Relation between the REM and the slice-based approach

This section demonstrates that the present formulation based on the REM can be easily reduced to the formulations of other upper bound limit analysis approaches proposed by Michalowski (1995) and Donald and Chen (1997), respectively, where slice techniques and translational failure mechanics are used.

We herein purposely divide the failing mass of the soil into rigid elements in the same way as the case of inclined slices (or 2D wedges) considered in the upper bound limit analysis approach by Donald and Chen (1997). As shown in Figure 2.13, the rigid elements below the assumed failure surface ABCDE are fixed with zero velocities and thus called base elements. The index k denotes the element number, φ_kis the internal friction angle on the base inter-face (the interinter-face between element k and the base element below) and

φ__k the internal friction angle at the left interface (the interface between elements k and k – 1) of the kth element, respectively. α_kis the angle of inclination of the kth element base from the horizontal direction (anti-clockwise positive) and β_k is the inclination angle of the kth element’s left interface from the vertical direc-tion (anti-clockwise positive). Suppose the kth element has a velocity V_k (mag-nitude denoted as V_k, with v_xkand v_yk in x and y directions, respectively) in the global coordinate system. Note here that, due to the assumption of a transla-tional collapse mechanism, the rotation velocity of the kth element equals zero.

As shown in Figure 2.13(b), the direction cosine matrix of the base inter-face of the kth element with respect to the base element can be written as

(2.58) The relative velocity of the base interface, Vk′, can be expressed as

(2.59) As shown in Figure 2.13(b), the element k has the tendency to move leftward with respect to the base element. According to the Mohr–Coulomb failure criterion (or yield criterion for perfect plasticity material) and the associated flow rule, the relationship between the normal velocity magnitude (Δv_n) and tangential velocity magnitude (Δv_s) jumps across the discontinuity and can be written as

E

k

k

y

x

D

(2)

(1) (2)

(1) V_k

ΔVk

ΔVnk

ΔVk ΔVsk φk

φk

φk αk

αk−1

αk n1

n₁

s₁ βk s1

C

B

base element

base element (a)

(b) (c)

A

k−1

k−1 k

Figure 2.13 Failure mechanism similar to traditional slice techniques.

Using eq. (2.60), we have

(2.61) Thus, the following relationships can be obtained:

(2.62) vxk= Vkcosðαk− φkÞ

vyk= Vksinðαk− φkÞ vnk

vsk = tan φk

Similarly, we can get

(2.63) From Figure 2.13(c), the direction cosine matrix of the left interface of the kth element with respect to the (k – 1)th element can be written as

(2.64) Similarly, the relative velocity of the left interface of the kth element, ΔVk, can be given in the form

(2.65) From eq. (2.60), we can get

(2.66) where the case with a negative sign in the above equation coincides with the case where the (k – 1)th element has a tendency to move upward with respect to the kth element shown in Figure 2.13(c) with the dashed lines. It is noted that this case is identical to Case 1 defined in the method proposed by Donald and Chen (1997), and similarly the case with the positive mark in the above equation corresponds to Case 2 as discussed in Donald and Chen’s method.

Putting eqs (2.62), (2.63) and (2.65) into eq. (2.66), we can get the following relationship:

(2.67) With above eq. (2.67), and according to eq. (2.62), we can express v_xkand v_ykin terms of V_{k − 1}

(2.68)

Together with eq. (2.63), we put eq. (2.68) into (2.65) and then we have:

(2.69) ΔVk= Vk− 1sinðα_k− φk− αk− 1+ φk− 1Þ

cos½ðαk− φkÞ− ðβk φkÞ

vxk= Vk− 1cos½ðαk− 1− φk− 1Þ− ðβk φkÞ

cos½ðαk− φkÞ− ðβk φ_kÞ cosðαk− φkÞ vyk= Vk− 1cos½ðα_{k− 1}− φk− 1Þ− ðβk φkÞ

cos½ðαk− φkÞ− ðβk φkÞ sinðαk− φkÞ Vk= Vk− 1cos½ðαk− 1− φk− 1Þ− ðβk φ_kÞ

cos½ðαk− φkÞ− ðβk φ_kÞ

Δvnk

Δvsk = ± tan φk

ΔVk= Δvnk

Δvsk

 

= cos β_kðvxk− vx, k− 1Þ + sin βkðvyk− vy, k− 1Þ

− sin βkðvxk− vx, k− 1Þ+ cos βkðvyk− vy, k− 1Þ

 L^ð1Þ= −cos β_k − sin βk

sin β_k − cos βk

 

v_{x, k− 1}= Vk− 1cosðα_{k− 1}− φk− 1Þ vy, k− 1= Vk− 1sinðαk− 1− φk− 1Þ

In the method proposed by Donald and Chen (1997), the velocities of 2D wedges can be determined by a hodograph:

(2.70)

φ__k defined in the present method, respectively. It should be noted that δ in their formulations equal to −β_k in the present formulation, since the direction definition of δ (clock-wise positive) is opposite to that of β_k used in the present method (anti-clockwise positive).

Substituting V_k–1, V_k, ΔV_k, α_k–1, φ_k–1, α_k, φ_k, _

φ_k and β_k into eq. (2.70), and keeping the consistency between corresponding cases in the two approaches, eq. (2.70) arrives at exactly the same form of eqs (2.67) and (2.69) in the proposed method.

In the method proposed by Michalowski (1995), vertical slices were employed. For vertical slices, β_k equals to zero, and eqs (2.67) and (2.69) can be reduced to the following two equations.

(2.73)

(2.74) It is noted that the above equations correspond to the case where the (k – 1)th element moves downward with respect to the kth element, that is, ΔV_nk/ΔVsk= tan _

φ_k. In such a case, the velocity relationships in the present method are identi-cal to those under the translational failure mechanism in the method proposed by Michalowski (1995).

It has been proved above that the present formulations in the REM reduce to exactly the same formulations of the methods proposed by

ΔVk= Vk

Donald and Chen (1997) and Michalowski (1995) if the same slices with the same translational failure mechanism are used. In other words, the upper bound limit analyses using slices (or 2D wedges) may be viewed as a special and simple case of the formulation of the present method.

As shown in Figure 2.14, Kim et al. (1999) have studied the slope in nine cases with different depth factors D and slope inclinations β. In this study, we only take one case to investigate the feasibility of the present method, for example, consider the slope with depth factor D=2, H=10 m and β =45°, and with soil properties γ =18 kNm⁻³, c′ =20 kNm⁻³and φ′ =15°. To assess the effects of pore water pressure, two locations of a water table with H_w= 4 and 6 m are considered in this study. Figure 2.15 shows three rigid finite element meshes (coarse, medium and fine meshes) used in the analysis, for the case of a water table H_w = 6 m. The relations between the number of rigid elements used in the mesh and calculated factor of safety, for the case of a water table H_w=4 m and H_w=6 m, are shown in Table 2.8.