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International Journal of Impact Engineering 34 (2007) 1637–1654

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Reproducing kernel collocation method applied to thenon-linear dynamics of pipe whip in a plane

Amit Shawa, D. Roya, S.R. Reidb,�, M. Aleyaasinb

aDepartment of Civil Engineering, Indian Institute of Science, Bangalore 560012, IndiabDepartment of Engineering, University of Aberdeen, Fraser Noble Building, Kings College, Aberdeen AB24 3UE, UK

Received 16 March 2006; received in revised form 30 August 2006; accepted 2 September 2006

Available online 2 January 2007

Abstract

A windowed collocation method, based on a moving least squares reproducing kernel particle approximation of

functions, is explored for spatial discretization of the strongly non-linear system of partial differential equations governing

large, planar whipping motion of a cantilever pipe subjected to a follower force pulse (the blow-down force) normal to the

deflected centreline at its tip. This problem was discussed by Reid et al. [An elastic–plastic hardening–softening cantilever

beam subjected to a force pulse at its tip: a model for pipe whip. Proc R Soc London A1998;454:997–1029] where a

space–time finite difference discretization was employed to solve the governing partial differential equation of motion. It

was shown that, despite the deflected shape predictions being accurate, numerical solutions of these equations might

exhibit problematic (possibly spurious) steep localized gradients. The resolution of this problem in the context of structural

mechanics is novel and is the subject of this paper. In particular, it is demonstrated that it is possible to reduce significantly

such spurious and localized numerical instabilities through a windowed collocation approach with a suitable choice of the

window size. The collocation procedure presently adopted is based on the moving least squares reproducing kernel particle

method. Material and structural non-linearity in the beam (pipe) model is incorporated via an elastic–plastic-

hardening–softening moment–curvature relationship. The projected ordinary differential equations are then integrated in

time through a fifth order, explicit Runge–Kutta method with adaptive step sizes.

The whipping motion of a pipe is often associated with the formation of one or more so-called moving plastic ‘hinges’

(regions). Depending on the magnitude and duration of the follower load pulse, plastic hinges may form at intermediate

positions along the pipe. In such cases numerical solutions, especially those for curvatures and their derivatives, obtained

through central difference or classical finite element methods may exhibit considerable fluctuations in space and time and

may even ‘blow up’ following the formation of a kink at the location of the plastic hinge. However, it is shown that the

presently adopted windowed collocation strategy has a far superior numerical performance, arresting these fluctuations to

a great extent. Indeed, as demonstrated in this paper, there is a band of window sizes for which such fluctuations may

nearly vanish.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Pipe whip; Collocation methods; Travelling plastic hinges; Large deflections; Elastic-plastic-hardening-softening beams

ee front matter r 2006 Elsevier Ltd. All rights reserved.

mpeng.2006.09.004

ing author.

esses: royd@civil.iisc.ernet.in (A. Shaw), steve.reid@abdn.ac.uk (S.R. Reid).

ARTICLE IN PRESSA. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–16541638

1. Introduction

Modelling the dynamic response of non-linear structural systems often involves a numerical solution of thegoverning non-linear partial differential equations (PDEs) under prescribed boundary and initial conditions.When such solutions have a very ‘regular’ behaviour, virtually any of the traditional methods (e.g. Galerkin’sprojection via trial functions of polynomial or sinusoidal types, use of piecewise shape functions within aRayleigh–Ritz framework as in finite-element methods, collocation methods based on Lagrangianinterpolation functions, finite-difference approximations etc.) are good enough for an accurate computationof the response. However, in certain practical applications of structural dynamics, as in the pipe whip problemto be taken up in this study, it is necessary to take into account sharply localized ‘bursts’ or apparentdiscontinuities in the numerical solution. Such phenomena are observed particularly for problems underimpact/impulsive loading conditions or with strong material/structural elasto-plastic softening. Moreover,since such local variations develop under dynamic conditions, they may move along the spatial coordinates astime progresses, a typical feature of dynamic structural plasticity.

It is known that the use of a classical finite element meshing procedure for problems involving very largedeformations and/or sharp changes in the gradients is not appropriate [1]. Simple extensions of traditionalapproaches, such as a finite-element analysis with dynamical re-meshing [2], adaptive finite difference schemes,or mesh-free (spectral) methods [3,4], orthogonal collocations on finite elements [5] have mostly been found tobe insufficiently accurate and computationally inefficient. Moreover, a global support of the continuous basisfunctions in many of the spectral methods renders them vulnerable to Gibb’s phenomenon and a consequentloss of numerical stability as the number of terms in the functional representation increases, especially nearlocations of sharp gradient changes.

An attractive alternative is provided by element-free or mesh-free methods, which completely avoid therequirements of mesh generation for problems of complicated geometries or those with large deformations.The use of such an approach for one of these challenging problems is illustrated in this paper.

2. Background

Several mesh-free methods have been developed over the last two decades. For reference, these include thediffuse element method (DEM) by Nayroles et al. [6], the element free Galerkin method (EFG) by Belytschkoet al. [7], Barry and Saigal [8], Kaljevic and Saigal [9], Lu et al. [10], the partition of unity finite element method(PUFEM) by Babuska and Melenk [11], Melenk and Babuska [12], the h-p clouds by Duarte and Oden [13],the moving least-square reproducing kernel method (MLSRK) by Liu et al. [14,15], Liu et al. [26], Chen et al.[16,17], the mesh-less local boundary integral equation method (LBIE) by Zhu et al. [18], the smooth particlehydrodynamics method ([19]), the mesh-less local Petrov–Galerkin method (MLPG) by Atluri et al. [25] andseveral others. A notably common feature of all these methods is that only nodes characterize the domain ofinterest and consequently the approximating functions are determined entirely in terms of their nodal values.Han and Meng [20] have provided a detailed analysis of errors in the discretizations via reproducing kernelprinciples.

The present study attempts to adapt a moving least-square reproducing kernel particle collocation methodfor studying the short-time, transient dynamic response of an elastic–plastic-hardening–softening (EPHS)cantilever beam driven by a follower transverse force at its free end, i.e. pipe-whip. Due to elasto-plasticmaterial behaviour and/or section collapse in the presence of large displacements or rotations, plastic ‘hinges’(kinks or curvature discontinuities) may form dynamically along the deflected profile of the beam. As such, thepresent problem is prototypical of a moving (near-) singularity in structural dynamics. Unlike the sinccollocation approach employed for the pipe-whip problem earlier by Reid and Roy [21], the presentcollocation strategy uses compactly supported basis (shape) functions and it does not need any fictitiousextension of functional approximations beyond the boundary to satisfy the Dirichlet and Neumann boundaryconditions at the fixed and free ends, respectively. While the spatial discretization (projection) of the governingnonlinear, hyperbolic PDE is achieved through a reproducing kernel functional representation; the numericalintegration (in time) of the projected system of non-linear ODEs is performed using a fifth-order Runge–Kuttamethod with adaptive time-step sizes.

ARTICLE IN PRESSA. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–1654 1639

A sample set of numerical results is presented for the elastic–plastic dynamical response of a cantilever pipeand compared with the central difference strategy adopted earlier by Reid et al. [22]. These results indicate theclearly superior numerical performance of the present collocation method, especially in restricting spuriousspace–time fluctuations of the curvature profiles in regions of severe plastic deformations. Indeed, a ratherremarkable observation is that there is a small band of choice for the dilation parameter, which controls thesize of the moving window function in the collocation method, such that these fluctuations are virtuallyeliminated.

3. The reproducing kernel collocation method for the pipe-whip problem

3.1. Mathematical background

The reproducing kernel particle method (RKPM) has been developed and applied for large deformationanalysis of non-linear structures through a weak formulation [17]. The usual steps are to construct the weakform of the governing equations and to evaluate numerically the integrals by employing either a backgroundmesh over a course grid to generate quadrature points or by the trapezoidal rule. The use of a backgroundmesh does not make the approach truly meshless. On the other hand integration by using the trapezoidal rulemay produce inaccurate and unstable results and also requires the use of nodal volumes. This phenomenongave impetus to the development of point collocation methods using MLSRKPM shape functions. The keyidea in a point collocation approach is to satisfy the governing partial differential equation at each of thepoints covering the domain of interest. Let the domain O be bounded in Rn with boundary G. Then the givenproblem may be written as

Lu ¼ f in O, (1)

u ¼ g in Gg, (2)

qu

qh¼ h in Gh, (3)

where L is the (possibly nonlinear) differential operator, u ¼ u(x), xAO the unknown (response) function, f(x)the forcing function, Gg and Gh are, respectively, portions of the boundary where Dirichlet and Neumannconditions are specified. It is now required to satisfy Eq. (1)–(3) at each collocation (nodal/grid) point in thedomain with approximated ua(x) as

LuaðxiÞ ¼ f ðxiÞ in O, (4)

uaðxiÞ ¼ gðxiÞ in Gg, (5)

quaðxiÞ

qh¼ hðxiÞ in Gh. (6)

Introducing the discrete approximation of ua(x), the point collocation method finally gives rise to thefollowing system of (possibly nonlinear) discretized equations:

cðUÞ ¼ b, (7)

where, C is a discretized algebraic (and possibly nonlinear) operator on the Np-dimensional vectorU ¼

Dui ¼ uðxiÞ : i 2 ½1;Np�� �

, which is the discretized response vector, and b is the known right-hand sidevector corresponding to the discretization of f(x). Once ui, i ¼ 1,2,y,Np are computed from Eq. (7), ua at anypoint in the domain (including the nodal points) may be computed as

uaðxiÞ ¼XNp

j¼1

cjðxiÞuj. (8)

When the differential operator L is linear, the algebraic operator C(U) is also linear with respect to theargument vector and thus Eq. (7) may be written as KU ¼ b, K 2 RNp�Np being the coefficient matrix. The

ARTICLE IN PRESSA. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–16541640

matrix K is generally non-symmetric in a point collocation method. Since the above approach does not requireany integration over any sub-domain and generates a strong solution of the governing equations, the issue of abackground mesh does not arise. Finally, it is also noted that if L is a differential operator involving both x

and time, t, then the discretized equationC(U) ¼ b would be a system of ordinary differential equations and U

may be found using an appropriately chosen numerical integration scheme.

3.2. Governing equations for cantilever pipe whip

The MLSRKPM is applicable to a large class of linear and nonlinear problems in structural mechanics.However the primary objective of this article is to apply and explore the method in the context of highly non-linear motions of a tubular cantilever beam under a follower force. As in [22], this problem is modelled using abeam theory with hardening and softening bending constitutive equations. The coupled equations of motionfor the axial and transverse displacement components, u(x,t) and w(x,t) is given as [22]

m u::¼

qqsðN cos yÞ þ

qqsðQ sin yÞ þ px, (9)

m w::¼

qqsðN sin yÞ �

qqsðQ cos yÞ þ py, (10)

where m is the mass per unit length, px(s,t) and py(s,t) are spatially and temporally varying, external appliedforce densities along the two Cartesian co-ordinate directions, and dots denote time derivatives. Moreover,Q(s,t) is the shear force at a section defined by the co-ordinate variables, measured tangentially along thedeflected shape of the beam centreline at time t, and related to bending moment M(s,t) by

Q ¼qM

qs. (11)

y(s,t) denotes the angle made by the deformed beam centreline with respect to the positive X-direction. Theseare shown in Fig. 1. The spatial co-ordinate variables x and s are related by

ds

dx¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðw0Þ2 þ ð1þ u0Þ2

q¼ qðxÞ, (12)

where w0 ¼ dw/dx and u0 ¼ du/dx. The quantities y, sin y and cos y in Eqs. (9) and (10) are expressed in termsof u and w by

sin y ¼w0

q; cos y ¼

1þ u0

q; y ¼ tan�1

w0

1þ u0

� �. (13)

The axial strain, e0 and the curvature, k, of the centre line of the deformed beam element can be expressed interms of u and w by

�0 ¼ds� dx

dx¼ qðxÞ � 1, (14)

k ¼dyds¼ð1þ u0Þw00 � w0u00

q3. (15)

If the Euler–Bernoulli assumption of plane sections remaining plane is adopted, strain at any point, which is adistance z away from the centroidal axis of the cross-section, is given by

� ¼ �0 þ zk. (16)

3.3. Constitutive relations: material/structural non-linearity

An EPHS relationship was proposed by Reid et al. [22] to describe the material/structural non-linearitycharacteristic of a pipe. It is assumed that the non-linear M–k relations remain unaffected by changes in the

ARTICLE IN PRESS

initial beam profile at t = t0

G (tip mass)

beam profile at t> t0

Q(t)

Q(t)

pX

pW

X

G

W

XO

W

Q

MN

pX

ds

pWM

Q

+

+

M

sQ

s

ds

N + ds

ds

�

(a)

(b)

N

s

Fig. 1. (a) A cantilever beam with a lumped end mass, G, loaded by a follower shear force Q(t) at the free end; and (b) an infinitesimal

beam element in the deformed configuration.

A. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–1654 1641

axial force N, even in the softening range. While this is done for purposes of simplicity, it turns out to be areasonable approximation for tubular beams wherein the maximum absolute value of N remains very muchbelow its elastic limit Ne. Thus the EPHS M–k relations takes the general form

M ¼ F ðkÞ ¼

f eðkÞ koke;

f hðkÞ keokokcr; Dk40;

f sðkÞ kcrokokcol; Dk40:

8><>: (17)

A typical M–k curve, valid for the first loading stage, is shown in Fig. 2. Here the terms ‘loading’ atany section of the beam are taken to imply an increase in the value of k with time, i.e. _k40. As is clearlyseen, the M–k curve consists of four different sections. These correspond to the initial elastic deformation.This is followed by a hardening phase, which is dominated by the material plastic hardening. Forincreased curvature, this is followed by a plastic softening phase representing the effects of the ovalization andcollapse of the pipe section. Beyond that, the M– k relation reduces to a flat line parallel to the k-axis andthis corresponds to the residual moment carrying capacity of the collapsed beam section. An explicitform of the function F(k) may be determined from a four-point bending experiment. Alternatively, for thicktubular beams that soften gradually, the curve may be obtained from an elastic–plastic numericalanalysis based on shell equations as described in Reid et al. [22]. Herein, the following form of the function

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M

Md

Me

Mmax

Mr

unloading curve

�e �d �cr �col �

Fig. 2. A typical elastic–plastic hardening–softening (EPHS) moment–curvature relationship following Reid et al. [22].

A. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–16541642

F(k) is made use of [23]:

M ¼ F ðkÞ ¼

EIk koke;

Me þ Aðk� keÞ � Bðk� keÞ2 keokokcr; Dk40;

Mmaxf1� aðk� kcrÞ2 kcrokokcol; Dk40:

8><>: (18)

In the above equations, A, B and a are constants obtained via a best fit of an experimentally or numericallydetermined M– k curve. Constants A, B satisfy the conditions

Mjk¼kcr¼Mmax and

dM

dkjk¼kcr

¼ 0. (19)

This leads to

A ¼2ðMmax �MeÞ

kcr � ke; B ¼

Mmax �Me

ðkcr � keÞ2. (20)

The softening parameter, a, affects the local slope of the softening branch of the M– k curve and is thusresponsible for determining the post-peak-load moment carrying capacity of the beam section. The thinner thepipe, the higher is a and the steeper is the fall of the softening branch

The discussion above is valid only for the loading states, i.e. _k40. Since the present study deals with thedynamical response of a pipe, M– k relations during the unloading states, i.e. _ko0, need to be duly accountedfor. As seen in Fig. 2, the elastic unloading curve is considered to be parallel to the loading one and reversedyielding follows the dashed line. Thus, assuming isotropic plastic behaviour, the M– k relations duringunloading takes the form

M ¼ F ðkÞ ¼

Md þ EIðk� kdÞ koke;

�f hðkd þ jk� kejÞ keokokcr; Dk40;

�f sðkd þ jk� kejÞ kcrokokcol; Dk40:

8><>: (21)

In the above equations, kd and Md are, respectively, the curvature and bending moments of the beam sectionat the instant when unloading is initiated, and kg ¼ kd�2Md/EI is the reverse yielding curvature. The

ARTICLE IN PRESSA. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–1654 1643

curvature at collapse, kcol may be obtained from the equation of the softening branch as

kcol ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMmax �Mr

aMmax

r. (22)

4. Discretization and the numerical technique

A spatial discretization of the governing PDEs is performed based on the MLSRKPM [15] interpolationfunction. Following this, any sufficiently smooth real valued function u(x) may be approximated as

uaðxÞ ¼XNP

i¼1

Cðx� xiÞHTðx� xiÞfaðx� xiÞui, (23a)

where for any multi-index notation a, HT(x�xi) ¼ {(x�xi)a}|a|pp is a set of monomial basis functions, aiAR+

is the dilation parameter associated with the ith node (particle), ffi¼D fðx� xiÞg is a set of finitely and

compactly supported basis functions (also called window or weight functions) with fi centred at xi,fa(x) ¼ f(x/a) is the a-dilated basis function, C(x�xi) the correction function, p the degree of polynomial andNP the number of particles. In this paper, the correction function CðziÞ (where zi ¼ x� xi) has been taken asthe following polynomial of degree 4:

CðziÞ ¼ b0ðxÞ þ b1ðxÞðziÞ þ b2ðxÞðziÞ2þ b3ðxÞðziÞ

3þ b4ðxÞðziÞ

4. (23b)

This is done recognizing that the non-linear equations of motion for beams (Eqs. (9) and (10)) do notcontain higher than fourth order derivatives in u and w with respect to x. For the ith node, the non-linearhyperbolic equations of motion may be written in terms of the independent variable x as

mu::

i ¼AE

q2i

qi;x 1þ u0i� �

þAE

q2i

ðqi � 1Þ u00 �1þ u0i

qi

qi;x

� �þ

1

q3i

Mi;xxw0i �1

q4i

Mi;xqi;xw0i

þ1

q3i

Mi;xðw00i �

w0iq

qi;xÞ, ð24Þ

mw::

i ¼AE

q2i

qi;xw0i þAE

q2i

ðqi � 1Þ w00i �w0iqi

qi;x

� ��

1

q3i

Mi;xxð1þ u0iÞ þ1

q4i

Mi;xqi;x 1þ u0i� �

�1

q3i

Mi;x u00i �1þ u0i

qqi;x

� �, ð25Þ

for i ¼ 1,y,NP.Using Eqs. (17)–(22), the first and second derivatives of the function M(x,t) are expressed as

Mi;x ¼ F i;kki;x, (26)

Mi;xx ¼ F i;kkðki;xÞ2þ F ;kki;xx, (27)

where ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðw0iÞ

2þ ð1þ u0iÞ

2q

, (28)

qi;x ¼w0iw

00i þ ð1þ u0iÞu

00i

qi

, (29)

qi;x ¼w0iw

000i þ ðw

00i Þ

2þ ðu00i Þ

2þ ð1þ u0iÞu

000i

q2i

�q2

i;x

qi

, (30)

ARTICLE IN PRESSA. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–16541644

k;x ¼ð1þ u0iÞw

000i � w0iu

000i

q3i

�3

q4i

qi;x ð1þ u0Þw00i � w0iu00i

, (31)

ki;xx ¼ð1þ u0iÞw

0000 þ u00iw000

i � w00iu000

i � w0iu0000

i

q3i

�6

q4i

qi;x ð1þ u0iÞw000

i � w0iu000

i½ �

�3

q4i

qi;xx ð1þ u0iÞw00

i � w0iu00

i½ � þ12

q5q2

i;x ð1þ u0iÞw00

i � w0iu00

i½ � ð32Þ

Spatial derivatives of independent variables u and w can be expressed as

u0i ¼XNP

j¼1

Nj;xðxiÞuj ; w0i ¼XNP

j¼1

Nj;xðxiÞwj, (33)

u00i ¼XNP

j¼1

Nj;xxðxiÞuj ; w00i ¼XNP

j¼1

Nj;xxðxiÞwj, (34)

u000i ¼XNP

j¼1

Nj;xxxxðxiÞuj ; w000i ¼XNP

j¼1

Nj;xxxðxiÞwj, (35)

u0000i ¼XNP

j¼1

Nj;xxxxðxiÞuj ; w0000i ¼XNP

j¼1

Nj;xxxxðxiÞwj (36)

Shape functions and their derivatives can be obtained following the same manner as in Liu et al. [14,15]. It isfurther noted that the following identity for axial force has been used while deriving Eqs. (24) and (25).

Ni ¼ AEðqi � 1Þ, (37)

where N denotes the total axial force, independent of z. This approximation is acceptable as long as themagnitude of the axial force remains far smaller than the axial force Ne at the elastic limit.

For the end-loaded cantilever in Fig. 1(a), one has pxi ¼ pyi ¼ 0. The PDEs in Eqs. (24) and (25) are subjectto the following boundary and initial conditions:

u1ðtÞ ¼ w1ðtÞ ¼ w01ðtÞ ¼ 0;

MNPðtÞ ¼ 0; QNP

ðtÞ ¼ P;

((38)

uið0Þ ¼ u:ið0Þ ¼ wið0Þ ¼ w

:ið0Þ ¼ 0. (39)

For purposes of numerical integration of the non-linear ODEs, an adaptive fifth order Range–Kutta schemehas been adopted in the present study.

5. Numerical results

5.1. Cantilever pipe under a follower force

The results presented below are all in the context of large, short time (transient) dynamic deflections oftubular cantilever beams subjected to a follower force at its free end. In this example, the magnitude of thefollower force Q(t) ¼ Q is assumed to be constant in all the numerical work to follow. The magnitude of theforce is equal to 8200N, which is the mean value of an actual blow-down force pulse measured in a pipe-whiptest. The structural and material properties given in Table 1 have been used for all numerical calculations. Forthe pipe described in Table 1 the maximum plastic bending moment Mmax may be obtained via the algebraicrelation MmaxE0.96Mp and kcr is approximately given by kcr ¼ 0:9m�1.

Reid et al. [22] obtained numerical solutions of the EPHS pipe whip problem under a constant followerpulse at the free end using an explicit central difference scheme, applied along both space and time

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Table 1

Typical pipe parameters from pipe-whip test ([22]Reid et al. 1998)

Sr. no. Parameter Notation Value (with units)

1 Outer diameter of the pipe D 0.0508m

2 Length of the pipe L 2.73m

3 Mass per unit length of the pipe m 2.022 kgm�1

4 Wall thickness H 0.00158m

5 Young’s modulus E 200GPa

6 Yield stress sy 295MPa

7 Ultimate stress sb 420MPa

8 Maximum elastic bending moment Mc 672Nm

9 Fully plastic bending moment Mp 1142Nm

10 Maximum elastic curvature ke 0.0448m�1

11 Critical curvature kcr 0.9m�1

12 Second moment of area of the cross-section about the neutral axis I 7.5� 10�8m4

13 Tip mass G 1.04 kg

-2

-1.6

-1.2

-0.8

-0.4

00 0.5 1 1.5 2

x (m)

CD

RKPM

t = 10 ms

t = 20 ms

t = 30 ms

t = 40ms

2.5

Fig. 3. Snapshots of the dynamic deflection profiles of a cantilever beam via central difference (CD) and point collocation based on

MLSRKPM shape function with a ¼ 0.001, L ¼ 2.73m, Q ¼ 8200N, ke ¼ 0.0448m�1 and a ¼ 1.2Dx.

A. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–1654 1645

co-ordinates. Procedures presented in that study have been verified against experimental results and hencethey serve as an important benchmark for a numerical validation of the present algorithm. Accordingly it isworthwhile to compare the results of the present approach with some of those by Reid et al. [22].

5.2. Discussion on the deformation plots

Consider a cantilever beam of length 2.73m, Q ¼ 8200N, a ¼ 0.001 and ke ¼ 0.0448m�1. Fig. 3 showssnapshots of deflected shapes of the beam at t ¼ 10, 20, 30 and 40ms predicted by the point collocation basedon the MLSRKPM approximation and the central difference approach. In Figs. 4(a)–(c), comparisons ofsnapshots of curvature profiles, obtained for both the MLSRKPM collocation and the central-differencemethods, are presented. A constant value of the dilation parameter a ¼ 1.2Dx has been chosen for theMLSRKPM collocation method.

While deflection snapshots predicted by both the methods appear to match very well, this is not the case forthe curvature profiles, especially when parts of the beam have entered the softening regime. From acomparison of the curvature profiles for t415ms, it is clear that the central difference strategy results in

ARTICLE IN PRESS

0 0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2 2.5 3

x (m)

CD (t=5 ms) RKPM (t=5 ms)CD (t=10 ms) RKPM (t=10 ms)CD (t=15 ms) RKPM (t=15 ms)CD (t=20 ms) RKPM (t=20 ms)CD (t=25 ms) RKPM (t=25 ms)kcr

κcr = -0.9 m-1

κcr = -0.9 m-1

κcr = -0.9 m-1

(a)

-3.1

-2.6

-2.1

-1.6

-1.1

-0.6

-0.1

0.4

x (m)

κ (m

-1)

-3.1

-2.6

-2.1

-1.6

-1.1

-0.6

-0.1

0.4

κ (m

-1)

-3.1

-2.6

-2.1

-1.6

-1.1

-0.6

-0.1

0.4

κ (m

-1)

CD (t=35 ms)

RKPM (t=35 ms)

kcr

(b)

x (m)

CD (t=50 ms)

RKPM (t=50 ms)

kcr

(c)

Fig. 4. Snapshots of the dynamic curvature profiles of a cantilever beam via central difference (CD) and point collocation based on

RKPM shape function with a ¼ 0.001, L ¼ 2.73m, Q ¼ 8200N, ke ¼ 0.0448m�1; (a) t ¼ 5–25ms; (b) t ¼ 40ms; (c) t ¼ 50ms and

a ¼ 1.2Dx.

A. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–16541646

considerably more fluctuations of the curvature graph over the region of acute plastic deformation. With thepresent value of the dilation parameter, these fluctuations are arrested to a great extent by the MLSRKPMcollocation and it is thus safe to observe that such fluctuations are manifestations of local instabilities(ill-conditioning of the Jacobians of local transformations) in the numerical algorithm.

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As the curvature at or around a section approaches and/or exceeds kcr corresponding to the maximum valueof bending moment (Mmax in Fig. 2), the material resistance to flexure at such sections reduces, while theresistance remains considerably higher at sections far away (i.e. sections which are still experiencing hardeningor elastic material behaviour). At this stage, if one linearizes the (projected) nonlinear ODEs about a typicalsuch response function, the corresponding system matrix would obviously be ill-conditioned. Thus thetemporal propagation of numerical errors over the softening region would be far higher than that elsewhere.Indeed, this observation encourages the adoption of an explicit Runge–Kutta integration strategy inpreference to an implicit strategy (such as the Newmark method), which would necessitate the inversion oflinearized system matrices over each time step. Thus, as structural degradation increases with time, the extentand levels of the spurious fluctuations also increase noticeably.

It is noted that the notion of a ‘‘traveling hinge’’ which has been described by Jones [23] and others in the analyticalsolutions of certain small deflection rigid-plastic beam problems is evident herein. Both the MLSRKPM and central-difference methods predict a movement of the plastic zone (‘hinge’) towards the fixed end (root) as time progresses.

The relative performances of the two methods in dealing with local instabilities are further brought out inFigs. 5 and 6 that show the effects of discretization on computed curvatures and their derivatives. It is readily

-2.5

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curv

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CD

RKPM

(b)

-3

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-1

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0

0.5

CD

RKPM

(c)

Fig. 5. Snapshots of the dynamic curvature profiles of a cantilever beam via central difference (CD) andMLSRKPM interpolation scheme

for different discretizations; (a) Ntot ¼ 40; (b) Ntot ¼ 60; (c) Ntot ¼ 80 with a ¼ 0.001, L ¼ 2.73m, Q ¼ 8200N, ke ¼ 0.0448m�1 and

a ¼ 1.2Dx.

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5

10

15

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x (m)

der

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RKPM

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2

0 0.5 1 1.5 2.5 32

x (m)

0 0.5 1 1.5 2.5 32

Fig. 6. Snapshots of the profile of derivative of curvature k0 of a cantilever beam via central difference (CD) and MLSRKPM

interpolation scheme for different discretizations; (a) Ntot ¼ 40; (b) Ntot ¼ 60; (c) Ntot ¼ 80 with a ¼ 0.001, L ¼ 2.73m, Q ¼ 8200 N,

ke ¼ 0.0448m�1 and a ¼ 1.2Dx.

A. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–16541648

observed that the central-difference method exhibits higher levels of spurious fluctuations in the computedapproximations with respect to the MLSRKPM collocation as the spatial step size decreases (i.e., as the totalnumber of nodes Ntot increases), especially in regions of acute plastic deformations. This loss of accuracy,associated with the high frequency bursts, is traceable to the well-known Gibbs phenomena in Taylor orFourier expansions.

As is clearly verified in the results presented above, locally unstable and numerically spurious fluctuations inthe computed response tend to amplify as the number of discretization points as well as the order ofderivatives in the shape functions increase. While these results establish the MLSRKPM collocation as asignificantly better numerical tool to reduce locally developed ill-conditioning in the present context, thefluctuations have not yet vanished.

5.3. Effects of the dilation parameter and the number of particles on convergence

In all the numerical work above, the dilation parameter a has been taken as 1.2Dx. A question that naturallyarises here is whether it is possible to use a as a control parameter to adjust the window sizes of the compactly

ARTICLE IN PRESSA. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–1654 1649

supported basis functions so that spurious oscillations in the computed solution are either absent or minimallypresent. While a mathematical framework to establish an optimal parameter value is yet to be developed, it isclearly seen from above that an adjustment of the value of a can achieve what is desired. This is adequatelybrought out in Figs. 7–10 with an increased value of the dilation parameter at a ¼ 2.5Dx. Figs. 7(a)–(f) showthe convergence of snapshots of the curvature profiles at t ¼ 40 and 50ms with an increasing Ntot. It is clearthat with the present value of the dilation parameter, the locally arising Gibbs phenomena are nearlyeliminated irrespective of the choice of Ntot as well as the level of material degradation. At t ¼ 50ms,convergences in snapshots of the curvature profiles with increasing Ntot via both the central-difference method

CD

RKPM

CD

RKPM

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t = 40 ms t = 50 ms

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t = 40 ms t = 50 ms

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RKPM

t = 40 ms t = 50 ms

2 0 0.5 1 1.5 2.5

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2

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2

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2

0 0.5 1 1.5 2.5

x (m)

2

(a) (b)

(c) (d)

(e) (f)

Fig. 7. Snapshots of the curvature profile of a cantilever beam via central difference (CD) and RKPM interpolation scheme for different

discretizations; (a), (b) Ntot ¼ 40; (c), (d) Ntot ¼ 60; (e), (f) Ntot ¼ 80 with a ¼ 0.001, L ¼ 2.73m, Q ¼ 8200N, ke ¼ 0.0448m�1 and

a ¼ 2.5Dx.

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-1)

Np = 20

Np = 40

Np = 60

Np = 80

Np = 20

Np = 40

Np = 60

Np = 80

(b)

Fig. 8. Convergence in curvature profile with varying number nodes for a cantilever beam with a ¼ 0.001, L ¼ 2.73m, Q ¼ 8200N,

ke ¼ 0.0448 m�1, Dt ¼ 10 ms, t ¼ 50ms, 2.5Dx, (a) central-difference method and (b) point collocation based on RKPM.

A. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–16541650

and the MLSRKPM collocation are presented in Fig. 8. It is clear from Figs. 7 and 8 that the MLSRKPMcollocation, as compared with the central-difference method shows better convergence with increasing Ntot

without the problem of spatially localized ill-conditioning of the tangent system matrices.

5.4. Relative numerical errors

It is also of interest to have an understanding of the relative errors while applying a numerical method withincreasing number of nodes. Four different levels of discretization have presently been considered, viz. level 0with Ntot,0 ¼ 20 nodes; level 1 with Ntot,1 ¼ 40 nodes; level 2 with Ntot,2 ¼ 60 nodes; and level 3 withNtot,3 ¼ 80 nodes. The snapshot of the relative error for two different levels may be defined asdij(x) ¼ |ua(x,Ntot,i)�ua(x,Ntot,i)|, where ua(x,Ntot,i) denotes the approximation to the function u(x,t) throughthe numerical method with Ntot ¼ Ntot,i for a fixed t with i,jA[0,3]. The relative error snapshots d01(x), d12(x),and d23(x) for the central difference method and the MLSRKPM have been plotted in Figs. 9(a), (b), 10(a) and(b), respectively, at t ¼ 40 and 50ms.

While the relative errors in the MLSRKPM collocation are, in general, much smaller than those in thecentral-difference method, it is also observed that choosing a very small spatial step size may result in a loss ofaccuracy. However, even in such cases, the MLSRKPM does not show any numerical instabilities if thewindow size is suitably adjusted.

5.5. Would arbitrarily large dilation parameters work?

Another point that needs to be borne in mind for problems with such extreme and localized material/structural non-linearities is that the dynamics at or near a location with high plastic deformations occur quitedifferently and nearly independently of another location far away from it (another feature of a moving-boundary

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abso

lute

err

or

d01

d12

d23

(b)

Fig. 9. Relative errors in curvature snapshots by MLSRKPM collocation method for different discretizations of a cantilever beam with

a ¼ 0.001, L ¼ 2.73m, Q ¼ 8200N, ke ¼ 0.0448m�1, Dt ¼ 10ms, a ¼ 2.5Dx, (a) t ¼ 40ms and (b) t ¼ 50ms.

A. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–1654 1651

problem). This observation is at variance with the usually accepted premise that the dynamics of anytwo spatially separated points in the deterministic (non-random/non-stochastic) vibration of a structuralelement are perfectly correlated, i.e. given a numerical scheme of solution as well as initial and boundaryconditions, the response of one point uniquely determines that of the other. This is essentially the resultof the invertibility (non-singularity) of the system matrix obtained as a result of discretization. Thecompact supports of the basis functions in the MLSRKPM collocation essentially helps in enforcing thisinvertibility while accounting for a possible loss of dependence in the dynamics of two points well separated inspace. It might therefore be anticipated that there could be a loss of accuracy of the numerical solution as thedilation parameter is increased arbitrarily, especially when portions of the beam undergo large plasticdeformations.

That this is indeed the case is brought out in Fig. 11, wherein snapshots of curvature plots areprovided at t ¼ 50ms for different sizes of the dilation parameter a, starting from a ¼ 2.5Dx. It is clear that ahigh value of a introduces a spurious smoothness in the curvature profiles and the numerical solutioncompletely misses the peaks. Moreover, it may erroneously predict the formation of a second plastic hinge atthe tip, which is not consistent with either the physics of the problem or the experimental observationsreported by Reid et al. [22].

Following the work of Han and Meng [20], it is observed that the errors in the MLSRKPM functionalapproximation for the displacement variables is presently of the order of O(rl), where r the characteristicsupport size of the window functions, as defined in Han and Meng [20] and l ¼ 4 is the polynomial degree of

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abso

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abso

lute

err

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d01

d12

d23

(b)

Fig. 10. Relative errors in curvature snapshots by the central-difference method for different discretizations of a cantilever beam with

a ¼ 0.001, L ¼ 2.73m, Q ¼ 8200N, ke ¼ 0.0448m�1, Dt ¼ 10ms, 2.5Dx, (a) t ¼ 40ms and (b) t ¼ 50ms

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x (m)

curv

atu

re (

m-1

)

a = 2.5dxa = 10dxa = 20dxa = 30dx

Fig. 11. Comparison of curvature profile for different values of the dilation parameter a at t ¼ 50ms.

A. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–16541652

the spline function. It is then conceivable that errors in temporal integration (via a fifth order Runge Kuttamethod with a time step Dt5r) are negligibly small with respect to the spatial discretization errors.Accordingly, no numerical results on the convergence of time histories of a given point have been provided inthis study.

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6. Concluding remarks

A moving-least-square-reproducing-kernel-particle method (MLSRKPM) was adapted for the spatialdiscretization via collocation of the highly non-linear partial differential equations (PDEs) representingelastic–plastic non-linear motion of a cantilever pipe under a follower load at its tip. Numerical integration ofthe system of the projected ordinary differential equations (ODEs) was subsequently carried out using a fifthorder Runge–Kutta scheme with a variable step size. Due to possible plastic structural softening of the pipe,solutions of such PDEs may not have a regular behaviour and steep gradient changes in the response and, inparticular, its derivatives are not uncommon.

The MLSRKPM collocation adopted in this study was shown to be well suited to negotiate such problems.Fifth order spline functions were employed to obtain the compactly supported basis functions whose supportsizes may be controlled through a dilation parameter. These basis functions are not interpolating, butconstitute a partition of unity. The material/structural non-linearity in the cantilever beam was incorporatedvia an elastic–plastic-hardening–softening (EPHS) moment–curvature relationship, established earlier by Reidet al. [22]. A typical phenomenon associated with the deformation of such beams is the dynamic formation ofplastic ‘hinges’ (plastic zones). For a large magnitude load pulse acting uniformly (in time) at the tip,deformation was generally seen to occur via the formation of a hinge at an intermediate position in the beam.However, following its formation, the plastic region tends to move towards the fixed end (root).

In a previous investigation, Reid et al. [22] employed a central-difference scheme both in space and time tointegrate the governing PDEs. Since the curvature profile in the plastic region may have highly localized peaks,the central-difference technique shows numerical instability and a lack of convergence as the number of nodesfor spatial discretization increases. Moreover, as the plastic region grows in size and moves to the root as timeprogresses, use of a truly meshless method, as adopted in this paper, appears to be a better choice ([21]).Indeed, a major observation of this paper has been that spurious and high frequency bursts in the numericalcomputations of curvature profiles, especially over regions with acute plastic deformations, may besignificantly arrested, and even nearly eliminated, through a proper choice of the dilation parameter.

Finally it is useful to observe that the pipe-whip problem is characterized by large and localizeddisplacement and rotations of (parts of) the beam. While the instability associated with response localizationappears to have been addressed through the MLSRKPM-based scheme, the modeling of deformations ispresently via a set of PDEs in terms of displacement functions alone. However for more complex problemsinvolving torsion (e.g. out-of-plane whipping of bent pipes), recourse may have to be taken to moreadvanced beam theories, such as Simo’s beam theory (Simo and Vu Quoc [27]; Jelenic and Saje [24]),wherein additional equations for modeling rotations as additional dependent functions are introduced. Indoing so, one must account for the fact that evolutions of rotation in time and space take place on a specialorthogonal (SO(3)) manifold. In fact, a specific advantage of this theory is that the equations of motioninvolve just first order spatial derivatives in the translational and rotational variables and hence occurrences oflocalized numerical instabilities while computing higher order derivatives are far less likely. The numericalperformance of the MLSRKPM in the context of such models is yet to be seen and the authors are presentlyexamining this.

Acknowledgements

The authors wish to express their thanks to the UK Engineering and Physical Science Research Council(EPSRC Grant Ref. GR/R55245/01) for their financial support to the North West Centre for VirtualPrototyping. The second author also wishes to record his thankfulness to the Indian Space ResearchOrganization for partially supporting this work through Grant Ref. ISTV/MCV/DR/157.

References

[1] Belytschko T, Glaum LW. Applications of higher order stretch theories to nonlinear finite element analysis. Comput Struct

1979;10:175–82.

[2] Sereno C. Metodo dos elementos finites movies: aplicacoes em engenharia quimica, PhD thesis, FEAP, 1989.

ARTICLE IN PRESSA. Shaw et al. / International Journal of Impact Engineering 34 (2007) 1637–16541654

[3] Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless methods: an overview and recent developments. Comput

Methods Appl Mech Eng 1996;139:3–47.

[4] Yagawa G, Furukawa T. Recent developments of mesh-free method. Int J Num Methods Eng 2000;47:1419–43.

[5] Gardini L. Use of orthogonal collocation on finite elements with moving boundaries for fixed bed catalytic reactor simulation.

Comput Chem Eng 1985;9:1–17.

[6] Nayroles B, Touzot G, Villon P. Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech

1992;10:307–18.

[7] Belytschko T, Lu YY, Gu L. Element-free Galerkin methods. Int J Num Methods Eng 1994;37:229–56.

[8] Barry W, Saigal S. A three-dimensional element-free Galerkin elastic and elastoplastic formulation. Int J Num Methods Eng

1999;46:671–93.

[9] Kaljevic I, Saigal S. An improved element-free Galerkin method. Int J Num Methods Eng 1997;40:2953–74.

[10] Lu YY, Belytschko T, Gu L. A new implementation of the element free Galerkin method. Comput Methods Appl Mech Eng

1994;113:397–414.

[11] Babuska I, Melenk JM. The partition of unity method. Int J Num Methods Eng 1997;40:727–58.

[12] Melenk JM, Babuska I. The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech

Eng 1996;139:289–314.

[13] Duarte CA, Oden JT. An h-p adaptive method using clouds. Comput Methods Appl Mech Eng 1997;139:237–62.

[14] Liu WK, Jun S, Zhang YF. Reproducing kernel particle methods. Int J Num Methods Fl 1995;20:1081–106.

[15] Liu WK, Jun S, Li S, Adee J, Belytschko T. Reproducing kernel particle methods for structural dynamics. Int J Num Methods Eng

1995;38:1655–79.

[16] Chen JS, Pan C, Wu TC. Large deformation analysis of rubber based on a reproducing kernel particle method. Comput Mech

1997;19:211–27.

[17] Chen JS, Pan C, Wu CT, Liu WK. Reproducing kernel particle methods for large deformation analysis of non-linear structures.

Comput Methods Appl Mech Eng 1996;139:195–227.

[18] Zhu T, Zhang J, Atluri SN. A meshless local boundary integral equation (LBIE) method for solving nonlinear problems. Comput

Mech 1998;22:174–86.

[19] Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron

Soc 1977;181:275–389.

[20] Han W, Meng X. Error analysis of the reproducing kernel particle method. Comput Methods Appl Mech Eng 2001;190:6157–81.

[21] Reid SR, Roy D. A Gaussian sinc-collocation approach for a whipping cantilever with a follower shear force at its tip. Int J Num

Methods Eng 2003;58:869–92.

[22] Reid SR, Yu TX, Yang JL. An elastic–plastic hardening–softening cantilever beam subjected to a force pulse at its tip: a model for

pipe whip. Proc R Soc London Ser A 1998;454:997–1029.

[23] Jones N. Structural impact. Cambridge: Cambridge University Press; 1989.

[24] Jelenic G, saje M. A kinematically exact space finite strain beam model—finite element formulation by generalized virtual work

principle. Comput Methods Appl Mech Eng 1995;120:131–61.

[25] Atluri S N, Kim HG, Cho JY. A critical assessment of the truly meshless local Petrov–Galerkin (MLPG) methods. Comput Mech

1999;24:348–72.

[26] Liu WK, Li S, Belytschko T. Moving least square reproducing kernel methods (I) methodology and convergence. Comput Methods

Appl Mech Eng 1996;143:422–33.

[27] Simo JC, Vu-Quoc L. On the dynamics in space of rods undergoing large motions—a geometrically exact approach. Comput

Methods Appl Mech Eng 1988;66:125–61.