Visco-elasto-plastic Earthquake Shear Hysteretic Response of Geomaterials

Visco-elasto-plastic Earthquake Shear Hysteretic Response of Geomaterials Zamila Harichane Associate Professor Geomaterials laboratory, Civil Engineering Department, University of Chlef, Chlef 02000, Algeria e-mail: z_harichane@yahoo.fr Azeddine Chehat Research Scholar Geomaterials laboratory, University of Chlef, Chlef 02000, Algeria e-mail: a_chehat@yahoo.fr Amina Sadouki Research Scholar Geomaterials laboratory, University of Chlef, Chlef 02000, Algeria e-mail: amina_sadouki@yahoo.fr ABSTRACT Seismic ground motion records show that accelerations and, consequently, induced stresses and strains, are cyclic. In order to highlight this phenomenon, we studied the nonlinear hysteretic shear stress-strain behaviour of one or more materials constituting a soil profile under earthquake loading. The soil profile is modelled as one-dimensional shear beam with base ground motion which was recorded at Keddara s rock station during the May 21, 2003 Boumerdes earthquake. The hardening function is represented by the widely used model in soil dynamic, the hyperbolic model. The influence of the nature of geomaterials on the hysteretic behaviour is studied by choosing different types of geomaterials. KEYWORDS: Geomaterial, hysteretic response, earthquake loading, Von Mises criterion, hyperbolic model. INTRODUCTION Local soil conditions have a great influence on the ground response during earthquakes. Observations of damages caused by many historical earthquakes (El Asnam, 1980; Mexico City, 1985; Loma Prieta, 1989; Northridge, 1994; Kobe, 1995; Boumerdes, 2003) showed that geotechnical aspects can strongly influence the performance of structures during an earthquake. The geotechnical materials between the ground surface and the rock influence the structural damage according to two ways [1]: (1) by modifying the manner in which the ground vibrates at a particular site; the geotechnical materials cause an amplification or an attenuation of the seismic waves, and (2) through the process of the ground failure in which a soil mass shows permanent - 1567 -

Vol. 16 [2011], Bund. M 1568 deformations. These two damage aspects must be considered in the evaluation of the seismic risk of a region. Though an important development was done, the geotechnical aspect of earthquake engineering remains relatively new. The first procedures of evaluation of site responses used the linear approximations and equivalent linear of the soil stress-strain behavior [2]. Although these types of approximations brought important progress in our comprehension of the soil response, as well as the development of much of useful and practical methods of analysis, the new procedures which represent the soil behavior more realistically continue to be developed and used. Particularly, nonlinear site response analyses which take into account hysteretic energy dissipations, the pore water pressure generation and the accumulation of the permanent deformation are now available. These analyses can provide improved representations of seismic site response and can also be used to evaluate the risk of ground failure in terms of permanent deformations, rather than the old ones based on the safety factor. The tendency towards the consideration of the nonlinear effects on the site response and the ground failure are in the center of the development of the geotechnical aspect of earthquake engineering. A nonlinear analysis is generally performed by using a discrete model such as finite elements or mass-dashpot models, and carrying out a step by step integration in time domain of the equation of motion. For this type of analysis, to obtain reliable results, the characteristics of the soil stress-strain relationship must be suitably represented. Depending on the site particularity, the one or two-dimensional nonlinear site response analyses can be used. The nonlinear analyses use a more realistic model to represent the soil behavior under cyclic loadings. Primarily, a nonlinear model traces the evolution of the hysteresis loops generated in soil under a cyclic loading in a sequential manner, while the equivalent linear model approaches only representative stiffness and damping after the whole sequence of the cyclic loadings. A more realistic representation of the nonlinear soil behavior cyclically loaded gives to the nonlinear analyses a significant advantage compared to the equivalent linear seismic response analyses for high levels of seismic vibration where the nonlinear effects are dominant. In general, equivalent linear site response analyses are considered unreliable for levels of soil vibration beyond 0.4g [3] or if the computed maximum shear strains exceed approximately 2%. However, the nonlinear site response analyses are also subjugated with limitations. The material models used in the nonlinear site response analyses often require parameters which are not easily accessible by the experimentation. The objective of this work is to study the nonlinear hysteretic shear stress-strain response of geomaterials under seismic loadings. To do this, we study the behavior of one or more materials constituting a soil profile. Then, the shear stress-strain response of a site located in the Metidja basin (in Algeria), close to the Algiers capital, is computed. The soil profile of the studied site is excited at its base by the accelerogram which was recorded at the Keddara s rock station at the time of the main shock of the 6.8 magnitude Boumerdes earthquake which shook the area on May 21, 2003. ELASTOPLASTIC STRESS-STRAIN RESPONSE An elastoplastic model based on the incremental theory of plasticity and the Von Misès criterion [4] is used. A simple 1-D shear stress-strain formulation is used [5].

Vol. 16 [2011], Bund. M 1569 We used a function of work hardening according to the well known hyperbolic model, given by the following general relation: ε τ = (1) a + bε where 1/a is the initial tangential Young s modulus and 1/b is the asymptotic value of failure stress. The equation (1) can be written as bellow [6]: ε τ = 1 R (2) f + ε E q The factor R f < 1 is called the failure ratio and q f is the failure stress. Figure (1) shows the work hardening function for the hyperbolic model. i f Figure 1: Work hardening function We inject the hyperbolic model like as work hardening function to determine the seismic response of different types of geomaterials: soft clay, limestone and other geomaterials. We can compute the stress-strain response at different locations in the selected soil profile. The spatial modeling is done by using linear finite element model with two nodes. The soil column with height h is divided into N elements. Each element e i is characterized by its thickness

Vol. 16 [2011], Bund. M 1570 h i, mass density ρ i and its shear modulus G i. In condensed form, the equation of motion to be solved for each temporal increment is: [ M ]{ u& } [ K]{ u} = [ M ]{ I} u& g + (3) where u& & g is the earthquake base excitation. [M] and [K] are the mass and stiffness matrix, respectively. VISCO-ELASTO-PLASTIC FORMULATION The visco-elasto-plastic formulation is incorporated by introducing damping matrix in equation (3). The last one is assumed as a linear combination of mass matrix and stiffness matrix according to Rayleigh s model as below. [ C] α [ M ] + β[ K] = (4) The α and β coefficients can take different expressions. According to Woodward et al. (1996) [7] one writes ω 1 ω2 ω1 ω2 α = 2ω1ω 2ξ ; β = 2ξ 2 2 2 2 (5) ω1 ω2 ω1 ω2 ξ is the damping ratio for each layer j of the soil profile (figure 2-a). ω 1 and ω 2 are, respectively, the first and second circular frequencies which are computed from the equation of motion obtained from equilibrium of soil element in free vibrations (figure 2-b) 2 ( y,t) u ρ i = τ 2 t y ( ( y,t) ) xy (6) Where τ xy is the shear stress in a soil element ( y,t) τ xy ( y,t) = G( y) (7) y After introduction of the shear modulus variation and the boundary conditions as equations (8) and (9), respectively G B y 1 1 2 b (8) h 2 3 5 ( y) = G. ; B = 0,,,, 1 u ( 0,t) = 0 ; G( y) ( y,t) u x y= h = 0 (9)

Vol. 16 [2011], Bund. M 1571 where G b is the shear modulus at the base of the soil profile, the equation (6) is reduced to the resolution of the Bessel equation. Natural frequencies are then obtained from the solutions of the Bessel function of first order [8]. (a) (b) (c) Figure 2: Schematization of (a) soil profile, (b) forces acting on a soil element and (c) discrete model The element stiffness matrix is: G j G j h j h j K = j (10) G j G j h j h j [ ] So, after introduction of damping matrix, the equation (3) becomes [ M ]{ u& } [ C]{ u& } + [ K]{ u} = [ M ]{ I}u& + (11) Equation (11) is solved by using Newmark s algorithm by mean of the iterative Newton s method.

Vol. 16 [2011], Bund. M 1572 NUMERICAL RESULTS Parametric study The seismic ground motion records show that the induced accelerations, and thus stresses and strains are cyclic and fast. In order to highlight this phenomenon, we study the behavior of a homogeneous soil layer with mechanical parameters shown in Table 1. We chose to study the behavior of different geomaterials: soft clay, limestone, and other geomaterial. We expose in this application, a parametric study on the influence of the nature of geomaterial on the behavior curve which describes shear stresses according to shear strains. For the case study, the soil profile is modeled by a soil column (Figure 3), with height equal to 10m (thickness of the soil profile). This mono layer system is excited at its base rock by the accelerogram which was recorded at the Keddara s rock station during the May 21, 2003 Boumerdes earthquake (Figure 4). Table 1: Mono layer soil profile characteristics Type of geomaterial Mass density ρ (kg/m 3 ) Shear modulus (MPa) Poisson s ratioν Yield elastic strain (%) Soft clay 2000 0.5 0.3 10-5 / 10-7 Limestone 2000 2200 0.3 10-5 / 10-7 Intermediate geomaterial 2000 2.0 0.3 10-5 / 10-7 Figure 5 schematizes the elastoplastic behavior of soft clay material according to the Keddara s time history excitation, while the figure 6 schematizes the visco-elasto-plastic behavior of the same geomaterial. The stress stage which corresponds to a permanent strain in the figure 5 expresses the strong nonlinear behavior of soft clay material. It is noted that the soft clay behavior is linear (for very small deformations), then there' is a fall of the shear modulus until the failure for the great shear strains. The hysteretic behavior of the soft clay is shown in Figure 6 by the hysteretic loop. The appearance of this one depends on the yield elastic strain. The Visco-elasto-plastic stress-strain behavior of limestone material is shown in Figure 7. It is clear that limestone shear modulus keeps its value along the excitation time which means linear elastic behavior. The intermediate geomaterial between soft clay and limestone presents an elastoplastic behavior (figure 8).

Vol. 16 [2011], Bund. M 1573 Figure 3: One dimensional soil profile 300 200 Acceleration (cm/s 2 ) 100 0-100 -200-300 -400 0 5 10 15 20 25 30 Time (s) Figure 4: Keddara s acceleration time history ( record )

Vol. 16 [2011], Bund. M 1574 1.5E+03 1.0E+03 Stress (kg/m 2 ) 5.0E+02-5.0E+02-2.00 0.00 2.00 4.00 6.00 8.00 Srain (%) Figure 5: Elastoplastic curve of soft clay for γ elas = 10-5 1.5E+03 1.1E+03 Sress (kg/m²) 7.0E+02 3.0E+02-1.0E+02-5.0E+02-0.02 0.02 0.06 0.10 0.14 0.18 Figure 6: Visco-elasto-plastic curve of soft clay for γ elas = 10-7

Vol. 16 [2011], Bund. M 1575 3.0E+05 2.0E+05 1.0E+05-1.0E+05-2.0E+05-3.0E+05-0.03-0.02-0.01 0.00 0.01 0.02 0.03 Figure 7: Visco-elasto-plastic curve of limestone material 1.5E+05 1.0E+05 5.0E+04-5.0E+04-1.0E+05-1.5E+05-0.40-0.20 0.00 0.20 0.40 Figure 8: Visco-elasto-plastic curve of intermediate geomaterial

Vol. 16 [2011], Bund. M 1576 Stress-strain behaviors of Keddara s dam geomaterials In this application we study the behaviors of geomaterials which constitute the keddara s dam in Boudouaou city at east of Algiers. The characteristics of each geomaterial are presented in table 2 [9]. Each stress-strain curve in figure 9 is obtained by computing the response of a soil layer of 10m thickness excited at its base by the Keddara s rock station record. 3.0E+03 Limestone 3.0E+03 Shist 1.5E+03 1.5E+03 Stress (kg/m 2 ) -1.5E+03 Stress (kg/m 2 ) -1.5E+03-3.0E+03-4.0E-06 4.0E-06-3.0E+03-2.0E-06 2.0E-06 4.0E+03 Clay 6.0E+03 Sand 2.0E+03 3.0E+03 Stress (kg/m 2 ) -2.0E+03 Stress (kg/m 2 ) -3.0E+03-4.0E+03-6.0E+03-1.5E-04 1.5E-04-2.0E-04 2.0E-04 Figure 9: Visco-elasto-plastic stress-strain response of Keddara s dam geomaterials

Vol. 16 [2011], Bund. M 1577 Table 2: Characteristics of Keddara s dam geomaterials Type of geomaterial Mass density ρ (kg/m 3 ) Shear modulus (MPa) Limestone 2700 9.6 Schist 2460 15.4 Clay 1950 0.5 Sand 2040 0.9 All curves in figure 9 are obtained for yield elastic strain γ elas = 10-5 %. This figure shows that the stress-strain behavior of limestone and schist are elastic linear. But clay geomaterial presents relatively non linear hysteretic behavior. Whereas the stress-strain behavior of sand is rather viscoelastic linear. Hysteretic site response In this application we study the hysteretic shear stress-strain response of a site located in the Metidja Basin (in Algeria), close to the Algiers capital to base excitation corresponding to Keddara s rock station record. Characteristics of the soil profile of the studied site (Bab-Ezzouar site) are shown in table 3 [10].The stress-strain responses at the studied site are computed at different locations in the representative soil profile in figure 10 and are shown in figures 11 to 15. Table 3: Mechanical parameter of the soil profile of Bab-Ezzouar site Layer number Thickness layer (m) Soil layer Nature Mass density (kg/m 3 ) Shear modulus (MPa) Damping (%) Yield stress σ elas (kg/m 2 ) 1 3.00 Filling 2000 180 13 13589 2 3.00 3 7.80 4 2.50 Gravely brown clay Marley compact beige clay Reddish clayey silt 2170 269 13 6123 2100 612 13 2691 2115 1360 13 1800 Base rock - Sandstone - - - -

Vol. 16 [2011], Bund. M 1578 (a) (b) Figure 10: Schematization of (a) soil profile of Bab-ezzouar site and (b) Lamped- mass model 2.5E+05 1.0E+06 1.5E+05 6.0E+05 5.0E+04-5.0E+04 2.0E+05-2.0E+05-1.5E+05-6.0E+05-2.5E+05-0.02-0.01 0.00 0.01 0.02-1.0E+06-0.50-0.25 0.00 0.25 0.50 Figure 11: Stress-strain response at point A (z = -15.06m) for (a) γ elas = 10-5, (b) γ elas = 10-7

Vol. 16 [2011], Bund. M 1579 1.5E+05 1.0E+05 5.0E+04-5.0E+04-1.0E+05-1.5E+05-0.03-0.02 0.00 0.02 0.03 2.5E+05 1.3E+05-1.3E+05-2.5E+05-0.05-0.03 0.00 0.03 0.05 Figure 12: Stress-strain response at point B (z = -09.90m) for (a) γ elas = 10-5, (b) γ elas = 10-7 8.0E+04 8.0E+04 4.0E+04-4.0E+04 4.0E+04-4.0E+04-8.0E+04-0.04-0.02 0.00 0.02 0.04-8.0E+04-0.05-0.03 0.00 0.03 0.05 Figure 13: Stress-strain response at point C (z = -04.50m) for (a) γ elas = 10-5, (b) γ elas = 10-7

Vol. 16 [2011], Bund. M 1580 1.2E+05 3.5E+04 6.0E+04 1.8E+04-6.0E+04-1.8E+04-1.2E+05-1.00-0.50 0.00 0.50 1.00-3.5E+04-0.02-0.01 0.00 0.01 0.02 Figure 14: Stress-strain response at point D (z = -01.50m) for (a) γ elas = 10-5, (b) γ elas = 10-7 2.5E+03 4.0E+03 1.3E+03 2.0E+03-1.3E+03-2.0E+03-2.5E+03-0.002-0.001 0.000 0.001 0.002-4.0E+03-0.003-0.002 0.000 0.002 0.003 Figure 15: Stress-strain response at free surface (z = 00.00) for (a) γ elas = 10-5, (b) γ elas = 10-7 Figures 11 to 15 show that stresses and strains increase with depth. Stress-strain curves are more visible for yield elastic strain equal to 10-7 than for 10-5 %, for the same geomaterial. Reddish clayey silt stress-strain curve presents more non linear behavior than for gravelly clay and compact clay ones.

Vol. 16 [2011], Bund. M 1581 CONCLUSION We have firstly, presented in this work a Visco-elasto-plastic formulation of the shear hysteretic behavior of different geomaterials. Then, a parametric study is carried in order to investigate the influence of different types of geomaterials on the earthquake stress-strain response. Lastly, we computed and analyzed the earthquake hysteretic stress-strain responses of a site located in the basin of Metidja in Algeria. From the obtained results, we have observed that the limestone and schist behaviors are linear elastic; the initial values of shear modulus are preserved during the loading. But the soft clay behavior is strongly non linear, because this last one loses its rigidity (shear modulus degradation) until the flow. Geomaterials located between the clay and limestone range, show an elastoplastic behavior. The sand behavior is linear viscoelastic. The Visco-elasto-plastic model describes adequately the nonlinear stress-strain relation during an earthquake loading. The shear modulus degradation is well presented by the onedimensional model because, for a certain range of deformations, this last remains constant (elastic behavior) but as soon as the deformation becomes important, the elastic prediction is not correct any more. REFERENCES 1. S. L. Kramer and J.P. Stewart (2004) Geotechnical aspects of seismic hazards; In Earthquake engineering from engineering seismology to performance based engineering, CRC Press. 2. P. B. Schnabel, J. Lysmer and H. B. Seed (1972) Shake: a computer program for earthquake response analysis of horizontally layered sites, Californian University, Rep. EERC 72-12, Berkeley. 3. K. Ishihara (2003) Soil behaviour in earthquake geotechnics, Clarendon Press, Oxford, New York. 4. F. Dunne and N. Petrinic (2006) Introduction to computational plasticity, Oxford University Press. 5. A. Chehat (2011) Modélisation du comportement non linéaire des géomatériaux sous sollicitations sismiques, Master Thesis, University of Chlef, Algeria. 6. D. Wulfsohn and B. A. Adams (2002) Elastoplastic soil mechanics; In Advances in Soil Dynamics, St. Joseph, Mich.: ASAE, 2, 1-116. 7. P. K. Woodward and D. V. Griffith (1996) Influence of viscous damping in the dynamic analysis of an earth dam using simple constitutive models, Computers and Geotechnics, 19(3), 245-263. 8. M. Krasnov, A. Kissélev and G. Makarenko (1978) Recueil de problemes sur les equations differentielles ordinaires, Mir Ed., Mosscou.

Vol. 16 [2011], Bund. M 1582 9. S. Louadj, R. Bahar, E. Vincens and N. Laouami (2008) Analysis of the seismic behavior of Keddara Dam using strong motion records, The Word Conference on Earthquake Engineering, Beijing China, 8. 10. R. Bahar (2006) Rapport Geotechnique: CMA CGM Algérie, D-Geot 20/05, Département Géotechnique, Société de Construction et d Engineering, Algérie. 2011 ejge