Note to testers

VPI software is currently in beta testing stage.

If you wish to test the software and helps us move forward, please leave a comment on this page or contact us at one of the following addresses:

Dave Cebon - dc29@hermes.cam.ac.uk
Riccardo Isola - riccardo.isola@nottingham.ac.uk

Thanks,

The VPI team.

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permanent behaviour

Permanent deformation in UGMs is mainly related to resilient stresses or strains and number of load cycles. The effects of these two factors are often separated and the models for predicting permanent deformations can assume the form of:
                                                                                                               (1)
Amongst the numerous models that are available in the literature, Hornych (2004) shows that the Gidel model is particularly suitable to describe the variation of permanent deformations with stress level:
                                                                                              (2)
The link between permanent deformation and number of load cycles can instead be expressed by the simple Sweere model:
                                                                                                                                                (3)
or by the slightly more elaborated Hornych model:
                                                                                                                          (4)
All these models are derived empirically by fitting curves to laboratory data and do not take into account the effect that moisture content has on the permanent behaviour of the granular material. In theory, a different set of fitting parameters would need to be calculated for each value of moisture content considered. The ME-PDG, instead, uses a modified version of the Tseng and Lytton model that relates permanent strains to resilient strains, number of cycles and moisture content:
                                                                                                                      (5)
where β, ρ and ε₀/ε_r are functions of water content.

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Resilient Behaviour

In dealing with granular materials, the elastic properties are often defined by the resilient modulus (which replaces the modulus of elasticity to indicate the nonlinearity of the behaviour) and Poisson’s ratio. A common approach in dealing with the stress dependency of granular materials stiffness is the K-θ model, which expresses resilient modulus as a function of the sum of the principal stresses, or bulk stress.
                                                                                                                                (1)
Thanks to its simplicity, this model has been widely adopted for analysis of stress dependence of material stiffness. However, it has the drawbacks of assuming a constant Poisson’s ratio and of ignoring the effect of other stress parameters apart from the bulk stress. For this reason, several more elaborated versions of this model have been suggested such as, for instance, the so called “Universal Model”, first suggested by Uzan and then adopted by the ME-PDG:
                                                                                             (2)
This model includes the 3D effect of shear, or deviator, stress by means of the octahedral stress. Most of the models of this family have empirical origins and are mainly used in the US.
A more theoretical approach is taken by those models that characterise the non-linear behaviour by decomposing stresses and strains into volumetric and shear components. In this case, resilient modulus and Poisson’s ratio are replaced by bulk and shear moduli:
                                                                                                                                                  (3)
                                                                                                                                                (4)
where
                                                                                                                                 (5)
                                                                                                                                            (6)
                                                                                                                                 (7)
                                                                                                                            (8)
In 1980 Boyce used this type of approach to develop a theoretical nonlinear elastic model for the stress-strain relationship of granular materials:
                                                                                                                         (9)
                                                                                                                         (10)
this model has then been employed by many other researchers, particularly in mainland Europe. It is important to note that it is possible to express resilient modulus and Poisson’s ratio in terms of bulk and shear moduli:
                                                                                                                                   (11)
                                                                                                                                          (12)
This can be useful considering that our current approach to including the effect of moisture content in granular materials makes use of the resilient modulus.

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UGMs - Introduction

Over the years, many researchers have studied the complex behaviour of granular materials, using laboratory and in situ testing techniques. An extensive literature review was carried out by Lekarp et al. (2000) to collect findings from previous research and summarize in two companion papers the state of knowledge on resilient and plastic properties of granular materials. In this work the authors point out that these properties are affected by numerous factors such as stress, density, grading, moisture content, stress history, particle shape and load frequency, nonetheless the effect of stress parameters is certainly dominant.
If granular materials are simulated by means of Layered Elastic algorithms or using the Method of Equivalent Thickness, this stress dependency is usually dealt with through iterative processes (Figure 1):
Figure 1: Iterative determination of resilient modulus
The complexity of the problem meant that a large number of models can be found in the literature. An interesting review of available models for the prediction of permanent deformations in UGMs has also been conducted as part of the SAMARIS project by Hornych et al. in 2004.
Hereafter we present a selection of the main approaches to resilient and permanent behaviour of UGMs that might be implemented in the WLPPS.

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Results 2

Normalised Aggregate Force and Fourth-Power Forces
Figure 1 is a comparative plot of the fleet normalised aggregate forces generated by the four simulation methods.
Figure 1: Comparison of fleet normalised aggregate force histories for each of the four simulated vehicle fleets
The ‘reference’ and ‘target’ time histories agree very well, while the time histories from the phase shifted and randomised QCM fleets appear to follow a similar pattern. This comes down to the inherent dynamics of each model; the ‘reference’ and ‘target’ histories were generated by pitch-plane models, the randomised QCMs and phase-shifted models used QCMs that lack pitch interaction and wheelbase filtering.
SRI Curves
Figure 2 and Figure 3 are comparative plots of the SRI curves generated by each of the four simulated fleets and a re-plotting of the region 0.8<SRI<1, respectively. From Figure 2 and Figure 3 it can be seen that the ‘target’ and the phase-shifted SRI curves agree well, as expected.
Due to the fact that vehicles with similar suspension characteristics correlate well with each other, all curves show a sharp peak close to the maximum value of one and a secondary peak in the region of SRI=0.85, corresponding respectively to those vehicles that are entirely air sprung and those that have leaf-sprung trailers.
Figure 2: Comparison plot of SRI curves for each of the four simulation methods
Figure 3: Comparison plot of SRI curves re-plotted for the range 0.8<SRI<1
Computation Time and Summary of Results
Table 1 provides a summary of computation time and accuracy metrics for each simulation method.

Method	Simulation Time (One Run) [sec]	Simulation Time (20 years, @ 1 per week) [days]	R2 of SRI Curves Relative to ‘Reference’	R2 of SRI Curves Relative to ‘Target’	Correlation of ATFs to ‘Reference’ Fleet	Correlation of ATFs to ‘Target’ Fleet
Reference	147600	1777	1	-	1	-
Target	26000	313	0.91	1	0.99	1
Random QCM	24000	290	0.76	0.76	0.75	0.74
Phase Shifted QCM	90	1.5	0.53	0.89	0.52	0.48

Table 1: Summary of simulation time, goodness-of-fit, and correlation of normalised aggregate forces for all four simulated fleets
Although the phase-shifted method includes the overhead time of generating the ‘target’ SRI curve first (i.e., from measured data or more realistic simulations), this is a ‘one-time’ cost and the phase-shifted models still represent a significant decrease in computation time. For example, if one was simulating 20 years worth of traffic in weekly intervals, 1040 separate traffic calculations would be required. Using the randomised pitch-plane models of the ‘target fleet’ would take approximately one month of CPU time. Conversely, the phase-shifted method would require only 1.5 days of CPU time, accounting for the overhead of generating the ‘target’ SRI distribution; a 99.5% reduction in computing time.
Given the substantial computational benefit of the phase-shifting method, and the excellent agreement of the SRI statistics, it is believed that the phase-shifted QCMs are still the best available method for simulating dynamic tyre forces for whole-life pavement performance calculations in which the effects of millions of axle loads need to be simulated over the lifetime of the road surface.

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Spatial repeatability 2

Spatial repeatability arises because trucks are similar in weights, dimensions, and dynamic characteristics and travel at similar speeds. As a result, each vehicle will apply its peak forces at approximately the same places along the pavement surface (Cole, Cebon 1992).
Cole et al. defined the spatial repeatability index (SRI) as the correlation coefficient between a dynamic tyre force histories,
                                                                                                         (1)
where x and y are dynamic tyre forces histories,
m_x and m_y are the mean forces of x and y, respectively,
and σ_x and σ_y are the standard deviations of x and y, respectively (Cole, Cebon 1992; Cole et al. 1996).
An alternate measure of spatial repeatability, suggested by Cole, is the fleet normalised aggregate tyre force,
                                                                                                                              (2)
where
F_i,jk is the tyre force at the i-th road point due to the k-th axle of the j-th vehicle,
m is the number of axles on each vehicle,
N_V is the total number of vehicles,
and is the mean of the double sum in the numerator (Cole et al. 1996).
The fleet normalised aggregate force gives a spatial (time-domain) picture of the cumulative pattern of traffic loading.

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Examples 2

Example 1 – Full depth crack
A full-depth crack allows the water to reach the top of the granular layer, saturating it. In order to model how the moisture spreads from this point throughout the layer, we keep the top left element of the structure saturated (Figure 1 to Figure 3).
Figure 1: Moisture infiltration through a full-depth crack – t=0
Figure 2: Moisture infiltration through a full-depth crack – t=4
Figure 3: Moisture infiltration through a full-depth crack – t=8
Example 2 – Pipe leak
Similarly to the case of a full depth crack, the effect of a leaking pipe can be simulated by keeping a node internal to the structure constantly saturated (Figure 4 to Figure 6).
Figure 4: Moisture infiltration from a leaking pipe – t=0
Figure 5: Moisture infiltration from a leaking pipe – t=4
Figure 6: Moisture infiltration from a leaking pipe – t=8
Example 3 – Micro-cracks
The presence of micro-cracks over a certain length of pavement could be modelled by keeping the degree of saturation of some nodes underneath the asphalt layer constant but lower than 1 (Figure 7 to Figure 9).
Figure 7: Moisture infiltration through micro-cracks– t=0
Figure 8: Moisture infiltration through micro-cracks– t=4
Figure 9: Moisture infiltration through micro-cracks– t=8

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Variably Saturated flow

Numerous numerical models have been developed for simulating water movements in materials with various degrees of saturation. Clement et al (1994) reviewed many of them underlining the most common shortcomings. In most applications, the pressure-based form of the variably saturated flow equation through homogeneous and isotropic porous media is used (Cooley, 1983; Huyakorn et al., 1984):
                                                                                        (1)
where:
S_s is the specific storage of the medium [L^-1]
Ψ is the pressure head [L]
θ is the water content
η is the porosity
K(Ψ) is the hydraulic conductivity [L T^-1]
t is time [T]
This equation describes the movement of water in soils under the following assumptions:
1. The dynamics of the air phase do not affect those of the water phase;
2. The density of water is only a function of pressure;
3. The spatial gradient of the water density is negligible.
Finite elements solutions to this approach often incur mass-balance problems in unsaturated media (Celia et al., 1990).
The finite-difference algorithm developed by Clement et al. has the advantages of being computationally simple while remaining capable of modelling a wide variety of problems, including infiltration into dry soils. In this approach, a mixed form of Equation 1 is solved by calculating the pressure heads Ψ at each node for a given time step (n+1) using a modified Picard iteration, where at each iteration (m+1) we solve a system of linear equations defined as:
                                                                                                                                                                                                                                   (2)
where
 
 
 
                                                                                                              (3)
where
Θ is the moisture content
K is the hydraulic conductivity
C is the water capacity
Δx is the horizontal step
Δz is the vertical step
Δt is the time step
S_s is the specific storage
η is the porosity
As can be seen, all the coefficients used for the current iteration (m+1) are calculated from the results of the previous iteration (m). Θ, K and C vary for each node and are functions of Ψ, therefore need to be re-calculated after each iteration.
As discussed earlier, the link between Θ and Ψ is given by the Soil-Water Characteristic Curve (SWCC). The hydraulic conductivity of the materials can be expressed as a function of the pressure heads (Van Genuchten, 1980) by:
                                                                                               (4)
Finally, by definition, the water capacity represents the slope of the SWCC and therefore can be expressed as:
                                                                                                                                               (5)
Equation 2 applies to all interior nodes, while at boundary nodes it needs to be modified to consider the appropriate boundary conditions.
In the examples shown hereafter we consider a pavement structure like the one in Figure 1, where the water table depth is 2m from the top of the unbound layer. In normal conditions this system is in equilibrium and the water content does not vary with time.
Figure 1: Variably saturated flow – pavement structure
The boundaries considered are as follows (see also Figure 2):
► Top and right-hand side nodes: Neumann boundaries (no flow);
► Bottom nodes: Dirichlet boundaries (constant moisture content);
► Left-hand side nodes: symmetry.
Figure 2: Variably saturated flow – boundary conditions

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fenv calculation 2

The inputs are organised in three hierarchical levels of detail, level 1 being the most accurate where most parameters are directly measured and level 3 being the simplest where only few parameters need to be known and are used to estimate everything else.
As noted before, the relationship between suction and degree of saturation for a soil is described by the Soil-Water Characteristic Curve (SWCC). Figure 1 shows various SWCCs for soils with different characteristics.
Figure 1: Soil-Water Characteristic Curves
The minimum inputs required (input level 3) to calculate the SWCC for a given soil are the plasticity index, PI, the percentage by weight passing the sieve #200, P200, and the effective grain size corresponding to 60% passing by weight, D60. As can be seen, these are all granulometric parameters of the material. P200 and PI are used for soils that contain a certain fraction of fines and have, therefore, a plasticity index larger than zero. They deliver the family of curves on the right hand side of Figure 1, where it is possible to see that thanks to the fine fraction the degree of saturation varies gradually with suction.
D60, instead, is used for coarser materials that have PI=0. In this case the SWCCs show a sharp change in degree of saturation with suction due to the lack of capillarity.
These curves and the determination of the parameters that define them are thoroughly described in the ME-PDG, Part 2 - Chapter 3, therefore won’t be reported in detail here.
The depth of the water table, which can be considered to vary seasonally or to be constant throughout the year, determines the values of suction at any point in the pavement, which are then used to calculate the degree of saturation by means of the SWCC. From the degree of saturation it is then possible to calculate F_u, the environmental factor for unfrozen or fully recovered material.
If the temperatures are below freezing point, the environmental factor takes the form of F_f, which has a constant value much larger than 1 since frozen materials are much stiffer.
Finally, when the soil is thawing the environmental factor becomes F_r (for recovering material) and is given a value lower than F_u. This value is at a minimum at the beginning of the recovering period, when the material is weakest, and tends gradually to F_u as the material returns to its normal, unfrozen state.
An example of how these factors appear with time for a number of nodes in a pavement is given in Figure 2.
Figure 2: Evolution of F_env with time
In Figure 2, it can be seed that on day 1 some nodes are frozen, some are thawing and the rest are unfrozen. As time passes, the coefficients for the frozen material remain constant for as long as the material stays frozen, while for the recovering material they increase up to the normal unfrozen value. Node 12, for instance, stays frozen until day 3 with F_f = 75, then starts thawing and the value of the environmental coefficient drops to 0.6. As the thawing goes on this value starts to increase and will eventually reach the value of F_u. Similarly, node 17 is already thawing at day 1 and at day 6 the thawing ends. At that point the material has completely recovered and the coefficient has reached a value of F_u of 0.9.

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