Joe Sandoval, HES Vice-Chair, Candidate for the Masters in History
Megan Bell, Candidate for the Bachelors of History
Skyler Biek, Candidate for the Bachelors of Mechanical Engineering
Paul Buck, Candidate for the Bachelors of Mechanical Engineering
Student Alumni Leadership Team:
Samantha Abbott, Bachelors of History, 2012
Akihiko Ohnaka, Bachelors of Mechanical Engineering, 2012
Faculty Leadership Team:
Roger L. Martínez, Ph.D., Assistant Professor of History
Michael Calvisi, Ph.D., Assistant Professor of Mechanical and Aerospace Engineering
1420 Austin Bluffs Parkway, Department of History, Columbine Hall 2046
Colorado Springs, CO 80918 USA. Office Phone: 001-719-255-4070
Email: firstname.lastname@example.org Web: www.uccshes.org
UCCS Historical Engineering Society
The Trebuchet Challenge
Table of Contents Executive Summary 2 Overview of the UCCS Historical Engineering Society (HES) 3
Leadership Team 3
Mission Statement of the HES 3 How the Project Started 3 Goals of the Project 4 History of the Trebuchet 4 Action Plan: A Five Stage Process 6 Work Completed on Stages 1 and 2 7
The Design Review Committee and Safety 9
Current Work: Stage 3
Proposed Construction and Demonstration Budget 10
Future Work: Stages 4 and 5 12
Appendix A: Complete CAD Drawings of the Proposed Trebuchet Appendix B: Final Design and Stress Analysis Overview Appendix C: Letter from the Design Review Committee Appendix D: University Faculty and Administration Letters of Support Appendix E: UCCS Club Registration Information
UCCS Historical Engineering Society:
The Trebuchet Challenge Executive Summary
The UCCS Historical Engineering Society (HES), an official UCCS student-driven organization with faculty advisers, endeavors to build a 20-percent scale model “Conrad Kyeser Trebuchet” (circa 1405 c.e.) as a demonstration project to inform the public on the engineering feats of the medieval world. A trebuchet is a medieval siege weapon, similar to a catapult, which uses a counterweight to launch projectiles to destroy castle walls and fortifications. Trebuchets were the “super” weapons of the Middle Ages, especially from the 12th through 15th centuries. During this time period, trebuchets were developed via cultural exchange between Christian European kingdoms and Islamic caliphates in the Mediterranean world.
The goals of “The Trebuchet Challenge” project are to promote and enhance students’ historical research, design, engineering, and leadership skills. In the interest of fulfilling these goals, and others, we hope to complete construction of the trebuchet during fall 2012. Now in the process of completing a five-stage research-to-demonstration work process, the HES has already completed its historical research, developed detailed computer-aided design (CAD) drawings, and evaluated the safety of the design using a design review panel composed of three UCCS engineering professors.
However, in order to construct and demonstrate the trebuchet (Stages 4 and 5 of the project) we require approximately $3,745 in financial and material donations. Over 50 percent of our budget is dedicated to purchasing an expensive oak beam that will serve as the machine’s 12-foot throwing arm. For historic, strength, and safety reasons, it is absolutely essential that a hardwood such as oak be employed for the throwing arm. At this time, the HES has already raised $2,300 (about 2/3 of the required funds) and is almost ready to begin constructing the machine.
The HES seeks funding from private, small business, organizational, and corporate donors in the Colorado Springs region. Funders are encouraged to contribute at varying sponsorship levels— $250 or more as a sponsor or up to $249 as a donor. Those entities that choose to donate will receive recognition at every event at which the trebuchet is present. Forms of recognition include, but are not limited to, donor name on display at every event, donor name on flyers advertising upcoming events, and donor name on display beside the trebuchet at the UCCS Heller Center for the Humanities, where the trebuchet will be kept when not in use. The two future stages of the project can only be made possible by generous donations from our community and without their help the realization of our goals and efforts will be impossible.
For more information about sponsoring or donating to this ambitious project, please contact Megan Bell, a member of the Student Leadership Team, at email@example.com. Additional information can be found at our website, http://www.uccshes.org or http://uccshes.wordpress.com.
Mr. Putnam is a junior in the Department of Mechanical and Aerospace Engineering. His interest in the project is derived from his previous experience building smaller scale trebuchet models and he believed this particular project would be both an enjoyable and interesting challenge.
Joe Sandoval - Vice Chair
Mr. Sandoval is a graduate school candidate in the Department of History with an interest in the Spanish colonial and medieval European history. His particular interest in the project is related to medieval siege engines.
Megan Bell, Candidate for the Bachelors of History Skyler Biek, Candidate for the Bachelors of Mechanical Engineering Paul Buck, Candidate for the Bachelors of Mechanical Engineering Student Alumni Leadership Team Akihiko Ohnaka, Bachelors of Mechanical Engineering, 2012
Samantha Abbott, Bachelors of History, 2012
Faculty Leadership Team Dr. Roger L. Martinez - Primary Faculty Leader
Dr. Martinez is an Assistant Professor in the Department of History. Trained as a medieval European historian, with an emphasis on the intercultural relations of Jews, Christians, and Muslims in medieval Spain, he is very interested in medieval science and technology.
Dr. Michael Calvisi - Co-Faculty Leader
Dr. Calvisi is an Assistant Professor in the Department of Mechanical and Aerospace Engineering. His research interests are in theoretical and computational fluid dynamics with an emphasis on multi-phase flows and bio-fluid mechanics.
Mission Statement of the HES
The Historical Engineering Society’s mission is to promote living history, to research historical engineering methods, to advance students’ collaboration and research skills, and to foster collaborations with the local community for the purpose of education and outreach.
How the Project Started During Dr. Roger Martinez’s fall 2010 course, HIST 1020: The Medieval World, one of the thematic areas he touched upon in his lectures and readings was the development of medieval weaponry, specifically the trebuchet. To illustrate medieval advancements in science, experimentation, and cross-cultural diffusion, Dr. Martinez presented students with materials from Lynn White, Jr.’s influential text, Medieval Technology and Social Change (1962), as well as showed film clips from NOVA’s Medieval Siege. After discussions with several students, especially Mr. Jeremy Rivera, it became apparent to Dr. Martinez that many students were interested in reconstructing a trebuchet so that they could explore the historical and engineering issues that medieval peoples encountered as they designed and developed these siege engines. Recognizing the pedagogical value of energizing students and faculty around a historical reconstruction project, Dr. Martinez (History) and Dr. Calvisi (MAE) initiated a series of informational meetings (late fall 2010 and winter 2011) to gauge student interest, as well as to develop a process for developing a trebuchet. (See the attached “Proposed Project Process” document.) These initial meetings attracted approximately 15 to 20 students and five faculty members’ interest, and initiated the creation of the UCCS Historical Engineering Society (HES). Since these initial meetings, the HES team (with a core of about eight students) has methodically researched designs and developed a safe design for a reproduction trebuchet.
The Goals of the Project After collecting information on the full extent of student and faculty commitment to the project, the participating faculty, students, and staff established the following primary goals of the project:
To build a historically-accurate model of a medieval trebuchet.
To expose participating students, faculty, and staff to medieval history.
To hone participating students’ historical research and evaluation skills through the study of primary and secondary sources that discuss and depict trebuchet.
To promote participating students’ understanding of the nature of cross-cultural diffusion of technology during the medieval era.
To allow participating students and faculty to explore medieval engineering and construction techniques.
To foster participating students’ leadership and long-term planning and development skills.
To provide participating students with an opportunity to enhance their writing and speaking skills via the preparation and delivery of written work products and oral presentations.
To promote university camaraderie through the development and exposition of a working trebuchet.
Collectively, we believe these goals will enhance students’ intellectual and professional training, while offering all participants a unique opportunity to explore medieval history in a very tangible manner.
History of the Trebuchet The trebuchet, a mechanical device used to hurl large stones to destroy castle walls and fortifications, were the “super” weapons of the Middle Ages. During this time period, trebuchets were developed via cultural exchange between Christian European kingdoms and Islamic caliphates in the Mediterranean world. As such, trebuchets evolved from human-powered, or “traction”, devices to gravity-assisted, or “counterweight”, devices. The HES is currently planning to build a counterweight trebuchet.
The earliest visual depiction of a counterweight trebuchet, in fact, is found in a medieval Islamic manuscript authored by Mardi al-Tarsusi, who will be discussed shortly. This manuscript presents a counterweight trebuchet that was conceptualized in 1187 c.e.
Mardi al-Tarsusi’s Counterweight Trebuchet, circa 1187 c.e.
According to John Norris, the first recorded use of the counterweight trebuchet came at Cremona, Italy, in 1199.1 The advantages of the trebuchet, when compared to a normal torsion catapult (the weapon of choice during the Roman Era), were staggering. Some trebuchets were built with 50-foot launching arms and used counterweights of 20,000 pounds, which meant the machine could launch a 300-pound stone up to 300 yards with great accuracy.2
While the counterweight trebuchet appeared in the late twelfth century, the trebuchet’s original date of development is not known. Before the development of the counterweight-powered machine, there was another form of the device, known as the traction trebuchet that relied on momentum created by men to launch projectiles. It is believed that Islamic civilizations were designing, building, and employing traction trebuchet as early as the 690s.3 Whereas the counterweight trebuchet operated off of momentum provided by the dropping of a massive weight, the traction trebuchet used momentum created when a group of men pulled on a set of ropes at the non-launching arm of the machine. The arm would pivot in the center and launch the projectile through the air.
One member of the HES, Mr. Russell Creger, prepared a short-research paper on the history of the trebuchet and in it we learn about the importance of Islamic contributions to the development of these devices. Below, we have adapted the following selection from his paper.
One of the most important medieval sources that discusses the construction of trebuchets is Mardi al-Tarsusi’s Instruction of the Masters on the means of Deliverance in Wars from Disasters, and the Unfurling of the Banners of Information: On Equipment and Engines which Aid in Encounters with Enemies. Mardi al-Tarsusi’s battle “manual” evaluated different kinds of trebuchets that were made by Turks, Franks, and Arabs. Mardi al-Tarsusi prepared the manual for his patron, Saladin, the late 12th century ruler of Egypt and Syria. Saladin, who was a contemporary of King Richard the Lionheart of England, was known to Europeans due to his excellence on the battlefield and his successful re-capture of the city of Jerusalem from the Crusaders in 1187 c.e. Mardi al-Tarsusi’s manual discusses the trebuchet building materials, their design, and how to launch projectiles effectively. On this later point, al-Tarsusi states:
“If the shooter stands directly under the pouch [of the sling], the stone will be very high and [the range] will be short, and it may possible fall on the men [i.e., the pulling crew]. If [the shoorter] moves out from the pouch toward the end of the beam by a distance of one span [of a hand= 22-24 cm], the launch will be farther. The most one should move out from the beam is two spans [44-48 cm], [and] no more, for, if one goes beyond this, the reach is 60 ba [ca. 120 m], and the shortest is 40 ba [ca. 80 m]. Another principle which determines the farness or the shortness of the distance [of the shot] is the flexibility or dryness of the beam. When the beam is flexible, but not excessively so, it has a farther range and is more effective. When it is dry, it is less so. The shooter should have his feet wide apart, grasp the pouch with his hands, and sit down while he pulls the pouch each time. The best and most proper wood to make the beam is cherry woods. If there is none of this kind, it must be of a closely-knotted wood of intermediate [quality] such as cedar or the like.”4 Through this remarkable description al-Tarsusi explains how minute variations in trebuchet operations and the actual properties of the wood could effect ways the trajectories of projectiles.
The counterweight trebuchet was used well into the 16th century and reported was used by Hernando Cortez in Mexico.5 The trebuchet served to connect siege weaponry from the fall of the Roman Empire and the use of torsion catapults, to the 16th century use of black powder artillery weapons.
Action Plan: A Five-Stage Process At the beginning of the project, the HES developed and began implementing a five-stage work process. The key elements of this work process include:
Stage 1. Historical Research and Seek Corporate/Community Support,
Stage 2. Design and Evaluation of Potential Trebuchet Models,
Stage 3. Secure Financial Resources and Technical Expertise to Begin Scale Construction,
Stage 4. Finish Construction and Safety Testing, and
Stage 5. Promotion of Trebuchet Demonstration and the First Annual Competition.
The following chart depicts the most important tasks for each stage.
Work Completed on Stages 1 and 2
Since the inception of the project, the HES has focused its attention on Stages 1 and 2. During spring 2011 and fall 2011, a core group of student leaders including Mr. Ben Putnam, Mr. Akihiko Ohnaka, Ms. Samantha Abbott, and Mr. Joe Sandoval, undertook two major tasks in order to realize their endeavor to build a functional reproduction trebuchet. These primary tasks included:
Conducting historical research on different types of medieval trebuchets and selecting a design to replicate, and
Developing initial hand and computer-based concept sketches and drawings of the proposed reproduction trebuchet.
Through our research, the HES decided the most desirable reproduction to build would be based on the trebuchet known as the “Conrad Kyeser of Eichstatt” which is depicted in the fifteenth century manuscript titled, Bellifortis. This manuscript, referenced as Codicil MS. philos. 63, and attributed to the year 1405 c.e., is now held by the Niedersächsische Staats- und Universitätsbibliothek, Göttingen (Germany). On folio 30r of the manuscript, a detailed drawing of the Conrad Kyeser trebuchet is depicted, and presented below.
15th Century Depiction of the Conrad Kyeser of Eichstatt
(hereafter referred to as the “Conrad Kyeser Trebuchet)
The HES selected the Conrad Kyeser Trebuchet because the medieval depictions of the machine including critical structural measurements for the machine. As Paul E. Chevedden noted in his article, “The Invention of the Counterweight Trebuchet: A Study in Cultural Diffusion,” the manuscript reported important, but incomplete information. On these issues, Chevedden stated:
The main beam measures 54 "workfeet," or 15.55 m, with a throwing arm of 46 "workfeet," or 13.248 m. The distance from the axle of the beam to the axle of the hinged counterweight box is 8 "workfeet," or 2.304 m, dividing the beam in the ratio 5.75:1. A ratio of 6:1 is designated for the beam of a trebuchet in the Innsbruck manuscript of Bellifortis (Fig. 4), indicating that the dimensions given here may have been miscalculated. The trestle frame is composed of two linked supporting trusses, each forming an equilateral triangle with base and sides measuring 46 "workfeet," or 13.248 m. The main axle is placed at the apex of the trusses 11.47 m above the ground.6 With the benefit of understanding the dimensions of key machine members in relationship to each other, specifically the throwing arm, main beam, and frame, the HES believed it could reverse engineer the entire design of the Conrad Kyeser Trebuchet.
After completing its initial historical research tasks, this spring 2012 the team aggressively pursued the design process for the building the trebuchet. These key efforts include:
Converting the initial concept sketches into detailed computer-assisted design (CAD) drawings and identifying the building materials for the trebuchet.
Conducting engineering stress analyses to ensure the trebuchet would function in a safe manner, and
Securing a secure work site for constructing the trebuchet.
Working as a design team led by Mr. Putnam and Mr. Ohnaka (now a graduate of UCCS), the HES labored to develop and consider all of the major components of the reproduction trebuchet. Mr. Putnam crated and revised all of our intricate CAD drawings and Mr. Ohnaka conducted computer-based stress analyses of the designs. Mr. Sandoval formed relationships with local lumber mills so as to determine who might provide the best materials for constructing the trebuchet, as well as locating a securing location on campus to build the device. The final HES design is depicted below. Complete CAD drawings and the stress analysis can be found in Appendices A and B, respectively.
Final HES Trebuchet Design
The key measurements of the reproduction trebuchet, which is a 20% scale design of the original Conrad Kyeser Trebuchet, are:
the throwing arm is 12 feet in length,
the A-frame trestle support is 7 feet in height, and
the counterweight box is can accommodate between 200 and 400 pounds of weight.
It is anticipated that our reproduction trebuchet will initially launch five-pound projectiles.
The Design Review Committee and Safety Although the HES will be conducting a major safety review in Stage 4, before operating the fully-constructed trebuchet, during the research and design phases we also brought our work before a Design Review Committee. In order to validate the quality and safety of the trebuchet’s design, on March 9, 2012, Mr. Putnam and Mr. Ohnaka presented their work to a Design Review Committee consisting of Dr. Peter Gorder, Dr. Steven Tregresser, and Dr. Michael Calvisi of the Department of Mechanical and Aerospace Engineering. During this review, the panel first evaluated the CAD drawings and found several shortcomings of the design, which required the HES to make modifications to its trebuchet plans.
On April 6, 2012, the Design Review Committee re-convened to review the design changes implemented by Mr. Putnam and Mr. Ohnaka. On the whole, the committee found that the HES have resolved those design and safety issues they had previously noted. Overall, the Design Review Committee concluded that the HES has developed a solid and safe design for a prototype trebuchet, albeit there remain a few important issues that should be resolved prior to its construction. (See attached letter from the Design Review Committee in Appendix C.)
Current Work: Stage 3
Proposed Construction and Demonstration Budget After a careful review of the design specifics, the materials needed, and the estimated material expenses, the HES anticipates it will need approximately $4,000 to construct the reproduction Conrad Kyeser Trebuchet.
The following budget includes our current list of building materials and their associated prices.
Trebuchet Wood and Steel Components A-frame (Douglas Fir Pine)
Throwing Arm (White Oak) 1- 4 X 6 X 144 inches (Approximately a $2,000 expense)
Strength Plate (Steel) 2- 4 X 54 X 0.125 inches
Other Elements (Douglas Fir Pine)
Projectile Slide 1- 3 X 36 X 144 inches
Rigid Support Beam 1- 4 X 6 X 120 inches
--Continues on the following pages--
Counterweight Box (Douglas Fir Pine)
Bottom 1- 18 X 18 X 0.75 inches
Left/Right sides 2- 15 X 18 X 0.75 inches
Front/Back 2- 14.25 X 18 X 0.75 inches
Total Wood and Steel Budget = $2,500
16 L-Brackets $2.57 each
3-boxes of 50 pan head wood screws $12.57 each
6-boxes of 5 8inch hex head bolts $11.14 each
1/ box of 100 ¼ 20 hex nuts $3
12 square feet of sheet metal (steel) $130
Long axle $110
Short axle $40
6-Shaft Clamps $11.00 each
Metal Pulley $35
Custom-fabricated quick release $250
Total Hardware Budget = $750
Rope (sling material and safety items) $250
Wood Sealant $90
Lubricating Oil (olive oil) $30
Projectiles (biodegradable) $40
Plastic Tarps (storage) $25
Total Other Materials Budget = $435
Eye protection $20
Total Safety Budget = $60
Grand Total Budget = $3,745
Fundraising The HES intends to secure most of its funding from private, small business, organizational, and corporate donors in the Colorado Springs region. We intend to target a wide range of funders, including: construction and engineering companies, building supply companies, local financial institutions, and defense contractors.
Funders are encouraged to contribute at varying levels— $250 or more as a sponsor or up to $249 as a donor. Sponsors will be acknowledged on a special plaque that accompanies the trebuchet and all sponsors/donors will be noted on our promotion materials (website, publications, and notices).
Details Regarding Contributions
All funds collected will be deposited in the HES’ university account (speed type account # 42000064) and therefore will be properly overseen by the university.
Checks can be made payable to the The University of Colorado – Colorado Springs.
All sponsors and donors will receive a letter from UCCS indicating the amount and date of their donation.
Future Work: Stages 4 and 5
During fall 2012-spring 2012, the HES will move on to Stage 4 (Finish Construction and Safety Testing), and Stage 5 (Promotion of Trebuchet Demonstration and the First Annual Competition).
Some of the key items that we will need to review during these stages are:
Use of modern hardware as safety precaution
Secure construction location
Safety checklist for construction
Secure and appropriate locations for demonstration with spectator seating separated from demonstration field by ropes
Dry run initial launch to ensure proper functioning
Safety checklist for proper launch
Biodegradable launch materials to ensure environmental responsibility
Post-launch clean-up of launch materials and spectator seating area
Checklist for routine maintenance post launch and while on display at (storage location)
Appendix A Detailed CAD Drawings of
the Proposed Trebuchet Appendix B UCCS Historical Engineering Society Final Design and Stress Analysis Overview March 2, 2012 Akihiko Ohnaka
The objective of this report is to review the design of the trebuchet model from an engineering standpoint. This study analyzes the dimensions and material selection using the COMSOL multiphisics solid mechanics module. The design is validated to ensure that it has enough strength for the task with the given conditions and suggests changes where necessary. The assumption made for this study is that the counterweight is 400lbs and the projectile weighs 5lbs. For the actual design, a counterweight of 200lbs will be used initially but 400lbs may be used in future for extended range.
II. Trebuchet Design Figure 1: Throwing arm dimensions
The overall design is 20% of full scale which requires the frame height to be 7’and the throwing arm to be 12’ long. The design with regard to the throwing arm’s strength is extremely conservative. Some additional tapering from both sides may be required to improve performance. The material chosen to be used for the construction is white oak for the throwing arm and Douglas fir for the frame.
III. Stress Analysis and Findings
Figure 1.1: Stress analysis for the entire trebuchet (static configuration)
Throwing Arm There are two major parts of stress analysis for the throwing arm. The first stress analysis is the static stress. This analysis shows how much stress the throwing arm experiences when the arm is in the initial resting position with the counterweight. This is compared to the yield stress, which is the maximum stress the material can endure before it permanently deforms or fails. The counterweight was assumed to be 400lbs for all the analysis, even though 200lbs will be used initially. The second stress analysis is the dynamic stress analysis. When the arm is rotating fast with the 5-pound projectile, it generates a tension force called “centrifugal force” acting along the sling that pulls the tip of the throwing arm. In addition, the counterweight undergoes centrifugal force that can increase load beyond its weight. The arm must be strong enough to endure not only at rest with the weight, but also support the forces while it is launching.
Edge constraint at the tip of the throwing arm
Roller constraints both ends of the axle (2inches)
Boundary force 400lbs along –y direction
Number of elements 79313
Degree of freedom 358446
Static stress analysis For the static analysis simulation, it is assumed that the counterweight weighs 400lbs, the axle is cast iron, 4 feet long and 1.5 inches in diameter and the throwing arm is oak. The axle is resting on the A-frame horizontally and a trigger system is holding the throwing arm to keep it from rotating. Figures 2 and 2-2 show how much the arm is deformed under this condition. The deformation in the figure is exaggerated 200 times for illustrative purposes. The actual deformation is not that significant. The maximum displacement occurs at the end of the throwing arm where the counterweight is attached. According to the analysis with these boundary conditions, the end actually deforms only 0.114 inches
Figure 2: Maximum deformation of the axle and the tip of the throwing arm with static condition. The deformation in the figure is exaggerated 200 times
Figure 3: Simulated maximum compression, shear, tensile stresses for static condition (counterweight = 400lbs)
For the static condition, the throwing arm experiences three kinds of stress. The first stress occurs due to forces squeezing the material, called “compression”. Since the throwing arm will be bending in an “upside down U shape” after the counterweight is attached, the material under the throwing arm (inside the “U shape”) will be compressed. The second stress is caused by forces acting parallel to the surface of the material. This is called “shear”. The shear stress also causes strain. The third stress comes from stretching of the material, called “tensile” stress. Figure 3 shows where the maximum stress occurs for these stresses in the simulation.
Since the tension, compression and shear stresses are unable to be calculated separately with COMSOL, von Mises stress was chosen to be measured. It was assumed that von Mises stress was purely tensile stress at the top, compression at the bottom, and shear in the middle.
Table 1: Comparison of compression, shear and tensile stress value vs. simulated static stresses
Table 1 compares the stress properties of three different materials to the simulation. The factor of safety is the ratio of the maximum allowable to the maximum simulated stress. For example the compression factor of safety for white oak is 41.8MPa/1.18MPa = 35.4. Considering buildings commonly use a factor of safety of 2 (Wikipedia) for each structural material, it is apparent that this design is very conservative from a stress standpoint. Even the lowest factor of safety is more than 35.4, using white oak and assuming that the location where the stress is measured is the location is where the material is being compressed.
Since this source did not specifically tell whether these stress values were yield stresses or ultimate, comparisons with other sources may be necessary. Shear stress may have a component which acts perpendicular to the grain direction, however, the source does not list the values.
Figure 3-2: Simulated static stress on the axle (cast iron diameter 1.5inch) due to 400lbs counterweight(arrows indicate the locations of the measured points)
Figure 3-2 shows the stress that the axle experiences when the 400lb counterweight is attached to the throwing arm. It is assumed that there is some trigger system preventing the tip of the arm from rotating the projectile. The maximum stress on the axle under this condition is 47.5MPa. Considering that cast iron’s yield strength is 310MPa, the factor of safety is 6.5 which indicates that it is a safe design with this diameter.
Figure 4: Schematic to visualize how the dynamic force is being applied
Dynamic stress analysis Figure 4 shows the forces that exist when the projectile is fired. When the object moves on a curved path, like a projectile which is fired by a trebuchet, there is a force called “centripetal force” (blue arrow) that acts on the object to maintain its path. As Figure 4 shows, the centripetal force acting on the projectile causes a reaction force called “centrifugal force” (red arrow) that pulls the tip of the throwing arm in that direction. Also, there is another centrifugal force generated by the counter weight acting on the bottom counter weight hole downward. Dynamic stress analysis determines if the material is strong enough to stand up to this centrifugal force with an assumption of a 5-pound projectile and 400lbs counterweight. The velocity of the projectile at the release point is estimated at 63ft/s from previous studies. Centrifugal force due to the projectile was calculated at 88lbs/394N. It was assumed that the projectile is released when the throwing arm is in the vertical potion and the sling is at 45 degrees from the horizontal line. Figure 4 shows the schematic of the release point. Centrifugal force due to the counterweight was calculated using equation below:
m is the mass of the counter weight 13slags=400lbs/32.2ft/s^2
r is the length between the axle hole and where the counterweight is attached and
is the angular velocity of the throwing arm at the releasing point 3.30rad/s from numerical data using the Lagrange equation
Using the values above, the estimated centripetal force becomes
F=566lbs/4297N which was added on to the 400lb counterweight itself.
Thus total force pulling downward at the releasing moment becomes
Mesh type free-triangular
Number of elements 79467
Degree of freedom359079
Roller constraints for both sides (4inches)
Boundary loads x-direction 4297N
Edge load -278.6N in y-direction 278.6N in x-direction (sqrt(278.6^2+ 278.6^2)=394N))
Figure 5: Exaggerated (scale 10) deformation due to reaction to centripetal force
Figure 5 shows that maximum deflection (1.79 inches at the tip of the arm) due to dynamic forces under these boundary conditions.
Figure 6: Simulated von Mises stresses for dynamic condition (Counter weight =400lbs projectile weight = 5lbs)
Table 2: Comparison of compression, shear and tensile stress value vs. simulated dynamic stresses 
Simulated Maximum stress
(von mises stress)
Douglas fir coast
white oak bur
Compression parallel to grain
Shear parallel to grain
Tension parallel to grain (over cup)
Factor of safety
With respect to:
For the dynamic analysis, the boundary condition was more complicated than for the stress analysis. Therefore, the maximum von Mises stress was compared to each stress property. The maximum stress point was near the axle hole. The lowest factor of safety was 1.22 with respect to shear stress for Douglas fir.
A-frame Stress Analysis For this analysis, it was assumed that a total weight of 523lbs (400lbs for the counterweight and 123 lbs for the throwing arm) is applied to 2 grooved surfaces just as the axle rests on the frame. Compared to both the dynamic and static stresses on the throwing arm, the stresses and deformation that the A-frame experiences are geometrically smaller. Maximum stress is 0.00224MPa and maximum deflection is 8.45e-6inches.
Figure 7: Simulated maximum stress for the A-frame. A total of 533lbs was applied on the grooved surfaces (123lbs for the throwing arm and 400lbs for the counterweight)
Figure 8: Maximum deformation of the A-frame due to the simulated weight (523lbs)
Figures 7 and 8 shows the maximum von Mises stresses and deformation due to force from counterweight and the throwing arm’s weight (total523lbs). Compared from the static stress analysis, they are significantly small, therefore, the design can be concluded to be a safe design.
III. Conclusions Given the counterweight was assumed to be 400lbs which was double the initial wight(200lbs) and that the static factors of safety for the throwing arm are consistently beyond 30 for each property (tensile, compression and shear), the design for the throwing arm is concluded to be a safe design. The axle rod which was assumed to be cast iron has a factor of safety of 6.5 and maximum deformation of 0.035inches. For dynamic analysis, factor of safety was calculated each stresses (shear, compression, and tensile) against the maximum von Mises stress which is a combination on three stresses at the release moment. As consequence, they were significantly smaller than static condition (1.22to17 ). Especially, for shear was almost one (1.22 with Douglas fir). This study does not, however, take into account construction failure (fastener selection or assembly) thus; it may require another stress analysis for the frame construction once the final construction methods are determined.
David W. Green, Jerrold E. Winandy and David E. Kretschmann, Mechanical Properties of Wood. https://webmail.uccs.edu/Session/427514-Dopdg0jt15dCDbcvXB8B/MessagePart/INBOX/2318-02-B/MECHANICAL%20PROPERTIES%20OF%20WOOD%20_USFS.pdf
Timothy A. Philpot. Mechanics of Materials.
Appendix C Letter from Design Review Committee
Appendix D University Faculty and Administration Letters of Support Appendix E UCCS Club Registration Information
1 John Norris, Medieval Siege Warfare (Stroud: Tempus, 2006), p. 199.
2 John Norris, Medieval Siege Warfare (Stroud: Tempus, 2006), p. 199.
3 Christopher Gravett, Medieval Siege Warfare (Oxford : Osprey, 2002), p. 49.
4 W.T.S. Tarver , W.T.S, "The Traction Trebuchet: A Reconstruction of an Early Medieval Siege Engine," Technology and Culture Vol. 36, no. (1995), p. 148.
5 John Norris, Medieval Siege Warfare (Stroud: Tempus, 2006), p. 206.
6 Chevedden, Paul E. “The Invention of the Counterweight Trebuchet: A Study in Cultural Diffusion,” Dumbarton Oaks Papers, Vol. 54 (2000), pp. 74-75.