Rotation Formulations for Dynamic Relaxation – with Application in 3D Framed Structures with Large Displacements and Rotations

Jian-Min Li

PhD student Institute of Building Structures and Structural Design (ITKE), University Stuttgart Stuttgar, Germany quantumamin@gmail.com

Jan Knippers Prof. Dr.-Ing. Institute of Building Structure and Structural Design (ITKE), University Stuttgart Stuttgart, Germany j.knippers@itke.uni-stuttgart.de

Summary Dynamic relaxation (DR) has been widely used as a powerful method for structural analysis and form-finding. Compared with the direct stiffness method, one advantage of DR is that there is no need to compose a global stiffness matrix; for DR, every node of the structure can be seen as a single particle constrained by forces generated by the interactive mechanism of the structural members. The equations of motion for particles are much easier to consider than the direct stiffness approach, especially when structures undergo large displacements and rotations.

However, most applications of DR are restricted in the field of cable, membrane, hanging chain structures. The common feature behind these structures is that only three degree of freedom (3-DOF) nodes are used. And for 3-DOF nodes, no bending moment but translational forces can be applied.

In this paper, we propose the rotation formulations for DR. The description of motion of one particle is then separated into two parts, the translational and the rotational. Therefore, additional rotation degree of freedom is introduced into the computation system and we can use DR to simulate structure systems composed of 6-DOF nodes.

We further show how we apply the 6-DOF DR, with the beam mechanism, to the simulation of framed structures. Validity is examined through examples of structures which undergo large displacements and rotations.

Keywords: six degree of freedom node; Dynamic Relaxation; large rotations; large displacements; 3D beam; framed structures

1. Related works Few attempts have been made to extend DR to have rotational degree of freedom. Wakefield uses rotation displacements to describe 3D rotations in DR [1]. In his derivation, Johnson and Brotton’s method [2] was used to get the sway angles.

Williams proposed to solve global translational displacements by solving the nodal displacement individually by assuming the objective node is not restrained and the adjacent nodes are fixed [3] and [4]. The velocity terms are not used in his derivation. Instead, he uses displacements to play the role of velocities.

Besides, there were other attempts to simulate bending within the 3-DOF framework. However, by the natural limitation of the 3-DOF node, these approaches can not deal with torsion effect and unsymmetrical section properties [5].

2. Dynamic relaxation

When a structure undergoes large displacements and rotations, the Newton-Raphson method (NR), which is commonly used as the iteration solver for nonlinear FEM, might diverge due to the

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However, NR is proved to be more efficient to deal with structure systems with less elements. But, it will lose its advantage when it goes to a system with large element number. Because the computation time needed for DR is proportional to N4/3 (N is the number of elements in the system), and for NR it is proportional to N7/3 [6].

2.1. Translation formulations for DR

Dynamic relaxation uses the concept of particles and constrained forces to simulate structure systems. It builds the inter relationship between the displacement, velocity and acceleration of one particle in a central difference form as in eq. (1) and (2), where n denotes the time point tnt . These equations can be further arranged as in eq. (3) and (4), in which the unknown displacement and velocity of the current time step are expressed as functions of known values of the previous time step. Calculating repeatedly, the motion of the system can be thereby simulated in time history.

Using Newton’s third law as in eq. (5), force can be introduced into the system. We substitute the acceleration term in eq. (4) with the force term to get eq. (6).

2.2 Rotation formulations for DR

Rotation matrix is used to describe the orientation of one particle/node. The matrix is composed of three unit vectors pointing the local x, y and z directions as in eq. (7). Similarly, here we can build relationship between the current and previous orientations, Rn+1 and Rn-1, as we did for displacements [7]. Their relationship is built up through eq. (8) where R is a transformation matrix which is defined by a rotation vectors ∆ e

as in eq. (9). The mathematical meaning of the

transformation matrix is that after rotating the previous local coordinates Rn-1 with ∆ degree around vector e

, Rn-1 will meet the current local coordinates Rn+1 [8].

This rotation vector can be approximated by the product of the mean angular velocity and the time interval as in eq. (10), in which we have applied the core meaning of central difference. Eq. (8) can be further expressed as eq. (11).

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Similar to the relationship between the velocity and acceleration in eq. (4), we can also express the current angular velocity in terms of the previous angular velocity and angular acceleration as in eq. (12). In contrast to eq. (5), which describes translational mechanism, eq. (13) describes rotational mechanism which can be derived by applying virtual power to a rotating body [9], where T denotes torque acting on the particle and I denotes the moment of inertia of the particle.

Since, in this research, we care about only the final equilibrium state not the dynamic process, we can assign the mass and moment of inertia for our own convenience. Therefore, we assign a scalar mass in eq. (6) and an isotropic rotational inertia here. The later assignment further reduces eq. (13) to eq. (14). Finally, we replace the angular acceleration in eq. (12) with the torque term to get eq. (15). Table 1 is presented here as a concise comparison between the translational and rotational formulations of DR.

3. Beam mechanism Our beam model is visualized as two particles set apart with distance as the beam length and having concentrated mass and given rotational inertia. The particles will shift and rotate freely if there is no any constrained force or torque. Once we add the constrained forces to the system, the particles’ movements will be coupled together. A simplest example is two balls oscillating on the two ends of a spring.

In the following sub sections, we introduce how we calculate the internal forces and torques acting on the particles. These forces and torques are caused by the relative movement between the particles. First, we calculate the relative displacements and included angles and then use them to calculate the corresponding forces and torques.

Noteworthily, using particles and constrained forces to simulate the coupled movements might be not so common in structural analysis but it has been a basic method to study molecule dynamics.

3.1 Orientations of beam ends

In sec.2, we give the description of motion of nodes/particles including its orientation definition. However, in order to calculate the beam deformation, there is need to define additional orientations for beam ends, as in eq. (16). Where Ra,n-1 and Ra,n+1 are the previous and current orientations of the beam-end- a and R is the transformation matrix of the node where beam-end- a is situated, as defined in eq. (11).

Table 1: translation and rotation formulations of DR

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Herein we have implied the rigid joint assumption with using the same transformation matrix to update the nodal and beam end’s orientations such that the beam end is able to rotate with the node simultaneously. The initial beam ends’ orientations of one beam, Ra,0 and Rb,0 , are equal to the beam’s initial orientation. Correspondingly, the initial nodal orientation is an identity matrix.

3.2 Included angles

The included angles, as in eq. (17)-(20), are used to compute bending moments and shears in a deformed beam. Their corresponding definitions are shown in the fig.1. These formulations are suitable for small relative rotations between the two ends of one beam. One should bear in mind that the orientations used here are the orientations of the ends of the beam instead of the nodes’ /particles’ orientations.

3.3 Internal forces

3.3.1 Axial forces and bending moments

The formulations for axial force, eq. (21), and bending moments of internal forces, eq. (22)-(24) are the same as the terms in the stiffness matrix of a finite beam element. They can be derived by applying virtual work method to a continuous deformable body [10].

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The internal forces and torques derived above are defined in local beam ends’ coordinates. Before further computing resultant torques, we have to be transformed them to the global coordinates with eq. (25).

3.3.3 Shear forces

The missing terms of shear forces in sec. 3.3.1 can be derived by using the equation of zero resultant force and torque of one beam. Solving eq. (26) and (27), we can get shears as in eq. (28). The reason for not using the shears defined in the stiffness matrix is we use local beam ends’ coordinates to transform bending moments instead of using local beam coordinates, which is the case for FEM. Therefore, we need to derive shear forces in a way to ensure zero resultant force and torque of one element. Finally, the applied translational force for one node, from one beam end, is the sum of the axial force and shear force as in eq. (29).

4. Implementation

4.1 Algorithm

a. Input initial geometry and mechanical properties

b. Compute length extension and included angles as defined in eq. (17)-(20).

c. Compute beam axial force and torque defined in eq. (21)-(24) and transform them to global coordinates. Use eq. (28) and (29) to compute the beam shear force and nodal translational force.

d. Compute resultant force and torque for each particle.

e. Check if the resultant force and torque are small than given values. If it is true for every particle, go to step g. Else, go to step f.

f. Use eq. (3), (6), (8) and (15) to update the position, orientation and velocity of each node, use eq. (16) to update each beam end’s orientation, and back to step b.

g. Terminate the calculation and output the result.

4.2 Mass and moment of inertia

Observing eq. (6) and (15), one can find that less the mass and moment of inertia, larger the increments of translational and angular velocities. And generally, faster the velocities, less time needed to attain equilibrium. However, too small values of mass and inertia will lead the calculation divergent.

A simple model as in fig.2 is introduced here to determine the minimized mass and rotational inertia at the same time. The structure, composed of two elements, is initially bended and will attain its straight and unstressed geometry finally. By try and error, we can find the minimised mass and the minimised rotational inertia, which can then be applied to a larger system with more elements and

Fig.2: simple model for determining mass and moment of inertia

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4.3 Damping

4.2.1 Velocity damping

Damping is used to reduce system’s kinetic energy until a final static equilibrium is attained. The velocity damping defined here is by multiplying a factor smaller than 1.0, to the translational velocities or angular velocities in every iteration loop.

4.2.2 Mixture of kinetic damping and velocity damping

Kinetic damping is proved to be a very efficient way to attain the equilibrium. Setting system’s velocities to zero once the system passes through the kinetic energy peak is how it works [4].

Here we mix two damping methods together. For translational velocities, we use kinetic damping and for angular velocities, we use velocity damping. Thereby, translational and angular velocities are treated separately. We set the translational velocities to zero once the system passes through the peak of translational kinetic energy, while we run out the angular kinetic energy gradually by applying velocity damping.

5. Numerical examples

5.1 Straight beam subjected to end bending

This example was seen in Ibrahimbegovic’s work [11]. A straight cantilevered beam composed of 10 elements is shown in fig.3 with material properties: E (elastic modulus) = 79577, (Poisson’s ratio) = 0. The beam is subjected to a pure end torque Tz = 2.5. The exact solution of this problem is EIlTz / , where l denotes total length which is 10. That is when Tz =2.5, is equal to /4. The result from the presented method is compared with Ibrahimbegovic’s, the analytic and the result from Sofistik, a FEM commercial software, as in table 2.

5.2 Curved beam subjected to end force

The curved beam proposed by Bathe and Bolourchi as in fig. 4 has a 45-degree bent geometry with radius 100 in. The material properties are: E = 107 psi, = 0. The beam composed of 8 elements with equal length lies in the x-y plane and is subjected to a concentrated load in z-direction. Comparison is made between Bathe’s [12], Crisfield’s [13] and Sofistik’s results, as in table 3.

Table 2: displacement of straight beam under end moment x y z

present -0.9950 3.7311 0 Ibrahimbegovic -0.9962 3.7290 0 FEM commercial -0.9945 3.7302 0 Analytic -0.9968 3.7292 0

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Fig.3: definition of the straight beam and its deformation

6. Conclusion and further work We have proposed the rotation formulations for dynamic relaxation, with which we are able to consider structure systems with 6-DOF per node. That is the bending and torsion effects can be considered in the DR framework. We further expressed how we integrate the rotation and translation formulations of DR with the beam mechanism. Validity has been examined through examples considering structures undergo large displacements and rotations.

To build a more robust algorithm, we will improve the way we optimise the mass and rotational inertia such that fine tuning is no more necessary.

References [1] Wakefield, D.S., Dynamic Relaxation Analysis of Pretensioned Networks Supported by

Compression Arches, PhD thesis,1980

[2] Johnson D., and Brotton, D.M., “A Finite Deflection Analzsis for Space Structuraes”, Proc. Int. Conf. on Space Structures, Univ. of Surrez, 1966.

[3] William, C.J.K., Private Report in the Context of Partial Supervision and Work on Grid Shells, Bath University,1999

[4] Adriaenssens, S.M.L., Stressed Spline Sturctures, PhD thesis, 2000

[5] Adriaenssens, S.M.L., Barnes, M.R. and Williams, C.J.K., “A new Analytic and Numerical basis for the Form-finding and Analysis of Spline and Grid-shell”, Computing Developments in Civil and Structural Engineering, 1999, pp. 83-90

[6] Sauve, R.G.., “Advances in Dynamic Relaxation Techniques for Nonlinear Finite Element Analysis”, Journal of Pressure Vessel Technology, Vol. 117, 1995, pp. 170-176

[7] Krzsl, P., Explicit Newmark/Verlet algorithm for Time Integration of the Rotational Dynamics of Rigid Bodies, International Journal for Numerical Methods in Engineering, vol. 62, 2005, pp. 2154-2177

[8] "Rotation matrix", http://en.wikipedia.org/wiki/Rotation_matrix

Table 3: displacement of Bathe’s curved beam x y z

present 13.685 -23.838 53.703 Bathe and Bolourchi 13.4 -23.5 53.4 Crisfield 13.63 -23.87 53.71 FEM commercial 13.559 -23.552 53.294

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[9] Wittenburg, J., Dynamics of Multibody Systems, Springer, 2008

[10] McGuire, W., Matrix structural analysis, Wiley, 2000

[11] Ibrahimbegović, A., Computational Aspects of Vector-like Parameterization of Three-dimensional Finite Rotations, International Journal for Numerical Methods in Engineering, vol. 38, 1995, pp. 3653-3673

[12] Bathe, K. J., "Large displacement analysis of three-dimensional beam structures", International Journal for Numerical Methods in Engineering, vol. 14, 1979, pp. 961-986

[13] Crisfield, M.A., "A consistent co-rotational formulation for non-linear three-dimensional beam-elements",Computer Methods in Applied Mechanics and Engineering, Vol 81,1990, pp.131–150