The effect of stand-off distance on the failure of fully clamped circular mild steel plates subjected to blast loads

Engineering Structures 29 (2007) 2723–2736 www.elsevier.com/locate/engstruct

The effect of stand-off distance on the failure of fully clamped circular mild steel plates subjected to blast loads N. Jacob, G.N. Nurick, G.S. Langdon ∗ Blast Impact and Survivability Research Unit (BISRU), Department of Mechanical Engineering, University of Cape Town, Rondebosch 7701, South Africa Received 6 July 2006; received in revised form 24 January 2007; accepted 24 January 2007 Available online 6 March 2007

Abstract The effect of stand-off distance and charge mass on the response of fully clamped circular mild steel plates, of radius 53 mm, subjected to blast loads travelling along tubular structures is reported. The procedure consists of creating a blast load using plastic explosive mounted onto the end of mild steel tubes. The stand-off distance is varied, from 13 to 300 mm, using different tube lengths. The plate responses range from large inelastic deformation to complete tearing at the boundary. Different loading regimes are observed, depending on the stand-off distance between the explosive charge and the plate, and are classified according to the plate response. At stand-off distances less than the plate radius (13–40 mm), the blast load is considered to be focused (localized). For stand-off distances greater than the plate radius (100–300 mm), the loading is considered uniformly distributed over the entire plate area. Theoretical and empirical analyses are performed, to enable predictions of the mid-point deflection. Appropriate modifications are introduced to account for the effect of stand-off distance on plate deformation. The modified analyses show satisfactory correlation with experimental results. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Blast loading; Scaling; Plates

1. Introduction The response of fully clamped metal plates subjected to uniform and localized blast loads has been studied for many years. Experimental work on beams, plates and shells has been widely reported. Nurick and Martin [1] presented an overview of theoretical and experimental results, up to 1989, that dealt mostly with uniformly loaded plates. In subsequent years, mild steel plates subjected to localized blast loads are reported by Nurick et al. [2–4]. The failure of circular plates subjected to uniform blast loads was investigated by Teeling-Smith and Nurick [5]. It was reported that permanent midpoint deflection increased with increasing impulse, resulting in thinning at the boundary. Further increases in impulse led to partial tearing along the plate boundary, followed by complete tearing. The mid-point deflection decreased as impulse was increased beyond the threshold of complete tearing, as the failure tended towards ∗ Corresponding author. Tel.: +27 0 21 650 5347.

E-mail address: [email protected] (G.S. Langdon). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.01.021

complete shear at the boundary edge. The effect of different edge fixations on mild steel plates subjected to uniform blast loads was reported by Thomas and Nurick [6]. Fully clamped plates were compared to built-in plates machined from 20 mm thick steel. For plates without tearing, the mid-point deflections were identical, that is within experimental error, regardless of edge fixation. Tearing along the boundary occurred at lower impulses for built-in plates compared to clamped plates, as the clamps did not fully prevent in-plane movement of the plate and hence delayed tearing. Schleyer et al. [7] also examined the effect of boundary clamping on the response of mild steel plates subjected to dynamic loading. The loading took the form of a triangular pressure pulse applied over 50 ms and was not impulsive considering the long load duration with respect to the natural period of the test plates. In-plane movement along the clamped edge was observed, confirming the observations reported in [6]. Nurick et al. [8] investigated the onset of thinning for different diameter circular mild steel plates clamped between frames with different edge conditions. The clamps featured sharp edges or fillet radii of 1.5 and 3.2 mm. Observations from

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Notation A0 D H I M0 R R0 S V0 h λ Φc ζc γ µ δ ρ σd σ0 σ01

Loaded area Plate diameter Plate thickness Impulse σ0 H 2 /4 Plate radius Charge radius Stand-off distance Initial velocity Charge height Jones’ dimensionless number Nurick and Martin damage number (circular plates) Loading parameter (circular plates) Stand-off distance parameter Mass per unit area (ρ H ) Midpoint deflection Material density Damage stress Static yield stress of material Dynamic yield stress of material

the experiments showed that thinning occurred for all plate diameters with sharp edged clamps, however, plates secured using frames with fillets allowed larger deflections to occur before the onset of thinning and tearing. Plates restrained with sharp edged clamps exhibit sharp indentation within the necked region due to the clamp followed thereafter by stretching and thinning. In the case of curved edge boundary, the thinning is similar to that observed in a uniaxial tensile test. Experiments on fully clamped circular mild steel plates subjected to localized blast loads were reported by Nurick and Radford [2]. The plate deformation was characterised by an inner dome superimposed on a larger global dome, later reported in Reference [3] for built-in circular plates and [4,9] for quadrangular plates subjected to localised loads. At higher impulses, thinning at the central area and boundary of the plate was observed. Tearing in the central area of the plate occurred with further increases in impulse after the onset of thinning. The tearing observed was characterized by a cap torn away from the plate. Tearing at the boundary was observed for larger load diameter — plate diameter ratios. All of the work by Nurick et al. [2–6,8,9] was performed using plastic explosive sited 13 mm away from the plate structure, using a polystyrene pad to prevent spalling. The effect of stand-off distance was not considered. Although there are numerous empirical relationships relating stand-off distance to blast overpressure for free-field explosions, the relationship between stand-off distance and plate deformation due to air blast is not widely reported. There are no established relationships for predicting either the response of plate or the characteristics of the blast loading when the explosion is partially confined.

Experimental studies using air blast loading to understand the effect of stand-off distance on plate deformation were reported by Akus and Yildirim [10]. Square steel plates were loaded with air pressure waves generated by detonating C4 explosive at different stand-off distances. The results indicated that the maximum permanent midpoint deflection occurred at the closest stand-off distance, followed by a rapid decrease as stand-off distance increased until an asymptote value was reached (for a given charge mass). Marchand and Alfawakhiri [11] examined the loading on steel buildings subjected to explosions and suggested a guide for the assumption of a uniform blast load over a structure. It was assumed that if the stand-off distance exceeded one-half of the largest dimension of the structure, then the loads could be reasonably averaged over the structure (provided the charge is centred on the structure). Despite these two studies [10,11] there remains a paucity of systematically generated experimental data demonstrating the effect of stand-off distance on structural loading and response. This paper presents new experimental results for fully clamped circular plates subjected to blast loads detonated at various stand-off distances. The stand-off distances vary from 13 to 300 mm, while the plate and load geometry are kept constant. The range of stand-off distances provides a spectrum of experimental data that is used to study the effect of standoff on plate response. The paper also proposes a modification to existing dimensionless analysis to incorporate the effect of stand off distance into the response of blast loaded circular plates. This work will be useful to those involved in research into the response of structures to explosive loading, a subject that has become increasingly important with heightened public awareness of potential explosive threats to civilian safety. It is also applicable to the study of explosion loading of underground facilities linked to the atmosphere via tunnels. 2. Failure modes of thin plates subjected to blast loads It is important when investigating the effect of stand-off distance upon plate response that load localization causes a change in failure mode. This section describes the failure modes observed in blast-loaded structures. The failure modes of structures subjected to uniformly distributed blast loading were first classified by Menkes and Opat [12] for beams, as shown in Fig. 1. The different modes of failure were defined as: large inelastic deformation of the entire beam (Mode I), tearing (tensile failure) at the supports (Mode II) and transverse shear failure of the beam at the supports (Mode III). Similar failure modes were observed by TeelingSmith and Nurick [5] for circular plates subjected to uniform blast loads and for square plates by Nurick and Shave [9]. Nurick, Gelman and Marshall [8] redefined Mode I failure to describe necking partially (Mode Ia) and completely (Mode Ib) around the boundary. Nurick and Shave [9] redefined Mode II failure to describe partial tearing along the boundary (Mode II*), failure with increasing mid-point deflection for increasing impulse (Mode IIa) and reducing mid-point deflection with increasing impulse (Mode IIb).

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Table 1 Summary of modes of failure defined for plates subjected to uniform and localized blast loads [2,3,8,9]

(a) Mode I failure — large inelastic deformation.

Modes of failure

Description

Uniform loading

Localized loading

Mode I

Large inelastic deformation Large inelastic deformation with necking around part of the boundary Large inelastic deformation with necking around the entire boundary Large inelastic deformation with thinning in the central area Large inelastic deformation with partial tearing around part of the boundary Partial tearing in the central area Tensile tearing at the boundary Tearing with increasing midpoint deflection with increasing impulse with complete tearing at the boundary Tearing with decreasing midpoint deflection with increasing impulse with complete tearing at the boundary Complete tearing in the central area — capping Transverse shear failure at the boundary Tearing at centre with ‘petals’ of material folded away from blast location

3

3

Mode Ia

Mode Ib

Mode Itc Mode II* (b) Mode II failure — tensile tearing at the supports.

Mode II*c Mode II Mode IIa (c) Mode III failure — shear failure at the supports. Mode IIb

Mode IIc Mode III Petalling

(d) Photograph of petalling failure in plates [2].

3

3

3

3 3

3 3

3

3

3

3 3 3

distances. The response of the plates is compared to those loaded uniformly and locally.

Fig. 1. Failure modes of plates and beams.

3. Dimensional analysis The range of failure modes observed for localized blast loading are large inelastic deformation (Mode I) to complete tearing at the boundary (Mode II). Shearing at the boundary (Mode III) was not observed for thin plates, but shear has been observed in thicker plates [3]. The failure modes were further refined in [2] for central localized blast load, as the plate deformation was concentrated in the central (loaded) portion of the plate. At higher impulses, a circular ring of thinning was observed in the central region of the plate (Mode Itc) followed by partial (Mode II*c) and complete tearing (Mode IIc) in the central portion as impulse was increased. If the impulse is further increased beyond the onset of Mode IIc, the plate tears in a “petalling” fashion. A summary of the modes of failure, contrasting locally and uniformly loaded plates, is listed in Table 1. This paper is limited to discussing the Mode I failure of plates subjected to blast loads at various stand-off

Theoretical predictions of plates subjected to impulsive loading have been reported for many years [1]. Jones [13] proposed a dimensionless impulse λ and an analytical solution to predict large inelastic deformation of fully clamped circular plates loaded by a uniformly distributed velocity. Nurick and Martin [1] proposed modified dimensionless numbers for both uniformly and locally loaded circular plates, based on Johnson’s damage number [14]. In this paper, an extension modification to both the Nurick and Martin [1] and Jones [13] dimensionless number is introduced to include a stand-off distance parameter. 4. Experimental procedure The test specimens are 244 mm by 244 mm by 1.9 mm thick. The specimens are clamped in a test rig, comprising two

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N. Jacob et al. / Engineering Structures 29 (2007) 2723–2736 Table 2 Summary of experimental details Test parameters

Details

Plate thickness (mm) Plate diameter (mm) Stand-off distance (mm) Charge diameter (mm) Charge mass (g)

1.9 106 13, 25, 30, 40, 50, 75, 100, 150, 200, 250, 300 34 4, 5, 7, 9, 11, 13, 15

Fig. 2. Photograph of mild steel tubes used in the experiments.

(244 mm × 244 mm) frames made from 20 mm thick mild steel plating. A tube of required length, selected from the range shown in Fig. 2, is screwed onto the front clamping frame and the rear clamping frame has a hole of diameter 106 mm, the same as the internal diameter of the tube as shown in Fig. 3. Therefore, each test specimen has a circular exposed area with a diameter of 106 mm. In the case of the smallest (13 mm) stand-off distance, the explosive charge is placed on a 13 mm thick polystyrene pad mounted directly onto the plate. For stand-off distances ranging from 25 to 300 mm, mild steel tubes of internal diameter 106 mm are used. This technique is different from previous work [2–6,8,9] because it employs tubes to improve the spatial uniformity of the blast loading incident on the target plate. There are no empirical equations relating the influence of stand-off distance to blast loading that is constrained to propagate in this way. The test rig is attached to the ballistic pendulum using four spacer rods. The oscillation amplitude of the pendulum is used to determine the impulse imparted to the plates, given

Fig. 4. Photograph of test rig attached to the ballistic pendulum.

the mass of the pendulum and the geometry in accordance with past work [2–6,8,9]. The rods allow the plate to deform without coming in contact with the I-beam of the pendulum. A photograph of the experimental set-up is shown in Fig. 4. Plastic explosive PE4 is moulded into a disc with a diameter of 34 mm and sited at the open end of the tube, as shown in Fig. 5. A 1 g leader of explosive is used to attach the detonator to the main charge, thus the total mass of explosive is the sum of the disc and the 1 g leader. The inclusive charge masses range from 4 to 15 g. The stand-off distance is equal to the length of the tube. The experimental details are given in Table 2.

Fig. 3. Schematic of the experimental test rig.

N. Jacob et al. / Engineering Structures 29 (2007) 2723–2736

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Table 3 Summary of material properties of mild steel test specimens, obtained from quasi-static tensile tests Density (kg/m3 )

7691

Static yield stress (MPa) Sheet A Sheet B Sheet C Average static yield stress (MPa)

238 241 241 240

Strain at failure Sheet A Sheet B Sheet C Mean average failure strain

0.417 0.406 0.408 0.410

Static ultimate tensile stress (MPa) Sheet A Sheet B Sheet C Mean average (MPa)

357 354 357 356

Fig. 5. Photograph of disc shaped PE4 explosive of diameter 34 mm attached to a circular polystyrene pad of diameter 110 mm and thickness of 13 mm.

The extensive series of experiments required that the blast test specimens be cut from three sheets of steel. Uniaxial tensile tests, for each sheet (A, B and C), are conducted at different quasi-static strain rates of 8.33 × 10−4 , 4.17 × 10−3 and 2.08 × 10−2 s−1 . From this data, the static yield stress is calculated using the Cowper–Symonds relation [13], given in Eq. (1): σ01 =1+ σ0



ε˙ ε˙0

1/η (1)

where σ01 is dynamic yield stress, σ0 is static yield stress, ε˙ is strain rate, ε˙ 0 and η are material constants; typical values for mild steel are given as ε˙ 0 = 40.4 s−1 and η = 5 [13]. A summary of the material properties of the mild steel plates is given in Table 3. It is evident from Table 3 that the static yield stress (σ0 ) and ductility are very consistent across sheets, with a mean average static yield stress of 240 MPa (1% variation) and failure strain of 0.41 (2.5% variation). 5. Results and discussion The experimental data are presented in Table 4. 5.1. Experimental observations In most cases, Mode I type failure (large inelastic deformation) of the plates is observed. Midpoint deflection increases with increasing impulse for a given stand-off distance.

The permanently deformed profiles differ according to the distance between the plastic explosive and the test specimens. The plate profile resembled a uniform dome shape, as shown in Fig. 6, for stand-off distances greater than 100 mm. The profile shape is similar to that reported by Teeling-Smith and Nurick [5] for uniformly loaded circular plates. The load distribution is therefore assumed to be uniform over the plate area for stand-off distance ranging from 100 to 300 mm. In cases of closer stand-off distances (ranging from 13 to 40 mm), the plate profile resembles a smaller inner dome superimposed atop a larger global dome, as shown in Fig. 7. This deformation profile concurs with experimental observations reported by Nurick et al. [2,15] for localized blast loaded circular plates using disc shaped plastic explosive mounted directly onto the test plate. The load distribution is considered as localized to the centre of the plate. The influence of stand-off distance on plate profile shape is illustrated in Fig. 8, for a particular charge mass (5 g). For stand off distances between 100 and 300 mm, the profile shape does not vary (as shown in Fig. 8 by the closed packed nature of the bottom five plates), resembling a large global dome with no evidence of load localization. As stand-off distance is decreased to 50 and 75 mm, increased deformation near the boundary is observed, but without the inner dome at the centre. The measured mid point deflections for stand-off distances of 75 mm and above show little variation for a given charge mass. At stand off distances below 50 mm, an inner dome atop a global dome becomes more evident, the deformation localizes in the plate centre and the midpoint deflection increases with decreasing stand-off distance.

Fig. 6. Photograph of plate profile showing large global dome (Mode I failures). Plate number NJ230405b, I = 12.03 N s, S = 150 mm.

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Table 4 Experimental results Test number

Stand-off distance (mm)

Charge mass (g)

Impulse (N s)

Midpoint deflection (mm)

Midpoint deflection–plate thickness ratio

Failure mode

NJ180405a NJ250405a NJ250405b NJ250405c

13

5 4 5 6

10.99 8.65 10.99 12.44

22.21 18.35 22.54 26.45

11.11 9.18 11.27 13.23

Mode I Mode I Mode I Mode Itc

NJ160405f NJ160405g NJ160405h NJ160405i

25

4 5 7 9

8.68 10.33 14.02 17.20

14.78 17.34 21.67 28.10

7.39 8.67 10.83 14.05

Mode I Mode I Mode Ib Mode II∗ +Mode Itc

NJ160405a NJ160405b NJ160405c NJ160405d NJ160405e

30

5 7 8 9 4

10.51 14.32 15.90 17.16 8.43

15.57 22.77 24.23 23.80 12.64

7.78 11.38 12.12 11.90 6.32

Mode I Mode Ib Mode Ib Mode Ib Mode I

NJ150405a NJ150405b NJ150405c NJ150405d NJ150405e

40

5 7 9 8 4

11.40 14.37 17.97 16.21 8.51

12.34 19.00 22.26 19.61 10.71

6.17 9.50 11.13 9.80 5.35

Mode I Mode Ib Mode Ib Mode Ib Mode I

NJ300405a NJ090405a NJ090405b NJ090405c NJ090405d

50

4 5 7 9 11

9.11 10.59 14.11 17.94 21.17

8.39 9.84 13.26 18.12 21.58

4.19 4.92 6.63 9.06 10.79

Mode I Mode I Mode Ib Mode Ib Mode II

NJ300405b NJ080405a NJ080405b NJ080405c NJ080405d NJ080405e

100

4 5 7 9 11 13

9.57 10.79 14.76 17.97 21.86 25.36

5.92 8.05 11.66 14.48 17.03 19.79

2.96 4.02 5.83 7.24 8.52 9.90

Mode I Mode I Mode I Mode I Mode I Mode Ib

NJ060405a NJ060405b NJ060405c NJ060405d NJ060405e NJ060405f NJ230405a NJ230405b NJ230405c NJ230405d NJ230405e NJ230405f

150

5 7 9 11 13 15 5 5 5 7 7 7

11.83 16.15 19.00 25.19 28.73 32.44 11.47 12.03 10.56 15.30 15.11 15.30

8.31 11.61 14.37 17.18 20.05 – 7.83 8.50 7.87 11.76 12.18 12.28

4.15 5.80 7.18 8.59 10.02 – 3.91 4.25 3.94 5.88 6.09 6.14

Mode I Mode I Mode I Mode I Mode Ib Mode II* Mode I Mode I Mode I Mode I Mode I Mode I

NJ300405c NJ050405a NJ050405b NJ050405c NJ050405d NJ050405e NJ050405f NJ220405a NJ220405b NJ220405c NJ220405d NJ220405e NJ220405f NJ220405g

200

4 5 7 9 11 13 15 5 5 7 5 7 7 7

9.64 13.06 17.35 22.41 25.18 30.69 31.67 11.91 11.67 16.18 11.38 15.66 15.26 14.85

5.96 7.98 11.29 13.55 16.05 18.55 20.01 7.65 7.82 10.72 8.07 11.38 11.14 11.77

2.98 3.99 5.65 6.78 8.02 9.28 10.00 3.82 3.91 5.36 4.04 5.69 5.57 5.89

Mode I Mode I Mode I Mode I Mode Ib Mode Ib Mode Ib Mode I Mode I Mode I Mode I Mode I Mode I Mode I

NJ290305a NJ290305b

250

11 13

24.05 26.33

15.30 17.03

7.65 8.52

Mode Ib Mode Ib

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N. Jacob et al. / Engineering Structures 29 (2007) 2723–2736 Table 4 (continued) Test number

Stand-off distance (mm)

Charge mass (g)

Impulse (N s)

Midpoint deflection (mm)

NJ290305c NJ290305d NJ290305e NJ290305f NJ290305g

250

15 17 16 5 15.5

30.24 29.92 32.10 11.85 33.47

18.92 – – 7.37 19.53

NJ310305a NJ310305b NJ310305c NJ310305d NJ310305e NJ210405a NJ210405b NJ210405c NJ210405d NJ210405e NJ210405f NJ300405d

300

5 11 9 13 15 5 5 5 7 7 7 4

14.95 24.82 21.63 30.24 33.03 13.32 13.20 12.24 16.74 16.30 15.65 10.96

7.32 14.97 13.06 18.17 18.98 7.01 7.38 7.87 10.43 10.27 10.41 5.66

Midpoint deflection–plate thickness ratio 9.46

Failure mode

3.69 9.76

Mode Ib Mode II Mode II* Mode I Mode Ib

3.66 7.49 6.53 9.09 9.49 3.50 3.69 3.94 5.21 5.14 5.20 2.83

Mode I Mode I Mode I Mode Ib Mode Ib Mode I Mode I Mode I Mode I Mode I Mode I Mode I

– –

Fig. 7. Photograph of typical plate profile for smaller stand- off distances (13–40 mm). Plate number NJ250405c, I = 12.44 N s, S = 13 mm (Mode I failures).

Fig. 8. Photograph of sequential layout of test specimens for increasing stand-off distance and constant charge mass, 5 g (Mode I failures). (Plate numbers top to bottom — NJ250405b, NJ160405g, NJ160405a, NJ150405a, NJ090405a, NJ020705b, NJ080405a, NJ230405c, NJ050405a, NJ290305f and NJ210405a).

5.2. Relationship between stand-off distance and midpoint deflection

5.3. Relationship between impulse and midpoint deflection

The measured midpoint deflection, for a given charge mass, decreases for increasing stand-off distance as shown in Fig. 9. This behaviour concurs with previous work on free field explosions, as blast peak overpressure decreases with increasing stand-off distance [16–20]. It is shown in Fig. 9 that, for a specific charge mass, there is a sharp decrease in midpoint deflection with increasing stand-off distance, for stand-off distances ranging from 13 to 50 mm. The percentage difference in midpoint deflection between stand-off distances 25 and 50 mm, is consistently 40% with a variation of ±4%, between charge masses of 4 g and 9 g. The midpoint deflections are similar (within ±1 mm) for stand-off distances ranging from 75 to 300 mm for a given charge mass.

The midpoint deflection increases with increasing impulse for a given stand-off distance as shown in Fig. 10. For standoff distances ranging from 13 to 50 mm there is significant variation in midpoint deflection as the charge is moved closer to the test plate. This behaviour concurs with results reported by Akus and Yildirim [10] for square plates subjected to blast loads of varying charge masses and stand-off distances. The change in midpoint deflection with increasing impulse is greatest for the 13 mm stand-off distance, and decreases with stand-off distance (ranging from 25 to 300 mm). The change is most noticeable in the stand off distance range of 13–50 mm. The loading at stand-off distances below 50 mm is considered to be focused (localized) with plate deformations showing an inner

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Fig. 9. Graph of measured midpoint deflection versus stand-off distance.

dome atop a global dome. The stand-off distance of 50 mm forms the lower bound of localized loading. A transition at stand-off distances 50 and 75 mm indicates a range where plate deformation exhibits aspects of both localized and uniform loading, namely a large global dome (associated with uniform loading) and increased deformation closer to the plate boundary (associated with localised loading). The loading condition within the envelope bounded by stand-off distances 75 and 300 mm is considered uniform loading. 5.4. Dimensional analyses 5.4.1. Empirical analysis Johnson [14] proposed a damage number α for impulsively loaded plates: α=

ρv 2 σd

(2)

where, ρ is material density, v is impact velocity and σd is damage stress. However, Eq. (2) does not consider method of impact, target geometry or boundary conditions. Nurick and Martin [15] introduced modifications that include plate geometry and loading conditions. The modified dimensionless damage number, φc , for circular plates is given as:   I 1 + ln RR0 φc = (3) π R H 2 (ρσ0 )1/2 where I is impulse, σ0 is static yield stress, R is plate radius, R0 is charge radius, ρ is material density, H is thickness, and ζc = 1 + ln RR0 , is referred to as the loading parameter. It should be noted that damage stress is replaced by static yield stress, σ0 . Nurick and Martin [15] proposed an empirical relationship between permanent midpoint deflection–thickness ratio, Hδ , and damage number for circular plates, Eq. (4): δ = 0.425φc . H

(4)

Mid-point deflection–thickness ratio versus dimensionless impulse φc is plotted for the experimental data in Table 4 and shown in Fig. 11. The confidence envelope for Eq. (4) is reported as 90% for ±1 displacement–thickness ratio and 99% for ±2 displacement–thickness ratio, in the range of 16 plate thicknesses for over 100 datapoints [21]. The statistical confidence level is the level at which the actual probability of an event is better than some amount — in this case, the 90% confidence level of ±1 displacement–thickness ratio is applied meaning that there should be a 90% probability that δ H will be within ±1 displacement–thickness ratio of the linear relationship proposed in Eq. (4). It is evident from Fig. 10 that the data points for stand-off distance 13 mm fall within the ±1 plate thickness confidence lines but that the other stand-off distances do not. The results also show a similar distribution to that presented in other work [2,5,9]. Data points for stand-off distances greater than 25 mm do not follow the linear relationship proposed in Eq. (4) which is from Reference [15]. This is because the influence of stand-off distance is not incorporated into the damage number. To account for the effect of stand-off distance on plate response, a stand-off distance parameter ζ S is introduced, and is a function of stand-off distance and charge radius: ζ S = 1 + ln



S R0

 (5)

where, S — stand-off distance, R0 — charge radius. Eq. (3) is rewritten as φcS =

Iγ π R H 2 (ρσ

0)

1/2

(6)

where, φcS — modified dimensionless impulse and γ is new loading parameter:

γ =

1 + ln



R R0

1 + ln



S R0

 .

(7)

N. Jacob et al. / Engineering Structures 29 (2007) 2723–2736

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Fig. 10. Graph of measured midpoint deflection versus impulse, grouped according to stand-off distance (from 13 to 300 mm).

Fig. 11. Graph of measured midpoint deflection–thickness ratio versus Nurick and Martin damage number φc (Eq. (3)), showing the empirical prediction of Eq. (4).

The stand-off distance parameter ζ S is only incorporated into Eq. (3) in cases where the stand-off distance is more than the charge radius (S ≥ R0 ). For S < R0 , Eq. (3) is unchanged. A schematic showing the two criteria for using the modified loading parameter is shown in Fig. 12.   R γ = ζ = 1 + ln for (S < R0 ) R0   1 + ln RR0 ζ   for (S ≥ R0 ) γ = = (8) ζS 1 + ln S R0

where, γ is new loading parameter, incorporating charge (R0 — charge radius) and plate (R — plate radius) geometry and stand-off distance (S). Now it is shown that by applying the conditions described in Eq. (8) (and substituting into Eq. (6)) for the 79 experiments conducted in this investigation, then all the data points plotted in the graph shown in Fig. 13 fall within the ±1 plate thickness

confidence lines of Eq. (4) with a confidence level of 90% (indicating good repeatability, in agreement with [21]). 5.4.2. Theoretical analysis Jones [13] proposed a dimensionless number λ to predict large inelastic deformation of fully clamped circular plates loaded impulsively by a uniformly distributed velocity, V0 . λ=

µV02 R 2 M0 H

(9)

where, µ is mass per unit area (=ρ H ), R is plate radius, M0 = σ0 H 2 /4 and V0 is initial velocity and H is thickness. Jones dimensionless number (Eq. (9)) can be written in terms of impulse: λ=

4I 2 π 2 R 2 H 4 ρσ0

(10)

where I is impulse, σ0 is static yield stress, R is plate radius, ρ is material density and H is thickness.

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(a) Case 1 (S < R0 ).

(b) Case 2 (S ≥ R0 ). Fig. 12. Schematic showing the two loading parameter criteria for Eq. (8).

Fig. 13. Graph of measured midpoint deflection–thickness ratio versus modified dimensionless impulse φcS .

The large permanent transverse displacement of the centre of the plate is given by: δ [1 + 2λ/3]1/2 − 1 = . H 2

(11)

The graph of mid-point deflection–thickness ratio versus λ from Eq. (11) for the data from this investigation is shown in Fig. 14. The results show a separation of data points according to stand-off distance similar to the relationship between midpoint deflection and impulse shown in Fig. 9. It is interesting to note that only the data for stand off distance

of 13 mm correlates with the prediction of Eq. (11). For all other stand offs, Eq. (11) overestimates the midpoint deflection. This is because λ does not incorporate the influence of stand-off distance on plate response. The new loading parameter γ , from Eq. (7) is now incorporated into λ, based on the relationship between λ and φc . Nurick and Martin [15] have shown that, for uniform loading (that is, when ln( RR0 ) is equal to zero):  2 4I 2 I λ= 2 2 4 =4 = 4φc2 π R H ρσ0 π R H 2 (ρσ0 )1/2

(12)

N. Jacob et al. / Engineering Structures 29 (2007) 2723–2736

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Fig. 14. Graph of midpoint deflection–thickness ratio versus Jones dimensionless number λ (Eq. (9)).

Fig. 15. Graph of midpoint deflection–thickness ratio versus modified Jones dimensionless number λ S .

where, φc — Nurick and Martin dimensionless impulse (Eq. (3)). Based on the relationship given by Eq. (10), the loading parameter γ (Eq. (7)) is incorporated into Jones’ dimensionless number as follows: λ S = λγ 2 =

4I 2 γ 2 π 2 R 2 H 4 ρσ0

.

(13)

The analytical solution proposed in [13] shown in Eq. (11) is rewritten as: δ (1 + 2λ S /3)1/2 − 1 = . (14) H 2 It should be noted that the same loading parameter criteria applied to modified Nurick and Martin number is applied to modified Jones dimensionless number. The effect of stand-off distance is only taken into consideration in cases where the stand-off distance is greater than the load radius (S > R0 ). The graph of midpoint deflection versus modified Jones’ number λ S

given in Fig. 15 shows a convergence of all data points onto a curve. The equation of the best fit curve through the data points is given as: δ = 0.22λ0.5 (15) S . H The equation of the best fit curve, Eq. (15), through the data points shown in Fig. 15 concurs satisfactorily with the empirical relationship proposed by Nurick and Martin [15] written in terms of Jones’ dimensionless number, λ  1/2 δ λ = 0.425φc = 0.425 = 0.213λ1/2 . (16) H 4 The predicted midpoint deflection calculated using Eq. (15) significantly over estimates the plate deflection. This is because the analytical solution does not take strain rate sensitivity of mild steel into account. The empirical relationship, Eq. (16), is derived from experimental data obtained from blast loaded mild steel plates that strain harden at high strain rates.

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Fig. 16. Graph of dynamic yield stress versus impulse using Eqs. (17) and (19).

Jones [13] proposes replacing the static yield stress σ0 with the dynamic yield stress σ01 to account for material strain rate sensitivity. Hence Jones’ dimensionless number λ, Eq. (10), is rewritten: λ=

4I 2 π 2 R 2 H 4 σ01 ρ

λ1S =

.

(17) σ01

is given by:


1/2

(18)

= nσ0

where  n =1+

V0 δ √ 3 2ε˙0 R 2

1/η (19)

ε˙0 = 40.4 and η = 5 are material constants for mild steel, assuming the Cowper–Symonds rate sensitivity relationship for mild steel [13]. If Eq. (11) is substituted into Eq. (19) to eliminate deflection, δ, then:   1/η  1/2 V0 H 1 + 2λ − 1 3    (20) n =1+ √   2 6 2ε˙0 R where V0 =

I π R 2 Hρ

4I 2 γ 2 . π 2 R 2 H 4 ρσ01

(22)

Hence the analytical prediction equation (11) is rewritten as:

The dynamic yield stress σ01

The new Jones’ dimensionless number incorporating strain rate sensitivity of mild steel and effect of stand-off distance is written as:

and λ =

4I 2 . π 2 R 2 H 4 σ0 ρ stress σ01 is determined

Hence the dynamic yield using Eqs. (17) and (19). The graph of dynamic yield stress σ01 versus impulse for all tests conducted in this investigation is plotted in Fig. 16. The graph shows that the dynamic yield stress increases with increasing impulse. The equation of the best fit curve through the data points is given by: σ01 = 304I 0.25 where σ01 has units of MPa and I has units of N s.

(21)

1 + 2λ1S /3 δ = H 2

−1

.

(23)

The experimental data are plotted in the graph of midpoint deflection–thickness ratio versus modified Jones’ dimensionless number λ1S as shown in Fig. 17. The experimental results show satisfactory correlation with the analytical solution given by Eq. (22). It should be noted that experimental results for stand-off distance 13 mm fall outside the ±1 deflection/thickness (90%) confidence lines. This is due to the localization of the strain distribution in the plate profile, meaning that the dynamic flow stress is not constant across the whole plate — the value calculated using Eq. (21) is an average for the plate. Eq. (21) will lead to an overestimate of the flow stress for plates with highly localized deformation, lowering the predictions of midpoint deflection, in the same way shown in Fig. 17. The analysis indicates that by incorporating dynamic yield stress and a stand-off distance parameter, the influence of strain rate sensitivity of mild steel and stand-off distance can be sufficiently accounted for with satisfactory correlation between analytical predictions and experimental data. 6. Concluding remarks The experimental results presented in this paper, for a specific specimen and load geometry, provide an insight into the relationship between stand-off distance and response of circular plates subjected to blast loads travelling through tubular structures. It is shown that stand-off distance influences the type

N. Jacob et al. / Engineering Structures 29 (2007) 2723–2736

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Fig. 17. Graph of midpoint deflection–thickness ratio versus modified Jones’ dimensionless number λ1S , showing the experimental data and the empirical prediction of Eq. (23).

of loading condition applied to a structure. This is illustrated by different plate profiles observed at various stand-off distances. An inner dome superimposed on a larger global dome is seen at stand-off distances ranging from 13 to 40 mm indicating a localisation of the blast load on the plate for stand-off distance less than plate radius. In the case of stand-off distance greater than 100 mm, the plate profile is a global dome indicating uniform distribution of the blast load. At stand-off distances of 50 and 75 mm, it appears that a transition in loading condition occurs. This is represented by plate response showing aspects of uniform and localized loading. The results show that midpoint deflections decrease significantly as stand-off distance increases from 13 to 50 mm for a given charge mass. In the case of stand-off distances ranging from 75 to 300 mm, similar midpoint deflections are obtained. A new loading parameter written as function of charge radius, plate radius and stand-off distance is incorporated into empirical and theoretical dimensionless impulse numbers and predictions proposed in the literature [13,15]. The results are plotted in the graph of midpoint deflection–thickness ratio versus modified dimensionless numbers. The experimental data fall within the ±1 deflection–thickness ratio (90%) confidence lines of the empirical relationship proposed in Reference [15] for predicting midpoint deflection. The same loading parameter, γ , along with dynamic yield stress is applied to Jones’ dimensionless number to account for the influence of stand-off distance and strain rate sensitivity of mild steel on plate deformation. The analytical solution proposed by Jones [13] with the modified Jones’ number λ1s shows satisfactory correlation with experimental results. Further work to include different tube diameters (plate specimen sizes) and different load diameters (and tube diameter/load diameter ratios) are in the planning stage. Numerical simulations are underway to attempt to better

understand the phenomena reported herein and those results will be reported in due course. Acknowledgements The authors wish to thank P. Park-Ross, L. Watkins and G. Newins at the University of Cape Town for their technical assistance, and Dr. S. Chung Kim Yuen for his experimental assistance. The financial support of the National Research Foundation (South Africa) and the 1851 Royal Commission (UK) is gratefully acknowledged. References [1] Nurick GN, Martin JB. Deformations of thin plates subjected to impulsive loading — A review; Part I — Theoretical consideration. International Journal of Impact Engineering 1989;8(2):159–70. [2] Nurick GN, Radford AM. Deformation and tearing of clamped circular plates subjected to localised central blast loads. In: Recent developments in computational and applied mechanics: A volume in honour of John B. Martin. Barcelona (Spain): International centre for numerical methods in engineering (CIMNE); 1997. p. 276–301. [3] Chung Kim Yuen S, Nurick GN. The significance of the thickness of a plate when subjected to localised blast loads. In: Proc 16th int symp military aspects of blast and shock. 2000. p. 491–9. [4] Jacob N, Chung Kim Yuen S, Bonorchis D, Nurick GN, Desai SA, Tait D. Quadrangular plates subjected to localised blast loads — An insight into scaling. International Journal of Impact Engineering 2004;30(8–9): 1179–208. [5] Teeling-Smith RG, Nurick GN. The deformation and tearing of circular plates subjected to impulsive loads. International Journal of Impact Engineering 1991;11(1):77–92. [6] Thomas BM, Nurick GN. The effect of boundary conditions on thin plates subjected to impulsive loads. In: Plasticity 1995 — The 5th international symposium on plasticity and its current application. 1995. p. 85–8. [7] Schleyer GK, Hsu SS, White MD, Birch RS. Pulse pressure loading of clamped mild steel plates. International Journal of Impact Engineering 2003;28:223–47.

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[8] Nurick GN, Gelman ME, Marshall NS. Tearing of blast loaded plates with clamped boundary conditions. International Journal of Impact Engineering 1996;18(7–8):803–27. [9] Nurick GN, Shave GC. The deformation and tearing of thin square plates subjected to impulsive loads — An experimental study. International Journal of Impact Engineering 1996;18(1):99–116. [10] Akus Y, Yildirim OR. Effects of changing explosion distance and explosive mass on deformation of shock loaded square plates. In: 11th International conference on machine design and production. 2004. [11] Marchand KA, Alfawakhiri F. Blast and progressive collapse — Facts for steel buildings, vol. 2. Amer Inst Steel Construction Inc.; 2004. [12] Menkes SB, Opat HJ. Tearing and shear failure in explosively loaded clamped beams. Experimental Mechanics 1973;13:480–6. [13] Jones N. Structural impact. Cambridge (UK): Cambridge University Press; 1989. [14] Johnson W. Impact strength of materials. London (UK): Edward Arnold (Pubs) Limited; 1972.

[15] Nurick GN, Martin JB. Deformation of thin plates subjected to impulsive loading — A review; Part II — Experimental studies. International Journal of Impact Engineering 1989;8(2):170–86. [16] Kinney KF. Explosive shocks in air. Macmillan; 1962. [17] Baker WE. Explosions in air. Austin: University of Texas; 1973. [18] Smith PD, Hetherington JG. Blast and ballistic loading of structures. Butterworth and Heinemann; 1994. [19] Structures to resist the effects of accidental explosions. TM5-1300, NAVFAC P-397, AFR 88-22. 1990. [20] Wharton RK, Formby SA, Merrifield R. Air-blast TNT equivalence for a range of commercial blasting explosives. Journal of Hazardous Materials 2000;A79:31–9. [21] Nurick GN. An empirical solution for predicting maximum central deflections of impulsively loaded plates. In: Harding J, editor. Mechanical properties of materials at high rates of strain. Oxford: Inst of Physics; 1989. p. 457–64.