Modelling granular soil to predict pressures on integral bridge abutments

Presented here is a granular soil model created to investigate the soil pressures which develop behind integral bridge abutments. The problem is introduced along with a brief summary of the fundamental behaviour, the model produced, and the initial validation. The paper looks at the initial outcome of the final validation procedure where experimental behaviour of an abutment is compared to the modelled behaviour, and the influence of the Young’s modulus profile applied is investigated. The results show that the model behaves as expected producing lateral stresses close to those measured experimentally. Comparisons show that the Young’s modulus profile adopted influences the resultant lateral stresses sufficiently to warrant further investigation. 2 THE FUNDAMENTAL BEHAVIOUR The basis of the developed numerical model was the work of Xu (2005). Xu carried out radial strain controlled cyclic triaxial tests implementing a stress path typical of that for an element of soil, represented by Leighton Buzzard sand, behind an integral bridge. The element considered was a typical mid-height element at 4m depth and the vertical cell pressure was kept constant at 80kPa to model the overburden stress to which the element would be subjected. The applicable radial strain range was estimated using both Finite Element analysis and a geostructural mechanism (Bolton & Powrie 1988) which ensured that the element was under typical loadings as found in service. The specimens were brought to an initial at rest state prior to cycling commencing. Various strain ranges and densities were considered to ensure that the results were applicable to a range of bridges with different soil conditions. The results of Xu showed that the typical relationship between horizontal stress and radial strain was that of a hardening law. As the sample was compressed the horizontal stress increased, and conversely upon triaxial extension the horizontal stress returned to the active state. When this was repeated for the same radial strain range in the following cycle the maximum horizontal stress was found to increase, as shown in Figure 1. Other key findings by Xu include: − The soil densifies until a critical value is reached, at which point it dilates − The soil stiffens, and peak horizontal stress increases, with cycles regardless of whether densification or dilatation occurs − Axial strain varies dependant on cycles Xu (2005) also investigated the same situation but replaced the sand with spherical glass ballotini. Under the same loading condition no build up of horizontal stress was found to occur. This allowed the researchers to conclude that the stress build up was primarily due to readjustment of the soil fabric due to rolling/sliding effects of non-spherical material close to the active state (Clayton et al. 2006). This is a significant finding as it concerns the fundamental behaviour of the soil. Figure 1. Typical Curve of Deviator Stress and Earth Pressure Coefficient K against Local Radial Strain for Leighton Buzzard Sand by Xu (2005). Figure 2. Development of the Hardening Parameter, γ, against cycles of Radial Strain. 3 BASIS OF CONSTITUTIVE MODEL The fundamental behaviour described above was used to establish the model. It was noted that stiffness and axial strain was proportional to density until dilation occurred but then continued with a similar trend post-dilation. This indicated that the influence of rolling/sliding could not be underestimated. The work of Xu (2005) had quantified the densification of the soil but this was not possible with the effects of rolling/sliding and the two mechanisms could not be separated. To compensate for this the concept of a Hardening Parameter (γ) was adopted. This was effectively a parameter which traced the density until the dilation point and then continued on the same slope beyond. This is shown in Figure 2. As the hardening parameter relied on density it was essential to investigate the densification behaviour. It was found that the rate at which it densifies under cycles of constant strain has two distinct slopes. The point at which the change between slopes was noted was designated as the Critical value of Relative Density. Therefore Equation 1 gives the change in relative density, ΔDr, as: ( ) ε γ Δ = Δ , r f r D (1) where γ = hardening parameter; and Δεr = change in radial strain. Similarly the critical value of relative density was found to be reliant on the radial strain range, giving: ( ) ε Δ = r f rcrit D (2) where Drcrit = critical relative density and Δεr = change in radial strain. Xu used Hooke’s Law to derive the secant horizontal Young’s modulus for each cycle. This was found to be represented by a logarithmic curve of the form given in Equation 3: ( ) B r A h E + Δ = ε ln ' (3) where E’h = Young’s modulus; Δεr = change in radial strain; and A and B = coefficients reliant on the hardening parameter and whether radial extension or compression is being considered. Similarly the relationship between radial and axial strain was found to be of the form: ( ) ( ) ( ) ε ε ε ε Δ + Δ + Δ = Δ r Z r Y r X a 2 3 (4) where Δεa = change in axial strain; Δεr = change in radial strain; and X, Y and Z are coefficients reliant on the hardening parameter and whether radial extension or compression is being considered. This relationship between radial and axial strain, although created based on data from Xu (2005) at a specific overburden stress (80kPa), was assumed to hold true at any value of overburden pressure. These relationships could then be used in conjunction with Hooke’s Law to calculate the stresses in the soil. However, before this could be tested the onset of failure needed to be considered. Values of the stress invariants p’ and q’ were plotted giving the stress path intended by Xu. The failure criterion from Critical State Soil Mechanics was considered. The failure surface of gradient M, where M is a function of the estimated angle of internal friction, was then superimposed on the plotted values. It was found from this that the soil failure coincided with the surface and therefore the decision was made to adopt this failure criterion. 4 VALIDATION BY TRIAXIAL TEST The next stage was to implement the constitutive model described above into the commercial finite difference method (FDM) package FLAC (Itasca 2005). There is considerable expertise in FLAC within the Mott MacDonald group and FLAC has the built in programming pseudolanguage ‘Fish’ (FLAC-ish) making it possible to program user-defined constitutive models. A model of a triaxial test was created using a single axi-symmetric element, and the constitutive model was assigned with the properties and initial loading conditions based on the experiments by Xu (2005), with an overburden of 80kPa to simulate a typical element at 4m depth, and at rest horizontal stresses. The analysis was run for many cycles, and a typical example of the lateral stress output is shown in Figure 3, showing that the model worked as expected. Figure 3. Typical Curve of Lateral Stress against Radial Strain from FLAC analysis of the triaxial test using the granular constitutive model. 5 VALIDATION USING EXPERIMENTAL DATA The triaxial test modelling showed that the mathematical model as implemented and programmed within FLAC worked for a single element. The next stage of validation was to investigate how the constitutive model behaved when applied to a larger system consisting of an abutment structure and retained soil. An initial series of analyses were carried out and are reported on in this paper in which the results were to be compared with experimental laboratory data for an abutment. Further research is under way to more fully explore the capabilities of the model and carry out parametric studies. 5.1 Tapper and Lehane experiments Tapper & Lehane (2004) carried out centrifuge testing using a 1:20 scale model of a stiff abutment wall with a pinned base. Medium dense uniform siliceous sand was used as the backfill. Three rotation amplitudes of the wall were investigated and the results show that a steady state in terms of pressure coefficient behind the wall was not reached even after 1000 cycles, which is similar to the findings of Xu (2005) in his triaxial tests. Therefore, the work of Tapper and Lehane was regarded as appropriate for validation purposes. The dimensions of the prototype on which the centrifuge model was based were used in the FLAC validation model rather than the centrifuge test specimen itself. The prototype consisted of a stiff wall pinned at the base and propped at the top, with the prop moved by a distance Δ either side of a zero position to imitate the thermally induced bridge deck movements. The retained height H of the granular material behind the wall was 4m, and the prototype was 10.6m long, ensuring that the remote vertical boundary would not influence the stress development at the back of the abutment.