Scaling the response of circular plates subjected to large and close-range spherical explosions. Part I: Air-blast loading

This two-part paper addresses scaling of the dynamic response of clamped circular plates subjected to close-range and large spherical blast loadings.     

Scaling the response of circular plates subjected to large and close-range spherical explosions. Part II: Buried charges

This paper discusses scaling of the dynamic response of clamped circular plates subjected to close-range, large spherical blast loadings that are flush buried in dry sand. As a continuation of part I which dealt with air blasts, similarity is obtained by using replica scaling for all geometrical parameters, while the blast effect is scaled by using the Hopkinson scaling law.     

Fracture of aluminium foam core sacrificial cladding subjected to air-blast loading

The effect of core density and cover plate thickness on the blast response of sacrificial cladding panels has been investigated through blast loading experiments and finite element modelling on structures with steel cover plates and aluminium foam cores. A range of foam core densities were examined, with 10%, 15% and 20% nominal relative densities. The cover plate thickness greatly influenced the response of the sacrificial cladding. Cover plates that were 2 mm thick exhibited significant permanent deformations and variable percentage crush across the section, whereas the 4 mm thick cover plates were more rigid causing the core to compress uniformly. Considerable fracture of the foam was observed after blast testing, particularly for the lower density foams. The effect of bonding the cover plate to the core was also examined. Numerical simulations of the experiments were performed using ABAQUS/Explicit to provide insight into the response mechanism. It was shown through the finite element simulations that tensile fracture of the foam occurred during the unloading phase of response and that adhesion of the cover plate to the foam caused higher levels of cracking. This was consistent with the experimental observations.     

Effect of distance from the support on the penetration mechanism of clamped circular polycarbonate armor plates

A 1.91-mm thick circular polycarbonate plate of 115 mm diameter was impacted by a spherical steel projectile of 6.98 mm diameter at its center. Subsequent impacts were made at 10, 20, 30, 40, and 50 mm radii of the plate. Dent dimensions for the damaged plate were measured using optical microscope. For a constant projectile velocity of 138 m s⁻¹ which was below the perforation limit of the plate under investigation, a maximum thickness reduction close to the edge support was observed. The experimental work was modeled into explicit finite-element analysis program LSDYNA for simulations. LSDYNA was able to predict the dent depth and reduction in plate thickness at impact points precisely. In this research, the effect of the impact location distance from the supports on the damage mechanism of circular polycarbonate armor plates is investigated. The target plate was subjected to constant velocity projectile impacts starting at the plate midpoint and varying the impact distance from midpoint towards the clamped edge. Failure of plate is predicted close to the constrained boundary under uniform conditions.     

Theoretical and experimental analysis of plastic response of isotropic circular plates subjected to underwater explosion loading

Dynamic response analysis of structures subjected to underwater explosion (UndEx) loading has been always an interesting field of study for ship designers and metal forming specialists. Understanding the deformation and rupture mechanism of simple structures plays an important role in successful designing of a reliable structure under this kind of loading. In this paper, first the major parameters of the UndEx phenomenon (peak overpressure, impulse per unit area, dimensionless damage parameter Φ, etc.) are discussed and determined by means of available experimental relations mostly offered by R. Cole. After that, the maximum deflection of a fully clamped circular plate has been calculated using a theoretical procedure assuming two different conditions: (1) neglecting the effect of strain rate, (2) considering the strain rate sensitivity of the material. Some experiments have been conducted on 5010 aluminum circular plates, using C4 as explosive. In order to simulate reality, a fixture was designed so that the plates are loaded in air‐backed condition. Plates were fixed on top of the fixture, so the fully clamped condition which was assumed as the boundary condition was achieved. The test specimens were measured, not only their maximum deflection but also their thickness at different radii were determined. The results are compared to experimental‐based predictions offered by Nurick and Rajendran who has conducted similar experiments. The results show reasonable agreement with theoretical predictions, especially when strain rate effects are considered. In addition, two new material constants (D, q) for this special aluminum alloy are introduced.     

Damage prediction of clamped circular plates subjected to contact underwater explosion

Contact explosion damage prediction of circular steel plates is of interest to the naval architects and warship designers. During contact explosion, a fraction of the explosive energy goes for plastic deformation and fracture of the plate. Experiments were conducted on air backed circular high strength low alloy (HSLA) steel plates to establish this fraction. There was a large amount of plastic deformation preceding fracture in the contact exploded plates. The deformation contour was found to be spherical showing maximum absorption of energy for the depth of bulge attained. The radius of penetration of the test plate was obtained by applying the terminal strain to fracture and the non-uniform strain distribution in the target plate. A case history is presented where the current predictions are compared with the existing empirical methodology and experimental data.     

Axisymmetric analytical solutions for a heterogeneous multi-ferroic circular plate subjected to electric loading

In the present article, responses of an electrically loaded heterogeneous circular plate of transversely isotropic multi-ferroic materials are investigated. The axisymmetric problem is solved by adopting the direct displacement method developed by the second author. The corresponding generalized displacement functions are obtained through a step-by-step integration procedure. The coupling magneto-electro-elastic fields in the plate, which exactly satisfy the upper and lower boundary conditions and approximately meet the cylindrical boundary condition, are explicitly expressed. Numerical calculations are performed for two particular plates to investigate the influence of material heterogeneity.     

The effect of welded boundaries on the response of rectangular hot-rolled mild steel plates subjected to localised blast loading

A series of experiments examining the effects of welded boundaries on the localised blast load response of mild steel plates is presented in this short communication. Two types of welding are examined, TIG and MIG welding. The welded boundary response is compared with clamped boundary and straight and chamfer machined boundary responses. The overall trend of midpoint deflection–thickness ratio versus dimensionless damage number for all the plates follows previously reported empirical trends. The tearing threshold (Mode II*) in terms of the damage number is lowest for the machined plates followed by the welded plates with the clamped plates having the highest resistance to tearing.     

空中近距爆炸下加筋板架的毁伤模式仿真研究

为研究导弹战斗部近距爆炸下舰船舷侧板架结构的毁伤模式及其转化规律,通过有限元模拟,研究了柱状炸药在矩形加筋板架中心正上方爆炸时加筋板架的毁伤模式及不同毁伤模式之间转化的临界爆距,分析了比例距离r和损伤因子Df对毁伤模式的影响,得到了关于D0f.5和r的毁伤模式判别图,提出了不同毁伤模式的判别方法,给出了预测不同毁伤模式之间转化的临界爆距的计算公式。结果表明,对于给定的药量和板架结构,若(rD0f.5)>0.023 3 m/kg1/3,则随爆距的减小,加筋板架依次经历模式Ⅰ(仅出现塑性大变形)、模式Ⅱ(长边中间出现拉伸撕裂)和模式Ⅲ(中央出现破口)这3种毁伤模式;若(rD0f.5)0.023 3 m/kg1/3,则随爆距的减小,加筋板架仅依次经历模式Ⅰ和模式Ⅲ。     

Interactive mechanisms between the internal blast loading and the dynamic elastic response of spherical containment vessels

The interactive mechanisms between internal blast loading and dynamic elastic response of spherical containment vessels are studied in this paper. The blast loading history in containment vessels can be divided into three periods, i.e. the primary-shock period, the shock-reflection period and the pressure-oscillation period. It is shown that the initial response of the containment vessel depends on both the impulse and the shape of the primary-shock depending on the ratio of the loading period to the breathing mode period. However, during the shock-reflection period, the response of the containment vessel can be coupled with the reflected shock waves in the vessel, especially when the dominant frequency of reflected shock waves is close to the breathing mode frequency of the vessel. During the pressure-oscillation period, the dynamic loading is mainly the oscillation of the internal pressure due to the oscillatory volume change of the vessel, which couples dissipatedly with the vibration of the vessel leading to reduced vibration amplitudes. The effects of the influential non-dimensional parameters on the resonant interaction in shock-reflection period are discussed, based on which guidelines are recommended for avoiding the strain growth in the shock-reflection period in the design of spherical containment vessels.     

均匀热载荷作用下功能梯度圆板的非线性振动

基于经典板理论,研究了热载荷作用下功能梯度圆板的大幅振动问题.在经典板理论下利用物理中面概念,导出了功能梯度圆板的非线性运动方程.利用Ritz?Kantorovich方法消去时间变量,将非线性运动方程转换成了一组关于空间变量的非线性常微分方程.采用打靶法数值求解所得方程,并利用数值结果研究了热载荷作用下功能梯度圆板静态...     

Analysis of RC structures subjected to air-blast loading accounting for strain rate effect of steel reinforcement

A new method to calculate the dynamic strength of reinforced concrete (RC) structures, with emphasis on the embedded response of mild steel reinforcement, subjected to air-blast loading is derived. The proposed method is based on a single-degree-of-freedom (SDOF) system with appropriate modification for strain-rate effect. Approximate formulae are used and explained in detail. A numerical example is presented to show how the derived formulae are used for the analysis of concrete structures subjected to air-blast loading.     

Nonlinear behavior of functionally graded circular plates with various boundary supports under asymmetric thermo-mechanical loading

The equilibrium equations of the first-order nonlinear von Karman theory for FG circular plates under asymmetric transverse loading and heat conduction through the plate thickness are reformulated into those describing the interior and edge-zone problems of the plate. A two parameter perturbation technique, in conjunction with Fourier series method is used to obtain analytical solutions for nonlinear behavior of functionally graded circular plates with various clamped and simply-supported boundary conditions. The material properties are graded through the plate thickness according to a power-law distribution of the volume fraction of the constituents. The results are verified with known results in the literature. The load–deflection curves for different loadings, boundary conditions, and material constant in a solid circular plate are studied and discussed. It is shown that the behavior of FG plates with clamped or simply-supported boundary conditions are completely different. Under thermo-mechanical loading, snap-through buckling behavior is observed in simply-supported FG plates which are immovable in radial direction. Moreover, it is found that linear theory is inadequate for analyzing FG and also homogenous plates with immovable boundary supports in radial direction and subjected to thermal loading, even for deflections that are normally considered small.     

Three-dimensional solution of axisymmetric bending of functionally graded circular plates

Based on three-dimensional theory, this paper investigates the axisymmetric bending of transversely isotropic and functionally graded circular plates subject to arbitrarily transverse loads using the direct displacement method. The material properties can arbitrarily vary along the thickness of the plate. The transverse load is expanded in the Fourier–Bessel series and superposition principle is then used to obtain the total response based on the results of each item of the series. For one item of the series of the load, we assume the distributions of the displacements in the radial direction and therefore only the distributions of the displacements in thickness direction are required to find. If the material properties vary in an exponential law, the exact solutions can be obtained for elastic simple support and rigid slipping support, which are satisfied on the every point of the boundaries. Moreover, the analytical solutions are also presented for simply supported and clamped conditions, which are satisfied using Saint Venant principle. Simultaneously, through the layerwise method a semi-analytical solution is proposed for the case of arbitrary variation of the material properties. Finally the numerical examples are presented to verify the proposed method.     

多次水下爆炸钢制圆板应变与挠度增长规律分析

在海上战争中,水面舰艇以及潜艇可能会受到多枚水中兵器的攻击,产生的累计损伤问题影响着水面舰艇以及潜艇的持续作战能力和生命力。对膨胀直径为390 mm,厚度为5 mm空气背衬钢制圆板,在50 g药量下,进行持续多次水下爆炸加载试验,对板中心应变、挠度以及厚度减薄率进行测量,试验结果分析表明:第一次爆炸加载时,局部空化使钢板产生的应变是冲击波加载的1倍,当钢板被炸弯后,再次爆炸加载局部空化的二次加载现象不明显。此外,气泡脉动及气泡射流会使钢板发生明显的弹性应变,应变值是初始冲击波作用的10%。对中心挠度和厚度减薄率的数据分析发现,钢板厚度减薄率是随着中心挠度呈线性增加的。该研究对舰船持续作战能力及生命力的评估提供了一定的试验支撑。     

Similitude invariants and scaling laws for buckling experiments on anti-symmetrically laminated plates subjected to biaxial loading

Establishing the similitude between the model and prototype rigorously is a necessary step in designing an experiment efficiently. So far, to the best of the authors’ knowledge, no one has ever derived the similitude invariant for anti-symmetric cross- and angle-ply laminated plates subjected to biaxial loading before. This research paper is the first to establish the similitude invariant of anti-symmetric cross- and angle-ply laminated plates by applying the similitude transformation to the governing differential equations of buckling directly. Then the scaling laws for buckling loads of laminated plates subjected to biaxial loads are derived. But in reality, either due to the complexity of the scaling laws or to economize on costly experiments, it may not be feasible to construct the model conforming to the scaling laws completely, therefore partial similitude is investigated theoretically and approximate scaling laws are recommended. The buckling loads of the prototype predicted from the scaling laws are then compared with the available theoretical values. The complete similitude cases show exact agreement between results predicted from the scaling laws and the available analytical solutions. For partial similitude cases, the models distorted in stacking sequences, number of plies, and material properties are studied and the approximate scaling laws which yield good agreement are recommended.     

Nonlinear bending of FGM plates subjected to combined loading and resting on elastic foundations

A nonlinear bending analysis is presented for a simply supported, functionally graded plate resting on an elastic foundation of Pasternak-type. The plate is exposed to elevated temperature and is subjected to a transverse uniform or sinusoidal load combined with initial compressive edge loads. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The formulations are based on a higher-order shear deformation plate theory and general von Kármán-type equation that includes the plate-foundation interaction and thermal effects. A two step perturbation technique is employed to determine the load–deflection and load–bending moment curves. The numerical illustrations concern nonlinear bending response of functional graded plates with two constituent materials resting on Pasternak elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The results reveal that the characteristics of nonlinear bending are significantly influenced by foundation stiffness, temperature rise, transverse shear deformation, the character of in-plane boundary conditions and the amount of initial compressive load. In contrast, the effect of volume fraction index N becomes weaker when the plate is supported by an elastic foundation.     

ALE方法模拟爆炸载荷作用下球壳反直观行为

本文利用任意拉格朗日与欧拉（ALE）方法,模拟了球壳在爆炸载荷作用下的动力反直观行为,得到了球壳中点位移时程曲线以及最终变形模式等动力响应。在研究中,考虑了材料的应变率效应,并对球壳的矢高和厚度对其反直观行为的影响进行了参数分析。结果表明,只有炸药距球壳距离在特定范围内,球壳才能发生反直观行为;当炸药当量一定时,球壳发生反直观行为的爆心距区域段随球壳矢高增加而减少,矢跨比大于一定值后,将不会发生反直观行为     

Buckling response of laminated composite stiffened plates subjected to partial in-plane edge loading

This article presents the buckling analysis of laminated composite stiffened plates subjected to partial in-plane edge loading. The finite element method is used to carry out the analysis. The eight-noded isoparametric degenerated shell element with C⁰ continuity and first-order shear deformation and a compatible three-noded curved beam element are used to model the plate skin and the stiffeners, respectively. The eigen value analysis is carried out to track the buckling load. The convergence study is performed for some specific problems and the results are compared with the available results in the literature. It is observed that the convergence of results is very fast for this finite element model. Effect of different parameters like orientation of fibers, number of layers, and loading types are considered in the present investigation. It is also observed that all these parameters have significant effect on the buckling response of the composite stiffened plate.     

The behaviour of square steel plates subjected to a circular blunt ended load

The behaviour of clamped and freely rotating square steel plates grade St52-3N with inplane displacements restrained at the support subjected to a circular blunt ended load at the centre, has been studied both experimentally and numerically. Based on this information a phenomenological design model has been proposed to calculate the energy absorption at plugging as well as the corresponding displacement when taking finite displacements into account.