Quasi-static analysis of structural impact damage

ELSEVIER 0143-974X(94)00002-6

J. Construct. Steel Research 33 t1995) t51 177 © 1995 Elsevier Science Limited

Printed in Malta. All rights reserved 0143-974X/95/$9"50

Quasi-static Analysis of Structural Impact Damage

N o r m a n Jones

Impact Research Centre, Department of Mechanical Engineering, The University of Liverpool, PO Box 147, Liverpool, L69 3BX, U K

A B S T R A C T

This article explores the accuracy of quasi-static methods of analysis for predicting the inelastic behaviour of structural impact problems.

The quasi-static behaviour of fully clamped beams when struck at the mid-span by a rigid mass, which produces either infinitesimal displacements or finite transverse displacements for sufficiently large impact loads, is examined and compared with the corresponding exact theoretical predictions of dynamic rigid-plastic analyses. Some comparisons are also made with experimental results. The quasi-static behaviour of beam grillages, pipelines and axially impacted tubes when struck by rigid masses is also discussed. Comparisons are made between the quasi-static theoretical predictions and experimental results and some observations are offered on the adequacy of quasi-static methods for design purposes.

1 I N T R O D U C T I O N

This article examines the response of engineering structures and components subjected to dynamic loads which produce large plastic strains. 1'2 Studies on this topic are relevant to the structural crashworthiness field, the design of energy-absorbing systems and for many safety calculations and hazard assessments throughout engineering. These structural impact problems are difficult to study because they often involve complex structural geometries which undergo large time-dependent deformations and inelastic strains. Moreover, the material properties are usually time-dependent but often not well specified, while, to add further complication, the dynamic external loading characteristics may not be known accurately.

The characteristics controlling the dynamic response of a structure may be quite different to those features which influence the behaviour of the

151

152 N. Jones

corresponding static problem. For example, transverse shear effects can dominate the dynamic behaviour of structural impact problems and even give rise to failure, while transverse shear effects would be unimportant for the corresponding static case. 3 The magnitudes of the transverse shear forces in a statically loaded structure are limited by the requirement to equilibriate the finite external loads which would be of the order of the static collapse load. However, impulsive loadings, for example, could cause infinite transverse shear stresses to be developed during impact in an ideal structural model. In practice, the magnitude of the transverse shear force would be limited by the requirements of the yield condition, but could be sufficiently severe to cause a transverse shear failure. 4

The complexity of structural impact problems has motivated the development of simple methods of analysis which, in fact, are often adequate from an engineering viewpoint. For example, elastic wave propagation effects are generally ignored in numerical solutions and in theoretical methods of analysis because the structural response duration is usually several orders of magnitude larger than the elastic stress wave transit times. In fact, elastic effects are often disregarded entirely because the elastic strain energies in many severely loaded structural systems are negligible compared with the inelastic energies. This simplification has led to the development of the rigid-plastic methods of analysis which have been used to predict successfully the response of many dynamic structural problems. 1'2 Numerical schemes may be used for more complex structural impact problems, or when more accurate estimates are required. The user, however, must have a high degree of competence and knowledge in non-linear mechanics in order to achieve reliable results. Moreover the accuracy of the numerical results for many problems will suffer from the paucity of information on the dynamic loading characteristics and the lack of experimental data on the material properties under dynamic conditions particularly for the strain-rate-sensitive behaviour at large strains and rupture.

It has been observed that theoretical quasi-static methods of analysis have been used successfully to predict the behaviour of a range of low-velocity structural impact problems. For example, the deformation modes produced during the dynamic axial progressive buckling of thin- walled tubes with circular or square cross-sections are similar to those which develop in the same tubes when loaded statically. In fact, good agreement has been obtained between the predictions of theoretical quasi-static methods of analysis and experimental results on metal tubes struck by relatively large masses travelling with axial impact velocities up to about 10 m/s. 5 The static deformation profiles have been used for these

Quasi-static analysis of structural impact damage 153

dynamic calculations, but all inertia forces in the tubes have been neglected though the influence of material strain-rate sensitivity must be considered for strain-rate-sensitive materials. Other low-velocity impact problems which have been examined using quasi-static methods of analysis include the drop mass impact loading of beam grillages 6 and the solid wedge dynamic indentation of pipelines. 7 In both cases, a theoretical rigid-plastic method of analysis, which used the static deformation profiles, gave good agreement with the corresponding experimental results.

A simple illustration of the quasi-static method of analysis is given in the next section for a fully clamped beam struck by a mass at the mid-span. The same problem is studied in Section 3 but when the important influence of finite transverse displacements is retained in the basic equations. Some guidelines on the accuracy of quasi-static methods of analysis are obtained by comparing the predictions with an exact dynamic plastic analysis. The quasi-static behaviour of laterally impacted grillages and pipelines is examined in Sections 4 and 5, respectively. Section 6 studies the quasi-static response (dynamic progressive buckling) of axially loaded thin-walled tubes. Several observations on the analysis of structural impact problems are offered in Section 7 and the article is completed with some conclusions.

2 QUASI-STATIC RESPONSE O F BEAMS SUBJECTED TO I M P A C T LOADS

A simple illustration of the quasi-static method of analysis is given for a fully clamped beam struck by a mass at the mid-span. 8 Parkes used a rigid, perfectly plastic analysis to study the dynamic response of the fully clamped beam in Fig. 1 which is struck transversely at the mid-span by a rigid mass G travelling with an initial velocity V0. The theoretical analysis assumes that the plastic yielding of the material is controlled by the magnitude of the bending moment and that the influences of the axial membrane force and the transverse shear force on plastic yielding may be neglected.

It transpires that the response consists of two phases of motion. For the first phase of motion, two plastic hinges develop at the impact point and travel outwards towards the respective supports where they arrive simultaneously for the particular case of an impact at the mid-span which is examined here. This completes the first phase of motion which is then followed by a second phase of motion with stationary plastic hinges at the mid-span and at both supports. The permanent deformed profile is

154 N. Jones

- - - . l _ .

I

_ _ I_ _1 I. _

- [ - - i N x

Fig. 1. Fully clamped rigid, perfectly plastic beam having a span 2L and a mass m per unit length and struck at the mid-span by a mass G travelling with an initial velocity Vo.

reached at the end of the second phase of motion when the initial kinetic energy of the striking mass has been absorbed by the travelling and stationary plastic hinges during the first phase of motion and by the stationary plastic hinges in the final phase of motion.

The maximum permanent transverse displacement when the mass strikes at the mid-span, as shown in Fig. 1, is 8

We =G 2 Vo2 {~/(1 + ~ ) + 2 log e(1 +~)}/24mMo, (1)

where ~ = mL/G and m is the mass per unit length of a beam with a span 2L and a plastic bending moment capacity Mo for the cross-section.

The theoretical procedure developed by Parkes s may be used to obtain the predictions presented in Figs 2-5. It is evident from Fig. 2(a) that the maximum permanent transverse displacements acquired during the first phase of motion (W1) are much smaller than those accumulated throughout the second phase (I4"2) for mass ratios G/2mL larger than about 5 to 10. This observation occurs because the duration of the first phase of motion (tl) for large mass ratios is very short compared with the duration of the second phase of motion (t2), as shown in Fig. 2(b).

It is evident from Fig. 3(a) that the impact force underneath the striker (P) is infinitely 1 large immediately after impact. However, it decreases quickly with time for all mass ratios as the plastic hinges propagate away from the impact position and is close to the static collapse force (Pc) throughout the second phase of motion for mass ratios larger than about 10, as shown in Fig. 3(b). Some of these results are replotted against the dimensionless time in Fig. 4. This figure reveals the very short duration of the high impact forces particularly for large mass ratios, as illustrated in Fig. 4(a) for a mass ratio of 5.

l In fact, the magnitude of P cannot exceed the force 2Qo =2z0BH when the transverse shear force is retained in the yield condition unlike the theoretical study in Ref. 8.


1,0 W

0'75

05

0 2 5 [ ~

,b 2'0

(a)

'0 6

2mL

1"0"

T

075

O5

025 tl TV-! lb

2'0 i0 6

2mL (b) Fig. 2. (a) Dimensionless permanent transverse displacements at the end of the first phase of motion (W1) and acquired during the second phase of motion (Wz) for the beam in Fig. 1. Wr is the total permanent transverse displacement given by eqn (1). (b) Dimension- less time duration of the first (tl) and second (t2) phases of motion for the beam in Fig. 1.

T=GVoL/4Mo is the response duration.

The external energy which is imparted to the beam in Fig. 1 during the first and second phases of motion (E~ =~P dW) is shown in Fig. 5(a) and 5(b), respectively. A significant amount of energy is imparted to a beam during the first phase of motion when the mass ratio is small. However, most of the external energy is imparted during the second phase of motion for large mass ratios notwithstanding the large impact forces immediately after impact in Figs 3(a) and 4(a).

The theoretical predictions in Figs 2-5 for the impact problem in Fig. 1 show that the second phase of motion with stationary plastic hinges at the mid-span and both supports dominates the response for mass ratios (G/2mL) larger than about 10 (i.e., ~>0.05). For example, the permanent transverse displacement acquired during the second phase of motion is about 97% of the total in Fig. 2(a) when G/2mL = 10, while the

156 N. Jones

_P 20

15

10

62 6~ 0:6 0'B 1:o (Q) -C

1.0

Pc 0"75

(}5

025

/

10 20 301__f L (b) 2~-2mL

Fig. 3. (a) The variation of the dimensionless force at the mid-span of the beam in Fig. 1 with the position ~ of the travelling plastic hinges during the first phase of motion. Pc is the static plastic collapse pressure given by eqn (2). (b) Dimensionless force at the mid-span

of the beam in Fig. 1 during the second phase of motion.

corresponding external energy imparted to the beam is about 91% of the initial kinetic energy (K) from Fig. 5(b). These observations suggest that the rigid-plastic analysis of many practical engineering problems may be simplified by disregarding the transient or travelling plastic hinge phase of motion.

Now, the static plastic collapse load for a fully clamped beam subjected to concentrated force at the mid-span is 1

Pc = 4Mo/L. (2)

The external work done by Pc, for a transverse displacement Wq underneath the concentrated load is Pc Wq, which, for a quasi-static method of analysis, must equal the initial kinetic energy of the mass in Fig. 1


2O P

- - i

Pc 15

~=0I

t~) or)

0I 02 03 3~t

(a) r

_P Pc

20

15

10 ~=I0

IT) ~o 30

3~i {b) T

Fig. 4. Variation of the dimensionless impact force with dimensionless time for the beam in Fig. 1 (tl is the duration of the first phase of motion and T is the response duration).

(a) ~=0.1; (b) ~= 10.

(K = GV~/2). Thus,

Wq = G V 2 L /8Mo (3)

which is independent of the mass of a beam. Equation (1) reduces to eqn(3) for large mass ratios (i.e., ~<<1). The quasi-static theoretical prediction of eqn (3) may also be obtained by equating the initial kinetic energy (K) to the energy absorbed by stationary plastic hinges which form at the mid-span and supports as in the second phase of motion in the theoretical analysis a leading to eqn (1). In this circumstance, the travelling plastic hinge phase of motion in the Parkes solution s makes a negligible contribution to the theoretical predictions for We, as shown in Fig. 2 for large mass ratios. In fact, it may be seen from Fig. 3(b) that the reaction force (P) underneath the striker mass equals the static collapse pressure

158 N. Jones

E~e K

110) ~= 0 m05

075 2-CmL = 0'5

0-5

0-25 L = 5

0 0 02 0-4 0-6 0'8 10

(a} k

1.0

0.7505 ~ ~

0.25

0 10 20 30

Ib) 2mk

Fig. 5. (a) Dimensionless external energy (Ee) imparted to the beam in Fig. 1 as a function of the dimensionless position (4, 0 ~< ~ ~< L) of the travelling plastic hinges during the first phase of motion. K= GV2/2 is the initial kinetic energy of the mass G. (b) Dimensionless external energy (Ee) imparted to the beam in Fig. 1 during the second phase of motion.

(Pc) given by eqn (2) when G/2mL>>I throughout the second or modal phase of motion.

A comparison is presented in Fig. 6 between the theoretical predictions of a complete dynamic analysis (eqn (1)) and a quasi-static method (eqn (3)). It is evident that the quasi-static procedure overpredicts the maximum permanent transverse displacements of a dynamic analysis by 6.6% when G/2mL = 5 and that this difference increases to only 10.8% for a striker which weighs three times the beam mass (G/2mL=3). The quasi-static method of analysis overpredicts the maximum permanent transverse displacement because all of the initial kinetic energy is absorbed in three stationary plastic hinges (i.e., W1 =0, Wz = Wf), whereas, in the complete dynamic analysis, the travelling plastic hinges cause plastic energy to be absorbed throughout the entire span as well as at stationary

Quasi-static analysis of structural impact dama#e 159

1'0

Wq 0'75

05i

0"25

10 20 30 ±-_0_ 2~-2mL

Fig. 6. Ratio of the maximum permanent transverse displacements for the rigid, perfectly plastic beam in Fig. 1 according to an exact dynamic analysis (Wf) (eqn (1)) and a quasi-static analysis (Wq) (eqn (3)). This curve is also valid for a cantilever beam of length

L struck by a mass G at the tip with 1/2~c = G/mL in the abscissa.

plastic hinges (i.e., WI 50, WI+ W2 = Wf). This difference in the energy- absorbing mechanisms is particularly significant when the striker masses are smaller than the total beam mass. In this case, the travelling plastic hinge phase of motion in the theoretical solution leading to eqn (1) is important, as shown in Fig. 2. In fact, for G/2mL = 0.05 almost all of the initial kinetic energy is absorbed during the first phase of motion with very little remaining to be absorbed during the second or modal phase, as indicated in Fig. 5. It may be shown that the theoretical analysis 8 leading to eqn (1) predicts that W1 ~ We and W2 ~ 0 when G/2mL-,O. Thus, it is clear that quasi-static methods of analysis would be wholly inappropriate when G/2mL < 1, approximately.

It is anticipated that similar results to those in Figs 2-6 would be obtained for a cantilever beam struck by a mass at the tip. In fact, it is observed in Refs 1 and 8 that the equations for a fully clamped beam of length 2L struck by a mass G at the mid-span also describe the behaviour of a cantilever beam of length L struck by a mass G at the tip when Mo and G are replaced by Mo/2 and 2G, respectively. Thus, eqn (1) for a cantilever beam becomes

Wf = G 2 Vo2 {~c/(1 + ct¢)+2 logdl +~¢)}/3mMo, (4)

where

~c = mL/2G, (5)

which was first obtained by Parkes. 9

160 N. Jones

The static collapse load for an end-loaded cantilever beam I is Pc =Mo/L so that the quasi-static method discussed earlier for a fully clamped beam predicts that Pc Wq = GV2/2, or

Wq = GLV~/2Mo, (6)

which may also be obtained from eqn (4) when ~c ~0. This equation may also be obtained from eqn (3) with Mo and G replaced by Mo/2 and 2G, respectively. It is straightforward to show that the results in Fig. 6 are again obtained except 1/20~ in the abscissa is replaced by 1/20~c = G/mL which is, in fact, the mass ratio for a cantilever beam.

A rigid-plastic method of analysis for the beam impact problem in Fig. 1 is only valid provided the external dynamic energy imparted to the beam (GV2/2) is much larger than the maximum possible strain energy which could be absorbed by the beam in a wholly elastic manner (an upper bound of this is taken as (ag/2E) × beam volume). This is discussed in Ref. 1 where it is found that the energy ratio

er = EGV 2 /a 2 2LBH (7)

must be larger than unity for the impact problem in Fig. 1, where E is Young's modulus, a0 is the uniaxial yield stress and B is the beam breadth. Equation (3) may now be recast into the form

Wq/H = (ao/E)(L/H) 2 Er, (8a)

o r

E~ = (E /ao )(H /L )2 (Wq/H). (8 b)

It is evident from eqn (8a) that the requirement of large energy ratios E r

could be incompatible with the assumption of infinitesimal displacements which has been incorporated into the derivations of the various equations in this section. Thus, the influence of finite displacements, or geometry changes, must be examined and is discussed in the next section.

Q U A S I - S T A T I C R E S P O N S E O F BEAMS S U B J E C T E D TO I M P A C T L O A D S P R O D U C I N G F I N I T E D I S P L A C E M E N T S

The theoretical predictions in the previous section were obtained for fully clamped and cantilever beams subjected to impact loads which produced


only infinitesimal displacements. However, it is well known that the influence of finite transverse displacements, or geometry changes, exercise a significant role during the response of fully clamped beams struck by masses which produce maximum permanent transverse displacements greater than about one-half of the beam thickness, x'4 The plastic yielding of a beam undergoing small transverse displacements is controlled by the bending moment with the transverse shear force customarily taken as a reaction force which does not contribute to plastic flow. As the transverse deflections increase, the influence of the bending moment and the transverse shear force diminishes, while the axial membrane force increases until the response approaches that of a plastic string for which plastic flow is controlled by the membrane force alone.

A theoretical study is reported in Ref. 10 which compares the predictions of a quasi-static method of analysis with a dynamic plastic solution for the beam impact problem in Fig. 1. Unlike the analysis in the previous section (eqns (2), (3) and (6)), a quasi-static solution must now cater for the increase of the static plastic collapse load of a beam with increase in transverse displacements. Thus, from Refs 1 and 11, for example, the transverse displacement (W) of a fully clamped, rigid, perfectly plastic beam subjected to a static concentrated load (P) at the mid-span, as shown in Fig. 7(a), is given by the expressions

P/Pc = 1 + ( W / H ) 2 W / H <~ 1 (9a)

p

la)

{b) Fig. 7. (a) Static concentrated load acting at the mid-span of a fully clamped beam.

(b) Transverse displacement profile; O: plastic hinges.

162 N. Jones

and

P/Pc = 2 W/H W/H > 1, (9b)

where H is the depth of a beam with a rectangular cross-section and Pc is given by eqn (2). This exact theoretical solution is obtained for a transverse displacement profile having stationary plastic hinges at both supports and at the mid-span underneath the concentrated load, as shown in Fig. 7(b).

A quasi-static analysis for the beam impact problem in Fig. 1 predicts that the maximum permanent transverse displacement Wq is given by the expression

fo v~ P dW= GV~/2, or

~0 ~q Pc {I+(W/H)2}dW=GV2/2 when Wq/H<~I (lOa)

and

P~ {1 +(W/H) 2 } dW+P~ (2W/H)dW=GV~/2

if W~/H > 1. (lOb)

Equations (10a) and (10b) my be integrated to give the quasi-static predictions

Wq/H+(W~/H)a/3=GV2L/8Mo H when Wq/H<~I (lla)

and

(Wq/H)2=GV2L/8MoH-1/3 if Wq/H>I. (llb)

Equation (l la) with VCq---,0 reduces to eqn (3) for infinitesimal displacements, as expected, so that the cubic term in eqn (lla) arises from the strengthening influence of finite transverse displacements. It is evident from eqn (lla) that this phenomenon leads to a one-third increase in the dimensionless external energy absorbed by a beam with Wq = H, while a comparison of eqns (3) and (l ib) shows that it is nearly four times larger when Wq/H=4.


Now, a theoretical solution for the dynamic beam problem in Fig. 1 is presented in Ref. 10 when the influence of finite-deflection effects is retained in the basic equations. This analysis has a first phase of motion with a stationary plastic hinge forming immediately underneath the striker mass G, which remains in continuous contact with the beam, and two travelling plastic hinges which propagate away from the mid-span towards the fully clamped supports. This phase of motion is completed when the travelling plastic hinges reach the supports which occurs simultaneously for an impact at the mid-span. The remaining kinetic energy is absorbed in a final phase of motion with stationary plastic hinges at both supports and at the mid-span which is the same as the single mode form in Fig. 7(b) which was used to obtain eqns (9a) and (9b) for static loads.

The percentage difference between the maximum permanent transverse displacements of a quasi-static theoretical procedure (Wq) (eqns (1 la) and (1 lb)) and the dynamic analysis (Wd) in Ref. 10 for the problem in Fig. 1 is defined as

ew =(Wq - W.) lO0/Wd (12)

The comparisons between the two analyses in Fig. 8 reveal that the error introduced by a quasi-static procedure is less than 10% for mass ratios (G/2mL) larger than about 2-3 and that the error is less than 1% for mass ratios larger than about 16-33, depending on the dimensionless impact velocity.

The differences according to eqn (12) between the quasi-static and exact theoretical predictions in Section 2 for the beam problem in Fig. 1 subjected to striking masses, which produce infinitesimal displacements, are shown as vertical dotted lines in Fig. 8. It is interesting to observe that the influence of finite deflections, or geometry changes, expands the range of validity of a quasi-static method of analysis. This is particularly noticeable for the larger mass ratios. Thus, an error of less than 1% is associated with a quasi-static method of analysis for infinitesimal displacements and mass ratios larger than about 33 which reduces to 16.56 for large dimensionless impact velocities when the influence of finite deflections is considered.

The theoretical predictions in Fig. 8 for infinitesimal displacements are independent of the dimensionless impact velocity. The curves associated with the finite-deflection case in Fig. 8 are virtually independent of the dimensionless impact velocity except for small dimensionless values when the beam behaviour changes from a bending response to a predominantly membrane one.

164 N. Jones

.s; I ,~07 823 16.56

-- _ewe 10% - --e w, 4%

- - - - a , ( 2%

%, I %

I I ] I

Ill I q,.z

65 fi

2mL

Fig. g. Percentage difference between the theoretical predictions of a quasi-static analysis (eqns (11)) and a dynamic rigid-plastic analysis 1° for the beam impact problem in Fig. 1 (2 = m V~ L2/4Mo H). ( ) Finite-displacement effects retained in analyses (eqns (11) and

Ref. 10; (- - - -) infinitesimal displacements (eqns (1) and (3)).

The theoretical predictions in Fig. 8 are replotted in Fig. 9 which shows that all of the curves are fairly close to each other for a wide range of dimensionless impact velocities 2. The error curve associated with the infinitesimal displacement case in Section 2 is virtually indistinguishable from the finite displacement curve for /l=0.008 drawn in Fig. 9. It is evident that the error of a quasi-static method of analysis is less than 5.5% for mass ratios larger than 3-6 depending on the magnitude of the dimensionless impact velocity. These results reveal that, for a given mass ratio, the error of a quasi-static method of analysis decreases for an increase in 2 which is due to the influence of finite displacements or geometry changes, as found in Fig. 8. The results in Fig. 10 show more clearly that, for a given mass ratio, the dimensionless maximum permanent transverse displacement Wf/H increases with an increase in the dimensionless impact velocity 2.

Experimental test results on fully clamped strain-rate-insensitive aluminium alloy beams struck at the mid-span by a mass travelling with an initial velocity Vo, as shown in Fig. 1, are reported in Refs 4 and 12. The


15

= "2

\\ \ \ r~,;oo~

°o- 1~ 2'0 ~o , :o 5..5_ 2mL

Fig. 9. Percentage difference between the theoretical predictions of a quasi-static analysis (eqns (11)) and a dynamic rigid-plastic analysis l° for the beam impact problem in Fig. 1

(2=mV 2 L2/4MoH)

15 I i I I

Il l ~ S ~ / Wf/H= 5"0

/I /! \~ zJ~ / I I~'~ ~,/" WflH =3"2

,t.t /1',

O0 10 20 30 40

2mL

Fig. 10. Comparison between the theoretical predictions of a quasi-static analysis ( ) (eqns (11)) and a dynamic rigid-plastic analysis from Ref. 10 ( . . . . ) for the beam impact

problem in Fig. 1 (2=mV 2 L2/4MoH).

166 N. Jones

H 3

2

1

0 I 5 10 15 20

A

Fig. 11. Comparisons between the theoretical quasi-static predictions (eqns (11)) for the maximum permanent transverse displacements of the beam impact problem in Fig. 1 and the corresponding experimental results in Refs 4 and 12 on aluminium alloy beams (A=GV2oL/8MoH): ( ) eqns (11); (O) experimental results on fiat beams; 4 (A) experimental results 12 with ao =(ay +~rm)/2 (Cry and o- m are yield and ultimate stresses,

respectively).

dimensionless maximum permanent transverse displacements at the mid- span are plotted in Fig. 11 against the dimensionless initial kinetic energy (A) and compared with the theoretical quasi-static predictions of eqns (11). The agreement between the theoretical predictions and both sets of experimental results in Fig. 11 illustrates the accuracy which may be achieved with quasi-static methods of analysis.

The mass ratios (G/2mL) for the beam tests in Ref. 4 varied from 235 to 471, while all the tests in Ref. 12 had a mass ratio of 283. Equation (7) was used to estimate the energy ratio (Er) for the test results reported in Fig. 11. It transpired that those in Ref. 4 had energy ratios, ranging from 0.82, for the smallest value of Wr/H in Fig. 11, to 22-9, while those in Ref. 12 were much larger and varied from 117 to 153. It is noted in Ref. 1 that reasonable agreement has been obtained between the theoretical predictions of a rigid-plastic method of analysis and experimental results for energy ratios as small as three. The energy ratio given by eqn (7) was developed by assuming that the entire volume of material in a beam became plastic at the exhaustion of elastic behaviour. In practice, permanent transverse displacements would occur when material plastic flow occurred only within restricted regions of a beam (e.g. in the vicinity of the hinge positions in Fig. 7(b)).

4 Q U A S I - S T A T I C B E H A V I O U R O F A BEAM G R I L L A G E

The quasi-static method of analysis which is discussed in the previous two sections is a surprisingly simple theoretical procedure which may be used


to predict the response of many structural members subjected to impact loads. In particular, this procedure was used in Ref. 6 to obtain the maximum permanent transverse displacement of a fully clamped beam grillage consisting of one main member with two cross-members struck by a mass at the centre, as indicated in Fig. 12.

The theoretical rigid-plastic analysis for a concentrated load at the mid-span involves a fairly complex transverse displacement profile when the influence of finite deflections, or geometry changes, is retained in the basic equations, as well as bending of the two cross-members in the plane of the beam grillage. In fact, 13 plastic hinges develop in the grillage with one at every support (6), two in each cross-member at the intersection with the main member (4), one in the main member at each intersection with a cross-member (2), and one underneath the striker mass. These plastic hinges remain stationary for a quasi-static analysis and a straightforward numerical solution gives a relationship P(W) between the concentrated force (P) and the associated maximum permanent transverse displacement (W). As outlined for the quasi-static analyses in the previous two sections, the maximum permanent transverse displacement of a grillage is obtained from the expression to w" P(W)dW= GV2/2, where Vo is the impact velocity of the striker mass G.

Figure 13 illustrates the typical agreement which is obtained between the theoretical quasi-static predictions and the corresponding experimental results for the permanent transverse displacements at the mid-span and at the junctions of the grillage in Fig. 12. The ratio between the striker mass and the grillage mass is 49 for these experiments. 6 Similar agreement was obtained for other mass ratios between 35 and 49. These theoretical

I /

-

A

Fig. 12. The idealised permanent transverse displacement profile for a beam grillage struck transversely at the mid-span by a heavy mass G travelling with an initial velocity V0. This

profile is also used for a quasi-static analysis. 6

168 N. Jones

Wof

_ z~ z~ ..~f

Vo (m/s}

Fig. 13. Dimensionless permanent transverse displacement versus the initial striker velocity (Vo) for aluminium alloy beam grillages having the geometry shown in Fig. 12. ( ) Quasi-static analysis6; (A) experimental results. 6 Woe and WBr are the permanent transverse

displacements at the mid-span and at the junctions, respectively.

predictions of the quasi-static me thod of analysis for a grillage are consistent with the compar isons made in Figs 8 and 9 for beams struck by large masses for which the error is less than 1% when the mass ratio is larger than 33, approximately. The energy ratios range from 31 to 77 for the experimental test results on the grillages in Fig. 13.

5 Q U A S I - S T A T I C B E H A V I O U R O F A P I P E L I N E

The idealised pipe impact problem in Fig. 14 is of interest in several industries. For example, the integrity of pipelines struck by objects

. . . . . . . . . . . . . r . . . . . . . . . . . . . . . . I ,t ] * I i

c . . . . . . . . . . . . i

Fig. 14. A mass G travelling with an initial velocity 1/o and striking a pipeline which is fully clamped across a span 2L.


dropped from overhead cranes during maintenance operations could be modelled as a pipeline supported across a span 2L and struck by a solid wedge indenter. This particular arrangement could also be used to investigate the response of a horizontal thin-walled bracing member in an offshore structure when struck by the bow of a ship.

A quasi-static method of analysis was developed in Ref. 7 in order to predict the maximum permanent transverse displacement and extent of the damaged region for the ductile metal pipeline in Fig. 14. The static finite-deflection rigid-plastic analysis for the problem was obtained by assuming that the loading edge of the solid wedge-shaped indenter, which is perpendicular to the pipeline axis, flattens the contact zone of the initially circular pipeline cross-section. Thus, the pipeline cross-section on the impact plane is straight immediately underneath the striker with inextensional deformations following a new circular profile in the remain- der of the cross-section. The shape of the deformed pipeline cross-section is similar on other transverse planes on both sides of the impact plane except that the magnitude of the local deformations decreases with increase of axial distance away from the impact plane until the pipeline is undisturbed beyond a certain axial length which is predicted by the theoretical analysis. The damaged region on the loaded surface spreads both circumferentially and axially with increasing transverse displacements of the indenter. Eventually, the local denting deformations, which are described above, weaken the pipeline sufficiently for it to start deforming in a global or beam-like manner. For further loading, the pipeline then continues to deform with simultaneous local and global deformations.

A straightforward numerical procedure is used to obtain the static lateral force (P)-transverse displacement (W) relation (P(W)). Thus, the quasi-static maximum permanent transverse displacement for the impact case is given by j'o wq P(W) d W = GV2/2, where V0 is the impact velocity of the striker mass G in Fig. 14.

A comparison between the theoretical quasi-static predictions for the maximum permanent transverse displacements and the corresponding experimental results is shown in Fig. 15. This agreement is typical for the pipelines in Ref. 7 which have various diameter-to-thickness ratios between 11 and 60 and which were struck at the mid-span and one- quarter span positions. However, if the influence of material strain rate sensitivity for the mild steel specimens was taken into account then it would appear that the experimental results in Fig. 15 would be larger than the quasi-static predictions. This discrepancy may be due to the fact that the idealised boundary conditions in Fig. 14, which were used in the quasi-static analysis, may not have been achieved in the experimental tests, as discussed in Ref. 7. The mass ratios (G/pipeline mass) for the

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80 w,

(mm) 60

¢0

Z0

0 500 1000 1500 2OOO

Gv°2 CNml 2

Fig. 15. Compar ison of the theoretical quasi-static predictions for the maximum permanent transverse displacements of the pipeline in Fig. 14 with the experimental results on mild steel pipelines for impacts at the one-quarter span position. ( ) Theoretical quasi-static predictions 7 using the static yield stress (try) as the flow stress; (A) experimental results, 7 D/H = 30 and 2L/D = 10, where D = 60 mm is the outside diameter of the pipeline.

experimental results in Fig. 15 were 11-8 which increased to 16.9 for the three largest values of GV2/2. The energy ratios, when defined in a similar manner to eqn (7) ranged from 4.3 for the smallest values of Wf to 14.9 for the largest deformations.

6 Q U A S I - S T A T I C B E H A V I O U R O F AXIALLY L O A D E D T U B E S

The axial impact behaviour of thin-walled tubes has been studied exten- sively because the associated plastic deformation profile absorbs impact energy efficiently)' 13 A thin-walled tube collapses progressively from one end when subjected to a sufficiently large static axial load which produces plastic deformations. A single wrinkle, or buckle, develops and grows until sufficient external energy has been imparted to complete the formation and, then, for further energy input, another wrinkle begins to form in the immediately adjacent material. This progressive buckling process continues until all external work input has been absorbed plastically.

It turns out that the axial impact loading of thin-walled tubes produces a deformation pattern with the wrinkles developing from one end in a similar progressive manner to the static loading case. In fact, most of the theoretical studies published on the axial impact behaviour of thin-walled tubes hav¢ been developed using quasi-static methods of analysis. These ~theoretical methods usually neglect the variation of the axial force about a mean value which is caused by the cyclic change in resistance associated with the progressive formation of the wrinkles. Thus, the axial plastic


collapse force (Pro) is taken as constant and is equal to the mean value of the maximum and minimum forces associated with the development of a wrinkle. The plastic energy absorbed in crushing a tube is, therefore, Pm A, where A is the permanent axial crushing displacement. The theoretical quasi-static prediction for the permanent axial displacement (Aq) of a similar tube struck axially by a mass G travelling with an initial velocity Vo is obtained from the energy conservation relation P~ Aq = GV2o/2.

This quasi-static method of analysis 5 has given good agreement with experimental results which have been obtained for the axial impact loading of both circular and square tubes. The experimental results in Fig. 16 for some square steel tubes lie between the quasi-static theoretical predictions for perfectly plastic and strain-rate-sensitive materials. The mass ratios (G/mass of tube) for the specimens in Fig. 16 are large and range from 319 to 1473, while the impact velocities are smaller than 10 m/s, approximately. The energy ratios defined in a manner smilar to eqn (7) are also large and range from 266 to 447 for the specimens in Fig. 16.

Now, as the axial impact velocity is increased for circular tubes, the dynamic progressive (quasi-static) buckling discussed above gives way to a phenomenon known as dynamic plastic buckling. The wrinkles form and grow simultaneously throughout the entire length of a circular tube undergoing dynamic plastic buckling which is different from the deformation mode for the dynamic progressive buckling behaviour which is

p~ lk N )

3C

20

f f

1o

® ÷

÷ + + ÷

®

o ~ g Vo (m/s)

Fig. 16. The variation of the mean dynamic progressive buckling load (Pro) with the axial impact velocity (Vo) for thin-walled square steel tubes. ( O) Quasi-static rigid, perfectly plastic analysis; 5 ( ®) quasi-static rigid-plastic analysis with material strain

rate effects; 5 (+) experimental results. 5

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observed at the lower impact velocities and discussed above. The transverse inertia of the tube wall cannot be neglected in this case so that a complete dynamic analysis is required. 1'2

A recent experimental study ~4 has examined the transition with increasing impact velocity from the static buckling mode (progressive or Euler) to dynamic plastic buckling for aluminium alloy circular tubes struck axially by rigid masses. It was found for the particular tubular specimens in Fig. 17, that the quasi-static Euler buckling occurs for impact velocities up to about 12 m/s and that dynamic plastic buckling develops for velocities larger than 50 m/s, approximately. Mixed modes of dynamic progressive buckling and dynamic plastic buckling were observed for intermediate velocities. The mass ratios in these experiments ranged from over 500 for the quasi-static Euler buckling to about 30 for the transition to dynamic plastic buckling, as indicated in Fig. 17. A mass ratio of 30 is well within the range of validity of the quasi-static methods of analysis in Sections 2 and 3, but a quasi-static response was not obtained for the experimental tests reported in Ref. 14 with Vo > 50 m/s, approximately, as indicated in Fig. 17. It is evident, therefore, that the transition from the Euler buckling which is a quasi-static response, to dynamic plastic buckling depends on the impact velocity as well as the mass ratio.

The energy ratios are identical for all types of buckling along a given curve in Fig. 17 and equal to 105, approximately, for an energy input of 1 kJ. This energy ratio is defined in a similar way to eqn (7) and is equal to 0"5 GV2/(a2rcDLH/2E), where tr0 is the static yield stress, E is the elastic

G 31'6 ~ O k J (kg) 10

316 1-0 ~ o

0316 /

0"1 LO.8kJ/ 0-0316

001 0 SO 100 I'30 200 Vo(m/s)

Fig. 17. The effect of striker mass and striker velocity on the failure mode of aluminium alloy 6061 T6 cylindrical tubes loaded axially (H= 1-65 mm, outside diameter=25.4 mm, L = 101"6 mm, mass ~ 0.034 kg). Experimental results: 14 (n) Euler (global) buckling; (A) mixed dynamic progressive and dynamic plastic buckling; (O) dynamic plastic buckling.


modulus and D, L and H are the diameter, length and wall thickness of a tubular specimen. This energy ratio is obviously approximate, as noted in Section 3, and neglects the influence of material strain-rate sensitivity.

7 D I S C U S S I O N

This article has focused on quasi-static methods of analysis for a selection of common engineering structures struck by rigid masses travelling with an initial impact velocity. However, the impact velocities associated with quasi-static responses were relatively low in some cases and up to about 12 m/s for the axial impact of a tube which is equivalent to a drop height of 7.34 m (24-1 ft). The largest impact velocity which may be studied with a quasi-static method of analysis is uncertain but it depends on the mass ratio and on the type of structural problem.

It is evident from the experimental results in Fig. 17 for a particular axially loaded tube that inertia forces become important for impact velocities larger than about 12m/s and cause a transition to another response mode. However, the comparisons between the theoretical predictions in Figs 6 and 8 reveal that a quasi-static method of analysis is very accurate even for quite large impact velocities provided a beam is struck by a heavy mass. The theoretical predictions in Figs 6 and 8 do not contemplate any transition to alternative deformation modes. Neverthe- less, it is possible for this particular beam impact problem, which is shown in Fig. 1, that transverse shear effects ~ -3,15,16 may become important for' high impact velocities so that another deformation mode may intervene and render meaningless the results in Figs 6 and 8 for high impact velocities.

Engineering structures and components may be subjected to many different dynamic loadings besides the mass impact loadings which have been examined in this article. Various pressure-time histories may be generated during accidents as well as explosive or blast loadings. However., the quasi-static methods of analysis in this article cannot be used to study' the behaviour of structures subjected to pressure-time histories because the external energy input into a structure, which equals J'o J'A P dA dw for a pressure pulse p(t) distributed over an area A and having a duration r, is unknown and cannot be calculated until a dynamic analysis is available. Thus, these structural problems are inherently dynamic. It is clear that a pressure pulse with a given peak pressure could produce quite different permanent transverse deformations depending on the pulse length and the pulse shape. A pressure could be large enough to produce large permanent transverse displacements when applied statically, but, if the same pressure

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acted only for a short time, then the inertia forces would restrain the structural deformations so that there would be a very small energy input by the time that the pressure pulse was released leading to very small permanent transverse displacements.

It was noted in Refs 1 and 17 that the transverse velocity of a structure cannot be reversed in a simple rigid-plastic analysis (known as the phenomenon of unloading in the theory of plasticity). Thus, a rectangular- shaped pressure pulse with a duration longer than the time when dw/dt = 0 does not produce any further deformation and the structure remains rigid regardless of the pulse length. It was observed for this particular loading acting on a beam or rectangular plate that the maximum permanent transverse displacement is twice the corresponding static value when finite deflections are considered. 1'1v Incidentally, this factor agrees with the well-known result that the maximum displacement of an elastic spring- mass system is also twice the static value for long pressure pulses having a rectangular-shaped pressure-time history.18 Thus, for long pressure pulses a static analysis could be used with the maximum permanent transverse displacement multiplied by a factor the value of which depends on the pulse shape and on the geometry of a structural member.

Another special case occurs for the class of structural problems which undergo the phenomenon of pseudo-shakedown. L ~ 7.19 This phenomenon occurs when certain structures (e.g. beams, frames, plates, etc.) suffer finite deflection or geometry change effects when subjected to repeated dynamic pressure pulses which eventually produce a permanent transverse displacement profile. This permanent profile would be the same as that obtained if the peak value of the dynamic pressure pulse had been applied statically. Thus, a static analysis could be used to predict a pseudo-shakedown state.

A dynamic pressure loading having a high peak value and a short duration may be idealised as an impulsive velocity loading Vo. In this particular case, the initial external energy would be known. For example, if the fully clamped beam in Fig. 1 was subjected to a blast loading which could be idealised as a uniformly distributed impulsive velocity Vo, then the initial external kinetic energy is mLV~. If this kinetic energy was equated to the work done at the plastic hinges which would form at the mid-span and the supports for the transverse displacement profile in Fig. 7(b), then the maximum permanent transverse displacement is Wi = mL 2 vZ/4Mo, which is 50% larger than the exact value for infinitesimal displacements. ~ This difference occurs because travelling plastic hinges develop in a fully clamped beam when the magnitude of a rectangular pressure pulse is greater than three times the corresponding static collapse pressure. Two plastic hinges develop within the span during


the first phase of motion and remain stationary, while the pressure pulse is active. The distance of these plastic hinges from the mid-span is related to the magnitude of the pressure pulse and they form at the supports for an impulsive loading when the influence of the transverse shear force on plastic yielding is neglected. Two other stationary plastic hinges form at the supports. When the pulse is removed the two plastic hinges within the span then travel towards the mid-span and coalesce into a single hinge at the mid-span during a second phase of motion. The transverse displacement profile is then similar to that in Fig. 7(b) with stationary plastic hinges during a final phase of motion which ceases when all of the initial kinetic energy has been absorbed plastically. Mode form methods of analysis and various other procedures have been developed to assist with the simplification of these dynamic plastic problems. 1'2'2°

Now, as noted in previous sections, it is necessary to cater for the strengthening influence of material strain-rate effects even when the structural impact problem is studied using a quasi-static method of analysis. The values of the material constants used in the constitutive equations (e.g. Cowper-Symonds relation) are obtained frequently from strain-rate tests conducted with small strains, whereas the strains produced in many structural impact and structural crashworthiness problems are large. Thus, it is important to select carefully the material constants, or even use alternative constitutive equations (e.g. a modified Cowper-- Symonds relation 21), although the proper choice is hampered by a paucity of experimental data for large strains.

8 C O N C L U S I O N S

Various structural impact problems have been examined in this article using quasi-static methods of analysis and comparisons have been made with experimental results in most cases. The quasi-static analyses are much simpler than full dynamic plastic analyses but they do give good agreement with experimental results in which the striker masses are much larger than the corresponding structural mass. The impact velocities are relatively low and up to about 12 m/s for the axial impact of the tubes studied in this article.

It is not clear how high the impact velocities may reach before a quasi-static method of analysis becomes inaccurate. The range of validity is a function of the striker mass to structural mass ratio and the type of structure and may be truncated by the intervention of another deformation mode. Further studies are required to identify the range of validity for

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quasi-static methods of analysis so that guidelines could be produced for designers.

A C K N O W L E D G E M E N T S

The author wishes to record his appreciation to Dr W. Q. Shen from the Impact Research Centre at the University of Liverpool for assistance with several figures and to Mrs M. White and Mr H. Parker, both of the Department of Mechanical Engineering at the University of Liverpool, for their assistance with the preparation of this manuscript.

REFERENCES

1. Jones, N., Structural Impact. Cambridge University Press, Cambridge, UK, 1989.

2. Jones, N., Recent studies on the dynamic plastic behaviour of structures. Applied Mechanics Reviews, 42(4) (1989) 95-115.

3. de Oliveira, J. G. & Jones, N., Some remarks on the influence of transverse shear on the plastic yielding of structures. Int. J. Mech. Sci., 20 (1978) 759-65.

4. Liu, J. & Jones, N., Experimental investigation of clamped beams struck transversely by a mass. Int. J. Impact Engng, 6(4) (1987) 303-35.

5. Abramowicz, W. & Jones, N., Dynamic progressive buckling of circular and square tubes., Int. J. Impact Engng, 4(4) (1986) 243-70.

6. Jones, N., Liu, T., Zheng, J. J. & Shen, W. Q., Clamped beam grillages struck transversely by a mass at the centre. Int. J. lmpact Engng, 11(3) (1991) 379-99.

7. Jones, N. & Shen, W. Q., A Theoretical Study of the Lateral Impact of Fully Clamped Pipelines. Proc. Inst. Mech. Engrs., 206(3) (1992) 129-46.

8. Parkes, E. W., The permanent deformation of an encastre beam struck transversely at any point in its span. Proc. Inst. Civil Engrs, 10 (1958) 277-304.

9. Parkes, E. W., The permanent deformation of a cantilever struck transversely at its tip. Proc. Roy. Soc., London, 225(A) (1955) 462-76.

10. Shen, W. Q. & Jones, N., A comment on the low speed impact of a clamped beam by a heavy-striker. Mechanics, Structures and Machines, 19(4) (1991) 527-49.

11. Haythornthwaite, R. M., Beams with full end fixity. Engineering, 183 (1957) 110-12.

12. Yu, J. & Jones, N., Further experimental investigations on the failure of clamped beams under impact loads., Int. J. Solids and Structures, 27(9) (1991) 1113-37.

13. Johnson, W. & Reid, S. R., Metallic energy dissipating systems. Applied Mechanics Reviews, 31 (1978) 277-288; 39 (1986) 315-19.

14. Murase, K. & Jones, N., The transition from progressive plastic buckling to dynamic plastic buckling, Reports of the Faculty of Science and Technology, Meijo University, Nagoya, Japan, 31 (1991) 81-7. The variation of modes in


the dynamic axial plastic buckling of circular tubes. Plasticity and Impact Mechanics, ed. N. K. Gupta, Wiley Eastern, New Delhi, 1993, pp. 222-37.

15. Jones, N. & Song, B. Q., Shear and bending response of a rigid-plastic beam to partly distributed blast-type loading. J. Structural Mechanics, 14(3) (1986) 275-320.

16. Jones, N., On the dynamic inelastic failure of beams. In Structural Failure, ed. T. Wierzbicki & N. Jones, 1989, pp. 133-59.

17. Shen, W. Q. & Jones, N., The pseudo-shakedown of beams and plates when subjected to repeated dynamic loads, Trans. ASME, J. Applied Mechanics, 59(1) (1992) 168-75.

18. Biggs, J. M., Introduction to Structural Dynamics. McGraw-Hill, New York, 1964.

19. Jones, N., Slamming damage. J. Ship Research, 17(2) (1973) 80-6. 20. Symonds, P. S., Kolsky, H. & Mosquera, J. M., Simple elastic-plastic method

for pulse loading--comparisons with experiments and finite-element solutions. In Mechanical Properties at High Rates of Strain, ed. J. Harding. Inst. of Phys. Conf. Series No. 70, Bristol and London, 1984, pp. 479-86.

21. Jones, N., Some comments on the modelling of material properties for dynamic structural plasticity. In Mechanical Properties of Materials at High Rates of Strain, ed. J. Harding. Inst. of Physics. Conf. Series No. 102, Bristol and New York, 1989, pp. 43545.

Quasi-static analysis of structural impact damage

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