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Deformation and tearing of clamped circular work-hardening plates under impulsive loading
,NRRN*TIONIIL,O”RNI~~~
Pressure Vessels and Piping International Journal of Pressure Vessels and Piping 75 (1998) 67-73
Deformation and tearing of clamped circular work-hardening plates under impulsive loading H. M. Wen* Applied
Mechanics
Division,
Department
of Mechanical
Engineering,
UMIST,
P.O. Box 88, Manchester
M60 IQD,
UK
Received 27 January 1998; accepted 16 February 1998
Abstract An approximate theory is presented in this paper to predict the deformation and tearing of clamped circular work-hardening plates subjected to uniformly distributed impulsive loads. Based on a power law stress-strain relationship throughout, various equations are obtained and an effective strain failure criterion is suggested for the rupture of the plates. It is shown that the theoretical predictions are in good agreement with the experimental observations Emterms of the maximum permanent transverse displacements and the critical input impulses which cause the tensile tearing failure of the plates under impulsive loading when material strain rate sensitivity is taken into account. It is also shown that, to a first approximation, the theory developed for circular plates is applicable to rectangular plates. 0 1998 Elsevier Science Ltd. All rights reserved
Nomenclature
m M, MO n NE 4 r R VO
W WO Wof
WP
W
Z Y Yc
Width of rectangular plate Constant, defined in Eq. (17) Kinetic energy imparted by an input impulse to a plate Critical energy dissipated by plastic shear Equivalent static force Transverse shear force Critical transverse shear force Thickness of plate Input impulse Critical input impulse causing tensile tearing failure of a plate Length of rectangular plate Mass of plate Radial bending moment per unit length Circumferential bending moment per unit length Work-hardening index, defined in Eq. (5) Radial membrane force per unit length Constant, defined in Eq. (17) Radial coordinate Radius of plate Initial impulsive velocity Transverse displacement of plate Transverse displacement of plate at the centre Critical transverse displacement at which plate tensile tearing failure occurs Maximum permanent transverse displacement including the shear sliding at the supports Total transverse displacement Transverse coordinate Shear strain Critical shear strain of plate material
* Tel.: 0044 0161 236 2403; fax: 0044 0161 200 4537 0308-0161/98/$19.00 0 1998 Elsevier Science Ltd. All rights reserved PII: SO308-0161(98)00023-4
Effective strain, defined by Eq. (Al) Radial membrane strain Mean strain rate Radial strain Radial bending strain Circumferential strain Circumferential bending strain Radial curvature of plate Circumferential curvature of plate Density of plate material Dynamic flow (yield) stress Constant, defined in Eq. (5) Uniaxial ultimate tensile strength Uniaxial static yield stress Shear sliding at the supports Critical shear sliding at the supports duy Transverse
shear stress
Critical shear stress
1. Introduction The deformation and failure of clamped metal plates under uniformly distributed impulsive loadings have been investigated by several authors [l-3]. Three major failure modes have been identified in the experimental study of fully clamped plates subjected to impulsive loads [4,5] which are similar to those observed by Menkes and Opat [6] for fully clamped beams under uniformly distributed blast loading. Thesemodesare classifiedas (i) large inelastic deformations, (ii) tensile tearing and (iii) transverse shear failure. More recently, Wen et al. [7] developed a
68
H.M.
Wem’hternational
Journal
of Pressure
quasi-static procedure to predict the deformation and failure of a clamped beam struck transversely at any point by a mass travelling at low velocities. A similar procedure has been proposed in Refs. [8,9] to construct failure maps for fully clamped metal beams and circular plates under impulsive loadings using a hybrid model (i.e., r.p.p. for the bending-membrane solution and a power law for the effects of local shear). This procedure, which has been further refined in Ref. [lo] to predict the deformation and tearing of impulsively loaded clamped work-hardening beams, is now used to solve the problem of clamped work-hardening plates subjected to uniformly distributed impulsive loading. Based on a power law stress-strain relationship throughout, various equations are obtained and the tearing failure of the plates is predicted by an effective strain failure criterion, although many assumptions and simplifications are introduced.
2. Quasi-static
theoretical
75 (1998)
67-73
U’
I 0 Fig. 1. Schematic
E diagram
of the power
law stress-strain
relationship.
are the radial membrane strain, radial bending strain and circumferential bending strain, respectively. z is the transverse coordinate. It is further assumed that for a work-hardening plate the stress-strain relation obeys the following power law, as shown in Fig. 1,
analysis
According to the principle of momentum conservation, one obtains [8-lo]
v,= f
m
‘
Vessels and Piping
where a, = IJ~,,/E~is a constant and crU, E, are the uniaxial ultimate tensile strength and the maximum uniform tensile strain corresponding to uU, respectively. It is, therefore, easy to show that the radial (M,) and circumferential (Me) bending moments per unit length are evaluated by the following expressions’
Llll
where V, and m are the initial impulsive velocity and the mass of a plate, respectively. Ek is the kinetic energy imparted by an input impulse (I) to the plate. It is assumed that the displacement profile of a impulsively loaded circular plate up to the initial rupture (tensile tearing) takes the following form [9]
where W is the transverse displacement of the plate and W, is the transverse displacement at the centre. r and R are the radial coordinate and the radius of the plate, respectively. It should be mentioned here that different displacement profiles have been used in the literature, see Refs. [3,11]. From Eq. (3), one obtains d2W Kr=-2=- dr
2w, R=
law Kg=---=- rar
2w, R*
Similarly, one obtains (7) where N, is the radial membrane force per unit length. On the other hand, the transverse shear stress (7) at the shear sliding interfaces at the supports may take the following form [9]
@a> or equivalently
@b)
(4b)
where TV, F, and F, ( = 2na,RHl&) are the critical shear stress, transverse shear force and the critical shear force, respectively. A, A, are respectively the shear sliding and the critical shear siding when shear failure occurs at the supports and are evaluated by
(4c)
A=s,Hfi
and E, = f, + f,b = -
(6a,b)
M,=Me=
+
ZKr
when neglecting the in-plane displacement. E,, erb and eeb
I For the derivation
(94
of M,, MB and N,. see Appendix
A.
H.M.
Wenhternational
Journal
of Pressure
75 (1598)
69
67-73
while
and
(9b) in which y, yC are the transverse shear strain at the shear sliding interfaces and the critical shear strain of the plate material, respectively. The structural behaviour of a clamped circul.ar workhardening plate subjected to uniformly distributed impulsive loads can be solved by following the sameprocedure in Refs. [9,10], i.e., (10) where F, = 6E&3Wt can be viewed as an equival.ent static load and W, = W, + A is the total transversedisplacementof the plate including shearsliding at the supports.Hence, the problem of a clamped circular plate under impulsive loading is transformed into that of the plate subjectedto an equivalent static force at the centre. Substituting Eqs. (4), (6) (7) into Eq. (10) and rearranging gives Fe =
Vessels and Piping
2”+‘(n+
l)na,RH (n + 2)
(114
W,=W,=W,+A
(lab)
where W, is the maximum permanent transverse displacement including the shearsliding (A) at the supportswhich is evaluated by Eq. (8b). E, = Ec,(Fe/Fu)‘“+“‘n is the energy dissipated by shear deformation at the supports with EC,= F,A,l(n + 1) being the critical value of E, when plate shearfailure occurs. X= oU/aYis the ratio of uniaxial ultimate tensile strength to yield stress. Similarly, one obtains
I2of
2
?r2pX~yR2H4
= (n
R 2 n+~ 0 i? ‘Of +
Yc 2(n+ l)e;+’ n+
x
; (
)[
&2+1pp+ (n-t21
&(n+l)
Of
25(n+2)
H
(17t
‘l+‘$
1
-7
OfI (15)
where p, Z,r are the density of the plate material and the critical input impulse causing plate failure due to tensile tearing, respectively. e,f = 2(Wof/R)2 can be viewed as the effective rupture strain of the plate material when the effects of transverse shear are taken into consideration with Wof being determined by the following equation’
and II F,=F,=F,,
k \ ( AC/
(1 lb)
after using W, = W, + A. Rearranging Eq. (1 lb) yields
which when combined with W, = W,,+ A gives (16) (13)
where W, is determinedby Eq. (1 la). Eq. (13) is an approximate equivalent static force-displacement relationship for a clamped circular work-hardening plate subjected to uniformly distributed impulsive loading. By following the same procedure in Refs. [9,10], one obtains
(14a)
where Ef is the uniaxial rupture strain of the plate material.
3. Comparison with the experimental data and discussion The equationsobtained in the preceding section are compared with the experimental results which were reported in Refs. [4,.5,11,12] for the clamped mild steel and aluminium alloy plates subjected to uniformly distributed impulsive loadings. For rate-sensitive materials, the dynamic flow (yield) ’ For its derivation,
see Appendix
A
70
H.M.
Wenllntemational
Journal
stress (cd) can be determined by the well-known Symonds empirical equation, i.e.,
of Pressure
Cowper-
(17) where D = 40.4 SC’, 6500 s-l and 4 = 5, 4 are empirical constants which are chosen to describe the rate sensitive behaviour of mild steel and aluminium alloy, respectively. The mean strain rate (Q may be approximated as [9] 2wovo
&“=m
(1W
when the plate fails in tensile tearing, Eq. (18a) can be rewritten as
(18b) after substituting W,,, in Eq. (16) for W, in Eq. (18a) and using eof = 2( W,,,/R)2. The theory developed above for clamped circular workhardening plates may also be applicable to the problem of clamped rectangular work-hardening plates subjected to uniformly distributed impulsive loads. Eq. (14a) can be rewritten in the following form
Vessels and Piping
75 (1998)
67-73
with Eqs. (16- 18) may be applicable to the deformation and rupture (tensile tearing) of an impulsively loaded clamped rectangular work-hardening plate with length L and width B (B I L) by replacing R in Eqs. (16-20) with &zY. Fig. 2 shows the comparison of the present theoretical predictions with the experimental data for the maximum permanent transverse displacements of clamped mild steel circular and square plates3 which were reported in Refs. [4,5]. It is evident from Fig. 2 that Eq. (19b) is in good agreement with the experimental results when material strain rate sensitivity is taken into consideration. Also shown in Fig. 2 is the theoretical predictions of eqn (15b) in Ref. [9] with LT~being replaced by (Td.It can be seen that these two approaches are close to each other. Comparison is also made of the present theory with the experiments[ 121 in Figs 3 and 4 on the clamped rectangular mile steel and aluminium alloy4 plates, respectively. It is clear from Figs 3 and 4 that Eq. (19b) agrees well with the experimental data for the maximum permanent transverse displacements of the clamped rectangular mild steel and aluminium alloy plates under impulsive loading when strain rate effects are taken into account. Fig. 5 shows the comparison of Eq. (20) with the experimental results for the clamped circular and square mild steel plates which were subjected to uniformly distributed impulsive loads [4,5]. It is demonstrated in
-.A.m2(a,lp) m(+p> n+l
W --.c= H
I2
2~5,
while W,=W,+A
(19b)
where m = prR2H is the mass of the clamped plate and p is the density of plate material. Similarly, Eq. (15) can be recast as
(194
Fig. 5 that the present theory is in good agreement with the experimental observations when strain rate sensitive behaviour is accounted for. Also shown in Fig. 5 is the theoretical predictions of eqn (17a) in Ref. [9]. Again, these two approaches are close to each other.
4. Conclusions An approximate quasi-static procedure has been given in this paper to predict the deformation and tearing of fully clamped plates subjected to uniformly distributed impulsive loading. Based on a power law stress-strain relationship throughout, various equations have been obtained and compared with the available experimental data. In particular, the tensile tearing failure of the plates examined is predicted by
(20) To a first approximation,
Eqs. (19) and (20) together
3 yC, n are taken to be 0.8 and 0.06 for the mild steel material, respectively. 4 yC, n are taken to be 0.5 and 0.075 for the 6061-T6 aluminium alloy, respectively.
U.M.
0 Fig. 2. Comparison Eq. (19b); ---, experiments[ll]
Wen/lntemational
025
Xoumal
of Pressure
05
Vessels and Piping
1
’
a75
75 (1998)
-- I
’
1
71
67-73
125
mm
of the theoretical predictions with the experimental data for clamped circular and square mild steel plates under impulsive loading. -, eqn (15b) in Ref. [9] when material strain rate sensitivity is taken into account. A, experiments[5]; 0, experiments[4]; 0,
an effective strain failure criterion which takes into account the effects of transverse shear on the radial tensile strain. It is found that the present theoretical predictions are in good agreement with the experimetnal observations in terms of the maximum permanent transverse displacements and the critical input impulses causing tensile tearing failure of fully clamped circular plates when material strain rate sensitivity is taken into consideration. It is also shown that, to a first approximation, the theory developed for circular plates can be applied to rectangular plates.
Appendix
A. An effective strain failure criterion
For a clamped circular metal plate subjected to uniformly distributed impulsive loads, it is reasonable to assume that E, = - (er + to), yrz # 0 and all other strain components areequal to zero. The effective strain (E,) can be expressed as
1 (Al)
wp H 75
5
0 Fig. 3. Comparison of the theoretical predictions with the experimental results for clamped rectangular steel plates under impulsive loading. , Eq. (19b) with cY being replaced by Ed. A, 0 and D represent 0.064, 0.098 and 0.173 in thick steel plates, respectively[lZ].
01
0.2
03
a4
05
mm
Fig. 4. Comparison of the theoretical predictions with the experimental data for clamped rectangular 6061-T6 aluminium alloy plates subjected to impulsive loading. , Eq. (19b) with nY being replaced by ud. A, 0 and D represent 0.123, 0.188 and 0.244 in thick aluminium alloy plates, respectively[l2].
12
H.M.
Wedinternational
Journal
of Pressure
Vessels and Piping
75 (1998)
0
0 0
001
ON,
003
004
005
67-73
,H ii
Fig. 5. Comparison of the theoretical predictions with the experimental observations for clamped circular and square mild plates under impulsive loading. -, Eq. (20); - - -, eqn (17a) in Ref. [9] when material strain rate sensitivity is taken into consideration. A, experiments[5]; 0, experiments[4]. Open and halfsolid symbols indicate no tensile tearing and tensile tearing failure, respectively.
Substituting E, = - (E, + eg), yZo = ~0~= 0 into Eq. (Al) and rearranging gives
q4(~)4+6(g3(f) +3(%)2(;)2] +$
(A41 (A53
after using Eqs. (4a-4d), z = H/2 and Y = R at the supports. Upon substitution of yrZ = y into Eq. (A2), one obtains
where W,r is the critical transverse displacement at which the plate ruptures. Appendix
B. Derivation
of M,, MB and N,
The radial bending moment (M,) per unit length can be expressed as H M, = 2 2 ~,(zK,)~z dz I 0 (A3) after using Eqs. @a), (9) and (11). If a plate fails in tensile tearing, it means that its ductility has been exhausted. In other words, the effective strain must reach the uniaxial rupture strain (ef) of the plate material. Therefore, from Eq. (0) one obtains
(A51
Substituting Eq. (4a) into the above equation, one obtains
Rearranging the above equation gives M,= Eq. (6b) for MB can be similarly derived. The radial membrane force per unit length (NJ can be expressed as H
N,=
CT(E >” dz
0 o
m
(A@
H.M.
Wenhtemational
Journal
of Pressure Vessels and Piping 75 (1998) 67-73
Substituting E, = 1/2(8W/&)’ and rearranging gives [6]
G49) [7]
after using Eq. (3). [8]
References [9] [l]
[2]
[3] [4]
[5]
Shen W Q, Jones N. Dynamic response and failure of fully clamped circular plates under impulsive loading. Int. J. Impact Eng., 1993;13:259-278. Bodner S R: Symonds P S. Experiments on viscoplastic response of circular plates to impulsive loading. J. Mech. Phys. Solids, 1979;27:91-113. Nurick G N, Martin .I B. Deformation of thin plates subjected to impulsive loading-a review. Int. J. Impact Eng., 1989;8:159-186. Teeling-Smith R G, Nurick G N. The deformation and tearing of thin circular plate subjected to impulsive loading. Int. J. Impact Eng., 1991;11:77-91. Olson M D, Nurick G N, Fagnan J R. Deformation and rupture of blast
[lo]
[l l]
[12]
13
loaded square plates-predictions and experiments. Int. J. Impact Eng., 1993;13:279-291. Met&es S B, Opat H J. Broken beams. Exptl. Mechs., 1973;13:480486. Wen H M, Reddy T Y, Reid S R. Deformation and failure of clamped beams under low speed impact loading. Int. J. Impact Eng., 1995;16(3):435-454. Wen H M, Yu T X, Reddy T Y. Failure maps of clamped beams under impulsive loading. Mech. Struct. & Mach., 1995;23(4):353-372. Wen H M. Yu T X, Reddy T Y. A note on the clamped circular plates under impulsive loading. Mech. Struct. & Mach., 1995;23(3):331342. Wen H M. Deformation and tearing of clamped work-hardening beams subjected to impulsive loading. Int. J. Impact Eng., 1996;18(4):425-433. Nurick G N, Pearce H T, Martin J B. The deformation of thin plates subjected to impulsive loading. In: Bevilacqua L et al, editors. Inelastic behaviour of plates and shells. Berlin/Heidelberg: Springer-Verlag, 1986:598-616. Jones N, Uran T 0, Tekin S A. The dynamic plastic behaviour of fully clamped rectangular plates. Int. J. Solids and Structures, 1970;6:1499-1512.
Pressure Vessels and Piping International Journal of Pressure Vessels and Piping 75 (1998) 67-73
Deformation and tearing of clamped circular work-hardening plates under impulsive loading H. M. Wen* Applied
Mechanics
Division,
Department
of Mechanical
Engineering,
UMIST,
P.O. Box 88, Manchester
M60 IQD,
UK
Received 27 January 1998; accepted 16 February 1998
Abstract An approximate theory is presented in this paper to predict the deformation and tearing of clamped circular work-hardening plates subjected to uniformly distributed impulsive loads. Based on a power law stress-strain relationship throughout, various equations are obtained and an effective strain failure criterion is suggested for the rupture of the plates. It is shown that the theoretical predictions are in good agreement with the experimental observations Emterms of the maximum permanent transverse displacements and the critical input impulses which cause the tensile tearing failure of the plates under impulsive loading when material strain rate sensitivity is taken into account. It is also shown that, to a first approximation, the theory developed for circular plates is applicable to rectangular plates. 0 1998 Elsevier Science Ltd. All rights reserved
Nomenclature
m M, MO n NE 4 r R VO
W WO Wof
WP
W
Z Y Yc
Width of rectangular plate Constant, defined in Eq. (17) Kinetic energy imparted by an input impulse to a plate Critical energy dissipated by plastic shear Equivalent static force Transverse shear force Critical transverse shear force Thickness of plate Input impulse Critical input impulse causing tensile tearing failure of a plate Length of rectangular plate Mass of plate Radial bending moment per unit length Circumferential bending moment per unit length Work-hardening index, defined in Eq. (5) Radial membrane force per unit length Constant, defined in Eq. (17) Radial coordinate Radius of plate Initial impulsive velocity Transverse displacement of plate Transverse displacement of plate at the centre Critical transverse displacement at which plate tensile tearing failure occurs Maximum permanent transverse displacement including the shear sliding at the supports Total transverse displacement Transverse coordinate Shear strain Critical shear strain of plate material
* Tel.: 0044 0161 236 2403; fax: 0044 0161 200 4537 0308-0161/98/$19.00 0 1998 Elsevier Science Ltd. All rights reserved PII: SO308-0161(98)00023-4
Effective strain, defined by Eq. (Al) Radial membrane strain Mean strain rate Radial strain Radial bending strain Circumferential strain Circumferential bending strain Radial curvature of plate Circumferential curvature of plate Density of plate material Dynamic flow (yield) stress Constant, defined in Eq. (5) Uniaxial ultimate tensile strength Uniaxial static yield stress Shear sliding at the supports Critical shear sliding at the supports duy Transverse
shear stress
Critical shear stress
1. Introduction The deformation and failure of clamped metal plates under uniformly distributed impulsive loadings have been investigated by several authors [l-3]. Three major failure modes have been identified in the experimental study of fully clamped plates subjected to impulsive loads [4,5] which are similar to those observed by Menkes and Opat [6] for fully clamped beams under uniformly distributed blast loading. Thesemodesare classifiedas (i) large inelastic deformations, (ii) tensile tearing and (iii) transverse shear failure. More recently, Wen et al. [7] developed a
68
H.M.
Wem’hternational
Journal
of Pressure
quasi-static procedure to predict the deformation and failure of a clamped beam struck transversely at any point by a mass travelling at low velocities. A similar procedure has been proposed in Refs. [8,9] to construct failure maps for fully clamped metal beams and circular plates under impulsive loadings using a hybrid model (i.e., r.p.p. for the bending-membrane solution and a power law for the effects of local shear). This procedure, which has been further refined in Ref. [lo] to predict the deformation and tearing of impulsively loaded clamped work-hardening beams, is now used to solve the problem of clamped work-hardening plates subjected to uniformly distributed impulsive loading. Based on a power law stress-strain relationship throughout, various equations are obtained and the tearing failure of the plates is predicted by an effective strain failure criterion, although many assumptions and simplifications are introduced.
2. Quasi-static
theoretical
75 (1998)
67-73
U’
I 0 Fig. 1. Schematic
E diagram
of the power
law stress-strain
relationship.
are the radial membrane strain, radial bending strain and circumferential bending strain, respectively. z is the transverse coordinate. It is further assumed that for a work-hardening plate the stress-strain relation obeys the following power law, as shown in Fig. 1,
analysis
According to the principle of momentum conservation, one obtains [8-lo]
v,= f
m
‘
Vessels and Piping
where a, = IJ~,,/E~is a constant and crU, E, are the uniaxial ultimate tensile strength and the maximum uniform tensile strain corresponding to uU, respectively. It is, therefore, easy to show that the radial (M,) and circumferential (Me) bending moments per unit length are evaluated by the following expressions’
Llll
where V, and m are the initial impulsive velocity and the mass of a plate, respectively. Ek is the kinetic energy imparted by an input impulse (I) to the plate. It is assumed that the displacement profile of a impulsively loaded circular plate up to the initial rupture (tensile tearing) takes the following form [9]
where W is the transverse displacement of the plate and W, is the transverse displacement at the centre. r and R are the radial coordinate and the radius of the plate, respectively. It should be mentioned here that different displacement profiles have been used in the literature, see Refs. [3,11]. From Eq. (3), one obtains d2W Kr=-2=- dr
2w, R=
law Kg=---=- rar
2w, R*
Similarly, one obtains (7) where N, is the radial membrane force per unit length. On the other hand, the transverse shear stress (7) at the shear sliding interfaces at the supports may take the following form [9]
@a> or equivalently
@b)
(4b)
where TV, F, and F, ( = 2na,RHl&) are the critical shear stress, transverse shear force and the critical shear force, respectively. A, A, are respectively the shear sliding and the critical shear siding when shear failure occurs at the supports and are evaluated by
(4c)
A=s,Hfi
and E, = f, + f,b = -
(6a,b)
M,=Me=
+
ZKr
when neglecting the in-plane displacement. E,, erb and eeb
I For the derivation
(94
of M,, MB and N,. see Appendix
A.
H.M.
Wenhternational
Journal
of Pressure
75 (1598)
69
67-73
while
and
(9b) in which y, yC are the transverse shear strain at the shear sliding interfaces and the critical shear strain of the plate material, respectively. The structural behaviour of a clamped circul.ar workhardening plate subjected to uniformly distributed impulsive loads can be solved by following the sameprocedure in Refs. [9,10], i.e., (10) where F, = 6E&3Wt can be viewed as an equival.ent static load and W, = W, + A is the total transversedisplacementof the plate including shearsliding at the supports.Hence, the problem of a clamped circular plate under impulsive loading is transformed into that of the plate subjectedto an equivalent static force at the centre. Substituting Eqs. (4), (6) (7) into Eq. (10) and rearranging gives Fe =
Vessels and Piping
2”+‘(n+
l)na,RH (n + 2)
(114
W,=W,=W,+A
(lab)
where W, is the maximum permanent transverse displacement including the shearsliding (A) at the supportswhich is evaluated by Eq. (8b). E, = Ec,(Fe/Fu)‘“+“‘n is the energy dissipated by shear deformation at the supports with EC,= F,A,l(n + 1) being the critical value of E, when plate shearfailure occurs. X= oU/aYis the ratio of uniaxial ultimate tensile strength to yield stress. Similarly, one obtains
I2of
2
?r2pX~yR2H4
= (n
R 2 n+~ 0 i? ‘Of +
Yc 2(n+ l)e;+’ n+
x
; (
)[
&2+1pp+ (n-t21
&(n+l)
Of
25(n+2)
H
(17t
‘l+‘$
1
-7
OfI (15)
where p, Z,r are the density of the plate material and the critical input impulse causing plate failure due to tensile tearing, respectively. e,f = 2(Wof/R)2 can be viewed as the effective rupture strain of the plate material when the effects of transverse shear are taken into consideration with Wof being determined by the following equation’
and II F,=F,=F,,
k \ ( AC/
(1 lb)
after using W, = W, + A. Rearranging Eq. (1 lb) yields
which when combined with W, = W,,+ A gives (16) (13)
where W, is determinedby Eq. (1 la). Eq. (13) is an approximate equivalent static force-displacement relationship for a clamped circular work-hardening plate subjected to uniformly distributed impulsive loading. By following the same procedure in Refs. [9,10], one obtains
(14a)
where Ef is the uniaxial rupture strain of the plate material.
3. Comparison with the experimental data and discussion The equationsobtained in the preceding section are compared with the experimental results which were reported in Refs. [4,.5,11,12] for the clamped mild steel and aluminium alloy plates subjected to uniformly distributed impulsive loadings. For rate-sensitive materials, the dynamic flow (yield) ’ For its derivation,
see Appendix
A
70
H.M.
Wenllntemational
Journal
stress (cd) can be determined by the well-known Symonds empirical equation, i.e.,
of Pressure
Cowper-
(17) where D = 40.4 SC’, 6500 s-l and 4 = 5, 4 are empirical constants which are chosen to describe the rate sensitive behaviour of mild steel and aluminium alloy, respectively. The mean strain rate (Q may be approximated as [9] 2wovo
&“=m
(1W
when the plate fails in tensile tearing, Eq. (18a) can be rewritten as
(18b) after substituting W,,, in Eq. (16) for W, in Eq. (18a) and using eof = 2( W,,,/R)2. The theory developed above for clamped circular workhardening plates may also be applicable to the problem of clamped rectangular work-hardening plates subjected to uniformly distributed impulsive loads. Eq. (14a) can be rewritten in the following form
Vessels and Piping
75 (1998)
67-73
with Eqs. (16- 18) may be applicable to the deformation and rupture (tensile tearing) of an impulsively loaded clamped rectangular work-hardening plate with length L and width B (B I L) by replacing R in Eqs. (16-20) with &zY. Fig. 2 shows the comparison of the present theoretical predictions with the experimental data for the maximum permanent transverse displacements of clamped mild steel circular and square plates3 which were reported in Refs. [4,5]. It is evident from Fig. 2 that Eq. (19b) is in good agreement with the experimental results when material strain rate sensitivity is taken into consideration. Also shown in Fig. 2 is the theoretical predictions of eqn (15b) in Ref. [9] with LT~being replaced by (Td.It can be seen that these two approaches are close to each other. Comparison is also made of the present theory with the experiments[ 121 in Figs 3 and 4 on the clamped rectangular mile steel and aluminium alloy4 plates, respectively. It is clear from Figs 3 and 4 that Eq. (19b) agrees well with the experimental data for the maximum permanent transverse displacements of the clamped rectangular mild steel and aluminium alloy plates under impulsive loading when strain rate effects are taken into account. Fig. 5 shows the comparison of Eq. (20) with the experimental results for the clamped circular and square mild steel plates which were subjected to uniformly distributed impulsive loads [4,5]. It is demonstrated in
-.A.m2(a,lp) m(+p> n+l
W --.c= H
I2
2~5,
while W,=W,+A
(19b)
where m = prR2H is the mass of the clamped plate and p is the density of plate material. Similarly, Eq. (15) can be recast as
(194
Fig. 5 that the present theory is in good agreement with the experimental observations when strain rate sensitive behaviour is accounted for. Also shown in Fig. 5 is the theoretical predictions of eqn (17a) in Ref. [9]. Again, these two approaches are close to each other.
4. Conclusions An approximate quasi-static procedure has been given in this paper to predict the deformation and tearing of fully clamped plates subjected to uniformly distributed impulsive loading. Based on a power law stress-strain relationship throughout, various equations have been obtained and compared with the available experimental data. In particular, the tensile tearing failure of the plates examined is predicted by
(20) To a first approximation,
Eqs. (19) and (20) together
3 yC, n are taken to be 0.8 and 0.06 for the mild steel material, respectively. 4 yC, n are taken to be 0.5 and 0.075 for the 6061-T6 aluminium alloy, respectively.
U.M.
0 Fig. 2. Comparison Eq. (19b); ---, experiments[ll]
Wen/lntemational
025
Xoumal
of Pressure
05
Vessels and Piping
1
’
a75
75 (1998)
-- I
’
1
71
67-73
125
mm
of the theoretical predictions with the experimental data for clamped circular and square mild steel plates under impulsive loading. -, eqn (15b) in Ref. [9] when material strain rate sensitivity is taken into account. A, experiments[5]; 0, experiments[4]; 0,
an effective strain failure criterion which takes into account the effects of transverse shear on the radial tensile strain. It is found that the present theoretical predictions are in good agreement with the experimetnal observations in terms of the maximum permanent transverse displacements and the critical input impulses causing tensile tearing failure of fully clamped circular plates when material strain rate sensitivity is taken into consideration. It is also shown that, to a first approximation, the theory developed for circular plates can be applied to rectangular plates.
Appendix
A. An effective strain failure criterion
For a clamped circular metal plate subjected to uniformly distributed impulsive loads, it is reasonable to assume that E, = - (er + to), yrz # 0 and all other strain components areequal to zero. The effective strain (E,) can be expressed as
1 (Al)
wp H 75
5
0 Fig. 3. Comparison of the theoretical predictions with the experimental results for clamped rectangular steel plates under impulsive loading. , Eq. (19b) with cY being replaced by Ed. A, 0 and D represent 0.064, 0.098 and 0.173 in thick steel plates, respectively[lZ].
01
0.2
03
a4
05
mm
Fig. 4. Comparison of the theoretical predictions with the experimental data for clamped rectangular 6061-T6 aluminium alloy plates subjected to impulsive loading. , Eq. (19b) with nY being replaced by ud. A, 0 and D represent 0.123, 0.188 and 0.244 in thick aluminium alloy plates, respectively[l2].
12
H.M.
Wedinternational
Journal
of Pressure
Vessels and Piping
75 (1998)
0
0 0
001
ON,
003
004
005
67-73
,H ii
Fig. 5. Comparison of the theoretical predictions with the experimental observations for clamped circular and square mild plates under impulsive loading. -, Eq. (20); - - -, eqn (17a) in Ref. [9] when material strain rate sensitivity is taken into consideration. A, experiments[5]; 0, experiments[4]. Open and halfsolid symbols indicate no tensile tearing and tensile tearing failure, respectively.
Substituting E, = - (E, + eg), yZo = ~0~= 0 into Eq. (Al) and rearranging gives
q4(~)4+6(g3(f) +3(%)2(;)2] +$
(A41 (A53
after using Eqs. (4a-4d), z = H/2 and Y = R at the supports. Upon substitution of yrZ = y into Eq. (A2), one obtains
where W,r is the critical transverse displacement at which the plate ruptures. Appendix
B. Derivation
of M,, MB and N,
The radial bending moment (M,) per unit length can be expressed as H M, = 2 2 ~,(zK,)~z dz I 0 (A3) after using Eqs. @a), (9) and (11). If a plate fails in tensile tearing, it means that its ductility has been exhausted. In other words, the effective strain must reach the uniaxial rupture strain (ef) of the plate material. Therefore, from Eq. (0) one obtains
(A51
Substituting Eq. (4a) into the above equation, one obtains
Rearranging the above equation gives M,= Eq. (6b) for MB can be similarly derived. The radial membrane force per unit length (NJ can be expressed as H
N,=
CT(E >” dz
0 o
m
(A@
H.M.
Wenhtemational
Journal
of Pressure Vessels and Piping 75 (1998) 67-73
Substituting E, = 1/2(8W/&)’ and rearranging gives [6]
G49) [7]
after using Eq. (3). [8]
References [9] [l]
[2]
[3] [4]
[5]
Shen W Q, Jones N. Dynamic response and failure of fully clamped circular plates under impulsive loading. Int. J. Impact Eng., 1993;13:259-278. Bodner S R: Symonds P S. Experiments on viscoplastic response of circular plates to impulsive loading. J. Mech. Phys. Solids, 1979;27:91-113. Nurick G N, Martin .I B. Deformation of thin plates subjected to impulsive loading-a review. Int. J. Impact Eng., 1989;8:159-186. Teeling-Smith R G, Nurick G N. The deformation and tearing of thin circular plate subjected to impulsive loading. Int. J. Impact Eng., 1991;11:77-91. Olson M D, Nurick G N, Fagnan J R. Deformation and rupture of blast
[lo]
[l l]
[12]
13
loaded square plates-predictions and experiments. Int. J. Impact Eng., 1993;13:279-291. Met&es S B, Opat H J. Broken beams. Exptl. Mechs., 1973;13:480486. Wen H M, Reddy T Y, Reid S R. Deformation and failure of clamped beams under low speed impact loading. Int. J. Impact Eng., 1995;16(3):435-454. Wen H M, Yu T X, Reddy T Y. Failure maps of clamped beams under impulsive loading. Mech. Struct. & Mach., 1995;23(4):353-372. Wen H M. Yu T X, Reddy T Y. A note on the clamped circular plates under impulsive loading. Mech. Struct. & Mach., 1995;23(3):331342. Wen H M. Deformation and tearing of clamped work-hardening beams subjected to impulsive loading. Int. J. Impact Eng., 1996;18(4):425-433. Nurick G N, Pearce H T, Martin J B. The deformation of thin plates subjected to impulsive loading. In: Bevilacqua L et al, editors. Inelastic behaviour of plates and shells. Berlin/Heidelberg: Springer-Verlag, 1986:598-616. Jones N, Uran T 0, Tekin S A. The dynamic plastic behaviour of fully clamped rectangular plates. Int. J. Solids and Structures, 1970;6:1499-1512.