Finite element simulation of pole vaulting
Abstract
Pole vaulting is one of the most spectacular disciplines in athletics. The evolution of world record heights is strongly influenced by the development of advanced poles and subsequent materials. Employing advanced, loadadjusted composites has resulted in a steady increase of the world record height. This study provides a framework for finite element simulations of pole vaulting with focus on the initial and boundary conditions as well as finite element choices. The influence of the pole bending stiffness on the achievable height is systematically simulated. Higher effective bending stiffness leads to higher pole vaulting heights. However, if a certain stiffness is exceeded, the vaulter will not be able to bend the pole enough which leads to failed attempts.
Keywords
Pole vaulting Simulation Finite element method1 Introduction
Pole vaulting is an astonishing sport where athletes use poles as an auxiliary equipment to clear the bar at heights of 5–6 m and more. Risen from a purpose driven technique, e.g., overcoming castle walls during siege, crossing irrigation ditches [1] and transport [2], pole vaulting has become one of the most spectacular and hightech disciplines at the Olympic games. It is a dynamic, elegant and fascinating sport.
To clear the bar, pole vaulters need to transfer the kinetic energy of the approach into potential energy via the pole. Two factors determine the success of a vault: (1) physical abilities of the athlete (speed, strength and springiness) and (2) properties of the pole (low weight and ability to store and return energy in the most efficient way).
In the early 20th century, bamboo poles were utilized which are light but also very stiff and, thus, not able to transfer the kinetic energy from the approach in an efficient manner. In the 1940s, aluminum and steel led to new world records—however, these materials show high stiffness as well and therefore did not significantly improve the vaulting height. In the beginning of the 1960s, lightweight fiberreinforced plastics (with lower bending stiffness) became increasingly popular. They allow large elastic deflections. As a consequence, they are able to transfer more energy resulting in a more pronounced catapult effect. In the following years, the world record steadily rose due to improved fiber arrangements and vaulting techniques. In the 1990s, a new plateau of nearly 6.15 m was reached. This indicates that human physiology and current materials may have reached a limit and enhancements of the pole’s material are necessary to further improve the world record. The aim of this work is to provide a finite element (FE) model allowing for simulation aided pole design.
2 Model specifications
FE simulations of the pole vaulting process were conducted with the commercial FE code abaqus 6.14. The initial boundary value problem of the dynamic equation of motion was solved with an implicit time integration method under consideration of finite deformations. Exemplary FE models of the pole and the vaulter were obtained and generated in abaqus/CAE. To capture the kinematics of the pole vault procedure, a setting was employed that couples pole and vaulter allowing for a relative motion.
2.1 Finite element model of the vaulter
A comprehensive overview of the mechanics of pole vaulting can be found in Frère et al. [4] describing the process in four steps: runup, takeoff, pole bending and pole straightening. A crucial part in the jumping process simulation is the description of the vaulter. Muscle work of the vaulter increases the performance [5, 6]. In addition, the moment exerted on the pole by the vaulter influences the vaulting performance [7]—elite vaulters bend the pole such that its effective length^{1} relative to the original length is reduced by ca. 30% [8]. As stated by Ekevad and Lundberg [9], a modeling approach representing the vaulter by a point mass with a fixed position relative to the pole is not sufficient. The complex motion in combination with muscle work needs to be considered.
Properties of segments of vaulter model (data from [11])
Symbol  Segment  Length/mm  Mass/kg 

A  Forearm  363  3.52 
B  Upper arm  343  4.48 
C  Head  332  6.48 
D  Trunk  526  39.8 
E  Thigh  447  16.0 
F  Shank  521  9.76 

Angle \(\varphi\) between ground and a line connecting the tips of the pole,

Height \(h_{\mathrm {p}}\) of upper pole tip,

Angle \(\theta _{0}\) of the rotated coordinate system \(x_{2}\)–\(y_{2}\),

Angle \(\theta _{1}\) between the coordinate system \(x_{2}\)–\(y_{2}\) and segment A, and

Angles \(\theta _{2}\) to \(\theta _{6}\) between the segments A to F.
We employed two variables during the pole vault to trigger the relative motion of the point mass: the pole angle \(\varphi\) from takeoff to the instance of maximum pole bending, and the relative height \(h_{m,\mathrm {rel}}\) of the point mass from maximum pole bending till pole release^{4}. This is due to motion of the point mass relative to the pole that cannot be implemented as time based. It depends on properties of the pole such as the pole’s length and stiffness—these properties, however, are to be varied during the simulations.
2.2 Finite element model of the pole
Modern, elite vaulting poles are manufactured out of lightweight materials consisting mainly of glass fiberreinforced plastics. The fiberglass pole is rolled on a mandrel and, subsequently, wrapped to stiffen the pole. The perfect pole differs for athletes, depending on their physical properties, abilities and vaulting technique and plays an essential role in the vaulter’s performance.
Summary of the properties of the pole
Property  Symbol  Value  Ref. 

Length  L  4200 mm  [14] 
Outer radius  R  21 mm  
Wall thickness  t  2.5 mm  
Density  \(\rho\)  1887.5 kg \(\mathrm m^{3}\)  
Poisson’s ratio  \(\nu\)  0.285  
Modulus of elasticity  E  37.5 GPa  
Flex number  15 
2.3 Coupling of pole and vaulter
The point mass needs to be connected to the pole such that it can move relatively to the upper pole tip according to the vaulter’s motion in different phases of the vault. Moreover, the connection fails once the pole is stretched and the vaulter would release it to clear the bar. An IAAF requirement is that the vaulter must hold the pole in the grip area (no higher than 0.1524 m from the top of any pole or no lower than 0.4572 m from the top of the pole).
abaqus’ connector elements allow constraints involving relative motion of the connected parts via Lagrange multipliers as additional solution variables and also failure of the connection [15]. Through the connector element, a relative position of the point mass was given as a constraint on the upper tip of the pole. Moreover, the connector elements were employed as a sensor to measure the position \([x_{\mathrm {p}},y_{\mathrm {p}}]^{T}\) of the upper pole tip and the height \(h_{m}\) of the mass center. A user subroutine UAMP was used in each increment of the calculation to call the quantities measured by the sensors and enforce relative displacement components [16].
2.4 Initial and boundary conditions of pole vaulting
Regarding the boundary conditions of the simulation, the pole was supported at the lower end fixing all translational degrees of freedom. This represents the contact to the planting box. Furthermore, a gravitational acceleration of \(g=9.81\,\mathrm{m\,s}^{2}\) was assumed.
Initial conditions of the simulations
Initial condition  Symbol  Value 

Takeoff angle  \(\alpha\)  \(18^\circ\) 
Takeoff velocity  \(v_{0}\)  8 m \(\mathrm s^{1}\) 
Horizontal component  \(v_{0,x}\)  7.608 m \(\mathrm s^{1}\) 
Vertical component  \(v_{0,y}\)  2.472 m \(\mathrm s^{1}\) 
Due to the ‘hard contact’ boundary condition at the lower end, such an initial condition results in artificial oscillations of the pole with large amplitudes. All modes of the pole, which is initially at rest, are excited by the impulse. To circumvent these artificial oscillations, we started the simulation at the instance right after the pole was planted and when it started to bend as the vaulter’s mass compressed it. Then, in addition to the initial velocity of the point mass, also a velocity profile \(v_{0,\mathrm {pole}}\) on the pole was necessary that induced the bending. This profile was obtained from a preliminary simulation.
 1.
Zero velocity at the lower end,
 2.
Velocity at the upper tip equals the preliminary simulation,
 3.
Maximum velocity node position matches the preliminary simulation, and
 4.
Maximum velocity value equals the preliminary simulation.
3 Results and discussion
The velocity of the point mass is displayed in Fig. 9. The velocity \(v_{x}\) in the horizontal direction decreases in the beginning as the kinetic energy of the approach is transferred into strain energy by deforming the pole. Subsequently, the velocity \(v_{y}\) in the vertical direction increases when the pole recovers to catapult the vaulter. After releasing the pole, the horizontal velocity \(v_{x}\) is constant since no forces act on the point mass in that direction. Contrary, the upward vertical velocity \(v_{y}\) of the vaulter decreases linearly under the effect of gravity. When the vertical velocity \(v_{y}\) decreases to zero, the maximum height \(h_{\mathrm {max}}\) and accordingly the maximum potential energy is reached. The energy transformations correspond well to the experimentally based descriptions of Dillmann and Nelson [21]. The simulated velocities correspond well in trend and magnitude to the measured ones of Frère et al. [4] and AnguloKinzler et al. [22].
3.1 Finite element choices: continuum vs. beam elements
The aim of this contribution was to allow for more detailed material models taking into account the heterogeneity of fiberreinforced plastics. If the pole’s heterogeneous microstructure is to be taken into account, continuum elements^{6} need to be applied. They are computationally more costly, but distributions of field variables can be identified precisely and they allow to account for a sophisticated microstructured material model at a later stage.
4 Conclusion
This study provides a framework for FE simulations of pole vaulting, suitable for investigations on the pole’s material. The vaulter was accounted for as a point mass to save computational effort in control that was necessary in other studies. The motion of the vaulter relative to the pole was implemented with an algorithm based on a reference vault of an elite pole vaulter. The focus was on the proper choice of boundary and initial conditions to avoid unphysical oscillations. An initial velocity profile of the pole was determined by running a presimulation with reduced pole weight and overcomes unnatural simulation outcomes.
Beam elements as well as continuum elements can be used for the discretization of the pole yielding similar results. Continuum elements give a more detailed insight in distributions of field variables such as stresses and strains. Furthermore, they provide the option to study and incorporate complex computationally expensive material models considering the micromechanical structure of the pole.
The simulation results resemble measurements of the pole vault process in trend and magnitude. Additionally, we examined the influence of the effective bending stiffness of the pole on the achievable height in pole vaulting using an isotropic NeoHookean (continuum elements)/linear elastic (beam elements) material model and varying the elastic modulus. We observed that up to a certain stiffness, higher effective bending stiffness leads to higher pole vaulting heights. However, overly stiff poles would not bend enough to catapult the vaulter forward leading to failed vault attempts. This naturally occurring phenomenon is well reflected in our model.
Footnotes
 1.
The effective length of the pole is referred to as the end to end distance.
 2.
The results of our simulations were extracted from the motion of one vaulter. To receive more general results for a specific pole, the variation of the motion of each vaulter as well as between different vaulters should be taken into account.
 3.
The rotation of the pole to the side was not taken into account as the influence of movements out of the 2D plane was marginal [12].
 4.
Ekevad and Lundberg [11] control the vaulter motion by the angle \(\varphi\). This leads to instabilities from the instance of maximum pole bending to pole release where the vaulter movement changes, based on the almost constant angle \(\varphi\). Ekevad and Lundberg [11] tried to overcome this drawback by calculating an average rate of the change of the angle \(\varphi\) in the first half and applying this for the second half. Their approach is nonphysical: the stretching of the pole is not coupled to the relative motion of the vaulter.
 5.
This only holds true as long as the pole is modeled with constant crosssection and homogeneous material.
 6.
abaqus’ C3D8R continuum elements (8 nodes with 3 displacement degrees of freedom per node and linear shape functions) were applied in this case [15]. These elements possess one Gaussian integration point. Reduced integration was used to avoid shear locking. For such elements, artificial hourglass deformation modes may occur. The occurrence of these modes was controlled via the artificial strain energy that has to be small compared to the strain or internal energy.
 7.
abaqus’ B21 beam elements were employed which are located in 2D space and use linear shape functions [15]. They possess two nodes with three degrees of freedom each (two displacements and one rotation) and two integration points. Timoshenko beam formulation was utilized.
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