# Safety Factor Calculations

## Introduction#

The main purpose of safety factor calculations is to examine whether a well design will withstand a given load or not, defined via a load case. Load cases are divided based on the differential pressure, $\Delta p = p_i - p_e$, i.e. internal- minus external- pressure, and are called burst loads if $\Delta p>0$ and collapse loads if $\Delta p<0$.

In addition to calculations of either a burst or a collapse safety factor, an axial and a triaxial safety factor are calculated. The triaxial safety factor is directly related to the von Mises (VME) stress, which is used to predict yielding of materials under complex loading. All calculations herein are done in accordance with the latest API-revision, API Technical Report 5C3, seventh edition, June 2018.

NOTE!
In this documentation we denote any tubular as tubing. All calculations however encompass any tubular, such as tubings, casings, liners, tie-backs etc.

### Printable Version#

Oliasoft Technical Documentation - Safety Factor Calculations

## Axial Safety Factor#

The axial safety factor is directly calculated as the ratio between the yield of the tubing, corrected for compression asymmetry and temperature deration, and the axial stress. In other words, given a load case, defined by internal- and external- pressure profiles, and a temperature profile, calculate the resulting axial stress, $\sigma_a$ , and correct the yield of the tubing for compression asymmetry and temperature. Denote this corrected yield by $Y_{cor}$. Then the axial safety factor is:

$\text{SF}_{axial} = \frac{Y_{cor}}{\sigma_a} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)$

NOTE!
We remark that this safety factor comes in four different flavours, depending on whether bending stress from dogleg is included or not, and whether bending stress from buckling is included or not.

## Burst#

For a thin-walled tubing, the burst rating is given by Barlow's formula:

$p_b = \text{Tol}\left(\frac{2Y_p t}{d_o} \right) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2)$

where $Tol$ is the wall thickness tolerance, usually $\text{Tol} = 0.875, Y_p$ is the minimum yield strength of the tubing, $t$ is the tubing thickness, and $d_o$ is the tubing outside diameter. We notice that this quantity can be calculated a priori, and is independent on the particular load.

A burst load case is defined by three quantities, i.e. internal- and external- load case pressure profiles, $p_i$ and $p_e$, and a load case temperature profile, $T$. Let $\Delta p = p_i - p_e$, denote the differential pressure, which is assumed positive.

The burst safety factor is calculated in two steps. First, the minimum hoop yield stress of the tubing is corrected for asymmetry and derated for temperature. This gives the burst rating, $p_b$, using Barlow's formula, equation 2. Then, the burst safety factor is calculated as:

$\text{SF}_{burst} = \frac{p_b}{\Delta p} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(3)$

## Collapse#

The collapse instability problem is more complex than burst, and the rating depends both on the axial load and internal pressure, in addition to the tubing dimensions. Theoretically, collapse of a tubing can happen in one of four possible ways, called modes, and both the axial load, the yield, and the internal pressure determine which collapse mode is relevant.

NOTE!
limitations have been identified. Currently, a more accurate probabilistic model has been developed. However, to use this new approach, a lot more information about the tubing involved is required, which mostly is not available. Therefore, the standard model is available by default, in accordance with API 5C3. Oliasoft is working with tubing vendors to develop required data points for meaningful use of the revised model.

The transition points between the modes are determined from a set of numerical constants, which are statistically calculated from a collection of collapse tests on tubings. These constants are expressed as:

$A_c = 2.8762 + 0.15489\cdot10^{-3}\cdot X + 0.44809\cdot 10^{-6}\cdot X^2 - 0.16211\cdot X^3 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(4)$

$B_c = 0.026233 + 0.73402\cdot 10^{-4}\cdot X \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(5)$

$\xi_c = \frac{B_c}{A_c} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(6)$

$\varkappa_c = \frac{3\xi_c}{2 + \xi_c} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(7)$

$C_c = -3.2125 + 0.030867\cdot X - 0.15204\cdot 10^{-5}\cdot X^2 + 0.77810\cdot 10^{-9} \cdot X^3 \;\;\;\;\;\;\;\;\;\;\;\;\;(8)$

$F_c = 3.237\cdot 10^5 \frac{\varkappa_c^3}{\left(\varkappa_c - \xi_c\right)\left(1 - \varkappa_c\right)^2X} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(9)$

$G_c = F_c\xi_c, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(10)$

where $X$ has dimension Mpa (Mega Pascal), and the defining expression depends on whether the tubing is in effective tension, i.e. if $\sigma_a + p_i > 0$, where $\sigma_a$ is the axial stress and $p_i$ is the internal pressure of the tubing, or compression. Explicitly,

$X = \begin{cases} Y_{pa}, \quad &\sigma_a + p_i \geq 0 \\ Y_p, \quad &\sigma_a + p_i < 0 \end{cases} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(11)$

where $Y_p$ is the minimum hoop yield stress corrected for temperature and anisotropy, and $Y_{pa}$ is called the combined loading equivalent grade and is given by:

$Y_{pa} = \left(\sqrt{1 - \frac34 \left(\frac{\sigma_a + p_i}{Y_p} \right)} - \frac{\sigma_a + p_i}{2Y_p}\right) Y_p \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(12)$

We note that $Y_{pa} = Y_p$ if $\sigma_a + p_i = 0$.

As noted, a tubing can collapse in four different modes, and these are denoted Yield strength collapse, Plastic collapse, Transition collapse, and Elastic collapse. The transition points, and collapse ratings for each of these are discussed below. First, define:

$D_{th} = \frac{2d_o}{d_o - d_i}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(13)$

where $d_o$ and $d_i$ are the outer and inner diameter of the tubing, respectively. This ratio is compared to the values described below, which in turn determines the collapse mode.

### Yield Strength Collapse#

The yield strength collapse rating is given by:

$p_y = \frac{2X\left(D_{th} - 1\right)}{D_{th}^2} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(14)$

and the transition point, $P_{y\rightarrow p}$, from yield strength collapse to plastic collapse is given by:

$a_{y\rightarrow p} = \sqrt{(A_c - 2)^2 + 8\left(B_c + \frac{C_c}{X}\right)} + (A_c - 2)$

$b_{y\rightarrow p} = 2\left(B_c + \frac{C_c}{X}\right) \;\;\;\;\;(15)$

$P_{y\rightarrow p} = \frac{a_{y\rightarrow p}}{b_{y\rightarrow p}}$

### Plastic Collapse#

The plastic collapse rating is given by:

$p_p = X\left(\frac{A_c}{D_{th}} - B_c\right) - C_c\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(16)$

and the transition point, $P_{p\rightarrow t}$, from plastic collapse to transition collapse is given by:

$a_{p\rightarrow t} = X\left(A_c - F_c\right)$

$b_{p\rightarrow t} = C_c + X\left(B_c - G_c\right)\;\;\;(17)$

$P_{p\rightarrow t} = \frac{a_{p\rightarrow t}}{b_{p\rightarrow t}}$

### Transition Collapse#

The transition collapse rating is given by:

$p_t = X\left(\frac{F_c}{D_{th}} - G_c\right) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(18)$

and the transition point, $P_{t\rightarrow e}$, from transition collapse to elastic collapse is given by:

$a_{t\rightarrow e} = 2 + \frac{B_c}{A_c}$

$b_{t\rightarrow e} = 3\frac{B_c}{A_c}\;\;\;(19)$

$P_{t\rightarrow e} = \frac{a_{t\rightarrow e}}{b_{t\rightarrow e}}$

### Elastic Collapse#

The elastic collapse rating is given by:

$p_t = \frac{323708.854758}{D_{th}\left(D_{th} - 1\right)^2} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(20)$

To determine which collapse regime the tubing, under the current conditions, is in, $D_{th}$ is compared to the numbers $P_{y\rightarrow p}$, $P_{p\rightarrow t}$, and $P_{t\rightarrow e}$. In other words, the collapse mode is:

\text{Collapse mode} = \left\{\begin{alignedat}{2} & \text{Yield}, \quad && D_{th} < P_{y\rightarrow p} \\ & \text{Plastic}, \quad & P_{y\rightarrow p} \leq\,\, &D_{th} < P_{p\rightarrow t} ,\\ &\text{Transition}, \quad & P_{p\rightarrow t} \leq\,\, &D_{th} < P_{t\rightarrow e}, \\ &\text{Elastic}, \quad && D_{th} \geq P_{t\rightarrow e}\end{alignedat}\right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(21)

A collapse load case is defined by three quantities, i.e. internal- and external- load case pressure profiles, $p_i$ and $p_e$, respectively, and a load case temperature profile, $T$. Let $\Delta p = p_e - p_i$, denote the differential pressure (the negative of the burst differential pressure), which is assumed positive. Then, collapse safety factors along the tubing is calculated as follows:

1. Calculate the axial stress without bending from dogleg, corresponding to the load case.

2. Identify the collapse mode and rating from the formalism above. Denote the rating by $p_c$.

3. The collapse safety factor is then given by:

$\text{SF}_{collapse} = \frac{p_c}{\Delta p} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(22)$

NOTE!
We note that the collapse safety factor comes in two flavours, depending on whether bending stress from buckling is included or not.

## Triaxial stress analysis#

Triaxial stress analysis is less intuitive than the other three above described cases. The analysis is based on von Mises stress, which is especially suited to predict yielding of ductile materials, such as steel, under complex loading.

If we ignore torque on the tubing, the VME stress is:

$\sigma_{\text{VME}} = \frac{1}{\sqrt{2}}\left( (\sigma_a - \sigma_t)^2 + (\sigma_t - \sigma_r)^2 + (\sigma_r - \sigma_a)^2 \right)^2\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(23)$

where $\sigma_a$ is the axial stress, $\sigma_r$ is the radial stress, and $\sigma_t$ is the tangential/hoop stress. Assume the tubing has inside- and outside- area, $A_i$ and $A_o$, respectively, is exposed to internal- and external- pressure profiles, $p_i$ and $p_e$, respectively, and is exposed to a temperature profile, $T$. Then, the radial and tangential stresses, as a function of area $A$, is given from Lamès equations, i.e.:

$\sigma_r(A) = \frac{p_iA_i - p_eA_o}{A_o - A_i} - \frac{(p_i - p_e)A_iA_o}{(A_o - A_i)A}, \qquad A\in [A_i, A_o], \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(24)$

$\sigma_t(A) = \frac{p_iA_i - p_eA_o}{A_o - A_i} + \frac{(p_i - p_e)A_iA_o}{(A_o - A_i)A}, \qquad A\in [A_i, A_o]. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(25)$

When calculating the triaxial safety factor, the highest value of the VME stress should be used. The VME stress is highest either at the inner- or outer- wall of the tubing, and there the radial and tangential stresses are reduced to:

$\sigma_r(A_i) = -p_i := \sigma_{r,i}, \qquad \sigma_r(A_o) = -p_o := \sigma_{r,o}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(26)$

$\sigma_t(A_i) = \frac{p_i(A_i + A_o) - 2p_eA_o}{A_o - A_i} := \sigma_{t,i}, \qquad \sigma_t(A_o) = \frac{2p_iA_i - p_e(A_i + A_o)}{A_o - A_i} := \sigma_{t,o}. \;\;\;\;\;\;(27)$

The peak VME stress can now calculated as follows:

1. Calculate the axial stress without bending from dogleg.

2. Calculate the bending stress on the inside and outside of the tubing due to dogleg. On the inside of the bend (compression) subtract the bending stress from the axial stress, while on the outside of the bend (tension) add the bending stress to the axial stress.

3. Calculate the radial- and tangential- stresses on the inside and outside of the tubing.

4. Calculate the VME stress at four locations, inside and outside of pipe at the inside and outside of bend, and report the highest of these and the location.

In triaxial safety factor calculations, a boosted version of the VME stress is used, i.e. the stresses involved are boosted according to asymmetry and anisotropy. Call this boosted VME-stress $\text{VME}_{cor}$. The triaxial safety factor is then given by:

$\text{SF}_{triax} = \frac{Y_p}{\text{VME}_{cor}}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(28)$

where $Y_p$ is the temperature derated yield stress.

NOTE!
The triaxial safety factor comes in two flavours, depending on whether bending stress from buckling is included or not.

### VME Plots#

Since both the radial- and tangential- stresses primarily are functions of the differential pressure, a useful visualisation of load cases is to plot the differential pressure versus effective axial load, i.e. the axial load corrected for internal pressure, together with the VME - ellipse at the onset of yield, i.e. when the stress equals the yield strength of the tubing. We note that this is an approximation, since the VME - stress depends on three inputs, i.e. radial-, tangential-, and axial- stresses. However, it is a good tool to visualize an applied load and design limits. It should be stressed that it is the calculated safety factors that first and foremost should be consulted.

Since the design limit plot is an approximation, some choices have to be made, whether to honor e.g. safety factors, or pressures, etc. Oliasoft's default approach is to honor the safety factors, and we do this the following way:

a. Adjust the axial load (e.g. temperature deration) to get the correct position relative to the axial limits.

b. Use this effective axial load to work out the effective differential pressure such that burst and collapse safety factors are honored.

Note, the collapse limits account for axial load, but assume zero internal pressure.

The end result is that the plot will be correct for burst and axial limits, and nearly correct for collapse. It will not be correct for triaxial (impossible together with the other three).

In the figure above, we show the design limit plot for an L80, 7'', #29 casing. Included in the figure are two load cases, one burst and one collapse. The other parts of the plot are:

i) The VME-limit, obtained by setting the VME-stress equal to the yield strength of the casing (green ellipse), adjusted to a triaxial design factor of $DF_{triax} = 1.25$.

ii) The black line is the burst limit of the casing, with a burst design factor of $DF_{burst} = 1.1$.

iii) The red lines are the axial limits of the casing, with a compression design factor $DF_{axial, comp} = 1.3$ and a tension design factor $DF_{axial, ten} = 1.6$.

iv) The blue line is the collapse limit of the casing, with a constant rating for compression, and a varying rating for tension, as the rating is dependent on the axial load. The collapse mode is constant along the blue line.

NOTE!
We remark that connection lines are not included in the plot above. If desired, these are added exactly as the design limits for the casing.

## References#

[1] Calculating performance properties of pipe used as casing or tubing. Technical report 5C3, seventh edition, American Petroleum Institute, June 2018.