How to understand formula for bending of a rectangular rod?

[Lotto]

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:

 

[Baluncore]

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.

 

[erobz]

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.

 

[kuruman]

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.

 

[Andy Resnick]

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?