Eccentric Stiffeners Research Articles

Isogeometric-based mapping modeling and buckling analysis of stiffened panels

Modeling and analysis of stiffened panels are two key technologies in the design of aerospace thin-walled structures. For the stiffened panels with complex geometry, classical finite element analysis (FEA) and conventional isogeometric analysis (IGA) based on explicit geometry usually require time-consuming and labor-intensive geometric processing, and additional coupling matrices to be ready for analysis. In this study, a new method for modeling and buckling analysis of stiffened panels is proposed, which provides a more efficient and simpler way. During the modeling process, the stiffeners are treated as curves on surfaces, which is not explicitly defined using the control-point-based representation of curves, but implicitly defined using parameter curves in the parametric space of the surface. Mapping modeling provides more accurate geometric description and transfer the complex modeling problems (three-dimensional space) of stiffeners on free-form surface into simple modeling problems in the regular parametric space (two-dimensional space). During the buckling analysis process, a new mapped stiffener element based on mapping modeling is proposed, which can model the section of the eccentric stiffener without changing the geometry. The precise normal information of the Non-Uniform Rational B-Splines (NURBS) surface can ensure that the stiffeners are perpendicular to the skin. In addition, the coupling of the stiffener and the skin is automatic, without any additional coupling matrix. This buckling analysis framework realizes the complete integration of modeling and analysis. Furthermore, for the stiffened panels with cutouts, the trimmed surface analysis (TSA) method is extended to be used for the numerical integration of the trimmed stiffeners, which means that no additional geometric process is required. Finally, four numerical examples of different types of stiffened panels are constructed, involving metal, trimmed surface, classical grid-stiffener, free-form surface, variable-stiffness composites, and curvilinear grid-stiffener. Several numerical examples of static and buckling analysis of stiffened panels with high fidelity demonstrate the effectiveness of the proposed framework.

Nonlinear thermo-mechanical analysis of ES double curved shallow auxetic honeycomb sandwich shells with temperature-dependent properties

It is the first time the nonlinear thermo-mechanical analysis (buckling and postbuckling) of imperfect eccentrically stiffened (ES) double curved shallow shells made of auxetic material under uniform external pressure, axial compressive and thermal loads has been studied by adopting analytical method. The double curved shallow sandwich shell is constructed by an auxetic honeycomb core layer with negative Poisson’s ratio and two isotropic homogeneous skin layers. In the present study, the outer surface of the structure is reinforced by a system of FGM stiffeners and taking the influence of pores in the stiffeners into account to increase the load capacity of the structure. Besides, the material properties of the shells and stiffeners are both temperature-dependent. The first-order shear deformation theory is applied to derive equilibrium and compatibility equations of double curved shallow auxetic shell, considering both the geometrical nonlinearity in the Von Karman sense and initial geometrical imperfections and shells placed on the Winkler – Pasternak foundation. The Galerkin and stress function methods are proposed to determine the critical load value and the deflection– load curve by taking the root form of the two-term rotation angles to give the Galerkin integral an equilibrium equation. Several comparisons with results in other literature are made to examine the reliability and accuracy of the method used in the present study. Finally, the effects of thermal environment, reinforced stiffeners, elastic foundation, pores and parameters of auxetic core layer on the buckling and postbuckling loads are evaluated in this paper. The numerical and graphical results show that the reinforced stiffeners, auxetic core, temperature and porous distribution have significantly influenced on the buckling and postbuckling loads of ES double curved shallow auxetic honeycomb sandwich shells.

Determination of Capacities of Eccentric Stiffeners Part 2: Analytical Studies

Analytical investigations were performed on column specimens with an effort to evaluate the effective stiffener capacity of eccentric stiffeners when used within moment connections of beams connecting to column flanges. First, analytical models were developed for the experimental column specimens presented in the companion paper Part 1: Experimental Studies (Alvarez Rodilla and Kowalkowski, 2021). These models were utilized to calibrate the finite element methodology and to develop consistent comparisons between the experimental and analytical results.Finite element models were then developed for larger column sizes more regularly used in practice. In all, 148 models were developed to represent W14×68, W14×120, W14×176, W14×233, W24×131, and W24×229 column specimens. The column models utilized a similar setup to experiments and were subjected to three different loading conditions described as (1) single tension with load pulling away from the column specimen; (2) single compression with load applied toward the column specimen; and (3) double compression with loads applied on both flanges, directly opposite of each other. Parameters also included the unsupported length and stiffener thickness for single compression, the stiffener thickness for double compression, and the weld size/loading plate thickness for single tension. For all test methods and each set of parameters, a group of four column specimens consisted of (1) one modeled without stiffeners, (2) one modeled with concentric stiffeners, (3) one modeled with stiffeners at an eccentricity of 2 in., and (4) one modeled with stiffeners at an eccentricity of 4 in.

Determination of Capacities of Eccentric Stiffeners Part 1: Experimental Studies

Forty column specimens were experimentally tested with an effort to identify the effective stiffener capacities of eccentric stiffeners when used within moment connections of beams connecting to column flanges. Three test methods were performed described as (1) single tension with load pulling away from the column specimen, (2) single compression with load applied toward the column specimen, and (3) double compression with load applied toward the column specimen and with a reaction plate directly opposite from the applied load. The sizes were selected with a range of slenderness ratios for the web and flange and to study multiple concentrated load limit states using AISC Specification Section J10 (AISC, 2016b). For each column size and test method, four column specimens were tested: (1) without stiffeners, (2) with stiffeners concentric with the applied load, (3) with stiffeners at a low eccentricity of 2 in. or 3 in., and (4) with stiffeners at a high eccentricity of 4 in. or 6 in. For column specimens tested without stiffeners and for compression tests, the maximum loads compared favorably to those predicted for web local crippling and were significantly higher than those predicted for web compression buckling (double compression tests) and web local yielding. For single tension tests, the maximum load always exceeded that predicted for the limit states of flange bending and web local yielding.

Dynamic analysis of stiffened bi-directional functionally graded plates with porosities under a moving load by dynamic relaxation method with kinetic damping

The principal purpose of this study is investigating the dynamic analysis of porous bi-directional functionally graded (FG) plates reinforced by eccentrically outside stiffeners and subjected to a moving load with a constant velocity. The materials are assumed to be graded in two directions and their effective properties are computed by the rule of mixtures. The FG plates are assumed to have both even and uneven distribution of porosities over the plate cross-section. Using appropriate kinematic relations, the displacements of the plate mid-plane are compatible with those of the stiffeners. The governing differential equations of porous bi-directional FG plates are derived through Hamilton's principle based on the first order shear deformation theory (FSDT) and Von Karman relations for large deflections. Moreover, dynamic relaxation method with kinetic damping (K-DR) coupled with Newmark integration technique are used to solve the plate's time-varying nonlinear equations. The effects of some numerical aspect ratios such as volume fraction, boundary conditions, porosity coefficients and distribution patterns and the existence of stiffeners on dynamic behaviors are investigated. The results show that the stiffness of the porous bi-directional FG plates is highly improved with the aid of eccentric stiffeners; hence, better dynamic behaviors are provided.

An Edge-Based Smoothed Finite Element Method for Analyzing Stiffened Plates

In this paper, an edge-based smoothed finite element method with the discrete shear gap using triangular elements (ES-DSG3) is presented for static, free vibration and sound radiation analyses of plates stiffened by eccentric and concentric stiffeners. In the present model, the ES-DSG3 for the plate element with the isoparametric thick-beam element is employed to formulate stiffened plate structures. The deflections and rotations of the plates and the stiffeners are connected at tying positions. By using Rayleigh integral, sound radiation of stiffened plates subjected to a point load can be obtained. The edge-based gradient smoothing technique is employed to perform the related numerical integrations over the edge-based smoothing domains. Compared with the original DSG3 model, the present ES-DSG3 model is relatively softer as a result of the edge-based gradient smoothing technique. From several numerical examples, it is observed that the ES-DSG3 can produce more accurate numerical solutions than the original DSG3 for stiffened plates.

Geometrically nonlinear dynamic response of eccentrically stiffened circular cylindrical shells with negative poisson's ratio in auxetic honeycombs core layer

Present research aims to analyse the nonlinear dynamic behavior of eccentrically stiffened (ES) circular cylindrical shells with negative Poisson's ratios in auxetic honeycombs core layer on elastic foundations and subjected to blast and mechanical loads. This study considers a three – layer circular cylindrical shells in which the core layer is the auxetic material with negative Poisson's ratio, and the external layers are reinforced by a system of stiffeners. Based on the analytical solution, Reddy's first order shear deformation theory with the geometrical nonlinear in von Karman and Airy stress functions method, Galerkin method and the fourth-order Runge–Kutta method, out explicit expressions can be determined: fundamental frequency, dynamic response and frequency-amplitude curves. Numerical results are provided to explore the effects of geometrical parameters, material properties, elastic foundations, imperfections, eccentrically stiffeners, mechanical and blast loads on the nonlinear dynamic behavior: fundamental frequency, dynamic response and frequency-amplitude curves.

Dynamic stiffness formulation for the vibrations of stiffened plate structures with consideration of in-plane deformation

A dynamic stiffness method is presented for the vibrations of plate structures that are reinforced by eccentric stiffeners. The model incorporates both out-of-plane and in-plane deformations of the plates and the stiffeners. Based on the relationship between the forces and displacements along the common edges of the plate or beam elements, the dynamic stiffness formulae for the plate and the beam elements are derived, respectively. The globally assembled dynamic stiffness matrix is then obtained using the finite element method so that the dynamics of built-up stiffened plates can be readily addressed by using the present method. Compared to the conventional finite element model, the dynamic stiffness model can provide very accurate solutions using only one element over each uniform plate and beam member, regardless of its geometry.

Free Vibration Characteristics of a Finite Ring-Stiffened Elliptic Cylindrical Shell

A theoretical method is employed to study the free vibration characteristics of a finite ring-stiffened elliptic cylindrical shell. Vibration equations of the elliptic cylindrical shell are derived based on Flügge shell theory, and the effects of the ring stiffeners are evaluated via “smeared” stiffener theory whereby the properties of the stiffeners are averaged over the shell surface. The displacements of the shell are expanded in double Fourier series in the axial and circumferential directions, and the circumferential curvature is expanded in single Fourier series in the circumferential direction. The partial differential characteristic equations with variable coefficients are converted into a set of linear equations with constant coefficients which couple with each other about the circumferential modal parameters. Then, the natural frequencies of the finite ring-stiffened cylindrical shell are obtained. To verify the accuracy of the present method, the finite ring-stiffened elliptic cylindrical shell is degenerated into two models: one of which is a ring-stiffened circular cylindrical shell and the other of which is an elliptic cylindrical shell without ring stiffeners. The present results of the two degenerated shells show good agreements with available results from the literature. The effects of main parameters, including the ellipticity, the shell length ratio, the stiffener's interval, the stiffener's depth, and the stiffener's eccentricity, on the free vibration of the ring-stiffened elliptic cylindrical shell are examined in detail. The ellipticity makes the difference between the symmetric and antisymmetric modal frequencies of the shell. The stiffeners have a greater influence on the free vibration at relatively higher order circumferential modal parameters. The circumferential modal parameters corresponding to the fundamental frequency are affected by the ellipticity, shell length, stiffeners' interval, and depth. The eccentricity of the ring stiffeners has a weak effect on the vibration of the structure.

Analytical investigation on the free vibration behavior of rotating FGM truncated conical shells reinforced by orthogonal eccentric stiffeners

ABSTRACTThis article presents an analytical investigation on the free vibration behavior of rotating functionally graded truncated conical shells reinforced by stringers and rings with the change of spacing between stringers. Using the Donnell shell theory, smeared stiffeners technique, and taking into account the influences of centrifugal force and Coriolis acceleration, the governing equations are derived. These variable coefficient partial differential equations are studied by the Galerkin method. The sixth-order polynomial equation of natural frequency is obtained. Numerical results show effects of stiffener and input parameters on the frequency of shell.

Nonlinear thermal vibration of eccentrically stiffened Ceramic-FGM-Metal layer toroidal shell segments surrounded by elastic foundation

The eccentrically stiffened Ceramic-FGM-Metal layer toroidal shell segments which applied for heat-resistant, lightweight structures in aerospace, mechanical, and medical industry and so forth are the new structures. Thus, the nonlinear vibration of eccentrically stiffened (ES) Ceramic-FGM-Metal (C-FGM-M) layer toroidal shell segments surrounded by an elastic medium in thermal environment is investigated in this paper. Based on the classical shell theory with the geometrical nonlinear in von Karman-Donnell sense, Stein and McElman assumption and the smeared stiffeners technique, the governing equations of motion of ES-C-FGM-M layer toroidal shell segments are derived. The dynamical characteristics of shells as natural frequencies, nonlinear frequency-amplitude relation, and nonlinear dynamic responses are considered. Furthermore, the effects of characteristics of geometrical ratios, ceramic layer, elastic foundation, pre-loaded axial compression and temperature on the dynamical behavior of shells are studied.

Nonlinear stability of shear deformable eccentrically stiffened functionally graded plates on elastic foundations with temperature-dependent properties

Abstract This article investigates the nonlinear stability of eccentrically stiffened moderately thick plates made of functionally graded materials (FGM) subjected to in-plane compressive, thermo-mechanical loads. The equilibrium and compatibility equations for the moderately thick plates are derived by using the first-order shear deformation theory of plates, taking into account both the geometrical nonlinearity in the von Karman sense and initial geometrical imperfections, temperature-dependent properties with Pasternak type elastic foundations. By applying the Galerkin method and using a stress function, the effects of material and geometrical properties, temperature-dependent material properties, elastic foundations, boundary conditions, and eccentric stiffeners on the buckling and post-buckling loading capacity of the eccentrically stiffened moderately thick FGM plates in thermal environments are analyzed and discussed.

Nonlinear vibration and dynamic response of imperfect eccentrically stiffened shear deformable sandwich plate with functionally graded material in thermal environment

This paper presents an analytical approach to investigate the nonlinear dynamic response and vibration of shear deformable imperfect eccentrically stiffened sandwich plate with functionally graded material (FGM) on elastic foundation using both of the first-order shear deformation plate theory and stress function with full motion equations (not using Volmir's assumptions). The thick sandwich plates are assumed to rest on elastic foundation and subjected to mechanical loads in thermal environment. Numerical results for dynamic response of the eccentrically stiffened thick sandwich plates are obtained by Runge–Kutta method. The results show the influences of geometrical parameters, material properties, imperfections, the elastic foundations, eccentric stiffeners, mechanical loads and temperature on the nonlinear dynamic response and nonlinear vibration of functionally graded sandwich plates. The numerical results in this paper are compared with the results reported in other publications.

Vibration and Buckling Analysis of Curvilinearly Stiffened Plates Using Finite Element Method

A finite element method based approach is developed for studying the static, vibration, and buckling behaviors of curvilinearly stiffened plates in the presence of in-plane compressive and tensile stresses. The first-order shear deformation theory is employed for both the plate and the Timoshenko beam modeling. Interpolation functions are used to build the displacement mapping between the stiffener and the plate nodes to allow the stiffener to be placed anywhere within the plate. One of the advantages of the present method is that the plate need not be remeshed while the stiffener configuration is changed; another advantage is that the results obtained by the present method with a much fewer number of elements match well with the results obtained by using a commercial finite element method software. Several numerical examples are solved to study both the static and dynamic behaviors of stiffened plates. The effects of boundary conditions, stiffener eccentricity, stiffener curvature, stiffener-plate geometry parameters, in-plane load condition, stiffener-plate cross-section area ratio, and stiffness ratio on the static and dynamic behaviors of a curvilinearly stiffened plate are investigated. Results have shown that the behavior of the natural frequency parameter as a function of applied in-plane stress could be affected by the plate thickness, in-plane load condition, stiffener-plate cross-section area ratio, and stiffness ratio during compression only, but not when subjected to in-plane tension.

Effect of Triangular Web Profile on the Shear Behaviour of Steel I-Beam

Shear buckling occurred in the instability modes of steel beams when it slender. This paper develops a three-dimensional finite element model using LUSAS 14.3 to study on the effect ofthe triangularsteel beamweb profile (T WP) inshearbuckling behaviourof differentthicknesscompared to that of thenormalflatbeam (FW). RI All specimens are cantilever beam which are fixedat one ends. The flange is constant with variable web thickness. Eigenvalue buckling analysis was used in analysing the buckling load of the flat plate model and triangular web profile (T WP). Results showed that the web thickness gave a significant impact on the shear RI buckling of the T WP. In addition, the corrugation thickness of web was also effective in increasing the shear RI buckling capacity of the profile. triangular web profile. The web profile is a modified beam which was identified as TS 400-2, two beams with section ofa trapezoid web profile steel section whichis the 1.7 metres length and six square web panels with 380 x 380 eccentric stiffeners of the trapezoidal section; that is mm. From the results of the tests, it was concluded that eliminated and changed to the slanting stiffeners. Beam local buckling failure modes were observed in most of the with triangular web profile steel sections are considerably test girders, which stretched over one cycle of the increase the flexural strength and stiffness of the steel trapezoidal form. The ultimate shear capacity of trapezoid beams according to Hashim (1). The flexural strength of web plate girder is higher than the elastic design capacity, the girder was also contributed by the girder's flanges. which only accounts for the web shear buckling limit. The flexural strength of a steel girder with a corrugated Usman (7) also carried out finite element analysis to study web plate is provided by the flanges with almost no the critical shear by using the Eigenvalue-buckling contribution from the web and with no interaction analysis. In this work, a new formula for critical shear between flexure and shear behaviour (2, 3). stress was proposed and the new proposed equation was Luo (4) has proposed three types of shear buckling compared to the experimental data collected in patterns that has been named as local buckling, global UniversitiTeknologi Malaysia (UTM). The formula of buckling and zonal buckling. Luo(5) also has critical shear stress that was proposed is Q (design shear recommended a formula to predict the ultimate strength of capacity) which is calculated based on the proposed trapezoid web under patch loading using non-linear finite design formula. It was derived while neglecting the element method. Several factors that influence the ultimate contribution of the flanges. The calculation of the shear strength were investigated; which includes strain- force capacity were derived from the following equations hardening models, initial imperfections, variation of yield and based on the support arrangement for each test: loading position, load distribution length and variation of

Damage tolerant evaluation of cracked stiffened panels under fatigue loading

This paper presents the methodologies for damage tolerant evaluation of stiffened panels under fatigue loading. The two major objectives of damage tolerant evaluation, namely, the remaining life prediction and residual strength evaluation of stiffened panels have been discussed. Concentric and eccentric stiffeners have been considered. Stress intensity factor for a stiffened panel has been computed by using parametric equations of numerically integrated modified virtual crack closure integral technique. Various methodologies for residual strength evaluation, namely, plastic collapse condition, fracture toughness criterion and remaining life approach have been described. Effect of various stiffener sizes and stiffener type (concentric and eccentric stiffeners) on remaining life and residual strength has been studied under constant amplitude load. From the studies, it has been observed that the predicted life is significantly higher with concentric and eccentric stiffener cases compared to the respective unstiffened cases. The percentage increase in life is relatively more in the case of concentric stiffener compared to that of eccentric stiffener case for the same stiffener size and moment of inertia. From the studies, it has also been observed that the predicted residual strength using remaining life approach is lower compared to other methods, namely, plastic collapse condition and fracture toughness criterion and hence remaining life approach will govern the design. It is noted that residual strength increases with the increase of stiffener size.

Transmission loss of periodically stiffened laminate composite panels: Shear deformation and in-plane interaction effects

This paper investigates the transmission loss of symmetric and asymmetric laminate composite panels periodically reinforced by composite stiffeners. A comprehensive model based on periodic structure theory is developed. First order shear deformation theory is used and the coupling of the in-plane motion of the panel with its out-of-plane motion is taken into account. Stiffeners interact with the panel through three forces (two in-plane, one out-of-plane) and a torsion moment. Three types of cross sections are investigated for the composite stiffeners: I-shaped, C-shaped, and omega-shaped cross-sections. The model is validated numerically by comparison with the finite element/boundary element method. Experimental validations are also conducted. In both cases, excellent agreement is obtained. Numerical results show that the in-plane coupling effect is increased by increasing the panel thickness and the stiffener's eccentricity. The in-plane coupling effect is also found to increase with frequency.

Dynamic and acoustic response of bidirectionally stiffened plates with eccentric stiffeners subject to airborne and structure-borne excitations

An analytical method based on the modal expansion technique was developed to predict the vibro-acoustic response of both unidirectionally and bidirectionally stiffened flat panel. This paper presents the response to diffuse acoustic field (DAF) and turbulent boundary layer (TBL) excitations in terms of their joint acceptance. Numerical results for the dynamic and acoustic responses are compared with finite element method (FEM) and boundary element (BEM) results for stiffened panel with complex and eccentrically shaped stiffeners subject to point force excitation. A theoretical prediction of the transmission loss (TL) is also compared with laboratory measurements conducted on flat panels representing aircraft models as well as with hybrid statistical energy analysis (SEA)–FEM periodic model. The results confirm that the stiffened panel has the same acoustic response as the skin without stiffeners at frequencies where the structural wavelengths are equal to the spacing between the stiffeners. In addition, the transmission loss is lowered by the presence of the stiffeners at some particular region of frequencies below the critical frequency with respect to the unstiffened panel.

A semi-analytical method to predict the vibroacoustics response of composite and isotropic stiffened panels.

The vibroacoustics behavior of aircraft-type stiffened panels is classically analyzed using deterministic methods such as the finite element and boundary element method at low frequencies or energy based methods at higher frequencies. In the present work, a general semi-analytical method based on modal expansion technique is developed to predict the vibration and acoustics radiation of both metallic and composite flat stiffened panels. Both unidirectional and bidirectional stiffened panels with eccentric stiffeners and various shapes are analyzed using the same matrix formulation. The presented model is also able to predict the response of both regular and irregular stiffened panels. The contributions of the force and moment modal coupling at each beam location are accounted for together with the effect of interaction between ribs in the case of orthogonal stiffened plate. The response to various types of excitations (point force, diffuse acoustic field, and turbulent boundary layer) are presented in terms of their joint acceptance. The model is numerically validated by comparison with the FEM/BEM and hybrid FEM/SEA methods for various configurations and excitations. Excellent agreement is found.

An improved formulation of space stiffeners

This paper presents a displacement based finite element model for predicting the constraint torsion effect of stiffeners. In structural modelling, the plate/shell and the stiffeners are treated as separate elements where the displacement compatibility transformation between these two types of elements takes into account the constraint torsional warping effect in the stiffeners. The development is based on a general beam theory which includes flexural-torsion coupling, constrained torsion warping, and shear-centre location. The virtual work principle includes the second order terms of finite beam rotations. For finite element analysis, cubic Hermitian polynomials are used as shape functions of the straight space frame element with two nodes. Elastic stiffness and geometric stiffness matrices for an arbitrary cross-section are evaluated in a closed form, and load correction stiffness for eccentric stiffener loads are considered. To demonstrate the importance of torsion warping constraints and to illustrate the accuracy of this formulation, finite element solutions are presented and compared with available solutions.