Importance of Constitutive Model in assessing 3-D Bearing Capacity of Square Footing on Clayey Soil using Finite Element Analysis

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1 SEC18: Proceedings of the 11th Structural Engineering Convention Jadavur University, Kolkata, India, December 19-21, 218 Paer No _R2 Imortance of Constitutive Model in assessing 3-D Bearing Caacity of Square Footing on Clayey Soil using Finite Element Analysis Arindam Dey 1*, Mousumi Mukherjee 2, Rana Acharyya 3 1, 3 Deartment of Civil Engineering, IIT Guwahati, Assam, INDIA arindamdeyiitg16@gmail.com, r.acharyya@iitg.ac.in 2 School of Engineering, IIT Mandi, Himachal Pradesh, INDIA mousumi.ju6@gmail.com ABSTRACT The classical bearing caacity estimation techniques commonly adots lane-strain condition and mostly confines to Mohr-Coulomb (MC) elastic-erfectly lastic model for reresenting the soil behaviour. However, being a lane-strain adatation, MC model cannot be exlicitly used to delineate the stress-strain resonse and bearing caacity of an arbitrarily shaed footing resting on semi-infinite medium. Further MC model cannot reresent the ost-eak strain-softening behaviour. In this regard, this article addresses the finite-element based bearing caacity estimation of a square footing resting on the surface of a semi-infinite clayey medium. The footing is reresented by linear-elastic element, while the clayey foundation soil has been reresented by both MC and Modified Cam-Clay (MCC) models. Comarison of the load-dislacement resonse of the uniformly loaded footing, obtained from considering the MC and MCC models, clearly delineates the shortcoming of the MC and necessity of adoting MCC behaviour in a 3-dimensional geotechnical roblem. Keywords: 3-D bearing caacity; Constitutive model; Finite Element Analysis 1. INTRODUCTION Assessment of bearing caacity has been one of the most oular roblems in geotechnical since its early days. The classical works related to the estimation of bearing caacity confined to stri footings resting on semi-infinite medium (Terzaghi 1943; Meyerhof 1951; Vesic 1973). However, it is worth mentioning that in contrast to stri footings, isolated square or rectangular, or combined, footings to be more common for building foundations. As such footings, when loaded, reresent 3-dimensional stress distribution, their characteristic resonse and failure mechanism is suosed to be different from the lane-strain mechanism observed for stri footing. In this regard, there have been modification in the basic theories of bearing caacity for stri footings to accommodate the shae effect. Exerimental and numerical aroaches have been resorted to estimate the bearing caacity of square footings on cohesionless or cohesive soils (Cerato and Lutenegger 26; Acharyya and Dey, 217; Acharyya et al. 218). In all the cases, it has been a common ractice to utilize Mohr-Coulomb (MC) elasticerfectly lastic model for reresenting the soil behavior. In order to cater the shae effect of footings on the bearing caacity, it is a common ractice to utilize suitable shae factors governed by the asect ratio of the footing. However, being a lane-strain adatation, MC model cannot exlicitly be used to delineate the stress-strain resonse and bearing caacity of an arbitrarily shaed footing resting on semiinfinite medium. Moreover, the MC model fails to address any ost-eak strain-softening feature in the load-dislacement characteristics of a loaded footing. Owing to the deficiency of M-C model in highlighting the develoment of intermediate rincial stresses, the 3- dimensional resonse of such foundations cannot be elucidated. This necessitates the alication of an advanced generalized 3Dconstitutive model that can accommodate both strain hardening and strain softening behaviour of the foundation soil. Alication of Modified Cam-Clay (MCC) model, within a numerical

2 SEC18: Paer No _R2 framework, can be an effective means of simulating the 3D bearing caacity roblems. Alication of MCC model for geotechnical roblems has already been into ractice (Abed and Vermeer 24; Lim et al. 21; Wang and Bienen 216; Oh and Vanaalli 218). In this regard, this aer addresses the estimation of bearing caacity of a square footing resting on the surface of a semi-infinite clayey medium. The analysis has been conducted with the aid of finite element analysis. The footing is reresented by linearelastic element, while the foundation soil, the clayey medium, has been reresented by both MC and MCC models. The influence of the chosen constitutive model on the loaddeformation resonse and the develoed failure mechanism has been clearly elucidated through the study. 2. DESCRIPTION OF CONSTITUTIVE MODELS The generalized 3D nonlinear elasto-lastic constitutive relation can be reresented by the following incremental form d C d (1) kl kl where and kl define the effective stress and strain tensor, resectively and the elasto-lastic tangent stiffness tensor is given by E P Q E C mn mn rs rskl kl E kl H Q E P ab abcd cd where E kl C kl (2) is the isotroic elastic stiffness tensor, H is the hardening modulus, P and Q are the directions of outer normal to the lastic otential (g) and yield surface (f), resectively, P g and Q f. Simulations have been carried out emloying two different constitutive models, Modified Cam-Clay (MCC) and Mohr-Coulomb (MC). For the resent work, focus has been restricted only to the associative formulation of these two models enforcing same mathematical exression for the lastic otential and the yield surface ( g f). The MCC model is a critical state concet-based hardening (softening) model with following exression for the yield surface 2 q f ( ) 2 (3) M In the above exression, q,, M, are the q shear stress, effective mean ressure, shear stress ratio at critical state and reconsolidation ressure. In MCC model, a logarithmic relation is assumed between secific volume and in virgin isotroic comression () v ln v v (4) where is the isotroic comression index. During unloading or reloading, a different logarithmic relation is followed ln v v (5) where is the isotroic swelling index. The isotroic hardening (softening) relation is defined in terms of evolution of volumetric strain d v d The bulk modulus ( ) K deends on with lastic and (6) v K (7) The yield condition of the elastic-erfectly lastic MC model consists of following six yield functions in terms of rincial stresses sin f ccos 1a sin f ccos 1b sin f ccos 2a sin f ccos 2b sin f ccos 3a sin f ccos 3b (8) In case of soil behaviour rediction through MC model, the elastic resonse is assumed linear

3 SEC18: Paer No _R2 with constant elastic arameter values. The constitutive model arameters for MC and MCC model and their values used for FEM simulation have been listed in Table 1. Fig. 1 resents the stress-strain and volumetric redictions of single element triaxial test for both the models at 2 kpa confining ressure with the arameter set given in Table 1. It can be observed from Fig. 1(a) that the redicted maximum shear stress from two models nearly becomes same (4 kpa) at a large axial strain, i.e., at 3%. For the MC model, however, the maximum shear stress is achieved at a very low axial strain level (around 1%) and only a slight variation on this onset strain may arise due to change in the elastic roerties. The MCC model atly catures the comressive volumetric resonse of the normally consolidated soil at 2 kpa confining ressure [Fig. 1(b)]; whereas, mostly a dilative reose has been redicted by the MC model. It is imortant to note that the volumetric strain in MCC model nearly becomes constant when the maximum shear stress has been achieved, i.e, it corresonds to a critical state marking the failure of the material. The excessive dilation rediction of the MC model may have significant imact on the manifestation of the failure mechanism in the bearing caacity simulation that is further exlored in the successive section. Further, such excessive dilation can only be handled by incororation of non-associative flow rule in the model (Wood, 24) which is ket out of the scoe of this aer. Table 1 MCC and MC model arameter values considered for FEM simulation Model Parameter Values MCC.77 MC.6 M 1.2 c E 2 kpa & 6 MPa (a) (b) Fig. 1 Triaxial test redictions for MCC and MC model at 2 kpa confining ressure with the arameter set given in Table 1 5. FINITE ELEMENT MODELING In the resent study, the bearing caacity of a square footing, resting on normally consolidated clayey foundation soil, has been assessed with the aid of three-dimensional finite element simulation using PLAXIS 3D AE.1. This software ackage is sufficed by several features to deal with comlex geotechnical roblems related to deformation, stability and flow through geotechnical media. Additionally, the availability of a large number of varying constitutive models for reresenting the stressstrain-time behaviour of soil makes it a very useful investigative tool to study the resonse of geotechnical entities (PLAXIS Material model manual 215). Fig. 2 deicts the tyical FE model used for the resent study. Following the rocedure described in Acharyya et al. (218), the size of the model domain (14 m in length and breadth with a height of 6 m) has been so decided that the existing lateral boundaries do not influence the stress or strains generated beneath the loaded footing. Standard fixity has been alied to the FE model such that the lateral boundaries are allowed only vertical deformation, while the

4 SEC18: Paer No _R2 bottom boundary is considered non-yielding. The model geometry has been discretised with the aid of 1-noded tetrahedral elements roviding a second-order interolation of dislacements, accomanied by a 4-oint Gaussian integration scheme. Various tyes of meshing schemes can be adoted in PLAXIS, governed by the mesh coarseness factor. For the resent case, mesh convergence study had been carried out (Acharyya et al. 218) and, accordingly, an otimal fine mesh has been chosen, associated with local refinements adjacent to the footing. Further details of the FE analysis can be found in PLAXIS 3D Reference Manual (215). on MC soil. It can be observed that the ultimate bearing caacity (q u) is aroximately 38 kpa. As er Terzaghi (1943), the ultimate bearing caacity of a square footing resting on the surface of a cohesionless semi-infinite medium can be ascertained as qu.4 BN. Considering the bearing caacity factor for, N q u can be comuted as 36 kpa. The excellent match between the theoretical and numerical magnitudes aids in the validation of the develoed FE model. Fig. 3 Load-settlement resonse of footing resting on MC soil Fig. 2 Tyical meshing, standard fixities and basic element used in the FE model The rigid square footing of width 2 m, made of M2 concrete, is reresented by Linear Elastic (LE) element (unit weight γ = 25 kn/m 3, elastic modulus E = 22 GPa, Poisson s ratio ν =.15) comrising 1-noded tetrahedral elements, and accomanied by an interface element (at the boundary of soil and concrete material) ensuring no-sliage condition. In order to comrehend the influence of constitutive behaviour of soil on the bearing characteristics of a square footing, the soil is reresented by two tyes of constitutive behaviour, namely Mohr-Coulomb (M-C) and Modified Cam Clay (MCC) models. 6. RESULTS AND DISCUSSIONS The load bearing resonse of the footing resting on the clayey medium has been assessed through load-settlement lots, along with the dislacement and stress distributions within the foundation. Fig. 3 shows the load-settlement resonse obtained at the mid-oint of the footing resting Fig. 4 shows the tyical failure deformations of the footing resting on MC soil having stiffness 6 MPa. It can be observed that the deformation attern closely follows a general shear failure, wherein the soil elements beneath the footing is ushed downwards in the form of a yramid, which induces an outward and uward radial shear in the adjacent soil elements towards the edge of the footing. This behavioural resonse is similar to the formation of the sli lines beneath a footing undergoing a general shear failure following a lane-strain condition, conforming to a vertical section assing through the centre of the footing. Fig. 5 shows the load-settlement resonse of the footing resting on MCC soil. It can be observed that in comarison to the footing resting on the MC soil, the ultimate bearing caacity enhances to aroximately 68 kpa, accomanied by a higher failure settlement. Fig. 6 highlights the deformation attern of the soil, which exhibits volumetric comression, and is markedly different from the dilative attern observed for MC soil. It can be noted that the MCC soil does not exhibit heaving at the edges of the footing, which was evident for MC soil. Further scrutiny shows that in the MCC soils, outward radial shears initiates beneath the

5 SEC18: Paer No _R2 footing itself, thus leading to a mixed elastic wedge and radial shear zone beneath the footing, while in the case of MC soil, the radial shear zone was distinctively outside the elastic wedge. Fig. 7 illustrates the deformation iso-surfaces develoed in the MC and MCC soils at failure. It can be clearly observed that the deformation attern for the footing on MC soil is mostly confined to the edges of the footing, while, in the MCC soil, it is distributed to the entire foundation soil beneath the footing. Further, the volume of soil articiating in resisting the alied stresses is significantly higher in the case of MCC soil in comarison to the MC soil, thus the bearing caacity of footing on MCC soil is distinctly higher. However, it is worth mentioning that the failure deformation is also significantly higher for footing on MCC soil. understanding the volume of foundation soil influenced by the alied stress, similar to the ressure bulb as obtained with the Boussinesq s theory for elastic conditions. It can be observed that in comarison to the isobars develoed in the MCC soil, the isobars san to wider area beneath the footing resting on MC soil. Further, it can be noted that the vertical stress contours are very distinctly develoed for MC soil owing to the distinct formation of deformation atterns, as shown in Fig. 4, while the stress contours develoed in the MCC soil (Fig. 6) are inadvertently scrambled due to mixed elasticradial shear zones develoed beneath the footing at failure. Fig. 6 Failure deformation attern of MCC soil Fig. 4 Failure deformation attern of MC soil Fig. 5 Load-settlement resonse of footing resting on MCC soil Fig. 8 highlights the distribution of vertical stress beneath the footing resting on MC and MCC soil. Such distribution aids in Fig. 7 Deformation iso-surfaces develoed in MC and MCC soils at failure 7. CONCLUSION This article resents the imortance and influence of two different constitutive models,

6 SEC18: Paer No _R2 Mohr-Coulomb and Modified Cam-Clay, on the assessment of bearing caacity of a square footing resting on the surface of a normally consolidated clayey medium. It is vividly evident from the results that the loaddeformation and failure mechanisms of the MC and MCC soils are markedly different, which, in turn, largely affects the bearing caacity of the square footing resting on NC clayey soil. The excessive dilative behaviour of the MC model fails to reresent the volumetric comression of the NC clayey soil, which is justifiably reresented by the MCC model. Such findings call for the adotion of MCC model for finding the resonse of NC clayey soil and the suorted structures. Fig. 8 Vertical stress contours develoed beneath footing resting on MC and MCC soils REFERENCES [1] Abed, A.A., and Vermeer, P.A. Numerical simulation of unsaturated soil behavior Proceedings of the 1 st Euro Mediterranean Conference on Advances in Geomaterials and Structures, Tunisia, 24, [2] Acharyya, R., and Dey, A. Finite element investigation of the bearing caacity of square footings resting on sloing ground, INAE Letters, Vol. 2, 217, [3] Acharyya, R., Dey, A., and Kumar, B. Finite element and ANN-based rediction of bearing caacity of square footing resting on the crest of c-φ soil sloe International Journal of Geotechnical Engineering, 218, DOI: 1.18/ [4] Cerato, A.B., and Lutenegger, A.J. Bearing caacity of square and circular footings on a finite layer of granular soil underlain by a rigid base, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 132, 26, [5] Lim, A., Ou, C.-Y., and Hsieh, P.G. Evaluation of clay constitutive models for analysis of dee excavation under undrained conditions Journal of GeoEngineering, Vol. 5, 21, [6] Meyerhof, G.G. The Ultimate Bearing Caacity of Foundations, Geotechnique, Vol. 2, 1951, [7] Oh, W.T., and Vanaalli, S.K. Modeling the stress versus settlement behaviour of shallow foundations in unsaturated cohesive soils extending the modified total stress aroach Soils and Foundations, Vol. 58, 218, [8] PLAXIS B.V. Material Model Manual 3D- Version AE.1, PLAXIS, Delft, The Netherlands, 215. [9] PLAXIS B.V. Reference Manual 3D- Version AE.1, PLAXIS, Delft, The Netherlands, 217. [1] Terzaghi, K., Theoretical Soil Mechanics. John Wiley and Sons Inc., New York, USA, [11] Vesic, A.S. Analysis of ultimate loads of shallow foundation, Journal of Soil Mechanics and Foundation Division, ASCE, Vol. 99, 1973, [12] Wang, D., and Bienen, B. Numerical investigation of enetration of a largediameter footing into normally consolidated Kaolin clay with a consolidation hase Geotechnique, Vol. 66, 216, [13] Wood, D.M., Geotechnical Modeling. Taylor and Francis, Floriuda, USA., 24.