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Deformation and failure of blast-loaded stiffened plates
International Journal of Impact Engineering 24 (2000) 457}474
Deformation and failure of blast-loaded sti!ened plates N.S. Rudrapatna!, R. Vaziri",*, M.D. Olson" !MIL Systems, d200-1150, Morrison Drive, Ottawa, ON, Canada K2H 8S9 "Department of Civil Engineering, The University of British Columbia, Vancouver, BC, Canada V6T 1Z4 Received 23 May 1998; received in revised form 10 November 1999
Abstract Numerical results for clamped, square sti!ened steel plates subjected to blast loading are presented. The "nite element formulation, which includes the e!ects of geometric and material nonlinearities as well as strain rate sensitivity, forms the basis for the numerical analysis. Failure is predicted using an interactive failure criterion comprising bending, tension and transverse shear. A node release algorithm is developed to simulate the progression of rupture. The analysis is continued in the post-failure phase to capture the free #ight deformation of the torn plate. The predicted failure modes for a blast-loaded sti!ened plate are presented and compared with previously published experimental data. Furthermore, the results of the numerical analysis are used to understand the experimentally observed localized tearing of sti!ened plates. ( 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Menkes and Opat [1] conducted experiments on clamped aluminium beams and were the "rst to distinguish the three blast-load induced failure modes: mode I * large inelastic deformation; mode II * tensile tearing at the support and mode III * transverse shear failure. Similar failure modes were also observed in blast loaded circular [2] and square plates [3,4]. Nurick et al. [3] further divided mode II into three sub-modes to account for the geometry of the plate: mode IIH * partial tearing; and modes IIa and IIb * complete tearing of the plate with increasing and decreasing central deformation, respectively. Recently, Rudrapatna modelled these failure modes for square, unsti!ened plates [5,6].
* Corresponding author. Tel.: #1-604-822-2800; fax: #1-604-822-6901. E-mail address: [email protected] (R. Vaziri) 0734-743X/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 4 - 7 4 3 X ( 9 9 ) 0 0 1 7 2 - 4
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Experimental data on sti!ened plates subjected to blast loading of su$cient intensity to cause failure is sparse. Nurick et al. [7] conducted experiments on fully built-in steel sti!ened plates. A series of 89]89]1.6 mm square sti!ened plates; each sti!ened with a single rectangular sti!ener along its centreline was used in the study. Four di!erent sizes of sti!eners (2, 4, 5, and 9 mm deep and common width of 3 mm) were considered in their study. Most of the tests reported were in mode I range. The results indicated that in mode I failure, the midpoint de#ection increases with an increase in the applied impulse. This behaviour, which is quite similar to that of unsti!ened plates [3,5], was true for all the sti!ened plates tested. A limited number of results were reported for mode II failure. Nurick et al. [7] reported that tearing initiated at the middle of the plate boundary parallel to the sti!ener (mode IIB) for plates with 3]2-mm and 3]4-mm sti!eners, while localized tearing of the plate at the sti!ener (mode IIS) was observed in all the 3]9-mm sti!ened plates and in one 3]5-mm sti!ened plate. Some experimental results for mode I failure have also been reported by Houlston and Slater [8] and Nurick [9]. Nurick et al. [7] and Fagnan [10] used a strain-based model to predict mode II failure numerically. The solution used nonlinear "nite element analysis with a plastic hinge model proposed by Jones [11,12] to calculate the maximum strain at the support. They de"ned mode II failure as the instant when the maximum strain reached the rupture strain obtained from a uniaxial static tensile test. Good correlation was reported in mode I failure domain while mode II failure was limited to predicting the initiation of failure only. Furthermore, shear e!ects on failure were neglected in their analysis. The current study is based on a novel approach that incorporates the shear and tensile interaction. The model is not only capable of predicting the onset of mode II failure, but also the failure progression, mode III failure and post-failure analysis, which, to the authors knowledge have not been previously reported in the literature. The objective of this work is to develop a simple design/analysis tool to provide fundamental information on the structural response of sti!ened panels to blast loading.
2. Theory 2.1. Finite element formulation Finite element analysis is performed using a special purpose program called NAPSSE (nonlinear analysis of plate structures using super elements), developed in-house [13,14]. Super plate and beam elements are used to model the large de formation, elastic}plastic transient response of sti!ened plate structures. The displacement "elds of the elements are represented by polynomials as well as continuous analytical functions specially formulated so that only a few elements are needed to model the deformational response of the entire structure. A description of the theoretical basis for modelling the deformation and failure behaviour of unsti!ened plates subjected to blast loads is presented in [5]. Here, we only describe the algorithmic modi"cations that were made to model the response of sti!ened plates. A double-node concept is used to de"ne two sets of nodes at the beam}plate junction, one each for beam and plate. To provide an estimation of the interfacial force, that is essential for predicting
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Fig. 1. Spring model for sti!ened plate structures (quarter-plate model).
the mode IIS failure a series of springs are introduced to connect the beam and plate element nodes (Fig. 1). In a manner similar to that employed for unsti!ened plates [5], the clamped boundary conditions are modelled by introducing a bed of continuously distributed transverse springs along the supported edges of the plate. This provides a simple means of computing the shear force, and hence the transverse shear stresses, along the plate boundaries that would otherwise have been intractable using the Kirchho! plate bending theory incorporated in NAPSSE. Details of this formulation are available in [6]. 2.2. Failure criteria 2.2.1. Mode II failure It is impractical to use "nite elements to model the highly concentrated plastic deformation of the plate near the boundary. A rigorous three-dimensional analysis would be necessary to predict the required information accurately and this is not realistic for preliminary design/analysis work. To simplify the analysis, an approximation is made in determining the bending strain at the boundary. We assume that a plastic hinge forms at the boundary and the rotation of this plastic hinge is equal to the maximum slope of the plate along a line perpendicular to the boundary, which usually occurs at a short distance away from the boundary. The total strain is made up of membrane and bending contributions. For a plate boundary oriented in the y-direction, the total strain in the x-direction can be calculated as [5,15}17]:
PC
A BD
1 Le Lu 1 Lw 2 hh e " # dx# x , x ¸ 2l Lx 2 Lx e 0
(1)
where the "rst term is the membrane strain approximated by averaging the membrane strain over the "rst element adjacent to the boundary, and the second term is the bending strain associated with the rotation of the plastic hinge. In Eq. (1) ¸ is the length of the element in the x-direction, e u and w are the in-plane and out-of-plane displacements of the mid-plane, h is the plate thickness, l is the plastic hinge length, and h is the plastic hinge rotation. The latter is assumed to correspond x to the maximum slope near the boundary of the de#ected plate and is determined by interpolating across each "nite element using the shape functions available in NAPSSE [13].
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In the case of sti!ened plates, plastic hinge lines are assumed to form in the plate along the supported boundary and also along the sti!eners. The hinge rotation at the boundary is taken to be the maximum slope in the plate adjacent to the "xed boundary and the hinge length is assumed to be l"2h. However, for the plastic hinge at the sti!ener, the hinge rotation is taken to be the algebraic di!erence between the analogous slope in the plate and the rotation of the sti!ener. Only the "rst term in Eq. (1) is used to calculate the membrane strain at the plate boundary. It is suggested that the failure in sti!ened plates initiates at the middle of the clamped boundary parallel to the sti!ener and terminates at the sti!ener end (where the sti!ener meets the boundary). This means that the sti!ener end would be the last section to fail. The sti!ener end section can then be treated like a clamped beam for the purposes of failure analysis. A clamped beam is considered to be in a membrane state once the central displacement exceeds the beam thickness. Jones [11,12] calculated the bending strain using a value of l"¸/4, where ¸ is the length of the beam. In the absence of a theory to calculate the plastic hinge length for the sti!ener end section, the above value is used to estimate the in#uence of the bending strain on the failure function. 2.2.2. Mode III failure In the present model, the introduction of transverse springs at the support boundary and at the interface between the plate and the sti!ener provides a direct means of computing the transverse shear forces at these locations. The relevant computational procedures are described in [5,6]. In the case of sti!ened plates, the sti!ener end would be the last section to fail as described before (i.e., the entire structure is supported by the sti!ener end section just before complete failure). The shear contribution towards the failure function needs to be evaluated appropriately, for the successful completion of the failure analysis. The shear stress for the sti!ener end is obtained via the reaction force estimated at the boundary through the overall structural equilibrium and dividing it by the total cross-sectional area. This value of shear stress is used in calculating the failure function. 2.2.3. Modes II}III interaction The two distinct failure models that have been incorporated into NAPSSE are the linear interaction criterion (LIC) and the quadratic interaction criterion (QIC):
G
f,
De6 D#Dq6D, LIC, e6 2#q62, QIC,
(2)
where e6 and q6 are direct strain and transverse shear stress normalized with respect to their ultimate values [5]. Failure is deemed to occur when the failure function f, reaches a value of one. 2.2.4. Post-failure analysis The limited experimental results [7] indicate that the failure of sti!ened plates in mode II (mode IIB-tensile tearing) starts at the middle of the plate boundary parallel to the sti!ener. An increasing portion of the boundary would fail simultaneously with increasing impulse. Therefore, a means of accounting for failure progression is necessary in the failure analysis of sti!ened plates. Since tearing is a very complicated process to model some simplifying assumptions need to be made. In particular, it is assumed that fracture occurs instantaneously through the plate thickness.
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The node release algorithm developed for the failure analysis of unsti!ened square plates [5,6] is further modi"ed to simulate the failure progression in sti!ened plates. In this algorithm, the entire element side is released once failure is identi"ed at the corner and midside nodes of that element (i.e., when failure function f"1). Failure progression is simulated by successive releasing of elements as failure occurs. Once all the nodes on the boundary are released, the sti!ened plate separates from the boundary and #ies freely. Post-failure analysis involves treating the ruptured plate as a free}free plate with prescribed initial motion. The "xed boundary conditions are replaced by free boundary conditions and the analysis is continued until the deformation of the free #ying plate reaches a steady state.
3. Modelling the experiments A series of di!erent sti!ener sizes are considered in the current study. All sti!eners are rectangular in shape with 3-mm width. Four di!erent depths (H"2, 4, 5 and 9 mm) are used in the analysis. The plates are designated as 3]2-, 3]4-, 3]5- and 3]9-mm sti!ened plate for all future reference, in view of sti!ener sizes. The con"guration of the plate and the "nite element model used for the analysis are shown in Fig. 2. The boundaries of the sti!ened plate were assumed to be rigidly clamped and various "nite element grids were used to represent a quarter of the plate due to symmetry. The material parameters used are as follows [7]: elastic modulus, E"197 GPa; tangent modulus, E "250 MPa; Poisson's ratio, l"0.3; static yield stress, p "265 MPa; rupture strain, T 0 e "0.18; density, o"7830 kg/m3; static ultimate shear stress, q "178.8 MPa. 361 6-5 To compare with the available experimental results [7] blast wave loading is considered. The pressure loading from the explosive charge is assumed to be a rectangular pulse uniformly distributed over the plate surface. The duration of loading is assumed to be 15 ls, equal to the approximate explosive burn time. The pressure intensity is then calculated to correspond to the measured impulse (area under the pressure-time curve).
4. Results and discussion For the purposes of numerical convergence and accuracy, several numerical simulations were carried out using di!erent grid sizes for the quarter plate. Unless stated otherwise, all the results reported here are for a 2]2 grid (Fig. 2b) with a time-step size of 0.5 ls and are considered to be accurate enough for engineering design purposes. 4.1. Mode I A typical mode I time history of the sti!ener central displacement (point C) and kinetic energy in the system is shown in Fig. 3, for a 3]2-mm sti!ened plate under 5 Ns impulse. The central deformation increases monotonically and reaches a peak value around 120}125 ls. This time to peak displacement is slightly lower than that of an unsti!ened plate of the same size which occurs around 135}140 ls [5]. All the input energy is absorbed at a time which coincides with the time to
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Fig. 2. Con"guration and "nite element model of a one-way sti!ened plate: (a) con"guration of plate; and (b) "nite element model of quarter-plate (2]2 grid).
reach peak displacement. The plate vibrates in an elastic manner about the permanent deformation position once energy absorption is complete. Temporal variations of strain ratio, stress ratio and failure function (e6 , q6 and f ), for the midpoint of the boundary parallel to the sti!ener and for the sti!ener centre (points A and C in Fig. 2b) are shown in Figs. 4 and 5, respectively. The results are for a 3]2-mm sti!ened plate under 5 Ns impulse and is typical for impulses which cause mode I deformation. The strain ratio at A increases monotonically to a peak value around 90 ls and remains more or less constant thereafter, although deformation continues. The time to peak strain is less than the time to reach peak displacement. This is true with all sti!ener sizes.
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Fig. 3. Time history of central displacement and kinetic energy of 3]2-mm sti!ened plate in mode I failure.
Fig. 4. Strain ratio, stress ratio and failure function variation with time at the midpoint of the boundary parallel to the sti!ener.
The stress ratio at A increases until 15 ls and drops when the applied load drops to zero. It reaches another lower maximum when the inertial forces in the system become signi"cant. Thereafter, it drops and oscillates around the zero mean value. The corresponding variation of failure function at A is also shown in Fig. 4. An initial sharp peak occurs at the end of the loading phase due to high initial shear stress ratio. The failure function drops when the applied load drops to zero. It increases again and reaches a maximum value when the inertial force of the system is substantial. At this stage, the plate has undergone a signi"cant amount of stretching and bending, thus contributing to the failure function via the strain ratio. The time to peak failure function occurs around 90 ls, which also coincides with peak strain. This time to peak failure function and the relative sizes of these peaks change with the applied impulse but remains the same for all sti!ener sizes.
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Fig. 5. Strain ratio, stress ratio and failure function variation with time at the sti!ener centre.
Fig. 6. Comparison of strain ratio and stress ratio at the threshold impulse to mode II failure under uniformly distributed load.
Similar variation of these parameters with time for the sti!ener centre (point C) is shown in Fig. 5. Both strain and stress ratios have much smaller magnitudes compared to their corresponding values at the boundary point. This is true with all sti!ener sizes under the uniformly applied load. 4.2. Mode II As the applied impulse is increased, the threshold impulse to cause rupture of sti!ened plates is reached. The threshold impulses (I H ) are 9.5 and 13.3 Ns according to LIC and QIC predictions, m2 respectively, using a 2]2 grid for both analyses. The changes in strain ratio and stress ratio at A and C as a function of sti!ener depth are shown in Fig. 6. Experimental investigation [7] showed
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Fig. 7. Comparison of failure function at the threshold impulse to mode II failure under uniformly distributed load using LIC and QIC.
that failure initiates at either of these points for di!erent sti!ened plates. Hence, these points are chosen to study the rupture initiation numerically. The plotted values at these two locations correspond to those obtained just prior to the end of the loading phase. Fig. 6 shows that although there is some change in the strain ratio with a change in the sti!ener size, these changes are insigni"cant. However, the stress ratio exhibits a much larger change with sti!ener size. For the same loading, a 3]9-mm sti!ened plate develops 25% more shear force at the sti!ener/plate interface than a 3]2-mm sti!ened plate at the same location. Fig. 7 shows the corresponding variation of the failure function with sti!ener depth at the threshold impulse to failure using LIC and QIC. The values shown correspond to the time just before the end of the loading phase and hence do not approach a value of one. From the "gure it is clear that the failure function at A has a much higher value than that at C. The failure function at A shows a slight increase with sti!ener depth, while the function at C increases monotonically. The gap between them decreases linearly with the increase in sti!ener depth. Table 1a and b present the time to "rst element failure (t ), strain ratio (e6 ), stress ratio (q6) and */ displacement at the sti!ener centre (* ) at the time of "rst failure for all sti!ened plates. #* The applied impulse is just enough to cause mode II failure in all the plates. The failure starts at the middle of the boundary parallel to the sti!ener (mode IIB) for all the sti!ened plates. From the table, it is evident that the mode II threshold is independent of sti!ener size. Also, the contribution of strain and stress ratios to the failure function is almost the same for all sizes of sti!eners. Although the time to failure is also the same, the central displacement at failure decreases monotonically with the sti!ener size. The failure at this impulse occurs due to a combination of stress and strain ratios much after the loading phase is over. This is typical of mode II failure which was also observed in the unsti!ened plates [5]. This strain domination is evident when the strain contribution to reach failure is observed which is characteristic of mode II failure. However, the failure function also has
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Table 1 Stress ratio, strain ratio, time to "rst element failure and sti!ener central displacement at the threshold impulse to mode II failure for all sti!ener sizes using a 2]2 grid I H (Ns) .2
q6
e6
t
(a) LIC 2 4 5 9
9.5 9.5 9.5 9.5
0.427 0.460 0.469 0.466
0.574 0.540 0.536 0.537
61.0 57.5 56.0 54.5
4.48 3.82 3.49 2.50
(b) QIC 2 4 5 9
13.3 13.3 13.3 13.3
0.553 0.604 0.607 0.600
0.835 0.798 0.796 0.801
64.0 55.5 54.5 53.0
6.64 5.05 4.66 3.51
H (mm)
*/
(ls)
* (mm) #*
Table 2 Stress ratio, strain ratio, time to "rst element failure and sti!ener central displacement at the threshold impulse to mode II failure for all sti!ener sizes using a 3x3 grid and LIC for failure prediction H (mm)
I H (Ns) .2
q6
e6
t
2 4 5 9
9.5 9.5 9.5 9.5
0.462 0.467 0.465 0.455
0.544 0.540 0.564 0.546
52.5 51.5 51.5 52.0
*/
(ls)
* (mm) #* 3.67 3.26 3.08 2.31
signi"cant contribution from stress ratio at the threshold impulse (46% of failure function is due to the shear stress ratio at the threshold impulse in sti!ened plates, as opposed to 50% in an unsti!ened plate [5,6] when LIC is used for the failure analysis). A purely strain based criterion would not capture this signi"cant e!ect and thus lead to a higher threshold to failure [7]. For comparison, the results in Table 2 were obtained using a 3]3 grid under 9.5 Ns of impulse, and LIC to detect rupture. 4.3. Post-failure analysis For illustration purposes, only the results for the 3]2- and 3]9-mm sti!ened plates are presented. The post-failure analysis of the plate with other sti!ener sizes is left for future study. Once su$cient experimental evidence is gathered, they can be used to correlate with the predicted results. The results reported here are obtained using QIC unless otherwise mentioned. 4.3.1. Mode IIH Fig. 8 shows the time history of central displacement, side pull-in and the kinetic energy of a 3]2-mm sti!ened plate under 13.3 Ns impulse. The failure "rst occurs at the midpoint on the
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Fig. 8. Time history of central displacement, side pull-in and kinetic energy of a 3]2-mm sti!ened plate in mode IIH failure.
Fig. 9. Central displacement, side pull-in and kinetic energy versus time for a 3]2-mm sti!ened plate in mode IIH failure.
boundary parallel to the sti!ener. Subsequently, the element side is released and the analysis is continued. The remaining energy is absorbed by the plate with no further failure. Thus, a partial failure regime is identi"ed. A plot of the central displacement, side pull-in and the kinetic energy versus time is shown in Fig. 9 for an applied impulse of 14.1 Ns. The failure starts at A (shown by the solid circle in the "gure). Subsequently failure propagates on the same side of the sti!ened plate (AO in Fig. 2b) and then switches to the boundary adjacent to the sti!ener (OB in Fig. 2b). These failure initiations are shown by the solid diamond and triangle symbols, respectively. The remaining element sides are released and the sti!ener end only (point B in Fig. 2b) supports the plate. The input energy is not su$cient to cause complete failure. Figs. 10a and b show the predicted 3-D pro"le of the deformed
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Fig. 10. 3-D deformation pro"le of 3]2-mm sti!ened plate in mode IIH failure (two views of the deformed plate computed at an impulse of 14.1 Ns).
plate. The two views of the plate clearly show that the sti!ener end remains attached to the boundary. 4.3.2. Mode IIa The temporal variation of the sti!ener centre displacement, side pull-in and the kinetic energy of the system are shown in Fig. 11 for the 3]2-mm sti!ened plate under an applied impulse of 16 Ns. The same trend continues at higher loads with failure always starting at the middle of the boundary parallel to the sti!ener (point A), and terminating at B. The solid symbols in the "gure are the instances at which each of the failure occurs. At this stage, all the nodes are released and the plate separates from the boundary. Temporal integration is further carried out to account for any deformation occurring during the free-#ight of the torn plate until a steady state is reached. The plate enters modes IIb and III failure regimes, as the applied impulse is increased. The failure is dominated by shear in these two failure modes, which is re#ected by the insigni"cant deformation of the plate.
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Fig. 11. Temporal variation of central displacement, side pull-in and kinetic energy of 3]2-mm sti!ened plate in mode IIa failure.
Fig. 12. Central displacement versus impulse for 3]2-mm sti!ened plate using QIC.
4.4. Permanent central displacement A plot of the sti!ener central displacement versus impulse is shown in Figs. 12 and 13 respectively, for the 3]2- and 3]9-mm sti!ened plates along with the experimental data [7]. Overall, the plate with thick sti!ener exhibits less permanent displacement compared to the plate with thin sti!ener. The predicted mode I displacements match well with the experimental data in the case of 3]2-mm sti!ened plates. The predicted mode II failure always occurred at the boundary with the initiation site being at point A for both plates. Only one experiment with 3]2-mm sti!ened plate showed rupture at the
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Fig. 13. Central displacement versus impulse for 3]9-mm sti!ened plate using QIC.
boundary parallel to the sti!ener, while no results are available for the 3]9-mm sti!ened plate and the results do not o!er a comprehensive comparison with the numerical predictions. Experimentally, mode II failure in 3]2-mm sti!ened plates occurred around 14}15 Ns impulse. This value compares favourably with the results of 13.3 Ns obtained using the QIC. The predicted threshold impulse for mode IIa is around 15 Ns. There is a monotonic increase in the sti!ener central displacement with the increase in impulse until the threshold impulse to mode II failure is reached. Similar to the unsti!ened plates, the predicted central displacement of sti!ened plates also show an increasing trend for mode IIH, mode IIa and decreasing trend for mode IIb. 4.5. Non-uniform load distribution The numerical analysis always predicted failure starting at the middle of the clamped boundary parallel to the sti!ener (mode IIB) for all the sti!ened plates analysed. This matched well with the experimental results for plates with thin sti!eners (2 and 4 mm deep). However, for a plate with deep sti!ener (5 and 9 mm deep) the experiments showed localized tearing at the sti!ener/plate interface [7]. In an attempt to explain this discrepancy, it is worth noting that in the experiments the explosive was laid out on a 12-mm thick polystyrene pad in two concentric square annuli, which were interconnected by two perpendicular strips of explosive called cross-leaders. A short tail of explosive holding the detonator was then attached to the centre of the cross-leaders. The mass of explosive for the tail and the cross-leaders was constant for all the tests. Increasing only the mass of the annuli explosive increased the impulse that was applied to the #at side of the plate. Since there is an extra mass of explosive along with the detonator at the centre of the plate, it was thought that this might have introduced nonuniformity in the load distribution. Therefore, a series of analyses with a nonuniform load distribution over the plate were undertaken.
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Fig. 14. Variation of failure function with sti!ener depth under non-uniformly distributed load (Nu-2).
Two di!erent nonuniform load distributions are considered. The element next to plate centre is subjected to a higher load magnitude compared to the remaining elements. Speci"cally, in a 2]2 grid (or four elements), the element next to the plate centre is subjected to twice the load applied to the other three elements while in the case of a 3]3 grid (or nine elements), the load applied to the element next to the plate centre is three times that applied to the other eight elements. However, in so doing the total impulse applied to the structure is kept the same. These are termed as Nu-2 and Nu-3, respectively, depending on whether the central element is subjected to two or three times the load applied to the other elements. Once again, the plate response is computed over a range of impulses for all sti!ener sizes. 4.5.1. Mode II Fig. 14 shows the variation of failure function as a function of the sti!ener depth under the non-uniform loading Nu-2. The plot is for a load corresponding to 5 Ns impulse and is analysed using a 2]2 grid and LIC. In addition, Fig. 14 also shows the changes in failure function with sti!ener depth for the "ner grid (3]3). The impulse is typical of that causing mode I failure. The values correspond to the time just before the end of the loading phase. The failure function at the midpoint on boundary (A) in both "gures remains almost constant, thus independent of sti!ener size. The same is not true with the failure function at the sti!ener centre (C). The localized load tends to increase the shear stress at the sti!ener centre. This e!ect is more pronounced with thick sti!eners (3]9-mm sti!ener) where the shear stress at the centre is more than that at the boundary. The crossover occurs around 7 and 6 mm sti!ener depth for the coarse and "ne grid, respectively, with load distribution Nu-2. Even though the failure function at the sti!ener centre is higher, it is well below the threshold to cause failure under the applied load (5 Ns). If the applied load is increased then failure is identi"ed either at the boundary (A) or at the sti!ener centre (C) depending on which of the failure functions reaches a value of one "rst. Figs. 15 and 16 show the variation of permanent central displacement versus impulse for 3]5and 3]9-mm sti!ened plates, respectively, under both uniform and nonuniform load distributions.
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Fig. 15. Comparison of sti!ener central displacement versus impulse for 3]5-mm sti!ened plate under various spatial load distributions.
Fig. 16. Comparison of sti!ener central displacement versus impulse for 3]9-mm sti!ened plate under various spatial load distributions.
The importance of load distribution is evident from the results. In the above two "gures, the nonuniform load distribution leads to a higher permanent displacement than a corresponding uniform load distribution. The threshold to cause mode II failure is reached as the applied impulse is increased. The threshold is 13 Ns using QIC for all sti!ened plates with load distribution Nu-2. The threshold impulse to cause mode II failure is, once again, the same for all sti!ener sizes, and failure at the boundary (point A) is predicted at this impulse. However, a transition occurs, wherein failure starts as localized tearing of the plate at the sti!ener centre with load distribution Nu-3 for plates with deep sti!eners (5 and 9 mm deep).
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The localized tearing of the plate at the sti!ener centre (mode IIS) is predicted for the 3]5- and 3]9-mm sti!ened plates at 16 and 10 Ns, respectively, using LIC. This matches well with the experimental observation (solid diamond symbols in Figs. 15 and 16). Since the aim of the nonuniform load analysis was to determine the threshold impulse at which failure mode transition occurred and thus explain the observed experimental phenomena, post-failure analysis was not carried out in this case. 5. Conclusions A numerical study on the failure of blast loaded, clamped, square, sti!ened plates has been presented. The nonuniform variation of shear force (and hence shear stresses) along the boundary is captured using "ctitious transverse spring supports. This enabled the model to consider the e!ect of shear stresses on the failure. The predicted results compare well with the available experimental data and clearly demonstrate the in#uence of shear on the failure mechanism not only for mode III, but also for mode II. The results con"rm the importance of the interaction e!ects of tensile and bending strain on tearing and shear failure. Through the use of springs at the sti!ener/plate interface the model captures the in#uence of the interfacial shear force on the localized tearing failure mode. The predicted impulse for the onset of failure at the boundary (mode IIB) is found to be independent of the sti!ener size. The e!ect of larger sti!ener is only in reducing the magnitude of plate displacement. Under uniform load distribution, tearing starts at the middle of the clamped boundary parallel to the sti!ener for all sti!ener sizes. The lack of comprehensive experimental data does not allow for a comparison of predicted results. However, the results do show the expected qualitative behaviour. The spatial distribution of the blast load plays a signi"cant role in determining the failure mode as identi"ed by the model. An apparent transition of failure from the middle of the clamped boundary to the localized tearing at the sti!ener centre shows the sensitivity of the plate response to load distribution. The model allows parametric investigations to determine the e!ects of changing panel dimensions, sti!ener sizes and shapes, material properties and rates of loading. The model and structural support philosophy that it embraces can be developed further to analyse unsti!ened and sti!ened shell structures, as well as plates with multiple sti!eners. Acknowledgements This research has been supported by the Canadian Department of National Defence, through a contract from the Defence Research Establishment Su$eld, Alberta, and the Natural Sciences and Engineering Research Council of Canada.
References [1] Menkes SB, Opat HJ. Tearing and shear failures in explosively loaded clamped beams. Exp Mech 1973;13:480}6. [2] Teeling-Smith RG, Nurick GN. The deformation and tearing of thin circular plates subjected to impulsive loads. Int J Impact Engng 1991;11(1):77}91.
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[3] Nurick GN, Shave GC. The deformation and tearing of thin square plates subjected to impulsive loads * an experimental study. Int J Impact Engng 1996;18(1):99}116. [4] Olson MD, Nurick GN, Fagnan JR. Deformation and rupture of blast loaded square plates * predictions and experiments. Int J Impact Engng 1993;13(2):279}91. [5] Rudrapatna NS, Vaziri R, Olson MD. Deformation and failure of blast-loaded square plates. Int J Impact Engng 1999;22(4):449}67. [6] Rudrapatna NS. Analysis of the deformation and failure of blast loaded unsti!ened and sti!ened plates. Ph.D. Thesis, Department of Civil Engineering. The University of British Columbia, Vancouver, Canada, 1997. [7] Nurick GN, Olson MD, Fagnan JR, Levin A. Deformation and tearing of blast loaded sti!ened square plates. Int J Impact Engng 1995;16(2):273}91. [8] Houlston R, Slater JE. A summary of experimental results on square plates and sti!ened panels subjected to air-blast loading. Presented at the 57 Shock and Vibration Symposium, New Orleans, Louisiana, USA, 1986. p. 14}16. [9] Nurick GN, Conolly AG. Response of clamped single and double sti!ened rectangular plates subjected to blast loads. In: Bulson PS, editor. Structures under shock and impact III. Southampton, UK: Computational Mechanics Publications, 1994. p. 207}23. [10] Fagnan JR, 1996. Failure analysis of sti!ened and unsti!ened mild steel plates subjected to blast loading. Ph.D. Thesis, Department of Civil Engineering. The University of British Columbia, Vancouver, Canada. [11] Jones N. Plastic failure of ductile beams loaded dynamically. J Engng Ind Trans ASME 1976;98:131}6. [12] Jones N. On the dynamic inelastic failure of beams. In: Wierzbicki T, Jones N, editors. Structural failure. New York: Wiley, 1989. p. 133}59. [13] Koko TS. Super "nite elements for nonlinear static and dynamic analysis of sti!ened plate structures. Ph.D. Thesis, Department of Civil Engineering. The University of British Columbia, Vancouver, Canada, 1990. [14] Jiang J, Olson MD. Documentation for program NAPSSE Version 1.0. Department of Civil Engineering, The University of British Columbia, Vancouver, Canada, 1990. [15] Nonaka T. Some interaction e!ects in a problem of plastic beam dynamics, Parts 1}3. J Appl Mech 1967;34:623}43. [16] Jones N, Shen WQ. Criteria for the inelastic rupture of ductile metal beams subjected to large dynamic loads. In: Jones N, Weirzbicki T, editors. Structural crashworthiness and failure. Amsterdam: Elsevier, 1993. p. 131}64. [17] Shen WQ, Jones N. Dynamic response and failure of fully clamped circular plates under impulsive loading. Int J Impact Engng 1993;13:259}78.
Deformation and failure of blast-loaded sti!ened plates N.S. Rudrapatna!, R. Vaziri",*, M.D. Olson" !MIL Systems, d200-1150, Morrison Drive, Ottawa, ON, Canada K2H 8S9 "Department of Civil Engineering, The University of British Columbia, Vancouver, BC, Canada V6T 1Z4 Received 23 May 1998; received in revised form 10 November 1999
Abstract Numerical results for clamped, square sti!ened steel plates subjected to blast loading are presented. The "nite element formulation, which includes the e!ects of geometric and material nonlinearities as well as strain rate sensitivity, forms the basis for the numerical analysis. Failure is predicted using an interactive failure criterion comprising bending, tension and transverse shear. A node release algorithm is developed to simulate the progression of rupture. The analysis is continued in the post-failure phase to capture the free #ight deformation of the torn plate. The predicted failure modes for a blast-loaded sti!ened plate are presented and compared with previously published experimental data. Furthermore, the results of the numerical analysis are used to understand the experimentally observed localized tearing of sti!ened plates. ( 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Menkes and Opat [1] conducted experiments on clamped aluminium beams and were the "rst to distinguish the three blast-load induced failure modes: mode I * large inelastic deformation; mode II * tensile tearing at the support and mode III * transverse shear failure. Similar failure modes were also observed in blast loaded circular [2] and square plates [3,4]. Nurick et al. [3] further divided mode II into three sub-modes to account for the geometry of the plate: mode IIH * partial tearing; and modes IIa and IIb * complete tearing of the plate with increasing and decreasing central deformation, respectively. Recently, Rudrapatna modelled these failure modes for square, unsti!ened plates [5,6].
* Corresponding author. Tel.: #1-604-822-2800; fax: #1-604-822-6901. E-mail address: [email protected] (R. Vaziri) 0734-743X/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 4 - 7 4 3 X ( 9 9 ) 0 0 1 7 2 - 4
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Experimental data on sti!ened plates subjected to blast loading of su$cient intensity to cause failure is sparse. Nurick et al. [7] conducted experiments on fully built-in steel sti!ened plates. A series of 89]89]1.6 mm square sti!ened plates; each sti!ened with a single rectangular sti!ener along its centreline was used in the study. Four di!erent sizes of sti!eners (2, 4, 5, and 9 mm deep and common width of 3 mm) were considered in their study. Most of the tests reported were in mode I range. The results indicated that in mode I failure, the midpoint de#ection increases with an increase in the applied impulse. This behaviour, which is quite similar to that of unsti!ened plates [3,5], was true for all the sti!ened plates tested. A limited number of results were reported for mode II failure. Nurick et al. [7] reported that tearing initiated at the middle of the plate boundary parallel to the sti!ener (mode IIB) for plates with 3]2-mm and 3]4-mm sti!eners, while localized tearing of the plate at the sti!ener (mode IIS) was observed in all the 3]9-mm sti!ened plates and in one 3]5-mm sti!ened plate. Some experimental results for mode I failure have also been reported by Houlston and Slater [8] and Nurick [9]. Nurick et al. [7] and Fagnan [10] used a strain-based model to predict mode II failure numerically. The solution used nonlinear "nite element analysis with a plastic hinge model proposed by Jones [11,12] to calculate the maximum strain at the support. They de"ned mode II failure as the instant when the maximum strain reached the rupture strain obtained from a uniaxial static tensile test. Good correlation was reported in mode I failure domain while mode II failure was limited to predicting the initiation of failure only. Furthermore, shear e!ects on failure were neglected in their analysis. The current study is based on a novel approach that incorporates the shear and tensile interaction. The model is not only capable of predicting the onset of mode II failure, but also the failure progression, mode III failure and post-failure analysis, which, to the authors knowledge have not been previously reported in the literature. The objective of this work is to develop a simple design/analysis tool to provide fundamental information on the structural response of sti!ened panels to blast loading.
2. Theory 2.1. Finite element formulation Finite element analysis is performed using a special purpose program called NAPSSE (nonlinear analysis of plate structures using super elements), developed in-house [13,14]. Super plate and beam elements are used to model the large de formation, elastic}plastic transient response of sti!ened plate structures. The displacement "elds of the elements are represented by polynomials as well as continuous analytical functions specially formulated so that only a few elements are needed to model the deformational response of the entire structure. A description of the theoretical basis for modelling the deformation and failure behaviour of unsti!ened plates subjected to blast loads is presented in [5]. Here, we only describe the algorithmic modi"cations that were made to model the response of sti!ened plates. A double-node concept is used to de"ne two sets of nodes at the beam}plate junction, one each for beam and plate. To provide an estimation of the interfacial force, that is essential for predicting
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Fig. 1. Spring model for sti!ened plate structures (quarter-plate model).
the mode IIS failure a series of springs are introduced to connect the beam and plate element nodes (Fig. 1). In a manner similar to that employed for unsti!ened plates [5], the clamped boundary conditions are modelled by introducing a bed of continuously distributed transverse springs along the supported edges of the plate. This provides a simple means of computing the shear force, and hence the transverse shear stresses, along the plate boundaries that would otherwise have been intractable using the Kirchho! plate bending theory incorporated in NAPSSE. Details of this formulation are available in [6]. 2.2. Failure criteria 2.2.1. Mode II failure It is impractical to use "nite elements to model the highly concentrated plastic deformation of the plate near the boundary. A rigorous three-dimensional analysis would be necessary to predict the required information accurately and this is not realistic for preliminary design/analysis work. To simplify the analysis, an approximation is made in determining the bending strain at the boundary. We assume that a plastic hinge forms at the boundary and the rotation of this plastic hinge is equal to the maximum slope of the plate along a line perpendicular to the boundary, which usually occurs at a short distance away from the boundary. The total strain is made up of membrane and bending contributions. For a plate boundary oriented in the y-direction, the total strain in the x-direction can be calculated as [5,15}17]:
PC
A BD
1 Le Lu 1 Lw 2 hh e " # dx# x , x ¸ 2l Lx 2 Lx e 0
(1)
where the "rst term is the membrane strain approximated by averaging the membrane strain over the "rst element adjacent to the boundary, and the second term is the bending strain associated with the rotation of the plastic hinge. In Eq. (1) ¸ is the length of the element in the x-direction, e u and w are the in-plane and out-of-plane displacements of the mid-plane, h is the plate thickness, l is the plastic hinge length, and h is the plastic hinge rotation. The latter is assumed to correspond x to the maximum slope near the boundary of the de#ected plate and is determined by interpolating across each "nite element using the shape functions available in NAPSSE [13].
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In the case of sti!ened plates, plastic hinge lines are assumed to form in the plate along the supported boundary and also along the sti!eners. The hinge rotation at the boundary is taken to be the maximum slope in the plate adjacent to the "xed boundary and the hinge length is assumed to be l"2h. However, for the plastic hinge at the sti!ener, the hinge rotation is taken to be the algebraic di!erence between the analogous slope in the plate and the rotation of the sti!ener. Only the "rst term in Eq. (1) is used to calculate the membrane strain at the plate boundary. It is suggested that the failure in sti!ened plates initiates at the middle of the clamped boundary parallel to the sti!ener and terminates at the sti!ener end (where the sti!ener meets the boundary). This means that the sti!ener end would be the last section to fail. The sti!ener end section can then be treated like a clamped beam for the purposes of failure analysis. A clamped beam is considered to be in a membrane state once the central displacement exceeds the beam thickness. Jones [11,12] calculated the bending strain using a value of l"¸/4, where ¸ is the length of the beam. In the absence of a theory to calculate the plastic hinge length for the sti!ener end section, the above value is used to estimate the in#uence of the bending strain on the failure function. 2.2.2. Mode III failure In the present model, the introduction of transverse springs at the support boundary and at the interface between the plate and the sti!ener provides a direct means of computing the transverse shear forces at these locations. The relevant computational procedures are described in [5,6]. In the case of sti!ened plates, the sti!ener end would be the last section to fail as described before (i.e., the entire structure is supported by the sti!ener end section just before complete failure). The shear contribution towards the failure function needs to be evaluated appropriately, for the successful completion of the failure analysis. The shear stress for the sti!ener end is obtained via the reaction force estimated at the boundary through the overall structural equilibrium and dividing it by the total cross-sectional area. This value of shear stress is used in calculating the failure function. 2.2.3. Modes II}III interaction The two distinct failure models that have been incorporated into NAPSSE are the linear interaction criterion (LIC) and the quadratic interaction criterion (QIC):
G
f,
De6 D#Dq6D, LIC, e6 2#q62, QIC,
(2)
where e6 and q6 are direct strain and transverse shear stress normalized with respect to their ultimate values [5]. Failure is deemed to occur when the failure function f, reaches a value of one. 2.2.4. Post-failure analysis The limited experimental results [7] indicate that the failure of sti!ened plates in mode II (mode IIB-tensile tearing) starts at the middle of the plate boundary parallel to the sti!ener. An increasing portion of the boundary would fail simultaneously with increasing impulse. Therefore, a means of accounting for failure progression is necessary in the failure analysis of sti!ened plates. Since tearing is a very complicated process to model some simplifying assumptions need to be made. In particular, it is assumed that fracture occurs instantaneously through the plate thickness.
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The node release algorithm developed for the failure analysis of unsti!ened square plates [5,6] is further modi"ed to simulate the failure progression in sti!ened plates. In this algorithm, the entire element side is released once failure is identi"ed at the corner and midside nodes of that element (i.e., when failure function f"1). Failure progression is simulated by successive releasing of elements as failure occurs. Once all the nodes on the boundary are released, the sti!ened plate separates from the boundary and #ies freely. Post-failure analysis involves treating the ruptured plate as a free}free plate with prescribed initial motion. The "xed boundary conditions are replaced by free boundary conditions and the analysis is continued until the deformation of the free #ying plate reaches a steady state.
3. Modelling the experiments A series of di!erent sti!ener sizes are considered in the current study. All sti!eners are rectangular in shape with 3-mm width. Four di!erent depths (H"2, 4, 5 and 9 mm) are used in the analysis. The plates are designated as 3]2-, 3]4-, 3]5- and 3]9-mm sti!ened plate for all future reference, in view of sti!ener sizes. The con"guration of the plate and the "nite element model used for the analysis are shown in Fig. 2. The boundaries of the sti!ened plate were assumed to be rigidly clamped and various "nite element grids were used to represent a quarter of the plate due to symmetry. The material parameters used are as follows [7]: elastic modulus, E"197 GPa; tangent modulus, E "250 MPa; Poisson's ratio, l"0.3; static yield stress, p "265 MPa; rupture strain, T 0 e "0.18; density, o"7830 kg/m3; static ultimate shear stress, q "178.8 MPa. 361 6-5 To compare with the available experimental results [7] blast wave loading is considered. The pressure loading from the explosive charge is assumed to be a rectangular pulse uniformly distributed over the plate surface. The duration of loading is assumed to be 15 ls, equal to the approximate explosive burn time. The pressure intensity is then calculated to correspond to the measured impulse (area under the pressure-time curve).
4. Results and discussion For the purposes of numerical convergence and accuracy, several numerical simulations were carried out using di!erent grid sizes for the quarter plate. Unless stated otherwise, all the results reported here are for a 2]2 grid (Fig. 2b) with a time-step size of 0.5 ls and are considered to be accurate enough for engineering design purposes. 4.1. Mode I A typical mode I time history of the sti!ener central displacement (point C) and kinetic energy in the system is shown in Fig. 3, for a 3]2-mm sti!ened plate under 5 Ns impulse. The central deformation increases monotonically and reaches a peak value around 120}125 ls. This time to peak displacement is slightly lower than that of an unsti!ened plate of the same size which occurs around 135}140 ls [5]. All the input energy is absorbed at a time which coincides with the time to
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Fig. 2. Con"guration and "nite element model of a one-way sti!ened plate: (a) con"guration of plate; and (b) "nite element model of quarter-plate (2]2 grid).
reach peak displacement. The plate vibrates in an elastic manner about the permanent deformation position once energy absorption is complete. Temporal variations of strain ratio, stress ratio and failure function (e6 , q6 and f ), for the midpoint of the boundary parallel to the sti!ener and for the sti!ener centre (points A and C in Fig. 2b) are shown in Figs. 4 and 5, respectively. The results are for a 3]2-mm sti!ened plate under 5 Ns impulse and is typical for impulses which cause mode I deformation. The strain ratio at A increases monotonically to a peak value around 90 ls and remains more or less constant thereafter, although deformation continues. The time to peak strain is less than the time to reach peak displacement. This is true with all sti!ener sizes.
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Fig. 3. Time history of central displacement and kinetic energy of 3]2-mm sti!ened plate in mode I failure.
Fig. 4. Strain ratio, stress ratio and failure function variation with time at the midpoint of the boundary parallel to the sti!ener.
The stress ratio at A increases until 15 ls and drops when the applied load drops to zero. It reaches another lower maximum when the inertial forces in the system become signi"cant. Thereafter, it drops and oscillates around the zero mean value. The corresponding variation of failure function at A is also shown in Fig. 4. An initial sharp peak occurs at the end of the loading phase due to high initial shear stress ratio. The failure function drops when the applied load drops to zero. It increases again and reaches a maximum value when the inertial force of the system is substantial. At this stage, the plate has undergone a signi"cant amount of stretching and bending, thus contributing to the failure function via the strain ratio. The time to peak failure function occurs around 90 ls, which also coincides with peak strain. This time to peak failure function and the relative sizes of these peaks change with the applied impulse but remains the same for all sti!ener sizes.
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Fig. 5. Strain ratio, stress ratio and failure function variation with time at the sti!ener centre.
Fig. 6. Comparison of strain ratio and stress ratio at the threshold impulse to mode II failure under uniformly distributed load.
Similar variation of these parameters with time for the sti!ener centre (point C) is shown in Fig. 5. Both strain and stress ratios have much smaller magnitudes compared to their corresponding values at the boundary point. This is true with all sti!ener sizes under the uniformly applied load. 4.2. Mode II As the applied impulse is increased, the threshold impulse to cause rupture of sti!ened plates is reached. The threshold impulses (I H ) are 9.5 and 13.3 Ns according to LIC and QIC predictions, m2 respectively, using a 2]2 grid for both analyses. The changes in strain ratio and stress ratio at A and C as a function of sti!ener depth are shown in Fig. 6. Experimental investigation [7] showed
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Fig. 7. Comparison of failure function at the threshold impulse to mode II failure under uniformly distributed load using LIC and QIC.
that failure initiates at either of these points for di!erent sti!ened plates. Hence, these points are chosen to study the rupture initiation numerically. The plotted values at these two locations correspond to those obtained just prior to the end of the loading phase. Fig. 6 shows that although there is some change in the strain ratio with a change in the sti!ener size, these changes are insigni"cant. However, the stress ratio exhibits a much larger change with sti!ener size. For the same loading, a 3]9-mm sti!ened plate develops 25% more shear force at the sti!ener/plate interface than a 3]2-mm sti!ened plate at the same location. Fig. 7 shows the corresponding variation of the failure function with sti!ener depth at the threshold impulse to failure using LIC and QIC. The values shown correspond to the time just before the end of the loading phase and hence do not approach a value of one. From the "gure it is clear that the failure function at A has a much higher value than that at C. The failure function at A shows a slight increase with sti!ener depth, while the function at C increases monotonically. The gap between them decreases linearly with the increase in sti!ener depth. Table 1a and b present the time to "rst element failure (t ), strain ratio (e6 ), stress ratio (q6) and */ displacement at the sti!ener centre (* ) at the time of "rst failure for all sti!ened plates. #* The applied impulse is just enough to cause mode II failure in all the plates. The failure starts at the middle of the boundary parallel to the sti!ener (mode IIB) for all the sti!ened plates. From the table, it is evident that the mode II threshold is independent of sti!ener size. Also, the contribution of strain and stress ratios to the failure function is almost the same for all sizes of sti!eners. Although the time to failure is also the same, the central displacement at failure decreases monotonically with the sti!ener size. The failure at this impulse occurs due to a combination of stress and strain ratios much after the loading phase is over. This is typical of mode II failure which was also observed in the unsti!ened plates [5]. This strain domination is evident when the strain contribution to reach failure is observed which is characteristic of mode II failure. However, the failure function also has
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Table 1 Stress ratio, strain ratio, time to "rst element failure and sti!ener central displacement at the threshold impulse to mode II failure for all sti!ener sizes using a 2]2 grid I H (Ns) .2
q6
e6
t
(a) LIC 2 4 5 9
9.5 9.5 9.5 9.5
0.427 0.460 0.469 0.466
0.574 0.540 0.536 0.537
61.0 57.5 56.0 54.5
4.48 3.82 3.49 2.50
(b) QIC 2 4 5 9
13.3 13.3 13.3 13.3
0.553 0.604 0.607 0.600
0.835 0.798 0.796 0.801
64.0 55.5 54.5 53.0
6.64 5.05 4.66 3.51
H (mm)
*/
(ls)
* (mm) #*
Table 2 Stress ratio, strain ratio, time to "rst element failure and sti!ener central displacement at the threshold impulse to mode II failure for all sti!ener sizes using a 3x3 grid and LIC for failure prediction H (mm)
I H (Ns) .2
q6
e6
t
2 4 5 9
9.5 9.5 9.5 9.5
0.462 0.467 0.465 0.455
0.544 0.540 0.564 0.546
52.5 51.5 51.5 52.0
*/
(ls)
* (mm) #* 3.67 3.26 3.08 2.31
signi"cant contribution from stress ratio at the threshold impulse (46% of failure function is due to the shear stress ratio at the threshold impulse in sti!ened plates, as opposed to 50% in an unsti!ened plate [5,6] when LIC is used for the failure analysis). A purely strain based criterion would not capture this signi"cant e!ect and thus lead to a higher threshold to failure [7]. For comparison, the results in Table 2 were obtained using a 3]3 grid under 9.5 Ns of impulse, and LIC to detect rupture. 4.3. Post-failure analysis For illustration purposes, only the results for the 3]2- and 3]9-mm sti!ened plates are presented. The post-failure analysis of the plate with other sti!ener sizes is left for future study. Once su$cient experimental evidence is gathered, they can be used to correlate with the predicted results. The results reported here are obtained using QIC unless otherwise mentioned. 4.3.1. Mode IIH Fig. 8 shows the time history of central displacement, side pull-in and the kinetic energy of a 3]2-mm sti!ened plate under 13.3 Ns impulse. The failure "rst occurs at the midpoint on the
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Fig. 8. Time history of central displacement, side pull-in and kinetic energy of a 3]2-mm sti!ened plate in mode IIH failure.
Fig. 9. Central displacement, side pull-in and kinetic energy versus time for a 3]2-mm sti!ened plate in mode IIH failure.
boundary parallel to the sti!ener. Subsequently, the element side is released and the analysis is continued. The remaining energy is absorbed by the plate with no further failure. Thus, a partial failure regime is identi"ed. A plot of the central displacement, side pull-in and the kinetic energy versus time is shown in Fig. 9 for an applied impulse of 14.1 Ns. The failure starts at A (shown by the solid circle in the "gure). Subsequently failure propagates on the same side of the sti!ened plate (AO in Fig. 2b) and then switches to the boundary adjacent to the sti!ener (OB in Fig. 2b). These failure initiations are shown by the solid diamond and triangle symbols, respectively. The remaining element sides are released and the sti!ener end only (point B in Fig. 2b) supports the plate. The input energy is not su$cient to cause complete failure. Figs. 10a and b show the predicted 3-D pro"le of the deformed
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Fig. 10. 3-D deformation pro"le of 3]2-mm sti!ened plate in mode IIH failure (two views of the deformed plate computed at an impulse of 14.1 Ns).
plate. The two views of the plate clearly show that the sti!ener end remains attached to the boundary. 4.3.2. Mode IIa The temporal variation of the sti!ener centre displacement, side pull-in and the kinetic energy of the system are shown in Fig. 11 for the 3]2-mm sti!ened plate under an applied impulse of 16 Ns. The same trend continues at higher loads with failure always starting at the middle of the boundary parallel to the sti!ener (point A), and terminating at B. The solid symbols in the "gure are the instances at which each of the failure occurs. At this stage, all the nodes are released and the plate separates from the boundary. Temporal integration is further carried out to account for any deformation occurring during the free-#ight of the torn plate until a steady state is reached. The plate enters modes IIb and III failure regimes, as the applied impulse is increased. The failure is dominated by shear in these two failure modes, which is re#ected by the insigni"cant deformation of the plate.
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469
Fig. 11. Temporal variation of central displacement, side pull-in and kinetic energy of 3]2-mm sti!ened plate in mode IIa failure.
Fig. 12. Central displacement versus impulse for 3]2-mm sti!ened plate using QIC.
4.4. Permanent central displacement A plot of the sti!ener central displacement versus impulse is shown in Figs. 12 and 13 respectively, for the 3]2- and 3]9-mm sti!ened plates along with the experimental data [7]. Overall, the plate with thick sti!ener exhibits less permanent displacement compared to the plate with thin sti!ener. The predicted mode I displacements match well with the experimental data in the case of 3]2-mm sti!ened plates. The predicted mode II failure always occurred at the boundary with the initiation site being at point A for both plates. Only one experiment with 3]2-mm sti!ened plate showed rupture at the
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Fig. 13. Central displacement versus impulse for 3]9-mm sti!ened plate using QIC.
boundary parallel to the sti!ener, while no results are available for the 3]9-mm sti!ened plate and the results do not o!er a comprehensive comparison with the numerical predictions. Experimentally, mode II failure in 3]2-mm sti!ened plates occurred around 14}15 Ns impulse. This value compares favourably with the results of 13.3 Ns obtained using the QIC. The predicted threshold impulse for mode IIa is around 15 Ns. There is a monotonic increase in the sti!ener central displacement with the increase in impulse until the threshold impulse to mode II failure is reached. Similar to the unsti!ened plates, the predicted central displacement of sti!ened plates also show an increasing trend for mode IIH, mode IIa and decreasing trend for mode IIb. 4.5. Non-uniform load distribution The numerical analysis always predicted failure starting at the middle of the clamped boundary parallel to the sti!ener (mode IIB) for all the sti!ened plates analysed. This matched well with the experimental results for plates with thin sti!eners (2 and 4 mm deep). However, for a plate with deep sti!ener (5 and 9 mm deep) the experiments showed localized tearing at the sti!ener/plate interface [7]. In an attempt to explain this discrepancy, it is worth noting that in the experiments the explosive was laid out on a 12-mm thick polystyrene pad in two concentric square annuli, which were interconnected by two perpendicular strips of explosive called cross-leaders. A short tail of explosive holding the detonator was then attached to the centre of the cross-leaders. The mass of explosive for the tail and the cross-leaders was constant for all the tests. Increasing only the mass of the annuli explosive increased the impulse that was applied to the #at side of the plate. Since there is an extra mass of explosive along with the detonator at the centre of the plate, it was thought that this might have introduced nonuniformity in the load distribution. Therefore, a series of analyses with a nonuniform load distribution over the plate were undertaken.
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Fig. 14. Variation of failure function with sti!ener depth under non-uniformly distributed load (Nu-2).
Two di!erent nonuniform load distributions are considered. The element next to plate centre is subjected to a higher load magnitude compared to the remaining elements. Speci"cally, in a 2]2 grid (or four elements), the element next to the plate centre is subjected to twice the load applied to the other three elements while in the case of a 3]3 grid (or nine elements), the load applied to the element next to the plate centre is three times that applied to the other eight elements. However, in so doing the total impulse applied to the structure is kept the same. These are termed as Nu-2 and Nu-3, respectively, depending on whether the central element is subjected to two or three times the load applied to the other elements. Once again, the plate response is computed over a range of impulses for all sti!ener sizes. 4.5.1. Mode II Fig. 14 shows the variation of failure function as a function of the sti!ener depth under the non-uniform loading Nu-2. The plot is for a load corresponding to 5 Ns impulse and is analysed using a 2]2 grid and LIC. In addition, Fig. 14 also shows the changes in failure function with sti!ener depth for the "ner grid (3]3). The impulse is typical of that causing mode I failure. The values correspond to the time just before the end of the loading phase. The failure function at the midpoint on boundary (A) in both "gures remains almost constant, thus independent of sti!ener size. The same is not true with the failure function at the sti!ener centre (C). The localized load tends to increase the shear stress at the sti!ener centre. This e!ect is more pronounced with thick sti!eners (3]9-mm sti!ener) where the shear stress at the centre is more than that at the boundary. The crossover occurs around 7 and 6 mm sti!ener depth for the coarse and "ne grid, respectively, with load distribution Nu-2. Even though the failure function at the sti!ener centre is higher, it is well below the threshold to cause failure under the applied load (5 Ns). If the applied load is increased then failure is identi"ed either at the boundary (A) or at the sti!ener centre (C) depending on which of the failure functions reaches a value of one "rst. Figs. 15 and 16 show the variation of permanent central displacement versus impulse for 3]5and 3]9-mm sti!ened plates, respectively, under both uniform and nonuniform load distributions.
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Fig. 15. Comparison of sti!ener central displacement versus impulse for 3]5-mm sti!ened plate under various spatial load distributions.
Fig. 16. Comparison of sti!ener central displacement versus impulse for 3]9-mm sti!ened plate under various spatial load distributions.
The importance of load distribution is evident from the results. In the above two "gures, the nonuniform load distribution leads to a higher permanent displacement than a corresponding uniform load distribution. The threshold to cause mode II failure is reached as the applied impulse is increased. The threshold is 13 Ns using QIC for all sti!ened plates with load distribution Nu-2. The threshold impulse to cause mode II failure is, once again, the same for all sti!ener sizes, and failure at the boundary (point A) is predicted at this impulse. However, a transition occurs, wherein failure starts as localized tearing of the plate at the sti!ener centre with load distribution Nu-3 for plates with deep sti!eners (5 and 9 mm deep).
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The localized tearing of the plate at the sti!ener centre (mode IIS) is predicted for the 3]5- and 3]9-mm sti!ened plates at 16 and 10 Ns, respectively, using LIC. This matches well with the experimental observation (solid diamond symbols in Figs. 15 and 16). Since the aim of the nonuniform load analysis was to determine the threshold impulse at which failure mode transition occurred and thus explain the observed experimental phenomena, post-failure analysis was not carried out in this case. 5. Conclusions A numerical study on the failure of blast loaded, clamped, square, sti!ened plates has been presented. The nonuniform variation of shear force (and hence shear stresses) along the boundary is captured using "ctitious transverse spring supports. This enabled the model to consider the e!ect of shear stresses on the failure. The predicted results compare well with the available experimental data and clearly demonstrate the in#uence of shear on the failure mechanism not only for mode III, but also for mode II. The results con"rm the importance of the interaction e!ects of tensile and bending strain on tearing and shear failure. Through the use of springs at the sti!ener/plate interface the model captures the in#uence of the interfacial shear force on the localized tearing failure mode. The predicted impulse for the onset of failure at the boundary (mode IIB) is found to be independent of the sti!ener size. The e!ect of larger sti!ener is only in reducing the magnitude of plate displacement. Under uniform load distribution, tearing starts at the middle of the clamped boundary parallel to the sti!ener for all sti!ener sizes. The lack of comprehensive experimental data does not allow for a comparison of predicted results. However, the results do show the expected qualitative behaviour. The spatial distribution of the blast load plays a signi"cant role in determining the failure mode as identi"ed by the model. An apparent transition of failure from the middle of the clamped boundary to the localized tearing at the sti!ener centre shows the sensitivity of the plate response to load distribution. The model allows parametric investigations to determine the e!ects of changing panel dimensions, sti!ener sizes and shapes, material properties and rates of loading. The model and structural support philosophy that it embraces can be developed further to analyse unsti!ened and sti!ened shell structures, as well as plates with multiple sti!eners. Acknowledgements This research has been supported by the Canadian Department of National Defence, through a contract from the Defence Research Establishment Su$eld, Alberta, and the Natural Sciences and Engineering Research Council of Canada.
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