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Impact loading of ductile rectangular plates
Thin-Walled Structures 50 (2012) 68–75
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Impact loading of ductile rectangular plates Norman Jones Impact Research Centre, Department of Engineering, University of Liverpool, Liverpool, UK
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 July 2011 Received in revised form 23 September 2011 Accepted 23 September 2011 Available online 25 October 2011
In many industries, rigid-plastic methods of analysis are a useful design aid for safety calculations, hazard assessments, security studies and forensic investigations of ductile structures, which are subjected to large dynamic loads producing an inelastic response. This paper examines the behaviour of a rectangular plate struck at the centre by a rigid mass impact loading. A theoretical method has been developed previously for arbitrarily shaped plates which retains the influence of finite transverse displacements, or geometry changes. It is used in this paper to predict the maximum permanent transverse displacements and response duration of plates having boundary conditions characterised by a resisting moment mM0 around the entire boundary, where m¼ 0 and 1 give the two extreme cases of simply supported and fully clamped supports, respectively. The theoretical predictions are compared with some experimental data recorded on fully clamped metal rectangular plates having a range of aspect ratios and struck by masses travelling with low impact velocities up to nearly 7 m/s and which produce large ductile deformations without failure. The theoretical analysis gives reasonable agreement with the corresponding experimental data for masses having blunt, conical and hemispherical impact faces. For sufficiently large initial impact energies, the projectile would perforate a plate and, for completeness, a useful design equation is presented which predicts perforation energies larger than all of the test data, as expected. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Rectangular plate Square plate Circular plate Mass impact Rigid-plastic analysis Large permanent transverse displacements Response duration Metal plate Simply supported Fully clamped Experiments Perforation
1. Introduction Simple and reliable theoretical methods are still valuable for design purposes, particularly for preliminary design and hazard assessments, and for forensic investigations after accidents. A theoretical rigid-plastic method was developed in [1], which retained the influence of large transverse displacements (i.e., geometry changes, or membrane effects) and which has been used to predict the maximum permanent transverse displacements, or damage, of ductile beams, circular and rectangular plates when subjected to a pressure pulse causing plastic strains. It was shown how this method can be simplified with an approximate yield condition to predict useful design equations, which circumscribe and inscribe the predictions of an exact yield criterion. This method was also used to examine the impulsive, or blast, loading of rectangular plates, and good agreement was found with experimental results recorded on ductile metal plates having various aspect ratios [2–4]. The method was extended to obtain the response of circular plates [4–6] and square plates [6] when struck by a solid mass at the centre, and again good agreement was reported with the maximum permanent transverse displacements observed in experimental tests on ductile metal plates. This paper extends the above theoretical method to obtain the maximum permanent transverse displacements, or ductile damage,
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for a rectangular plate struck by a rigid mass at the centre. It turns out that a relatively simple equation was obtained which gives good agreement with experimental data recorded on ductile metal rectangular plates having a range of aspect ratios from 0.4 to 1 and reported in [7]. Thus, design equations are now available for predicting the maximum permanent transverse displacements, or damage, for circular plates and rectangular plates (including square plates and beams) subjected to pressure pulses (including the limiting case of an impulsive loading) or central solid mass impacts. Moreover, these equations have been tested against experiments on ductile metal plates and are therefore suitable for design purposes, safety calculations, security studies and hazard assessments. The next section of this paper outlines the theoretical method which is used in Section 3 to examine the behaviour of a ductile rectangular plate subjected to a mass impact loading at the mid-span. Section 4 discusses briefly the experimental details of the data obtained on mild steel rectangular plates struck by a mass at the plate centre which produces large ductile deformations without any failure. Sections 5 and 6 contain a discussion and conclusions, respectively.
2. Theoretical method for dynamic loading of plates A theoretical procedure was developed in [1], to study the response and predict the permanently deformed profile of an
N. Jones / Thin-Walled Structures 50 (2012) 68–75
Nomenclature
Nr, Ny
a d m
R S T V V0 W
defined in the Appendix diameter of projectile dimensionless moment resistance at supports, m¼0 and 1 for simply and fully clamped supports, respectively Cowper Symonds exponent (Eq. (14)) time transverse displacement Cartesian coordinates (Fig. 2) surface area of a plate width of a rectangular plate (Fig. 2) Cowper Symonds coefficient (Eq. (14)) energy ratio (Eq. (13)) mass of a projectile or striker plate thickness dimensionless initial kinetic energy (Eq. (15)) length of a rectangular plate (Fig. 2) plastic collapse moment per unit length (s0H2/4) radial and circumferential bending moments per unit length plastic collapse force per unit length (s0H)
q t w x, y A 2B D Er G H K 2L M0 Mr, My N0
Wf
a b
g, gc ef e_ Z kr , ky l
m r s0 , s00 su O
arbitrarily shaped ductile plate, when subjected to large static or dynamic loads which produce plastic strains. The material is assumed to be rigid, perfectly plastic with a yield stress s0 and the plate has a uniform thickness H. The governing equations can be simplified for an impact loading and written in the form Z Z € W _ mw €w _ dA ¼ fðM r þ wNr Þk_ r þ ðMy þ wNy Þk_ y g dA GW A
A
þ
n Z X m¼1
þ
Cm
v Z X
u¼1
Cu
69
radial and circumferential membrane forces per unit length radius of circular plate span response time volume of material initial impact or impulsive velocity transverse displacement at centre of rectangular and circular plates (Fig. 2) maximum permanent transverse displacement defined by Eq. (5a) aspect ratio (Eq. (5b)) mass ratios (Eqs. (5c) and (17) for rectangular and circular plates, respectively) engineering rupture strain in tension strain rate (Eq. (14)) pressure pulse ratio (Appendix) radial and circumferential changes of curvature rV 20 L2 =M0 for a rectangular or square plate, Eq. (A.4) for a circular plate mass per unit surface area of a plate density of plate material static and dynamic flow stresses static ultimate tensile stress dimensionless initial kinetic energy (Eq. (12))
of four generalised stresses ðM r ,M y ,N r ,Ny Þ, which can be related by the limited interaction surface shown in Fig. 2 of [6]. However, if a deformation profile consists only of rigid regions separated by plastic hinges, then the exact yield condition in Fig. 1 governs plastic flow at the hinge lines. A square yield condition circumscribes the exact yield condition (maximum normal stress yield criterion), while another one which is 0.618 times as large would
_ ðM r þwN r Þ @w=@r dC m m
_ u dC u Q r ðwÞ
ð1Þ
where G is an impact mass, and m is the mass per unit surface area _ of a plate. The transverse displacement of a plate is w, while w € are the associated velocity and acceleration. W is the and w transverse displacement at the plate centre which is immediately underneath a striking mass. The terms on the left hand side of Eq. (1) are the work rate due to the inertia forces, where A is the surface area of a plate. The first term on the right hand side of Eq. (1) is the energy dissipated in any continuous deformation fields. The second term gives the energy dissipated in n plastic bending hinges, each having an _ angular velocity ð@w=@rÞ m across a hinge of length Cm. The final term is the plastic energy absorption in n transverse shear hinges, _ u and a length Cu. Eq. (1) each having a velocity discontinuity ðwÞ ensures that the external work rate equals the internal energy dissipation. The general method has been used to study the dynamic plastic response of beams, and of circular, square and rectangular plates subjected to dynamic pressure pulses and also blast loadings [1–4], and for beams, circular and square plates struck at the mid-span by a rigid mass [4–6]. It is used in this paper to examine a rectangular plate struck by a rigid mass at the centre, and, since large ductile deformations are studied without any material failure, or perforation, the last (transverse shear) term in Eq. (1) is not considered further. Thus, the yield condition consists
Fig. 1. Yield conditions at the plastic hinge lines (including the supports for ma 0) which develop within the rectangular plate in Fig. 2.
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N. Jones / Thin-Walled Structures 50 (2012) 68–75
inscribe it, thus providing upper and lower bounds on the exact yield condition according to a corollary of the limit theorems [4]. 3. Mass impact loading of a rectangular plate The kinematically admissible transverse velocity field in Fig. 2 is assumed to govern the response of the rectangular plate (2L 2B) in Fig. 3, which has a moment resistance mM0 around the supports (m ¼0 and 1 give the simply supported and the fully clamped cases, respectively, but intermediate cases (0 omo1) are included in the analysis). It is assumed that the mass strikes the centre of a rectangular plate, has a negligible cross-section when compared with the plate dimensions, B and L, and produces a response characterised by the transverse velocity profile in Fig. 2, which can be written as _ ð1x=LÞ, _ ¼W w
0 r x rL,
0 r y rBx=L
ð2aÞ
and _ ð1y=BÞ, _ ¼W w
0 r x r yL=B,
0 r y rB
ð2bÞ
for the quadrant with positive values of x and y. From considerations of double symmetry, the behaviour of this quadrant is similar to the other three quadrants.
The transverse velocity profile in Fig. 2 requires plastic hinges along the diagonals and at the supports (unless m¼0) of a rectangular plate. The interior regions remain rigid for a rigid, perfectly plastic material so that the integral for continuous deformations in Eq. (1) does not contribute to the internal energy dissipation. Thus, Eq. (1) becomes Z Z r X € W _ mw €w _ dA ¼ GW ðM þ NwÞy_ m dlm ð3Þ A
lm
m¼1
for an initially flat, rigid, perfectly plastic plate which deforms into a number of rigid regions separated by r stationary straight line plastic hinges, each of length lm. y_ m is the relative angular velocity across a straight line hinge, w is the transverse displacement along a hinge, and N and M are the membrane force and bending moment, respectively, which act on a plane which passes through a hinge and is transverse to the mid-surface of a plate. Eq. (3) for the impact problem in Fig. 3, when using Eqs. (2) for the transverse velocity field and the square yield condition in Fig. 1, yields the governing equation: € þ a2 W ¼ ð1þ mÞHa2 =2 W
ð4Þ
where 2
a2 ¼
12M0 ð1 þ b Þ
mHL ð1 þ 6gÞb2 2
,
b¼
B G and g ¼ L 4mBL
ð5a2cÞ
Now, when satisfying the conservation of linear momentum at t¼0 requires _ þ 4m GV 0 ¼ GW
Z
L 0
Z
Bx=L
_ ð1x=LÞ dx dy þ 4m W
0
Z
B 0
Z
Ly=B
_ ð1y=BÞ dx dy, W
0
or _ ¼ V 0 =ð1 þ1=3gÞ W
ð6Þ
where V0 is the initial velocity of the striking mass. Eq. (4) has the general solution W ¼ A sin at þ B cos atð1þ mÞH=2,
ð7Þ
where the constants of integration A and B are obtained from the _ given by Eq. (6). initial conditions at t ¼0, namely w¼0 and W Thus, W¼
V0
að1 þ 1=3gÞ
sin at þ
ð1þ mÞH ð1 þmÞH cos at 2 2
ð8Þ
Now differentiating Eq. (8) with respect to time gives the _ from which the duration of motion, T, is transverse velocity W _ ¼ 0, or obtained when W Fig. 2. Pyramidal-shaped transverse velocity field for a rigid, perfectly plastic rectangular plate struck at the centre as shown in Fig. 3: (a) plan view of plastic hinge lines which develop along diagonals and at boundaries (except for the simply supported case when m¼ 0 on the four boundaries). (b) Side view (in y direction) of pyramidal-shaped transverse velocity field at time t.
tan aT ¼
2V 0 ð1þ 1=3gÞð1 þ mÞaH
ð9Þ
Finally, Eqs. (8) and (9) give the dimensionless maximum permanent transverse displacement ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v )ffi 3 u( Wf ð1 þ mÞ 4u 12 bO g ð1þ 6 g Þ t 1þ 15: ð10Þ ¼ 2 2 H ð1 þ b Þð1þ mÞ2 ð1 þ 3gÞ2 If the striking mass, G, is heavy compared with the plate mass, then g b1 and Eq. (10) reduces to ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v( ) u u Wf 2bO ¼ t 1þ 1 ð11Þ 2 H ð1 þ b Þ where
Fig. 3. Rectangular plate having a length 2L and a breadth 2B struck at the centre by a rigid mass G travelling with an impact velocity V0. The bending resistance around the four supports is mM0, where 0r mr 1 and M0 is the plastic limit moment per unit length of the plate cross-section.
O¼
GV 20 4s0 H3
ð12Þ
for the fully clamped case with m¼1. Eq. (11) for a square plate with b ¼1 reduces to the corresponding equation developed in
N. Jones / Thin-Walled Structures 50 (2012) 68–75
Ref. [6]. The theoretical predictions for a simply supported rectangular plate struck by a rigid mass are given by Eqs. (9) and (10) with m¼0. The theoretical predictions in this section have been obtained using the circumscribing square yield condition in Fig. 1. Another solution with the flow stress s0 replaced by 0.618s0 would give a ‘‘lower bound’’ on the exact yield condition in Fig. 1, but lead to an ‘‘upper bound’’ on the maximum permanent transverse displacement predicted by Eq. (10). Thus, the two predictions with circumscribing and inscribing yield curves provide ‘‘lower’’ and ‘‘upper’’ bounds on the maximum permanent transverse displacements.
4. Experimental details The plates in the theoretical analysis reported in Section 3 undergo wholly ductile deformations without any material failure or perforation. Some of the data for the perforation study reported in Ref. [7] on rectangular plates were obtained for lower impact velocities than those required for perforation and gave rise to wholly ductile behaviour of a rectangular plate without failure. This data is reported in Table 1. The uniaxial tensile characteristics of the H¼4 mm thick mild steel plate are s0 ¼262 MPa, su ¼ 359 MPa and ef ¼ 37.5%. The projectiles have a 10.16 mm diameter (d) body, and the impact faces are blunt, hemispherical or conical (901 included angle). A drop hammer rig was used for the tests and the impact mass (G) was 11.8 kg for the blunt-tipped projectiles and 19 kg for the hemispherically tipped and conically tipped projectiles. Thus, the mass ratios g for the test specimens in Table 1 range from 9.4 to 60.5 according to Eq. (5c) with a material density of 7850 kg/m3. The initial impact velocity (V0) was recorded using a laser Doppler velocimeter [8] and the maximum permanent transverse displacements were measured at the plate centre. The energy ratios for the experimental data in Table 1 vary between 9.6 and 59.0 when it is defined as the ratio of the initial kinetic energy, GV 20 =2, and an upper estimate on the elastic strain energy which is approximated as ðs20 =2EÞV, or
71
5. Discussion The theoretical predictions in Section 3 are made particularly simple by using the square yield condition in Fig. 1, which bounds the maximum normal stress yield criterion. A theoretical solution using the exact maximum normal stress, or von Mises, yield criterion, is possible with the method of successive approximations (e.g., see [4]), or some other method of analysis. It appears that no exact theoretical analyses have been published for the mass impact loading of rectangular plates undergoing finite displacements, but the theoretical analyses using an exact yield surface for impulsively loaded beams and circular plates are bounded by the lower and upper bound predictions using circumscribing and inscribing yield criteria, as shown in [1]. The theoretical analysis in Section 3 idealises the striking mass, G, as a concentrated point loading. A theoretical analysis was reported in [6] for a fully clamped circular plate struck by a blunt faced cylindrical mass having a radius a. The theoretical rigid perfectly solution was expressed in terms of a parameter r ¼a/R. Fig. 7.24 [4, 2nd Edition] shows a reduction of about 10% in the maximum permanent transverse displacement for strikers having r ¼0.12. Thus, the retention of r for the ductile plastic behaviour of plates appears to produce a relatively modest effect, particularly when considering various other simplifications in the analysis, such as the idealised impact loading characteristics and approximations introduced in the modelling of the boundary conditions. Studies on the perforation of ductile plating reveal that the dimensionless perforation energy is a fairly strong function of the parameter c ¼d/H, as indicated in [11], because the plate area resisting shear plugging through the plate thickness is roughly proportional to d¼2a. Nevertheless, it is not likely that c would strongly influence the response of plating having a wholly ductile inelastic behaviour without failure, unless r or c are relatively large. Eq. (10) is valid for any value of the dimensionless moment resistance at the supports, m, which lies within the range 0om o1. In particular, Eq. (10) with m¼0 for simple supports 2
1=2
gives W f =H ¼ ½f1 þ8bO=ð1 þ b Þg 1=2 for g b1, which when the dimensionless kinetic energy O b1 for large external load2
EGV 2 Er ¼ 2 0 , s0 V
ð13Þ
where V¼H 2B 2L is the total volume of the plate material. Further details of the test arrangement are reported in Refs. [7,9,10]. Table 1 Experimental data for rectangular plates struck at mid-span. H ¼4 mm, s0 ¼262 MPa, G ¼ 11.8 kg for blunt-tipped projectiles and G ¼ 19 kg for hemispherically tipped and conically tipped projectiles [7]. Shape
b
2B 2L (mm2)
V0 (m/s)
O
Wf/H
Blunt Blunt Blunt Blunt Blunt Blunt Hemisph’l Hemisph’l Hemisph’l Hemisph’l Hemisph’l Conical Conical Conical Conical
0.4 0.5 0.66 0.66 1 1 0.4 0.5 0.66 1 1 0.5 0.66 1 1
100 250 100 200 100 150 100 150 100 100 200 200 100 250 100 200 100 150 100 100 200 200 100 200 100 150 100 100 200 200
6.49 6.49 6.26 6.42 6.42 6.57 6.86 6.72 6.57 6.42 6.86 6.57 6.64 5.60 6.64
7.42 7.42 6.90 7.25 7.25 7.60 13.34 12.79 12.23 11.67 13.32 12.23 12.51 8.89 12.51
2.10a 2.03 2.20 1.83 1.84 2.12 2.74a 2.66 2.51 2.36 2.91 2.51a 2.40a 2.01 2.76a
a Indicates that the plating has cracked underneath the projectile impact location.
1=2
ings, becomes W f =H f2bO=ð1 þ b Þg . This expression is identical to that predicted by Eq. (11) for O b1. In other words, the distinction between the permanent behaviour for plates having simply supported and fully clamped boundary conditions diminishes as the transverse displacement increases and the plate tends towards membrane behaviour. In practice, as long as the axial restraint is maintained, then the moment restraint at the boundary does not have a significant effect on the final displacement for sufficiently large values of O or Wf/H. It is evident from Eq. (11) for a fully clamped rectangular plate with g b1 that the theoretical predictions for Wf/H collapse on to a single curve irrespective of the plate aspect ratio b when 2 2bO=ð1þ b Þ is used in the abscissa, as shown in Fig. 4. A yield stress 0.618s0 inscribes the exact yield condition in Fig. 1 and gives rise to the ‘‘upper bound’’ predictions according to Eq. (11) in Fig. 4. The experimental data in Table 1 is also plotted in Fig. 4 for values of 0.4 r b r1 and for three shapes of the projectile impact faces. First of all, it is evident that most of the experimental data is bounded by the ‘‘upper’’ and ‘‘lower’’ bound predictions of Eq. (11). Secondly, a trend is discerned which indicates that the experimental data migrates from the ‘‘lower bound’’ curve towards the ‘‘upper bound’’ curve as b decreases for both the hemispherically tipped and blunt-tipped projectiles. However, it should be noted that both b ¼0.4 plates had some cracking underneath the impact site. The experimental results for the conical projectiles are not distinct from those for the two other projectile nose shapes and remained fairly close to the
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N. Jones / Thin-Walled Structures 50 (2012) 68–75
velocities in Table 1 are not expected to be as large as those in [12] for blast loadings. It should be remarked that large plastic strains cause a reduction in the material strain rate effect [14], when compared with relatively small strain problems. For example, the constants D ¼40.4 s 1 and q¼5 are often used in the Cowper Symonds equation 1=q s00 e_ ¼ 1þ ð14Þ s0 D
Fig. 4. Maximum dimensionless permanent transverse displacements at the centre of fully clamped rectangular plates versus the dimensionless initial kinetic energy. ———— : Eq. (11), circumscribing yield condition. – – – – – –: Eq. (11), inscribing yield condition. — — — 1,2: Eq. (10) with m ¼1 and g ¼ 9.4 for circumscribing and inscribing yield conditions, respectively. Experimental data from Table 1: x, J, B; b ¼ 0.4, 0.5, 0.66, respectively. &, ’; b ¼1 for 200 mm 200 mm and 100 100 mm square plates, respectively. The superscripts 1, 2 and 3 denote projectiles with hemispherical, blunt and conical (901 included angle) impact faces, respectively.
‘‘lower bound’’ predictions of Eq. (11); all of these plates, except one, had cracked underneath the projectile impact location. It is noted here that the ‘‘upper’’ and ‘‘lower’’ bound predictions of Eq. (11) with b ¼1 bound the experimental data for the fully clamped mild steel square plates reported in Fig. 7 of Ref. [6]. Eq. (10) retains the influence of the mass ratio, g, whereas Eq. (11) assumes that g b1. As noted in the previous section, g ¼9.4 is the smallest value and would have the greatest effect on the predictions of Eq. (10). Eq. (10) with g ¼9.4 is plotted in Fig. 4 for circumscribing and inscribing square yield curves. The differences between these curves and the g b1 cases are quite small and the values of g for all of the experimental data would lie between g ¼ 9.4 and g b1. It is possible that the transverse displacement and velocity profiles in Fig. 2 and described by Eqs. (2) are less satisfactory as b becomes smaller and approaches b-0 for a ‘‘beam’’ having a span 2B. Thus, the experimental data for rectangular plates with b ¼0.4 lie near to the ‘‘upper’’ bound predictions of Eq. (11) and are only just bounded by the theoretical predictions. The theoretical analysis might not be suitable for rectangular plates with even smaller aspect ratios b when another kinematically admissible transverse profile becomes necessary, although there are no experimental data available for smaller values of b to test this hypothesis. This contrasts with the uniform blast loading of rectangular plates using the same general theoretical method [1] which has an associated velocity profile that gives the response of a beam when b-0. The rectangular plates in Ref. [7] are made from mild steel which usually is a strain rate sensitive material, although the dynamic material characteristics are not available for this particular plate material. The dynamic stress state in a rectangular plate is complex and is not uniaxial. In this regard, it is interesting to observe in Fig. 11 of [12] that the maximum permanent transverse deflections of impulsively loaded rectangular plates made from aluminium alloy 6061-T6, which is an essentially strain rate insensitive material, and mild steel, which is a strongly strain rate sensitive material, differ by about 10–15%, whereas the difference would be nearer 30–40% for a similar comparison on beams [3,13] which have a less complex stress state. Nevertheless, if the data for the strain rate effects of the plate material were known and incorporated into Eq. (11), as shown in [4] for blast loaded plates, then the theoretical curves in Fig. 4 would be lower, although the strain rate effects for the low impact
to characterise the strain rate behaviour of mild steel. In this case, the dynamic flow stress is s00 ¼ 2s0 for a uniaxial strain rate e_ ¼ 40:4s1 . Now for a strain of 0.05, then the data of Marsh and Campbell [15], which is discussed in Ref. [14], reveals that D E1300 s 1 and q¼5, with even larger values of D for larger strains. Thus, the strain rate sensitive behaviour of mild steel varies with strain as embodied in the modified Cowper Symonds equation developed in Ref. [14]. Unfortunately, there does not appear to be any experimental data which has been published for the large ductile deformation behaviour of aluminium alloy rectangular plates struck by a mass. However, the impact behaviour of fully clamped square plates was studied by Langseth and Larsen [16] and the experimental data is reported in Fig. 5. The ordinate in Fig. 5, K, is the dimensionless kinetic energy, or from Eq. (11) W f =H ¼ ð1 þ KÞ1=2 1
ð15Þ
where K¼ O for a square plate (b ¼1). A fully clamped square plate loaded impulsively over the entire surface area can also be cast into the same form of Eq. (15) [1,4], where in this case K ¼ l/6 and l ¼ rV 20 L2 =M 0 when V0 is the uniformly distributed initial impulsive velocity. Thus, both the central impact and uniform impulsive loading cases for a square plate can be cast in the form of Eq. (15) and the ‘‘upper’’ and ‘‘lower’’ bound curves in Fig. 5 for inscribing and circumscribing yield curves apply to both cases. It is evident from the comparisons in Fig. 5 that the simple rigid, perfectly plastic theoretical predictions of Eq. (15), with the appropriate values of K, provide a reasonable design estimate for the experimental values of the strain rate insensitive
Fig. 5. Dimensionless kinetic energy, K, versus the dimensionless maximum permanent transverse displacement according to Eq. (15) for aluminium alloy square plates. ———: Eq. (15), circumscribing yield surface for mass impact and impulsive velocity loadings. – – – –: Eq. (15), inscribing yield surface with s0 replaced by 0.618s0. Experimental results: & AA 6061-T6, uniform impulsive loading [2,3]. ’ AA 6082-T6, mass impact loading [16]. B AA 5083-H112, mass impact loading [16].
N. Jones / Thin-Walled Structures 50 (2012) 68–75
73
aluminium alloy square plates. The experimental data from Ref. [16] (’, B) was obtained for square plates having flanges around the edges rather than fully clamped supports. This might lead to larger values of Wf/H when compared with the data (&) obtained with fully clamped boundaries, as observed for the perforation of square plates [11] and as shown in Fig. 5. K could be regarded as the dimensionless energy required to produce a given dimensionless maximum permanent transverse deformation Wf/H which could be identified with damage. A similar theoretical method has been used to study the impact behaviour of fully clamped circular plates 1=2
W f =H ¼ f1 þð6Ogc =pÞð1 þ6gc Þð1 þ3gc Þ2 g
1
ð16Þ
where
gc ¼ G=mpR2
ð17Þ
which for gc b1 gives [4,6] W f =H ¼ ð1þ 4O=pÞ1=2 1:
ð18Þ
A comparison in Fig. 6 with the experimental data from [17] reveals that the experiments are almost bounded by Eq. (18). It is also interesting that the experimental data of Tian and Jiang [18] on steel circular plates with gc ¼ 1.33 in Fig. 7 are bounded by the predictions of Eq. (16) for inscribing and circumscribing yield conditions. For comparison purposes, the predictions of Eq. (18) with gc b1 are also given in Fig. 7. The energy ratios defined by Eq. (13), but when applied to a circular plate, are between about 10–15 for the experimental data [18] in Fig. 7. However, the mass ratio for these tests is gc ¼1.33 according to Eq. (17), which for the mass impact behaviour of a beam [19], would lead to a difference of about 10 per cent when compared with an exact solution. Unfortunately, no similar studies have been reported on plates, so that the error is not known for the impact behaviour of a circular plate with gc ¼1.33. The experimental data [18], which are plotted in Fig. 7, were obtained using a steel plate having a low material strain rate sensitivity since the flow stress at a strain rate of 17 s 1 increases only 5 per cent when compared with the corresponding static flow stress. This should be compared with an average strain rate of about 7 s 1 for those test specimens in Fig. 7 with Wf/HE3, according to the equation given in the footnote on p. 376 of Ref. [4] (p. 367 of 2nd ed. [4]). Thus, material strain rate effects are not expected to exercise a significant influence on the response.
Fig. 7. Comparison of the theoretical predictions of Eq. (16) with experimental results on fully clamped steel circular plates struck by a mass at the centre. ———: Eq. (16), circumscribing yield surface with gc ¼1.33. – – – –: Eq. (16), inscribing yield surface with s0 replaced by 0.618s0 and gc ¼1.33. J, x, D : experimental results [18] for H¼ 3.6 mm, 1.8 mm and 0.9 mm, respectively. – – – , – – –: Eq. (18) for circumscribing and inscribing yield conditions with gc b1, respectively.
It was observed earlier that Eq. (15) gives the theoretical predictions for a fully clamped square plate struck by a rigid mass at the plate centre when K ¼ O, while Eq. (15) with K ¼ l/6 is valid for a fully clamped square plate subjected to a uniform impulsive velocity loading. Moreover, the theoretical predictions for a fully clamped circular plate can be cast in the form of Eq. (15) provided K ¼4O/p and l/6 for the mass impact (with gc b1) and impulsive loading (see Appendix) cases, respectively. The experimental values in Fig. 8 were obtained from five different studies in four laboratories and using four materials. Two sets of data (x, &) cluster around the predictions of Eq. (15) for a circumscribing yield condition, while the remaining data are near to the predictions for an inscribing yield condition. Florence’s [20] (J) were obtained for an impulsively loaded circular plate having simple supports rather than fully clamped ones considered here. This difference would lead to a larger value of Wf/H for a given value of K. In fact, the more complete theoretical analysis which was developed in Ref. [21] for a simply supported circular plate loaded impulsively does provide close bounds on Florence’s data for the maximum permanent transverse displacements. Similarly, the square plate specimens [16] (’, B) have flanges around the edges rather than fully clamped supports. Again this would cause larger values of Wf/H for a given value of K, as noted earlier and in Ref. [11]. Thus, all of the experimental data (’, B, J) clustered around the predictions of Eq. (15) for an inscribing yield condition do not have fully clamped boundary conditions. Particular attention was given to achieving fully clamped boundary conditions for the plate specimens (x, &) in Fig. 8. The experimental study reported in [7] has examined the threshold conditions for the perforation of plates which occurs at impact energies larger than those associated with the data for the wholly ductile deformations reported in Table 1. An empirical perforation equation
Op ¼ ðp=4Þðd=HÞ þ ðd=HÞ1:53 ðS=dÞ0:21 Fig. 6. Comparison of the theoretical predictions of Eq. (18) with experimental results on fully clamped aluminium alloy circular plates struck by masses at the centre. ———: Eq. (18), circumscribing yield surface. – – – –: Eq. (18), inscribing yield surface with s0 replaced by 0.618s0. J: Experimental results [17] (H¼9.53 mm, R¼ 101.6 mm, a¼11.905 mm (radius of cylindrical body of projectile), G¼ 20 kg, s0 ¼ 442.6 MPa, gc ¼24.24).
ð19Þ
was presented in [22] for circular plates, struck by cylindrical projectiles having blunt impact faces and travelling at low velocities. It turns out that this equation gave good estimates for the dimensionless perforation energy, Op, of rectangular mild steel plates when the span, S, is taken as the plate breadth 2B [7].
74
N. Jones / Thin-Walled Structures 50 (2012) 68–75
Fig. 8. Comparison of the theoretical predictions of Eq. (15) for circular plates struck by a mass (x, K¼ 4O/p) or loaded with a uniform impulsive loading (J, K¼ l/6) and for square plates struck by a mass (’, B, K¼ O) or loaded with a uniform impulsive loading (&, K¼ l/6). ——— : Eq. (15), circumscribing yield condition. – – – –: Eq. (15), inscribing yield condition. l ¼ rV 20 R2 =M 0 and l ¼ rV 20 L2 =M0 for circular and square plates, respectively. Experimental data. &, ’, B ; defined in Fig. 5. x; defined in Fig. 7. J; Aluminium alloy 6061-T6 simply supported circular plates, uniform impulsive velocity loading [20].
The perforation energy of mild steel plates struck by projectiles with cylindrical bodies having blunt, conical and hemispherical impact faces, and travelling at impact velocities up to 12 m/s, were studied in [7]. It transpired that projectiles with blunt impact faces perforated mild steel plates with the least energy, while projectiles having hemispherical impact faces required the greatest energy and those projectiles with conical impact faces (901 included angle) required a perforation energy, which was intermediate between the two values, but closer to the hemispherical case. Eq. (19) for 2B¼100 mm predicts Op ¼ 8.72, while 2B¼200 mm gives Op ¼9.77. These values are larger than all of the experimental results for the square and rectangular plates struck by blunt-tipped 2 projectiles in Table 1. The factor 2bO=ð1 þ b Þ in Eq. (11), which is used for the abscissa in Fig. 4, is 0.69O, 0.80O and 0.92O for b ¼0.4, 0.5 and 0.66, respectively, and yields the corresponding values of 2 2bOp =ð1 þ b Þ ¼ 6:01, 6:97 and 8:01. As anticipated, these values lie above all of the corresponding rectangular plate data struck by blunttipped projectiles, which are identified with a superscript 2 in Fig. 4. Moreover, the experimental results in [7,9] reveal that the threshold values of the dimensionless perforation energy, Op, are higher for projectiles having hemispherical and conical impact faces in this series of experimental tests. Eq. (19) for the fully clamped steel circular plates [18] in Fig. 7 with S¼ 2R predicts that projectiles with blunt impact faces would perforate the plates when Op ¼35.5. This value is well above the values associated with a large ductile deformation response.
subjected to either static or dynamic loadings which produce large plastic strains and finite transverse displacements, has been used to study the behaviour of a rectangular plate struck by a mass at the centre. A relatively simple equation has been derived for the maximum permanent transverse displacements and response duration, which can be used for preliminary design purposes and accident investigations. The theoretical predictions have been developed for a plate having a resisting moment mM0, 0omo1, on the entire boundary. In the particular case of m¼1 for the fully clamped case, the theoretical predictions were compared with some experimental data recorded on fully clamped rectangular plates having aspect ratios between 0.4 and 1 (square) and struck transversely by masses with cylindrical bodies travelling up to about 7 m/s. The theoretical predictions provide reasonable estimates of the corresponding experimental data. The theoretical procedure has now been used to provide relatively simple design equations for the response of ductile beams, and circular, square and rectangular plates subjected to dynamic pressure pulses (including blast and impulsive velocity loadings) and mass impact loadings which produce permanent deformations and large plastic strains. The accuracy of these cases have been supported with experimental data obtained from various laboratories over the years. For completeness, an empirical equation has been used to predict the initial impact energy required to perforate a plate by a blunt-faced missile. This equation predicts perforation energies which lie above all of the corresponding experimental data for large-mass and low-velocity impacts on steel plates.
Acknowledgments The author is grateful to Mrs. J. Jones for her secretarial assistance and to Mrs. I. Arnot for her assistance with the figures.
Appendix. Impulsive loading of a fully clamped circular plate The particular case of a rigid, perfectly plastic simply supported circular plate, which is subjected to a rectangular shaped pressure pulse of magnitude p0 and duration t, is developed in Section 7.7 of Ref. [4]. The limit of an infinitely large pressure pulse with a vanishingly short duration, but a finite impulse, gives the impulsive velocity loading case. Now the analysis for fully clamped supports follows closely the analysis in Section 7.7 [4], except the first term on the right hand side _ =RÞ2pR for a fully clamped boundary which of Eq. (7.103) gives M 0 ðW is added to the right hand side of Eq. (7.106). Eq. (7.107) is again obtained except b is replaced by b ¼ 24M 0 ðZ1Þ=mR2 , where Z ¼ p0 =pc and pc ¼ 12M0 =R2 is the static plastic collapse pressure for a circular plate having a square yield condition. The solution of the differential equation with the same initial conditions follows the first (0rt r t) and second (t rt rT) phases of the procedure outlined in Section 7.7 [4]. Motion ceases at the time T, where tanaT ¼
Z sin at 1Z þ Z cos at
ðA:1Þ
leading to a dimensionless maximum permanent transverse displacement W f =H ¼ f1 þ 2ZðZ1Þð1cos atÞg1=2 1,
6. Conclusions A theoretical rigid, perfectly plastic method, developed previously for initially flat plates having an arbitrarily shaped boundary and
2
2
ðA:2Þ
where a ¼ 24M 0 =mHR . The impulsive loading case is obtained by taking Z b1, t-0 and p0 t ¼ mV 0 , where V0 is the initial uniformly distributed impulsive velocity over the entire plate surface. Thus, Eq. (A.2)
N. Jones / Thin-Walled Structures 50 (2012) 68–75
reduces to 1=2
W f =H ¼ ð1þ l=6Þ
1
ðA:3Þ
where
l ¼ mV 20 R2 =M0 H:
ðA:4Þ
References [1] Jones N. A theoretical study of the dynamic plastic behavior of beams and plates with finite-deflections. International Journal of Solids and Structures 1971;7:1007–29. [2] Jones N, Baeder RA. An experimental study of the dynamic plastic behavior of rectangular plates. In: Symposium on plastic analysis of structures, vol. 1, Published by Ministry of Education, Polytechnic Institute of Jassy, Civil Engineering Faculty; 1972: pp. 476–97. [3] Jones N. A literature review of the dynamic plastic response of structures. The Shock and Vibration Digest 1975;7(8):89–105. [4] Jones N. Structural impact. Cambridge University Press; 1989. p. 575, Paperback edition, 1997. Chinese edition translated by Ping Jiang and Lili Wang, Sichuan Education Press, Chengdu, 1994. 2nd edition (in press). [5] Jones N, Kim S-B, Li QM. Response and failure analysis of ductile circular plates struck by a mass. Transactions of the ASME, Journal of Pressure Vessel Technology 1997;119(3):332–42. [6] Jones N. On the mass impact loading of ductile plates. Defence Science Journal, Defence Research and Development Organisation, India 2003;53(1):15–24. [7] Jones N, Birch RS, Duan R. Low velocity perforation of mild steel rectangular plates with projectiles having different shaped impact faces. ASME, Journal of Pressure Vessel Technology 2008;130(3):031206-1–8. [8] Birch RS, Jones N. Measurement of impact loads using a laser Doppler velocimeter. Proceedings of the Institution of Mechanical Engineers 1990;204(C1):1–8. [9] Jones N, Birch RS. Low velocity perforation of mild steel circular plates with projectiles having different shaped impact faces. ASME, Journal of Pressure Vessel Technology 2008;130(3):031205-1–11.
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[10] Birch RS, Jones N, Jouri WS. Performance assessment of an impact rig. Proceedings of the Institution of Mechanical Engineers 1988;202(C4):275–85. Corrigenda 204(C1), p. 8, 1990. [11] Jones N, Paik JK. Impact perforation of aluminium alloy plates. International Journal of Impact Engineering, doi:10.1016/j.ijimpeng.2011.05.007. In press. [12] Jones N, Uran TO, Tekin SA. The dynamic plastic behavior of fully clamped rectangular plates. International Journal of Solids and Structures 1970;6: 1499–512. [13] Jones N, Griffin RN, Van Duzer RE. An experimental study into the dynamic plastic behavior of wide beams and rectangular plates. International Journal of Mechanical Sciences 1971;13(8):721–35. [14] Jones N. Some comments on the modelling of material properties for dynamic structural plasticity. In: Harding J, editor. International conference on the mechanical properties of materials at high rates of strain. Institute of Physics conference series no. 102; 1989. p. 435–45. [15] Marsh KJ, Campbell JD. The effect of strain rate on the post-yield flow of mild steel. Journal of the Mechanics and Physics of Solids 1963;11:49–63. [16] Langseth M, Larsen PK. Dropped objects’ plugging capacity of aluminium alloy plates. International Journal of Impact Engineering 1994;15(3):225–41. [17] Wen HM, Jones N. Experimental investigation into the dynamic plastic response and perforation of a clamped circular plate struck transversely by a mass. Proceedings of the Institution of Mechanical Engineers 1994;208(C2): 113–37. [18] Tian C, Jiang P. Experimental investigation of the scaling laws for circular plates struck by projectiles. In: Zheng Z, Tan Q, editors. Proceedings of the IUTAM symposium on impact dynamics, Peking University Press; 1994: pp. 440–5. [19] Jones N. Quasi-static analysis of structural impact damage. Journal of Constructional Steel Research 1995;33(3):151–77. [20] Florence AL. Circular plate under a uniformly distributed impulse. International Journal of Solids and Structures 1966;2:37–47. [21] Jones N. Impulsive loading of a simply supported circular rigid-plastic plate. Journal of Applied Mechanics 1968;35(1):59–65. [22] Wen HM, Jones N. Semi-empirical equations for the perforation of plates struck by a mass. In: Bulson PS, editor. Proceedings of the 2nd international conference on structures under shock and impact. Computational Mechanics Publications, Southampton and Boston and Thomas Telford, London 1992; pp. 369–80.
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Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Impact loading of ductile rectangular plates Norman Jones Impact Research Centre, Department of Engineering, University of Liverpool, Liverpool, UK
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 July 2011 Received in revised form 23 September 2011 Accepted 23 September 2011 Available online 25 October 2011
In many industries, rigid-plastic methods of analysis are a useful design aid for safety calculations, hazard assessments, security studies and forensic investigations of ductile structures, which are subjected to large dynamic loads producing an inelastic response. This paper examines the behaviour of a rectangular plate struck at the centre by a rigid mass impact loading. A theoretical method has been developed previously for arbitrarily shaped plates which retains the influence of finite transverse displacements, or geometry changes. It is used in this paper to predict the maximum permanent transverse displacements and response duration of plates having boundary conditions characterised by a resisting moment mM0 around the entire boundary, where m¼ 0 and 1 give the two extreme cases of simply supported and fully clamped supports, respectively. The theoretical predictions are compared with some experimental data recorded on fully clamped metal rectangular plates having a range of aspect ratios and struck by masses travelling with low impact velocities up to nearly 7 m/s and which produce large ductile deformations without failure. The theoretical analysis gives reasonable agreement with the corresponding experimental data for masses having blunt, conical and hemispherical impact faces. For sufficiently large initial impact energies, the projectile would perforate a plate and, for completeness, a useful design equation is presented which predicts perforation energies larger than all of the test data, as expected. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Rectangular plate Square plate Circular plate Mass impact Rigid-plastic analysis Large permanent transverse displacements Response duration Metal plate Simply supported Fully clamped Experiments Perforation
1. Introduction Simple and reliable theoretical methods are still valuable for design purposes, particularly for preliminary design and hazard assessments, and for forensic investigations after accidents. A theoretical rigid-plastic method was developed in [1], which retained the influence of large transverse displacements (i.e., geometry changes, or membrane effects) and which has been used to predict the maximum permanent transverse displacements, or damage, of ductile beams, circular and rectangular plates when subjected to a pressure pulse causing plastic strains. It was shown how this method can be simplified with an approximate yield condition to predict useful design equations, which circumscribe and inscribe the predictions of an exact yield criterion. This method was also used to examine the impulsive, or blast, loading of rectangular plates, and good agreement was found with experimental results recorded on ductile metal plates having various aspect ratios [2–4]. The method was extended to obtain the response of circular plates [4–6] and square plates [6] when struck by a solid mass at the centre, and again good agreement was reported with the maximum permanent transverse displacements observed in experimental tests on ductile metal plates. This paper extends the above theoretical method to obtain the maximum permanent transverse displacements, or ductile damage,
E-mail addresses: [email protected], [email protected] 0263-8231/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2011.09.006
for a rectangular plate struck by a rigid mass at the centre. It turns out that a relatively simple equation was obtained which gives good agreement with experimental data recorded on ductile metal rectangular plates having a range of aspect ratios from 0.4 to 1 and reported in [7]. Thus, design equations are now available for predicting the maximum permanent transverse displacements, or damage, for circular plates and rectangular plates (including square plates and beams) subjected to pressure pulses (including the limiting case of an impulsive loading) or central solid mass impacts. Moreover, these equations have been tested against experiments on ductile metal plates and are therefore suitable for design purposes, safety calculations, security studies and hazard assessments. The next section of this paper outlines the theoretical method which is used in Section 3 to examine the behaviour of a ductile rectangular plate subjected to a mass impact loading at the mid-span. Section 4 discusses briefly the experimental details of the data obtained on mild steel rectangular plates struck by a mass at the plate centre which produces large ductile deformations without any failure. Sections 5 and 6 contain a discussion and conclusions, respectively.
2. Theoretical method for dynamic loading of plates A theoretical procedure was developed in [1], to study the response and predict the permanently deformed profile of an
N. Jones / Thin-Walled Structures 50 (2012) 68–75
Nomenclature
Nr, Ny
a d m
R S T V V0 W
defined in the Appendix diameter of projectile dimensionless moment resistance at supports, m¼0 and 1 for simply and fully clamped supports, respectively Cowper Symonds exponent (Eq. (14)) time transverse displacement Cartesian coordinates (Fig. 2) surface area of a plate width of a rectangular plate (Fig. 2) Cowper Symonds coefficient (Eq. (14)) energy ratio (Eq. (13)) mass of a projectile or striker plate thickness dimensionless initial kinetic energy (Eq. (15)) length of a rectangular plate (Fig. 2) plastic collapse moment per unit length (s0H2/4) radial and circumferential bending moments per unit length plastic collapse force per unit length (s0H)
q t w x, y A 2B D Er G H K 2L M0 Mr, My N0
Wf
a b
g, gc ef e_ Z kr , ky l
m r s0 , s00 su O
arbitrarily shaped ductile plate, when subjected to large static or dynamic loads which produce plastic strains. The material is assumed to be rigid, perfectly plastic with a yield stress s0 and the plate has a uniform thickness H. The governing equations can be simplified for an impact loading and written in the form Z Z € W _ mw €w _ dA ¼ fðM r þ wNr Þk_ r þ ðMy þ wNy Þk_ y g dA GW A
A
þ
n Z X m¼1
þ
Cm
v Z X
u¼1
Cu
69
radial and circumferential membrane forces per unit length radius of circular plate span response time volume of material initial impact or impulsive velocity transverse displacement at centre of rectangular and circular plates (Fig. 2) maximum permanent transverse displacement defined by Eq. (5a) aspect ratio (Eq. (5b)) mass ratios (Eqs. (5c) and (17) for rectangular and circular plates, respectively) engineering rupture strain in tension strain rate (Eq. (14)) pressure pulse ratio (Appendix) radial and circumferential changes of curvature rV 20 L2 =M0 for a rectangular or square plate, Eq. (A.4) for a circular plate mass per unit surface area of a plate density of plate material static and dynamic flow stresses static ultimate tensile stress dimensionless initial kinetic energy (Eq. (12))
of four generalised stresses ðM r ,M y ,N r ,Ny Þ, which can be related by the limited interaction surface shown in Fig. 2 of [6]. However, if a deformation profile consists only of rigid regions separated by plastic hinges, then the exact yield condition in Fig. 1 governs plastic flow at the hinge lines. A square yield condition circumscribes the exact yield condition (maximum normal stress yield criterion), while another one which is 0.618 times as large would
_ ðM r þwN r Þ @w=@r dC m m
_ u dC u Q r ðwÞ
ð1Þ
where G is an impact mass, and m is the mass per unit surface area _ of a plate. The transverse displacement of a plate is w, while w € are the associated velocity and acceleration. W is the and w transverse displacement at the plate centre which is immediately underneath a striking mass. The terms on the left hand side of Eq. (1) are the work rate due to the inertia forces, where A is the surface area of a plate. The first term on the right hand side of Eq. (1) is the energy dissipated in any continuous deformation fields. The second term gives the energy dissipated in n plastic bending hinges, each having an _ angular velocity ð@w=@rÞ m across a hinge of length Cm. The final term is the plastic energy absorption in n transverse shear hinges, _ u and a length Cu. Eq. (1) each having a velocity discontinuity ðwÞ ensures that the external work rate equals the internal energy dissipation. The general method has been used to study the dynamic plastic response of beams, and of circular, square and rectangular plates subjected to dynamic pressure pulses and also blast loadings [1–4], and for beams, circular and square plates struck at the mid-span by a rigid mass [4–6]. It is used in this paper to examine a rectangular plate struck by a rigid mass at the centre, and, since large ductile deformations are studied without any material failure, or perforation, the last (transverse shear) term in Eq. (1) is not considered further. Thus, the yield condition consists
Fig. 1. Yield conditions at the plastic hinge lines (including the supports for ma 0) which develop within the rectangular plate in Fig. 2.
70
N. Jones / Thin-Walled Structures 50 (2012) 68–75
inscribe it, thus providing upper and lower bounds on the exact yield condition according to a corollary of the limit theorems [4]. 3. Mass impact loading of a rectangular plate The kinematically admissible transverse velocity field in Fig. 2 is assumed to govern the response of the rectangular plate (2L 2B) in Fig. 3, which has a moment resistance mM0 around the supports (m ¼0 and 1 give the simply supported and the fully clamped cases, respectively, but intermediate cases (0 omo1) are included in the analysis). It is assumed that the mass strikes the centre of a rectangular plate, has a negligible cross-section when compared with the plate dimensions, B and L, and produces a response characterised by the transverse velocity profile in Fig. 2, which can be written as _ ð1x=LÞ, _ ¼W w
0 r x rL,
0 r y rBx=L
ð2aÞ
and _ ð1y=BÞ, _ ¼W w
0 r x r yL=B,
0 r y rB
ð2bÞ
for the quadrant with positive values of x and y. From considerations of double symmetry, the behaviour of this quadrant is similar to the other three quadrants.
The transverse velocity profile in Fig. 2 requires plastic hinges along the diagonals and at the supports (unless m¼0) of a rectangular plate. The interior regions remain rigid for a rigid, perfectly plastic material so that the integral for continuous deformations in Eq. (1) does not contribute to the internal energy dissipation. Thus, Eq. (1) becomes Z Z r X € W _ mw €w _ dA ¼ GW ðM þ NwÞy_ m dlm ð3Þ A
lm
m¼1
for an initially flat, rigid, perfectly plastic plate which deforms into a number of rigid regions separated by r stationary straight line plastic hinges, each of length lm. y_ m is the relative angular velocity across a straight line hinge, w is the transverse displacement along a hinge, and N and M are the membrane force and bending moment, respectively, which act on a plane which passes through a hinge and is transverse to the mid-surface of a plate. Eq. (3) for the impact problem in Fig. 3, when using Eqs. (2) for the transverse velocity field and the square yield condition in Fig. 1, yields the governing equation: € þ a2 W ¼ ð1þ mÞHa2 =2 W
ð4Þ
where 2
a2 ¼
12M0 ð1 þ b Þ
mHL ð1 þ 6gÞb2 2
,
b¼
B G and g ¼ L 4mBL
ð5a2cÞ
Now, when satisfying the conservation of linear momentum at t¼0 requires _ þ 4m GV 0 ¼ GW
Z
L 0
Z
Bx=L
_ ð1x=LÞ dx dy þ 4m W
0
Z
B 0
Z
Ly=B
_ ð1y=BÞ dx dy, W
0
or _ ¼ V 0 =ð1 þ1=3gÞ W
ð6Þ
where V0 is the initial velocity of the striking mass. Eq. (4) has the general solution W ¼ A sin at þ B cos atð1þ mÞH=2,
ð7Þ
where the constants of integration A and B are obtained from the _ given by Eq. (6). initial conditions at t ¼0, namely w¼0 and W Thus, W¼
V0
að1 þ 1=3gÞ
sin at þ
ð1þ mÞH ð1 þmÞH cos at 2 2
ð8Þ
Now differentiating Eq. (8) with respect to time gives the _ from which the duration of motion, T, is transverse velocity W _ ¼ 0, or obtained when W Fig. 2. Pyramidal-shaped transverse velocity field for a rigid, perfectly plastic rectangular plate struck at the centre as shown in Fig. 3: (a) plan view of plastic hinge lines which develop along diagonals and at boundaries (except for the simply supported case when m¼ 0 on the four boundaries). (b) Side view (in y direction) of pyramidal-shaped transverse velocity field at time t.
tan aT ¼
2V 0 ð1þ 1=3gÞð1 þ mÞaH
ð9Þ
Finally, Eqs. (8) and (9) give the dimensionless maximum permanent transverse displacement ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v )ffi 3 u( Wf ð1 þ mÞ 4u 12 bO g ð1þ 6 g Þ t 1þ 15: ð10Þ ¼ 2 2 H ð1 þ b Þð1þ mÞ2 ð1 þ 3gÞ2 If the striking mass, G, is heavy compared with the plate mass, then g b1 and Eq. (10) reduces to ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v( ) u u Wf 2bO ¼ t 1þ 1 ð11Þ 2 H ð1 þ b Þ where
Fig. 3. Rectangular plate having a length 2L and a breadth 2B struck at the centre by a rigid mass G travelling with an impact velocity V0. The bending resistance around the four supports is mM0, where 0r mr 1 and M0 is the plastic limit moment per unit length of the plate cross-section.
O¼
GV 20 4s0 H3
ð12Þ
for the fully clamped case with m¼1. Eq. (11) for a square plate with b ¼1 reduces to the corresponding equation developed in
N. Jones / Thin-Walled Structures 50 (2012) 68–75
Ref. [6]. The theoretical predictions for a simply supported rectangular plate struck by a rigid mass are given by Eqs. (9) and (10) with m¼0. The theoretical predictions in this section have been obtained using the circumscribing square yield condition in Fig. 1. Another solution with the flow stress s0 replaced by 0.618s0 would give a ‘‘lower bound’’ on the exact yield condition in Fig. 1, but lead to an ‘‘upper bound’’ on the maximum permanent transverse displacement predicted by Eq. (10). Thus, the two predictions with circumscribing and inscribing yield curves provide ‘‘lower’’ and ‘‘upper’’ bounds on the maximum permanent transverse displacements.
4. Experimental details The plates in the theoretical analysis reported in Section 3 undergo wholly ductile deformations without any material failure or perforation. Some of the data for the perforation study reported in Ref. [7] on rectangular plates were obtained for lower impact velocities than those required for perforation and gave rise to wholly ductile behaviour of a rectangular plate without failure. This data is reported in Table 1. The uniaxial tensile characteristics of the H¼4 mm thick mild steel plate are s0 ¼262 MPa, su ¼ 359 MPa and ef ¼ 37.5%. The projectiles have a 10.16 mm diameter (d) body, and the impact faces are blunt, hemispherical or conical (901 included angle). A drop hammer rig was used for the tests and the impact mass (G) was 11.8 kg for the blunt-tipped projectiles and 19 kg for the hemispherically tipped and conically tipped projectiles. Thus, the mass ratios g for the test specimens in Table 1 range from 9.4 to 60.5 according to Eq. (5c) with a material density of 7850 kg/m3. The initial impact velocity (V0) was recorded using a laser Doppler velocimeter [8] and the maximum permanent transverse displacements were measured at the plate centre. The energy ratios for the experimental data in Table 1 vary between 9.6 and 59.0 when it is defined as the ratio of the initial kinetic energy, GV 20 =2, and an upper estimate on the elastic strain energy which is approximated as ðs20 =2EÞV, or
71
5. Discussion The theoretical predictions in Section 3 are made particularly simple by using the square yield condition in Fig. 1, which bounds the maximum normal stress yield criterion. A theoretical solution using the exact maximum normal stress, or von Mises, yield criterion, is possible with the method of successive approximations (e.g., see [4]), or some other method of analysis. It appears that no exact theoretical analyses have been published for the mass impact loading of rectangular plates undergoing finite displacements, but the theoretical analyses using an exact yield surface for impulsively loaded beams and circular plates are bounded by the lower and upper bound predictions using circumscribing and inscribing yield criteria, as shown in [1]. The theoretical analysis in Section 3 idealises the striking mass, G, as a concentrated point loading. A theoretical analysis was reported in [6] for a fully clamped circular plate struck by a blunt faced cylindrical mass having a radius a. The theoretical rigid perfectly solution was expressed in terms of a parameter r ¼a/R. Fig. 7.24 [4, 2nd Edition] shows a reduction of about 10% in the maximum permanent transverse displacement for strikers having r ¼0.12. Thus, the retention of r for the ductile plastic behaviour of plates appears to produce a relatively modest effect, particularly when considering various other simplifications in the analysis, such as the idealised impact loading characteristics and approximations introduced in the modelling of the boundary conditions. Studies on the perforation of ductile plating reveal that the dimensionless perforation energy is a fairly strong function of the parameter c ¼d/H, as indicated in [11], because the plate area resisting shear plugging through the plate thickness is roughly proportional to d¼2a. Nevertheless, it is not likely that c would strongly influence the response of plating having a wholly ductile inelastic behaviour without failure, unless r or c are relatively large. Eq. (10) is valid for any value of the dimensionless moment resistance at the supports, m, which lies within the range 0om o1. In particular, Eq. (10) with m¼0 for simple supports 2
1=2
gives W f =H ¼ ½f1 þ8bO=ð1 þ b Þg 1=2 for g b1, which when the dimensionless kinetic energy O b1 for large external load2
EGV 2 Er ¼ 2 0 , s0 V
ð13Þ
where V¼H 2B 2L is the total volume of the plate material. Further details of the test arrangement are reported in Refs. [7,9,10]. Table 1 Experimental data for rectangular plates struck at mid-span. H ¼4 mm, s0 ¼262 MPa, G ¼ 11.8 kg for blunt-tipped projectiles and G ¼ 19 kg for hemispherically tipped and conically tipped projectiles [7]. Shape
b
2B 2L (mm2)
V0 (m/s)
O
Wf/H
Blunt Blunt Blunt Blunt Blunt Blunt Hemisph’l Hemisph’l Hemisph’l Hemisph’l Hemisph’l Conical Conical Conical Conical
0.4 0.5 0.66 0.66 1 1 0.4 0.5 0.66 1 1 0.5 0.66 1 1
100 250 100 200 100 150 100 150 100 100 200 200 100 250 100 200 100 150 100 100 200 200 100 200 100 150 100 100 200 200
6.49 6.49 6.26 6.42 6.42 6.57 6.86 6.72 6.57 6.42 6.86 6.57 6.64 5.60 6.64
7.42 7.42 6.90 7.25 7.25 7.60 13.34 12.79 12.23 11.67 13.32 12.23 12.51 8.89 12.51
2.10a 2.03 2.20 1.83 1.84 2.12 2.74a 2.66 2.51 2.36 2.91 2.51a 2.40a 2.01 2.76a
a Indicates that the plating has cracked underneath the projectile impact location.
1=2
ings, becomes W f =H f2bO=ð1 þ b Þg . This expression is identical to that predicted by Eq. (11) for O b1. In other words, the distinction between the permanent behaviour for plates having simply supported and fully clamped boundary conditions diminishes as the transverse displacement increases and the plate tends towards membrane behaviour. In practice, as long as the axial restraint is maintained, then the moment restraint at the boundary does not have a significant effect on the final displacement for sufficiently large values of O or Wf/H. It is evident from Eq. (11) for a fully clamped rectangular plate with g b1 that the theoretical predictions for Wf/H collapse on to a single curve irrespective of the plate aspect ratio b when 2 2bO=ð1þ b Þ is used in the abscissa, as shown in Fig. 4. A yield stress 0.618s0 inscribes the exact yield condition in Fig. 1 and gives rise to the ‘‘upper bound’’ predictions according to Eq. (11) in Fig. 4. The experimental data in Table 1 is also plotted in Fig. 4 for values of 0.4 r b r1 and for three shapes of the projectile impact faces. First of all, it is evident that most of the experimental data is bounded by the ‘‘upper’’ and ‘‘lower’’ bound predictions of Eq. (11). Secondly, a trend is discerned which indicates that the experimental data migrates from the ‘‘lower bound’’ curve towards the ‘‘upper bound’’ curve as b decreases for both the hemispherically tipped and blunt-tipped projectiles. However, it should be noted that both b ¼0.4 plates had some cracking underneath the impact site. The experimental results for the conical projectiles are not distinct from those for the two other projectile nose shapes and remained fairly close to the
72
N. Jones / Thin-Walled Structures 50 (2012) 68–75
velocities in Table 1 are not expected to be as large as those in [12] for blast loadings. It should be remarked that large plastic strains cause a reduction in the material strain rate effect [14], when compared with relatively small strain problems. For example, the constants D ¼40.4 s 1 and q¼5 are often used in the Cowper Symonds equation 1=q s00 e_ ¼ 1þ ð14Þ s0 D
Fig. 4. Maximum dimensionless permanent transverse displacements at the centre of fully clamped rectangular plates versus the dimensionless initial kinetic energy. ———— : Eq. (11), circumscribing yield condition. – – – – – –: Eq. (11), inscribing yield condition. — — — 1,2: Eq. (10) with m ¼1 and g ¼ 9.4 for circumscribing and inscribing yield conditions, respectively. Experimental data from Table 1: x, J, B; b ¼ 0.4, 0.5, 0.66, respectively. &, ’; b ¼1 for 200 mm 200 mm and 100 100 mm square plates, respectively. The superscripts 1, 2 and 3 denote projectiles with hemispherical, blunt and conical (901 included angle) impact faces, respectively.
‘‘lower bound’’ predictions of Eq. (11); all of these plates, except one, had cracked underneath the projectile impact location. It is noted here that the ‘‘upper’’ and ‘‘lower’’ bound predictions of Eq. (11) with b ¼1 bound the experimental data for the fully clamped mild steel square plates reported in Fig. 7 of Ref. [6]. Eq. (10) retains the influence of the mass ratio, g, whereas Eq. (11) assumes that g b1. As noted in the previous section, g ¼9.4 is the smallest value and would have the greatest effect on the predictions of Eq. (10). Eq. (10) with g ¼9.4 is plotted in Fig. 4 for circumscribing and inscribing square yield curves. The differences between these curves and the g b1 cases are quite small and the values of g for all of the experimental data would lie between g ¼ 9.4 and g b1. It is possible that the transverse displacement and velocity profiles in Fig. 2 and described by Eqs. (2) are less satisfactory as b becomes smaller and approaches b-0 for a ‘‘beam’’ having a span 2B. Thus, the experimental data for rectangular plates with b ¼0.4 lie near to the ‘‘upper’’ bound predictions of Eq. (11) and are only just bounded by the theoretical predictions. The theoretical analysis might not be suitable for rectangular plates with even smaller aspect ratios b when another kinematically admissible transverse profile becomes necessary, although there are no experimental data available for smaller values of b to test this hypothesis. This contrasts with the uniform blast loading of rectangular plates using the same general theoretical method [1] which has an associated velocity profile that gives the response of a beam when b-0. The rectangular plates in Ref. [7] are made from mild steel which usually is a strain rate sensitive material, although the dynamic material characteristics are not available for this particular plate material. The dynamic stress state in a rectangular plate is complex and is not uniaxial. In this regard, it is interesting to observe in Fig. 11 of [12] that the maximum permanent transverse deflections of impulsively loaded rectangular plates made from aluminium alloy 6061-T6, which is an essentially strain rate insensitive material, and mild steel, which is a strongly strain rate sensitive material, differ by about 10–15%, whereas the difference would be nearer 30–40% for a similar comparison on beams [3,13] which have a less complex stress state. Nevertheless, if the data for the strain rate effects of the plate material were known and incorporated into Eq. (11), as shown in [4] for blast loaded plates, then the theoretical curves in Fig. 4 would be lower, although the strain rate effects for the low impact
to characterise the strain rate behaviour of mild steel. In this case, the dynamic flow stress is s00 ¼ 2s0 for a uniaxial strain rate e_ ¼ 40:4s1 . Now for a strain of 0.05, then the data of Marsh and Campbell [15], which is discussed in Ref. [14], reveals that D E1300 s 1 and q¼5, with even larger values of D for larger strains. Thus, the strain rate sensitive behaviour of mild steel varies with strain as embodied in the modified Cowper Symonds equation developed in Ref. [14]. Unfortunately, there does not appear to be any experimental data which has been published for the large ductile deformation behaviour of aluminium alloy rectangular plates struck by a mass. However, the impact behaviour of fully clamped square plates was studied by Langseth and Larsen [16] and the experimental data is reported in Fig. 5. The ordinate in Fig. 5, K, is the dimensionless kinetic energy, or from Eq. (11) W f =H ¼ ð1 þ KÞ1=2 1
ð15Þ
where K¼ O for a square plate (b ¼1). A fully clamped square plate loaded impulsively over the entire surface area can also be cast into the same form of Eq. (15) [1,4], where in this case K ¼ l/6 and l ¼ rV 20 L2 =M 0 when V0 is the uniformly distributed initial impulsive velocity. Thus, both the central impact and uniform impulsive loading cases for a square plate can be cast in the form of Eq. (15) and the ‘‘upper’’ and ‘‘lower’’ bound curves in Fig. 5 for inscribing and circumscribing yield curves apply to both cases. It is evident from the comparisons in Fig. 5 that the simple rigid, perfectly plastic theoretical predictions of Eq. (15), with the appropriate values of K, provide a reasonable design estimate for the experimental values of the strain rate insensitive
Fig. 5. Dimensionless kinetic energy, K, versus the dimensionless maximum permanent transverse displacement according to Eq. (15) for aluminium alloy square plates. ———: Eq. (15), circumscribing yield surface for mass impact and impulsive velocity loadings. – – – –: Eq. (15), inscribing yield surface with s0 replaced by 0.618s0. Experimental results: & AA 6061-T6, uniform impulsive loading [2,3]. ’ AA 6082-T6, mass impact loading [16]. B AA 5083-H112, mass impact loading [16].
N. Jones / Thin-Walled Structures 50 (2012) 68–75
73
aluminium alloy square plates. The experimental data from Ref. [16] (’, B) was obtained for square plates having flanges around the edges rather than fully clamped supports. This might lead to larger values of Wf/H when compared with the data (&) obtained with fully clamped boundaries, as observed for the perforation of square plates [11] and as shown in Fig. 5. K could be regarded as the dimensionless energy required to produce a given dimensionless maximum permanent transverse deformation Wf/H which could be identified with damage. A similar theoretical method has been used to study the impact behaviour of fully clamped circular plates 1=2
W f =H ¼ f1 þð6Ogc =pÞð1 þ6gc Þð1 þ3gc Þ2 g
1
ð16Þ
where
gc ¼ G=mpR2
ð17Þ
which for gc b1 gives [4,6] W f =H ¼ ð1þ 4O=pÞ1=2 1:
ð18Þ
A comparison in Fig. 6 with the experimental data from [17] reveals that the experiments are almost bounded by Eq. (18). It is also interesting that the experimental data of Tian and Jiang [18] on steel circular plates with gc ¼ 1.33 in Fig. 7 are bounded by the predictions of Eq. (16) for inscribing and circumscribing yield conditions. For comparison purposes, the predictions of Eq. (18) with gc b1 are also given in Fig. 7. The energy ratios defined by Eq. (13), but when applied to a circular plate, are between about 10–15 for the experimental data [18] in Fig. 7. However, the mass ratio for these tests is gc ¼1.33 according to Eq. (17), which for the mass impact behaviour of a beam [19], would lead to a difference of about 10 per cent when compared with an exact solution. Unfortunately, no similar studies have been reported on plates, so that the error is not known for the impact behaviour of a circular plate with gc ¼1.33. The experimental data [18], which are plotted in Fig. 7, were obtained using a steel plate having a low material strain rate sensitivity since the flow stress at a strain rate of 17 s 1 increases only 5 per cent when compared with the corresponding static flow stress. This should be compared with an average strain rate of about 7 s 1 for those test specimens in Fig. 7 with Wf/HE3, according to the equation given in the footnote on p. 376 of Ref. [4] (p. 367 of 2nd ed. [4]). Thus, material strain rate effects are not expected to exercise a significant influence on the response.
Fig. 7. Comparison of the theoretical predictions of Eq. (16) with experimental results on fully clamped steel circular plates struck by a mass at the centre. ———: Eq. (16), circumscribing yield surface with gc ¼1.33. – – – –: Eq. (16), inscribing yield surface with s0 replaced by 0.618s0 and gc ¼1.33. J, x, D : experimental results [18] for H¼ 3.6 mm, 1.8 mm and 0.9 mm, respectively. – – – , – – –: Eq. (18) for circumscribing and inscribing yield conditions with gc b1, respectively.
It was observed earlier that Eq. (15) gives the theoretical predictions for a fully clamped square plate struck by a rigid mass at the plate centre when K ¼ O, while Eq. (15) with K ¼ l/6 is valid for a fully clamped square plate subjected to a uniform impulsive velocity loading. Moreover, the theoretical predictions for a fully clamped circular plate can be cast in the form of Eq. (15) provided K ¼4O/p and l/6 for the mass impact (with gc b1) and impulsive loading (see Appendix) cases, respectively. The experimental values in Fig. 8 were obtained from five different studies in four laboratories and using four materials. Two sets of data (x, &) cluster around the predictions of Eq. (15) for a circumscribing yield condition, while the remaining data are near to the predictions for an inscribing yield condition. Florence’s [20] (J) were obtained for an impulsively loaded circular plate having simple supports rather than fully clamped ones considered here. This difference would lead to a larger value of Wf/H for a given value of K. In fact, the more complete theoretical analysis which was developed in Ref. [21] for a simply supported circular plate loaded impulsively does provide close bounds on Florence’s data for the maximum permanent transverse displacements. Similarly, the square plate specimens [16] (’, B) have flanges around the edges rather than fully clamped supports. Again this would cause larger values of Wf/H for a given value of K, as noted earlier and in Ref. [11]. Thus, all of the experimental data (’, B, J) clustered around the predictions of Eq. (15) for an inscribing yield condition do not have fully clamped boundary conditions. Particular attention was given to achieving fully clamped boundary conditions for the plate specimens (x, &) in Fig. 8. The experimental study reported in [7] has examined the threshold conditions for the perforation of plates which occurs at impact energies larger than those associated with the data for the wholly ductile deformations reported in Table 1. An empirical perforation equation
Op ¼ ðp=4Þðd=HÞ þ ðd=HÞ1:53 ðS=dÞ0:21 Fig. 6. Comparison of the theoretical predictions of Eq. (18) with experimental results on fully clamped aluminium alloy circular plates struck by masses at the centre. ———: Eq. (18), circumscribing yield surface. – – – –: Eq. (18), inscribing yield surface with s0 replaced by 0.618s0. J: Experimental results [17] (H¼9.53 mm, R¼ 101.6 mm, a¼11.905 mm (radius of cylindrical body of projectile), G¼ 20 kg, s0 ¼ 442.6 MPa, gc ¼24.24).
ð19Þ
was presented in [22] for circular plates, struck by cylindrical projectiles having blunt impact faces and travelling at low velocities. It turns out that this equation gave good estimates for the dimensionless perforation energy, Op, of rectangular mild steel plates when the span, S, is taken as the plate breadth 2B [7].
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N. Jones / Thin-Walled Structures 50 (2012) 68–75
Fig. 8. Comparison of the theoretical predictions of Eq. (15) for circular plates struck by a mass (x, K¼ 4O/p) or loaded with a uniform impulsive loading (J, K¼ l/6) and for square plates struck by a mass (’, B, K¼ O) or loaded with a uniform impulsive loading (&, K¼ l/6). ——— : Eq. (15), circumscribing yield condition. – – – –: Eq. (15), inscribing yield condition. l ¼ rV 20 R2 =M 0 and l ¼ rV 20 L2 =M0 for circular and square plates, respectively. Experimental data. &, ’, B ; defined in Fig. 5. x; defined in Fig. 7. J; Aluminium alloy 6061-T6 simply supported circular plates, uniform impulsive velocity loading [20].
The perforation energy of mild steel plates struck by projectiles with cylindrical bodies having blunt, conical and hemispherical impact faces, and travelling at impact velocities up to 12 m/s, were studied in [7]. It transpired that projectiles with blunt impact faces perforated mild steel plates with the least energy, while projectiles having hemispherical impact faces required the greatest energy and those projectiles with conical impact faces (901 included angle) required a perforation energy, which was intermediate between the two values, but closer to the hemispherical case. Eq. (19) for 2B¼100 mm predicts Op ¼ 8.72, while 2B¼200 mm gives Op ¼9.77. These values are larger than all of the experimental results for the square and rectangular plates struck by blunt-tipped 2 projectiles in Table 1. The factor 2bO=ð1 þ b Þ in Eq. (11), which is used for the abscissa in Fig. 4, is 0.69O, 0.80O and 0.92O for b ¼0.4, 0.5 and 0.66, respectively, and yields the corresponding values of 2 2bOp =ð1 þ b Þ ¼ 6:01, 6:97 and 8:01. As anticipated, these values lie above all of the corresponding rectangular plate data struck by blunttipped projectiles, which are identified with a superscript 2 in Fig. 4. Moreover, the experimental results in [7,9] reveal that the threshold values of the dimensionless perforation energy, Op, are higher for projectiles having hemispherical and conical impact faces in this series of experimental tests. Eq. (19) for the fully clamped steel circular plates [18] in Fig. 7 with S¼ 2R predicts that projectiles with blunt impact faces would perforate the plates when Op ¼35.5. This value is well above the values associated with a large ductile deformation response.
subjected to either static or dynamic loadings which produce large plastic strains and finite transverse displacements, has been used to study the behaviour of a rectangular plate struck by a mass at the centre. A relatively simple equation has been derived for the maximum permanent transverse displacements and response duration, which can be used for preliminary design purposes and accident investigations. The theoretical predictions have been developed for a plate having a resisting moment mM0, 0omo1, on the entire boundary. In the particular case of m¼1 for the fully clamped case, the theoretical predictions were compared with some experimental data recorded on fully clamped rectangular plates having aspect ratios between 0.4 and 1 (square) and struck transversely by masses with cylindrical bodies travelling up to about 7 m/s. The theoretical predictions provide reasonable estimates of the corresponding experimental data. The theoretical procedure has now been used to provide relatively simple design equations for the response of ductile beams, and circular, square and rectangular plates subjected to dynamic pressure pulses (including blast and impulsive velocity loadings) and mass impact loadings which produce permanent deformations and large plastic strains. The accuracy of these cases have been supported with experimental data obtained from various laboratories over the years. For completeness, an empirical equation has been used to predict the initial impact energy required to perforate a plate by a blunt-faced missile. This equation predicts perforation energies which lie above all of the corresponding experimental data for large-mass and low-velocity impacts on steel plates.
Acknowledgments The author is grateful to Mrs. J. Jones for her secretarial assistance and to Mrs. I. Arnot for her assistance with the figures.
Appendix. Impulsive loading of a fully clamped circular plate The particular case of a rigid, perfectly plastic simply supported circular plate, which is subjected to a rectangular shaped pressure pulse of magnitude p0 and duration t, is developed in Section 7.7 of Ref. [4]. The limit of an infinitely large pressure pulse with a vanishingly short duration, but a finite impulse, gives the impulsive velocity loading case. Now the analysis for fully clamped supports follows closely the analysis in Section 7.7 [4], except the first term on the right hand side _ =RÞ2pR for a fully clamped boundary which of Eq. (7.103) gives M 0 ðW is added to the right hand side of Eq. (7.106). Eq. (7.107) is again obtained except b is replaced by b ¼ 24M 0 ðZ1Þ=mR2 , where Z ¼ p0 =pc and pc ¼ 12M0 =R2 is the static plastic collapse pressure for a circular plate having a square yield condition. The solution of the differential equation with the same initial conditions follows the first (0rt r t) and second (t rt rT) phases of the procedure outlined in Section 7.7 [4]. Motion ceases at the time T, where tanaT ¼
Z sin at 1Z þ Z cos at
ðA:1Þ
leading to a dimensionless maximum permanent transverse displacement W f =H ¼ f1 þ 2ZðZ1Þð1cos atÞg1=2 1,
6. Conclusions A theoretical rigid, perfectly plastic method, developed previously for initially flat plates having an arbitrarily shaped boundary and
2
2
ðA:2Þ
where a ¼ 24M 0 =mHR . The impulsive loading case is obtained by taking Z b1, t-0 and p0 t ¼ mV 0 , where V0 is the initial uniformly distributed impulsive velocity over the entire plate surface. Thus, Eq. (A.2)
N. Jones / Thin-Walled Structures 50 (2012) 68–75
reduces to 1=2
W f =H ¼ ð1þ l=6Þ
1
ðA:3Þ
where
l ¼ mV 20 R2 =M0 H:
ðA:4Þ
References [1] Jones N. A theoretical study of the dynamic plastic behavior of beams and plates with finite-deflections. International Journal of Solids and Structures 1971;7:1007–29. [2] Jones N, Baeder RA. An experimental study of the dynamic plastic behavior of rectangular plates. In: Symposium on plastic analysis of structures, vol. 1, Published by Ministry of Education, Polytechnic Institute of Jassy, Civil Engineering Faculty; 1972: pp. 476–97. [3] Jones N. A literature review of the dynamic plastic response of structures. The Shock and Vibration Digest 1975;7(8):89–105. [4] Jones N. Structural impact. Cambridge University Press; 1989. p. 575, Paperback edition, 1997. Chinese edition translated by Ping Jiang and Lili Wang, Sichuan Education Press, Chengdu, 1994. 2nd edition (in press). [5] Jones N, Kim S-B, Li QM. Response and failure analysis of ductile circular plates struck by a mass. Transactions of the ASME, Journal of Pressure Vessel Technology 1997;119(3):332–42. [6] Jones N. On the mass impact loading of ductile plates. Defence Science Journal, Defence Research and Development Organisation, India 2003;53(1):15–24. [7] Jones N, Birch RS, Duan R. Low velocity perforation of mild steel rectangular plates with projectiles having different shaped impact faces. ASME, Journal of Pressure Vessel Technology 2008;130(3):031206-1–8. [8] Birch RS, Jones N. Measurement of impact loads using a laser Doppler velocimeter. Proceedings of the Institution of Mechanical Engineers 1990;204(C1):1–8. [9] Jones N, Birch RS. Low velocity perforation of mild steel circular plates with projectiles having different shaped impact faces. ASME, Journal of Pressure Vessel Technology 2008;130(3):031205-1–11.
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[10] Birch RS, Jones N, Jouri WS. Performance assessment of an impact rig. Proceedings of the Institution of Mechanical Engineers 1988;202(C4):275–85. Corrigenda 204(C1), p. 8, 1990. [11] Jones N, Paik JK. Impact perforation of aluminium alloy plates. International Journal of Impact Engineering, doi:10.1016/j.ijimpeng.2011.05.007. In press. [12] Jones N, Uran TO, Tekin SA. The dynamic plastic behavior of fully clamped rectangular plates. International Journal of Solids and Structures 1970;6: 1499–512. [13] Jones N, Griffin RN, Van Duzer RE. An experimental study into the dynamic plastic behavior of wide beams and rectangular plates. International Journal of Mechanical Sciences 1971;13(8):721–35. [14] Jones N. Some comments on the modelling of material properties for dynamic structural plasticity. In: Harding J, editor. International conference on the mechanical properties of materials at high rates of strain. Institute of Physics conference series no. 102; 1989. p. 435–45. [15] Marsh KJ, Campbell JD. The effect of strain rate on the post-yield flow of mild steel. Journal of the Mechanics and Physics of Solids 1963;11:49–63. [16] Langseth M, Larsen PK. Dropped objects’ plugging capacity of aluminium alloy plates. International Journal of Impact Engineering 1994;15(3):225–41. [17] Wen HM, Jones N. Experimental investigation into the dynamic plastic response and perforation of a clamped circular plate struck transversely by a mass. Proceedings of the Institution of Mechanical Engineers 1994;208(C2): 113–37. [18] Tian C, Jiang P. Experimental investigation of the scaling laws for circular plates struck by projectiles. In: Zheng Z, Tan Q, editors. Proceedings of the IUTAM symposium on impact dynamics, Peking University Press; 1994: pp. 440–5. [19] Jones N. Quasi-static analysis of structural impact damage. Journal of Constructional Steel Research 1995;33(3):151–77. [20] Florence AL. Circular plate under a uniformly distributed impulse. International Journal of Solids and Structures 1966;2:37–47. [21] Jones N. Impulsive loading of a simply supported circular rigid-plastic plate. Journal of Applied Mechanics 1968;35(1):59–65. [22] Wen HM, Jones N. Semi-empirical equations for the perforation of plates struck by a mass. In: Bulson PS, editor. Proceedings of the 2nd international conference on structures under shock and impact. Computational Mechanics Publications, Southampton and Boston and Thomas Telford, London 1992; pp. 369–80.