SME
Advanced Solid Mechanics (Solid Mechanics II) · Problem type
Plane strain: the out-of-plane stress
Use this for “A long steel part is held so εz = 0. With σₓ = 100 MPa, σᵧ = 50 MPa and ν = 0.3, find σz.”
Diagram
Walls hold both z faces, so εz = 0 and σz = ν(σₓ + σᵧ) = 0.3 × (+100 MPa + +50 MPa) = 45 MPa.
This is an example. Type your own values to start a new problem.
σz (MPa)
=
ν
(
σₓ (MPa)
+
σᵧ (MPa)
)
Formulas
σz
=
ν
×
(σₓ
+
σᵧ)
+45 = 0.3 × (100 + 50)
Assumptions
•
Plane strain: the body is long or held at its ends, so εz = 0 (dams, tunnels, long pipes).
•
Setting εz = 0 in Hooke’s law gives σz = ν(σₓ + σᵧ); tension is +.
•
Linear elastic and isotropic; steel: ν = 0.3.
Step-by-step
WE KNOW
σₓ = +100 MPa, σᵧ = +50 MPa, ν = 0.3
FIND
normal stress on z (σz)
Step 1 · Find normal stress on z (σz)
Use σz = ν × (σₓ + σᵧ)
With
εz
=
σz −
ν
(σₓ + σᵧ)
E
held
at
0,
the
bracket
must
be
0.
σz = 0.3 × (100 + 50)
σz = 0.3 × 150
σz = +45 MPa
Check
Put the numbers back in. Both sides should match.
+45 = 0.3 × (100 + 50) ✓
Topic check
See what you know: 5 questions on this topic
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