3. Maximum shear stress theory

We again do the simple tension test to find out the critical value of shear stress in the body. We increase the distributed load σ as shown in Figure 2. The load at which the body fails is σy. The state of stress is given in (1). The principal stress components for this state of stress are λ1 = σ11, λ2 = 0, and λ3 = 0.

In this initial chunk, we begin our exploration of the Maximum Shear Stress Theory by revisiting the simple tension test. In this procedure, we apply a tensile load to a material, gradually increasing it until the material ultimately fails. At a specific critical load, denoted as σy, the failure occurs. The state of stress is characterized by three principal stress components, where only the first component λ1 equals the applied tensile stress σ11, while the remaining components λ2 and λ3 are zero. This indicates that under simple tension, there is essentially unidirectional stress present.

Think of a rubber band being gradually stretched. As you pull it, it stores energy and can withstand a certain amount of tension. However, once you exceed its capacity (its critical load), it snaps. Similarly, in the context of materials under tension, there is a limit (σy) beyond which they fail.

Thus, maximum shear stress which is obtained by turns out to be (4). This will become equal to the critical shear stress value τ when σ = σy.

Here, we derive the maximum shear stress value from the stress components. The maximum shear stress is determined based on how stress resolves within the material under tension. When the applied tensile stress σ11 equals the yield stress σy at failure, we find that the maximum shear stress reaches a critical state denoted as τ. This critical value indicates the point at which the material will start to deform or fail due to shear forces.

Imagine twisting a piece of playdough. As you apply torque, there’s a limit to how much twist it can take before it starts to deform. The point at which it can no longer maintain its shape under shear stress is akin to the critical shear stress τ in this theory.

Once τ is obtained through simple tension test, it holds even for general loading scenario. The maximum shear stress for a general loading case with principal stress components λ1, λ2, and λ3 is given by (5).

After establishing the shear stress from a basic tension test, this value can be applied to more complex loading situations. The theory posits that the maximum shear stress value remains applicable even when loads are applied in various directions or magnitudes. The equations used assess the principal stress components to determine an overall maximum shear stress for these different scenarios.

Consider a bridge that experiences both vertical loads from vehicles and lateral forces from wind. Even though these forces vary, the principles governing shear stress still apply, helping engineers ensure their designs can withstand complex load interactions without failing.

We can also obtain the critical value of shear stress τ from torsion test instead of tension test. Figure 3 shows a circular beam subjected to equal and opposite torques at the ends.

In addition to using a tension test to establish critical shear stress, we can also perform a torsion test. In a torsion test, we twist a circular beam at both ends applying equal and opposite torques. This testing method allows us to measure how the beam responds to twisting forces and thus derive the critical shear stress, τ, specific to torsional loading.

Think about opening a jar. As you twist the lid, you can feel the resistance. If you twist too hard, the lid can break or strip. This sensation mirrors what happens in torsion tests, where the material's resistance to twisting correlates directly to its critical shear stress.

We can find the maximum shear stress for this state of stress by drawing the Mohr’s circle. As e is a principal axis, we can draw the Mohr’s circle directly.

Mohr’s Circle is a graphical representation that helps visualize the state of stress at a point. By constructing this circle based on the principal stresses, engineers can easily identify the maximum shear stress. In the case of torsion or combined loading scenarios, it provides a clear picture of how different stress states interact and where maximum shear stress occurs.

Imagine laying a piece of rope out on a flat surface. If you twist the ends of the rope, creating a spiral, Mohr's Circle helps visualize the stresses within the rope at any point, simplifying how we assess its performance and failure risk.

Therefore, the maximum shear stress τ is simply τθz, the value of shear stress in the cross-sectional plane. We can write this shear stress τ in terms of the applied torque T as follows.

Here we establish a direct relationship between the applied torque T and the resulting shear stress τ on the beam. The equation showcases how the applied torque translates into stress distributed across the cross-section of the beam, helping engineers understand the expected shear reaction when torques are applied.

Similar to turning the handle of a door, where the applied force creates stress on the hinge (the pivot point), in a beam under torsion, the torque creates shear stress that can lead to failure if it exceeds the critical value.

Thus, the critical value of shear stress τy can be written as (8) where T* is the critical torque value at failure. Once τ is obtained from torsion test, for the general loading scenario, one can write (9).

Finally, we define conditions that must be satisfied in order to prevent failure of the material subjected to shear stress. This includes a critical shear stress value τy that should not be exceeded, coinciding with the critical torque T*. This establishes a safeguard for engineers to ensure structures remain safe under load conditions.

Think of a warning light in your car. It illuminates when you're close to exceeding safe limits for speed or engine temperature, reminding you to slow down or pull over. Similarly, the conditions set in shear stress theories act as warnings that alert engineers when they are approaching critical stress limits.

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