Pipelines are used to transfer liquid and gas products from one place to other place where is in distant from source points. For example a pipeline between oil well to a refinery carries extracted oil to a refining facility for being refined. The refined oil may be transferred to a shipping terminal by other pipelines. Pipelines can be aboveground or underground. For analysis and designing a pipeline, piping stress engineer should have knowledge about piping behavior and design criteria mentioned in proper codes such as B31.4 or B31.8. For instance for designing a buried pipeline which is based on B31.4, engineers must know how soils interacts with pipes as well as proper criteria such as allowable stresses which are defined in the code. One of the major differences between pipeline and process piping is pipe materials. Pipes in pipeline industry are made of high strength steel because high internal pressure in pipelines leads to a thick pipe. Using a normal piping material increases the pipe thickness which is costly in transportation, fabrication and erection. Therefore in pipelines high yield strength materials are used. The consequence of using high-yield materials is a reduction in safety factor because the margin between yield and ultimate stress is low comparing to its corresponding value in a process pipes. So a stress engineer must pay attention to local stresses in discontinuity area such as branches. However, fortunately pipelines are straight in the most of its rout. One of the major problems on which an engineer my face is pressure and thermal elongation resulted from high pressure and big length of pipelines. Depending on the situation and the elongation value an engineer may find an economic solution. Increase in flexibility, using anchor blocks, using flexible supports and providing a system enabling equipment to move a portion of the pipeline movement are some possible solutions. 

Pipelines and Buried Pipes


2-Buried pipe behavior 

For analysis of buried pipes, an engineer should have knowledge of pipe-soil interaction because pipes are placed in soil and so the pipes behave differently comparing to aboveground piping systems which are not in touch with soil. This interaction introduces some new concepts and parameters. In order to reach a safe design of a buried pipe, an engineer must know types, sources and calculation methods of all forces exerted on the pipe. These forces define the behavior of buried pipes such as reaction to thermal and pressure elongations. By referring to a proper code in which allowable stresses, deformations and forces are defined and by using the pipe-soil interaction knowledge, engineers design buried piping systems to maintain operability and integrity.

 The figure 1 demonstrates a pipeline system which includes both underground and aboveground parts. 

Figure 1: Restrained and unrestrained parts of a Buried pipe 

The pipe is connected to two pumps which are far away from each other. The pipe which connected to the pumps at points D and D', goes underground at points C and C' respectively. Because of a friction between pipe and soil, the buried pipe between points A and A’ does not tend to move. So points A and A' play an anchor role in the system and are known as virtual anchor points. Consequently the system comprises two parts: restrained and unrestrained parts. Behavior of each part is important: stresses in restrained part are determining factor in the system design whereas the pipe end elongations in unrestrained part, at points B and B', are important factors in the system design. Moreover generally, depending on pipe location, a buried pipe may be under external pressure due to live surface loads, buoyancy loads, and other loads. In a safe design all these sources of stresses and strains should be considered. Therefore engineers should consider all factors which affect all parts of a buried system including restrained, unrestrained, underground and aboveground parts.

Depending of virtual anchor locations, a buried pipe may or may not include a restrained part. The following parts shed a light on behavior of restrained and unrestrained parts of an aboveground and underground pipe. 

3-Restrained AG/UG piping system

3-1-Developed stress 

The pressurized pipe shown in figure 2 is heated and anchored between two points A, B. The developed stresses in the pipe wall are as follows:

Figure 2: A restrained pressurized pipe under a temperature change 

3-1-1-Range stress due to thermal load


3-1-2-Longitudinal stress 


Note: Referring to B31.4-2012, the above longitudinal stress developed in a restrained system, includes range stress.

3-1-1-Combined stress

Figure 3: Stresses on an element of a restrained pipe 

Referring to figure 3, combined stress is calculated as following:


Note: B31.4-2012 considerers an allowable stress for combined stress in a restrained system.

3-1-4-Effective stress 

Referring note (2) of table 403.3.1-1 in B31.4-2012, an effective stress is “sum of the stress caused by temperature change and from circumferential, longitudinal, and radial stresses from internal design pressure and external loads in pipe installed under railroads or highways.”

An allowable stress has been mentioned for the effective stress in B31.4-2012.

3-1-5-Elastic stability for AG restrained system 

Referring 403.3.9.3 of B31.4-2012, system elastic stability should be checked.

3-1-6-Criteria stresses

The criteria stresses for various load cases are mentioned in Table 403.3.1-1 of B31.4-2012. Obviously, for stress calculation and its criteria engineer must refer to proper codes on which the system must be designed. 

4-Unrestrained AG/UG piping system

4-1-Developed stress 

4-1-1-Range stress due to thermal load

If an aboveground or unrestrained part of an underground pipe which is under temperature change faces a resistance, some internal loads are developed in the pipe. These internal loads including axial, shear forces and bending moments create stresses which are known as thermal expansion or contraction stress. It is calculated in such a way all temperature range of the system, from minimum to maximum system temperature is covered. Addressing codes B31.3 or B31.4, because of shakedown; if system can go beyond  at the beginning of thermal cycles (follow up does not occurs), the allowable range stress which is defined in the code satisfies integrity of the system. 

Because of complexity of rang stress calculation, a finite element method is used for the object. The only difference between AG and UG pipe is that the pipe-soil interaction should be considered in UG analysis. The soil-pipe interaction is modeled as springs with specific stiffness which are calculated based on the interaction characteristics. For being ensured that the system sustains its integrity, a proper code should be used for both range stress calculation and its criteria. For example B31.4-2012 suggests the following formula for the range stress.




4-1-2-Longitudinal stress 

This stress comprises longitudinal stresses due to internal pressure, sustained stresses resulting from system weight and stresses due to pressure thrust.


4-1-3-Effective stress 

Referring note (2) of table 403.3.1-1 in B31.4-2012, an effective stress is “sum of the stress caused by temperature change and from circumferential, longitudinal, and radial stresses from internal design pressure and external loads in pipe installed under railroads or highways.”

An allowable stress has been mentioned for the effective stress in B31.4-2012.

4-1-4-Criteria stresses

The criteria stresses for various load cases are mentioned in Table 403.3.1-1 of B31.4-2012. Obviously, for stress calculation and its criteria engineer must refer to proper codes on which the system must be designed. 

4-2-Pressure and thermal pipe elongation

4-2-1-Free Pressure and thermal elongation of an AG piping system

 As shown if figure 4, the pipe elongation resulting from temperature and internal pressure are   and        respectively. 

Figure 4: Free thermal and pressure elongation of an AG pipe

Applying Hook’s law for element M:




From substituting equations (9) and (10) in (8):



From equations (11) and (12):


Equation (13) can be written as:


In which



4-2-2-Pressure and thermal elongation of an AG piping system with a resistant force

In case of any resistance against thermal and pressure pipe movements, a resistant force (Q) is developed as shown in figure 5:

Figure 5: Thermal and pressure elongation of an AG pipe facing a force



From equations (17), (18) and (13):


4-2-3-Free Pressure and thermal elongation of an unrestrained UG piping system

4-3-2-1-Virtual anchor length

As mentioned early, virtual anchor concept plays an important role in UG piping analysis. A friction force which is developed between soil and pipe causes the pipe gets fixed from a point known as virtual anchor. The virtual anchor is defined as point A in figure 6. For calculation of virtual anchor length, the anchored force which is defined in equation (14) should balance the axial friction force between pipe and soil as follows:

Figure 6: Free body diagram of unrestrained part of an UG piping system



In which f is defined as soil longitudinal resistance force per unit length of pipe.

4-2-3-2-Free movement

Referring to figure 7, end free movement of an unrestrained buried pipe is calculated as follows:

Figure 7: Diagram of unrestrained part of an UG piping system


From equations (22) and (21):


Comparing equations (19) and (23) show that free elongation of a buried pipe is half of the corresponding free elongation in AG pipe. So regardless of soil longitudinal resistance force (f), the free elongation of an UG pipe decreases to half of its corresponding value in AG. The stress at end point of the UG pipe reaches to        whereas the stress in restrained portion is:


Please note that in the restrained part of a buried pipe, there is not any pulling force due to pressure. So no any tensile stresses due to internal pressure appear in the pipe wall. 

4-2-3-3- Pressure and thermal elongation of an unrestrained UG piping system with a resistant force

Similar to AG piping systems, if the end pipe of an unrestrained UG pipe connects to another system, a resistant force is developed in the pipe. This force leads to a reduction in the elongation as shown in figure 8:

Figure 8: Diagram of unrestrained part of an UG piping system facing a force



From equations (25) and (26):


5- Bends in a buried pipeline

Bends are places where pipes direction change. So bends are likely to experience maximum pipe deformations and consequently are susceptible to be overstressed. As shown in figure 10, if the bend radius is lower than a specific value, the friction force developed between soil and pipe cannot balance the anchored force. Therefore in both sides of the bend two virtual anchored are created as well as a tendency of movement at bends. For ensuring the system works in a safe area, this movement must be analyzed and studied by a stress engineer. For avoiding any expensive analysis, designer should use bends which their radiuses are large enough. In this case, shown in figure 9, with a good estimation, the whole pipeline can be assumed as a straight pipe mathematically. However other design criteria should be met at all locations of a buried pipeline. 

Figure 9: A buried pipe line with bends

5-1-Calculation of 

Figure 10: Plan view of a buried bend

As mentioned early, if the developed friction force between soil and pipe balances the anchor force, the bend belongs to the restrained area of the buried pipe. For calculation of the bend minimum radius, figure 10 is referred:



6-Pipe-Soil interaction


For safe design of a buried pipe, stress engineer should be familiar with all types of loads exerted on the pipe such as soil weight, live surface loads, buoyancy loads, settlement, earthquake, blasting loads and etc. There are some limitations for these loads, stresses and strains which must be satisfied by a stress engineer. Moreover after a pipe moves in soil, some resistance loads are developed. These loads are modeled as springs with a specific stiffness which is function of the pipe-soil interaction characteristics. At this section the interaction is studied. 

6-2-Internal pressure

Obviously a circumferential membrane stress, known as hoop stress, is developed in pipe wall resulting from an internal pressure. If pipe is free to move longitudinally, a longitudinal membrane stress which is half of hoop stress appears in pipe wall. The wall thickness calculation formula which is presented in [10] is different from the one mentioned in B31.4-2012 as follows:

Equation (6-30) is presented in [10]:


Equation (31) is presented by B31.4-2012: 


For allowance consideration in above formula, some code requirements which are mentioned in 403.2.2 of B31.4-2012 should be met by stress engineers. Referring 403.2.4 of B31.4-2012, mill tolerance is not required to be considered if pipe materials are according to mandatory appendix I of B31.4-2012.

As an important rule, stress engineer must refer to the code on which the piping system has been designed. All code requirements which are minimum design requirements must be consistently satisfied. 

6-3-Vertical loads: dead soil load, live surface loads and impact loads

Weight of soil prism on a pipe exerts an external dead load on the buried pipe. Also live loads such as train or trucks cause pressure loads on the pipe. The pressure impact loads resulting from falling big objects on the earth surface where a buried pipe has been placed can be categorized in this category. 

6-3-1-Dead loads

As shown in figure 11, [10] introduces the following formula to calculate the dead loads:

Figure 11: Dead loads on a buried pipe


In which


In equation (32),  buoyancy force has been included.

Note: In undisturbed and unsaturated soil in which pipes are jacketed, the exerted load reduces in                    .

In which:                 is soil cohesion. 

6-3-2-Live loads

Live loads including truck loads on highways, train loads on railways and airplane gear assembly loads on runways leads to a vertical load on buried pipes. Two following methods for its calculation have been suggested by [10].


6-3-2-1-Directly from table 4.1.1 ALA

The values of the vertical loads which have been tabulated in table 4.1.1 of [10] are function of pipe depth and road types. A safety factor which covers impact effects of the loads should be applied referring to table 4.1-2 of [10].


6-3-2-2-Formula (section 4-1 of [7])

Another proposed method is a formula in which exerted pressure on the pipe is calculated regarding the local live load value and its offset to pipe center.

Figure 12: A concentrated surface load with an offset distance from a buried pipe


A surface load may lead to a vertical or lateral soil displacement around the pipe. For more information please refer to [10].

6-3-3-Impact loads

If a heavy object falls on surface of earth where a pipe is buried, a vertical load exerts on the pipe. For calculation of maximum load on the soil surface, the following formula has been given by [10]. Equation (34) can be used to calculate its developed pressure on the pipe.


5-2 of [7] suggests a following value for G for large strains near impact area:


Apart from the vertical load, a falling object causes a penetration in soil which may be calculated by the formula bellow (refer to [10]): 


Impact loads causes wave propagation which may create a deformation on a buried pipe which is far away from the surface impact load. 5-4 of [10] suggests some formula for that.

6-4-Effects of external loads on a buried pipe


The vertical surface loads may cause ovality, shown in figure 13, in a buried pipe. Iowa suggests the formula bellow for its calculation (4-2 of [10]):

Figure 13: Ovality of a buried pipe resulting from an external load

Subscript L and c in equation (39) implies lining and coating respectively.

6-4-2-Through-wall bending


As shown in figure 14, a vertical load on pipe creates a circumferential bending moment which is the source of bending stresses in circumferential direction. 

Figure 14: Circumferential local stress 


6-4-3-Crushing of side walls


Pipe having a     ratio greater than 100 needs to be studied more for buckling in the pipe wall as demonstrated in figure 15.

Figure 15: Pipe side walls crushing

Addressing to 403.2.5 of B31.4-2012, a    ratio greater than 100, reduction in thickness and reduction in yield stress may lead to ovality, denting and buckling.


6-4-4-Ring buckling


A high vertical load exerting on top of a pipe may cause a buckling on the top of pipe as shown in figure 16. An allowable load has been defined by [10] as follows:

Figure 16: Ring buckling on top of a pipe








In some places where a pipe is buried bellow a water table, it is possible the weight of water which is displaced by the pipe is greater than sum of all vertical loads. In this case an upward force named as buoyancy force tends to pull up the pipe. Whereas some parts of the pipe which are not under the buoyancy force behave like anchor. So as figure 17 shows, the pipe is under stress. 

Figure 17: Buoyancy force on a buried pipe


6-5-1-Bending stress calculation resulting from buoyancy


The real boundary conditions at points A and B, shown in figure 17 is somewhere between fully restrained supports and pinned ones (figure 18). 

Figure 18: Maximum bending moments for fixed and pinned restraints 



Buoyancy force increases when pipe is empty. So it is better to calculate stress when a pipe is empty. For balancing of the force, concrete weights, gravel filled blanket and etc. may be used. 


6-6-Fluid transients


A sudden change in a fluid rate stemming from valve closing is one of the major factors of a rapid change in fluid pressure, known as shock. This increased pressure is propagated along the pipe very fast until reaching to an elbow or bend where the pipe direction changes. Because fluid momentum changes at elbow, a force is exerted on the bend. This phenomenon has two major effects which should be noticed by engineers: Firstly increased internal pressure may lead to pipe burst and secondly a pressure thrust force exerting at bends or elbows which tends to separate welded or mechanical joints and besides to dislocate the pipe. For burst avoiding 403.3.4 of B31.4-2012 put some limitations for increased pressure (10%). Figure 19 shows how a thrust block takes this force.

Figure 19: Using thrust block in UG piping systems 

[10] introduces some formula enabling engineers to calculate valve closing time and pressure rise stemming from a rapid valve closing in which closing time is smaller than the calculated closing time.


6-7-Other sources of loads on a buried pipe


[10] introduces the following sources which may cause some stresses on a buried pipe. Because of complexity engineers need a FEM software and field information to calculate some of them.

- Soil displacement resulting from fault, earthquake, settlement, landside, frost heave or thaw settlement

- Mine subsidence

- Earthquake

- Blasting

- In-Service relocation


6-8-Soil resistance against a moving buried pipe


Soil resistance against a buried pipe exists when a pipe starts to move. As shown in figure 20, this soil behavior can be simulated as a spring with a specific stiffness which is calculated as follows:

Figure 20: Soil resistance against pipe movement 

[10] gives formula for maximum soil resistance and the corresponding displacement for the following conditions. CAESAR II uses the formula in modeling of buried pipes.


6-8-1-Axial soil force

Because of friction between pipe and soil, a resistant axial load which is against longitudinal pipe movement is developed. As mentioned before this force causes a reduction in pipe elongation. Virtual anchor length is calculated based on this force.


6-8-2-Lateral soil force

The soil which is in front of a buried pipe resists against pipe lateral movement. This force is used to calculate minimum bend radius explained in section 5-1.


6-8-3-Vertical uplift soil force

This resistance occurs when a buried pipe starts to move upward.


6-8-4-Vertical bearing soil force

This force showing bearing loads of soil is created when a buried pipe starts to move downward.

7-Anchor Block in Underground Pipe Line


As mentioned in section 4-2-3-3, at a pipeline-equipment interface such as pig lunching station, there is a thermal and pressure movement at the end of an unrestrained underground pipe. If the pipe is not flexible enough to compensate the movement, a big thrust force is developed at the nozzle connection. For ensuring integrity and operability of the system, the nozzle loads should be calculated by stress engineers. Imposing a restriction on the pipe movement which creates the thrust load by a putting a thrust block makes the equipment isolated from the pipe and consequently the nozzle would face up less nozzle loads, which are possible to be handled by the equipment.

Anchor blocks are also used to resist the hydrostatic thrust loads which are developed by internal pressure in fittings by which pipe direction change (such as bends, wyes, tees); pipe cross section area change (such as reducers) or pipeline is terminated (such as bulkheads). If joints of a pipe are not adequately strong and restricted, such as bell and spigot joints in concrete pipes or adhesive bonded joints in GRP pipes, these forces may disengage joints or end up significant cracks. [1]

7-2- Hydrostatic Thrust 

Hydrostatic thrust in buried dead ends, outlets, lateral branches and reducers is a function of internal pressure P and cross-sectional area at the pipe joints, as defined by the following equations and shown in figure (21).

Figure 21: Parameters for Hydrostatic thrust calculation for Wye, Tee and Birfucation 

Branch Wye     



Dead Ends        





The resultant thrust at bends and bifurcations is also a function of the deflection angle Δ and is given by: [4]





7-3- Anchor force in pipelines

A pipe line anchor block consists of an anchor flange in a large concrete block, as shown in figure 22. This anchor block generally is located at proper locations to prevent the pipeline movement from causing damage to the connecting pipes and equipment. The anchor force can be determined as mentioned in equation (20).

Figure 22: Anchor flange in a concrete block  

7-4- Thrust blocks design

For buried pipe lines, the horizontal thrust is resisted by frictional drag due to fitting or pipe dead weight or passive resistance of the soil against this movement. For unbalanced uplift thrust, the resistance load is the dead weight of the fitting plus earth cover and fluid weight in the pipe. If these resistances are not adequate, it must be compensated by increasing the dead weight of the pipe by use of gravity type thrust blocks or increasing the supporting area on the bearing of the fitting, figure 23. [4] 

Figure 23: Passive resistance of soil against pipe movement        

The adequate thrust block size may be selected from typical standard drawings. They may be designed based on thrust forces and transverse lateral or bearing resistance of the soil. The shear resistance of passive soil wedge behind the thrust block should be checked to confirm that it can support the weight of the thrust block without significant pipe settlement.

8-Criteria for design of buried pipes

For making sure that a buried pipe system works safely, stress engineer should have knowledge about stress and deflection calculation as well as their criteria which are mentioned in codes. Moreover there are other code requirements which must be satisfied. Engineering practices may be applicable for some issues on which codes are silent. For getting allowable values of some analysis parameters which have been mentioned in this paper, reader must refer to a proper reference.

Table 403.3.1-1 of B31.4-2012 and appendix A of [10] may be used by engineers to find out the allowable values if the system is designed based them. This paper is not intended to be used as design reference. Proper codes must be used by engineers.