The objective of this section is to briefly review the basic approaches used to predict the blast parameters required in blast-resistant design. These approaches are often divided into empirical, semi-empirical and numerical (or computational) methods [143]. Empirical methods are basically correlations with experimental data, which implies that this approach is limited by the extent of the underlying experimental database. The accuracy of all empirical methods diminishes as the blast becomes increasingly near-field and approaches close-in detonations. Semi-empirical methods are based on simplified models of physical phenomena. They attempt to model the underlying physics in a simplified way.
These methods rely on extensive data and case studies and their accuracy is generally better than that provided by the empirical methods. Numerical (or computational) methods are based on the mathematical equations that describe the basic laws of physics governing a problem. These principles typically include conservation of mass, momentum and energy. In addition, the material behaviour of the air is described by constitutive relations (known as equations of state). These models are commonly called computational fluid dynamic (CFD) models. Note that semi-empirical methods are developed primarily by defence-related agencies and the distribution are therefore restricted to the government and its contractors (see e.g. [153, 154]). This thesis is therefore limited to empirical and CFD methods which are readily available to the public.
The empirical method are presented inPart IIof this thesis, while the CFD method are discussed in more detail inPart III.
2.5. Prediction of blast loads 41 Of the many available references in the literature (see e.g. [3, 5, 15, 142, 143, 150]) the most reliable and referenced works dates back to a few U.S. Army publications. That is, the technical report by Kingery and Bulmash [4] and the Army Technical Manual TM 5-1300 [16]. The latter was updated and replaced by UFC 3-340-02 in recent years and provides detailed information and procedures for the design of structures to resist the effects of explosions.
However, many of the blast properties predicted by this manual also dates back to the former publication. The most common reference in predicting blast loading is therefore the work by Kingery and Bulmash [4], in which experimental data from idealized conditions (i.e., spherical and hemispherical high-explosive detonations) were gathered and curve-fitted to higher-order polynomial equations for the necessary blast parameters using the Hopkinson-Cranz scaling laws for a large range of TNT equivalent charges (1 kg< WTNT<
400,000 kg). As already mentioned in Section 1.2.1, the data used by Kingery and Bulmash [4] contained limited data for blast parameters at scaled distances less than 0.40 m/kg1/3, and some of the parameters were therefore extrapolated at smaller distances using the available data and theoretical considerations.
These experiments and empirical equations, together with the Friedlander equation in Eq. (2.1), form the basis for various simplified tools to predict blast loading from a given explosive weight at a known distance from the target. The most common and widely used tool is known as the Conventional Weapons Effects Program (ConWep, formerly TM 5-855-1) [17]. These simplified tools are often called empirical methods in the literature and present an idealized representation of blast loads for design purposes. Due to the idealized nature, these methods have significant advantages compared to other methods in terms of time consumption and are therefore frequently used in so-called quick assessments. This is a typical starting point of a blast-load analysis, providing useful insight in the performance of structures and may be used for a first optimization before more elaborate analyses and methods may be considered.
The empirical equations by Kingery and Bulmash [4] are given for spherical free airbursts in Figure 2.17a and hemispherical surface bursts in Figure 2.17b.
It is observed that the blast properties in Figure 2.17b are similar to those in Figure 2.17a. However, in a surface burst the parameters will be larger in magnitude due to the instantaneous reflection from the ground (see Figure 2.14).
Note that the parameters in terms of specific impulses i, positive duration td+and time-of-arrivalta are scaled using Hopkinson-Cranz scaling [14]. That is, scaled by the cube-root of the charge mass. Also note that Figures 2.17a and 2.17b do not provide a value for the exponential decay coefficient b in Eq. (2.1). This may be found by solving the implicit non-linear equation in Eq. (2.3) since the value of the reflected specific impulse ¯ir+, peak reflected overpressurepr,max and duration ¯td+ of the positive phase is known. As the loading becomes increasingly near-field (Z ≤2.0), the use of the empirical
10−2 10−1 100 101 102
Scaled distance, Z [m/kg1/3] pr,max [kPa]
Scaled distance, Z [m/kg1/3] pr,max [kPa]
Figure 2.17: Blast parameters for a charge of TNT, detonated in free air at sea level, given by Kingery and Bulmash [4].
equations requires that the non-uniform spatial distribution of the loading is included in the calculation. This can be done by considering both the incident overpressurepso,maxand the angle of incidenceαwhen determining the reflected pressure acting on the structure (see Section 2.1 and Figures 2.4, 2.7 and 2.10).
Intermediate values of the peak incident overpressure pso,max in Figure 2.7 may be found by interpolation between adjacent curves. If the charge shape is spherical or hemispherical, the reflecting surface is considered plane and of infinite size, and the scaled distance is defined as a near- or far-field detonation (Z ≥0.5), the simplified methods based on Figures 2.17a and 2.17b are in
general found to provide good estimates of the blast properties [33].
However, at scaled distances defined as a close-in detonation (Z ≤0.5), the use of simplified methods is questionable due to the increasing complexity of the load in the vicinity of the structure [38, 39]. That is, the empirical methods do not consider potential interactions between the fireball and the blast overpressure. Moreover, these methods are not valid for blast environments involving complex geometries. Complex geometries and structures of finite surfaces involves clearing, shielding and confinement between neighboring buildings. This introduces the need for more elaborate methods (e.g. semi-empirical or numerical methods), since the reflected pressure will be relieved by a rarefaction wave generated during the diffraction of the reflected shock around
2.5. Prediction of blast loads 43 the boundary of the reflecting surface (see e.g. [16,155]). This requires advanced numerical methods based on computational fluid dynamics (CFD) to estimate the loading for contact detonations or complicated geometries [38]. Thus, when using the empirical methods, it is necessary with a proper understanding of the blast phenomenon and underlying limitations of the method to ensure that the blast properties are valid estimates.
Finally, although the effects of the negative phase are usually neglected in the design of hardened structures (e.g. reinforced concrete or similar), this phase may be of importance when considering the response of flexible structures where the overall motion will be affected by the timing of the negative phase.
The most commonly used negative phase parameters seem to be those given in [16, 27]. These parameters are presented in Figures 2.18a and 2.18b for spherical and hemispherical charges, respectively. Note thatpr,max, ¯td+and ¯ir+
from Figures 2.17a and 2.17b are included for comparison to the corresponding negative phase parameters (pr,min, ¯td− and ¯ir−). It is observed that although there is a relatively large difference between peak reflected pressure pr,max and peak negative pressurepr,min, the corresponding impulses (¯ir+ and ¯ir−) approaches the same order of magnitude forZ >1. Studies by Rigby et al. [30]
also suggest that the angle of incidenceαhas limited influence of the negative phase parameters (pr,min, ¯td− and ¯ir−).
Scaled distance, Z [m/kg1/3] pr,max [kPa]
Scaled distance, Z [m/kg1/3] pr,max [kPa]
Figure 2.18: Parameters for the positive and negative phase from a charge of TNT detonated in free air at sea level [4, 16].
3
Airblast experiments
This chapter is mainly based on the first paper published in International Journal of Impact Engineering [156]. It presents an experimental investigation on the influence of stand-off distance on the dynamic response of thin aluminium and steel plates subjected to airblast loading and covers the experimental work related to Part II of this thesis. The experimental results provide a set of data which can be used to evaluate the performance of current computational methods in predicting the structural response of thin ductile plates exposed to blast loading.
3.1 Introduction
As discussed in Section 1.2.4, experiments involving high-explosive detonations are necessary to investigate the inherent complexity in such blast environments (e.g. highly non-uniform spatial and temporal pressure distributions and the interaction between the fireball and the blast overpressure in the vicinity of the target). A series of airburst detonations was therefore carried out to obtain knowledge and an improved understanding of near-field blast events. The experimental results presented in this chapter will also serve as a basis of comparison for the numerical simulations in Chapter 5.
The blast intensity and pressure distribution were varied by detonating small-scale spherical charges of plastic explosive at various stand-off distances relative to the centre of the plates. Piezoelectric pressure sensors were used for pressure recordings and synchronized with two high-speed cameras in a stereovision setup to capture the dynamic response using a finite element-based three-dimensional digital image correlation (3D-DIC) technique. Material tests were also performed to determine the materials’ behaviour at large plastic strains.