Generalized analytical virtual-work thermoelastic framework for straight and curved beams under arbitrary boundary conditions and temperature-dependent materials

A comprehensive understanding of thermoelastic behavior in beam-type structural components is essential for reliable performance in modern civil, mechanical, and aerospace systems exposed to non-uniform and evolving thermal environments. This study introduces a unified and fully generalized analytical framework for predicting thermally induced displacements in both straight and circularly curved beams under arbitrary boundary conditions, including semi-rigid, elastic, and fully fixed supports. By extending the principle of virtual work, closed-form displacement functions are derived for beams subjected simultaneously to uniform temperature rise and through-thickness thermal gradients, while rigorously incorporating temperature-dependent material properties. The proposed formulation captures the markedly different thermoelastic responses of statically determinate and indeterminate systems, revealing that the former exhibit geometry-governed behavior independent of sectional stiffness, whereas the latter are highly sensitive to mechanical properties and support rigidity. A generalized six-spring end-support model is developed, enabling seamless representation of a broad spectrum of boundary conditions and offering new insight into how partial fixity and foundation stiffness modulate thermal deformation and support reactions. The analytical solutions are further extended to curved beams, demonstrating consistent trends between straight and arched geometries and showing that boundary constraint severity governs whether thermal loads induce deformation or solely internal stress. Parametric investigations confirm that temperature-dependent elastic modulus and thermal expansion can alter predicted displacements by up to 60%, underscoring the necessity of realistic material modeling for high-temperature applications. The proposed framework unifies disparate thermoelastic beam formulations into a single adaptable model, providing a powerful benchmark for numerical methods and a practical tool for the design of thermally resilient structural systems.