a kinematically exact space finite strain beam model - finite … · 2007-03-28 · a kinematically...

Computer methods in applied

mechanics and englneerlng

Comput. Methods Appl. Mech. Engrg. 120 (199.5) 131-161 ELSEVIER

A kinematically exact space finite strain beam model - finite element formulation by generalized virtual work principle

G. JeleniC, M. Saje* Faculty of Architecture, Civil Engineering and Surveying, University of Ljubljana. SLO-61OW Ljubljana. Jamova 2, Slovenia

Received 30 June 1993

Revised 6 April 1994

Abstract

The present paper presents a novel finite element formulation for static analysis of linear elastic spatial frame structures

extending the formulation given by Simo and Vu-Quoc [A geometrically-exact rod model incorporating shear and torsion-warping

deformation, Int. J. Solids Structures 27 (3) (1991) 371-3931, along the lines of the work on the planar beam theory presented by

Saje [A variational principle for finite planar deformation of straight slender elastic beams, Internat. J. Solids and Structures 26

(1990) 887-9001. We apply exact non-linear kinematic relationships of the space finite-strain beam theory, assuming the Bernoulli

hypothesis and neglecting the warping deformations of the cross-section. Finite displacements and rotations as well as finite

extensional, shear, torsional and bending strains are accounted for in the formulation. A deformed configuration of the beam is

described by the displacement vector of the deformed centroid axis and an orthonormal moving frame, rigidly attached to the

cross-section of the beam. The position of the moving frame relative to a fixed reference frame is specified by an orthogonal

matrix, parametrized by the rotational vector which rotates the moving frame from an arbitrary position into the deformed

configuration in one step. Also, the incremental rotational vector is introduced, which rotates the moving frame from the

configuration obtained at the previous iteration step into the current configuration of the beam. Its components relative to the

fixed global coordinate system are taken to be the rotational degrees of freedom at nodal points. Because in 3-D space both the

axial and the follower moments are non-conservative, not the variational principle but the principle of virtual work has been

introduced as a basis for the finite element discretization. Here we have proposed the generalized form of the principle of virtual

work by including exact kinematic equations by means of a procedure, similar to that of Lagrangian multipliers. This makes

possible the elimination of the displacement vector field from the principle, so that the three components of the incremental

rotational vector field remain the only functions to be approximated in the finite element implementation of the principle. Other

researchers, on the other hand, employ the three components of the incremental rotational vector field and the three components

of the incremental displacement vector field. As a result, more accurate and efficient family of beam finite elements for the

non-linear analysis of space frames has been obtained. A one-field formulation results in the fact that in the present finite

elements the locking never occurs. Any combination of deformation states is described equally precisely. This is in contrast with

the elements developed in literature, where, in order to avoid the locking, a reduced numerical integration has to be applied,

which unfortunately, diminishes the accuracy of the solution. Polynomials have been chosen for the approximation of the

components of the rotational vector. In this case the order of the numerical integration can rationally be estimated and the

computer program can be coded in such a way that the degree of polynomials need not be limited to a particular value. The

Newton method is used for the iterative solution of the non-linear equilibrium equations. In an non-equilibrium configuration, the

tangent stiffness matrix, obtained by the linearization of governing equations using the directional derivative, is non-symmetric

even for conservative loadings. Only upon achieving an equilibrium state, the tangent stiffness matrix becomes symmetric. Thus,

obtained tangent stiffness matrix can be symmetrized without affecting the rate of convergence of the Newton method. For

non-conservative loadings, however, the tangent stiffness matrix is always non-symmetric. The numerical examples demonstrate

capability of the present formulation to determine accurately the non-linear behaviour of space frames. In numerical examples the

out-of-plane buckling loads are determined and the whole pre-and post-critical load-displacement paths of a cantilever and a

right-angle frame are traced. These, in the analysis of space beams standard verification example problems, show excellent

accuracy of the solution even when employing only one element to describe the displacements of the size of the structure itself,

the rotations of 2n, and extensional strains much beyond the realistic values of linear elastic material.

* Corresponding author.

0045-7825/95/$09.50 0 1995 Elsevier Science S.A. All rights reserved SSDI 0045-7825(94)00056-S

132 G. JeleniC, M. Saje I Comput. Methods Appl. Mech. Engrg. 120 (1995) 131-161

1. Introduction

The finite element modeling of the behaviour of frame structures has gained wide attention both in science and industry. A number of papers has recently been published, presenting new concepts and new algorithms for modeling highly flexible spatial frame structures [l, 2, 4, 6, 7, 10, 12, 13, 16-20, 2.51. These structures are made of elastic-plastic materials with high yield strength, have low mass and are very flexible, so that large displacements, rotations, and sometimes large strains have to be considered. An extensive account of previous work on the subject was given by Saleeb et al. [15]. The present paper presents a novel finite element formulation for static analysis of linear elastic spatial frame structures extending the formulation given by Simo [17] and Simo and Vu-Quoc [18], along the lines of the recent work on the planar beam theory presented by Saje [14]. As in [17, 181, we apply exact non-linear kinematic relationships of the space finite-strain beam theory, assuming the Bernoulli hypothesis and neglecting the warping deformations of the cross-section. Finite displacements and rotations as well as finite extensional, shear, torsional and bending strains are accounted for in the formulation.

A deformed configuration of the beam is described by the displacement vector of the deformed centroid axis and an orthonormal moving frame, rigidly attached to the cross-section of the beam. The position of the moving frame relative to a fixed reference frame is specified by an orthogonal matrix, here, as in [17, 181, parametrized by the rotational vector which rotates the moving frame from an arbitrary position into the deformed configuration in one step. The rotational vector is, for this type of problem, an optimal choice of the parametrization, because it has a simple geometric meaning, does not introduce singularity at any magnitude of the rotation, the set of parameters is minimal, and its components, the nodal rotations around the fixed axes, are convenient degrees of freedom for production computer programs. Other parametrizations employ other parameters, e.g. the Euler angles, the Euler parameters or the Rodrigues parameters [3, 231. A particular selection of the parameters has a direct influence on the algorithm and the structure of the tangent stiffness matrix (see [15] for the discussion). Once the rotational vector has been determined, the associated exact value of the rotation matrix is obtained by the aid of the Rodrigues formula. Due to the non-linearity of the problem, the solution procedure must be incremental and iterative; therefore the incremental rotational vector is introduced, which rotates the moving frame from the configuration obtained at the previous iteration step into the current configuration of the beam. Its components relative to the fixed global coordinate system are taken to be the rotational degrees of freedom at nodal points.

Because in 3-D space both the axial and the follower moments are non-conservative, not the variational principle but the principle of virtual work has been introduced as a basis for the finite element discretization. We have here proposed the generalized form of the principle of virtual work by including exact kinematic equations by means of a procedure, similar to that of Lagrangian multipliers. This makes possible the elimination of the displacement vector field from the principle, so that the three components of the incremental rotational vector field remain the only functions to be approximated in the finite element implementation of the principle. The ambiguities concerning the order of polynomial approximations for physically different variables, i.e. translational and rotational variables, are thus avoided. In [4, 10, 13, 15-181, on the other hand, the three components of the incremental rotational vector field and the three components of the incremental displacement vector field have to be approximated. As a result more accurate and efficient family of beam finite elements for the non-linear analysis of space frames has been obtained.

A one-field formulation results in the fact that in the present finite elements the locking never occurs. Any combination of deformation states is described equally precisely. This is in contrast with the elements developed in [17, 181 for example, where, in order to avoid the locking, a reduced numerical integration has to be applied, which unfortunately, diminishes the accuracy of the solution.

We have chosen polynomials for the approximation of the components of the rotational vector. In this case the order of the numerical integration can rationally be estimated and the computer program can be coded in such a way that the degree of polynomials need not be limited to a particular value. M-point Gaussian integration is recommended and shown to be sufficient for elements with (M - I)th degree interpolation polynomials.

G. Jelenid, M. Saje I Comput. Methods Appl. Mech. Engrg. 120 (1995) 131-161 133

The Newton method is used for the iterative solution of the non-linear equilibrium equations. As in [6,13,18] but different than in [2,4,10,15,16], in an iteration step, in an non-equilibrium configuration, the tangent stiffness matrix, obtained by the linearization of governing equations using the directional derivative, is non-symmetric even for conservative loadings. Only upon achieving an equilibrium state, the tangent stiffness matrix becomes symmetric. Simo [21] shows that thus obtained tangent stiffness matrix can be symmetrized without affecting the rate of convergence of the Newton method, and gives a rigorous mathematical justification for the symmetrization. For non-conservative loadings, however, the tangent stiffness matrix is always non-symmetric.

The numerical examples demonstrate capability of the present formulation to determine accurately the non-linear behaviour of space frames. in numerical examples the out-of-plane buckling loads are determined and the whole pre- and post-critical load-displacement paths of a cantilever and a right-angle frame are traced. These, in the analysis of space beams standard verification example problems, show excellent accuracy of the solution even when employing only one element to describe the displacements of the size of the structure itself, the rotations of 21r, and extensional strains much beyond the realistic values of linear elastic material.

2. Kinematics

2.1. Basic dejinitions and kinematic assumptions

Let the line of the centroids of cross-sections of the undeformed beam element be a straight line, and let it coincide with the z axis of the fixed Cartesian coordinate system (x, y, z) with g, , g,, g, as the unit base vectors (Fig. 1). The position vector of a material particle (O,O, z) on the line of centroids is denoted by r,,(z). The cross-sections of the undeformed beam in the coordinate plane z = const are perpendicular to the line of centroids. Their normals coincide with the base vector g,. The remaining two base vectors, g, and g,, are taken to be directed along the principal axes of inertia of the cross-section. The geometric shape of the cross-section is assumed to be arbitrary and constant along the axis of the beam. Only for the sake of clearness the cross-section plotted in Fig. 1 is rectangular.

In the deformed state, the line of centroids is a space curve. The position vector of a material particle

/ 'Y

XI

,g$Ejgg- k L

z

Fig. 1. The reference coordinate system.

Fig. 2. The moving basis and the deformed configuration of the beam

134 G. JeleniC, M. Saje i Comput. Methods Appl. Mech. Engrg. 120 (1995) 131-161

(0, 0, z) on the deformed line of centroids is denoted by r(z) (Fig. 2). According to the Bernoulli hypothesis the plane cross-sections suffer only rigid rotation during deformation and remain plane after deformation and preserve their shape and area. It is therefore convenient to introduce an orthonormal

basis G,(Z), G2(z), G3( z o a cross-section at z, termed the moving basis, such that G, is normal to the ) f rotated cross-section, and G, and G, lie in the plane of the rotated cross-section. G, and G, are taken to be directed along the principal axes of inertia of the cross-section. As a result of shear deformations of the beam, the cross-sections are not perpendicular to the line of centroids. Due to the orthonormality of the moving basis, the base vectors satisfy the following equations

G,G,=S, (i,j=l,2,3); G,=G, xG,. (I)

8, denotes Kronecker’s delta [22]. The position vector R of an arbitrary material particle (x, y, z) of the deformed beam may now be written as

W, Y, 2) = 4~) + xG, (2) + yG,@) . (2)

The deformed configuration of the beam is thus completely defined by (i) the position vector of the deformed line of centroids, and (ii) the orientation of the moving basis with respect to the fixed coordinate system.

Because the fixed and the moving bases are orthonormal, they are, for any z, related by an orthogonal transformation A by the relation

Go = A(Z)g; =

Here, A is a linear operator in Pi?‘, satisfying the relations

n’n=nn’=z; detA=l; AP’=A’.

Its components are often represented by direction cosines

A= [a:; ;;I ;#[~,%~ =;,i ,l,;;],

(4)

in the form

where ‘Y;, denotes the angle between the base vectors g, and G,. The matrix of components of the operator A will be referred to here as the rotation matrix. Note that A belongs to the Lie group of proper orthogonal transformations SO(3).

2.2. Parametrization of the rotation matrix

Since the rotation matrix is a proper orthogonal matrix in %! 3, its nine components can be expressed by only three independent parameters. This is called the parametrization of the rotation matrix. There is a number of choices for the parametrization, the Euler angles, the quaternion parameters, and the rotational vector being the most usual [23]. Here, as in Simo [ 171 and Simo and Vu-Quoc [18], we employ the rotational vector.

Assume that the deformation process moves the beam through a sequence of deformed configurations. Let the moving basis of the cross-section z at the kth configuration be specified by the base vectors G Ik’ (i = 1,2,3). The rotational vector 6 is introduced as the vector

6=6e, (5)

which rotates the base vectors Gjkl into the base vectors G/‘+” at the k + 1st configuration. In Eq. (5), e is the unit vector of the rotational axis, and 6 is the rotational norm or the length of the rotational vector. The representation of the rotational vector with respect to the fixed basis is

G. JeleniC, M. Saje I Compuf. Methods App/. Mech. Erzgrg. 1.20 (1995) B-161 135

4 6=%&r, +fi~&+~~&= 4 ; 11 I9 = (aI= <I?; + s; + ,y2. (6)

4

Similarly, the rotational vector 6* rotates the base vectors G; tk+” into the base vectors Gjk+21 at the k + 2nd configuration. The compound rotational vector, which rotates the basis from kth to k + 2nd configuration, is generally not a sum of the two rotational vectors. The composition of space rotations is therefore not represented by the sum of the corresponding rotational vectors. This fact is sometimes emphasized by the use of the term rotational pseudo vector. In accord with the standard notations, we will, however, call it the rotational vector.

Using the rotational vector, the rotation matrix A is determined by the expression [3, 231

the rotational vector 6

Note that for an arbitrary vector u E 6% 3 the following identity can be established

6xv=&.

An expansion of trigonometric functions in Eq. (7) in MacLaurin’s series yields

. = exp 0 .

(8)

(9)

(10)

Thus, the rotation matrix may alternatively be expressed by the exponentiation of the skew-symmetric matrix associated with the rotational vector. Note that, as a consequence of the exponentiation of the skew-symmetric matrix 0 being equal to A E SO(3), the skew-symmetric matrix associated with the rotational vector belongs to the Lie algebra SO(~) associated with the Lie group SO(3) [8]. As exp 0 is an orthogonal matrix, its inverse equals its transpose and the following expression can be derived from Eqs. (10) and (7)

(exp 0))’ = (exp 0)’ =exp(-0) . (11)

Relating the bases of the two consecutive configurations, k and k + 1, as in Eq. (3), and employing Eq. (lo), yields

Gjk+‘l = exp @Gikl . (12)

In Eq. (12) 0 is the skew-symmetric matrix associated with the rotational vector, which rotates the base vectors Gi”’ into the base vectors Gik+rl. The application of Eq. (3) for the configurations k and k + 1 gives

Glk’ = Alklg, ,

$+I, = A[k+‘lg, I I .

(13)

By inserting Eqs. (13) into Eq. (12), we obtain

Alk+ ‘1 = exp @/ilk1 (14)

Eq. (14) gives the rule for the determination of the rotation matrix at the k + 1st configuration using the rotation matrix of the configuration k and the rotational vector 6. The algorithm for the numerical solution of the beam problem, which is addressed in Section 5, considers the configuration k to be known and the configuration k + 1 to be unknown. Thus, by the help of Eqs. (14) and (8), the rotation matrix at z at the current configuration k + 1 of the beam is determined by three components of the

136 G. Jelenid, M. Saje I Comput. Methods Appl. Mech. Engrg. 120 (1995) 131-161

rotational vector 19 at z. These components, relative to the global fixed coordinate system, will play a role of rotational parameters (the degrees of freedom) of the beam.

In terms of the rotational vector 6, Eqs. (7) and (8) give the exact value of the current rotation matrix. Using truncated MacLaurin’s series of various order in Eq. (lo), approximate values of the rotation matrix are obtained and corresponding simplified theories are derived. For example, a so called lst-order theory is obtained if small rotations are assumed so that quadratic and higher order terms in Eq. (10) may be neglected. An approximate rotation matrix in the form

is then derived. Second-order theory uses the relation

Al”“‘& ( z+@+1@2 API, 2 )

The rotation matrices derived this way are, however, non-orthogonal, consequently the moving basis loses its orthonormality. This introduces an error in the analysis, which results in the fact that the solution may not converge to an exact solution of the problem or does not converge at all.

2.3. Configuration space

As already mentioned, the deformed configuration of the beam is specified by the position vector of the line of centroids, and the orientation of the moving basis with respect to the fixed basis. The position vector is an element of Euclidean linear vector space 6%‘. The orientation of the moving basis is represented by the rotation matrix, which is an element of the Lie group SO(3). Accordingly, the set of all possible configurations of the beam is defined by

%Y = {(r, A) Ir: [0, L]+%?,A: [O, L]+SO(3)} .

This set is here referred to as the configuration space. The quantities r and A (or 6) are termed the kinematic quantities of the beam. In Section 2.2 we showed by Eq. (7) that the rotation matrix was related to the three real parameters, the components of the rotational vector 19. Thus, the Lie group SO(3) of rotation matrices is three-parametric, i.e. it may be viewed as being a three-dimensional non-linear differentiable manifold. The configuration space %’ is therefore not a linear vector space. In order to perform any numerical calculations, the configuration space has to be linearized. This is considered in Section 5.2.

3. Strains, stress resultants and constitutive relations

In this section we introduce strain and stress measures which will later on be employed for the determination of the internal virtual work. First, we introduce the translational strain vector y and the rotational strain vector K at a cross-section z. These vectors are referred to the moving basis by the equations

YI Y=~/,G,+Y~G~+EG~= 1’2 ,

(i (l-9

E

Kl

K = K,G, + K&, + K&, =

11

K2 . (16)

K3

Here, ‘yr and yZ are shear strains in the directions of base vectors G, and G, of the moving basis, K, and ~~ are corresponding bending strains, F is extensional strain, and ~~ is torsional strain, all at material particle z of the line of centroids of the beam. Note that only when both, the shear and the extensional strain, are zero, the bending strains K, and ~2 represent the components of the bending curvature vector

G. JeleniC, M. Saje I Comput. Methods Appl. Mech. Engrg. 120 (1995) 131-161 137

of the deformed line of centroids in the directions of base vectors G, and G,, and K~ is then its torsional curvature about G,. The strain vectors y and K will be termed the deformation quantities of the beam.

The deformation quantities y and K are related to the kinematic quantities r and A (or S) by the equations given in Tables 1 and 2 of [18]

0 y= A’r’- 0 , (i 1 (17)

K = Ii’U ; (18)

J2 = A’At ; (19)

We emphasize that these relations are, within the basic kinematic assumption (2), mathematically exact. The vector o, introduced in Eq. (18), is the rotational strain vector K given with respect to the fixed basis. Note that o is a function of A, or via Eqs. (7) and (8), a function of 6. The prime (‘) denotes the derivative with respect to z. In Eq. (19) the skew-symmetric matrix J2 is associated with the vector w.

Next, stress resultants over a cross-section at z are introduced. We define a stress resultant termed a cross-sectional force N, and a stress-couple resultant, designated a cross-sectional moment M. These vectors are referred to the moving basis by the equations

N = N,G, + N2G2 + N,G, = ,

M = M,G, + M,G, + M,G, = .

(20)

(21)

The components N,, N2, N3, M,, M,, and M, of N and M have a clear physical meaning. N, and N2 are shear forces in the directions of the base vectors G, and G,, respectively. M, and M, are bending moments. Nj is the normal force, and M, is the torsional moment at the cross-section. Notice that N3 is, strictly speaking, not an axial force, because the tangent to the centroid axis does not coincide with the normal to the cross-section.

A linear elastic material is assumed. The constitutive law for elastic material is taken to be given by a linear relation between the stress resultants and strains as

Here, C, and C, are constitutive matrices of the relations between translational strains and cross- sectional forces, and rotational strains and cross-sectional moments, respectively. Both matrices are taken to be diagonal with constant, stress independent coefficients. In Eqs. (22) and (23) E and G denote elastic and shear moduli of material. A 1 and A, are the shear areas in the principal directions 1 and 2 of the cross-section; J, and I2 are the cross-sectional inertial moments about the principal directions 1 and 2; A is the cross-sectional area and J, is the torsional inertial moment of the cross-section.


4. Equilibrium equations and their weak form

4.1. Principle of virtual work

Consider a straight beam of initial length L, subjected to prescribed external distributed force and moment vectors per unit of the undeformed length of the centroid line, rz and m, and to external point loads S and point moments M at the boundaries z = 0 and z = L. The following principle of virtual work may then be stated

L L

(Sy ‘A’ + SKIM) dz = (Sr’n + 66’m) dz + Gr”‘S” + 66”‘M” + i3rL’SL + S6LfM” , (24)

where 6y and SK are variations of deformation quantities, i.e. of the strain vectors y and K; 6r and 613 are variations of kinematic quantities, the position vector and the rotational vector, r and 6; 6r” and 6rL are variations of the position vector at z = 0 and z = L; So and SL are external point loads at z = 0 and z = L; 619~ and MJL are variations of the rotational vector at z = 0 and z = L; M” and ML are external point moments at z = 0 and z = L. In Eq. (24) the deformation quantities, y and K, need be expressed by the kinematic quantities r and 19 via Eqs. (17)-( 19). So the principle, defined by Eq. (24), is a function of two vector functions, r(z) and 6(z).

4.2. Generalized principle of virtual work

Eqs. (17)-(19) may be looked upon as being two sets of constraining equations for the set of four vector variables, r, 6, y and K. Assume that the set of equations (18) and (19) is exactly satisfied so that K can accurately be expressed by 6. Eq. (17) will then be the only constraining equation

0 r’-A 0

0 -Ay=O (25)

1

for the remaining variables r, 6 and y, which will now become mutually independent. Following the method similar to that of Lagrangian multipliers in constrained problems of calculus of variation, an independent continuous and at least once differentiable vector function, a multiplier a(z), with the components relative to the fixed basis, is introduced, by which the constraint (25) is scalarly multiplied and integrated along the axis of the beam. The resulting equation

~~t(r’-~{~]-&)dz=O

is then varied with respect to r, 6, y and a, and the term with a’ Sr’ is partially integrated. obtain

(26)

Thus, we

=o. (27)

In Appendix A we show, that the variations S/1 and SK are related to the variations 60 and 66 by the equations

6A=60/1, (28)

8K =li’t+‘. (29)

By adding Eq. (27) to Eq. (24), taking into account Eqs. (28) and (29), and rearranging terms, we obtain the equation


I.

i

L

Sy’(N - A’a) dz - 6r’(n + a’) dz + 0

i:~~t[~‘-~([‘ilj+~)]dz

0 + ~~~‘~-66’m-a’60/1

(11 ii 0 +y dz 1

- [Sr’ys” + a”) + 68”h” + srLt(SL -a”) + stLhfL] = 0 ) (30)

in which the variations 6r, 67, 66 (or equivalently, 60) and Sa are arbitrary and independent vector fields. In such case the coefficients at the variations should be zero for Eq. (30) to be satisfied. Equating the coefficient at 6~’ to zero, gives

N - A’a = 0. (31)

Inserting Eq. (22) into Eq. (31), yields

y = c,‘n’a . (32)

From the second term of Eq. (30) we obtain

n+a’=O. (33)

Integration of Eq. (33) with respect to z gives the multiplier a in terms of n

i

Z a(z) = a” - n(i) 4’ . (34) ,I

Here a” = a(0). Eq. (34) f urnishes also the physical meaning of vector a: it is a negative resultant force of external distributed force vector n, and as such, is equal to the cross-sectional stress resultant at z with respect to the fixed basis. Eq. (34) indicates that the law of distribution of a along the z axis is completely determined by its boundary value a(0) and the law of distribution of n. Thus, the function a(z) has been expressed by three parameters, the components of its boundary value a(0). The variation of Eq. (34) gives

au(z) = 6a” - ,,* &z(S) dl . i (35)

In Eq. (35) the variation 6n is taken to be a nonzero vector to account for the possible dependence of the force vector on the kinematic and deformation variables 6 and K.

The fourth term in Eq. (30) has first to be modified. Using Eq. (9), the following identity is deduced

(36)

Inserting Eqs. (31), (33), (35), (32) and (36) into Eq. (30), gives

The first integral in Eq. (37) may be integrated to obtain

I

L

&z”‘r’ dz = 6aot 0 I

I.

r’ dz = Ga”‘r(,~ = 6a”‘(rL - r”) . 0

Employing Eqs. (23), (25) and (38) in Eq. (36), yields

(37)

(38)

140 G. Jelenid, M. Saje I Comput. Methods Appl. Mech. Engrg. 120 (1995) 131-161

- Gr”‘(S” + a’) - GrL’(SL - a”) - 66”fA40 - c%YLfM = 0. (39)

In Eq. (39) the only function is the rotational vector 19(z). The remaining kinematical quantity, the position vector r, is included in Eq. (39) only through its boundary values, r” and rL. Eq. (39) represents the weak formulation of the equilibrium equations of the beam and is here termed the generalized principle of virtual work.

For the approximation of the distribution of the rotational vector along the beam, polynomial interpolation functions Z,(z) of degree M - 1 are employed. The distribution of the rotational vector 19 along the centroid axis of the beam element of initial length L is expressed by the interpolation equation

(40)

where IY,,, (m = 1,2, . . . , M) are nodal rotational vectors at the interpolation nodes (Fig. 3). From (40) one derives

66(z) = 2 Z,(z)S8, )

m=l

SS’(z) = 5 Z~(z)68, . t?l=l

Application of Eqs. (41) in Eq. (39) gives

-Grof(So+ao)-GrLt(SL-aL)=O,

(41)

(42)

By zeroing factors at independent variations 6a”, Sr’, 6rL and 68, (m = 1,2, . . . , M) in Eq. (42), by specifying the relation between the position vectors of the centroid axis in undeformed and deformed configuration, r. and r, and its displacement vector, U, in the form

u(z) = r(z) - To(z) = r(z) - zg, = r(z) - z

and making use of the notation

S”=$ ; I?“=&

1 2 3 M-l M

0 0 3 L

Fig. 3. The beam finite element. Nodal points 1,2. , M.

(43)

G. JeleniC, M. Saje / Comput. Methods Appl. Mech. Engrg. 120 (1995) 131-161 141

(Fig. 3), we derive the following system of discretized kinematical and equilibrium equations of the beam

m=l,2,...,M:

(44)

(45)

i

MD for m = 1 MZ= 0 form=2,...,M-1

ML for m = M

$1 + a() = 0 (46)

SL-aLd=O. (47)

It may be shown that for the planar case the formulation presented here reduces to that proposed by Saje [14]. In the planar case, only one function, i.e. the rotation of the centroid line about the normal to the plane of deformation of the beam, is present in the weak formulation of the equilibrium equations.

4.3. Generalized configuration space

By introducing the vector field a(z) into the principle of virtual work, the configuration space has been expanded and is defined by

%*={(a,r,A)~a:[0,L]+9233, r:[0,L]+~~,n:[0,L]~S0(3)}.

However, as we have shown in Section 4.2, by some mathematical manipulations, the vector functions a(z) and T(Z) can be eliminated from the principle of virtual work, so that only their boundary values a’, r” and rL remain in the principle. Considering that (i) the position vector r can be determined in terms of a, u0 and A by integrating Eq. (25) using kinematical equations (32) and (43), and (ii) the vector a(z) can be obtained from a0 and n employing Eq. (34), the associated set of all possible configurations of the beam is defined by

%l ={(an,u0,A)~a0:2=0~~,“, u”:z=0~~33,A:[0,L]~S0(3)}.

This set is, in a correspondence with the term the ‘generalized’ virtual generalized configuration space, %i.

4.4. Non-conservative loading

(48)

work, referred to as the

So far we have not precisely defined the type of loading. This will be done in this section. Let us assume that a load is composed from two additive parts. The first one represents a load with

fixed direction and prescribed intensity, defined with respect to the fixed basis. The other one is a follower type of load with prescribed intensity and defined with respect to the moving basis. The two parts will be designated by superscripts C and F, i.e. ()’ for fixed (or Constant) and ()’ for Follower loads.

In keeping with the load definitions in the principle of virtual work, Eq. (24), the total load has to be defined with respect to the fixed basis. Employing the transformation relation between the fixed and the moving bases, Eq. (3), and our load assumptions, the total point and distributed loads may be expressed as


ttl=i?lC+Ai?lF.

Application of Eq. (53) in (34) gives

a(z) = a0 - [n”( [) + A( {)n”( S)] dJ .

(49)

(50)

(51)

(52)

(53)

(54)

5. Finite element formulation

5.1. Componential form of discretized equilibrium equations

For later convenience, Eqs. (44)-(47) will be rewritten in a componential form. We shall use the summation convention that if in some expression a certain index occurs more than once, this index is the summation index. The range of indices i, j, k, p, q, r, s, t is taken to be 1, 2, 3; the range of indices m, n is 1,2,. . . , M. The Kronecker delta S~mn and the permutation symbol e,,, [22] are also used.

Application of the summation convention and the rules of vector algebra in Eqs. (44)-(47) and (49)-(54) yields

i= 1,2,3: L

Li=6i,L+u,L-uy- i( ‘i!f

a

‘i, + GA, - Apka, >

dz = 0

m=1,2,...,M; i = 1,2,3:

MZ: = J1: {L[eikPak( & fL+% + A,, )

- (rnc + Aipmi) + I,&,EJp~P 4 1

- S,,(M;C + A;M;‘) - S,,&M;C + Al”,M;“) = 0

i = 1,2,3:

Syc + A:),sfF + a: = 0

i = 1,2,3:

S~C+A~S~F-a~=O.

(56)

(57)

(58)

(59)

In these equations ai are the components of the vector a with respect to the fixed basis, which are determined from Eq. (55) by the expression

a, = a’.’ - oz (nc + A,jnr) dt . I (60)

The quantities GA, and EJ, have the following meaning:

G;, G\, G;, E3A (61) EJ; EJ, EJ, GJ, .

Eqs. (56)-(59) constitute a system of 3(M + 3) non-linear equations for 3(M + 3) unknowns. These are:


l three components of the displacement vector at z = 0: U:), Uk!, Ui; l three components of the displacement vector at z = L : I/f, uk, r/i; l three components of the multiplier a at t = 0: a:, ai, ai; l the components of two functions of z, the rotation matrix A,(z) and the rotational strain vector

K;(Z), which are expressed, using Eqs. (14), (8), (B-(19) and (40), by 3M components %l’ q&, GM3 (m = 1,2,. . . > M) of the rotational vector at M nodal points on the finite element. (The subscripts 1,2,3 denote components with respect to the fixed base vectors g,, g,, g,; the remaining subscript, m, in expression am,,, denotes the node number) (Fig. 3).

5.2. Linearization of discretized equilibrium equations

The system (56)-(59) is solved iteratively, employing the Newton method. In an iteration step, Eqs. (56)-(59) are linearized, yielding a system of 3(M + 3) linear equations

i=1,2,3:

3 Aa:’ + $ C3L. l3L

hj AA,, + -‘AUPL AU; =

I XJ; XJ;

m=l,2 ,..., M; i=l,2,3:

aM** 2 Aao + aM**

&Z:) ’ ank, ‘A@ + -=a~, = -MzI? aK

k

i = 1,2,3:

Sf” AAyk + Aa: = -Sy’ - AykSy -a:

i= 1,2.3:

-

L

SfF AAfi. - Aa: - 2 AA, = -Sf-” - AiSiF + a,” . 4

L, (62)

(63)

(64)

(65)

Here, -L, are the unbalanced displacements and -Mz,T, -Syc - AykStF - a: and -St’ - AhSk’ + a: are the unbalanced forces and are evaluated by employing the known values of as, A,,(z), K~(z), Up and U,! of previous iteration step. The terms on the left-hand side of Eqs. (62)-(65) yield the tangent stiffness matrix of the beam. Consider first the linearization of the rotation matrix A! The rotation matrices in two consecutive deformed configurations are related by Eq. (14). Denoting the two rotation matrices by A and A + AA and employing Eq. (10) in Eq. (14) gives

A+AA=(l+@+&82+.. .+;tT@“+... A. > (66)

Here, by the use of bold Greek letter A it is emphasized that AA is a finite increment in A (by contrast a linear part of the finite increment will be denoted by the normal Greek letter A). Notice that the matrix AA is not an orthogonal matrix. The linearization of Eq. (66) with respect to 0 gives the linear part of AA

A_4 = @A . (67)

The linear part of AK is obtained by the linearization of Eqs. (18) and (19). After some mathematical manipulation the relations

AK=AA’~~-A~A~ (68)

Ati = AA’ A’ + A’ AA’ (69)

are obtained, where A0 is a skew-symmetric matrix associated with the vector ho (as in Eq. (19)). Inserting Eq. (67) into Eq. (69) and proceeding as in Appendix A, Eq. (68) takes a simple form


AK=A~~~‘. (70)

The corresponding componential form of expressions given in Eqs. (67) and (70) is derived by first considering the componential form of Eqs. (8) and (40)

eij = -ej,ktik , (71)

e.=z,fimi. (72)

Then, inserting Eqs. (71) and (72) into Eqs. (67) and (70), gives

AA, = - e,J,,,&j~,,,, > (73)

AK, = I,&,,I’&,, . (74)

Eqs. (73) and (74) give the required linearizations of the rotation matrix and the rotational strain, A+, and AK;. By putting the two expressions (73) and (74) in Eqs. (62)-(65), a system of linear equations 1s obtained for the following unknown quantities: AU:, AU:, AU:, AU:, AU;, AU;, Aa:, Aa:, Aa:, amI, I9 ml23 am3 (m=l,...,M).

For the sake of convenience we introduce the notations

aM;*: (DAkp>n, + aK, tDKk)nj

a*a au. d=~(DA,,),i a*finj 84,

(75)

(76)

(77)

and

(78)

(79)

Here, (DAij)mp and (DK~)~~ denote the component representation of the operators DA and DK, defined as the directional (or Frechet) derivatives of A and K in the direction of 6. Employing the notations (75)-(79) in Eqs. (62)-(65), yields

i=1,2,3:

a*L. -f$Aay +&

I ak8niir,j+-$AU;+%AUj=-Li W-0

m=l,2 ,..., M; i-1,2,3:

dM*? -$ Aa;’ +

,g*M*! 2 finni = -M;,!

I a*%, (81)

i=l,2,3:

(DA~~)ijS”,‘~,j + Aa: = -SF - AI:,Sf’ -a:

i=1,2,3:

(DA~),iS~F$Mi - Aa! - $$ TY,,~ = -SF’ - AiS;’ + af- .

nt

(82)

Note that the coefficients at the rotations IY,,~ are defined by Eqs. (75)-(79). The terms on the left-hand side of Eqs. (80)-(83) will constitute the directional tangent stiffness matrix as opposed to the covariant

G. JeleniC, M. Saje ! Comput. Methods Appl. Mech. Engrg. 120 (1995) 131-161 145

tangent stiffness matrix [21] (see Section 5.4). The detailed form of the coefficients of the system of Eqs. (80)-(83) is given in Appendix B. A numerical integration is needed for the determination of these quantities. Gaussian integration has been used.

5.3. The update procedure

Assume that, in iteration i, an approximation of the deformed configuration is known. By inserting the known values of U!{‘) , $I’}, a;(i), “l’li’ and ,$ . mto the system of Eqs. (80)-(83) and solving it, gives AU t, AU;, Aaf and ?$‘ni. New, improved approximations for U(:. , Uk and a: in iteration i + 1 are obtained by adding the incremental values

u, (Ur+r) = uO(i) k +AlJ;

uL(i+l) k =U;“‘+AU,L; k=1,2,3.

o(i+l) ak

= a:(‘) + Aat

New components of the rotation matrix at iteration step i + 1 are determined by Eq. (14)

(84)

A”+11 = exp @Ali) , (85)

where the skew-symmetric matrix 0 associated with the rotational vector 6 is given in terms of i$! by Eqs. (8) and (40). The update of the rotation matrix, Eq. (85), is, in contrast to Eqs (84), not additive. This is a direct consequence of the non-linearity of the generalized configuration space YE. By the linearization of the non-linear generalized configuration space, viewed as a hypersurface in a multi- dimensional space, the solution space is mapped onto the tangent hyperplane to the hypersurface. Once the incremental solution for the rotational vector is obtained, the corresponding rotation matrix is found by projecting the rotational vector from the tangent hyperplane back to the hypersurface. This projection is defined by Eq. (85). In fact, this update is the only possible one that gives an updated configuration which belongs to the generalized configuration space ‘%‘i. The mathematical background of such updating procedure is found in the theory of the Lie groups and the associated Lie algebras, which are tangent spaces to the Lie groups. The relation between the Lie algebras and the associated Lie groups is defined by the exponential functor [8].

When the rotation matrix is determined, the rotational strain vector K at iteration i + 1 is obtained by Eqs. (18) and (19) [18]

fjn”+11 = n(i+l)' A .

{i+l)’

Alternatively, it can be expressed by the rotation matrix, and the rotational strain vector at iteration i, and the rotational vector, by the equation [ll]

(86)

The derivation of Eq. (86) is given in Appendix C. the form

A!‘+11 sin 6 Ik = 6, - elPs 6 i$ + eirqerps

1 - cos I3

I?*

In the componential form Eqs. (85) and (86) take

(;+I) Kk =4)

where

6 = (+YJ”’

(88)

(89)

146 G. Jelenid, M. Saje I Cornput. Methods Appl. Mech. Engrg. 120 (1995) 131-161

3 = L%, (90)

(91)

5.4. Element tangent stiffness matrix and unbalanced force vector

In the system of equations (80)-(83) the unknowns ammi (WI = 2,. . . , A4 - 1; j = 1,2,3) are the components of the rotational vector at the internal nodal points of an element (Fig. 3). They are termed internal degrees of freedom of the element. The unknowns AU:, 19,~, AU;, and tiMj, defined at the boundaries, z = 0 and z = L, are termed external degrees of freedom. The internal degrees of freedom need not to appear in the assembled global tangent stiffness matrix and are thus eliminated on the element level. The unknowns Aay are also considered to be internal degrees of freedom, eliminated on the element level, because they do not need to be continuous across boundaries, as the derivative of a, with respect to z is not present in the weak formulation (39).

Eliminating the internal degrees of freedom, eni, and Aa; (m = 2, . . . , A4 - 1; j = 1,2,3) from Eqs. (80)-(83) gives

kAx=Af (92)

(93)

where k is the directional tangent stiffness matrix and Af the unbalanced nodal force vector of a beam finite element. Ax denotes the vector of unknowns. Eq. (92) presents the relation between the elemental unbalanced forces and the kinematic unknowns in a local coordinate system of the element. The corresponding relation in a global coordinate system is obtained by the coordinate transformation.

Inspection of k shows that, as in [18], the directional tangent stiffness matrix in (92) is, at a non-equilibrated configuration and even for conservative loading, not symmetric. Yet, at an equilibrium configuration, the directional tangent stiffness matrix becomes symmetric, if the loading is conservative. For non-conservative loadings the directional tangent stiffness matrix is non-symmetric.

Only recently, Simo [21] showed that the linearization using the directional derivative, as in Section 5.2, gave the correct tangent stiffness matrix in the Riemannian sense only at the equilibrium configuration. If the directional derivative is replaced by the covariant derivative with respect to the metrics of the non-linear configuration space, the (covariant) tangent stiffness matrix is obtained, which is symmetric also away from the equilibrium for the conservative problems. Fortunately, he shows that the covariant differentiation need not be performed so that the actual metrics of the non-linear configuration space need not be computed, because the correct tangent stiffness matrix, in the Riemannian sense, is obtained simply by the symmetrization of the tangent stiffness matrix derived by the directional differentiation.

5.5. Global tangent stiffness matrix and unbalanced force vector of structure

The global tangent stiffness matrix and the unbalanced force vector of the structure are obtained by a routine assemblage procedure of the finite element method. Upon imposing displacement and rotation boundary conditions, the linearized system of equilibrium equations takes the form

KM(=AF. (94)

Here K is the global tangent stiffness matrix and M the global unbalanced nodal force vector of the structure. AX is the vector of global unknowns.

G. JeleniC, M. Saje I Cornput. Methods Appl. Mech. Engrg. 120 (1995) 131-161 147

5.6. Solution algorithm

Assume that, after an iteration step i has been completed, the current values of the boundary displacement vector, the rotation matrix, the multiplier vector and the rotational strain vector in all elements are known. With the introduction of these values into Eqs. (Bl)-(B9) and (56)-(59) and elimination of the internal degrees of freedom, the tangent stiffness matrices and the unbalanced force vectors of elements are obtained. After the transformation into the global coordinate system, the assemblage of the global tangent stiffness matrix and the global unbalanced force vector and the solution of the system of linear equations (94), the incremental nodal displacements AU:: (i = 1,2,3; n = 1, . . , N) and nodal rotational vectors 8:: (i = 1,2,3; n = 1, . . . , N) in the global coordinate system, are obtained. The index ( )o stands for ‘global’ and N is the number of the nodes of a structure. After transforming these quantities back to the local element coordinate system, the internal degrees of freedom Aa)’ and Fiji (m = 2,3, . . . , M - 1; i = 1,2,3) of elements are determined, based on the values of the incremental boundary displacements and rotational vectors. New approximations for U]‘, U,” and a: for each element are then obtained by Eqs. (84). New approximations for the distribution of A, and K; within elements follow from Eqs. (87)-(91). Once these quantities are calculated, they are used for the evaluation of Eqs. (Bl)-(B9) and (56)-(59) again. After the elimination of the internal degrees of freedom and the transformation of newly obtained tangent stiffness matrix and the unbalanced force vector into the global coordinate system for each element, new global tangent stiffness matrix and new unbalanced force vector of the structure are assembled. New corrections to the unknowns are obtained by the solution of Eq. (94) and the iteration procedure outlined above can be repeated. The iteration is repeated until the required accuracy for unknowns and / or unbalanced forces is achieved.

In the first loading zero step values are assumed as starting approximations for the quantities I/j’, Uf-, a:’ and K,, and the identity matrix for the rotation matrix A,, of an element. In the subsequent loading steps the converged values at the end of the previous loading step are taken as starting approximations in the first iteration step.

Once the iteration has been completed and the unknowns determined, distributions of displacements and internal forces over elements can be computed. The distributions of the position vector of the centroid line over the element and the corresponding deformed shape of a beam is obtained by the numerical integration of differential equation (17) for r. Gaussian integration is applied. Once the deformed shape is known, the distribution of internal forces is easily obtained by the numerical solution of the equilibrium equations.

6. Numerical examples

The following examples will demonstrate high accuracy and excellent performance of the proposed finite elements. Only three-dimensional examples are considered because the planar version of the present formulation is completely coincident with the one given in [14], where its performance for planar problems has been established. Here we consider three problems: (1) a lateral buckling of an in-plane-rigid cantilever, and a lateral buckling and complete post-buckling behaviour, (2) of a simply supported right-angle frame, and (3) of a fixed supported right-angle frame. The example (1) represents the so-called initial instability problem, where, due to the assumption of the complete in-plane shear and bending rigidity of the beam, it remains undeformed until the occurrence of the buckling. The example (2) shows the ability of the present procedure to describe large displacements and truly large space rotations of a beam. Large displacements and rotations, combined with large strains of the centroid axis, are considered in the third example. We investigate the influence of the degree of interpolation polynomials and the number of elements in the mesh on the accuracy of the numerical solution. Finite elements with lst, 2nd, . . . ,llth degree interpolation polynomials, here termed elements E,, E,, . . . , E,, , have been examined.

Element E,, has 3(n + 4) degrees of freedom (see Section 5.2). This favourably compares with the two-field finite elements where the displacements and the rotations have to be interpolated, as in [18],


where an element using nth degree polynomial for the displacements and rotations, has 6(n + 1) degrees of freedom. For n = 1,2,3, . . ,8, elements E,, E,, E,, , E, have 15, 18, 21, . . . ,36 d.o.f., compared to 12, 18, 24, . . . ,54 d.o.f. of the two-field elements. Only in the case of the linear element, the two-field elements have less d.o.f. than the present elements. In all other cases our elements have equal (for II = 2) or less (for y1 23) d.o.f. for an equal degree of interpolation.

The lowest reasonable order of Gaussian integration can be estimated by inspecting the integrands of Eqs. (Bl)-(B6) of Appendix B and Eqs. (56)-(57) for an undeformed or only slightly deformed configuration, such as, e.g. in the case of the initial instability. Then we may take that A,, = S, and K, = 0. If we further neglect the presence of the distributed forces and moments, it follows that a,(z) = a: = const. Then the only functions of z in (Bl)-(B6) and (57) are interpolation polynomials and their derivatives with respect to z. Under these assumptions the integrand in Eq. (B6) appears to be the one having the polynomial of the highest degree, 2(M - 1). To make the integration of such a polynomial accurate, the M-point Gaussian integration is required for an element E,_, with M nodal points [9]. This is defined to be the minimal order of Gaussian integration of the present finite elements. The numerical experimentations have shown that the order of Gaussian integration bigger than M for element E,_, , has some minor influence on accuracy of the solution only for low-order elements. Therefore, only the minimal order of Gaussian integration is used throughout the present examples.

As observed from Eq. (Bl), for very large values of the shear and axial stiffness, the elements aL,/&zr of the tangent stiffness matrix of an element tend to zero. This may result in ill-conditioning of the stiffness matrix for element E,, where the internal rotational degrees of freedom are absent, and the results may strongly depend on the way the numerical integration and the static condensation are performed.

The iteration at each loading step is terminated when the Euclidean norm of the vector of nodal unknowns and/or of the vector of unbalanced forces is less than a prescribed small value. The double precision arithmetic was used in calculations (8 bytes per real number).

6.1. Lateral buckling of a cantilever subjected to point load

The inextensible, shear- and in-plane bending-stiff, straight cantilever is subjected at its free end to the vertical force F. The out-of-plane buckling load (in the sequel termed the critical load), F,,, is sought. Geometric and material data are shown in Table 1. The analytic solution for the critical load was provided by Timoshenko and Gere [24] and is expressed by the formula

(95)

where k = 4.01259 934 is evaluated according to the method, presented in [24]. Inserting data from Table 1 into Eq. (95), yields the value of the critical force, accurate to 9 digits

F,., = 0.100 314 984 kN .

In numerical calculations the inextensibility, shear- and in-plane bending rigidity were approximated by having taken large values (1015) for stiffnesses GA,, GA,, EA, EJY (see Table 1). The Newton iterations were performed such that the Euclidean norm of the solution vector was smaller than lo-*“. Gaussian integration of order n + 1 for an element E, was employed.

The present formulation uses exact kinematic and static equations. Since, in the present numerical example, we deal with in-plane stiff beam, which remains undeformed prior to the buckling load, the integration of the tangent stiffness matrix and the unbalanced force vector is exact (see the introduction to Section 6 for discussion). Thus, the error introduced is due only to the degree of interpolation polynomials and the number of finite elements in a mesh. By increasing the degree of polynomials and/or the number of elements, the solution should converge to the exact one.

A variety of finite elements and element meshes have been applied. The cantilever has been modeled by 1, 2, 5, 10 and 20 elements of type E,, E,, . . , E,. Due to large axial and shear stiffnesses employed, element E, is, as discussed in introduction to numerical examples, not reliable and has not

G. JeleniC, M. Saje I Comput. Meihods Appl. Mech. Engrg. 120 (1995) 131-161 149

Table 1

Lateral buckling of cantilever. Critical load F_

Lx GA, = 1015 kN/cmZ

-. GA, = 1Ol5 kN/cm*

Y” EA = 10” kN/cmZ

EJ, = 1250 kN/cm*

EJ, = 10” kN/cm*

GJt = 50 kN/cm’

L = 100 cm

e d.o.f. 11. = 1 n_ = 2 n,, =5 n, = 10 n, = 20

E, 0.112 124 266 0.101386 932 0.100348719 0.100 317 169 0.100 315 122

E, 300 0.101249 574 0.100349516 0.100315 175 0.100 314 987 0.100314984

E, 90 0.100 407 873 0.100315903 0.100 314 984

E, 105 0.100 320 908 0.100315004 0.100314984

E, 48 0.100 315 382 0.100 314 984

E, 54 0.100 315 000 0.100 314 984

E, 30 0.100 314 984

Analytical solution [24] 0.100314984

e = type of element.

n, = number of elements in mesh.

been used. The critical load is obtained iteratively from the condition that the tangent stiffness matrix is singular at the critical state. The results for the critical force are presented in Table 1. By employing only one element E,, the relative error of the computed buckling load is roughly 12%; by increasing the number of elements E,, the error decreases and, for 20 elements E,, we have six digit accurate solution. The convergence is much faster though, if, instead of increasing the number of elements, the degree of interpolation polynomials is increased. As observed from Table 1, 20 elements E,, 5 elements E, or E,, 2 elements E, or E, or only one element E, give the buckling load which is accurate to 9 digits. The second column of Table 1 shows the number of degrees of freedom of particular mesh corresponding to this accurate critical load. Note that by employing one element E,, only 30 d.o.f. are required, compared to 300 d.o.f. of the mesh with 20 elements E,. We may then conclude that our numerical solution is convergent to the exact one.

6.2. Lateral buckling of a simply supported right-angle frame subjected to in-plane point moments Post-buckling behaviour

This example was first introduced by Argyris et al. [l] and later on reanalyzed by Simo and Vu-Quoc [18], Saleeb et al. [15], Nour-Omid and Rankin [13] and Yang and Kuo [25], among others. The geometry, boundary conditions, material and cross-section data and the position of the fixed coordinate system are illustrated in Fig. 4. Due to the symmetry of the frame and the load about the (y, z) plane through the apex, only one half (the left one) of the frame is considered. At the support 0 the displacement along axis x and the rotation about axis z are permitted. The apex A is allowed to move along the y and z axes and to rotate about the x axis. The frame is subjected to in-plane moment M at 0, as shown in Fig. 4. Since the shape of the cross-section is taken to be a thin rectangular with a thickness-to-height ratio of l/50, the frame is prone to out-of-plane torsional buckling, which makes the problem complicated.

A number of finite elements and element meshes have been applied. The frame member OA (here also termed the leg of the frame) has been modeled by 1, 2, 4 and 8 finite elements E,, E,, . . . , E,. The minimal order of Gaussian integration is used as defined in Section 6. The Newton iterations are terminated when the Euclidean norm of the solution vector is smaller than 10m7. In the course of

150 G. Jelenit, M. Suje I Compur. Methods Appl. Mech. Engrg. i.Zf (‘is%) 131-lhl

/-

Al = 21.6 mm* Jt = 1350 mm4 E = 71240 N/mm*

A? = 21.6 mm* Jz = 0.54 mm4 G = 27191 N/mm2

A = 18 mm2 Jt = 2.16 mm4 L = 240 mm

Fig. 4. Lateral buckling of a simply supported right-angle frame subjected to end point moments: (a) geometry, material data and

boundary conditions; (b) static model.

analysis the critical, i.e. the buckling moment is obtained first. Then, the complete out-of-plane post-buckling behaviour of the frame is traced.

The critical moment. The analytical solution for the critical moment exists in literature provided that the shear, axial and in-plane bending stiffnesses of the cross-section are infinitely large. The critical moment is then [24]

MC,= * L = + 622.2 Nmm .

The above-mentioned stiffnesses are finite in the present problem. Numerical experimentations, however, show, that, for the present data, the influence is minor, as it effects the result for the buckling moment at most on the 5th digit. Hence, the numerical results are compared to the analytical ones to 4 digits only. The numerical results, predicted by the present method, are given in Table 2. The 4-digit-accurate buckling moment is obtained when using 8 elements E,, 4 elements E,, 2 elements E, or one element E, (Table 2). One element E, does not supply a solution. As indicated in Table 2 and also found out in Section 6.1, the convergence is faster if, instead of increasing the number of low-order elements in the mesh, the number of elements is retained and the degree of element increased.

To show the superiority of the present method over other methods published in literature, the present results for the buckling moment are compared in Table 3 to the results presented in

[2, 13, 15,181.

Post-buckling behaviour of the frame. Once the moment M reaches the critical value M,,, the response of the frame has two paths. The primary path represents the (x, y)-plane response, while the secondary path follows the buckled configuration of the frame. The two paths are separated by the bifurcation point at M,, = + 622.2 Nmm. In the following, only the secondary path will be described in detail.

In order to show a high capability of the present finite elements to accurately describe the behaviour of the frame at large displacements and rotations, the post-buckling behaviour of the frame has been

Table 2

Simply supported right-angle frame. Critical moment M,, (Fig. 4)

Number of elements 1 2 4 8

Element type E, _ rF 792.3 7 656.3 5 630.3

EZ T 686.1 ? 626.9 z 622.5 5 622.2

E, 7 626.3 T 622.3 5 622.2

E, 5 622.4 i- 622.2

E, + 622.2


Table 3

Comparison of various solutions for critical moment M,,

Reference Parametrization Strains

of rotations

Interpolation Number of

elements

PI

[I31

1151

ItsI

Present

Semitangential

nodal rotations

Nodal rotational

vector

Rotational

vector

Rotational

vector

Rotational

vector

Analvtical solution 1241

Small Technical theory

Small Technical theory

Small

Finite

Finite

Linear displacements

constant strains

Quadratic for displacements

and rotational vector

Only for components of

rotational vector:

quadratic

cubic

4th degree

5th degree

10

10

10

10

i 624.77

i 626.1

+ 627.37

F 626

T 622.2

-+ 622.2

i 622.2

T 622.2

T 622.2

Technical theory: small strains relative to the deformed system; lateral displacements are approximated by 3rd degree polynomial,

axial and torsional displacements are linear.

traced by employing only one element E,. This is far less elements than reported in the literature (to our best knowledge, the results published by now have been obtained by the use of ten finite elements). 7-point Gaussian integration is used. The iteration tolerance for both, the vectors of unknowns and unbalanced forces, is lo-‘.

The transition from the primary path to the secondary one at the critical point is achieved by the introduction of a small perturbation force (approximately 0.06 N), acting at the point A in the positive direction of the z axis. Once the frame laterally buckles, the force is removed. Due to the presence of the perturbation force, the corresponding buckling moment is slightly smaller than 622.2 Nmm. Then the frame starts rotating about the x axis. The plots of the horizontal displacement of the support 0, and the vertical and lateral displacements of the apex A in terms of the moment M, are depicted in Figs. 5-7. Points 1,2,3, . . . ,9, marked on these plots, are of a special interest. The equilibrium states, specified by the segment of the path between points 1 and 2, are unstable equilibrium states. Any small perturbation of the configuration would return the frame back to the primary path. In contrast, the equilibrium states between points 2 and 3 are stable. Note from Figs. 5 and 7 that, in some configuration between points 2 and 3, the horizontal displacement U,, of the left support and the lateral displacement of the apex A reach their maximum values. Observe also that the size of the two displacements is roughly of the same order as the length of the undeformed legs of the frame.

Fig. 5. Lateral buckling of a simply supported right-angle frame subjected to end point moments: applied moment versus

horizontal displacement of the support 0.


5.6

Fig. 6. Lateral buckling of a simply supported right-angle frame subjected to end point moments: applied moment versus vertical

displacement of the apex A.

Point 4 presents the equilibrium state with the frame twisted half a circle (r) around the x axis. The point A thus returns into the (x, y)-plane once more. Fig. 6 shows that the vertical displacement of the apex A is now maximal. The equilibrium state with the frame returning into its original position at the bifurcation, where it has twisted full circle (2~) around the x axis, is represented by point 5. It is important to emphasize, however, that in this state the moment load is not directed outwards anymore, but inwards. The three curves in Figs. 5-7 intersect the moment axis exactly at the value of the critical moment, but with the reverse sign. This again shows that the lateral buckling moment of the frame considered is identical in magnitude for the pair of moments acting either in the directions as shown in Fig. 4 or in the opposite directions. By decreasing the moment from the value -622.2 to 0, the in-plane primary path between points 5 and 0 is followed, returning eventually the frame back to the initial undeformed state.

Once in the undeformed state, the magnitude of the moment is further increased, yet in the negative direction. Simultaneously a small increasing perturbation force, acting at the apex A in the lateral direction, is applied (this time even smaller magnitude suffices, approximately 0.0006 at the critical state). When the frame buckles, the perturbation force is removed. The influence of the perturbation force on the moment-displacement diagrams (Figs. 5-7) is localized on the segment O-6 and is only minor. Then the frame starts rotating around the x axis. With the exception of the segment of the path between points 0 and 1, where the influence of the lateral perturbation force is noticed, the diagrams showing the displacements U, and V, versus the moment M, are completely coincident with the ones

Fig. 7. Lateral buckling of a simply supported right-angle frame subjected to end point moments: applied moment versus lateral displacement of the apex A.

G. JeleniC, M. Saje I Comput. Methods Appl. Mech.

obtained during the first rotation cycle around the x axis, but direction. The post-buckling path, describing the variation of moment M, is fully symmetric relative to the moment axis.

101 loading steps were applied to calculate the response

Engrg. 120 (1995) 131-161 153

they have to be followed in the opposite the displacement W, with respect to the

path between points 0 and 5 (the first

revolution, see Figs. S-7), and 94 steps for the second revolution. The variable-arc-length algorithm with the iteration within a hypersphere was employed [5]. The arc-length is here defined as the Euclidean norm of translational degrees of freedom. The initial arc-length was chosen to be 0.01 mm. The arc-length in subsequent loading steps is taken such that the previous arc-length is scaled by the factor nopt/ni, h w ere ni is the number of iterations at the previous loading step, and nopt is a desired (optimal) number of iterations, here taken to be 6. If, in a loading step, the equilibrium is not achieved

in 2nopt iterations, the arc-length is halved.

6.3. Lateral buckling of a fixed supported right-angle frame subjected to in-plane point moment. Post-buckling behaviour

The geometry, boundary conditions, material and cross-section data are given in Fig. 8. This example is practically identical to the one presented in Section 6.2 except for the support 0, which is now fixed. Due to the fixed support, the extensional strain in the legs becomes large and exceeds the bearing capacity of some realistic linear elastic material. We present the example to show the capability of the present method of accounting correctly also for large strains. Neither analytical nor numerical results are available for the present example in literature, so no comparison of the present results is given.

As in Section 6.2 the critical, out-of-plane buckling moment is computed first. Then the complete post-buckling behaviour of the frame is traced.

For the determination of the critical moment, M,,, elements E,-E, have been used. Due to the symmetry of the frame only one half of the frame is considered. The left leg of the frame has been modelled by 1, 2, 4, and 8 finite elements. The iteration tolerance for the vector of unknowns is lo-‘. For the post-critical analysis one element E,, is used. The minimal order of Gaussian integration is employed throughout the calculations. The iteration tolerance for the vector of unknowns and the vector of unbalanced forces is lo-‘.

The critical moment. The analytical solution for the critical moment is not available. In Tables 4 and 5 we present our numerical results and compare the solutions, obtained by various elements and meshes. In the present example the critical moment very much depends on the direction of its application. Because the node 0 is fully supported, the leg is in tension for the moment directed outwards (as in Fig. 4, in the sequel called the positive moment), and in compression otherwise. The compression gives rise to the combination of compressive and torsional buckling which results in the critical moment which is nearly three times smaller in magnitude than in the case of the positive moment. The numerical

Ai = 21.6 mm2 Jr = 1350 mm4 E = 71240 N/mm2

-42 = 21.6 mm2 J2 = 0.54 mm4 G = 27191 N/mm2

A = 18 mm2 Jt = 2.16 mm4 L = 240 mm

Fig. 8. Lateral buckling of a fixed supported right-angle frame subjected to end point moments: (a) geometry, material data and boundary conditions; (b) static model.


Table 4

Fixed supported right-angle frame. Positive critical moment

M,, (Fig. 8)

Number of elements 1 2 4 8

Element type

4 _ 6391 2738 2266

E? 4132 2436 2162 2131

ES 2588 2166 2130 2129

E, 2223 2132 2129

E, 2149 2129

E, 2133 2129

E, 2129

Table 5

Fixed supported right-angle frame. Negative critical moment

MC, (Fig. 8)

Number of elements 1 2 4 8 Element type

E, - 1023 -857.2 -813.9 EL -857.1 -818.8 -801.2 -799.8

E, -832.1 -800.5 -799.7

E, -800.8 -799.8 -799.7

ES -799.9 -799.7

E, -799.7

results for the positive and negative moments, given to 4 digits, are displayed in Tables 4 and 5, respectively. The magnitude of the positive critical moment M,, = 2129 Nmm is obtained using 8 elements E,, 4 elements E4, 2 elements E, or E,, or one element E,. The value of the negative critical moment M,, = -799.7 Nmm is found by using 4 elements E, or E,, 2 elements E, or one element E,. In Sections 6.1 and 6.2 we show that our formulation gives solutions which converge to exact solutions. Thus, the two values 2129 and -799.7 may be considered to be accurate values for the critical moment, precise to 4 digits.

Post buckling behaviour of the frume. We consider the post buckling behaviour due to the positive moment first. The positive moment and a small perturbation load at the apex A in the z direction are increasing until the moment reaches the critical value. Then the perturbation load (approximately 0.0002 N) is removed. The plots of the vertical and lateral displacements of the apex A versus the moment M are shown in Figs. 9 and 10. The equilibrium state corresponding to the critical moment, M,,, is marked by point 1. Along the segment l-2 the system is rather stiff. At point 2, M e 2800 Nmm. Then, as illustrated in Fig. 10, the system softens and at point 3 (M G 2851 Nmm) becomes unstable. The equilibrium states, specified by the segment of the path between points 3 and 4, are unstable. Along this segment of the path the moment changes its direction; its magnitude at point 4 takes the value M e - 2204 Nmm. Point 5 represents the equilibrium state with the frame twisted half a circle (rr) around the x axis. The point A is now in the (x, y)-plane. The vertical displacement of the apex A is here maximal (Fig. 9). Then the path makes a loop through points 6 and 7. Point 8 represents the

Fig. 9. Lateral buckling of a fixed supported right-angle frame subjected to end point moments: applied moment versus vertical

displacement of the apex A.

G. JeleniC, M. Saje I Comput. Merhods Appl. Mech. Engrg. 120 (1995) 131-161 155

-160

A

M

13 14 t 2 3 Y Y

15--l

-- 1000

-80 0 80

S-:9

i

160

1 ‘, 0 w,

Fig. 10. Lateral buckling of a fixed supported right-angle frame subjected to end point moments: applied moment versus lateral displacement of the apex A.

bifurcation point for the negative moment. The deformation then follows the segment 8-O of the primary in-plane path for the negative moment until the moment vanishes, when the frame finally returns back to its initial undeformed configuration.

For the post-buckling behaviour of the frame subjected to the negative moment. the moment-vertical displacement of the apex curve is identical to the one shown in Fig. 9, but now taken in the reverse order. This path is marked by points 0,9,10,11, . . . , 15,0. Observe that the moment-lateral displacement curves of the two loading cases are symmetric with respect to the moment axis.

148 loading steps were applied to calculate the response path between revolution, see Figs. 9 and 10) and 139 steps for the second revolution. arc-length algorithm are identical to those given in Section 6.2.

8. Conclusions

A finite element formulation for the numerical treatment of static linear elastic spatial finite-strain beam structures has been proposed, which extends the formulation given in the papers by Simo [17] and Simo and Vu-Quoc [18] along the lines of the previous work on planar beam theory presented by Saje [14]. The following may conclude our presentation.

points 0 and 8 (the first The data concerning the

(1)

(2)

(3)

Exact non-linear kinematic rehnionships of the spatial finite-strain beam theory have been applied, which assume the Bernoulli hypothesis of plane cross-sections remaining plane and undistorted during the deformation. Finite displacements and rotations as well as finite extensional, shear, torsional, and bending strains, are accounted for. A configuration of the beam is described by the displacement vector of the deformed centroid axis and the orthonormal moving frame, rigidly attached to the cross-section of the beam. The position of the moving frame relative to the global fixed reference frame is specified by an orthogonal matrix. This is parametrized by the use of the rotational vector. For the purposes of the iterative solution by the Newton method the incremental rotational vector is introduced, which rotates the moving frame from the previous into the current iteration configuration of the beam. Its components relative to the fixed coordinate system are taken to be the rotational degrees of freedom at the finite element nodes. Because in 3-D space both the axial and the follower moments are non-conservative, the principle of virtual work is introduced, and not a variational principle, as the basis of the finite element discretization. A novel feature of the present formulation is the introduction of a

1.56 G. Jelenid, M. Saje I Comput. Methods Appl. Mech. Engrg. 120 (1995) 131-161

generalized principle of virtual work in which the displacements, rotations, strain vectors, and multipliers (i.e. stress resultants in the reference coordinate system) are independent variables. By eliminating the displacements, the strain and the multiplier vector fields from the generalized principle of virtual work, the three components of the incremental rotational vector field remain the only functions in the principle, which have to be approximated within a finite element domain. This approach is different than in [17, 181 and in [4, 10, 13, 15, 161, where the three components of the incremental rotational vector and the three components of the incremental displacement vector have to be approximated within the element. As a result, an accurate, efficient, and robust family of beam finite elements for the non-linear analysis has been obtained.

(4) The absence of locking is an important advantage of the present finite elements. The elements describe, with equal precision, both extensible and inextensible beams as well as shear-, torsional- and/or bending-stiff, or shear-, torsional- and/or bending-flexible ones. This is in contrast to the elements developed in e.g. [17, 181 where, in order to avoid the locking at critical combinations of cross-sectional data, the reduced numerical integration has to be used. Yet, by the use of the reduced integration, the elements become less accurate, which requires more elements in the mesh than would be necessary and convenient from the structural analyst point-of-view.

(5) Polynomial interpolations for the approximation of the components of the rotational vector have been chosen. The present finite elements have two boundary rotational vectors and an arbitrary number of rotational vectors at internal nodes as the rotational degrees of freedom. The latter are, together with the boundary value of the multiplier at z = 0, eliminated on the element level. Finite elements with polynomials of lst, 2nd, . . . and up to 1 lth degree have been employed in the presented numerical examples. In the computer program, though, the degree is not limited to some particular value. No approximation is necessary regarding the distribution of displacements along the axis of the beam. When the loading is non-conservative, the degree of polynomials should be higher in order to describe the changing direction of the loading better. M-point Gaussian integration is recommended, since it is proven to be sufficient for elements with (M - 1)th degree interpolation polynomials. Element E, with 1st degree polynomial was found to be unreliable in the case of very large axial and shear stiffnesses. In such cases element E, should be avoided.

(6) The Newton iterative method has been used for the solution of the discretized non-linear equilibrium equations. A special update algorithm for the rotations is employed, which assures the rotation matrix to remain orthogonal. In an iteration step, the element tangent stiffness matrix, linearized by employing the directional derivative, is found to be non-symmetric even for conservative loadings, where the unsymmetric part depends solely on the unbalanced forces. Only upon achieving an equilibrium configuration, the unbalanced forces vanish and the tangent stiffness matrix becomes symmetric. The tangent stiffness matrix, obtained by employing the covariant derivative with respect to the Riemannian metric of the rotational group SO(3) instead of the directional one, is in contrast, for conservative problems, symmetric also away from the equilibrium state. It is calculated simply by the symmetrization of the non-symmetric tangent stiffness matrix. This significant result was derived by Simo [21]. Nour-Omid and Rankin [13] showed that the quadratic rate of convergence, which is a characteristic of the Newton method, is retained whether either the unsymmetric or the symmetric matrix is used. For non-conservative loadings stiffness matrix is always non-symmetric.

(7) In the numerical examples we have demonstrated the capability of the present formulation of determining the out-of-plane buckling load and of tracing the whole pre- and post-critical load-displacement path of a cantilever and a right-angle frame. The comparisons of our results to analytic and numeric solutions show excellent accuracy of solution and quadratic convergence even when employing only one finite element to describe the displacements of the size of the structure itself and the rotations of 2~.

(8) The present paper is concerned with the static analysis. Because the displacement functions are not included in the present generalized principle of virtual work, the formation of mass matrices

G. JeleniC. M. Saje I Comput. Methods Appl. Mech. Engrg. 120 (1995) 131-161 157

is not straightforward. A use of simple, diagonal mass matrices, in combination with the present sophisticated elements, may not be effective enough [26]. The refined, possibly consistent mass matrices, still remain to be derived. The extension of the present formulation to beams made of non-linear materials, defined e.g. by a strain energy function, is feasible.

Acknowledgment

This work was supported by the Ministry of Science and Technology of the Republic of Slovenia under Contract P2-3149. The support is gratefully acknowledged. The writers would like to thank Prof. Ales Zaloinik for helpful discussions.

Appendix A. Variations of the rotation matrix A and the rotational strain vector K

The rotation matrices in two consecutive deformed configurations are related by Eq. (14). Denoting the two rotation matrices by A and A + AA and employing MacLaurin’s series Eq. (14) gives

A+AA=exp@A= 1+@+$8’+&0’+.*. A. > (Al)

Here the bold Greek letter A emphasizes that AA is a finite increment of A. Note that the summation is not a group operation of SO(3). Therefore AA is not an element of SO(3), i.e. it is non-orthogonal. Introducing the notation 60 for the variation of the skew-symmetric matrix, associated with the variation of the rotational vector 613, inserting it into Eq. (Al) and neglecting non-linear terms, gives the corresponding variation of the rotation matrix

The variation of Eqs. (18) and (19) gives

SK =SA'a +A'Sw (A3)

Ml =6ArA'+A'8A'. 644)

By the help of Eq. (A2) and taking into account that AA' = I, d2 = A'A' and 0’ = -0, Eq. (A4) can be written as

6~=8@‘+60n-nso

or alternatively

(A5)

sw=s6’+s6 XLC).

Inserting Eqs. (A2) and (A6) into (A3), yields

646)

SK =(S@A)'o +A%Y+A'(S8 X co). (A7)

Utilizing Eq. (9) and the skew-symmetric nature of 0, we are left with


Appendix B. Coefficients of the system of Eqs. (80)-(83)

dLi aL, aa, aL, aL, ~=~~=~skj=~=- I

L 4 / k I

k I ” mA,k dz

a*aL L

--I-=e

a*anj I IPI 0 I,A,,,n~ dz

For further reference see also Eqs. (56), (57), (61) and (75)-(79).

Appendix C. The derivation of Eq. (86)

Employing Eqs. (14), (10) and (7) at iteration i + 1, yields

(

sin 6 A”+“= I+--@+ 1 -cos6

?Y* 02

> n’i’ .

Inserting Eq. (14) into Eq. (19) and considering Eqs. (11) and (19), gives

0”“’ = (exp 0)’ exp(-O) + exp OJ2’i’ exp(-O) .

031)

WY

033)

(B4)

(B5)

( w (J37)

038)

039)

(Cl)

The axial vector o “+‘) of the skew-symmetric matrix 0 (‘+l) IS expressed as a sum of two axial vectors

WP+l) = oj~+U + U;i+l) . (C2)

o;i+li and m;~+ll are the axial vectors of skew-symmetric matrices nj”” and a:‘“‘, satisfying the relations

G. JeleniC, M. Saje I Compur. Methods Appl. Mech. Engrg. 120 (1995) 131-161

f2j”” = (exp 0)’ exp(-0)

fl”“’ = exp 0L3{ii exp(-O) .

The vector o ii+‘) was derived by Simo and Vu-Quoc [18, Appendix B.21 and takes the form

Putting Eqs. (10) and (7) into Eq. (C4), we obtain

fJ’i+l) ( sin 6 1 - cos 6

> (

sin 6 02 a’(’ z----o+ 1 - cos 6 2 = I+------u

6 O+ ?Y2 lY2 o2

>

159

(C3)

(Cd)

(C5)

(C6)

Taking into account the relations

OKtO = -6’wO

0” zz -6’60 = -620

fin”102 + f3*an’;} = __6tW{‘)e _ ataa’i) ,

which are easily proved by performing indicated multiplications, the expression (C6) takes the form

fl”+“=cos$jJ2((‘)+ 6 2 l-cos6 8’~“’ o+ sin6

6 _(@a(!) - &‘@). (C7)

The skew-symmetric matrix 00 ‘i’ - 0 ‘i)O is associated to the axial vector 6 x m(I). Then Eq. (C7) may be written in the form

02 ([+I) = cos qjo”)

l-cos6 8)‘U{‘) sin 6 + 6 ----iIT+- I3 6 6 x wti) . (C8)

According to Eq. (C2), the vector cc) “+l) is obtained by the summation of Eqs. (C5) and (CS)

(C9)

Next the expression for A ‘i)~(i+ll in terms of 19 and mii) . 1s derived. Employing Eqs. (18), (14), (lo), (7), (11) and (C9), we obtain

_y(l_yy_$@&&q?@8r sy l-~~s~o(9x8’)

-~cos6&“‘-~ sin 6 sin 6 1 -

cos 6 6 (‘)

poi)- 6’0~ 6 -7 sin 6 6 sin 6 O(6 x UC’))

+ l-;y (1 sy> y’@*$+ 1-lgcy+2B,

1 - cos 6 1 - cos 6 +

6* 6* 02(6 X 8’)


1 - cos I3 +

a2 cos 60*&J til +

l-cosl9 l-cos6 S’o(‘) 029

7?* ~ *

6 6

1 - cos 6 sin 6 +

a2 6 02(6 x 0P) .

Taking into account that 06 = 6 x 19 = 0, and the following series of formulae

08’=6x6’,

&f’) = 6 x U(i) )

0(6X8’)=6X(iiX6’),

@(?I x cd’)) = 6 x (6 x co{‘)) )

026’ = 6 x (6 x S’),

o*o{‘) = 6 x (6 x 0”‘) ,

@*(I9 x 8’) = 036’ = -7P(ly x 8’) ,

@(+j x ,(i)) = @3oiiI = 42(i) x &‘) ,

in Eq. (ClO), we are left with

One part of the last term in Eq. (Cll) may alternatively be written as

6 x (9 x w”‘) = (6’ &‘$9 - (6’6)o”’ .

We now have

AwKII+l) _(l_%&z)~8+ ~-~~~~~ I syg,+,,,&~

1 - cos 6 _ 2

+a )

6X6’- l -,“y 6 (&J4)g + 1 -,“ys I9 (&+p

sin6 6’6’ 1 7 -

62 a+

sin 6 - cos I9 _ +j ill’--

$+2 9x6’+0(‘)

Employing Eq. (18), Eq. (86) is obtained

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(CW

(Cll)

cc=)

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