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To solve this problem, let's go through the required calculations step by step. The beam is subject to distributed and point loads, so we will calculate the following:

1. **Reactions at Supports**: Calculate the reactions at support \( A \) (using equilibrium equations) to account for the distributed load \( w_{AB} \), and the point loads \( P_C \) and \( P_E \).
  
2. **Moment of Inertia (I)**: Calculate the moment of inertia for the cross-sectional shape shown in the second figure.

3. **Maximum Bending Moment**: Using shear and moment diagrams, calculate the maximum bending moment \( M \) at the critical locations along the beam.

4. **Bending Stress (Tension and Compression)**: Apply the bending stress formula:
   \[
   \sigma = \frac{M \cdot c}{I}
   \]
   where \( \sigma_T \) is the maximum tension bending stress and \( \sigma_C \) is the maximum compression bending stress.

### Step-by-Step Solution:

#### Step 1: Calculate Reactions at Supports
Using the beam dimensions and loads given:
- Distributed load: \( w_{AB} = 940 \, \text{lb/ft} \) over \( L_{AB} = 7 \, \text{ft} \).
- Point loads: \( P_C = 2170 \, \text{lb} \) at \( L_{AB} + L_{BC} = 10 \, \text{ft} \).
- Point load: \( P_E = 1600 \, \text{lb} \) at the end of the beam \( x = 15 \, \text{ft} \).

Use the sum of moments about \( A \) and the sum of forces in the vertical direction to determine the reaction forces.

#### Step 2: Calculate Moment of Inertia (I)
Given the dimensions of the cross-section:
- \( b_1 = 7 \, \text{in.}, \, b_2 = 2 \, \text{in.}, \, b_3 = 4 \, \text{in.} \)
- \( d_1 = 7 \, \text{in.}, \, d_2 = 11 \, \text{in.}, \, d_3 = 2 \, \text{in.} \)

Calculate the moment of inertia \( I \) about the neutral axis \( y \)-\( z \) using the parallel-axis theorem if necessary.

#### Step 3: Calculate Maximum Bending Moment \( M \)
Using the shear and moment diagrams:
1. Integrate the shear forces to find the bending moment at each section.
2. Identify the location where the bending moment is maximum (likely under the largest load or between load applications).

#### Step 4: Calculate Bending Stress
Using the bending stress formula \( \sigma = \frac{M \cdot c}{I} \):
- \( \sigma_T \): The maximum tension bending stress occurs at the farthest point from the neutral axis on the tension side.
- \( \sigma_C \): The maximum compression bending stress occurs at the farthest point from the neutral axis on the compression side.

Given the answers:
- \( \sigma_T = 1020.57 \, \text{psi} \)
- \( \sigma_C = 4877.73 \, \text{psi} \)

If you'd like further breakdowns on any of these steps, let me know!

### Follow-Up Questions:
1. How do we calculate the centroidal location in the cross-section?
2. How does the distributed load affect the bending moment along different points?
3. What is the significance of the moment of inertia in determining bending stress?
4. How would adding additional loads change the maximum bending stress?
5. Why is it important to consider both tension and compression in bending stress calculations?

### Tip:
Remember that the location of maximum bending stress is determined by both the magnitude of the moment and the distance from the neutral axis.

A flanged wooden shape is used to support the loads shown on the beam. The dimensions of the shape are shown in the second figure. Assume LAB = 7 ft, LBC = 3 ft, LCD = 2 ft, LDE = 3 ft, PC = 2170 lb, PE = 1600 lb, wAB = 940 lb/ft, b1 = 7 in., b2 = 2 in., b3 = 4 in., d1 = 7 in., d2 = 11 in., d3 = 2 in. Consider the entire 15-ft length of the beam and determine: (a) the maximum tension bending stress σT at any location along the beam, and (b) the maximum compression bending stress σC at any location along the beam.

Learn how to calculate the maximum tension and compression bending stress in a flanged wooden beam supporting multiple loads. This detailed solution covers bending stress calculations using moment of inertia and shear/moment diagrams, ideal for undergraduate engineering students.

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