Modelling sliding interfaces with friction and gapping in geotechnics M. Hertweck, T. Schanz
Institute of Geotechnical Engineering, FedeW JWzWe o/
ABSTRACT
The numerical modelling of the interaction soil-mass/structure or between different soil-masses which are separated by geomembrans or geotextils is a well-known problem in geotechnics. One example for the use of sidelines in Geotechnical calculations is discussed by a multi-layer liner for steep walls in a waste disposal. The global bearing of the system is mainly influenced by the interaction of the different components which could be described by the friction angle. So for realistic simulation between the different materials the effects of friction, sliding, separation and closure have to be simulated. The mechanical properties of the materials and interfaces (slidelines) used in the FEM-code are determined by routine laboratory tests. For calibration of the slidelines the measured stress-strain-relations are back calculated with the numerical model. A set of characteristic values describing the properties of the interfaces is obtained. With this set the overall behaviour of a multi-layer liner is described as function of the shear contact at the interfaces.
INTRODUCTION
The ability to treat technical problems where adjacent parts of the construction are able to slide and separate independently along material interfaces is very important in the field of Geotechnical engineering. It is necessary to describe the soil-structure interaction in a real way. This behaviour is crucial in many fields as for example at the backside of a retaining wall, the roughness of a foundation and the contact between an underground construction and the surrounding soil/rock-body. The stress-strain performance of the whole mechanical system depends very strong on the capability of the used (numerical-) model to describe the boundary conditions in these region correctly.
The common practice today is to replace the interfaces by thin layered elements and to decrease their material properties in order to allow no stiffness nor strength. Opening of the interface related to large deformations and correct modelling of measured frictional effects is impracticable by these simulations.
A general slideline capability is implemented in the FEM-code [1] used that includes both sliding and tied interfaces with arbitrary mesh refinement on adjacent sides of the interface. The algorithms are based on a penalty formulation. While there is contact, linear interface springs are inserted into the stiffness matrix to preclude penetration. The parameters used in the code have only an implicit meaning in Geotechnical engineering and are not determinate directly. The characteristics of the used Finite Element code NIKE2D are given in detail elsewhere [1,2]. The program is running on a mainframe of the "Rechenzentrum derETHZ"bu\ pre- and post processing are done on a workstation.
The influence of the parameters to calibrate the sliding interfaces and their effects on the global bearing behaviour of a whole structure is
shown for an actual waste deposit project [3]. A multi-layered barrier on a steep rock slope is analysed in its response as a function of modelling its structural components. A mineral waste deposit sealing may not lose its function during external effects, such as deformation of the sealing because of load- and self settlements in the waste deposit. The concept of the steep slope barrier which has already been constructed at present in Germany is shown in Fig. 1.
One effect of the steep wall sealing barrier is the separation between rock and sealing system which allows relative deformations to avoid admissible loads onto the sealing system. The other effect is to reduce large shear forces from the gabions which result form the vertical settlement of the domestic waste and reach 30 % of the total height. By avoiding cracks and breaking up (due to plastification and/or tension) in the barrier the overall safety of the construction is warranted. Another relative deformation is the vertical settlement of the mineral barrier between the rock and the gabions. These vertical movements must be allowed to avoid horizontal cracks resulting in unwanted permeability.
, Slideline "Geotextil" ^Slideline "Geomembrane"
Gabions Domestic waste
Rock face
Figure 1: Detail of the sealing system [3].
At the point where the steep slope system bends to be horizontal another problem may occur, in that ultimate bearing capacity could be attempted under the gabions. At stiff materials the load is concentrated. Plastic zones are built which reduce the permeability.
THEORY
In this chapter the theory of the interface implementation would be described. Six different types of slidelines may be used in the newest version of the used code [2]: sliding only (type #1), tied (type #2), sliding with separation (type #3), sliding with separation and interface friction (type #4), sliding with separation and tie break (type #5) and single surface contact (type #6).
slaveslideline with slavenodes
r S 1 r~ — — i
r ^ s • •''•'/
j— — - v.
masterslideline with masternodes
Figure 2: Typical slideline [2].
In the two dimensional calculation, the contact surface appears in Fig. 2 as two lines referred to as the master- and slave-lines. The nodes at the two boundaries are called slave and master nodes. Each type is defined by the coordinates of these points, its penalty function scale factor, additionally #4 a friction coefficient and #5 a plastic strain to fail tie break.
All types include arbitrary mesh refinement on adjacent sides of the interface. Each time step internal logic identifies a master segment for each slave node and a slave segment for each master node and updates this information. Wherever there is contact, an interface spring is used to preclude penetration. For each penetrating node one spring
is inserted into the stiffness matrix that couples the penetrating node to two adjacent nodes on the contact surface. Except in the case of type #4 or #2 these springs act normal to the contact surface and there are no effects on sliding. The test for penetration determines the penetration depth and the determination of contact points. Initialisation of the penalty method in two dimensions involves the following steps. The general interface treatment is as follows:
Step 1- The closest master node, rim, for each slave node, ns, is located and the master segments that includes nm is checked to identify the segment, if any, containing ns.
Step 2: On the master segment the position of the slave nodes is located.
Step 3: Determine if ns has penetrated the master segment. If so, compute and add an interface force to the load vector and, if the global stiffness matrix is being reformed, add in an
interface stiffness matrix.
Step 4: Repeat Steps 1 -3 for the master nodes.
So for each master and slave segment the element that contains the segment in its connectivity can be determined and the segment stiffness is calculated from the bulk modulus and the thickness. Penetrating nodes are projected back on the surface. For a slideline of type #3 the following Fig. 3 illustrates the situation when penetration of
node m through segment jk is detected.
Figure 3: Contact of node m with segment of jk.
With the stiffness matrix K and the internal nodal force F"* local equilibrium relation can be written as
= ps - ps (1)
where
AuS is the incremental displacement vector containing the penalty spring degrees of freedom (AuS = [Av^, Aw^ AVJ, Aw^AVk, AwJ)
K^ is the spring stiffness
ps is the spring internal force and
ps is the external force arising from internal states of stress in the interface element.
Hereby the 6x6 spring stiffness matrix KS is defined as (1-a)sc o/» M a\c* —o(/ —\l — ayo c^ (1-a)sc -as asc asc -ac -(\-afsc (1-a)as* -(1-a)asc
(1-afc' -(1-a)asc (l-a)ac'
symmetric
(2)
where c = cos 0, s = sin 0 and K is the penalty stiffness. The spring internal force ps is defined by
-5 C -(l-fl)c as -ac (3)
where 8 is the amount of penetration of node m through segment jk and is assumed to be negative. A slideline becomes active for distance less
then 10-3 L (Fig. 3) away from a segment of length L. Nodal interface forces do not develop until penetration happens. This preliminary insertion of the spring has been proofed to improve stability although it can slow convergence. For all active slideline nodes and segments the spring stiffness K^and force F^are calculated and built in the global finite elements equations. So the profile of the stiffness matrix changes with evolution of the system. The interfaces are treated like another element class. Each time step, and each time the stiffness matrix is reformed during a time step, the equation solver is initialised with a new block structure to account for the changing connectivities of the sliding interface stiffness.
The penalty stiffness K is calculated from the material properties of the elements including the contact area. It may be changed by a scale factor to be defined in the input. If noticeable penetration occurs this scale factor has to be increased but too high values lead to convergence problems for the global iterations. This class of slidelines is normally confronted with fear of ill-conditioning, of erratic behaviour and of trial-and-error procedures to determine a well fitting penalty size. Beyond a certain size of K there will be large differences of the magnitude of individual stiffness terms in the system matrix and oscillation of the solution.
LABORATORY TEST
The mechanical behaviour of the materials used for numerical simulation are established by different laboratory tests. The elastic parameters, Young's modulus and poisson ratio, were find out by triaxial and oedometer tests, the failure parameters c and 9 are determined through standard triaxial tests. The material was installed from the wet side of proctor curve, two percent over water content optimum. These are the same conditions like the real installation for the
sealing system. The samples are not saturated during the tests. The parameters are shown in table 1.
mineral barrier Clay (CL) cpH 19.4 c fkN/m^] 32.9 E [kN/m2] 8555 V 0.34 Table 1: Parameters of the mineral material used.
To find out the failure criteria for the interaction zone between geomembrane and soil, direct- and ring shear tests were done. From direct shear test (Fig. 4) we yield the peak terms for wall friction angle 8 and for adhesion a.
Figure 4: Construction of direct shear test.
In the ring shear tests were great distance can be done without changing the size of the sliding surface we yield the rest-strength of the interaction zone. The results were shown in table 2. In both shear instruments the shear box below was filled by a steel dummy on which the geomembrane was fixed. The surface of the geomembrane was lying in given shear plane of the apparatus. The Clay (CL) was installed by given density in the upper box. After consolidation the test was
running. The lower shear box was fixed and the upper shear box was moving with constant deformation velocity.
direct shear rinq shear Soeak PI 12.1 9.6 aoeak [kN/m2] 21.3 4.6 Brest PI 12.2 8.9 arest[kN/m2] 14.4 2.7 Table 2: Parameters for slideline geomembrane/mineral barrier. NUMERICAL SIMULATION
For all the calculations we use 4 noded elements with 2x2 Gauss integration. The rock, gabions, household waste and the drainage layer are modelled with a perfect elastic material law. The relevant parameters are the Young's modulus E and the Poisson's ratio v. The mineral barrier is modelled with an ideal elastic perfect plastic law described by the shear modulus G and the bulk modulus K and a Drucker-Prager failure criteria. Table 3 shows the used parameter com-binations for all simulations.
qabiones mineral barrier household waste qeo membrane rock E [kN/m^l 80000. 5000. 100000. G [kN/m2] _ 2000. . . -K [kN/m2] _ 3333. _ _ -v r-i 0.3 _ 0.1 0.4 C [kN/m2] . 30.0 . _ -4> M _ 20.0 -. -T fkN/m^] 15.0 20.0 8.0 22.0 Table 3: Material parameter used in the FE-Code.
A detailed description of the direct-shear apparatus was given in chapter 3. For this problem it is necessary to divide the lower and upper
shear box for getting same conditions in numerical simulations and shear apparatus. The FE-mesh (Fig. 5) consists of 210 nodes forming 142 elements. The nodes at the bottom are fixed, all other are free. The frame of the apparatus is simulated as ideal elastic with the parameter of steel divided by slidelines in different parts. The lower part of the sample, connected to the upper one by one slideline, is a geomembrane (simulated elastic), the upper part the mineral barrier material using an ideal elastic perfect plastic law.
Figure 5: Deformed FE-mesh of the direct shear apparatus. The peak values are already reached for small deformations depending of the consolidation pressure. The measured and calculated results are drawn in Fig. 6.
In the program we used there is no possibility to simulate the adhesion and the friction on the same sliding type. Therefor we used an overall friction angle with condition to neglect adhesion. The input parameter |Tis defined by |T = 1.5 tan (overall friction angle). During the numerical simulation and the calibration of the direct shear test, the spring stiffness was obtained to avoid penetration from one element into another along the slideline. After slideline calibration the obtained parameters were used for modelling of the steep slope sealing system.
100-_ 80-CM 3 60-tn 40-cti 0) s=. " 20-| 0 A direct shear/calculated # direct shear/ measured
Regression i i . i i i i ' i ' ' « ' i ' ' ' ' i ' ' ' 0 50 100 150 200 250
normal stress [kN/m2]
Figure 6: Measured and calculated results of direct shear test. The multi-layered sealing system was modelled by 707 nodes forming 556 elements (Fig. 7). There are two vertical slidelines with sliding, gaps and friction and two horizontal and one vertical with only sliding. At the left and the bottom border elements are fixed, at the right side they are allowed to move just vertically. The materials of the barrier and the household waste are shown with the geometry in the following picture. The overall dimensions of the mesh are H = 30m and B = 30m.
It is a detail of the total landfill (60 m height) to simulate the staged con-struction of the barrier and the filling with waste.
Three points of the mesh were monitored for all calculations: no. 27 at the top of the gabions, no. 115 in the mineral barrier and no. 119 in the drainage under the gabions. The stresses and settlements are always related to the ones without friction. For two different frictional coefficients between gabions and mineral barrier, which was hold
constant the friction between rock and barrier was varied and the specific settlements and stresses were calculated.
,27
115
119
Figure 7: FE-mesh of the sealing system (slidelines, tracked points). variation of shearstrength -1.25 £ -1.00 -0.75 ^ 0.5 1.0 interface friction /
Figure 8: Relations for low frictional gabion-mineral barrier interaction.
1.00-varlotlon of shearstrength • 1.50 0.50 0.0 0.5 1.0 1.5 interface friction p* (/z^=1.0) [-] 2.0
Figure 9: Relations for high frictional gabion-mineral barrier interaction. DISCUSSION
For the range of shear parameters discussed in this paper there is an remarkable influence of the interface friction on the stresses and strains at crucial parts of the construction. This influence increases with decreasing friction on the one hand between gabions and mineral barrier and on the other hand between mineral barrier and rock. The influence of frictional coefficients on the strain- is stronger then on the stress-distribution. The steepest rise of the settlement curve lies in the realistic range of the interface friction. The settlements of the gabions decrease with increasing friction in both sidelines. When friction between gabions and barrier increases the mineral barrier is loaded and the drainage is unloaded. For the stresses under the gabions (node 119) there seems to be a limit value of friction when its influence is not important any more. This value lies by m = ^2 approximately. The stresse in the mineral barrier (node 115) increases with a constant rise at the same value where is no influence for the stresses in node 119.
For low and high friction between gabions and mineral barrier the qualitative form of the curves of the stresses is totally different. The reason for this behaviour has to be analysed in future.
CONCLUSION
In this paper the behaviour of sidelines in Geotechnical engineering was analysed by laboratory tests and numerical studies. The overall behaviour of the chosen structure depends very strong on the ffictional characteristics of the different interfaces. For the practical range of shear parameters the influence on the settlements is more important then on the stresses. For further studies it seems to be useful first to back calculate known stress-strain relations from former states of the construction to get a reliable set of input parameters for additional states to predict. This would be possible, if we recieved results of the measurements which are done in the mineral barrier for the waste disposal.
REFERENCES
1. Hallquist, J.O., Goudreau, G.L., Benson, D.J., 'Sliding interfaces with contact impact in large scale lagrangian computations' Computer Methods in Applied Mechanics and Engineering , Vol. 51, pp. 107-137, 1985.
2. Engelmann, B., 'NIKE2D, a non-linear, implicit, two-dimensional finite element code for solid mechanics', Lawrence Livermore National Laboratory, UCRL-MA-105413, 1991.
3. Amann, P. and Hertweck, M., 'Untersuchung und Anwendung von Steilwandbarrieren fur Deponien in Steinbruchen' Felsbau, Vol.11/3, 1993.
