Multimode Nonlinear Vibration Analysis of Stiffened Functionally Graded Double Curved Shells in a Thermal Environment

The motivation of the current work is to develop a multi-modal analysis of the nonlinear response of stiffened double curved shells made of functionally graded materials under thermal loads. The formulation is based on the first order shear deformation shell theory in conjunction with the von Kármán geometrical nonlinear strain-displacement relationships. The nonlinear equations of motion of stiffened double curved shell based on the extended Sanders’s theory were derived using Galerkin’s method. The resulting system of infinite nonlinear ordinary differential equations, that includes both cubic and quadratic nonlinear terms, was solved using a nonlinear dynamic software XPPAUT to obtain the force-amplitude relationship. The effect of both, longitudinal and transverse stiffeners, was considered using the Lekhnitsky’s technique and the material properties are temperature dependent and vary in the thickness direction according to the linear rule of mixture. In order to obtain accurate natural frequency in thermal environments, critical buckling temperature differences are carried out, resulting in closed form solutions. The effect of temperature’s variation as well as power index, functionally graded stiffeners, geometrical parameters, temperature depended materials and initial imperfection on the nonlinear response of the stiffened shell are considered and discussed. This dissertation showed that the nonlinear study of problems of thin-walled structures with even stiffeners is of paramount importance. It was also found that the difference between single-mode and multi-mode analyses could be very significant for nonlinear problems in a thermal environment. Hence, multimode vibration analysis is necessary for structures of this nature

Contribution to the study of the mechanical behavior of curved structures made of advanced materials

Doubly curved shallow shells (DCSSs), frequently encountered in advanced engineering such as aerospace, civil, and mechanical engineering, present substantial challenges in predicting mechanical responses due to their complex geometry and material properties. Moreover, research on functionally graded doubly curved shallow shells (FG DCSs) is very limited, with most studies relying on analytical methods, highlighting the need for a novel and efficient approach to improve predictive analysis. Therefore, to address this gap, the aim of this research work is to develop an efficient and simple finite element model to investigate the bending deflection, stress distribution, and free vibration behavior of FG DCSSs. A new eight-node quadrilateral isoparametric element, named SQ8-IFSDT, with five degrees of freedom per node, is formulated based on improved first-order shear deformation theory (IFSDT). The present IFSDT simplifies the assumptions related to transverse shear stresses, replacing the conventional shear correction factor. As a result, it accurately predicts the parabolic shear stress distribution across the thickness of the shell while maintaining free traction conditions on both surfaces. In the present study, five types of DCSSs, namely flat plates, cylindrical shells, spherical shells, hyperbolic paraboloid shells, and elliptical paraboloid shells, are considered for the analysis. The material properties of FG DCSS change continuously across the thickness according to a power-law function. A variety of comparative studies is conducted to assess the accuracy and robustness of the developed finite element model. A comparison study shows that the proposed model is: (a) accurate and comparable with the literature; b) of fast rate of convergence to the reference solution; c) excellent in terms of numerical stability; and d) valid for both thin and thick FG DCSs. Moreover, comprehensive numerical results are presented and discussed in detail to examine the effects of material properties, power-law index, radius-to-thickness ratio, radius-to-side ratio, radii of curvature, loading, vibration modes, and boundary conditions on the bending and free vibration response of FG DCSSs. Finally, the outcomes of this research provide a robust benchmark for the design, testing, and manufacture of DCSSs and will inform future investigations into shell structures

Recent Advances in Theoretical and Computational Modeling of Composite Materials and Structures

The advancement in manufacturing technology and scientific research has improved the development of enhanced composite materials with tailored properties depending on their design requirements in many engineering fields, as well as in thermal and energy management. Some representative examples of advanced materials in many smart applications and complex structures rely on laminated composites, functionally graded materials (FGMs), and carbon-based constituents, primarily carbon nanotubes (CNTs), and graphene sheets or nanoplatelets, because of their remarkable mechanical properties, electrical conductivity and high permeability. For such materials, experimental tests usually require a large economical effort because of the complex nature of each constituent, together with many environmental, geometrical and or mechanical uncertainties of non-conventional specimens. At the same time, the theoretical and/or computational approaches represent a valid alternative for designing complex manufacts with more flexibility. In such a context, the development of advanced theoretical and computational models for composite materials and structures is a subject of active research, as explored here for a large variety of structural members, involving the static, dynamic, buckling, and damage/fracturing problems at different scales

Vibration and post-buckling of a functionally graded beam subjected to non-conservative forces

Vibration and post-buckling of beams made from functionally graded materials (FGM) subjected to uniformly and tangentially compressing follower forces are studied in this paper. Based on the accurately and geometrically nonlinear theory for extensible beams, the dynamic governing equations for FGM beams under non-conservative load are formulated. By using a shooting method to solve the non-linearly differential equations numerically, the responses of post-buckling and free vibration in the vicinity of post-buckling configuration are obtained, in which the hinged-fixed boundary conditions of beam are considered. Effects of material gradient parameter on the critical buckling, post-buckling and lower frequencies of the FGM beam are discussed in details

Nonlinear Thermoelastic Static Vibration and Buckling Behaviour of Functionally Graded Shell Panel

Functionally graded material (FGM) has created the interest of many researchers due to its tailor-made material properties. FGMs are the advanced form of composites that exhibit an inhomogeneous character especially designed for the high-temperature applications such as aircraft engines, rocket heat shields, thermal barrier coatings, heat exchanger tubes, etc. This material has been developed by taking the gradual variation of metal and ceramic constituents in a very efficient manner to suit the needs of the engineering structure. The effective material properties of the FGM follow the rule-based grading of two counterparts as discussed above, metals and ceramics. Shell structures made of FGMs are subjected to different kind of loading during their service life, and the structural responses (deformations, buckling/post-buckling, and linear/nonlinear natural frequencies) are affected considerably due to that. In this regard, a general nonlinear mathematical model has been developed for the FGM doubly curved shell panel using Green-Lagrange geometrical nonlinear kinematics in the framework of the higher-order shear deformation theory. The effective material properties of FGM shell panels are evaluated using Voigt’s micromechanical model via the power-law distribution. The material properties each constituent are assumed to be temperature-dependent. In addition, to achieve the true flexure of the structure, all the nonlinear higher-order terms are included in the mathematical model. The system governing equation of the FGM structure is obtained using the variational principle, and the direct iterative method is employed to compute the desired nonlinear responses in conjunction with suitable isoparametric finite element steps. The convergence behaviour of the proposed numerical model has been checked and validated further by comparing the responses with those available published literature. The linear and the nonlinear flexural, free vibration and the buckling responses of the FGM single/doubly curved shell panels are examined under thermo-mechanical loading. Finally, the effects of different geometrical and material parameters, support conditions, loading types on the deflection, frequency and critical buckling load parameter of the FGM single/doubly curved shell panels are examined and discussed in detail. This is also believed that the present study would be beneficial to the analysis and the design of FGM structure and/or structural component for real-life problems

Mechanics of Micro- and Nano-Size Materials and Structures

For this reprint, we intend to cover theoretical as well as experimental works performed on small scale to predict the material properties and characteristics of any advanced and metamaterials. New studies on mechanics of small-scale structures such as MEMS/NEMS, carbon and non-carbon nanotubes (e.g., CNTs, Carbon nitride, and Boron nitride nanotubes), micro/nano-sensors, nanocomposites, macrocomposites reinforced by micro-/nano-fillers (e.g., graphene platelets), etc., are included in this reprint

Multistable Shell Structures

Multistable structures, which possess by definition more than one stable equilibrium configuration, are capable of adapting their shape to changing loading or environmental conditions and can further improve multi-purpose ultra-lightweight designs. Whilst multiple methods to create bistable shells have been proposed, most studies focussed on free-standing ones. Considering the strong influence of support conditions on related stability thresholds, surprisingly little is known about their influence on multistable behaviour. In fact, the lack of analytical models prevents a full understanding and constitutes a bottle-neck in the development process of novel shape-changing structures. The relevance becomes apparent in a simple example: whilst an unsupported sliced tennis ball can be stably inverted without experiencing a reversion, fixing its edge against rotation erodes bistability by causing an instantaneous snap-back to the initial configuration. This observation reveals the possibility to alter the structural response dramatically by a simple change of the support conditions. This dissertation explores the causes of this behaviour by gaining further insight into the promoting and eschewing factors of multistability and aims to point out methods to exploit this feature in optimised ways. The aforementioned seemingly simple example requires a geometrically nonlinear perspective on shells for which analytical solutions stay elusive unless simplifying assumptions are made. In order to captures relevant aspects in closed form, a novel semi-analytical Ritz approach with up to four degrees of freedom is derived, which enforces the boundary conditions strongly. In contrast to finite element simulations, it does not linearise the stiffness matrix and can thus explore the full solution space spanned by the assumed polynomial deflection field. In return, this limits the method to a few degrees of freedom, but a comparison to reference calculations demonstrated an excellent performance in most cases. First, the level of influence of the boundary conditions on the critical shape for enabling a bistable inversion is formally characterised in rotationally symmetric shells. Systematic insight is provided by connecting the rim to ground through sets of extensional and rotational linear springs, which allows use of the derived shell model as a macro-element that is connected to other structural elements. It is demonstrated that bistability is promoted by an increasing extensional stiffness, i.e. bistable roller-supported shells need to be at least twice as tall compared to their fixed-pinned counterparts. The effect of rotational springs is found to be multi-faceted: whilst preventing rotation has the tendency to hinder bistable inversions, freeing it can even allow for extra stable configurations; however, a certain case is emphasised in which an increasing rotational spring stiffness causes a mode transition that stabilises inversions. In a second step, a polar-orthotropic material law is employed to study variations of the directional stiffness of the shell itself. A careful choice of the basis functions is required to accurately capture stress singularities in bending that arise if the radial Young’s modulus is stiffer than its circumferential equivalent. A simple way to circumvent such singularities is to create a central hole, which is shown not to hamper bistable inversions. For significantly stiffer values of the radial stiffness, a strong coupling with the support conditions is revealed: whilst roller-supported shells do not show a bistable inversion at all for such materials, fixed-pinned ones feel the most disposed to accommodate an alternative equilibrium configuration. This behaviour is explained via simplified beam models that suggest a new perspective on the influence of the hoop stiffness: based on observations in free-standing shells, it was thought to promote bistability, but it is only insofar stabilising, as it evokes radial stresses; if these are afforded by immovable supports, it becomes redundant and even slightly hindering. Finally, combined actuation methods in stretching and bending that prescribe non-Euclidean target shapes are considered to emphasise the possibility of multifarious structural manipulations. When both methods are geared to each other, stress-free synclastic shape transformations in an over-constrained environment, or alternatively, anticlastic shape-changes with an arbitrary wave number, are achievable. Considering nonsymmetric deformations offers a richer buckling behaviour for certain in-plane actuated shells, where a secondary, approximately cylindrical buckling mode as well as a ‘hidden’ stable configuration of a higher wave number is revealed by the presented analytical model. Additionally, it is shown that the approximately mirror-symmetric inversion of cylindrical or deep spherical shells can be accurately described by employing a simpler, geometrically linear theory that focusses on small deviations from the mirrored shape. The results of this dissertation facilitate a versatile practical application of multistable structures via an analytical description of more realistic support conditions. The understanding of effects of the internal stiffness makes it possible to use this unique structural behaviour more efficiently by making simple cross-sectional adjustments, i.e. by adding appropriate stiffeners. Eventually, the provided theoretical framework of emerging actuation methods might inspire novel morphing structures.Friedrich Ebert Foundation (Friedrich-Ebert-Stiftung) Corpus Christi College, Cambridge Department of Engineering, Cambridg

Fundamental solutions for beams, plates, and shells under thermomechanical actions

As the engineering profession moves from prescriptive or “deemed-to-satisfy” approaches towards design methodologies based on quantification of performance, sophisticated modelling tools are increasingly needed, especially when complex combinations of demand and capacity are encountered. Recourse is invariably made to advanced computational tools to provide high fidelity solutions to large and complex problems, such as the response of structural systems or components to thermomechanical actions. Software packages based on the finite element method are most commonly used for such analyses. There are some essential prerequisites to effective use of advanced computational software for complex nonlinear problems, which are often ignored, particularly in professional practice. These include a thorough understanding of the underlying mechanics of the problem under consideration; a good appreciation of the approximation methods for modelling the problem properly (e.g. the choice between elements, continuum or structural, low or high order interpolation, degree of mesh refinement necessary and so on); and perhaps most importantly ensuring that the software is reliable and is able to reproduce established fundamental solutions to an acceptable degree of accuracy. This thesis attempts to address most of these issues but focusses primarily on the last mentioned prerequisite and provides a range of novel and unprecedented fundamental solutions for beams, plates, and shallow shells subject to moderate or extreme thermomechanical loads such as those resulting from a fire. Geometric and material nonlinearities are included in the proposed formulations along with the most common idealised boundary conditions. Thermally induced deformations generate large displacements and require the solutions to account for geometric nonlinearity, while material nonlinearity arises from the degradation of the material at elevated temperatures. In the context of structural performance under extreme thermal action (such as fire), a finite element procedure is employed to analytically characterise generic temperature distributions through the thickness of a structural component arising from different types of fire exposure conditions including: a “short hot” fire leading to a high compartment temperature over a relatively short duration; and a “long cool” fire with lower compartment temperatures, but over a longer duration. Results have shown that despite the larger area under the long cool fire time-temperature curve, which traditionally represented the fire severity, the effect of the short hot fire on the nonlinear responses of beams, plates, and shallow shells is more pronounced. Also, the effect of temperature-dependent material properties is found to be more pronounced during the short hot fire rather than the long cool fire. Comparison studies have confirmed that while the current numerical and theoretical approaches for analysing of thin plates and shells are often computationally intensive, the proposed approach offers an adequate level of accuracy with a rapid convergence rate for such structures. The solutions developed can be used to: verify software used for modelling structural response to thermomechanical actions; help students and professionals appreciate the fundamental mechanics better; provide relatively quick solutions for component level analyses; and visualise internal load paths and stress trajectories in complex structural components such as composite shells that can help engineers develop deeper insights into the relevant mechanics. The formulations developed are versatile and can be used for other applications such as laminated composite or orthotropic shallow shells. A very significant by-product of developing such fundamental solutions is their potential use in the development of highly accurate hybrid elements for very efficient modelling of large problems. While this has not been fully developed and implemented in the current work, the requisite theoretical framework has been developed and reported in one of the appendices, which can be used to develop such elements and implement on an appropriate software platform

Vibration, Buckling and Parametric Instability of Delaminated Composite Panels in Hygrothermal Environment

The present investigation deals with free vibration, static and dynamic stability performance of bidirectional delaminated composite flat and curved panels with inplane periodic loading in hygrothermal environment.The dynamic instability under in-plane periodic forces for delaminated woven fiber composite panels are studied in varying environmental conditions of temperature and moisture using finite element method (FEM). Numerical analysis by FEM and experimental studies are conducted on free vibration and buckling response of bidirectional delaminated composite panels in hygrothermal environment. The influences of various parameters such as hygrothermal conditions, area and strip delaminations, boundary conditions, ply orientations, stacking sequence, curvatures, static and dynamic load factors on the free vibration, static and parametric instability characteristics of bidirectional composite panels are considered in the present study. A finite element model is developed having 8-noded isoparametric element with 5 degrees of freedom (DOF) per node for the vibration, static and dynamic instability characteristics of delaminated bidirectional composite flat and curved panels under hygrothermal environment utilizing first order shear deformation theory (FSDT). Principal instability zones are located by solutions of Mathieu-Hill equations using Bolotins approach. Based on principle of minimum potential energy, the elastic stiffness matrix, geometric stiffness matrix due to hygrothermal and applied loads, mass matrix and load vectors are formulated. Provision for area and strip delamination modeling is also made in the numerical analysis using multi-point constraint algorithm. A general formulation for vibration, buckling and dynamic stability characteristics of bidirectional delaminated composite flat and curved panels under in-plane periodic forces is presented.The materials utilised for casting of specimens are bidirectional Glass fiber, epoxy as resin, hardener, polyvinyl alcohol as a releasing agent and Teflon film for introducing artificial delaminations. The material constants are calculated from the tensile tests of coupons under varying temperature and moisture conditions as per appropriate ASTM standards. For free vibration testing, FFT analyzer with PULSE Labshop software is used. A test set up is fabricated for vibration test of composite plates under different boundary conditions.The Universal testing machine INSTRON 8862 is used for determination buckling loads experimentally. A good matching is observed between predicted and test results for free vibration and buckling of delaminated composite panels in hygrothermal field. The natural frequencies and buckling loads are observed to decrease with increase in delamination at elevated temperature and moisture conditions under different boundary conditions. However, an increment in the fundamental frequencies is found at sub-zero temperatures up to cryogenic range as against ambient conditions because of development o