give answer in 2 step with explanation at the end of each step and final answer at the end:As a finale for the homework assignments in this course, Lyle the pig and his
twin brother, Kyle*, have deci

Question

give answer in 2 step with explanation at the end of each step and final answer at the end:As a finale for the homework assignments in this course, Lyle the pig and his
twin brother, Kyle*, have decided to perform a circus act by standing on the
stacked beams depicted below. If Lyle and Kyle each weigh 2 kN, determine
the support reactions atA, B,C, D, and E.
*For reference, Kyle is the one wearing the top hat. This is not pertinent to the
problem, but a pig in a top hat is always interesting.
1.5m 1.5m |
Cg (od 4
010 C10
CH J:
0 .
A 2 ®
2m | 3m 3m 2m

VIPSolutions · Accepted Answer

## Step 1: Analyze the Top Beam (Loads Transferred to D and E)

### Forces on Top Beam:
- Lyle (2 kN) stands 1.5 m from the left.
- Kyle (2 kN) stands 1.5 m from the right.
- The top beam rests on supports D (left) and E (right).

Let:
- Reaction at D: $ R_D $
- Reaction at E: $ R_E $

#### Sum of Vertical Forces (Top Beam):
$$
R_D + R_E = 2\, \text{kN} + 2\, \text{kN} = 4\, \text{kN}
$$

#### Moment About D:
Taking moments about D (distance between D and E is 3 m):

$$
\text{Moment at D:} \quad 2 \times 1.5 + 2 \times (3 - 1.5) - R_E \times 3 = 
$$
$$
3 + 3 - 3R_E = 
$$
$$
6 = 3R_E \implies R_E = 2\, \text{kN}
$$

#### Substitute Back:
$$
R_D + 2 = 4 \implies R_D = 2\, \text{kN}
$$

**Explanation:**  
Both supports D and E take equal load (2 kN each) because the pigs are symmetrically placed.

---

## Step 2: Analyze the Bottom Beam (Reactions at A, B, C)

### Loads on Bottom Beam:
- Vertical reactions at D and E from above (2 kN each, downwards).
- Pin support at A (vertical reaction $ R_A $).
- Pin support at B (vertical reaction $ R_B $).
- Roller at C (vertical reaction $ R_C $).

#### Sum of Vertical Forces (Bottom Beam):

$$
R_A + R_B + R_C = 2\, \text{kN} + 2\, \text{kN}
$$
$$
R_A + R_B + R_C = 4\, \text{kN}
$$

#### Moment About A:

Positions:
- D is 2 m from A.
- E is 5 m from A.
- B is 5 m from A.
- C is 8 m from A.

Taking moments about A (clockwise positive):

$$
2 \times 2 + 2 \times 5 - R_B \times 5 - R_C \times 8 = 
$$
$$
4 + 10 - 5R_B - 8R_C = 
$$
$$
14 = 5R_B + 8R_C
$$

#### Moment About B:

Relative to B (A is 3 m left, D is 3 m left, E is at B, C is 3 m right):

$$
-2 \times 3 - 2 \times  + R_A \times 3 + R_C \times 3 = 
$$
$$
-6 + 3R_A + 3R_C = 
$$
$$
3R_A + 3R_C = 6 \implies R_A + R_C = 2
$$

Now, we have:

1. $ R_A + R_B + R_C = 4 $
2. $ 14 = 5R_B + 8R_C $
3. $ R_A + R_C = 2 \implies R_A = 2 - R_C $

Substitute $ R_A $ into the first equation:

$$
(2-R_C) + R_B + R_C = 4 \implies R_B = 2
$$

Now use in second equation:

$$
14 = 5 \times 2 + 8R_C \implies 14 = 10 + 8R_C \implies 8R_C = 4 \implies R_C = .5
$$

Now $ R_A = 2 - R_C = 2 - .5 = 1.5 $

---

## **Final Answers**

### **Support Reactions:**
- **A:** $ \boxed{1.5\, \text{kN}} $
- **B:** $ \boxed{2\, \text{kN}} $
- **C:** $ \boxed{.5\, \text{kN}} $
- **D:** $ \boxed{2\, \text{kN}} $
- **E:** $ \boxed{2\, \text{kN}} $

---

### **Summary Table**

| Support | Reaction (kN) |
|---------|---------------|
|   A     |     1.5       |
|   B     |     2.       |
|   C     |     .5       |
|   D     |     2.       |
|   E     |     2.       |

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