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A further study of plastic shear failure of impulsively loaded clamped beams
International Journal of Impact Engineering 24 (2000) 613}629
A further study of plastic shear failure of impulsively loaded clamped beams T.X. Yu*, F.L. Chen Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Received 20 March 1999; received in revised form 29 June 1999
Abstract As early as in 1973, Menkes and Opat (Exp Mech 1973; 13:480}6) conducted an experimental investigation on the dynamic plastic response and failure of fully clamped metal beams subjected to a uniformly distributed impulsive loading and identi"ed three basic failure modes: large inelastic deformation (Mode I), tensile tearing (Mode II) and transverse shear failure at the supports (Mode III). A rigid-plastic analysis was later carried out by Jones (Trans ASME J Eng Ind 1976; 98 (B1): 131}6), in which an elementary failure criterion was adopted to estimate the threshold impulsive velocities at the onset of Mode II or Mode III failure. A deep understanding of these three basic failure modes is of fundamental importance to failure analyses of various structures under intense dynamic loading. The present paper re-examines the plastic shear failure (Mode III) of impulsively loaded clamped beams, with focus on two e!ects: (i) the interaction between the shear force and bending moment; and (ii) the weakening of the sliding sections during the failing process. A dimensional analysis is "rst performed to obtain a general form of the threshold impulsive velocity, which overlooks succeeding concrete analyses. The elementary failure criterion is then modi"ed to incorporate the sliding sections' weakening e!ect. Interaction between the shear force and the bending moment at the supporting ends is considered by using circular yield curve (Robinson, Int J Solids Struct 1973; 9:819), Hodge's curve (J Appl Mech 1957; 24:453}6), or a yield condition based on slip-line solutions. By taking into account the variation with time of the shear force and the bending moment over the failing cross-sections, the plastic deformation and failure process of the beams are traced and the ratio of plastic shear dissipation to the total plastic dissipation is thus calculated. This is followed by a discussion on a shear strain failure criterion. Finally, the predictions from various approaches are compared with each other as well as with relevant experimental results. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Shear failure; Rigid-plastic beam; Dynamic failure criterion; Interactive yield condition; Weakening e!ect
* Corresponding author. Tel.: #852-2358-8652; fax: #852-2358-1543. E-mail address: [email protected] (T.X. Yu). 0734-743X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 4 - 7 4 3 X ( 9 9 ) 0 0 0 3 8 - X
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1. Introduction Study of structural failure under intense dynamic loading is of importance for the safety and hazard assessments in many engineering "elds. Simple structural members, such as beams, plates and thin shells, may fail due to di!erent modes under su$ciently intense dynamic loads [1]. Several experimental and theoretical studies [2}8] have examined the onset of these failure modes in order to estimate the structural capacities of sustaining impact loading. As early as 1973, Menkes and Opat [2] conducted an experimental investigation on the dynamic plastic response and failure of fully clamped metal beams subjected to a uniformly distributed impulsive loading and identi"ed three basic failure modes, i.e. large inelastic deformation (Mode I), tensile tearing (Mode II) and transverse shear failure at the supports (Mode III). Later, a theoretical rigid-plastic analysis provided by Jones [3}5] showed reasonable agreement with the experimental results [2] on the threshold impulsive velocities at the onset of Mode II or Mode III failure. In this analysis, square yield criteria relating either the axial force to the bending moment or the transverse shear force to the bending moment, and elementary failure criteria were adopted. It is evident that the foundation of the simple formulations employed, especially those for Mode III failure, needs to be further validated. In the 1990s, an energy density criterion was proposed by Shen and Jones [8] which caters to the simultaneous in#uence of the bending moment, the axial force as well as the transverse shear in the structural response. According to this criterion rupture occurs in a rigid-plastic structure when the absorption of plastic work per unit volume, h, reaches a critical value; and further if the ratio of the plastic work absorbed through shearing deformation to the total plastic work done by all the stress components, b, reaches a critical value, b "0.45, a structure su!ers a Mode III failure. The merit of the energy density criterion is that it may be universally applicable for a large class of dynamic structural problems. Nevertheless, it has been noted that some problems associated with this criterion remain unanswered, e.g. (i) in a rigid-plastic analysis, the calculation of the energy density depends upon the length of a plastic hinge; (ii) in FE calculations, the energy density is mesh-dependent; and (iii) it is unclear how to determine the critical ratio b . It was suggested that b "0.45, but this value disagrees with 0.857 from the theoretical prediction in [3], refer to Fig. 11 in [8]. A clearance is needed. Transverse shear failure becomes increasingly important and dominant when intensity of the dynamic loading increases. Until now, several analyses [3,8}12] have been concerned with this failure mode by employing the concept of `plastic shear slidinga or `plastic shear hingea. In a rigid, perfectly plastic beam, it is generally assumed that a plastic shear hinge has an in"nitesimal length, that is, transverse shear sliding is treated as a transverse-displacement discontinuity which is the idealization of abrupt change in transverse displacement across a short segment of a beam [5]. On the other hand, Wang and Jones [13] found that in a rigid-linear hardening beam subjected to impulsive loading, the location of the shear hinge travels a "nite distance ¸ during the failure process. A penetrating understanding of these basic failure modes is of fundamental importance to the failure analyses of various structures under intense dynamic loading. Two important e!ects, however, have been ignored in all previous investigations: (i) the interaction between the shear force and bending moment; and (ii) the weakening of the sliding sections during the failure process. Thus, the present paper is aimed at re-examining the shear failure problem of impulsively loaded beams
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by various approaches, but focusing on these two factors. In addition to further comments and modi"cations on the elementary failure criterion, the interaction between the shear force and the bending moment at the sliding sections will be considered by adopting circular yield curve [14], Hodge's curve [15], or a yield condition based on slip-line "eld solutions [16]. The variation of the shear force and the bending moment over the sliding cross-sections during the beam's deformation process will also be taken into account.
2. Mode III failure of impulsively loaded clamped beams: general consideration Consider a clamped beam of length 2¸ and of rectangular cross-section with thickness H and width B, subjected to uniformly distributed initial velocity over the entire span, as shown in Fig. 1. For a simply supported beam subjected to an impulsive velocity, Jones et al. [11,4] found that dimensionless ratio l"Q ¸/2M with M and Q denoting the fully plastic bending moment and the fully plastic shear force of the cross-section, respectively, is an important parameter; di!erent transverse shear response modes occur when l41 or l'1. Obviously, this parameter mainly depends on the geometry of the beam. Nevertheless, it is of importance merely for some special beam cross-sectional shapes, such as wide-#anged I-beam or sandwich beam, where l&1. As for the rectangular cross-sectional beam considered here, l"¸/H 1. For instance, in the experiments conducted by Menkes and Opat [2], l ranged from 5.33 to 21.4. It was observed that, unlike tearing failure (Mode II) which followed the large de#ection of the beam (Mode I), transverse shear failure (Mode III) was characterized by insigni"cant #exural deformation at most cross-sections. Therefore, shear failure occurs at the supports in the early stage of the beam's response and generally exhibits a type of localized behaviour, while the Mode III thresholds does not depend on the length of the beam. Thus, with the length of the beam being excluded from the theoretical description, if a rigidplastic model is adopted, the maximum shear sliding at the supports, * , can be expressed as
a function of impulsive velocity < , yield stress >, density o and thickness H of the beam: * "* (< , >, o, H). (1)
From the n-theorem of dimensional analysis, we have
< * "f ((j).
"f H (>/o
(2)
Here non-dimensional parameter j"o< /> is termed the damage number which is commonly used to specify the severity of impact, according to Johnson [17]. Hence, if we adopt a criterion for
Fig. 1. An impulsively loaded clamped beam.
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shear failure in form * "kH (3)
with k being a constant determined by experimental measurement or theoretical consideration, the threshold impulsive velocity for transverse shear failure (i.e. Mode III failure) can be expressed as
> . o
< "f\(k)
(4)
It transpires that for rectangular cross-sectional beam with ¸/H 1, the threshold impulsive velocity for a Mode III failure does not depend on geometry of the beam but only on material properties.
3. Elementary failure criterion (EFC) With the aid of the velocity "eld shown in Fig. 2 where sliding occurs at the supports of the beam having a "nite shear strength Q , Jones [3] employed a simple rigid-plastic procedure to estimate the threshold velocity for the onset of a mode III failure as
2Q . (5) < " 3oBM In this simple theoretical analysis, an elementary failure criterion for complete severance at the supports of a beam
* "
R
< dt"H Q
(6)
was adopted. Condition (6) can be regarded as an extreme case of Eq. (3) with k"1. If von Mises yield criterion is adopted, then for a beam of rectangular cross-section, M ">BH/4 and Q ">BH/(3, so that 2(2 < " 3
> > "0.943 . o o
Fig. 2. Velocity "eld for the impulsively loaded clamped beam with shear sliding at supports.
(7)
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4. Modi5ed elementary failure criterion It is observed in a few recent experiments [18,19] that in most cases the realistic shear failure is caused by a combination of shear sliding and crack propagation. In these cases, the elementary failure criterion (6) can be straightforwardly modi"ed to the form (3). The factor k(0(k41) slightly depends on the material properties, geometric constraint of the structure as well as loading conditions. One may easily conjecture that the value of k for a tough material is larger than that for a brittle material [20]. However, for a wide range of situations k can be approximately taken as a constant, say k"0.3. Using modi"ed elementary failure criterion (3) and following the same derivation given by Jones [3], the modi"ed threshold velocity for the onset of a Mode III failure is obtained as
2kQ < " 3oBM which can be rewritten as 2(2 < " 3
(8)
k> k> "0.943 o o
(9)
for beams of rectangular cross-section. These two expressions predict a lower value of the threshold velocity < by a factor of (k. If k takes 0.3, (k"0.548, and consequently < "0.517(>/o. Both Eqs. (5) and (8) are based on the assumption that during the beam's deformation process, the bending strength M and shear strength Q of the sliding section A (see Fig. 2) remain constants M and Q and the same situation occurs at D. In other words, the internal force state of sections A and D is always located at the corner G (M , Q ) on the square yield locus (Fig. 3(a)). This is, undoubtedly, a quite rough approximation, refer to the con"gurations sketched in Fig. 4. The sketch suggests a weakening e!ect of shear sliding * on the bending and shear strengths of
Fig. 3. Yield loci on the m}q plane: (a) a square yield locus for intact sections and a rectangular yield locus for the sliding sections; (b) a yield circle for intact sections and yield ellipses for the sliding sections; (c) interactive yield curves based on slip-line "eld solutions for the sliding sections.
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Fig. 4. Shear sliding takes place (a) over a cross-section; (b) over a "nite width.
sections A and D, that is, M "fM and Q "fQ , where the weakening factor f depends on the shear sliding: f"f(*). If the weakened bending moment M and the weakened shear force Q are incorporated into the deformation mechanism shown in Fig. 2, then the conservation of linear momentum of segment AB demands 2R Q dt < !< " , oBHz
(10)
while the conservation of angular momentum requires 6R (M #M ) dt < !< " . oBHz
(11)
It is noted that at interior hinges B and C, M"M and Q"0, so that the mid-portion of the beam, BC, moves at a constant velocity < . Eqs. (10) and (11) therefore result in the length of segment AB, 3R (M #M ) dt 3M R (f#1) dt 6M z" " 5 . (12) Q Q R Q dt R f dt Eq. (12) indicates that two interior hinges B and C travel towards the mid-point as f decreases from 1 during the beam's plastic deformation process. This is di!erent from the prediction that hinges B and C remain stationary until < "0 based on the assumption of constant bending and shear strengths at the sliding sections A and D. Substituting Eq. (12) into Eq. (10) yields
R f dt 2Q d* "< "< ! . 3oBHM R (f#1) dt dt
(13)
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For rectangular cross-sectional beam, by introducing non-dimensional variables, * d, , H
t q, H/<
(14)
Eq. (13) is recast into non-dimensional form dd 8 (O f dq) "1! . (15) dq 9j O (f#1) dq where j is Johnson's damage number mentioned before. Fig. 4 shows two con"gurations of the plastic shear deformation region. (a) The shear sliding is treated as a stationary, strong discontinuity, as introduced by Nonaka [9], Symonds [10] and Jones [3]; then by assuming a statically admissible stress "eld at sliding sections A and D, we "nd f"1!*/H"1!d. (b) A "nite width of shear sliding is considered. Experiments [2,18,19] and some theoretical models indicate that this width w is usually very small. For example, in the plastic shear band model shown in Section 7, only half of its total width, i.e. e, lies outside the clamp "xture, so that the e!ective width of plastic shear deformation region of the beam, w, could be taken as e"((3/8)H"0.216H. Referring to Fig. 4(b), the maximum shear force then occurs over straight line E F provided *50.049H. Since "E F"+H!*, it is seen that f+1!*/H"1!d. Consequently, for both con"gurations (a) and (b), Eq. (15) can be rewritten as
O (1!d) dq 8 dd "1! . (16) O 9j [(1!d)#1] dq dq Eq. (16) is an integral-di!erential equation, which can be transformed into a standard set of "rst-order di!erential equations ready to be integrated numerically by means of a second-order Runge}Kutta procedure to obtain the maximum sliding as a function of j, d (j). The calculated
results are depicted in Fig. 5, where the variation of the "nal value of the weakening factor, f , with j is also exhibited. Recalling failure criterion (3), that is d "k, the threshold impulsive velocity
for shear failure of the beam is thus determined as a function of parameter k, j (k). If the value of
Fig. 5. The variation of the "nal value of the weakening factor and shear sliding with impulsive loading parameter j.
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k in criterion (3) is speci"ed, a threshold value of j can be determined accordingly. For example, if k"0.3, we "nd j "0.221, and accordingly < "0.447(>/o. In addition, it can be seen from Fig. 5 that when j approaches a critical value j "0.369, the maximum shear sliding increases from 0.7H to 1.0H rapidly. Thus, a critical impulsive velocity is identi"ed as
< "
j > > "0.607 . o o
(17)
It is 0.644 times the value given by Eq. (7) but is 1.17 times that by Eq. (9) for k"0.3. The di!erences re#ect the in#uence of the reductions of M and Q during the beam's deformation process.
5. Interaction between transverse shear and bending moment at sliding sections On bending moment-shear force plane, interactive yield curves, rather than the square ones, provide better representations of the yield condition. Although interaction between shear force and bending moment is intricate, several interactive yield curves for beams of rectangular cross-section have been proposed from various postulations. 5.1. Circular yield curve (CYC) A typical one of these interactive yield curves is a circle m#q"1
(18)
with m"M/M and q"Q/Q , as shown in Fig. 3(b). This was found by Robinson [14] from an approximation to Ilyushin}Shapiro's yield surface [21,22] and is the special case of m(1!q) #n#q"1 with n"0, which was derived from a uniform stress distribution in a rectangular cross-section [23]. According to the associated #ow rule, the generalized velocity vector (< Q , uM ) must be normal to the yield curve, that is, M u (3 Hu dq " "! (19) Q < 4 < dm where u denotes the angular velocity about the generalized plastic hinge A. Di!erentiating Eq. (18) gives dq (1!q ) "! . (20) dm q For any given value of dq /dm , q can be determined from this equation, and then m can be calculated from yield condition (18) as follows: m "(1!q ) so that the location of the internal force state on the yield circle can be determined.
(21)
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Applying the velocity "eld shown in Fig. 2, Eqs. (10) and (11) provide
8k < "< 1! 9jl
(22)
and < !< 32k < " u" , z H 27(3jl
(23)
where
k,
O
q dq, v,
O
(m #1) dq. Substitution of Eqs. (22) and (23) into Eq. (19) results in dq 8k "! . dm 27jl!24kl Eq. (22) is then recast into dd 8k "1! . dq 9jl
(24)
(25)
(26)
With the help of (24), (25), (20) and (21), Eq. (26) is numerically integrated by means of a second-order Runge}Kutta procedure starting from initial conditions d"0, q "1, m "0 at q"0. The results reveal that on the yield circle, the internal force state of sections A and D moves from point R counterclockwise in the early stage and approaches point S in the later stage. The prediction on the "nal shear sliding obtained from Eq. (26), as marked by `CYCa in Fig. 6(a), is larger by around 25% than that given by Jones [3] based on the assumption of constant shear force and bending moment at sections A and D. To reach d "k"1, j"j "0.706 and correspond ingly < "0.840(>/o; or to reach d "k"0.3, j"j "0.212 and correspondingly
< "0.460(>/o. In the foregoing analysis, the weakening e!ect of the shear sliding on the interactive yield condition is neglected. If this weakening e!ect is incorporated, as discussed in the later part of Section 4, the bending and shear strengths of the sliding sections may be expressed as M "fM and Q "fQ , with weakening factor f"1!d. Then, the yield condition (18) is modi"ed as m q # "1 (27) f f
and accordingly Eqs. (20) and (21) are replaced by
dq "! dm
1!
q f q
(28)
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Fig. 6. A comparison with the result given in Ref. [3] on the predictions of the "nal shear sliding from various interactive yield curves with the weakening e!ect of the shear sliding incorporated: (a) circular yield curve (CYC) and Hodge's yield curve (HYC); (b) yield curve based upon slip-line "eld solutions (SLF).
and
q m "f 1! f
(29)
whilst Eqs. (24)}(26) remain valid. As a result, the internal force state moves along the yield curve while the yield condition of sliding sections A and D progressively shrinks on the q}m plane: RPPPP P2, as sketched in Fig. 3(b). Predictions on the "nal shear sliding are marked by `CYC-Wa in Fig. 6(a). Yet when j approaches a critical value j "0.396, the "nal shear sliding increases from 0.65H to 1.0H rapidly, so that a critical impulsive velocity is identi"ed as < "(j >/o"0.629(>/o. 5.2. Hodge's yield curve (HYC) With the help of a variational method, Hodge [15] obtained another interactive yield curve between shear force and bending moment for beams of rectangular cross-section as q"ln ((1# # )/
(30a)
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and m"((1# !q)/
(30b)
with serving as a parameter. It can be well approximated by [24] m# q# q"1. (31) This yield curve closely circumscribes the yield circle (18), but inscribes the square yield condition. Similar to the analysis in Section 5.1, by taking account of weakening e!ect of the sliding sections, another critical impulsive velocity is obtained as < "(j >/o"(0.415>/o" 0.644(>/o. 5.3. Slip-line xeld solutions (SLF) Based on slip-line "elds constructed in the root region of cantilever beams under various loading conditions, Yu and Hua [16] obtained an interactive yield curve between m and q at the beam's root, which can be "tted by a single approximation m!2.106q#4.632q!1.526q!1"0
(32)
with error less than 1%, as depicted in Fig. 3(c). Adopting this interactive yield, we can get clear deformation patterns for the plastic shear failure of the dynamically loaded beams. Eqs. (20) and (21) and (27)}(29) are accordingly replaced by (1#2.106q !4.632q #1.526q ) dq " , 1.053!4.632q #2.289q dm m "(1#2.106q !4.632q #1.526q )
(33) (34)
and
m q q q !2.106 #4.632 !1.526 !1"0, f f f f
dq [1#2.106(q /f)!4.632(q /f)#1.526(q /f)] " , dm f[1.053!4.632(q /f)#2.289(q /f)] q q q . m "f 1#2.106 !4.632 #1.526 f f f
(35) (36) (37)
Hence, the beam's shear failure progress can also be traced as in Section 5.1. Fig. 7 exhibits the history of the shear force and the bending moment at sliding section A. Similar variation trends are obtained as adopting circular or Hodge's yield curves in these sections. From the prediction on the "nal shear sliding shown in Fig. 6(b), it is found that when j approaches a critical value j "0.410, the "nal shear sliding increases from 0.65H to 1.0H rapidly, so that a critical impulsive velocity is also identi"ed as < "(j >/o"0.641(>/o. For a speci"ed j,
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Fig. 7. The history of shear force and bending moment at end A (sliding section) based on SLF.
this interactive yield condition results in a slightly smaller prediction on the "nal shear sliding compared with that from circular or Hodge's yield curves. The reason is that the slip-line solution is an upper bound while the latter two are lower bounds of the true yield condition.
6. Plastic dissipation at sliding sections With regard to the deformation mechanism shown in Fig. 2, the increment of plastic dissipation merely due to shear deformation at sliding sections A and D is dE "Q d*"q Q H dd, while the increment of plastic dissipation due to bending is dE "M
m kQ H(dq!dd) < !< dt" . 3l z
(38)
(39)
Hence, the history of plastic dissipation can be traced by using each of the foregoing approaches. The ratio of plastic dissipation merely due to shear deformation to the total dissipation due to both bending and shear, de"ned by E E b, " (40) E E #E can then be calculated. Fig. 8(a) shows the variation of the ratio b with time for three representative values of j. For any speci"ed values of j(4j ), b decreases quickly from 1.0 in the very early stage after impact, then it approaches gradually to its "nal value b at < "0, when the shear sliding ceases. In other words, shear sliding dominates the plastic dissipation in the very early stage, but this stage swiftly transits to the stage of shear-bending deformation. In correspondence to each of the critical values j obtained from various approaches, the critical values of b , denoted by b , based on these approaches can be obtained and summarized in Table 1. A b &j curve calculated from CYC with
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Fig. 8. (a) The variation history of ratio b with time; (b) the variation of "nal value of b with impulsive loading parameter j (CYC-W).
the weakening e!ect of shear sliding being considered is shown in Fig. 8(b). It can be seen that when j(j , the calculated values of b are notably less than b , that is, at the end of dynamic response of the beam, shear sliding only dissipates a small portion of the total energy, so at the supports shear failure will not take place. Based on a comparison with the experimental results conducted by Menkes and Opat [2], Shen and Jones [8] suggested that the critical value of b at the onset of Mode III failure, b , could be estimated as 0.45. On the other hand, as shown in Table 1, our study indicates that the values of b calculated from the square yield condition range from 0.85 to 0.94, around double of 0.45, but those calculated from the interactive yield curves range from 0.44 to 0.66, fairly close to b "0.45 suggested in Ref. [8].
7. Shear strain failure criterion The shear sliding * at the supports of a beam subjected to impulsive load may be expressed as [25] *"ce
(41)
where shear sliding * is an accumulative quantity over the shear band, c is the average shear strain and e is the half-width of the shear band as de"ned in Fig. 9. As mentioned in Section 4, although the shear band has a total width of 2e, only half of its width lies outside the clamp "xture; so the e!ective width of plastic shear deformation region of the beam should be e, see Ref. [26] for details. From the elementary beam theory, it can be deduced [26] that (3 M H. e" " 8 2Q Substituting Eq. (42) into Eq. (41) yields (3 cH. *" 8
(42)
(43)
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Fig. 9. Shear bands located at the supporting ends.
Thus, when transverse shear failure occurs at the supports, a critical shear sliding * is reached, i.e. (3 * " c H, 8
(44)
where c is the rupture shear strain of the beam material, which may be taken a typical value of 0.5 for aluminum alloy and 0.8 for mild steel [25]. Hence, if expressed in uniform form (3), it is found that k"((3/8)c "0.108 and 0.173, respectively. As in Ref. [26], it is assumed that the displacement pro"le of a clamped beam after impulsively loading as
= [1!(x/¸)] for a41, (45) = [1!(x/¸)? ] for a51, where a"I/I with I being the critical input impulse at which the beam fails by tensile tearing. By exploiting the principle of equivalent work done, the critical input impulse I required to cause transverse shear failure of the beam can be deduced as (refer to Eq. (35b) in Ref. [26]) ="
I 4!3(H/¸) c " # o>BH 4(H/¸) (1#n) (H/¸)
(46)
so that the threshold impulsive velocity is found to be
< "
4!3(H/¸) c (H/¸) > # 16 o 4(1#n)
(47)
in which n is the work-hardening index of the material. It slightly depends on the rupture shear strain c and the thickness-to-length ratio H/¸. As H/¸P0, it has a limit value < "0.5
> . o
(48)
It is interesting to note that this limit threshold velocity is very close to that given by Eq. (9) for k"0.3.
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Table 1 A summary of the results Critical energy ratio b
Criterion/approach
Critical impulsive parameter j
Threshold impulsive velocity/(>/o
Dimensional analysis Elementary failure criterion (EFC) Modi"ed elementary failure criterion
[ f\(k)] 8/9"0.889 8k/9"0.889k (0.267 for k"0.3) 0.369
f \(k) 2 (2/3"0.943 2 (2k/3 (0.517 for k"0.3) 0.607
0.706 (for k"1.0) 0.212 (for k"0.3) 0.396
0.840 (for k"1.0) 0.460 (for k"0.3) 0.629
0.444 (k"1.0)
0.762 (for k"1.0) 0.230 (for k"0.3) 0.415
0.873 (for k"1.0) 0.480 (for k"0.3) 0.644
0.438 (k"1.0)
0.843 (for k"1.0) 0.252 (for k"0.3) 0.410
0.918 (for k"1.0) 0.502 (for k"0.3) 0.641
0.477 (k"1.0)
0.25 0.659 0.923 0.743 0.605
0.5 0.812 0.961 0.862 0.778
* 0.45
EFC incorporating the weakening e!ect of the shear sliding Circular yield curve (CYC) CYC incorporating the weakening e!ect of the shear sliding Hodge's yield curve (HYC) HYC incorporating the weakening e!ect of the shear sliding Slip-line "eld solutions (SLF) SLF incorporating the weakening e!ect of the shear sliding Shear strain failure criterion Energy density criterion [8] Experimental results [2]
(H"0.250 in) (H"0.187 in) (H"0.250 in) (H"0.375 in)
(H"0.250 in) (H"0.187 in) (H"0.250 in) (H"0.375 in)
* 0.857 0.857 0.938
0.470
0.519
0.658
The material parameters for aluminium 6061 T6 used in the experiments were o"2.734;10 kg/m and >"286.8 MPa.
8. Concluding remarks Plastic shear failure at the supports (Mode III) of clamped beams loaded impulsively is re-examined by several approaches; two e!ects, i.e. (i) the interaction between the shear force and bending moment, and (ii) the weakening of the sliding sections during the failing process, are emphatically considered. Table 1 summarizes all the predictions on the threshold impulsive velocity from these various approaches and compares them with the experimental results reported in Ref. [2]. It is concluded that although the estimate on the threshold velocity obtained by an elementary failure criterion [3] agrees well with the relevant experimental results, by adopting interactive yield conditions, such as circular yield curve, Hodge's yield curve or a yield curve based on slip-line "eld
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solutions, the present studies provide more reasonable and better predictions. It is also noted that most approaches result in lower values of threshold velocity compared with the experimental results. This is partly attributed to material's hardening behaviour of aluminium used in the experimental tests. The critical ratios of the plastic shear dissipation to the total plastic dissipation at the sliding sections from these various approaches are also summarized in Table 1. It is found that those from the square yield condition are much larger than 0.45, which was suggested in [8], but those from the interactive yield curves are fairly close to this value. Whilst the critical ratio b "0.45, as suggested by Shen and Jones [8] based on a comparison to experimental measurements [2], can be accepted as an empirical criterion for the onset of shear failure at the supports, the present analysis has laid a rational theoretical foundation to understand the complex variation of the ratio b in the shear deformation/failure process of impulsively loaded beams.
Acknowledgements The work described in this paper was conducted as a part of the research projects funded by Hong Kong Research Grant Council (RGC) under CERG project HKUST 811/96E. This support is gratefully acknowledged.
References [1] Yu TX, Chen FL. Failure modes and criteria of plastic structures under intense dynamic loading: a review. AEPA'1998, Metals and Materials, Vol. 4, 1998. p. 219}26. [2] Menkes SB, Opat HJ. Broken beams. Exp Mech 1973;13:480}6. [3] Jones N. Plastic failure of ductile beams loaded dynamically. Trans ASME J Eng Ind 1976;98(B1):131}6. [4] Jones N. Structural impact. Cambridge: Cambridge University Press, 1989. paper back edition 1997. [5] Jones N. On the dynamic inelastic failure of beams. In: Wierzbicki T, Jones N, editors. Structural failure. New York: Wiley, 1989. p. 133}59. [6] Du!y TA. Dynamic rupture of shells. In: Wierzbicki T, Jones N, editors. Structural failure. New York: Wiley, 1989. p. 161}92. [7] Teeling-Smith RG, Nurick GN. The deformation and tearing of thin circular plates subjected to impulsive loads. Int J Impact Eng 1991;11:77}91. [8] Shen WQ, Jones N. A failure criterion for beams under impulsive loading. Int J Impact Eng 1992;12: 101}21. [9] Nonaka T. Some interaction e!ects in a problem of plastic beam dynamics, Part 1: interaction analysis of a rigid-perfectly plastic beam. ASME J Appl Mech 1967;34:623}30. [10] Symonds PS. Plastic shear deformation in dynamic load problems. In: Heyman J, Leckie FA, editors. Engineering Plasticity. Cambridge: Cambridge University Press, 1968. p. 647}64. [11] Jones N, Oliveira JG. The in#uence of rotatory inertia and transverse shear on the dynamic plastic behaviour of beams. ASME J Appl Mech 1979;46:303}10. [12] Li QM, Jones N. Blast loading of fully clamped beams with transverse shear e!ects. Mech Struct Mach 1995;23:59}86. [13] Wang LL, Jones N. An analysis of the shear failure of rigid-linear hardening beams under impulsive loading. Acta Mech Sini (English Ser) 1996;12:336}48.
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[14] Robinson M. The e!ect of transverse shear stresses on the yield surface for thin shells. Int J Solids Struct 1973;9:819. [15] Hodge PG. Interaction curves for shear and bending of plastic beams. J Appl Mech 1957;24:453}6. [16] Yu TX, Hua YL. Plastic collapse of cantilever beams under bending, shearing and tension/compression. In: Abe T, Tsuta T, editors. Proceedings of AEPA'96. Oxford: Pergamon, 1996. p. 845}50. [17] Johnson W. Impact strength of materials. London: Edward Arnold, 1972. [18] Jouri WS, Jones N. The impact behaviour of aluminium alloy and mild steel double-shear specimens. Int J Mech Sci 1988;30:153}72. [19] Chan LC. Process modelling of "ne-blanking. PhD Thesis, Dept of Manufacturing Eng, Hong Kong Polytechnic University, 1997. [20] Zhao YP, Fang J, Yu TX. Dynamic plastic shear failure analysis for an in"nitely large plate with centred cylinder under impulsive loading. Int J Solids Struct 1994;31:1585}95. [21] Ilyushin AA. Plasticite. Paris: Eyrolles, 1956 [in French]. [22] Shapiro GS. On yield surfaces for ideally plastic shells. In: Problems of continuum mechanics. Philadelphia: SIAM, 1961. p. 414}8. [23] Sobotka Z. Theorie plasticity. I}II. Praha: CSAV, 1954}1955. [24] Landgraf G. Aufstellung von Fliessbedingungen fur Schnittgrossen von Flachentragwerken. ZAAM 1968;48:317. [25] Bai YL, Johnson W. Plugging: physical understanding and energy absorption. Met Technol 1992;9:182}90. [26] Wen HM, Yu TX, Reddy TY. Failure maps of clamped beams under impulsive loading. Mech Struct Mach 1995;23:453}72.
A further study of plastic shear failure of impulsively loaded clamped beams T.X. Yu*, F.L. Chen Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Received 20 March 1999; received in revised form 29 June 1999
Abstract As early as in 1973, Menkes and Opat (Exp Mech 1973; 13:480}6) conducted an experimental investigation on the dynamic plastic response and failure of fully clamped metal beams subjected to a uniformly distributed impulsive loading and identi"ed three basic failure modes: large inelastic deformation (Mode I), tensile tearing (Mode II) and transverse shear failure at the supports (Mode III). A rigid-plastic analysis was later carried out by Jones (Trans ASME J Eng Ind 1976; 98 (B1): 131}6), in which an elementary failure criterion was adopted to estimate the threshold impulsive velocities at the onset of Mode II or Mode III failure. A deep understanding of these three basic failure modes is of fundamental importance to failure analyses of various structures under intense dynamic loading. The present paper re-examines the plastic shear failure (Mode III) of impulsively loaded clamped beams, with focus on two e!ects: (i) the interaction between the shear force and bending moment; and (ii) the weakening of the sliding sections during the failing process. A dimensional analysis is "rst performed to obtain a general form of the threshold impulsive velocity, which overlooks succeeding concrete analyses. The elementary failure criterion is then modi"ed to incorporate the sliding sections' weakening e!ect. Interaction between the shear force and the bending moment at the supporting ends is considered by using circular yield curve (Robinson, Int J Solids Struct 1973; 9:819), Hodge's curve (J Appl Mech 1957; 24:453}6), or a yield condition based on slip-line solutions. By taking into account the variation with time of the shear force and the bending moment over the failing cross-sections, the plastic deformation and failure process of the beams are traced and the ratio of plastic shear dissipation to the total plastic dissipation is thus calculated. This is followed by a discussion on a shear strain failure criterion. Finally, the predictions from various approaches are compared with each other as well as with relevant experimental results. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Shear failure; Rigid-plastic beam; Dynamic failure criterion; Interactive yield condition; Weakening e!ect
* Corresponding author. Tel.: #852-2358-8652; fax: #852-2358-1543. E-mail address: [email protected] (T.X. Yu). 0734-743X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 4 - 7 4 3 X ( 9 9 ) 0 0 0 3 8 - X
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1. Introduction Study of structural failure under intense dynamic loading is of importance for the safety and hazard assessments in many engineering "elds. Simple structural members, such as beams, plates and thin shells, may fail due to di!erent modes under su$ciently intense dynamic loads [1]. Several experimental and theoretical studies [2}8] have examined the onset of these failure modes in order to estimate the structural capacities of sustaining impact loading. As early as 1973, Menkes and Opat [2] conducted an experimental investigation on the dynamic plastic response and failure of fully clamped metal beams subjected to a uniformly distributed impulsive loading and identi"ed three basic failure modes, i.e. large inelastic deformation (Mode I), tensile tearing (Mode II) and transverse shear failure at the supports (Mode III). Later, a theoretical rigid-plastic analysis provided by Jones [3}5] showed reasonable agreement with the experimental results [2] on the threshold impulsive velocities at the onset of Mode II or Mode III failure. In this analysis, square yield criteria relating either the axial force to the bending moment or the transverse shear force to the bending moment, and elementary failure criteria were adopted. It is evident that the foundation of the simple formulations employed, especially those for Mode III failure, needs to be further validated. In the 1990s, an energy density criterion was proposed by Shen and Jones [8] which caters to the simultaneous in#uence of the bending moment, the axial force as well as the transverse shear in the structural response. According to this criterion rupture occurs in a rigid-plastic structure when the absorption of plastic work per unit volume, h, reaches a critical value; and further if the ratio of the plastic work absorbed through shearing deformation to the total plastic work done by all the stress components, b, reaches a critical value, b "0.45, a structure su!ers a Mode III failure. The merit of the energy density criterion is that it may be universally applicable for a large class of dynamic structural problems. Nevertheless, it has been noted that some problems associated with this criterion remain unanswered, e.g. (i) in a rigid-plastic analysis, the calculation of the energy density depends upon the length of a plastic hinge; (ii) in FE calculations, the energy density is mesh-dependent; and (iii) it is unclear how to determine the critical ratio b . It was suggested that b "0.45, but this value disagrees with 0.857 from the theoretical prediction in [3], refer to Fig. 11 in [8]. A clearance is needed. Transverse shear failure becomes increasingly important and dominant when intensity of the dynamic loading increases. Until now, several analyses [3,8}12] have been concerned with this failure mode by employing the concept of `plastic shear slidinga or `plastic shear hingea. In a rigid, perfectly plastic beam, it is generally assumed that a plastic shear hinge has an in"nitesimal length, that is, transverse shear sliding is treated as a transverse-displacement discontinuity which is the idealization of abrupt change in transverse displacement across a short segment of a beam [5]. On the other hand, Wang and Jones [13] found that in a rigid-linear hardening beam subjected to impulsive loading, the location of the shear hinge travels a "nite distance ¸ during the failure process. A penetrating understanding of these basic failure modes is of fundamental importance to the failure analyses of various structures under intense dynamic loading. Two important e!ects, however, have been ignored in all previous investigations: (i) the interaction between the shear force and bending moment; and (ii) the weakening of the sliding sections during the failure process. Thus, the present paper is aimed at re-examining the shear failure problem of impulsively loaded beams
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by various approaches, but focusing on these two factors. In addition to further comments and modi"cations on the elementary failure criterion, the interaction between the shear force and the bending moment at the sliding sections will be considered by adopting circular yield curve [14], Hodge's curve [15], or a yield condition based on slip-line "eld solutions [16]. The variation of the shear force and the bending moment over the sliding cross-sections during the beam's deformation process will also be taken into account.
2. Mode III failure of impulsively loaded clamped beams: general consideration Consider a clamped beam of length 2¸ and of rectangular cross-section with thickness H and width B, subjected to uniformly distributed initial velocity over the entire span, as shown in Fig. 1. For a simply supported beam subjected to an impulsive velocity, Jones et al. [11,4] found that dimensionless ratio l"Q ¸/2M with M and Q denoting the fully plastic bending moment and the fully plastic shear force of the cross-section, respectively, is an important parameter; di!erent transverse shear response modes occur when l41 or l'1. Obviously, this parameter mainly depends on the geometry of the beam. Nevertheless, it is of importance merely for some special beam cross-sectional shapes, such as wide-#anged I-beam or sandwich beam, where l&1. As for the rectangular cross-sectional beam considered here, l"¸/H 1. For instance, in the experiments conducted by Menkes and Opat [2], l ranged from 5.33 to 21.4. It was observed that, unlike tearing failure (Mode II) which followed the large de#ection of the beam (Mode I), transverse shear failure (Mode III) was characterized by insigni"cant #exural deformation at most cross-sections. Therefore, shear failure occurs at the supports in the early stage of the beam's response and generally exhibits a type of localized behaviour, while the Mode III thresholds does not depend on the length of the beam. Thus, with the length of the beam being excluded from the theoretical description, if a rigidplastic model is adopted, the maximum shear sliding at the supports, * , can be expressed as
a function of impulsive velocity < , yield stress >, density o and thickness H of the beam: * "* (< , >, o, H). (1)
From the n-theorem of dimensional analysis, we have
< * "f ((j).
"f H (>/o
(2)
Here non-dimensional parameter j"o< /> is termed the damage number which is commonly used to specify the severity of impact, according to Johnson [17]. Hence, if we adopt a criterion for
Fig. 1. An impulsively loaded clamped beam.
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shear failure in form * "kH (3)
with k being a constant determined by experimental measurement or theoretical consideration, the threshold impulsive velocity for transverse shear failure (i.e. Mode III failure) can be expressed as
> . o
< "f\(k)
(4)
It transpires that for rectangular cross-sectional beam with ¸/H 1, the threshold impulsive velocity for a Mode III failure does not depend on geometry of the beam but only on material properties.
3. Elementary failure criterion (EFC) With the aid of the velocity "eld shown in Fig. 2 where sliding occurs at the supports of the beam having a "nite shear strength Q , Jones [3] employed a simple rigid-plastic procedure to estimate the threshold velocity for the onset of a mode III failure as
2Q . (5) < " 3oBM In this simple theoretical analysis, an elementary failure criterion for complete severance at the supports of a beam
* "
R
< dt"H Q
(6)
was adopted. Condition (6) can be regarded as an extreme case of Eq. (3) with k"1. If von Mises yield criterion is adopted, then for a beam of rectangular cross-section, M ">BH/4 and Q ">BH/(3, so that 2(2 < " 3
> > "0.943 . o o
Fig. 2. Velocity "eld for the impulsively loaded clamped beam with shear sliding at supports.
(7)
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617
4. Modi5ed elementary failure criterion It is observed in a few recent experiments [18,19] that in most cases the realistic shear failure is caused by a combination of shear sliding and crack propagation. In these cases, the elementary failure criterion (6) can be straightforwardly modi"ed to the form (3). The factor k(0(k41) slightly depends on the material properties, geometric constraint of the structure as well as loading conditions. One may easily conjecture that the value of k for a tough material is larger than that for a brittle material [20]. However, for a wide range of situations k can be approximately taken as a constant, say k"0.3. Using modi"ed elementary failure criterion (3) and following the same derivation given by Jones [3], the modi"ed threshold velocity for the onset of a Mode III failure is obtained as
2kQ < " 3oBM which can be rewritten as 2(2 < " 3
(8)
k> k> "0.943 o o
(9)
for beams of rectangular cross-section. These two expressions predict a lower value of the threshold velocity < by a factor of (k. If k takes 0.3, (k"0.548, and consequently < "0.517(>/o. Both Eqs. (5) and (8) are based on the assumption that during the beam's deformation process, the bending strength M and shear strength Q of the sliding section A (see Fig. 2) remain constants M and Q and the same situation occurs at D. In other words, the internal force state of sections A and D is always located at the corner G (M , Q ) on the square yield locus (Fig. 3(a)). This is, undoubtedly, a quite rough approximation, refer to the con"gurations sketched in Fig. 4. The sketch suggests a weakening e!ect of shear sliding * on the bending and shear strengths of
Fig. 3. Yield loci on the m}q plane: (a) a square yield locus for intact sections and a rectangular yield locus for the sliding sections; (b) a yield circle for intact sections and yield ellipses for the sliding sections; (c) interactive yield curves based on slip-line "eld solutions for the sliding sections.
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Fig. 4. Shear sliding takes place (a) over a cross-section; (b) over a "nite width.
sections A and D, that is, M "fM and Q "fQ , where the weakening factor f depends on the shear sliding: f"f(*). If the weakened bending moment M and the weakened shear force Q are incorporated into the deformation mechanism shown in Fig. 2, then the conservation of linear momentum of segment AB demands 2R Q dt < !< " , oBHz
(10)
while the conservation of angular momentum requires 6R (M #M ) dt < !< " . oBHz
(11)
It is noted that at interior hinges B and C, M"M and Q"0, so that the mid-portion of the beam, BC, moves at a constant velocity < . Eqs. (10) and (11) therefore result in the length of segment AB, 3R (M #M ) dt 3M R (f#1) dt 6M z" " 5 . (12) Q Q R Q dt R f dt Eq. (12) indicates that two interior hinges B and C travel towards the mid-point as f decreases from 1 during the beam's plastic deformation process. This is di!erent from the prediction that hinges B and C remain stationary until < "0 based on the assumption of constant bending and shear strengths at the sliding sections A and D. Substituting Eq. (12) into Eq. (10) yields
R f dt 2Q d* "< "< ! . 3oBHM R (f#1) dt dt
(13)
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For rectangular cross-sectional beam, by introducing non-dimensional variables, * d, , H
t q, H/<
(14)
Eq. (13) is recast into non-dimensional form dd 8 (O f dq) "1! . (15) dq 9j O (f#1) dq where j is Johnson's damage number mentioned before. Fig. 4 shows two con"gurations of the plastic shear deformation region. (a) The shear sliding is treated as a stationary, strong discontinuity, as introduced by Nonaka [9], Symonds [10] and Jones [3]; then by assuming a statically admissible stress "eld at sliding sections A and D, we "nd f"1!*/H"1!d. (b) A "nite width of shear sliding is considered. Experiments [2,18,19] and some theoretical models indicate that this width w is usually very small. For example, in the plastic shear band model shown in Section 7, only half of its total width, i.e. e, lies outside the clamp "xture, so that the e!ective width of plastic shear deformation region of the beam, w, could be taken as e"((3/8)H"0.216H. Referring to Fig. 4(b), the maximum shear force then occurs over straight line E F provided *50.049H. Since "E F"+H!*, it is seen that f+1!*/H"1!d. Consequently, for both con"gurations (a) and (b), Eq. (15) can be rewritten as
O (1!d) dq 8 dd "1! . (16) O 9j [(1!d)#1] dq dq Eq. (16) is an integral-di!erential equation, which can be transformed into a standard set of "rst-order di!erential equations ready to be integrated numerically by means of a second-order Runge}Kutta procedure to obtain the maximum sliding as a function of j, d (j). The calculated
results are depicted in Fig. 5, where the variation of the "nal value of the weakening factor, f , with j is also exhibited. Recalling failure criterion (3), that is d "k, the threshold impulsive velocity
for shear failure of the beam is thus determined as a function of parameter k, j (k). If the value of
Fig. 5. The variation of the "nal value of the weakening factor and shear sliding with impulsive loading parameter j.
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k in criterion (3) is speci"ed, a threshold value of j can be determined accordingly. For example, if k"0.3, we "nd j "0.221, and accordingly < "0.447(>/o. In addition, it can be seen from Fig. 5 that when j approaches a critical value j "0.369, the maximum shear sliding increases from 0.7H to 1.0H rapidly. Thus, a critical impulsive velocity is identi"ed as
< "
j > > "0.607 . o o
(17)
It is 0.644 times the value given by Eq. (7) but is 1.17 times that by Eq. (9) for k"0.3. The di!erences re#ect the in#uence of the reductions of M and Q during the beam's deformation process.
5. Interaction between transverse shear and bending moment at sliding sections On bending moment-shear force plane, interactive yield curves, rather than the square ones, provide better representations of the yield condition. Although interaction between shear force and bending moment is intricate, several interactive yield curves for beams of rectangular cross-section have been proposed from various postulations. 5.1. Circular yield curve (CYC) A typical one of these interactive yield curves is a circle m#q"1
(18)
with m"M/M and q"Q/Q , as shown in Fig. 3(b). This was found by Robinson [14] from an approximation to Ilyushin}Shapiro's yield surface [21,22] and is the special case of m(1!q) #n#q"1 with n"0, which was derived from a uniform stress distribution in a rectangular cross-section [23]. According to the associated #ow rule, the generalized velocity vector (< Q , uM ) must be normal to the yield curve, that is, M u (3 Hu dq " "! (19) Q < 4 < dm where u denotes the angular velocity about the generalized plastic hinge A. Di!erentiating Eq. (18) gives dq (1!q ) "! . (20) dm q For any given value of dq /dm , q can be determined from this equation, and then m can be calculated from yield condition (18) as follows: m "(1!q ) so that the location of the internal force state on the yield circle can be determined.
(21)
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Applying the velocity "eld shown in Fig. 2, Eqs. (10) and (11) provide
8k < "< 1! 9jl
(22)
and < !< 32k < " u" , z H 27(3jl
(23)
where
k,
O
q dq, v,
O
(m #1) dq. Substitution of Eqs. (22) and (23) into Eq. (19) results in dq 8k "! . dm 27jl!24kl Eq. (22) is then recast into dd 8k "1! . dq 9jl
(24)
(25)
(26)
With the help of (24), (25), (20) and (21), Eq. (26) is numerically integrated by means of a second-order Runge}Kutta procedure starting from initial conditions d"0, q "1, m "0 at q"0. The results reveal that on the yield circle, the internal force state of sections A and D moves from point R counterclockwise in the early stage and approaches point S in the later stage. The prediction on the "nal shear sliding obtained from Eq. (26), as marked by `CYCa in Fig. 6(a), is larger by around 25% than that given by Jones [3] based on the assumption of constant shear force and bending moment at sections A and D. To reach d "k"1, j"j "0.706 and correspond ingly < "0.840(>/o; or to reach d "k"0.3, j"j "0.212 and correspondingly
< "0.460(>/o. In the foregoing analysis, the weakening e!ect of the shear sliding on the interactive yield condition is neglected. If this weakening e!ect is incorporated, as discussed in the later part of Section 4, the bending and shear strengths of the sliding sections may be expressed as M "fM and Q "fQ , with weakening factor f"1!d. Then, the yield condition (18) is modi"ed as m q # "1 (27) f f
and accordingly Eqs. (20) and (21) are replaced by
dq "! dm
1!
q f q
(28)
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Fig. 6. A comparison with the result given in Ref. [3] on the predictions of the "nal shear sliding from various interactive yield curves with the weakening e!ect of the shear sliding incorporated: (a) circular yield curve (CYC) and Hodge's yield curve (HYC); (b) yield curve based upon slip-line "eld solutions (SLF).
and
q m "f 1! f
(29)
whilst Eqs. (24)}(26) remain valid. As a result, the internal force state moves along the yield curve while the yield condition of sliding sections A and D progressively shrinks on the q}m plane: RPPPP P2, as sketched in Fig. 3(b). Predictions on the "nal shear sliding are marked by `CYC-Wa in Fig. 6(a). Yet when j approaches a critical value j "0.396, the "nal shear sliding increases from 0.65H to 1.0H rapidly, so that a critical impulsive velocity is identi"ed as < "(j >/o"0.629(>/o. 5.2. Hodge's yield curve (HYC) With the help of a variational method, Hodge [15] obtained another interactive yield curve between shear force and bending moment for beams of rectangular cross-section as q"ln ((1# # )/
(30a)
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623
and m"((1# !q)/
(30b)
with serving as a parameter. It can be well approximated by [24] m# q# q"1. (31) This yield curve closely circumscribes the yield circle (18), but inscribes the square yield condition. Similar to the analysis in Section 5.1, by taking account of weakening e!ect of the sliding sections, another critical impulsive velocity is obtained as < "(j >/o"(0.415>/o" 0.644(>/o. 5.3. Slip-line xeld solutions (SLF) Based on slip-line "elds constructed in the root region of cantilever beams under various loading conditions, Yu and Hua [16] obtained an interactive yield curve between m and q at the beam's root, which can be "tted by a single approximation m!2.106q#4.632q!1.526q!1"0
(32)
with error less than 1%, as depicted in Fig. 3(c). Adopting this interactive yield, we can get clear deformation patterns for the plastic shear failure of the dynamically loaded beams. Eqs. (20) and (21) and (27)}(29) are accordingly replaced by (1#2.106q !4.632q #1.526q ) dq " , 1.053!4.632q #2.289q dm m "(1#2.106q !4.632q #1.526q )
(33) (34)
and
m q q q !2.106 #4.632 !1.526 !1"0, f f f f
dq [1#2.106(q /f)!4.632(q /f)#1.526(q /f)] " , dm f[1.053!4.632(q /f)#2.289(q /f)] q q q . m "f 1#2.106 !4.632 #1.526 f f f
(35) (36) (37)
Hence, the beam's shear failure progress can also be traced as in Section 5.1. Fig. 7 exhibits the history of the shear force and the bending moment at sliding section A. Similar variation trends are obtained as adopting circular or Hodge's yield curves in these sections. From the prediction on the "nal shear sliding shown in Fig. 6(b), it is found that when j approaches a critical value j "0.410, the "nal shear sliding increases from 0.65H to 1.0H rapidly, so that a critical impulsive velocity is also identi"ed as < "(j >/o"0.641(>/o. For a speci"ed j,
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Fig. 7. The history of shear force and bending moment at end A (sliding section) based on SLF.
this interactive yield condition results in a slightly smaller prediction on the "nal shear sliding compared with that from circular or Hodge's yield curves. The reason is that the slip-line solution is an upper bound while the latter two are lower bounds of the true yield condition.
6. Plastic dissipation at sliding sections With regard to the deformation mechanism shown in Fig. 2, the increment of plastic dissipation merely due to shear deformation at sliding sections A and D is dE "Q d*"q Q H dd, while the increment of plastic dissipation due to bending is dE "M
m kQ H(dq!dd) < !< dt" . 3l z
(38)
(39)
Hence, the history of plastic dissipation can be traced by using each of the foregoing approaches. The ratio of plastic dissipation merely due to shear deformation to the total dissipation due to both bending and shear, de"ned by E E b, " (40) E E #E can then be calculated. Fig. 8(a) shows the variation of the ratio b with time for three representative values of j. For any speci"ed values of j(4j ), b decreases quickly from 1.0 in the very early stage after impact, then it approaches gradually to its "nal value b at < "0, when the shear sliding ceases. In other words, shear sliding dominates the plastic dissipation in the very early stage, but this stage swiftly transits to the stage of shear-bending deformation. In correspondence to each of the critical values j obtained from various approaches, the critical values of b , denoted by b , based on these approaches can be obtained and summarized in Table 1. A b &j curve calculated from CYC with
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Fig. 8. (a) The variation history of ratio b with time; (b) the variation of "nal value of b with impulsive loading parameter j (CYC-W).
the weakening e!ect of shear sliding being considered is shown in Fig. 8(b). It can be seen that when j(j , the calculated values of b are notably less than b , that is, at the end of dynamic response of the beam, shear sliding only dissipates a small portion of the total energy, so at the supports shear failure will not take place. Based on a comparison with the experimental results conducted by Menkes and Opat [2], Shen and Jones [8] suggested that the critical value of b at the onset of Mode III failure, b , could be estimated as 0.45. On the other hand, as shown in Table 1, our study indicates that the values of b calculated from the square yield condition range from 0.85 to 0.94, around double of 0.45, but those calculated from the interactive yield curves range from 0.44 to 0.66, fairly close to b "0.45 suggested in Ref. [8].
7. Shear strain failure criterion The shear sliding * at the supports of a beam subjected to impulsive load may be expressed as [25] *"ce
(41)
where shear sliding * is an accumulative quantity over the shear band, c is the average shear strain and e is the half-width of the shear band as de"ned in Fig. 9. As mentioned in Section 4, although the shear band has a total width of 2e, only half of its width lies outside the clamp "xture; so the e!ective width of plastic shear deformation region of the beam should be e, see Ref. [26] for details. From the elementary beam theory, it can be deduced [26] that (3 M H. e" " 8 2Q Substituting Eq. (42) into Eq. (41) yields (3 cH. *" 8
(42)
(43)
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Fig. 9. Shear bands located at the supporting ends.
Thus, when transverse shear failure occurs at the supports, a critical shear sliding * is reached, i.e. (3 * " c H, 8
(44)
where c is the rupture shear strain of the beam material, which may be taken a typical value of 0.5 for aluminum alloy and 0.8 for mild steel [25]. Hence, if expressed in uniform form (3), it is found that k"((3/8)c "0.108 and 0.173, respectively. As in Ref. [26], it is assumed that the displacement pro"le of a clamped beam after impulsively loading as
= [1!(x/¸)] for a41, (45) = [1!(x/¸)? ] for a51, where a"I/I with I being the critical input impulse at which the beam fails by tensile tearing. By exploiting the principle of equivalent work done, the critical input impulse I required to cause transverse shear failure of the beam can be deduced as (refer to Eq. (35b) in Ref. [26]) ="
I 4!3(H/¸) c " # o>BH 4(H/¸) (1#n) (H/¸)
(46)
so that the threshold impulsive velocity is found to be
< "
4!3(H/¸) c (H/¸) > # 16 o 4(1#n)
(47)
in which n is the work-hardening index of the material. It slightly depends on the rupture shear strain c and the thickness-to-length ratio H/¸. As H/¸P0, it has a limit value < "0.5
> . o
(48)
It is interesting to note that this limit threshold velocity is very close to that given by Eq. (9) for k"0.3.
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Table 1 A summary of the results Critical energy ratio b
Criterion/approach
Critical impulsive parameter j
Threshold impulsive velocity/(>/o
Dimensional analysis Elementary failure criterion (EFC) Modi"ed elementary failure criterion
[ f\(k)] 8/9"0.889 8k/9"0.889k (0.267 for k"0.3) 0.369
f \(k) 2 (2/3"0.943 2 (2k/3 (0.517 for k"0.3) 0.607
0.706 (for k"1.0) 0.212 (for k"0.3) 0.396
0.840 (for k"1.0) 0.460 (for k"0.3) 0.629
0.444 (k"1.0)
0.762 (for k"1.0) 0.230 (for k"0.3) 0.415
0.873 (for k"1.0) 0.480 (for k"0.3) 0.644
0.438 (k"1.0)
0.843 (for k"1.0) 0.252 (for k"0.3) 0.410
0.918 (for k"1.0) 0.502 (for k"0.3) 0.641
0.477 (k"1.0)
0.25 0.659 0.923 0.743 0.605
0.5 0.812 0.961 0.862 0.778
* 0.45
EFC incorporating the weakening e!ect of the shear sliding Circular yield curve (CYC) CYC incorporating the weakening e!ect of the shear sliding Hodge's yield curve (HYC) HYC incorporating the weakening e!ect of the shear sliding Slip-line "eld solutions (SLF) SLF incorporating the weakening e!ect of the shear sliding Shear strain failure criterion Energy density criterion [8] Experimental results [2]
(H"0.250 in) (H"0.187 in) (H"0.250 in) (H"0.375 in)
(H"0.250 in) (H"0.187 in) (H"0.250 in) (H"0.375 in)
* 0.857 0.857 0.938
0.470
0.519
0.658
The material parameters for aluminium 6061 T6 used in the experiments were o"2.734;10 kg/m and >"286.8 MPa.
8. Concluding remarks Plastic shear failure at the supports (Mode III) of clamped beams loaded impulsively is re-examined by several approaches; two e!ects, i.e. (i) the interaction between the shear force and bending moment, and (ii) the weakening of the sliding sections during the failing process, are emphatically considered. Table 1 summarizes all the predictions on the threshold impulsive velocity from these various approaches and compares them with the experimental results reported in Ref. [2]. It is concluded that although the estimate on the threshold velocity obtained by an elementary failure criterion [3] agrees well with the relevant experimental results, by adopting interactive yield conditions, such as circular yield curve, Hodge's yield curve or a yield curve based on slip-line "eld
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solutions, the present studies provide more reasonable and better predictions. It is also noted that most approaches result in lower values of threshold velocity compared with the experimental results. This is partly attributed to material's hardening behaviour of aluminium used in the experimental tests. The critical ratios of the plastic shear dissipation to the total plastic dissipation at the sliding sections from these various approaches are also summarized in Table 1. It is found that those from the square yield condition are much larger than 0.45, which was suggested in [8], but those from the interactive yield curves are fairly close to this value. Whilst the critical ratio b "0.45, as suggested by Shen and Jones [8] based on a comparison to experimental measurements [2], can be accepted as an empirical criterion for the onset of shear failure at the supports, the present analysis has laid a rational theoretical foundation to understand the complex variation of the ratio b in the shear deformation/failure process of impulsively loaded beams.
Acknowledgements The work described in this paper was conducted as a part of the research projects funded by Hong Kong Research Grant Council (RGC) under CERG project HKUST 811/96E. This support is gratefully acknowledged.
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