5. Example

Let us now discuss an example which demonstrates how we can use these theories to design a beam. Consider a circular beam which is subjected to equal and opposite torques T and bending moments M at its ends, as shown in Figure 5.

In this example, we are going to explore how failure theories can be applied practically to design a beam. A circular beam is used as our subject, experiencing equal twisting forces (torques) and bending forces at each end. This scenario is a classic example of how combined loads can affect structural elements, which is vital for engineers to understand to prevent failures in real-world applications.

Imagine using a simple plastic ruler. If you twist one end while pushing down on the middle, you're applying forces similar to the torques and bending moments. Understanding how it bends (twists) and at what point it might break helps in designing stronger rulers or similar items.

We want to first find the state of stress corresponding to this loading in the cylindrical coordinate system. We can consider the effects of bending and torsion separately and then superimpose them. As the bending will be pure bending, we will only get non-zero σ due to it.
Due to torsion, only τ and τ will be non-zero and will be given by.

To analyze how the beam reacts under the applied loads, we need to determine the state of stress first. This involves calculating the stress resulting from bending and torsion separately, as these two effects combine to give the total state of stress on the beam. The stress due to bending is represented by the normal stress (σ), while the shear stress (τ) arises from the torsional effect. By working in cylindrical coordinates, we can express these stresses mathematically.

Think about trying to stretch and twist a piece of clay. When you pull and twist it, you're applying different stresses on it. We need to understand each of these stresses individually before we try to predict how the clay will break or bend completely.

The state of stress obtained by superposition can be written as.

Once we find the stress from bending and torsion, we can combine these two states using the principle of superposition, which allows us to add the different stress contributions to find the total state of stress acting on any point in the beam. This is a crucial process in structural analysis as it enables engineers to understand how the material will behave under composite loading conditions.

Think of cooking where you might mix ingredients. Each ingredient adds flavor, and by combining them, you get the final dish. Similarly, by adding stresses from bending and torsion, we understand the overall stress on the beam.

To get the condition for failure, we can apply a suitable theory of failure. Let us think of applying maximum shear stress theory. We can find the maximum shear stress by drawing the Mohr’s circle for the given state of stress.

To ensure the beam does not fail, we need to check the maximum shear stress against a critical value using the maximum shear stress theory. We can visualize this concept through Mohr’s circle, a graphical representation that helps us determine the maximum and minimum shear stresses at any point based on the combined states of stress we previously derived. This helps engineers assess the safety of the design under expected load conditions.

Consider a tire. The maximum shear stress helps us understand how much weight or pressure the tire can handle before it blows out. Just like checking that pressure, engineers use Mohr's circle to ensure that the beam can handle expected stress without failing.

The factor of safety finally reflects on the RHS in this particular case which limits our operational value of torque and moment while designing as we don’t want the operational load to reach anywhere near the critical load.

In engineering, it's essential to design structures with a safety margin. This is achieved through a factor of safety, which is a multiplier applied to the expected loads. By ensuring that the operational conditions remain below critical values, engineers can protect against unexpected overloads, material flaws, or other unforeseen circumstances that might lead to failure.

Building a bridge might use a factor of safety to ensure that even heavy traffic won't exceed the load it can bear. It’s like driving a car; manufacturers rate tires for certain weights, and we use a factor of safety to ensure they last longer than just the rated limit.

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