Journal of Computational and Nonlinear Dynamics JANUARY 2010, Vol. 5 / Copyright 2010 by ASME

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1 A. L. Schwab Laboratory for Engineering Mechanics, Delft University of Technology, Mekelweg 2, NL-2628 CD Delft, The Netherlands J. P. Meijaard Laboratory of Mechanical Automation and Mechatronics, Faculty of Engineering Technology, University of Twente, P.O. Box 27, NL-7500 AE Enschede, The Netherlands Comarison of Three- Dimensional Flexible Beam Elements for Dynamic Analysis: Classical Finite Element Formulation and Absolute Nodal Coordinate Formulation Three formulations for a flexible satial beam element for dynamic analysis are comared: a Timoshenko beam with large dislacements and rotations, a fully arametried element according to the absolute nodal coordinate formulation (ANCF), and an ANCF element based on an elastic line aroach. In the last formulation, the shear locking of the antisymmetric bending mode is avoided by the alication of either the two-field Hellinger Reissner or the three-field Hu Washiu variational rincile. The comarison is made by means of linear static deflection and eigenfreuency analyses on stylied roblems. It is shown that the ANCF fully arametried element yields too large torsional and flexural rigidities, and shear locking effectively suresses the antisymmetric bending mode. The resented ANCF formulation with the elastic line aroach resolves most of these roblems. DOI: 0.5/ Corresonding author. Contributed by the Design Engineering Division of ASME for ublication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscrit received October 28, 2008; final manuscrit received March 2, 2009; ublished online December 8, Assoc. Editor: Subhash C. Sinha. Introduction Several finite element method FEM formulations for satial finite beam elements for use in multibody system dynamic rograms can be found in literature. A common aroach is using a small dislacement formulation with resect to a reference frame moving with the average overall motion of the beam,2. In order to reduce the number of degrees of freedom, a limited number of modes for the deformation are chosen 3. The linear contribution to the stiffness matrix due to nominal or actual stresses can be included by adding a geometric stiffness matrix 4. A different way to describe the motion of the elements is by using nodal coordinates that describe the configuration of the elements with resect to an inertial reference frame. This aroach is more in line with traditional nonlinear finite element formulations used in statics. A convenient element formulation was develoed by Van der Werff and Jonker 5, imlemented in the rogram SPACAR 6 and further extended thereafter 7 9. This formulation defines a number of generalied deformations for each element that are invariant under roer Euclidean dislacements, so arbitrary rigid-body motions can exactly be described. A descrition with indeendently interolated dislacements and rotations was given by Simo and Vu-Quoc 0. Reduced numerical integration was used to revent shear and membrane locking. Extensions to higher-order interolations and hysical nonlinearities can easily be included. More recently, another aroach using nodal coordinates, the absolute nodal coordinate formulation ANCF, was roosed by Shabana. This finite element formulation describes the osition of a material oint within the element by interolations based on the Cartesian absolute coordinates of the nodal oints and on gradients of the ositions with resect to material coordinates describing a reference configuration. The use of gradients allows the exact reresentation of the inertia arameters of the beam as a rigid body and avoids redundancy of rotational arameters. At the cost of an increased comlexity of the stiffness, the resulting element mass matrix is constant, which allows an efficient evaluation of the accelerations 2. Poisson locking in the continuum mechanics formulation of the ANCF, which results in an overestimate of the flexural rigidity, and a oor descrition of the deflection of a cantilever beam were noted in Ref. 3. An elastic line model with a reduced shear stiffness was introduced to imrove the beam element. Two other aroaches to imrove on the descrition of varying bending moments in the element were roosed in Ref. 4. In the first, terms were added to the dislacement interolation and in the second, an elastic line model was used, which cannot describe torsion and can be used to model cables. An elastic line model with constraints on the deformation of the cross section was develoed in Ref. 5, which aroximates classical beam models. The urose of the resent aer, which elaborates on Ref. 6, is to make a comarison between the finite element formulation for a two-noded satial beam element as described in Ref. 7 and several corresonding absolute nodal coordinate formulations. In order to overcome the deficiencies of the continuum mechanics formulation 7 and the elastic line formulation 8, a modification of the latter is taken into the comarison that includes deformations of the beam cross section and uncoules the shear deformation and antisymmetric bending by means of the Hellinger Reissner 9 or Hu Washiu 20 variational rincile. All discussed beam elements can be used for multibody system roblems in which large rigid-body motions and small or large elastic vibrations need to be modeled. The ANCF element was esecially develoed for the large vibration roblems. However, a first reuirement is that roblems with small vibrations can be accurately calculated. Therefore, for the comarison of the element formulations with each other and analytic solutions, only linearied eigenfreuencies and linearied static deflections for a single beam are considered. Journal of Comutational and Nonlinear Dynamics JANUARY 200, Vol. 5 / 000- Coyright 200 by ASME Downloaded 2 Jan 200 to Redistribution subject to ASME license or coyright; see htt://

2 O y x x e The organiation of the aer is as follows. After this introduction, a beam element based on the Timoshenko beam with large dislacements and large rotations is described. Then the ANCF elements with their modifications are described. Next, results on the eigenfreuencies and deflections due to a static load for a single beam element are resented and discussed. The article ends with conclusions. 2 Classical FEM Beam Element The finite beam element in the multibody dynamics rogram SPACAR is a shear flexible beam based on the elastic line concet. This suoses that the beam is slender and the cross section is doubly symmetric. Large dislacements and rotations are allowed, but the deformations must remain small. The resentation of the element mainly follows 7. The configuration of the element Fig. is determined by the ositions and orientations of the two end nodes, by which it can be couled to and interact with other elements. The ositions of the end nodes and are given by their coordinates x and x in a global inertial system Oxy. The change in orientation of a node with resect to the reference orientation is determined by an orthogonal rotation matrix R, which can be arametried by a choice of arameters, denoted by, such as Euler angles, modified Euler angles, Rodrigues arameters, and Euler arameters. We use Euler arameters with a constraint, but this choice is immaterial to the descrition of the roerties of the element. For a beam element, orthogonal triads of unit vectors e x,e y,e and e x,e y,e rigidly attached to the nodes and, resectively, are defined. The unit vector e x is erendicular to the average wared crosssectional lane of the beam in the sense of Cower 2, and e y and e are in the rincial directions of the cross section. In the absence of shear deformations, e x is tangent to the elastic line of the beam. The change in orientation of the triads is determined by the rotation matrix as e = R e where e is a unit vector in the reference configuration. 2. Elastic Forces. The elastic forces are derived with the elastic line concet. The element has 6 degrees of freedom as a rigid body, while the nodes have 2 degrees of freedom. Hence the deformation that is determined by the end nodes of the element can be described by six indeendent generalied strains, which are functions of the ositions and orientations of the nodes and the geometric arameters of the element. With l=x x and l the length of the initial undeformed beam, we define the six generalied strains as = l T l l elongation 3 = l T e, e y e x w 2 = l e T e y e y T e /2 v Fig. 4 = l T e u ϕ FEM beam ϕ y ϕ x torsion e x e y bending in the x-lane e x 5 = l T e y, 6 = l T e y bending in the xy-lane 2 These generalied strains, which may be comared with what Argyris called natural modes 22, are invariant under arbitrary rigidbody motions, so they truly measure the amount of strain in the element. If we grou the ositions and orientations of the nodes in a vector x= x,,x, and denote the vector of generalied deformations by ε, then we can write for generalied strains 2 symbolically i = D i x k, i =,...,6, k =,...,2 3 The energetically dual uantities of the generalied strains ε are the generalied stresses. The hysical meaning of these stresses is found by euating the internal virtual work of the elastic forces T ε to the external virtual work f T x of the nodal forces. Substitution of the virtual generalied strains derived from E. 3 results in i i = i D i,k x k = f k x k, x k 4 with the virtual nodal dislacements and rotations x T = x T, T, x T, T, and a subscrit after the comma to denote artial derivatives. From E. 4 we derive the force euilibrium conditions for the element as f k = D i,k i 5 In the case of small deformations, the generalied stresses have a clear hysical meaning. As the deformed and undeformed geometries are nearly the same, we consider the undeformed situation in which the beam central axis coincides with the global x-axis. For the rotational arameters we choose the small rotations about the three coordinate axes x, y, and. The Jacobian of the generalied strains then takes the values l l 0 0 D l i,k = l l l 6 The euilibrium nodal force system according to E. 5 is then given by f =, 6 5, 3 4, M = 2 l, 3 l, 5 l f =, 5 6, 4 3, M = 2 l, 4 l, 6 l 7 From this result we interret that is the normal force, 2 l is the torsion moment, and 3 l, 4 l, 5 l, and 6 l are the bending moments at the nodes and. If for each beam element the strains remain small by dividing the overall beam in sufficiently many elements, then the usual linear stress-strain relation can be alied, which results for the generalied stresses and strains in i = S ij j, i, j =,...,6 8 where the stiffness S ij =diag S,S 2,S 3,S 4 is given by S = EA/l, S 2 = S t /l 3 3 EI y S 3 = 4+ 2+, = 2EI y + l symm. 4 + GAk l 2 3 EI S 4 = 4+ y 2+ y, y = 2EI + y l symm. 4 + y GAk y l 2 9 Here, E is the modulus of elasticity Young s modulus, G is the shear modulus, A is the area of the cross section, S t is the torsional stiffness, I y and I are the area moments of inertia of the cross / Vol. 5, JANUARY 200 Transactions of the ASME Downloaded 2 Jan 200 to Redistribution subject to ASME license or coyright; see htt://

3 section with resect to the rincial axes, and k y and k are the shear coefficients according to Cower 2. Note that the inclusion of the shear deformation is done by a slightly modified stiffness matrix 23. This tying of the shear deformation to the bending by means of the statics of the beam revents roblems of shear locking. Finally the element stiffness matrix is obtained by taking artial derivatives of the nodal forces f with resect to the nodal coordinates, resulting in a tangent stiffness matrix K ij = D k,i S kl D l,j + D k,ij k 0 which consists of two arts. The last art is the geometric stiffness matrix, which, evaluated in the undeformed and unstressed geometry, is identical to ero, and the first art is the linear stiffness matrix K ij = D k,i S kl D l,j 2.2 Mass Matrix. The derivation of the consistent mass formulation for the flexible satial beam element is based on the elastic line concet. To arrive at a Timoshenko beam the rotary inertia of the cross section will be included by a searate interolation of the angular rates of the cross section along the elastic line. The first art of the mass matrix is obtained by neglecting the contribution of the rotary inertia and only taking the mass distribution along the elastic line into account. The interolation for the ositions on the elastic line for finite deformation is a third-order olynomial r = x le x x le x 2 where =x/l and x,0 x l is a material coordinate along the beam axis, and where the rincial directors e x and e x are transformed from their initial values according to E.. The kinetic energy together with the mass matrix can be obtained from the integral T = 2 m ṙ 0 T ṙd = 2 ẋt M ẋ 3 where m is the total mass of the beam. If the rotations at the nodes are arametried by and, this results in a mass matrix where 56I 22lA 54I 3lB M = 420 m 4l 2 A T A 3lA T 3l 2 A T B 4 56I 22lB symm. 4l 2 B T B A = e x /, B = e x / 5 The inertia forces can, in general, be written as the sum of mass accelerations+convective terms f in = M x ẍ + h x,ẋ 6 These convective terms h, which are uadratic in the seeds, arise only for a nonconstant mass matrix and can be derived from the mass matrix as h i = M ij M jk ẋ j ẋ k x k 2 x i 7 Clearly the mass matrix M is not constant and the corresonding convective terms are where l 22Ȧ 3Ḃ h = 420 m l 2 A T 4Ȧ 3Ḃ l 3Ȧ 22Ḃ l 2 B T 3Ȧ +4Ḃ 8 Ȧ = A/, Ḃ = B/ 9 The second art of the mass matrix takes into account the rotary inertia of the cross section. Interolation of angular rates along the elastic line is more convenient than interolation of the orientation of the cross section. The angular rates, exressed in a body-fixed reference frame, are interolated linearly as = + 20 where and are the angular rates at nodes and exressed in a body-fixed reference frame. The mass matrix can be obtained from the kinetic energy integral = T 2 = T dj = 2 ẋt M 2 ẋ 2 2 =0 where dj is the mass moment of inertia matrix along the bodyfixed axes of an infinitesimal small section of length ld. The body-fixed angular rates at the nodes exressed in terms of the nodal rotational arameters and are = P, = Q 22 where the matrices P and Q have the same functional deendence on the rotation arameters. For a constant cross section with a sectional mass moment of inertia matrix dj=mjd, the second art of the mass matrix, which takes into account the rotary inertia of the cross section, is M 2 = m 2P T J P 0 P T symm. 2Q J Q T where the corresonding convective terms h 2 can be derived from this mass matrix by alication of E. 7. For slender beams the rotary inertia contributions are usually small comared with the regular inertia terms. If the rincial dimension of the cross section is h, then J is of the order h 2 and the rotary inertia contributions are of the order h/l 2 comared with the regular inertia terms 4 and 8. Therefore the linear interolation in E. 20 is sufficiently accurate. 3 ANCF Beam In this section a satial beam element with two nodes according to the absolute nodal coordinate formulation will be resented. In the first art we follow mainly the descrition by Yakoub and Shabana 7 and Shabana and Yakoub 8. A distinguishing oint in the ANCF is the use of sloe vectors to describe the orientation of the cross section in the nodes, where the sloe vectors are not necessarily unit vectors. This leaves more room for the cross section to deform and change shae. It is exected 7,8 that this tye of descrition, together with a threedimensional continuum mechanics aroach, leads to more accurate results. A well-known major advantage of this descrition is that it leads to a constant mass matrix. Unfortunately, the exressions for the elastic forces are more comlex. The configuration of the beam element Fig. 2 is determined by the osition and sloe vectors of the two end nodes and. Each node is defined by one vector for the osition r and three Journal of Comutational and Nonlinear Dynamics JANUARY 200, Vol. 5 / Downloaded 2 Jan 200 to Redistribution subject to ASME license or coyright; see htt://

4 O y r x r r y w x r x v x u x w v u r r r y r x ij = 2 r k,ir k,j ij = 2 r T,x r,x r T,x r,y r T,x r, r T,y r,y r T,y r, symm. r T, r, 33 From this we identify six indeendent strain comonents, which we write in the form of a strain vector such that the vector dot roduct 2 T reresents the elastic energy. The comonents of the strain vector are now Fig. 2 ANCF beam = 2 r,x T r,x, 4 = r,x T r,y 2 = 2 r,y T r,y, 5 = r T,y r, vectors for the sloes r x, r y, and r, where every vector is exressed in a global inertial system Oxy. Thus the element has 24 nodal coordinates given by the vector e = r T,r T x,r T y,r T,r T,r T x,r T y,r T T 24 The location of an arbitrary oint r in the beam is determined by the interolation r = S x,y, e 25 where S is the element shae function and e is the vector of nodal coordinates. The shae function is obtained using olynomials that are in this case cubic in x and linear in y and, where x is a material coordinate along the beam axis and y and are the two other erendicular directions. The element shae function matrix S is now defined as S = S I,S 2 I,S 3 I,S 4 I,S 5 I,S 6 I,S 7 I,S 8 I 26 where I is the 3 3 identity matrix and the olynomials S = , S 2 = l S 3 = l, S 5 = , S 4 = l S 6 = l S 7 = l, S 8 = l 27 with the nondimensional coordinates = x/l, = y/l, = /l 28 and l the initial length of the beam. The initial undeformed configuration where the beam central axis coincides with the global x-axis is r 0 = Se 0 29 with the initial nodal coordinates e 0 as e 0 = 0 T,e T x,e T y,e T,le T x,e T x,e T y,e T T 30 with fixed triads e x,e y,e of the global inertial system Oxy. Indeed, substitution of e 0 in E. 29 leads to the identities r 0 = x,y, T. 3. Elastic Forces, Fully Parametried Element. The elastic forces are derived from a general continuum mechanics aroach. We start from the dislacements u of an arbitrary oint of the beam exressed in the global Oxy coordinate system as given by u = r r 0 3 Substitution of these dislacements in the Green Lagrange strain tensor ij = 2 u i,j + u j,i + u k,i u k,j, i, j,k = x,..., 32 where artial derivatives are denoted by u x,y = u x / y, etc., leads to the strain tensor exressed in the absolute coordinates r and their derivatives as 3 = 2 r, T r,, 6 = r T, r,x 34 and the energetically dual stress vector comonents are the three normal stresses, 2, and 3, and the three shear stresses 4, 5, and 6. The virtual work of the elastic stresses now can be written as W = V T dv 35 Under the assumtion of a Saint-Venant Kirchhoff material model, a homogeneous isotroic elastic material, the stress vector is related to the strain vector as = E where the nonero elastic coefficients E are given by 2G E ij = 2, i, j =,...,3 36 E kk = G, k =4,...,6 37 Here, G is the shear modulus and is Poisson s ratio. Euating the virtual work of the elastic stresses with the virtual work of the external nodal forces Q as in V T dv = Q T e 38 yields the elastic nodal forces exressed in terms of the nodal dislacements Q = V / e T E dv 39 The tangent stiffness matrix is obtained by lineariing the elastic forces with resect to the nodal dislacements K = V / e T E / e dv + V 2 / e 2 T E dv 40 The second matrix is the geometric stiffness matrix, which, evaluated in the undeformed and unstressed geometry, is identical to ero, whereas the first matrix is the linear stiffness matrix K = V / e T E / e dv 4 Finally, we will evaluate the linear stiffness matrix in the undeformed configuration. With the notion that the sloes in the initially undeformed configuration are identical to the global directions r,x 0 = e x, r,y 0 = e y, r, 0 = e 42 and / Vol. 5, JANUARY 200 Transactions of the ASME Downloaded 2 Jan 200 to Redistribution subject to ASME license or coyright; see htt://

5 r,x = S,x, r,y = S,y, r, = S, 43 e 0 e 0 e 0 the artial derivatives of the strain vector evaluated at this initial undeformed configuration become S 2,y = S,x S 3, e 0 S,y + S 2,x S 2, + S 3,y S 3,x + S, 44 where S i,j stands for the ith row of S,j. Substitution into E. 4 and evaluating the integral over the volume of the beam lead to the desired linear stiffness matrix K 0 = V / e 0 T E / e 0 dv Mass Matrix. The absolute nodal coordinate formulation leads to inertia forces that can be exressed as minus the roduct of a constant mass matrix times the accelerations of the nodal coordinates. There are no inertia forces that are uadratic in the velocities. The constant mass matrix is defined as M = V S T SdV 46 The above integral defines a mass matrix that only deends on the mass distribution and the dimensions of the beam and, under the assumtion of a consistent shae function, catures all linear and rotary inertia effects. 3.3 Elastic Forces, Elastic Line Aroach. The fully arametried beam element of Sec. 3. suffers from shear locking for antisymmetric bending, as will be shown in Sec. 4. As an alternative, we roose to develo an element in which the elastic line concet is used, along lines as in Ref. 8. For the interolation of oints of the beam we will still use the same cubic form in the longitudinal direction and linear form in the transverse direction as in the fully arametried element 25 and 26, but all deformations extension, shear, torsion, and bending will be evaluated on the elastic line. First we define the sloes on the elastic line as r x = r,x x,0,0, r y = r,y x,0,0, r = r, x,0,0 47 With these sloes we define nine generalied deformations x = 2 r x T r x, y = 2 r y T r y, = 2 r T r y = r y T r xy = r x T r y, x = r x T r x = 2 r T r y r y T r, y = r T r x, = r y T r x 48 where a rime denotes a derivative with resect to x. The first six deformations are the usual Green Lagrange strains: the first four, x, y, and, and y, reresent the extension of the beam and the deformation of the cross section, and xy and x are the transverse shear deformations. For small strains, x reresents the torsion, and y and the bending deformations. Definitions of the bending deformations better in agreement with Timoshenko s beam theory are y =r x T r and = r x T r y, but these are not used, because they would lead to constant curvature for small deformations, instead of a linear variation for the definition in E. 48, and revent antisymmetric bending altogether. The strain energy of the beam can be written as the sum of four distinct arts, as in W e =W l +W t +W b +W s. The individual strain energies are for extension and couled deformation of the cross section W l = 2 l A ie ij j d, i, j =,..., with the strains i= x, y,, y and the elasticity coefficients E ij as in E. 37 ; for torsion and bending W t = 2 l S t x 0 2 d, W b = 2 l EI y y EI 2 d and for the transverse shear deformation 50 W s = 2 l GAk y xy GAk 2 x d 5 The elastic forces are determined, in the same manner as in Sec. 3., from euating the variation of the elastic energy to the virtual work of the nodal forces W e = W e / e e = Q et e 52 Then the linear stiffness matrix is found by lineariing the elastic forces with resect to the nodal coordinates at the undeformed reference configuration K = Q e / e 0 53 Because the elastic energy is a direct sum of contributions due to extension, torsion, bending, and shear, the same holds for the stiffness matrix. The individual contributions to the stiffness matrix are as follows: For the extension for the torsion K l = i,e 0 l 0 T E ij j,e 0 Ad, i, j =,...,4 54 K t = S t x,e 0 l 0 T x,e 0 d 55 for the bending K b = EI y y,e 0 l 0 T y,e 0 d + EI,e 0 l 0 T,e 0 d 56 and for the shear deformation K s = GAk y xy,e 0 l 0 T xy,e 0 d + GAk x,e 0 l 0 T x,e 0 d 57 resulting in a total linear stiffness matrix K = K l + K t + K b + K s 58 The artial derivatives of the generalied strains with resect to the nodal coordinates in the initially undeformed configuration take on even simler forms as in Es. 42 and 44 since the only variable is now the elastic line coordinate x. With the cubic/linear interolation on the elastic line according to Es. 26 and 47, adding the contribution of the shear deformation according to E. 5 will result in shear locking for antisymmetric bending, as will be shown in Sec. 4. Therefore we roose to add the transverse shear stiffness by means of a variational rincile in which we are free to define not only the osition field but also the deformation field or the stress distribution Hellinger Reissner aroach. In the Hellinger Reissner 9 variational aroach we are free to define both the osition Journal of Comutational and Nonlinear Dynamics JANUARY 200, Vol. 5 / Downloaded 2 Jan 200 to Redistribution subject to ASME license or coyright; see htt://

6 field and the stress distribution. The strain field is related to the stress field by the constitutive relation. This can be advantageous if the stress distribution, usually from an engineering oint of view, is known beforehand. The general form of this rincile is V T ε r W c dv Q r =0 59 where ε r is the strain as calculated from the osition field and W c is the comlementary elastic energy density, while Q contains all external forces and inertia forces. To resolve the shear locking roblem we consider the case of ure shear deformation. We assume that the shear forces will vary linearly over the elastic line of the element. The strain energy of the shear deformation is the sum of the shear in the y- and the -direction. For the shear forces in the -direction we assume a linear shear stress distribution according to x = N 60 with the shae function N =, 6 and the shear stresses at the nodes = x, x T 62 The Hellinger Reissner shear strain functional is now W s x = V T x W c x dv 63 where W c is the comlementary shear stress energy according to W c x = 2 2Gk x 64 with shear factor k to account for the fact that the shear stress is not uniformly distributed over the cross section 2. Substitution of the linear shear stress distribution from E. 60 results in the shear strain functional W s = V T N T x T 2Gk N T N dv 65 with the shear strain distribution x according to E. 48. For this shear strain functional we seek a stationary value with resect to the generalied shear stress arameters, resulting in W s T N = V T x N T N Gk dv =0 Integration over the volume yields W x Al H Gk = 0 with the shear strain terms W x e = N Al 0 T x e d 68 which are in general nonlinear functions in the nodal coordinates e, and the constant coefficient matrix /3 /6 H = 69 /6 /3 From this we can solve for the generalied shear stress arameters = S W x, S = Gk Al H = Gk Al 2 4 Substitution in the original shear energy function yields the shear energy according to the Hellinger Reissner rincile W s = 2 W x T S W x 7 The shear strain energy for shear forces in the y-direction W sy can be derived in the same manner resulting in a total shear strain energy function of W s = 2 W xy T S y W xy + 2 W x T S W x 72 The strain energy for the beam with the adated shear stiffness according to the Hellinger Reissner rincile now becomes W e = W l + W s + W t + W b 73 The shear stiffness matrix according to the Hellinger Reissner rincile, which relaces E. 57, is found in the same manner as in E. 53 resulting in K s = W xy,e 0 T S y W xy,e 0 + W x,e 0 T S W x,e Hu Washiu Aroach. The shear locking roblem can be resolved even more when one has a greater freedom in choosing the various interolations. In the Hu Washiu variational aroach 24 we are free to define not only the osition field r and the stress field but also the strain field ε. The general form of this rincile is V W ε + T ε r ε dv Q r =0 75 where ε r is the strain as calculated from the osition field and W ε is the elastic otential energy density, while Q contains all external forces and inertia forces. The variation has to be taken for the stress field, the strain field, and the osition field. The weak solution is found for arbitrary admissible variations of the stress field, strain field ε, and osition field r. When one takes the strain field in accordance with the stress field one regains the Hellinger Reissner rincile. For the shear deformation in the -direction, we have = x, ε= x, and r is as in E. 25. In the case of ure shear deformation we assume that the shear strain will vary linearly over the elastic line of the element x = N 76 with the shae function N from E. 6 and the shear strains at the nodes = x, x T The elastic shear strain energy is 77 W s = Gk 2 x dv 78 2 V Take for the stress field two Dirac functions, one at each end of the beam, which means that the kinematic conditions are only enforced at both ends. With the Dirac or imulse function ˆ the stress field then becomes x = I with the imulse shae function I = ˆ 0, ˆ and the shear stresses at the nodes = x, x T 8 For the strain ε r as calculated from the osition field we use the exression for the shear strain on the elastic line r T x r from E. 48. Then, variation of the stresses gives the kinematic conditions, namely, that the shear strains at the nodes are determined by / Vol. 5, JANUARY 200 Transactions of the ASME Downloaded 2 Jan 200 to Redistribution subject to ASME license or coyright; see htt://

7 = r x T r =0, r x T r = 82 Next, variation of the shear strains gives the constitutive behavior, which, after integration over the volume, results in = GAk lh 83 The variation of the osition field, which in this case finally comes down to the variation of the nodal coordinates e, results in the element euilibrium euations Q = T e 84 Substitutions of E. 82 in E. 83 and of E. 83 in E. 84 give the nodal elastic shear forces Q e s in terms of the nodal coordinates. The linear shear stiffness matrix, which relaces E. 57, is found in the same manner as in E. 53, resulting for the shear in the -direction in K s = GAk l T H 85 e 0 e 0 The contribution of the shear in the y-direction to the linear shear stiffness matrix is found in the same manner as above. 4 Results and Discussion To investigate the erformance of the three different element tyes, classical FEM, ANCF fully arametried, and ANCF with an elastic line aroach and a Hellinger Reissner or a Hu Washiu variational method, a number of tests on linear roblems have been erformed. The first is a static loading on a cantilevered beam. Since the ultimate goal for these elements is the use in a multibody dynamics environment, an eigenfreuency analysis on a beam with various boundary conditions has been erformed as a second test. Moreover, the eigenfreuency analysis yields coordinate-free results and therefore makes a comarison easy and objective. The beam is modeled by one finite element and has an undeformed length l and a uniform suare cross section of width and height h=0.02l. The material is isotroic and linearly elastic with modulus of elasticity E, Poisson s ratio, and volumetric mass density. The solutions deend on Poisson s ratio, and throughout a value of =0.3 is used. The shear factors for a suare cross section are k y =k =0 + / The torsional stiffness is given by S t =Gk x I, with G=E/ 2 +, the shear distribution factor k x = , and the olar area moment of inertia I =I y +I. The beam is initially aligned along the x-axis with the rincial axes of the cross section along the y-axis and -axis. For the static test a cantilevered beam loaded at the ti by a bending moment M y and a transverse force F is considered. The boundary conditions for the classical FEM element are straightforward: All three dislacements and three rotations at node are suressed. For the ANCF elements the three dislacements of node are suressed together with all three dislacements of the two cross-sectional sloe vectors r y and r in node. This is as if the cross section is welded on all sides to the suort. The generalied alied forces Q e for the ANCF element associated with the nodal coordinates e due to the alied bending moment M y and transverse forces F can be found by a virtual work aroach. It is assumed that stress distributions x y, and x y, in the cross section are such that the only nonero static result is a moment about the y-axis as in M y = x da and a transverse force F = x da. For instance, a linear normal stress x =c and a constant shear stress x =c 2 will do. Then the virtual work of these stresses in the cross section must be eual to the virtual work of the alied generalied force Q e as in Table Dimensionless transverse ti dislacement w, crosssectional rotation csy, and elastic line rotation ely for a cantilevered beam of length l loaded by a moment M y at the ti for four different element tyes: classical FEM, ANCF fully arametried, ANCF with a Hellinger Reissner variational method, and ANCF with a Hu Washiu variational method. Dislacements are nondimensionalied by M y l 2 / EI y and rotations by M y l/ EI y. The shear factor =2EI y / GAk l 2 and a common factor = 2 + / are used. Method w csy ely Classical FEM 2 ANCF-f 2 ANCF-HR ANCF-HW W = r x x + r x da = Q e T e 86 where the virtual dislacements in the cross section r = r x, r y, r are interolated according to E. 25. This results in an external generalied force in node as Q e = 0,0,F,0,0,0,0,0,0,M y,0,0 87 Clearly, the rotation of the cross section about the y-axis is determined by the x-dislacement of the sloe vector r. The transverse ti dislacement w and the rotations at the ti about the y-axis of the cross section csy and of the elastic line ely resulting from the static tests are resented in Tables and 2. Note that the two rotations differ due to transverse shear deformation. For the classical FEM element they are defined as csy = y and ely = w/ x and for the ANCF element as csy = r x and ely = r x. The classical FEM element yields the exact solution for both loading cases. The ANCF fully arametried element shows, for the moment loading Table, in all results an offset by a Poisson factor = 2 + /. This is so, because the anticlastic deformation of the cross section cannot be described by the continuum osition field. This Poisson factor is also resent in the results for the transverse force loading, Table 2, but here the leading term in the dislacement is also off by a factor of 3 4, which is due to the shear locking. The effect of transverse shear is resent but off by a factor k since the transverse shear stress distribution is not taken into account with the ANCF fully arametried element aroach. The elastic line aroach together with a Hellinger Reissner variational method gives some imrovement in that the Poisson factor is gone but the element still suffers from a oor rediction of the deflection. This is due to the weak inclusion of the connection between the deflection of the center line and the shear deformation at the clamed end, and some additional transverse shear, O h/l 2, which is resent in all solutions. Alication of the Hu Washiu variational rincile yields better Table 2 Dimensionless transverse ti dislacement w, crosssectional rotation csy, and elastic line rotation ely as in Table but now for loading by a transverse force F. Dislacements are nondimensionalied by F l 3 / EI y and rotations by F l 2 / EI y. Method w csy ely Classical FEM ANCF-f ANCF-HR ANCF-HW k k Journal of Comutational and Nonlinear Dynamics JANUARY 200, Vol. 5 / Downloaded 2 Jan 200 to Redistribution subject to ASME license or coyright; see htt://

8 Table 3 Dimensionless eigenfreuencies = / B,T,L for the free vibration of a comletely free beam modeled by one element for four different element tyes: classical FEM, ANCF fully arametried, ANCF elastic line aroach with a Hellinger Reissner variational rincile, and ANCF elastic line aroach with a Hu Washiu variational method together with the analytical solution for an Euler Bernoulli beam. The eigenfreuencies are nondimensionalied by B = EI/ Al 4 for bending modes, T = G/ l 2 for torsion mode, and L = E/ l 2 for longitudinal Lg and cross-sectional Cs modes. Mode Analytic Classical FEM ANCF-f ANCF-HR ANCF-HW B st B2nd T Lg st Lg 2nd Lg 3rd Cs, Cs 3, Cs 5, Cs 7, Cs Cs results. The leading term in the transverse dislacement at the ti for a transverse force loading is now correct, because the shear deformations at the nodes are used to calculate the contribution in the stiffness matrix due to shear. The transverse shear correction shows an offset by a factor of 4. For the constant moment loading the solution still suffers form surious transverse shear deformation. The dynamic tests are eigenfreuency analyses on a beam modeled by one element with various boundary conditions. The first case is a fully free beam, which is somewhat idealied but has a number of advantages: There are no kinematic boundary conditions and it demonstrates the ossibility of the element to describe the six rigid-body motions. The nonero dimensionless eigenfreuencies for the various elements are resented in Table 3 together with the analytic solution for an Euler Bernoulli beam. The ero eigenvalues are not reorted since all element tyes describe the rigid-body modes exactly. Owing to the symmetric cross section some freuencies come in airs. The first observation that we make is that the ANCF elements yield 2 further eigenfreuencies, of which 2 are longitudinal modes and 0 are cross-sectional modes with a high freuency. These cross-sectional modes are deicted in Fig. 3, where the asect ratio of the cross section versus the beam length is exaggerated for a better illustration of the modes. The first bending mode in the ANCF fully arametried is too high by a factor / + / 2.6, which is exected because the anticlastic deformation of the cross section cannot be described by the continuum osition field. Note that this result is also resent in the dynamic resonse of the midoint deflection of the endulum beam from Fig. 8 in Ref. 7. The torsional eigenfreuency is also too high, because the factor k x is not included. The ANCF fully arametried element gives a large value for the second bending mode, because this mode is couled to a transverse shearing deformation, a henomenon that is referred to as shear locking. The modified ANCF with the elastic line aroach gives a far more realistic value. This second bending mode for all ANCF elements is illustrated in Fig. 4. The shear locking is evident in the fully arametried element. The elastic line aroach with the Hellinger Reissner method yields a correct eigenfreuency but incorrect rotations of the cross section at the nodes. The Hu Washiu method yields not only a correct eigenfreuency but also correct rotations of the cross section at the nodes. Note that the rotations between the nodes along the elastic line are incorrect. All ANCF elements give identical and good aroximations for the longitudinal eigenfreuencies, because the interolation is cubic instead of linear, as for the classical FEM element. The second boundary condition case is a simly suorted beam. The boundary conditions for the classical FEM beam are straightforward. Both nodes are suorted in the y-lane and in node the horiontal dislacement and the rotation along the central axis are suressed. This leaves six degrees of freedom, namely, u c = y,,u, x, y, 88 where the dislacements of a osition vector r are denoted by u,v,w. For the ANCF element we aly the same boundary conditions, where the rotation along the central axis in node is,2 Ω = ,4 Ω = Ω = Ω = Ω = Ω = Ω = Ω = Fig. 3 The ten cross-sectional eigenmodes together with their dimensionless eigenfreuencies for the free vibration of a comletely free beam modeled by one ANCF fully arametried element. See also Table / Vol. 5, JANUARY 200 Transactions of the ASME Downloaded 2 Jan 200 to Redistribution subject to ASME license or coyright; see htt://

9 f, Ω = f, Ω = HR, Ω = HR, Ω = HW, Ω = HW, Ω = Fig. 4 The second bending eigenmode together with the dimensionless eigenfreuency for the free vibration of a comletely free beam modeled by one element for three ANCF element tyes: fully arametried, elastic line aroach with a Hellinger Reissner method and elastic line aroach with a Hu Washiu method. See also Table 3. now suressed by fixing the y-dislacement of the -sloe vector: v. Dislacements of a sloe vector r x are denoted by u x,v x,w x, etc. The remaining degrees of freedom, 8 in total, are u c = u x,v x,w x,u y,v y,w y,u,w,u,u x,v x,w x,u y,v y,w y,u,v,w 89 The results of this analysis are resented in Table 4. The ANCF fully arametried element shows the same drawbacks as in the revious case. The Hellinger Reissner aroach resolves the shear locking in the second bending mode but still yields incorrect cross-sectional rotations at the nodes, as can be seen in Fig. 5. The torsional and longitudinal eigenfreuencies show similar behavior as in the free-free case. Also the cross-sectional modes are nearly identical in freuency, because the boundary conditions for the simly suorted beam ut almost no restriction on the deformation of the cross section. This is not the case for a cantilevered beam, the third tye of boundary condition considered. Here the same boundary conditions are taken as in the static test, namely, restricting all dislacements of the built-in node and fixing the two sloe vectors r y and r. The remaining degrees of freedom, 5 in total, are u c = u x,v x,w x,u,v,w,u x,v x,w x,u y,v y,w y,u,v,w 90 These results are resented in Table 5 and Figs. 6 and 7. The results for the longitudinal and torsion eigenfreuencies are as in the simly suorted case. For bending, we see the same kind of henomena as before, esecially the shear locking in the second Fig. 5 The second bending eigenmode together with the dimensionless eigenfreuency as in Fig. 4 but now for a simly suorted beam. See also Table 4. bending mode for the ANCF fully arametried element; see Fig. 6. The ANCF element with Hellinger Reissner method still gives rather high values for the two bending eigenfreuencies. This can be attributed to the difficulty in rescribing the condition of ero rotation of the cross section for the clamed end. Because the Hellinger Reissner formulation relaxes the rigidity against shear deformation, shear can still occur at the ends of the beam for asymmetric bending; the shear is only small in an average sense. The Hu Washiu aroach gives more realistic values and modes. The cross-sectional modes are now fewer, seven instead of ten, and their freuencies have shifted somewhat comared with those from the simly suorted case. This is due to the restriction of the deformation of the cross section at the built-in node. This is also the reason for the relatively low eigenfreuency of the first double cross-sectional mode Fig. 7 because the cross-sectional deformation is now couled to the global dislacement. 5 Conclusions In this aer some formulations for a flexible satial beam have been comared. In general, the classical FEM formulation gives good results for the linearied case. Some shortcomings in the satial beam formulation given in Ref. 7 were found, esecially that it yields too large torsional and flexural rigidities and that shear locking effectively suresses the asymmetric bending mode. An ANCF element with an elastic line aroach, along similar lines as in Ref. 8, has been roosed. This formulation yields better torsional and flexural rigidities. The shear locking of the asymmetric bending mode can be avoided by the aid of the Hellinger Reissner variational rincile. The roblem of the Table 4 Dimensionless eigenfreuencies as in Table 3 but now for a simly suorted beam Mode Analytic Classical FEM ANCF-f ANCF-HR ANCF-HW B st B2nd T Lg st Lg 2nd Lg 3rd Cs, Cs 3, Cs Cs Cs Cs Cs Cs Journal of Comutational and Nonlinear Dynamics JANUARY 200, Vol. 5 / Downloaded 2 Jan 200 to Redistribution subject to ASME license or coyright; see htt://

10 Table 5 Dimensionless eigenfreuencies as in Table 3 but now for a cantilevered beam Mode Analytic Classical FEM ANCF-f ANCF-HR ANCF-HW B st B2nd T Lg st Lg 2nd Lg 3rd Cs, Cs 3, Cs 5, Cs roer imosition of the boundary conditions at clamed ends is alleviated by the judicious alication of the Hu Washiu variational rincile. As a direction of future research, it is desirable to develo the ANCF satial beam based on the elastic line concet further. Acknowledgment Thanks to Johannes Gerstmayr, Luis Maueda, and Ahmed Shabana for discussion and technical comments. References f, Ω = HR, Ω = HW, Ω = Fig. 6 The second bending eigenmode together with the dimensionless eigenfreuency as in Fig. 4 but now for a cantilevered beam. See also Table 5. f, Ω = HR, Ω = HW, Ω = Fig. 7 The first double cross-sectional eigenmode together with the dimensionless eigenfreuency as in Fig. 4 but now for a cantilevered beam. See also Table 5. Fraeijs de Veubeke, B., 976, The Dynamics of Flexible Bodies, Int. J. 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Y., 200, Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Theory, ASME J. Mech. Des., 23, Reissner, E., 953, On a Variational Theorem for Finite Elastic Deformations, J. Math. Phys., 32, Washiu, K., 982, Variational Methods in Elasticity and Plasticity, Pergamon, New York. 2 Cower, G. R., 966, The Shear Coefficient in Timoshenko s Beam Theory, ASME J. Al. Mech., 33, Argyris, J. H., 966, Continua and Discontinua, an Aerçu of Recent Develoments of the Matrix Dislacement Methods, Matrix Methods in Structural Mechanics, J. S. Premienniecki, R. M. Bader, W. F. Boich, J. R. Johnson, and W. J. Mykytow, eds., Wright-Patterson Air Force Base, Dayton, OH, Premieniecki, J. S., 968, Theory of Matrix Structural Analysis, McGraw- Hill, New York. 24 Fraeijs de Veubeke, B., 965, Dislacement and Euilibrium Models in the Finite Element Method, Stress Analysis, Recent Develoments in Numerical and Exerimental Methods, O. C. Zienkiewic and G. S. Holister, eds., Wiley, New York, Cha. 9, rerinted in International Journal for Numerical Methods in Engineering, 52, 200, Timoshenko, S. P., and Goodier, J. N., 987, Theory of Elasticity, McGraw- Hill, New York / Vol. 5, JANUARY 200 Transactions of the ASME Downloaded 2 Jan 200 to Redistribution subject to ASME license or coyright; see htt://