Shell course of a storage tank tends to expand due to temperature difference and head pressure. The tank base is not possible to move freely because of the base restriction. So the displacement of that location can’t reach to its free movement. This takes part in a discontinuity area which is extended about , from tank base. Apparently all nozzles in that area are affected as follows:

Nozzle Rotation: The rotation of shell in the discontinuity area makes all nozzles rotate. Depending on the attached piping flexibility, some loads are developed at the nozzle and shell junction.

Radial Displacement: As mentioned earlier, due to chocking effects in the discontinuity area, the nozzles tend to move in radial direction due to thermal expansion and head pressure. At the bottom of tank where the chocking exists, some parts of circumferential membrane stress capacity go to the corresponding local stresses. This increased capacity can be calculated as a new allowable stress which is used when a local tensile membrane stress exists. It does not give credit to a compression membrane stress because the tank may be empty or half.
Figure 1 shows two extreme scenarios which may exist: fully restrained and semirestrained.
API 650 Requirements for Bottom Nozzles of Storage Tanks: Nozzle Movements, Local Stresses on Nozzles and Allowable Nozzle Loads
Figure 1: Tank bottom restriction types
In a fully restrained boundary condition a bending moment is created in such a way it leads to a smaller rotation comparing to a semirestrained system. The real one is somewhere between them. The big rotation is our main concern because it leads to big nozzle loads.
1Radial and rotational movements of nozzle calculation
Figure 2 demonstrates final nozzle location stemming from thermal and head pressure of tank. These movements are input of CAESAR II.
Figure 2: Radial and rotational movement of nozzle in discontinuity area
It is assumed the tank has a different thickness at each shell courses. So the free pressure displacement along tank is uniform unless at discontinuity area. Free pressure and thermal movement can be calculated as follows:
(1)
(2)
If each shell courses is designed based on its head pressure, so has nearly the same value along the tank height. A friction force between tank bottom and the ground prevents it expanding freely. In reality it expands a percent of the free expansion. The expansion can be assumed as in which .
(3)
Where
(4)
11Calculation of
Tank shell is chocked at its bottom. So similar to the chocking model in which an infinite beam on an elastic foundation is used to simulate pipe shell, a semiinfinite beam on an elastic foundation is used to develop beam deflection and slope functions. Referring to case 8 in table 8 of [7], the following functions for are calculated as follows:
Figure 3: Semifinite beam on an elastic foundation
(5)
(6)
(8)
From equations (9) and (8):
(9)
From substituting factors, equations (8) and (10), in equations (5) and (6):
(11)
has been calculated in the chocking model.
12Radial and rotational movements of nozzle of a tank with a nonuniform shell thickness
Referring to figure 2:
(13)
has been calculated in chocking model as
From (3) and (13) w is calculated:
(14)
Δ is substituted from equation (4) to (14):
(15)
From equations (12), (13):
(16)
13Radial and rotational movements of tank nozzle with a uniform shell thickness
When the tank wall thickness is uniform along the tank height, pressure elongation of tank shell is according to figure 4. Therefore equations (15) and (16) should be modified based on the shell pressure elongation profile.
Figure 4: Pressure movements of tank shell with a uniform shell thickness
(17)
(18)
(19)
is calculated by the bellow formula:
(20)
For calculation of w and θ, the shell profile should be considered in equations (15) and (16). Figure 5 is used for the calculation:
Figure 5: Radial and rotational movement of nozzle of tank with uniform shell thickness
(21)
(22)
In API 650, B is assumed 1, so equations (21) and (22) are reduced to:
(23)
(24)
From equations (3), (23) and (24):
(25)
(26)
(27)
is the same as equation (16).
2Stiffness calculation of tank nozzle
In piping stress softwares which are based on beam elements, flexibility of pipe nozzle connection to tank shell cannot be calculated. If the connection is modeled as rigid, the resulting nozzle loads increase significantly which is not realistic. So a piping stress engineer should consider flexibility of the connection in the softwares. In CAESAR II the stiffness can be defined as restraints stiffness at proper nodes. Literally stiffness is expressed as a relation between loads and their corresponding deformations. API 650 introduces a method to calculate stiffnesses of nozzle connections for radial force, longitudinal and circumferential bending moments. For a tank nozzles placed in a discontinuity area, radial and longitudinal deformations are correlated in such a way any radial force ends up not only a radial deformation, but a rotation and vice versa.
Referring to figure 6, the relation between loads and deformations are as follows:
Figure 6: Loads and displacements of tank nozzles
(28)
(29)
(30)
In above formula, shows effects of the longitudinal moment on radial movement and shows effects of radial force on rotational movement.
are extracted by using figures P2A to P2L shown in appendix of API 650 for reinforcement on neck and reinforcement on shell. A stress engineer must be aware that tank shell has usually less stiffness related to a pressure vessel. Although this less stiffness leads to a less nozzle loads but it has some adverse effects: one is low allowable stresses and another is followup strain in which strain concentration at nozzles grows even in a stress range which is confirmed by ASME codes. Some finite element softwares such as FE Pipe give the strain concentration factor in results.
3Allowable nozzle loads calculation
Passing a storage tank nozzle loads has been a nightmare for the most stress engineers. Low allowable tank nozzle loads as well as tank settlements make the task challenging. If a tank is designed based on API 650 the allowable nozzle loads in a discontinuity area can be calculated by the formula mentioned in appendix P of API 650. For a better understanding of concepts behind the formula which may help engineers to deal with a problem coming up in the nozzle loads analysis, basis of the allowable nozzle loads calculations are mentioned as follows.
As mentioned before, all nozzles in discontinuity area tends to rotate and they are checked based on their locations (height): closer to tank bottom, more being chocked and rotated. The chocked area can sustain more local stresses comparing to an area which is out of the discontinuity area. The reason is a circumferential membrane stress due to chocking balances a part of hoop stress. This balanced stress named as can be calculated as following (figure 7):
Figure 6: Loads and displacements of tank nozzles
The tank shell tends to be elongated in the discontinuity area but because of being chocked, it cannot reach to its maximum pressure deformation. So a compressive stress is developed in shell element as shown in figure 7. Before being chocked, the element circumferential membrane stresses were limited to in which is the shell allowable stress. But after the chocking, the local stresses can increase to which results to a higher allowable stress. is defined as following:
31Calculation
Figure 8: Pipe radius change under internal pressure and internalexternal pressure
(31)
As shown in figure 8, shell radius reaches to its maximum value in absence of external pressure: . When external pressure applies, the radius decreases to a middle state shown in figure 8. produce respectively.
(32)
So,
(33)
In state 1:
(34)
In state 2:
(35)
(36)
From equations (31) and (36):
(37)
Refer to figure 7: and in which is pressure fraction of w calculated in equation (15):
(38)
From equations (37) turns to:
(39)
32Allowable nozzle loads
The allowable nozzle loads are based on maximum local stress at nozzle connection. Refer to the chocking model, external loads on the nozzle cause circumferential local membrane stresses and longitudinal local bending stresses. Theses stresses are calculated based on the following formulas:
Longitudinal local bending stress:
(40)
Longitudinal local bending stress:
(41)
Regarding Kellogg’s model: for radial force and for bending moments. is the nozzle radius. The equation (41) changes to the following formula:
(43)
(42)
API 650 represents three dimensionless parameters as follows:
(44)
(45)
(46)
In which
(47)
(48)
are allowable coefficients for respectively. Referring to appendix P of API 650 figures P4A and P4B are used to obtain . Parameters (44), (45) and (46) must be less than one and can be written as follows. The allowable stress in stress combination is considered instead of .
(49)
(50)
(51)
The above formulas are very similar to equations (42) and (43). Only some coefficients are different. Therefore the formulas in API 650 are based on circumferential local membrane stress achieved in Kellogg’s method.
Figure 9: Allowable nozzle loads in point A, B, C and C’
As mentioned earlier, equation (39), chocking effect results in an increase in allowable load which is function of nozzle height. It doesn’t give any credit to circumferential compressive stresses because tank may be empty or half. Referring to figure (9), stresses should be cautiously studied separately in different points A, B, C and C'.
Figure 10: Stresses on an element at point A
Regarding figure 10, stresses in an element A may be combined as bellow:
: Local longitudinal bending stress caused by radial force, P, at point A
: Local longitudinal bending stress caused by longitudinal moment, ( ), at point A
: Local circumferential membrane stress caused by radial force, P, at point A
: Local circumferential membrane stress caused by longitudinal moment, ( ), at point A
API 650 uses old revision of ASME BPVC section VIII Div 2 in which the stress criteria as mentioned as follows:
(52)
As mentioned before, K’ is assumed 1.1 by API 650 instead of 1.5. It leads to a safer design. In equation (52) only primary membrane stress is incorporated.
(53)
can be estimated by the way explained bellow instead of using its precise value given by equation (45) where reaches to . Figure 11 shows how is calculated. The curve can be assumed as a straight line.
(54)
With substituting in equation (54):
With equations (53), (49), (50) and (55):
(55)
(56)
The above equation is extracted and valid for stresses at point A. Applying a similar method for points B, C and C’ result in the following equations:
(59)
(58)
(57)
The following notes should be considered when using those formulas:

The minimum value of which is 0.1 occurs outside of the discontinuity area where the chocking effect diminishes and so the whole stress capacity is used for hoop stress.

When the circumferential membrane stresses due to external loads are compressive, hoop stress doesn’t provide any margin for allowable loads. So should not be considered in equation (53) and equation (53) changes to:
(60)
Piping stress engineers may use the nomograms which are mentioned on figures P5A and P5B in appendix P of API 650.
4Tank settlement
Settlement of storage tanks are usually considerable comparing to settlement of other equipment. Settlement highly depends on tank dimensions and soil conditions. Depending on the connected piping flexibility, it may cause some problems to connecting pipes and nozzles. During settlement a connected pipe tends to be pulled down by the tank. If the piping system has enough flexibility to accommodate the settlement, it will not lead to a dangerous situation. Moreover, usually pipes connecting to storage tanks have a heavy isolating valve near nozzles. So stress engineer should find a solution to deal with not only piping flexibility but sustained stresses.
41Solutions
411Reducing of settlement
One of the best solutions is settlement reduction which can be achieved by connecting pipes after hydro test when the tank is not empty, preferably halffull tank. Therefore only settlement that the pipes incorporate is long term settlement. Civil engineers are expected to give the long term settlement to stress engineers.
412Supporting
Obviously type of supports which are placed at near nozzles affects stresses caused by settlement in following ways:

Common foundation: Pipes which are supported on tank foundation have the same displacement as the tank settlement has. Consequently it results in a lower stress.

Adjusting support: If settlement profiles versus time are available, some adjustable supports can be used. The supports should be adjusted in field.

Spring Hangers: Spring hangers make the connected pipes more flexible and consequently it leads to lower stresses at nozzles. Most of the time it is considered the best way because it sustains the heavy valve weight.
413Piping Routes
Stress engineers may resort to this method to make the piping system more flexible. The solution may bring up two problems: firstly this method requires enough room especially for large bore pipes. Moreover it makes the pipe more vulnerable to vibration and dynamic loads by reducing natural frequency of the piping system. Secondly this solution leads to an increase in sustained stress because rout change implies adding some piping components. As said before, heavy isolating valves deteriorate the situation more.
To sum up, stress engineers should be aware of the solutions and select the best one depending on the situation.
42 Category of stresses caused by settlement
It is important to know which stress category should be assigned for stresses caused by settlement. Generally primary stress is more dangerous and critical than secondary stress because the primary one is not reduced after going beyond yielding points. So primary stresses should be checked against lower allowable stresses. Regarding the mentioned explanation, putting settlement stress in primary category makes intensity stress more difficult to satisfy its stress criteria. The nature of settlement loading is similar to secondary loading because it disappears after reaching to yielding point. It points out settlement stress is not as dangerous as a primary load such as weight. However it is not cyclic, it may be considered in a secondary loading category. Depending on the situation and mentioned items stress engineer may make the best decision.
The boundary condition:
(7)
From applying the boundary condition in equation (5):
(10)
(12)