14.3 Evaluating Settlement Limits, Cosine Curve Analysis & Remedial Action

Mathematical Derivation of Optimum Cosine Curve Analysis

When survey measurements of shell perimeter elevations are taken at $N$ equidistant stations around a tank, the raw data reflects a combination of three simultaneous physical phenomena:

1. Uniform Settlement: The overall average downward displacement of the foundation.
2. Planar (Rigid Body) Tilt: The rotation of the tank foundation as an undistorted, rigid flat plane.
3. Out-of-Plane Deflection ($S_i$): The true structural distortion, representing the localized vertical deviation of each station away from the planar tilt plane.

To isolate the damaging out-of-plane deflection from benign planar tilt, API Standard 653 Annex B.2.2 employs a mathematical harmonic regression technique known as Optimum Cosine Curve Fitting.

+-------------------------------------------------------------------------+
|               MEASURED SETTLEMENT VS. FITTED COSINE CURVE               |
|                                                                         |
|   Elevation                                                             |
|      ^                                                                  |
|      |          Station 4 (Low Point)                                   |
|      |             * Measured Raw Elevation (U_i)                       |
|      |            /|                                                    |
|      |           / | S_i = Out-of-Plane Deflection                      |
|      |          /  v                                                    |
|      |  ~~~~~--o---o------------------------o---~~~~~ Fitted Cosine     |
|      |        /     \                      /         Plane (u_i)        |
|      |       /       \                    /                             |
|      |      *         \                  /                              |
|      |                 *                *                               |
|      |                  \              /                                |
|      |                   *------------*                                 |
|      +------------------------------------------------------------->    |
|      0 deg (Pt 1)     90 deg (Pt 4)     180 deg (Pt 7)     360 deg      |
|                                 Circumferential Station                 |
+-------------------------------------------------------------------------+

The Mathematical Model (Annex B.2.2)

The theoretical elevation ($u_i$) of any station $i$ located at an angular position $\theta_i$ along a flat tilted plane is given by the sinusoidal relationship:

$u_i = a + b \cos \theta_i + c \sin \theta_i$

Where:

• $\theta_i$ is the circumferential angle of station $i$, defined as $\theta_i = \frac{2 \pi (i - 1)}{N}$ radians (with station 1 established at $0^\circ$).
• $a$ is the mean elevation of all survey points, representing uniform settlement.
• $b$ and $c$ are harmonic Fourier coefficients representing the orthogonal components of planar tilt.

Least-Squares Fitting Formulation

Applying the principle of least squares to minimize the sum of the squared deviations $\sum (U_i - u_i)^2$ across $N$ equally spaced stations, the coefficients are calculated directly via closed-form summation:

$a = \frac{1}{N} \sum_{i=1}^{N} U_i$

$b = \frac{2}{N} \sum_{i=1}^{N} \left( U_i \cos \theta_i \right)$

$c = \frac{2}{N} \sum_{i=1}^{N} \left( U_i \sin \theta_i \right)$

Once constants $a$, $b$, and $c$ are determined, the planar tilt elevation $u_i$ is calculated for each station. The out-of-plane settlement ($S_i$) at station $i$ is the exact mathematical difference between the measured elevation ($U_i$) and the fitted plane elevation ($u_i$):

$S_i = U_i - u_i$

• A positive value of $S_i$ indicates that station $i$ has settled less than the tilt plane (a high spot or crest).
• A negative value of $S_i$ indicates that station $i$ has settled more than the tilt plane (a low spot or trough).

Validity of the Fit ($R^2 \ge 0.9$)

API 653 B.2.2.4 does not accept any cosine curve that a solver happens to return. The optimum cosine curve is considered valid only if $R^2 \ge 0.9$, where

$R^2 = \frac{S_{yy} - SSE}{S_{yy}}$

• $S_{yy}$ is the sum of the squares of the differences between the average measured elevation and the measured elevations.
• $SSE$ is the sum of the squares of the differences between the measured and predicted elevations.

When out-of-plane settlement is concentrated in one or two local areas, a least-squares fit tends to under-predict the local deflection and is therefore not conservative; $R^2$ will typically fall below 0.9. In that case take more measurements than the Figure B.1 minimum, use a more rigorous curve-fitting procedure, or abandon the rigid-tilt-plane route entirely and evaluate the tank under B.2.2.5 (and its permissible limit in B.3.2.2) instead.

Maximum Permissible Out-of-Plane Shell Deflection ($S_{\max}$)

Under API 653 Annex B.2.2, out-of-plane deflections must not exceed the structural capacity of the cylindrical shell. An excessive out-of-plane deflection induces membrane shear and circumferential bending stresses that can cause shell buckling, weld cracking, or floating roof jamming.

The Governing Formulations

API 653 Annex B.2.2 defines the maximum permissible out-of-plane settlement deflection ($S_{\max}$) through the following fundamental relationship:

In US Customary Units:

$S_{\max} = \frac{11 \cdot L^2 \cdot Y}{2 \cdot E \cdot H} \quad \text{(feet)}$

To express the permissible settlement in inches, the formula is multiplied by $12$:

$S_{\max} = \frac{66 \cdot L^2 \cdot Y}{E \cdot H} \quad \text{(inches)}$

Where:

• $S_{\max}$ = maximum permissible out-of-plane settlement deflection at any station (inches or feet).
• $L$ = arc distance between adjacent survey stations along the shell circumference ($L = \frac{\pi D}{N}$) in feet.
• $Y$ = specified minimum yield strength ($S_y$) of the shell plate material in the course under consideration (psi). For older carbon steels where material specifications are unknown, API 653 permits assuming $Y = 30,000\text{ psi}$.
• $E$ = Young's Modulus of Elasticity for carbon steel, taken as $29,000,000\text{ psi}$ ($29 \times 10^6\text{ psi}$).
• $H$ = maximum total height of the tank shell in feet.

In SI Metric Units:

$S_{\max} = \frac{11 \cdot L^2 \cdot Y}{2 \cdot E \cdot H} \quad \text{(meters or millimeters)}$

Where $L$ and $H$ are in meters, $Y$ and $E$ are in megapascals (MPa), with $E = 200,000\text{ MPa}$.

Engineering Basis and Safety Margins

The $S_{\max}$ formulation is derived from thin-shell elastic beam theory and finite-element analysis of cylindrical shells resting on elastic foundations. The constant incorporates a factor of safety ensuring that the secondary bending stresses induced by foundation deflection do not exceed the yield strength ($Y$) of the steel, thereby preventing permanent plastic deformation and elastic shell buckling.

Permissible Edge Settlement Evaluation (API 653 Annex B.2.3 and B.3.4)

Edge settlement is localized subsidence of the bottom plates immediately adjacent to the shell. API 653 Annex B.2.3 defines the geometry and the measurement rules; Annex B.3.4 supplies the acceptance criteria. Keeping the two apart is essential, because the figures that define the settlement are not the figures that give the allowable.

• B.2.3.1 — Figure B.6 illustrates edge settlement and defines the two measured quantities: $B$, the settlement depth, and $R$, the radial width of the settled area.
• B.2.3.2 — locating the breakover point where the settled area begins requires judgment: lay a straight edge on the unsettled bottom and observe where the plate separates from it. If the bottom is cone-up or cone-down, $B$ must be measured from a projection of the unsettled bottom, not from a level line taken from the breakover point to the shell. That correction is exactly what Figure B.7 shows.
• B.2.3.3 — which allowable applies depends on the bottom welds inside the settled area:
  • $B_{ew}$ applies where a bottom lap weld is essentially parallel ($\pm 20^\circ$) to the shell.
  • $B_e$ applies where there are no bottom welds, only butt welds, or lap welds essentially perpendicular ($\pm 20^\circ$) to the shell.

+-------------------------------------------------------------------------+
|                    EDGE SETTLEMENT GEOMETRY & LIMITS                    |
|                                                                         |
|         Shell Course                                                    |
|             ||                                                          |
|             ||  Corner Fillet Weld                                      |
|         ====++======================== Original Bottom Plane            |
|               \                       /                                 |
|                \                     /                                  |
|                 \     Depression    /                                   |
|                  \_________________/                                    |
|                  <-------- R ------->                                   |
|                     Radial Settled                                      |
|                     Width (ft)                                          |
|                                                                         |
|   B = Measured Maximum Settlement Depth (inches)                        |
|   R = Measured Radial Settlement Width (feet)                           |
|                                                                         |
|   ALLOWABLE (Annex B.3.4):                                              |
|     Figure B.11 -> Bew  lap welds ~PARALLEL to shell (conservative)     |
|     Figure B.12 -> Be   no welds / butt welds / ~PERPENDICULAR laps     |
|     Figure B.13 -> lap weld at an ARBITRARY angle (interpolate)         |
+-------------------------------------------------------------------------+

Selecting the Allowable Curve (Annex B.3.4)

1. Measure $B$ and $R$ as defined by Figure B.6, applying the Figure B.7 correction on cone-up or cone-down bottoms.
2. Read the allowable from the correct figure:
   • Figure B.11 gives $B_{ew}$ for settled areas containing bottom lap welds approximately parallel to the shell (B.3.4.1).
   • Figure B.12 gives $B_e$ for areas with no welds, butt welds, or lap welds approximately perpendicular to the shell (B.3.4.3).
   • Figure B.13 covers a lap weld at an arbitrary angle (B.3.4.4). Interpolate between the two curves using

$B = B_e - (B_e - B_{ew})\sin\theta$

where $\theta$ is the angle of the weld to a tank centerline. 3. Screening shortcut (B.2.3.4): because $B_{ew}$ is more conservative than $B_e$, the simplest approach is to evaluate every settled area against $B_{ew}$ first. If all areas pass, the settlement is acceptable and no further evaluation is necessary; only the areas that fail need to be re-evaluated separately against $B_e$ or the interpolated allowable.

The 75 % Rule Is an Examination Trigger, Not a Knock-down Factor

A frequent exam trap is to treat 75 % as a multiplier applied to the allowable. It is not — the allowable is read straight off Figure B.11 or Figure B.12. The 75 % value sets an NDE trigger. Per B.3.4.1 and B.3.4.3, where measured settlement $B$ reaches or exceeds 75 % of the allowable:

• all shell-to-bottom welds and bottom welds in the settled area should be examined visually and by magnetic particle or liquid penetrant methods; and
• per B.3.4.2, welds within 300 mm (12 in.) of either side of the breakover area (see Figure B.6) should be examined visually, with any suspect area examined by MT or PT.
• All indications should be repaired, or evaluated for risk of brittle fracture and/or fatigue failure, before the tank is returned to service.

Tanks with settlement below 75 % of the allowable may be returned to service. Tanks above the allowable require repair or a detailed analysis of the bottom and the bottom-to-shell junction.

B.3.4.5 adds the timing rule: settlement occurs slowly and most of it is presumed to have happened in the first few years of service, so typical practice compares measured edge settlement directly against $B_{ew}$ and $B_e$ without adding an allowance for further settlement. The exception is an active mechanism such as erosion of the pad adjacent to the tank, which will keep producing settlement until the pad is repaired; where significant additional settlement is expected, an experienced engineer should evaluate the settlement projected to the next inspection against the B.3.4 limits, analogous to a corrosion allowance.

Internal Bottom Depressions and Bulges (Annex B.2.5)

Settlement depressions or bulges located remote from the shell (outside the 3-inch Critical Zone) are evaluated under API 653 Annex B.2.5 using empirical plastic membrane strain models.

The Parabolic Depressed Area Criterion

For localized dish-shaped depressions that are smooth and gradual without sharp creases or ridges, the maximum allowable settlement depth ($B_B$) is governed by the radius of the settled area:

$B_B = 0.37 \cdot R$

Where:

• $B_B$ = maximum permissible depth of the bottom depression in inches.
• $R$ = radius of the inscribed circle that fits within the settled depression area, measured in feet.

+-------------------------------------------------------------------------+
|             INTERNAL BOTTOM DEPRESSION INSCRIBED CIRCLE                 |
|                                                                         |
|          +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+                         |
|         /          Irregular Settled           \                        |
|        /               Depression               \                       |
|       |             . - - - - - - .              |                      |
|       |           /                 \            |                      |
|       |          |    Inscribed      |           |                      |
|       |          |   Circle (R)      |           |                      |
|       |           \                 /            |                      |
|       |             ' - - - - - - '              |                      |
|        \                                        /                       |
|         \~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~/                        |
|                                                                         |
|        Permissible Settlement Depth: B_B = 0.37 * R (inches)            |
+-------------------------------------------------------------------------+

Restrictive Conditions on Annex B.2.5 Evaluation

The $B_B = 0.37 \cdot R$ criterion is valid only under strict baseline conditions:

• No Welds in Severe Bending: Lap welds within the depression must not exhibit localized sharp peaking, buckling, or angular creasing.
• Sufficient Remaining Thickness: The bottom plates within the depression must maintain a remaining thickness meeting the minimum bottom thickness requirements ($MRT$) of API 653 Section 4.4. If severe pitting or top-side corrosion is present, localized membrane stretching under hydrostatic head can rupture thinned plates.
• Floating Roof Support Interference: The depression must not cause floating roof support legs to tilt or bind during touchdown.

Remedial Engineering Actions & Step-by-Step Worked Examples

When foundation settlement exceeds permissible limits ($S_i > S_{\max}$, $B > B_{\text{allowable}}$, or $B > B_B$), the tank cannot be returned to service without formal remediation approved by the Storage Tank Engineer.

Primary Settlement Remediation Techniques

1. Hydraulic Shell Jacking & Foundation Re-leveling:
   • Multi-point synchronized climbing jacks are attached to the shell via welded temporary lifting brackets.
   • The shell is raised in fractional increments ($< 1/4\text{ in.}$ differential between adjacent jacks) to prevent shell buckling.
   • Non-shrink cementitious grout or crushed stone ballast is packed beneath the raised shell and annular plates to re-establish a uniform planar foundation.
2. Sub-Bottom Mudjacking & Polyurethane Compaction Grouting:
   • Small injection ports are drilled through internal bottom plates in settled zones.
   • High-density expanding polyurethane structural resin or cement-bentonite slurry is injected at high pressure into the subgrade void space.
   • The expanding grout lifts the depressed floor plates back to nominal grade, eliminating water-trapping dish pockets.
3. Bottom Plate Cutout and Renewal:
   • If edge settlement has caused permanent plastic deformation or sharp creasing across the Critical Zone, the damaged annular plates and sketch plates must be cut out and replaced with new steel plates.

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Step-by-Step Worked Numerical Examples

Example 1: Evaluating Shell Out-of-Plane Settlement ($S_{\max}$)

Problem: A reconstructed storage tank has a diameter $D = 120\text{ ft}$ and total shell height $H = 48\text{ ft}$. The lowest shell course is constructed of ASTM A36 carbon steel with a specified minimum yield strength $Y = 36,000\text{ psi}$. The foundation survey is conducted using $N = 16$ equidistant stations. The mathematical cosine curve analysis reveals a maximum out-of-plane deflection at Station 5 of $S_5 = 0.85\text{ inches}$. Does this out-of-plane settlement satisfy API 653 Annex B?

Step 1: Calculate the arc distance between adjacent stations ($L$) $L = \frac{\pi \cdot D}{N} = \frac{\pi \cdot 120}{16} = 23.562\text{ ft}$ Verify spacing limit: $L = 23.562\text{ ft} \le 32\text{ ft}$ (Code compliant).

Step 2: Calculate maximum permissible out-of-plane deflection ($S_{\max}$) in inches Using the API 653 Annex B.2.2.4 formula in inches: $S_{\max} = \frac{66 \cdot L^2 \cdot Y}{E \cdot H}$

Substitute the known parameters ($E = 29,000,000\text{ psi}$): $S_{\max} = \frac{66 \cdot (23.562)^2 \cdot 36,000}{29,000,000 \cdot 48}$ $S_{\max} = \frac{66 \cdot 555.168 \cdot 36,000}{1,392,000,000} = \frac{1,319,079,168}{1,392,000,000} = 0.948\text{ inches}$

Step 3: Compare measured deflection to code limit $|S_5| = 0.85\text{ in.} \le S_{\max} = 0.948\text{ in.}$ Conclusion: Acceptable. The measured out-of-plane deflection is within the permissible limit. No shell leveling or jacking is required.

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Example 2: Evaluating Internal Bottom Depression ($B_B$)

Problem: An internal floor survey of an out-of-service crude tank reveals a smooth, localized dish depression located $25\text{ ft}$ from the shell. The largest circle that can be inscribed inside the settled boundary has a measured radius $R = 8.0\text{ ft}$. The measured maximum depression depth at the center is $B = 2.50\text{ inches}$. There are no sharp creases, and ultrasonic testing confirms bottom plate thickness exceeds minimum MRT. Is this depression acceptable per API 653 Annex B.2.5?

Step 1: Calculate permissible settlement depth ($B_B$) $B_B = 0.37 \cdot R = 0.37 \cdot 8.0 = 2.96\text{ inches}$

Step 2: Compare measured depth to permissible depth $B = 2.50\text{ in.} \le B_B = 2.96\text{ in.}$ Conclusion: Acceptable. The internal depression satisfies the parabolic floor criteria and does not require mudjacking or plate replacement.