A Consistent Co-rotational Formulation for Non-linear, Three-dimensional, Beam-elements

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 81 (1990) 131-150 NORTH-HOLLAND

A CONSISTENT CO-ROTATIONAL FORMULATION FOR NON-LINEAR, THREE-DIMENSIONAL, BEAM-ELEMENTS

M.A. CRISFIELD Department of Aeronautics, Imperial College of Science & Technology, London, United Kingdom

Received 14 December 1988 Revised manuscript received 12 September 1989

This paper describes a co-rotational formulation for three-dimensional beams in which both the internal force vector and tangent stiffness matrix are consistently derived from the adopted 'strain measures'. The latter relate to standead beam theory but are embedded in a continuously rotating frame. A set of numerical examples show that the element provides an excellent numerical performance.

1. Introduction

Three-dimensional beam elements have been proposed by a number of authors [1-16]. In the non-linear context, as considered here, the formulation is complicated by the non-vectorial nature of rotational variables [17]. Under rotations including a significant rigid-body component, many elements produce over-stiff solutions due to 'self-straining'. As a consequence, a number of authors have introduced so-called co-rotational elements or co-rotational theories [3, 7-10, 18-22]. The phrase 'co-rotational' is used in a number of different contexts but, in the present paper, it will be taken to refer to the provision of a single element-frame that continuously rotates with the element. Co-rotational elements of this kind appear to have been first proposed by Belytschko and co-workers [3, 18, 19] and Oran [9, 20, 21] although the latter author included 'beam-column' terms which somewhat obscured the co-rotational basis.

The co.rotational formulation seems to offer a non-linear framework in which standard linear formulations are used with respect to the rotating frame and non-linearity is introduced via the rotation of this frame. This argument has led some authors to simply apply transformation matrices to linear tangent stiffness matrices [7, 10]. However, this procedure does not correctly account for the variation of these transformation matrices. This was recognised by Oran [9] who, in a two-dimensional context, derived an elegant and consistent tangent stiffness formulation. Similar principles were applied to a three-dimensional formulation [20] although the latter was restricted to both small increments and small local rotations. Belytschko's work involved 'explicit integration' in a dynamic context [3, 18, 19] and so the issue of a consistent tangent stiffness matrix was not directly addressed although Belytschko did schematically outline the procedure for such a formulation and showed [18] that, contrary to some arguments [23], it was possible to derive a tangent stiffness matrix using co-rotational procedures (rather than updated Lagrangian, in which for a certain period the reference configuration is fixed).

0045-7825/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

132 M.A. Crisjield, A consistent co-rotational formulation

In an important paper, Rankin and Brogan [10] derived a general framework for three- dimensional large-rotational problems (both beams and shells) within which one could embed a 'standard element'. Special care was taken of the non-vectorial nature of large-rotations and a number of useful updating formulae were derived. However, the issue of consistent deriv- ations for the out-of-balance force-vector and tangent stiffness matrices was not addressed. In a more recent paper (first seen by the author since the original draft of this paper), Rankin and Nour-Omid [24] have addressed these matters although the direct applications are limited to linear analysis.

2. Theory

2.1. Some basic relationships involving large-rotations

Before presenting the element formulation, we will introduce a number of formulae for large-rotations that will be used later. If no specific reference is given, the derivation of these formulae can be found in [17].

The key formula is apparently due to Rodriguez and involves an orthogonal rotation matrix R, which can be used to rotate a vector x0 into a vector x~, so that

x, = R(O)xo . (1)

The matrix R can be derived from the 'pseudo-vector' O, where

o = o2 = o~i, + od, + o~i~ = oi, 0 = [o'o] ''~ (2) 03

and i is a unit vector. The rotation matrix R is given by

sin 0 (1 - cos 0) R = 1 + ~ S(O) + 02 S(O)S(O), (3)

where S(#) is the skew-symmetric matrix given by

s (o)= o3 o o3 -o,~ o,

(4)

Throughout the paper, the notation S will be reserved for such skew-symmetric matrices. For small-rotations, R becomes I + S(O).

For compound rotations, if x~ = R~(fl)x o is followed by x 2 = R2(ff2)x~,

In order to compute R n without adopting the matrix multiplication of (5), it is useful to define

M.A. Crisfield, A consistent co-rotational formulation 133

a scaled pseudo-vector oJ, where

2 tan(O/2) O. ¢o = oai = 2 t a n ( O / 2 ) i = 0 (6)

Substitution from (6) into (3) gives

1 R----" l + [S(¢[O) "l" 1S(¢jo)2] , (7)

1 +

where S(co) is of the same form as S ( O ) in (4). (If oJ is replaced by 0, (7) provides the approximate relationship due to Hughes and Winget [25]). With the pseudo-vector in the form of (6), we can directly c o m u t e R12(~12 ) appropriate to (5) using

4~O 1 "l" ~2 -- ½tfO1 X ~2 • (8 ) t ~12 1 - ~ l l ~ O 2

The adoption of such vectorial updating reduces the required computer storage. The scaled pseudo-vector of (6) becomes infinite at 0 = 180 ° (and multiples thereof). This

difficulty can be overcome by adopting a sine-scaling [11] but, with a view to the provision of unique updating with angles beyond 180 °, it is better to adopt Euler parameters or normalized quartonians [14, 17,26-29], so that

q ] = [ sin(O/2)i q - - q + q f f i [ ~ cos(0/2) ] " (9)

For angles of magnitude less than 180 °, we can obtain the tangent-scaled pseudo-vector, ~o of (6) from

o , = 2 q / ¢ . (10)

The quartonian can then be updated [28] via

q~2 - q~q~ - q~tq2 + q~q2 + q2q~ - q l x q2 . (11)

(It can be shown that substitution of (10) into (8) is consistent with (11).) In place of (7), the rotation matrix, R, can be expressed in terms of the quartonian via [28]

R ffi ( ~2 _ qtq) I + 2qqt + 2~S(q) . (12)

In some circumstances, we require to compute the quartonian or pseudo-vector from the rotation matrix R. This computation involves the eigenvectors of R. Here, we follow Simo and Vu-Quoc [14] and adopt an algorithm due to Spurrier [26], which obtains the quartonian from the rotation matrix via

a = max(Tr(R), Rll, R22, R33)

134 M.A. Crisfield, A consistent co-rotational formulation

and

then if a = T r ( R ) = R, , + R22 + R33, (14)

q-'- ½(1 + a) 112 , (15)

q, = 2(Rkj -- R j k ) / q , for i= 1, 3, (16)

with i, j, k as the cyclic permutation of 1, 2, 3 and qi as the ith component of the vector q in (9).

If a # Tr(R) but instead ffi R . , (17) then

qi = [½a + 14(1 - T r ( S ) ) ] '/2 , (18 )

= ~ ( R k j - Rjk)/q~, (19)

qt = X4(R,, + R,,) /q, for I= j, k . (20)

For rotations of magnitude less than 180 ° , the tangent-scaled pseudo-vector can then be obtained from (10).

When applying the principle of virtual work, we will require the variation of (1) which is given by

Sx = s ( ~ 0 ) x = - S ( x ) 8 0 , (21)

where the last relationship stems from the skew-symmetric nature of 5 (see (4)). The notation in (21) is a little unconventional because, following (21), the pseudo-vector O could not be updated via 0, = 0 o + 80 (with subscript n for 'new' and o for 'old') but rather from R(0n) = [l + $(68)]R(0o) or via (8) or (11).

2.2. The element and nodal base-vectors and the local 'strains'

In the following description of the finite element theory, we will initially refer to the most basic implementation. Sophistications will be discussed later. Apart from the matrix $, which will be reserved for skew-symmetric matrices of the form of (4), the notation will be independent of that given previously.

Figure 1 shows the three-dimensional beam element with the element base-vectors el-e3 (colmans of £), the left-hand side nodal base vectors t~-t 3 (columns of T), and the right-hand side nodal base vectors q~-q3 (columns of Q). T and Q are rotation matrices associated with pseudo-vectors a and/3, respectively. The initial values of these matrices would be input as geometry and the associated pseudo-vectors, a 0 and ~ , can then be computed from (13)-(20), (10) and (6). The non-additive changes in a and/3 (Sa and 8j0) are the structural rotation variables so that as the analysis proceeds, we can use (8) tO update scaled versions of the pseudo-vectors and hence T and Q (via (7)). Alternatively, we can use (2) and (9) to obtain the quartonian equivalent of, say, 8a, then update the quartonian equivalent of a using (11) before obtaining T from (12).

M. A. Crisfield, A consistent co.rotational formulation 135

Current configuration

Initial configuration

(b)

q2

z,l, Fig. 1. The element. (a) Geometry. (b) Current base-vectors and local slopes.

The nodal displacement variables would be updated in a conventional manner using, for example at node 1 (Fig. 1),

dl = dl + Adi . (22)

The element base vector e 1 passes between nodes 1 and 2 and is therefore given by

1 2 (X2!-I- d21)t(x21 -!- d21 ) el-----~-(X21 +d21 ) , I n - tn

where 1~

(23)

is the new 'length' between nodes and the 21 convention implies, for example,

X2, = X, - X , , (24)


with X~ containing the initial co-ordinates of node 1. The base-vector e 2 should in some sense be an average of t 2 and q2 but orthogonal to e~ (with a similar procedure for e3). ~re will postpone discussion of the details until we have defined the 'local strains'.

The local axial 'strain' is simply the relative displacement in the el direction, i.e.,

u, = I. - l o = sqrt((X21 + d2,)t(X21 + d2 , ) ) - sqrt(X2,X21), (25)

where 1, and l o are new and original straight-line 'lengths' between nodes. In practice (25) is badly conditioned and it is better to adopt the mid-point formula [18],

2 (X21 + ½d2~)td2~. (26) ut = (1. + 1o)

Using (26), the axial force in the bar can be computed from

E A N~-" to u i • ( 2 7 )

The local rotational 'strains' are the local rotations 0zl - Or6 (collectively, #t) (Fig. 1) which can be computed from

2 sin 011 = --tae 2 + tt2e3 , 2 sin 012 ffi -t'2e, + e~t l ,

2 sin 0~3 ' = - t 3 e ~ + e~t~, t t 2 sin 014 = -q3¢2 + q2¢3 , (28)

2 sin 015 ' ' ffi - q 2 e s + e2q I , 2 sia 016 ffi -q~e I + e~q I .

(Apart from the sine terms, which could probably, with little loss of accuracy, be replaced by the angles, (28) can be obtained from the anti-symmetric part of EtT for the first three and EtQ for the last three.) The local 'bending moments' M are conjugate to 0 t with

M tffi (M1, M 2, M3, M4, Ms, Me) , (29)

where M 4 = - M I is the torque (Fig. 1). The moments M are related to the slopes, 0t, via

M = D(0 z - 8zo ) (30)

where 8to are the initial local slopes and

1 ° = y °

m I

GJ 0 0 - G J 0 0 0 4 E I 2 0 0 2EI 2 0 0 0 4 E l 3 0 0 2 E l 3

- G J 0 0 GJ 0 0 0 2 E I 2 0 0 4 E l 2 0 0 0 2 E l 3 0 0 4E13.

(31) i


The second-moment of area, 12 in (31), is assumed to be 'associated' with rotations about t2 and q2, while the inertia I 3 relates to rotations about t 3 and q3-

We must now detail the previous, rather loose, definitions for e 2 and e 3. In order to introduce a frame-invariant choice we firstly define an 'average nodal rotation matrix', so that

s.,,-- s( a+p ) . (32)

A better definition for this matrix is given by

where

AR( IJ -- a )T (a )= AR(a-- P)Q(p) ,

AR(p - 0~) -- O T t .

(33)

(34)

Although pseudo-vectors are not additive, (/3 - a ) will only be moderately large and hence AR(½(/3- a)) can be used as a reasonable representation of the rotation from T to the 'average configuration' R,v. (Further details on the precise method for the computation of R will be given later.)

In general the first column of R,v, r~, will not coincide with the inter-nodal, el, and hence in order to obtain the E matrix (with columns e~-e3), the mean-rotation matrix, Rav -- [r I , r 2, r3], must be 'rotated' through the angle between r~ and e~ onto e~. This can be achieved [22] by rotating the r vectors (columns of R,v) through a vector angle,

r I X e 1 p-cos-'(r',e,) Ir, x e,I ' (35)

so that, for example, e 2 = R ( p ) r 2. However, with a view to subsequent differentiation for the virtual work, the author has used a 'mid-point rule' (see Fig. 2) to obtain the approximations

r~e, (e, + r,) (36) e2--r2 2

t r3• 1

e3 - r3 2 (el + rl) . (37)

Using this approximation,

t (38)

As illustrated in Fig. 2, with 30 ° for the angles related to the inner products, the lack of orthogonality in (38) is 1.9 °. Thirty degree angles are fairly large for 'local slopes' (after removal of the element rigid-body rotation). For slopes of 15 °, the lack of orthogonality is reduced to 0.25 °. Using (36) and (37), it can be shown that the lack of orthogonality in e~e 3 is even smaller. As the mesh is refined, so that the local angles tend to zero, full orthogonality is recovered.

138 M. A. Crisfield, A consistent co.rotational formulation

-(r t or) (e, + r,)

y ~

'2 I i i~--.- e~ via eqn (36)

j r 1 / ~ *

• 1

Fig. 2. Illustration of (33) with cos-t(r ' te t)- 30 °.

2.3. Computation o f the internal force vector, qi, using virtual work

The principal of virtual work requires that

where ql 8p = N 8u t + M t 80t ,

8p t = (Sd',, 8at, ~d~, 8fit)

(39)

(40)

are the virtual nodal 'displacements'. A consistent derivation of qj requires the variation of ut and 0 t. From (23) and (26),

= t 8d2, ffi etl(Sd2 - 6dl) . 8u~ et (41)

In order to obtain 60 t from (28), we require terms such as 6t 2, which can be obtained from (21) so that

= - s ( t , ) , s q , = - S ( q , ) SIJ , (42)

where it should be emphasized that 6a and 6~1 are non-additive to a and/3 even when a and II are small.

We also require 8et-~e 3. From (23) and (41),

where

8d2t (X2~ + d2~) 6u,-- A 6d2t, (43) 6e~ = I. 12

1 A = T [ I - e,etll •

tn (44)


In order to obtain 6e 2 and 6e:3, we require 6r~-6r~. From (33), we can find

6r~ -- S(an)ri - (R.v.Rta.o - l ) r i , (45)

where R,v o, the old R,v matrix, is that in (33) (a function of a and ~1), while the new Ray matrix, Ra, . , is that following the application of 66 and 8/I (as in (42)). It can be shown that

S(611) = ½S(6a + 6p) + ~$(fl - a ) S ( 6 a - 8/I) + ~S(6fl - 6a )S (p - a ) (46)

and that the pseudo-vector 611 is given by

s n = ½(aa +8#) + ~ ( # - a) x (s,~ - a # ) . (47)

The second term should be very small because not only are ( / I - a ) (a local difference after the removal of the overall rigid-body rotation) and (66 - 6 # ) small, but, in addition, the two terms are often nearly parallel thus making the cross-product even smaller. Hence, the author has considered only the first term so that, in conjunction with (45),

)(aa +a#) 8r, ----- -$(r, ~ . (48)

From (36), (37), (43) and (48), 8e2 and 8e 3 can be obtained from

where se~ = n(r,)' ~t, ; 8e, = n(r~)* sp ,

B t = [ B t , t , , B 2 , - B 1 , B ' 2 ] ,

(49)

(50)

r~e 1 1 Bl(r~) = ~ A + ~ Ar,(e, + r , ) ' ,

B2(r~) = 5(2 Q r~e 1 $(rl ) _ 1 $(r~)el(el + r,)t - 4'" 4

(52)

With the aid of (42), (43), (48) and (49), the variations of 8 t (28) can now be obtained as

where

with

sot = F' ~p , (53)

1 1 F - 2 cos 0t ~ " " ' 2 cos 06

f~ = B ( r 3 ) t 2 - B ( r 2 ) t , + h~ ,

f6] , (54)

J'2 - S(r2)t~ + h2 ,

I"3 - B(r3)t l + h3 , :, = B(r,)e,- B(r,~)q~ + h, , (55)

fs -- B(r2)ql + hs , f 6 - B(r3)qx ÷ h6

140 M.A. Crisfield, A consistent co.rotational formulation

and htl - {0 t, (-S(t3)e 2 + S(t2)e3) t, 0 t, 0t},

h~ = {(At2) t, ( - S ( t 2 ) e 1 + S(tl)e2) t, - (At2) t, 0t},

h~ = {(At3) t, (-S(t3)ea + S(t~)e3) t, -(At3) t, 0t},

h 4 = {0 t, 0 t, 0 t, ( - S ( q3)e2 + S( q2)eff} , (56)

h~ = {(Aq2)', 0', -(Aq2) t, (-S(q2)el + S(qt)e2)t} ,

h~-- {(Aq3) t, 0 t, -(Aq3) t, ( - S ( q 3 ) e 1 ~" S(ql)£3)t} •

Also, (41) can be re-expressed as

' ' o'18p (57) 8u t = gt Sp= [ - e l , 0t, el,

and hence, from (39), the internal force vector, q~, is given by

q, = + Ng, (58)

with F from (54) and g from (57).

2.4. The tangent stiffness matrix

The tangent stiffness matrix follows from the variation of (58). In particular,

8q, - g t 8p - (K,t + K~,) 8p. (59)

The 'standard' tangent stiffness matrix, Kt~ is derived by making variations on N and M. From (58), (27), (30), (53) and (57), it is given by

EA g , t = lo gg t + F D F t " ( 6 0 )

In forming the geometric stiffness matrix, K~,, it is useful to define a modified moment vector m, where

mr-'-[ MI M6 ]=( ro t , m2, . . , m6) (61) 2 cos 0, " ' " 2'cos 06 ' "

The geometric stiffness matrix K., involves the variations of F and g in (58) and is given by

K~, = K~,~ + F Diag(M, tan O,)F' + m,[K.2(r2, t 3 - q3) + K,,2(r3, q2 - t2)]

+ m2K.2(r 2, t ,) + m3K.2(r3, t ,) + msK.2(r2, tit)

+ m6K.2(r3, q,) + K.3 + Kt,.3 + g.4 + K . s . (62)

M.A. Crisfieid, A consistent co-rotational formulation 141

In describing the various matrices, it is useful to work with submatrices,

FKll K12 K13 K142

K=IK I, Lg, g,,.l

(63)

and to adopt the convention that if a submatrix is unmentioned, it is zero. Then Kor 1 c o m e s from the differentiation of g (see (57)) and involves

K n = Kss = -K13 = -K31 = N A , (64)

where A has been defined in (44). The K,.2(r~, z) terms come from the variation of the B(r~)z terms in (55) with z being fixed.

K¢2(r~, z) involves the submatrices

r~el (2(e',z) + ztr,)A (65) K~I = - g x 3 = -K31 = g33 = U + U' + ~

where t zt(el + rl) ,

1 A ~ A + r~el Azetl + Ar~e I , (66) U = - ~ " ~ 2/.

4K n = 4K,4 = -4Ks2 = -4Ks4 = -AzetlS(r,) - AriztS(rl) - zt(e, + rx)AS(ri) , (67)

(68)

8K22 = 8K24 = 8K42 = 8K44 = -(rtlel)S(z)S(rl) + S(rl)zetlS(r,) + S(r,)e,ztS(rt)

- (e 1 + rt)tzS(el)S(r,) + 2S(z)S(r , ) . (69)

Note that the submatrices K22 etc. are non-symmetric. The K.3 terms come from (55) via terms such as B ( r 2 ) 8 t I and involve

with K2 = -B(r2)[m4S(ts) + m2S(tl)] + B(r3)[m4S(t2)- m3S(tl)] ,

(70)

(71)

F'4 = B(r2)[m4S( q3) - msS( ql)l - B(ra)[m4S( q2) + m6S( qx)l " (72)

The matrix K¢4 comes from terms such as S(St3)e 2 from 6h~ (see (56)) and in relation to (63) has only non-zero K22 and K44 (non-symmetric) submatrices, where

K22 = m4[S(e2)S(t 3) - S(e3)S(t2)] + m2[-S(el)S( t2) + S(e2)S(t~)] + m3[_S(e l )S( t3)+ S(e3)S(tl)], (73)

K44 = -m4[S(e2)S( qa) - S(ea)S( q2)] -I- m5[-S(ex)S( q2) + S(e2)S( ql)]

+ m6[-S(e l )S(q3) + S(e3)S(qt)] . (74)

142 M. A. Crisfield, A consistent co-rotational formulation

The matrix K¢5 has

K12 = - K 3 2 -- -(m2AS(t2) + m3AS(t3)],

KI4 = -K34 = - (msAS( 4/2) + m6AS( q3)] ,

K21 = - K 2 3 -- Kt12,

(75)

K, , , = - / : , 3 =

Rows 1 and 3 come from terms such as A 8t 2 from 8h 2 (see (56)), while rows 2 and 4 come from terms such as -S(t2)8e 1 in 8h 2 (see (56)).

Finally, K~r5 comes from terms such as 8At 2 in h 2 (see (56)) and involves

with Kll = K33 = - K 1 3 = - K 3 1 --~ Avetl + e,vtA + (etlv)A,

1 v - 7" {m2t2 + m3t3 + msq2 + m6q3} •

~n

(76)

(77)

It will be noted that the 'geometric stiffness matrix' K¢ is non-symmetric. This observation is consistent with that of Simo [13] and Simo and Vu-Quoc [14] who adopted a different formulation. However, numerical experiments have shown that for the conservative problems addressed here, the tangent stiffness matrix becomes symmetric as the iterative procedure reaches equilibrium. This observation is again consistent with the theory of Simo and Vu-Quoc. In common with the latter authors, the present author ha~ found that the excellent convergence characteristics exhibited by the method (when used in conjunction with the full Newton-Raphson procedure) are only marginally impaired if the tangent stiffness matrix is artificially symmetrised. This will be demonstrated.

2.5. Implementation

The updating of nodal rotation matrices T and Q has been performed indirectly using the quartonian update in (11). For the computation of the 'average rotation matrix' Ray in (33), the following procedure has been implemented:

A L G O R I T H M (a) Obtain AR(~ - a) ffi QT t. (b) Use (13)-(20) and (10) to obtain ( / 3 - a) in tangent-scaled form from AR. (c) Obtain the unscaled ( / 3 - a) from the tangent scaled ( / 3 - a) using (6). (d) Compute (/3 - a)12. (e) Tangent scale ( , 8 - a)12. (f) Compute AR((~- a)/2) from (7). (g) Compute Ray from (33).

This procedure will not work for I/3 - al ~> 180 °. However, p - a is the relative rotation (in


pseudo-vector form) of one end of the element with respect to the other. Hence no difficulties are likely (or have been encountered) on this account.

A reasonable approximation involves omitting steps (c) and (e) in the above algorithm. For angles up to 30 ° or so, the tangent scaled pseudo-vector is very close to the unscaled pseudo-vector.

2.6. Discussion

As a result of comments in [30], as well as those from a referee, the following somewhat tentative discussion may be relevant. The tangent stiffness matrix, put forward in this paper, is consistent in relation to variables, 6a say, for which (with the subscript o for 'old' and n for 'new'):

r(an) = T(ao + Sa') = p + s(~a)]T(ao). (78)

However, except when a o is small 8a # 8a' , but rather

Bet = C(a) 6 a ' , (79)

where the precise form of C(a) is given in [14]. To obtain true Newton-based quadratic convergence, it might appear necessary for the stiffness matrix to be consistent with regard to the 'additive' (in the limit) variables 6a'. This could require the inclusion of additional terms involving the derivative of C(a). However, it would appear that these additional terms will vanish in the limit as the out-of-balance forces tend to zero. Hence, such terms may not be required for quadratic convergence. The issue of quadratic convergence will be raised again in the following section on applications.

3. Applications

The following examples all involve the use of the full Newton-Raphson iterative method in conjunction with a tangential, incremental predictor. The quoted number of iterations do not include the latter. The adopted convergence criterion is

," = II ell/11 ell < ~ , (8o)

where [I g[[ is the Euclidean norm of the out-of-balance forces and [[ q[[ is the Euclidean norm of the total applied forces.

Example 1. Cantilever subject to an end-moment

The first example does not test the full three-dimensional behaviour but, in two-dimensions, is a severe test of inextensional bending. The initially-straight cantilever (Fig. 3) is subject to an end moment of

M * = ~ M L = 1, (81) 2~r El


IVI* ,, 1.0

M* =0.3

ML M* -- 2~ 'E I

-- " - ) M*=O

Fig. 3. Initial and deformed geometries for cantilever subject-to-end moment (5 elements).

which forces the beam tO curl into a complete circle (Fig. 3). Rankin and Brogan [10] showed that with their co-rotational formulation this could be achieved with ten elements. Here, we have followed Simo and Vu-Quoc [14] and used five elements (Fig. 3). The solution was obtained using ten equal steps of AM* = 0.1. At each step, convergence was achieved to r (eq. (80)) < 10 -4 in three iterations. Bathe and Bolourchi [2], also using five elements, found inaccuracies beyond M * = 0.5.

Example 2. Forty-five degree bend

This example [2] involves a genuinely three-dimensional response and has been used by a number of authors [2, 4, 14]. Figure 4 shows the initial geometry and the response under increasing load. The bend is of unit cross-section and E was taken as 107. It was modelled using eight elements (as in [14]). Table 1 compares the present solution with those given by a number of authors for the tip-geometry at loads of 300, 450 and 600. Table 2 compares the incremental/iterative performances. Such a comparison cannot be definite because of different convergence criteria but, nonetheless, gives an overall trend. For the present solutions, the convergeace criterion F of (80) was set to 10 -3. It will be noticed that the introduction of an artificially symmetrised tangent stiffness matrix (Table 2) has had almost no effect on the convergence characteristics.

Table 3 compares the residual norms obtained with the present method with those quoted


y

I P=600

= 300

P =0

Fig. 4. Initial and deformed geometries for 45 ° bend.

Jt

by Simo and Vu-Quoc [14], for the fifth increment of solutions obtained using eight equal load increments of 75. Both sets of figures relate to solutions obtained with an artificially symmetrised tangent stiffness matrix. (Very similar figures were obtained with the full non-symmetric stiffness matrix.) I,ittle should be read into the slightly faster convergence of the present solution--choosing one increment at random is hardly scientific. However, both sets of results show a very fast convergence rate as equilibrium is approached. Considering the residual norms for the last three iterations (with I as the last), both methods give

llsll, kllsllT- , II¢ II,- =kI161l, -2, (82)

Table 1 Tip geometry (x, y, z) for 45 ° bend (initially, 70.71, 0.0, 29.29)

Load level

300 450 600

Present Bathe and

Bolourehi [2] Simo and

Vu-Quoc [14] Cardona and

Geradin [4]

58.53, 40.53, 22.16 59.2, 39.5, 22.5

58.84, 40.08, 22.33

58.64, 40.35, 22.14

51.93, 48.79, 18.43

52.32, 48.39, 18.62

52.11, 48.59, 18.38

46.84, 53.71, 15.61 47.2, 53.4, 15.9

47.23, 53.37, 15.79

47.04, 53.50, 15.55


Table 2 Incremental/iterative performance for 45 ° bend (No. of iterations/increment)

Load level

300 450 600

Present non-symmetric

Present symmetric

Simo and Vu-Quoc [14]

Bathe and Bolourchi [2]

Cardona and Geradin [4]

8 5 3

9 5 3

13 8 6

60 equal increments

6 equal incs., average 7.8 its/inc.

with a of the order of 3.5 implying at least quadratic convergence. However one should probably not obtain a convergence rate from just these few iterations. Yet, on proceeding further, the present solution appeared to be affected by rounding error and little further improvement was gained so that a meaningful convergence rate could not be derived. Applying (82) (with [[gl[i below 10 -4) to the last three iterations of the other seven increments generally gave a convergence rate, a, greater than two.

Example 3. Lateral buckling of a cantilever right-angled frame under end-load

The third example also involves a full three-dimensional response and relates to the behaviour of the right-angled frame of Fig. 5 when subject to a load P in the x-direction. Young's modulus was taken as 71240. This problem has previously been analysed by Argyris et al. [1] and Simo and Vu-Quoc [14].

For the present analysis, the frame was modelled using ten elements, with five in each leg. Buckling was artificially induced by applying a very small perturbation load of 2 x 10 -4 in the z-direction at the tip. This produced an initial z-deflection of about half the thickness of the frame.

Table 3

Comparison of convergence rates for 45 ° bend (Table gives Ilgll at end of each iteration)

Iteration number Present Simo and Vu-Quoc [14]

0 83475 75 1 12.9 14700 2 948 423 3 0.23 x 10 -~ 1400 4 0.193 x 10 -2 0.844 5 0.37 × 10 -6 0.661 × 10 - t 6 m 0.19 x 10 -4

M. A. Cri~eld, A consistent co-rotational formulation 147

Y

I

I

Fig. 5. Initial and deformed geometries for right-angled frame.

f S

/

! ! ! !

! ! ! I ! !

!

! !

! I f

The computed response between the applied load and the lateral (z-direction) tip-deflection is shown in Fig, 6. The present solution was obtained using the arc-length method [31], while Simo and Vu-Quoc used displacement control. The kink in their curve (Fig. 6), just prior to 'buckling', relates to the removal of the 'perturbation load'. This load was not removed in the present analysis but (see Fig. 6), the current perturbation load was significantly smaller than that used by Simo and Vu-Quoc. Figure 5 shows the deflected shape of the frame at the final point in Fig. 6.


1.5

I A

1.3

1.2

1.1

Simo and Vu-Ouoc [13 ]

Present solution

1.0

0.9

0.8 I I I I I I I I I I I I 0 5 10 15 20 25 30 35 40 45 50 55 60

Tip z • displacement

Fig. 6. Load/tip z-displacement for right-angled frame.

4. Conclusions and future developments

The paper has described a co-rotational formulation for three-dimensional beams in which the 'strains' relate to conventional small-deflection beam theory but are embedded in a continuously rotating frame. In contrast to many previous formulations, the internal force vector and tangent stiffness matrix are consistently computed from these 'strains'. It has been demonstrated that, as a consequence, the element gives an excellent numerical performance with a convergence rate, when coupled with the Newton-Raphs0n method, that appears close to quadratic. However, machine rounding error makes an absolute confirmation difficult to establish.

The tangent stiffness matrix is found to be non-symmetric. However, for the conservative problems analysed, symmetry is recovered as an equilibrium state is reached. It is saown that the convergence characteristics are only marginally reduced if the tangent stiffness matrix is artificially symmetrised.

Further work could involve a study of the geometric stiffness matrix in order to discover which terms may be reasonably neglected without serious cost to the performance. Extensions to non-conservative loadings are also required. Because the element is based on a single moving framework, the axial-strain terms could be enhanced by the addition of 'shallow shell' or 'Green-like terms' (as in [19]). Equivalent co-rotational formulations have been developed for beam-elements including shear-deformation [29, 32], but further work i~ required.

M. A. Crisfield, A consistent co-rotational formulation 149

Acknowledgement

The author would like to acknowledge the help of Garry Cole of Kingston Polytechnic and, in addition, the constructive and thoughtful comments of the referees.

References

[11

[21

[31

[41

[51

[61

[71

[81

[9] [101

[11]

[121

[13]

[14]

[151

[16]

[17] [18]

[19]

[2O] [21]

J.H. Argyris, H. Baimer, J.St. Doltsinis, P.C. Dunne, M. Haase, M. Kleiber, G.A. Malejannakis, H.-P. Mlejenek, M. Miiller and D.W. Scharpf, Finite element method--The natural approach, Comput. Methods Appl. Mech. Engrg. 17/18 (1979) 1-106. K-J. Bathe and S. Bolourchi, Large displacement analysis of three-dimensional beam structures, Internat. J. Numer. Methods Engrg. 14 (1979) 961-986. T. Belytschko and L. Schwer, Large displacement, transient analysis of space frames, Internat. J. Numer. Methods Engrg. 11 (1977) 65-84. A. Cardona and M. Oeradin, A beam finite element non-linear theory with finite rotations, Internat. J. Numer. Methods Engrg. 26 (1988) 2403-2438. E.N. Dvorkin, E. Onate and J. Oliver, On a nonlinear formulation for curved Timoshenko beam elements considering large displacement/rotation increments, Internat. J. Numer. Methods Engrg. 26 (1988) 1597- 1613. M. Epstein and D. Murray, Three-dimensional large deformation analysis of thin walled beams, Internat. J. Solids and Structures 12 (1976) 867-876. K.-M. Hsiao, H.J. Horng and Y.-R. Chen, A corotationai procedure that handles large rotations of spatial beam structures, Comput. & Structures 27 (6) (1987) 769-781. I.M. Kani and R.E. McConnel, Collapse of shallow lattice domes, ASCE J. Struct. Div. 113 (8) (1987) 1806-1819. C. Oran, Tangent stiffness in space frames, ASCE J. Engrg. Mech. Div. 99 (ST6) (1973) 987-1001. C.C. Rankin and F.A. Brogan, An element independent corotational procedure for the treatment of large rotations, in: L/H. Sobei and K. Thomas, eds., Collapse Analysis of Structures (ASME, New York, 1984) 85-100. C.C. Rankin and B. Nour-Omid, The use of projectors to improve finite element performance, Symp. on Advances and Trends in Computational Struct. Mechanics and Fluid Dynamics, Washington, October 1988. G. Shi and S.N. Atluri, Eiasto-plastic large deformation analysis of space-frames: a plastic-hinge and stress-based explicit derivation of tangent stiffnesses, Internat. J. Numer. Methods Engrg. 26 (1988) 589-615. J.C. Simo, A finite strain beam formulation. The three-dimensional dynamic problem, Part I, Comput. Methods Appl. Mech. Engrg. 49 (1985) 55-70. J.C. Simo and L. Vu-Quoc, A three-dimensional finite strain rod model. Part II: Computational aspects, Comput. Methods Appl. Mech. Engrg. 58 (1986) 79-!16. W. Wunderlich and H. Ob~echt, Large spatial deformations of rods using generalised variational principles, in: W. Wunderlich, E. Stein and K.-J. Bathe, eds., Non-linear Finite Element Analysis in Structural Mechanics (Springer, Berlin, 1981) 185-216. R.K. Wen and J. Rahimzadeh, Nonlinear elastic frame analysis by finite element, ASCE J. Struct. Div. 109 (8) (1983) 1952-1971. J. Argyris, An excursion into large rotations, Comput. Methods Appl. Mech. Engrg. 32 (1982) 85-155. T. Belytschko and J. Hseih, Non-linear transient finite element analysis with convected co-ordinates, Internat. J. Numer. Methods Engrg. 7 (1973) 255-271. T. Belytschko and L.W. Glaum, Applications of higher order corotational stretch theories to nonlinear finite element analysis, Comput. & Structures 10 (1979) 175-182. C. Oran, Tangent stiffness in plane frames, ASCE J. Struct. Div. 99 (ST6) (1973) 973-985. C. Oran and A. Kassimali, Large deformations of framed stuctures under static and dynamic loads, Comput. & Structures 6 (1976) 539-547.


[22]

[231

[24]

[25]

[26]

[27]

[28]

[291

[30] [31]

[32]

D.N. Bates, The mechanics of thin walled structures with special reference to large rotations, Department of Civil Engineering, Imperial College of Science and Technology, London, 1987. S.C. Tang, K.S. Yeung and C.T. Chart, On the tangent stiffness matrix in a convected coordinate system, Comput. & Structures 12 (1980) 849-856. C.C. Rankin and B. Nour-Omid, The use of projectors to improve finite element performance, Symp. on Advances and Trends in Computational Struct. Mechanics and Fluid Dynamics, Washington, October 1988. T.J.R. Hughes and J. Winget, Finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis, Internat. J. Numer. Methods Engrg. 15 (1980) 1413-1418. R.A. Spunder, Comment on "Singularity-free extraction of a quartonian from a discrete cosine matrix", J. Spacecraft 15 (4) (1978) 255. J.F. Besseling, Large-rotations in problems of structural mechanics, in: P.G. Bergan et al., eds., Finite Element Methods for Nonlinear Problems, Europe-US Symposium, Trondheim, Norway (Springer, Berlin, 1986) 25-39. K.W. Spring, Euler parameters and the use of quartonian algebra in the manipulation of finite rotations: A review, in: Mechanism and Machine Theory 21 (1986) 365-373. G. Cole, Consistent co-rotational formulation for geometrically non-linear beam elements with special reference to I.~rge rotations, Ph.D. Thesis, submitted to Department of Civil Engineering, Kingston Polytech- nic, England, 1~,~0. C.C. Rankin, Private communication, December 1988. M.A. Crisfield, A fast incremental/iterative solution procedure that handles "snap-through" 13 (1-3) (1981) 55-62. M.A. Crisfield and G. Cole, Co-rotational beam elements for two and three-dimensional non-linear analysis, Proc. IUTAM/IACM Symp. on Discretization Methods in Structural Mechanics, Vienna, June 1989 (Spring- er, Berlin, 1990) to be published.

Documents

A Consistent Co-rotational Formulation for Non-linear, Three-dimensional, Beam-elements