A wide flange section for a 4 m long column (hinged at both ends) has the following properties:

Cross-sectional area = 6,000 mm²
Radius of gyration, rx = 150mm
Radius of gyration, ry = 100mm
Modulus of Elasticity, E = 200,000 MPa

Determine the value of Euler critical stress.

step by step no words$$  $$

Given:
- $ A = 6,000 \, \text{mm}^2 $
- $ r_x = 150 \, \text{mm} $
- $ r_y = 100 \, \text{mm} $
- $ E = 200,000 \, \text{MPa} $
- $ L = 4 \, \text{m} = 4000 \, \text{mm} $

Step-by-step solution:

1. **Calculate Effective Length (hinged at both ends):**  
   \[
   L_{\text{eff}} = L = 4000 \, \text{mm}
   \]

2. **Calculate Slenderness Ratio (we use the smaller $ r_y $ for conservative calculation):**  
   \[
   \lambda = \frac{L_{\text{eff}}}{r_y} = \frac{4000}{100} = 40
   \]

3. **Euler’s Critical Stress Formula:**
   \[
   \sigma_{\text{cr}} = \frac{\pi^2 E}{\lambda^2}
   \]

4. **Substitute values:**
   \[
   \sigma_{\text{cr}} = \frac{\pi^2 \times 200,000}{40^2}
   \]

5. **Calculate $ \sigma_{\text{cr}} $:**
   \[
   \sigma_{\text{cr}} = \frac{9.8696 \times 200,000}{1600}
   \]
   \[
   \sigma_{\text{cr}} = \frac{1,973,920}{1600}
   \]
   \[
   \sigma_{\text{cr}} = 1233.7 \, \text{MPa}
   \]

**Answer:**
\[
\sigma_{\text{cr}} \approx 1233.7 \, \text{MPa}
\]

A wide flange section for a 4 m long column (hinged at both ends) has the following properties:

Cross-sectional area = 6,000 mm²
Radius of gyration, rx = 150mm
Radius of gyration, ry = 100mm
Modulus of Elasticity, E = 200,000 MPa

Determine the value of Euler critical stress.

Step-by-step solution to determine Euler critical stress for a 4-meter long hinged column with a cross-sectional area of 6,000 mm², radius of gyration ry = 100 mm, and modulus of elasticity E = 200,000 MPa. Ideal for engineering students studying mechanics of materials.

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