Contact Between Rolling Beams and Flat Surfaces

7/23/2019 Contact Between Rolling Beams and Flat Surfaces

1/24

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng2014; 97:683706Published online 17 December 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4611

Contact between rolling beams and flat surfaces

Alfredo Gay Neto1, *, , Paulo M. Pimenta1 and Peter Wriggers2

1Polytechnic School at University of So Paulo, So Paulo, Brazil2Leibniz Universitt Hannover, Hannover, Germany

SUMMARY

This work presents a new approach to model the contact between a circular cross section beam and a flatsurface. In a finite element environment, when working with beam elements in contact with surfaces, it iscommon to consider node or line to surface approaches for describing contact. An offset can be included innormal gap function due to beam cross section dimensions. Such a procedure can give good results in fric-tionless scenarios, but the friction effects are not usually properly treated. When friction plays a role (e.g.,

rolling problems or alternating rolling/sliding) more elaboration is necessary. It is proposed here a methodthat considers an offset not only in normal gap. The basic idea is to modify the classical definition of tan-gential gap function in order to include the effect of rigid body rotation that occurs in a rolling scenario and,furthermore, consider the moment of friction force. This paper presents the new gap function definition andalso its consistent linearization for a direct implementation in a Newton-Raphson method to solve nonlinearstructural problems modeled using beam elements. The methodology can be generalized to any interactioninvolving elements with rotational degrees of freedom. Copyright 2013 John Wiley & Sons, Ltd.

Received 21 July 2013; Revised 5 November 2013; Accepted 6 November 2013

KEY WORDS: contact; rotation; rolling; sliding; beam

1. INTRODUCTION

Many engineering applications present components that are naturally modeled using beam or shell

elements, instead of solids. Contact interaction between these components in such a modeling has

to be considered. As an example, which is the focus of the present paper, one can think of an inter-

action between beam-like structures and surfaces. Some practical examples are present in offshore

engineering, as long pipelines interaction with the seabed or a rope rolling and sliding in the seabed

as a part of an anchoring system of a ship. The fact is that many practical problems present rolling or

sliding or both situations occurring in different regions of circular cross section beam-like structures.

The rolling/sliding status may even change along time in some transient dynamics problems.

Aiming to model such a kind of problem using a FEM environment with beam elements, leads

to an issue to consider properly the rolling effect with the classical contact formulations. One could

think of modeling everything using solid elements. Then, not only a huge number of degrees of free-

dom (DOFs) can be achieved, but also very difficult to find convergence nonlinear models have to be

solved, once the nature of rolling includes successive changes of contact points, which is usually a

difficult task when solving a nonlinear contact model. This motivates the improvement of a contact

formulation to be used together with beam elements or, more generally, with structural elements

containing rotation DOFs. For that, it is necessary to include rotations DOFs in a proper way into

a contact formulation. Once this is done, structural elements can be used to represent the structure

*Correspondence to: Alfredo Gay Neto, Polytechnic School at University of So Paulo, So Paulo, Brazil.E-mail: [email protected]

Copyright 2013 John Wiley & Sons, Ltd.


2/24

684 A. G. NETO, P. M. PIMENTA AND P. WRIGGERS

Figure 1. Example of representation of a cylindrical cross section beam with four nodes. For contact detec-tion and description purposes, each node can be faced as a rigid sphere, which movement is described by the

node degrees of freedom.

involving such a kind of more complex contact kinematics, preserving the usual small number of

DOFs when compared to solid models.

In this work the focus will be given to the contact problem between a beam with circular cross

section and a flat rigid surface, which is a quite simple geometry. The here developed idea can nat-

urally be expanded to more complex geometries and contact pairs, such as beam to beam, shell to

shell and beam to shell interactions.

The contact between a beam and a surface is a classical problem and can be treated using the

node to surface approach. Furthermore, it is possible to consider a line to surface description for

a list of references in classical contact formulations one can refer to Wriggers (2002) [1]. When the

nodes or lines used in the contact formulation to represent the slave points are from a beam or shell

element, it is possible to modify the gap function including the size of the element as an offset. Such

a procedure can be seen in Benson and Hallquist (1990) [2] for shell contact and was also addressedby Wriggers and Zavarise (1997) [3] when treating the beam to beam contact frictionless modeling.

Zavarise and Wriggers (2000) [4] visited the same problem again, but assuming a frictional contact.

However, usually the moments of friction forces are not considered, assuming implicitly that radii

of contact beams are small. The inclusion of such effect in this kind of interaction is still an open

subject, yet to be incorporated in computational contact mechanics formulations.

The present work introduces a new technique to modify the definition of the gap function in a

node to surface approach. The main idea is to include in the gap function the information of the dis-

placements and rotations of the underlying body. The aim is to represent phenomena that physically

are described by changes in contact points (such as a rolling contact) only with the displacements

and rotations of each node of the modeled bodies in contact.

Each node in a structural element can be geometrically represented as a small body. In cylindri-

cal cross section beam elements, for example, each node can be seen as a small cylinder or, in an

approximated description, as a small radius sphere (Figure 1), equal to beam radius. For contactdescription purposes it is possible to assume that the geometry of each body related to each node

of a structural element can be described as a rigid body. With that, it is possible to write for each

node an equation to describe the movement of any generic point of the assumed rigid body, only

Here, the rigid body assumption implicitly is consistent with a rigid cross section of a beam or constant thickness of ashell assumption. To generalize this idea to more elaborated structural element theories considering cross section radiusor shell thickness variations, the same assumption of rigid body could be addressed in the first moment but, after, thecurrent radius of the sphere would have to be updated during the solution of the model, according to the modificationsoccurred. This contribution would change the weak form of the contact theory as well as some extra contribution wouldappear in tangent operator of the model.

Copyright 2013 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng2014; 97:683706

DOI: 10.1002/nme


3/24

CONTACT BETWEEN ROLLING BEAMS AND FLAT SURFACES 685

containing information of displacements and rotations DOFs of the node. With this equation, it is

possible to describe the movement of exterior points of each sphere, which can be assumed to be the

actual points of contact, instead of the center of the sphere (the slave point itself). Such a procedure

can be done with any shape of rigid body but, definitely the simplest choice is a sphere. In this work,

the sphere approximation will be assumed from now on.

In order to include the sphere movement in the gap function to describe the contact kinemat-

ics, it is necessary to study, first, the kinematics of each sphere in 3D space. This is done in thenext section.

2. KINEMATICS OF SPHERE

Once each node will be assumed to represent a rigid sphere, it is necessary to adopt a rotation param-

eterization to describe the rigid body movement. This is a classical subject and will not be discussed

deeply here. Information about the topic can be found in works from Spring (1986) [5] and Altmann

(2005) [6], regarding many possible options to describe rotations.

The present formulation assumes description of movement of the sphere using an updated

Lagrangian framework, there is, the movement will be divided into steps and, from one step to

the other, it is necessary to describe a rotation and displacement of the sphere in order to map its

movement. Figure 2 shows a scheme of this description for a given example of sphere. The idea isto deal with small (but finite) rotations in each step and, because on that the updated Lagrangian

technique was a natural choice.

On the basis of Figure 2, Rodrigues rotation parameters were adopted for writing a rotation tensor

between the current and the next configuration (Q/(Refer to Pimenta & Campello (2001) [7] forapplications of Rodrigues parameters with beams and Pimenta, Campello, & Wriggers (2011) [8]

for shells dynamics and Gay Neto, Martins, & Pimenta (2013) [9] for models with beams involving

contact).

Rodrigues parameters are related to the Eulers parameters, which can be used to define the rota-

tion tensor. One has to calculate a rotation pseudo-vector to evaluate Q. Eulers parameters will

be here briefly described and also their relation with Rodrigues parameters, for completeness of the

present formulation. Let D e be the Eulers rotation pseudo-vector, which magnitude is the

rotation angle. The mapping of a generic vector vsuffering a rotation and leading to the vector v

0

can be done using the rotation tensorQ, such that v0

D Qv.The Rodrigues rotation parameter is defined by D 2 tan. =2/ and the Rodrigues rotation

pseudo-vector (/ is given by: D e. It is possible to obtain the following expressionfor the rotation tensor using Rodrigues parameters, see, e.g., Campello, Pimenta, and Wriggers

Figure 2. Updated Lagrangian description for a sphere. A framework accompanying the rigid body motionof a sphere can suffer displacements and rotations. This scheme shows different configurations: referenceconfiguration, current configuration .i /, and next configuration .i C1/. The transformation between the

current and the next configuration is referred as.


DOI: 10.1002/nme


4/24


(2003) [10]:

Q D IC g./

A C

1

2A2

(1)

with ADskew

andg./D 4

4C2.

It is also possible to define the angular velocity skew-symmetric tensor given by: D

PQ.t /QT .t /. The relation between P and ! Daxial ./is given by:

! D P (2)

Using some algebra, it is possible to obtain the following expression to the operator :

Dg./

IC

1

2A

(3)

Furthermore, it is useful to calculate the time derivative of , given by:

P D 1

2g./

h P

PA

i (4)

Going back to sphere description for each node, the only important geometric parameter thataffects the model is the external radius of the cross section of the beam, which defines the size of

the sphere that will be adopted to represent the movement of each node in central axis of the beam.

Thus, the present formulation is valid for tubular cross section beams. Then, the external radius has

to be considered to define external surface of the beam from each node.

If one considers an offset radius in each point of the beam for correcting the classical gap func-

tions of a node to surface or line to surface element, this will not be enough to represent the correct

kinematics of the problem. The only exception is when the problem is frictionless, when only that

offset is enough. Assuming that contact point is not exactly located at the centroid of the cross sec-

tion leads to friction force non-null moments related to that point. This always occurs and can be

very important in rolling problems when the friction plays a role not permitting sliding. The action

line of normal force crosses the center of the considered cross section and, naturally, results in null

moment related to that pole. The friction, however, does not have the same property. Neglecting the

friction force moment does not permit a good physical description of rolling phenomenon.To explain the basic idea of the present formulation kinematics one can refer first to a classical

(and simple) 2D problem, illustrated in Figure 3. A circular rigid body with pure sliding in a flat

surface is described in Figure 3a. In this problem, one can see that the point of contact of the body

C.i /does not change with the movement but, once the body translates, that point moves to anotherposition. In the second example in Figure 3b) when there is sliding and rolling together, the situation

is completely different: the contact point in the beginningC.i /changes intoC.i C 1/, which is adifferent point of the circular cross section body, once it presented a rotation given by '. Then,

aiming to describe these different contact interactions using a tangential gap function, the change

in the contact point presented in the second scenario has to be considered to describe it as different

from the first scenario.

Figure 3. Example of movement of a circular rigid body representing the cross section of a beam. (a) Puresliding and (b) sliding and rolling.


DOI: 10.1002/nme


5/24


For Figure 3a, it would be correct to say simply that tangential gap is given by the vector defined

by successive positions ofC.i / and C.i C1/ in the plane of contact. However, for the scenariodescribed in Figure 3b, it is possible to adapt the tangential gap definition including the rotation

. Describing for simplicity the tangential gap as a scalar variable for the 2D problem, gap func-tion could be calculated simply bys r, already considering in this definition the amount ofrolling given by the radius r multiplied by the angle and the change in the contact point total

distance, given bys .If one has pure rolling situation,s Dr, this new definition of tangential gap would give a

null result, which is correct, once the tangential gap function has to describe the amount of sliding

between two points according to its classical definition (see, e.g., Wriggers (2002) [1]).

The formulation of contact described next generalizes this basic idea for 3D rotations of a sphere

in contact with a flat surface in 3D problems.

3. CONTACT FORMULATION

First of all, it is useful to define some nomenclature used from now on in the text:

Cis the point of contact between the flat surface and the sphere in configuration (iC 1). O is the centroid of the sphere in configuration (i C 1).

fis the friction force in configuration (iC 1). Mfis the friction force moment in configuration (iC 1) related to the pointO , calculated by

Mf D .C O / f. f is the friction force pseudo-moment in configuration (i C 1) related to the pointO (ener-

getically conjugated with the rotation parameter time derivative see Pimenta, Campello, and

Wriggers (2008) [11] for more details about pseudo-moment and its implications to the beam

formulation).

vc is the velocity of point C.

vo is the velocity of point O.

! is the angular velocity of a sphere representing a node movement as described previously.

Let Pfbe the power of the friction force acting in the contact between the flat surface and the

sphere. It is possible to write

Pf D f vc D f voC ! .C O/D f voC f ! .C O/D f voC Mf ! (5)

with f D T Mf. Equation 5 leads to the following weak form:

Wf D f xoC T Mf

(6)

This weak form is convenient to work together with structural models. Once xo and are

respectively the displacements and rotations of the nodes of the structure, these can be elected as

slave points from now on. Then, one can deal directly with the DOFs of the beam (or if wanted,

for a shell) when developing this weak form and its tangential operator. From now on, the pseudo-

vector will be represented simply by, only for a cleaner notation purpose. It remains with the

physical interpretation of rotation between configurations.i /and.i C 1/.The major issue to describe correctly the rolling behavior using only that beam element informa-

tion is related to the kinematics of the cross section and the subsequent changes that rolling causes

in the contact point. For representing that, special defined gap functions can be constructed using

only information of the DOFs of each node of the beam to describe the contact situation. Let r bethe radius of the cross section of the beam and then defining the radius of the sphere associated

to the node of interest. Figure 4 shows the master surface (rigid and flat) with the defined normal

(n) and tangential (t1/ and (t2/ directions. The sphere oriented by the node DOFs changes is rep-resented in configuration .i C1/. The previous contact point of configuration .i / C.i/ turns intoC0.i C1/because on rigid body rotation of the sphere. The new point of contact is C.i C1/, definedin configuration.i C 1/, as shown in Figure 4.


DOI: 10.1002/nme


6/24


Figure 4. Geometric description to define the tangential gap function.

The amount or rolling between configurations .i / and .i C1/ is given by the arc length s,also described in the Figure 4, assuming implicitly the simplest movement history connecting

configuration.i /to .i C 1/.Assuming that the magnitude of rotation is small from one configuration .i /to the next.i C 1/it

is possible to write an approximation to represent a vector amount of rolling, given by: r0

r .

Once r D rn and r

0

D rQ

n, one can write: r

0

.r/ D r

Q

I

n. Using thisexpression, it is possible to quantify the amount of rolling related to the displacement of the cen-

ter of the sphere from O.i / to O.i C1/. If one projects this vector describing the amount ofrolling in the master surface plane, it would be possible to define the tangential gap simply by:

O(i+1)O.i / C rh

Q I

ni

t1

t1C r

hQ I

ni

t2

t2.

However, this information has to be propagated to the next step properly. For that, it is convenient

to define a fictitious point of initial contact between the sphere and the flat plane xf.iC1/ , which has

to be updated after each step of the updated Lagrangian scheme:

xf.iC1/ D xf.i/ C rh

Q I

ni

t1

t1C r

hQ I

ni

t2

t2 (7)

This fictitious point of initial contact is physically associated to the information of the total

amount of accumulated rolling that occurred until the configuration .i C1/. The idea is to sub-tract this amount from the difference between the centers of the spheres in order to define tangentialgap function. Finally, with this information, one can define the so called elastic gaps for each sphere

related to the configuration.i C 1/of an updated Lagrangian description:

gt.iC1/ D gt1.iC1/t1C gt2.iC1/t2 (8)

gt1.iC1/ D

xo xf.iC1/

t1 gst.i/

t1 (9)

gt2.iC1/ D

xo xf.iC1/

t2 gst.i/

t2 (10)

gn.iC1/ D

xo xf.iC1/

n r (11)

The vectorgst is the accumulated sliding tangential gap, which is null in the beginning gst.0/

D 0

and has to be updated when sliding occur. The name elastic gaps is related to the classical and

commonly mentioned analogy between the Coulomb friction model with sticking/slipping and the

structural material models of elasticity/plasticity. This analogy will be assumed in the present model

together with a penalty method formulation, which is also used from now on (the reader can refer

to Wriggers (2002) [1] for more information about different methodologies of imposing contact

constraints, as Penalty method, Lagrange multiplier methods, Augmented Lagrange, etc.).

All this procedure for defining the gap function is done for creating a fictitious initial point of con-

tact, considering the possibly rolling on its composition. The only geometric approximation made


DOI: 10.1002/nme


7/24


is related to the actual arc of rolling, considered as the cord of the same arc, projected in the master

surface. This is valid for small rotations in each step of the updated Lagrangian formulation. The

asymptotic tendency is to represent the actual arc of rolling, as the step between two configurations

tends to zero. The expression (7) actually does not make a special update for the axial or transversal

directions of the cylindrical beam element and, everything occurs as if the rolling was of a sphere

with a flat surface in each node of the beam. This simplifies a lot the formulation. Rotations of nodes

causing penetration due to bending could make the sphere assumption a bad choice. However, theserotations naturally do not occur due to contact restriction in the problem, which constrains the rota-

tions of the beam in order to keep the structure in the contact plane or above it, obeying the inequality

constraint. Thus, the sphere rigid body can be used as a good approximation for representing the

rolling of a cylinder rigid body in the contact situation.

As usual, in any Coulomb friction model, one has to check the condition of sliding or sticking.

This can be made using a tangential penalty approach. Let t be the tangential penalty stiffness andn the normal penalty stiffness. Then, one can define the trying friction force:

ftry Dt.gt1t1C gt2t2/ (12)

If this trying friction force norm is less than the maximum allowed friction magnitude, given

by fslide Djngnj, then one has the sticking situation. Otherwise, there is slipping and it isnecessary to update the slide accumulated tangential gap by:

gst.iC1/ D gst.i/

C nT (13)

With

D 1

t

ftry fslide (14)and the direction of sliding nTcan be calculated by:

nT Dget.iC1/

get.iC1/

(15)

with the elastic tangential gap used to calculate the friction try force given by:

get.iC1/ D gt.iC1/ gst.i/

(16)

Weak form and tangential operator sticking status

In sticking situation, the friction force is given by Equation (12). Then, one can obtain the following

weak form:

Wstickf Dt.gt1t1C gt2t2/ xoC rtT .gt2t1 gt1t2/ (17)

Including the normal contribution one obtains

W

stick

Dt.gt1t1C gt2t2/ C ngnn xoC rt

T

.gt2 t1 gt1t2/ (18)The tangent operator is given by the linearization of the expression (18):

Wstick D t .gt1t1C gt2t2/ C ngnn xoC rtT .gt2t1 gt1t2/

C rtT .gt2t1 gt1t2/

(19)

Each one of the contributions can be given by:

gt1 Dh

..xo/T/ xT

f.i/

i Ct1

t1

T

hr

T

skew

Qn

t1

i (20)


DOI: 10.1002/nme


8/24


gt2Dh

xTo xTf.i/

i Ct2

t2

T

hr

T

skew

Qn

t2

i (21)

gnD

xTo xf.i/T Cn

n

(22)

To calculate the contribution given by T

, one can use the result from Equation (4):

PT

D 1

2g./

h. P / PA

iTD

1

2g./

h. P /

T

C PAi

(23)

Operating in a generic vector v, one has

PT

v D 1

2g./

h. P /

T

v C PAvi

D 1

2g./

h

T

vT P v Pi

(24)

Analogously, it is possible to write a linearization that defines the operator O1.v/.

T

v

D

1

2g./

h

T

vT V

i DO1.v/ (25)

It is possible to write the tangent operator in a ready to implement matrix form, separating eachcontribution for each term. Once rotations of the cross section of the beam were considered, some

contribution appears related to the rotation parameters and also some cross terms between and

xo DOFs of the cross section of the beam. Once the flat surface is supposed to have no meshed

bodies, but it is defined only as a rigid body surface, its contribution will not be considered from this

point on in the tangent operator. Then, only the slave point DOFs should be altered by this approach.

Wstick D

xTo T

K1C K2C K3C K4C K5

xo

(26)

With:

K1D " t t1tT1 trt1tT1skew Qn

0 0# (27)

K2D

" t t2t

T2

trt2tT2

skew

Qn

0 0

# (28)

K3D

nnn

T 0

0 0

(29)

K4D

0 0

0 rtO1.gt2t1 gt1t2/

(30)

K5D

" 0 0

rtT

t1tT2

t2tT1

r2t

Th

t1tT2

skew

Qn

t2tT1

skew

Qn

i#(31)

Weak form and tangential operator sliding status

Noting that when one has contact, the normal gap function gn becomes negative; in the slidingsituation, the friction force can be written as:

fslide D ngnnT (32)


DOI: 10.1002/nme


9/24


and also the friction force moment related to the circular cross section centroid is given by:

Mslidef Drngn.n nT/ (33)

Using the weak form general expression (6) with the friction force (32) and the friction moment

(33), one obtains the weak form for the status when sliding occurs, which has the contributions of

forces and moments of friction contact interaction for the slave point xo:

Wslidef D ngnnT xoC rngnT .n nT/ (34)

Including the normal direction, one obtains

Wslide DngnnT C ngnn xoC rngnT .n nT/ (35)

The tangent operator is given by the linearization of the weak form (35) and can be written as:

Wslide DngnnT ngnnT C ngnn xoC

C rn

gn

T

.n nT/ C T gn.n nT/ C

T gnNnT

(36)

The linearizations necessary to calculate the tangent operator aregn,

T

, andnT. Thefirst two are already calculated and used in the sticking status tangent operator. The last one has to

be evaluated. Such a procedure for that is showed next:

nT D

gt1.iC1/t1C gt2.iC1/t2gt1.iC1/t1C gt2.iC1/t2

! (37)

which can be developed by (see, e.g., Zavarise and Wriggers (2000) [4] for algebraic details):

nT D 1gt1.iC1/t1C gt2.iC1/t2 .I nT nT/

gt1.iC1/t1

C

gt2.iC1/t2

(38)

with

gt1.iC1/t1

Dgt1t1

gt2.iC1/t2

Dgt2t2(39)

By using the results from Equations (20) and (21) together with (38), analogously to that done to

evaluate the sticking status tangent operator, one can write for the sliding situation the matrix form

ready to implement tangent operator:

Wslide D

xTo T

K3C K6C K7C K8C K9C K10C K11C K12

xo

(40)

with K3already calculated in Equation (29) and the other contributions given by:

K6D nnTnT 0

0 0

(41)

K7D

" .ngn/

1gt1.iC1/ t1Cgt2.iC1/ t2 .I nT nT/

t1t

T1

C t2tT2

0

0 0

# (42)

K8D

240 r.ngn/ 1gt1.iC1/ t1Cgt2.iC1/ t2 .I nT nT/

t1t

T1

C t2tT2

skew

Qn

0 0

35

(43)


DOI: 10.1002/nme


10/24


K9D

0 0

0 rngnO1.n nT/

(44)

K10D

" 0 0

rnT .n nT/ n

T 0

# (45)

K11D

264

0 0

.rngn/ T N

1gt1.iC1/ t1Cgt2.iC1/ t2

.I nT nT/

t1tT1 C t2t

T2

0

375(46)

K12D

264

0 0

0

r2ngn

T

N

1gt1.iC1/ t1Cgt2.iC1/ t2

.InT nT/

t1tT1

Ct2tT2

skew

Qn

375(47)

Equations (18), (26), (35), and (40) are ready to implement in a FEM code. Once the beam nodes

that have contact with the flat surface are set-up as slave points, these equations represent direct

contributions to force vector and tangent operator of the problem.As usual, in contact problems numerical implementation, it is important to monitor the contact

status along the evolution of the simulation. For that, one can use the evolution of the sign of normal

gap function. If a slave point loses contact and then starts having contact again, it is important to

set up the accumulated tangential gap and fictitious initial point of contact as null, as if the contact

would have beginning at that time of contact recovering. Otherwise, a false time history of contact

force interaction could be registered.

4. NUMERICAL EXAMPLES

This section shows numerical examples with applications using the proposed contact formulation.

The first is related to transient dynamics of a very simple problem of a disc sliding and rolling in a

flat surface; the second example is a static nonlinear problem of a beam under rolling contact condi-

tions suffering large displacements and large rotations, and the third and fourth examples are related

to transient dynamics involving impact problems. All examples were chosen to identify situations in

which the rolling or alternating rolling/sliding plays a role to demonstrate the usage of the proposed

methodology.

The beam model used in all examples is the presented by Gay Neto, Martins, and Pimenta (2013)

[9] for statics and complemented by Gay Neto, Pimenta, and Martins (2013) [12] for dynamics. For

dynamics, one could naturally apply the proposed contact formulation to the Pimenta, Campello,

and Wriggers (2008) [11] beam element. All these referenced works assume a geometrically-exact

beam formulation able to consider large rotations in 3D problems. For more details about the beam

formulation, the reader can refer to the mentioned works. Once the focus here is to discuss the

contact problem and its details, no further comments are made about the beam formulation.

It is important to emphasize that the proposed here contact modeling can be used together with

any beam theory to model the structure. The only requirement is the correct treatment of the largerotations in a geometrically nonlinear approach.

4.1. Dynamics of a cylinder initially sliding and turning into rolling

This numerical example presents a dynamic problem of classical mechanics illustrated in Figure 5.

A rigid disc of unit length has initial angular velocity given by ! D !0k and its center has aninitial velocity given by V0D V0i . There is friction between the disc and a flat surface defined bya friction coefficient.

A specific initial condition can be imposed such that, in the beginning, there is sliding, which

changes the angular velocity of disc and the velocity of pointO along time. A sticking condition is


DOI: 10.1002/nme


11/24


Figure 5. Numerical example 1: description of the problem.

then achieved and the disc starts rolling with no sliding. The analytical solution of this problem is

straightforward and can be found using the Newton Law applied for both translation and rotation of

the rigid disc. Ifr is the disc radius and g is the gravitational field magnitude (defined in negativeydirection), the results are that during the sliding, one can write the following expressions for !.t/and V0.t /:

!.t /D

!0C

2g

r t

k (48)

V0.t/D.V0C gt/ i (49)

The kinematic condition of rolling is given by kV0.t/kDk!.t /k r , which is achieved in timegiven bytroll :

troll D!0r V0

3g (50)

After start rolling, the angular velocity of the disc and the center of disc velocity remain for any

time value. With the numerical values given by!0 D 127.03,V0 D 2.355,r D0.2, D 0.25, andgD9.81, one can calculatetroll D3.13.

A simulation was performed with the proposed contact method applied to the beam model utilized

in [9]. Only one element discretization and the appropriate boundary conditions to simulate the disc

problem were considered. For dynamics, the Newmark trapezoidal rule was utilized, as mentioned

in [12]. The time step considered is constant and equal to 1E-3. The penalty parameters considered

arenD 1E8and t D1E7. The amount of penetration registered is of order 1E-4. The total massof the considered disc in the simulation is 1E3.

Figures 69 show the time evolution of the displacement of the center of the disc, total rotation

of the disc , magnitude of velocity of the center of the disc, and magnitude of angular velocity ofthe disc. It is possible to see clearly that the proposed rolling model could represent very well the

large amount of sliding and the transition from sliding to rolling as well. It is of interest to mention

that the chosen value of tangential penalty stiffness causes very small displacements in the sticking

situation, once it is a very high value. It is almost impossible to see such an oscillation in the pre-

sented plots during the sticking steady state (Figures 8 and 9). If one diminishes this penalty value,

the condition of sticking would be relaxed and it would be possible to visualize an oscillation in

velocity of the center of the disc and in the angular velocity of the disc along time during the steady

state rolling behavior, with mean equal to the obtained by the analytical solution. Obviously, this

oscillation goes to zero if one increases the value oft , as in the results here shown.


DOI: 10.1002/nme


12/24


Figure 6. Numerical example 1: x displacement of pointO as a function of time.

Figure 7. Numerical example 1: angular coordinate measuring rotation of cylinder as a function of time.

Figure 8. Numerical example 1: magnitude of velocity of the pointO of the disc as a function of time.


DOI: 10.1002/nme


13/24


Figure 9. Numerical example 1: magnitude of the angular velocity of the disc as a function of time.

Figure 10. Numerical example 2: geometric description.

Table I. Time series for they direction imposeddisplacement at point B.

Time y direction imposed displacement at point B

0 0

1 02 +53 04 +5

4.2. Cantilever beam with rolling and sliding

In this example, it was considered a quasi-static nonlinear model of a scenario of an initially straight

beam connecting the points A and B (Figure 10). The cross section of the beam was assumed to

be circular with diameter D D 0.3, and the length of the beam is L D 10. The material propertiesare Young Modulus E = 1E7, Poisson ratio D 0, and density D 1E5. The gravitational fieldconsidered has magnitude 9.81 in negative direction of. The coefficient of friction in the interfacewas assumed to beD0.5.

The displacements and rotations are imposed to be zero in point A(fixed point) and, in point B,

theydirection movement is imposed using the time series described in Table I. The other DOFs in Bare free, including all the rotations. Imposing this transversal ( y direction) displacement in point Bcauses the beam to roll and slide or alternate it, depending on the analyzed cross section and on the

amount of imposed displacement. Naturally, to make normal contact forces non-null, in a first step

of solution, the structures own weight was considered and equilibrated by normal contact forces.

This first step is considered in a time history evolution of loading from time D0to timeD1. Then,in timeD1, the transversal displacements start being imposed progressively, according to Table I.

Once working with penalty method, a study of influence of penalty parameters was done in the

model. To get good results with small values of penetration in normal direction (of maximum order


DOI: 10.1002/nme


14/24


of 1E-4), the chosen values of parameters were n D 1E 9and t D1E 7. Quadratic (three nodes)beam elements were used in a 100 elements uniform mesh, which showed to present good results.

For a comparison, the same conditions were simulated using ANSYSTM 14 solver, however, using

solid elements to represent the beam. Then, the external surface of the whole beam was defined as

slave surface, and a defined plane flat region just below the cylinder was defined as the master

surface. The same boundary conditions and imposed displacements were considered as in the beam

model. Second order (SOLID186) elements were used in a wedge shaped elements sweep mesh (seeANSYSTM (2012) [13] for details about the formulation). For achieving convergence, it was neces-

sary to use the Augmented Lagrange formulation, with manual stiffness factor D1.0and permittingthe stiffness updating automatically in each iteration, according to the evolution of the simulation.

It is important to emphasize that this setup was not straightforward and many other parameters and

combinations with other algorithms were performed, with many difficulties to achieve convergence,

due to the extremely nonlinear behavior that this problem presents, once the contact points in the

solid model change every time when there is rolling. A good mesh quality in the external surface of

the beam was required in order to capture smoothly the changes in contact points (or at least to try to

do that). Here, the results presented that 200 divisions were considered along the length of the beam

and 32 divisions along the circumferential direction of the cross section. With this discretization, the

final mesh, which results are here shown, has 45,363 nodes and 13,825 elements. To input the own

weight and the sequence of imposed displacements in the beam tip 3385 iterations were necessary,

of course, due to the large number of sub-steps to perform the rolling contact properly. This took a

time of many hours of simulation in a 4 core XeonTM processor computer with 8Gb RAM memory.

The tendency of the presented formulation when the beam radius (and consequently the sphere

radius associated with each node) tends to zero is to go to the classical node to surface treatment,

with no rolling properly treated. A third simulation considering this assumption was performed for

comparisons with the proposed formulation in this paper, just to check the influence of the rolling

in the results. Results related to this check (r D 0) simulation are named beam model (no rolling).The results considered using the here presented and proposed model with the rolling treated prop-

erly are simply named beam model. Both simulations solved using beam models can give results

in some seconds or no more than few minutes using the same computer 4 core XeonTM processor

computer with 8Gb RAM memory.

Figure 11 shows the deformed beam structure at timeD2.0, there is, after the first lateral displace-

ment imposition. It is possible to see that the proposed model captured well the rolling behavior thatoccurred in a large amount of the beam, giving very good agreement with ANSYSTM solution. The

beam model with no rolling could not capture correctly the scenario, once rolling plays an important

role. This is clearly shown in Figure 11.

Figure 11. Numerical example 1: deformed shape of the beam structure at timeD 2.0(for the solid model(ANSYSTM/the central line passing through the centroids of cross sections were plot).


DOI: 10.1002/nme


15/24


Figure 12. Numerical example 2: force reaction to impose the y direction displacement at point B of thebeam (fy/ from timeD1.0to timeD2.0.

Figure 13. Numerical example 2: moment of torsion in the beam structure at timeD2.0as a function of thereference configurationx coordinate (undeformed structure length coordinate).

Furthermore, one can check in Figure 12 the force reaction at point B in ydirection because of theimposed displacement. This can be faced as a global friction action in the problem. Again, the pro-

posed contact model agrees very well with ANSYSTM solid model results. The non-smoothness

behavior of ANSYSTM results is due to the mesh discretization, which can never capture smooth

changes in contact points during rolling. For the proposed method, this smoothness is straight-

forward once the contact point changing is automatically taken into account in the definition of

tangential gap. The beam model with no rolling gives no proper results, once considered larger fric-

tion actions due to a false sliding identified instead of rolling. Actually, the rolling diminishes the

amount of friction work when compared with the no rolling assumption, even with more movement

of the whole beam.

Looking now to the comparison between the torsion moment along length of the beam between

the beam model and the beam model (no rolling), one can notice in Figure 13 that all the torsion

present comes only from the moment of friction forces, and simply neglecting these effects leads to

incorrect null torsion predictions.

Looking forward in time history results, after a cycle of movement, in timeD 4.0, one can lookat the deformed structure comparison between the models in Figure 14. Again, the beam model


DOI: 10.1002/nme


16/24


Figure 14. Numerical example 2: deformed shape of the beam structure at time = 4.0 (for the solid model

(ANSYSTM/the central line passing through the centroids of cross sections were plot).

Figure 15. Numerical example 2: x coordinate of point B of the beam (xB/as a function of time.

represented very well the rolling behavior when compared with the solid model (ANSYSTM/,and the beam model (no rolling) could not capture the important contribution of rolling in the

cyclic behavior, giving improperly results. The Figure 15 gives the additional information of the

x coordinate of the point B as a function of time. It is possible to see clearly that the proposedmodel can represent very well the solid behavior, but the beam model (no rolling) could not deal

with that.

A last plot in Figure 16 shows the hysteresis of the friction problem behavior in cyclic loading

situation. One can see the force reaction at point B in y direction due the imposed displacementfrom timeD 2.0to timeD 4.0. The comparison of beam model and the solid model (ANSYSTM/is very good. The beam model (no rolling), however, could not represent well again the friction

forces, giving improperly results. Note the interesting fact that the work of friction force is larger

in this cycle than in the rolling situation, for the same imposed displacement in the tip of the

beam. This could naturally be expected once rolling is a natural way of movement that does not

dissipate energy.

Figure 17 shows some post-processing images to illustrate the movement of the beam. It is pos-

sible to see the solid model and a post-processed as solid result regarding the beam model. This can

be useful to see the regions of rolling or sliding in the model.


DOI: 10.1002/nme


17/24


Figure 16. Numerical example 2: force reaction to impose the y direction displacement at point B of thebeam (fy/ from timeD2.0to timeD4.0.

Figure 17. Numerical example 2: top view of the deformed structure for different time values. On left col-umn side are the present contact algorithm solved using beam model and rendered (post-processed) as a

solid structure. On right side column are the solid model solutions made using ANSYSTM solver.


DOI: 10.1002/nme


18/24


4.3. Puck sliding on a flat surface with successive lateral impacts

This example presents a transient dynamics problem involving impacts. The idea is to represent the

movement of a cylindrical disc, named puck, on a rigid and flat surface. The surface is a squared

region with edges . The puck radius is given byr and its movement is described using the coordi-nate system shown in Figure 18. Puck is initially located in the centroid of the squared region and

oriented according to Figure 18, with null velocity and null spin. It cannot roll over the surface, but

only slide, once it is laid laterally (see Figure 18 showing the top view of the proposed scenario).

The puck is initially subjected to two simultaneous time history loadings, described in Figure 19,

being one force and one moment, injecting kinetic energy in the system causing translation and

spinning to the puck. The square region is limited by some lateral surfaces, named walls, which

limit the puck movement range. Puck was modeled by one only finite element beam subjected to

boundary conditions to keep its central point in the planex. The only free rotation is in ydirection.Interaction between puck and the flat surface was modeled just imposing the y direction movementequal to zero in the nodes of the beam representing the puck, which represents a frictionless behav-

ior with no possibility of status changing, there is, the puck never leaves the flat surface. The walls

were considered to be rigid and the puck interaction with them was addressed considering the here

proposed methodology for contact interaction.

Figure 18. Numerical example 3: geometric description.

time

Fz

0.1

1E4

time

My

0.1

5E3

Figure 19. Numerical example 3: loading time history in each node of the beam representing the puck. Thebeam has three nodes, being the total load given by 3 times the shown in these plots.


DOI: 10.1002/nme


19/24


The numerical parameters used in the example were chosen such that both loadings have a very

short duration, going to zero prior to the first puck impact in one of the walls. Then, it is possible

to analyze the system behavior and kinetic energy variation only due to impacts occurrences. All

the results were generated using the following parameters: axial stiffness of puck EAD1E8; bend-ing stiffness of puck EI D 1E8; torsion stiffness of puck GJ D 1E8; mass of the puck 1E3; puckradiusr D 0.25; puck height is unitary; total simulation time 10; time-step 1E-4; normal stiffness

penalty factornD1E8; and tangential stiffness penalty factor t D 1E7. The maximum registeredpenetration was less than 7E-3. The solver used in this example was the same used in Example 1.

Some structural parameters here described are not important for the contact method, such as the

stiffness of the puck, modeled as a beam. Actually, the beam behaves as a rigid body in this example;

however, these parameters were here provided for completeness. Once dealing with impact prob-

lems, such stiffness parameters can be important due to numeric difficulties, once high frequency

vibration modes are exited during impacts.

Two different values of friction coefficient between puck and the walls were considered:D0.0and D 0.4. The idea is to see a comparison and show that friction plays a role as a way for thesystem transform kinetic energy from rotation into translation and also the opposite, and that the

proposed contact modeling can capture that.

Figure 20a shows the centroid of puck trajectory for D 0.0 case. The disc simply stays ina steady state straight line movement, going and returning in the same path, suffering successive

impacts in only two walls. The coordinates of puck centroid along time can be seen in Figure 21a,

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

x

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

z

x

(a) (b)

Figure 20. Numerical example 3: trajectory of puck center in the surface (a) frictionless interaction with thewalls (b) frictional interaction with the walls.

0

5

10

15

20

25

30

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10

xand

z

Time

x z

0

0.5

1

1.5

2

2.5

3

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10xandz

Time

x z

(a) (b)

Figure 21. Numerical example 3: translational and angular coordinates describing puck center and itsorientation along time (a) frictionless interaction with the walls (b) frictional interaction with the walls.


DOI: 10.1002/nme


20/24


showing that angular coordinate only increases uniformly due to the constant angular speed afterinstant 0.1, when the applied loading moment My becomes zero (Figure 19). The time evolution ofvelocity components of puck, such as the angular speed can be seen in Figure 22a, with a typical

alternating pattern due to the successive impacts. Kinetic energy due to puck spin remains during the

successive impacts for the whole analysis period, as one can see in Figure 23a, which also shows

0

5

10

15

20

25

30

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8 9 10

VxandVz

TimeVx Vz

-10

-5

0

5

10

15

20

25

30

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10

VxandVz

TimeVx Vz

(a) (b)

Figure 22. Numerical example 3: translational speeds of puck center and angular speed of puck along time(a) frictionless interaction with the walls (b) frictional interaction with the walls.

0

5000

10000

15000

20000

25000

0 1 2 3 4 5 6 7 8 9 10

KineticEnergy

Time

Total Translation Rotation

0

5000

10000

15000

20000

25000

0 1 2 3 4 5 6 7 8 9 10

KineticEnergy

Time

Total Translation Rotation

(a) (b)

Figure 23. Numerical example 3: kinetic energy evolution along time (a) frictionless interaction with thewalls (b) frictional interaction with the walls.

Figure 24. Numerical example 4: top view of the initial beam configuration. The black line represents thetrace of the vertical plane in which the beam impacts during its movement.


DOI: 10.1002/nme


21/24


that the total kinetic energy of the system (composed by the rotation and translation movements)

remains constant, only achieving minimum values during the impacts, when for some very small

period the normal to the contact surface component of velocity goes to zero.

Looking now at results considering D 0.4for the interaction between puck and the walls onehas a completely different behavior. The same initial kinetic energy is supplied to the system, but just

after the first impact, some spin turns into translational movement inx direction, because of friction

impulse during impact. Then, in other impacts, one has again the friction playing a role as a mech-anism of changing the kinetic energy from spin into translation, and also the opposite. However,

always some sliding occurs during contact and a fraction of the kinetic energy is dissipated.

Figure 20b shows the trajectory in plane x, and Figure 21b shows the evolution of the coordinatesof the centroid of puck. Furthermore, Figure 22b shows the evolution of the velocity components,

clearly totally different from the frictionless example. Now, part of the spin turns into translational

speed and vice-versa during each impact. The most interesting is to analyze the kinetic energy evo-

lution, shown in Figure 23b. Clearly there is a loss of energy in each impact occurrence, and in

the long term, it is possible to say that the system spin energy goes to zero and it stays only with

translational kinetic energy. All the dissipation is due to sliding friction occurred during successive

impacts once no other damping sources were considered in the model.

4.4. The flying beamThis example aims to show a mechanical system subjected to impacts with more than one surface,

involving essentially 3D behavior. The structure simulated is a straight beam with circular cross

time

Fy

1.0

1E3

Figure 25. Numerical example 4: loading time history applied in the center of the beam.

Figure 26. Numerical example 4: top view of the beam trajectory. In red bullets and blue crosses one cansee both beam tips centers successive positions and, at some equally spaced instants, a black bar linking the

points showing instantaneous configurations of the beam.


DOI: 10.1002/nme


22/24


section, shown in Figure 24 by a top view. The centroid of the beam is located at coordinates

(0,0,r/, with r being the external radius of the cross section. The beam is laying on a horizontalplane, located inD0. There is another contact surface constructed in vertical direction and shownin Figure 24, with which there is no initial contact.

The beam numerical data is given by: axial stiffness EAD3.6E8; bending stiffness EID9.2E9;torsional stiffness GJ D 9.2E9; shear stiffness GA D 1.8E8; mass per unit length 1.3E-3;r D25;

and lengthL D1000. Environment data is given by the following: gravitational field g D 9.81k;the vertical contact plane contains the point (0,1000,0) and its normal direction is (-1,-1,0); friction

coefficient in all contact regions is D 0.8; normal stiffness penalty factor n D 1E3; and tan-gential stiffness penalty factort D1E3. Simulation total time is 20 units and time step 1E-3. Thebeam is uniformly meshed using 10 elements, with quadratic shape functions for displacements and

rotations DOFs.

Initially, the beam is statically subjected to the gravitational field, and the contact with the hori-

zontal surface plays a role, only equilibrating the weight of the structure. After, a transient dynamics

Figure 27. Numerical example 4: isometric view of the beam trajectory. In red bullets and blue crosses onecan see both beam tips centers successive positions and, at some equally spaced instants, a black bar linking

the points showing instantaneous configurations of the beam.

0

5

10

15

20

25

30

35

40

45

50

55

60

-2500

-2250

-2000

-1750

-1500

-1250

-1000-750

-500

-250

0

250

500

750

1000

1250

1500

0 2 4 6 8 10 12 14 16 18 20

xandy

Time

x y z

Figure 28. Displacements time history of beam point initially located at position (0,-L=2, r).


DOI: 10.1002/nme


23/24


simulation starts being performed, because of the application of a loading in the centroid of the

beam, according to Figure 25 time history. Because of friction effect, some angular speed appears

in the axial direction of the beam (x/. When the beam suffers the first impact with the vertical plane,only its tip touches that plane, and the angular speed makes the beam suffer a vertical force in upward

direction, due to sliding friction in the vertical wall. This tip leaves the horizontal plane, loosing that

contact condition. The movement continues until the opposite beam tip touches the vertical plane,

and the same phenomenon is observed. After, beam starts shaking a lot due to excited modes in suc-cessive impacts. It stays alternating touching in the horizontal plane in both tips, sometimes flying

for a while, but always returning to the horizontal plane, due to gravitational field action. Figures 26

and 27 show the trajectories of both beam tip points, marked in bullets and crosses in both plots.

At some instants, a bar connecting both tips is shown to identify the orientation of the beam, for

clarifying the history of movement. Displacements time histories from both tip points and middle

point of the beam are shown in Figures 2830.

0

5

10

15

20

25

30

35

40

45

50

55

60

-3000

-2750

-2500

-2250

-2000

-1750

-1500

-1250

-1000

-750

-500

-250

0250

500

750

1000

1250

1500

0 2 4 6 8 10 12 14 16 18 20

z

xandy

Time

x y z

Figure 29. Displacements time history of beam point initially located at position (0,+L=2, r).

0

5

10

15

20

25

30

35

40

45

50

55

60

-2500

-2250

-2000

-1750

-1500

-1250-1000

-750

-500

-250

0

250

500

750

1000

0 2 4 6 8 10 12 14 16 18 20

z

xandy

Time

x y z

Figure 30. Displacements time history of beam point initially located at position (0,0,r).


DOI: 10.1002/nme


24/24


5. CONCLUSION

A new methodology was presented to include the rigid body motion of rotation in structural ele-

ments contact scenarios. The formulation here developed applies for a beam element contact with

a flat surface; however, the same concepts could be used in more complex interactions involving

beam to beam or shell interactions with structural or solid elements. It was shown that the proper

definition of the tangential gap function could take into account correctly the rotations occurredin the numerical examples shown, even in situations of large amount of sliding, alternating slid-

ing/rolling, and impact problems. The model can naturally be used in static nonlinear or transient

nonlinear dynamic models. The moment of friction forces and kinematics can be addressed when

there is rolling situation. This is not possible with previous contact formulations that consider only

an offset in the normal gap due to the thickness effect of structural elements. Then, the ideas and

concepts here discussed open a field of development to be continued in many situations in order to

consider properly the rigid body rotation of structural elements in contact interactions when they

are being considered in the finite element model.

ACKNOWLEDGEMENTS

The authors acknowledge FAPESP (Fundao de Amparo Pesquisa do Estado de So Paulo) for the support

under the grants 2012/09912-0 and 2012/21167-8 (PostDoc). The second author acknowledges the supportby CNPq (Conselho Nacional de Desenvolvimento Cientfico e Tecnolgico) under the grant 301279/2009-8.

REFERENCES

1. Wriggers P.Computational Contact Mechanics. John Wiley and Sons Ltd.: England, 2002. 440 p.

2. Benson DJ, Hallquist JO. A single surface contact algorithm for the post-buckling analysis of shell structures.

Computer Methods in Applied Mechanics and Engineering1990; 78:141163.

3. Wriggers P, Zavarise G. On contact between three-dimensional beams undergoing large deflections.Communications

in Numerical Methods in Engineering1997; 13:429438.

4. Zavarise G, Wriggers P. Contact with friction between beams in 3-D space. International Journal for Numerical

Methods in Engineering2000; 49:9771006.

5. Spring KW. Euler parameters and the use of quaternion algebra in the manipulations of finite rotations: a review.

Mechanism and Machine Theory1986; 21:365373.

6. Altmann SL.Rotations, Quaternions, and Double Groups. Dover Books on Mathematics: New York, 2005.7. Pimenta PM, Campello EMB. Geometrically nonlinear analysis of thin-walled space frames.Proc. of the Second

European Conference on Computational Mechanics, II ECCM, Krakow, 2001.

8. Pimenta PM, Campello EMB, Wriggers P. An exact conserving algorithm for nonlinear dynamics with rotational

dofs and general hyperelasticity. Part 2: shells. Computational Mechanics2011; 48:195211.

9. Gay Neto A, Martins CA, Pimenta PM. Static analysis of offshore risers with a geometrically-exact 3D beam model

subjected to unilateral contact. Published online inComputational MechanicsJuly 2013. DOI: 10.1007/s00466-013-

0897-9.

10. Campello EMB, Pimenta PM, Wriggers P. A triangular finite shell element based on a fully nonlinear shell

formulation. Computational Mechanics2003; 31:505518.

11. Pimenta PM, Campello EMB, Wriggers P. An exact conserving algorithm for nonlinear dynamics with rotational

dofs and general hyperelasticity. Part 1: rods. Computational Mechanics2008; 42:715732.

12. Gay Neto A, Pimenta PM, Martins CA. Loop formation in catenary risers on installation conditions: a comparison of

statics and dynamics.Proceedings of the 32nd International Conference on Ocean, Offshore and Arctic Engineering

OMAE 2013, Nantes, 2013.

13. Ansys Inc. Mechanical APDL Theory Reference 2012. version 14.

Contact Between Rolling Beams and Flat Surfaces

Documents

Transcript of Contact Between Rolling Beams and Flat Surfaces