Problem 4 - An Introduction to the Mechanics of Solids in SI Units

Question

Show that if a general state of stress is to be described in cylindrical coordinates, the requirement that \(\Sigma \mathbf{F}=0\) leads to the following three equations: Prob. \(4.4\) $$ \begin{aligned} &\frac{\partial \sigma_{r}}{\partial r}+\frac{1}{r} \frac{\partial \tau_{r \theta}}{\partial \theta}+\frac{\partial \tau_{z r}}{\partial z}+\frac{\sigma_{r}-\sigma_{\theta}}{r}=0 \\ &\frac{\partial \tau_{r \oplus}}{\partial r}+\frac{1}{r} \frac{\partial \sigma_{\theta}}{\partial \theta}+\frac{\partial \tau_{\theta z}}{\partial z}+2 \frac{\tau_{r \theta}}{r}=0 \\ &\frac{\partial \tau_{\pi}}{\partial r}+\frac{1}{r} \frac{\partial \tau_{\theta z}}{\partial \theta}+\frac{\partial \sigma_{z}}{\partial z}+\frac{\tau_{x c}}{r}=0 \end{aligned} $$

Step-by-Step Solution

Answer

The solution involves using the force equilibrium condition \(\Sigma \mathbf{F}=0\) in the context of stress in cylindrical coordinates. You express the forces according to the stress tensor (involving both normal and shear stress components), apply the force equilibrium, leading to three equations. These equations correspond to the absence of a resultant force in the radial, tangential, and axial directions.

Step 1: Express the Force Equilibrium

In order to apply the force equilibrium condition \(\Sigma \mathbf{F}=0\), express the forces as per the stress tensor in cylindrical coordinates. The stress tensor has six elements: radial stress \(\sigma_{r}\), circumferential or tangential stress \(\sigma_{\theta}\), axial stress \(\sigma_{z}\), and the shear stresses \(\tau_{r\theta}\), \(\tau_{\theta z}\) and \(\tau_{z r}\). Each force on the infinitesimal element is determined by the corresponding stress multiplied by the area to which it applies.

Step 2: Apply the Equilibrium Condition

Next, apply the equilibrium condition, \(\Sigma \mathbf{F}=0\), to each direction separately, i.e., radial, circumferential, and axial. This ends up forming a system of three equations corresponding to the balance of forces in these three directions, i.e., there should be no resultant force left in any of these directions.

Step 3: Work out the Equations

The resulting equations from applying \(\Sigma \mathbf{F}=0\) for all the directions form the equations of equilibrium. These need to be simplified and written in their detailed form, allowing to demonstrate the three equations given in the original problem.

Key Concepts

Stress Tensor

In cylindrical coordinates, stress is represented by a stress tensor. A stress tensor in this context has distinct elements that simplify understanding how internal forces distribute within a cylindrical system. The stress tensor is pivotal because it encapsulates:

• Radial stress, denoted by \(\sigma_r\), acting perpendicular to the surface of a cylindrical element.
• Circumferential or tangential stress, \(\sigma_\theta\), which acts along the circumference of the cylindrical surface.
• Axial stress, \(\sigma_z\), directed along the length of the cylinder.
• Shear stresses, \(\tau_{r\theta}, \tau_{\theta z}, \) and \(\tau_{zr}\), which denote forces parallel to the surface and between different planes.

The arrangement and understanding of these stress components help in analyzing mechanical stability and behavior of structures under load. Understanding the tensor allows one to set up equations necessary for equilibrium analysis, which is crucial in structural engineering.

Force Equilibrium

Force equilibrium is a fundamental principle in mechanics, particularly relevant when analyzing structures. It involves setting the sum of all forces in a system to zero, expressed mathematically as \(\Sigma \mathbf{F}=0\). This principle ensures that a structure, under load, remains static and stable. In cylindrical coordinates, to maintain equilibrium, forces must balance in three primary directions:

• Radial direction: Here, the sum of radial and shear forces clients to zero.
• Circumferential direction: Circumferential and tangential forces counterbalance for stability along the circular plane.
• Axial direction: Longitudinal forces, including shear components, align to maintain steadiness along the cylinder's length.

Applying force equilibrium appropriately aligns forces in all directions, assisting in deriving the equations of equilibrium and ensuring mechanical stability in the analyzed system.

Equations of Equilibrium

Equations of equilibrium are mathematical representations that ensure a mechanical system is at rest. In the context of cylindrical coordinates, these equations result from applying the force equilibrium concept to an element of the structure. They serve to describe how stresses and forces interact and are distributed across an infinitesimal cylindrical element: - **Radial Equation:** Describes the balance of radial stress and shear forces along the cylinder radius. - **Circumferential Equation:** Manages the interplay between circumferential stress around the cylinder and tangential shear stresses. - **Axial Equation:** Accounts for axial stress and shears along the cylinder's height, ensuring forces along the axis are balanced. Expressed as differential equations, they guide engineers in designing safe and efficient structures. In practice, evaluating these equations allows one to account for variations in stress due to changes in shape, load, or material properties, crucial for verifying the soundness of engineering designs.