Introduction
You may have tried to validate stress transfer functions calculated in Fatigue against member end force transfer functions generated in Wave Response and found that the transfer functions do not match. In this post, we will take a look at how SACS calculates member force transfer functions and why they are different from the stress transfer functions reported by Fatigue.
When Wave Response performs a transfer function analysis, SACS calculates the dynamic response of the structure due to a series of deterministic waves defined in the Seastate input, typically using Generate Transfer Function (GNTRF) lines. Wave Response then generates an equivalent static loads corresponding to each wave and phase specified in the transfer function wave definitions. As part of this analysis, users have the option to generate member end force transfer function reports and plots using the Transfer and Response Function Plot (PLTTF) and Plot Selection Member Force (PSMF) lines. Wave Response generates these plots using the modal responses.
After the Wave Response analysis is performed a linear static analysis of the equivalent static loads is performed to generate a common solution file containing the solution of each wave and phase specified in the transfer function wave definitions. This common solution file is used by the Fatigue program to calculate the stress transfer functions, RMS stresses, and ultimately the fatigue damage. All stress transfer functions are calculated using the fatigue input and the results from the common solution file.
It is important to note that the transfer functions for Wave Response are based upon the modal response of the structure whereas the transfer functions for Fatigue are based upon the solution of the equivalent static loads. Because the two approaches will result in different internal loads, the member force and member stress transfer functions reported by the two programs is not comparable.
Let’s take a look at an example for a simple connection to see the how this difference occurs in practice.
Example
For this example we will focus on a simple T joint made of a 41.25x1 in tubular chord with 42x1.375 in tubular cans and a 16x0.625 in brace.
The transfer function will be limited to two waves at 6.0 s and 5.0 s to reduce the amount of data that we need to work with for this example. Realistically many more waves would be required to accurately define the transfer function.
First, we will perform a modal analysis to determine the natural period and mode shapes of the structure. We can also determine the modal internal forces in the members by solving the linear static analysis of the displaced shape.
SACS already calculates the linear static analysis in a modal analysis to generate the modal reaction reports, but we normally don’t have access to internal forces. However if we run Post on the mode shape (dynmod.*) file we can generate a SACS Report Database to view the internal loads in R_POSTMEMBERRESULTS.
Below is an excerpt of the member results table with the internal forces.
| MemberName | LoadConditionName | Distance | AxialLoad | YYBendingMoment | ZZBendingMoment |
|---|---|---|---|---|---|
| 301-0000 | 1 | 0 | 2.92196E-40 | 0 | 1293.467875 |
| 301-0000 | 2 | 0 | -2978.828 | -122.7281016 | 0 |
| 301-0000 | 3 | 0 | -45.06422656 | -1650.703875 | 0 |
Note that the load conditions correspond to the natural frequencies of the structure and I am only reporting the internal forces at distance zero because we are only interested in the transfer function at the interface of the brace and the chord.
In order to calculate a modal transfer function we need to linearly combine the modal responses for each mode to determine the member force transfer function for each wave. We can get the modal responses from the modal coordinates reported in the Wave Response neutral chart file.
| Mode | Wave @ 5 s | Wave @ 6 s |
|---|---|---|
| 1 | 1.93E-02 | 2.23E-02 |
| 2 | 8.73E-09 | 1.68E-08 |
| 3 | 6.31E-07 | 1.21E-06 |
Note that we only have one modal response for each mode which represents the transfer function (i.e. maximum response vs minimum response).
Because wave load is primarily in the out-of-plane direction and the natural frequency is in that direction as well, the majority of the internal load will be out-of-plane bending about the minor axis (ZZBendingMoment). We can calculate the transfer function for the out-of-plane bending moment with the following equation.
Where:
is the modal response for mode n
is the internal bending moment in the beam for mode n
The resulting values are 24.982 kip-ft and 28.860 kip-ft for the 5 second wave and 6 second wave respectively which exactly corresponds to the reported member end force transfer function for moment about the z-axis.
Next, we will perform the transfer function generation analysis with Wave Response. We have defined two waves with 18 crest positions 36 equivalent static load cases are generated. We can inspect the reported internal loads for each of the equivalent static load conditions by running Post on the common solution file (saccsf.*) and generating a SACS Reports Database. The internal loads are located R_POSTMEMBERRESULTS.
| MemberName | LoadConditionName | AxialLoad | YYBendingMoment | ZZBendingMoment |
|---|---|---|---|---|
| 301-0000 | 1 | 0 | 0.000742 | -1.389913208 |
| 301-0000 | 2 | 0 | 0.000469 | -0.87899646 |
| 301-0000 | 3 | 0 | 0.000135 | -0.252711517 |
| 301-0000 | 4 | 0 | -0.00022 | 0.409227844 |
| 301-0000 | 5 | 0 | -0.00055 | 1.025679077 |
| 301-0000 | 6 | 0 | -0.00081 | 1.521213379 |
| 301-0000 | 7 | 0 | -0.00098 | 1.836553467 |
| 301-0000 | 8 | 0 | -0.00103 | 1.934891235 |
| 301-0000 | 9 | 0 | -0.00096 | 1.804689819 |
| 301-0000 | 000A | 0 | -0.00078 | 1.460782349 |
| 301-0000 | 000B | 0 | -0.0005 | 0.943273193 |
| 301-0000 | 000C | 0 | -0.00017 | 0.314461029 |
| 301-0000 | 000D | 0 | 0.000185 | -0.347203613 |
| 301-0000 | 000E | 0 | 0.000513 | -0.961402466 |
| 301-0000 | 000F | 0 | 0.000774 | -1.45034436 |
| 301-0000 | 000G | 0 | 0.000939 | -1.75799292 |
| 301-0000 | 000H | 0 | 0.000992 | -1.856880249 |
| 301-0000 | 000I | 0 | 0.000921 | -1.72503064 |
| 301-0000 | 000J | 0 | 0.000147 | -0.275235779 |
| 301-0000 | 000K | 0 | 0 | 0 |
| 301-0000 | 000L | 0 | -0.00011 | 0.196785355 |
| 301-0000 | 000M | 0 | -0.00022 | 0.413073456 |
| 301-0000 | 000N | 0 | -0.00031 | 0.579588318 |
| 301-0000 | 000O | 0 | -0.00036 | 0.676827271 |
| 301-0000 | 000P | 0 | -0.00037 | 0.692759216 |
| 301-0000 | 000Q | 0 | -0.00033 | 0.626284973 |
| 301-0000 | 000R | 0 | -0.00026 | 0.484986359 |
| 301-0000 | 000S | 0 | -0.00015 | 0.28506955 |
| 301-0000 | 000T | 0 | 0 | 0 |
| 301-0000 | 000U | 0 | 0.0001 | -0.187336151 |
| 301-0000 | 000V | 0 | 0.000214 | -0.401042175 |
| 301-0000 | 000W | 0 | 0.000302 | -0.565854065 |
| 301-0000 | 000X | 0 | 0.000352 | -0.659247437 |
| 301-0000 | 000Y | 0 | 0.000361 | -0.675728638 |
| 301-0000 | 000Z | 0 | 0.000326 | -0.609803833 |
| 301-0000 | 000a | 0 | 0.000252 | -0.471911255 |
Here we do not have the results reported as a transfer function so we need to calculate the transfer function ourselves. The formula to determine the transfer function for the out-of-plane bending moment is:
Where:
is the internal bending moment in the beam for crest position i
is the wave height
The resulting values are 20.391 kip-ft and 25.651 kip-ft for the 5 second wave and 6 second wave respectively which does not match the reported member end force transfer function for moment about the z-axis. This change is due to the difference in the member end forces in the modal response analyses versus the equivalent static load analysis.
There are a few reasons why the internal forces in the equivalent static load solution do not match the dynamic analysis. SACS calculates the equivalent static load from the original static wave loading on the structure () plus the inertia load (
). This can cause a change in the internal loading. Additionally, the modal response is determined from a linear system, so any non-linear elements in the equivalent static load analysis like pile/soil interaction can produce different results.
Conclusion
We have shown why the transfer functions in the Wave Response analysis do not necessarily match the transfer functions in the Fatigue analysis because the transfer functions in the Wave Response analysis are determined from the modal response and the transfer functions in the Fatigue analysis are determined form the equivalent static load conditions.