A new equation to predict the footings settlement on sand based on the fnite element method

In the design of shallow foundations normally used in rural buildings on sand, the settlement criterion is more critical than the bearing capacity of the soil. Likewise, it has also been found that widely used methods to estimate the footings settlement on sand, generates differences with the observed in full scale field tests. The aim of this study was to find a new equation based on finite element method (FEM) easy to apply to estimate the footings settlement on sand. This new equation considers the effect of the depth of the founding level, the footing breadth, the equivalent soil stiffness, the Poisson’s ratio and the net increase in the effective stress on settlement values. To obtain this equation, a three-dimensional finite element model was generated and subsequently validated using actual footings settlement measured during field tests. The results of settlement predictions extracted from this method are slightly better than those obtained by other methods, but this equation has the advantage of being easier and faster to apply, which implies a savings in computation time. Additional key words: granular soils; rural buildings; shallow foundations; vertical displacement.

tred, loaded square footing on sand, has been performed.The interaction soil-footing is taken into account (Breysse et al., 2005) and the soil is considered as an elastic-perfectly plastic anisotropic material (Mabrouki et al., 2010;Loukidis & Salgado, 2011;Oh & Vanapalli, 2011), which implies a better approach to actual soil behaviour.
The aim of this study is to find a new equation fairly simple to use based on finite element method (FEM), which allows estimating the footing settlement on sand.

Development and verification of finite element model
In this study, the commercial finite element program ANSYS v.10 was used.Both the footing and soil were modelled using twenty-noded 3D solid element SOLID95 (Moaveni, 2008).
Fig. 1 shows a finite element mesh of a sandy soil submitted to an axially vertical, centred, loaded, 1.8-mbreadth square footing and the surrounding landfill surcharge.The lateral and bottom boundaries of the mesh were located 6 m horizontally and 10 m vertically from the centre of the footing base, and their movements were restricted in perpendicular directions.
The meshing on the soil took place gradually, from the limits of the soil model where the larger elements

Introduction
In the design of shallow foundations normally used in the rural buildings on sand, the settlement criterion is more critical than the bearing capacity of soils.On sandy soils, the elastic settlement is the most relevant result to be considered (Shin & Das, 2011).
Most of the available methods to calculate the settlement value can be classified in two categories (Shin & Das, 2011): 1. Empirical or semi-empirical methods based on observed settlements of structures and full scale prototypes.These values are correlated with the results of in-situ test data to quantify, in an indirect way the soil parameters, which can be useful to predict the settlement values.Among these procedures is the Burland & Burbidge's method (Burland & Burbidge, 1985), which provides more reasonable estimation of settlement in shallow foundations on sands, however it is difficult to determine the overconsolidation ratio and the preconsolidation pressure from field exploration (Shin & Das, 2011).Sivakugan & Johnson (2004) established that, although this method is a substantially improved technique to estimate settlement, its results are conservative.
2. Methods based on theoretical relationships derived from the theory of elasticity, which has the advantage of considering three-dimensional deformation of soil and simplify its behaviour, considering it as elastic material.The expressions for settlement predictions contain the term of equivalent soil stiffness, which introduces some uncertainty in it.Among these procedures is the methods proposed by Steinbrenner modified by Fox (Das, 2006) and more recently, by Mayne & Poulos (1999), which appears to give better elastic settlement predictions than the above theoretical method (Das & Sivakugan, 2007).
However, soil is far from an elastic, homogeneous and isotropic material, as these methods consider, which generates differences between observed and predicted settlement.These discrepancies can be attributed to the complexity of the proposed formulations and the current inability to estimate a reliable modulus of elasticity of the soil.These require reviewing and upgrading traditional foundation design procedures and tools, using new experimental and theoretical findings.
In this study a three-dimensional nonlinear finite element analysis (Comodromos et al., 2009;Eid et al., 2009;Li & Zhang, 2009), of an axially, vertical, cen- The symmetric nature of the foundation allowed the generation of one half of the model only, reducing the computational effort (Moaveni, 2008).
The models were built in two steps: one corresponding to the generation of the initial state and to the simulation of digging the excavation and the other corresponding to the formwork installation and the footing loading.
The footings were considered to be rigid and rough, as it most often is in reality and were modelled as elastic with a much greater stiffness than the soil (footing stiffness [E f ] = 3 × 10 7 kPa, unit weight of concrete [γ f ] = 25 kN m -3 , Poisson's ratio of concrete [ν f ] = 0.2).The soils were modelled with a non-associated flow rule (Loukidis et al., 2008) and an anisotropic behaviour, which led to variable stiffness with depth, following a linear Drucker-Prager yield criterion, which assumes that soil is an elastic-perfectly plastic material.
Although the soils are not linearly elastic and perfectly plastic for the entire range of load applied (Ti et al., 2009), along with the effects of stress and strain level on soil stiffness, in this study they has been considered as an elastic-perfectly plastic material.The advantages of the Drucker-Prager model applied to foundations cases like the one analysed here, in which the nonlinearity of the load-settlement responses are not large owing to the relatively small width of footings used on medium dense to dense sands (Loukidis et al., 2008), are remarkable.This makes it a favourable option as soil model.
The interface between the footing and the surrounding soil was modelled using a rigid-to-flexible, faceto-face, eight-noded element without thickness or stiffness, compound by an TARGE170 element beneath the footing and a CONTA174 element over the soil surface (Potts & Zdravkovic, 1999), which introduces some nonlinearity in model (Moaveni, 2008).
The global stiffness matrix of this model was solved by an iterative process known as the Newton-Raphson iterative method, which makes small successive load increments until the difference between the applied load and the obtained ones by solving the governing equation systems of the model is lower than a reference value, which is sufficiently small to assume that the analysis has converged, giving the solution.This method allows reducing computation time (Potts & Zdravkovic, 1999).
To verify the finite element model adopted, finite element analysis of five cases of actual footings test, published by Burland & Burbidge (1985) over sandy soils, was carried out.
Table 1 shows the data from these five field tests, where B is the footing breadth, D is the depth of the founding level and q N is the net increase in the effective stress.Soils compactness, according to the criteria proposed by Terzaghi et al. (1996), were obtained from penetration test data (N av ).The parameters of the soils obtained by Tiznado & Rodriguez-Roa (2011) in their works were used as representative of medium-dense sandy soils, while the parameters proposed by Al-Shayea & Mohib (2011) were chosen as representative of dense sandy soils.Table 1 also gathers the geotechnical parameters used in each type of soil, where γ is the unit weight of the dry soil, E s is the equivalent soil stiffness at the founding level, c is the cohesion value, ν is Poisson's ratio, Ø is the effective internal friction angle and Ψ is the dilatancy angle.To account for the variation in soil properties with depth, the equivalent soil stiffness was assumed to increase linearly according to E sz = E s + E sinc (z-z 0 ), where E sinc is the increase of the soil stiffness per unit of depth z and z 0 is the founding level.

Development of a new method to predict the footings settlement on sand
The finite element model has been applied to nine footings in depth and three footings in surface with different sizes, submitted to four different loads, resting on 18 types of sandy soils, yielding 864 settlement values, which has allowed developing a mathematical model that estimates the settlement values of a square footing submitted to centred loads.
Table 2 shows the geotechnical parameters of the 18 soils used in this study and the reference used to set their values.To analyse the influence of E s , ν, D, q N and B on settlement (δ), three different footing breadths, 1.2, 1.5 and 1.8 m, and four different footing dephts, 0, 0.45, 0.65 and 0.85 m, were used in the analysis.The footing thickness was equal to the depth of the foundation, except for a depth of 0, where 0.45 m was used.
To perform the model, a non-associated flow rule (Ø ≠ Ψ ) and anisotropic behaviour of the soils were considered.Pore pressure was neglected.
The net increases in effective stress tested in these soils were equal to 100, 150, 200 and 250 kPa.The applied stress was always lower than one-third of the ultimate soil pressure.Thus, the load-settlement curve studied was linear, where E s at founding level, remains as a constant value.

Development and verification of finite element model
Table 3 shows the measured settlement in footing field tests (Burland & Burbidge, 1985) (δ m ), the predicted ones by FEM (δ FEM ), the obtained through the method of Mayne & Poulos (1999) (δ MP ) and the Steinbrenner's modified by Fox method (Das, 2006) From settlement values gathered in Table 3, a statistic analysis of relative error values of predicted settlements by different methods, compared with the measured ones, has been performed [e.g.(δ FEM -δ m ) / δ m ].The average and the variance of these relative errors, lead to conclude that the predicted results from the finite element model here proposed (average relative error ARE = 30.5%),reveal a good agreement with the settlement obtained by other analytical methods (AREs of 34.4% and 26.5% for Mayne & Poulos and Steinbrenner & Fox methods, respectively).Therefore this model is as good predictor of measured settlement as

Development of a new method to predict the footings settlement on sand
Tables 4, 5 and 6 show predicted settlements from foundations analyses on each type of soils, depending on the footing breadth.
Taking into consideration the δ values obtained from foundation analyses, a equation to predict these settlements is proposed: where δ is the footing settlement in depth, δ s is the settlement values at the surface, and I D , I q and I B represent the influence of D, q N and B on δ, respectively.

Settlement values at the surface δ s
Keeping in mind the δ s values given in Table 4, which correspond to a surface square footing with B equal to 1.2 m and q N equal to 100 kPa, a linear regression analysis of these values depending on ν, was performed to obtain six different straight lines, one for each type of soil.These lines correspond to the general Eq.[2], where the α and β parameters, that define each line, are described in Fig. 2, for each type of soil analysed.The coefficient of determination (R 2 ) in all cases was greater than or equal to 0.99.
where the α and β parameters represent the influence of E s on δ s .

Influence factor I D
From δ values collected in Tables 4, 5 and 6, for each B, E s , ν and q N , it is possible to appreciate that the settle- ment decreases hyperbolically with D, which implies that increasing D leads to a greater difference between δ and δ s .The main reason for this observation is the lateral movement of soil particles under the footing edge while it settles, which is more difficult in depth because of the overloading of lateral landfill surrounding the footing.
Tables 4, 5 and 6 show that increasing E s leads to a greater uniformity in the recorded δ for the different D tested, which means that there are very few differences between δ and δ s .The reason for this is that stiffer soils resist better lateral movement of soil particles under the footing edge while it settles.Likewise, To consider the influence of D on δ, the ratios of δ s and δ corresponding to the different B, E s , ν, D and q N applied, were calculated.The values of these ratios remain within a narrow range in each type of soil and for each B, q N and D considered, independent of ν used (variance lower than 0.01).As a result, it is possible to set an average ratio independent of ν for each one of these cases, which appear in Table 7.
From these average ratios, eighteen linear regression analyses depending on D/B, were performed, which correspond to the different B and E s analysed, for each one of q N applied.The coefficient of determination (R 2 ), in all analysed cases, was greater or equal to 0.99.The mathematical fit of this linear regression analysis corresponds to the following equation: where λ is the rate of decrease of δ with D/B, shown in Table 7.
Finally, the I D factor corresponds to the following equation: Influence factor I q As can be seen in Tables 4, 5 and 6, when a certain footing size is considered, the increase in q N leads to greater δ.From these settlement values, it is possible to compare the ratio of the pair of δ for any q N applied (δ q ) with those obtained for q N equal to 100 kPa (δ 100 ).
The values of these ratios, for surface footings, reveal a good agreement among them for any ν used in each type of soil, B and q N considered.Moreover, because the λ parameter hardly changes for any D studied in each soil, these ratios are close to the ones between δ s , regardless of the D tested (variance < 0.01).In this sense, considering a q N , the obtained ratios for each type of soil, are also very similar to each other regardless of the size of footing studied (variance < 0.01).As a result, it is possible to consider an average constant value of these ratios in each type of soil and for each q N used, independent of ν, D and B used.This decision also relies on the fact that errors in predicting δ q , considering these average ratios, are lower than 1.8% with regard to the obtained by finite element analyses.The values of these average ratios for each type of soil, depending on q N -P a , where P a is the atmospheric pressure (100 kPa), are shown in Table 8.
The mathematical fit of these average ratios obtained by a linear regression analysis depending on q N -P a leads to the generation of six different straight lines, one for each type of soil, corresponding to Eq. [5].The coefficient of determination (R 2 ) in all cases was greater than or equal to 0.99.
where f q defines the slope of the straight lines for each soil analysed and represents the influence of E s on δ for any q N used, in units of kilopascal.Fig. 4 outlines the f q parameter curve versus E s .Thus, the I q factor behaves following the Eq.[6]: Influence factor I B Tables 4, 5 and 6 describe that when a constant value of q N is considered, an increase in footing size leads to increased settlements on each type of soil.
The ratios of the pairs of δ for a certain B (δ B ) compared with those obtained when B is equal to 1.2 m (δ 1.2 ) were calculated for each ν, D and q N used in each E s studied.These ratios are very similar each other (variance < 0.01), which allows to set an average value for this ratio in each type of soil and for each footing size, because the errors in predicting δ B through this average ratio, with regard to the obtained by finite ele- Equivalent soil stiffness E s (MPA) A new equation to predict the footing settlement on sand based on the FEM ment analyses, are lower than 1.8%.Table 9 shows these average ratios.
The mathematical fit of these ratios depending on B by an exponential function, corresponds to the equation of Terzaghi et al. (1996): where the exponent f s represents the influence of E s on δ for any B analysed and can be calculated by the Eq.
[8], for B greater than 1.2 m: Table 9 gathers the f s values obtained from Eq. [8] which are very close for each E s analysed (variance < 0.007).This means that δ B calculated through this parameter, hardly change.Thus, it is possible to consider an average value of f s for each type of soil, because the errors in predicting δ B using this average parameter, compared with the obtained from finite element analyses, are lower than 2%.Fig. 5 shows the average f s values depending on E s .Therefore, the I B factor corresponds to the Eq.[9]:

Model validation
After analysing all of the factors involved in calculating the footing settlement on different sandy soils, it is possible to replace the Eqs.[2], [4], [6] and [9] in the main Eq.[1] to obtain the following: To verify the applicability of this new equation, numerical solutions of the five foundation field tests cases (Burland & Burbidge, 1985) used to verify the model generated in this study, were calculated (δ A ) and subsequently compared with the measured settlement in field tests and with the obtained ones through the methods proposed by Burland & Burbidge (1985) (δ BB ), Mayne & Poulos (1999) and Steinbrenner modified by Fox (Das, 2006).
Table 10 shows the settlement obtained through this new Eq.[10], the measured ones in footings field tests and the calculated ones through the analytical methods.Likewise, the Table 10 gathers the average and the variance of relative error values of predicted settlements by different methods, compared with the measured ones [e.g.(δ A -δ m ) / δ m ].This statistic analysis shows that the analytical method of Burland & Burbidge (1985) overestimates the results.On the other hand, the new Eq.[10] predicts the settlement extracted from field tests cases, with an accuracy (ARE = 31.1%)similar to the methods proposed by Mayne & Poulos (1999) and Steinbrenner modified by Fox (Das, 2006) (AREs of 34.2% and 26.5% respectively), which leads to confirm the validity of the new equation to predict the settlement at the centre of the base of an axially vertical, centred, loaded square footing (B = 1.2 to 1.8 m), resting on dense to medium-dense drained sand.
However, this new Eq.
[10] has the advantage of the speed and ease in its application; since unlike the other analytical methods used, it is possible to obtain, in a direct way, the values of the influence parameters, function of the equivalent soil stiffness (E s ) that define the new equation, which implies a savings in computation time.

Figure 1 .
Figure 1.Finite element mesh for footing field test.

Figure 2 .
Figure 2. Values of α and β with regard to E s .

Figure 3 .
Figure 3. Values of λ average with regard to E s .

Figure 5 .
Figure 5. Values of f s with regard to E s .Equivalent soil stiffness E s (MPA)

Table 1 .
Analyses of Burland & Burbidge (1985)'s case records and soils geotechnical parameters used in model verification

Table 2 .
Soils geotechnical parameters used to develop the new method

Table 3 .
Settlement prediction by different methods and statistic analysis of average relative errors (ARE)

Table 4 .
Settlement from foundations analyses for B = 1.2 m (mm)

Table 5 .
Settlement from foundations analyses for B = 1.5 m (mm)

Table 6 .
Settlement from foundations analyses for B = 1.8 m (mm) 68 A new equation to predict the footing settlement on sand based on the FEM an increase in Poisson's ratio, within the range considered in this study, leads to stiffer soils and lower δ.

Table 7 .
Average ratio between δ s and δ and λ values

Table 8 .
Average ratios between δ q and δ 100 in each type of soil

Table 9 .
Values of f s parameter for each type of soil and footing breadth

Table 10 .
Predicted settlement using the new equation [10] versus other methods and statistic analysis of average relative errors (ARE)