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This problem involves a composite beam made of wood and steel, with dimensions and material properties given, including the modulus of elasticity and allowable stress for each material. The objective is to determine the largest permissible bending moment about the horizontal axis.

To solve this, we need to follow these steps:

### Step 1: Equivalent Section for the Composite Beam
Since the beam consists of wood and steel, we convert the steel section to an equivalent wood section using the **modular ratio**, which is the ratio of the modulus of elasticity of steel to that of wood:
\[
n = \frac{E_{steel}}{E_{wood}} = \frac{29 \times 10^6 \text{ psi}}{2 \times 10^6 \text{ psi}} = 14.5
\]
This means that the steel is 14.5 times stiffer than the wood.

### Step 2: Moment of Inertia of the Equivalent Section
We will convert the dimensions of the steel plates into equivalent wood by multiplying their width by the modular ratio, and then compute the moment of inertia of the composite section.

- **For the wood sections**: The total width of the wood is \(2 \times 2 \text{ in} = 4 \text{ in}\).
- **For the steel plates**: The equivalent width of the steel plates is \(2 \times 14.5 = 29 \text{ in}\), meaning each plate becomes \(14.5 \text{ in}\) wide in the equivalent wood section.

We calculate the moment of inertia, \(I\), for this transformed section (assuming symmetry and using the parallel axis theorem if necessary).

### Step 3: Calculate Maximum Allowable Moment
Once we have the moment of inertia of the equivalent section, we can find the maximum bending moment using the formula for the maximum bending stress:
\[
M_{max} = \frac{\sigma_{allowable} \cdot I}{c}
\]
where:
- \(\sigma_{allowable}\) is the allowable stress,
- \(I\) is the moment of inertia,
- \(c\) is the distance from the neutral axis to the outermost fiber (half the height, since the beam is symmetric).

We will calculate \(M_{max}\) for both the wood and steel, considering the allowable stresses for each material (2000 psi for wood, 22,000 psi for steel), and choose the smallest one as the limiting moment.

Would you like to proceed with the detailed calculations, or do you have any specific questions so far?

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Here are five questions to deepen your understanding:
1. How is the modular ratio used to transform composite sections in bending problems?
2. What role does the moment of inertia play in determining the bending capacity of a beam?
3. Why is it important to consider both wood and steel allowable stresses when calculating the maximum bending moment?
4. How does the parallel axis theorem apply when calculating the moment of inertia for composite beams?
5. In what other situations might composite beams be used, and how are they beneficial?

**Tip:** In problems involving composite materials, always transform the stiffer material to the equivalent properties of the softer material for easier calculations.

Three wooden beams and two steel plates are securely bolted together to form the composite member shown. Using the data given, determine the largest permissible bending moment when the member is bent about a horizontal axis.

Determine the largest permissible bending moment for a composite beam made of wood and steel. This solution involves key concepts like the modular ratio, moment of inertia, and bending stress, with data on the modulus of elasticity and allowable stress for each material.

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