FEM/LET/MET
FEM/LET/MET
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FEM Plot
Edit Mesh
This option is only available if a license has been issued by Dynatest.
This option cannot be used with jointed pavements.
With this option backcalculation may be carried out either with the Finite Element Method (FEM), Linear Elastic Theory (LET) or the Method
of Equivalent Thickness (MET). FEM makes use of a modified version of an axial symmetric finite element program, originally developed by
Wilson at University of California (Duncan et al., 1968). LET makes use of the Waterways Experiment Station’s program (WESLEA) and MET
is similar to the method used in the Elmod option "Basin fit", but with a simpler use of adjustment factors. Where Elmod applies adjustment
factors directly on the calculated responses, MET makes use of the traditional Odemark adjustment factors directly on the layer thicknesses.
The moduli backcalculated by the two methods may, therefore, be different.
It is important to notice that this option is for backcalculation only, not for calculation of remaining life or overlay design. The moduli
calculated with this option are for the test conditions, without any adjustment for temperature or seasonal effects. Therefore temperatures are
not checked. The program may, however, be used to calculate the stresses and strains under a single load. If all the points of a section are
evaluated using the Fit all points option, the moduli will be stored in the database and can be used for calculation of remaining life and
needed overlay design in Elmod.
When the option FEM/LET/MET is selected on the Estimating Moduli screen, the main input and output screen is shown, for example as in
Figure 1. The default data shown on this screen, refers to the first drop and the first section selected on the Moduli screen in Elmod. The
chainage will be from the lowest to the highest value for the first section, and the two numbers in the From drop and the to input boxes at the
top of the screen will both be equal to the first drop number selected on the Moduli screen. Both chainage and drop numbers may be
changed by the user. If, for example, the From drop value is changed to 1, then the FWD test data used will be the average values for all of
the drops 1 to 4.
When opening this option, the test data corresponding to the first chainage and the first drop number will automatically be imported. The
contact stress is shown in a text box as is the radius of the loaded area (Plate radius). For backcalculation none of these values should be
changed. For forward calculation of pavement response, different values of contact stress, radius of loaded area and an offset distance input,
which can be set in Options, may be entered.
The measured deflections are shown in a column next to the column giving the distances to the geophones. In Figure 1 a deflection of 640.0
is measured at the center geophone (distance 0). In SI units deflections are in μm, distances in mm and moduli are in MPa. The contact
stress is in MPa, but output stresses (response) are in kPa and output strains are in μstrain. The corresponding US units are: deflections in
mil, distances in inch and moduli in ksi. Contact stress is ksi, but output stress is psi.
The layer thicknesses are those corresponding to the section and the moduli are the default values entered in the Structures form (in
Elmod), next to the thicknesses. If no default moduli have been entered in Elmod, the program will suggest some values. Poisson’s ratio is
0.35 for all layers, as a default, but may be changed. Default values are also used for the minimum (Emin) and maximum modulus (Emax), of
each layer.
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FEM/LET/MET
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Figure 1. Main input and output screen for FEM/LET/MET.
The column with the heading n indicates the non-linearity of the materials. With FEM all layers may be non-linear, with LET none of the layers
and with MET only the subgrade may be non-linear. With a negative non-linearity (n is negative, cohesive materials), n will be the exponent in
the relationship:
where C is a constant equal to the value in the E column,
σ is the major principal stress, and
1
p is atmospheric pressure.
With FEM the vertical stress, σ , may be used instead of σ , by checking the Use zigz with FEM box on the Options screen.
z
1
With a positive non-linearity (granular materials), n is the exponent of:
Where θ is the hydrostatic stress (mean normal stress or first stress invariant, 1/3 times the bulk stress).
The deflections may be calculated by pressing the command buttons: FEM, LET or MET. Before FEM can be used, however, the finite
element mesh has to be set up. This may be done by pressing the command button Generate FEM mesh with, where the number of
columns and rows is given in the text box. A maximum of 100×100 columns and rows may be used. The program suggests a distance to the
outermost node. This may be selected or overwritten. If no mesh is generated, the program will create a mesh based on default values.
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Figure 2 Input box for maximum distance in finite element mesh.
The mesh is saved in the active Microsoft Access database in the tables "Column" and "Row". The default mesh may be edited in the
database.
Backcalculation
The layer moduli, the non-linearity and the depth to bedrock may be varied in the backcalculation. The checkbox next to the parameter
determines whether a parameter is varied or kept constant. In Figure 1, for example, the moduli of the three layers are all varied, whereas the
non-linearities are kept constant (equal to 0 for the first two layers and -0.01 for the third). In this example no thickness has been entered for
the subgrade which is, therefore, treated as semi-infinite. With LET all layers are treated as linear elastic and with MET only the subgrade
may be non-linear, as mentioned above. With FEM the depth to bedrock will not be varied.
The option buttons below FEM, LET or MET are used to select the backcalculation method.
Fit point
This option will calculate the moduli for only the current point, and the results will not be saved. This option can be used for manually
calibrating the seed values and the backcalculation method.
Fit and save all points
This option will calculate all points for the section and save the results.
Save point
This will save results for the current point only. The idea is to use this after a calculation for all points to adjust the results for one or few single
points. This could be by changing seed moduli or adjusting layer thicknesses. The procedure is following to select the point, use the Fit point
option and click Save point.
Notice following:
-
You can only save single point results afterwards you have made a full analysis for the section, using any
backcalculation routine from FEM/LET/MET or deflection basin fitting from the core module.
You cannot change number of layers in the analysis.
You may change thickness of layers. This will be recognized in the design, and also be listed correctly in exports and
reports from the PLOT program. It will not change the structural input as defined, so thickness plot in the PLOT features
will not reflect the change, as well as thickness is reset next time you open FEM/LET/MET module.
In the example in Figure 3, Linear Elastic Theory was chosen. The resulting calculated deflections are shown in the column under LET. The
deflections are also shown in an output grid which may be activated by selecting Show → Details, see figure 4. Here the deflections are
shown at the top of each layer in the structure. The two output lines at the bottom of the grid show the difference between the measured and
calculated deflections, first the percentage difference ({calculated-measured}/measured * 100%) and then the absolute difference (calculatedmeasured), in μm for SI units and mil with US units. As a default, the program will attempt to minimize the Root Mean Square (RMS) of the
absolute difference. The check box minimize % in Figure 3 may be used to minimize with respect to relative difference (percentage).
Figure 3 Example of backcalculation using LET.
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Figure 4 Details of deflections
The two text boxes at the top of Figure 3, next to RMS show of the Root Mean Square (RMS) difference as absolute difference (abs) and
percentage difference (%). In the example the absolute RMS was 4.04 μm and the percentage was 1.80. It may be noticed in Figure 3 that
the modulus of layer two is lower than the modulus of the subgrade, a frequent problem when the subgrade is assumed to be linear elastic.
Plotting deflections and surface moduli
Select Show → Graph to plot the deflections and the surface moduli, for measured as well as for calculated values.
In Figure 5 moduli have also been calculated with MET, assuming a non-linear elastic subgrade. The results from MET are compared to
those from LET below:
Layer
Modulus LET
Modulus MET
1
2147
1858
2
119
175
3
127
112
With MET the subgrade was non-linear n = -0.02. The modulus at the top of the subgrade, corresponding to the FWD load, is not shown
during back-calculation but can be obtained by selecting Show → Responses
Figure 5. Plot of measured, LET and MET values.
The response is given for each interface and each layer. The top line is for interface number 1 and layer number 1 (1/1), the next line for
interface number 1 and layer number 2 (1/2), etc. The tangential stress (zigt), the radial stress (zigr) and the vertical stress (zigz) are given
followed by the corresponding strains. The last column is the modulus at the position where stresses and strains are calculated. For a nonlinear subgrade it may be recalled that the modulus to be used for calculation of deflections is approximately (1-2n) times the modulus used
for strain.
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Figure 6. Response under FWD load.
Additional controls
A number of additional input boxes may be used to control the backcalculation. The input box Use first may be used to limit the number of
geophones used for the backcalculation. Sometimes, particularly on weak pavements, the deflections recorded by the outer geophones are
judged to be unreasonable, and it may be preferable to leave them out. If for example only the first five geophones were to be used, the
number 5 would be entered in the Use first input box.
The menu Calibrate is used to calibrate FEM and MET against LET, which provides the mathematical exact solution for a layered linear
elastic system. First the response is calculated using LET (with an appropriate subgrade modulus if the subgrade is non-linear). Then FEM is
used on the linear elastic case, and appropriate calibration factors are determined to get the same response as with LET. Then FEM is used
on the non-linear case and with the calibration factors to determine the response in the linear case, and finally MET is used on the non-linear
case and the calibration factors needed for MET to give the same response as FEM are determined.
Pressing the button Fit point will result in a basin fit with the method selected (FEM, LET or MET), but the results will not be stored in the
database. To store results in the database the button Fit all points must be used. This will process all of the data points within the given
chainage range. Similarly for responses. It is normally advisable to manually modify the moduli until a reasonably good agreement is obtained
between measured and calculated deflections, and then store these values as seed values using the Save seed values button.
When FEM is used the results of the last iteration are stored in the tables "Displacements" and "Stress" in the database.
When a new test point is selected with Select point the thicknesses and moduli are refreshed with the seed values.
More options are available from the Options menu:
Figure 7. Options
If a value (integer) is entered in Alternate field optimization of the RMS differences will be attempted for the number of times entered here. If
1 is entered, the program will first minimize with respect to the absolute difference, it will then use the resulting parameters (moduli, nonlinearity or depth to bedrock) as seed values for a minimization with respect to relative difference (%), and finally redo the absolute
minimization starting from the results of the relative.
MET will use a standard set of adjustment factors, unless a value is given for the adjustment factor, in which case this will be used for all
layers.
If a value is given for the Ei/Ei+1 (the ratio of two consecutive moduli), the equivalent thickness will not be calculated if the ratio is less than
this value. Below this ratio the actual layer thickness is used directly. This is used only with MET
The input for Factor on E1 for response is not used for backcalculation, but only for calculation of response. In calculating the response, the
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modulus of the first layer will be multiplied by the value entered here. This allows, for example, for a lower asphalt modulus under a slow
moving rolling wheel load than under the FWD impact load.
MET will as default use a point load for calculation of response off the centerline of the load, unless the box for adjusting for point load is
checked.
The number of iterations used by the Finite Element Method is set in an input box, and a checkbox use zigz with FEM may be used to let
the modulus be calculated based on the vertical stress rather than on the major principal stress.
The value given in the Search width input box controls how much the parameters are varied. A large value will search with larger steps.
Use tail for C and n seeds calculates seed values for these based on the deflections from the outer-most geophones.
References
Duncan, J.M., Monismith, C.L. & Wilson, E.L. "Finite Element Analysis of Pavements", Highway Research Record 228, 1968.
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