What force needs to be considered for finding the strain in body?

Question

I know that for a masless body, the tension forces are equal and opposite on any element in the body, but I was wondering what forces should we consider if the tension forces on any element in the body are not equal, for example consider a rod of mass $M$ hanging in the vertical plane and hinged at one of the ends, let the length of the rod be L, now consider length of dx at distance of $L/3$ its mass is given by $\lambda dx$ where $\lambda$is the linear mass denisty,

The forces on the element are

1. Tension force due to the part of the rod above the element, in the upward direction

2. The weight of the element in downwards direction

3. Tension force due to the part of the rod below it, in the downward direction

We know that the area of cross section is uniform throughout, so which forces are needed to be considered for finding stress ? And for finding the change in the length of the element , knowing the Young's Modulus, what should be considered the original length of the element ? Here, strain = $\delta$÷$dx$, the $\delta$ x can be found then after we have calculated the stress, representing the change in element's length.

Answer 1

We know that the area of cross section is uniform throughout, so which forces are needed to be considered for finding stress ?

For a vertically hanging rod of Mass $M$, length $L$ and uniform cross section area $A$, that is not supporting any load other than its own weight, the force (and stress) is a maximum at the top and equals $Mg$ and zero bottom of the bar. So the engineering tensile stress $\sigma$ at any distance $x$ along the length of the the bar is

$$\sigma = \frac{F}{A}=\frac{Mgx/L}{A}$$

Where $x=0$ at the bottom of the rod and $x=L$ at the top of the bar.

From there you can calculate engineering strain using Young's Modulus, the assumption of linear elastic behavior. Constant cross sectional area neglects any poisson effect.

Hope this helps.