A torque 𝐓 applied to a solid rod made of an elastoplastic...

A torque $\mathbf{T}$ applied to a solid rod made of an elastoplastic material is increased until the rod is fully plastic and then removed. ( $a$ ) Show that the distribution of residual shearing stresses is as represented in the figure. (b) Determine the magnitude of the torque due to the stresses acting on the portion of the rod located within a circle of radius $c_{0}.$

Key Concepts

Residual Stress Distribution

Residual stresses are the stresses that remain locked into a material after the removal of external loads, particularly due to plastic deformation. When a solid rod is loaded to the point of full plasticity and then unloaded, a non-uniform distribution of residual shear stresses persists through the cross-section. These stresses are responsible for the internal moment even in the absence of external loads, and their proper determination is key to assessing the post-yield behavior of structural components.

Elastoplastic Behavior

Elastoplastic materials exhibit an elastic response up to a yield limit, after which they deform plastically. In the elastic region, stress and strain are linearly related; once the yield point is reached, additional deformation occurs without an increase in stress. This concept is fundamental to understanding how materials behave under loads that exceed their elastic capacities and is crucial in analyzing the transition to fully plastic deformation, as in the case of a rod under increasing torsional load.

Torsional Loading and Shear Stress Distribution

Torsion in cylindrical shafts induces shear stresses that vary with the distance from the center of the shaft. In the elastic regime, the shear stress increases linearly with radial distance. However, when the material yields and plastic zones develop, the shear stress reaches a maximum (yield stress value) over parts of the cross-section. Understanding this transition from elastic to plastic stress distribution is crucial for predicting the behavior under torsional loads and for determining the residual stress pattern after unloading.

Torque Calculation from Shear Stress

The torque transmitted by a shaft is determined by integrating the shear stresses over the entire cross-sectional area. Specifically, the contribution of an elemental area to the torque is proportional to the product of the local shear stress and the radial distance from the center. In scenarios where the stress distribution changes (for example within a sub-region defined by a particular radius), integrating these contributions accurately is essential for evaluating the net torque acting on that portion of the material.

Transcript

00:07 In problem 4 .83, we have a rod bent into the shape of a circle, held in place with two pegs at point d and e, and there's applied force at b down.

00:20 And since it is a three -force rigid body, all three forces must be concurrent, which means that the lines of action of all three forces must intersect.

00:36 Now since force e and d are at frictionless pegs, it means that their force is going to be normal to the surface they're at.

00:48 And since it is a circle, their force is going to point towards the middle of the circle, which means that all three forces intersect at the middle, the exact middle of the circle.

01:20 And knowing that, we can make some triangles to figure, out what the length of c is.

01:28 We know that a given in the problem is 0 .02 meters, and they're asking us to find the length of c between points e and b.

01:49 So if i draw out the problem as a bunch of triangles it looks like this.

02:15 Points c, d, e, and b.

02:30 I'm going to make a couple other triangles that'll be helpful.

02:36 I'll call this point a to make a right triangle between c e and a.

02:42 And then we also know that this right triangle has the sides of a.

02:51 0 .02 meters.

03:03 Now since these two sides of the right triangle are the same, means that these angles must be 45 degrees.

03:13 We can find the distance d .e.

03:16 By using the pythagorean theorem...