Elastoplasticity - see picture please

[Xand888]

Homework Statement

The stress-strain curve of the steel bars may be idealized as elastoplastic with yield stress σy. Each bar has cross sectional area A.
Determine the yield load PY and the plastic load Pp.

Homework Equations

σ = F/A

The Attempt at a Solution

See picture with my attempt

- I think I need to relate the angle theta to the elongation, but I'm not sure...
Help would be greatly appreciated! (First post)

 

[KataKoniK]

Thank you for your post. I understand that you are trying to determine the yield load and plastic load for steel bars with a stress-strain curve that can be idealized as elastoplastic. I can offer some guidance to help you solve this problem.

First, let's define the yield stress, σy, as the stress at which the material begins to exhibit plastic deformation. This means that the stress-strain curve transitions from a linear elastic region to a nonlinear plastic region. In this case, the yield stress is also equal to the plastic limit stress, σpl, which is the maximum stress the material can withstand before permanent deformation occurs.

To determine the yield load, PY, we can use the equation σ = F/A, where σ is the stress, F is the applied force, and A is the cross-sectional area. In this case, we can rearrange the equation to solve for F, which gives us F = σy * A. Therefore, the yield load is equal to the yield stress multiplied by the cross-sectional area of the steel bar.

To determine the plastic load, Pp, we need to find the point on the stress-strain curve where the material reaches its plastic limit stress, σpl. This can be done by finding the intersection of the stress-strain curve with a horizontal line at σpl. Once we have this point, we can use the same equation as before, F = σ * A, to determine the plastic load.

I hope this helps you solve the problem. If you need further assistance, please don't hesitate to ask. Good luck!

 

[Mene]

I can provide some insight into the concept of elastoplasticity and how it relates to the problem at hand. Elastoplasticity is a mechanical property of materials that describes their ability to deform elastically (i.e. return to their original shape) up to a certain point, after which they undergo plastic deformation (i.e. permanent deformation). This is seen in the stress-strain curve, where the initial linear portion represents elastic deformation and the subsequent nonlinear portion represents plastic deformation.

In this problem, we are dealing with steel bars that have a yield stress (σy) and a cross-sectional area (A). To determine the yield load (PY) and plastic load (Pp), we can use the equation σ = F/A, where σ is the stress, F is the applied load, and A is the cross-sectional area. The yield load can be calculated by setting σ equal to the yield stress (PY = σy * A) and the plastic load can be calculated by setting σ equal to the maximum stress on the stress-strain curve (Pp = σmax * A).

In order to determine the maximum stress on the stress-strain curve, we can use the angle theta (θ) and the elongation (ΔL) to calculate the strain (ε) using the equation ε = ΔL/L, where L is the initial length of the bar. Then, the maximum stress can be calculated using the equation σmax = E * ε, where E is the Young's modulus of the material. This will give us the maximum stress on the stress-strain curve, which we can use to calculate the plastic load.

I hope this helps to provide some clarification on how to approach this problem. It is important to understand the concept of elastoplasticity and how it relates to the stress-strain curve in order to solve this problem accurately. If you have any further questions, please do not hesitate to ask.