This paper presents a closed form solution of E. Reissner's shallow spherical shell equations for the thermal bowing of a curved circular mirror, which is kinematically supported and subjected to a generalized nonuniform thermal load distribution. A variety of nonsymmetric functional forms of thermal moment and thermal force distributions can be constructed from the assumed form of the thermal load distribution. Several test cases of the closed form solution are compared with independent, Nastran based, finite element solutions. The comparison shows excellent correlation between the two methods in all cases. However, the closed form approach is significantly more efficient, convenient and cost-effective. The closed form solution is applied to the development of normalized thermal deflection curves and related RMS figure errors in curved circular mirrors. The solution and the data are also applicable for the thermal bowing component of the thermal distortion of light-weighted curved circular mirrors.
Pravin K. Mehta,
"Nonsymmetric Thermal Bowing Of Curved Circular Mirrors", Proc. SPIE 0748, Structural Mechanics of Optical Systems II, (3 June 1987); doi: 10.1117/12.939831; https://doi.org/10.1117/12.939831