Response of blast-loaded plates

In blast-resistant design of protective structures the focus is usually on the structural response due to the positive phase of the blast loading (see e.g. [15–

17,40–42]). This is particularly the case for small values of the scaled distance Z, resulting in high magnitudes of overpressure. In such events the structure is expected to deform in the same direction as the incoming blast wave, i.e., in the intuitive direction. Nurick and Martin [43,44] presented a comprehensive literature review of thin plates subjected to blast loading. These studies included theoretical considerations, experimental techniques and experimental results for relatively large permanent displacements. Nurick and Martin [44]

also suggested a non-dimensional empirical analysis in an attempt to compare experimental results from various studies using different loading parameters, plate dimensions and materials. This approach has proven to be a useful guideline to predict the maximum deflection of impulsively loaded plates.

The dynamic elasto-plastic structural response under pulse loading may be

1.2. Previous work 9 divided into three categories depending on the intensity of the loading and the permanent mid-point deflection (see Figure 1.3 and [45,46]). If the structural component oscillates on both sides of its original configuration with a positive permanent deflection this is called Type I. However, if the pulse is more intense the structural component will oscillate only on the positive side of the original configuration (Type II). Finally, the structural component may first deform in the positive direction and then rebound to the negative side of the original configuration (Type III). The two first types of response are intuitive as the final deflection is positive (i.e., in the same direction as the external loading), while the latter type confounds intuition as the permanent deflection is negative (i.e., in the direction opposite to the external loading).

This phenomenon was first reported during numerical studies by Symonds and Yu [47] and called counter-intuitive behaviour (CIB). They noted that this behaviour was extremely sensitive to the structural and loading parameters, and concluded that the response pattern was strongly dependent on the peak deflection and the corresponding rotation in the plastic hinges at which reverse motion starts. Thus, CIB only occurred within a narrow range of structural and loading conditions during the transition from elastic to moderate plastic deformations and is frequently referred to as reversed snap buckling (RSB). The unexpected nature of this behaviour has received much attention during the years [48–50], and is still a topic of interest in the literature [51]. Theoretical and numerical investigations have managed to associate the phenomenon with chaotic and complex vibrations [50, 52], and this insight has motivated experiments to evaluate both theoretical and numerical investigations [46,53,54].

The experiments found in the literature observing CIB due to RSB mainly consider projectile impacts where there is no negative loading phase [46–48] or blast events where only the positive phase is considered [51].

While the effect of the positive phase on blast-loaded structures seems to be well understood, the current literature indicates that the research on the influence of the negative phase is rather sparse. The U.S. Army Technical Manual [16] states that the negative phase may influence the response of flexible structures in some blast loading situations, without going into any details of the governing parameters in such events. After reviewing the existing methods in representing the negative phase, Rigby et al. [30] discussed the consequences of choosing an inappropriate model by using an elastic SDOF system. It was shown that the dynamic response was highly dependent on an accurate description of the negative part of the pressure-time history. Bryant et al. [31]

used a cubic representation of the negative phase and investigated its influence on the response of blast-loaded reinforced concrete panels and flexible metal wall systems. The negative phase was found to either mitigate or dominate the structural response depending on the timing and pressure magnitude relative to the dynamic response of the structure. Krauthammer and Altenberg [22]

dz

t Type I Type II

Type III

Figure 1.3: Typical response of blast-loaded plates in terms of mid-point deflection-time histories. Dashed lines illustrates the permanent deflections for each type of response.

followed the recommendations in [16] and used a piecewise linear representation of the pressure-time history to investigate the influence of the negative phase on glass panels. Their numerical study indicated that the negative phase dominated the dynamic response for relatively small pressures, i.e., when the peak reflected overpressure was of similar magnitude to the peak negative overpressure. The influence of the negative phase was also found to depend on the dynamic characteristics of the flexible panel relative to the explosive load. In particular, the negative phase was found to dominate the response if it occurred during the elastic rebound. Enhancement of the elastic rebound after maximum deflection was also observed experimentally by Galiev [54] for blast-loaded aluminium plates.

It is evident that depending on the blast intensity the dynamic response of flexible structures may become significantly different. Menkes and Opat [55]

reported failure modes on clamped aluminium beams subjected to blast loading using sheet explosives (see Figure 1.4). By monotonically increasing the impulse they identified three different damage modes, i.e., large inelastic deformation (Mode I in Figure 1.4a), tensile tearing at supports (Mode II in Figure 1.4b) and transverse shear at supports (Mode III in Figure 1.4c). Teeling-Smith and Nurick [56] found the same failure modes for clamped circular plates subjected to impulsive loading, and reported that the magnitude and shape of the deformed plates depend on the intensity of the loading. These failure modes were also observed for square plates by Olson et al. [57]. However, a slight change in the interpretation was needed to account for tensile tearing at the supports as failure was first observed at the centre of the boundary before progressing towards the corners with increasing impulse. Subsequent work by

1.2. Previous work 11 Nurick et al. [58,59] extended these failure modes by including necking at the boundary for Mode I, and some geometric additions to Mode II by including the amount of tearing at the boundary (called Mode II* in the literature).

Experimental evidence was used to show a significant effect of the boundary conditions when predicting tearing. Similar results were also reported by Wierzbicki and Nurick [60].

(a)

(b)

(c)

Figure 1.4: Failure modes for impulsively loaded beams and plates [55, 59]: (a) Mode I - Large inelastic deformation, (b) Mode II - Tensile tearing at supports and (c) Mode III - Shear failure at supports.

It is observed that the Type I and Type II responses in Figure 1.3 resemble the Mode I response in Figure 1.4a, and that the plate will respond in a ductile manner and experience a permanently deformed shape (see Figure 1.4a) when subjected to imposed velocities less than a certain value (see Jones [61]).

However, when the imposed velocities are equal to this critical value, the plate will fail due to tearing at the supports (see Figure 1.4b). If the impulsive velocities are further increased beyond this critical value, failure will occur and the plastic deformation of the plate will become more localized near the supports until another critical velocity is reached. At this second critical velocity transverse shear failure will occur at the supports (see Figure 1.4c).

Thus, as the blast intensity increases and the loading becomes increasingly

impulsive, structural elements exposed to intensive blast loading must have adequate shear capacity to ensure that they do not fail in Mode III if Mode I or Mode II failure is required.

Even though blast events often involve plated structures with holes, the current literature indicates that there is limited research on the influence of holes on the dynamic response of these types of structures. Holes may be induced in plated structures for various applications and reasons, e.g. due to perforations in combined blast and fragmentation events [62] and pre-formed holes in façade systems or passive mitigation systems [63,64]. Rakvåg et al. [65] investigated the response of medium-strength steel plates exposed to pressure pulse loading and the influence of pre-formed holes with different geometries. Although the plates experienced large deformations, there were no signs of failure other than some localization of plastic strain in the extremities of the holes. Schleyer et al. [66]

also studied the inelastic deformation of mild steel plates with pre-formed holes subjected to a pressure pulse loading, and evaluated the capabilities of energy-based analytical solutions in predicting the experimental observations.

The predictions by the approximate methods were found to be in acceptable agreement with the experimental data, which indicated that such methods may provide design guidelines for blast-loaded plates. Veldman et al. [67] studied the response of pre-pressurized aluminium plates subjected to blast loading. The test panels were reinforced with aluminium extrusions attached to the plates using rivet joints. Consistent failure of the rivet joints motivated tests without reinforcement and only the drilled holes at the rivet locations. It was found that the crack initiation emerged from these holes due to stress concentrations.

Complex failure patterns were also observed for thin steel sheets supported on a cellular metal foundation [68], where the cracks originated at laser welded joints.