In this paper, we develop two alternative formulations for the rotational constraint between the tangents to connected beams with large deformations in 3-D space. Such a formulation is useful for modeling bonded/welded connections between beams. The first formulation is derived by consistently linearizing the variation of the strain energy and by assuming linear shape functions for the beam elements. This formulation can be used with both the Lagrange multiplier and the penalty stiffness method. The second non-consistent formulation assumes that the contact normal is independent of the nodal displacements at each iteration, and is updated consistently between iterations. In other words, we ignore the contribution due to the change of the contact normal in the linearization of the contact gap function. This assumption yields simpler equations and requires no specific assumption regarding the shape functions for the underlying beam elements. However, it is limited to the penalty method. We demonstrate the performance of the presented formulations in solving problems using implicit time integration. We also present a case showing the implications of ignoring this rotational constraint in modeling a network of beams.

The beam-to-beam contact formulation was originally discussed
by

In all the above mentioned papers, a pointwise contact between the fibers was
assumed without taking into account the moment transfer at the point of
contact. Although the effect of a rotational contact might be negligible in
some cases, it is an important factor for a number of applications, e.g. for
modeling welded connections between beams or for considering joints in a
fiber network

The mentioned beam-to-beam contact formulations are generally search for a
contact point based on the shortest distance between elements. In case of
parallel or almost parallel beams, these methods will not work, as a unique
contact point cannot be found. To handle this problem, instead of
point-to-point methods, point-to-line and line-to-line algorithms are used.
For example Durville suggested defining an intermediate geometry between the
contacting beams and using a master/slave type of procedure between each beam
and this intermediate geometry

Although the term contact usually refers to an inequality constraint, when inseparable/adhesive connections between bodies are considered, an equality constraint has to be satisfied. This case is especially applicable when studying beams connected in complex assemblies or for modeling multiple fiber-to-fiber contacts in simulating wires or fiber-based composite materials.

In this paper, we present the formulation for a constraint between the tangents to the beams at their point of connection. This constraint keeps the angle between the tangents constant, or in other words, avoids relative rotation of the tangents. We use the term rotational constraint in the rest of this article to refer to this equality constraint. This rotational constraint formulation can easily be expanded into a rotational contact formulation by considering a criterion which can activate or deactivate the contributions from this constraint during the system evolution.

Consequently, we first present a weak formulation for both the Lagrange multiplier and the penalty stiffness method. Lagrange multiplier and penalty stiffness are two standard methods, successfully used with FEM for contact analyses. Next, we suggest two alternative formulations for the rotational constraint between beams. In the first method, based on the assumption of linear shape functions for the beams, a rotational gap is defined, and using the variational methods and by linearizing the nonlinear equations, the consistent tangent stiffness matrix is derived for both the Lagrange multiplier and the penalty stiffness methods. In the second method a non-consistent approach is used. In this method the contact normal is updated consistently between successive iterations, however the contributions from the change of the contact normal are ignored when linearizing the gap function. In other words, we assume that the contact normal is independent of the deformation at each iteration. We present this simpler formulation with the penalty stiffness method. We show the efficiency of both formulations by solving several examples and demonstrate the effect of including the rotational constraint in the simulation of a network of randomly oriented, interconnected fibers.

The weak formulation of the rotational constraint is presented in this section and will be used in the derivation of the tangent stiffness matrices, using both presented formulations, in the following section.

The total potential energy,

Definition of parameters in

In penalty stiffness method, it is assumed that a high-stiffness spring is
connected between the contacting points. In this case, the elastic energy can
be written as:

We want to solve the Eq. (

Note that while with the first (consistent approach) formulation presented in
Sect. (

We can define the rotational gap function,

The contact between the beam elements is pointwise.

No pair of beam elements can be in contact at more than one point. (If two beam elements have more that one contact point, we need to divide at least one of these elements into two or more elements, to ensure no more than one single contact point between each pair of elements. Note that it is possible for an element to have several contact points with different elements.)

In this approach, in addition to the assumptions mentioned above, we assume
that the beam elements have linear shape functions, both for their geometry
and displacements (this is the same assumption as in

Using the results obtained in Eqs. (

To derive the tangent stiffness matrix of the rotational constraint using a
penalty stiffness method, we substitute Eqs. (

Projection of the change in the rotational degree of freedom of the
beams,

In this approach, in contrast to the consistent formulation presented above,
we do not assume a particular form of the shape functions for the beam
elements. Instead we assume that the contact normal and contact location are
constant (independent of the deformations) in a single iteration. However,
the contact normal and location can (and most often do) vary from one
iteration to the next. For this reason we consistently update them between
iterations. Using the assumption of independence of contact normal and
location from the deformations (in a single iteration), the contribution from
the change of contact normal to the linearization of the contact gap function
vanishes. This simplifies the formulation significantly. Also, we will use a
penalty stiffness method to enforce the rotational constraint. Using these
assumptions we can write the incremental change of the rotational gap,

In this section, we verify the correctness and benchmark performance of the suggested methods by solving different examples. In the first example, we consider a case where the points on each beam, corresponding to the contact point, do not tend to move in the normal and tangential directions with respect to each other and only the rotational constraint acts. We will compare the results with an ideal case in which a common node exists instead of the contact point. The goal of this example is to evaluate the correctness and performance of the suggested formulations. In the second example, we consider a case where all the normal, tangential, and rotational constraints are active. Here we can compare the performance of the two suggested formulations, as well as, evaluate the effect of including the rotational constraint. In the third example, a small structure consisting of intertwined beams is considered. The main focus of this example is to study how inclusion of the rotational constraint and addition of extra beams to the structure affect the stiffness of the whole structure. In the last example, we analyze a random network of interconnected fibers with and without rotational constraint. The selection of this fourth example is motivated by the application to the study of the strength and stiffness of 3-D fiber networks, where the fibers may have multiple contact points along their length and the corresponding simulations will benefit from the presented formulation.

In the first, second, and third examples, we consider beams with a circular
cross-section with a radius of 0.25, a length of 10 (and 5 in the third
example), and an elasticity modulus of

The proposed formulations are implemented in MATLAB R2015a

The code
and examples can be downloaded from:

Comparison of the deformed shape of beams with a contact point versus beams connected over a common node.

Effect of rotational penalty stiffness on the rotational gap.

Energy imbalance reduction during the analysis of Example 1.

Comparison of the deformed shape of beams with and without the rotational constraint in Example 2.

Energy imbalance reduction during the analysis of Example 2.

In this example we consider two crossing perpendicular beams. The
translational degrees of freedom are constrained at the two ends of one beam,
while the other beam is given a prescribed in-plane displacement at the two
ends, which would cause a rigid body rotation of 15

In this example, we consider two crossing perpendicular beams which are fully
constrained at one end. We apply an in-plane force of

The change in the energy imbalance during the solution process is shown in
Fig.

Different models used in Example 3.

Effect of rotational penalty stiffness on the stiffness of different models of Example 3.

Effect of rotational penalty stiffness on the stiffness of different
models of Example 3.

In this example, we start with a structure consisting of 4 intertwined beams
of length 10, as shown in Fig.

We then modified our model, by adding beams of length 5 that passed through
the intertwined beams in order to get the models shown in Fig.

Initial configuration of the random network of Example 3.

Total reaction force versus prescribed displacement in Example 3, with and without rotational constraint.

In this example, we generated a random fiber network of size

Geometric data of the random networks.

All the contact points are assumed to be inseparable (bonded constraints)
throughout the analysis. We assumed that no new contacts are established
during the deformation. A total prescribed displacement of 3 units
corresponding to 15 % of the network length is applied to the model, as
shown in Fig.

Two alternative methods were suggested in order to handle the rotational constraint between beam elements. In the consistent method, we followed the derivation methodology encountered in earlier works dealing with normal and frictional contact between beams. This formulation generally requires an assumption for the shape functions. For the sake of simplicity, we assumed linear shape functions for the underlying beam elements. The non-consistent method does not require predefined shape functions for the beam elements, as it assumes that the contact normal is independent of the nodal deformations in each iteration (the change of contact normal at each iteration does not contribute to the linearization of the gap function and as a result to the tangent stiffness matrix), and it is updated between iterations. Despite large differences in their mathematical formulations, the two methods generally show similar convergence rates and stability for the chosen examples. However, the non-consistent method offers the advantages of simpler derivations and an easier implementation. In the considered examples where the beams are well constrained, the inclusion of the rotational constraint does not generally affect the convergence rate of the problems.

Including the rotational constraints in the problem of a random fiber network significantly affected the stiffness of the network, due to the addition of constraints in the system. It clearly shows that ignoring rotational constrains can yield inappropriate results in such cases. In addition, the presence of rotational constraints improves the stability of the non-linear solution procedure by effectively preventing rigid body rotation, in particular, in the fibers which have few contacts along their length.

If

We consider the unit vector

We consider the unit vector

Consider the unit vector

In this section, we will derive the rotational shape functions for a linear
and an Euler–Bernoulli beam element in their local coordinate system.
Rotation matrices are then used to transfer these shape functions to the
global coordinate system. Shape functions interpolate the solution between
the discrete values at the nodes, as follows:

The linear shape functions for a beam element are as follows:

The shape functions for an Euler–Bernoulli beam element are given below.

HRM derived the formulation, implemented it into a code and used it to create and analyze the models in numerical examples 1–3, performed all the verification case studies, and wrote the article. AK initiated and supervised the work, implemented the formulation into a large-scale computational framework used in numerical example 4 and edited the article.

The authors declare that they have no conflict of interest.

Funding from the WoodWisdom ERA-NET program (PowerBonds project) is gratefully acknowledged. We also thank Anders Eriksson and Elsiddig Elmukashfi for valuable discussions and for comments that greatly improved the manuscript. Edited by: Amin Barari Reviewed by: three anonymous referees