Advances in Cryogenic Engineering pp 490-499 | Cite as

# Stress Analysis of Nonhomogeneous Superconducting Solenoids

## Abstract

Recent advances in the design of large, high-field, superconducting magnets have shown a need for a better understanding of the structural behavior of such coils under load. This paper describes an analysis technique and related computer program developed to study axisymmetrically loaded solenoid coils constructed from interwoven layers of conductor, insulation, and reinforcing materials. Proposed extensions to this study are described that will be used to evaluate the effects of asymmetric and out-of-plane coil loads.

## Notation

- \(\delta _{Ai}^{\left( n \right)}\)
function of material properties and geometry

*C*(^{n})function of material properties, applied load and geometry

*N*number of outermost layer

*p*_{n}interface pressure between layers — 1 and

*n**P*_{n}winding preload in

*n*th layer*q*_{n}interface pressure induced by winding preload in

*n*th layer- R
_{k}, R_{n} inner radius of layer

*W*_{k}length of preloaded layer

*Xn*electromagnetic body force in

*n*th layer

## Greek Symbols

- \(\delta _{11}^{\left( n \right)} ,\,\delta _{k1}^{\left( n \right)} ,\,\delta _{Rn}^{\left( n \right)}\)
radial deflection at inner surface of

*n*th layer- \(\delta _{12}^{\left( n \right)} ,\,\delta _{k2}^{\left( n \right)} ,\,\delta _{R_{n + 1} }^{\left( n \right)}\)
radial deflection at outer surface of nth layer

- \(\delta _{Ai}^{\left( n \right)}\)
radial deflection due to unknown interface pressures

*p*_{ n }and*p*_{n+1}acting on*n*th layer- \(\delta _{Bi}^{\left( n \right)}\)
radial deflection due to differential temperature in the

*n*th layer- \(\delta _{Ci}^{\left( n \right)}\)
radial deflection due to electromagnetic body force in the

*n*th layer- \(\delta _{Di}^{\left( n \right)}\)
radial deflection due to applied internal or external pressure (layers n = 1 or

*n = N*only)- θ
angle in hoop direction

- \(\sigma _{{\rm R}_{{\rm 11}} }^{\left( {\rm n} \right)} ,\,\sigma _{Rk_1 }^{\left( n \right)} ,\sigma _{R_n }^{\left( n \right)} \)
radial stress at inner surface of

*n*th layer- \(\sigma _{\,12}^{\left( n \right)} ,\sigma _{Rk2}^{\left( n \right)} ,\sigma _{R_{n + 1} }^n \)
radial stress at outer surface of

*n*th layer- \(\sigma _{\theta 11}^{\left( n \right)} ,\,\sigma _{\theta k_1 }^{\left( n \right)} ,\,\sigma _{\theta _n }^{\left( n \right)} \)
hoop stress at inner surface of

*n*th layer- \(\sigma _{\theta 1}^{\left( n \right)} ,\,\sigma _{\theta k_2 }^{\left( n \right)} ,\,\sigma _{\theta _{n + 1} }^{\left( n \right)} \)
hoop stress at outer surface of

*n*th layer

## Subscripts

*i*used to denote inner (

*i*= 1) or outer (*i*= 2) surface of layer*k*number of layer with applied winding preload

*n*layer number (

*n*= 1 for inner layer)

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## References

- 1.L. M. Lontai and P. G. Marston, in:
*Proceedings of Intern. Symposium on Magnet Technology*, Conf. 650922, NTIS, Springfield, Virginia (1965), p. 723.Google Scholar - 2.A. J. Middleton and C. W. Trowbridge, in:
*Proceedings of Second Intern. Conference on Magnetic Technology*, Rutherford Laboratory, Didcot, England (1967), p. 140.Google Scholar - 3.W. F. Westendorp, “Balancing of Magnetic Stresses and Winding Stresses in Superconducting Coils,” General Electric Company Rept. No. 71-C-161, Schenectady, New York (June 1971).Google Scholar
- 4.F. B. Seely and J. O. Smith,
*Advanced Mechanics of Materials, 2nd ed*., John Wiley and Sons, New York (1952).Google Scholar - 5.N. E. Johnson, “The Structural Analysis of Non-Homogeneous Solenoids Using the STANSOL Computer Program,” Rept. No. MRI-C2754-TR-3, Mechanics Research, Inc., Oak Ridge, Tennessee (June 23, 1975).Google Scholar