Toggle Sidebar
MODELS

New

Image

We have a digital Universal testing machine which is two cross heads and working on hydrulic system and on that machine we fo many type of testing like tensile test, flattening test, expansion test etc. For expansion test of tube specimen we use two type of conical tool to expand as per requirement. One is 60 degree and other one is 45 degree. Currently we are use manually method means epanding little then measure outer diameter again expanding and measuring outer diameter until doesn't get required expansion (like 20 percent of original diameter or 25 percent of original diameter or any other) to expand tube. But recently in audit a assessor say that the method is incorrect because expanding again and again is not correct. We have make a equation of relation between movement of cross head and expansion requirements. So give me a equation to calculate the axis displacement between two cross heads of UTM Machine to expand a tube. For example if a specimen which outer diameter is 20 mm and wall thickness is 4 mm and i want to expand it 20 percent ( the final diameter is 24mm) using 60 degree cone then how much displacement required?

To calculate the displacement of the crosshead of the Universal Testing Machine (UTM) when expanding the tube using a conical tool (in this case, a 60-degree cone), we need to consider the relationship between the initial and final outer diameters and how much the conical tool displaces along the axis of the tube to achieve this expansion.

### Key parameters:
- **Initial outer diameter of the tube (D₀)**: 20 mm
- **Final outer diameter of the tube (D₁)**: 24 mm (this is 20% larger than the original diameter)
- **Wall thickness of the tube (t)**: 4 mm
- **Cone angle (θ)**: 60 degrees (half-angle of 30 degrees per side of the cone)

### Concept:
When the conical tool is pressed into the tube, it pushes outward, increasing the diameter. The axial displacement (\(d\)) of the tool depends on the geometry of the cone, which can be derived using trigonometric relations.

We use the tangent function, as the displacement forms a right triangle where the angle of expansion is half the cone angle (since the cone expands in both radial directions), and the difference in radius corresponds to the change in diameter.

### Steps:

1. **Radius increase:**
   - The initial radius of the tube is \( r_0 = \frac{D_0}{2} = 10 \text{ mm} \).
   - The final radius of the tube is \( r_1 = \frac{D_1}{2} = 12 \text{ mm} \).
   - The increase in radius \( \Delta r = r_1 - r_0 = 12 - 10 = 2 \text{ mm} \).

2. **Cone angle relation:**
   The displacement of the conical tool (\( d \)) is related to the increase in radius and the angle of the cone. For a 60-degree cone, the half-angle is \( 30^\circ \), so we use the tangent of this angle:
   \[
   \tan(30^\circ) = \frac{\Delta r}{d}
   \]
   Solving for \( d \):
   \[
   d = \frac{\Delta r}{\tan(30^\circ)}
   \]
   \[
   d = \frac{2}{\tan(30^\circ)}
   \]

3. **Calculating displacement:**
   Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577 \), the displacement is:
   \[
   d = \frac{2}{0.577} \approx 3.46 \text{ mm}
   \]

### Result:
To expand the tube from 20 mm to 24 mm in diameter using a 60-degree conical tool, the crosshead of the UTM must move approximately **3.46 mm**.

### General Formula:

If you want a general equation for the displacement \( d \) for any expansion, the formula is:

\[
d = \frac{\frac{D_1}{2} - \frac{D_0}{2}}{\tan\left(\frac{\theta}{2}\right)}
\]

Where:
- \( D_0 \) = Initial diameter
- \( D_1 \) = Final diameter (after expansion)
- \( \theta \) = Full cone angle of the tool (e.g., 60° for this case)

---

Would you like more details or have any questions? Here are some follow-up queries that might expand on this explanation:

1. How does the wall thickness influence the calculation of displacement?
2. How would the displacement change if using a 45-degree cone instead?
3. Can this equation be adapted for non-circular tube profiles?
4. What are the implications of over-expanding a tube in terms of material stress?
5. How does this method of calculation compare with empirical data from repeated tests?

**Tip:** Always verify theoretical calculations by conducting trial runs to account for material variability.

We have a digital Universal testing machine which is two cross heads and working on hydrulic system and on that machine we fo many type of testing like tensile test, flattening test, expansion test etc. For expansion test of tube specimen we use two type of conical tool to expand as per requirement. One is 60 degree and other one is 45 degree. Currently we are use manually method means epanding little then measure outer diameter again expanding and measuring outer diameter until doesn't get required expansion (like 20 percent of original diameter or 25 percent of original diameter or any other) to expand tube. But recently in audit a assessor say that the method is incorrect because expanding again and again is not correct. We have make a equation of relation between movement of cross head and expansion requirements. So give me a equation to calculate the axis displacement between two cross heads of UTM Machine to expand a tube. For example if a specimen which outer diameter is 20 mm and wall thickness is 4 mm and i want to expand it 20 percent ( the final diameter is 24mm) using 60 degree cone then how much displacement required?

Learn how to calculate the crosshead displacement in a Universal Testing Machine (UTM) for tube expansion tests. This problem involves calculating the axial displacement for a tube with an initial diameter of 20 mm expanded to 24 mm using a 60-degree conical tool. The solution covers key concepts in trigonometry and geometry, making it ideal for engineering students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation