Linear Static Behavior Research Articles

A study on the static behavior of functionally graded I-shaped beams

<abstract> <p>Functionally graded materials are well-known for their ability to minimize abrupt stress transitions that are typical of laminated composites, as well as for being very suitable to operate in adverse high-temperature environments. They can act as thermal barriers if a proper selection of the constituent materials is considered. These materials and structures have attracted the attention of many researchers; however, previous research efforts have been focused on the systematic study of rectangular-shaped cross-section profiles. The use of beams characterized by other cross-sections, which are commonly used in metallic construction, may benefit from the continuous, smooth materials mixture that is typically found within functionally graded composites. Hence, this work aims to investigate the behavior of symmetrical, I-shaped, cross-section beams made from these advanced composites by performing a set of parametric analyses. These beams are modeled after solid finite elements to be able to describe the materials mixture evolution through the beam thickness with greater detail. Thick and moderately thick beams are submitted to static loading while considering different boundary conditions, and their linear static behavior is analyzed. As expected, the stress profiles are highly influenced by the materials' mixture profiles. Significant shear stress was found in the I-shaped cross sections.</p> </abstract>

Effect of Curvilinear Reinforcing Fibers on the Linear Static Behavior of Soft-Core Sandwich Structures

The present research deals with the linear static behavior of soft-core sandwich plates and shells. The external skins are reinforced by curvilinear fibers. Their curved paths are described by a general mathematical law that allows the definition of arbitrary placements. The mechanical behavior of these structures is modeled through several Higher-order Shear Deformation Theories (HSDTs) including the zig-zag effect, based on an Equivalent Single Layer (ESL) approach. The solution of the governing equations is achieved numerically by means of the Generalized Differential Quadrature (GDQ) method. A huge number of parametric investigations is proposed in graphical and tabular forms to highlight the influence of the fiber orientation on the static response. The results prove that the structural behavior is affected by such parameters. Thus, the desired structural behavior can be modified by means of a proper choice of the fiber orientation.

Form Follows Force: A theoretical framework for Structural Morphology, and Form-Finding research on shell structures

With the springing up of freeform architectures, the key problem to structural engineers is to generate structural forms with high structural efficiency subject to the architectural space constraints during the conceptual design process. In this research, a theoretical framework for Structural Morphology has been proposed, that provides an effective solution to the problem. To enrich the proposed framework, systematic Form-Finding research on shell structures is conducted.

Form Follows Force: A theoretical framework for Structural Morphology, and Form-Finding research on shell structures

Form Follows Force: A theoretical framework for Structural Morphology, and Form-Finding research on shell structures

Linear Static Behavior of Damaged Laminated Composite Plates and Shells.

A mathematical scheme is proposed here to model a damaged mechanical configuration for laminated and sandwich structures. In particular, two kinds of functions defined in the reference domain of plates and shells are introduced to weaken their mechanical properties in terms of engineering constants: a two-dimensional Gaussian function and an ellipse shaped function. By varying the geometric parameters of these distributions, several damaged configurations are analyzed and investigated through a set of parametric studies. The effect of a progressive damage is studied in terms of displacement profiles and through-the-thickness variations of stress, strain, and displacement components. To this end, a posteriori recovery procedure based on the three-dimensional equilibrium equations for shell structures in orthogonal curvilinear coordinates is introduced. The theoretical framework for the two-dimensional shell model is based on a unified formulation able to study and compare several Higher-order Shear Deformation Theories (HSDTs), including Murakami’s function for the so-called zig-zag effect. Thus, various higher-order models are used and compared also to investigate the differences which can arise from the choice of the order of the kinematic expansion. Their ability to deal with several damaged configurations is analyzed as well. The paper can be placed also in the field of numerical analysis, since the solution to the static problem at issue is achieved by means of the Generalized Differential Quadrature (GDQ) method, whose accuracy and stability are proven by a set of convergence analyses and by the comparison with the results obtained through a commercial finite element software.

Linear static behavior of curved beams coupled with strings representing also fiber-reinforced masonry arches

A linear elastic analytical model of curved beam with constant curvature coupled with curved strings has been developed. To model the connections between the strings and the beam, a continuous distribution of tangential and orthogonal linear elastic springs have been introduced. To evaluate the behavior of the system, a numerical integration of the field equations has been performed. A parametric analysis, capable of pointing out the influence on the behavior of the system of both the stiffness of the tangential connection between strings and beam and of the geometrical characteristic of the beam, has been performed. The results of the proposed model have been compared with those obtained from a finite element model. They show that the strings, in some cases, do not work as expected.

Mono-dimensional formulation of axial-symmetric spherical shells and characterization of the linear static behaviour

In spherical shells, due to the axial-symmetry of the system, all the quantities involved in the elastic problem only depend on the curvilinear abscissa along the meridian lines. Hence, the mathematical domain of the model becomes mono-dimensional. However, in the classical model of axial-symmetric spherical shells, the kinematic problem is not the adjoint of the static problem. With the aim to clarify this aspect, a mono-dimensional linear model of axial-symmetric spherical shells has been proposed. The adopted mono-dimensional approach furnishes a model where the kinematic problem is the adjoint of the static problem. This model, able to describe the linear static behaviour of the shell, can be considered as a curved beam posed on an elastic variable soil. Specifically, the axis of the generic curved beam coincides with a meridian line and the elastic soil is related to the characteristics of the ring beams along the parallel lines. Several internal constraints and simplifying assumptions have been introduced. The comparison among approximate analytical models and a Finite Element one confirm the effectiveness of the proposed mono-dimensional approach. The most approximate model admits a closed-form solution. It appears to be perfectly capable of describing the classical oscillatory-damped behavior of beams on elastic soil. For this reason it has been used to introduce a criterion to classify spherical shells as long or short shells. Graphical abaci, that permit a handy classification, are then proposed, depending only on the geometrical characteristics of the shell.

Semi-analytical approach for the study of linear static behaviour and buckling of shells with single constant curvature

A model of linear, internally constrained shell with single, constant curvature is used to describe the behaviour of existing structures, such as barrel shells. A linear, elastic, isotropic material is considered. Observing that in the shell two families of mono-dimensional interacting beams can be recognized: straight longitudinal beams and transversal arches, a non-conventional semi-analytical approximate solution, which uses the method of separation of variables, is proposed. By using an integral procedure, reduced differential, ordinary equations, capable of describing the behaviour of the shell, are obtained. Both linear static behaviour and longitudinal buckling of the shell are investigated. The approximate solution proposed leads to results that match those of a finite element model and permits to give a description of shells similar to that of beams on elastic soil. With regard to the linear static behaviour of the shell, a “short” and a “long” characterization is proposed and original graphical abaci are obtained with the purpose of facilitating the classification. An extensive study is then performed in order to analyse the buckling of the shells.

Approximate Internal Load Prediction in Composite Structures with Locally Buckled Panels

DOI: 10.2514/1.30465 Most aerospace structural designs are complex assemblies of a large number of panel components. In designing such a structure, it is customary to divide the structure into many regions and design them independently/semiindependently. A coarse finite element model of the entire structure is first used to determine loads in each component, and subsequently the components are designed using local constraints and performance requirements for these fixed loads. This two-level approach works well if a linear static behavior is considered. For cases in which local components display postbuckling behavior, the internal load distribution in the structure is altered requiring nonlinear global analysis, which is computationally expensive. This paper focuses on developing a computationally efficient approximate methodology for predicting internal load distribution in a structure considering some or all panels to be buckled. In this paper, a semi-analytical approach based on the Rayleigh–Ritz method and the perturbationapproachisdevelopedtocomputethepostbuckledstiffnessofbuckledpanels.Aniterativeprocedureis suggestedtopredicttheinternalloaddistributionwithsuccessivebucklingduringloading.Threedifferentexamples are presented. A simply supported flat composite panel subjected to different edge compression is used to verify the perturbation approach used for the calculation of the postbuckled panel stiffness. A two-panel arrangement with different thicknesses subjected to end compression and a 48-panel composite wing structure subjected to bending load are used as cases for structures with buckled panel components. In all cases, the present approach agrees with full nonlinear analysis in predicting the internal load distribution.

Stacking layer sequence effects for glass fibre/epoxy resin cross-ply laminates

We describe the effect of the stacking sequence on the distribution of the deformation and stress fields through the thickness of the laminate. The laminates are made of epoxy resin and glass fibres, having the ratio of 0 to 90° plies equal to 1. This stacking sequence is applied to obtain composite laminates with different thicknesses. The study presented in this paper is focused on the linear static behavior of the cross-ply laminates by means of finite-element models developed using the commercial codes ANSYS 6.1 and LUSAS 15.3. The numerical models represent three-point bending loading cases. A viscoelastic analysis of the laminates based on the correspondence principle is also performed. The results from the finite-element models show good agreement with those obtained using the classical laminate theory, while the viscoelastic storage modulus and loss factors’ distribution indicate the optimal stacking sequences for structural dynamics applications, such as spring leaves for novel car suspension systems.

Floating-roof steel tanks under harmonic settlement: FE parametric study and design criterion

Large vertical steel tanks for fluid storage are usually constructed on soft foundations, so it is not surprising that the tank wall will settle unevenly with the settlement of the foundation, thus inducing deformations and stresses in the tank. This work investigates the linear static behavior of floating-roof tanks under harmonic settlement through finite element (FE) analyses. The influences of the radius-to-thickness ratio, the height-to-radius ratio and the wind girder stiffness on the structural behavior are first analyzed. Comparisons between the circumferential stresses in the wind girder and the vertical stresses in the tank bottom are then made. The displacement and the stress along the tank height are also discussed, and the concept of tank division along its height is presented. Finally, a design approximation for the radial displacement at the tank top is developed based on FE results, and a settlement criterion based on the top radial displacement is proposed which can be used in practical design.

Efficient non-linear idealisations of aircraft fuselage panels in compression

Aircraft fuselages are complex assemblies of thousands of components and as a result simulation models are highly idealised. In the typical design process, a coarse FE model is used to determine loads within the structure. The size of the model and number of load cases necessitates that only linear static behaviour is considered. This paper reports on the development of a modelling approach to increase the accuracy of the global model, accounting for variations in stiffness due to non-linear structural behaviour. The strategy is based on representing a fuselage sub-section with a single non-linear element. Large portions of fuselage structure are represented by connecting these non-linear elements together to form a framework. The non-linear models are very efficient, reducing computational time significantly.

A least squares finite element formulation for plane elasto-statics based on p-version with applications to laminated composites

This paper presents a weighted p-version least squares finite element formulation (LSFEF) for two-dimensional plane elasto-statics and its application for simulating linear static behavior of laminated composites. The second order partial differential equations of equilibrium in displacements are recast into a series of first order coupled partial differential equations using stresses as auxiliary variables. The primary (displacements: u and v) and the secondary (stresses: σ xx , τ xy and σ yy ) variables are interpolated using equal order, C 0 continuity, p-version hierarchical approximation functions. The p-version hierarchical approximation functions and the corresponding nodal variable operators are derived directly from the Lagrange family of interpolation functions by first constructing the one dimensional p-version hierarchical approximation functions and the corresponding nodal variable operators for three node equivalent configurations in the ξ and η directions and then taking their products. The least squares procedure yields a system of linear, symmetric simultaneous algebraic equations. The formulation provides inter-element continuity of displacements as well as stresses. Even though the element approximation is of type C 0, in the limit when the error functional I converges to zero (i.e., when I → 0) the formulation behaves more like a C 1 type. In addition, for an inadequate mesh with an inadequate order of approximation for the elements, the element error functional values provide a mechanism for h and/or p-adaptive refinements. This is a unique feature of the LSFEF as compared to variational based formulations. Many other unique features of the formulation are discussed and illustrated in the paper. A numerical example is presented to illustrate the effectiveness of the formulation in simulating accurate inter-lamina stress behavior as well as accurate stress values and stress gradients at and near singularities. The results obtrained from the present formulation are compared with the results obtained from the p- version variational formulation and with those reported in the literature.

Application of Volterra series to modelling of rainfall-runoff systems and flow in open channels

ABSTRACT The method presented combines a black box analysis with a conceptual model approach. To describe the nonlinear behaviour of the systems, a model in the form of a second-order approximation of a cascade of nonlinear reservoirs was used. Such a model is equivalent to the first two terms of the Volterra series. Applications of the Volterra model based on a nonlinear cascade to the modelling of flow in open channels and of surface runoff systems indicate that the proposed model can be used to represent systems with nonlinear dynamic and linear static behaviour.

Matrix Formulation of Curved Girders

Exact solutions to the linear static behavior including warping torsion of thin-walled spatial curved beam structures are obtained by the displacement formulation through the use of minimum potential energy. The generalized displacement field functions are represented by the exact solutions of the coupled homogeneous differential equations governing the static behavior, which also permits the evaluation of exact work-equivalent nodal forces due to external loadings. The obtained total torsional moment is separated through an analytical procedure into St. Venant and warping torque. Several examples are presented and comparisons made.