Let's break down and solve **Problem 13** step by step: --- ### **Given Data** - **Young’s modulus, \( E \) = 70,000 N/mm²** - **Poisson’s ratio, \( \nu \) = .3** - **Compressive yield strength, \( \sigma_y \) = 500 N/mm²** - **n = .6**, **α = .8** - **Need:** Failure load for the column --- ### **Step 1: Understanding the Problem** This is a column failure problem. The failure load can be found using a formula that blends yield and buckling criteria (often known as the interaction formula). Since the cross-section is not fully shown here, we’ll proceed with the data given and assume the required cross-sectional area **A** is known or will be symbolically carried. --- ### **Step 2: Formula for Failure Load** A common empirical formula for the failure load \( P_f \) is: \[ P_f = \alpha \cdot A \cdot \sigma_y \left[ 1 - n \left(\frac{\sigma_{cr}}{\sigma_y}\right) \right] \] Where: - \( A \) = Cross-sectional area (mm²) - \( \sigma_y \) = Compressive yield strength (N/mm²) - \( \sigma_{cr} \) = Euler's critical stress (N/mm²) - \( \alpha, n \) = Given constants #### **Euler's Critical Stress:** \[ \sigma_{cr} = \frac{\pi^2 E}{(\lambda)^2} \] Where \( \lambda \) = slenderness ratio, which depends on length, radius of gyration, and boundary conditions (not given here). --- ### **Step 3: Plug in Known Values** #### **If Area \( A \) is Not Given:** Since the cross-section is not visible, let's denote area as \( A \) and proceed symbolically. You can plug in the value once you know the cross-sectional area. #### **Substitute the Values:** \[ P_f = .8 \cdot A \cdot 500 \left[ 1 - .6 \left(\frac{\sigma_{cr}}{500}\right) \right] \] --- ### **Step 4: Find \( \sigma_{cr} \) (If Slenderness Ratio is Given)** If you know the column's effective length \( L \) and radius of gyration \( r \): \[ \sigma_{cr} = \frac{\pi^2 \cdot 70,000}{(\lambda)^2} \] \[ \lambda = \frac{L}{r} \] --- ### **Step 5: Final Answer (General Form)** \[ \boxed{ P_f = .8 \cdot A \cdot 500 \left[ 1 - .6 \left(\frac{\sigma_{cr}}{500}\right) \right] \text{ N} } \] Where: - \( A \) = Cross-sectional area in mm² (get from Fig. P.13) - \( \sigma_{cr} = \dfrac{\pi^2 E}{(L/r)^2} \) --- ## **What You Need To Do Next** 1. **Find the cross-sectional area \( A \)** from Fig. P.13. 2. **Determine the slenderness ratio \( \lambda = L/r \)** if column length and radius of gyration are given. 3. **Calculate \( \sigma_{cr} \) and substitute all values into the formula.** --- ### **If you provide the cross-section details or the slenderness ratio, I can give you the numerical answer!** --- #### **Summary Table** | Symbol | Meaning | Unit | |--------|----------------------------------|--------------| | \( E \)| Young's modulus | N/mm² | | \( \sigma_y \)| Yield strength | N/mm² | | \( A \)| Cross-sectional area | mm² | | \( \lambda \)| Slenderness ratio | Dimensionless| | \( \alpha, n \)| Empirical constants | - | | \( \sigma_{cr} \)| Critical buckling stress | N/mm² | --- > **Let me know the cross-sectional details or column length to get the final numerical answer!**