uniformly tapered bar 720p | Summary and Q&A

TL;DR

The content explains how to calculate the total elongation in a uniformly tapered bar by using unitary method and integration.

Key Insights

• 🍫 The formula for calculating the total elongation in a uniformly tapered bar is Delta L = 4P * L / (Pi * D1 * D2 * E).
• 🍉 The unitary method is used to find the diameter of a small element in terms of the initial and final diameters.
• ☺️ The area of a small element is determined using the formula ax = (Pi * (D1 - k*X)^2) / 4.
• 🤢 Integration is used to find the total elongation by integrating the small element deformations over the entire length of the bar.

Transcript

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Questions & Answers

Q: How can the total elongation in a uniformly tapered bar be calculated?

The total elongation can be calculated by considering small elements of the bar and determining the deformation in each element using the formula Delta L = P * DX / (Pi * (D1 - k*X)^2 * E). The small element deformations are then integrated over the entire length of the bar to obtain the total elongation.

Q: What is the significance of using the unitary method in finding the small element deformation?

The unitary method allows us to express the deformation in a small element as D subscript X = D1 - k*X, where D1 is the initial diameter, D2 is the final diameter, L is the length, and X is the distance from the initial point. This method simplifies the process of finding the deformation in each small element.

Q: How is the area of a small element determined in the calculation of total elongation?

The area of a small element is determined using the formula ax = (Pi * (D1 - k*X)^2) / 4, where ax is the area at a certain distance X from the initial point. This area is then used in the formula for determining the deformation in the small element.

Q: Can you explain the integration process in finding the total elongation?

The small element deformation formula, Delta DX = P * DX / (Pi * (D1 - k*X)^2 * E), is integrated from 0 to L to find the total elongation. By applying the upper and lower limits of integration and simplifying the expression, the result Delta L = 4P * L / (Pi * D1 * D2 * E) is obtained.

Summary & Key Takeaways

• The content explains the concept of a uniformly tapered bar and how to calculate the total elongation in such a bar.

• To find the total elongation, a small element of the bar is considered, and the deformation in that element is determined using the formula: Delta L = P * DX / (Pi * (D1 - k*X)^2 * E).

• The deformation in the entire bar is then found by integrating the small element deformations from 0 to L, resulting in the formula: Delta L = 4P * L / (Pi * D1 * D2 * E).