How to understand formula for bending of a rectangular rod?

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Discussion Overview

The discussion revolves around understanding the formula for calculating Young's modulus in the context of bending a rectangular rod. Participants explore the implications of applying additional forces to a rod that is already bent, considering the principles of superposition and the relationship between force, deflection, and material properties.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants note that deflection is linear with force, suggesting that each force causes a proportional deflection as long as the deflection remains small.
  • One participant presents a formula for Young's modulus, ##E =\frac{F l^3}{4yab^3}##, and questions whether to use the total force and total deflection or just the additional force and change in deflection when calculating ##E## after applying an additional force to an already bent rod.
  • Another participant suggests that superposition can be used if the deflections are not too large, implying a method for calculating the effects of multiple forces.
  • A participant raises a hypothetical scenario to reason through the problem, questioning how to account for the rod's shape when no load is applied and how to incorporate the effects of both forces when they are applied.
  • A later reply seeks clarification on whether the forces are applied at the same location, indicating a consideration of the mechanics involved in the bending process.

Areas of Agreement / Disagreement

Participants express differing views on how to approach the calculation of Young's modulus when multiple forces are involved, indicating that the discussion remains unresolved with multiple competing perspectives on the application of the formula.

Contextual Notes

Limitations include assumptions about the linearity of deflection, the applicability of superposition, and the potential need to consider the rod's weight in the calculations.

Lotto
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TL;DR
If we have a rod shown on the picture below, we can calculate Young's modulus as ##E =\frac{F l^3}{4yab^3}##, where ##a## is a width and ##b## is a height of the rod.

Now my question is: If the rod is already bended with a certain ##y## as on the picture and if we apply an another additional force ##F'## and the total bend would be ##y'##, can we use in the formula to calculate ##E## only ##F'## and ##y'-y##? Or do we have to use ##F+F'## and ##y'##?
The picture:
picture.png
 
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Notice that the deflection is linear with force.
Each force causes a proportional deflection.
So long as the deflection is small, the order of application is not critical.
 
Lotto said:
TL;DR Summary: If we have a rod shown on the picture below, we can calculate Young's modulus as ##E =\frac{F l^3}{4yab^3}##, where ##a## is a width and ##b## is a height of the rod.

Now my question is: If the rod is already bended with a certain ##y## as on the picture and if we apply an another additional force ##F'## and the total bend would be ##y'##, can we use in the formula to calculate ##E## only ##F'## and ##y'-y##? Or do we have to use ##F+F'## and ##y'##?

The picture:
View attachment 358226
You can use superposition if the deflections are not too large.
 
You can reson this one out. Suppose that there is not load, i.e. both ##F## and ##F'## are zero. What does the rod look like? Now add ##F## and ##F'##. How would you calculate the height? You may or may not have to ignore the bending of the rod under its weight.
 
Lotto said:
Now my question is: If the rod is already bended with a certain ##y## as on the picture and if we apply an another additional force ##F'## and the total bend would be ##y'##, can we use in the formula to calculate ##E## only ##F'## and ##y'-y##? Or do we have to use ##F+F'## and ##y'##?
Quick question- are F and F' applied at the same location?
 
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