Consider the floor beam given in Figure 4-2 below. Due to spacing issues, this beam cannot be laterally braced against lateral-torsional buckling between the supports. Therefore, it must remain in Zone 1 where $ L_b < L_p $. In this case, $ L_b $ is the beam's total span length.

a. First, determine the minimum $ r_y $ (radius of gyration, found in Table 1-1 of AISC) required considering that $ L_p = 1.76 \sqrt{\frac{r_y F_y}{E}} $, where $ E = 29000 \, \text{ksi} $ and $ F_y = 50 \, \text{ksi} $. (Hint: set $ L_p $ as the unbraced length of the beam, $ L_b $.)

b. Choose the lightest possible W-shape section, designing as if the beam is in Zone 1 and meets the minimum $ r_y $ requirement from part (a). Check moment, shear, and deflection.

Figure 4-2: Simply supported floor beam supporting a distributed dead load of 1 k/ft, a distributed live load of 1.2 k/ft, and two equal concentrated live loads of 6 kips.

a. To determine the minimum required radius of gyration (ry) considering the unbraced length (Lb), we can use the formula Lp = 1.76 * ry * Fy * E, where E is the modulus of elasticity (29000 ksi) and Fy is the yield strength (50 ksi). Since the beam cannot be laterally braced against lateral-torsional buckling between the supports, Lp is equal to Lb, the total span length of the beam. Rearranging the formula, we have ry = Lb / (1.76 * Fy * E).

b. The lightest possible W-shape section that meets the minimum required ry from part (a) needs to be chosen. This involves selecting a W-shape section with dimensions that satisfy the minimum ry requirement while minimizing the weight of the beam. By referring to the AISC tables, we can find the W-shape section that meets the required ry value. The selected section should be able to handle the moment, shear, and deflection imposed by the distributed dead load, distributed live load, and concentrated live loads.

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