The solution of the consolidation equation is given by the series in the figure. By evaluation of the series, determine the average degree of consolidation Uavg to the nearest 0.001 for time factors 0.15, 0.6, 0.8 and infinity. Verify your computations by referring to the table provided for Uavg and T.
The solution of the consolidation equation is given by the series in the figure. By evaluation of the series, determine the average degree of consolidation Uavg to the nearest 0.001 for time factors 0.15, 0.6, 0.8 and infinity. Verify your computations by referring to the table provided for Uavg and T.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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Question
The solution of the consolidation equation is given by the series in the figure. By evaluation of the series, determine the average degree of consolidation Uavg to the nearest 0.001 for time factors 0.15, 0.6, 0.8 and infinity. Verify your computations by referring to the table provided for Uavg and T.
![The equation presented on the educational website represents a mathematical expression used to calculate the average utilization \( U_{\text{avg}} \) as a percentage. Here is the transcription of the equation:
\[
U_{\text{avg}} (\%) = 100 \left[ 1 - \sum_{n=0}^{\infty} \frac{8}{(2n + 1)^2 \pi^2} e^{-\frac{(2n + 1)^2 \pi^2}{4} T} \right]
\]
**Explanation:**
- **\( U_{\text{avg}} (\%) \):** Represents the average utilization as a percentage.
- **100:** This factor scales the result to a percentage.
- **Summation (\(\sum\)):** The expression contains an infinite series summing over \( n \) from 0 to infinity.
- **\( \frac{8}{(2n + 1)^2 \pi^2} \):** The term inside the summation, where \( n \) represents the index of summation.
- **Exponential Term \( e^{-\frac{(2n + 1)^2 \pi^2}{4} T} \):** This adds an exponentially decaying factor to each term in the series. The variable \( T \) typically represents time or another parameter affecting the decay.
- **Brackets \([ \cdots ]\):** Encapsulate the entire series to ensure that subtraction from 1 happens after the calculation of the series.
This equation might be useful in contexts like heat transfer, vibrations, or signal processing, where such forms are common for describing transient responses or distributions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40578013-3e17-4421-b4af-d1f1eb4da799%2F8fca0a19-ec3a-468b-b674-8c7e479574f5%2Fnk5n7d_processed.png&w=3840&q=75)
Transcribed Image Text:The equation presented on the educational website represents a mathematical expression used to calculate the average utilization \( U_{\text{avg}} \) as a percentage. Here is the transcription of the equation:
\[
U_{\text{avg}} (\%) = 100 \left[ 1 - \sum_{n=0}^{\infty} \frac{8}{(2n + 1)^2 \pi^2} e^{-\frac{(2n + 1)^2 \pi^2}{4} T} \right]
\]
**Explanation:**
- **\( U_{\text{avg}} (\%) \):** Represents the average utilization as a percentage.
- **100:** This factor scales the result to a percentage.
- **Summation (\(\sum\)):** The expression contains an infinite series summing over \( n \) from 0 to infinity.
- **\( \frac{8}{(2n + 1)^2 \pi^2} \):** The term inside the summation, where \( n \) represents the index of summation.
- **Exponential Term \( e^{-\frac{(2n + 1)^2 \pi^2}{4} T} \):** This adds an exponentially decaying factor to each term in the series. The variable \( T \) typically represents time or another parameter affecting the decay.
- **Brackets \([ \cdots ]\):** Encapsulate the entire series to ensure that subtraction from 1 happens after the calculation of the series.
This equation might be useful in contexts like heat transfer, vibrations, or signal processing, where such forms are common for describing transient responses or distributions.
Transcribed Image Text:**Table 1: Degree of Drainage Values**
| \( U_{\text{avg}} \) (%) | \( T \) | \( U_{\text{avg}} \) (%) | \( T \) | \( U_{\text{avg}} \) (%) | \( T \) | \( U_{\text{avg}} \) (%) | \( T \) |
|--------------------------|---------|--------------------------|---------|--------------------------|---------|--------------------------|---------|
| 0 | 0 | 26 | 0.0531 | 52 | 0.212 | 78 | 0.529 |
| 1 | 0.00008 | 27 | 0.0572 | 53 | 0.221 | 79 | 0.547 |
| 2 | 0.0003 | 28 | 0.0615 | 54 | 0.230 | 80 | 0.567 |
| 3 | 0.00071 | 29 | 0.0660 | 55 | 0.239 | 81 | 0.588 |
| 4 | 0.00126 | 30 | 0.0707 | 56 | 0.248 | 82 | 0.610 |
| 5 | 0.00196 | 31 | 0.0754 | 57 | 0.257 | 83 | 0.633 |
| 6 | 0.00283 | 32 | 0.0803 | 58 | 0.267 | 84 | 0.658 |
| 7 | 0.00385 | 33 | 0.0855 | 59 | 0.276 | 85 | 0.684 |
| 8 | 0.00502 | 34 | 0.0907 | 60 | 0.286 | 86 | 0.712 |
| 9 | 0.00636 | 35 | 0.0962 |
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