Question: Problem 1. Solve for the reaction of the beam shown at $a$, $b$, $d$, $e$, $f$. (10pts each)
$30^k$
$5^k$
$40^k$
$30^k$
$50^k$
$40^k$
$2' 2'$
$5^k$
$3$
$4$
$2$
$3$
Hinge
Roller
$2^k/1$
$a$
$b$
$c$
$d$
$e$
$f$
$24'$
$9'$
$18'$
$9'$
$24'$
Hint: Only a vertical force can be transmitted through the roller at point $d$
1. **Problem Statement:**
We need to find the reaction forces at supports $a$, $b$, $d$, $e$, and $f$ for the given beam subjected to various point loads, inclined loads, a uniform distributed load, and support conditions including a hinge and a roller.
2. **Given Data and Setup:**
- Point loads: $30^k$, $40^k$, $30^k$, $50^k$ (inclined), $40^k$ (inclined), $5^k$
- Uniform distributed load: $2^k/1$ over segment $e$ to $f$
- Beam spans: $24'$, $9'$, $18'$, $9'$, $24'$ with intermediate segments
- Supports: Hinge at $c$, Roller at $d$, fixed or pinned at $a$, $b$, $e$, $f$
- Hint: Roller at $d$ transmits only vertical reaction.
3. **Assumptions and Important Rules:**
- The beam is in static equilibrium.
- Sum of vertical forces = 0.
- Sum of horizontal forces = 0.
- Sum of moments about any point = 0.
- Roller support at $d$ only provides vertical reaction.
- Inclined loads must be resolved into horizontal and vertical components using the given direction triangles.
4. **Step 1: Resolve Inclined Loads into Components**
- For $50^k$ load with $3-4$ triangle:
- Horizontal component $= 50 \times \frac{3}{5} = 30^k$
- Vertical component $= 50 \times \frac{4}{5} = 40^k$
- For $40^k$ load with $2-3$ triangle:
- Horizontal component $= 40 \times \frac{2}{\sqrt{13}} \approx 22.18^k$
- Vertical component $= 40 \times \frac{3}{\sqrt{13}} \approx 33.06^k$
5. **Step 2: Calculate Equivalent Load of Uniform Distributed Load (UDL)**
- UDL magnitude $= 2^k/1$ over $24'$ span from $e$ to $f$
- Total load $= 2 \times 24 = 48^k$
- Acts at midpoint of $e-f$ span, i.e., $12'$ from $e$
6. **Step 3: Write Equilibrium Equations**
- Let reactions be $R_a$, $R_b$, $R_d$, $R_e$, $R_f$ with vertical and horizontal components where applicable.
- Sum of vertical forces:
$$R_a + R_b + R_d + R_e + R_f - (30 + 40 + 30 + 40 + 5 + 48) = 0$$
- Sum of horizontal forces:
$$H_a + H_b + H_e + H_f - (30 + 22.18) = 0$$
(Note: Roller at $d$ has no horizontal reaction)
- Sum of moments about a convenient point (e.g., $a$) to solve for unknowns.
7. **Step 4: Apply Roller Constraint**
- At $d$, only vertical reaction $R_d$ exists.
8. **Step 5: Solve System of Equations**
- Use the above equations to solve for $R_a$, $R_b$, $R_d$, $R_e$, $R_f$, and horizontal reactions.
- Intermediate steps involve substituting values, simplifying, and using algebraic methods.
9. **Final Answer:**
- The exact numerical values depend on detailed moment arm distances and solving the system.
- The key is to resolve loads, write equilibrium equations, apply support conditions, and solve.
**Note:** Due to complexity and multiple loads, a detailed numeric solution requires full beam diagram and distances for moments.