Buckling Optimization of Curved Grid Stiffeners through the Level Set Based Density Method

Stiffened structures have great potential for improvingmechanical performance,and the study of their stability is of great interest.In this paper,the optimization of the critical buckling load factor for curved grid stiffeners is solved by using the level set based density method,where the shape and cross section(including thickness and width)of the stiffeners can be optimized simultaneously.The grid stiffeners are a combination ofmany single stiffenerswhich are projected by the corresponding level set functions.The thickness and width of each stiffener are designed to be independent variables in the projection applied to each level set function.Besides,the path of each single stiffener is described by the zero iso-contour of the level set function.All the single stiffeners are combined together by using the p-norm method to obtain the stiffener grid.The proposed method is validated by several numerical examples to optimize the critical buckling load factor.

Nonlinear Bend Stiffener Analysis Using A Simple Formulation and Finite Element Method

Flexible marine risers are commonly used in deepwater floating systems. Bend stiffeners are designed to protect flexible risel against excessive bending at the connection with the hull. The structure is usually analyzed as a cantilever beam subjected to an inclined point load. As deflections are large and the bend stiffener material exhibits nonlinear stress-strain characteristics, geometric and material nonlinearities are important considerations. A new approach has been developed to solve this nonlinear problem. Its main advantage is its simplicity; in fact the present method can be easily implemented on a spreadsheet. Finite element analysis using ABAQUS is performed to validate the method. Solid elements are used for the bend stiffener and flexible pipe. To simulate the near inextensibility of flexible risers, a simple and original idea of using truss elements is proposed. Through a set of validation studies the present method is found to be in a good agreement with the finite element analysis. Further, parametric studies are performed by using both methods to identify the key parameters and phenomena that are most critical in design. The most important finding is that the common practice of neglecting the internal steel sleeve in the bend stiffener analysis is non-conservative and therefore needs to be reassessed.

Key Techniques and Applications of Adaptive Growth Method for Stiffener Layout Design of Plates and Shells

The application of the adaptive growth method is limited because several key techniques during the design process need manual intervention of designers. Key techniques of the method including the ground structure construction and seed selection are studied, so as to make it possible to improve the effectiveness and applicability of the adaptive growth method in stiffener layout design optimization of plates and shells. Three schemes of ground structures, which are comprised by different shell elements and beam elements, are proposed. It is found that the main stiffener layouts resulted from different ground structures are almost the same, but the ground structure comprised by 8-nodes shell elements and both 3-nodes and 2-nodes beam elements can result in clearest stiffener layout, and has good adaptability and low computational cost. An automatic seed selection approach is proposed, which is based on such selection rules that the seeds should be positioned on where the structural strain energy is great for the minimum compliance problem, and satisfy the dispersancy requirement. The adaptive growth method with the suggested key techniques is integrated into an ANSYS-based program, which provides a design tool for the stiffener layout design optimization of plates and shells. Typical design examples, including plate and shell structures to achieve minimum compliance and maximum bulking stability are illustrated. In addition, as a practical mechanical structural design example, the stiffener layout of an inlet structure for a large-scale electrostatic precipitator is also demonstrated. The design results show that the adaptive growth method integrated with the suggested key techniques can effectively and flexibly deal with stiffener layout design problem for plates and shells with complex geometrical shape and loading conditions to achieve various design objectives, thus it provides a new solution method for engineering structural topology design optimization.

Buckling analysis of functionally graded material(FGM) sandwich truncated conical shells reinforced by FGM stiffeners filled inside by elastic foundations

An analytical solution for buckling of an eccentrically stiffened sandwich truncated conical shell is investigated. The shell consists of two functionally graded material （FGM） coating layers and a core layer which are metal or ceramic subjected to an axial compressive load and an external uniform pressure. Shells are reinforced by stringers and rings, in which the material properties of shells and stiffeners are graded in the thickness direction following a general sigmoid law distribution. Two models of coated shell-stiffener arrangements are investigated. The change of the spacing between stringers in the meridional direction is taken into account. A couple set of three-variable- coefficient partial differential equations in terms of displacement components are solved by the Galerkin method. A closed-form expression for determining the buckling load is obtained. The numerical examples are presented and compared with previous works.

THE LAYOUT OPTIMIZATION OF STIFFENERS FOR PLATE-SHELL STRUCTURES

The plate-shell structures with stiffeners are widely used in a broad range of engineering structures. This study presents the layout optimization of stiffeners. The minimum weight of stiffeners is taken as the objective function with the global stiffness constraint. In the layout optimization, the stiffeners should be placed at the locations with high strain energy/or stress. Conversely, elements of stiffeners with a small strain energy/or stress are considered to be used inefficiently and can be removed. Thus, to identify the element efficiency so that most inefficiently used elements of stiffeners can be removed, the element sensitivity of the strain energy of stiffeners is introduced, and a search criterion for locations of stiffeners is presented. The layout optimization approach is given for determining which elements of the stiffeners need to be kept or removed. In each iterative design, a high efficiency reanalysis approach is used to reduce the computational effort. The present approach is implemented for the layout optimization of stiffeners for a bunker loaded by the hydrostatic pressure. The numerical results show that the present approach is effective for dealing with layout optimization of stiffeners for plate-shell structures.

A NEW DISPLACEMENT-TYPE STABILITY EQUATION AND GENERAL STABILITY ANALYSIS OF LAMINATED COMPOSITE CIRCULAR CONICAL SHELLS WITH TRIANGULAR GRID STIFFENERS

In this paper, based on the theory of Donnell-type shallow shell, a new displacement-type stability equations is first developed for laminated composite circular conical shells with triangular grid stiffeners by using the variational calculus and generalized smeared-stiffener theory. The most general bending stretching couplings, the effect of eccentricity of stiffeners are considered. Then, for general stability of composite triangular grid stiffened conical shells without twist coupling terms, the approximate formulas are obtained for critical external pressure by using Galerkin's procedure. Numerical examples for a certain C/E composite conical shells with inside triangular grid stiffeners are calculated and the results are in good agreement with the experimental data. Finally, the influence of some parameters on critical external pressure is studied. The stability equations developed and the formulas for critical external pressure obtained in this paper should be very useful in the astronautical engineering design.

Rapid Modeling and Finite Element Analysis of Missile Stiffener

Stiffener is the important missile structure to ensure the structural strength of the missile. In order to improve the design efficiency and the quality of the missile stiffener, the methods of missile stiffener rapid modeling and analysis are proposed. First, the problems of traditional manual modeling of the stiffener are analyzed. According to the problems and actual requirement of modeling, volume decomposition method is used to divide the stiffener into the upper section, the lower section and the web in order for feature analysis and parameter extraction. Then based on the parameters the basic unit decomposed above is created for Boolean operation to establish the stiffener. Finally, a rapid stiffener modeling and analysis program were developed based on UG Open API, the modeling and analysis result validates the feasibility of the method.

Mechanical and thermal postbuckling of FGM thick circular cylindrical shells reinforced by FGM stiffener system using higher-order shear deformation theory

The postbuckling of the eccentrically stiffened circular cylindrical shells made of functionally graded materials （FGMs）, subjected to the axial compressive load and external uniform pressure and filled inside by the elastic foundations in the thermal environments, is investigated with an analytical method. The shells are reinforced by FGM stringers and rings. The thermal elements of the shells and stiffeners in the fundamental equations are considered. The equilibrium and nonlinear stability equations in terms of the displacement components for the stiffened shells are derived with the third-order shear deformation theory and Leckhniskii smeared stiffener technique. The closed-form expressions for determining the buckling load and postbuckling load-deflection curves are obtained with the Galerkin method. The effects of the stiffeners, the foundations, the material and dimensional parameters, and the pre-existent axial compressive and thermal load are considered.

Nonlinear stability of functionally graded material(FGM)sandwich cylindrical shells reinforced by FGM stiffeners in thermal environment

In this paper, Donnell＇s shell theory and smeared stiffeners technique are improved to analyze the postbuckling and buckling behaviors of circular cylindrical shells of stiffened thin functionally graded material （FGM） sandwich under an axial loading on elastic foundations, and the shells are considered in a thermal environment. The shells are stiffened by FGM rings and stringers. A general sigmoid law and a general power law are proposed. Thermal elements of the shells and reinforcement stiffeners are considered. Explicit expressions to find critical loads and postbuckling load-deflection curves are obtained by applying the Galerkin method and choosing the three-term approximate solution of deflection. Numerical results show various effects of temperature, elastic foundation, stiffeners, material and geometrical properties, and the ratio between face sheet thickness and total thickness on the nonlinear behavior of shells.

Influence of local stiffeners and cutout shapes on the vibration and stability characteristics of quasi-isotropic laminates under hygro-thermo-mechanical loadings

The perforated stiffened panel is generally found as a sub-component of sophisticated structures.The fundamental purpose of this panel is to withstand against buckling under complicated loading and environmental conditions.Hence,an accurate knowledge of critical buckling behaviour of stiffened panels is very much essential for a reliable and lightweight structural design.In this paper,the focus is on quasi-laminated panels with different cutout shapes of various sizes and their responses to hygrothermal environments under nonlinearly varying edge loads and is compared with the locally stiffened panels.Towards this,the modelling of the panel and stiffener is done by adopting nine-noded heterosis plate elements and three noded beam elements respectively.The stiffener formulation is suitably modified in order to take the torsional effect also into consideration along with the effect of shear deformation.Initially,the plate and the stiffener elements are treated separately,and then the displacement compatibility is maintained between them by using the transformation matrix.For a given loading and geometric discontinuity,the stress distribution within the perforated panel is highly non-uniform in nature and hence a dynamic approach has been used to calculate buckling loads by adopting two sets of boundary conditions,one set for pre-buckling stress analysis and the second set for buckling analysis.Four different quasi-isotropic stacking sequences are deliberated in this work by varying different ply-orientation in each scheme.The study also addresses the effect of various parameters such as nonlinear loads,hygro-thermal loads,cutout size and shapes,position of cutout,stiffener parameters,stacking sequences,thickness of plate and boundary conditions.

Layout design of thin-walled structures with lattices and stiffeners using multi-material topology optimization

In this paper,the thin-walled structures with lattices and stiffeners manufactured by additive manufacturing are investigated.A design method based on the multi-material topology optimization is proposed for the simultaneous layout optimization of the lattices and stiffeners in thin-walled structures.First,the representative lattice units of the selected lattices are equivalent to the virtual homogeneous materials whose effective elastic matrixes are achieved by the energy-based homogenization method.Meanwhile,the stiffeners are modelled using the solid material.Subsequently,the multi-material topology optimization formulation is established for both the virtual homogeneous materials and solid material to minimize the structural compliance under mass constraint.Thus,the optimal layout of both the lattices and stiffeners could be simultaneously attained by the optimization procedure.Two applications,the aircraft panel structure and the equipment mounting plate,are dealt with to demonstrate the detailed design procedure and reveal the effect of the proposed method.According to numerical comparisons and experimental results,the thin-walled structures with lattices and stiffeners have significant advantages over the traditional stiffened thin-walled structures and lattice sandwich structures in terms of static,dynamic and anti-instability performance.

Geometry and size optimization of stiffener layout for three-dimensional box structures with maximization of natural frequencies

Based on the growth mechanism of natural biological branching systems and inspiration from the morphology of plant root tips,a bionic design method called Improved Adaptive Growth Method(IAGM)has been proposed in the authors’previous research and successfully applied to the reinforcement optimization of three-dimensional box structures with respect to natural frequencies.However,as a kind of ground structure methods,the final layout patterns of stiffeners obtained by using the IAGM are highly subjected to their ground structures,which restricts the optimization effect and freedom to further improve the dynamic performance of structures.To solve this problem,a novel post-processing geometry and size optimization approach is proposed in this article.This method takes the former layout optimization result as start,and iteratively finds the optimal layout angles,locations,and lengths of stiffeners with a few design variables by optimizing the positions of some specific node lines called active node lines.At the same time,thick-nesses of stiffeners are also optimized to further improve natural frequencies of three-dimensional box structures.Using this method,stiffeners can be successfully separated from their ground structures and further effectively improve natural frequencies of three-dimensional box structures with less material consumption.Typical numerical examples are illustrated to validate the effectiveness and advantages of the suggested method.

An Optimization Approach for Stiffener Layout of Composite Stiffened Panels Based on Moving Morphable Components(MMCs)

An explicit topology optimization method for the stiffener layout of composite stiffened panels is proposed based on moving morphable components(MMCs).The skin and stiffeners are considered as panels with different bending stiffnesses,with the use of equivalent stiffness method.Then the location and geometric properties of composite stiffeners are determined by several MMCs to perform topology optimization,which can greatly simplify the finite element model.With the objective of maximizing structural stiffness,several typical cases with various loading and boundary conditions are selected as numerical examples to demonstrate the proposed method.The numerical examples illustrate that the proposed method can provide clear stiffener layout and explicit geometry information,which is not limited within the framework of parameter and size optimization.The mechanical properties of composite stiffened panels can be fully enhanced.

Concurrent topology optimization design of stiffener layout and cross-section for thin-walled structures

Stiffened plates or shells are widely used in engineering structures as primary or secondary load-bearing components.How to design the layout and sizes of the stiffeners is of great significance for structural lightweight.In this work,a new topology optimization method for simultaneously optimizing the layout and cross-section topology of the stiffeners is developed to solve this issue.The stilfeners and base plates are modeled by the beam and shell elements,respectively,significantly reducing the computational cost.The Giavotto beam theory,instead of the widely employed Euler or Timoshenko beam theory,is applied to model the stiffeners for considering the warping deformation in evaluating the section stiffness of the beam.A multi-scale topology optimization model is established by simultaneously optimizing the layout of the beam and the topology of the cross-section.The design space is significantly expanded by optimizing these two types of design variables.Several numerical examples are applied to illustrate the validity and effectiveness of the proposed method.The results show that the proposed two-scale optimization approach can generate better designs than the single-scale method.

Stiffeners layout design of thin-walled structures with constraints on multi-fastener joint loads

The purpose of this paper is to present an extended topology optimization method for the stiffeners layout design of aircraft assembled structures. Multi-fastener joint loads and manufacturing constraints are considered simultaneously. On one hand, the joint loads are calculated and constrained within a limited value to avoid the failure of fasteners. On the other hand, the manufacturing constraints of the material distribution in the machining directions of stiffeners are implemented by an improved piecewise interpolation based on a beveled cut-surface. It is proven that the objective function is strictly continuous and differentiable with respect to the piecewise interpolation. The effects of the extended method with two different constraints are highlighted by typical numerical examples. Compared with the standard topology optimization, the final designs have clearly shown the layout of stiffeners and the joint loads have been perfectly constrained to a satisfying level.

Effect of Orthogonal Stiffeners on the Stability of Axially Compressed Steel Jacking Pipe

Load conditions for steel pipe-jacking are complex during the construction stage. The stability of steel jacking pipe has been an increasingly important problem as jacking forces, pipe diameters and jacking distances increase. However, there are no standards for pipe reinforcement, for prevention of buckling, or for remedying pipe that buckles when being jacked axially. Past experience suggests that stiffeners can effectively reinforce the structure. This study analyzes the effect of different stiffeners on the stability of steel jacking pipe under axial compression using finite element analysis. The results suggest that the stability of steel jacking pipe can be significantly improved by using orthogonal stiffeners, in terms of engineering costs and construction space inside the pipe. Based on current engineering practice, the application of orthogonal stiffeners is discussed. This study provides a useful reference for the design and construction of steel jacking pipe.

Protective performance of shear stiffening gel-modified foam against ballistic impact:Experimental and numerical study

As one of the most widely used personal protective equipment(PPE),body armors play an important role in protecting the human body from the high-velocity impact of bullets or projectiles.The body torso and critical organs of the wear may suffer severe behind-armor blunt trauma(BABT)even though the impactor is stopped by the body armor.A type of novel composite material through incorporating shear stiffening gel(STG)into ethylene-vinyl acetate(EVA)foam is developed and used as buffer layers to reduce BABT.In this paper,the protective performance of body armors composed of fabric bulletproof layers and a buffer layer made of foam material is investigated both experimentally and numerically.The effectiveness of STG-modified EVA in damage relief is verified by ballistic tests.In parallel with the experimental study,numerical simulations are conducted by LS-DYNA®to investigate the dynamic response of each component and capture the key mechanical parameters,which are hardly obtained from field tests.To fully describe the material behavior under the transient impact,the selected constitutive models take the failure and strain rate effect into consideration.A good agreement between the experimental observations and numerical results is achieved to prove the validity of the modelling method.The tests and simulations show that the impact-induced deformation on the human body is significantly reduced by using STG-modified EVA as the buffering material.The improvement of protective performance is attributed to better dynamic properties and more outstanding energy absorption capability of the composite foam.

裂纹加筋和缺口加筋对钢加筋板极限强度的影响

This paper numerically evaluates the effect of the crack position on the ultimate strength of stiffened panels.Imperfections such as notches and cracks in aged marine stiffened panels can reduce their ultimate strength.To investigate the effect of crack length and position,a series of nonlinear finite element analyses were carried out and two cases were considered,i.e.,case 1 with thin stiffeners and case 2 with thick stiffeners.In both cases,the stiffeners have the same cross-section area.To have a basis for comparison,the intact panels were modeled as well.The cracks and notches were in the longitudinal and transverse direction and were assumed to be in the middle part of the panel.The cracks and notches were assumed to be through the thickness and there is neither crack propagation nor contact between crack faces.Based on the numerical results,longitudinal cracks affect the behavior of the stiffened panels in the postbuckling region.When the stiffeners are thinner,they buckle first and provide no reserved strength after plate buckling.Thus,cracks in the stiffeners do not affect the ultimate strength in the case of the thinner stiffeners.Generally,when stiffeners are thicker,they affect the postbuckling behavior more.In that case,cracks in the stiffeners affect the buckling and failure modes of the stiffened panels.The effect of notch was also studied.In contrast to the longitudinal crack in stiffeners,a notch in the stiffeners reduces the ultimate strength of the stiffened panel for both slender and thick stiffeners.

Analyzing the Form-Finding of a Large-Span Transversely Stiffened Suspended Cable System: A Method Considering Construction Processes

The precise control of the shape of transversely stiffened suspended cable systems is crucial. However, existing form-finding methods primarily rely on iterative calculations that treat loads as fixed known conditions. These methods are inefficient and fail to accurately control shape results. In this study, we propose a form-finding method that analyzes the load response of models under different sag and stress levels, taking into account the construction process. To analyze the system, a structural finite element model was established in ANSYS, and geometric nonlinear analysis was conducted using the Newton-Raphson method. The form-finding analysis results demonstrate that the proposed method achieves precise control of shape, with a maximum shape error ranging from 0.33% to 0.98%. Furthermore, the relationships between loads and tension forces are influenced by the deformed shape of the structures, exhibiting significant geometric nonlinear characteristics. Meanwhile, the load response analysis reveals that the stress level of the self-equilibrium state in the transversely stiffened suspended cable system is primarily governed by strength criteria, while shape is predominantly controlled by stiffness criteria. Importantly, by simulating the initial tensioning process as an initial condition, this method solves for a counterweight that satisfies the requirements and achieves a self-equilibrium state with the desired shape. The shape of the self-equilibrium state is precisely controlled by simulating the construction process. Overall, this work presents a new method for analyzing the form-finding process of large-span transversely stiffened suspended cable system, considering the construction process which was often overlooked in previous studies.

Vibrational power flow of a finite cylindrical shell with discrete axial stiffeners

The structural wave power flows in an elastic finite cylindrical shell with discrete axial stiffeners are studied when a simple harmonic force is applied on it. The equations of motion of the shell are derived by using Flügge equation and Hamilton variational principle, and the responses of the shell are obtained. By use of the basic definition of the power flow, the characteristics of axial propagation of the power flow supplied by input structure and carried by different shell internal forces of a forced shell are investigated. The effects of parameters, such as relative location of driving force and stringer, driving force type and structural damping on the vibrational power flows in the shell, are discussed. These provide some theoretical bases for vibration control and noise reduction of this kind of structure.