•
The octahedral planes make equal angles with the three principal axes (normal 1/√3 each).
•
σoct = I₁ ÷ 3 is the mean stress; it changes volume, not shape.
•
τoct measures distortion: von Mises yielding is τoct reaching (√2 ÷ 3)σY.
•
Tension is +; the formulas give the same values whatever order σ₁, σ₂, σ₃ are typed in.
Step-by-step
WE KNOW
σ₁ = +65 MPa, σ₂ = +15 MPa, σ₃ = −30 MPa
FIND
octahedral normal stress (σoct), octahedral shear stress (τoct)
Step 1 · Find octahedral normal stress (σoct)
Use σoct = (σ₁ + σ₂ + σ₃) ÷ 3
The
octahedral
normal
stress
is
the
mean
of
the
three
principal
stresses,
I₁
3
.
σoct
=
65 + 15 + (−30)
3
σoct
=
50
3
σoct = +16.6667 MPa
Step 2 · Find octahedral shear stress (τoct)
Use τoct = (1 ÷ 3) × √((σ₁ − σ₂)² + (σ₂ − σ₃)² + (σ₃ − σ₁)²)
Square the three differences of principal stresses, add them, take the root and divide by 3.
τoct
=
(
1
3
)
×
√(
(65 − 15)
2
+
(15
−
(−30))²
+
(−30 − 65)
2
)
τoct
=
1
3
×
√(
50
2
+
45
2
+
(−95)
2
)
τoct
=
1
3
×
√
2,500 + 2,025 + 9,025
τoct
=
1
3
×
√
13
,550
τoct
=
1
3
×
116.4
τoct = 0.3333 × 116.4
τoct = 38.8015 MPa
Check
Put the numbers back in. Both sides should match.
+16.6667
=
65 + 15 + (−30)
3
✓
38.8015
=
(
1
3
)
×
√(
(65 − 15)
2
+
(15
−
(−30))²
+
(−30 − 65)
2
)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.