1. **Stating the problem:** We need to design a compression member 3 m long using a pair of back-to-back angles with a 10 mm gusset plate at each end. 2. **Given:** Length $L=3$ m, Load $P=450000$ N, Modulus of Elasticity $E=200000$ MPa, Yield Strength $F_y=230$ MPa, Assumed stress $\sigma=100$ MPa, Gusset plate thickness $t_g=10$ mm. 3. **Step 1: Calculate required cross-sectional area $A$ using allowable stress:** $$A=\frac{P}{\sigma}=\frac{450000}{100}=4500\, \text{mm}^2$$ 4. **Step 2: Calculate slenderness ratio $\lambda$:** Assuming radius of gyration $r$ unknown yet, slenderness ratio $$\lambda=\frac{KL}{r}$$ where $K=1$ for pinned ends. 5. **Step 3: Find allowable stress $F_{cr}$ using AISC formula (Euler and inelastic buckling):** Calculate slenderness parameter $$\lambda_c=\sqrt{\frac{2\pi^2 E}{F_y}}=\sqrt{\frac{2\pi^2 \times 200000}{230}} \approx 41.3$$ 6. Calculate $F_{cr}$ using $$F_{cr}=\left\{\begin{matrix}F_y\quad & \lambda \leq \lambda_c\\ \frac{\pi^2 E}{\lambda^2} & \lambda > \lambda_c \end{matrix}\right.$$ 7. **Step 4: Estimate or select angle section and calculate radius of gyration $r$** We need to pick angles with $A \geq 4500$ mm$^2$ and sufficient $r$ so $F_{cr} > 100$ MPa. 8. **Step 5: Check final design:** Calculate $F_{cr}$ with assumed $r$ and check if $$P \leq A \times F_{cr}$$ 9. **Summary:** This is an iterative process: select pair of angles, compute $r$, calculate $F_{cr}$ and ensure member is safe under given load and assumptions. **Final note:** Without exact angle sizes, the design is conceptual. With actual angle dimensions, the calculations above guide ensuring the member resists the 450 kN load safely.