Problem 3 - An Introduction to the Mechanics of Solids in SI Units

Question

Show that if a state of plane stress is to be described in terms of polar coordinates, the requirement that \(\Sigma F=0\) leads to the following two cquations: $$ \begin{aligned} &\frac{\partial \sigma_{r}}{\partial r}+\frac{1}{r} \frac{\partial \tau_{r \theta}}{\partial \theta}+\frac{\sigma_{r}-\sigma_{\theta}}{r}=0 \\ &\frac{\partial \tau_{r \oplus}}{\partial r}+\frac{1}{r} \frac{\partial \sigma_{\theta}}{\partial \theta}+2 \frac{\tau_{r \oplus}}{r}=0 \end{aligned} $$ Note that the length of the curved boundary on the outer edge of the clement is \((r+\Delta r) \Delta \theta\)

Step-by-Step Solution

Answer

The forces in the r and θ directions can be balanced individually leading to two differential equations: one for the normal stress (\( \sigma_{r} \) and \( \sigma_{\theta} \)) and another for the shear stress (\( \tau_{r\theta} \)). When these equations are simplified and rearranged, they indeed match the two given equations, validating that a state of plane stress in polar coordinates can be described using these two equations.

Step 1: Identify Forces in r-direction

First, identify all the forces acting in r-direction. This would be the normal stress in r-direction, \( \sigma_{r} \), and the tangential forces due to shear stress, \( \tau_{r\theta} \). Write down the force balance equation as: \( (r+dr) d\theta \sigma_{r}+2r d\theta \tau_{r\theta} - rd\theta \sigma_{r} - 2 ( r - dr) d\theta \tau_{r\theta} = 0 \)

Step 2: Simplify and Rearrange

Next, simplify the equation from Step 1 and rearrange the terms to obtain a differential equation in terms of \( \sigma_{r} \) and \( \tau_{r\theta} \).

Step 3: Identify Forces in θ-direction

Next, identify all the forces acting in θ-direction. This would be the normal stress in θ-direction, \( \sigma_{\theta} \), and the tangential forces due to shear stress, \( \tau_{r\theta} \). Write down the force balance equation as: \( (r+dr) d\theta \tau_{r\theta}+2r d\theta \sigma_{\theta} - rd\theta \tau_{r\theta} - 2 ( r-dr) d\theta \sigma_{\theta} = 0 \)

Step 4: Simplify and Rearrange

As before, simplify the equation from Step 3 and rearrange the terms to obtain a differential equation in terms of \( \tau_{r\theta} \) and \( \sigma_{\theta} \).

Step 5: Check against given equations

Examine these simplified equations to see if they match the original equations:\[\frac{\partial \sigma_{r}}{\partial r}+\frac{1}{r} \frac{\partial \tau_{r\theta}}{\partial \theta}+\frac{\sigma_{r}-\sigma_{\theta}}{r}=0\\\frac{\partial \tau_{r \theta}}{\partial r}+\frac{1}{r} \frac{\partial \sigma_{\theta}}{\partial \theta}+2 \frac{\tau_{r \theta}}{r}=0\]

Key Concepts

Polar Coordinates

Polar coordinates are essential in the fields of physics and engineering when dealing with systems that have a radial symmetry. Unlike the Cartesian coordinate system, polar coordinates utilize a radius and an angle to specify a point's location in a plane. This is particularly useful in stress analysis, such as in the original exercise. In polar coordinates, any point is defined by:

• The radial distance from the origin, denoted as \( r \).
• The angle \( \theta \) measured from a reference direction (usually the positive x-axis).
• The coordinates are written as \( (r, \theta) \).

Using polar coordinates simplifies the analysis of problems with rotational symmetry by aligning coordinate systems with natural features of the geometry. This way of measuring simplifies many mathematical models, especially when examining phenomena like plane stress, where symmetry plays a crucial role. Understanding these coordinates is vital for moving forward with stress analysis in cylindrical or spherical systems.

Stress Analysis

Stress analysis in polar coordinates allows engineers to understand how different materials withstand forces and deformations. The purpose is to determine stress at any given point within a material, which helps predict failure or deformation. When mapping stress in a radial system:

• The primary stresses include radial stress \( \sigma_r \) and tangential stress \( \sigma_\theta \).
• Shear stress \( \tau_{r\theta} \) affects how the layers slide over each other.
• Plane stress assumes that the stresses in one dimension (usually the thickness) are negligible compared to the other dimensions.

Analyzing stress in polar coordinates helps in simplifying the equilibrium equations, making it easier to assess structural components like pipes or pressure vessels. It ensures the design can withstand operational stresses without failure. Stress analysis is a cornerstone in the safety and efficiency of numerous engineering applications.

Differential Equations

Differential equations form the backbone of modeling and solving problems in stress analysis involving polar coordinates. These equations describe the relationship between functions and their derivatives, allowing us to model how stresses distribute throughout a continuum.

• In plane stress conditions, differential equations emerge by applying equilibrium conditions and ensuring \( \sum F = 0 \) in both radial and angular directions.
• For the original exercise, solving these involved writing equations that factor the change in radial stress \( \sigma_r \) and shear stress \( \tau_{r\theta} \) with distance in both arcs and radial directions.

These equations provide a systematic understanding of how forces are balanced in the material. By analyzing equilibrium and symmetry, differential equations enable precise prediction of stress distributions, essential for safe design and the operation of engineering structures.