Classical layout optimization is a well-established structural design tool, however it considers only truss structures composed solely of axially loaded bars, limiting its applicability to many structures. To address this limitation, this study introduces a novel approach that takes into account moment-resisting beams in the optimization framework. The interaction between moment, shear, and axial forces is simplified for inclusion in the optimization problem, solved effectively via a sequential conic programming scheme. Various numerical examples show that the proposed approach identifies structures with lower-volume than the classical layout optimization in problems involving multiple load cases or pre-existing members. Additionally, the method can also be used to solve problems that classical layout optimization cannot address, including constrained design spaces and point moment loads. The findings indicate that the proposed approach provides greater flexibility and efficiency in designing hybrid truss and beam structures, paving the way for versatile structural solutions.

This paper details the development of a rapid inverse approach to determine the yield and location of an explosion through trilateration of empirical laws for blast wave arrival time. A rigorous sensitivity analysis of measurement uncertainty is first performed. From this, a probabilistic framework is proposed that utilizes Monte Carlo sampling of datasets to mitigate the effects of the variability and uncertainties typically present in blast events. Subsequently, the trilateration method is successfully applied to two existing datasets. Analysing well-controlled small-scale laboratory experiments, charge mass is predicted within 6.3% of the true yield, and position within 3.65 charge radii of the true centre. Social media footage of the 2020 Beirut explosion is then used to assess performance against data collected under in-field conditions. The predicted yield of 0.52 kt[Formula: see text] shows good agreement with the literature, and charge position is predicted to within the radius of the crater. Trilateration is shown to be able to rapidly and reliably determine explosive yield and centre, despite large levels of sensor noise. The sub-second computation time of this approach offers the possibility to better model and predict the damage and injury patterns immediately after an explosion, facilitating more effective disaster response planning.This article is part of the theme issue 'Frontiers of applied inverse problems in science and engineering'.

A significant amount of scientific effort has been dedicated to measuring and understanding the effects of explosions, leading to the development of semi-empirical methods for rapid prediction of blast load parameters. The most well-known of these, termed the Kingery and Bulmash method, makes use of polylogarithmic curves derived from a compilation of medium to large scale experimental tests performed over many decades. However, there is still no general consensus on the accuracy and validity of this approach, despite some researchers reporting consistently high levels of agreement. Further, it is still not known whether blast loading can be considered deterministic, or whether it is intrinsically variable, the extent of this variability, and the range and scales over which these variations are observed. This article critically reviews historic and contemporary blast experiments, including newly generated arena tests with RDX and PETN-based explosives, with a view to demonstrating the accuracy with which blast load parameters can be predicted using semi-empirical approaches.

As the detonation product cloud from a high explosive detonation expands, an arresting flow is generated at the interface between these products and the surrounding air. Eventually this flow forms an inward-travelling shock wave which coalesces at the origin and reflects outwards as a secondary shock. Whilst this feature is well known and often reported, there remains no established method for predicting the form and magnitude of the secondary shock. This paper details an empirical superposition method for modelling the secondary shock, based on the physical analogy of the secondary loading pulse resembling the blast load from a smaller explosive relative to the original. This so-called dummy charge mass is determined from 58 experimental tests using PE4, PE8, and PE10, utilising Monte Carlo sampling to account for experimental uncertainty, and is found to range between 3.2–4.9

When explosives detonate in a confined space, repeated boundary reflections result in complex shock interactions and the formation of a uniform quasi-static pressure (QSP). For fuel-rich explosives, mixing of partially oxidized detonation products with an oxygen-rich atmosphere results in a further energy release through rapid secondary combustion or 'afterburn'. While empirical formulae and thermochemical modelling approaches have been developed to predict QSP, a lack of high-fidelity experimental data means questions remain around the deterministic quality of confined explosions, and the magnitude and mechanisms of afterburn reactions. This article presents experimental data for RDX- and PETN-based plastic explosives, demonstrating the high repeatability of the QSP generated in a sealed chamber using pressure transducers and high-speed infrared thermometry. Detonations in air, nitrogen and argon atmospheres are used to identify the contribution of afterburn to total QSP, to estimate the duration of afterburn reactions and to speculate on the flame temperature associated with this mechanism. Computational fluid dynamic modelling of the experiments was also able to accurately predict these effects. Understanding and quantifying explosions in complex environments are critical for the design of effective protective structures: the mechanisms described here provide a significant step towards the development of fast-running engineering models for internal blast events.

Buried charges pose a serious threat to both civilians and military personnel. It is well established that soil properties have a large influence on the magnitude and variability of loading from explosive blasts in buried conditions. In this study, work has been undertaken to improve techniques for processing pressure data from discrete measurement apparatus; this is performed through the testing of truncation methodologies and the area integration of impulses, accounting for the particle size distribution (PSD) of the soils used in testing. Two experimental techniques have been investigated to allow for a comparison between a global impulse capture method and an area-integration procedure from a Hopkinson Pressure Bar array. This paper explores an area-limiting approach, based on particle size distribution, as a possible approach to derive a better representation of the loading on the plate, thus demonstrating that the spatial distribution of loading over a target can be related to the PSD of the confining material.

The response of plates subjected to blast loads is of considerable scientific interest. The loading imparted to a structure following a close-in detonation of a high explosive is typically high in magnitude, near-impulsive, spatially non-uniform, with localised variability and a high dependence on factors such as charge shape, position, and composition. The resulting structural response may induce large displacements in materials whose properties may not be fully characterised. In order to properly account for the effects of such intrinsic and extrinsic uncertainties, modelling approaches must balance the competing demands of accuracy and low computational demand. This article applies the extended Hamilton's principle to rigid-plastic thin plates subjected to impulsive blast loads to derive the governing equation of motion without a prior assumption of the initial specific impulse distribution. Closed-form solutions to predict the plastic response are derived for rectangular and circular plates. The analytical models for uniform specific impulses are found to be in good agreement with high-fidelity numerical simulations performed using LS-DYNA and experimental data available in the literature.

Structural engineers are often tasked with the challenge of enhancing the resilience of buildings and structures to withstand blast loading from the detonation of energetic materials. Such loading typically occurs during terrorist attacks and from accidental explosions in nearby chemical and explosive storage facilities. However, existing methods for predicting the governing blast loading on a structure, through consideration of multiple explosion scenarios, are unsuitable for most practising structural engineers. This paper will address the need for fast-running tools for predicting blast loads on structures in the far-field by presenting the semi-empirical EMBlast method that calculates free-field and reflected pressure-time histories for both the positive and negative phase. For finite size target surfaces, this method also accounts for clearing effects, a phenomenon that results in gradually reducing the reflected pressures on the front face of a building, to the lower free-field pressures experienced by the sides and roof. To validate the predictions of the EMBlast method in the far-field, computational fluid dynamic analyses are performed over a long-range of scaled distances. Furthermore, the EMBlast predictions are also compared with the results from published blast tests, existing empirical methods and computational fluid dynamic simulations identified in the literature.

The modelling of energetic materials is usually performed within an Eulerian framework, inherently well suited to simulate fluid-like behaviour of the reaction products and the propagation of pressure waves in the surrounding fluid media (e.g. air or water). However, the Lagrangian framework becomes more attractive in the effort to reproduce the unreacted mechanical response of the material, especially as we move towards trying to capture better the response of damaged explosives. In this paper the implementation of the History Variable Reactive Burn (HVRB) model in the Lagrangian explicit software LS-Dyna is presented. The HVRB reaction rate parameters are verified against simple ratestick tests before being used to simulate the detonation of hemispherical charges from an ignition point located at the pole of the hemisphere. The numerical results are directly compared against high-speed video from the experimental tests, illustrating reasonable agreement given the simplicity of the HVRB formulation. This paves the way for developing deeper understanding by incorporating more complex reactive burn models, e.g. Damage Initiated Reaction (DMGIR), which can consider the role of mechanically induced damage upon the response of explosives.

Accurate modelling of free-field detonations needs to account for both the initial energy release and gas generation, and the subsequent reaction of the initial detonation products with external oxygen. An upper limit for the extent of ‘afterburn’ can be ascertained from comparison of contained blasts in reactive (air) and inert (nitrogen) bath gases. The peak quasistatic pressures (QSP) and the gas phase products were determined in a 0.276 m3 blast chamber following detonations of the plastic explosive PE4. The experimental observations were compared to predictions based on standard models, CEA and EXPLO5. The best agreement between models and experiment, for both products and QSPs, was obtained from the Springall-Roberts treatment of detonation products in nitrogen, and complete combustion of these in air.

The well-known ‘ground structure’-based truss layout optimization method has recently been extended to allow accurate modelling of distributed self-weight. By incorporating equally stressed catenaries in the ground structure, non-conservative errors caused by neglecting bending effects within members carrying their own weight are eliminated. However, in cases where the self-weight of a structure has a favourable role in supporting the applied loads, solutions that include convoluted arrangements of overlapping elements may often be generated. To address this, an enhanced layout optimization formulation is proposed that explicitly allows inclusion of favourable unstressed masses, such as counterweights. Frictional supports are also modelled and the cost of abutments and anchorages taken account of in the formulation. The efficacy of the proposed methodology is demonstrated through application to benchmark examples and to the conceptual design of a simplified long-span bridge structure, considering both ground anchored and self-anchored alternatives.

The understanding of blast loads is critical for the development of infrastructure that protects against explosions. However, the lack of high-quality experimental work on the characterisation of such loads prevents a better understanding of many scenarios. Blast loads are typically characterised by use of some form of pressure gauge, from which the temperature can be inferred from a pressure measurement. However, such an approach to temperature measurement is limited; it assumes ideal gas laws apply throughout, which may not be the case for high temperature and pressure scenarios. In contrast, infrared radiation thermometers (IRTs) perform a measurement of temperature based upon the emitted radiance from the target object. The IRTs can measure fast changes in transient temperature, making them seemingly ideal for the measurement of a fireball's temperature. In this work, we present the use of a high-speed IRT for the measurement of early-stage explosive development and fireball expansion within a confined blast, with the temperature of the explosive fireball measured from its emitted radiance. The temperature measured by the IRT was corroborated against the temperature inferred from a pressure gauge measurement; both instruments measured the same temperature from the quasi-static pressure (QSP) point onwards. Before the QSP point, it is deduced that the IRT measures the average temperature of the fireball over a wide field-of-view (FOV), as opposed to that inferred from the singular shocks detected by the pressure gauge. Therefore, use of an IRT, in tandem with a pressure gauge, provides a potential invaluable measurement technique for the characterisation the early stages of a fireball as it develops and expands.

The ability to accurately determine blast loading parameters will enable more fundamental studies on the sources of blast parameter variability and their influence on the magnitude and form of the loading itself. This will ultimately lead to a better fundamental understanding of blast wave behaviour, and will result in more efficient and effective protective systems and enhanced resilience of critical infrastructure. This article presents a study on time of arrival as a diagnostic for far-field high explosive blasts, and makes use of the results from a large number of historic tests and newly performed experiments where the propagating shock front was filmed using a high-speed video (HSV) camera. A new method for optical shock tracking of far-field blast tests is developed and validated, and full-field arrival time results are compared against those determined from the historic data recorded using traditional pressure gauges. Arrival time variability is shown to be considerably lower than peak pressure and peak specific impulse, and is shown to decrease exponentially with increasing scaled distance. Further, the method presented in this article using HSV cameras to determine arrival time yields further reductions in variability. Finally, it is demonstrated that the method can be used to accurately determine far-field TNT equivalence of high explosives.

The ability to measure the structural and material response to air-blast loading is vital to developing a proper understanding of near-field blast loading and response. Computational modelling has advanced significantly but, until recently, experimental techniques lagged behind. This paper discusses recent advances in these experimental techniques. The first part describes a bilateral test programme between the UK and South Africa. The high-speed imaging and digital image correlation system at Cape Town gives repeatable and accurate impulse distributions across a central strip of a panel, useful for model validation. Flexural wave behaviour was observed from the transient velocity and displacement profiles, giving good insights into the mechanics of plate response from blast loads. The second part demonstrates the value of high-speed stereo-imaging for measuring the transient response of blast loaded fibre reinforced polymer panels and sandwich structures. The peak displacements. elastic rebounds and transient oscillations provide valuable insights into the damage propagation within these types of structures. The final part of the paper describes some of the continued developments since the success of those early trials, resulting in a new optical diagnostics for blast capability at the University of Sheffield. The imaging system operates at higher frame rates and can cover a wider region of interest on the structure. Ultra-high speed imaging is also shown to be a useful tool for visualising detonations fronts in explosive charges and the expanding fireball.

Loading of vehicle undercarriages from the detonation of shallow-buried explosives remains a serious threat to life in conflict and post-conflict zones. One method to protect lightly-armoured vehicles is to retrofit them with applique armour, which must be strong enough to provide adequate protection, but light enough to maintain vehicle manoeuvrability. A key performance metric of this armour is its deformation under loading, which must be limited to avoid impact upon vehicle occupants. The high-strength steel Armox 440T is commonly used due to its high load capacity, strength-to-weight ratio, ductility and low cost: as other protection systems are developed, it would be of great benefit to compare their deformation against an Armox 440T benchmark. However, no definitive benchmarking study has been published to date, mainly due to the difficulties in ensuring repeatable loading from complex buried detonations. This paper presents experiments which underpin such a benchmarking study, building on the authors' previous work to establish a methodology which produces very consistent loading from shallow-buried detonations. Tests were conducted with a range of explosive masses and plate thicknesses, with target plates secured in a purpose-designed frame to produce simple, consistent boundary conditions. Plate deformations captured by stereo high-speed digital image correlation were compared to a commonly-used low-cost peak deflection method. High-speed digital image correlation was found to make highly reproducible displacement measurements with a standard deviation of 2% of the mean. The low-cost method provided slightly higher variability up to 5% of the mean value, and measurements of peak deformation were systematically 20% higher, but in a consistent manner, with a low unit cost and without risk to expensive test equipment. The low-cost method therefore allowed the development of a multivariate regression relationship between deformation, charge size and plate thickness, which provides a benchmark for the assessment of future protection solutions.

Transmissible loads are external loads defined by their line of action, with actual points of load application chosen as part of the topology optimization process. Although for problems where the optimal structure is a funicular, transmissible loads can be viewed as surface loads, in other cases such loads are free to be applied to internal parts of the structure. There are two main transmissible load formulations described in the literature: a rigid bar (constrained displacement) formulation or, less commonly, a migrating load (equilibrium) formulation. Here, we employ a simple Mohr’s circle analysis to show that the rigid bar formulation will only produce correct structural forms in certain specific circumstances. Numerical examples are used to demonstrate (and explain) the incorrect topologies produced when the rigid bar formulation is applied in other situations. A new analytical solution is also presented for a uniformly loaded cantilever structure. Finally, we invoke duality principles to elucidate the source of the discrepancy between the two formulations, considering both discrete truss and continuum topology optimization formulations.

Research into the characterisation of blast loading on structures following the detonation of a high explosive commonly assumes that the charge is spherical. This has the advantage of simplifying experimental, analytical and computational studies. In practice, however, designers of protective structures must often consider explosive threats which have other geometric forms, which has significant influence on the loading imparted to structures very close to the explosion source. Hitherto, there has been little definitive experimental investigation of the ‘near-field’ blast load parameters from non-spherical explosive charges and studies that have been conducted are usually confined to measurement of the total impulse imparted to a target. Currently, a detailed understanding of the development of loading on a target, necessary to fully inform the design process and appraise the efficacy of predictions from computational models, is lacking. This article, the first part of a wider investigation into these geometrical effects, details work conducted to address this deficiency. Results are presented from an experimental study of loading from detonations of cylindrical charges, set with the longitudinal axis parallel to an effectively rigid target, instrumented to facilitate the capture of the spatial and temporal evolution of the loading at different radial and angular offsets from the charge. These results are compared against loads from spherical charges and the effect of charge shape is identified. Significant differences are observed in the mechanisms and magnitude of loading from cylindrical and spherical charges, which is confirmed through the use of numerical analysis. The overall study provides insights which will assist the future design of effective protection systems.

In this paper, we introduce geometry optimization into an existing topology optimization workflow for truss structures with global stability constraints, assuming a linear buckling analysis. The design variables are the cross-sectional areas of the bars and the coordinates of the joints. This makes the optimization problem formulations highly nonlinear and yields nonconvex semidefinite programming problems, for which there are limited available numerical solvers compared with other classes of optimization problems. We present problem instances of truss geometry and topology optimization with global stability constraints solved using a standard primal-dual interior point implementation. During the solution process, both the cross-sectional areas of the bars and the coordinates of the joints are concurrently optimized. Additionally, we apply adaptive optimization techniques to allow the joints to navigate larger move limits and to improve the quality of the optimal designs.

While it is well known that detonation of shallow-buried high explosive charges generally results in above-surface loading which is greatly amplified compared with the same detonation in air, uncertainty persists as to the mechanisms leading to this effect. The work presented in this paper is a systematic investigation into the mechanisms of load transfer in buried blast events. This paper details the results from a parametric study into the mechanisms and magnitudes of load transfer following a shallow-buried explosion, where spatial and temporal load distributions are directly measured on a rigid surface using an array of Hopkinson pressure bars. In particular, the investigation has looked at the influence of both geometrical confinement and geotechnical conditions on the loading. The parametric study was separated into four main threads: the influence of physical confinement; gravimetric moisture content; stand-off distance and depth of burial; and soil material/particle size distribution. This study allows a direct observation of the contributions of each of these distinct parameters, and in particular the ability to discern how each parameter influences the temporal form and spatial distribution of the loading.