Structural response to transverse impact

loading. Some orders of magnitude

C. Casapulla

1 & A. Maione

1

1Department of “Costruzioni e Metodi matematici in Architettura”,

University of Naples “Federico II”, Italy.

Abstract

Structural response to transverse impact loading is generally regulated by many

parameters, such as the geometrical and mechanical properties of structural

elements and the amount of kinetic energy transferred to them. In this paper

some simple models of a cantilever beam subjected to impact loading are

analysed in order to characterise the influence of such parameters and to provide

some orders of magnitude. Firstly, by carrying out the analysis within elastic

field, a simple two-degree of freedom model is used for the prediction of the

structural response while retaining the global motion of the structure and the

local deformation in the contact region. Intermediate cases between the two

limiting ones generally accepted are also provided. Moreover, a rigid-plastic

model accounting for shear deformations is herein studied in order to evaluate

the transient and stationary phases of motion and to identify the critical values of

the parameters characterising different failure modes of the structure. Solutions

provided by other authors are discussed in terms of energy dissipation.

1 Introduction

Structural collapse under dynamic loading conditions may take place for several

reasons. Examples are the effects due to seismic loading with large impulsive

content (the so-called “near field” earthquakes) and exceptional actions such as

blast and impact, which are today of international interest.

In this paper the attention is restricted to the study of transverse impact problems

for ductile structures mainly aiming at providing a first classification of possible

dynamic responses depending on such characterising parameters as structural

properties and the initial kinetic energy of the system.

Generally speaking, the main difficulties in the analysis of transverse impact on a

ductile structure are:

1) to define the impact conditions when the transient phase can have a

dominant role with respect to the global structural response. In particular,

when the initial kinetic energy is much larger than the maximum possible

strain energy that can be absorbed elastically, elastic effects may be

disregarded and the study of the transient phase requires the analysis of

models with travelling and stationary plastic zones (i.e. plastic hinges for

beams), which are basically discontinuity interfaces ([1], [2], [3], [4], [5],

[6], [7]).

2) to identify the critical values of the parameters characterising different

failure modes of the structure. With regard to ductile beams, it is well

known [8] that three basic failure modes may take place under impulsive

loading: large inelastic deformation (Mode I), tensile tearing (Mode II) and

transverse shear failure (Mode III). In particular, within the Mode III, it is of

great importance to investigate the possible transverse shear propagation in

the element, both perfectly plastic and strain hardening plasticity models

being adopted ([9], [10], [11], [12], [13], [14], [15]).

3) to distinguish between local and global response to impact. The former is

generally represented by the well-known Hertzian local indentation model

and is characterised by dominant shearing effects. The latter, occurring

when bending effects are prevalent, is commonly studied in elastic field by

means of natural frequency analysis or, more generally and simply, by

means of single degree of freedom approximation ([16], [17]). The

intermediate cases require more complex analysis.

In this paper the above aspects of structural impact are discussed with reference

to the simple case of a cantilever beam struck at its tip by a mass moving at some

initial speed. However, as this model has largely been studied in literature ([1],

[2], [3], [7], [14]), the main contribution of the present work is to gather and

compare the different solutions and to propose simpler methods of analysis

aimed at providing some orders of magnitude of this kind of problem.

In particular:

a) in elastic field, a simple two-degree of freedom model, with two lumped

masses and two elastic springs, is herein used for the prediction of the

structural response while retaining the global motion of the structure and the

local deformation in the contact region. It is fully accepted [16] that the

problem depends on the mass of the colliding body in comparison to the

mass of the structure and the deformability of the contact region in

comparison to the structural compliance. The intermediate cases between

the limits described above are also included.

b) the rigid-plastic model accounting for shear deformations is herein studied

in order to evaluate the transient and stationary phases of motion. Solutions

provided by other authors ([1], [2]) are compared in terms of energy

dissipation and some orders of magnitude of critical parameters defining the

possible failure modes are evaluated according to existing theoretical rigid-

plastic models.

Ms

Meq

Kc

Keq

X2

X1

Ms, V0

2 Two-degree of freedom model

Transverse impact problems for a flexible structure generally involve dynamic

deformation of the structure in addition to local deformation of the contact

region. In some cases, the local deformation absorbs a significant portion of the

impact energy and structural vibrations are negligible. In other problems, the

global motion of the structure plays a major part in the response to impact. The

two limiting cases have already been described [16] as follows.

On the one hand, a light mass striking a stiff structure results in little motion of

the structure during impact. In this case the response to impact is accurately

represented by the Hertzian local indentation model for which the contact period

is brief, the contact force is large and the local deformation is predominant (there

is little energy lost to structural vibrations).

On the other hand, a heavy missile striking a compliant structure results in

substantial structural deformations that limit the contact force: consequently, the

indentation is small, the contact period is relatively long (roughly half the period

of the lowest mode) and the impact energy is mainly lost to structural vibrations.

In this case the response to impact can be represented by a single degree of

freedom approximation of the lowest mode shape of the structure, provided that

the impact conditions do not imply the inelastic behaviour.

Here, in order to investigate also the intermediate cases between the limit ones

described above, a simple two-degree of freedom model has been analysed to

represent the behaviour of a cantilevered beam subjected to transverse impact at

its tip by a rigid mass Ms moving at some initial speed V0 (Figure 1). This

mechanical model used for the prediction of the structural response is aimed at

reducing everything to a simple mathematical form, while still retaining the

essential physics of the real phenomena, such as the global motion of the

structure, the kinetics of the striking mass and the local deformation in the

contact region.

Accounting for impact conditions implying elastic response only, the

assumptions of elastic impact and elastic behaviour of the structure represented

by the two elastic springs (with stiffness Kc and Keq, respectively) are made.

On the other hand, the equivalency between the discrete model (represented by

the lamped mass Meq and the spring stiffness Keq) and the actual structure is

based on the Rayleigh-Ritz approximation of the fundamental mode shape of the

beam expressed as:

( ) ( ) ( )φωtsinxLxCx,tu −−= 323 (1)

with phase angle ϕ and amplitude C.

Figure 1: Two-degree of freedom model.

γ

µ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90 100

Keq/Kc=0.1

Keq/Kc=0.01

Keq/Kc=100 Keq/Kc=10

Keq/Kc=1

For this mode approximation the equivalent mass Meq, the spring stiffness Keq

and the modal frequency ωeq for free vibrations of the beam are, as is well

known:

eqeqeqeqeq /MKω;EI/LKM;.M === 332360 (2)

where M, EI and L are the global mass, the cross-sectional bending stiffness and

the length of the beam, respectively.

Thus, the equilibrium equations for the two masses can be written in this form:

( )( ) 0

0

122

1211

=+

=−+

-XXKXM

-XXKXKXM

cs

ceqeq

&&

&&

(3)

which can be solved with the classical modal frequency analysis, putting Vo as

initial condition for the velocity of the striking mass.

Basically, the parameters that influence the structural response are the energy,

mass and stiffness ratios, which are respectively defined as:

ceqeqsdk /KKκ;/MMμ;/EEη === (4)

where Ek is the initial kinetic energy of the striking mass and Ed is the maximum

elastic energy that could be stored in the beam.

The results, plotted in Figures 2, 3 and 4, are referred to the following

dimensionless variables:

)minc(c)maxc(ckcont /ttτ/FFλ;/EEγ === (5)

where Econt is the contact energy, Fc and tc are the contact force and the duration

of contact, respectively, while Fc(max) and tc(min) are the latter parameters referred

to the local deformation only.

First of all, the relation between local and global deformation does not depend on

the impact velocity but only on the mass and stiffness ratios. In fact, Figure 2

shows that, the smaller µ is and the stiffer the beam is in comparison to the

contact region, the larger is the contact energy absorbed. As predicted above, this

means that very stiff cantilevers struck by light masses will involve negligible

structural vibrations while the local indentation is prevalent. On the contrary,

when the structure is compliant in comparison to the contact region and the

striking mass is larger (say more than 10 times) than the beam mass, the global

motion will be mainly involved.

Figure 2: The contact energy as a function of the mass ratio.

τ

κ 0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

κ

λ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10 110 210 310 410 510 610 710 810 910 1010

Ek/Ed=0.01

Ek/Ed=0.1

Ek/Ed=1

Ek/Ed=10

However, it is worth noticing that if the initial kinetic energy is very large the

beam has to be stiff enough to avoid that the maximum elastic energy Ed is

exceeded. Therefore, if for example Ec/Ek=10, only Keq/Kc>10 must be

considered.

Figure 3: The contact period for variable stiffness ratios.

Even the contact period is not affected by impact speed. In fact, it only increases

when κ decreases, as shown in Figure 3, and is also independent of µ. At most,

when local deformation is negligible t→T/2 (T is the fundamental period of

structural vibration) and is very short for local deformation only.

On the other hand, impact velocities influence local indentations and

consequently contact forces. High-speed impact on stiff beams involves large

contact forces, independently of the mass ratio (Figure 4).

Figure 4: The contact force for variable stiffness ratios.

However, all these results provide a good approximation of the exact solutions

only when the global motion of the beam is prevalent, that is when the lowest

modes are involved (the correlation between the equivalent parameters of the

model and the real ones is based on the fundamental motion of the beam).

Therefore the outlined procedure is not rigorously valid for cases with a

prevalence of local effects which involve higher modes, since they are not

considered in the analysis.

3 Rigid-plastic models. Mechanisms of energy dissipation

As mentioned above, a rigid mass striking the tip of a cantilever beam causes the

genesis of deformation waves which transfer kinetic energy from the impact

zone to the whole structural element. It is well known that if kinetic energy

exceeds, by at least one order of magnitude, the maximum of energy elastically

storable, so that the rigid-plastic analysis is possible, the transient phase of

response involves the propagation of plastic deformations only.

The evaluation of the amount of plastic dissipation in the transient phase and of

the governing parameters is necessary in order to identify the conditions under

which a simplified analysis is possible, neglecting the aforementioned phase.

For impact loading conditions, the importance of the shear effects in the early

stage of the response is generally recognised. The analysis provided by Jones [1]

for a simplified yield condition, describes a plastic shear hinge as strong

discontinuity in transverse velocity at the cantilever tip. Instantaneously, a

stationary bending hinge is also localised at a fixed distance x0 = 3Mp/Tp. The

progress of bending deformation causes the acceleration of the portion of the

beam between the hinge and the tip, so that the discontinuity in velocity due to

shear vanishes in a relatively short time. At the end of this stage, the ratio

between the plastic work due to shear Ls and the initial kinetic energy K0 is

related to the ratio between the mass of the portion of beam subjected to

acceleration in the initial phase, that is mx0 (m is the beam mass per unit length)

and the striking mass Ms. In fact, it is:

( )mL/M

x

L

mx

MK

Ls

s

s =

+

=

+

= µµ

with2

1

1

21

1

00

0

(6)

L being the whole length of the beam.

If0

2

x

Lµ>>1, the shear dissipation Ls is negligible. This is either the case of strong

shear resistance (the limiting case Tp→∞, assumed by Parkes, as will be shown

later, implies L/x0→∞) or of large mass ratio µ. On the other hand, if 0

2

x

Lµ<<1,

Ls plays an important role. In particular, when strong shear dissipation occurs,

the larger L/x0 is (always assuming L/x0>1), the smaller is µ, associated with

increasing impact velocity.

After the closing-up of sliding hinge, the following phase is characterised by the

motion of the bending one toward the built-in end, where the residual kinetic

energy is dissipated through plastic rotation.

If infinite shear resistance is assumed, as in Parkes’ analysis [2], the only

mechanism of energy dissipation in the transient phase is that related to the

travelling hinge which propagates bending deformations. This is an unloading

process for the region on the plastic side of the hinge, as the motion occurs from

the rigid segment to a plastic zone.

0 10 20 30 40 50 60 70 80 90

100%

ν = 0 (Parkes) ν = 0.05 ν = 0.1 ν = 0.25 ν= 0.33

Sliding Bending (early stage) Bending (travelling hinge) Bending (modal phase)

If the two models are compared in terms of dissipation mechanisms, µ becomes

the main parameter that influences the importance of the transient phase with

respect to the modal one. Figure 5(a) shows that if the mass ratio is 10, then

about 90% of the kinetic energy is dissipated in the modal phase, irrespective of

the value of the shear resistance, whose dimensionless index is assumed to be the

ratio ν=Mp/(TpL).

The opposite situation occurs when the mass ratio is small (µ=0.1 in Figure

5(b)): in this case, in fact, the energy dissipation in the modal phase is less than

10%. The differences between the two models lie in the nature of the

mechanisms of dissipation involved in the transient phase (sliding rather than

bending) and they increase with lower values of shear resistance.

(a)

(b)

Figure 5: Dissipation mechanisms for different values of shear resistance.

(a): mass ratio µ=10; (b): µ=0.1.

It should be noted that the parameter ν for a rectangular cross-section is

connected with the ratio H/L so that small values of ν mean slender beams, while

large values of ν correspond to thick ones. From Figure 5(b) it results that for a

thick beam struck by a little mass (µ=0.1), the role of the plastic work due to

shear is very relevant. The limiting case is the one with ν = 0.33 (that is for x0=L)

corresponding to a thick beam with H/L=2/3, when the shear work is about 85%.

The prevalence of sliding deformation over the bending one results in a local

feature of transient response that requires the investigation of the conditions for

shear failure in order to verify the possibility of developing the subsequent phase

dominated by the propagation of the bending plastic hinge, with global

redistribution of deformations.

Basic failure modes of impulsively loaded beams have been identified since the

1973 by experimental investigation performed by Menkes and Opat [8]. These

0

10

20

30

40

50

60

70

80

90

100%

ν = 0 (Parkes) ν = 0.05 ν = 0.1 ν = 0.25 ν = 0.33

were classified as: large inelastic deformation (Mode I), tensile tearing (Mode II)

and transverse shear failure (Mode III). However, with reference to the Mode III,

Jones ([1], [10]) developed a simplified theoretical procedure to define the

threshold velocities required for a transverse shear failure, in good agreement

with the experimental results. According to this procedure, sliding permanent

deformations consist of a discontinuity ∆s in the vertical displacements

underneath the mass, as the length of the transverse shear hinge is neglected.

This elementary criterion, later developed also by Yu and Chen [13], implies that

the severance of the structure due to shear occurs when:

KHs =∆ (7)

where H is the thickness of the cross-section and 0 < Κ ≤1 is a constant that

generally depends on the mechanical properties, geometrical constraints and

loading features. For a wide range of impact conditions, however, experimental

measurements and theoretical studies allow to assume K=0.3.

For the cantilever beam under impact loading herein analysed, eqn (7) becomes:

KH

mx

MT

VM

sp

ss =

+

=

0

2

0

21

1

2∆ (8)

Thus, the threshold impact velocity sV0 can be found as a function of the

mechanical properties, mass ratio and shear resistance. In particular, assuming

2/AT sp σ≅ ( sσ is the yield stress), the critical velocity can be expressed as

follows:

+

=

−

µν

µν

ρ

σ

3

231

0H

LKV ss

(9)

where ρ is the material density and ν has been introduced above.

This elementary failure criterion is also related to the damage number defined by

Johnson [3] as ( ) s

sV σρλ2

0= and it is commonly used in order to judge the

severity of the impact. Thus, for the purpose of evaluating the influence of the

governing parameters, irrespective of the mechanical ones, a dimensionless

damage index can be introduced, which for a rectangular cross-section is

expressed by:

µ

µ+=

λ=

σ

ρ=α

−

5.1

2)H/L(5.1

KV

K

1s

s0

(10)

being ( ) ( )L/HLT/M pp 2≅=ν .

As is evident in Figure 6, for a rectangular cross-sectional beam, the damage

index defined by eqn (10) is essentially independent of the geometrical features

of the element when the mass ratio is µ >1. In this case, it will be related only to

the mechanical properties. Instead, with decreasing mass ratios, i.e. µ <1, the

dimensionless threshold velocity strongly increases together with decreasing

values of the ratio L/H. Thus, the dynamic behaviour of thick beams to impact

loading with high velocity is dominated by sliding deformations which can cause

a local failure in the early stage of the response, before the setting-up of a global

mechanism.

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

1.5 6.5 11.5 16.5 21.5 26.5 L/H

µ=0.05

µ=0.1

µ=1 µ=10

α

31.5

Figure 6: Damage index for a rectangular cross-section.

4 Conclusions

To the aim of providing some orders of magnitude for the dynamic response of

ductile structures under impact loading, a cantilever beam struck at its tip by a

mass moving at some initial velocity has been analysed in this paper, by means

of two simple models. Firstly, providing that the impact conditions imply the

elastic response only, a two-degree of freedom model has been used to evaluate

the parameters that influence both the local and the global response, such as the

stiffness, mass and energy ratios. The results have been shown in easy-going

curves that readily give the magnitude of the problem for all the cases included

between the two limiting ones described in the text. The second model herein

used is the classical rigid-plastic one, already studied by many authors. Different

mechanisms of energy dissipation, accounting for both bending and shear hinges,

have been investigated and it has been shown the great influence of the mass

ratios and shear resistance on the transient phase of motion. Properly graphs

highlight some solutions existing in literature in terms of energy dissipation and

threshold velocity of impact involving a transverse shear failure.

Acknowledgements

The authors wish to thank prof. P. Jossa for his scientific support and helpful

guidance throughout this work.

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