slightly over-estimated below the raft and the pile toes where mostly vertical strains occur.
4.4 Raft-piles load split
The load percentage split between raft and piles resulting from the analyses is similar, as shown in the Table 3. This is in good agreement with the correla-tion derived from the study of several piled raft foun-dation case histories that shows that the raft contribu-tion is usually dependent on geometrical parameters such as the raft area, the pile group area, the pile di-ameter and the pile spacing (Mandolini 2003).
Table 3 Load split between raft and settlement-reducer plies
RAFT PILES GSA 35% 65%
MIDAS elastic 31% 69% MIDAS elasto-
plastic 32% 68%
Finally, it is also worth observing that the piles are
generally working with a factor of safety lower than 2, which is usually expected for settlement-reducing piles.
5 CONCLUSIONS
This paper presented the comparison of the results of two FE programs used to study the behavior of a piled raft foundation during the design development phase. In both GSA and MIDAS GTS the soil-pile interaction has been modeled by a series of springs with pre-defined stress-displacement curves for the shaft and the base. Under the same assumptions on material properties, GSA and MIDAS provide close-ly similar results. However, the FE analysis in MIDAS allows for a more realistic stress and strain distribution in the soil as plasticity is accounted for according to the Mohr-Coulomb failure criterion.
The results show similar absolute settlements but a reduction in differential settlements and an increase in the pile axial force is noticeable when MIDAS elasto-plastic is used. The load split between the piles and the raft is comparable in the two programs and is consistent with published literature correlations.
Finally, a constitutive model which adopts a con-stant value of soil stiffness for each layer intrinsically
neglects the effects of effective stress and strain vari-ation induced in the soil. Should such model be cho-sen, the designer shall carefully select the stiffness to adopt and verify the strain plot resulting in the analy-sis.
ACKNOWLEDGEMENT
The Authors would like to acknowledge the sup-port of ARUP colleagues Alessandro Baliva, Angelo Mussi and Anton Pillai during the piled raft design development phase and of Dr. Brian Simpson for re-viewing this paper.
REFERENCES
Allievi L., Ferrero S., Mussi A, Persio R. & Petrella F. 2013. Structural and geotechnical design of a piled raft for a tall building founded on granular soil. Proceedings of the 18th International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013. American Petroleum Institute, 1993. Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms – Load and Resistance Factor Design, API Recommended practice 2A-LRFD, 1st edition. API, Northwest Washington, DC 20005. Oasys Ltd, Oasys Geo Suite v 19.1 GSA Software Manual, London 2010. Houlsby, G.T., 1992. How the dilatancy of soils affects their be-haviour. Deformation of soils and displacements of structures. Eu-ropean conference on soil mechanics and foundation engineering, 10, Florence, 26-30 May 1991. Proceedings, Vol. 1-5 Mandolini, A. 1999. Small-strain stiffness and settlement predic-tion for piled foundations. Pre-failure deformation characteristics of geomaterials. Jamiolkowski, Lancellotta & Lo Presti (edt.), 2001. Mandolini, A. 2003. Design of piled rafts foundations: practice and development. Proceedings Geotech. Int. Seminar on Deep Foundations on Bored and Auger Piles, Ghent, Van Impe & Hageman eds., 59-80 MIDAS GTS 2012 Scientific Manual Tomlinson, M. J. 1995. Pile design and construction practice. Ad-dison Wesley Longman Ltd., Harlow, UK. Tradigo F., Pisanò F., Di Prisco C. & Mussi A. 2015. Non-linear soil–structure interaction in disconnected piled raft foundations. Computers and Geotechnics 63 (2015) 121–134. Vaziri H, Simpson B, Pappin J W & Simpson L, 1982. Inte-grated forms of Mindlin's equations. Geotechnique Vol. 32 No.3 Septem-ber Wehnert, M. & Vermeer, P.A. 2004. Numerical analyses of load tests on bored piles. Proceedings of the 9th Symposium on Numer-ical Models in Geomechanics (NUMOG IX) A.A. Balkema Pub-lishers, Leiden, 505-51125-27
Comparison of advanced numerical methods for geomechanical problems with large deformations
Comparaison des méthodes numériques avancées pour les problèmes géomécaniques avec grande déformation
C. Moormann*1, S. Fatemizadeh1, J. Aschrafi1 1 University of Stuttgart, Institute for Geotechnical Engineering, Stuttgart, Germany
* Corresponding Author
ABSTRACT Since first application in the middle of the last century, Finite Element Method (FEM) has become a very powerful tool ingeotechnical engineering for analysing small deformation problems like deep foundations, excavations, tunnels etc. with increasing com-plexity. It is evident that this method has many shortcomings solving geotechnical problems with large deformations and/ or large displa-cements (e.g. pile installation effects, slope stability, piping or erosion effects). Especially contact problems and large finite element meshdistortions may lead to inaccurate results or even to a non-convergent solution of the problem. Because of this reason it is necessary to usemore advanced modelling techniques like the Arbitrary Lagrangian-Eulerian (ALE) Method, the Coupled Eulerian-Lagrangian (CEL)Method or the Material Point Method (MPM), which is further developing for geotechnical applications currently at University of Stuttgart(IGS). These methods can overcome mesh distortion and contact problems and have the potential to improve the understanding of complexsoil-structure interaction and failure mechanism of soil under large deformation conditions. The objective of this paper is to give a brief in-troduction on the governing equations and solution procedures of all methods mentioned before. Following, these methods will be compa-red and their potential will be shown for a large deformed strip footing and a landslide problem considering plane strain conditions. Con-cluding, recommendations on the applicability of all these methods for other geomechanical problems are given.
RÉSUMÉ Depuis la première application au milieu du 20ème siècle, la méthode des éléments finis (FEM) est devenue, avec une com-plexité croissante, un outil très performant en géotechnique pour l'analyse des petits problèmes de déformation, comme par exemple pourles fondations profondes les travaux de terrassement, les tunnels, etc. Il est évident que cette méthode présente encore de nombreux pointsfaibles pour résoudre les problemes géotechniques de grandes déformations et/ou de grands déplacements (par ex. les effets de l’installationde pieux, la stabilité des pentes, systemes de canalisation ou les effets de l’érosion. Surtout les problèmes de contact et la distortion degrands éléments finis de maillage peuvent conduire à des résultats imprécis ou même à une solution non-convergente du problème. C’estpour cette raison qu’il est nécéssaire de se servir de techniques de modélisation plus dévelloppées comme la méthode arbitraire LagrangeEulerian- (ALE), la méthode Couplé Euler-Lagrange (CEL) ou la Méthode du point matériel (MPM), qui est actuellement en continu dedéveloppement pour les applications géologiques à l'Université de Stuttgart (IGS). Ces méthodes peuvent avoir raison des distortions demailles ainsi que des problèmes de contact et ont le potentiel d’amériorer la compréhension de l’interaction complexe de la structure du solainsi que le méchanisme de fracture du sol sous des conditions de grande déformation. L’objectif de cet articel est de donner une courte in-troduction sur les équations qui dominent et sur les procédures de solution de toutes les méthodes mentionnées avant. Ensuite, ces méthodesseront comparées et leurs potentiels seront démontrés sur une grande semelle filante déformée et un problème de glissement de terrain, entenant compte des conditions de déformation plane. En conclusion, les recommandations sur l'applicabilité de toutes ces méthodes pourd'autres problèmes géomécaniques sont données.
1 INTRODUCTION
In the last decades the Finite Element Method (FEM) has become a standard tool in many fields of engi-neering sciences. This method has been improved and applied successfully in different applications
(Hughes 1987). FEM is used widely in the field of geotechnical engineering as well (Potts & Zdrav-ković 2001). Despite all these improvements, FEM is not able to model large deformations which are common in the field of geotechnical engineering (Figure 1) and suffers from what is called the “mesh
Geotechnical Engineering for Infrastructure and Development
4026
distortion”. By using the updated Lagrangian formu-lation, errors are introduced in the mapping process (Więckowski 2001) and remeshing is computational-ly expensive.
Here three methods namely Arbitrary Lagrangian-Eulerian (ALE) method, the Coupled Eulerian-Lagrangian (CEL) method and the Material Point Method (MPM) are introduced as powerful methods which are able to overcome this shortcoming. A strip footing and a landslide problem are analyzed using these three methods to show their capabilities and the results are compared. At the end some concluding remarks are presented.
Figure 1. Distortion of the finite element mesh under large de-formations (Beuth 2012).
2 NUMERICAL METHODS FOR LARGE DEFORMATIONS
In order to overcome the limitations of FEM simulat-ing large deformations, many different numerical methods are introduced which can be categorized in three groups.
In the first group the methods try to use the ad-vantages of the both Eulerian and Lagrangian de-scriptions of motion and avoid their drawbacks. ALE (Hirt et al. 1974) and CEL (Noh 1964) can be con-sidered in this group.
Second group contains the meshless methods. Smoothed Particle Hydrodynamics (Monaghan 1988) and Element-free Galerkin method (Belytschko et al. 1994) are amongst others in this group.
The last group includes mesh-based particle meth-ods. Particle-in-cell method (Harlow 1964), Fluid-implicit Particle method (Brackbill & Ruppel 1986)
and MPM (Sulsky et al. 1995; Sulsky & Schreyer 1996) can be categorized in this group.
ALE, CEL and MPM are discussed briefly in the next subsections.
2.1 Arbitrary Lagrangian-Eulerian (ALE) method
The Arbitrary Lagrangian-Eulerian (ALE) method is an effective method to analyze problems including large deformations. This method was first introduced by Hirt et al. (1974) to solve fluid dynamics or later metal forming problems (Khoei et al., 2008). This method uses a finite mesh with nodes, which may move with the material (Lagrangian description) or be held fixed in space (Eulerian description). The mesh is partially attached on material and can be-come independent where necessary. Initially, La-grangian and Eulerian bodies entirely overlap each other and have the same mesh discretization. Mesh motion is constrained to the material motion only where necessary (at free boundary). Otherwise, mate-rial motion and mesh motion are independent. The ALE method can overcome the mesh distortion while representing the boundary conditions correctly.
According to Hirt et al. (1974) one time step of an ALE calculation is based on three phases. It starts with a classical explicit Lagrangian calculation phase, followed by solving the equilibrium of mo-ments with another standard Lagrangian analysis. The next step is the optional adaptive meshing phase. The mesh is adapted independently from the underly-ing material, i.e. there are no more critical mesh dis-tortions. In a final Eulerian phase the solution ob-tained from the Lagrangian phase is remapped onto the new relocated mesh (advection phase), which was developed through the adaptive meshing algorithm.
The interested reader is referred to Hirt et al. (1974) and Benson (1992).
2.2 Coupled Eulerian-Lagrangian (CEL) method
Originally, the Coupled Eulerian-Lagrangian (CEL) method was introduced by Noh (1964) and further developed by Benson (1992) and Benson & Okazawa (2004). This method, which attempts to capture the strength of the Lagrangian and the Eulerian descrip-tions of motion, is implemented in the commercial code ABAQUS/ Explicit. For geotechnical problems, a Lagrangian mesh is used to discretize the structure
highly distorted mesh
while an Eulerian mesh is used to discretize the sub-soil. The Eulerian material is tracked as it flows through the Eulerian mesh computing its Eulerian volume fraction (EVF). If an Eulerian element is completely filled with material, its EVF is 1. If there is no material in the element, its EVF is 0 (Dassault Systéms, 2013). The interface between structure and subsoil could be represented using the boundary of the Lagrangian domain. On the other hand, the Eulerian mesh, which represents the soil that may experience large defor-mations, has no problems regarding mesh distortion. According to Noh (1964) initially the pressure on the interface of the Eulerian cells is integrated to calcu-late the force acting on the Lagrangian nodes at the interface. Following, the motion of the Lagrangian domain is calculated and the portion of the domain at the time is determined using the new posi-tion of the Lagrangian domain. Finally, the discre-tized Eulerian equations are solved to obtain the new pressure.
In ABAQUS contact between Lagrangian and Eu-lerian material is enforced using a penalty contact method (Dassault Systéms, 2013). It uses a master-slave relationship in which the Lagrangian material usually is defined as master and the Eulerian as slave surface. On the master surface so called seeds are created, whereas on the slave surface there are so called anchor points. Between the seeds and anchors a force will be created, depending on the distance
and a contact stiffness . This stiffness is de-pendent on the Lagrangian material on one side and on the Eulerian material on the other side.
(1)
Qui et al. (2011) or Henke (2009) showed the po-
tential of the CEL method for geotechnical applica-tions undergoing large deformations. The interested reader is referred to Noh (1964), Benson (1992), Benson (1995) and Benson (2000).
2.3 Material Point method (MPM)
For the first time Sulsky et al. (1995) applied fluid implicit particle (FLIP) method from fluid to solid mechanics and called it Material Point Method (Sulsky & Schreyer 1996). Bardenhagen et al. (2000)
added a frictional contact algorithm to this method. Since the first formulation of MPM, it is applied on different applications that show the capability of it to simulate large deformations.
MPM uses two different discretizations, one is the space discretization based on the background fixed mesh, and the other one is the material discretization based on the material points or “particles”. During the simulation, material points that carry all the physical information (strains, stresses, etc.) can move through the background mesh. With this prop-erty extremely large deformations can be simulated.
MPM working procedure in one time step con-sists of three phases: first the initialization phase where all the data are mapped from the particles to the background mesh using the nodal shape func-tions. Next is the Lagrangian phase where the equa-tions of motion are solved. At last is the convective phase where the updated nodal values are mapped back to the particles. The positions of particles are updated and finally background mesh is brought back to its original position.
The weak formulation, space and time integration used in MPM is the same as FEM, but in MPM par-ticle based integration is done i.e.
(2)
where is the total number of particles, is the mass of particle and is the shape function of node evaluated at particle .
Using particles to discretize the material (contin-uum), is done by approximating the density field with the help of the Dirac delta function
(3)
where is an arbitrary position vector and is
the position vector at particle . For the complete formulation of the governing equations please refer to Sulsky et al. (1995) or Jassim (2013) and for the complete formulation of the contact algorithm please refer to Bardenhagen et al. (2000).
4027
distortion”. By using the updated Lagrangian formu-lation, errors are introduced in the mapping process (Więckowski 2001) and remeshing is computational-ly expensive.
Here three methods namely Arbitrary Lagrangian-Eulerian (ALE) method, the Coupled Eulerian-Lagrangian (CEL) method and the Material Point Method (MPM) are introduced as powerful methods which are able to overcome this shortcoming. A strip footing and a landslide problem are analyzed using these three methods to show their capabilities and the results are compared. At the end some concluding remarks are presented.
Figure 1. Distortion of the finite element mesh under large de-formations (Beuth 2012).
2 NUMERICAL METHODS FOR LARGE DEFORMATIONS
In order to overcome the limitations of FEM simulat-ing large deformations, many different numerical methods are introduced which can be categorized in three groups.
In the first group the methods try to use the ad-vantages of the both Eulerian and Lagrangian de-scriptions of motion and avoid their drawbacks. ALE (Hirt et al. 1974) and CEL (Noh 1964) can be con-sidered in this group.
Second group contains the meshless methods. Smoothed Particle Hydrodynamics (Monaghan 1988) and Element-free Galerkin method (Belytschko et al. 1994) are amongst others in this group.
The last group includes mesh-based particle meth-ods. Particle-in-cell method (Harlow 1964), Fluid-implicit Particle method (Brackbill & Ruppel 1986)
and MPM (Sulsky et al. 1995; Sulsky & Schreyer 1996) can be categorized in this group.
ALE, CEL and MPM are discussed briefly in the next subsections.
2.1 Arbitrary Lagrangian-Eulerian (ALE) method
The Arbitrary Lagrangian-Eulerian (ALE) method is an effective method to analyze problems including large deformations. This method was first introduced by Hirt et al. (1974) to solve fluid dynamics or later metal forming problems (Khoei et al., 2008). This method uses a finite mesh with nodes, which may move with the material (Lagrangian description) or be held fixed in space (Eulerian description). The mesh is partially attached on material and can be-come independent where necessary. Initially, La-grangian and Eulerian bodies entirely overlap each other and have the same mesh discretization. Mesh motion is constrained to the material motion only where necessary (at free boundary). Otherwise, mate-rial motion and mesh motion are independent. The ALE method can overcome the mesh distortion while representing the boundary conditions correctly.
According to Hirt et al. (1974) one time step of an ALE calculation is based on three phases. It starts with a classical explicit Lagrangian calculation phase, followed by solving the equilibrium of mo-ments with another standard Lagrangian analysis. The next step is the optional adaptive meshing phase. The mesh is adapted independently from the underly-ing material, i.e. there are no more critical mesh dis-tortions. In a final Eulerian phase the solution ob-tained from the Lagrangian phase is remapped onto the new relocated mesh (advection phase), which was developed through the adaptive meshing algorithm.
The interested reader is referred to Hirt et al. (1974) and Benson (1992).
2.2 Coupled Eulerian-Lagrangian (CEL) method
Originally, the Coupled Eulerian-Lagrangian (CEL) method was introduced by Noh (1964) and further developed by Benson (1992) and Benson & Okazawa (2004). This method, which attempts to capture the strength of the Lagrangian and the Eulerian descrip-tions of motion, is implemented in the commercial code ABAQUS/ Explicit. For geotechnical problems, a Lagrangian mesh is used to discretize the structure
highly distorted mesh
while an Eulerian mesh is used to discretize the sub-soil. The Eulerian material is tracked as it flows through the Eulerian mesh computing its Eulerian volume fraction (EVF). If an Eulerian element is completely filled with material, its EVF is 1. If there is no material in the element, its EVF is 0 (Dassault Systéms, 2013). The interface between structure and subsoil could be represented using the boundary of the Lagrangian domain. On the other hand, the Eulerian mesh, which represents the soil that may experience large defor-mations, has no problems regarding mesh distortion. According to Noh (1964) initially the pressure on the interface of the Eulerian cells is integrated to calcu-late the force acting on the Lagrangian nodes at the interface. Following, the motion of the Lagrangian domain is calculated and the portion of the domain at the time is determined using the new posi-tion of the Lagrangian domain. Finally, the discre-tized Eulerian equations are solved to obtain the new pressure.
In ABAQUS contact between Lagrangian and Eu-lerian material is enforced using a penalty contact method (Dassault Systéms, 2013). It uses a master-slave relationship in which the Lagrangian material usually is defined as master and the Eulerian as slave surface. On the master surface so called seeds are created, whereas on the slave surface there are so called anchor points. Between the seeds and anchors a force will be created, depending on the distance
and a contact stiffness . This stiffness is de-pendent on the Lagrangian material on one side and on the Eulerian material on the other side.
(1)
Qui et al. (2011) or Henke (2009) showed the po-
tential of the CEL method for geotechnical applica-tions undergoing large deformations. The interested reader is referred to Noh (1964), Benson (1992), Benson (1995) and Benson (2000).
2.3 Material Point method (MPM)
For the first time Sulsky et al. (1995) applied fluid implicit particle (FLIP) method from fluid to solid mechanics and called it Material Point Method (Sulsky & Schreyer 1996). Bardenhagen et al. (2000)
added a frictional contact algorithm to this method. Since the first formulation of MPM, it is applied on different applications that show the capability of it to simulate large deformations.
MPM uses two different discretizations, one is the space discretization based on the background fixed mesh, and the other one is the material discretization based on the material points or “particles”. During the simulation, material points that carry all the physical information (strains, stresses, etc.) can move through the background mesh. With this prop-erty extremely large deformations can be simulated.
MPM working procedure in one time step con-sists of three phases: first the initialization phase where all the data are mapped from the particles to the background mesh using the nodal shape func-tions. Next is the Lagrangian phase where the equa-tions of motion are solved. At last is the convective phase where the updated nodal values are mapped back to the particles. The positions of particles are updated and finally background mesh is brought back to its original position.
The weak formulation, space and time integration used in MPM is the same as FEM, but in MPM par-ticle based integration is done i.e.
(2)
where is the total number of particles, is the mass of particle and is the shape function of node evaluated at particle .
Using particles to discretize the material (contin-uum), is done by approximating the density field with the help of the Dirac delta function
(3)
where is an arbitrary position vector and is
the position vector at particle . For the complete formulation of the governing equations please refer to Sulsky et al. (1995) or Jassim (2013) and for the complete formulation of the contact algorithm please refer to Bardenhagen et al. (2000).
Moormann, Fatemizadeh and Aschrafi
Geotechnical Engineering for Infrastructure and Development
4028
3 DISCUSSION
In the previous sections the shortcomings of FEM simulating large deformations was discussed. After a short overview on the available numerical methods to simulate large deformations, ALE, CEL and MPM were briefly introduced as powerful methods which are able to simulate large deformations. These meth-ods which are based on continuum formulation have been developed in the last decades and applied on different examples in different fields of engineering.
In order to show the capabilities of these methods also in the field of geotechnical engineering next two benchmark problems will be studied.
4 BENCHMARK PROBLEMS
To demonstrate the capabilities of the three discussed methods to be used in geotechnical engineering, two benchmark problems are studied. The examples are in 2D and the plain strain conditions are considered. First the strip footing example and then the slope sta-bility problem are solved. Finally the results are compared.
4.1 Example 1: Strip footing
The strip footing is a single foundation with 6 m width placed on the soil domain of 10 m height and 40 m width. The symmetry conditions are used, so the half of the domain is simulated.
The soil on which the footing is located has a density of =1800 kg/m3, a cohesion of cu=100 kN/m2, a friction angle of =0°, Young’s modulus of Eu=300×cu kN/m2, Poisson’s ratio of =0.49, and a dilation angle of =0°. Tresca model was adopted in the simulation of the problem. The footing pres-sure was increased incrementally from zero, each in-crement consisting of 10 kN/m2 increase. Figure 2 shows the geometry and boundary conditions of the problem.
The commercial software ABAQUS was used to solve the strip footing problem for the ALE and CEL approaches. At the beginning the initial stress state was established by using a gravity load.
The CEL model is a 3D model with symmetry conditions on the left, front and back side. Unfortu-
nately ABAQUS is limited to 3D models when us-ing an Eulerian model. Here the depth is set to 1 m. Like the ALE model, at first there is a step to estab-lish the initial stress state, followed by the pressure acting on the footing.
Figure 2. The geometry and boundary conditions of the problem.
The MPM simulation was done based on own code using 2D linear triangular elements with ten particles per element. A damping coefficient of 0.75 is used to converge to quasi static equilibrium. To overcome the volumetric locking which appears in low order elements Enhanced Volumetric Strain method was used. Figure 3 shows the total dis-placements of the particles resulting from the MPM simulation when the failure happens.
Figure 3. Total displacement of the particles from MPM simula-tion at failure state.
In Figure 4, total displacement of a node on the right hand side of the footing as a function of footing pressure can be seen from the calculations with
q
20 m
10 m
3 m
ALE, CEL and MPM. They are compared with the analytical solution according to Hill (1950):
(4)
Figure 4. Comparison of the load displacement curves from the ALE, CEL and MPM with the analytical solution.
Figure 3 shows that MPM is able to capture the shape of the failure body in soil under the strip foot-ing. Results obtained from ALE and CEL simulations also depict the same figures as MPM. It can be seen in Figure 4 that CEL delivers the most accurate re-sults with regard to the analytical solution while MPM and ALE overpredict the solution. ALE is the less accurate method in this example, but it still can give an idea about the ultimate bearing capacity of the strip footing.
4.2 Example 2: Slope stability
Next the problem of slope stability is analyzed. The slope has a height of 10 m, a base length of 21 m and an inclination angle of 45°. The soil of the slope is assumed to behave according to the elastic-perfectly plastic Mohr-Coulomb model. The slope consists of non-cohesive sand. Figure 5 shows the geometry and boundary conditions of the problem.
The unit weight of the soil was increased incre-mentally from zero, each increment consisting of 2 kN/m3 increase till 160 kN/m3.
For the ALE Model there is a rigid body at the left side and the bottom. At the left side there is a fric-tionless penalty contact between the soil and the rigid body, i.e. the soil is “constraint”, but still with the
possibility to flow with the slope. There is also a fric-tionless kinematic contact with the right side of the slope and the rigid body at the bottom. It shall ensure that the soil can move after a slope failure. To ensure that the mesh is moving with the material within these large deformations, the ALE constraint “Follow underlying material” was used.
Figure 5. The geometry and boundary conditions of the problem.
The CEL model is a 3D model with a depth of 1 m and also a symmetry condition in depth. As there is no Lagrangian part in this model, this can be consid-ered as complete Eulerian simulation. Unfortunately the flow of the material cannot be tracked in ABAQUS. Therefore it is not possible to give a load-displacement curve. Figure 6 shows the shear strains from CEL simulation.
The results obtained from the ALE simulation is not shown here while ALE is not suitable for very large deformations as the case in this example. This method can be used with moderate deformations like the strip footing problem.
In the other hand MPM and CEL are able to simu-late the slope failure. Due to the similarity, the graphs from MPM are not shown here. The difference in the application of boundary conditions and discretization leads to a slight difference in the results, but the same pattern in the diagrams obtained from the CEL and MPM simulations demonstrates their applicability for this example.
5 CONCLUSION AND OUTLOOK
In this study the shortcoming of the FEM to simulate large deformations was demonstrated. ALE, CEL and
45°
10 m
11 m
20 m
4029
3 DISCUSSION
In the previous sections the shortcomings of FEM simulating large deformations was discussed. After a short overview on the available numerical methods to simulate large deformations, ALE, CEL and MPM were briefly introduced as powerful methods which are able to simulate large deformations. These meth-ods which are based on continuum formulation have been developed in the last decades and applied on different examples in different fields of engineering.
In order to show the capabilities of these methods also in the field of geotechnical engineering next two benchmark problems will be studied.
4 BENCHMARK PROBLEMS
To demonstrate the capabilities of the three discussed methods to be used in geotechnical engineering, two benchmark problems are studied. The examples are in 2D and the plain strain conditions are considered. First the strip footing example and then the slope sta-bility problem are solved. Finally the results are compared.
4.1 Example 1: Strip footing
The strip footing is a single foundation with 6 m width placed on the soil domain of 10 m height and 40 m width. The symmetry conditions are used, so the half of the domain is simulated.
The soil on which the footing is located has a density of =1800 kg/m3, a cohesion of cu=100 kN/m2, a friction angle of =0°, Young’s modulus of Eu=300×cu kN/m2, Poisson’s ratio of =0.49, and a dilation angle of =0°. Tresca model was adopted in the simulation of the problem. The footing pres-sure was increased incrementally from zero, each in-crement consisting of 10 kN/m2 increase. Figure 2 shows the geometry and boundary conditions of the problem.
The commercial software ABAQUS was used to solve the strip footing problem for the ALE and CEL approaches. At the beginning the initial stress state was established by using a gravity load.
The CEL model is a 3D model with symmetry conditions on the left, front and back side. Unfortu-
nately ABAQUS is limited to 3D models when us-ing an Eulerian model. Here the depth is set to 1 m. Like the ALE model, at first there is a step to estab-lish the initial stress state, followed by the pressure acting on the footing.
Figure 2. The geometry and boundary conditions of the problem.
The MPM simulation was done based on own code using 2D linear triangular elements with ten particles per element. A damping coefficient of 0.75 is used to converge to quasi static equilibrium. To overcome the volumetric locking which appears in low order elements Enhanced Volumetric Strain method was used. Figure 3 shows the total dis-placements of the particles resulting from the MPM simulation when the failure happens.
Figure 3. Total displacement of the particles from MPM simula-tion at failure state.
In Figure 4, total displacement of a node on the right hand side of the footing as a function of footing pressure can be seen from the calculations with
q
20 m
10 m
3 m
ALE, CEL and MPM. They are compared with the analytical solution according to Hill (1950):
(4)
Figure 4. Comparison of the load displacement curves from the ALE, CEL and MPM with the analytical solution.
Figure 3 shows that MPM is able to capture the shape of the failure body in soil under the strip foot-ing. Results obtained from ALE and CEL simulations also depict the same figures as MPM. It can be seen in Figure 4 that CEL delivers the most accurate re-sults with regard to the analytical solution while MPM and ALE overpredict the solution. ALE is the less accurate method in this example, but it still can give an idea about the ultimate bearing capacity of the strip footing.
4.2 Example 2: Slope stability
Next the problem of slope stability is analyzed. The slope has a height of 10 m, a base length of 21 m and an inclination angle of 45°. The soil of the slope is assumed to behave according to the elastic-perfectly plastic Mohr-Coulomb model. The slope consists of non-cohesive sand. Figure 5 shows the geometry and boundary conditions of the problem.
The unit weight of the soil was increased incre-mentally from zero, each increment consisting of 2 kN/m3 increase till 160 kN/m3.
For the ALE Model there is a rigid body at the left side and the bottom. At the left side there is a fric-tionless penalty contact between the soil and the rigid body, i.e. the soil is “constraint”, but still with the
possibility to flow with the slope. There is also a fric-tionless kinematic contact with the right side of the slope and the rigid body at the bottom. It shall ensure that the soil can move after a slope failure. To ensure that the mesh is moving with the material within these large deformations, the ALE constraint “Follow underlying material” was used.
Figure 5. The geometry and boundary conditions of the problem.
The CEL model is a 3D model with a depth of 1 m and also a symmetry condition in depth. As there is no Lagrangian part in this model, this can be consid-ered as complete Eulerian simulation. Unfortunately the flow of the material cannot be tracked in ABAQUS. Therefore it is not possible to give a load-displacement curve. Figure 6 shows the shear strains from CEL simulation.
The results obtained from the ALE simulation is not shown here while ALE is not suitable for very large deformations as the case in this example. This method can be used with moderate deformations like the strip footing problem.
In the other hand MPM and CEL are able to simu-late the slope failure. Due to the similarity, the graphs from MPM are not shown here. The difference in the application of boundary conditions and discretization leads to a slight difference in the results, but the same pattern in the diagrams obtained from the CEL and MPM simulations demonstrates their applicability for this example.
5 CONCLUSION AND OUTLOOK
In this study the shortcoming of the FEM to simulate large deformations was demonstrated. ALE, CEL and
45°
10 m
11 m
20 m
Moormann, Fatemizadeh and Aschrafi
Geotechnical Engineering for Infrastructure and Development
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MPM as three powerful methods which are able to overcome this shortcoming was introduced. At the end two benchmark problems from the field of geotechnical engineering were analyzed.
The results obtained from both benchmark prob-lems demonstrate the usability of the methods to study the problems including large deformations in the field of geotechnical engineering. These three methods predict the prefailure conditions as well as failure situations accurately. How a failure mass looks like, as in the strip footing problem or in the slope stability problem, can be predicted with these methods, as shown above. Postfailure behavior of the soil which includes very large deformations is cap-tured with CEL and MPM. After failure in the slope, in which pattern the soil deforms itself is predicted with these methods precisely.
In the future work more advanced material models will be used to achieve more accurate results and more attention will be paid to increase the efficiency of the methods in terms of computational cost.
Figure 6. Shear strains from CEL simulation.
REFERENCES
Bardenhagen, S.G. Brackbill, J.U. & Sulsky, D. 2000. The materi-al-point method for granular materials, Computer Methods in Ap-plied Mechanics and Engineering 187, 529–541. Belytschko, T. Lu, Y.Y. & Gu, L. 1994. Element-free Galerkin methods, International Journal for Numerical Methods in Engi-neering 37(2), 229–256. Benson, D.J. 1992. Computational methods in Lagrangian and Eu-lerian hydrocodes, Computer Methods in Applied Mechanics and Engineering 99, 235-394. Benson, D.J. 1995. A multi-material Eulerian formulation for the efficient solution of impact and penetration problems, Comput. Mech. 15, 558-571. Benson, D.J. 2000. An implicit multi-material Eulerian formula-tion, Int. J. Numer. Meth. Engrg. 48, 475-499. Benson, D.J. & Okazawa, S. 2004. Contact in a multi-material eu-lerian finite element formulation, Computer Methods in Applied Mechanics and Engineering 193, 4277-4298. Brackbill, J.U. & Ruppel, H.M. 1986. FLIP: A low dissipation par-ticle-in-cell calculations of fluid flows in two dimensions, Journal of Computational Physics 65(2), 314–343. Dassault Systéms 2013. ABAQUS, Version 6.12 Documentation. Harlow, F.H. 1964. The particle-in-cell computing method for flu-id dynamics, Methods in Computational Physics 3, 319–343. Henke, S. 2009. Herstellungseinflüsse aus Pfahlrammung im Kai-mauerbau, Dissertation, Veröffentlichungen des Instituts für Geo-technik und Baubetrieb der TU Hamburg-Harburg 18. Hill, R. 1950. The Mathematical Theory of Plasticity, Clarendon Press, Oxford. Hirt, C.W. Amsden, A.A. & Cook, J.L. 1974. An arbitrary La-grangian-Eulerian computing method for all flow speeds, Journal of Computational Physics 14(3), 227-253. Hughes, T.J.R. 1987. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Inc., Eng-lewood Cliffs, New Jersey. Jassim, I.K. 2013. Formulation of a Dynamic Material Point Method (MPM) for Geomechanical Problems, Institute of Geotechnical Engineering of University of Stuttgart, Germany. Khoei, A. Anahid, M. Shahim, K. & Dormohammadi, H. 2008. Arbitrary lagrangian-eulerian method in plasticity of pressure-sensitive material: application to powder forming processes, Com-put. Mech. 42, 13–38. Monaghan, J.J. 1988. An introduction to SPH, Computer Physics Communications 48(1), 89–96. Noh, W.F. 1964. CEL: A time-dependent, two-space-dimensional, coupled Eulerian-Lagrangian code, Methods in Computational Physics 3, 117-179. Potts, D.M. & Zdravković, L. 2001. Finite Element Analysis in Geotechnical Engineering, Application, Thomas Telford, London. Sulsky, D. & Schreyer, H.L. 1996. Axisymmetric form of the ma-terial point method with applications to upsetting and Taylor im-pact problems, Jo. Computer Methods in Applied Mechanics and Engineering 139, 409–429. Sulsky, D. Zhou, S.J. & Schreyer, H.L. 1995. Application of a par-ticle-in-cell method to solid mechanics, Computer Physics Com-munications 87, 236–252. Więckowski, Z. 2001. Analysis of granular flow by the material point method. European conference on computational mechanics, Cracow, Poland.
0.00
0.80
1.60
� = 0 kPa
� = 40 kPa
� = 160 kPa
4031
Proceedings of the XVI ECSMGEGeotechnical Engineering for Infrastructure and DevelopmentISBN 978-0-7277-6067-8
MPM as three powerful methods which are able to overcome this shortcoming was introduced. At the end two benchmark problems from the field of geotechnical engineering were analyzed.
The results obtained from both benchmark prob-lems demonstrate the usability of the methods to study the problems including large deformations in the field of geotechnical engineering. These three methods predict the prefailure conditions as well as failure situations accurately. How a failure mass looks like, as in the strip footing problem or in the slope stability problem, can be predicted with these methods, as shown above. Postfailure behavior of the soil which includes very large deformations is cap-tured with CEL and MPM. After failure in the slope, in which pattern the soil deforms itself is predicted with these methods precisely.
In the future work more advanced material models will be used to achieve more accurate results and more attention will be paid to increase the efficiency of the methods in terms of computational cost.
Figure 6. Shear strains from CEL simulation.
REFERENCES
Bardenhagen, S.G. Brackbill, J.U. & Sulsky, D. 2000. The materi-al-point method for granular materials, Computer Methods in Ap-plied Mechanics and Engineering 187, 529–541. Belytschko, T. Lu, Y.Y. & Gu, L. 1994. Element-free Galerkin methods, International Journal for Numerical Methods in Engi-neering 37(2), 229–256. Benson, D.J. 1992. Computational methods in Lagrangian and Eu-lerian hydrocodes, Computer Methods in Applied Mechanics and Engineering 99, 235-394. Benson, D.J. 1995. A multi-material Eulerian formulation for the efficient solution of impact and penetration problems, Comput. Mech. 15, 558-571. Benson, D.J. 2000. An implicit multi-material Eulerian formula-tion, Int. J. Numer. Meth. Engrg. 48, 475-499. Benson, D.J. & Okazawa, S. 2004. Contact in a multi-material eu-lerian finite element formulation, Computer Methods in Applied Mechanics and Engineering 193, 4277-4298. Brackbill, J.U. & Ruppel, H.M. 1986. FLIP: A low dissipation par-ticle-in-cell calculations of fluid flows in two dimensions, Journal of Computational Physics 65(2), 314–343. Dassault Systéms 2013. ABAQUS, Version 6.12 Documentation. Harlow, F.H. 1964. The particle-in-cell computing method for flu-id dynamics, Methods in Computational Physics 3, 319–343. Henke, S. 2009. Herstellungseinflüsse aus Pfahlrammung im Kai-mauerbau, Dissertation, Veröffentlichungen des Instituts für Geo-technik und Baubetrieb der TU Hamburg-Harburg 18. Hill, R. 1950. The Mathematical Theory of Plasticity, Clarendon Press, Oxford. Hirt, C.W. Amsden, A.A. & Cook, J.L. 1974. An arbitrary La-grangian-Eulerian computing method for all flow speeds, Journal of Computational Physics 14(3), 227-253. Hughes, T.J.R. 1987. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Inc., Eng-lewood Cliffs, New Jersey. Jassim, I.K. 2013. Formulation of a Dynamic Material Point Method (MPM) for Geomechanical Problems, Institute of Geotechnical Engineering of University of Stuttgart, Germany. Khoei, A. Anahid, M. Shahim, K. & Dormohammadi, H. 2008. Arbitrary lagrangian-eulerian method in plasticity of pressure-sensitive material: application to powder forming processes, Com-put. Mech. 42, 13–38. Monaghan, J.J. 1988. An introduction to SPH, Computer Physics Communications 48(1), 89–96. Noh, W.F. 1964. CEL: A time-dependent, two-space-dimensional, coupled Eulerian-Lagrangian code, Methods in Computational Physics 3, 117-179. Potts, D.M. & Zdravković, L. 2001. Finite Element Analysis in Geotechnical Engineering, Application, Thomas Telford, London. Sulsky, D. & Schreyer, H.L. 1996. Axisymmetric form of the ma-terial point method with applications to upsetting and Taylor im-pact problems, Jo. Computer Methods in Applied Mechanics and Engineering 139, 409–429. Sulsky, D. Zhou, S.J. & Schreyer, H.L. 1995. Application of a par-ticle-in-cell method to solid mechanics, Computer Physics Com-munications 87, 236–252. Więckowski, Z. 2001. Analysis of granular flow by the material point method. European conference on computational mechanics, Cracow, Poland.
0.00
0.80
1.60
� = 0 kPa
� = 40 kPa
� = 160 kPa
Finite element analysis of a large span precast arch
Analyse par Eléments Finis d’une arche préfabriquée à longue travée pour le dédoublement de l’A465, Section 3
Kyla Nunn1, Sachin Kumar 2, Aled Phillips3
1 Arup Geotechnics, Cardiff, UK 2 Arup Geotechnics, Solihull, UK 3 Arup Geotechnics, Cardiff, UK
ABSTRACT The A465 “Heads of the Valleys” dualling involves upgrading a single three lane to a dual two lane highway. The Section 3between Brynmawr and Tredegar involves extensive offline construction, which includes a high embankment to carry the A465 across a narrow steep-sided valley. An underpass formed of precast reinforced concrete arch units is incorporated within the embankment to contain the River Ebbw and provide access to the Carno Reservoir. The varying ground conditions across the valley and high vertical and somewhatunbalanced loading from the geometry of the embankment, needed detailed consideration in terms of the underpass design particularly in terms of soil-structure effects. This was undertaken using Finite Element (FE) analysis. The aim of the FE analysis was to demonstrate overall stability and allow structural design of the arch and its foundations. The sensitivity of the analyses to various design input parameters is discussed in the paper. The arch structure has been constructed and filling of the embankment above was completed in January 2015. Arch movements have been monitored and compared with the predicted movements from the FE analysis and this paper discusses some of the keyconclusions from the comparisons made.
RÉSUMÉ Le dédoublement de l'A465 "Heads of the valleys" consiste à améliorer une trois voies en une autoroute de deux fois deux voies. La section entre Brynmawr et Tredegar comprend des travaux ‘hors route’ importants qui incluent un talus élevé pour supporter l’A465 à travers une vallée étroite et à pente abrupte. Un passage souterrain formé d’une arche en segments de ciment renforcés préfabriqués estincorporée dans le talus pour contenir la rivière Ebbw et fournir un accès au réservoir Carno. A cause des conditions de sols variées à travers la vallée et les charges élevées et déséquilibrées résultant de la géométrie complexe du talus, le design du passage souterrain a du considérerles effets de l’interaction sol-structure. Ceci a été entrepris utilisant une analyse par Eléments Finis (EF). L’objectif des analyses a été dedémontrer la stabilité d’ensemble and permettre le design de la structure de l’arche et de ses fondations. La sensibilité des analyses des différents paramètres entrants est discutée dans ce rapport. La structure de l’arche a été construite en Janvier 2015 et le remplissage du talus mentionné est presque finit. Les mouvements de l’arche ont été contrôlés et comparés avec les mouvements prédits par l’analyse EF et ce rapport présente certaines des conclusions majeurs des comparaisons entreprises.
1 BACKGROUND AND PURPOSE
The A465 Heads of the Valley Section 3 involves im-provement works of the existing trunk road between Brynmawr and Tredegar from three lane to a dual two lane carriageway. The scheme is promoted by the Welsh Government who appointed Carillion, with
Arup as their designers, under an ECI (Early Contrac-tor Involvement) Design and Build contract to deliver the scheme.
The scheme includes a new offline section within which the route crosses a steep sided valley (see Fig-ure 1) on an embankment up to a maximum height of 33m. At the base of the valley there is the River Ebbw and an access road to a nearby reservoir.
