Component-wise models for static, dynamic and aeroelastic analyses of metallic and composite aerospace structures

In the framework of structural mechanics, the classical beam theories that are commonly adopted in many applications may be affected by inconsistencies, because they are not able to foresee higher-order phenomena, such as elastic bending/shear couplings, restrained torsional warping and 3D strain effects. Depending on the problem, those limitations can be overcome by using more complex and computationally expensive 2D and 3D models or, alternatively, by adopting refined beam models, to which many scientists have dedicated their research over the last century. % One of the latest contributions to the development of advanced models, including variable kinematic beam theories, is the Carrera Unified Formulation (CUF), which is the main subject of the research discussed in this thesis. According to CUF, the 3D displacement field can be expressed as an arbitrary expansion of the generalized displacements. Depending on the choice of the polynomials employed in the expansion, various classes of beam models can be implemented. In this work, for instance, Taylor-like and Lagrange polynomials are adopted. The former choice leads to the so-called TE (Taylor Expansion) beam models, whereas LE (Lagrange Expansion) beam models with only pure displacement variables are obtained by interpolating the problem unknowns by Lagrange polynomials. The strength of CUF lies in the fact that, independently of the choice of the polynomials, the governing equations are written in terms of fundamental nuclei, which are invariant with the theory class and order. In this thesis, both strong and weak form governing equations for arbitrarily refined CUF models are derived. Subsequently, exact closed-form and approximate solutions are sought. Exact solutions of any beam model with arbitrary boundary conditions are found by formulating a frequency-dependant Dynamic Stiffness (DS) matrix and by using the Wittrick-Williams algorithm to carry out the resulting transcendental eigenvalue problem for free vibration analysis. Conversely, a linear eigenvalue problem is also derived by approximating the strong form governing equations by Radial Basis Functions (RBFs). On the other hand, weak form solutions are discussed by Finite Element Method (FEM), which still deserves important attentions due to its versatility and numerical efficiency. The various problems of the mechanics are addressed, including static, free vibration and dynamic response problems. Based on CUF and the proposed numerical methods, advanced methodologies for the analysis of complex structures, such as aircraft structures and civil engineering constructions, are developed. Those advanced techniques make use of the Component-Wise (CW) and the Multi-Line approaches. The CW method exploits the natural capability of the LE CUF beam models to be assembled at the cross-section level. This characteristic allows the analyst to use only CUF beam elements to model each component (e.g., stringers, panels and ribs) of the structure and purely physical surfaces are employed to construct the mathematical models. In the ML framework, on the other hand, each component of the structure is modelled via TE beam elements of arbitrary order. Compatibility of displacements between two or more components is then enforced through the Lagrange multipliers method. The second part of this thesis deals with aeroelasticity. In particular, the Vortex (VLM) and the Doublet Lattice Methods (DLM) are employed and extended to CUF to develop aeroelastic models. VLM is used to model the steady contribution in the aerodynamic model, whereas DLM provides the unsteady contribution in the frequency domain. The infinite plate spline approach is adopted for the mesh-to-mesh transformation. Finally, the g-method is described as an effective means for the formulation of the flutter stability problem. Particular attention is given to the extension of this methodology to exact DS solutions of CUF beams. Simplified, discrete, dynamic gust response analysis by refined beam models is also discussed. In this work, vertical gusts and one-minus-cosine idealization is addressed. Accordingly, gust loads in terms of time-dependent load factors are formulated. Subsequently, the mode superposition method is briefly introduced in order to solve the linear dynamic response problem in the time domain by using both weak and strong form solutions of CUF models. In the final part of the work, extensions of 1D CUF models for Fluid-Dynamics problems are carried out. CUF approximation of laminar, incompressible, Stokes flows with constant viscosity was introduced in a recent thesis work and it is here extended to the hierarchical p-version of FEM, which makes use of Legendre-like polynomials to interpolate the generalized unknowns along the 1D computational domain. Finally, the structural, aeroelastic and fluid-dynamics formulations are validated by discussing some selected results. In particular, regarding structures, the efficiency of the various numerical approaches when applied to CUF is investigated and simple to complex problems are considered, including metallic and composite wings. The aeroelastic analyses show that classical beam models are not adequate for the flutter detection, and at least a third-order beam model is required. Contrarily, classical beam models can be quite accurate in dynamic gust response analysis if no coupling phenomena occur, i.e. when the response is dominated by only pure bending modes. Regarding fluid-dynamics, it is demonstrated that CUF models can reproduce the results by finite volume codes for both simple Poiseuille and complex non-axisymmetric fluids in cylinders. In general, the capability of the proposed CUF models to provide accurate results with very low computational efforts is firmly highlighted. Similar analyses are possible only by using 3D models, which usually require a number of degrees of freedom that is some two order of magnitude higher

Analysis of reinforced and thin-walled structures by multi-line refined 1D/beam models

This paper focuses the attention on the use of appropriate combinations of refined one-dimensional (1D) beam theories to analyze thin-walled, reinforced structures. The cross-section of a slender body is seen as the sum of different sub-domains. Each sub-domain is subsequently used as the cross-section of a beam discretization. Displacement variables are then expanded around the beam axis of each subdomain by using refined 1D models which are based on the Carrera Unified Formulation. The order of the beam elements can vary in different sub-domains. This subdivision has been called "multi-line" as opposed to the "one-line" approach of classical beam theories. 1D compatibility conditions of the displacements at selected points of the sub-domain interface boundaries are imposed by using Lagrange multipliers. Various problems have been analyzed to highlight the advantages and disadvantages of the present multi-line approach. It is concluded that the multi-line approach appears very effective in the case of thin-walled sections made by locally connected walls as well as in the case of reinforced structures

Evaluation of the accuracy of classical beam FE models via locking-free hierarchically refined elements

It is well known that the classical 6-DOF (Degrees of Freedom) beam theories that are incorporated in commercial finite element (FE) tools are not able to foresee higher-order phenomena, such as elastic bending/shear coupling, restrained torsional warping and three-dimensional strain effects. In this work, the accuracy of one-dimensional (1D) finite elements based on the classical theories (Euler-Bernoulli and Timoshenko theories as well as a 6-DOF model including torsion) is evaluated for a number of problems of practical interest and modelling guidelines are given. The investigation is carried out by exploiting a novel hierarchical, locking-free, finite beam element based on the well-known Carrera Unified Formulation (CUF). Thanks to CUF, the FE arrays of the novel beam element are written in terms of fundamental nuclei, which are invariant with respect to the theory approximation order. Thus, results from classical as well as arbitrarily refined beam models can be formally obtained by the same CUF beam element. Linear Lagrange shape functions are used in this paper to interpolate the generalized unknowns and shear locking phenomena are avoided by adopting an MITC (Mixed Interpolation of Tensorial Components) scheme. Different sample problems are addressed, including rectangular and warping-free circular cross-sections as well as thin-walled beams. The results from classical theories and the 6-DOF model are compared to those from higher-order refined beam models, both in terms of displacement and stress fields for various loading conditions. The discussion focuses on the limitations of the commonly used 1D FEs and the need for refined kinematics beams for most of the problems of common interest. The research clearly depicts CUF as a valuable framework to assess FE formulations such as the 6-DOF model herein considered, which is one of the most known and used finite element for the analysis of structures

Accurate response of wing structures to free-vibration, load factors and non-structural masses

Based on the Carrera Unified Formulation (CUF), this work extends variable kinematic finite beam elements to include load factors and non-structural masses for the static and vibration analyses of complex, metallic wing structures. According to CUF, variable kinematic beam theories are formulated in an automatic and hierarchical manner by expressing the displacement field as an arbitrary expansion through generic cross-sectional functions. Both Taylor-like and Lagrange polynomials are used in this paper to develop refined beam kinematics, and the related theories are referred to as TE and LE, respectively. The generalized unknowns of TE models are the beam axis displacements and the N-order displacement derivatives, N being a free parameter of the analysis. Classical beam theories are clearly particular cases of the linear (N=1) TE model. On the other hand, LE models have only pure translational displacements as unknowns. By exploiting this characteristic of LE, a Component-Wise (CW) approach is implemented and used for the analysis of multi-component reinforced-shell structures. Numerical applications are developed by classical finite element procedures, and both static response and free vibration analyses are addressed. Various configurations of a benchmark wing are considered, and the capabilities of the present methodologies when dealing with higher-order effects due to deformable cross-sections and geometrical discontinuities (e.g. underside windows) are evaluated. The attention is focused on the applicability of the present refined beam models to problems involving complex, external inertial loadings. The results are compared to finite element solutions from commercial tools, including full 3D models and models obtained by assembling 2D shell and 1D finite elements

Gasdynamics of rapid and explosive decompressions of pressurized aircraft including active venting

In this paper, a zero-dimensional mathematical formulation for rapid and explosive decompression analyses of pressurized aircraft is developed. Air flows between two compartments and between the damaged compartment and external ambient are modeled by assuming an adiabatic, reversible transformation. Both supercritical and subcritical decompressions are considered, and the attention focuses on intercompartment venting systems. In particular, passive and active vents are addressed, and mathematical models of both swinging and translational blowout panels are provided. A numerical procedure based on an explicit Euler integration scheme is also discussed for multi-compartment aircraft analysis. Various numerical solutions are presented, which highlight the importance of considering the opening dynamics of blowout panels. The comparisons with the results from the literature demonstrate the validity of the proposed methodology, which can be also applied, with no lack of accuracy, to the decompression analysis of spacecraft

Reliability design optimisation of classic composite plates using a CUF-based layerwise approach

Uncertainties in the manufacturing process of structures may arise at any moment of the fabrication chain. In the case of composite structures, such uncertainties may appear in the material elastic properties as a result of the microscale features of such material or even from manufacturing flaws, such as misalignments, fibre waviness, etc. [1] when assembling the final product. As a result of these defects, the structural response of the final structure might be compromised. Therefore, a reliability analysis is needed. In this work, a reliability-based design optimization (RBDO) [2] regarding the linearized buckling behavior of a straight-fibre composite laminate is carried out concerning homogeneous material elastic properties variation. In order to perform such analyses, Carrera Unified Formulation (CUF) [3] is used, according to which structural theories with low-order accuracy to layerwise models can be implemented in a hierarchical and unified manner. These analyses are then used to build a surrogate model based on Polynomial Chaos Kriging (PCK) [4], which substitutes the finite element model and thus accelerates the optimization process. The final scope of the work is to show that layerwise models can help to broaden the design space that other structural approaches may have shrunk, while subjected to the manufacturing constraints that the industry has imposed through the years [5]. References [1] A. Pagani, A.R. Sanchez-Majano. Influence of fibre misalingments on buckling performance of variable stiffness composites using layerwise models and random fields. Mechanics of Advanced Materials and Structures 2020. DOI: https://doi.org/10.1080/15376494.2020.1771485 [2] M. Moustapha, B. Sudret. Surrogate-assisted reliability-based design optimization: a survey and a unified modular framework. Structural and Multidisciplinary Optimization 60, 2157–2176 (2019). DOI: 10.1007/s00158-019-02290-y [3] E. Carrera, M. Cinefra, M. Petrolo, E. Zappino. Finite Element Analysis of Structures through Unified Formulation. Wiley & Sons. 2014. ISBN: 978-1-119-94121-7. [4] R. Schobi, S. Marelli, B. Sudret, UQLab user manual – Polynomial chaos Kriging, Report # UQLab-V1.3-109, Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Switzerland, 2019 [5] G.H.C. Silva, A.P. do Prado, P.H. Cabral, R. De Breuker, J.K.S. Dillinger. Tailoring of a Composite Regional Jet Wing Using the Slice and Swap Method. Journal of Aircraft, 1–15. (2019) DOI:10.2514/1.c03509

Unified theory of one-dimensional structures and flows with applications to biomedical engineering and coupled problems

Advanced theories for structures and viscous flows are discussed in this work. In the first part, onedimensional structural beam theories are formulated by employing the Carrera Unified Formulation (CUF). According to CUF [1], the primary mechanical variables are expressed as an arbitrary expansion series of the generalized unknowns. In this manner, by using an index notation, the governing equations are formulated in terms of fundamental nuclei, whose mathematical expressions are formally independent of the theory order. Advanced beam theories with higher-order kinematics can be, therefore, implemented in an automatic and straightforward manner without the need of ad-hoc assumptions. The finite element method is used to obtain numerical solutions, and the enhanced capabilities of the refined CUF-based beam models are widely demonstrated by comparison with literature results and commercial codes. Various problems are considered, and particular emphasis is given to biomedical engineering applications. Attention is focussed on the extension of CUF to computational fluid-dynamics in the second part of the present work. Similarly as in the structural formulation, CUF is used here to develop, in a unified manner, advanced hierarchical one-dimensional theories for the analysis of Stokes flows with arbitrary accuracy. The accuracy and the numerical efficiency of the present methodology in dealing with laminar, incompressible, viscous, steady flows with arbitrary velocity/pressure fields are established by comparisons with state-of-theart finite volumes tools and analytical solutions (see Fig. 1). The 1D CUF fluid-mechanics models are subsequently coupled with 1D CUF structural theories for the fluid-structure analysis of internal flows within deformable structures [2]. The advanced capabilities of the devised tool are widely supported by the results, which provide enough confidence for future research in this direction