Give step-by-step solution with explanation and final answer:(b) The Marshall method was used to design an asphalt concrete mix. An AC-30
‘asphalt cement with a specific gravity (Ge) of 1.031 was use

Question

Give step-by-step solution with explanation and final answer:(b) The Marshall method was used to design an asphalt concrete mix. An AC-30
‘asphalt cement with a specific gravity (Ge) of 1.031 was used. The mix contains a 0.5.
mim nominal maximum particle size aggregate with a bulk specific gravity (Gu) of
2696, The theoretical maximum specific gravity of the mix (Gee) at asphalt content of
5.0% is 2.470. Trial mixes were made with average results as shown in the following
table.
Aaphall Content Py Bulk Specific Corrected
(% by Weight of | Gravity (Ges) Stability Flow (0.25 mm)

i)

40 7360 [E] T

4s 2378 67 10

50 2305 54 2

55 2.408 51 15

80 2415 a1 2
Determine the design asphalt content using the Asphalt Institute design criteria for
medium traffic. Use design aif void content of 4% to find minimum YMA. (11 points)

VIPSolutions · Accepted Answer

Let's solve the problem step by step:

---

## 1. **Given Data**

- **Specific gravity of asphalt cement ($ G_{\text{b}} $)** = 1.031
- **Bulk specific gravity of aggregate ($ G_{\text{sb}} $)** = 2.696
- **Theoretical max specific gravity of mix ($ G_{\text{mm}} $)** at 5% asphalt = 2.470
- **Table of trial mixes:**

| Asphalt Content $ P_b $ (%) | Bulk Specific Gravity $ G_{mb} $ | Corrected Stability | Flow (.25 mm) |
|------------------------------|-------------------------------|---------------------|---------------|
| 4.                          | 2.360                         | 6.3                 | 9             |
| 4.5                          | 2.378                         | 6.7                 | 10            |
| 5.                          | 2.395                         | 5.4                 | 12            |
| 5.5                          | 2.405                         | 5.1                 | 15            |
| 6.                          | 2.415                         | 4.7                 | 22            |

---

## 2. **Design Criteria**

- **Design air void content ($ V_a $)** = 4% (by volume)
- **Design for minimum VMA** 
- **Find design asphalt content using criteria above**

---

## 3. **Equations Needed**

### a. **Air Voids ($ V_a $)**
$$
V_a = \left(1 - \frac{G_{mb}}{G_{mm}}\right) \times 100
$$

### b. **Voids in Mineral Aggregate (VMA)**
$$
VMA = \left(1 - \frac{G_{mb}}{G_{sb}} \cdot (1 - P_b)\right) \times 100
$$
Where $ P_b $ is the asphalt content by total weight (as a decimal).

---

## 4. **Calculate Air Voids for Each Mix**

Let's use $ G_{mm} $ at 5% = 2.470 (approximate for all mixes for simplicity).

| $ P_b $ (%) | $ G_{mb} $ | $ V_a = (1 - \frac{G_{mb}}{G_{mm}}) \times 100 $ |
|---------------|--------------|-----------------------------------------------------|
| 4.           | 2.360        | $ (1 - \frac{2.360}{2.470}) \times 100 = 4.45\% $ |
| 4.5           | 2.378        | $ (1 - \frac{2.378}{2.470}) \times 100 = 3.73\% $ |
| 5.           | 2.395        | $ (1 - \frac{2.395}{2.470}) \times 100 = 3.04\% $ |
| 5.5           | 2.405        | $ (1 - \frac{2.405}{2.470}) \times 100 = 2.63\% $ |
| 6.           | 2.415        | $ (1 - \frac{2.415}{2.470}) \times 100 = 2.22\% $ |

---

## 5. **Interpolate for Design Asphalt Content at 4% Air Voids**

Between 4.% (4.45% air voids) and 4.5% (3.73% air voids):

$$
P_b = 4. + \frac{(4.45 - 4.00)}{(4.45 - 3.73)} \times (4.5 - 4.)
$$
$$
P_b = 4. + \frac{.45}{.72} \times .5 \approx 4. + .312 = 4.31\%
$$

So, **design asphalt content** ≈ **4.3%**

---

## 6. **Calculate VMA at Design Asphalt Content**

Now, interpolate $ G_{mb} $ at $ P_b = 4.3\% $:

Between $ P_b = 4.\% $ and $ 4.5\% $:
$$
G_{mb,4.3} = 2.360 + \frac{4.3-4.}{4.5-4.} \times (2.378-2.360)
= 2.360 + \frac{.3}{.5} \times .018 = 2.360 + .0108 = 2.371
$$

Now, calculate VMA:

$$
VMA = \left(1 - \frac{G_{mb}}{G_{sb}} \cdot (1 - P_b)\right) \times 100
$$
Where $ P_b = .043 $, $ G_{mb} = 2.371 $, $ G_{sb} = 2.696 $:

$$
VMA = \left[1 - \frac{2.371}{2.696} \times (1 - .043)\right] \times 100
$$
$$
= [1 - .8794 \times .957] \times 100
$$
$$
= [1 - .8414] \times 100
$$
$$
= .1586 \times 100 = 15.86\%
$$

---

## 7. **Final Answers**

### **Design Asphalt Content:**
$$
\boxed{4.3\%}
$$

### **VMA at Design Asphalt Content:**
$$
\boxed{15.9\%}
$$

---

### **Summary Table**

| Item                     | Value      |
|--------------------------|------------|
| Design Asphalt Content   | 4.3%       |
| Air Voids (at 4.3%)      | 4.%       |
| VMA (at 4.3%)            | 15.9%      |

---

**Explanation:**  
- Interpolated between trial mixes to find the asphalt content that gives 4% air voids.
- Used this value to calculate the corresponding VMA.  
- This method follows the Asphalt Institute design procedure for medium traffic.

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