https://file.aiquickdraw.com/mathbot/file/17273609922428eqgud8d.jpg

The problem involves a rigid bar ABC supported by a cable and subjected to a thermal expansion problem. We are asked to compute the displacement of point C after the bar undergoes thermal expansion due to a temperature rise of 60°C.

### Given Data:
- Diameter of the bar \( d = 36 \, \text{mm} \).
- Length of bar \( AB = 2 \, \text{m} \), \( BC = 1.4 \, \text{m} \).
- Load \( P = 100 \, \text{kN} \) at point C.
- Modulus of elasticity \( E = 200 \, \text{GPa} = 200 \times 10^3 \, \text{MPa} \).
- Coefficient of thermal expansion \( \alpha = 12 \times 10^{-6} \, ^\circ \text{C}^{-1} \).
- Temperature rise \( \Delta T = 60^\circ \text{C} \).

### Step-by-Step Approach:

1. **Thermal Expansion of Bar:**
   The thermal expansion of a rod is given by:
   \[
   \Delta L = L \cdot \alpha \cdot \Delta T
   \]
   where:
   - \( L \) is the length of the bar,
   - \( \alpha \) is the coefficient of thermal expansion,
   - \( \Delta T \) is the change in temperature.

2. **Calculating the Expansion for Sections:**
   The total expansion will occur over the entire length \( AB + BC = 2 \, \text{m} + 1.4 \, \text{m} = 3.4 \, \text{m} \).

   The thermal expansion for the entire bar is:
   \[
   \Delta L = 3.4 \, \text{m} \times 12 \times 10^{-6} \, ^\circ \text{C}^{-1} \times 60^\circ \text{C}
   \]
   Let me calculate this.The total thermal expansion of the bar due to the temperature rise of 60°C is \( 0.002448 \, \text{m} \) (or 2.448 mm).

### Step 3: Displacement of Point C

Since the rigid bar is pinned at point A and supported by a cable at B, the displacement at point C will depend on the geometry of the system. The bar will expand along its length, but the displacement at C will also involve the rotational effect of the expansion since the bar is pinned at A.

For such cases, the displacement at C can be determined using similar triangles or geometry based on the lengths of \( AB \) and \( BC \).

Let me calculate the displacement of C based on the geometry.The displacement of point C due to the thermal expansion of the bar is \( 0.001008 \, \text{m} \) (or approximately 1.008 mm).

### Summary:
- **Thermal expansion of the bar**: \( 2.448 \, \text{mm} \).
- **Displacement of point C**: \( 1.008 \, \text{mm} \) due to the geometry and constraints of the system.

Would you like further clarification or additional details? Here are some related questions for deeper understanding:

1. How is the modulus of elasticity related to the displacement caused by thermal expansion?
2. How would the displacement change if the temperature increase were different?
3. What would happen if the bar had a different material with a lower thermal expansion coefficient?
4. How would adding a fixed support at point C affect the displacement of the bar?
5. Can we neglect the load of 100 kN when calculating the displacement due to thermal expansion?

**Tip:** When working with thermal expansion problems in structures, always take note of the constraints, as they significantly influence how expansion translates into displacement.

Compute the displacement of point C if the temperature rises by 60°C. A rigid bar ABC is pinned at A and supported by cables at point B. The rod is pin connected. Diameter of bar = 36 mm. Load P = 100 kN is applied at C. E = 200 GPa. Coefficient of thermal expansion = 12 x 10^-6 /°C.

Learn how to calculate the displacement of a rigid bar's point C caused by thermal expansion due to a temperature rise of 60°C. Includes details on the thermal expansion formula, bar geometry, and key mechanical properties like modulus of elasticity and load effects.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation