Direct stress and elongation

[Modern Algebr]

A prismatic bar AB of length L, cross-sectional area A ,modulus of elasticity E ,and weight W hangs vertically under its own weight
a>derive a formula for the downward displacement d(c) of point C which is located at a distance l from the lower end of the bar

The answer is W(L^2-l^2)/(2EA)

how do we get it?

Thanks for any hints or answers!

 

[aerospaceut10]

Well, so delta = PL/EASo P varies along the y direction...P = W*y/L, right?

and if we're doing incremental displacement changes, the L is your dy

d delta = integral W*y/LEA * dx

so you integrate from your l to L. Shouldn't your answer also have an L term in the denominator?

 

[oswaler]

To derive the formula for the downward displacement of point C, we can use the concept of direct stress and elongation. Direct stress is the force per unit area acting on a material, while elongation is the increase in length of a material under stress.

First, let's consider the forces acting on the prismatic bar AB. The weight W of the bar is acting downward, while the reaction force R at the support point A is acting upward. Since the bar is in equilibrium, we can say that the sum of these forces is equal to zero:

W + R = 0

Since the bar is hanging vertically, we can also say that R is equal to the weight W of the bar:

R = W

Next, let's consider the elongation of the bar. We know that elongation is directly proportional to the applied stress and the original length of the material. In this case, the applied stress is the weight W and the original length is the length of the bar L. Therefore, we can say that the elongation of the bar is given by:

ΔL = WL/EA

We can further break down the elongation into two components: the elongation of the entire bar (ΔL) and the elongation of the portion of the bar below point C (Δl). We can express this relationship as:

ΔL = Δl + (L-l)

Substituting this into our equation for elongation, we get:

WL/EA = Δl + (L-l)

Solving for Δl, we get:

Δl = WL/EA - (L-l)

Since Δl is the elongation of the portion of the bar below point C, we can say that it is also equal to the downward displacement of point C, which is what we are trying to find. Therefore, we can rewrite the equation as:

d(c) = WL/EA - (L-l)

Simplifying, we get:

d(c) = W(L-l)/EA

This is the formula for the downward displacement of point C. To get the final answer, we can substitute the values for the weight W, length L, distance from the lower end of the bar l, cross-sectional area A, and modulus of elasticity E into the equation. This will give us the final answer of:

d(c) = W(L^2-l^2)/(2EA)

I hope this explanation helps in understanding how we can