Curvilinear Stiffeners Research Articles

Thermal Buckling Analysis and Optimization of Curvilinearly Stiffened Plates with Variable Angle Tow Laminates

This paper presents results from thermal buckling analysis and optimization of stiffened composite panels with variable angle tow (VAT) laminates and curvilinear stiffeners. Considering the meshing difficulties for curvilinearly stiffened VAT laminates due to both spatially dependent fiber path orientation and arbitrarily shaped stiffeners in the traditional finite element analysis, the present work models the plate and stiffeners separately without the need to place finite element nodes along the stiffener/plate and stiffener/stiffener interfaces. The displacement compatibility conditions are enforced at these interfaces in the finite element analysis. Convergence and verification study results show that the present method can accurately predict the thermal buckling behaviors of VAT laminates and curvilinearly stiffened VAT laminates. Parametric studies show that either VAT laminates or curvilinear stiffeners alone can improve the buckling performance as compared with using straight fiber laminates and straight stiffeners. VAT laminates and curvilinear stiffeners are found to redistribute the in-plane stress resultants and tailor the buckling mode shape, respectively, to improve the total buckling response. Optimization studies using curvilinear stiffeners and VAT laminates together for maximizing buckling temperature for a stiffened plate are conducted.

Hybrid Optimization of Curvilinearly Stiffened Shells Using Parallel Processing

With the advances being made in additive manufacturing, it is becoming increasingly possible to fabricate a broad class of complex-shaped designs for practical applications. This manufacturing capability has allowed structural designers to implement the use of bio-inspired curvilinear stiffeners for achieving better designs of stiffened plate and shell structures. Curvilinear stiffeners have proven to be useful over straight stiffeners in some applications for achieving better structural efficiency. A framework has been developed that employs a hybrid optimization technique of using particle swarm optimization for stiffener shape optimization and gradient-based optimization as implemented in MSC.NASTRAN SOL 200 for optimization of stiffener cross section and shell thickness. Parallel processing has been used to save extensive wall-clock time. The framework has been employed for optimally designing cylindrical shells/panels stiffened by arbitrarily placed stiffeners, the motivation being lighter-weight fuselage and space launch vehicle designs. Structural optimization results have been presented for the optimal design of stiffened shells with the objective of weight minimization for shells subjected to both buckling and stress constraints. Optimization studies show that both the stiffener placement and the stiffener geometric curvature influence the shell buckling load, which can help to decrease the structural weight by optimizing the cross-sectional dimensions for stiffeners and panel thickness. In this paper, cylindrical shells, stiffened by four stiffeners, placed arbitrarily, are studied under compression and shear load cases. It is seen that arbitrarily placed curvilinear stiffeners lead to a potential 13% weight saving as compared to the use of equally spaced straight stiffeners when designed for the compression load case.

Application of 6-DOFs meshfree modeling to linear buckling analysis of stiffened plates with curvilinear surfaces

A buckling analysis of stiffened plates including curvilinear surfaces is carried out by an effective meshfree model. The buckling loads and modes computed by the present method are analyzed. Six degrees of freedom (6-DOFs) curved shell meshfree formulation in a convected coordinate system including a drilling rotation component is employed, which enables the assembly of curved shells for the modeling of more complex structures. By this formulation, the assembly of any arbitrary shape of geometry can be modeled in convected coordinates, while the 5-DOFs shell formulation suffers from the modeling of shell assemblies. Particularly, curved shells with straight stiffeners and plates with curvilinear stiffeners are considered. Furthermore, a twisted T-shaped structure where both web and flange have curvilinear geometry is analyzed. A meshfree discretization is employed, with which the reproducing kernel particle method is used as the meshfree interpolant. A boundary singular kernel method is adopted to precisely impose an essential boundary condition and to model folded shell geometries. The accuracy and effectiveness of the proposed method are demonstrated by several shell buckling problems for stiffened plate structures with curvilinear surfaces. The obtained meshfree results are compared with the linear and quadratic shell element results of finite element method ANSYS and discussed.

Hierarchical Nondeterministic Optimization of Curvilinearly Stiffened Panel with Multicutouts

The concept of a curvilinearly stiffened panel is promising for aerospace and aircraft structures with cutouts, since the stiffness distribution and loading path can be flexibly tailored in terms of cutouts. However, uncertainties from the manufacturing process can significantly reduce the practical load-carrying capacity of a curvilinearly stiffened panel. Traditional nondeterministic design optimization methods would suffer from a high computational burden because numerous discrete-continuous variables, in which thousands of finite element analyses are usually required, are involved. Therefore, a hierarchical nondeterministic optimization framework of a curvilinearly stiffened panel is established, containing two steps (global search and local search) and four main parts (layout design, nondeterministic design optimization, reshape layout design, and stress validation). In the first-step deterministic optimization, all variables are considered, and a rough result set will be obtained by a global search. In the second-step nondeterministic optimization, all discrete-continuous layout parameters are fixed in the initial stage, and the efficient adaptive-loop method is applied to significantly increase the computational efficiency, in which the variations of material properties and geometric dimensions are considered as uncertainties. The layout also will be reshaped to explore the potential of the load-carrying capacity. Finally, the stress constraint and crippling constraint are verified for the optimum design. An illustrative example indicates the advantage of the proposed framework not merely provides a competitive efficiency and robustness over other nondeterministic approaches but also improves the rationality of deterministic design optimization.

Service ORiented Computing EnviRonment (SORCER) for deterministic global and stochastic aircraft design optimization: part 1

With rapid growth in the complexity of large scale engineering systems, the application of multidisciplinary analysis and design optimization (MDO) in the engineering design process has garnered much attention. MDO addresses the challenge of integrating several different disciplines into the design process. Primary challenges of MDO include computational expense and poor scalability. The introduction of a distributed, collaborative computational environment results in better utilization of available computational resources, reducing the time to solution, and enhancing scalability. SORCER, a Java-based network-centric computing platform, enables analyses and design studies in a distributed collaborative computing environment. Two different optimization algorithms widely used in multidisciplinary engineering design-VTDIRECT95 and QNSTOP-are implemented on a SORCER grid. VTDIRECT95, a Fortran 95 implementation of D. R. Jones‟ algorithm DIRECT, is a highly parallelizable derivative-free deterministic global optimization algorithm. QNSTOP is a parallel quasi-Newton algorithm for stochastic optimization problems. The purpose of integrating VTDIRECT95 and QNSTOP into the SORCER framework is to provide load balancing among computational resources, resulting in a dynamically scalable process. Further, the federated computing paradigm implemented by SORCER manages distributed services in real time, thereby significantly speeding up the design process. Part 1 covers SORCER and the algorithms, Part 2 presents results for aircraft panel design with curvilinear stiffeners.

Comparison of curvilinear stiffeners and tow steered composites for aeroelastic tailoring of aircraft wings

A series of aeroelastic optimization problems are solved on a high aspect ratio wingbox of the Common Research Model, in an effort to minimize structural mass under coupled stress, buckling, and flutter constraints. Two technologies are of particular interest: tow steered composite laminate skins and curvilinear stiffeners. Both methods are found to afford feasible reductions in mass over their non-curvilinear structural counterparts, through both distinct and shared mechanisms for passively controlling aeroelastic performance. Some degree of diminishing returns are seen when curvilinear stiffeners and curvilinear fiber tow paths are used simultaneously.

Flutter Analysis of Laminated Curvilinear-Stiffened Plates

This paper proposes the use of curvilinear stiffeners as a mechanism to control supersonic panel flutter. To account for transverse shear deformation, the plate and stiffeners are modeled according to the first-order shear deformation theory and Timoshenko beam theory, respectively. The Chebyshev polynomials are the basis of the deflection and rotation functions in the Ritz method. The aeroelastic load is formulated according to the first-order high-Mach-number approximation to linear potential flow theory. The minimum potential energy and Hamilton’s principle are used to solve the problem. Plots of frequency versus aerodynamic pressure and damping versus aerodynamic pressure are used to determine the critical aerodynamic pressure, and hence to predict flutter of various isotropic and composite, stiffened, and unstiffened plates. The results for the flutter of straight-stiffened plates are validated through comparison with published papers. Several numerical examples are discussed, for which parametric studies for the shape and number of stiffeners and fiber orientation are performed. The flutter mode shapes are also presented. The present study attests that the critical dynamic pressure can be enhanced, and flutter can be successfully suppressed by changing the stiffener’s shape and fiber orientation.

Vibration Analysis of Curvilinearly Stiffened Composite Panel Subjected to In-Plane Loads

This paper presents an efficient finite element approach to study the prestressed vibration mode results of a curvilinearly stiffened composite panel subjected to various in-plane loads. The present method models the plate and the stiffener separately, which allows the stiffener element nodes to not coincide with the plate shell-element nodes. The stiffness and mass matrices of a stiffener are transformed to those of the plate through the displacement compatibility conditions at the plate–stiffener interface via finite element interpolation. Convergence and validation studies have been conducted to verify the present method in the finite element vibration analysis by using the examples from the existing literature. Prestressed vibration mode results are examined for a stiffened composite panel with arbitrarily shaped composite stiffeners in the presence of the in-plane normal and shear loads. Numerical results show the possible benefits of using curvilinear stiffeners to improve the vibration response by increasing both the fundamental frequency and the buckling load through changing the vibration and buckling mode shapes with a negligible or even no weight penalty. The stiffener depth ratio is found to increase both the fundamental frequency and the buckling load, but only up to a certain value. Any further increase in the stiffener depth ratio causes stiffener buckling before plate buckling, and it leads the free and prestressed vibration mode results to behave as the vibration mode results for a plate with simply supported boundary conditions along the stiffeners.

Optimization of Curvilinearly Stiffened Panels with Single Cutout Concerning the Collapse Load

The motivation of this study is to meet the requirement of cutout reinforcements in aerospace industry by using the design flexibility of curvilinearly stiffened panels. The reason is that the load path and tension field caused by curvilinear stiffeners can increase the load-carrying capacity of stiffened panels. The inherent superiority of such structures was examined in detail by comparison with those having straight stiffeners. Due to the large computational burden caused by post-buckling analysis, the optimum design of curvilinearly stiffened panels with single cutout was obtained based on surrogate model. On this basis, variable symmetrization and reduction methods were developed for different design cases, aimed at reducing the number of dependent variables in the optimization. Illustrative examples show the superiority of curvilinearly stiffened panels for cutout reinforcement, and demonstrate the effectiveness of the proposed optimization framework.

An artificial neural network residual kriging based surrogate model for curvilinearly stiffened panel optimization

We have performed a design optimization of a stiffened panel with curvilinear stiffeners using an artificial neural network (ANN) residual kriging based surrogate modeling approach. The ANN residual kriging based surrogate modeling involves two steps. In the first step, we approximate the objective function using ANN. In the next step we use kriging to model the residue. We optimize the panel in an iterative way. Each iteration involves two steps-shape optimization and size optimization. For both shape and size optimization, we use ANN residual kriging based surrogate model. At each optimization step, we do an initial sampling and fit an ANN residual kriging model for the objective function. Then we keep updating this surrogate model using an adaptive sampling algorithm until the minimum value of the objective function converges. The comparison of the design obtained using our optimization scheme with that obtained using a traditional genetic algorithm (GA) based optimization scheme shows satisfactory agreement. However, with this surrogate model based approach we reach optimum design with less computation effort as compared to the GA based approach which does not use any surrogate model.

Optimization design of unitized panels with stiffeners in different formats using the evolutionary strategy with covariance matrix adaptation

Curvilinear stiffener concept has been introduced to aircraft panel structures most recently for possible further weight reduction during optimization design. However, due to enlarged design space and high design complexity, more computational time is needed to optimize curvilinearly stiffened panels. Considering the requirement for both lighter structure and less design time, optimization designs of a unitized panel with stiffeners in three different formats, i.e. curvilinear, oblique, and evenly distributed straight, are conducted under various loading conditions to figure out which stiffener format should be selected in a real design environment. Four single loading cases with different compression-shear ratios and a set of multiple load cases are considered. Evolution strategy with covariance matrix adaption is combined with sequential quadratic programming to seek out the optimum structure with minimum mass taking into account buckling and strength constraints. Comparative analysis of the optimization results indicates that evenly distributed straight format is suitable for compression-dominant loading conditions whereas curvilinear format become superior in case that shear loading is considerable or multiple biaxial compression and shear loads are applied. Besides, curvilinear format adds more design flexibilities to the stiffeners, which enables them to play a more important role in the optimized structures. Furthermore, evolution strategy with covariance matrix adaption is found to be more efficient than particle swarm optimization for this optimization problem. This study can provide a useful guidance for future optimization design of aircraft structures.

Efficient Optimization of Cylindrical Stiffened Shells with Reinforced Cutouts by Curvilinear Stiffeners

An efficient optimization framework of cylindrical stiffened shells with reinforced cutouts by curvilinear stiffeners is proposed in this study. First, an adaptive method to determine the near field around the cutout and far field away from the cutout is presented. Then, a novel hybrid model is established to reduce the computational efforts of postbuckling analysis; the numerical implementation asymptotic homogenization method is used to smear out the stiffeners in the far field, and curvilinear stiffeners are adopted to improve the loading path and thus local stiffness of the near field, which can provide a type of flexible stiffener configurations for cutout reinforcement. After that, the optimization of curvilinear stiffeners is performed by a novel bilevel strategy based on the hybrid model. In the first level, a stiffener distribution function is used to reduce the number of active variables, and then stiffener layout, stiffener number, and section profile are optimized simultaneously. In the second level, the stiffener number and section profile are held constant, and local optimization is then performed for each curvilinear stiffener location. An illustrative example demonstrates the effectiveness of the proposed framework, when compared with traditional optimizations.

Free Vibration of Curvilinearly Stiffened Shallow Shells

The free vibration of curvilinearly stiffened shallow shells is investigated by the Ritz method. Based on the first-order shear deformation shell theory and three-dimensional (3D) curved beam theory, the strain and kinetic energies of the stiffened shells are introduced. The stiffener can be placed anywhere within the shell, without the need for having the stiffener and shell element nodes coincide. Numerical results with different geometrical shells and boundary conditions and different stiffener locations and curvatures are analyzed to verify the feasibility of the presented Ritz method for solving the problems. The results show good agreement with those using other methods, e.g., using a converged set of results obtained by Nastran.

Experimental Validation of the EBF3PanelOpt Vibroacoustic Analysis of Stiffened Panels

Vibroacoustic experiments are performed at the Structural Acoustics Loads and Transmission facility of NASA Langley Research Center to measure the sound radiation from and transmission loss of two panels with straight and curvilinear stiffeners, respectively. The focus of the work addressed in this paper is to validate the vibroacoustic analysis methods implemented in a multi-objective design optimization framework, EBF3PanelOpt. The stiffened test panels with six stiffeners have been designed using EBF3PanelOpt for turbulent boundary-layer excitation. To build the confidence in finite-element models, free-vibration tests of the manufactured stiffened panels are also performed. The excited experimental mode shape and eigenfrequencies match quite well with the finite element analysis results for the free-vibration test. A medium-density fiberboard and aluminum frame is used to clamp the stiffened panels in the Structural Acoustics Loads and Transmission loss window. Because of strong coupling of the stiffened test panels with the frame, a finite-element model of the stiffened panels with frame is developed. To capture the effect of the bolted joints in the test setup, bush elements with optimal values of rotational and translational stiffnesses are used. Measurements are also performed to estimate the damping in the stiffened panels with the frame. The numerical prediction of sound radiation from and transmission loss of stiffened panels is in fair agreement with the measured values for both the panels.

Multi-objective vibro-acoustic optimization of stiffened panels

A multi-objective vibro-acoustic design optimization of straight or curvilinearly stiffened panels excited by an acoustic diffuse field is performed. During design optimization, the panel mass and the radiated acoustic power are the two objectives to be minimized while satisfying constraints on buckling, von Mises stress and crippling. Based on the concept of plane wave propagation, a diffuse acoustic field is developed for use along with a finite element model. To represent the panel's structural behavior, the dynamic analysis of the panel is performed for the developed diffuse acoustic excitation and the radiated acoustic power is calculated using the velocities obtained from the dynamic analysis. A baseline design is obtained by optimization study with mass as an objective to be minimized while constraints are put on buckling, von Mises stress and crippling. The obtained baseline grid stiffened panel is used for a comparative study of a panel with curvilinear stiffeners in vibro-acoustics with diffuse sound field as the source of the excitation.

A framework combining meshfree analysis and adaptive kriging for optimization of stiffened panels

A framework is developed for structural optimization using an Element Free Galerkin (EFG) method for analyzing the structure, a kriging for surrogate model, and a Genetic Algorithm (GA) for finding the optimum design. The framework is tested for a plate with curvilinear stiffeners which are now possible with additive manufacturing such as 3-D printing. The efficiency and accuracy of the framework is compared with two other approaches: 1) MD.PATRAN, MD.NASTRAN and VisualDOC as implemented in EBF3PanelOpt, a Computational Design Environment being developed at Virginia Tech to optimally design panels with curvilinear stiffeners, and 2) a kriging, MD.PATRAN and MD.NASTRAN and genetic algorithms. All three approaches use optimization methods on both the shape and size design variables. The optimization scheme is a two-step optimization approach that divides the design variables into size and shape variables. First, the buckling parameter is maximized over the shape design variable, and then the mass is minimized over the size variable with constraint on buckling. The comparison between the three approaches shows the efficiency of the developed framework.

Curvilinearly T-Stiffened Panel-Optimization Framework Under Multiple Load Cases Using Parallel Processing

Future aerospace vehicles like hybrid wing/body, truss-braced wing, and “double bubble” would have pressurized noncircular fuselage structures and complex wing geometry. Traditional aircraft designs have led to the confidence and experience of designing such structures using the knowledge base built over the years and the resulting rules of thumb. However, there is a lack of experience of load calculations and design of complex, multifunctional, aircraft structural concepts for future aerospace vehicles. Designing such structures will require a physics-based optimization framework. Therefore, a new optimization framework, EBF3PanelOpt, is being developed. Commercial software MD-PATRAN (geometry modeling and mesh generation) and MD-NASTRAN (finite-element analysis) are integrated in EBF3PanelOpt framework using the Python programming environment to design stiffened panels with curvilinear stiffeners. Currently, EBF3PanelOpt optimizes the stiffened panel with curvilinear blade stiffeners, where the loads are applied only through the plate. During the optimization, the mass is minimized with the constraints on buckling, von Mises stress, and crippling or local failure of the stiffeners. EBF3PanelOpt is enhanced to have curvilinear T stiffeners with or without axial loads in addition to loads through plate. The panel/stiffener geometry is defined in a parametric fashion based on design variables that include variables for orientation and shape of the stiffeners, the thicknesses and heights of the webs and flanges of the stiffeners, and the plate-pocket thicknesses. This framework is supported with coarse-grained parallelism using Python to analyze multiple designs on the cluster. Using this framework, a vertical stabilizer skin panel of transport aircraft panel having two extreme load cases is optimized using with or without stiffener loads. When the equivalent uniform loads are applied only through the plate, the plate buckling becomes critical, but combined buckling of plate and stiffeners becomes critical when the loads are applied through both the plate and the stiffeners. When the uniform in-plane compressive loads are applied through both the plate and the stiffeners, the panel with straight T stiffeners is more optimal than the panel with curvilinear T stiffeners.

Fracture Analysis of a Curvilinearly Stiffener Panel Subjected to Multiple Combined Loadings

A global–local finite element analysis was performed to study the damage tolerance of curvilinearly stiffened panels, fabricated using the modern additive-manufacturing process that can create so-called unitized structures. In the first step, a buckling analysis of panels was performed to determine whether the panels satisfied the buckling constraint in an undamaged state. In the second step, stress distributions in the panel were analyzed to determine the location of the critical stress under combined shear and compression loadings. Then, a fracture analysis of the curvilinearly stiffened panel with a crack designed at the earlier-obtained location of the critical stress, which was the common location with the maximum magnitude of the principal stresses and von Mises stress, was performed under combined shear and tensile loadings. A mesh-sensitivity analysis was performed to validate the choice of the mesh density near the crack tip. All analyses were performed using the global–local finite element method using MSC.Marc, and the global finite element methods using MSC.Marc and ABAQUS. A negligible difference in results and 94% savings in the CPU time were achieved using the global–local finite element method over the global finite element method by using a mesh density of 8.4 element/mm ahead of the crack tip. To study the influence of different loads on basic modes of fracture, the shear and normal (tensile) loads were varied differently. It was observed that the case with the fixed shear load but variable normal loads, and the case with the fixed normal load but variable shear loads were mode I. Under the maximum combined-loading condition, the largest effective stress-intensity factor was much smaller than the critical-stress-intensity factor.

Optimum Design of Curvilinear Stiffened Panels with Stiffening Beams

The paper presents a framework for design and optimization of curvilinear stiffened panels with any cross-section shape of stiffener. Within this framework, the weight of stiffened panel applied complex loads, is minimized under the constraints of structural responses include global buckling, maximum stress and crippling. Then, a stiffened panel with two blade curvilinear stiffeners is optimized using this framework, and the optimum performs better than another existing design. Finally, it is concluded that stiffening beam is suitable for simulate stiffener sustains part of applied load directly.

EBF3PanelOpt: An optimization framework for curvilinear blade-stiffened panels

A new framework, EBF3PanelOpt, is being developed for design and optimization of complex, multifunctional, aircraft structural concepts like pressurized non-circular fuselage structures to be used in hybrid wing/body vehicles that are subjected to complex structural loading cases. This tool can be used to integrate materials and structural concepts to exploit emerging additive manufacturing processes that offer the ability to efficiently fabricate complex structural configurations. Commercial software packages, MD-Patran (geometry modeling and mesh generation), MD-Nastran (Finite Element Analysis), are integrated in the EBF3PanelOpt framework using Python programming environment to design stiffened panels with curvilinear stiffeners. Typically, these panels experience multiple loading conditions during the operations of these vehicles. EBF3PanelOpt has the capability to optimize flat/curved multi-sided panels with straight/curved edges having curvilinear, blade-type stiffeners under multiple loading conditions. The mass of the panel is minimized subjected to constraints on buckling, von Mises stress, and crippling or local failure of the stiffener using global optimization techniques or gradient based optimization techniques. The panel/stiffener geometry is parametrized using design variables that include variables for orientation and shape of the stiffeners, the thicknesses and height of the stiffeners, and the plate thickness. The plate can have uniform thickness or non-uniform thicknesses for the pockets created by the stiffeners. During optimization, constraints can be applied for each of the loading conditions by aggregating all the responses using Kreisselmeir–Steinhauser criteria or using worst response amongst all the responses or applying all the constraints. Initially, the flat rectangular panel is optimized for the single load-case to study the effectiveness of the panel thickness option. Then, the optimization of flat rectangular and cylindrical panels is carried out for three sample load cases of practical interest. This paper discusses the advantages and disadvantages of the proposed constraints' application.