The given problem is a statically determinate beam with two supports: a pinned support at A and a roller support at B. The beam is subject to the following loads:

• A uniformly distributed load of $w = 1.5 \, \text{k/ft}$ over 6 feet starting from A.
• A point load of $P = 17 \, \text{kips}$ acting 12 feet from A.

Objective:

We need to calculate the reactions at support A and support B. The beam is in static equilibrium, so we can apply the following equilibrium equations:

1. $\sum F_y = 0$ (Sum of vertical forces)
2. $\sum M_A = 0$ (Moment about point A)

Step 1: Calculate the total uniformly distributed load (UDL)

The total force due to the UDL is calculated by multiplying the load intensity by the length over which it acts: $F_{\text{UDL}} = w \times L = 1.5 \, \text{k/ft} \times 6 \, \text{ft} = 9 \, \text{kips}$ This load acts at the centroid of the distributed load, which is located at 3 feet from point A (half the length of the UDL).

Step 2: Apply equilibrium equations

Moment about point A

Taking moments about point A (clockwise moments are positive): $\sum M_A = 0 \implies (R_B \times 18 \, \text{ft}) - (9 \, \text{kips} \times 3 \, \text{ft}) - (17 \, \text{kips} \times 12 \, \text{ft}) = 0$ Substitute the known values: $18 R_B - 27 - 204 = 0$ $18 R_B = 231$ $R_B = \frac{231}{18} = 12.83 \, \text{kips}$

Vertical force equilibrium ($\sum F_y = 0$)

$\sum F_y = 0 \implies R_A + R_B = 9 \, \text{kips} + 17 \, \text{kips}$ Substitute $R_B = 12.83 \, \text{kips}$: $R_A + 12.83 = 26 \, \text{kips}$ $R_A = 26 - 12.83 = 13.17 \, \text{kips}$

Final reactions:

• Reaction at A ($R_A$) = 13.17 kips
• Reaction at B ($R_B$) = 12.83 kips

Do you have any questions about this solution? Would you like further details on any part of the process?

Here are some related questions to further explore this topic:

1. How do you determine the point of application for a uniformly distributed load?
2. What are the differences between a pin support and a roller support in terms of reactions?
3. How does changing the position of the point load affect the reaction forces?
4. Can you calculate the shear force and bending moment diagrams for this beam?
5. How would the reaction forces change if the length of the UDL was increased?

Tip: When calculating moments, always ensure that the distances are measured from the point where you're summing the moments. This helps in avoiding errors when applying the moment equilibrium equation.