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Numerical analysis of dynamic buckling of rectangular plates subjected to intermediate-velocity impact
International Journal of Impact Engineering 25 (2001) 147}167
Numerical analysis of dynamic buckling of rectangular plates subjected to intermediate-velocity impact Shijie Cui, Hong Hao*, Hee Kiat Cheong School of Civil and Structural Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Received 19 August 1999; received in revised form 6 December 1999; accepted 3 June 2000
Abstract This paper numerically investigates the dynamic buckling of thin imperfect rectangular plates subjected to intermediate-velocity impact loads. From numerical results obtained, a dynamic buckling and a dynamic yielding critical condition are de"ned, and the corresponding critical dynamic loads are estimated. Numerical model employed in the present study is validated by experimental data reported earlier. Results from parametric study indicate that initial imperfection and load duration have signi"cant in#uence on the dynamic buckling of the plates. The smaller the initial imperfection and the load duration, the higher the dynamic buckling critical loads of the plates. Moreover, di!erent hardening ratios of plate material also a!ect the elastic}plastic dynamic buckling properties of the plates. If the plate buckles plastically, the dynamic buckling load increases as the hardening ratio of plate material increases. Unlike thin plates under high-velocity impact that buckling always occur after load application, plates under intermediate-velocity impact analyzed in the present study all buckle during the loading phase. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Dynamic buckling of structures under impact loads can be divided into three categories according to the applied dynamic loads and structural responses, viz., high-velocity impacted buckling, low-velocity impacted buckling and intermediate-velocity impacted buckling. For the problem of high-velocity impacted buckling of structures, the dynamic load poses a very high amplitude and short duration (in an order of microseconds). Thus, this kind of the dynamic load
* Corresponding author. Tel.: #65-791-1744; fax: #65-791-0676. E-mail address: [email protected] (H. Hao). 0734-743X/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 4 - 7 4 3 X ( 0 0 ) 0 0 0 3 5 - X
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Nomenclature B [B] [C] +d, +dQ , +d$ , G h L M [M] q(t) q q q # q q # +R, t t T G w (x, y) d j & p +p, m
width of the plate (mm) strain}displacement transformation matrix of the plate material damping matrix nodal displacement vector of plate element nodal velocity vector of plate element nodal acceleration vector of plate element total energy dissipation due to plastic response (J) thickness of the plate (mm) length of the plate (mm) bending moment per unit width about the y-axis of the plate (N mm) mass matrix of the plate uniformly distributed load on plate edge (N/mm) peak value of the dynamic load (N/mm) dynamic buckling critical load amplitude (N/mm) experimental result of dynamic buckling critical load amplitude (N/mm) dynamic yielding critical load amplitude (N/mm) experimental result of dynamic yielding critical load amplitude (N/mm) equivalent external load vector of the plate time (s) dynamic load duration (s) periods of transverse-free vibration of the plate (s), i"1, 2,2, 5 initial imperfection function of the plate (mm) maximum initial imperfection of the plate (mm) Hardening ratio of the plate material, j "E /E, E and E are the elastic and & 2 2 plastic modulus of the plate material, respectively yield stress of the plate material (MPa) stress vector of the plate Rayleigh-type viscous damping ratio
can be simpli"ed as a pulse load, and this problem is also called `pulse bucklinga [1]. The experimental studies of this problem are usually realized by striking the structure with a moving mass or moving it towards a rigid surface (solid}solid impact). For the problem of low-velocity impacted buckling, the dynamic load duration is much longer while its amplitude is relatively lower than those of the high-velocity impact. It may be simpli"ed as a step load with a constant amplitude and in"nite duration. Under these simpli"cations, solving the dynamic buckling equation of structures is relatively straightforward. Most of the previous studies of dynamic buckling of structures are focused on these two types of the problem. The beginning of the studies of dynamic buckling of suddenly loaded plates may be traced back to that by Zizicas [2] in 1952, who, by neglecting the in-plane inertia e!ects, presented a theoretical solution for simply supported plate under in-plane time-dependent load. However, neither critical
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condition nor buckling criterion was investigated in this study. Birkgan and Vol'mir [3] also studied the dynamic buckling properties of plates under sudden loads using an electronic digital computer. Unlike Zizicas, they estimated the dynamic buckling loads of the plates and found that the high loading rates (high-velocity impact) resulted in a higher-order dynamic buckling mode of the plates in their in-plane loading direction. Ari-Gur et al. [4] investigated theoretically and experimentally the dynamic buckling of a rectangular plate impacted by a mass M moving in an in-plane direction. They obtained the critical in-plane strain at dynamic buckling of plate and discussed the e!ect of initial imperfection and the load duration on plate buckling properties. Passic and Herrmann [5] studied theoretically the nonlinear dynamic buckling behavior of imperfect rectangular plates under in-plane compressive step loads. Weller et al. [6] performed analytical studies by using the ADINA computer code to determine the dynamic load ampli"cation factor (DLF) of metal beams and plates subjected to axial in-plane impact compressive loads. More literature on dynamic buckling of plates under high- and low-velocity impacts can be found in the publications of Hac [7], Lindberg and Florence [1], Subramanian [8], Tam and Calladine [9], and review papers by Simitses [10] and Jones [11,12]. For the intermediate-velocity impacted buckling of structures, the dynamic load has a moderate amplitude and duration (in an order of milliseconds). It can neither be simpli"ed as a pulse load nor a step load with in"nite duration. In such a problem, the load duration is a very important parameter which will strongly a!ect the dynamic buckling properties of structures. The dynamic buckling characteristics of structures may be much di!erent from those of structures under the high- and low-velocity impacts. The studies of this problem for structures, especially for plates, however, are very limited in literature up to now. Karagiozova and Jones [13}15] investigated the dynamic elastic}plastic buckling of a twodegree-of-freedom model subjected to a rectangular pulse load and two triangular loads. In#uences of initial imperfection, dynamic load shape and duration, axial inertia and plastic reloading on the dynamic buckling behavior of the model were examined in these studies. They concluded that both the initial imperfection and the dynamic load duration strongly a!ect the dynamic buckling properties of the model, and load duration also has a signi"cant in#uence on the sensitivity of the initial imperfection. Recently, Cui et al. [16] reported an experimental study of the dynamic buckling of rectangular plates under in-plane intermediate-velocity impacts. The intermediate-velocity impacts were generated by #uid}solid slamming. The dynamic buckling and dynamic yielding properties were investigated based on the observation of elastic}plastic dynamic response characteristics of the plates. The e!ect of di!erent boundary conditions on the elastic}plastic dynamic buckling properties of plates was also examined in the study. This paper presents a numerical study of dynamic buckling of imperfect rectangular plates subjected to intermediate-velocity impact loads by using the computer code ABAQUS [17]. The emphasis of the study is focused on the parameters that a!ect the dynamic buckling of the plates. The primary objectives are to investigate the dynamic buckling mechanism of plates subjected to intermediate-velocity impact loads, to de"ne a suitable dynamic buckling criterion and to discuss the e!ects of initial imperfection, boundary conditions, dynamic load duration, and the hardening ratio of plate material on the dynamic buckling characteristics of the plates.
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2. Numerical model of plates For a plate under intermediate-velocity impact load (Fig. 1a), its equilibrium equation can be written as
[M]+d$ ,#[C]+dQ ,#
[B]2+p, d<"+R,,
(1)
4
where [M] and [C] are mass and viscous damping matrices; [B] is the strain}displacement transformation matrix; +d , is the nodal displacement vector and the dot represents taking
Fig. 1. A rectangular plate and its "nite element mesh.
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derivative with respect to time; +p, is the stress vector of the plate and +R, is the equivalent load vector. In the present study, compatible mass matrix is used in "nite element modeling. It should be noted that the e!ect of viscous damping is not important in dynamic responses of plates in this study because of the short duration of the impact load. However, for modeling completeness, Rayleigh-type viscous damping of 2.5% critical ratio (m "0.025) corresponding to the "rst two modes of the plate is speci"ed in the analysis. The equivalent load vector is obtained in "nite element modeling by using the shape function and the impact load q(t) acting at the loading edge of the plate. q(t) is assumed to be having a half-sine form as observed in test [16]. Thus, it can be de"ned by specifying its amplitude and duration. In the numerical analysis, large de#ections are considered by including the second-order terms in strain estimation:
*w *u 1 *w 1 *w !z , ! e " # V *x 2 *x 2 *x *x *w *v 1 *w 1 *w !z ! , e " # W *y 2 *y 2 *y *y
(2)
*u *v *w *w *w *w *w !2z c " # # ! , VW *y *x *x *y *x *y *x*y where u, v, w are the longitudinal, lateral and transverse displacements (in the x, y and z directions) of the plate; w is its initial transverse deformation. A bi-linear elastic}plastic model is employed to describe the nonlinear property of the material. The material properties of the model are: elastic modulus E"2.1;10 MPa, plastic modulus E "1.519;10 MPa and the yield stress p "289 MPa, respectively. 2 3. Dynamic response characteristics of plates Eighteen rectangular plates are analyzed in this numerical study. Table 1 lists the dimensions of the plates. In order to calibrate the numerical model, some of the plates are selected to have the same dimensions as those of the tested plates [16]. It should be noted that, in the experimental test of plates [16], impact duration depends on the dimensions and boundary conditions of the plate specimens, as well as the test set-up. Because the same set-up was employed in the test [16], for a particular plate with the prescribed dimensions and boundary conditions, the impact duration is basically unchanged even the impact load amplitude is di!erent. Thus, the dynamic load duration is also taken as a constant for each type of plates in the numerical analysis and is listed in Table 1. The initial imperfection of the plates is assumed following the same shape as their "rst mode of transverse-free vibration. Therefore, only the maximum value, d , is given in the table. In order to investigate the e!ect of di!erent boundary conditions on the dynamic buckling of plates, "ve types of boundary conditions are considered in this study as indicated in Table 1. The "rst three types, which are the same as those implemented in the experimental tests, are clamped on both top and bottom edges, and the two vertical sides of the plates are clamped (CCP), hinged (CSP) or free (CFP), respectively. The other two types of the boundary conditions are simply supported at four
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Table 1 Dimensions, initial imperfection and critical dynamic loads of plates
Plate no.
Dimension (mm) L B
h
t (s)
d (mm)
q (N/mm)
q (N/mm)
Boundary conditions
CCP01 CCP02 CCP03 CCP04 CCP05 CCP06 CSP01 CSP02 CSP03 CFP01 CFP02 CFP03 SSP01 SSP02 SSP03 SFP01 SFP02 SFP03
500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500
2.0 2.0 2.0 2.5 2.5 2.5 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
0.017 0.017 0.017 0.001 0.006 0.038 0.0175 0.0175 0.0175 0.019 0.019 0.019 0.022 0.022 0.022 0.038 0.038 0.038
0.11 0.10 0.12 0.15 0.15 0.15 0.11 0.10 0.13 0.11 0.10 0.12 0.10 0.12 0.14 0.10 0.12 0.14
198 202 192 1188 505 300 148 152 141 78 80 76 81 77 75 50 47 45
233 240 230 850 460 285 187 192 178 152 157 148 163 156 151 98 81.5 73.5
Clamped}clamped Clamped}clamped Clamped}clamped Clamped}clamped Clamped}clamped Clamped}clamped Clamped}hinged Clamped}hinged Clamped}hinged Clamped}free Clamped}free Clamped}free Hinged}hinged Hinged}hinged Hinged}hinged Hinged}free Hinged}free Hinged}free
325 325 325 325 325 325 325 325 325 325 325 325 325 325 325 325 325 325
Table 2 Transverse free vibration periods of plates T (s) Plate no.
T
T
T
T
T
CCP01 CCP04 CSP01 CFP01 SSP01 SFP01
0.0079 0.0063 0.0124 0.0226 0.0151 0.0524
0.0052 0.0042 0.0062 0.0082 0.0080 0.0129
0.0033 0.0026 0.0036 0.0068 0.0045 0.0073
0.0022 0.0017 0.0023 0.0042 0.0028 0.0057
0.0017 0.0014 0.0022 0.0041 0.0023 0.0047
edges (SSP) and hinged-free (SFP), viz., the top and the bottom edges are simply supported and the two vertical edges of the plates are free. Moreover, in order to simulate the elastic and plastic buckling, plates with di!erent thickness are analyzed, they are 2 and 2.5 mm thick plates, respectively. The "rst "ve periods of transverse-free vibration of a few typical plates are given in Table 2. In "nite element modeling, each plate is divided into 12 segments in each direction in x}y plane as shown in Fig. 1(b). Eight-node isoparametric element is used. Thus, there are a total of 144 elements and 481 nodes for each plate in the numerical calculation.
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In numerical analysis, a series of dynamic responses of the plates are calculated by loading each plate with the half-sine form impact load q(t) with duration given in Table 1 and di!erent amplitudes. Fig. 2 shows the displacement time histories at points A}D as indicated in Fig. 1(b) of plate CCP01 under several load amplitudes. Firstly, it can be seen from the "gure that the maximum displacement always occurs during the load duration t . The displacement responses of plate decay rapidly after the load application. The displacement responses for other plates, which are not shown here, have similar characteristics as those of CCP01 shown in Fig. 2. This indicates that the plates may buckle during the dynamic load duration. This dynamic response characteristic of the plates subjected to intermediate-velocity impacts is di!erent from that of plates under high-velocity impacts (solid}solid impacts). This is because the duration of an intermediate-velocity dynamic load is far longer than that of a high-velocity impact load. For the high-velocity impacted buckling, the load duration is usually in an order of microsecond, and the loading will be completed before the deformation of the plate is fully developed. So that dynamic buckling of plate always occurs after load application [1,4]. For the intermediate-velocity impacted buckling, however, the millisecond duration is su$cient to allow the bending deformation of plate to develop before the end of load application. Thus, the maximum response of the plate occurs during the load duration, and the dynamic buckling may take place before the dynamic load is completed. This observation is identical with that observed in experimental tests of the plates [16]. As can also be seen from Fig. 2, the maximum displacement response of the plate increases along with the increase of the dynamic load. When the dynamic load is small (e.g. q 4200 N/mm), the maximum displacement of the plate increases gradually. When the dynamic load is large (e.g. q '200 N/mm), the maximum transverse displacement response of the plate increases rapidly. This is more clearly illustrated in Fig. 3(a), in which the maximum displacement of the plate is plotted as a function of the dynamic load amplitude. As can be seen from the "gure, the curve of the maximum transverse displacement of the plate has a `kneea at a `critical dynamic loada. When the dynamic load is smaller than this critical load, the slope of the curve is #at, indicating the development of the maximum transverse displacement of the plate is steady as the dynamic load increases. When the dynamic load is larger than the critical load, the slope of the curve becomes very steep, indicating the maximum displacement of the plate increases rapidly for a small increment of the dynamic load. The above observation indicates that there exists a critical condition for the maximum displacement response of the plate in the process of increasing impact load from small to large. When the dynamic load is small, any increment in the dynamic load will cause steady increment in the displacement responses; when the dynamic load is large, however, any increment in dynamic load amplitude will result in a signi"cant increment in dynamic displacement responses of the plate. As will be discussed later, at q(t)"200 N/mm, the plate is still in elastic range. Thus, it is obvious that the change of the maximum transverse displacement of the plate from steady increase when the load is smaller than q(t)"200 N/mm to rapid increase when the load is larger than that is caused by dynamic buckling of the plate. According to this observation, a dynamic buckling criterion of plate is de"ned as follows: for an impacted plate with a prescribed aspect ratio, the point corresponding to the sudden rapid increase of the maximum transverse displacement is the dynamic buckling critical condition of the plate. The corresponding dynamic load is de"ned as the dynamic buckling critical load and denoted by q .
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Fig. 2. Displacement response time histories of CCP01 under half-sine impact loads with di!erent amplitudes (t "0.017 s, m "0.025).
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Fig. 3. The maximum displacement and bending moment at cross-section of the plate CCP01 versus dynamic load.
To further verify the buckling condition discussed above, Fig. 3(b) displays the variation of the maximum bending moment force per unit width about y axis at point 2 as shown in Fig. 1(b) versus di!erent dynamic load amplitudes. As can be seen from this "gure, when the dynamic load is small, the bending moment of the plate is nearly zero, implying the dynamic response is primarily in-plane compression and the plate is stable. When the impact load increases to the critical load as shown in Fig. 3(a), the bending moment begin to increase rapidly, indicating that the bending deformation of the plate becomes pronounced and dynamic buckling of the plate takes place. Fig. 4 shows the Mises stress time histories on the compressive side of the plate surface at points 1}3 as shown in Fig. 1(b) of plate CCP01 under di!erent dynamic loads. The variation of the maximum Mises stresses versus the loading amplitude is given in Fig. 5. As can be seen from Fig. 4, the peak value of Mises stress of plate also occurs within the load duration (t "0.017 s for CCP01) as expected since the maximum displacement response occurs during the load duration as discussed above. The Mises stress grows with the increase of the dynamic load as shown in Figs. 4 and 5. Before the dynamic buckling takes place, the dynamic load is small, and the transverse displacement of the plate is insigni"cant. As a result, the Mises stress of the plate is not large. After the dynamic buckling occurs, the maximum Mises stress of the plate increases sharply because of the rapid increase in the transverse deformation of the plate. When the dynamic load increases to a certain value, the maximum Mises stress reaches to the yield stress of plate material. This indicates that the plastic deformation begins to appear in the plate, and the dynamic response of the plate consists of both elastic and plastic deformation. It can be seen from Fig. 2, the residual plastic deformation can be observed when q "288 N/mm, and it becomes signi"cant when q "332 N/mm. In addition, after the plastic deformation occurs in the plate, the increase of the maximum Mises stress of the plate is slowed down as the dynamic load increases. This is because plastic deformation of the plate absorbs more impact energy. The critical point that
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Fig. 4. Mises stress time histories of CCP01 under di!erent dynamic load amplitudes (t "0.017 s).
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Fig. 5. Variation of the maximum Mises stress of CCP01 versus dynamic load amplitude.
Fig. 6. Maximum responses of plate CCP05 versus dynamic load amplitude (t "0.006 s, m "0.025).
when the maximum Mises stress of the plate reaches the material yield stress is de"ned as the critical yielding condition. The corresponding load is the dynamic yielding critical load, and is denoted by q . For the plates with thickness h"2.5 mm, the dynamic buckling characteristic is di!erent. Fig. 6 shows the variations of the maximum transverse displacement and the maximum Mises stress on the compressive surface of plate CCP05 versus dynamic load amplitude. When the dynamic
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load amplitude increases to approximately 460 N/mm, the maximum Mises stress of the plate reaches its yielding stress of material. At this level of the dynamic load, the slope of the maximum displacement curve changes drastically. This is because local plastic deformation appears in the plate although most parts of the plate remain elastic. When the dynamic load increases to q "505 N/mm, the slope of the maximum displacement curve shows another change. This is believed to be caused by dynamic buckling of the plate. Unlike the plate CCP01, the plate CCP05 with thickness h"2.5 mm buckles plastically. When a plate buckles elastically, its maximum Mises stress increases rapidly with dynamic load as shown in Fig. 5 for plate CCP01. However, when a plate buckles plastically, the increase of the maximum Mises stress is steady during the post-buckling process as shown in Fig. 6. This is because the plate CCP05 yields before buckling. After the dynamic yielding occurs, plastic deformation appears in the plate, and more impact energy is absorbed by the plastic deformation. Thus, although buckling will cause rapid increase of the stresses in the plate, the rate of stress increment will not be as signi"cant as that in an elastically buckled plate.
4. Determination of dynamic critical loads To estimate the dynamic buckling critical load and the yielding critical load, curves of the maximum displacement response and the maximum Mises stress of the plates are plotted with respect to the dynamic load. According to the above de"nitions of dynamic buckling and dynamic yielding critical conditions, the corresponding critical dynamic loads can be easily determined. Figs. 3, 5 and 6 are examples for plate CCP01 and plate CCP05. From these "gures, it can be obtained that q "198 N/mm, q "233 N/mm for plate CCP01, and q "505 N/mm and q "460 N/mm for plate CCP05. The curves for other plates, which are not shown here, are similar. The corresponding critical dynamic loads are listed in Table 1. It is obvious that the plates with thickness h"2.0 mm in the present study buckle elastically, while those with h"2.5 mm buckle plastically.
5. Comparison with the experimental results For comparison, results obtained in this numerical analysis and the corresponding test results for the critical load amplitudes obtained by Cui et al. [16], which are denoted as q and q , # # are listed in Table 3. It can be noted from the table that the numerical results of dynamic buckling critical loads of the plates agree well with the experimental results. The errors are about 2.63}4.96% for the 9 tested plates with an average error of 3.49%. The errors between numerical and experimental results for yielding critical loads of the plates are about 6.67}9.03% with an average error of 8.18%. The error between the experimental and the numerical results might be attributed to the facts that (1) the impact loads used in this numerical study are not exactly the same as the actual impact loads [16]; (2) small loading eccentricity is inevitable for each plate in the test, but no loading eccentricity is considered in the numerical study; and (3) the relationship between stress and strain of plate material used in this numerical analysis is bilinear, which is not exactly the same as the specimen material property.
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Table 3 Comparison with the experimental results
Plate no.
q # (N/mm)
q (N/mm)
Error %
q # (N/mm)
q (N/mm)
Error %
CCP01 CCP02 CCP03 CSP01 CSP02 CSP03 CFP01 CFP02 CFP03
192 196 184 152 158 148 76 77 74
198 202 192 148 152 141 78 80 76
3.13 3.06 4.35 2.70 3.95 4.96 2.63 3.90 2.70
214 222 212 172 180 165 140 144 138
233 240 230 187 192 178 152 157 148
8.88 8.11 8.49 8.72 6.67 7.88 8.57 9.03 7.25
mean
3.49
8.18
6. Dynamic buckling mode of plates Fig. 7 shows the distributions of the maximum displacements of plates CCP01, CSP01, SSP01, CFP01 and SFP01 along the longitudinal direction and under di!erent load amplitudes. Fig. 8 displays the three-dimensional deformations of the plates under the corresponding dynamic buckling critical loads. It can be found from these "gures that the dynamic buckling modes of the plates are dominated by their fundamental transverse-free vibration mode. This is because the initial geometric imperfection of the rectangular plates discussed in this study has the same shape as their fundamental transverse-free vibration mode; and the dynamic load duration is in an order of millisecond (from 0.017 to 0.038 s), which is closer to the fundamental transverse-free vibration periods of the tested plates than to the periods of higher transverse vibration modes as given in Table 2. When a plate is loaded by an intermediate-velocity impact, it deforms in its fundamental transverse vibration mode shape, namely, the initial imperfection shape, rather than the higher transverse vibration mode shape. Therefore, the buckling modes of the plates under intermediatevelocity impact follow their fundamental transverse-free vibration modes. This observation is the same as that obtained from experimental tests of the plates [16]. It should be noted that this observation is di!erent from the dynamic buckling mode of plate under high-velocity impact. For plates under high-velocity impact loads, their buckling modes may be governed by higher transverse vibration modes [1,3,18]. It can also be found from Fig. 7 that, for the plates with four edges supported, either "xed or simply supported (CCP01, CSP01 and SSP01), after dynamic buckling takes place, the transverse deformation shapes of the plates change to their second transverse vibration modes. While for the plates with two free edges (CFP01 and SFP01), the transverse deformation shapes always follow their fundamental transverse vibration modes even after dynamic buckling occurs. This indicates that, for the plates with four supported edges, their vibration modes in post-buckling process are di!erent from the original buckling modes of the plates. This is because of the e!ect of di!erent
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Fig. 7. Distributions of the maximum displacement of plates along the longitudinal direction under di!erent impact loads.
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Fig. 8. Three-dimensional views of the dynamic buckling mode of plates.
boundary conditions. For the plates with four supporting edges, the maximum transverse response occurs at a position near to the loading edge of the plate. With the increase of dynamic load, the location corresponding to the maximum transverse displacement move closer toward the loading edge. This characteristic can also be seen in Fig. 2. When the dynamic load is small, the maximum transverse response occurs at point B, but it occurs at point C or D if the dynamic load is large. As observed in the experimental tests of the plates, the e!ect of lateral supporting of plate strongly in#uences its transverse bending deformation [16]. Before dynamic buckling takes place, the maximum transverse response is small, and it occurs at a point closer to the center of the plate. In such cases, the e!ect of lateral supporting of plate is insigni"cant. Thus, the plate will vibrate and buckle in its fundamental transverse vibration mode. After dynamic buckling occurs, the transverse bending deformation of the plate increases sharply, and the e!ect of lateral supporting becomes pronounced. The lateral supports restrict the development of transverse bending deformation near the loading edge of the plate. As a result, more energy induced by the impact load is transferred to another part of the plate and makes it deform in the opposite direction. Therefore, the plate will vibrate in its second transverse vibration mode. For the plates with two free edges, there is no any lateral support, hence their transverse vibration shape remains in their fundamental transverse vibration mode. In addition, Figs. 7(d) and (e) show that the transverse vibration modes of plates CFP01 and SFP01 are, respectively, the same as that of "xed-ended columns as reported by Zhang et al. [19] and that of simply supported columns as presented by Cui et al. [20]. This indicates that the plates with two free lateral edges under intermediate-velocity impact behave as a `wide columna. Therefore, the dynamic buckling of plates with two free lateral edges and subjected to intermediate-velocity impact loads can be treated as a one-dimensional problem in the analysis. This conclusion was also drawn based on the experimental results reported by Cui et al. [16].
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7. E4ect of di4erent boundary conditions It can be noted from Table 1 that the dynamic buckling critical loads are quite di!erent for the rectangular plates with di!erent boundary conditions even though they have the same dimensions. The dynamic buckling critical load and the yielding critical dynamic load are the largest for the plates with clamped}clamped boundaries, and the smallest for the plate with simply supported-free boundaries. This indicates that, as expected, a plate will be more stable and have a higher ability to resist the plastic deformation if it is restrained more. The reason is that the lateral supports resist the transverse bending deformation of the plate and induce tensile strains. Therefore, the dynamic buckling critical load for the clamped}clamped plate is the largest. Because the lateral supports also transfer the impact energy to lateral strains, the critical plastic dynamic load is also larger for plates with more lateral restraints. This observation is the same as that obtained from the experimental tests by Cui et al. [16], which indicates again that strengthening the plate boundary constraints is a very e!ective way to enhance its ability to sustain dynamic buckling and plastic deformation.
8. E4ect of initial imperfections of plates To investigate the e!ect of initial imperfection of plate on its dynamic buckling critical load, six plates with di!erent initial imperfection magnitudes and four clamped edges are calculated with di!erent load duration t "0.001, 0.017 and 0.050 s. As an example, Fig. 9(a) shows the maximum
Fig. 9. E!ects of initial imperfection on dynamic buckling of plates.
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transverse displacement of the plates under di!erent dynamic load with duration t "0.017 s. It is obvious that, the smaller the initial imperfection, the larger the dynamic buckling load. Fig. 9(b) shows the variations of dynamic buckling loads of plates versus the initial imperfection. It shows that, for the cases of t "0.017 and 0.050 s, the critical buckling load of rectangular plate is very sensitive when the initial imperfection is small, i.e. d 40.2. The critical buckling load of plate decreases rapidly as the imperfection increases. When d '0.2, however, the relationship between q and d is basically linear, and the critical load decreases steadily with increase of the initial imperfection. For the case of t "0.001 s, however, the e!ect of initial imperfection of plate is pronounced for all the initial imperfections considered in the present study. The critical buckling load decays exponentially when d 40.2, and linearly when d '0.2 with a steep slope. These observations indicate that the load duration will in#uence the e!ect of imperfection on dynamic buckling of plate. When the duration is small, the critical buckling load of plate increases sharply as the imperfection decreases. These observations are the same as those made by Karagiozova and Jones [14].
9. E4ect of di4erent dynamic load duration As pointed out by Karagiozova and Jones [14], dynamic load duration is another important parameter that a!ects the dynamic buckling properties of structures. To inspect the in#uence of load duration on the dynamic buckling loads of plates, di!erent load duration (from t "0.001 to 0.050 s) is considered in this study. Fig. 10(a) presents the variations of the maximum transverse
Fig. 10. E!ects of dynamic load duration on dynamic buckling of plates.
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displacement of the plate with four clamped edges under di!erent dynamic loads. It shows that by maintaining the initial imperfection a constant (d "0.1 mm) and increasing the load duration, the maximum response curves have almost the same slopes in the `buckleda state. However, the buckling critical load is di!erent. A larger critical load is required to buckle a plate if its duration is short. This behavior is illustrated in Fig. 10(b), where relationships between the dynamic buckling critical loads and their duration are given for three di!erent initial imperfections. As can be seen, the dynamic buckling load of plates increases signi"cantly as load duration decreases. It should be noted that, as the load duration decreases, the dynamic buckling critical loads of plates with smaller imperfection increase faster than those of plates with larger imperfection, implying the dynamic load duration also in#uences the e!ect of initial imperfection. This observation indicates that, as observed by Karagiozova and Jones for the dynamic buckling of an idealized `spring-rigid bara model [14], the dynamic load duration has a signi"cant in#uence not only on the dynamic buckling critical load of plates, but also on the sensitivity of the initial imperfection. The sensitivity to the initial imperfection increases as the load duration decreases.
10. E4ect of hardening ratio of plate material For elastic}plastic dynamic buckling of plate, the hardening ratio of material may a!ect the dynamic buckling behavior of the plate. This e!ect is investigated in the present study by varying the plastic sti!ness, E , of the material. The variations of the maximum transverse displacement 2 response and the total energy dissipated by plasticity of plates with thickness h"2.5 mm and four clamped edges versus di!erent hardening ratios are given in Figs. 11(a) and (b), respectively. As can be seen from Fig. 11(a), unlike the curves showing in Fig. 10(a), for a constant value of load duration
Fig. 11. Variations of the maximum transverse response and total energy dissipated by plasticity of plates (h"2.5 mm, t "0.001 s).
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Fig. 12. Variations of dynamic buckling loads versus j . &
(t "0.001 s), the slopes of displacement responses with respect to the buckling loads are di!erent for di!erent values of hardening ratio, j . As indicated in the "gure, the plate will yield at about & q "850 N/mm where the slope of the displacement response changes. It is interesting to note that, although the plate yielding is independent of the hardening ratio, the buckling critical load depends on it. This is because plastic deformation is induced in the 2.5 mm thick plate before it buckles as discussed above. A higher dynamic buckling load is required to cause the plate to buckle if the hardening ratio is large. This is because the plates with di!erent hardening ratios may result in di!erent plastic strains and dissipate di!erent amount of energy. It is obvious that, for the plate materials having larger hardening ratios, smaller plastic strains develop and lower plastic deformation energy dissipates as shown in Fig. 11(b), thus the lateral displacements grow less rapidly. As a result, the slopes of the maximum transverse displacement response curves are smaller and larger impact loads are needed to buckle the plates. Fig. 12 shows the dynamic buckling critical loads of plates as a function of the hardening ratio. It is evident that the dynamic buckling load increases with the increase of the hardening ratio of plate material. Furthermore, as can be seen from the "gure that the in#uence of the hardening ratio on the dynamic buckling of plates also depends on the dynamic load duration. This draws the conclusions again that the e!ect of load duration is signi"cant. These observations are identical with that obtained from the study of the dynamic buckling of an idealized model by Karagiozova and Jones [14]. It should be noted that, for the present 2.0 mm thick plates, the hardening ratio has no in#uence on their dynamic buckling loads, because they all buckle before dynamic yielding. Therefore, the hardening ratio only a!ects the elastic}plastic buckling of the plates.
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11. Conclusions Dynamic buckling of rectangular plates subjected to intermediate-velocity impact loads has been numerically investigated. By observing the dynamic response characteristics of the plates, a dynamic buckling critical condition and a dynamic yielding condition have been de"ned. The results indicate that, for the present rectangular plates subjected to intermediate-velocity impact, dynamic buckling always occurs within the impact load duration. The dynamic buckling mode of the plates is the same as their fundamental transverse vibration mode. However, after the dynamic buckling takes place, the vibration shape of the plates with four supporting edges will change to their second transverse vibration mode because of the e!ect of lateral supporting boundaries. Strengthening the plate boundary constraints is a very e!ective way to enhance its stability. When two lateral edges of a plate are free, the lateral e!ect of the plate is insigni"cant and can be neglected, so that the dynamic buckling analysis can be simpli"ed to a one-dimensional problem. Initial imperfection has signi"cant in#uence on the dynamic buckling properties of plates. The dynamic buckling critical load of plate decreases rapidly as the initial imperfection increases, especially when the load duration is small. Dynamic load duration is another important parameter which strongly a!ects the dynamic buckling properties of plates under intermediate-velocity impact. Shorter load duration associates with higher dynamic buckling critical load. Besides, the dynamic load duration also has a signi"cant in#uence on the sensitivity of the initial imperfection. The in#uence of the initial imperfection increases as the duration decreases. Di!erent hardening ratios of plate material a!ect the elastic}plastic buckling of plates. The dynamic buckling load increases as the hardening ratio of plate material increases if the plate buckles plastically.
References [1] Lindberg HE, Florence AL. Dynamic pulse buckling * theory and experiment. Dordrecht: Martinus Nijho! Publishers, 1987. [2] Zizicas GA. Dynamic buckling of thin elastic plates. Trans ASME 1952;74(7):1257}68. [3] Birkgan AY, Vol'mir, An investigation of the dynamic stability of plates using an electronic digital computer. Sov Phys Dokl (English translation) 1961;5(6):1364}6. [4] Ari-Gur J, Singer J, Weller T. Dynamic buckling of plates under longitudinal impact. Israel J Tech 1981;19: 57}64. [5] Passic H, Herrmann G. E!ect of in-plane inertia on buckling of imperfect plates with large deformation. J Sound Vib 1984;95(4):469}78. [6] Weller T, Abramovich H, Ya!e R. Dynamic buckling of beams and plates subjected to axial impact. Comput Struct 1989;32(3}4):835}51. [7] Hac M. Dynamic stability of a de#ected non-linear rectangular plate. J Sound Vib 1989;129(2):327}34. [8] Subramanian S. Dynamic stability of viscoelastic plates subjected to randomly varying in-plane loads. Proceedings of Engineering Mechanics, vol. 1, ASCE, New York, NY, USA, 1995. p. 191}4. [9] Tam LL, Calladine CR. Inertia and strain-rate e!ects in a simple plate-structure under impact loading. Int J Impact Engng 1991;11(3):349}77. [10] Simitses GJ. Instability of dynamically-loaded structures. Appl Mech Rev 1987;40(10):1403}8. [11] Jones N. Recent studies on the dynamic plastic behavior of structures. Appl Mech Rev 1989;42(4):95}115.
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[12] Jones N. Recent studies on the dynamic plastic behavior of structures * an update. Appl Mech Rev 1996;49(10):S112}7. [13] Karagiozova D, Jones N. Dynamic buckling of a simple elastic}plastic model under pulse loading. Int J Non-linear Mech 1992;27(6):981}1005. [14] Karagiozova D, Jones N. Dynamic pulse buckling of a simple elastic}plastic model including axial inertia. Int J Solid Struct 1992;29(10):1255}72. [15] Karagiozova D, Jones N. Some observations on the dynamic elastic}plastic buckling of a structural model. Int J Impact Engng 1995;16(4):621}35. [16] Shijie Cui, Hee Kiat Cheong, Hong Hao, Experimental study of dynamic buckling of plates under #uid}solid slamming. Int J. Impact Engng 1999;22:675}91. [17] Hibbitt, Karlsson, Sorensen, Inc. ABAQUS/standard user's manual; ABAQUS theory manual; and ABAQUS/standard example problem manual, 1997. [18] Karagiozova D, Jones N. Multi-degrees of freedom model for dynamic buckling of an elastic}plastic structure. Int J Solid Struct 1996;33(23):3377}98. [19] Zhang Q, Li S, Zheng J. Dynamic response, buckling and collapsing of elastic}plastic straight columns under axial solid}#uid slamming compression * I. Experiments. Int J Solids Struct 1992;29(3):381}97. [20] Shijie Cui, Hee Kiat Cheong and Hong Hao, Experimental study on dynamic buckling of simply-supported columns under axial slamming. J Engng Mech ASCE 1999;125(5):513}20.
Numerical analysis of dynamic buckling of rectangular plates subjected to intermediate-velocity impact Shijie Cui, Hong Hao*, Hee Kiat Cheong School of Civil and Structural Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Received 19 August 1999; received in revised form 6 December 1999; accepted 3 June 2000
Abstract This paper numerically investigates the dynamic buckling of thin imperfect rectangular plates subjected to intermediate-velocity impact loads. From numerical results obtained, a dynamic buckling and a dynamic yielding critical condition are de"ned, and the corresponding critical dynamic loads are estimated. Numerical model employed in the present study is validated by experimental data reported earlier. Results from parametric study indicate that initial imperfection and load duration have signi"cant in#uence on the dynamic buckling of the plates. The smaller the initial imperfection and the load duration, the higher the dynamic buckling critical loads of the plates. Moreover, di!erent hardening ratios of plate material also a!ect the elastic}plastic dynamic buckling properties of the plates. If the plate buckles plastically, the dynamic buckling load increases as the hardening ratio of plate material increases. Unlike thin plates under high-velocity impact that buckling always occur after load application, plates under intermediate-velocity impact analyzed in the present study all buckle during the loading phase. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Dynamic buckling of structures under impact loads can be divided into three categories according to the applied dynamic loads and structural responses, viz., high-velocity impacted buckling, low-velocity impacted buckling and intermediate-velocity impacted buckling. For the problem of high-velocity impacted buckling of structures, the dynamic load poses a very high amplitude and short duration (in an order of microseconds). Thus, this kind of the dynamic load
* Corresponding author. Tel.: #65-791-1744; fax: #65-791-0676. E-mail address: [email protected] (H. Hao). 0734-743X/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 4 - 7 4 3 X ( 0 0 ) 0 0 0 3 5 - X
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Nomenclature B [B] [C] +d, +dQ , +d$ , G h L M [M] q(t) q q q # q q # +R, t t T G w (x, y) d j & p +p, m
width of the plate (mm) strain}displacement transformation matrix of the plate material damping matrix nodal displacement vector of plate element nodal velocity vector of plate element nodal acceleration vector of plate element total energy dissipation due to plastic response (J) thickness of the plate (mm) length of the plate (mm) bending moment per unit width about the y-axis of the plate (N mm) mass matrix of the plate uniformly distributed load on plate edge (N/mm) peak value of the dynamic load (N/mm) dynamic buckling critical load amplitude (N/mm) experimental result of dynamic buckling critical load amplitude (N/mm) dynamic yielding critical load amplitude (N/mm) experimental result of dynamic yielding critical load amplitude (N/mm) equivalent external load vector of the plate time (s) dynamic load duration (s) periods of transverse-free vibration of the plate (s), i"1, 2,2, 5 initial imperfection function of the plate (mm) maximum initial imperfection of the plate (mm) Hardening ratio of the plate material, j "E /E, E and E are the elastic and & 2 2 plastic modulus of the plate material, respectively yield stress of the plate material (MPa) stress vector of the plate Rayleigh-type viscous damping ratio
can be simpli"ed as a pulse load, and this problem is also called `pulse bucklinga [1]. The experimental studies of this problem are usually realized by striking the structure with a moving mass or moving it towards a rigid surface (solid}solid impact). For the problem of low-velocity impacted buckling, the dynamic load duration is much longer while its amplitude is relatively lower than those of the high-velocity impact. It may be simpli"ed as a step load with a constant amplitude and in"nite duration. Under these simpli"cations, solving the dynamic buckling equation of structures is relatively straightforward. Most of the previous studies of dynamic buckling of structures are focused on these two types of the problem. The beginning of the studies of dynamic buckling of suddenly loaded plates may be traced back to that by Zizicas [2] in 1952, who, by neglecting the in-plane inertia e!ects, presented a theoretical solution for simply supported plate under in-plane time-dependent load. However, neither critical
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condition nor buckling criterion was investigated in this study. Birkgan and Vol'mir [3] also studied the dynamic buckling properties of plates under sudden loads using an electronic digital computer. Unlike Zizicas, they estimated the dynamic buckling loads of the plates and found that the high loading rates (high-velocity impact) resulted in a higher-order dynamic buckling mode of the plates in their in-plane loading direction. Ari-Gur et al. [4] investigated theoretically and experimentally the dynamic buckling of a rectangular plate impacted by a mass M moving in an in-plane direction. They obtained the critical in-plane strain at dynamic buckling of plate and discussed the e!ect of initial imperfection and the load duration on plate buckling properties. Passic and Herrmann [5] studied theoretically the nonlinear dynamic buckling behavior of imperfect rectangular plates under in-plane compressive step loads. Weller et al. [6] performed analytical studies by using the ADINA computer code to determine the dynamic load ampli"cation factor (DLF) of metal beams and plates subjected to axial in-plane impact compressive loads. More literature on dynamic buckling of plates under high- and low-velocity impacts can be found in the publications of Hac [7], Lindberg and Florence [1], Subramanian [8], Tam and Calladine [9], and review papers by Simitses [10] and Jones [11,12]. For the intermediate-velocity impacted buckling of structures, the dynamic load has a moderate amplitude and duration (in an order of milliseconds). It can neither be simpli"ed as a pulse load nor a step load with in"nite duration. In such a problem, the load duration is a very important parameter which will strongly a!ect the dynamic buckling properties of structures. The dynamic buckling characteristics of structures may be much di!erent from those of structures under the high- and low-velocity impacts. The studies of this problem for structures, especially for plates, however, are very limited in literature up to now. Karagiozova and Jones [13}15] investigated the dynamic elastic}plastic buckling of a twodegree-of-freedom model subjected to a rectangular pulse load and two triangular loads. In#uences of initial imperfection, dynamic load shape and duration, axial inertia and plastic reloading on the dynamic buckling behavior of the model were examined in these studies. They concluded that both the initial imperfection and the dynamic load duration strongly a!ect the dynamic buckling properties of the model, and load duration also has a signi"cant in#uence on the sensitivity of the initial imperfection. Recently, Cui et al. [16] reported an experimental study of the dynamic buckling of rectangular plates under in-plane intermediate-velocity impacts. The intermediate-velocity impacts were generated by #uid}solid slamming. The dynamic buckling and dynamic yielding properties were investigated based on the observation of elastic}plastic dynamic response characteristics of the plates. The e!ect of di!erent boundary conditions on the elastic}plastic dynamic buckling properties of plates was also examined in the study. This paper presents a numerical study of dynamic buckling of imperfect rectangular plates subjected to intermediate-velocity impact loads by using the computer code ABAQUS [17]. The emphasis of the study is focused on the parameters that a!ect the dynamic buckling of the plates. The primary objectives are to investigate the dynamic buckling mechanism of plates subjected to intermediate-velocity impact loads, to de"ne a suitable dynamic buckling criterion and to discuss the e!ects of initial imperfection, boundary conditions, dynamic load duration, and the hardening ratio of plate material on the dynamic buckling characteristics of the plates.
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2. Numerical model of plates For a plate under intermediate-velocity impact load (Fig. 1a), its equilibrium equation can be written as
[M]+d$ ,#[C]+dQ ,#
[B]2+p, d<"+R,,
(1)
4
where [M] and [C] are mass and viscous damping matrices; [B] is the strain}displacement transformation matrix; +d , is the nodal displacement vector and the dot represents taking
Fig. 1. A rectangular plate and its "nite element mesh.
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derivative with respect to time; +p, is the stress vector of the plate and +R, is the equivalent load vector. In the present study, compatible mass matrix is used in "nite element modeling. It should be noted that the e!ect of viscous damping is not important in dynamic responses of plates in this study because of the short duration of the impact load. However, for modeling completeness, Rayleigh-type viscous damping of 2.5% critical ratio (m "0.025) corresponding to the "rst two modes of the plate is speci"ed in the analysis. The equivalent load vector is obtained in "nite element modeling by using the shape function and the impact load q(t) acting at the loading edge of the plate. q(t) is assumed to be having a half-sine form as observed in test [16]. Thus, it can be de"ned by specifying its amplitude and duration. In the numerical analysis, large de#ections are considered by including the second-order terms in strain estimation:
*w *u 1 *w 1 *w !z , ! e " # V *x 2 *x 2 *x *x *w *v 1 *w 1 *w !z ! , e " # W *y 2 *y 2 *y *y
(2)
*u *v *w *w *w *w *w !2z c " # # ! , VW *y *x *x *y *x *y *x*y where u, v, w are the longitudinal, lateral and transverse displacements (in the x, y and z directions) of the plate; w is its initial transverse deformation. A bi-linear elastic}plastic model is employed to describe the nonlinear property of the material. The material properties of the model are: elastic modulus E"2.1;10 MPa, plastic modulus E "1.519;10 MPa and the yield stress p "289 MPa, respectively. 2 3. Dynamic response characteristics of plates Eighteen rectangular plates are analyzed in this numerical study. Table 1 lists the dimensions of the plates. In order to calibrate the numerical model, some of the plates are selected to have the same dimensions as those of the tested plates [16]. It should be noted that, in the experimental test of plates [16], impact duration depends on the dimensions and boundary conditions of the plate specimens, as well as the test set-up. Because the same set-up was employed in the test [16], for a particular plate with the prescribed dimensions and boundary conditions, the impact duration is basically unchanged even the impact load amplitude is di!erent. Thus, the dynamic load duration is also taken as a constant for each type of plates in the numerical analysis and is listed in Table 1. The initial imperfection of the plates is assumed following the same shape as their "rst mode of transverse-free vibration. Therefore, only the maximum value, d , is given in the table. In order to investigate the e!ect of di!erent boundary conditions on the dynamic buckling of plates, "ve types of boundary conditions are considered in this study as indicated in Table 1. The "rst three types, which are the same as those implemented in the experimental tests, are clamped on both top and bottom edges, and the two vertical sides of the plates are clamped (CCP), hinged (CSP) or free (CFP), respectively. The other two types of the boundary conditions are simply supported at four
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Table 1 Dimensions, initial imperfection and critical dynamic loads of plates
Plate no.
Dimension (mm) L B
h
t (s)
d (mm)
q (N/mm)
q (N/mm)
Boundary conditions
CCP01 CCP02 CCP03 CCP04 CCP05 CCP06 CSP01 CSP02 CSP03 CFP01 CFP02 CFP03 SSP01 SSP02 SSP03 SFP01 SFP02 SFP03
500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500
2.0 2.0 2.0 2.5 2.5 2.5 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
0.017 0.017 0.017 0.001 0.006 0.038 0.0175 0.0175 0.0175 0.019 0.019 0.019 0.022 0.022 0.022 0.038 0.038 0.038
0.11 0.10 0.12 0.15 0.15 0.15 0.11 0.10 0.13 0.11 0.10 0.12 0.10 0.12 0.14 0.10 0.12 0.14
198 202 192 1188 505 300 148 152 141 78 80 76 81 77 75 50 47 45
233 240 230 850 460 285 187 192 178 152 157 148 163 156 151 98 81.5 73.5
Clamped}clamped Clamped}clamped Clamped}clamped Clamped}clamped Clamped}clamped Clamped}clamped Clamped}hinged Clamped}hinged Clamped}hinged Clamped}free Clamped}free Clamped}free Hinged}hinged Hinged}hinged Hinged}hinged Hinged}free Hinged}free Hinged}free
325 325 325 325 325 325 325 325 325 325 325 325 325 325 325 325 325 325
Table 2 Transverse free vibration periods of plates T (s) Plate no.
T
T
T
T
T
CCP01 CCP04 CSP01 CFP01 SSP01 SFP01
0.0079 0.0063 0.0124 0.0226 0.0151 0.0524
0.0052 0.0042 0.0062 0.0082 0.0080 0.0129
0.0033 0.0026 0.0036 0.0068 0.0045 0.0073
0.0022 0.0017 0.0023 0.0042 0.0028 0.0057
0.0017 0.0014 0.0022 0.0041 0.0023 0.0047
edges (SSP) and hinged-free (SFP), viz., the top and the bottom edges are simply supported and the two vertical edges of the plates are free. Moreover, in order to simulate the elastic and plastic buckling, plates with di!erent thickness are analyzed, they are 2 and 2.5 mm thick plates, respectively. The "rst "ve periods of transverse-free vibration of a few typical plates are given in Table 2. In "nite element modeling, each plate is divided into 12 segments in each direction in x}y plane as shown in Fig. 1(b). Eight-node isoparametric element is used. Thus, there are a total of 144 elements and 481 nodes for each plate in the numerical calculation.
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In numerical analysis, a series of dynamic responses of the plates are calculated by loading each plate with the half-sine form impact load q(t) with duration given in Table 1 and di!erent amplitudes. Fig. 2 shows the displacement time histories at points A}D as indicated in Fig. 1(b) of plate CCP01 under several load amplitudes. Firstly, it can be seen from the "gure that the maximum displacement always occurs during the load duration t . The displacement responses of plate decay rapidly after the load application. The displacement responses for other plates, which are not shown here, have similar characteristics as those of CCP01 shown in Fig. 2. This indicates that the plates may buckle during the dynamic load duration. This dynamic response characteristic of the plates subjected to intermediate-velocity impacts is di!erent from that of plates under high-velocity impacts (solid}solid impacts). This is because the duration of an intermediate-velocity dynamic load is far longer than that of a high-velocity impact load. For the high-velocity impacted buckling, the load duration is usually in an order of microsecond, and the loading will be completed before the deformation of the plate is fully developed. So that dynamic buckling of plate always occurs after load application [1,4]. For the intermediate-velocity impacted buckling, however, the millisecond duration is su$cient to allow the bending deformation of plate to develop before the end of load application. Thus, the maximum response of the plate occurs during the load duration, and the dynamic buckling may take place before the dynamic load is completed. This observation is identical with that observed in experimental tests of the plates [16]. As can also be seen from Fig. 2, the maximum displacement response of the plate increases along with the increase of the dynamic load. When the dynamic load is small (e.g. q 4200 N/mm), the maximum displacement of the plate increases gradually. When the dynamic load is large (e.g. q '200 N/mm), the maximum transverse displacement response of the plate increases rapidly. This is more clearly illustrated in Fig. 3(a), in which the maximum displacement of the plate is plotted as a function of the dynamic load amplitude. As can be seen from the "gure, the curve of the maximum transverse displacement of the plate has a `kneea at a `critical dynamic loada. When the dynamic load is smaller than this critical load, the slope of the curve is #at, indicating the development of the maximum transverse displacement of the plate is steady as the dynamic load increases. When the dynamic load is larger than the critical load, the slope of the curve becomes very steep, indicating the maximum displacement of the plate increases rapidly for a small increment of the dynamic load. The above observation indicates that there exists a critical condition for the maximum displacement response of the plate in the process of increasing impact load from small to large. When the dynamic load is small, any increment in the dynamic load will cause steady increment in the displacement responses; when the dynamic load is large, however, any increment in dynamic load amplitude will result in a signi"cant increment in dynamic displacement responses of the plate. As will be discussed later, at q(t)"200 N/mm, the plate is still in elastic range. Thus, it is obvious that the change of the maximum transverse displacement of the plate from steady increase when the load is smaller than q(t)"200 N/mm to rapid increase when the load is larger than that is caused by dynamic buckling of the plate. According to this observation, a dynamic buckling criterion of plate is de"ned as follows: for an impacted plate with a prescribed aspect ratio, the point corresponding to the sudden rapid increase of the maximum transverse displacement is the dynamic buckling critical condition of the plate. The corresponding dynamic load is de"ned as the dynamic buckling critical load and denoted by q .
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Fig. 2. Displacement response time histories of CCP01 under half-sine impact loads with di!erent amplitudes (t "0.017 s, m "0.025).
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Fig. 3. The maximum displacement and bending moment at cross-section of the plate CCP01 versus dynamic load.
To further verify the buckling condition discussed above, Fig. 3(b) displays the variation of the maximum bending moment force per unit width about y axis at point 2 as shown in Fig. 1(b) versus di!erent dynamic load amplitudes. As can be seen from this "gure, when the dynamic load is small, the bending moment of the plate is nearly zero, implying the dynamic response is primarily in-plane compression and the plate is stable. When the impact load increases to the critical load as shown in Fig. 3(a), the bending moment begin to increase rapidly, indicating that the bending deformation of the plate becomes pronounced and dynamic buckling of the plate takes place. Fig. 4 shows the Mises stress time histories on the compressive side of the plate surface at points 1}3 as shown in Fig. 1(b) of plate CCP01 under di!erent dynamic loads. The variation of the maximum Mises stresses versus the loading amplitude is given in Fig. 5. As can be seen from Fig. 4, the peak value of Mises stress of plate also occurs within the load duration (t "0.017 s for CCP01) as expected since the maximum displacement response occurs during the load duration as discussed above. The Mises stress grows with the increase of the dynamic load as shown in Figs. 4 and 5. Before the dynamic buckling takes place, the dynamic load is small, and the transverse displacement of the plate is insigni"cant. As a result, the Mises stress of the plate is not large. After the dynamic buckling occurs, the maximum Mises stress of the plate increases sharply because of the rapid increase in the transverse deformation of the plate. When the dynamic load increases to a certain value, the maximum Mises stress reaches to the yield stress of plate material. This indicates that the plastic deformation begins to appear in the plate, and the dynamic response of the plate consists of both elastic and plastic deformation. It can be seen from Fig. 2, the residual plastic deformation can be observed when q "288 N/mm, and it becomes signi"cant when q "332 N/mm. In addition, after the plastic deformation occurs in the plate, the increase of the maximum Mises stress of the plate is slowed down as the dynamic load increases. This is because plastic deformation of the plate absorbs more impact energy. The critical point that
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Fig. 4. Mises stress time histories of CCP01 under di!erent dynamic load amplitudes (t "0.017 s).
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Fig. 5. Variation of the maximum Mises stress of CCP01 versus dynamic load amplitude.
Fig. 6. Maximum responses of plate CCP05 versus dynamic load amplitude (t "0.006 s, m "0.025).
when the maximum Mises stress of the plate reaches the material yield stress is de"ned as the critical yielding condition. The corresponding load is the dynamic yielding critical load, and is denoted by q . For the plates with thickness h"2.5 mm, the dynamic buckling characteristic is di!erent. Fig. 6 shows the variations of the maximum transverse displacement and the maximum Mises stress on the compressive surface of plate CCP05 versus dynamic load amplitude. When the dynamic
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load amplitude increases to approximately 460 N/mm, the maximum Mises stress of the plate reaches its yielding stress of material. At this level of the dynamic load, the slope of the maximum displacement curve changes drastically. This is because local plastic deformation appears in the plate although most parts of the plate remain elastic. When the dynamic load increases to q "505 N/mm, the slope of the maximum displacement curve shows another change. This is believed to be caused by dynamic buckling of the plate. Unlike the plate CCP01, the plate CCP05 with thickness h"2.5 mm buckles plastically. When a plate buckles elastically, its maximum Mises stress increases rapidly with dynamic load as shown in Fig. 5 for plate CCP01. However, when a plate buckles plastically, the increase of the maximum Mises stress is steady during the post-buckling process as shown in Fig. 6. This is because the plate CCP05 yields before buckling. After the dynamic yielding occurs, plastic deformation appears in the plate, and more impact energy is absorbed by the plastic deformation. Thus, although buckling will cause rapid increase of the stresses in the plate, the rate of stress increment will not be as signi"cant as that in an elastically buckled plate.
4. Determination of dynamic critical loads To estimate the dynamic buckling critical load and the yielding critical load, curves of the maximum displacement response and the maximum Mises stress of the plates are plotted with respect to the dynamic load. According to the above de"nitions of dynamic buckling and dynamic yielding critical conditions, the corresponding critical dynamic loads can be easily determined. Figs. 3, 5 and 6 are examples for plate CCP01 and plate CCP05. From these "gures, it can be obtained that q "198 N/mm, q "233 N/mm for plate CCP01, and q "505 N/mm and q "460 N/mm for plate CCP05. The curves for other plates, which are not shown here, are similar. The corresponding critical dynamic loads are listed in Table 1. It is obvious that the plates with thickness h"2.0 mm in the present study buckle elastically, while those with h"2.5 mm buckle plastically.
5. Comparison with the experimental results For comparison, results obtained in this numerical analysis and the corresponding test results for the critical load amplitudes obtained by Cui et al. [16], which are denoted as q and q , # # are listed in Table 3. It can be noted from the table that the numerical results of dynamic buckling critical loads of the plates agree well with the experimental results. The errors are about 2.63}4.96% for the 9 tested plates with an average error of 3.49%. The errors between numerical and experimental results for yielding critical loads of the plates are about 6.67}9.03% with an average error of 8.18%. The error between the experimental and the numerical results might be attributed to the facts that (1) the impact loads used in this numerical study are not exactly the same as the actual impact loads [16]; (2) small loading eccentricity is inevitable for each plate in the test, but no loading eccentricity is considered in the numerical study; and (3) the relationship between stress and strain of plate material used in this numerical analysis is bilinear, which is not exactly the same as the specimen material property.
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Table 3 Comparison with the experimental results
Plate no.
q # (N/mm)
q (N/mm)
Error %
q # (N/mm)
q (N/mm)
Error %
CCP01 CCP02 CCP03 CSP01 CSP02 CSP03 CFP01 CFP02 CFP03
192 196 184 152 158 148 76 77 74
198 202 192 148 152 141 78 80 76
3.13 3.06 4.35 2.70 3.95 4.96 2.63 3.90 2.70
214 222 212 172 180 165 140 144 138
233 240 230 187 192 178 152 157 148
8.88 8.11 8.49 8.72 6.67 7.88 8.57 9.03 7.25
mean
3.49
8.18
6. Dynamic buckling mode of plates Fig. 7 shows the distributions of the maximum displacements of plates CCP01, CSP01, SSP01, CFP01 and SFP01 along the longitudinal direction and under di!erent load amplitudes. Fig. 8 displays the three-dimensional deformations of the plates under the corresponding dynamic buckling critical loads. It can be found from these "gures that the dynamic buckling modes of the plates are dominated by their fundamental transverse-free vibration mode. This is because the initial geometric imperfection of the rectangular plates discussed in this study has the same shape as their fundamental transverse-free vibration mode; and the dynamic load duration is in an order of millisecond (from 0.017 to 0.038 s), which is closer to the fundamental transverse-free vibration periods of the tested plates than to the periods of higher transverse vibration modes as given in Table 2. When a plate is loaded by an intermediate-velocity impact, it deforms in its fundamental transverse vibration mode shape, namely, the initial imperfection shape, rather than the higher transverse vibration mode shape. Therefore, the buckling modes of the plates under intermediatevelocity impact follow their fundamental transverse-free vibration modes. This observation is the same as that obtained from experimental tests of the plates [16]. It should be noted that this observation is di!erent from the dynamic buckling mode of plate under high-velocity impact. For plates under high-velocity impact loads, their buckling modes may be governed by higher transverse vibration modes [1,3,18]. It can also be found from Fig. 7 that, for the plates with four edges supported, either "xed or simply supported (CCP01, CSP01 and SSP01), after dynamic buckling takes place, the transverse deformation shapes of the plates change to their second transverse vibration modes. While for the plates with two free edges (CFP01 and SFP01), the transverse deformation shapes always follow their fundamental transverse vibration modes even after dynamic buckling occurs. This indicates that, for the plates with four supported edges, their vibration modes in post-buckling process are di!erent from the original buckling modes of the plates. This is because of the e!ect of di!erent
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Fig. 7. Distributions of the maximum displacement of plates along the longitudinal direction under di!erent impact loads.
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Fig. 8. Three-dimensional views of the dynamic buckling mode of plates.
boundary conditions. For the plates with four supporting edges, the maximum transverse response occurs at a position near to the loading edge of the plate. With the increase of dynamic load, the location corresponding to the maximum transverse displacement move closer toward the loading edge. This characteristic can also be seen in Fig. 2. When the dynamic load is small, the maximum transverse response occurs at point B, but it occurs at point C or D if the dynamic load is large. As observed in the experimental tests of the plates, the e!ect of lateral supporting of plate strongly in#uences its transverse bending deformation [16]. Before dynamic buckling takes place, the maximum transverse response is small, and it occurs at a point closer to the center of the plate. In such cases, the e!ect of lateral supporting of plate is insigni"cant. Thus, the plate will vibrate and buckle in its fundamental transverse vibration mode. After dynamic buckling occurs, the transverse bending deformation of the plate increases sharply, and the e!ect of lateral supporting becomes pronounced. The lateral supports restrict the development of transverse bending deformation near the loading edge of the plate. As a result, more energy induced by the impact load is transferred to another part of the plate and makes it deform in the opposite direction. Therefore, the plate will vibrate in its second transverse vibration mode. For the plates with two free edges, there is no any lateral support, hence their transverse vibration shape remains in their fundamental transverse vibration mode. In addition, Figs. 7(d) and (e) show that the transverse vibration modes of plates CFP01 and SFP01 are, respectively, the same as that of "xed-ended columns as reported by Zhang et al. [19] and that of simply supported columns as presented by Cui et al. [20]. This indicates that the plates with two free lateral edges under intermediate-velocity impact behave as a `wide columna. Therefore, the dynamic buckling of plates with two free lateral edges and subjected to intermediate-velocity impact loads can be treated as a one-dimensional problem in the analysis. This conclusion was also drawn based on the experimental results reported by Cui et al. [16].
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7. E4ect of di4erent boundary conditions It can be noted from Table 1 that the dynamic buckling critical loads are quite di!erent for the rectangular plates with di!erent boundary conditions even though they have the same dimensions. The dynamic buckling critical load and the yielding critical dynamic load are the largest for the plates with clamped}clamped boundaries, and the smallest for the plate with simply supported-free boundaries. This indicates that, as expected, a plate will be more stable and have a higher ability to resist the plastic deformation if it is restrained more. The reason is that the lateral supports resist the transverse bending deformation of the plate and induce tensile strains. Therefore, the dynamic buckling critical load for the clamped}clamped plate is the largest. Because the lateral supports also transfer the impact energy to lateral strains, the critical plastic dynamic load is also larger for plates with more lateral restraints. This observation is the same as that obtained from the experimental tests by Cui et al. [16], which indicates again that strengthening the plate boundary constraints is a very e!ective way to enhance its ability to sustain dynamic buckling and plastic deformation.
8. E4ect of initial imperfections of plates To investigate the e!ect of initial imperfection of plate on its dynamic buckling critical load, six plates with di!erent initial imperfection magnitudes and four clamped edges are calculated with di!erent load duration t "0.001, 0.017 and 0.050 s. As an example, Fig. 9(a) shows the maximum
Fig. 9. E!ects of initial imperfection on dynamic buckling of plates.
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transverse displacement of the plates under di!erent dynamic load with duration t "0.017 s. It is obvious that, the smaller the initial imperfection, the larger the dynamic buckling load. Fig. 9(b) shows the variations of dynamic buckling loads of plates versus the initial imperfection. It shows that, for the cases of t "0.017 and 0.050 s, the critical buckling load of rectangular plate is very sensitive when the initial imperfection is small, i.e. d 40.2. The critical buckling load of plate decreases rapidly as the imperfection increases. When d '0.2, however, the relationship between q and d is basically linear, and the critical load decreases steadily with increase of the initial imperfection. For the case of t "0.001 s, however, the e!ect of initial imperfection of plate is pronounced for all the initial imperfections considered in the present study. The critical buckling load decays exponentially when d 40.2, and linearly when d '0.2 with a steep slope. These observations indicate that the load duration will in#uence the e!ect of imperfection on dynamic buckling of plate. When the duration is small, the critical buckling load of plate increases sharply as the imperfection decreases. These observations are the same as those made by Karagiozova and Jones [14].
9. E4ect of di4erent dynamic load duration As pointed out by Karagiozova and Jones [14], dynamic load duration is another important parameter that a!ects the dynamic buckling properties of structures. To inspect the in#uence of load duration on the dynamic buckling loads of plates, di!erent load duration (from t "0.001 to 0.050 s) is considered in this study. Fig. 10(a) presents the variations of the maximum transverse
Fig. 10. E!ects of dynamic load duration on dynamic buckling of plates.
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displacement of the plate with four clamped edges under di!erent dynamic loads. It shows that by maintaining the initial imperfection a constant (d "0.1 mm) and increasing the load duration, the maximum response curves have almost the same slopes in the `buckleda state. However, the buckling critical load is di!erent. A larger critical load is required to buckle a plate if its duration is short. This behavior is illustrated in Fig. 10(b), where relationships between the dynamic buckling critical loads and their duration are given for three di!erent initial imperfections. As can be seen, the dynamic buckling load of plates increases signi"cantly as load duration decreases. It should be noted that, as the load duration decreases, the dynamic buckling critical loads of plates with smaller imperfection increase faster than those of plates with larger imperfection, implying the dynamic load duration also in#uences the e!ect of initial imperfection. This observation indicates that, as observed by Karagiozova and Jones for the dynamic buckling of an idealized `spring-rigid bara model [14], the dynamic load duration has a signi"cant in#uence not only on the dynamic buckling critical load of plates, but also on the sensitivity of the initial imperfection. The sensitivity to the initial imperfection increases as the load duration decreases.
10. E4ect of hardening ratio of plate material For elastic}plastic dynamic buckling of plate, the hardening ratio of material may a!ect the dynamic buckling behavior of the plate. This e!ect is investigated in the present study by varying the plastic sti!ness, E , of the material. The variations of the maximum transverse displacement 2 response and the total energy dissipated by plasticity of plates with thickness h"2.5 mm and four clamped edges versus di!erent hardening ratios are given in Figs. 11(a) and (b), respectively. As can be seen from Fig. 11(a), unlike the curves showing in Fig. 10(a), for a constant value of load duration
Fig. 11. Variations of the maximum transverse response and total energy dissipated by plasticity of plates (h"2.5 mm, t "0.001 s).
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Fig. 12. Variations of dynamic buckling loads versus j . &
(t "0.001 s), the slopes of displacement responses with respect to the buckling loads are di!erent for di!erent values of hardening ratio, j . As indicated in the "gure, the plate will yield at about & q "850 N/mm where the slope of the displacement response changes. It is interesting to note that, although the plate yielding is independent of the hardening ratio, the buckling critical load depends on it. This is because plastic deformation is induced in the 2.5 mm thick plate before it buckles as discussed above. A higher dynamic buckling load is required to cause the plate to buckle if the hardening ratio is large. This is because the plates with di!erent hardening ratios may result in di!erent plastic strains and dissipate di!erent amount of energy. It is obvious that, for the plate materials having larger hardening ratios, smaller plastic strains develop and lower plastic deformation energy dissipates as shown in Fig. 11(b), thus the lateral displacements grow less rapidly. As a result, the slopes of the maximum transverse displacement response curves are smaller and larger impact loads are needed to buckle the plates. Fig. 12 shows the dynamic buckling critical loads of plates as a function of the hardening ratio. It is evident that the dynamic buckling load increases with the increase of the hardening ratio of plate material. Furthermore, as can be seen from the "gure that the in#uence of the hardening ratio on the dynamic buckling of plates also depends on the dynamic load duration. This draws the conclusions again that the e!ect of load duration is signi"cant. These observations are identical with that obtained from the study of the dynamic buckling of an idealized model by Karagiozova and Jones [14]. It should be noted that, for the present 2.0 mm thick plates, the hardening ratio has no in#uence on their dynamic buckling loads, because they all buckle before dynamic yielding. Therefore, the hardening ratio only a!ects the elastic}plastic buckling of the plates.
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11. Conclusions Dynamic buckling of rectangular plates subjected to intermediate-velocity impact loads has been numerically investigated. By observing the dynamic response characteristics of the plates, a dynamic buckling critical condition and a dynamic yielding condition have been de"ned. The results indicate that, for the present rectangular plates subjected to intermediate-velocity impact, dynamic buckling always occurs within the impact load duration. The dynamic buckling mode of the plates is the same as their fundamental transverse vibration mode. However, after the dynamic buckling takes place, the vibration shape of the plates with four supporting edges will change to their second transverse vibration mode because of the e!ect of lateral supporting boundaries. Strengthening the plate boundary constraints is a very e!ective way to enhance its stability. When two lateral edges of a plate are free, the lateral e!ect of the plate is insigni"cant and can be neglected, so that the dynamic buckling analysis can be simpli"ed to a one-dimensional problem. Initial imperfection has signi"cant in#uence on the dynamic buckling properties of plates. The dynamic buckling critical load of plate decreases rapidly as the initial imperfection increases, especially when the load duration is small. Dynamic load duration is another important parameter which strongly a!ects the dynamic buckling properties of plates under intermediate-velocity impact. Shorter load duration associates with higher dynamic buckling critical load. Besides, the dynamic load duration also has a signi"cant in#uence on the sensitivity of the initial imperfection. The in#uence of the initial imperfection increases as the duration decreases. Di!erent hardening ratios of plate material a!ect the elastic}plastic buckling of plates. The dynamic buckling load increases as the hardening ratio of plate material increases if the plate buckles plastically.
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