Explicit Topology Optimization Design of Stiffened Plate Structures Based on the Moving Morphable Component(MMC)Method

This paper proposes an explicit method for topology optimization of stiffened plate structures.The present work is devoted to simultaneously optimizing stiffeners’shape,size and layout by seeking the optimal geometry parameters of a series of moving morphable components(MMC).The stiffeners with straight skeletons and the stiffeners with curved skeletons are considered to enhance the modeling and optimization capability of the current approach.All the stiffeners are represented under the Lagrangian-description framework in a fully explicit way,and the adaptive ground structure method,as well as dynamically updated plate/shell elements,is used to obtain optimized designs with more accurate analysis results.Compared with existing works,the proposed approach provides an explicit description of the structure.Thus,a stiffened plate structure with clear stiffener distribution and smooth geometric boundary can be obtained.Several numerical examples provided,including straight and curved stiffeners,hierarchical stiffeners,and a stiffened plate with a cutout,validate the effectiveness and applicability of the proposed approach.

Impact Damage Identification of Aluminum Alloy Reinforced Plate Based on GWO-ELM Algorithm

As a critical structure of aerospace equipment,aluminum alloy stiffened plate will influence the stability of spacecraft in orbit and the normal operation of the system.In this study,a GWO-ELM algorithm-based impact damage identification method is proposed for aluminum alloy stiffened panels to monitor and evaluate the damage condition of such stiffened panels of spacecraft.Firstly,together with numerical simulation,the experimental simulation to obtain the damage acoustic emission signals of aluminum alloy reinforced panels is performed,to establish the damage data.Subsequently,the amplitude-frequency characteristics of impact damage signals are extracted and put into an extreme learning machine(ELM)model to identify the impact location and damage degree,and the Gray Wolf Optimization(GWO)algorithm is employed to update the weight parameters of the model.Finally,experiments are conducted on the irregular aluminum alloy stiffened plate with the size of 2200 mm×500 mm×10 mm,the identification accuracy of impact position and damage degree is 98.90% and 99.55% in 68 test areas,respectively.Comparative experiments with ELM and backpropagation neural networks(BPNN)demonstrate that the impact damage identification of aluminum alloy stiffened plate based on GWO-ELM algorithm can serve as an effective way to monitor spacecraft structural damage.

A study of the perforation of stiffened plates by rigid projectiles

In the present paper, a four-stage perforation model that accurately predicts the residual velocity is developed by adopting an energy method. The four stages are plug formation, dishing formation, petal formation and projectile exit. In addition, some important experimental results are presented and analyzed to validate the present perforation model. In the experiments, high speed camera system is used to record the perforation process. Observations on target damage and measurements of initial velocities and residual velocities with the aid of the system are presented. Numerical simulations are carried out for projectiles against single and layered plates adopted in the experiments. The perforation process is studied and the deformation and failure modes are obtained. The predictions of numerical simulations and analytical model are found in reasonably good agreement with those of experiments, and can be used to predict the ballistic limit and residual velocity of stiffened plates perforated by rigid projectiles.

B-Spline Wavelet on Interval Finite Element Method for Static and Vibration Analysis of Stiffened Flexible Thin Plate

A new wavelet finite element method(WFEM)is constructed in this paper and two elements for bending and free vibration problems of a stiffened plate are analyzed.By means of generalized potential energy function and virtual work principle,the formulations of the bending and free vibration problems of the stiffened plate are derived separately.Then,the scaling functions of the B-spline wavelet on the interval(BSWI)are introduced to discrete the solving field variables instead of conventional polynomial interpolation.Finally,the corresponding two problems can be resolved following the traditional finite element frame.There are some advantages of the constructed elements in structural analysis.Due to the excellent features of the wavelet,such as multi-scale and localization characteristics,and the excellent numerical approximation property of the BSWI,the precise and efficient analysis can be achieved.Besides,transformation matrix is used to translate the meaningless wavelet coefficients into physical space,thus the resolving process is simplified.In order to verify the superiority of the constructed method in stiffened plate analysis,several numerical examples are given in the end.

Dynamic Response of Stiffened Plates with Holes Subjected to Shock Waves and Fragments

The power field of shock waves and fragments is analyzed and set up, and the damage modes of stiffened plates are put forward. According to the structural characters of the stiffened plates investigated and the properties of the shock waves and fragments, the experiments on the shock waves acting on the stiffened plates (penetrated and non-penetrated by fragments) are mainly conducted. The dynamic response rules of stiffened plates with holes under shock waves and fragments loading are obtained. The results show that the penetration of fragments into stiffened plates hardly affects their deformation produced by shock waves..

Buckling and large deflection behaviors of radially functionally graded ring-stiffened circular plates with various boundary conditions

The buckling and large deflection behaviors of axis-symmetric radially functionally graded （RFG） ring-stiffened circular plates are investigated by the dynamic relaxation （DR） method combined with the finite difference discretization technique. The material properties of the constituent components of the RFG plate are assumed to vary continuously according to the Mori-Tanalka distribution along the radial direction. The nonlinear governing equations are obtained in the incremental form based on the firstorder shear deformation plate theory （FSDT） and the von Karman relations for large deflection. In the buckling analysis, an external in-plane load is applied to the plate in- crementally so that, in each load-step, the incremental form of the governing equations can be solved by a numerical code prepared based on the DR method. After converging the DR code in the first increment, the latter load-step is added to the previous one, and the program is repeated again. The critical buckling load is determined from the compressive load-displacement curve obtained by solving the incremental form of the governing equations. Based on the present incremental form of formulation, a bending analysis can also be conducted if the whole load is applied simultaneously. Finally, a detailed parametric study is carried out to investigate the influences of various boundary conditions, grading indices, thickness-to-radius ratios, stiffener＇s positions and depths on the critical buckling load, and displacements and stresses resulted from the bending analysis. It is observed that the effect of the stiffener on the results is much greater in the functionally graded plate with higher material grading indices. The results also reveal that, by increasing the depth of the stiffer, the values of ascending the critical buckling load are approximately identical for both simply supported and clamped boundary conditions.

Inverse Load Identification in Stiffened Plate Structure Based on in situ Strain Measurement

For practical engineering structures,it is usually difficult to measure external load distribution in a direct manner,which makes inverse load identification important.Specifically,load identification is a typical inverse problem,for which the models(e.g.,response matrix)are often ill-posed,resulting in degraded accuracy and impaired noise immunity of load identification.This study aims at identifying external loads in a stiffened plate structure,through comparing the effectiveness of different methods for parameter selection in regulation problems,including the Generalized Cross Validation(GCV)method,the Ordinary Cross Validation method and the truncated singular value decomposition method.With demonstrated high accuracy,the GCV method is used to identify concentrated loads in three different directions(e.g.,vertical,lateral and longitudinal)exerted on a stiffened plate.The results show that the GCV method is able to effectively identify multi-source static loads,with relative errors less than 5%.Moreover,under the situation of swept frequency excitation,when the excitation frequency is near the natural frequency of the structure,the GCV method can achieve much higher accuracy compared with direct inversion.At other excitation frequencies,the average recognition error of the GCV method load identification less than 10%.

DYNAMIC BUCKLING OF STIFFENED PLATES UNDER FLUID-SOLID IMPACT LOAD

A simple solution of the dynamic buckling of stiffened plates under fluid-solid impact loading is presented.Based on large deflection theory,a discretely stiffened plate model has been used.The tangential stresses of stiffeners and in-plane displacement are neglected.Applying the (Hamilton's) principle,the motion equations of stiffened plates are obtained.The deflection of the plate is taken as Fourier series,and using Galerkin method,the discrete equations can be deduced,which can be solved easily by Runge-Kutta method.The dynamic buckling loads of the stiffened plates are obtained from Budiansky-Roth(B-R) curves.

Dimensional Analysis on the Perforation of Stiffened Plates by Projectiles

The phenomena attendant to the perforation of truncated oval shape projectile into multi-layered stiffened plates were investigated. Dimensional analysis was employed to give an empirical formula. Then a membership function was introduced to modify the empirical formula. The effects of initial velocities, base plate thicknesses, height and width of stiffener on residual velocities were explored. The predictions of the empirical formula are in reasonably good agreement with those of experiment and numerical results. All these results indicate that the empirical formula is capable of predicting the residual velocity of the projectile penetrating the multi-layered stiffened plates.

A General Model for Analysis of Sound Radiation from Orthogonally Stiffened Laminated Composite Plates

A general theoretical model is developed to investigate the sound radiation from an infinite orthogonally stiffened plate under point excitation force. The plate can be metallic or composite, and fluid loading is also considered in the research. The first order shear deformation theory is used to account for the transverse shear deformation. The motion of the equally spaced stiffeners is examined by considering their bending vibrations and torsional movements. Based on the periodic structure theory and the concepts of the equivalent dynamic flexibility of the plate, the generalized vibro-acoustic equation of the model is obtained by applying the Fourier transform method. The generalized model that can be solved numerically is validated by comparing model predictions with the existing results. Numerical calculations are performed to investigate the effects of the location of the excitation, the spacing of the stiffeners, the plate thickness, the strengthening form and the fiber orientation on the sound radiation characteristic of the orthogonaUy stiffened plate, and some practical conclusions are drawn from these parameter studies.

LARGE DEFLECTION DYNAMIC PLASTIC RESPONSE OF CLAMPED SQUARE PLATES WITH STIEEENERS

The large deflection dynamic plastic response of fully clamped square plates with stiffeners under blast load is analyzed in detail in this paper. Various relevant motion patterns and criterions are presented. The formulas of maximum permanent deformation of the plates with stiffeners are derived. The results of calculation are compared with those of experiment in [3], with good agreement shown in most cases.

NUMERICAL ANALYSIS OF DELAMINATION GROWTH FOR STIFFENED COMPOSITE LAMINATED PLATES

A study of postbuckling and delamination propagation behavior in delaminated stiffened composite plates was presented. A methodology was proposed for simulating the multi-failure responses, such as initial and postbuckling, delamination onset and propagation, etc. A finite element analysis was conducted on the basis of the Mindlin first order shear effect theory and the von-Krmn nonlinear deformation assumption. The total energy release rate used as the criteria of delamination growth was estimated with virtual crack closure technique (VCCT). A self-adaptive grid moving technology was adopted to model the delamination growth process. Moreover, the contact effect along delamination front was also considered during the numerical simulation process. By some numerical examples, the influence of distribution and location of stiffener, configuration and size of the delamination, boundary condition and contact effect upon the delamination growth behavior of the stiffened composite plates were investigated. The method and numerical conclusion provided should be of great value to engineers dealing with composite structures.

Constructal design method dealing with stiffened plates and symmetry boundaries

A new computational procedure for modelling the structural behavior of stiffened plates with symmetry boundary conditions is here presented.It uses two-dimensional finite elements as a way to decrease computational time without losing precision thanks to a relatively small number of elements applied for analyzing out-of-plane displacements(deflections)and stresses.Adding,the constructal design method was included in the procedure,together with the exhaustive search technique,with the scope to optimize the stress/strain status of stiffened plates by design changes.For the purpose,a reference plate without stiffeners was initially design and used as starting point.Part of the volume was reshaped into stiffeners:thickness was reduced maintaining unchanged weight,length and width.The main goal was to minimize strains and stresses by geometric changes.Results demonstrated that,thanks to this design procedure,it is always possible to find an adequate geometry transformation from reference plate into stiffeners,allowing significant improvements in mechanical behavior.

海洋管状结构轴向荷载下K型接头与外板局部节点柔度的数值分析与讨论

The Local Joint Flexibility(_(LJF))of steel K-joints reinforced with external plates under axial loads is investigated in this paper.For this aim,firstly,a finite element(FE)model was produced and verified with the results of several experimental tests.In the next step,a set of 150 FE models was generated to assess the effect of the brace angle(θ),the stiffener plate size(ηandλ),and the joint geometry(γ,τ,ξ,andβ)on the_(LJF)factor(f_(LJF)).The results showed that using the external plates can decrease 81%of the f_(LJF).Moreover,the reinforcing effect of the reinforcing plate on the f_(LJF)is more remarkable in the joints with smallerβ.Also,the effect of theγon the f_(LJF)ratio can be ignored.Despite the important effect of the f_(LJF)on the behavior of tubular joints,there is not available any study or equation on the f_(LJF)in any reinforced K-joints under axial load.Consequently,using the present FE results,a design parametric equation is proposed.The equation can reasonably predict the f_(LJF)in the reinforced K-joints under axial load.

3D numerical simulation on fluid-structure interaction of structure subjected to underwater explosion with cavitation

In the underwater-shock environment, cavitation occurs near the structural surface. The dynamic response of fluid-structure interactions is influenced seriously by the cavitation effects. It is also the difficulty in the field of underwater explosion. With the traditional boundary element method and the finite element method （FEM）, it is difficult to solve the nonlinear problem with cavitation effects subjected to the underwater explosion. To solve this problem, under the consideration of the cavitation effects and fluid compressibility, with fluid viscidity being neglected, a 3D numerical model of transient nonlinear fluid-structure interaction subjected to the underwater explosion is built. The fluid spectral element method （SEM） and the FEM are adopted to solve this model. After comparison with the FEM, it is shown that the SEM is more precise than the FEM, and the SEM results are in good coincidence with benchmark results and experiment results. Based on this, combined with ABAQUS, the transient fluid-structure interaction mechanism of the 3D submerged spherical shell and ship stiffened plates subjected to the underwater explosion is discussed, and the cavitation region and its influence on the structural dynamic responses are presented. The paper aims at providing references for relevant research on transient fluid-structure interaction of ship structures subjected to the underwater explosion.

Bandgap formation and low-frequency structural vibration suppression for stiffened plate-type metastructure with general boundary conditions

Metastructures with unique mechanical properties have shown attractive potential application in vibration and noise reduction.Typically,most of the metastructures deal with the vibration bandgap properties of infinite structures without considering specific boundary condition and dynamic behaviors,which cannot be directly applied to the engineering structures.In this research,we design a Stiffened Plate-type Metastructure(SPM)composed of a plate with periodic stiffeners and cantilever beam-type resonators subjected to general boundary conditions for low-frequency vibration suppression.The effects of boundary conditions and the number and orientation of the stiffeners on Locally Resonant(LR)type bandgap properties in SPM are further investigated.An analytical modeling framework is developed to predict the bandgap formations and vibration behaviors of SPMs in finite-size configuration.The governing equations of the SPM reinforced by various arrangements of stiffeners are derived based on the first-order shear deformation theory and Hamilton’s principle,and a Fourier series combined with auxiliary functions is employed to satisfy the arbitrary boundary conditions.Finite element analysis and experimental investigations of vibration behaviors for the SPM are carried out to validate the accuracy and reliability of the present analytical model.For practical designs of the SPMs with specific boundary conditions,it is found that there exist optimal numbers of stiffeners and resonators which can produce the significant LR-type bandgap behaviors.Furthermore,various arrangements of stiffeners and resonators are explored for different boundary conditions by breaking the requirement of spatially periodicity.It is shown that for the designed SPM,the vibration modes of its host structure should be considered to widen the frequency range in which the resonators transfer and store energy,and hence improve the performance of low-frequency vibration suppression.The present work can provide a significant theoretical guidance for the engineering application of metamaterial stiffened structures。

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.

A SEMI-ANALYTICAL METHOD FOR THE VIBRATION OF AND SOUND RADIATION FROM A TWO-DIMENTIONAL BEAM-STIFFENED PLATE

A semi-analytical method based on space harmonics to investigate the vibration of and sound radiation from an infinite, fluid-loaded plate is presented. The plate is reinforced with two sets of orthogonally and equally spaced beam stiffeners, which are assumed to be line forces. The response of the stiffened plate to a convected harmonic pressure in the wave-number space is obtained by adopting the Green＇s function and Fourier transform methods. Using the boundary conditions and space harmonic method, we establish the relationship between the stiffener forces and the vibration displacement of the plate. In this paper, the stiffener forces are expressed in terms of harmonic amplitudes of the plate displacement, which are calculated by using a numerical reduction technique. Finally, the Fourier inverse transform is employed to find expressions of the vibration and sound radiation in physical space. Agreements with existing results prove the validity of this approach and more numerical results are presented to show that this method converges rapidly.