1. Utilizing Beam Theory to look at warfare problems from the 15th
Cannon Vs. Wall
Utilizing Beam Theory to look at warfare problems from the 15th
Century

2. Early Cannon vs Trebuchet
1st decisive use of cannon was at Constantinople around 1450.
Cannon and Trebuchets were used congruently for several generations afterward.

3. Postulation
Historian Steven Turnbull postulates that early cannon were used because they could fire at lower angles causing greater overall damage to the walls.
So I decieded to use basic beam theory to model the effect of the angle of impact and wall thickness looking at the flexural stress on a beam.

4. Math First I modeled the trajectory of cannonball.
Simple ODE model F=MA Calculate using Euler’s method.

5. equations In x direction ax=-b*vx^2/M In y direction
ay=-Mg – (b*vy^2)/M
B is air resistance
b=MG/terminal velocity^2

6. Variables Angle of fire. Initial velocity. Height above the target.
Terminal Velocity(used to calculate air resistance).
Height and width of wall.

7. Force of Impact in X dir vs angle of fire.

8. Total Force Vs Angle of fire
Max total force is at Pi/4 or 45 degrees.

9. Beam Theory
Two kinds of stress, basic sheer stress caused by the x component of the force divided by the average area. Sigma=Fx/base width*cross section.
Second is Flexural stress. This is given by the equation Sigma= M*y/I.
Where M is the moment arm y is distance of moment from the fixed reaction point and I is the beams moment of inertia.

10. Moment of Inertia of a Beam
Beam moment of inertia is given by the equation base^3*height/12.
Assuming the cannonball hits the structure ¾ of the way from the base the moment lever increases the force.
Also stress put on the wall is relative to the size of the base^3. As Midevil walls were often only 1/8 as thick as they were tall.

11. Base Thickness vs Shear and Flexural Stress
Shear Flex
The Flexural Stress dominates for a thin wall

12. Ratio of wall Height/base vs flexural stress

13. Flexural Stress vs angle of fire for typical Medieval wall H/B =6.5

14. Flexural stress H=10 base from
Flexural stress H=10 base from .5 to 10 at 45 and 20 degree angle of fire

15. Result
Ability to fire with and angle of 20 degrees can nearly double applied stress from an angle of 45 degrees.

16. Fortresses