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Keywords:

  • virtual work;
  • structural mechanics;
  • constrained theories;
  • geometrically exact beam;
  • nonlinear beam elements

SUMMARY

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. FUNDAMENTAL PRINCIPLES OF A CONTINUOUS BODY
  5. GEOMETRICALLY EXACT BEAM
  6. FINITE ELEMENT FORMULATION
  7. NUMERICAL INVESTIGATIONS
  8. CONCLUSIONS
  9. APPENDIX
  10. REFERENCES

In the present work, a new director-based finite element formulation for geometrically exact beams is proposed. The new beam finite element exhibits drastically improved numerical performance when compared with the previously developed director-based formulations. This improvement is accomplished by adjusting the underlying variational beam formulation to the specific features of the director interpolation. In particular, the present approach does not rely on the assumption of an orthonormal director frame. The excellent performance of the new approach is illustrated with representative numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.

INTRODUCTION

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. FUNDAMENTAL PRINCIPLES OF A CONTINUOUS BODY
  5. GEOMETRICALLY EXACT BEAM
  6. FINITE ELEMENT FORMULATION
  7. NUMERICAL INVESTIGATIONS
  8. CONCLUSIONS
  9. APPENDIX
  10. REFERENCES

This work deals with nonlinear beam finite elements emanating from the geometrically exact Simo-Reissner beam model (Simo [1], Reissner [2]). The finite element discretization applied in the present work relies on the interpolation of the director field, in contrast to the use of rotational degrees of freedom or quaternions, see, for example, McRobie & Lasenby [3]. This type of finite element interpolation can be considered as a characteristic feature of continuum-based beam elements and has been applied, among others, in Bathe & Bolourchi [4] and Gruttmann et al. [5], see also Chapter 9 in Belytschko et al. [6]. The director interpolation has originally been employed in the finite element discretization of the Simo-Reissner beam theory in Romero & Armero [7] and Betsch & Steinmann [8, 9]. Because the director interpolation does not rely on rotational degrees of freedom, it retains the fundamental properties of frame indifference (or objectivity) and conservation of angular momentum in the finite element framework, as shown in [8, 10] . Moreover, it facilitates the straightforward design of structure-preserving time integrators as has been shown in Betsch & Steinmann [11, 9], Armero & Romero [12], and Leyendecker et al. [13].

In the director interpolation, the orthonormality of the director frame is typically relaxed to the nodal points of the finite element formulation. That is, the orthonormality of the directors is typically enforced in the nodal points, either by using three rotation parameters or by applying the method of Lagrange multipliers. We refer to [8] where the connection between the two alternative approaches is highlighted.

The lack of orthonormality of the director frame on element level can be viewed as discretization error that diminishes when the number of elements is increased. Consequently, the corresponding finite element formulation is still consistent with the underlying beam theory. However, as has been observed in [8], to achieve a certain level of accuracy, beam finite elements relying on the director interpolation typically require a larger number of elements when compared with the finite element formulations based on the interpolation of three rotation parameters. Further investigations in this direction can be found in Romero [10] and Bauchau & Han [14].

In the present work, we propose a new finite element implementation of the geometrically exact beam theory. The newly developed formulation relies on the director interpolation and accounts for the lack of orthonormality of the discrete director frame in a natural way. This is achieved by the formulation of the underlying variational equations using convected coordinates. Such a formulation, where covariant and contravariant directors are distinguished, inherently allows for skew coordinate systems. In case the directors are mutually orthonormal, the new formulation boils down to the original version of the geometrically exact beam theory. It turns out that the new approach yields a dramatically improved numerical performance when compared with the original director-based implementation developed in [8].

The quality of the new finite element formulation lies also in the notion that the beam is considered as a constrained three-dimensional continuum. Using the principle of virtual work and the symmetry condition of the stress vector, the variational beam equations are developed merely by kinematical assumptions. This approach guarantees complete consistency between all introduced kinematic and kinetic quantities. Additionally, with a variational formulation of the symmetry condition, the authors never leave the route of a variational formulation of the mechanical problem.

An outline of the rest of the paper is as follows. In Section 2, the fundamental principles of the underlying continuum theory are outlined in a concise manner. The geometrically exact beam theory in skew coordinates is derived in Section 3. In Section 4, we apply a spatial discretization based on finite elements. Representative numerical examples are given in Section 5. Eventually, conclusions are drawn in Section 6.

FUNDAMENTAL PRINCIPLES OF A CONTINUOUS BODY

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. FUNDAMENTAL PRINCIPLES OF A CONTINUOUS BODY
  5. GEOMETRICALLY EXACT BEAM
  6. FINITE ELEMENT FORMULATION
  7. NUMERICAL INVESTIGATIONS
  8. CONCLUSIONS
  9. APPENDIX
  10. REFERENCES

In this section, we summarize the variational form of the continuum mechanical theory that serves as a starting point for the derivation of the beam formulation in the subsequent section. In addition to that we present a derivation of the symmetry condition of the stress vector from the variational form of the law of interaction which is in this form completely new as far as it is known to the authors.

image

Figure 1. Schematic overview of the kinematics of the body manifold inline image.

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We consider a three-dimensional body inline image as a three-dimensional smooth manifold with boundary that can be covered by a single chart φ, see Figure 1. Hence, every material point inline image can be described by three coordinates inline image where inline image. A configuration inline image is an embedding of the body manifold into the Euclidean three-space inline image. Because the configuration maps the material points p to the Euclidean three-space, which is a vector space, the placement of a material point κ(p) can be represented by the position vector inline image. A motion inline image of the body is a differentiable parametrization of configurations with respect to time inline image. Thus, at a given instant of time t, the subset inline image is covered by the body manifold. Using the chart φ, the coordinate representation of the motion is the vector valued function

  • display math(1)

also denoted as the position field. Note that we are using the same symbol for the variables x as for the functions whose results they are. In the following sections, we will work mainly with the coordinate representation of the motion x(θk,t) and will treat motion and coordinate representation of the motion synonymously. A variational family of the motion of the body is a differentiable parametrization of motions inline image with respect to a parameter inline image. The actual time behavior is obtained for ϵ = ϵ0, that is, inline image. Insofar the variation of the position field x is given as

  • display math(2)

Furthermore, we introduce the covariant base vectors gi, its variation δgi, and its associated contravariant base vectors gi for i ∈ {1,2,3} according to

  • display math(3)

where partial derivatives (●) ∕ ∂θk are abbreviated by (●),k. The construction of the contravariant basis is valid for even permutations of {i,j,k} = {1,2,3}. The contravariant base vectors fulfill the reciprocity condition inline image. In the following, all vector triads will tacitly be equipped with a contravariant triad as defined in (3). In the remainder of this article, we call quantities with lower index (●)i covariant and quantities with upper index (●)i contravariant.

The virtual work principle states that the body inline image is in the dynamical equilibrium if and only if the virtual work δW vanishes for all virtual displacements δx at any instant of time t, that is,

  • display math(4)

Here, δWint represents the contribution of the internal virtual work, which is formulated in the body chart φ as

  • display math(5)

where d3θ = dθ1dθ2dθ3 and summation convention for repeated indices is applied. The stress vector ti(θk,t) can be recognized in the Cauchy stress tensor σ(x(θk),t) = g − 1 ∕ 2ti ⊗ gi. Hence, the stress vector ti is the traction in the current configuration that acts at the surface element gj × gkdθjdθk = gig1 ∕ 2dθjdθk for even permutations of the indices {i,j,k} = {1,2,3}. An analogous formulation of the internal virtual work (5) can be found, for example, in Chapter 16 of Antman [15].

The contributions of the inertia terms, the external forces df, and the perfect bilateral constraint forces dz are

  • display math(6)

where inline image denotes the total time derivative. We consider the mass distribution dm and the force distributions df and dz as measures allowing for Dirac-type contributions as well. The perfect bilateral constraint forces dz are defined by the principle of d'Alembert-Lagrange, stating that the virtual work of the perfect bilateral constraint forces integrated over the total mechanical system, that is the body inline image, vanishes for all admissible virtual displacements δxadm. Virtual displacements are called admissible if they do not violate the perfect bilateral constraints. Hence, the principle of virtual work can be expressed in the body chart as follows.

Theorem 1. (Principle of Virtual Work) A body inline image is in the dynamical equilibrium if and only if the virtual work expression vanishes for all admissible virtual displacements δxadm at any instant of time t, that is,

  • display math(7)

Besides the virtual work principle, the law of interaction for internal forces has to be respected (cf. Section 2 of Glocker [16]), which coincides in its variational form with the ‘Axiom of Power of Internal Forces’ formulated by Germain [17]. The variational form of the law of interaction states that the internal virtual work (5) of any subbody inline image is unaffected and therefore invariant with respect to all Euclidean transformations. The transformed position field takes the form x +  = Qx + c where Q ∈ SO(3) is a special orthogonal tensor satisfying inline image, where I is the second order identity tensor, and c is a vector of the Euclidean vector space inline image. The rotation Q and the translation c are homogenous for the whole subbody inline image. Hence, the virtual displacement of the transformed position field is δx +  = δQx + Qδx + δc. The transformed stress vector (ti) +  and the partial derivative of the virtual displacement are given by

  • display math(8)

Hence, we write the law of interaction for an arbitrary subbody inline image in the body chart φ as

  • display math(9)

where inline image. Because inline image is a skew-symmetric tensor, we introduce in the last line of (9) the associated axial vector inline image, defined by the cross product as inline image for all inline image. The law of interaction requires now the first term on the last line of (9) to vanish for all δQ and consequently for all δw. Because the considered mechanical system is a continuous body and the law of interaction has to be fulfilled for any subsystem inline image, the law of interaction in the body chart φ coincides with the symmetry condition of the Cauchy stress tensor.

Theorem 2. (Law of Interaction) As an internal force the stress vector ti has to fulfill the symmetry condition in every material point θk, that is,

  • display math(10)

GEOMETRICALLY EXACT BEAM

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. FUNDAMENTAL PRINCIPLES OF A CONTINUOUS BODY
  5. GEOMETRICALLY EXACT BEAM
  6. FINITE ELEMENT FORMULATION
  7. NUMERICAL INVESTIGATIONS
  8. CONCLUSIONS
  9. APPENDIX
  10. REFERENCES

In this section, we treat the theory of the geometrically exact beam, also known as the special Cosserat beam, introduced by [2] and [1]. The theory is developed by restricting the kinematics of the three-dimensional continuum to a beam-like kinematic. By inserting the reduced kinematics into the virtual work of the continuum, we deduce the virtual work of the beam. From the virtual work of the beam, we then obtain directly the equations of motion of the beam and the corresponding boundary conditions. Throughout our developments, we pay attention to distinguish properly between covariant and contravariant base vectors. As will be shown through the numerical examples presented in Section 5, the newly proposed approach yields a significant improvement over previously developed finite element formulations that do not rely on the distinction between covariant and contravariant base vectors.

Kinematical assumptions

We define the geometrically exact beam inline image as a structural element that has a special configuration, that is a reference configuration, where the beam is a three-dimensional body with the position field

  • display math(11)

In this configuration, as depicted in Figure 2, an appropriate chart φ gives us the body coordinates inline image, which are referred to as convected coordinates. The space curve ϕ0(s) = X(0,0,s) is the reference curve of the beam and is bounded by its ends s = s1 and s = s2 for s2 > s1. At every material point s of the reference curve ϕ0, we have attached a positively oriented orthonormal director triad {dk}. The two directors dα span the plane cross section of the beam. The area of the cross section is parametrized by the coordinates inline image. The director triad {dk} can be related to an orthonormal basis {ek} fixed in space by introducing the rotation tensor R0(s) ∈ SO(3) such that

  • display math(12)

According to Section 2, the beam is in its reference configuration, a three-dimensional body. This implies bijectivity of the position field X(θα,s). Because of the kinematical constraints (11), it can happen that interpenetration occurs when the beam is curved too much or the cross sections are too large, compare with Chapter 5 of Rubin [18]. As in the three-dimensional theory, we exclude for the reference configuration interpenetration that is expressed mathematically as X,1 ⋅ (X,2 × X,3) > 0 for all inline image. That is the reason why beam theories are generally related to slender bodies.

image

Figure 2. Reference and current configuration of the beam.

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The motion of the geometrically exact beam is the restricted position field

  • display math(13)

We call the space curve ϕ(s,t) = x(0,0,s,t), which is a bijective function of s and time t, the center line of the beam. At each material point of the center line ϕ, we have attached a positively oriented orthonormal director triad {dk}, which is related to the basis {ek} by introducing the rotation tensor R(s,t) ∈ SO(3) such that

  • display math(14)

The directors dα describe the current state of the cross section inline image. In the current configuration, we merely ask the center line ϕ to be a bijective function. That means, interpenetration of the cross sections may occur. Insofar, the beam is strictly speaking not a three-dimensional body anymore but a one-dimensional body. Because every material point can be addressed by the convected coordinates θk and the variations δx have to comply with the restricted kinematics (13), the dynamics of the constrained system can be derived by the principle of virtual work (7).

In the following, we introduce the effective curvature, the angular velocity, and the virtual rotation. All three objects describe the change of the directors when changing a single parameter as, for example, the parameter s. Using (14), the effective curvature inline image is obtained by

  • display math(15)

Because the present finite element formulation (see Section 4) relies on the nodal interpolation of the director frame, the directors generally do not span an orthornormal basis anymore. That is, on element level the metric coefficients dij = di ⋅ dj and dij = di ⋅ dj in general do not coincide with the Kronecker delta δij. In other words, on element level, the tensor R introduced in (14) generally does not belong to the special orthogonal group. Inspired by this observation, we relax the orthonormality condition on the director frame and consider metric coefficients that are merely assumed to be constant. Under this condition, the tensor inline image still remains skew-symmetric. This is shown in the Appendix. Hence, the skew-symmetric effective curvature inline image has an associated axial vector inline image so that

  • display math(16)

Later in this paper, we are interested in the contravariant components of the effective curvature, which can be written as (cf. Schade & Neemann [19])

  • display math(17)

where ϵijk are the alternating symbols and

  • display math(18)

Analogously to (16), we introduce the angular velocity inline image and its associated axial vector ω as

  • display math(19)

In the same manner, as for the effective curvature and the angular velocity, we obtain the virtual rotation inline image when changing the variational parameter ϵ.

  • display math(20)

The last expression of the virtual rotation vector is motivated by the following identity and is used in the sequel for the director formulation.

  • display math(21)

The velocity and acceleration fields are introduced by taking the total time derivative of the position field (13) and the kinematical relation introduced in (19).

  • display math(22)
  • display math(23)

With regard to (13) and (16), the partial derivatives of the position field assume the form

  • display math(24)

Using (13), (20) and (24), the variation of the position field and its partial derivatives are given by

  • display math(25)

In the Appendix, we verify the important identity

  • display math(26)

which is obtained by the fact that the derivative with respect to s and the variation commute, that is, (δdk),s = δ(dk),s.

Virtual work and equations of motion

With the kinematical assumption (13), we impose in fact infinitely many perfect bilateral constraints Φ(θα,s,t) = x − ϕ − θαdα = 0 on the system. The admissible virtual displacements δxadm are all variations of the reduced kinematics x, that is, δxadm = δx. Because the variations of x are admissible with respect to the constraints Φ = 0, all corresponding constraint forces are projected out and will not appear in the equations of motion. In the following, we reformulate the virtual work expression (7) by taking into account the reduced kinematics of the beam. In this way, we identify the kinematic quantities of the beam and the corresponding kinetic quantities, that is, the resultant contact force n and the resultant contact torque m. Furthermore, by assuming a hyperelastic material behavior, appropriate strain measures are identified.

Using (5), (25), and the property of the cross product of (A.2), the internal virtual work density can be written as

  • display math(27)

Employing the symmetry condition (10), we can rewrite the first term in (27) as follows.

  • display math(28)

Using the relation (28) and the Jacobi identity (A.1), we can manipulate (27) further to obtain

  • display math(29)

Because the kinematical quantities in the squared brackets depend merely on s, we split the integration over inline image in an integration over the cross section in the body chart inline image and an integration along s ∈ [s1,s2].

  • display math(30)

The integrated kinetic quantities n and m are the resultant contact force and the resultant contact torque of the current configuration defined by

  • display math(31)

with the surface element d2θ = dθ1dθ2. For the sake of clarity, the contributions due to external forces and inertia are developed in compact form in the Appendix.

Taking all the modified contributions of the virtual work (30), (A.13), and (A.18), the principle of virtual work (7) leads to

  • display math(32)

Using the identity (26) and integration by parts, the virtual work can be expressed as

  • display math(33)

This is the variational expression of the BVP in the current configuration of the geometrically exact beam as derived from the balance laws in Chapter 8 of Antman [15], that is,

  • display math(34)

with the boundary conditions inline image and inline image for s = {s1,s2}.

Constitutive law

We assume the constitutive law to be hyperelastic. Thus, there exists an elastic potential W(s) such that

  • display math(35)

We formulate the elastic potential as an additive split of two potentials

  • display math(36)

each of which depends on the strain measures γi and ki, respectively. The effective reference curvature is defined as inline image. The covariant strain

  • display math(37)

measures the difference between the deformation of the center line in the direction di and the deformation of the reference curve in direction di. When measuring the difference between the effective curvature and the effective reference curvature in the direction Dk,Dj and Dk,Dj, respectively, we obtain the covariant components inline image. Because these components are skew-symmetric, there is an associated axial vector with contravariant components

  • display math(38)

which is the second strain measure. In the following, we prove that we obtain the internal virtual work expression (30) when varying the elastic potential (36). Using (20) and (A.2), the variation of the first potential takes the form

  • display math(39)

where we recognize the resultant contact force inline image. By expansion with the reciprocity condition inline image and using (20), the variation of the second potential yields

  • display math(40)

Here, we identify the contact torque as inline image. Comparing (39) and (40) with (30) proves the choice of the strain measures and their corresponding elastic potentials.

Remark 3. The contact force n and the contact torque m are naturally represented by covariant directors di and by contravariant directors di, respectively. According to the property of the cross product, (3), and the definition of the generalized contact forces (31), these representations are completely reasonable.

We assume the following quadratic form as the elastic potential

  • display math(41)

with

  • display math(42)

where inline image and inline image contain the collection of the stiffness components inline image and inline image, respectively. Consequently, the contact force and the contact torque are

  • display math(43)

The elastic potential (41) differs from the elastic potential mentioned in Simo & Vu-Quoc [20] by the factor d1 ∕ 2, which is 1, in the case of an orthonormal frame {di}. As it is shown in Auricchio et al. [21], the small-strain constitutive law (43) is motivated by inserting a three-dimensional linear constitutive law into (31), omitting the quadratic terms of the strain, and integrating over the cross section inline image. Because d1 ∕ 2 is constant over the cross section inline image, the following law for the contact force is obtained

  • display math(44)

The derivation of the contact torque law works analogously.

Director description

In the subsequent subsection, we describe the rotational degrees of freedom using constraint directors. This formulation leads to a reparametrization of (32). Hence, we reformulate all virtual work contributions that include rotational degrees of freedom such as virtual rotations or curvatures. The internal virtual work due to the contact force is

  • display math(45)

With regard to the variation of (17), the internal virtual work of the contact torque takes the form

  • display math(46)

The virtual work contribution of the external torques is

  • display math(47)

For the reparametrization of the virtual work contribution of the dynamical forces, it is easier to start directly from the contribution in (6) than from (A.18). Hence,

  • display math(48)

with the abbreviation of the time constant inertia coefficients

  • display math(49)

As mentioned before, the director description coincides with the geometrically exact beam theory if the director frame remains an orthonormal frame, that is, the following perfect bilateral constraints have to be satisfied at any instant of time

  • display math(50)

By the principle of d'Alembert-Lagrange, this will lead to an additional Lagrange multiplier λij(s,t) in the virtual work expression. The virtual work principle of the geometrically exact beam reparametrized for the variation of directors takes the following form.

  • display math(51)

When we do not distinguish anymore between covariant and contravariant directors, we arrive at the virtual work expression as proposed in [8].

FINITE ELEMENT FORMULATION

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. FUNDAMENTAL PRINCIPLES OF A CONTINUOUS BODY
  5. GEOMETRICALLY EXACT BEAM
  6. FINITE ELEMENT FORMULATION
  7. NUMERICAL INVESTIGATIONS
  8. CONCLUSIONS
  9. APPENDIX
  10. REFERENCES

To achieve a numerical solution for the constrained problem under consideration, we subdivide the center line inline image of the beam into a set of finite elements inline image via

  • display math(52)

characterized by associated nodal points inline image, where the position vector of node inline image is denoted by inline image. Then we introduce a polynomial, finite dimensional approximation of the solution space, that is, of the configuration as well as of the director field, as follows

  • display math(53)

Using a standard Galerkin type approach, the space of admissible test functions is approximated analogously

  • display math(54)

where inline image are global, Lagrange-type shape functions.

Taking the definition of the strain measures (37) and (38) into account, we can write

  • display math(55)

for their discrete counterparts. Here, the discrete rotation matrix is given by inline image. Note that the matrix inline image in general is not a proper rotation matrix because of the lack of orthonormality of the discrete director frame caused by interpolation (53). Using the reciprocity condition, the values of the contravariant directors inline image can be extracted from the inverse discrete rotation matrix inline image. The internal virtual work of the contact force (45) reads

  • display math(56)

Because d − 1 ∕ 2,h = det (R − 1,h) and inline image, the internal virtual work of the contact torques (46) follows as

  • display math(57)

For the numerical implementation, it is convenient to collect the components inline image in the matrices [γh] and [kh], respectively. Using the constitutive laws of (43), the local, discrete contact force and torque are given by

  • display math(58)

The external as well as the dynamical contributions can be formulated analogously. Note that we apply a reduced integration, that is, a one-point Gauss quadrature for the two-node element to avoid locking effects. At last, we consider the approximation of the Lagrange multipliers associated to the constraints of orthonormality (50). In particular, these constraints are enforced at the nodal points inline image leading to the algebraic constraint equations ΦA = 0. The constraints (50) entail six independent constraint functions per node, which are collected in the vector

  • display math(59)

Correspondingly, the Lagrange multipliers are approximated by using Dirac deltas as basis functions inline image. Accordingly,

  • display math(60)

This procedure is in accordance with the developments in [8].

Remark 4. Instead of using the method of Lagrange multipliers for the enforcement of the nodal constraints of orthonormality, three nodal rotation parameters could be introduced. This procedure yields a significant size-reduction of the algebraic system of nonlinear equations to be solved. We refer to [8], Section 3.2, for further details. Note, however, that this approach merely reduces the number of unknowns and does not alter the numerical approximation properties of the finite element formulation at hand.

Remark 5. The discrete strain measures emanating from the present approach are frame-indifferent (or objective). This can be verified in a straightforward manner in complete analogy to Section 3 in [8].

NUMERICAL INVESTIGATIONS

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. FUNDAMENTAL PRINCIPLES OF A CONTINUOUS BODY
  5. GEOMETRICALLY EXACT BEAM
  6. FINITE ELEMENT FORMULATION
  7. NUMERICAL INVESTIGATIONS
  8. CONCLUSIONS
  9. APPENDIX
  10. REFERENCES

In this section, we evaluate the accuracy and performance of the newly proposed method. All examples are carried out in a three-dimensional setting, although some of them remain planar. Because the new contribution of the proposed formulation relies on the formulation of the internal virtual work, we demonstrate the performance using static benchmark tests and compare the results with the original director-based formulation in [8]. Throughout all the examples, two-node elements are used.

Planar equilibrium problem

image

Figure 3. Buckling test, the results of the new approach are marked with o, the results of the original approach with x.

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In this example, we consider the buckling of a hinged right angle frame. The geometry of the reference configuration is given in Figure 3, the material data for the used model are given as follows: inline image, inline image, inline image, and inline image, see also [20]. Dirichlet boundary conditions are applied to the translational degrees of freedom at both ends, such that rotation is possible. Four different loads inline image are applied to the frame as dead loads, using inline image, inline image, inline image, and inline image. The results of the original director formulation are added for comparison for inline image and inline image. In both loading situations, the new formulation reacts stiffer as the original one; this behavior is closer to the solution as will be shown in a subsequent convergence test, see Section 5.3. Using a standard Newton-Raphson iteration scheme, the snap-through of the buckling problem for the loading inline image and inline image could only be achieved for the new approach, since the original formulation diverges. An arc-length method would be necessary for the calculation of the results of the original approach. Thus, the new approach is more robust than the original director formulation.

Spatial cantilever problem

image

Figure 4. Cantilever problem, reference configuration of the center line and the director triads.

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The next example consists of a cantilever bending test, see Figure 4. The cantilever is curved in its stress-free reference configuration. In particular, inline image of a circle with radius inline image in z-direction is used as center line. Different constant loads are applied to the tip at the end of the beam, whereas the other end of the beam at inline image is completely clamped in all directions. The material data are as follows: inline image, inline image and inline image. Table 1 shows the corresponding results for different numbers of elements. In addition to that, the numerical results corroborate the frame-indifference of the present method (cf. [8]).

Table 1. Tip displacement in inline image direction.
Load levelOriginal approachNew approach
inline image8 El.16 El.32 El.8 El.16 El.32 El.
inline image000000
inline image61.330260.117759.902260.124659.903359.8510
inline image40.032338.934238.753938.796638.719538.7027
inline image38.376937.726437.582937.522837.531637.5351
inline image000000

As can be seen, the new formulation outperforms the original one for coarse meshes. Furthermore, both approaches converge to the same results.

Bending test

This example consists of a straight beam, clamped at one end, and a moment applied to it at the other end. The length of the beam is inline image and the material data are as follows: inline image, inline image, inline image and inline image. An analytical solution for a closed curve exists with inline image, see the snapshots in Figure 5.

image

Figure 5. Bending problem, configuration for inline image, inline image, inline image, inline image, and inline image.

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A convergence plot is shown in Figure 6, where the size of the elements is plotted versus the norm of the distance inline image between the numerical and the analytical solution of the tip displacement of the beam.

image

Figure 6. Convergence results for the bending problem.

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This convergence test shows once again that the new approach clearly outperforms the original formulation. We were able to run this example with a minimum number of two elements, whereas the iterative solution procedure (i.e. Newton's method) does not converge for the original formulation if less than 15 elements are employed.

Beam with slope discontinuity

This last example demonstrates the capabilities of the proposed approach for non-smooth, three-dimensional geometries, compare with Romero [22]. The structure consists of three, in their reference configuration straight beams of unit length, connected at right angels, see Figure 7. To model the connection of two beams at right angles, we apply six algebraic constraints in complete analogy to the description of shell intersections in [23], Section 6.1. The material data are as follows: inline image and inline image. Moreover, the beam has a quadratic cross section area of inline image.

image

Figure 7. Geometry of the structure and tip displacement versus applied load inline image.

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image

Figure 8. Errors in tip displacements versus h-refinement.

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The structure is fully clamped at one end, and two forces with magnitude inline image are applied in negative ex- and ez-direction to the other end. In Figure 7, the displacement of the tip is plotted versus the load inline image using a total of 12 elements of uniform length and distribution.

Analogous to the bending test, a convergence plot is given in Figure 8 to compare the results of the new approach with the original formulation. We established an error indicator using the distance inline image between the numerical solution of the tip displacement for different element numbers and a reference solution with 1920 elements. Note that the norm of the difference in the tip displacement between both approaches for the reference solution is below inline image.

Again, the new approach clearly outperforms the original formulation even for the complex three-dimensional problem at hand. The original approach diverges using a total of six elements for the chosen load increment size of inline image, whereas the new approach converges without problems.

CONCLUSIONS

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. FUNDAMENTAL PRINCIPLES OF A CONTINUOUS BODY
  5. GEOMETRICALLY EXACT BEAM
  6. FINITE ELEMENT FORMULATION
  7. NUMERICAL INVESTIGATIONS
  8. CONCLUSIONS
  9. APPENDIX
  10. REFERENCES

The present approach can be viewed as generalization of the method proposed in [24]. In this work, the director-based formulation of rigid body dynamics is treated. There, the discretization in time generally destroys the orthonormality of the director frame, which is inherently connected to the kinematic assumption of rigidity. It is shown in [24] that the lack of orthonormality of the director frame distorts the application of external torques and thus destroys the balance of angular momentum. This deficiency of the discrete rigid body formulation can be resolved by properly distinguishing between covariant and contravariant directors in much the same way as in the present work.

In particular, the newly proposed formulation of the beam curvature accounts in a natural way for the lack of orthonormality of the director frame caused by the nodal interpolation of the directors. The lack of orthonormality of the director frame can be regarded as a discretization error, which diminishes if the number of elements is increased.

Consequently, the present approach has an especially pronounced effect when course discretizations and low-order finite elements are used. The numerical results presented in Section 5 show indeed a significant improvement over the original implementation of the director-based beam finite element formulation. It has further been verified, both theoretically and numerically, that the present approach yields a consistent discretization of the geometrically exact (Simo-Reissner) beam model. It is also worth noting that in analogy to the original director-based implementation, the present method retains the property of frame-indifference (or objectivity) in the discrete setting.

APPENDIX

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. FUNDAMENTAL PRINCIPLES OF A CONTINUOUS BODY
  5. GEOMETRICALLY EXACT BEAM
  6. FINITE ELEMENT FORMULATION
  7. NUMERICAL INVESTIGATIONS
  8. CONCLUSIONS
  9. APPENDIX
  10. REFERENCES
Properties of the cross product

The cross product × as a skew-symmetric operator on inline image has some useful identities that are used frequently in this paper. In inline image, the cross product fulfills the Jacobi identity

  • display math(A.1)

The triple product is invariant under even permutation, that is,

  • display math(A.2)

The vector triple product fulfills Grassmann's identity

  • display math(A.3)

The quadruple product

  • display math(A.4)

and another useful identity, where we use the tilde to denote the skew-symmetric tensor to the associated axial vector, is

  • display math(A.5)
Skew-symmetry of inline image

For dij = di ⋅ dj = constant, we can show that inline image in the basis of di ⊗ dj is skew-symmetric. Because RR − 1 = I, it follows directly that

  • display math(A.6)

By applying di ⋅ (●)dj to (A.6) and the fact that the symmetric metric dij = constant, this leads to

  • display math(A.7)

Using (A.7),

  • display math(A.8)

which shows the skew-symmetry of inline image in the basis of di ⊗ dj. For the basis di ⊗ dj, the proof works analogously.

Proof of identity (26)

Variation and derivative with respect to s commute (δdk),s = δ(dk,s). By (16) and (20), this can be written as

  • display math(A.9)

Applying the product rule and using again (16) and (20) yields

  • display math(A.10)

By subtracting the left-hand side from the right-hand side, using the skew-symmetric property of the cross product and the Jacobi identity (A.1), we obtain

  • display math(A.11)

Because the right-hand side of (A.11) has to vanish for all directors inline image, we obtain the important identity

  • display math(A.12)
Virtual work contributions of external forces

Because the measure df allows for Dirac-type contributions as well, boundary terms do not vanish. Insofar,

  • display math(A.13)

where the generalized external forces inline image and inline image are the integrated quantities

  • display math(A.14)

We want to mention that contact forces on the lateral surfaces contribute on the boundary of inline image.

Virtual work contributions of inertia terms

For manipulating the inertia terms, it is convenient to introduce some abbreviations of integral expressions. With ϕc, in the following, the line of centroids is meant.

  • display math(A.15)

Using the fact that

  • display math(A.16)

and

  • display math(A.17)

we modify the virtual work expression as follows. The tilde denotes the skew-symmetric tensor to the associated axial vector.

  • display math(A.18)

REFERENCES

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. FUNDAMENTAL PRINCIPLES OF A CONTINUOUS BODY
  5. GEOMETRICALLY EXACT BEAM
  6. FINITE ELEMENT FORMULATION
  7. NUMERICAL INVESTIGATIONS
  8. CONCLUSIONS
  9. APPENDIX
  10. REFERENCES
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