Int. d. Impact Enqna Vol 12, No. I, pp. 101-121. 1992
0734-743X/92 $5,00+ 0.00 c, 1992 Pergamon Press pie
Prmted m Great Britain
A FAILURE CRITERION FOR BEAMS UNDER IMPULSIVE LOADING WEI QIN SHEN a n d NORMAN JONES Impact Research Centre, Department of Mechanical Engineering, The University of Liverpool, P.O. Box 147, Liverpool L69 3BX, U.K
(Received 28 June 1991; and m revised Jbrm 25 November 1991) Summary--An energy criterion is suggested in this paper for predicting the inelastic failure modes of beams subjected to large dynamic Ioadmgs. The experimental results of Menkes and Opat on the failure of impulsively loaded beams may be interpreted to show that the length of a plastic hinge varies with the ratio of the plastic shear work to the total plastic work dissipated at a hard point or other region of intense localized plastic strain. The variation of this parameter is obtained from a theoretical analysis for the impulsive loading of a rigid, perfectly plastic clamped beam with plastic yielding governed by an interaction yield surface which combines the influences of bending, shear and membrane tension. The Cowper-Symonds dynamic constitutive relation is used to study the influence of material strain rate sensitivity.
NOTATION c k rn P Pc Po q t t2 If w x x* B D Eext Ek Ektf Ep
Ep: Ep2 Epb E,p¢
Eps Eps i, Eps z Epsb E* G H I I* L M Mdo M^ MA1 Mc Mo M*
hinge length
I + (~m/D) TM mass per unit length of a beam load per unit length acting on a beam 4Mdo/L 2, fully plastic dynamic collapse load per unit length peak value of p material constant time tlme at the end of phase 2 time at failure or when motion ceases transverse displacement axial coordinate illustrated in Figs 6(a) and 7(a)
x/L beam breadth material constant in Eqn (3c) external work kinetic energy of a beam residual kinetic energy of a beam plastic work dissipated m a particular region Ep during phases I and 2, respectively total plastic work dissipated in a beam critical value of Ep plastic shear work dissipated in a particular region Eps during phases 1 and 2, respectively total plastic shear work dissipated in a beam
E/Mo mass of a striker beam or plate thickness i m p ~ unit area
l/x/pao H2 half length of a beam bending moment trdoBH2/4, dynamic fully plastic bending moment M at support A MA during phase 1 M at middle point C static fully plastic bending moment
M/Mdo lOl
102
W . Q . SHEN and N. JONES
N
Ndo N*
Q Qdo
OA QAI
Q*
tensile force
adoBH, dynamic fully plastic membrane force N/Ndo transverse shear force
adoBH/x~, dynamic fully plastic transverse shear force Q at support A QA during phase I
9_/Qdo residual momentum of a beam
Ttr/21BL
Vo v,
W* X1, X2, X3
Y,, ~, Y3, Y,
initial velooty of a dropping mass velocity of a beam at rupture transverse displacements at support A and at middle point C, respectively WA and Wc during phase 1, respectively WA and Wc at t r, respectively
W/H variables defined in Eqn (31) variables defined in Eqn (54) c/H, dimensionless length of a plastic hinge
EpJEp /~,
critical value of fl which distinguishes between failure modes 2 and 3
,/
2Mdo/QdoL = x ~ H / 2 L p,/mH = ado/pL 2
E
true strain or equivalent strain rupture strain mean value of r:
~f
~'m II Oa, On ).
~Lf
Po/Pc rotations at points A and B, respectively
f~bH/4Mdo OH/4Mdo location of a travelling hinge during phase 1 at tf
UL P ff (7 0 cYd
O'do
(p A Aa Aa
AW~ At A* ®
®b Op f~b, f~p ¢')
(")
density of materml static uniaxial true stress or equivalent stress initml yield stress in a static uniaxial tensile test dynamic uniaxml tensile true stress or equivalent stress mean dynamic uniaxial stress duration time of pulse
~b/Ob (or flUO p) total axial elongation A at support A A at point B
Wet- WA, time step
A/L critical density of plastic work dissipated per unit volume in a structure critical plastic work dissipated per unit length in beams and per unit area m plates, respectively plastic work dissipated per unit length in beams and per unit area in plates, respectively density of plastic shear work dissipated per unit length in beams
~( )l~t ~2( )/?t'1. I N T R O D U C T I O N
One- and two-dimensional structural members, such as beams, plates and thin shells, may fail as indicated in Fig. 1, under sufficiently large dynamic loads. Several recent experimental and theoretical studies I-1-4] have examined the onset of these failure modes in order to estimate the structural capacities which are required for design purposes. Three major failure modes were identified in the experimental study in [ 1] on impulsively loaded ductile metal beams. However, the first one is characterized by excessive permanent transverse displacements without any failure of the material. The second and third failure
A fadure criterion for beams under impulsive loading Vo
Vo
vo \ Hard point (o)
~
f
o
103
[]
[I
((~Foilure) ) areas
durearea
Ic)
(b)
FIG. 1. Structural failure. (a) A clamped beam subjected to impulsive loading. (b) A clamped beam struck by a mass. (c) A clamped cyhndrical shell subjected to impulsive partial loading
modes are concerned with tensile tearing dominant and transverse shear dominant failures, respectively. A theoretical rigid-plastic method [5], which retains the effects of finite displacements but neglects the influence of transverse shear in the yield condition, gives reasonable agreement with the corresponding experimental results I- 1] for the onset of a mode 2 failure which is taken to occur in the theoretical analysis when the maximum strain reaches the static uniaxial rupture strain of the material. A square yield criterion, which relates the membrane force and the bending moment, is used in this analysis. The magnitude of this yield condition may be adjusted to inscribe and circumscribe and, therefore, provide bounds on the exact parabolic yield curve. Another theoretical rigid-plastic procedure I-5] uses a square yield condition which relates the transverse shear force and the bending moment. This analysis predicts encouraging agreement with the experimental data for the onset of a mode 3 transverse shear failure reported by Menkes and Opat I-1] which is assumed to occur in the theoretical analysis when the amount of transverse shear sliding at the supports equals the beam thickness. However, in practice, a sharp distinction is not found between the mode 2 and mode 3 failures of structures. It will be shown in this article that the membrane force plays an important role in a mode 3 failure and that the effect of the transverse shear force is also significant for a mode 2 failure. It is found in [6] that the failure of a clamped beam with particular characteristics and struck by a heavy mass is described by a critical angle 0c = 0.34 radians between the two parts of a beam which rotate around the impact point. To obtain a more general failure criterion for impact loading, a method for calculating the maximum strain by means of an assumed plastic hinge length using the slip line theory method, which was employed in [5] for the impulsive loading case, is developed in [7, 8]. The theoretical method in [5, 7, 8] neglects the effects of the transverse shear forces in the yield condition and predicts the onset of cracking for a mode 2 response when the maximum strain reaches the rupture strain which is recorded in a uniaxial tensile test. However, some types of dynamic loads, such as impulsive loading and impact by a blunt striker, may cause the development of large transverse shear forces which may also influence the onset of a mode 2 tearing failure. Thus, the maximum strain failure criterion in I-5, 7, 8] requires further development for these particular cases. Finally, it should be noted that a structural member might continue to support loads after the initiation of a crack and before severance has occurred, as observed in [9]. An energy failure criterion is proposed in Section 2 which caters for the simultaneous influence of the transverse shear and axial membrane forces as well as the bending moment in the structural response. The influence of material strain rate sensitivity is explored using the Cowper-Symonds constitutive equation. A theoretical analysis for the dynamic response
104
W . Q . SHEN and N. JONES
of a clamped beam under impulsive loading which retains the simultaneous influence of the transverse shear and axial membrane forces and bending moment is presented in Section 3 and some results are discussed in Section 4. It is the objective of this work to develop an energy density failure criterion to incorporate into FEM Codes which are used to solve practical structural impact problems but which do not predict adequately the structural capacity when limited by material failure.
2. F A I L U R E C R I T E R I O N
It is remarked in Ref. [ 10] that a crack initiates at a point in a structure when the specific dissipation (density of plastic work) at that point reaches a critical value: 0
=
(1)
trddg
where er and crd are the true rupture strain and true dynamic stress in a uniaxial tensile test, respectively, and are assumed to equal the equivalent strain and equivalent stress in an actual structure. Figure 2 shows a potential failure region in a plate, where c and H denote the length of the plastic hinge region and the plate thickness, respectively. The critical dissipation of plastic work per unit of the original plate surface area in Fig. 2 is: ®P = O H
(2a)
®b = ®HB
(2b)
while
is the critical dissipation of plastic work per unit initial hinge length in a beam having a width B and a thickness H. It is observed from the experimental results on the uniaxial tensile specimens in [ 11 ] that er is essentially independent of strain rate for mild steel but increases slightly with increasing strain rate for an aluminium alloy. However other experimental results have found that the uniaxial rupture strain may decrease with strain rate for mild steel and increase or decrease for other materials, as discussed in [12, 13]. The variation of the true stress-true strain curves with true strain rate from [11] is demonstrated qualitatively in Fig. 3. A rigid, perfectly plastic material is employed for the current calculations since the strain rate will decrease with time during the dynamic response to yield a stress-strain curve which approaches perfect plasticity, while elastic deformations are assumed to be negligible for the class of problems which is examined in this article. Thus, let (3a)
0 p = crdogfH
for plates and O b =
~
(3b)
o'do~fHB
[ine
FIG. 2. A potential failure region in a structure.
A failure criterion for b e a m s under impulsive loading
105
O
t:as : Qa. :t'l 3 : el1 :Q1
oa
O0"
Ef
E
FIG. 3. Variation of true s t r e s s - s t r a i n curves with strain rate [ 11 ].
f We: 0 (o)
Wc:H
FIG. 4. Plastic hinge zones. (a) Wc = 0. (b) Wc = H.
for beams, where the mean dynamic flow stress in Fig. 3 is taken as adO=aO
[ (;my] 1+
(3C)
according to the Cowper-Symonds constitutive equation [4, 13, 14], where ao is the initial yield stress in a static uniaxial tensile test, D and q are material constants, ~m is the mean uniaxial strain rate and ef in Eqn (3) is assumed to be independent of~ [8, 13]. The mean strain rate in Eqn (3c) is estimated from ~m = --Ef
(4a)
tf
where tt is the time at failure. Equation (4a) for severely deformed structures with a final strain e = opel, where (p ~< 1, is replaced by
~m = -(p ef tr
(4b)
when assuming that (p = flb/®b (or f~P/®P) and where fib or f~P and tf are the actual dissipation density of plastic work in the critical region and the time when motion ceases, respectively. Thus, Eqn (3) becomes 0 p = kaoetH
(5a)
0 b = kaoefHB
(5b)
and
respectively, where k = 1 + (~m/D) l/q. Nonaka [ 15] used slip line theory to examine the plastic hinge zones in the fully clamped beam in Fig. 4(a) and (b), when the influence of the transverse shear stress on plastic
106
W.Q. SHEN and N. JONES
yielding is neglected. The mean lengths of the plastic hinges are H / 2 and H when Wc = 0 and Wc = H, respectively. Thus, let the plastic hinge length c = c~H
(6)
with 0.5 ~< a ~< 1 for the beams examined in this paper. The total dissipation of plastic shear work, total plastic work and critical total plastic work within one hinge having a length c are Ups = ~QsbC
(7a)
Ep = ~')bc
(7b)
Epc = ®bc
(7C)
and
respectively, where f~sb is the actual dissipation density of plastic shear work within a hinge having a length c. It will be shown later that for the hinges with a large plastic shear strain, the parameter ~ in Eqn (6) depends on the ratio fl - UPs Ep
(8)
and that ct decreases with an increase in ft. The ct-fl curve is constructed using the experimental results of Menkes and Opat [ 1 ] on impulsively loaded beams and will be given later in this paper. It was noted earlier that three failure modes were defined in i-1] and examined theoretically in 1-53 for impulsively loaded beams. It is suggested here and shown later in Fig. 11 that tic = 0.45 marks the critical transition between a mode 2 and a mode 3 failure, when using the experimental results of Menkes and O p a t [ I 3. An iterative method must be used to predict the failure modes for structural members when subjected to an impulse I or struck by a mass G having an initial velocity Vo. For known values of the quantities ao, ef, p, D, q, H, B, L and I or G and Vo, and assuming a value k 1 for k, one can calculate the values of Ep, fl, a, tf and Epc when motion ceases using a theoretical method or a numerical scheme. This gives a new value k 2. The iteration on k is continued until k i ~ k;_l, to give the associated value of q~. If ~o < l, then a mode 1 structural failure occurs (i.e. large ductile deformations). However, if ~0 > 1, then a mode 2 or a mode 3 structural failure has occurred. In this case, the value offl, a, tf and k are calculated using the same iterative procedure as above, except tf corresponds to the condition of Ep ~ Epc. The final value of fl indicates whether or not a mode 2 or a mode 3 failure governs the failure process. To the authors' knowledge, no theoretical solution is available for a fully clamped rigid-plastic, strain rate sensitive beam under impulsive loading when the influences of the transverse shear and axial membrane forces on yielding are taken into account, as well as the bending moment. A theoretical analysis is presented in the next section for the dynamic response of this problem.
3. DYNAMIC RESPONSE OF A CLAMPED BEAM UNDER IMPULSIVE LOADING
For simplicity, a rectangular shaped pressure pulse with a constant peak value Po for a short duration z is distributed over the entire span of a beam to model an actual explosive loading. Thus P = Po,
0 ~< t ~< r,
(9a)
p = 0,
t > ~
(9b)
and I = poz/B,
(9c)
A failure criterion for beams under impulsive loading p
L
p
2L
107
H
.I
M
M (I
FIG. 5. A clamped beam subjected to uniform distributed impulsive loading.
where p denotes the load per unit length acting on a beam and I is the impulse per unit area, as shown in Fig. 5. Let
rl = Po/Pc
(10)
denote a loading factor, where Pc = 4Moo/Lz and Mdo = adoBHZ/4 are, respectively, the fully plastic collapse load per unit length and fully plastic bending moment for a beam which is made from a strain rate sensitive material, where ado is given by Eqn (3c). It can be shown for a rigid, perfectly plastic beam that, as q approaches (2L/3H) 2, the transverse shear forces at the supports tend to the dynamic fully plastic shear force Qdo = adoBH/x/~. It transpires that the transverse shear forces at the supports may play an important role in the yielding behaviour of a beam under impulsive loading, especially at the start of motion, as discussed further in [4, 16]. An interaction yield surface combining the bending moment M, axial tensile force N and the transverse shear force Q [17-] i M , l x / 1 _ Q,Z + N , z + Q,Z = 1
(11)
is employed in the current calculations, where M* = M / M d o , Q* = Q/Qdo, N* = N/Ndo and Nao = adoBH. Equation (11)is shown to be statically admissible in Ref. [ 18].
3.1. Phase l : O 0 Eqns (39), (40) with A* = A~ and 0 B = 0c, (56), (58) and (59) give N*(1 + l/x/1 - O~Z) = 2(Y, - Y3)
(60)
while Eqns (41) and (56) and (57) produce O~,[2/~/1 - Q~2 _ M * / ( I - Q.2)] = 2 Y,/(Y2 - Y,). y
(61)
If M* = 0, then Eqns (40), (45) and (56)-(59) give N*[(Y~/x//3Q~, + ( H / 2 L ) ( Y 2 - 111)] = ( H / L ) ( Y 2 - I"1)(I"4 - Ya).
(62)
Thus, Q~,, M* and M~ could be expressed in terms of Y1, Y2, Y3 and 1I, from Eqns (37), (38), (60) and (61) for M~, > 0. Similarly, Q~, and M~ could be obtained from Eqns (38), (44) and (62) for M~, = 0. It is evident that the complete solution is obtained when solving Eqn (55) with the aid of the R u n g e - K u t t a method. The total plastic work and the plastic shear work dissipated during the third phase of motion may be calculated from Eqns (47) and (48) with 4" = 1. This phase of motion ends when the motion of the beam ceases with Ep < Epc for a mode 1 deformation. Alternatively, a mode 2 or a mode 3 failure occurs at the supports when Ep = Epc. The residual kinetic energy Ek,f and momentum Ttr when the beam fails at the supports are Ektf = 2
m(L-~)W~+-ml
2
[ WA + (IiVc -- WA)~ JoL
dx
and Ttf = 2 m ( L - ~)ff'c + 2m
I/VA+ (I'Vc -- if'A)
dx
or
E*tf = ( 4 k / 5 ) ( H / L ) [ I ? ¢ * 2 + ~*(I4,'.2 + W A W c
2I~2)/3]
(63)
and Tt~ = ( k a o / 6 l ) ( H / L ) 2 [ ff'~ + ~*( 14/* -- I,I/~)/2]
(64)
where E*tt = Ektr/Mo, and Tt~ = Ttf/2IBL.
4. RESULTS
The authors have measured the permanent transverse displacement differences AW* = W*f - W]f of the severed fully clamped aluminium alloy 6061T6 beams from the photographs in the original report by Menkes and Opat from which [ l l is abstracted. The dimensionless lengths of the plastic hinges ~ for each value of AW* are calculated
A fadure criterion for beams under impulsive loading
10t~" ~*°
115
H(mm)L(mm) 101-6 • 6-35 10l 6 •
t,75
°
9 53
.
10"1-6
* t,75 S0B
o 635
50B
% o
.\
o2 "\,
05
0
11o
o's
13
FIG. 9. ct-fl relation given by Eqn (65) and the calculated results from the experimental data in [1].
H=635mm AWl" L =I01 6ram
AW
I /I, ~
H=9 53mm L =101-6mm
m
I
' ,
,I, 0 t,
0
0 (o)
, nl
I
;
2
.'x.
I
.I I I
/_k"
I'2
l"
oJ2 o~, 1016 ~,08
I"
(b)
FIG. 10. Variation of the dimensionless displacement difference AW* with the dimensionless impulse I*. (a) H = 6.35 mm, L = 101.6 mm. (b) H = 9.53 ram, L = 101.6 mm. * experimental data I-1]; - current calculated result with D = 6500 s - l ; . . . . current calculated result with D = 1288000 s-J; - - . - - theoretical predictions for mode 1 from 1,5]; • theoretical predictions at the threshold from mode 2 to mode 3 from 1,5]; . . . . . current calculated threshold lines from mode 1 to mode 2 and from mode 2 to mode 3. from the theoretical analysis in Section 3 when t a k i n g p = 2686 k g / m 3, a o = 279.6 N / m m 2 [ 5 1 ef = 0.5 [ 1 1 ] , D = 6 5 0 0 s -1 a n d q = 4 [4, 14] a n d are p l o t t e d in Fig. 9 a g a i n s t the d i m e n s i o n l e s s ratio fl = Er,,/E p which is also o b t a i n e d from the n u m e r i c a l analysis. These results indicate that a simple straight line ct + 1.2fl = 1.3
(65)
m a y be t a k e n as an a p p r o x i m a t i o n for the ct - fl relationship. The theoretical p r o c e d u r e in Section 3, t o g e t h e r with the semi-empirical relation (65) for the d i m e n s i o n l e s s hinge length, has been used to e x a m i n e two series of the test results which were r e p o r t e d in Ref. [ 1 ]. T h e three zones labelled I, II, III in Figs 1 0 - 1 5 designate the failure m o d e s 1, 2 a n d 3, respectively, which are p r e d i c t e d a c c o r d i n g to the c u r r e n t energy failure criterion. T h e t r a n s i t i o n from a m o d e 1 to a m o d e 2 failure c o r r e s p o n d s to the m i n i m u m impulse required for r u p t u r e (Vf = 0 a n d E v = Epc ), while the t h r e s h o l d between a m o d e 2 a n d a m o d e 3 is a s s o c i a t e d with the impulse which causes r u p t u r e of a b e a m (l/f > 0 a n d Ep = Epc) with fie --- 0.45, which is suggested by the results in Fig. 11. T h e n u m e r i c a l p r e d i c t i o n s for the d i m e n s i o n l e s s impulses for the t r a n s i t i o n s between a m o d e 1 a n d a m o d e 2 failure a n d between a m o d e 2 a n d a m o d e 3 failure are c o m p a r e d in T a b l e 1 with the c o r r e s p o n d i n g e x p e r i m e n t a l results r e p o r t e d in I-1 ] a n d the theoretical rigid, perfectly plastic p r e d i c t i o n s in [ 5 ] .
116
W . Q . SHEN and N. JONES
H=6-35mm.
113
H=9-53 mm. 101.6m m
L=
10
rI
I
05
I, L =101.6mm ,'[I ,~III
~oCO/l
] j
0-5
I 11/
I
,III ~,I. I
i
13c
0
O't,' 08 , I'.2
o
r
0'2
o'~.
(n)
1o,.6 o.s I, (b)
FIG. I 1. Variation of fl versus the dimensionless impulse I*. (a) H = 6.35 mm, L = 101.6 mm; (b) H = 9.53 mm, L = 101.6 mm. - . . . . current calculated threshold lines from mode 1 to mode 2 and from mode 2 to mode 3; • theoretical predictions at the threshold from mode 2 to mode 3 from [5].
H:635 mm
~f
1
10 I
0
,
~f'
I L=1016mm II ~ I I I
O't*
,
,I
,
08 (o)
1
1.0 0
112
I
0
I
] L=1OI.6mm I .
, II
'i Ill
I,
I~
(J'2 0'4 06 (b)
0"8
I"
FIG. 12. Variation of the travelling hinge B position ~ at tf against the dimensionless impulse I*. (a) H = 6.35 mm, L = 101.6 mm; (b) H = 9.53 mm, L = 101.6 mm. current calculated result; - - . - - theoretical predictions for mode 1 from [5]; • theoretical predictions at the threshold from mode 2 to mode 3 from [5]; . . . . . current calculated threshold lines from mode 1 to mode 2 and from mode 2 to mode 3.
Eext-Ektf E~t 1.0
H = 9"53 m m L =I01-6 m m
H=6 35mm L =101 6 m m Ee~t-Ektf
J
i --7 II I Ill
1.0
[I~
05
I
I I
III
05 I"
I I I '
,
I
J '1
'
O.B
12
0
I"
(a)
I I
04 (b)
0.8 i ~
FIG. 13. Variation of the total plastic work dissipation expressed as a proportion of the initial kinetic energy versus the dimensionless impulse I*. (a) H = 6 . 3 5 m m , L = 101.6mm; (b) H = 9.53 mm, L = 101.6 ram. - current calculated result; - - . - - theoretical predictions for mode 1 from [5]; • theoretical predictions at the threshold from mode 2 to mode 3 from I-5]; ..... current calculated threshold lines from mode 1 to mode 2 and from mode 2 to mode 3.
A comparison displacement and compared theoretical
between the theoretical predictions
difference AW*
and the experimental
results for the
a t t h e t i m e t r v e r s u s t h e i m p u l s e I * is i l l u s t r a t e d i n F i g . 10
w i t h t h e s i m p l e t h e o r e t i c a l p r e d i c t i o n s o f Ref. [ 5 ] . A c u r v e u s i n g t h e c u r r e n t
procedure
is a l s o i n c l u d e d
in F i g . i 0 f o r D = 1 2 8 8 0 0 0 s - 1 f r o m 1-19] w h i c h
A failure criterion for beams under impulsive loading
H : 6 35mm . I L=101.6mm
10
IH=953mm I L :101.6mm
10
I I
I r
u Im
I
117
I 05
05
I ,[ 08
0.'t~
,
T
~
If
00
1'2
I"
0/+
1,08
(b)
(o)
FIG. 14. Variation of the d~mensionless m o m e n t u m loss versus the dimensionless ~mpulse I* (a) H = 6.35 mm, L = 101.6 m m ; (b) H = 9.53 mm, L = 101.6 m m current calculated result; - - - - - theoretical predictions for mode 1 from [5]; • theoretical predictions at the threshold from mode 2 to mode 3 from [5]; . . . . . current calculated threshold lines from mode 1 to mode 2 and from mode 2 to mode 3.
w;,
I
I
H=635mm L =101 6ram
05
wA; 05
III
'
0'4
'
08 '
I'2
i
i
I I
I I
H:953mrn L=lO16mm
~ I
/Li I"
0'2
t
I
I
i
O'L, I,06
(ta)
II
a, 08
I'
(b}
FIG. 15. Variation of the final dimensionless transverse displacement W~r with the dimensionless impulse I*. (a) H = 6.35 mm, L = 101.6 m m ; (b) H = 9.53 mm, L = 101.6 mm. - . . . . current calculated threshold lines from mode 1 to mode 2 and from mode 2 to mode 3.
TABLE 1. COMPARISON OF THE THEORETICAL DIMENSIONLESS THRESHOLD IMPULSES FOR MODE 2 AND MODE 3 RESPONSES FROM R E F
[51 WITH THE CURRENT THEORETICAL PREDICTIONS AND THE CORRESPONDING EXPERIMENTAL RESULTS [ 1 "1 ON ALUMINIUM 6061-T6 IMPULSIVELY LOADED BEAMS
I* Onset of mode 2 3
H (mm)
L (mm)
Experiments [ 1] (current predictions)
6.35 9.53 6.35 9.53
101.6 101.6 101.6 101.6
0.49"1" (0.49) 0.54 (0.54) 0.87 (0.82) 0.79 (0.76)
Simple theoretical predictions [ 5 ] Lower bound Upper bound 0.50 0.48 0.81 0.81
0.63 0.61 0.95 0.95
t l * = 0.58 is reported in Table 1 of Ref. [1], while I* ~ 0.49 in Fig. 3 of Ref. [1].
produces an increase in the yield stresses of about 20% for the associated strain rates of around 2000 s- 1. Figure 1 ! contains the/7 - I* curves, which reveals the importance of transverse shear effects in the evaluation of the mode 2 threshold impulse since a significant proportion of the initial kinetic energy is absorbed in transverse shear deformations. The theoretical procedure in [5] neglects the influence of transverse shear forces in the yield condition for the mode 2 threshold impulse but overestimates the importance of transverse shear effects for the transition to a mode 3 failure, as shown in Fig. 11. It is evident from Fig. 12 for the location of the travelling hinge B at tf that the theoretical analysis in [5]
W . Q . SHEN and N. JONES
118
El
E~.t 6_0 50 H=635mm , L:IOI 6ram
40 Ek,
E" 30
30
E;~t Et
o
20
40
80
Mode I
12o
I;0323
o/;bo
F2
20
, °Oo
o
t(~)
E;~y'
Los6
Node II
(o)
•
[=0638 (b]
-
flgs)
Mode III (c)
E;~b t(~)
1=0.961
FIG. 16. Partition of energy with time for a specimen with H = 6.35 mm and L = 101.6 mm. (a) Mode 1, I* = 0.323; (b) mode 2, 1" = 0.638; (c) mode 3, 1" = 0.961.
agrees with the numerical results for a mode 1 behaviour but it predicts that the position of the hinge is much closer to the supports at the transition between a mode 2 and a mode 3 behaviour. The dimensionless total plastic work expressed as a proportion of the total external work dissipated in a beam and the dimensionless linear momentum of a beam lost during the deformation process are presented in Figs 13 and 14, respectively. It may be shown from these dimensionless results that the maximum plastic work and the maximum loss of momentum occur at the transition between a mode 1 and a mode 2 behaviour with a rapid decrease in the mode 2 region. Although a large impulse is required to induce a mode 3 failure, it is evident that a significant amount of kinetic energy and linear momentum remain in a beam after failure, as found in the simplified theoretical analysis in [5] in which (1 - x/3H/L) times the initial kinetic energy remained in a beam after severance. The relationship between the dimensionless final transverse (shear) displacement at the support A with the dimensionless impulse I* is shown in Fig. 15. It is evident that an approximate mean value of 0.6 is associated with a mode 2 failure and with the onset of a mode 3 failure. However, the shear strain within the hinge region increases as the hinge length decreases from the onset of a mode 2 failure to the onset of a mode 3 failure (see Figs 9 and I1). A value of W~f = 1 was proposed in [ 5 ] for the threshold of a mode 3 failure, while ( H was suggested in [8] with ( ~< 1, as observed experimentally in [9]. A partition of the total energy in a beam is shown in Fig. 16, where
/: /o
E~ t = 2po
Ek =
Evb =
2
rn
w(z, x) dx
~2(t, x) dx
/o(
QA I'VA + (MA + Ma)
[( Wc -- WA)/~ ] + NA
dt
and Eps b =
2
L
QA I~VAdt.
It is observed from these calculations that energy conservation is maintained throughout
A failure criterion for beams under impulsive loading
X (a) 010[_ 0 ~ XO1~
f=3B htS X" 1'0 f= ~.74 I-.tS
0:5 0~5
X"
(b) 01
10
t = 15.5ItS
O0 W" 0"5
Xj I '0 t =It*04 ItS
Ol
0 h OX2~ O~h k !
0:5
X"
1.0 O2
1.0
0 ~ ............. 05
x"
X" ~kc
(STOP)
0!5
X"
I:0 ,
t=116.0ItS
0
10
IO
(RUPTURE)
02
f= 27091J.S "
I()
f= B60 ItS
Or,
02
X"
o'5
U0
kc
OZ~
0
X"
05
t= 53.51.tS
t= 19Bt+iJ.S 015
119
0'5
X"
1.0
(C)
t= 7-77l.J.S 00
0-'5
X"
I.'0
t= 2577itS
ols
000 ~
0.4 0.2 o ~
i7o
t=37771tS
0'5 X!
x"
X"
Ib
Xc ( RUPTURE) t--L? o2~s
0'5
X"
1.5
FIG. 17. Variation of the dimensionless plastic work density 2 with time t for a specimen with H = 6.35 mm and L = 101.6 mm. (a) Mode 1, I* = 0.323; (b) mode 2, I* = 0.638; (c) mode 3, I* = 0.961.
the response duration for the three failure modes, as expected, i.e. Eex t = Ek(t ) "Jr Epb(t )
0 ~< t ~< tf.
It transpires that the ratio Epb/Eex t decreases significantly from m o d e 1 to m o d e 3, while the ratio Epsb/Epb increases with increasing impulse. The simple theoretical procedure in [-5] predicts that Epb/Eex t = 1 and Epsb/Epb = 0 for all impulses producing a m o d e 1
120
W.Q. SHEN and N. JONES
response. However, Epb/Eex t = x / 3 H / L a n d Epsb/Epb = 3 / 4 are a s s o c i a t e d with the t h r e s h o l d impulse for a m o d e 3 response. It is interesting to note that the response d u r a t i o n is 268 Its a c c o r d i n g to the theoretical m e t h o d in I-5] for the m o d e 1 response of the b e a m studied in Fig. 16(a). This c o m p a r e s f a v o u r a b l y with the value of 270.9/is p r e d i c t e d by the present n u m e r i c a l m e t h o d . O n the o t h e r hand, the response d u r a t i o n at the t h r e s h o l d of a m o d e 3 response is 41.7/~s a c c o r d i n g to [ 5 ] which c o m p a r e s f a v o u r a b l y with 43.0 ~s in Fig. 16(c) for a dimensionless impulse slightly larger than the t h r e s h o l d value, as i n d i c a t e d in T a b l e 1. Finally, the v a r i a t i o n of the dimensionless density of the plastic w o r k d i s s i p a t i o n 2 with time t for the three failure m o d e s is s h o w n in Fig. 1 7 ( a - c ) where 2 = ~ b H / 4 M d o . It is interesting to observe the significant differences in the hinge length c, r u p t u r e time t r a n d the final travelling hinge l o c a t i o n ~tf for the three m o d e s of failure.
5. CONCLUSIONS An energy failure criterion is p r e s e n t e d in this p a p e r which p o s t u l a t e s that the d y n a m i c inelastic failure of b e a m s occurs when the plastic w o r k density reaches a critical value. The theoretical analysis considers the s i m u l t a n e o u s influence of the axial m e m b r a n e force, the transverse shear force a n d the b e n d i n g m o m e n t in the yield condition. F a i l u r e m o d e s have been e x a m i n e d from a theoretical v i e w p o i n t in this p a p e r for the a l u m i n i u m alloy 6061T6 b e a m s which were subjected to uniform impulsive velocities by M e n k e s a n d O p a t [ 1 ]. S o m e insight into the characteristics of the different failure m o d e s is o b t a i n e d from the n u m e r i c a l results a n d it a p p e a r s w o r t h w h i l e to d e v e l o p further the energy failure criterion suggested herein to e x a m i n e o t h e r b e a m p r o b l e m s a n d the failure of plates a n d shells which are g o v e r n e d by generalized stresses. Acknowledgements--The authors wish to thank Dr C. E. Nicholson of the Safety Engineering Laboratory, Sheffieldand the Health and Safety Executive for their support of this study through contract number 2516/R31.22. The authors are also indebted to the Impact Research Centre in the Department of Mechanical Engineering at the University of Liverpool and in particular to Mrs H. Jiang for some secretarial assistance and Mr F. J. Cummins for his assistance with the tracings.
REFERENCES 1. S. B. MENKESand H. J. OPAT, Broken beams. Exp. Mechs 13, 480-486 (1973). 2. R. G. TEELING-SMITHand G. N. NURICK,The deformation and tearing of thin circular plates subjected to impulsive loads. Int. J. Impact Engng 11, 77-91 (1991). 3 .T.A. DUFFEY, Dynamic rupture of shells, Structural Failure, T. WIERZBICKIand N. JONES (editors), John Wiley and Sons, New York, pp. 161-192 (1988). 4. N. JONES, Structural Impact. Cambridge University Press, Cambridge (1989). 5. N. JONES, Plastic failure of ductile beams loaded dynamically, Trans. ASME, J. Enyny Industry 98(B1), 131-136 (1976). 6. J. Yu and N. JONES, Numerical simulation of a clamped beam under impact loading, Comput. Struct. 32, 281-293 (1989). 7. J. H. LIu and N. JONES, Plastic failure of a clamped beam struck transversely by a mass, Inelastic Solids and Structures, M. KLEIBERand J. A. KONIG(editors), Pineridge Press, U.K., pp. 361-384 (1990). 8. N. JONES, On the dynamic inelastic failure of beams, Structural Failure, T. WIERZBICKJand N. JONES (editors), John Wiley and Sons, New York, pp. 133-159 (1988). 9. W. S. JOUR!and N. JONES,The impact behaviour of aluminium alloy and mild steel double-shear specimens. Int. J. Mech. Sci. 30, 153-172 (1988). 10. E. JENNINGS,K. GRUBBS,C. ZANISand L. RAYMOND,Inelastic deformation of plate panels. Ship Structure Committee Report, SSC-364, pp. B1-B7 (1991). l I. J. Yu and N. JONES,Further experimental investigations on the failure of clamped beams under impact loads. Int. J. Solids Struct. 27, 1113-1137 (1991). 12. K. KAWATAet al., High Velocity Deformation of Solids, K. KAWATAand J. SHIORI(editors), Springer Verlag, New York, pp. 1-15 (1977). 13. N. JONES, Some comments on modelling of material properties for dynamic structural plasticity. Mechanical Properties of Materials at High Rates of Strain, J. HARDING(editor), Institute of Physics Conference Series, No. 102, pp. 435-445 (1989).
A failure criterion for beams under impulsive loading
121
14. G. R. COWPER and P. S. SYMONDS,Strain hardening and strata rate effects In the Impact loading of cantilever beams. Techmcal Report No. 28 from Brown University to the Office of Naval Research under Contract No. 562(10), September 1957. 15. T. NONAKA, Some interaction effects in a problem of plastic beam dynamics, Part 2: Analysis of a structure as a system of one degree of freedom. J. Appl. Mech, 34, 631-637 (1967). 16. J. G. DE OLXVEIRAand N. JONES, Some remarks on the influence of transverse shear on the plastic yielding of structures. Int. J. Mech. Sci. 20, 759-765 (1978). 17. Z. SOBOTKA, Theorie PlasticitY, t.I-II, CSAV, Praha 1954 and 1955. 18. W. Q. SHEN and N. JONES, Interaction yield surfaces for the plastic behaviour of beams due to combined bending, tension and shear. Impact Research Centre Report, No. ES/55/90, Department of Mechanical Engineering, The University of Liverpool (1990). 19. N JONES, Some remarks on the strain-rate sensltwe behaviour of shells, Problems of Plasticity, A. SAWCZUK (editor), Vol. 2, Noordhoff, pp. 403-407 (1974).
APPENDIX A Examination of static admissibility
The statically admissible reqmrement for the theoretical solution in Section 3 is always satisfied for x >/¢, so that the region 0 ~< x ~< ~ is examined in this Appendix. 1. Phase 1.
Substituting Eqn (13b) into Eqn (14), integrating twice and taking mto account that l,Pc~ = po/m,
yields Q(x) = ( - p o + mWA1)(X-- x2/2¢1) + QAI and M(x) = ( - P o
+
ml~Ai )(x2/2 - x3/6¢1) +
QAI x --
MAI
or, when using Eqns (16) and (17) Q*(x) = Q*I(I - x*) 2
(A.I)
M*(x) = - - M * 1 + 3(1 + M I l l ( x * -- x .2 + x*S/3),
(A.2)
and
where x* = x/¢1. It transpires that Q*2(x) + M*2(x) ~< 1 is satisfied for 0 ~< x ~< ~I. Hence, the generalized stress field associated with the velocity field in Fig. 6 is staucally admissible. 2. Phases 2 and 3.
Integrating Eqns (14) and (27) twice yields Q(x) = [ X i - p o + m ~ ) d x + QA do
(A.3)
and M(x)=
(-po+mff)dxdx+Qax-MA
(A.4)
for phase 1, while Q(x) = f o~ mff dx + QA -- N aw tg---x
(A.5)
and M(x) =
m~' dx + QAx -- M h -- N ( w -- WA)
(A.6)
for phases 2 and 3, when satisfying the boundary conditions. The integrals in equations (A.3)-(A.6) are linear for a given value of ~ between 0 ~< x ~< ¢ and equal to zero at x = ¢ according to Eqn (15). It is shown in part 1 of this Appendix that static admissibility is satisfied for infinitesimal displacements, i.e. N = 0. It is evident that the velocity profiles during phases 2 and 3 are also statically admissible, because the only differences between equations (A.3) and (A.5), as well as (A.4) and (A.6) are the two terms ( - N aw/ax) and [ - N ( w - W^)] which are negative since N > 0, 3w/tgx > 0 and w > W^.
