A Koiter reduction technique for the nonlinear thermoelastic analysis of shell structures prone to buckling

Abstract

The multi-modal Koiter method is a reduction technique for estimating quickly the nonlinear buckling response of structures under mechanical loads requiring a fine discretization. The reduced model is based on a quadratic approximation of the full model using a few linear buckling modes and their second order corrections, followed by the projection of the equilibrium equations onto the modal subspace. In this article, the method is reformulated for geometrically nonlinear thermoelastic analyses of shell structures. The starting point is an isogeometric solid-shell discretization with an accurate modeling of thermal strains, temperature-dependent materials and general temperature distributions. The equilibrium path is defined as a displacement versus temperature amplifier curve, while mechanical loads are kept constant. The strain energy nonlinearity with respect to the temperature amplifier is the main obstacle in the definition of an accurate reduced model. This task is achieved by coherent asymptotic expansions in mixed form using independent stress variables at the integration points and accurate linear buckling modes obtained by a two-point mixed eigenvalue problem. Structures made of isotropic and multi-layered composite materials, including variable angle tow laminates, are considered to show the accuracy of the proposed method.

Nonlinear Thermal Buckling Analysis of Thin‐walled Structures: Solid‐shell Discretization, Continuation Method and Reduction Technique

“…A linear temperature distribution 𝑇 ̂(𝜁) = 𝑇 𝑀 + 2𝜁 ℎ 𝑇 𝐿 is assumed through the shell thickness. Additional tests are reported in [12].…”

“…A reduction technique based on the Koiter method was also proposed and applied to buckling problems. Tests on composite shells and thin-walled structures are available in [11,12].…”

“…Newton iterations can be then used to solve the extended nonlinear system of equations (12). A step size that changes adaptively as a function of the curve nonlinearity can be conveniently adopted to minimise the iterative burden and to describe the equilibrium path properly.…”

“…where 𝒖 𝑓 [𝜆] = 𝝀 𝒖 ̂ is the solution along the initial path tangent, 𝒗[𝒕] = ∑ 𝑡 𝑖 𝒗̇𝑖 𝒊 is a combination of buckling modes corresponding to the first 𝑚 linearized critical temperatures and 𝒘 = ∑ ∑ 𝑡 𝑖 𝑡 𝑗 𝒘 𝑖𝑗 + 𝝀 𝟐 𝒋 𝒊 𝒘 ̂ is the corresponding orthogonal second order correction. We refer to [12] for the consistent evaluation of the modal basis.…”

Geometrically nonlinear thermoelastic analysis of shells: modelling, incremental-iterative solution and reduction technique

“…A reduction technique based on the Koiter method was also proposed and applied to buckling problems. Tests on composite shells and thin-walled structures are available in [6,7].…”