Knockdown Factor Research Articles

Comparing the Buckling Strength of Spherical Shells With Dimpled Versus Bumpy Defects

AbstractWe investigate the effect of defect geometry in dictating the sensitivity of the critical buckling conditions of spherical shells under external pressure loading. Specifically, we perform a comparative study between shells containing dimpled (inward) versus bumpy (outward) Gaussian defects. The former has become the standard shape in many recent shell-buckling studies, whereas the latter has remained mostly unexplored. We employ finite-element simulations, which were validated previously against experiments, to compute the knockdown factors for the two cases while systematically exploring the parameter space of the defect geometry. For the same magnitudes of the amplitude and angular width of the defect, we find that shells containing bumpy defects consistently exhibit significantly higher knockdown factors than shells with the more classic dimpled defects. Furthermore, the relationship of the knockdown as a function of the amplitude and the width of the defect is qualitatively different between the two cases, which also exhibit distinct post-buckling behavior. A speculative interpretation of the results is provided based on the qualitative differences in the mean-curvature profiles of the two cases.

Image-Driven Intelligent Prediction of Buckling Behavior for Geometrically Imperfect Cylindrical Shells

Thin-walled cylindrical shells are key load-carrying components for aerospace structures. Axial compression is the most common load for cylindrical shells; however, critical buckling loads obtained experimentally are significantly lower than the theoretical values, owing to the presence of initial geometric imperfections (GIs). Previous studies often determined only the knockdown factor through the statistics of experimental data; it is an approximate lower limit of the load-carrying capacity and cannot accurately predict the buckling mode in real time. In this paper, an image-driven framework for the intelligent prediction of buckling load and mode based on measured GIs is proposed. First, through measured samples and the random field method, the GI data pool is obtained. Subsequently, two convolutional neural network (CNN) models are trained to predict the buckling load and mode for cylindrical shells, respectively. The active learning strategy is adopted to discern beneficial sample sets and enhance the prediction accuracy and training speed of the CNN models. A numerical example demonstrates that the proposed framework can effectively predict the buckling behavior of cylindrical shells with GIs. Furthermore, using the proposed prediction framework, it was observed that a common feature of unfavorable GIs is a large fluctuation in magnitude along the axial direction.

Probabilistic buckling of imperfect hemispherical shells containing a distribution of defects.

The buckling of spherical shells is plagued by a strong sensitivity to imperfections. Traditionally, imperfect shells tend to be characterized empirically by the knockdown factor, the ratio between the measured buckling strength and the corresponding classic prediction for a perfect shell. Recently, it has been demonstrated that the knockdown factor of a shell containing a single imperfection can be predicted when there is detailed a priori knowledge of the defect geometry. Still, addressing the analogous problem for a shell containing many defects remains an open question. Here, we use finite element simulations, which we validate against precision experiments, to investigate hemispherical shells containing a well-defined distribution of imperfections. Our goal is to characterize the resulting knockdown factor statistics. First, we study the buckling of shells containing only two defects, uncovering non-trivial regimes of interactions that echo existing findings for cylindrical shells. Then, we construct statistical ensembles of imperfect shells, whose defect amplitudes are sampled from a lognormal distribution. We find that a 3-parameter Weibull distribution is an excellent description for the measured statistics of knockdown factors, suggesting that shell buckling can be regarded as an extreme-value statistics phenomenon. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

Foreword to the special issue on new developments in structural stability.

Foreword to the special issue on new developments in structural stability.

Probing the stability landscape of cylindrical shells for buckling knockdown factors.

The buckling response of axially compressed cylindrical shells is well known for its imperfection sensitivity. Mapping out a stability landscape by localized probing has recently been proposed as a rational means for establishing shell buckling knockdown factors. Probing using a point force directed radially inwards and perpendicular to the cylinder wall is based on the insight that a localized single dimple exists as an edge state in the basin boundary of the stable prebuckling equilibrium. Here, we extend the idea of probing to bi-directional inwards and outwards forces to trigger both single-dimple and double-dimple edge states. We identify key features of the ensuing probing stability landscape and generalize these to derive three design curves of varying conservatism that are a function of the non-dimensional Batdorf parameter only. Interestingly, the most conservative of the three knockdown curves bounds a large dataset of experimental buckling results from below, despite being derived from probing features of geometrically perfect cylinders. Overall, the three design curves permit a more nuanced design approach than legacy knockdown factors, as different levels of conservatism can be chosen based on expected manufacturing quality. For instance, the most and least conservative of the three design guidelines differ by a factor of 3 for the most slender cylinder geometries, and the associated reduction in safety factor has profound implications for efficient structural design. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

3D-printing and cylinder buckling: challenges and opportunities.

Cylinder buckling is notoriously sensitive to small geometric imperfections. This is an underlying motivation for the use of knock-down factors in the design process, especially in circumstances in which minimum weight is a key design goal, an approach well-established at NASA, for example. Not only does this provide challenges in the practical design of this commonly occurring structural load-bearing configuration, but also in the carefully controlled laboratory setting. The recent development of 3D-printing (additive manufacturing) provides an appealing experimental platform for conducting relatively high-fidelity experiments on the buckling of cylinders. However, in addition to geometric precision, there are a number of shortcomings with this approach, and this article seeks to describe the challenges and opportunities associated with the use of 3D-printing in cylinder buckling in general, and probing the robustness of equilibrium configurations in particular. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

Reliability analysis and design of randomly imperfect thin cylindrical shells against post-critical drops

Thin-walled cylindrical shells are employed for high specific bending stiffness and light design. Elastic buckling of shell is a critical design consideration. The Knock Down Factors (KDFs) are stipulated for accounting the markedly reduced experimental critical load by multiplying the linearized critical load by the KDF. Key to such reductions is the nonlinearity induced modal interactions among the closely spaced buckling modes, over which, the geometric imperfections act in synergy. Being the lower bound of an extensive experimental dataset; the stipulated design KDFs are largely over-conservative; which can be relaxed by accounting for imperfections. With this as the eventual goal, this study first demonstrate the feasibility of modeling the KDFs as uniformly distributed Uncertain-but-Bounded (UBB) type variable. The UBB description is employed to obtain the reliability bounds and the design variables (radius-to-thickness R/t and L/R ratio) for achieving a target reliability index through Reliability Based Design (RBD). Repeated computations of KDFs for reliability assessment are bypassed by fitting the KDFs vs. R/t relations for given L/R. The approach is demonstrated on a thin cylindrical shell with a zero-mean, stationary, gaussian, stochastic imperfection fields. The Riks method is employed for the nonlinear finite element analysis of the shell using the code ABAQUS. The KDFs and reliability bounds are provided for varying imperfections and are compared with an existing study on refining the KDFs. Furthermore, the RBD presents the desired R/t for a given L/R for achieving a target reliability index and the respective KDF for a given imperfection level. The L/R dependency is shown to reduce the over-conservatism in the design stipulation.

Research on the Buckling Load of Clamped Spherical Caps Under External Pressure: Analyzed by the Fourier Series Model With Initial Geometric Imperfections

AbstractThe knockdown factor (KDF), which characterizes the difference between the actual buckling pressure and the classical theoretical pressure of clamped spherical caps under external pressure, is highly discrete. The mechanism that causes this discreteness is still not understood and has not been reproduced in numerical simulation. By scanning six clamped spherical caps, the geometric characteristics of the shells were analyzed, and a geometric model was established based on the Fourier series. The geometric model was simulated by the geometric and material double nonlinear buckling analysis method. The results were compared with the experimental data from this study and other references. 720 sets of clamped spherical caps under external pressure were simulated using the proposed Fourier series model and simulation method. The influence of the yield strength, geometrical parameter λ, dimensionless parameters radius-thickness ratio R/t, and the imperfection-thickness ratio e/t on KDF were studied, and the highly discrete characteristics of KDF were reproduced. The results showed that the proposed method has a better predictive effect on KDF, which is significantly improved over the “Eigemode imperfections” method. KDF is not only related to λ and e/t, but is also affected by the yield strength and R/t. The lower envelopes of KDF were obtained when e/t was less than 1.0 and 2.0. The NASA SP-8032 curve corresponds to the lower envelope of KDF when e/t is less than 8.0, and the curve is below the lower envelope of KDF when e/t is less than 1.0 and 2.0. As stipulated in the pressure vessel standard, the KDF obtained by NASA SP-8032 will be conservative for design conditions with e/t less than 1.0 or 2.0, and appropriate adjustment should be considered.

Comments on effective use of numerical modelling and extended classical shell buckling theory

This paper argues that in the understandable enthusiasm to use the now readily available high-fidelity numerical software allowing the analysis of the complexities of shell buckling, and especially the notorious case of the axially loaded cylinder, there has been insufficient consideration of classical shell buckling theory and its more recent reduced stiffness extension. If future design is to be able to effectively exploit the potential benefits of the use of advanced composite materials and/or the use of rib reinforcement, it is suggested that these numerically driven research programmes would benefit from more explicit integration within the framework of classically extended buckling theory. This suggestion is reinforced in what follows by highlighting the results from a longstanding research programme in which classically extended buckling theory has been demonstrably enhanced through its integration with reliable numerical experimentation. This enhancement has helped to resolve a number of important issues including: identification of the most critical forms of imperfection affecting the buckling loads; how the most critical knockdown factors are influenced by changes in the shells' geometric and material characteristics and how safe design might be more explicitly related to the prescribed levels of geometric tolerances. Recommendations are made as to how such integrated research programmes could in the future further enhance our ability to design materially efficient shells against buckling.

Numerical prediction and experimental analysis of the buckling loads of SMPC cylindrical shells under axial compression

The axial compressive buckling loads of shape memory polymer composite (SMPC) cylindrical shells are very sensitive to geometric imperfections and temperatures, but few studies have been conducted on this phenomenon. In this study, the buckling loads and geometric imperfection sensitivities of SMPC cylindrical shells at different temperatures are determined by means of numerical simulations and experimental analyses. The single perturbation displacement imperfection (SPDI), multiple perturbation displacement imperfection (MPDI) and linear buckling mode imperfection (LBMI) techniques are used to simulate the initial geometric imperfections and calculate the corresponding buckling loads and knock-down factors (KDFs) of SMPC cylindrical shells at different temperatures. In the experimental part, the [0/90/±45]s SMPC cylindrical shells are manufactured through the autoclave molding process. The load-bearing capacities at different temperatures are tested and compared with the numerical results. The results demonstrate that the buckling loads of the SMPC cylindrical shell obtained by numerical techniques are sensitive to temperature, while the KDFs are insensitive to temperature. Meanwhile, results indicate that brittle fracture is the main failure mode instead of buckling at low temperature, so there is a risk when using any numerical technique to design SMPC cylindrical shells in low-temperature region. At high temperatures, the SPDI method overestimates the KDFs, while the KDFs calculated by the MPDI and LBMI techniques are in good agreement with the experimental results. However, only the LBMI method can distinguish the influence of temperature on the post-buckling patterns, and the corresponding post-buckling pattern is more consistent with the experiment. In addition, the shape-recovery properties and repeatability of the SMPC cylindrical shells are good.

Hierarchical-level failure analysis for CFRC lattice stiffened panel

A hierarchical-level failure analysis method from carbon fiber reinforced composite (CFRC), unit-cell lattice to lattice stiffened component was developed to predict the failure load of lattice stiffened panels. A nondestructive method was applied to identify the composite parameters of lattice ribs. Axial compression experiments were carried out to reveal the failure pattern and load of unit-cell lattice and lattice-stiffened panels. Compared with the standard unidirectional laminate strength, these component failure analyses suggest knock-down factors (KDFs) to include the attenuation effect of the rib node intersection. The failure loads of CFRC lattice stiffened panels were predicted by the KDFs. The research lays a good foundation to design and manufacture large-sized CFRC lattice structures.

Influence of Spatially Distributed Out-of-Plane CFRP Fiber Waviness on the Estimation of Knock-Down Factors Based on Stochastic Numerical Analysis

The presence of waviness defects in CFRP materials due to fiber undulation affects the structural performance of composite structures. Hence, without a reliable assessment of the resulting material properties, the full weight-saving potential cannot be exploited. Within the paper, a probabilistic numerical approach for improved estimation of material properties based on spatially distributed fiber waviness is presented. It makes use of a homogenization approach to derive viable knock-down factors for the different plies on the laminate level for reference material and is demonstrated for a representative tension loadcase. For the stochastic analysis, a random field is selected which describes the complex inner geometry of the plies in the laminate model and is numerically discretized by the Karhunen–Loeve expansion methods to fit into an FE model for the strength analysis. Conducted analysis studies reveal a substantial influence of randomly distributed waviness defects on the derived knock-down factors. Based on a topological analysis of the waviness fields, the reduction of the material properties was found to be weakly negatively correlated related to simple geometrical properties such as maximum amplitudes of the waviness field, which justifies the need for further subsequent sensitivity studies.

Shape optimisation of stainless steel corrugated cylindrical shells for additive manufacturing

Axially compressed circular cylindrical shells with large diameter-to-thickness ratios are highly susceptible to local buckling, and their load-carrying capacities are known to be very sensitive to initial geometric imperfections. Hence, severe knock-down factors on their theoretical buckling loads are typically prescribed in design specifications, which greatly impair their structural efficiency. With the aim of enhancing load-bearing resistance and reducing sensitivity to imperfections, the shape optimisation and assessment of compressed free-form wavy cylindrical shells, the realisation of which is now viable through additive manufacturing, are the subject of the present study. The adopted optimisation framework employs the Particle Swarm Optimisation (PSO) algorithm, integrating computer-aided geometric design, nonlinear numerical simulations and imperfection sensitivity analyses. The structural performance of the optimised free-form wavy shells is analysed and compared to that of reference circular shells, as well as other types of non-circular shell profiles, including sinusoidally corrugated shells, Aster shells and stringer-stiffened shells. The optimised free-form wavy shell profiles are shown to exhibit increases in ultimate stress of up to 136% compared with the reference circular shell profiles; in general, greater benefits are achieved for more slender cross-sections. In future work, the proposed optimised shells will be manufactured in stainless steel by means of powder bed fusion (PBF), and their structural performance will be further verified through physical experiments.

Enhanced vibration correlation technique to predict the buckling load of unstiffened composite cylindrical shells

Recent advances applying the vibration correlation technique as a nondestructive experimental procedure for determining the in-situ buckling load of unstiffened and skin-dominated stiffened cylindrical shells are showing promising results. Previous studies associated the applicability and the convergence of the mentioned technique with the knockdown factor to be estimated. It is upon this basis that this paper proposes to exploit further this aspect towards a load factor for enhancing the buckling load estimations. The study considers existing validated finite element models for a systematic evaluation of the compliance of the vibration correlation technique and, based on such numerical results, it proposes a load factor for enhanced buckling load estimations. The concept is firstly verified for the numerical results, supporting its establishment. Subsequently, existing experimental results are reevaluated for an assessment of the devised load factor into the buckling load predictions. The appropriate magnitude of the load factors is determined through an iterative study grounded on numerical models that could be defined beforehand. Throughout the numerical- and experimental-based studies, the potential of the proposed load factor is demonstrated towards enhanced VCT buckling load estimations for unstiffened composite cylindrical shells.

An enhanced elastoplastic damage coupled model for compression characteristics analysis of continuous fiber-reinforced thermoplastic composite stiffened panel

In this work, the elastoplastic behaviors, damage characteristics, and debonding forms of continuous fiber-reinforced thermoplastic (CFRTP) composite stiffened panel under uniaxial compression load have been studied. The enhanced elastoplastic damage coupled model (EPDM) was established to model the mechanical properties of CFRTP composites which were based on generalized Hill yield criterion, the associated flow rules, the power hardening law, Chang-Chang failure criterion, and continuous damage mechanics (CDM). The forward Euler algorithm was implemented in Abaqus via user-defined subroutine VUMAT. A series of experiments including tensile, compression, and interlaminar shear tests of laminates were performed to validate the material model. Subsequently, the developed EPDM combined with cohesive zone model (CZM) was applied to simulate the nonlinear mechanical responses of CFRTP composite stiffened panel. Moreover, an axial compression test of CFRTP composite stiffened panel was also carried out, which confirmed that the simulation results were in agreement with the corresponding experimental results. Furthermore, based on the established finite element model, the buckling knockdown factor (KDF) and plastic region development of the CFRTP composite stiffened panel were also analyzed. Overall, the obtained results demonstrate the efficiency of the proposed EPDM for the compression characteristics analysis of the CFRTP composite structures.

Repeated uniform blast loading on welded mild steel rectangular plates

The present research paper deals with a combined experimental and numerical assessment of the deformation mode and failure mechanism of welded rectangular plates exposed to three successive uniform blast loading. To perform the experimental study, three different categories of 0 W (a plate without any weld seam), 1 W (a plate with a single weld seam), and 2 W (a plate with double weld seams) were conducted and mounted onto the ballistic pendulum apparatus. The effect of the number of blasts, the number of weld seams, and the explosive charge mass was investigated on the structural dynamic response. Unlike 0 W plates, plates with double weld seams experienced large plastic deformation along with small thinning happening at the long side clamped boundary. Therefore, despite increasing the deformation resistance, the welding lines tended to mitigate the damage tolerance of the tested mild steel plates. To gain more insight into the influence of two other key parameters like the position of the weld seam and its orientation on the dynamic response and energy absorption capacity, numerical simulations were also performed using the ABAQUS/Explicit finite element solver in conjunction with the inbuilt ConWep blast function code. For modeling the amplified blast wave produced by the detonation of the explosive charge, the scaling factor of 4.1 was adopted. Due to the welding process on the tested plates, the material degradation at the Heat-Affected Zone (HAZ) was also incorporated into the simulation by obtaining the knockdown factor from quasi-static tensile tests on unwelded and welded specimens. An excellent agreement between experiments and simulation results was achieved in terms of the impulse of the applied load and central permanent deflection. Furthermore, the present paper provides the primary experimental data and numerical simulation support for the blast resistance design of metallic structures using welded specimens.

Comparison of DLin-MC3-DMA and ALC-0315 for siRNA Delivery to Hepatocytes and Hepatic Stellate Cells.

Ionizable cationic lipids are essential for efficient in vivo delivery of RNA by lipid nanoparticles (LNPs). DLin-MC3-DMA (MC3), ALC-0315, and SM-102 are the only ionizable cationic lipids currently clinically approved for RNA therapies. ALC-0315 and SM-102 are structurally similar lipids used in SARS-CoV-2 mRNA vaccines, while MC3 is used in siRNA therapy to knock down transthyretin in hepatocytes. Hepatocytes and hepatic stellate cells (HSCs) are particularly attractive targets for RNA therapy because they synthesize many plasma proteins, including those that influence blood coagulation. While LNPs preferentially accumulate in the liver, evaluating the ability of different ionizable cationic lipids to deliver RNA cargo into distinct cell populations is important for designing RNA-LNP therapies with minimal hepatotoxicity. Here, we directly compared LNPs containing either ALC-0315 or MC3 to knock-down coagulation factor VII (FVII) in hepatocytes and ADAMTS13 in HSCs. At a dose of 1 mg/kg siRNA in mice, LNPs with ALC-0315 achieved a 2- and 10-fold greater knockdown of FVII and ADAMTS13, respectively, compared to LNPs with MC3. At a high dose (5 mg/kg), ALC-0315 LNPs increased markers of liver toxicity (ALT and bile acids), while the same dose of MC3 LNPs did not. These results demonstrate that ALC-0315 LNPs achieves potent siRNA-mediated knockdown of target proteins in hepatocytes and HSCs, in mice, though markers of liver toxicity can be observed after a high dose. This study provides an initial comparison that may inform the development of ionizable cationic LNP therapeutics with maximal efficacy and limited toxicity.

Test Studies on Geogrid–Soil Interface Behavior under Static and Dynamic Loads

Pullout tests on geogrids have been regarded as the most direct way to investigate geogrid–soil interaction. In the pullout tests on geogrids, either static or dynamic load is commonly used for applying the vertical loads. In order to investigate the influence of static and dynamic load on pullout test results, pullout tests are carried out to analyze the mechanical response of geogrids and soils under static and dynamic load from the large-scale pullout tester. The results show that frequency and amplitude have significant effects on the pullout test results under the dynamic load. The interface cohesion and friction coefficients under dynamic loads are smaller than those under static loads. The reinforcement effect of geogrids is reduced by dynamic load. Therefore, the strength of geogrids should be reduced when quasi-static analysis is used for reinforced structures. Knockdown factor is recommended for the corresponding reduction. The investigation results of this study may provide scientific references for regulating the design method of reinforced structures.

Stochastic analysis of geometrically imperfect thin cylindrical shells using topology-aware uncertainty models

Buckling of thin-shell structures is one of the most canonical problems in mechanics. In practice, the buckling load and its deviation from theoretical prediction is often handled through the development of knock-down factors in shell structure design. Uncertainty-informed analysis based design is an alternative path that has been aggressively pursued in recent years. In this study, we use a novel representation of the stochastically imperfect shell geometry to investigate the buckling behavior of imperfect cylindrical shells. The representations rest on a non-Gaussian random field model, obtained by translating a latent Gaussian field defined as the solution to a stochastic partial differential equation and allows for the construction of topology-aware spatially correlated imperfections on nonconvex domains. We perform finite element analysis of buckling for imperfect shells and gain new insights into how the interplay between random imperfections and topological features influences buckling behavior.

Knockdown factor of buckling load for axially compressed cylindrical shells: state of the art and new perspectives

Knockdown factor of buckling load for axially compressed cylindrical shells: state of the art and new perspectives