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# Stress distribution in a long solenoid

A very long, thick solenoid has a uniform distribution of circumferential current. The magnetic flux density and stress distribution in the solenoid has to be calculated.

Problem Type:
Axisymmetric problem of magneto-structural coupling.

Geometry:

Given:
Dimensions R1 = 1 cm, R2 = 2 cm;
Relative permeability of air and coil μ = 1;
Current density j = 105 A/m2;
Young's modulus E = 1.075·1011 N/m2;
Poisson's ratio ν = 0.33.

Problem:
Calculate the magnetic flux density and stress distribution in a solenoid.

Solution:
Since none of physical quantities varies along z-axis, a thin slice of the solenoid could be modeled. The axial length of the model is arbitrarily chosen to be 0.2 cm. Radial component of the flux density is set equal to zero at the outward surface of the solenoid. Axial displacement is set equal to zero at the side edges of the model to reflect the infinite length of the solenoid.

Results:
 Magnetic flux density distributionin a long solenoid: Mechanical stress distributionin a long solenoid:

Magnetic flux density and circumferential stress at r = 1.3 cm:
 Bz (T) σq (N/m) Reference* 8.796·10-3 97.407 QuickField 8.798·10-3 96.71

See the Coupl1MS.pbm and Coupl1SA.pbm problems for magnetic and structural analysis respectively.

* Reference: F.A. Moon, "Magneto-Solid Mechanics", John Wiley & Sons, N.Y., 1984, Chapter 4.

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