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The problem describes a simply supported beam made of Douglas Fir Larch, No. 1 grade timber, with a span of 15 feet. It is subject to a uniform distributed load of \( w = 0.2 \, \text{kip/ft} \) along its entire length, as well as two concentrated loads of \( P = 0.5 \, \text{kip} \) at its third points. We need to determine the smallest possible cross-section of the beam using the Allowable Stress Design (ASD) method. The maximum allowable bending stress for timber is given as \( F_b = 1,350 \, \text{psi} \).

### Steps to solve:

1. **Beam loading configuration:**
   - Length of the beam, \( L = 15 \, \text{ft} \).
   - Distributed load, \( w = 0.2 \, \text{kip/ft} \).
   - Concentrated loads \( P = 0.5 \, \text{kip} \) at one-third points, i.e., at \( L/3 = 5 \, \text{ft} \) and \( 2L/3 = 10 \, \text{ft} \).

2. **Reactions at the supports:**
   Since the beam is simply supported and symmetric, we can calculate the reactions at supports \( A \) and \( B \) due to both the distributed load and the concentrated loads. By symmetry, the reactions should be equal.

3. **Maximum bending moment:**
   We need to calculate the maximum bending moment, which will occur under the combined effect of the distributed load and the concentrated loads. The maximum moment will typically occur either at midspan or under the concentrated loads.

4. **Section modulus calculation:**
   Using the maximum moment from step 3, we will use the bending stress formula to find the required section modulus \( S \):
   \[
   M = F_b \cdot S
   \]
   where \( M \) is the maximum bending moment and \( F_b = 1350 \, \text{psi} \) is the allowable bending stress.

5. **Selecting the cross-section:**
   Based on the required section modulus, we will select the smallest cross-section of timber that satisfies the condition.

I'll now walk through these calculations. Let me know if you'd like a detailed breakdown of any specific steps!

A 15 ft long Douglas Fir Larch, No. 1 grade, simply supported beam has a uniformly distributed load of intensity (w) = 0.2 kip/ft through its entire length and two concentrated loads (P=0.5 kip) at its third points. Determine the smallest possible cross-section using the ASD method and make a selection from the structural lumber properties table for timber. Given maximum allowable bending stress Fb = 1,350 psi for timber.

This problem involves determining the smallest possible cross-section of a 15 ft long Douglas Fir Larch beam under a uniform distributed load and concentrated loads using the Allowable Stress Design (ASD) method. The problem provides insights into structural analysis concepts such as bending stress and moment calculations.

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