To solve Question 1, we need to calculate the load, moment, and various parameters for the beams, specifically the inner beams in the truss structure. Let's go through the steps outlined in the question.

Given Information:

• Span of truss: 40 ft
• On-center spacing (oc distance): 20 ft
• Beam bracing (Lb): 10 ft
• Dead load (D): 40 psf
• Live load (L): 20 psf
• Material: A36 steel (yield strength, $F_y = 36 \text{ ksi}$)

Steps to solve the question:

1. Apply Load Factors: $w_u = 1.2D + 1.6L$ Substitute $D = 40 \text{ psf}$ and $L = 20 \text{ psf}$ to find $w_u$: $w_u = 1.2 \times 40 + 1.6 \times 20 = 48 + 32 = 80 \text{ psf}$

2. Calculate the Load that the Inner Beams Support: $W_u = (w_u) \times \text{oc} \times L$ Given that $w_u = 80 \text{ psf}$, $\text{oc} = 20 \text{ ft}$, and $L = 40 \text{ ft}$: $W_u = 80 \times 20 \times 40 = 64000 \text{ lbs}$

3. Calculate the Ultimate Moment that the Inner Beams Support: $M_u = \frac{W_u \times L}{8}$ Substitute $W_u = 64000 \text{ lbs}$ and $L = 40 \text{ ft}$: $M_u = \frac{64000 \times 40}{8} = 320000 \text{ ft-lbs}$

4. Calculate the Nominal Moment: $M_n = \frac{M_u}{0.9}$ Substitute $M_u = 320000 \text{ ft-lbs}$: $M_n = \frac{320000}{0.9} \approx 355556 \text{ ft-lbs}$

5. Calculate $Z_{\text{min}}$: $Z_{\text{min}} = \frac{M_n}{36 \text{ ksi}}$ Convert $M_n$ from ft-lbs to in-lbs (1 ft = 12 in): $M_n = 355556 \times 12 = 4266667 \text{ in-lbs}$ Then calculate $Z_{\text{min}}$: $Z_{\text{min}} = \frac{4266667}{36} \approx 118519 \text{ in}^3$

6. Determine the Lightest Weight Shape for the Inner Beams Using the Chart:

   • Use a structural steel shape table to find a wide flange (W) beam with a section modulus $Z$ equal to or greater than $Z_{\text{min}} \approx 118519 \text{ in}^3$.

7. Calculate the Nominal Moment Capacity of the Beam: Use the formula provided: $M_n = M_p - \left( (M_p - M_r) \times \frac{(L_b - L_p)}{(L_r - L_p)} \right)$ where $M_p$ and $M_r$ are plastic and reduced moments, and $L_p$ and $L_r$ are lateral-torsional buckling limits for the selected shape.

8. Make Final Selection: After finding the section that satisfies all criteria, confirm its suitability based on moment capacity and other checks.

Would you like a detailed breakdown of each calculation or further explanation on using steel design tables for step 6?

Additional Questions for Further Understanding:

1. How does lateral-torsional buckling affect the beam design?
2. What is the role of load factors in structural design safety?
3. How can different dead and live load scenarios influence the beam size?
4. Why is it important to calculate both ultimate and nominal moments?
5. How would changes in the span or bracing distance affect the calculations?

Tip: Always cross-check your section modulus with steel tables to ensure accuracy in beam selection.