PhanthomJay said:
Since the applied force is constant, the internal stress, strain, and force varies linearly from 0 at the free end to a maximum at the applied force end; the average normal strain can then be computed by looking at the internal force at the midpoint of the beam, at x = L/2. What is the axial force at the midpoint of the beam?Abhishekdas said:
PhanthomJay said:
Yes, that's right...engineers don't like to do calculusAbhishekdas said:
...the average stress is [(F/A) + 0]/2 = (F)/(2A). If you look at the entire beam, F = Ma, so a= F/M; now cut a section through the midpoint (a Free body diagram of half the beam)... There is an unknown axial force there , call it P. Using Newton 2 for the free end to mid point part of the beam, where the mass of the beam section is a half of M, then P = (1/2)(M)a, or P = F/2, and stress = F/(2A).yes, indeed, it is total elongation divided by length of barAbhishekdas said:
strain is difficult to visualize...if I asked you what is the average force in the bar, would you agree it is F/2?
Oh, sure. But start from the free end for simplicity, P = Fx/L. P linearly varies with x. Then you get the average force by using the definition of average, which is 1/L times the integral of Fx/L (integrating from 0 to L) , which, from my rusting calculus skills, since F/L is a constant, is (F/L)(L2/2L = F/2, which is precisely the same result you get by adding the forces at the extreme ends and dividing by 2, which works for linearly varying equations only.
The speed of movement can affect the stress on a uniform elastic plank in several ways. As the speed increases, the plank will experience greater forces and accelerations, leading to higher levels of stress. Additionally, faster movement may result in more frequent and rapid changes in direction, causing the plank to bend and stretch more, further increasing stress.
The material of the plank is a crucial factor in determining the level of stress it can withstand during movement. Different materials have varying levels of elasticity, strength, and durability, which can impact how they respond to external forces and movements. A more elastic material may be able to absorb and distribute stress more effectively, while a weaker material may break or deform under the same conditions.
Yes, the shape of the plank can have a significant impact on how stress develops during movement. A plank with a more curved or irregular shape will experience greater bending and stretching forces, leading to higher levels of stress. On the other hand, a plank with a more uniform and flat shape may distribute stress more evenly, reducing the overall level of stress.
The force applied to the plank, such as weight or pressure, directly affects the level of stress it experiences. A higher force will result in a greater amount of stress, while a lower force will lead to less stress. Additionally, the direction and distribution of the force can also impact stress development. For example, a concentrated force in one area may cause more stress than a distributed force across the entire plank.
There are several ways to reduce stress on a moving uniform elastic plank. One approach is to decrease the speed of movement, as slower movements will result in lower levels of stress. Another method is to use materials with higher levels of elasticity and strength, which can better withstand external forces. Additionally, adjusting the shape or distribution of weight on the plank can also help distribute stress more evenly and decrease its overall effect.