Buckling Load Factor Research Articles

Automatic hyperparameter tuning of topology optimization algorithms using surrogate optimization

This paper presents a new approach that automates the tuning process in topology optimization of parameters that are traditionally defined by the user. The new method draws inspiration from hyperparameter optimization in machine learning. A new design problem is formulated where the topology optimization hyperparameters are defined as design variables and the problem is solved by surrogate optimization. The new design problem is nested, such that a topology optimization problem is solved as an inner problem. To encourage the identification of high-performing solutions while limiting the computational resource requirements, the outer objective function is defined as the original objective combined with penalization for intermediate densities and deviations from the prescribed material consumption. The contribution is demonstrated on density-based topology optimization with various hyperparameters and objectives, including compliance minimization, compliant mechanism design, and buckling load factor maximization. Consistent performance is observed across all tested examples. For a simple two hyperparameter case, the new framework is shown to reduce amount of times a topology optimization algorithm is executed by 90% without notably sacrificing the objective compared to a rigorous manual grid search.

Enhancing structural stability in civil structures using the bi-directional evolutionary structural optimization method

Topology optimization techniques are increasingly utilized in structural design to create efficient and aesthetically pleasing structures while minimizing material usage. Many existing topology optimization methods may generate slender structural members under compression, leading to significant buckling issues. Consequently, incorporating buckling considerations is essential to ensure structural stability. This study investigates the capabilities of the bi-directional evolutionary structural optimization method, particularly its extension to handle multiple load cases in buckling optimization problems. The numerical examples presented focus on three classical cases relevant to civil engineering: maximizing the buckling load factor of a compressed column, performing buckling-constrained optimization of a frame structure, and enhancing the buckling resistance of a high-rise building. The findings demonstrate that the algorithm can significantly improve structural stability with only a marginal increase in compliance. The detailed mathematical modeling, sensitivity analyses, and optimization procedures discussed provide valuable insights and tools for engineers to design structures with enhanced stability and efficiency.

Enhancement layout optimisation of grid structures with stability constraints

Grid structures have found widespread use in civil and mechanical engineering owing to their ability of covering long spans with an outstanding strength-to-weight ratio. However, their shallow structural depth and slender members make them susceptible to buckling failures. Therefore, it is crucial to enhance the stability of grid structures. An under-explored way to achieve this is by optimally configuring enhancement materials. This paper addresses the challenges of identifying the optimal enhancement layouts for grid structures by solving a density-based topology optimisation problem. The problem formulation is based on the SIMP approach, minimising compliance while satisfying constraints on material volume and structural stability. The stability constraint is expressed using buckling load factors obtained from a linear eigenvalue analysis. Two types of enhancement schemes, involving in-plane and out-of-plane elements, are developed for the single-layer base grid structures. The efficiency of the proposed method is evaluated by performing enhancement layout optimisation for a flat and three curved square grid structures, subject to a uniformly distributed vertical load and a half-span load. The optimisation results shed light on the impact of in-plane and out-of-plane enhancements on the stability of grid structures, providing insights on optimally configuring enhancement materials to achieve desired stability. In addition, comparative analysis between the case studies indicates that for curved structures, a higher curvature normally leads to greater achievable stability. In contrast, the flat grid structure exhibits the broadest range of customisable stability due to its primarily bending-dominant load-carrying mechanism. Furthermore, the outcomes of geometric non-linear analysis confirm the load-carrying capacity of the optimal structures, with non-linear critical loads typically showing a positive correlation with linear buckling loads.

A Level-Set-Based Density Method for Buckling Optimization of Structure with Curved Grid Stiffeners

Curved grid stiffeners, compared to straight stiffeners, offer greater flexibility in adjusting the force transmission paths and give better structural performance. In this paper, a level-set-based density method is employed to generate layouts of curved grid stiffeners so that the critical buckling load factor (BLF) of the stiffened structures is improved. During the optimization process, volume constraint is incorporated to control material utilization, and gradient constraints are employed to maintain uniformity in the width of the stiffeners. Finally, the proposed method is demonstrated through several numerical examples.

Buckling mode constraints for topology optimization using eigenvector aggregates

Buckling-constrained structural design problems have conventionally prioritized optimizing the buckling load factor with less consideration given to the buckling mode shape. In this work, mode shape constraints are imposed within a topology optimization problem using an eigenvector aggregate constraint that is a weighted sum of homogeneous quadratic functions of the linearized buckling eigenvectors. A generalized formulation of the eigenvector aggregate is introduced, extending previous work. A new adjoint-based derivative evaluation technique is derived that is valid even in the presence of repeated eigenvalues. Numerical examples, including a clamped beam, a compressed column, and a square plate, demonstrate the effectiveness of the proposed approach. The results show the ability of the eigenvector aggregate to handle repeated eigenvalues, enable design space exploration, and capture mode shape switching.

Optimization-free design of stiffened thin-walled structures guided by data-rich buckling modes

Regarding the buckling resistance design of thin-walled structures, the conventional iterative optimization approaches pose significant challenges for engineering professionals and create a high barrier for structural designers. This study introduces a novel approach for the anti-buckling design of stiffening ribs, which is based on the solved linear buckling modes rather than on the buckling load factors (BLFs) as in traditional iterative optimization methods. Specifically, the design process begins with a linear buckling analysis of the pre-stiffened part to extract the first-order eigenmode shape. Subsequently, normal vectors are calculated for both the pre-buckling geometric model and the buckled mode shape. By analyzing the normal deflection angle, a group of characteristic points is identified, and these points would be used to serve as guidelines for determining the layout of the stiffening ribs. We validate the effectiveness and efficiency of the proposed protocol through numerical examples spanning from 2D to 3D space considering different loading conditions. Our approach eliminates the need for tedious parametric modeling and high computational cost associated with conventional optimization-based methods. It is particularly well-suited for engineering applications that solely require linear buckling analysis.

A numerical study on thermal-structural characteristics of cryogenic common bulkhead propellant tanks for various tank lengths and core thicknesses

Common bulkhead (CBH) propellant tanks have the advantage that its structural and space efficiency are high. The tank is a form in which an oxidizer tank and a fuel tank are integrally combined, and the oxidizer and fuel are separated by the CBH. The CBH propellant tank in which two tanks are integrated can reduce unnecessary space between the two tanks. However, there are other difficult problems regarding effectively designing the CBH propellant tanks even though there are structural and space benefit. This paper is a study on the thermal-structural characteristics of CBH tanks containing LOX/LH2 propellant. In order to conduct the thermal-structural analysis, the thermal analysis model was constructed using 2-dimensional axisymmetric elements to reflect the shape features of the tank, and the structural analysis model was built using 3-dimensional elements. The thermal-structural analysis was performed on the length of the bottom tank and the foam core thickness of the CBH. In the structural analysis of the tank, when the thermal load was ignored and only the pressure load was applied, there was a difference in the buckling load factor depending on the thickness of the CBH foam core, but as the thermal load was taken into consideration, there was also a difference in the bottom tank length. The analysis results show that the thermal-structural characteristics of the CBH propellant tank were influenced by the length of the bottom tank as well as the thickness of the foam core.

Enhanced Optimization of Composite Laminates: Multi-Objective Genetic Algorithms with Improved Ply-Stacking Sequences.

This study introduces multi-objective genetic algorithms for optimizing the stacking sequence of lightweight composite structures. Notably, significant emphasis is placed on adhering to engineering design guidelines specific to stacking sequence design. These guidelines are effectively integrated into the optimization problem formulation as either constraints or additional objectives. To enhance the initialization process, a novel strategy is proposed based on mechanical considerations. The method is then applied to optimize a composite laminate in terms of weight, inverse reserve factor, and buckling load factor. Three laminates were considered, and the influence of the design and the material composition on their mechanical properties were studied. This research demonstrated that a new stacking sequence [906/454/06] resulted in improved optimum designs compared to the traditional stacking sequence comprising plies at 0°, 45°, and 90° angles. These outcomes can be deemed the optimum stacking sequence, making them valuable for future applications in unmanned aerial vehicle (UAV) structures.

Probabilistic buckling assessment and reliability of fiber-metal laminate, composite and aluminum cylindrical panels under compression with load and fabrication uncertainties

The objective of this article is to study the buckling behavior and reliability of fiber metal laminated (FML), composite and aluminum cylindrical panels under uniaxial compression taking into account fabrication and loading uncertainties. A 3D finite element modeling with ANSYS software has been implemented for this purpose. The panels are discretized using shell elements and the eigenvalue buckling analysis is conducted for the prediction of elastic buckling. The influences of load and fabrication uncertainties on the buckling load factor are studied with probabilistic analyses and the reliability of the panels is calculated. It is found that the thickness of aluminum layers is the most significant uncertain variable for the critical buckling load factor of FML panels, whereas the fiber misalignment angle of their composite layers is insignificant. As the metal volume fraction decreases, the sensitivity of the elastic buckling load factor to variations of the axial load distribution is reduced whereas its sensitivity to variations of the thickness of composite layers is increased. Consequently, the metal volume fraction is an important design parameter for the uncertain buckling behavior of the panels.

Simultaneous shape and topology optimization of inflatable soft robots

Simultaneous shape and topology optimization is used to design pressure-activated inflatable soft robots. The pressure loaded boundary is meshed conformingly and shape optimized, while the morphology of the robot is topology optimized. The design objective is to exert maximum force on an object, i.e. to produce soft “grippers”. The robot’s motion is modeled using nearly incompressible finite deformation hyperelasticity. To ensure stability of the robot, the buckling load factors obtained via linearized buckling analyses are constrained. The finite element method is used to evaluate the optimization cost and constraint functions and the adjoint method is employed to compute their sensitivities. The numerical examples produce pressure-driven soft robots with varying complexity. We also compare our simultaneous optimization results to those obtained via sequential topology and then shape optimization.

Composite anisogrid lattice toroidal shell: Application to a load-carrying rim of the spacecraft reflectarray antenna

A novel design of the composite transformable/foldable rim of the spacecraft reflectarray antenna is presented in the paper. The rim is made in the form of anisogrid lattice toroidal shell split into six parts that are jointed using hinges containing helical springs. The springs create the moments required to deploy the antenna in orbit. A dielectric flexible flat membrane with reflective radiating elements generating a radio signal with required directivity is stretched on the rim. An algorithm of the finite-element model generation is developed for the toroidal rim using a typical finite-element unit cell of lattice structure. The lattice toroidal shell is modelled as a three-dimensional frame composed of curvilinear ribs. The results of design analysis of the lattice rim delivering a minimal mass to its structure subject to constraints imposed on the fundamental frequency, values of stress, and the buckling load factor are presented in this work. The corresponding set of design parameters of lattice structure, including the angle of orientation of helical ribs, number of helical ribs of one orientation, and size of the ribs’ cross section are identified based on these results.

Efficient inverse-designed structural infill for complex engineering structures

Inverse design of high-resolution and fine-detailed 3D lightweight mechanical structures is notoriously expensive due to the need for vast computational resources and the use of very fine-scaled complex meshes. Furthermore, in designing for additive manufacturing, infill is often neglected as a component of the optimized structure. In this paper, both concerns are addressed using a so-called de-homogenization topology optimization procedure on complex engineering structures discretized by 3D unstructured hexahedrals.Using a rectangular-hole microstructure (reminiscent to the stiffness optimal orthogonal rank-3 multi-scale) as a base material for the multi-scale optimization, a coarse-scale optimized geometry can be obtained using homogenization-based topology optimization. Due to the microstructure periodicity, this coarse-scale geometry can be up-sampled to a fine single-scale physical geometry with optimized infill, with only a minor loss in structural performance and at a fraction of the cost of a fine-scale solution. The upsampling on 3D unstructured grids is achieved through stream surface tracing, aligning with the optimized local orientations. The periodicity of the physical geometry can be tuned, such that the material serves as a structural component and also as an efficient infill for additive manufacturing designs.The method is demonstrated through three examples of varying geometrical complexity. It achieves comparable structural performance to “brute force” state-of-the-art methods but stands out for its significant computational time reduction. By allowing multiple active layers, the mapped solution becomes more mechanically stable, leading to an increased critical buckling load factor without additional computational expense. The control of active layers also provides direct control over the internal structure, i.e., infill, ensuring that the infill is incorporated as a structural component and enhancing the manufacturability of the de-homogenization procedure. Furthermore, the proposed approach exhibits promising results, achieving volume fractions and weighted compliance values within 5% of the large-scale SIMP model, while demonstrating a computational efficiency improvement ranging from 10 times to over 250 times.

Buckling stability of variable stiffness composite cylinder shell based on automated fiber placement technology

Automated fiber placement technology (AFP) is the key technology to realize variable-stiffness design that have fiber orientation variation across its in-plane. However, defects such as fiber angle deviation and overlap are inevitable during AFP manufacturing. A finite element model has been developed for variable-stiffness cylinder shell that can investigate the effect of the overlap areas and fiber angle deviation on the buckling stability of variable stiffness cylinder shell. Furthermore, the influence of the design parameters of the curved fiber path and the process parameters such as the tow width, the number of tow feeding, and the overlapping strategies on the buckling performance of the structure was studied by parametric analysis. The results showed under the condition of compression-bending coupling, the buckling performance is improved by the variable stiffness design in which the fiber angle varies along the circumferential direction, the maximum critical buckling load factor of optimal variable-stiffness design [0 /〈30|80〉/ 90 /−〈30|80〉] s is 10771, which is 21.1% higher than that of the optimal normal stiffness design. It is found that increasing the course width reduces the buckling performance of the variable stiffness structure, when the tow width is kept constant. When the course width is kept constant, the buckling performance of the variable stiffness structure increases with the increase of the tow width.

I-section stainless steel portal frames: Loading tests and numerical modelling

Austenitic stainless steel is an attractive structural material due to its high ductility, appropriate toughness, significant strain hardening and good fire resistance. Also the laser welding technology for fabrication of structural stainless steel profiles has recently become increasingly popular. However, there is still only a limited amount of experimental data on laser-welded stainless steel members. And loading tests of whole structures made of these sections have still not been published. Therefore, loading tests were performed on two austenitic stainless steel frames made of laser-welded I-sections. The frames had pinned columns and were loaded both vertically and horizontally. The objective of the loading tests was to capture the effects of material nonlinearity on the global sway behaviour of the stainless steel frames. In fact, material nonlinearity leads to a gradual loss of stiffness, which results in higher deformations and thus a higher second-order effect. The resulting load-displacement diagrams of the experiment are tabulated, allowing their further use. A numerical model of the studied frames was developed and validated on the basis of the performed experiments. Based on the elastic buckling load factor and the modified elastic buckling load factor, a large influence of material nonlinearity on second-order effect has been demonstrated.

Buckling optimization of additively manufactured cellular structures using numerical homogenization based on beam models

Interest in components with detailed structures increased with the progress in advanced manufacturing techniques. Parts with lattice elements can provide improved global buckling stability compared to solid structures of the same weight. However, thin features are prone to local buckling. We present a two-scale optimization approach that simultaneously improves the local and global stability of parametrized graded lattice structures. Elastic properties and local buckling behavior are upscaled via homogenization based on geometric exact beam theory. To reduce computational effort, we construct a worst-case model for the homogenized buckling load factor, which acts as a safeguard against local buckling. We briefly discuss advantages and limitations by means of numerical examples.

Buckling Analysis of a Thin-Walled C-Section Channel under Mechanical and Thermal Load

Buckling analysis of thin-walled structures is essential to ensure their structural integrity under various loading conditions. The study aims to analyse buckling in thin-walled structures, particularly a C-section channel, under different loads, considering cases with and without holes, using FEM. It investigates hole impact on buckling, critical load, and mode shapes, offering insights for C-section channel design. A finite element method (FEM) is used to model the structure and simulate mechanical and thermal loading conditions. Additionally, the study investigates the critical buckling load factor and mode shapes for both cases with and without holes. The results show that the presence of holes significantly affects the buckling behaviour of the structure. The mode shapes indicate that the buckling behaviour of the structure is different for each case. The critical buckling load factor is lower for the C-section channel with holes, indicating a higher risk of buckling. The results of this study provide insights into the buckling behaviour of thin-walled structures and demonstrate the importance of considering the presence of holes in the design and optimization of C-section channels.

Optimization of Hybrid Composite Laminates with Distinct Ply Thicknesses Using Integer Programming

Optimization of constant-stiffness composite laminates with discrete values of fiber orientations and ply thicknesses is a nonconvex combinatorial problem. Recent studies show that this problem can be efficiently solved using the mixed integer program (MIP) by reformulating lamination parameters as functions of decision variables that represent fiber orientation and thickness choices. These formulations are limited to analyzing laminates having plies of the same thickness and material. This paper extends the previously employed lamination parameter reformulation to enable laminate designs with plies of different thicknesses and materials, thereby leading to MIP with cubic constraints. The linearization techniques are employed to convert cubic constraints of buckling load factors in terms of binary variables into sets of linear constraints that are efficiently solved with optimization algorithms for MIPs. It is shown that the use of distinct ply thicknesses in the optimization further reduces the mass of single-material laminates made with carbon-fiber-reinforced polymer when compared with previous studies using the same material. Additionally, the combination of plies made of glass-fiber-reinforced polymer and carbon-fiber-reinforced polymer with distinct thicknesses is shown to reduce the weight penalty when optimized for cost minimization in comparison to plies with uniform thickness.

Topology optimization for maximizing buckling strength using a linear material model

Buckling resistance has gained significant attention in topology optimization due to its profound implications for structural designs. Despite considerable research on buckling-constrained topology optimization, maximizing the critical buckling load factor (BLF) still remains a challenging topic. In this study, an innovative algorithm that utilizes a linear material interpolation scheme is introduced to maximize the buckling resistance of structures. The linear material model offers several advantages, such as obviating the need to select the penalization schemes and penalty values, facilitating straightforward sensitivity analysis, and removing the ambiguous physical meaning of penalization for the stress stiffness matrix. The accuracy of the linear material model for buckling analysis is systematically examined, and the avoidance of stress singularities in low-density regions is investigated. The effectiveness and efficiency of the proposed approach are supported by four buckling optimization design examples, which also demonstrate substantial improvements compared to the existing algorithms.

Buckling of non-Lévy-type rectangular thick plates:New benchmark solutions in the symplectic framework

The third-order shear deformation theories (TSDTs) are frequently employed to enhance the accuracy of mechanical behavior characterization for thick plates. However, the analytic buckling solutions based on the TSDTs are rare due to the major challenge in mathematical treatments. This study introduces an analytic symplectic superposition method to derive new benchmark buckling solutions of TSDT-based rectangular thick plates with non-Lévy-type boundary conditions (BCs), which are inaccessible using conventional analytic approaches. The governing equations are formulated in the Hamiltonian system. The symplectic framework-based mathematical techniques are manipulated, which shapes the characteristic of the method that a rational derivation is realized without any predetermined solution forms. The comprehensive buckling load factors and mode shapes of fully clamped plates and their variations are given, all of which are verified through comparison with the finite element method as well as literature results. Utilizing the obtained solutions, a comprehensive parametric analysis is conducted to elucidate the effects of BC, thickness-to-width ratio, and aspect ratio on critical buckling load factors. A comparison study is made between the results by the TSDT and those by the Kirchhoff theory and the first-order shear deformation theory, which demonstrates the superior accuracy of the former. The proposed rigorous framework offers a rational route to obtaining more benchmark buckling solutions.

Two-scale optimization of graded lattice structures respecting buckling on micro- and macroscale

Interest in components with detailed structures increased with the progress in advanced manufacturing techniques in recent years. Parts with graded lattice elements can provide interesting mechanical, thermal, and acoustic properties compared to parts where only coarse features are included. One of these improvements is better global buckling resistance of the component. However, thin features are prone to local buckling. Normally, analyses with high-computational effort are conducted on high-resolution finite element meshes to optimize parts with good global and local stability. Until recently, works focused only on either global or local buckling behavior. We use two-scale optimization based on asymptotic homogenization of elastic properties and local buckling behavior to reduce the effort of full-scale analyses. For this, we present an approach for concurrent local and global buckling optimization of parameterized graded lattice structures. It is based on a worst-case model for the homogenized buckling load factor, which acts as a safeguard against pure local buckling. Cross-modes residing on both scales are not detected. We support our theory with numerical examples and validations on dehomogenized designs, which show the capabilities of our method, and discuss the advantages and limitations of the worst-case model.