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Marston-Spangler theory for analysis of loads on sewer sanitary system pipe is established by Marston and it is applicable for most of the loading conditions that may encountered in site. Not only does this analysis approach based on the theory but also on experiments, and it is considered to be substantially reliable but conservative. Marston-Spangler theory for analysis of loads on sewer sanitary pipe is explored in this article.Contents:

**Marston-Spangler Load Analysis Theory for Sewer Sanitary System**

Following topics regarding load analysis for sewer sanitary system is discussed:
- Marston-Spangler theory and assumptions
- Types of loading conditions

**Marston-Spangler Load Analysis Theory and Assumptions**

The theory states that the load on the installed sewer sanitary pipe is equal to the weight of soil prism, which is termed as interior prism, on the pipe minus or plus the frictional shearing force transferred to the soil over the pipe by the trench wall side or exterior soil prisms on either side of interior prism.
The direction and magnitude of the frictional force is dependent on the settlement of the prism over the pipe in relation to the neighboring soil prisms.
Assumptions considered in this theory is that, the computed load equal to the load developed when maximum settlement is realized and Rankine theory is used to calculate lateral pressure which generates shearing force between the soil prism over the pipe and adjacent soil prisms.
Finally, the general form of the equation of Marston-Spangler load analysis theory can be seen below:
*W=CwB ^{2}*

**->Equation-1**

**Where:**

*W*: vertical load per unit length acting on sewer pipe due to gravity soil loads

*C*: dimensionless coefficient that combine the influence of the height of fill to trench width ratio, shearing force between interior prism and adjacent prisms and the direction and magnitude of interior prism settlement in relation to the adjacent prisms for embankment conditions.

*w*: soil unit weight

*B*: sanitary sewer trench breadth

**Types of Loading Conditions on Sewer Sanitary System**

Even though all factors required for the analysis of all types of sewer sanitary pipe placement conditions are included in the general form of the equation of Marston-Spangler theory, it is more suitable to categorize the placement conditions which may be encountered in the field, derive specific form of equations and generate graphs and tables for each type of installation condition.
Trench, Negative-projecting embankment, Positive-projecting embankment and induced trench condition, which is a specific installation condition, are types of sewer sanitary pipe placement methods considered while Marston-Spangler loads analysis is conducted. These are explained in the following sections separately:
**Trench Installation Condition**** ****for Sewer Pipes**

Trench condition is the case where the sanitary sewer pipe is placed in nearly narrow width trench excavated in undisturbed soil and covered with backfill materials to the original ground level as shown in Figure-1.
**Fig.1: Loads Imposed on Sewer Pipes in Trench Condition**

**The weight of the prism is computed according to the following formula:**

*W _{c}=C_{d}wB_{d}^{2} *

**-> Equation-2**

*W*: soil prism load

_{c}*w*: backfill material density

*B*: trench width at the top of the pipe

_{d}*C*: is computed according to the following equation:

_{d}**Where:**: friction coefficient between interior prism and adjacent undisturbed soil

*H*: height of backfill material above the pipe

*k*: Rankineâ€™s ratio of active lateral unit pressure to vertical unit pressure, and it can be computed using the following expression:

**Where:**: coefficient of internal friction of backfill material and it is always equal or greater than . It should be known that equation-2 provides total load imposed on the sanitary sewer pipe. If the pipe is rigid, then it will support all the load calculate using equation-2. However, if the pipe is flexible and soil on both sides of the pipe is compacted to an extent that experience deformation under vertical load, that is smaller than the load under which the pipe itself is deformed, then backfill material on either side of the pipe are anticipated to support a portion of the vertical load. So, the equation used to estimate loads on the pipe would be as follow:

*W _{c}=C_{d} wB_{c} B_{d}*

**-> Equation-5**

*B*is the outside width of the pipe. Finally, it is recommended to estimate the transition width of the trench because if the trench width is wider than the transition width for any reason, then it may be required to employ sanitary sewer pipe with greater strength. The

_{c}**transition width**is the trench width at which the maximum vertical load is realized and the extra increase of trench width would not influence the load.

**Positive Projecting Embankment Condition for Sewer Pipes**

In the case of positive projecting embankment condition, the pipe is covered above the original ground surface or the trench width is so large that the influence of trench wall friction does not exist and the top of the pipe is above the original ground surface.
The load exerted on sanitary sewer pipe in positive projecting embankment condition is equal to the weight of soil prism above the pipe plus or minus the shearing force extended from the sewer pipe side upward into the embankment.
When adequate embankment height is available, then it is likely that shearing force would not reach the top of the embankment but rather ends at some distance above the top of the pipe. The location at which shearing force ends is termed as plane of equal settlement as shown in Figure-2.
Moreover, Figure-2 illustrates different types of settlements which involves settlement of natural ground adjacent to the pipe (*s*), sewer pipe deflection (

_{g}*d*), settlement at the bottom of the pipe (

_{c}*s*), and compression of soil columns (

_{f}*s*) of height (

_{m}*pB*) that affect the load on the positive projecting sewer pipe. Furthermore, if the sanitary sewer pipe is pile supported in organic soil, it would be necessary to take down drag loads into consideration and the load imposed on the pipe is larger than that of interior prism over the pipe.

_{c}**Fig.2: Settlement which affects the Load Imposed on Positive Projecting Sanitary Sewer Pipe**

*W _{c}*

*=C*_{c}wB_{c}^{2}**->Equation 6**

**Where:**

*W*: load supported by sewer sanitary pipe

_{c}*w*: backfill material density

*B*: outside width of the pipe

_{c}*C*: can be estimated using one of the following expressions: If plane of equal settlement ends at a distance above the pipe, the means

_{c}*H>H*as can be observed in Figure-2, then the following formula is used to compute (

_{e}*C*) Otherwise, the equation provided below should be applied to estimate (

_{c}*C*): To compute (

_{c}*H*) in equation-7, the relationship between the deflection of the sewer sanitary pipe and relative settlement between interior soil prism and neighboring soil prisms should be considered. This relationship is called settlement ratio and may be evaluated as follow: Settlement ration design values have been established and advised to employ. These values can be found in Table-1.

_{e}**Table-1: Design Value for Settlement Ratio**

Sanitary sewer pipe type |
Soil condition |
Settlement ratio, r_{sd} |

Rigid | Rock or unyielding foundation | +1 |

Rigid | Ordinary foundation | +0.5 to +0.8 |

Rigid | Yielding foundation | 0 to +0.5 |

Rigid | Negative projecting installation | -0.3 to +0.5 |

Flexible | Poorly compacted fills on either side of pipe | -0.4 to 0 |

Flexible | Well compacted fill on either side of the pipe | 0 |

**) times projection ratio (**

*r*_{sd}*p*). The projection ratio is equal to the vertical distance that the sanitary sewer pipe projects above the original ground surface divided by outside vertical height of the pipe (

*B*). The value of (

_{c}'*C*) can be estimated for different values [(

_{c}*H/B*] and projection ratio (

_{c}) r_{sd}*p*) using Figure-3. It can be noticed that curved lines represent complete trench condition and projection condition whereas straight lines represent incomplete trench condition and projection condition. If the plane of equal settlement is located above the embankment, then the installation is termed as complete trench condition or complete projection condition according to the direction of the shearing force. However, If the plane of equal settlement is within embankment as shown in Figure-2, then the installation is

*incomplete trench condition*or*incomplete projection condition.***Fig.3: Determination of C_{c} Coefficient for Positive Projecting Sewer Pipes**

**Negative Projecting Embankment Condition for Sewer Pipes**

In the case of negative embankment condition, the pipe is placed in a trench which is narrow compared with pipe size and trench depth. In this installation condition, the top of the pipe is below the original ground surface and the fill material over the pipe exceeds the original ground level surface as shown in Figure-4.
**Fig.4: Load on Negative Projection Sewer Pipes and Different Types of Settlement**

*W _{c}=C_{n} wB_{d}^{2} -> Equation-10*

*W _{c}=C_{n} wB_{d} B_{d}^{'} -> Equation-11*

**Where:**

*W*: load on the pipe

_{c}*w*: backfill material density

*B*: trench breadth

_{d}*B*: average of trench width and outside diameter of the pipe

_{d}'*C*: can be estimated using one of the following expressions: If

_{n}*H>H*as shown in Figure-4, then the following formula is used to compute (

_{e}*C*) Otherwise, the equation provided below should be applied to estimate (

_{n}*C*): Similar to the positive projecting condition, it is required to specify the relationship between pipe deflection and relative settlement between interior prism and exterior prisms so as to evaluate (

_{n}*H*). the relationship that is called settlement ratio is expressed as follow:

_{e}**Where:**

*s*: the compression within the fill for the height of (

_{d}*p'B*), where p' negative projection ratio which is equal to the vertical distance from the top of the pipe to the original ground surface at the time of installation divided by trench width. If the natural ground surface is not levelled, then it is required to consider average vertical distance from the top of sanitary sewer pipe to the both side of the trench Finally, table-1 provides recommended design values for settlement ratio for negative projecting condition. The value of (

_{d}*C*) can be estimated for different values (

_{n}*r*) and projection ratio (

_{sd}*p*) value of 0.5, 1. 1.5, and 2. Values falling between the provided projection ratio (

*p*) should be found by interpolation.

**Fig.5: C_{n} values for Negative Projecting Condition Sewer Pipes**

**Induced Trench Condition for Sewer Pipes**

As can be noticed from the Figure-5, induced trench sewer pipe is initially placed as positive projecting sewer pipe. After that, an embankment, which is extended to a certain height above the ground surface is constructed over the pipe and properly compacted.
Then, a trench is dug over the sewer pipe as shown in Figure-5, and filled with compressible material. Finally. The remaining part of the embankment is completed.
The load on induced trench sewer pipe is estimated as follows:
**W _{c}=C_{n} wB_{c}^{2} -> **

**Equation-15**

**Where:**

*B*: is the width of infill which is the same as the width of the pipe

_{c}*C*: the same procedure used in the negative projecting condition is employed to estimate this coefficient.

_{n}**Fig.6: Induced Trench Sewer Pipe**

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