Problem 48 - An Introduction to the Mechanics of Solids in SI Units

Question

A circular plate (or short cylinder) of outer radius \(a\) has a small central hole of radius \(c\), where \(c \ll a .\) The plate is subjected to an outer pressure \(p\) as shown; there is no axial force. (a) Calculate the maximum normal stress existing in the plate in the limit as \(c / a \rightarrow 0\), that is, as the hole becomes microscopic in size. (b) Calculate the ratio of this maximum stress to the maximum stress which would exist in a solid circular plate of radius \(a\) loaded with an outer pressure \(p\) This ratio gives the stress concentration factor for a small hole in a plate which is subjected to hydrostatic (equal in all directions) stress in the plane of the plate.

Step-by-Step Solution

Answer

The maximum normal stress in the plate is \(-p\) and the stress concentration factor is 1.

Step 1: Calculate Normal Stress

Use the formula for finding the radial stress in a thick-walled cylinder to determine the maximum normal stress in the plate as the hole shrinks \(c / a \rightarrow 0\). The radial stress, \(\sigma\), is given by: \(\sigma = -(p*a^2)/(a^2 - c^2)\), noting that the outer pressure is negative, as it’s in the opposite direction as the positive radial direction.

Step 2: Limit as c/a Approaches 0

As \(c / a \rightarrow 0\), we find the limit of this expression by using L'Hopital's rule or simply acknowledging that as the denominator approaches \(a^2\), the stress approaches \(-p\).

Step 3: Calculate Stress for a Solid Plate

The stress in a solid plate of radius \(a\) with an outer pressure \(p\) is simply \(-p\) due to the pressure being applied everywhere on the plate equally.

Step 4: Calculate the Ratio

The ratio of the maximum stress in the plate with a hole to the maximum stress in a solid plate is the ratio of \(-p\) to \(-p\), which is simply 1.

Key Concepts

Normal Stress

Normal stress is a fundamental concept in the study of mechanics. It describes the stress acting perpendicular to a surface, like the walls of a thick-walled cylinder. In this context, when computing the normal stress in a circular plate with a small hole subjected to external pressure, we consider how the material bears the load. It becomes important to distinguish the pressure exerted outward due to compression, which is equal and opposite to the normal stress being calculated. The formula for normal stress in the plate with a hole is derived from the radial stress equations of thick-walled cylinders: \[ \sigma = -\frac{p \cdot a^2}{a^2 - c^2} \]Here, \(p\) is the outer pressure, \(a\) is the outer radius, and \(c\) is the radius of the hole. The negative sign conventionally implies the direction of pressure opposing the radial outward force. This formula lays groundwork for evaluating limits as the hole size approaches zero, simplifying the determination of stress concentrations.

Hydrostatic Stress

Hydrostatic stress refers to a condition of stress entirely equal in magnitude across all directions. It plays a subtle role in problems of stress analysis, much like our circular plate from the exercise. When subjected to an outer pressure, the plate experiences a uniform hydrostatic stress. This type of stress often simplifies analysis because the forces involved in each direction cancel one another, focusing the effect internally within the material. This explains why the ratio in our problem results in a stress concentration factor of '1'. Hydrostatic stress doesn't distort the shape, but it compels us to consider how normal stresses distribute around anomalies like holes, causing localized increases that propagate perpendicular to the line of action. Therefore, the overall geometric symmetry aids in achieving analytical solutions, as seen in this case.

Thick-Walled Cylinder

A thick-walled cylinder, like the circular plate from the exercise, is crucial in stress concentration analysis. Understanding this concept helps ascertain how stresses distribute in structures where the thickness is not negligible compared to the radius.These cylinders are different from thin-walled ones due to the substantial strain distribution across their walls, demanding precise comprehensive stress applications. Considering this, formulas for radial and circumferential stress — incorporating internal and external pressures — become essential. In this case, the particular focus is on the radial stress, which aligns with the described normal stress.Understanding how the radial stress formula: \[ \sigma = -\frac{p \cdot a^2}{a^2 - c^2} \]aligns with the thick-walled cylinder principles aids learners in handling more complex situations. Knowing how a minute hole (\(c \rightarrow 0\)) affects these distributions demonstrates fundamental engineering physics — where minute variations have a directed proportional effect on overall stress.