To analyze the beam given the provided loads, we will perform the following steps:

1. Determine Support Reactions: Using equilibrium equations to find the reactions at the supports (assuming a simply supported beam with supports at the ends).

2. Draw Shear Force Diagram (SFD) and Bending Moment Diagram (BMD): Using the calculated reactions, construct the SFD and BMD to understand how shear forces and moments vary along the length of the beam.

3. Calculate Maximum Shear Force and Bending Moment: Use the diagrams to find the critical points for design checks.

4. Check for Beam Capacity: Using the given material properties (A36 steel with $F_y = 248 \, \text{MPa}$), determine if the beam can safely carry the applied loads based on maximum shear and moment.

Step 1: Determine Support Reactions

Given Loads:

• Point load of 10 kN at 2 m from the left support (Point A).
• Point load of 15 kN at 4 m from the left support (Point B).
• Point load of 10 kN at 8 m from the left support (Point C).
• Uniformly Distributed Load (UDL) of 40 kN/m from 2 m to 8 m.

Let's denote the beam as having supports at $x = 0$ (left support, $R_A$) and $x = 10 \, \text{m}$ (right support, $R_B$).

Equilibrium Equations:

1. Sum of Vertical Forces: $R_A + R_B = 10 + 15 + 10 + (40 \times 6)$

2. Sum of Moments about Point A: $\sum M_A = 0 = R_B \times 10 - 10 \times 2 - 15 \times 4 - 10 \times 8 - (40 \times 6) \times (2 + 3)$

Let's calculate these values:

Step 2: Draw SFD and BMD

After calculating support reactions, the SFD and BMD can be drawn to understand the distribution of shear forces and moments along the beam.

Step 3: Calculate Maximum Shear and Bending Moment

Identify the critical points in the beam based on the shear and bending diagrams.

Step 4: Check Beam Capacity

Compare the maximum bending stress with the yield strength of A36 steel to ensure safety.

Let's start by calculating the support reactions.The reactions at the supports are:

• Reaction at the left support ($R_A$) = 139 kN
• Reaction at the right support ($R_B$) = 136 kN

Next Steps:

1. Shear Force Diagram (SFD): We will plot the shear force at different points along the beam.
2. Bending Moment Diagram (BMD): We will calculate and plot the bending moment at key points.
3. Maximum Shear and Moment: Find the maximum values for design checks.
4. Capacity Check: Compare the maximum bending moment with the beam's capacity using A36 steel properties.

Would you like to proceed with the diagrams and capacity check?

Related Questions:

1. How do you calculate the maximum bending moment for a given beam configuration?
2. What are the steps to plot shear force and bending moment diagrams?
3. How does the position of loads affect the beam's reaction forces?
4. What safety factors are used when designing beams in structural engineering?
5. How does material yield strength influence beam selection?

Tip: Always verify the support conditions and loading distribution before proceeding with calculations to ensure accurate results.