1. 2013 INTERNATIONAL VAN EARTHQUAKE SYMPOSIUM
In the Name of God
2013 INTERNATIONAL VAN EARTHQUAKE SYMPOSIUM
Dynamic Active Earth Pressure on Retaining Walls Using Limit Analysis theorem and Nonlinear Yield Criterion
Tohid, Akhlaghi
Associate Professor, Faculty of Civil Engineering, University of Tabriz, Tabriz, Iran.
Peyman, Hamidi
PhD student, Faculty of Civil Engineering, University of Tabriz, Tabriz, Iran .
Hamed, Norouzi
M.Sc in geotechnical engineering, manager of technical office and central laboratory, Khak & Rah Azma consulting Engineers, Urmia, Iran .
Hamid, Hashemi
M.Sc in earthquake engineering, manager of technical office, Khak & Rah Azma consulting Engineers, Tehran, Iran. 1
October 2013

2. Contents 2 Introduction Non-Linear Mohr–Coulomb Yield Criterion
Upper Bound Analysis for Dynamic Active Earth Pressure
Numerical Results and Comparisons
Conclusion
References 2

3. INTRODUCTION
The determination of lateral earth pressure of a fill on a retaining wall when frictional forces act on the back of the wall is one of the classical stability problems in soil mechanics. The main methods in the literature for earth pressure problems can be mainly classified into the following four categories:
limit equilibrium method
slip line method
limit analysis method
finite element or finite difference numerical techniques. 3

4. INTRODUCTION
Among them the limit analysis method is a powerful mathematical method that provides a rigorous upper or a lower bound to the exact collapse load.
According to the upper bound limit analysis, the loads, determined by equating the external rate of work to internal rate of dissipation in an assumed deformation mode (or velocity field) that satisfies: (1) velocity boundary conditions; and (2) strain and velocity compatibility conditions, are not less than the actual collapse load. The dissipation of energy in plastic flow associated with such a field can be computed from the idealized stress/strain rate relation (or the so-called flow rule). A velocity field satisfying the above conditions has been termed a kinematically admissible velocity field. 4

5. Figure 1: Load-displacement relationship for retaining walls.
INTRODUCTION
Fig. 1 shows a load displacement curve depicting the behavior of the soil under active and passive earth pressure. The points marked P0, Ppn and Pan represent the wall force at rest, passive collapse and active collapse, respectively. The subscripts p, a and n indicate passive, active and normal components of the force P, respectively.
Figure 1: Load-displacement relationship for retaining walls. 5

6. INTRODUCTION
It is seen in Fig. 1 that the loads for which the wall will not continually displace lie below the passive earth pressure and above the active earth pressure. Likewise, the upper bounds lie outside the region, above the passive earth pressure and below the active earth pressure. 6

7. INTRODUCTION
Experiments have shown that the strength envelopes of soils have the nature of nonlinearity and friction angle in most soils decreases with increasing confining pressures with the Mohr’s envelope curved. Therefore, this paper presents a limit analysis method with a nonlinear yield criterion in the framework of upper bound theorem of plasticity for calculating seismic active earth pressures.
To see the validity of the proposed method, numerical results using the extended Rankine’s earth pressure theory and the proposed method are presented and compared. 7

8. NON-LINEAR MOHR–COULOMB YIELD CRITERION
In general, a nonlinear yield criterion for the nonlinear failure envelope can be expressed as:
(1)
where σn is the normal stress and τ is the shear stress on the failure surface, and values of the three parameters c0, σt and m can be determined from test results. 8

9. NON-LINEAR MOHR–COULOMB YIELD CRITERION
As shown in Fig. 2, c0 is the initial cohesion, σt is the axial tensile stress at failure and m is a parameter that controls the curvature of the nonlinear envelope. It is noted that, when m = 1, Eq. (1) becomes well known linear Mohr-Coulomb yield criterion.
Figure2: Non-linear yield criterion in σn-τ space
If a stress level reaches the envelope in the (σn-τ) space, plastic flow or failure will occur. 9

10. NON-LINEAR MOHR–COULOMB YIELD CRITERION
A mobilized internal friction angle φt as an intermediate variable is introduced as:
(2)
Using Eq. (1), the normal stress σn and shear stress τ of the nonlinear yield criterion can be expressed as:
(3)
(4) 10

11. NON-LINEAR MOHR–COULOMB YIELD CRITERION
As illustrated in Fig. 2, the tangential line to the curve at the location of tangency point M, can be expressed by the following Eq.:
(5)
where ct is the intercept of the tangential line on the τ-axis. According to the laboratory experiments and research, ct can be determined by the following expression:
(6) 11

12. UPPER BOUND ANALYSIS FOR DYNAMIC ACTIVE EARTH PRESSURE
The rotational log-spiral failure mechanism for the present analysis is shown in Figure.
Figure 3: Rotational log-spiral failure mechanism for active earth pressure
In this mechanism the region ABC rotates as a rigid body about the center of rotation O with the material below the logarithmic-spiral surface remaining at rest. 12

13. UPPER BOUND ANALYSIS FOR DYNAMIC ACTIVE EARTH PRESSURE
Regarding the point that, the homogeneous soil masses will be rigid, the internal energy is only dissipated along the sliding surface.
The rate of work due to the soil mass weight Wsoil and horizontal inertia force khWsoil can be expressed as:
(7) 13

14. UPPER BOUND ANALYSIS FOR DYNAMIC ACTIVE EARTH PRESSURE
The rate of work due to the surcharge q and horizontal inertia force khq can be expressed as:
(8)
The rate of work due to the active earth pressure Pa and the adhesive force Pf can be expressed as:
(9) 14

15. UPPER BOUND ANALYSIS FOR DYNAMIC ACTIVE EARTH PRESSURE
It should be noted that the active earth pressure is assumed to act at the lower third of the wall length for the calculation of the work rate. For the rigid material considered in this paper, the internal energy is only dissipated along the sliding surface. The rate of energy dissipation can be expressed as:
(10)
The work rates of external forces are equal to the internal energy dissipation rate:
(11) 15

16. UPPER BOUND ANALYSIS FOR DYNAMIC ACTIVE EARTH PRESSURE
Substituting the Eqs into the equation 11 and rearranging them, we can obtain:
(12)
The above equation is obtained using the assumption that fracture surface follows the shape of a log-spiral. 16

17. NUMERICAL RESULTS AND COMPARISIONS
The most critical seismic earth pressure can be obtained by optimization for the walls with the sliding surface being a log-spiral surface. For the rotational fracture surface, upper bound solutions are obtained by maximizing Eq. (12) with respects to θh, θ0 and φt. Numerical results are summarized in Table 1.
Table 1. Comparisons of seismic active earth pressure from presented upper bound solution and extended Rankine method
Coefficient m
1.2
1.4
1.6
1.8
2.0
Upper bound solution presented in this paper (kN/m)
32.66
39.78
45.61
50.39
54.32
Extended Rankine method (kN/m)
35.28
41.41
46.48
50.66
54.12
kh=0.1, q=0, δ=0◦, Pf =0, β =90◦, γ=18 kN/m3, H=4 m, C0=9 kPa and σt=20.0 kPa 17

18. NUMERICAL RESULTS AND COMPARISIONS
To investigate the effects of nonlinear coefficient m on active earth pressures, an inclined and rough wall with a vertical surcharge is considered.
Firure4. Influence of nonlinear coefficient (m) and seismic coefficient (kh) on active earth pressure 18

19. NUMERICAL RESULTS AND COMPARISIONS
With the log-spiral fracture surface, the values of active earth pressures are shown in Fig. 5 with c0 varying from 3 kPa to 15 kPa.
Firure5. Influence of initial coefficient (c0) on active earth pressure 19

20. NUMERICAL RESULTS AND COMPARISIONS
The values of active earth pressures are shown in the Figure below with different values of q and for m = 1.5 and 2.0.
Firure6. Influence of surcharge (q) on active earth pressure 20

21. CONCLUSION
In this paper a simple method is presented to calculate seismic active earth pressures on retaining walls incorporating Mohr-Coulomb nonlinear yield criterion and upper bound limit analysis. Using the log-spiral rotational fracture surface, the effects of the nonlinear coefficient m, the seismic coefficient kh, the initial cohesion c0 and the vertical surcharge q have been investigated and discussed. It is found that all these parameters have significant influences on the active earth pressures. 21

22. CONCLUSION
It is found that the seismic active earth pressures calculated using the proposed method with a log-spiral rotational fracture surface is very close to solutions by the extended Rankine’s theory, with the absolute difference being 2.6 or less. 22

23. REFERENCES
- Chen, W.F., Limit Analysis and Soil Plasticity, Elsevier, Amsterdam.
- Lysmer, J., Limit analysis of plane problem in soil mechanics, Journal of Soil Mechanics Foundation Division, ASCE, pp
- Sloan, S.W., Upper bound limit analysis using finite elements and linear programming, International Journal for Numerical and Analytical Methods in Geomechanics, pp 263–282.
- Barker, R., 2004, Nonlinear Mohr envelops based on triaxial data, Journal of Geotechnical and Geoenvironmental Engineering, ASCE 130 (5), pp
- Hoek, E. and Brown, E.T., 1997, Practical estimate the rock mass strength, International Journal of Rock Mechanics and Mining Sciences, pp
- Drescher, A. and Christopoulos, C., 1988, Limit analysis slope stability with nonlinear yield condition, International Journal for Numerical and Analytical Methods in Geomechanics, pp
- Hanna, A. and Khoury, I., 2005, Passive earth pressure of overconsolidated cohesionless backfill, Journal of Geotechnical and Geoenvironmental Engineering, ASCE 131 (8), pp 23

24. Thank You