3. Graphical representation of the derived formulation

We need to see what equation(17) means. Let us think of a σ−τ plane and plot σ and τ for each α in this plane. The plane with σ on the x-axis and τ on the y-axis is shown in Figure 4.

In this chunk, we start by visualizing the relationship between the normal stress (σ) and shear stress (τ) on a two-dimensional plane. By plotting σ on the horizontal axis and τ on the vertical axis, we create a coordinate system where each point represents the state of stress for a specific angle α. This visual representation allows us to understand how σ and τ vary as we change the angle of interest.

Imagine a graph where you plot the performance of students in a class based on their study hours (σ) and their grades (τ). Each point on this graph tells us how well students perform based on the amount of time they study. Similarly, in the σ−τ plane, each point shows the relationship between normal and shear stresses.

From equation(17), we can see that by plotting all points, we will get a circle centered on the σ axis at . Let us start by plotting σ and σ on the σ axis.

In this step, we begin to plot all the points derived from the equation related to σ and τ. These points will reveal a circular formation in the σ−τ plane, with the center of this circle positioned on the σ axis. By starting with specific stress values on the σ axis, we can outline the circle that reflects all possible states of shear and normal stresses for varying angles. This circle is known as Mohr's Circle, and it helps visualize how shear and normal stresses interact.

Think of a spinning wheel where the center is fixed, and as the wheel turns, various points on the edge display different positions. Similarly, on Mohr's Circle, as we vary the angles α, different points on this circle represent stress conditions that the material might experience.

The circle has to pass through the point corresponding to the e plane: the circle is the locus of all (σ, τ) when α is varied and the point corresponding to the e plane is obtained for α= 0.

The radius of Mohr's Circle represents the maximum shear stress and is obtained by determining the distance from the center to any point on the circle, especially the points derived for specific planes. The radius allows us to visually and mathematically understand how shear stresses vary with respect to normal stresses as the plane angle changes. When α is set to 0, the point corresponding to the e plane (the normal direction) becomes a pivotal reference from which all stress states are compared.

Consider this as measuring the distance from the center of a merry-go-round to the edge where a child is sitting. As the merry-go-round spins, the distance (or radius) remains the same, but the position of the child changes. In Mohr's Circle, as we change the angle α, the positions of stresses change, but the key metrics (like the radius) help us maintain a proper understanding of their boundary limits.

Once we have obtained the radius and the center of the circle, we can draw the complete circle as shown in Figure 5.

With the center identified along with the radius calculated, we proceed to complete the representation of Mohr's Circle. This involves rotating and plotting a full circle based on the center and radius, visually encapsulating all possible normal and shear stress values corresponding to varying angles of the normal plane. The process signifies crucial states of stress that materials can undergo as conditions change, serving as a vital tool in structural analysis.

Think of drawing a circle on a piece of paper where you mark the center with a pencil and use a compass to draw around it. Just as you can identify any point along the edge based on your compass’s radius, Mohr's Circle lets engineers locate any stress state on that boundary using the predetermined parameters.

To find (σ, τ) for α-plane, rotate the radial line of e plane by 2α clockwise.

In this last chunk, we focus on utilizing Mohr's Circle to determine shear and normal stresses on any arbitrary plane inclined at an angle α. By taking the previously established radial line corresponding to the e plane and rotating it 2α degrees in a clockwise direction, we can easily find the corresponding point on Mohr's Circle that reveals the stresses for that specific orientation. This method provides an efficient means of deriving critical stress information without needing to resort to complex calculations frequently.

Imagine you are trying to find the angle of a slice cut into a pizza. If you know the angle that represents one slice (the e plane), you could find the angle representing a different slice (the α-plane) by simply tracking your finger around the pizza, essentially rotating to find where to cut.

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