1. **State the problem:** We need to design a W-shape column subject to a load of 600 kN with given unbraced lengths about major and minor axes and check the allowable stress using the formula $$Fb = 119 - 0.0034 \left(\frac{L}{r}\right)^2$$, assuming a stress of 100 MPa. 2. **Given data:** - Load $ P = 600 \text{ kN} = 600000 \text{ N} $ - Unbraced length about major axis $ L_x = 9 \text{ m} = 9000 \text{ mm} $ - Unbraced length about minor axis $ L_y = 3.5 \text{ m} = 3500 \text{ mm} $ - Stress $ \sigma = 100 \text{ MPa} = 100 \text{ N/mm}^2 $ - Formula for allowable bending stress $ Fb = 119 - 0.0034 \left( \frac{L}{r} \right)^2 $ 3. **Determine required radius of gyration $ r $ and moment of inertia properties for W shape:** We don't have radius of gyration $ r $ given; to design, we will use the allowable stress and moments. For stability, the smaller effective slenderness $ L/r $ dominates. 4. **Calculate slenderness ratios for major and minor axes:** We will assume the critical slenderness is the larger of the two ratios to use in $ Fb $. 5. **Calculate allowable stress $ Fb $ for each axis:** - For major axis: $$ Fb_x = 119 - 0.0034 \left( \frac{9000}{r_x} \right)^2 $$ - For minor axis: $$ Fb_y = 119 - 0.0034 \left( \frac{3500}{r_y} \right)^2 $$ 6. **Relationship between allowable stress and applied stress:** For safe design, $$ \sigma \leq Fb $$ Which means, $$ 100 \leq 119 - 0.0034 \left( \frac{L}{r} \right)^2 $$ 7. **Rearranging to find allowable slenderness ratio:** $$ 0.0034 \left( \frac{L}{r} \right)^2 \leq 119 - 100 = 19 $$ $$ \left( \frac{L}{r} \right)^2 \leq \frac{19}{0.0034} = 5588.24 $$ $$ \frac{L}{r} \leq \sqrt{5588.24} = 74.74 $$ 8. **Calculate required radius of gyration to satisfy slenderness limit:** For major axis: $$ r_x \geq \frac{L_x}{74.74} = \frac{9000}{74.74} = 120.42 \text{ mm} $$ For minor axis: $$ r_y \geq \frac{L_y}{74.74} = \frac{3500}{74.74} = 46.85 \text{ mm} $$ 9. **Conclusion:** The selected W shape must have radius of gyration $ r_x \geq 120.42 \text{ mm} $ about the major axis and $ r_y \geq 46.85 \text{ mm} $ about the minor axis to safely withstand the applied stress of 100 MPa using the given formula. This guides the selection of an appropriate W section satisfying these geometric properties.