Asymptotically correct isoenergetic formulation of geometrically nonlinear anisotropic inhomogeneous stiffened plates

Abstract

Stiffener Reinforced structures are widely used in many engineering disciplines like aerospace, marine, civil and automotive. By adding reinforcement to a structure in the form of a stiffener, the mechanical properties of the structure especially stiffness and fracture toughness are greatly improved without much increase in weight and cost. However, stiffened structures undergo a localized shift in the neutral plane due to the geometric discontinuities introduced by the stiffener. This shift necessitates careful attention when analyzing these structures. Conducting full-scale 3D finite element analyses (FEA) for such structures can be computationally expensive, particularly during the design optimization phase. To address this computational burden, reduced-order models are often favored. The literature employs various approaches utilizing reduced-order models for plates and beams to analyze stiffened structures efficiently. However, selecting appropriate reduced-order beam and plate models is crucial, as the accuracy and efficiency of the analysis heavily rely on this selection. Furthermore, ensuring compatibility between the beam and plate models and accounting for geometric discontinuities pose challenges in accurately modeling stiffened plates. Most of the approaches found in the literature are based on ad hoc and a priory assumptions and have their own advantages and shortcomings. This research addresses this challenge by developing a reduced-order model for the stiffened plates that captures their deformation behavior accurately with significantly reduced computational cost. The primary objective is to create a systematic and mathematically sound approach for analyzing various stiffened plate configurations, enabling efficient design optimization. The core of the work lies in establishing an asymptotically correct reduced-order plate theory for anisotropic plates. This is achieved by leveraging variational calculus and the concept of isoenergetics. The theory was derived using first principles avoiding any ad hoc and a priory assumptions. The derived model accurately captures the deformation characteristics while significantly reducing computational complexity compared to fullscale 3D finite element analysis (FEA). The framework developed for the single-layer anisotropic plates is then extended to handle more complex scenarios. The plate theory is modified to incorporate analysis of multilayered composite plates and functionally graded plates, reflecting real-world engineering structures with tailored properties. This allows for the analysis of plates with varying stiffness and strength profiles throughout their thickness. Finally, the developed the developed reduced order plate theory is equipped to handle stiffened plates, a crucial component in many engineering applications. The model can analyze both symmetric and asymmetric stiffener configurations, providing valuable insights into the influence of stiffener orientation, size, and number on the overall plate behavior. Key contributions of the present work are (a) First principles-based derivation of the reduced order 2D model from the 3D model energy (b) No dependency on the preassumed kinematics, (c) A systematic ordering scheme is employed utilizing the geometry of the structure and a bound on the maximum value of the strains. (d) The plane stress condition is a natural outcome of the present mathematical framework (e). The higherorder derivatives appearing during the dimensional reduction process were dealt with by a novel isoenergetic approach, reducing the computational complexities. Overall, this research work presents a powerful tool for engineers by providing a mathematically rigorous and computationally efficient framework for analyzing and optimizing stiffened plates. The developed reduced order model allows for a deeper understanding of plate deformation behavior under various loading conditions, ultimately leading to improved design decisions for a diverse range of engineering applications.