Problem 23 - An Introduction to the Mechanics of Solids in SI Units

Question

A thin-walled cylindrical tank of radius \(r\), thickness \(t\), and length \(L\) is supported at its ends. It is filled with a heavy liquid which is vented to the atmosphere. If the weight of the tank is negligible compared with the weight of the liquid, show that the maximum bending stress in the tank is independent of the radius of the tank.

Step-by-Step Solution

Answer

The maximum bending stress in the tank is independent of its radius. It is given by the equation \( \sigma = \dfrac{ \gamma L}{2 \pi t} \).

Step 1: Define the Variables

We first define our known variables: \(r\) is the radius, \(t\) is the thickness, and \(L\) is the length.

Step 2: Analyse the Forces involved

The total horizontal force which is the product of pressure \(p\), area of the side of the tank \(2 \pi rL\) is given by \(F = p(2 \pi rL)\) since \(p = \text{Weight Density} \times \text{Height} = \gamma r\), we replace the pressure \(p\) in the force equation with \( \gamma r\). Thus, our force formula becomes \(F = 2 \pi \gamma r ^2L\). This is the force exerted by the heavy liquid on the tank.

Step 3: Compute Moment of Inertia

The moment of inertia of a cylindrical shell about its central axis can be given as \( I = \pi r^3 t \). This, in essence, represents the resistance of the cylindrical tank to bending or deformation.

Step 4: Compute Bending Stress on the Tank

According to the bending stress equation \( \sigma = \dfrac{My}{I} \), where \(M\) is the moment, \(y\) is the distance from the neutral axis and \(I\) is the moment of inertia. Given that \(M = Fr/2\), and replacing the values for \(F\) and \(I\) from our previous equations, we get \( \sigma = \dfrac{ \gamma r^3 L}{2r} \div \dfrac{ \pi r^3 t }{r} \), which simplifies to \( \sigma = \dfrac{ \gamma L}{2 \pi t} \).

Step 5: Analyse the Result

We observe that the bending stress \( \sigma \) is indeed independent of the radius \(r\), as it does not appear in the simplified equation.

Key Concepts

Moment of Inertia

Understanding the concept of the moment of inertia is crucial when analyzing the structural stability of objects under various load conditions. In the context of cylindrical tanks, such as in our exercise, the moment of inertia is a measure of the object's resistance to bending when exposed to a force. Imagine bending a ruler with your hands; the moment of inertia is like the ruler's stubbornness to bend.

The moment of inertia can be mathematically expressed as a function of the geometry of the object. For a thin-walled cylinder under consideration, it is represented by the formula \( I = \pi r^3 t \), where \(r\) is the cylinder's radius and \(t\) is the thickness of its wall. In essence, it's a property intrinsic to the material distribution of the tank and is crucial in our calculation of bending stresses.

Role in Bending Stress Analysis

When calculating bending stresses, the moment of inertia comes into play as it directly relates to how much a structural member will deflect under a load. The higher the moment of inertia, the less it will bend, assuming the same applied force. Therefore, ensuring accuracy in calculating the moment of inertia is paramount to correctly estimating bending stresses in engineering practice.

Forces in Fluid Mechanics

In fluid mechanics, forces can arise from various sources, such as pressure differences, gravitational forces, and viscous stresses. In our problem, we're primarily concerned with the pressure exerted by a heavy liquid on the sides of a cylindrical tank.

The force due to the fluid pressure on a surface can be daunting to visualize, but it's similar to how you feel the shower curtain push against you when you change the water temperature. The pressure at any point in a stationary fluid varies with depth due to gravitational forces. Using our given formulas, we express the force as \(F = 2 \pi \gamma r^2L\), where \(\gamma\) is the weight density of the liquid, \(r\) is the tank radius, and \(L\) is the length of the tank. The total force applied on the tank is the product of this pressure and the area of the tank's side exposed to the liquid.

Impact on Structural Integrity

A keen understanding of the forces acting on a structure due to fluids is vital in predicting potential failure modes and designing with safety margins. As fluid mechanics continually influences structures, engineers must consider these forces in their calculations and the resulting stresses they induce.

Bending Stress Equation

When it comes to ensuring the structural safety of cylindrical tanks under load, the bending stress equation serves as an indispensable tool. This equation evaluates the stress experienced by a material subjected to bending, and is given by \( \sigma = \dfrac{My}{I} \), where \(M\) represents the bending moment, \(y\) is the distance from the neutral axis (the point where stress is zero), and \(I\) is the moment of inertia.

The bending moment \(M\) can be thought of as a measure of the load-induced bending experienced by the tank, typically calculated as the force times the distance from the point of interest to the location where the force applies. In our case, \(M = Fr/2\). By plugging in the appropriate values as shown in the exercise, we get the maximum bending stress independent of the tank's radius.

Ensuring Tank Safety

Engineers must consider the maximum bending stress that a tank can endure without failing. This exercise underscores that, for a given weight density of the liquid \(\gamma\) and wall thickness \(t\), the maximum bending stress is the same no matter how large or small the tank's radius is. Such knowledge is indispensable when deciding material selection and tank dimensions during the design process.