2.4. Final Simplification

We can now plug in the derivatives of basis vectors into equation (6) to get.

In this chunk, we start the final simplification process by substituting the derivatives of the basis vectors into the equation we had derived earlier, specifically equation (6). This is a crucial step because it allows us to reduce our complex expressions to something more manageable, leveraging the relationships we have established in the cylindrical coordinate system.

Think of this process as assembling the final pieces of a puzzle. Just like you fit in the last few pieces to see the complete picture, here we are fitting in the derivatives to finalize our mathematical expression.

Let us now consider the first set of terms, i.e., σ is evaluated at the center of the cylindrical element and hence can be taken out of the integral. The integral of dA then gives us A: the area of +z face.

In evaluating the first terms of the expression, we note that the stress component σ can be treated as a constant in relation to the area of the +z face. Since it is evaluated at the center of the element, we can simplify our integral by taking σ out of it. This simplifies our calculations, as we only need to calculate the area A of the +z face to continue our analysis. This method is commonly used in calculating forces where uniform conditions are assumed over a defined area.

Imagine you are calculating the total cost for materials to build a flat surface like a table. If the cost per square meter is the same (a constant value), you can easily multiply that cost by the area of the table to find your total. Here, σ is analogous to that 'cost per square meter' which is constant across our defined surface.

Considering the second term, the integration of ξdA will give us A times r of +z face. As we are measuring ξ from the centroid itself, r becomes zero.

In this chunk, we analyze how the second term behaves under integration. Since ξ represents a distance that is measured from the centroid of our cylindrical element, when we carry out the integration of ξ multiplied by the area dA, we find that it results in A times r. However, because we are measuring ξ from the centroid of the cylindrical element, the value of r effectively becomes zero, which simplifies our calculations further by negating this term.

This can be likened to measuring distances from a central point, like the distance from the center of a circle to any point on its edge. If you were measuring distances for more than one point and averaging them out, the average distance from the center would naturally come out to zero because positive and negative distances would cancel each other out.

The last set of terms can be similarly simplified to yield the total force on +z plane.

In this chunk, we complete the simplification process for the last set of terms. By applying the same logic as in previous steps, we derive an expression for the total force on the +z plane. This encapsulation of forces allows us to analyze the system comprehensively as it unifies the contributions from various aspects of the cylindrical coordinates and stresses.

Consider how you might calculate the total weight on a beam supported at various points. By summing the various weights (forces) acting on the beam, you can determine the total load it is supporting. Just like that, we are summing all the relevant forces acting on our cylindrical element to find the total force.

If we compare equations (3) and (4), we find that we have an overall negative sign for the −z plane and also that the stress components are to be evaluated at z- instead of z +.

This chunk emphasizes the importance of comparing the derived equations for both the +z plane and the -z plane. We notice an overall negative sign for the -z plane. This indicates that the direction of force due to stress acts opposite to that of the +z plane. Additionally, stress components at the -z plane are evaluated at a different height than for the +z plane. This difference highlights the balancing forces acting on our cylindrical element and ensures accurate modeling of the physical reality.

Think about balancing a seesaw. If one side has heavier weights, it will drop, indicating a force in that direction. The seesaw's behavior illustrates how forces act differently based on their positions: one side goes up while the other goes down. In our equations, we see a similar principle where forces on different planes affect the overall stability of our system.

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