Permanent Transverse Displacements Research Articles

Development of Hybrid Machine Learning Models for Predicting Permanent Transverse Displacement of Circular Hollow Section Steel Members under Impact Loads

The impact effect is a crucial issue in civil engineering and has received considerable attention for decades. For the first time, this study develops hybrid machine learning models that integrate the novel Extreme Gradient Boosting (XGB) model with Particle Swam Optimization (PSO), Grey Wolf Optimizer (GWO), Moth Flame Optimizer (MFO), Jaya (JA), and Multi-Verse Optimizer (MVO) algorithms for predicting the permanent transverse displacement of circular hollow section (CHS) steel members under impact loads. The hybrid machine learning models are developed using data collected from 357 impact tests of CHS steel members. The efficacy of hybrid machine learning models is evaluated using three performance metrics. The results show that the GWO-XGB model achieves high accuracy and outperforms the other models. The values of R2, RMSE, and MAE obtained from the GWO-XGB model for the test set are 0.981, 2.835 mm, and 1.906 mm, respectively. The SHAP-based model explanation shows that the initial impact velocity of the indenter, the impact mass, and the ratio of impact position to the member length are the most sensitive parameters, followed by the yield strength of the steel member and the member length; meanwhile, member diameter and member thickness are the parameters least sensitive to the permanent transverse displacement of CHS steel members. Finally, this study develops a web application tool to help rapidly estimate the permanent transverse displacement of CHS steel members under impact loads.

Analytical and empirical methods for the characterisation of the permanent transverse displacement of quadrangular metal plates subjected to blast load: Comparison of existing methods and development of a novel methodological approach

The behaviour of blast loaded structures has been extensively investigated over the past fifty years through experimental tests. These tests are quite challenging and require dedicated infrastructures to be efficiently and safely performed, however, the obtained data are useful to develop predictive approaches. Among them, analytical approaches are capable of efficient and satisfactory characterisation of blast events and the related effects on structures. In particular, among the analytical methods two categories can be identified: those methods exploiting fully analytical relationships, e.g., the Jones’ theory, and those based on model fitting to experimental results, e.g., the Nurick and Martin's method. More recently, numerical models have been proposed to define the response of structures to blast loads: the main numerical methods considered to assess the structural response to blast loading are the coupled Eulerian-Lagrangian, uncoupled Eulerian-Lagrangian and Analytical-Lagrangian analyses. In this context, this paper aims at establishing a detailed comparison of the main fully analytical and empirical methods available in the literature, exploiting consolidated experimental evidence and results from numerical simulations. The focus of this work is on the estimation of the permanent transverse deflection of a quadrangular, initially flat plate subjected to blast loading, considering both close-range and far-field explosions. Moreover, a modelling framework is herein presented, which serves as a fast and reliable predictive tool for estimating blast load effects on plates.

On the dynamic plastic response of steel membranes subjected to localised blast loading

Permanent plastic deformation is expected when close-in blasts due to e.g. detonation of Improvised Explosive Devices (IED's) hit thin metallic targets. A circular thin steel plate i.e. a membrane is studied in the present work subject to a general form of a localised blast loading. The spatial shape of the pulse is fixed and different temporal shapes are investigated. Dynamic analyses are conducted and the permanent transverse displacements are found for each case.For high amplitude pulse loads of short duration, it was found that the permanent transverse displacement can be found by replacing the load by an impulse without the loss of accuracy. Excellent correlation with numerical simulations obtained from ABAQUS/Explicit is achieved. The predicted final displacements for different pulse shapes are also found to be similar, thus where membrane action is dominant, the response is insensitive to pulse shape.

Impact Loading of Ductile Rectangular Plates

Abstract In many industries, rigid-plastic methods of analysis are a useful design aid for safety calculations, hazard assessments, security studies and forensic investigations of ductile structures, which are subjected to large dynamic loads producing an inelastic response. This paper examines the behaviour of a rectangular plate struck at the centre by a rigid mass impact loading. A theoretical method has been developed previously for arbitrarily shaped plates which retains the influence of finite transverse displacements, or geometry changes. It is used in this paper to predict the maximum permanent transverse displacements and response duration of plates having boundary conditions characterised by a resisting moment mM 0 around the entire boundary, where m =0 and 1 give the two extreme cases of simply supported and fully clamped supports, respectively. The theoretical predictions are compared with some experimental data recorded on fully clamped metal rectangular plates having a range of aspect ratios and struck by masses travelling with low impact velocities up to nearly 7 m/s and which produce large ductile deformations without failure. The theoretical analysis gives reasonable agreement with the corresponding experimental data for masses having blunt, conical and hemispherical impact faces. For sufficiently large initial impact energies, the projectile would perforate a plate and, for completeness, a useful design equation is presented which predicts perforation energies larger than all of the test data, as expected.

Pseudo-shakedown phenomenon for the mass impact loading of plating

The pseudo-shakedown phenomenon for the repeated and identical mass impact loadings on circular and rectangular plates, is studied using a rigid-plastic method of analysis. Comparisons are made with experimental data which has been published on the response of circular and square plates subjected to repeated mass impacts. Previous studies have explored the effect of only the current impact mass and ignored the accumulation of any masses from earlier impacts on a plate. This has led to a steady growth of the transverse displacements without the achievement of pseudo-shakedown. The effect of allowing the masses to accumulate on a plate surface, as, for example, in the loading of cargo in a hold, is shown to have an important influence on the behaviour. The asymptotic values of the associated expressions reveal that a pseudo-shakedown state is achieved for this case and has an associated modest value of the maximum permanent transverse displacement. The phenomenon of pseudo-shakedown is clarified and the equations are useful design tools for an estimate of the pseudo-shakedown behaviour of plating, which is loaded repeatedly with cargo and other masses in various practical situations.

Damage of plates due to impact, dynamic pressure and explosive loads

It is the purpose of this article to present design equations which can be used to predict the damage of ductile plating when subjected to mass impact, dynamic pressure or impulsive loadings. The external loadings are sufficiently severe to produce inelastic material behaviour and produce finite transverse displacement, or geometry change, effects. The damage is characterised as the final or permanent transverse displacement of a plate. The theoretical method predicts values for the maximum permanent transverse displacements which agree reasonably well with the corresponding experimental results generated on aluminium alloy circular, square and rectangular plates. Thus, the equations presented in this article are valuable for preliminary design purposes and for forensic studies, while the experimental data can be used for validating numerical schemes.

Dynamic inelastic response of strain rate sensitive ductile plates due to large impact, dynamic pressure and explosive loadings

This article studies the significant amount of literature that has been published recently on the dynamic response of plating subjected to large dynamic loadings. A theoretical method of analysis for mass impacts, dynamic pressure pulses and impulsive velocity or blast loadings on circular, square and rectangular plates is presented for the idealisation of a rigid, perfectly plastic material. This theoretical analysis caters for the influence of finite-displacements and is developed further to predict the response for plates made from a strain rate sensitive material. Relatively simple equations are presented for the maximum permanent transverse displacements which give good agreement with the corresponding experimental data, and therefore, can be used for design purposes, safety calculations, security studies, hazard assessments and forensic investigations.

Dynamic Performance of Simply Supported Rigid Plastic Circular Thick Steel Plates Subjected to Localized Blast Loading

Close-in explosive charges, such as improvised explosive devices, produce localized blast loadings that can potentially cause damage to property in military and civil structures and/or loss of life. Because the localized short-duration blast pulse affects most severely a small area of a plated structure, the plate’s boundary effects are not as influential as they would be when quasi-static or even a global blast loading is applied, and thus full plate action may not be used. Many common structural forms are composed of individual plated elements, and thus the investigation of localized blast loading effects on plates is an important aspect that leads to understanding the integral behavior. Typically, plates are made of ductile metallic materials, such as steel, which exhibit considerable postyield deformation capacity when subjected to such extreme dynamic loads. An analytical study of the dynamic plastic response of rigid plastic plated structures is the aim of the current study. A circular plate is studied in the present work, and a general form of a localized blast loading function with a spatial variation having a central radial zone with constant pressure and exponentially decaying profile outside the zone is assumed. Assuming that steel exhibits perfectly plastic behavior and considering transverse shear and rotatory inertia effects, using the approach developed previously, results for the permanent transverse displacements and response durations are found in terms of an applied impulsive velocity, which is a function of the applied localized blast load. It is found that the influence of transverse shear is only relevant for small values of ν, or plate radius-to-thickness ratio, which do not correspond to typical plate geometries found in plated structures subjected to blast loading. Both midpoint and support displacement are only significantly affected by transverse shear effects for ν≤5. The same argument is extended to inclusion of rotatory inertia. It is also seen that, for close-in blasts, the effects of transverse shear and rotatory inertia are irrelevant, even for small values of ν.

Damage of Plates due to Impact, Dynamic Pressure and Explosive Loads

It is the purpose of this study to present design equations which can be used to predict the damage of ductile plating when subjected to mass impact, dynamic pressure or impulsive loadings. The external loadings are sufficiently severe to produce inelastic material behaviour and produce finite transverse displacement, or geometry change, effects. The damage is characterised as the final or permanent transverse displacement of a plate. The theoretical method predicts values for the maximum permanent transverse displacements which agree reasonably well with the corresponding experimental results generated on aluminium alloy circular, square and rectangular plates. Thus, the equations presented in this article are valuable for preliminary design purposes and for forensic studies, while the experimental data can be used for validating numerical schemes.

The dynamic performance of simply-supported rigid-plastic circular steel plates subjected to localised blast loading

Near-field explosive charges, such as buried land mines, produce localised blast loadings which can potentially cause damage to property and/or loss of life in both military and civil structures. As the localised short duration blast pulse affects most severely a small area of a plated structure, boundary effects are not as significant as they are when a quasi-static or global blast loading is applied and full plate action may not be utilised.Many common structural forms are composed of individual plated elements and thus the investigation of localised blast loading effects on plates is a prerequisite to understanding the integral behaviour. Typically, plates are made of ductile materials such as steel, which exhibit considerable post-yield deformation capacity when subjected to such extreme dynamic loads. Analytical study of the dynamic plastic response of rigid-plastic plated structures is the aim of the present study.A circular plate is studied in the present work and a general form of a localised blast loading function with a spatial variation having a central radial zone with constant pressure and exponentially decaying profile outside the zone is assumed. Assuming that steel exhibits perfectly plastic behaviour and ignoring membrane action, transverse shear and rotatory inertia effects, the static plastic collapse pressure is initially found and the analysis is extended to take into account the inertial effects arising from dynamic loading. Results for the permanent transverse displacements are found for rectangular and linearly and exponentially decaying pulse loads.For high loads and/or loads of short duration, it was found that the permanent transverse displacement can be found by replacing the applied pulse load by means of an impulsive velocity without great loss of accuracy. Good correlation with numerical simulations obtained from ABAQUS/Explicit is achieved (within 15% accuracy) for plate geometries falling within the identified limits where membrane and shear effects are negligible.The predicted final transverse displacements are found to be dependent on the loading pulse shape, but the pulse-shape effects are eliminated by using the correlation parameters (effective impulse and pressure) advocated by Youngdahl (1971) [52] to give a single pulse shape-independent curve for the final plate deflection as a function of the effective pressure and effective impulse.

Impact loading of ductile rectangular plates

In many industries, rigid-plastic methods of analysis are a useful design aid for safety calculations, hazard assessments, security studies and forensic investigations of ductile structures, which are subjected to large dynamic loads producing an inelastic response. This paper examines the behaviour of a rectangular plate struck at the centre by a rigid mass impact loading. A theoretical method has been developed previously for arbitrarily shaped plates which retains the influence of finite transverse displacements, or geometry changes. It is used in this paper to predict the maximum permanent transverse displacements and response duration of plates having boundary conditions characterised by a resisting moment mM 0 around the entire boundary, where m=0 and 1 give the two extreme cases of simply supported and fully clamped supports, respectively. The theoretical predictions are compared with some experimental data recorded on fully clamped metal rectangular plates having a range of aspect ratios and struck by masses travelling with low impact velocities up to nearly 7 m/s and which produce large ductile deformations without failure. The theoretical analysis gives reasonable agreement with the corresponding experimental data for masses having blunt, conical and hemispherical impact faces. For sufficiently large initial impact energies, the projectile would perforate a plate and, for completeness, a useful design equation is presented which predicts perforation energies larger than all of the test data, as expected.

Low-velocity impact of pressurised pipelines

Experimental tests are reported on steel pipelines which have been struck by a relatively large rigid wedge-shaped mass travelling up to 10.4 m/s. A pipeline is supported across a span, is fully clamped at both ends and is struck at the mid-span and at the one-quarter span positions. Most of the pipelines are pressurised with a nitrogen gas. The initial impact energy produces large inelastic ductile deformations of the pipeline and, in some cases, failure. A method is introduced which idealises a deformed pipeline cross-section in order to estimate the local and global components of the total displacement from experimental measurements of the final cross-section. Comparisons are also made with several previously published experimental studies for which sufficient data are available to make the calculations. A clearer insight into pipeline behaviour is achieved than is possible only with values of the maximum permanent transverse displacements which have been reported in previous experimental studies. This information should assist with pipeline design, provide more rigorous validation for numerical schemes and contribute to a better understanding of pipeline failure. Recommendations are made on the accuracy and suitability of some well known empirical equations for predicting the permanent deformations of pressurised and empty pipelines caused by large rigid masses, relative to the pipeline mass, travelling with low initial impact velocities up to about 15 m/s.

Low-Velocity Perforation of Mild Steel Rectangular Plates With Projectiles Having Different Shaped Impact Faces

This article studies the perforation of mild steel square and rectangular plates struck normally by cylindrical projectiles having blunt, hemispherical, and conical impact faces. Experimental results are obtained in a drop hammer rig for the perforation of 4mm and 8mm thick plates struck by relatively heavy projectiles weighing between 11.9kg and 200kg and traveling at an initial velocity up to about 13m∕s. The plates were struck at the center and at several positions near the fully clamped supports. The effect of the aspect ratio on the perforation energies of rectangular plates is examined, and comparisons are made with the perforation behavior of fully clamped circular plates. The predictions of several empirical equations are compared with the corresponding experimental values of the perforation energies. Simple design equations are also presented for predicting the maximum permanent transverse displacements of square plates prior to any cracking or perforation.

Dynamic plastic behavior of a free–free beam striking the mid-span of a clamped beam with shear and membrane effects considered

Recently Yu et al. (Int. J. Solids Struct. 38 (2001) 261) made a study on the dynamic behavior of a flying free–free beam striking the tip of a cantilever beam using the rigid, perfectly plastic (r-p-p) material model. Later, also based on the r-p-p material model Yang and Yu (Mech. Struct. Mach. 29 (2001) 391) analyzed another impact problem of a free rotating hinge beam striking a cantilever beam. Both of these studies ignored the finite deflection effects on the plastic behavior of the colliding beams. However if the free–free beam strikes a clamped beam, the influence of finite-deflections, or, geometric changes, must be retained in the governing equation if the maximum permanent transverse displacement of the clamped beam exceeds the corresponding beam thickness. The problem becomes more interesting since the deformation mechanisms of the beam system and the partitioning of energy dissipation in the beams are significantly different from those predicted by ignoring the influence of membrane forces. Accordingly the failure modes of the structure are different. In the present paper, a theoretical model based on the r-p-p material idealization is proposed to simulate the dynamic behavior when the mid-point of a translating free–free beam impinging on the mid-span of a clamped beam with the beam axes perpendicular to each other. The plastic behavior of the beam system is explored with shear sliding and finite deflection effects taken into account. The final deflection, the dissipation of energy within the two beams after impact and the influence of the structural and material parameters are discussed. It is shown that membrane force plays an important role during the response process, especially when the deflection is of the same order as the thickness of the clamped beam.

Deformation and perforation of cylindrical shells struck normally by blunt projectiles

An approximate theory is proposed in this paper to predict the deformation and perforation of metallic cylindrical shells struck normally by blunt projectiles at velocities up to 229 m s −1. On the basis of the experimental observations of quasi-static load–displacement characteristics, the problem of a cylindrical shell impacted normally by a blunt projectile can be tackled through the solution of an equivalent clamped circular plate struck transversely by the same projectile. It is shown that the approximate theoretical predictions are in good agreement with the experimental results for cylindrical shells in terms of the maximum permanent transverse displacements and the critical impact velocities (ballistic limits) when material strain rate sensitivity is taken into account.

Deformation and Perforation of Spherical Shells by Normal Impact of Blunt-Ended Projectiles*

ABSTRACT An approximate quasi-static theory is developed to predict large plastic deformation and perforation of spherical shells subjected to impact by blunt-ended projectiles at normal incidence. Based on experimental observations for quasi-static load-displacement characteristics, the problem of a spherical shell under normal impact by a flat-ended missile may be analyzed through the solutions to that of an equivalent circular plate struck transversely by the same missile. It is shown that the approximate theoretical predictions with a = 30 (here a is an empirical constant) are in good agreement with experimental data, in terms of maximum permanent transverse displacements and dimple radii. Furthermore, a theoretical formula for ballistic limits of spherical shells under missile impact is presented and the range of applicability of the theory is discussed.

Deformation and tearing of clamped circular work-hardening plates under impulsive loading

An approximate theory is presented in this paper to predict the deformation and tearing of clamped circular work-hardening plates subjected to uniformly distributed impulsive loads. Based on a power law stress-strain relationship throughout, various equations are obtained and an effective strain failure criterion is suggested for the rupture of the plates. It is shown that the theoretical predictions are in good agreement with the experimental observations in terms of the maximum permanent transverse displacements and the critical input impulses which caused the tensile tearing failure of the plates under impulsive loading when material strain rate sensitivity is taken into account. It is also shown that, to a first approximation, the theory developed for circular plates is applicable to rectangular plates.

Large plastic deformation of spherical shells under impact by blunt-ended missiles

A procedure is given in this paper to estimate the large plastic deformation of spherical shells subjected to impact by blunt-ended missiles. Based on the experimental observations for the quasi-static load-displacement characteristics, the problem of a spherical shell under impact by a blunt missile can be analysed through the solutions to that of an equivalent circular plate struck transversely by the same missile. It is demonstrated that the approximate theoretical predictions using this method are in good agreement with the experimental data in terms of the maximum permanent transverse displacements and the dimple radii.

Response and Failure of Ductile Circular Plates Struck by a Mass

A theoretical analysis is presented in this paper to predict the dynamic plastic response of aluminium alloy and mild steel circular plates when struck normally by blunt solid cylindrical masses at the center. Reasonable agreement is obtained with experimental results for the permanent transverse displacements. Different failure mechanisms, which have been found in previous experimental studies on aluminium alloy and mild steel plates, are also illustrated by the proposed theory.

Deformation and tearing of clamped work-hardening beams subjected to impulsive loading

An approximate theoretical analysis, based on a power law stress-strain relationship throughout, is presented herein to examine the behaviour of clamped beams subjected to uniformly distributed impulsive loads, which represents an extension of the previous study. In particular, the rupture (tensile tearing) of the beams under impulsive loading is predicted by an effective strain failure criterion which takes into consideration the influence of the transverse shear on the axial tensile strain. It is found that the present theoretical predictions are in good agreement with the experimental observations in terms of the maximum permanent transverse displacement and the critical input impulse causing beam tensile tearing failure when material strain rate sensitivity is taken into account.