Lecture 5: Tennis Racquet Stresses Caused by Stringing – By Dr Paul Zarda

Dr. Paul Zarda of Sanford Florida notes:  This is Lecture 5 and it is the last lecture 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 developed a basic understanding of two different kinds of loads or stresses that a tennis racquet will carry when it is being strung.

Lecture 2 reviewed typical loads that a tennis racquet can withstand. It also reviewed actual Instron failure loads of some typical racquet frames.

Lecture 3 reviewed the development of numerical models, or simulation models, called finite element (FE) models, and how such practices can be used to assess loads and stresses developed in a tennis racquet due to a typical stringing of the racquet.

Lecture 4 reviews the theoretical determination of loads/stresses in a tennis racquet. It also presents, based on a FE model of Wilson tennis racquet frame, loads/stresses in the frame of the racquet due to string loads.

Dr. Zarda of Orlando notes that this lecture, Lecture 5, first reviews the different kind of loads that can develop when stringing a tennis racquet. The assessment of these loads is done based on a finite element model that was developed and presented in Lecture 3.  Following that discussion, the loads that are developed in a tennis racquet (a Wilson racquet is used and is typical of almost any racquet) when it is being strung in a stringing machine are reviewed.

The shape of the tennis racquet frame, and the stringing and string pattern of the frame, is important in determining the loads and stresses in the frame. The largest loads that the frame will see occur because of the string tension that is applied to the frame. 

Figure 1 shows the bending moment that can developed in a strung frame. These results are based on a finite element (FE) model of a Wilson tennis racquet that is discussed in Lectures 2 and 3. Here the tensions chosen are based on the formula presented in Lecture 4. In this case 60 lbs was chosen as the main string tension, and the cross string tension, per Equation 1 of lecture 4, was determined to be 40 lbs. For an elliptical racquet with equal string spacing, the tensions determined by Lecture 4 would produce a bending moment M close to zero. In this case, because of non-equal spacing of the Wilson frame (FE model is shown in Figure 1), the bending moment is not zero but varies with position along the racquet (measure by S shown in Figure 1, where S=0 is at 12 noon, and S=18 inches is at 6 o-clock). Note that the maximum bending moment is around 120 in-lbs and occurs at about 3 o-clock (S=9 inches). This bending moment is significantly less than the failure bending moment of 372 in-lbs discussed in Lectures 2 and 3. Also note that there is no discussion here of how this racquet was strung. It is assumed, for this discussion, that the strings are “dropped in” and their final in-racquet tensions are 60 lbs for the main and 40 lbs for the cross. While this final strung condition of a tennis racquet is difficult or even impossible to achieve in a practical way using a stringing machine, it does offer the reader an understanding of the minimal load/stress level that one might achieve. 

Another notable observation of Figure 1 is that the bending moment is not zero as predicted by the discussion in Lecture 4. This is caused by the fact that the shape of the racquet frame head is not a perfect ellipse and because the string spacing is not uniform. Even with the conditions implied in Lecture 4 not being met by the racquet shown in Figure 1, the bending moment of Figure 1 is likely at its lowest level per Lecture 4.

Dr. Zarda, Orlando and Sanford, notes that Figure 2 shows bending moment loads for the same Wilson tennis racquet under 2 additional loading conditions:  (i) main strings only, at 60 lbs, are in the racquet with no cross strings and no support of the racquet, and (ii) cross strings only, at 45 lbs, are in the racquet with no main strings and no support of the racquet. For reference, the condition of Figure 1, where the racquet is strung at 60 lbs main, 40 lbs cross, is also shown.

As Figure 2 shows, without adequate support, the main-string-only-loading condition will cause this Wilson frame to fail (the bending moment at 12 noon exceeds the failure bending moment of 372 in-lbs discussed/determined/presented in previous lectures). While the moment for the cross-string-only-loading does not show failure, it does show levels that are significantly higher than the moments shown in Figure 2 (and Figure 1) for the final strung racquet (60 main, 40 cross).

Figure 3 shows the FE-predicted bending moments during a typical stringing process. For the bending moments shown in this figure, the frame is supported at 12 and 6, and it can also have additional side supports at 10:30 – 1:30 and 7:30 – 4:30. These side supports come into play as strings are added during the stringing process. The legend shown in Figure 3 explain how and when the side supports are used or not used. The bending moment curves shown represent the bending moment in the tennis frame at various stages in the stringing process. This particular racquet is strung by first stringing the main strings (one side and then the other side), and then the cross strings from top down. The figure shows, for this 6 point support stringing machine (4 side supports and a top and bottom support), that the maximum bending moment (at any location and for any number of strings in the racquet) of approximately 210 in-lbs does not exceed the failure bending moment of  372 in-lbs (Wilson frame). This level of 210 in-lbs is almost 100% higher than the maximum bending moment (120 in-lbs) found in the final strung racquet (see Figure  1).

Lecture 4: Tennis Racquet Stresses Caused by Stringing – By Dr Paul Zarda

Dr. Paul Zarda of Orlando notes:  This is Lecture 4 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 developed a basic understanding of two different kinds of loads or stresses that a tennis racquet will carry when it is being strung.

Lecture 2 reviewed typical loads that a tennis racquet can withstand. It also reviewed actual Instron failure loads of some typical racquet frames.

Lecture 3 reviewed the development of numerical models, or simulation models, called finite element (FE) models, and how such practices can be used to assess loads and stresses developed in a tennis racquet due to a typical stringing of the racquet.

This lecture, Lecture 4, reviews a theoretical basis for predicting loads in a tennis racquet frame. It then uses that theoretical background to look at real loads in a strung tennis racquet.

Dr. Zarda of Sanford notes that the shape of the tennis racquet frame, and the stringing and string pattern of the frame, is important in determining the loads and stresses in the frame. The largest loads that the frame will see occur because of the string tension that is applied to the frame. A basic understanding of these loads and resulting stresses will now be discussed.  A formula for the tensioning of the strings can be derived to minimize stresses in the frame.

Consider, referring to Figure 1, a uniform load Wy (lbs/in) applied to an arch (like a Roman arch). For an arbitrary shape of the arch, loads will develop in the arch as shown in the Figure 1 insert: an axial force N (lbs), a shear force V (lbs), and a bending moment M (lb-in).  The bending moment M causes large stresses in the structure; minimizing M will reduce stresses in the frame.

It is well known that there is a shape for this arch that will minimize the bending moment M in the arch.  This shape is a parabolic shape involving L and h (see Figure 1) and such a shape will cause the bending moment M (and shear V) to go to zero, thus minimizing stresses in the arch and requiring the arch to only carry, and very efficiently only carry, the axial load N. Such a loading situation (axial load only) will minimize the forces/stresses in the structure.

For the double loading shown in Figure 2 (see description in Figure 2), is there a shape for that structure that will also minimize M? Yes, there is. This structure and loading can represent a strung tennis racquet, with Wx, Wy representing cross string and main string loading, respectively.  Although not well known, but now documented in a recent patent application, M & V will go to zero if the shape of the structure is a mathematical ellipse (minor axis a, major axis b, see Figure 2), and the Wx, Wy string loading is not arbitrary but chosen per Equation 1 in Figure 2. If there are Nx\Ny equally spaced cross\main strings, respectively, then, for a specified Ty main string tension, the cross string tension Tx is given by Equation 2 of Figure 2. Stringing the tennis racquet frame based on the tension formula of Equation 2 of Figure 2, will minimize the stresses (M and V will go to zero) for the tennis frame. 

Figure 3 shows the bending moment that can developed in a strung frame. These results are based on a finite element (FE) model of a Wilson tennis racquet that is discussed in Lectures 2 and 3. Here the tensions chosen are based on the formula presented in Figure 2. In this case 60 lbs was chosen as the main string tension, and the cross string tension, per Equation 1 of Figure 2, was determined to be 40 lbs. For an elliptical racquet with equal string spacing, the tensions determined by Equation 1 and 2 of Figure 2 would produce a bending moment M close to zero. In this case, because of non-equal spacing of the Wilson frame (FE model is shown in Figure 3), the bending moment is not zero but varies with position along the racquet (measure by S shown in Figure 3, where S=0 is at 12 noon, and S=18 inches is at 6 o-clock). Note that the maximum bending moment is around 120 in-lbs and occurs at about 3 o-clock (S=9 inches). This bending moment is significantly less than the failure bending moment of 372 in-lbs discussed in Lectures 2 and 3. Also note that there is no discussion here of how this racquet was strung. It is assumed, for this discussion, that the strings are “dropped in” and their final in-racquet tensions are 60 lbs for the main and 40 lbs for the cross. While this final strung condition of a tennis racquet is difficult or even impossible to achieve in a practical way using a stringing machine, it does offer the reader an understanding of the minimal load/stress level that one might achieve.

In conclusion, Dr. Zarda, Orlando and Sanford, notes another notable observation of Figure 3 is that the bending moment is not zero as predicted by the discussion of Figure 2. This is caused by the fact that the shape of the racquet frame head is not a perfect ellipse and because the string spacing is not uniform. Even with the conditions implied in Figure 2 not being met by the racquet shown in Figure 3, the bending moment of Figure 3 is likely at its lowest level by using Equation 2 of Figure 2.

The next and final lecture will look at loads/stresses that are caused using an actual stringing machine.

Lecture 3 - Tennis Racquet Stresses Caused by Stringing

Dr. Paul Zarda of Orlando and Sanford Florida notes:  This is Lecture 3 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 developed a basic understanding of two different kinds of loads or stresses that a tennis racquet will carry when it is being strung.

Lecture 2 reviewed typical loads that a tennis racquet can withstand. It also reviewed actual Instron failure loads of some typical racquet frames.

This lecture, Lecture 3, shows an overview of a finite element model of a tennis racquet. A finite element model is a simulation technique that can be used to assess the loads a tennis racquet might see and whether or not those loads might cause the tennis racquet to fail. The loads that are predicted with the finite element model can be compared to the actual failure loads discussed in Lecture 2 in order to assess if the tennis racquet might withstand a specific loading.

Dr. Paul Zarda of Orlando and Sanford notes that Figure 1 shows a typical tennis racquet that has been strung in the top left image. This is a Wilson tennis racquet that has a stringing pattern of 16 x 19, which means it has 16 main strings and 19 cross strings. The bottom right image of Figure 1 is a finite element model of that very tennis racquet. A description of finite element techniques can be found on the internet and will only be discussed here in an overview way.

Dr. Zarda of Orlando points out that Figure 2 shows more details of the racquet where both the frame and the strings are modeled as beam elements. A beam element is a work horse finite element that can accurately model structures that are beam like: a beam-like structure is a structure that is long in one direction compare to the other two (8 to 1 or more). The tennis racquet meets those conditions. Loads can be applied to this finite element beam model and the stresses in the tennis frame can be determined.

Figure 3 shows two different types of loads that were applied to the tennis racquet: a load in the 3-9 direction forcing the two sides of the frame together, and a load in the 12-6 direction forcing the top of the racquet toward the yoke. These loads can be both applied to the finite element model as well as to the actual Wilson tennis racquet. The resulting deflections for both of these loading conditions (test and FE simulation) can be compared. If they are in disagreement, attributes (for example, x-sectional properties) of the FE model (the simulation model) can be adjusted to correlate the simulation and test.

Figure 4 shows the correlated results of that simulation for the two loading conditions presented in Figure 3. The loading condition for the 3-9 load case is presented here. The deformed shape of the loading is exaggerated in the left image of Figure 4, and the un-exaggerated deformed shape of the simulation is shown in the image at the right of Figure 4. These results match the tested racquet.

This “tuned” FE model can now be used (future lectures) to predict the loads/stresses in the tennis racquet frame when the racquet is loaded during a typical stringing process.

Paul Zarda Diligently Serves All His Duties And Responsibilities

Paul Zarda is a highly accomplished professional known for his skills and expertise. Based in Orlando, he has been diligently fulfilling all of his duties and responsibilities. He is serving as the Manager of IMD System (Interactive Missile Design System). This system will serve as the future backbone for the conceptual and preliminary design, in Mechanical and System's Engineering, of Missile Design Systems. It will directly support Product and Process Centers, DTC, AM**3, and Common Seeker. It will also be presented to Lockheed Martin as a standard for the Corporation. IMD has contributed to $3.5 million dollars in savings (reported to the LM21 initiative).

In addition, Paul Zarda is responsible for the initial collaboration with UCF and FAMU in the area of IMD and IGD. This effort has lead to corresponding undergraduate and graduate courses and research. In his career span, he worked for many leading organizations. He is a former Chairman of the RaDEO Guidance Team, which received a very successful review by DARPA and WPAFB. This program has been on budget and on schedule.

Paul Zarda has the background, experience and leadership that has helped mature an Engineering Methods Group that is un-equaled throughout the corporation. This group is known both for their leadership and expertise in performing multi-disciplinary analyses and has significantly impacted analytical methods used throughout Lockheed Martin for the last twenty years. When he is not busy, he likes to enjoy and celebrate his family, including his wife Adrienne, their three children and their five grandchildren (and still counting); playing tennis as well as the design and analysis of tennis racquets; wood working in his home shop; and his recent association with 30,000 “alligator friends” at his home on Orlando’s Lake Jesup, Florida’s “highest density of gators” lake.

Paul Zarda Has Been Working With Lockheed Martin For Over Three Decades

Paul Zarda is an Orlando based seasoned professional with significant years of experience under his belt. At Lockheed Martin, he is the Manager of Engineering Methods Group and Distinguished Member Technical Staff. His responsibilities at Lockheed Martin Missiles and Fire Control provides direct support of the engineering community in the areas of Mechanical Design and Analysis, System’s Engineering, and Optical Design. He has over 30 years of experience in numerical analysis techniques both in the area of finite element and finite difference algorithms.

Paul Zarda is an expert in finite element techniques applied to stress and dynamic analyses within Lockheed Martin. His work has led to the development of various tools that increase an engineer’s productivity. He has taught over 20 courses in the CES program at Lockheed Martin as well as approximately 15 courses to over 300 students in UCF’s undergraduate and graduate program where he is Research Professor of Engineering in the Mechanical, Materials and Aerospace Engineering Department. He has been an invited speaker at many functions, including AIAA, ASME, University of Florida and the University of Central Florida (UCF) functions.

In addition to this, Paul Zarda also reviewed engineering processes at Lockheed’s Missile and Fire Control Facility in the area of conceptual and preliminary design of missile and fire-control systems, and this lead to a developed architecture and methodology to capture that knowledge. Today the Engineering Methods Group offers the IMD (Interactive Missile Design) and the IGD (Interactive Gimbal Design) systems for conceptual missile and fire control designs, respectively.

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 understand Racquet Stress

 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 Loaded Yardstick

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.

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.