Tennis Racquet Stresses Caused By Stringing (Part 1) - By Dr. Paul Zarda

Dr. Paul Zarda of Orlando and Sanford Florida notes:  This is Lecture 1 in a series of lectures to help understand stresses that develop in a tennis racquet when the racquet is being strung on a stringing machine.

 Lecture 1 develops a basic understanding of two different kinds of loads or stresses that a tennis racquet will carry when it is being strung. Future lectures will develop a more in-depth understanding of these stresses during an actual stringing process. The purpose of this lecture is to develop the basic understanding of two distinctly different types of stresses that occur in a tennis frame and also to develop the foundation on which future lectures will be based.

Figure 1:     A CAD and Finite Element Model (top left) of the Wilson Tennis Racquet (bottom right)

Figure 1 shows a typical tennis racquet that has been strung (bottom right) and the corresponding CAD model of the racquet (top left). This racquet has a stringing pattern of 16 x 19, which means it has 16 main strings and 19 cross strings. The main strings are strung at roughly 60 lbs, and the cross strings at roughly 40 lbs. This results in a 12-6 o-clock load of 960 lbs and a 3-9 o-clock load of 760 lbs. This is quite a lot of load that the racquet must carry even before it sees the additional load that occurs during ball impact.

Figure 2:  Understanding Axial Stress and Bending Stress will help

 Dr. Paul Zarda notes in Figure 2 the two fundamentally different ways in which a structure, like a tennis racquet, can carry load. Consider a wooden yardstick which almost everyone has held in their hand. The top image in Figure 2 shows an axial load that is applied along the longer dimension of that yardstick. The yardstick is supported on the left side and the load is applied on the right side. The stress developed in this case are given by the equation to the right of that figure: the stress is the axial load P divided by the x-sectional area (b x h) of the yardstick. For those unfamiliar with the concept of stress, it is noted that materials will fail when a stress level (here defined as P/A) reaches a maximum value for the material in question (in this case it would be the yield stress for the wood material of the yardstick).

The bottom image in Figure 2 shows another kind of stress that can develop: a bending stress. Here the yardstick is supported on its left side, and a load is applied to the right end. Unlike the previous case, the load here is lateral to the longer dimension of the yardstick. This loading will cause the yardstick to deflect laterally, and the largest stresses, called bending or flexural stresses, are generate at the root (left end).  Unlike the first case of axial stress which leads to a uniform stress over the x-section, the developed bending stress will not be uniform over the x-section. The formula for this bending stress is shown to the right of the bottom image of Figure 2. Note that this bending stress in this formula depends on the moment of the load about the base of the support (P * L) and geometry of the x-section. A bending stress will peak at outside dimension of a x-section and actually go through a zero level at the center of the x-section. In general, for the same load, carrying load via a bending stress will produce significantly higher stresses than carrying the load via an axial stress. This last statement will become more obvious as the stresses in a tennis frame are explored.

Figure 3:   Finite Element (FE) Model of a yardstick (green) showing a Bending Load and Axial Load

Figures 3, 4 and 5 show a more in-depth understanding of the axial stresses and bending stresses that develop in our yardstick test. Shown in Figure 3 is a simulation, called a finite element model, of the yardstick. Actually the model has 2 yardsticks in it: a yardstick that is loaded with an axial load, and a yardstick that is loaded with a bending-causing transverse load. The image on the right in Figure 3 shows an overall view of both yardsticks where the yardstick is supported at one end and it is loaded at the other. The image on the left in Figure 3 shows the close-up of the loading conditions. Note, in this Figure, there is also a “grid” seen that models the yardstick. This is a finite element grid of solid elements. Use of finite elements to simulate loads, supports, deflections and stresses produced in a structure can be found in the literature/internet and will not be covered here. The main observation to be made is that the loading conditions of an axial load and bending load can be simulated, quite accurately, by using finite elements. Those results will be discussed next.

Figure 4:   Deflection Contours for a Bending Loaded and an Axially 

Figure 4 shows the deflection contours for these 2 loading conditions. The plots for both deformed shapes (axial load and lateral bending load) are to the same level: red is the highest deformation and blue is the smallest deformation. Note that the deflections for the beam (red) are significantly larger than the deflections for the axial load (blue).

Figure 5:   Stress Contours for a Bending Loaded and an Axially Loaded Yardstick

Dr. Zarda, Orlando and Sanford Florida, notes that Figure 5 shows the maximum stresses for each of the cases: a bending load and an axial load. In this case the contours are not coordinated to the same level in both plots. The left image in figure 5 shows the stress levels in the bending-loaded yardstick. Note that the maximum bending stress is 3850 psi, and it occurs at the base support of the yardstick, and it occurs there on the outer fibers of the yardstick (there is a 3850 psi positive tension stress on one face, and there is a 3850 negative compression stress on the opposite face). In direct contrast, note the stress level for the axial case (same load) is 18 psi, and this is a uniform stress throughout the x-section and along the yardstick. The important observation here is that, for the same load, the axial stress levels are 2 orders of magnitude lower than the bending stress levels.

The stresses that can develop in a tennis racquet are a combination of axial stresses and bending stresses. The game plan, for a given string tension level, is to cause axial load/stresses and minimize bending load/stresses.  And although it will not be shown in this article, it is possible, for elliptical shaped racquets, for a controlled set of main string and cross string tensions, to cause only axial stresses and cause the bending stresses to zero (theoretically). But that’s another lecture.