ELSJWER

Computer methods in applied

mechanics and englnasrlng

Comput. Methods Appl. Mech. Engrg. 135 (1996) 211-228

Non-linear ‘master-slave’ relationships for joints in 3-D beams with large rotations

G. JeleniPb, M.A. Crisfieldb’* “Faculty for Architecture, Civil Engineering and Surveying, University of Ljubljana, Slovenia

bDepartment of Aeronautics, Imperial College of Science, Technology & Medicine, London, UK

Received 1 April 1994; revised 1 June 1995

Abstract

A novel approach is presented for the analysis of spatial beam elements with end releases, in which, for each joint in the structure, an additional (slave) set of kinematic variables is introduced, which is directly related to the existing (master) set of variables at that node. This relationship is established according to the nature of the joint to be modelled and takes into account that the sliding/rotation takes place along/around the axis that is rigidly attached to the structural node and is thus not fixed in space. In this way, the most interesting releases such as revolute, spherical, prismatic and cylindrical joints can be analysed accurately and efficiently. The main concepts are applicable to any spatial beam finite element, provided that a displacement vector and a rotation matrix are defined at both end nodes. The numerical examples, in which a highly deformable space frame with different joints is analysed, demonstrates the accuracy of the proposed procedure and its advantages compared to the ‘penalty’ technique.

1. Introduction

Many structures which are modelled with three-dimensional beam elements require joints at the nodes which follow the axes of the rotating system. Examples include deployable space structures, robots and rotating machinery. Two main methods of analysis are employed: those based on the finite element method and those originally developed in relation to rigid body dynamics [l]. The latter procedure can introduce special techniques that allow the introduction of flexible members [l-3] although some important potential pitfalls were pointed out by Kane et al. [3]. The provision for flexibility is increasingly being required and, in these circumstances, there is a strong case for modifying finite element systems so that they can be applied to the analysis of such structures [4]. The present paper follows this line and concentrates on the modelling of the joints or ‘hinges’. Although the work is, at present, restricted to static analysis, it is believed that the concepts can later be extended to the dynamic situation.

A large range of methods has been used for the modelling of joints. Lagrangian multipliers can be used to handle the non-linear constraint equations. The main disadvantage of this method is the introduction of a set of extra structural variables. In addition, Cardona and Geradin [5] have shown that problems can arise involving oscillations with the Lagrangian multipliers in a non-linear dynamic environment. A version of the augmented Lagrangian technique can be used to improve the convergence of the method (41 while with another version, referred to by Luenberger [6] as the ‘method

* Corresponding author.

0045-7825/96/$15.00 0 1996 Elsevier Science S.A. All rights reserved PIZ SOO45-7825(96)01017-l

212 G. J&xi?, M.A. Crisfield 1 Comput. Methods Appl. Mech. Engrg. 135 (1996) 211-22X

of multipliers’, the limitation of using extra variables can be overcome [7] at the cost of a s4ower convergence rate.

In relation to the finite element method, probably the simplest procedure is to adopt a form of ‘penalty’ approach which effectively introduces locally stiff joint elements. The limitations of such a technique will be illustrated in the paper; in particular there are problems with the choice of the penalty stiffnesses and not all of the joint types can be modelled in this way.

Many finite element formulations for joints are restricted to small deformations and, in this context, the ‘master-slave’ approach, in which dependent variables are written in terms of independent variables, can be very effective [S]. However, to cover the present class of structure, it is clear that non-linear constraints, which are associated with large rotations [9, lo], must be used. It is possible to adopt an incremental form of the linear master-slave procedure. However, to take full advantage of the large steps that can be used with the recently developed range of non-linear beam elements [ll-161, it is important that the master-slave procedure should involve a total formulation with a consistent linearisation. This is the main topic of the paper.

A large number of non-linear three-dimensional beam elements have recently been developed [11-171. It is intended that the present work should be usable with any of these elements. All that is required is that the element should have six degrees of freedom at each of the two end nodes (elements with internal nodes can also be handled, although it is probably best that they are eliminated at the element level). The six degrees of freedom should involve three translations and three ‘rotations’ associated with the definition of a nodal triad. For a direct implementation of the present work, the incremental rotation variables, that are obtained from the finite element solver should be components of the pseudo-vector associated with the skew symmetric matrix obtained from the variation of the rotation matrix [9] ( as in [ll-1.51). Other rotation variables can be used although, in these circum- stances, some modification will be required to the presented theory. Nonetheless, the concepts should remain valid.

The non-linear master-slave procedure has been developed for prismatic (sliding) joints, revolute joints (hinges), spherical joints and cylindrical joints where the adopted terminology follows that of Angeles [ 181.

2. Basis for the master-slave procedure

Suppose there is a node which is initially shared by a number of elements, one of which is not fully connected to the others. In the deformed configuration, therefore, the node is no longer completely shared and from Fig. 1 the following relations can be established:

Fig. 1. Master. slave and released nodal variables

G. JeleniC, M.A. Crisjield I Comput. Methods Appl. Mech. Engrg. 135 (1996) 211-228 213

d=d,,,+p (1)

Q=Q*Q, 9 (2)

where d, and Q, define the displacement vector and rotation matrix of the master node, and d and Q define the displacement vector and rotation matrix of the disconnected (at least partially) slave node. It should be noted that, following conventional beam theory assumptions, the origins of the vectors d,,,

and d coincide, with the gap in Fig. 1 being drawn only for the sake of illustrative clarity. The columns of the rotation matrices Q, and Q consist of orthonormal base vectors qk, qi, qi and

ql, q2, q3 (see Fig. 1):

Qm = [dn 4: dl

Q= [d d q31. In (2), the rotation matrix Q* is the matrix that defines the rotation of the master triad, q:, qi, qi, onto the slave triad, ql, q*, q3. The rotation matrices may be expressed in terms of various rotation parameters (see [9, lo]). In the following we will largely use the components of the pseudo-vector.

2.1. Axes of release

When modelling different types of joints, the master variables, d,,, and Q,, are generally not entirely independent of the slave variables, d and Q. Depending on the type of joint, some of the components of the displacement vectors, d,,, and d, and/or parameters of the rotation matrices, Q, and Q, can be the same. Different types of joints are defined by releasing displacements and/or rotations along/around chosen axes. However, in a geometrically non-linear environment, these axes rotate together with the structure which means that the releases should be defined either with respect to the triad q:, qk 45, or with respect to the triad ql, q2, q3. In the following, all the releases will be defined with respect to the master triad, qk, qk, q2.l

For translational joints, this means that the ‘difference vector’, p, between the master and slave variables (with components which are given in fixed spatial coordinates, defined by the triad e’, e*, e3) is, when transformed into coordinates defined by the master triad, equal to the vector of released displacements:

s=QT,p, (3)

where the vector of released displacements, s, has zero components in non-released directions. In a similar fashion, if from the ‘rotation difference matrix’, Q*, we extract the rotational

pseudovector /? * and transform it into coordinates defined by the master triad, the result must be equal to the rotational pseudo-vector of released rotations:

ce=QT,P* 7 (4)

which has zero components in non-released directions. To illustrate the concept further, we will initially concentrate on two basic types of joint. A prismatic

joint [18] allows sliding along a particular chosen axis. Let it be the axis that initially corresponds with the base vector e2 (See Figs. 1 and 2). When transformed into coordinates defined by the master triad, the ‘difference vector’, p from Fig. 1, must read:

0 s=Q;p= S2 ;

(i s = ISI = s2 .

0

‘The present developments are based on the assumption that the master slave triads are initially coincident. The required

kinematics is not always consistent with such an assumption. However, this problem could be overcome by defining an auxiliary

triad, initially coincident with the master triad and rigidly connected to the standard slave triad.

214 G. Jelenid, M.A. Crisfield i Comput. Methods Appl. Mech. Engrg. 135 (1996) 211-228

Fig. 2. Prismatic joint; the master triad and the released displacement

Fig. 3. Revoke joint; the nodal triads and the released rotation.

The concept is illustrated in Fig. 2.

A second basic type of joint is a revolute joint 1181 which allows rotation around one chosen axis. Let it be the axis that initially corresponds to the base vector e’ (see Figs. 1 and 3). The application of (4) then leads to the relationship:

where cp is illustrated in Fig. 3.

2.2. Relationships between the variations of the master, slave and released variables

The first step in the finite element calculation is the application of the principle of virtual work in order to obtain the internal force vector. To this end, we need to express the variations of the slave variables in terms of the master and released variables and their variations. In particular, we will aim to express Sp (associated with the variation of Q) in terms of S&,, (associated with the variation of Q,) and 6<0. The variations of the variables will be referred to as the degrees of freedom.

The variation of the rotational pseudovector, Sfl, relates to the variation of the rotation matrix in the following way (see Appendix A)

SQ = fWP)Q

The components of Sp will in future be referred to as the ‘spin’ variables. They are non-additive to /3 even when Sp tends to zero. S(Sfi) is a skew symmetric matrix in which the components of vector S/3 are arranged in the following manner

S(SP) = GX 0

[

-W, W,

-W,

-w, w, 0 I

Inserting (3) into (1) gives (note that the inverse of an orthogonal matrix equals its transpose)

d=d,,,+Q,s. (5)

Variation of this equation is (for the variation of a rotation matrix see Appendix A):

6d=6d,+6e,s+Q,6s=6d,+S(SP,)Q,s+Q,Ss,

where & is the rotational pseudovector associated with the rotation matrix Q,.

G. Jelenid, M.A. Crisjield I Comput. Methods Appl. Mech. Engrg. 135 (1996) 211-228 215

It is worth emphasising that the skew symmetric matrix multiplied by -1 equals its transpose, and that the skew symmetric matrix acts as a vector product operator, i.e. for every two vectors u, and u, in three-dimensional space, the following equation is valid

u, x u* = S(u,)u, = -S(u,)u, =

Therefore, 6d may be expressed as

w=W,,- S(Q,,dW,z+Q,~

or

-u, x u1 .

(6)

where

H = -S(Q,s) . (7)

In Eq. (6), the slave degrees of freedom are expressed in terms of the master and released degrees of freedom through the linear transformation matrix which contains master and released variables. The slave rotational degrees of freedom will be treated in a similar manner. As the rotational pseudovector associated with Q* is p*, the following relationship can be established [9, lo] (see also Appendix A)

Q*=expW*),

and (2) may be rewritten as

Q=evW*)Q,.

Inserting j?* from (4) into this equation and employing the equality exp S(Q,rp) = exp[Q,S(p)QL] =

Q, evS(dQi, which is easily confirmed by matrix multiplication using a MacLaurin series expansion from Appendix A, this equation becomes

Q =Q, expS(y?). (8)

Eq. (8) is chosen as the starting point for the application of the variations rather than Eq. (2), because we wish to find a relationship between the slave ‘spin’ variables S/? (associated with Q) and the master ‘spin’ variables Sj3, (associated with Q,) and the released ‘spin’ variables 6cp. In other words, we do not wish to introduce S/3*. Consequently, variation of (8) yields

SQ = W, exp S(V) + Q, ~[ev S&P)1 ,

which, after using the result from Appendix A, gives

s(sp)Q=s(sp,)Q,expS(4p)+Q,s(~)expS(~).

Inserting (8) into (9) and employing the properties of orthogonal matrices, we finally obtain

S(W) = S(Wm) + S(Q, s4p) 3

(9)

which can be written in terms of axial vectors associated with skew symmetric matrices as

W = W,, + Q, W

or more compactly

(10)

W = [Z Qml {‘k} . (11)

216 G. Jelenid. M.A. C&field I Comput. Methods Appl. Mech. Engrg. 1.35 (1996) 211-228

Eqs. (6) and (11) define the relationships between the slave and the ‘master and released’ degrees of freedom.

2.2. 3-D beam element with released variables

The beam element is shown in both its original and deformed configuration in Fig. 4. In relation to this figure, the following symbols have been introduced:

Left node Right node

Slave displacements Master displacements Differences

Released displacements Slave rotation matrices Slave rotational pseudovectors Master rotation matrices Master rotational pseudovectors Difference rotation matrices Difference rotational pseudovectors Pseudovectors of released rotations

The degrees of freedom for the beam are arranged in the following manner:

Slave degrees of freedom: 6p7= @d: s/3: 6d; s/3;)

Master degrees of freedom: aP;= Wl,, SK, x2 WrjL2)

Released degrees of freedom: 6~: = (8s: tiq: 6s; Sql.)

Master and released DOF: lip:, = (6s: @o: 8s: &o?‘ &IL, @3:, hi:, SPY,)

Before proceeding further, it should be re-emphasised that, in practice, not all of the released variables will in fact be ‘released’. Instead, some of them will be set to zero.

Using (6), (7) and (11) and the previously introduced symbols, the slave degrees of freedom can be expressed in terms of the master and released degrees of freedom in the following way

Fig. 4. Master, slave and released element variables

G. Jelenit, M.A. Crisfield I Comput. Methods Appl. Mech. Engrg. 135 (19%) 211-228 217

em, 0 0 0 Z H, 0 0'

0 em, 0 0 0 z 0 0

0 0 0 Z H2

0 0 0 Qm2 0 0 0 Z

(12)

where the indices ( )1 and ( )2 on the matrix H designate the corresponding node. Eq. (12) can be rewritten in compact form as

SP = @ 6p,, (13)

2.4. Element equilibrium and its linearisation

Application of the principle of virtual work to a general beam element leads to the relationship

gT 6P~ = (Qi - $1' 6P~ = O 3

where g is the residual or ‘out-of-balance’ force vector and qi and q, are, respectively, the internal and external nodal force vectors [19]. According to the definition of the principle of virtual work, the vector 6p, consists of infinitesimal and kinematically admissible variables and is thus equal to the vector Sp of (13). The introduction of (13) into this equation now leads to the work equation

gTI? 6p,, = 0.

This equation must hold for any virtual variables, i$r,, and hence we can obtain the equilibrium equation

tiTg=O,

where g is the ‘out-of-balance’ force vector for the original ‘slave’ element. The previous equation is non-linear. If we expand it using a Taylor series around the actual configuration and neglect all the higher-order terms, we obtain

ti=g+6z?=g+ti=6g=0. (14)

The term Sg in (14) is related to the iterative slave variables, Sp, via the conventional tangential relationship

%=&6p,

where K, defines the original element’s tangent stiffness matrix. If we substitute (13) into this equation and return to (14) we obtain

ti=K,fi Sp,,,, + 61?=g = -I?=g . (15)

The first term on the left-hand side of (15) gives the primary contribution to the modified tangent stiffness matrix while the term on the right-hand side gives the negative modified residual. It is shown in Appendix B that the second term on the left-hand side can be expressed in the form

6fiTg=B6p,,. (16)

Consequently, we can write

K SP,, = -gw (17)

with

218 G. JeleniC. M.A. Crisfield / Comput. Methods Appl. Mech. Engrg. 13.5 (1996) 21 l-228

K = ti I‘K,,ti + I?

g ,nr =I-Fg

It should be noted that K”, which is calculated in Appendix B, is non-symmetric away from equilibrium but for conservative loads is symmetric at equilibrium. This non-symmetry relates to the current use of the pseudo-vector terms (‘spin’ variables) as structural variables. However, as shown by Simo and Vu-Quoc [ 1 l] and Crisfield [ 121. there is no deterioration in the iterative convergence rate if an artificial symmetrisation is adopted. These conclusions related to the beam elements in [ll-1.51. It will later be shown that it also applies to the current master-slave relationship.

Mathematically, the non-symmetry of the tangent stiffness matrix when using ‘spin’ variables is caused by the linearisation procedure using the directional derivative [20], which does not require the rotation group in three-dimensional space to be a Riemannian manifold. However, if it is equipped with a specific Riemannian metric that stems from the nature of its tangent space, which is isomorphic to the

space of rotational pseudovectors, and if covariant differentiation with respect to this metric is performed, the resulting tangent stiffness matrix of a conservative problem is always symmetric [21].

The latter is equivalent to working with alternative iterative rotation variables.

2.5. Solution process

One could solve (17) in a standard manner to give an iterative Newton-Raphson change at the structural level of the form

SP,,, = -K ‘g,,,, (18)

However, the released degrees of freedom, 6p,, are only related to the individual elements and so can be solved for at the element level. At the element level, Eq. (17) can be partitioned to take the form

(19)

where the subvectors g,n and g, of g,,,, relate to the residuals associated with the master and released

degrees of freedom, respectively. Eq. (19) leads to the equation

(4, m - KJ,,‘L,,) SP,,, = -g,, + K,,KL,‘g, 3 (20)

which can be added into the structural equations, which only involve master variables, and can then be solved in the standard manner. The local released variables, 6p,, are later recovered, at the element level, via

6p, = -K,.‘g, - K,,‘K,n, &P,,, (21)

2.6. Updating the master, slave and released variables

The master translations are updated in the standard manner using

d ml.new =&I, + w,,,

d m2,new = 4n2 + W,z

The master rotation matrices are updated using Rodrigues’s formula which is related to the exponentia- tion of a skew symmetric matrix [9, lo] (see also Appendix A):

where S/3,,,, and i3&2 are the incremental rotational pseudovectors (‘spin’ variables) obtained directly

G. JeleniC, M.A. Crisjield I Comput. Methods Appl. Mech. Engrg. 135 (1996) 211-228 219

from the finite element solver as part of 6p,. The released variables are treated in a similar way. The released translations are updated through

Sl ,llew =sl+sss,

s2.new =s,+&s,

and the released rotations are indirectly updated through

expS(<o,,,,,) = exp S(M) expSM

exp GY~J = exp S(&) exp S(e) .

In these equations the quantities &s,, as,, f@i and 6% are obtained from the vector of the incremental released degrees of freedom, Sp,, via (21). To extract the released rotational pseudovectors, ql,new and

44 ,new from the last two equations, Spurrier’s algorithm [22], as given in [ll] can be employed. The algorithm is outlined in Appendix C. The slave variables are updated using (5) and (8) so that

d I,new = d ml.new + Q ml,newSl.new

d 2,new = d m2,new + Q m2.newS2,new

Q l,lEW =Q ml,new exp%bJ Q 2,new =Q m2,new exp S(%“W) .

3. Applications

Fig. 5 shows a right-angled frame and gives the adopted dimensions and geometric and material properties. The frame is loaded by means of a conservative force F of magnitude 27~. 10’ which acts at the connection of the two legs of the frame in the negative direction of the base vector e3. To test the proposed solution strategy, a number of different joint types have been inserted at the connection of the two legs.

The proposed non-linear master-slave procedure has been mounted in a research version of the finite element system LUSAS [23]. For the analysis, each leg of the structure was modelled with the aid of four first-order Timoshenko beam elements which are embedded within a co-rotational framework (see Cole [14] and Crisfield and Cole [13]). The mesh is shown in Fig. 6(a).

For each of the following examples, element 5 has been given different released freedoms in relation to the local axis corresponding to the current directions of the master triad at node 0. The latter initially corresponded with the local x, y, z axes of the element (see Fig. 6(b)). The full Newton- Raphson solution procedure was adopted and the iterations were terminated once the Euclidean norm of the iterative displacement change (considering only translational variables) was less than 10m5.

Az0.1

F = 28.10’ 4 b)

Fig. 5. Right-angle planar frame with perpendicular point load.

Fig. 6. Right-angle planar frame. (a) Finite element mesh; (b) local coordinate axes of element 5.

220 G. JeleniC, M.A. Crisfield I Comput. Methods Appl. Mech. Engrg. 13.5 (1996) 211-228

As a check on the results, the problems were re-analysed with the aid of an artificial small element that was inserted between elements 4 and 5. The local stiffnesses of this element were adjusted so as to simulate the behaviour of the required joints. This process can be seen as a form of penalty technique. In some circumstances, it was impossible to obtain converged solutions when this approach was used in conjunction with the first-order co-rotational Timoshenko elements [13,14]. In these circumstances, the higher-order mixed-type elements of Jelenid and Saje [15] were used instead.

EXAMPLE 1: Revolute joint. For this problem, the rotation around the axis that initially coincided with the local y axis of element 5 at node @ (see Fig. 6(b)) was released. Consequently, continuity of the rotation of elements 4 and 5 (see Fig. 6(a)), around the corresponding moving axis was not maintained. To help to illustrate this phenomenon, a dummy element was provided (shown as dashed in Fig. 7) which was rigidly attached to element 4 at node 0. In Fig. 7, the angle between the centroidal axes of element 5 and the dummy element is designated by cp’. In the case of a shear-stiff structure, this angle coincides with the angle between the cross sections of the two elements, i.e. with the released rotation, cp. For this problem, the final configuration was reached in five equal loading steps with the results for the displacement and the released rotation at node @ being given in Table 1, which also shows the required number of iterations for each load step. The number of iterations in Table 1 relates to the solution obtained with a symmetrised effective tangent stiffness matrix. An identical number of iterations was required at each step when the full non-symmetric formulation was adopted.

Fig. 7. Example 1: Initial and final configuration of the frame and the released rotation

Table 1

Revolute joint

d = me’ + ve’ + we3

Increment Loading factor Number of iterations u ” w v

1 0.2 5 -0.0147 0.0152 -0.1595 0.2441 2 0.4 5 - 0.0492 0.0509 -0.2861 0.4495 3 0.6 5 -0.0884 0.0915 -0.3752 0.6072 4 0.8 5 -0.1249 0.1291 -0.4364 0.7256 5 1 .o 5 -0.1565 0.1616 -0.4794 0.8151

G. JeleniC, M.A. Crisfield I Comput. Methods Appl. Mech. Engrg. 135 (1996) 211-228 221

For the equivalent ‘penalty based’ solution, the joint was simulated via a small element e of the length 0.0001 between elements 4 and 5. Rotation around the axis that initially coincided with the local y axis of element 5 was allowed by nullifying the corresponding moment of inertia of the element e (see figure in Table 2). All the other translations and rotations within this element were reduced by enlarging the remaining geometric properties by the factor of 100 with respect to those from Fig. 5. The results are tabulated in Table 2.

In the final configuration, the results for the two models differ by -0.4% for the w component of the displacement vector, by -1.2% for its u and u components and by -3.6% for the released rotation, 4p. The ‘penalty’ simulation required -84% more iterations to achieve equilibrium.

EXAMPLE 2: Prismatic joint. A prismatic joint was modelled by releasing the displacement along the axis that initially coincided with the local x axis of element 5 at node @ (see Fig. 6(b)). A dummy element was again added to emphasise that in the final configuration the released translation lies almost along element 5’s centroid axis (s’ in Fig. 8). To be precise, it takes place in a direction that lies along the normal to element 5’s cross section at node @ (s in Table 3). For the given geometric properties, the difference between s’ and s cannot be seen. The displacement of node @ and the magnitude of the

Table 2 Simulation of revolute joint

4

A = 10.0

A, = 10.0

A, = 10.0

It = 1.667.10-*

IV = 0.0

I, = 8.333. 1O-3

1. = 0.0001

Increment Loading factor Number of iterations U V w Q

1 0.2 9 -0.0147 0.0153 -0.1596 0.2448 2 0.4 10 -0.0494 0.0512 -0.2867 0.4540

3 0.6 9 -0.0892 0.0923 -0.3764 0.6187 4 0.8 10 -0.1262 0.1305 -0.4381 0.7458 5 1.0 8 -0.1583 0.1635 -0.4813 0.8442

Fig. 8. Example 2: Initial and final configuration of the frame and the released translation.

222 G. Jelenit. M.A. Crisfield I Comput. Methods Appl. Mech. Engrg. 135 (1996) 211-228

Table 3

Prismatic joint

Increment Loading factor Number of iterations U U w S

d = ue’ + IX? + we3

I 0.2 6 -0.0115 -0.0109 -0.1409 0.0222

2 0.4 7 -0.0480 -0.0335 -0.2848 0.0793 3 0.6 8 -0.1057 -0.0513 -0.4164 0.1495 4 0.8 6 -0.1716 -0.0594 -0.5219 0.2162 5 I .o 5 ~0.2352 -0.0610 ~0.6016 0.2738

released translation are given in Table 3 along with the required number of iterations. When the effective tangent stiffness matrix was symmetrised, the required number of iterations remained fixed for each increment except the last, when it was reduced by one. Table 4 shows how the iterative displacement norm and the residual force norm, expressed as percentages of the norm of total displacements and the norm of external forces (including reactions), change as the iterations proceed for the fourth increment.

To provide a basis for comparison, the joint was simulated in a similar manner to that adopted for the previous example. A small element e was again inserted between the two legs of the structure. The adopted axial stiffness related to a very small cross sectional area of A = lo-‘“, while all of the remaining geometric properties from Fig. 5 (shear areas and moments of inertia) were increased by a factor of 100 (see figure in Table 5). With these modifications it was impossible to obtain converged solutions when the corotational first-order Timoshenko elements [13,14] were used. Consequently, as an alternative, a similar small element was used in conjunction with a modelling involving finite-strain beam elements in which polynomials of the fifth degree were used to interpolate the distribution of the rotational pseudovector field [15]. The legs of the structure and the small element e have each been modelled using a single higher-order element. The displacement at node 0 and the elongation of element e are tabulated in Table 5 in conjunction with the required number of iterations.

A comparison of the results in Tables 3 and 5 shows that, apart from the results of the u component of the displacement vector (by far the smallest component), which differ by -4.6%, all of the remaining results, together with the Euclidean norm of the displacement vector, match one another up

Table 4

Relative iterative displacement and residual force norms during increment 4 of Example 2

Iteration Relative iterative displacement norm [%I Relative residual force norm [ %]

0 2.3659’10’ 6.0676’10’ I 3.6504. 10” 3.4051’ 10’ 2 4.3794 10” 1.4924.10’ 3 5.7152 lO_ 8.3236.10-’ 4 6.6135. 10 ’ 4.4372. lo-’ 5 1.4512. 10 -I 5.6078. lo-’ 6 4.8251 10 -” 2.5340 lo-”

G. JeleniC, M.A. Crisfield I Comput. Methods Appl. Mech. Engrg. 135 (1996) 211-228 223

Table 5 Simulation of prismatic joint (using higher-order elements [15])

A = 1(1-‘~

A, = 10.0

A, = 10.0

It = 1.667. lo-’

I, = 8.333. 1O-3

I, = 8.333. 1O-3

d=ue’+ve2+we3

Increment Loading factor Number of iterations L( u w s

1 0.2 5 -0.0121 -0.0117 -0.1442 0.0236 2 0.4 5 -0.0503 -0.0357 -0.2899 0.0834 3 0.6 5 -0.1091 -0.0541 -0.4202 0.1547 4 0.8 5 -0.1741 -0.0623 -0.5221 0.2202 5 1.0 5 -0.2355 -0.0638 -0.5977 0.2751

to a precision of -0.6%. In this instance, the better convergence rate for the simulated model cannot be used to draw any conclusions regarding the comparative performance of the master-slave and ‘penalty’ models but rather relates to the more sophisticated nature of the higher-order elements.

It is worth noting that while the proposed method is able to handle prismatic joints in arbitrary released directions (not only along the centroidal axis but also along the principal axes of inertia of the cross section), the ‘penalty’ technique, via beam elements, is not. Nullifying the corresponding shear stiffness itself does not enable transverse sliding, because there is still a bending stiffness that prevents it. If the latter is nullified too, not only transverse sliding but also the rotation around the principal axis of inertia associated with the nullified bending stiffness is released.

EXAMPLE 3: Cylindrical joint. A cylindrical joint allows translation along a particular chosen axis as well as rotation around the same axis, both quantities being independent from one another. In this example, the chosen axis is that which initially coincided with the local x axis of element 5 at node @ (see Fig. 6(b)). In this example, the deformed centroidal axis of the structure cannot be used to help illustrate the two releases and hence it has not been plotted. However, the results are tabulated in Table 6. From the column containing the required number of loading steps, it is apparent that this problem is more demanding than the previous ones.

As with the previous example, converged solutions could not be obtained for this problem when a ‘penalty type’ simulation was attempted in conjunction with linear corotational Timoshenko elements. Indeed, even when the more sophisticated, higher-order elements of [15] were used, considerable numerical difficulties were encountered. In particular, it was found that the cross-sectional area could not be reduced to the same level as before and that the bending moments of inertia could not be enlarged without introducing numerical problems. Finally, the cylindrical joint was satisfactorily simulated by reducing the cross-sectional area and torsional moment of inertia of the element e to 10-l’ and by enlarging the shear areas by the factor of 100 with respect to those from Fig. 5 (see figure in Table 7). However, using the simulated ‘penalty’ model, it was found that twenty load steps were needed to reach the final configuration. Table 7 tabulates the results for the five characteristic load levels used for the previous analyses.

The results for the u component of the displacement vector of node @ differ in the two models by almost 10%. However, this is by far the smallest of the components and scarcely contributes to the

224 G. JeleniC. M. A. Crisfield I Comput. Methods Appl. Mech. Engrg. 135 (1996) 21 l-228

Table 6

Cylindrical joint

d = ue’ + u.2 + ,,=3

Increment Loading factor Number of iterations U u w s cp

2 0.2 5 -0.0166 -0.0087 -0.1666 0.0245 0.2472

4 0.4 4 -0.0648 -0.0238 -0.3251 0.0839 0.4630

6 0.6 4 -0.1321 -0.0327 -0.4568 0.1524 0.6267

8 0.8 4 -0.2023 -0.0352 -0.5555 0.2155 0.7420

10 1.0 4 -0.2668 -0.0344 -0.6276 0.2693 0.8220

Table 7

Simulation of cylindrical joint (using higher-order elements [ 151)

z, A = lo-”

A, = 10.0

A, = 10.0

It = 10-t’

I, = 8.333. lo-

I. = 8.333. lo-’

w P s 1. = 0.0001

d = uel + uea + we3

Increment Loading factor Number of iterations U ” w s cp

4 0.2 6 -0.0174 -0.0097 -0.1699 0.0263 0.2481

8 0.4 6 -0.0675 -0.0263 -0.3300 0.0865 0.4637

12 0.6 7 -0.1357 -0.0360 -0.4602 0.1583 0.6262

16 0.8 5 -0.2049 -0.0387 -0.5556 0.2203 0.7313

20 1.0 4 -0.2670 -0.0378 -0.6240 0.2717 0.8180

difference in results for the Euclidean norm of the displacement vector, which is -0.45%. The two other components, u and W, and the released translation and rotation, s and cp, match up to a precision of -0.9%.

4. Conclusions

This paper has described the theory for a non-linear master-slave procedure that can be used for joints with three-dimensional beam elements and large rotations. The procedure is independent from

G. JeleniC, M.A. Crisfield I Comput. Methods Appl. Mech. Engrg. 135 (1996) 211-228 225

the original element formulation, although to be applied directly, the iterative rotation variables must be components of the non-additive pseudo-vector (‘spin’ variables).

To test the technique, alternative solutions have been obtained using a form of ‘penalty’ technique, in which a small element is used to simulate the joint with the aid of ‘artificial’ stiffnesses. Close agreement has been obtained. However, while no numerical difficulties were encountered with the master-slave procedure, considerable difficulties were encountered with the ‘penalty’ technique. In particular, a trial and error system was required to find satisfactory stiffnesses and, in addition, significantly more iterations were required. Indeed, when applied with the original first-order Timoshenko corotational elements (as used for the master-slave procedure), it was sometimes impossible to obtain converged solutions with the ‘penalty method. Besides, due to additional joint elements, the system of equations to be solved has more degreees of freedom. Another important drawback of the ‘penalty’ technique is that it cannot easily be used to model certain types of joints.

Although the present work has been limited to statics, work is in hand in extending the approach to dynamics where the real returns are likely to lie.

Acknowledgment

This work has been financially supported by Joint European Project TEMPUS - ACEM No. 2246-91 and by EPSRC.

Appendix A. Variation of the rotation matrix

It is known from the theory of large rotations [9, lo] that two consecutive rotations expressed in terms of their rotation martices R and R, (with index ( ), for ‘new’), can be related to each other via a rotational pseudovector 6 which twists the triad defined by R onto the triad defined by R, in the following manner:

R,=expS(6)R= sin 6

Z+7S(6)+ ’ -I:’ S(lt)S(S)]R , (A.11

where the second form is particularly useful for numerical implementation. In (A.l) S(6) is skew symmetric matrix associated with rotational pseudovector 8 in the following way:

If the new rotation matrix, R,, is understood as R + AZt and the exponential function expressed in the form of a MacLaurin’s series, Eq. (A.l) becomes

R+AR= Z+S(8)+$S(9)2++,S(b)3+-- R. 1

In the case of an infinitesimal rotational pseudovector, 66, the finite change in the rotation matrix, AR, becomes a variation of the rotation matrix, ZiR. After replacing 6 by 619 and bR by 8R in the previous equation, and neglecting higher-order terms, the following expression is obtained for the variation of the rotation matrix:

6R =S(W)R .

Appendix B. Derivation of B

Using the results from Appendix A and taking into accou_nt that the skew symmetric matrix acts as a vector product operator, variations of the submatrices of H (see (12) and (13)) read

226 G. JeleniC. M.A. Crisfield I Compui. Methods Appl. Mech. Engrg. 135 (1996) 211-228

wm, = S@Pm,>Q,,

%&?I* = WPm*)Qm7

After inserting these expressions into (16), a long but straightforward derivation, in which the vector product operator nature of the skew symmetric matrix and algebraic properties of single and double vector products are employed, finally gives

K”=

0 0 0 0 0 Q;,S(g,) 0 0 0 0 0 0 Q;,S(g,) 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

-Sk, IQ,, 0 0 0 0 Q,,w: -dQ,,s,z 0 0 0 0 0 0 0 0

0 0 -Sk>Q,z 0 0 0 0

where g, , g,, g, and g, are subvectors of the original element’s vector of residuals

EC= <g: g: s: d>.

Appendix C. Extracting the rotational pseudovector from the rotation matrix

To extract the components of rotational pseudovector 19 from the rotation matrix

R=

Spurrier’s algorithm [22], as presented in [ll] is utilised: (1) Calculate

a = max(R,,. R,,, R,,, R,, + R,, + R,,)

(2) If a = R,, + R,, + R,, then calculate

q,= -qe,,,R,,; i = 1,2,3,

where erlk is permutation symbol [24] and indices are summational. If a ZR,, + Rz2 + R,, but instead a = Ri, then calculate

4 = & CR,, - R,,) I

4, = + CR,, + R,,) I

G. JeleniC, M.A. Crisfield I Comput. Methods Appl. Mech. Engrg. 135 (1996) 211-228 227

1 qk = 4q, @ki + Rik) 7

where i, j and k are again the cyclic permutation of 1, 2 and 3. (3) Calculate the norm and the components of the rotational pseudovector 9 through

6 = 2 arccos 4

6 q=-

6 4i . sin -

2

The quaterion components 4, ql, q2, q3, introduced in the second step of the algorithm, are used here only as intermediate variables during the transition from the components of the rotation matrix to the components of rotational pseudovector. However, sometimes they are used to optimise the numerical procedure as they require less disk space than do the components of the rotation matrix.

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t11 PI

[31

141

PI

PI 171

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191 [lOI [I11

u21

1131

t141

[151

tW

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