Plasticity Model Required to Prevent Geotechnical Failures in Tectonic Earthquakes

Abstract

Although geotechnical engineering design must meet seismic design specifications, many geotechnical failures, including foundations, retaining walls, and slopes, have nevertheless occurred during tectonic earthquakes. The evaluation of the ultimate bearing capacity of the foundation, the active and passive earth pressure of the retaining wall, and the safety factor of the slope all need to use the shear failure band and soil plasticity models at the same time. In view of this, it is first proven that shear failure bands can only appear in the strain softening model. Secondly, it is shown through case studies that traditional evaluation methods for determining the foundation ultimate bearing capacity, the active earth pressure of the retaining wall, and the safety factor of slope stability all adopt both the shear failure band and the perfectly plastic soil model. Since the perfectly plastic soil model is incompatible with shear failure bands, this results in a large number of foundations, retaining walls, and slopes failing during tectonic earthquakes. Based on the research results, it is suggested that a soil strain softening model compatible with shear failure bands be adopted in the analysis of geotechnical engineering projects so as to ensure safety during tectonic earthquakes.

1. Introduction

According to the generalized soil stress–strain diagram shown in Figure 1, after the strain in firm or dense soil exceeds the point of maximum stress, the stress decreases with increasing strain (known as strain softening) and finally approaches the residual stress.

Figure 1.

Results of a typical triaxial compression test for strain softening soil and perfectly plastic soil. (a) Stress-strain relationships for a confining pressure. (b) Shear strength-confining pressure relationships for three different confining pressures.

The deformation of various structures under the action of external loads has previously been determined under Drucker stability postulates [1, 2], i.e., $d\mathit{\sigma }:{\mathrm{d}\mathbf{\epsilon }}_{\mathrm{p}}\ge 0,$ where $d\mathit{\sigma }$ is the stress increment tensor and ${\mathrm{d}\mathbf{\epsilon }}_{\mathrm{p}}$ is the incremental plastic strain tensor. For this reason, limit analysis methods have been proposed for analysis of foundation ultimate bearing capacity, the active earth pressure of retaining walls, and slope stability. Based on the requirements of limit analysis, the strain softening behavior of soil has been ignored and been replaced with a perfectly plastic model. Figure 1a is the stress–strain relationships of soil obtained from a typical triaxial compression test. Figure 1a shows that when the strain is greater than the strain corresponding to the peak strength, the soil strength should decrease with the increase of the strain, so the phenomenon of strain softening occurs. However, since the traditional analyses of soil mechanics and foundation engineering are carried out under the conditions of Drucker stability postulates, the experimental result of strain softening is modified to be perfectly plastic.

When foundations, retaining walls, and slopes are designed using limit analysis and safety factors are used in accordance with seismic design codes, geotechnical failures such as those shown in Figures 2–4 should not be prevalent during tectonic earthquakes. Therefore, it is necessary to further explore the suitability of the perfectly plastic model as used in limit analysis.

Figure 2.

Tilting failure of a new building [3] after the Meinong earthquake in 2016 (Tainan, Taiwan).

Figure 3.

Failure of retaining walls under different conditions [4]: (a) during no rain and no earthquake (Formosa expressway 3.1 K, Keelung, Taiwan); (b) during the rainy season (Lincoln County in New Taipei City, Taiwan); (c) during an earthquake (Quanjiafu Community in Taichung, Taiwan).

Figure 4.

Shear failure of slopes under different conditions: (a) normal conditions [5] (Formosa expressway 3.1 K, Keelung, Taiwan); (b) during the rainy season (Tai-8 route 109 K, Taichung, Taiwan); (c) during an earthquake (Jiufen-Ershan, Nantou, Taiwan).

2. Numerical simulation analysis

2.1 The elastic: plastic model

The yield function adopted is given in Eq. 1 [6]:

$$F=\sqrt{J_{2D}}-\left(\kappa +H{\gamma }_{\mathit{oct}}^P\right)=0$$E1

where $J_{2D}$ is the second invariant of deviatoric stress; $\kappa$ is the size of the initial yield surface; ${\gamma }_{\mathit{oct}}^p$ is the plastic octahedral shear strain; and H is the strain-hardening (or strain-softening) parameter. Differentiating Eq. 1 leads to:

$$\mathit{dF}={\left(\frac{\partial F}{\partial \mathit{\sigma }}\right)}^{T}d\mathit{\sigma }+{\left(\frac{\partial F}{\partial {\gamma }_{\mathit{oct}}^p}\right)}^{T}d{\gamma }_{\mathit{oct}}^p={\left(\frac{\partial F}{\partial \mathit{\sigma }}\right)}^{T}d\mathit{\sigma }-\mathit{Hd}{\gamma }_{\mathit{oct}}^p=0$$E2

where $d\mathit{\sigma }$ is the incremental stress vector; the transpose of vector $\frac{\partial F}{\partial \mathit{\sigma }}$ is ${\left(\frac{\partial F}{\partial \mathit{\sigma }}\right)}^T$; the transpose of vector $\frac{\partial F}{\partial {\gamma }_{\mathit{oct}}^p}$ is ${\left(\frac{\partial F}{\partial {\gamma }_{\mathit{oct}}^p}\right)}^T;$ and $d{\gamma }_{\mathit{oct}}^p$ is the incremental plastic octahedral shear strain. The flow rule is:

$$d{\mathit{\epsilon }}^p=\lambda \frac{\partial F}{\partial \mathit{\sigma }}$$E3

where $d{\mathit{\epsilon }}^p$ is the incremental plastic strain vector. Therefore, the elastic–plastic stress–strain matrix ${\mathit{D}}^{\mathit{ep}}$can be expressed as follows:

$${\mathit{D}}^{\mathit{ep}}={\mathit{D}}^e-{\mathit{D}}^p={\mathit{D}}^e-{\mathit{D}}^e{\left(\frac{\partial F}{\partial \mathit{\sigma }}\right)}^T\frac{\partial F}{\partial \mathit{\sigma }}{\mathit{D}}^e{\left[\frac{1}{\sqrt{6}}H+{\left(\frac{\partial F}{\partial \mathit{\sigma }}\right)}^T{\mathit{D}}^e\frac{\partial F}{\partial \mathit{\sigma }}\right]}^{-1}$$E4

where super indexes e and p mean elastic and plastic, respectively.

2.2 Conditions of stability

For strain-softening materials, under the action of external force, when the strain goes deep into the plastic range, to obtain the stable incremental finite-element solutions, Prevost [7] used the variational approach based on the internal strain energy due to the stresses of the existing state and the stress increments, thereby obtaining the failure mechanism; then, define “A failure mechanism develops when the incremental external work put into the system, plus the incremental work reduced by the plastic, strain-softening zone, equals or exceeds the work that may be absorbed by the surrounding unyielded and/or strain-hardening material.” The total volume of the body, V, can be divided into $V_{\mathit{ps}}$ and V- $V_{\mathit{ps}}$, where $V_{\mathit{ps}}$ is the total volume of the strain-softening regions. Under this consideration, the uniqueness of the incremental solutions is proved.

For physical materials, if the total external incremental applied energy is positive, the total induced internal incremental strain energy must be positive. On the other hand, if the total external incremental applied energy is negative, the total induced internal incremental strain energy must be negative. All stable numerical solutions must obey such a law; otherwise, the numerical solution will be unstable.

If ${\pi }_p,$ is the total potential energy in static analyses under a variational approach, then the total incremental potential energy, $d{\pi }_p$ is:

$$d{\pi }_p=\frac{1}{2}{\int }_{V}d{\mathit{\sigma }}^{T}d\mathit{\epsilon }\mathit{dV}-\sum d{\mathit{u}}^{T}d\mathit{F}$$E5

where $d{\mathit{\sigma }}^T$ is the transpose of the incremental stress vector, $d\mathit{\epsilon }$ is the incremental strain vector, $d{\mathit{u}}^T$ is the transpose of the incremental displacement vector, and $d\mathit{F}$ is the incremental force vector.

In the finite-element approximation, Eq. 5 can be expressed as

$$d{\pi }_p=\frac{1}{2}{\int }_V\left(d{\mathit{u}}^T{\mathit{B}}^T{\mathit{D}}^{\mathit{ep}}\mathit{B}d\mathit{u}\right)\mathit{dV}-\sum \left(d{\mathit{u}}^{T}d\mathit{F}\right)$$E6

where ${\mathit{B}}^T$ is the transpose of the strain–displacement matrix. The global stiffness matrix $\mathit{K}$ is:

$$\mathit{K}={\int }_V\left({\mathit{B}}^T{\mathit{D}}^{\mathit{ep}}\mathit{B}\right)\mathit{dV}$$E7

Substitution of Eq. 7 into 6 leads to:

$$d{\pi }_p=\frac{1}{2}d{\mathit{u}}^T\mathit{K}d\mathit{u}-d{\mathit{u}}^{T}d\mathit{F}$$E8

It can be seen from Eq. 8 that, for the condition of the prescribed displacements and since all of the forces are induced, both terms on the right-hand side of Eq. 8 will always have the same sign. This means that the condition of stability for the incremental solutions for the prescribed displacement case is guaranteed. When the forces are prescribed and if the total external incremental energy is positive, the stability of the incremental solutions is guaranteed only when the global stiffness matrix is positive definite. When the global stiffness matrix is negative definite, the solution will be unstable [6].

2.3 Conditions of uniqueness

For a strain-hardening material or perfectly plastic material, since these types of materials meet the Drucker stability postulates, the solution is unique because the two potential energy increments on the right-hand side of Eq. 8 are both positive.

Based on the need to prove a unique solution, first let us assume two sets of solutions, $d{\mathit{u}}_1$ and $d{\mathit{u}}_2$, satisfy the equilibrium condition at the same time. Under this assumption, both $d{\mathit{u}}_1$ and $d{\mathit{u}}_2$ should provide the same minimum total incremental potential energy. By substituting $d{\mathit{u}}_1$ and $d{\mathit{u}}_2$ into Eq. 8 and minimizing the total incremental potential energy with respect to $d{\mathit{u}}_1$ and $d{\mathit{u}}_2$, one obtains:

$$\mathit{K}d{\mathit{u}}_1=d\mathit{F}$$E9

$$\mathit{K}d{\mathit{u}}_2=d\mathit{F}$$E10

Eq. 9 minus Eq. 10 equals:

$$\mathit{K}\left(d{\mathit{u}}_1-d{\mathit{u}}_2\right)=\mathbf{0}$$E11

Therefore, one of the following two conditions must be true:

$$\mathit{d}{\mathit{u}}_{\mathbf{1}}\mathit{=}\mathit{d}{\mathit{u}}_{\mathbf{2}}$$E12

(when the determinant of $\mathit{K}$ is not equal to zero) or

$$\text{det}\left(\mathit{K}\right)\mathit{=}\mathbf{0}(\text{when}\mathit{d}{\mathit{u}}_{\mathbf{1}}\mathit{\ne }\mathit{d}{\mathit{u}}_{\mathbf{2}})$$E13

2.4 Finite element solutions

For the 5.08 cm × 2.54 cm plate shown in Figure 5 under plane strain conditions loaded at both ends and where the movement in the direction perpendicular to the loading is constrained, a uniform 50 × 25 mesh was used to analyze its behavior under uniformly prescribed loading conditions. The material properties used were: (1) the initial size of the yield surface, $\kappa$, was equal to 24 kPa, (2) Young’s modulus, E, was equal to 1200 kPa, (3) Poisson’s ratio, $\nu$, was equal to 0.3, (4) the shear modulus, G, was equal to 462 kPa, and (5) the strain softening parameter, H/2G, was equal to 0.0 to model perfectly plastic behavior and equal to −0.05 to model strain softening behavior. To obtain shear bands in a numerical solution close to 45 degrees, it will be better to use square with the five-node elements. For the plate problem analyzed, the sequence for the application of the uniform prescribed displacements in the main load steps are 0.1143 cm, 0.0635 cm, 0.0635 cm, 0.0635 cm, 0.0635 cm, and 0.0635 cm, respectively. Each main load step is further subdivided into five sub-steps of load. Therefore, after applying all the load, the width of the plate remains unchanged at 2.54 cm, and the length is shortened to 4.6482 cm.

Figure 5.

Finite element mesh, boundary conditions, and prescribed lateral displacements.

For elastic–perfectly plastic materials, when the determinant of the structural stiffness matrix $\mathit{K}$ is not equal to zero, then $d{\mathit{u}}_1=d{\mathit{u}}_2$. In this case, the results of finite element analysis including the deformed mesh, velocity vector distribution map, and contour lines of incremental strain energy density (see Figure 6) do not include shear failure bands and maintain the original symmetry because Drucker stability postulates continue to be satisfied.

Figure 6.

Results of the finite-element analysis of a plate using an elastic–perfectly plastic model [6]: (a) deformed mesh; (b) velocity vector distribution map; (c) contours of incremental strain energy density. Note: Dimensions for the plates shown in Figure 6b and c are 5.08 cm × 2.54 cm.

For elastic–plastic strain-softening materials, the determinant of the structural stiffness matrix $\mathit{K}$ is equal to zero. In this case, external agencies must be applied as specified displacements instead of specified loads to obtain the results of the finite element analysis, including the deformed mesh, velocity vector distribution map, and contour lines of incremental strain energy density (see Figure 7), such that they do include asymmetric shear failure bands.

Figure 7.

Results of the finite-element analysis of a plate using an elastic–plastic strain softening model [6]: (a) deformed mesh; (b) velocity vector distribution map; (c) contours of incremental strain energy density. Note: Dimensions for the plates shown in Figure 7b and c are 5.08 cm × 2.54 cm.

2.5 Loss of symmetry

It can be seen from Figure 7 that the condition for the loss of symmetry of the foundation soil or the soil specimen is the appearance of shear failure bands induced by plastic strain softening.

As far as the structural stiffness matrix $\mathit{K}$ is concerned, its determinant will be equal to zero because of the plastic strain softening, which will induce shear failure bands under prescribed displacements. In the elastic range, the entries of the structural stiffness matrix $\mathit{K}$ have a symmetric condition, that is $k_{\mathit{ij}}=k_{\mathit{ji}}$, and in the plastic strain-softening range, the condition for the determinant of $\mathit{K}$to be equal to zero is that the entries in the two adjacent rows of the structural stiffness matrix $\mathit{K}$ correspond, that is $k_{\mathit{mi}}=k_{\mathit{nj}}$ for n = m+1. The structure matrix is therefore asymmetric [3].

3. The ultimate bearing capacity of the strip foundation

3.1 Equation derivation

Before deriving the ultimate bearing capacity of a strip foundation, Terzaghi [8] assumed symmetric general shear failure bands in the foundation soil under the ultimate load (see Figure 8). The required foundation soil properties include the cohesion c, angle of internal friction ϕ, and unit weight γ. The cohesion $c=c_{\mathit{ult}}$ and internal friction angle $\varphi ={\varphi }_{\mathit{ult}}$ were obtained from the perfectly plastic curve shown in Figure 1b. Figure 8 shows that Terzaghi [8] divided the whole area enclosed by the general shear failure bands into active zone I, radial shear zones II and II1, and passive zones III and III1. The ultimate bearing capacity $Q_{\mathit{ult}}$ of the foundation was equal to twice the passive earth pressures P_p that act on the shear failure bands $\overline{\mathrm{ad}}$ and $\overline{\mathrm{bd}}$. The passive earth pressure P_p included the passive earth pressure P_p1 generated by the soil cohesion c, the passive earth pressure P_p2 was generated by the overburden pressure $\gamma D_f$ (i.e., q), and the passive earth pressure P_p3 was generated by the weight of the soil enclosed by the shear failure band $\overline{\text{adef}}$. Based on the balance of the vertical components of all forces, Terzaghi obtained the ultimate bearing capacity of the foundation as:

$$q_{\mathit{ult}}=2\left(P_{p1}+P_{p2}+P_{p3}+\frac{1}{2}\mathit{Bc}tan\varphi \right)/B={\mathit{cN}}_c+q{N}_q+\frac{1}{2}{\mathit{B\gamma N}}_{\gamma }$$E14

where $N_c$, $N_q$, and $N_{\gamma }$ are bearing-capacity factors: $N_q=\frac{a^2}{2{cos}^2\left({45}^{°}+\varphi /2\right)}$; $N_c=\left(N_q-1\right)cot\varphi$; $N_{\gamma }=\frac{tan\varphi }{2}\left(\frac{K_{\mathit{p\gamma }}}{{cos}^2\varphi }-1\right)$; $a=exp\left[\left(0.75\pi -\frac{\varphi }{2}\right)tan\varphi \right]$; $K_{\mathit{p\gamma }}$ =10.8, 14.7, 25.0, 52.0, and 141.0 when ϕ is equal to $0^{°}$, ${10}^{°}$, ${20}^{°}$, ${30}^{°}$, and ${40}^{°}$, respectively.

Figure 8.

The general shear failure bands set before deriving the ultimate bearing capacity equation of the strip foundation (reproduced from Terzaghi [8]).

3.2 The overestimation problems

3.2.1 Problems arising from adopting symmetric general shear failure band

When plastic strain softening induces structural instability, the asymmetric structural matrix results in asymmetric general shear failure bands (detailed in Figure 9). Thus, the adoption of symmetric general failure bands by Terzaghi before deriving the ultimate bearing capacity equation of the strip foundation should be improved by the asymmetric shear failure pattern. In this case, when the ultimate bearing capacity of the strip foundation is evaluated by Eq. 14, the evaluation result will be overestimated twice.

Figure 9.

Schematic diagram of the asymmetric general shear failure band of a strip foundation (reproduced from [3]).

3.2.2 Problems arising from adopting perfectly plastic model

When the perfectly plastic model is adopted, the shear strength parameters used are obtained from the experimental results of the ultimate shear strength shown in Figure 1, but the asymmetric general shear failure band is induced by the plastic strain softening effect; therefore, the shear strength will decrease from the peak value to the residual value as the strain softens. In other words, the cohesion $c$ will be reduced from the peak value $c_p$ to the residual value $c_r$, $c_r$ ≈0, and the angle of internal friction ϕ will also be reduced from the peak value ${\varphi }_p$ to the residual value ${\varphi }_r$. Therefore, when calculating the ultimate bearing capacity $Q_{\mathit{ult}}$ using the ultimate shear strength parameter, there will also be an overestimation problem [3].

4. The active Coulomb’s earth pressure of retaining wall

4.1 The adopted perfectly plastic model

In Figure 1a, the red solid line represents the stress–strain curve of real soil obtained from laboratory triaxial compression tests. It shows the transition from peak strength to residual strength due to plastic strain softening. However, the plastic strain softening curve of soil has generally been simplified as a perfectly plastic curve in the retaining wall literature, as shown in Figure 1a.

4.2 Numerical simulation for the formation of shear failure bands

Coulomb [9] proposed the theory of active earth pressure as applied to retaining walls using both a shear failure band and a perfectly plastic model. At that time Coulomb did not know whether the use of the perfectly plastic model could induce shear failure bands.

For perfectly plastic and strain softening models, the numerical simulation analysis results using the finite element method are presented in Figures 6 and 7, respectively. The results show that in the perfectly plastic model, the plate does not develop a shear failure band when the strain of the lateral compression of the plate goes deep into the plastic range. Only in the plastic strain softening model does the plate show the induced shear failure bands.

Figure 7 shows that the formation of the shear failure bands has two requirements: strain deep into the plastic range and strain softening. The strain softening originates from the volume expansion of the sliding failure block. In other words, the negative incremental strain energy in strain softening is the motivating force for inducing the shear failure band.

4.3 Case study of when Coulomb’s active earth pressure of retaining walls is not a maximum

In Figure 10, ΔABC is the potential active sliding failure block for a retaining wall, as proposed by Coulomb [9]. Here, H is the height of the retaining wall, γ is the unit weight of the soil, W is the weight per unit length of ΔABC, β is the inclination angle of $\overline{\mathrm{AC}}$, θ is the inclination angle of $\overline{\mathrm{AB}}$, ρ is the inclination angle of the potential active shear failure band $\overline{\mathrm{BC}}$, ρ−β is the angle between $\overline{\mathrm{CA}}$ and $\overline{\mathrm{CB}}$, R is the resultant shear resisting force acting on $\overline{\mathrm{BC}}$, internal friction angle ϕ is the intersection angle of R and the normal to $\overline{\mathrm{BC}}$, $P_a$ is the active earth pressure acting on $\overline{\mathrm{AB}}$, and wall friction angle δ is the angle of intersection between $P_a$ and the normal to $\overline{\mathrm{AB}}$. From Figure 10, it can be seen that the Coulomb’s active earth pressure $P_a$ of retaining walls does not consider the effects of strain softening and shear banding.

Figure 10.

Various forces acting on a retaining wall as proposed by Coulomb [9]: (a) Coulomb’s potential sliding failure block under active conditions; (b) the closed force polygon of W, R, and $P_a$.

We used a retaining wall height of H = 6 m, soil unit weight $\gamma =22$$\mathrm{kN}/{\mathrm{m}}^3$, internal friction angle $={50}^{°}$, cohesion c = 0 kPa, wall friction angle $\delta$= ${33.3}^{°}$, $\overline{\mathrm{AC}}$ inclination angle $\beta$ = $0^{°}$, and $\overline{\mathrm{AB}}$ inclination angle $\theta$ = ${105}^{°}$. When adopting the elastic-perfectly plastic model, the result of the analysis shows that the sliding failure plane inclination angle was $\rho ={72.83}^{°}$, the failure block weight was W = 228.46 kN, and the traditional Coulomb active earth pressure was $P_a=$ 98.19 kN.

4.3.1 Increase in the Coulomb’s active earth pressure of retaining walls from the strain softening effect

Coulomb’s active earth pressure of retaining walls is the maximum lateral earth pressure. Figure 11a shows that during the 921 Jiji earthquake, the shear-banding zone on the left bank of the downstream Shigang Weir caused the sandy gravel layer on the back of the retaining wall to change from its original dense state to a loose state after shear banding. Figure 11b shows that after shear banding, the sand and gravel separated and the retaining wall slid into the riverbed.

Figure 11.

Collapse of the retaining wall induced by shear banding during the 921 Jiji earthquake [10]: (a) different sandy gravel conditions on both sides of the failure plan; (b) the separated sandy gravel and retaining wall slipped to the riverbed.

As the failure block continued to slide along the same shear failure band, it continued to undergo brittle fracture and strain softening during the sliding process. Further, the angle of internal friction ϕ decreased from a peak value of ${50}^{°}$ to a residual value of ${33}^{°}$, and the wall friction angle δ decreased from ${33.3}^{°}$ to ${22}^{°}$. When the inclination angle β of $\overline{\mathrm{AC}}$, the inclination angle θ of $\overline{\mathrm{AB}}$, the internal friction angle ϕ, the wall friction angle δ, and the angle $\alpha$ shown in Figure 10b are all known, the inclination angle ρ of the potential active shear failure band $\overline{\mathrm{BC}}$ can be calculated using Eq. 15. Then the Coulomb’s active earth pressure $P_a$ of the retaining wall shown in Table 1 can be calculated using Eq. 16 [10]:

${\varphi }_p\to {\varphi }_r$	$\delta$	$P_a\left(\mathit{kN}\right)$	$∆P_a\left(\mathit{kN}\right)$
${50}^{°}\to {33}^{°}$	${33.3}^{°}\to {22}^{°}$	$98.19\to 146.51$; $\left(100\%\to 149.2\%\right)$	48.32

Table 1.

Analysis results of the increase of the Coulomb’s active earth pressure of the retaining wall induced by soil strain softening [10].

$$\mathit{cot}\left(\theta -\rho \right)+\mathit{cot}\left(\rho -\beta \right)=\mathit{cot}\left(\rho -\varphi \right)-\mathit{cot}\left(\rho +\alpha -\varphi \right)$$E15

$$P_a=W\frac{\mathit{sin}\left(\rho -\varphi \right)}{\mathit{sin}\left(180^{\circ }-\rho -\alpha +\varphi \right)}=\frac{1}{2}\gamma H^2\frac{\mathit{sin}\left(\theta -\beta \right)}{{\mathit{sin}}^2\theta }·\frac{\mathit{sin}\left(\theta -\rho \right)}{\mathit{sin}\left(\rho -\beta \right)}·\frac{\mathit{sin}\left(\rho -\varphi \right)}{\mathit{sin}\left(\rho +\alpha -\varphi \right)}$$E16

According to Table 1, the Coulomb’s active earth pressure of the retaining wall increased from $98.19\mathrm{kN}$ to $146.51\mathrm{kN}$ due to soil strain softening; therefore, the increment of the Coulomb’s active earth pressure of the retaining wall, $∆P_a$, was $48.32\mathrm{kN}$.

4.3.2 Increase in the Coulomb’s active earth pressure of retaining walls from the shear band tilting effect

Without the constraint of vertical pressure on the ground surface, a tectonic plate induces a shear-band tilting effect under lateral compression. Figure 12a shows the shear-band tilting effect of the shear band on the back of the retaining wall. Figure 12b shows the shear-band tilting force $P_s$ caused by shear banding shown in Figure 12a, the resultant force R_s of the resistance on the shear failure band of the retaining wall, and the increment of the Coulomb’s active earth pressure of the retaining wall $P_{\mathit{as}}$ [10].

Figure 12.

The mechanism of the shear-band tilting effect inducing an increment in the Coulomb’s active earth pressure of the retaining wall [10]: (a) shear-band tilting effect; (b) the action positions of $P_s$, $R_s$, and $P_{\mathit{as}}$ on the retaining wall and the soil behind the wall; (c) the closed force polygon of P_s, R_s, and $P_{\mathit{as}}$.

According to the closed force polygon shown in Figure 12c and the sine law, Eq. 17 can be used to calculate $P_{\mathit{as}}$ [10]:

$$P_{\mathit{as}}=P_s\frac{\mathit{sin}\left(\lambda +\varphi -\rho \right)}{\mathit{sin}\left(\rho +\alpha -\varphi \right)}$$E17

Following the case study of the retaining wall discussed in Section 4.3.1, when the elastic-perfectly plastic model was adopted, the calculated inclination angle of the sliding failure plane was $\rho ={72.83}^{°}.$, the failure block weight was $W=228.46\mathrm{kN}$, and the Coulomb’s active earth pressure was $P_a=98.19\mathrm{kN}$.

When one side of the failure block at the back of the retaining wall was uplifted by the shear-band tilting force $P_s$, if $P_s=0.5W$, the increment in the Coulomb’s active earth pressure of the retaining wall $P_{\mathit{as}}$ can be calculated as $48.81\mathrm{kN}$. Thus, the increment rate of Coulomb’s active earth pressure caused by the shear-band tilting effect was 49.7% [10].

5. The shear-band slope stability analysis method

Considering the lateral compression of tectonic plates, Hsu [3] simulated the formation of shear failure bands, as shown in Figure 7. Figure 7a and b demonstrate that when the strain goes deep into plastic range, the localization of deformations of the tectonic plate will appear due to the loss of ellipticity caused by strain softening and further develop shear failure bands and shear-band tilting slopes.

Since the amount of shear banding accumulates from all previous tectonic earthquakes, the degree of brittle fracture in rocks of multi-step shear-band tilting slopes (Figure 13) will increase and large-scale shear failure of slopes will occur.

Figure 13.

Five-steps appeared in the Tsaoling shear failure planes during the Jiji earthquake [11, 12].

The Tsaoling area had three shear failures before the 1999 Jiji earthquake, in 1861, 1941, and 1979 [13]. Given that slope shear failure continued to occur at the same location, it can be deduced that the existing slope stability analysis methods did not consider the shear banding effect. Therefore, the Tsaoling landslide area had a total length of 2324.5 m and an average thickness of 24 m as shown in Figure 13. Although Hung et al. [14] obtained the shear strength parameters for the sliding plane, since the shear failure plane shown in Figure 13 is the interface between silty sandstone and shale, the shear strength parameters they obtained were low because only the horizontal ground vibration was considered. To improve this, a new shear-band slope stability analysis method is provided below.

5.1 The shear-band slope stability analysis method

Figure 14 shows a sliding block with two sliding surfaces, the lower sliding surface has a relatively gentle slope and the upper sliding surface has a relatively steep slope. Since the relatively steep sliding surface is caused by the uplifting action of shear banding, Figure 14 shows that the shear textures within an overall shear band include principal displacement shear D, thrust shear P, Riedel shear R, conjugate Riedel shear R’, and compression textures S. Thus, the hanging wall shown in Figure 14 will continue to be lifted during the shear banding of tectonic earthquakes, whereas the foot wall will remain stationary [12].

Figure 14.

Shear banding that causes the hanging wall to rise [12].

First, based on the consideration of the shear banding and ground vibration effects of tectonic earthquakes, a new slope stability analysis model as shown in Figure 15 is presented; in which a shear banding force, $P_s$, is acting on the top of the upper sliding surface and is perpendicular to it. The horizontal and vertical ground vibration forces acting on the upper sliding block are $k_h{W}_1$ and - $k_v{W}_1$, respectively, and the horizontal and vertical ground vibration forces acting on the lower sliding block are $k_h{W}_2$ and $-k_v{W}_2$, respectively.

Figure 15.

The proposed slope stability analysis model with dual shear failure planes [12].

As for the shear-band tilting force $P_s$, due to the shear banding effect, it is assumed that tension cracks exist at both ends of the right side of the upper sliding block and the left side of the lower sliding block, and the relatively steep upper sliding surface was caused by the uplifting effect of shear banding from previous tectonic earthquakes. Under these conditions, the shear-band tilting force $P_s$ will be approximately equal to half of the upper sliding block weight $W_1/2$ [12].

Next, since the groundwater table is lower than the sliding surface when the Tsaoling landslide occurs, it will not affect the slope stability analysis results. Therefore, as shown in Figure 13, the required parameters or material properties for the slope stability analysis of the Tsaoling landslide area include:

For the upper sliding block, the sliding surface length $L_1.$, inclination angle ${\alpha }_1$, weight W₁, normal component of W₁ on the upper sliding surface $N_1$, adhesion $c_{\alpha 1}$, friction angle ${\delta }_1$, the resultant force from the friction and adhesion resistances $R_1$, and shear banding force $P_s$.

For the lower sliding block, the sliding surface length $L_2.$, inclination angle ${\alpha }_2$, weight $W_2$, normal component of $W_2$ on the lower sliding surface $N_2$, adhesion $c_{\alpha 2}$, friction angle ${\delta }_2$, and the resultant from the friction and adhesion resistances $R_2$

As for the vertical interface between the upper and lower sliding blocks, under static equilibrium the active earth pressure of the upper sliding block $P_{a1}$ and its angle of intersection with the horizontal plane ${\varphi }_1$ need to equal the active earth pressure of the lower sliding block $P_{a2}$ and its angle of intersection with the horizontal plane ${\varphi }_2$, respectively; therefore, the factor of safety that satisfies the relationships of $P_{a1}=P_{a2}=P_a$ and ${\varphi }_1={\varphi }_2=\varphi$ is the factor of safety FS desired in the analysis.

For the forces acting on the upper and lower sliding blocks shown in Figure 16 under static equilibrium, the closed force polygons of all forces acting on the upper sliding block and lower sliding block are presented in Figure 16a and b, respectively.

Figure 16.

The closed force polygons for the upper and the lower sliding blocks [12].

For the upper sliding block, the closed force polygon shown in Figure 16a can be used as supplemented by the horizontal and vertical force balances (i.e., $\sum F_h=0$ and $\sum F_v=0$). The active earth pressure for the upper sliding block $P_{a1}$ is thus given in Eq. 18 as:

$$P_{a1}=A^{\ast }/\left[\mathit{cos}{\varphi }_m+\mathit{sin}{\varphi }_m\mathit{tan}\left({\alpha }_1-{\delta }_{1m}\right)\right]$$E18

where

$$A^{\ast }=\begin{array}{l}\left[W_1\left(1-k_v\right)-P_s\mathit{cos}{\alpha }_1-c_{\alpha 1m}{L}_1\mathit{sin}{\alpha }_1\right]\mathit{tan}\left({\alpha }_1-{\delta }_{1m}\right)+W_1{k}_h+P_s\mathit{sin}{\alpha }_1\\ -c_{\alpha 1m}{L}_1\mathit{cos}{\alpha }_1\end{array}$$E19

Next, for the lower sliding block, the closed force polygon shown in Figure 16b can be used as supplemented by the horizontal and vertical force balances of $\sum F_h=0$ and $\sum F_v=0$, respectively. The active earth pressure of the lower sliding block $P_{a2}$ is given by Eq. 20 as:

$$P_{a2}=B^{\ast }/\left[\mathit{cos}{\varphi }_m+\mathit{sin}{\varphi }_m\mathit{tan}\left({\alpha }_2-{\delta }_{2m}\right)\right]$$E20

where

$$B^{\ast }=-\left[W_2\left(1-k_v\right)-c_{\alpha 2m}{L}_2\mathit{sin}{\alpha }_2\right]\mathit{tan}\left({\alpha }_2-{\delta }_{2m}\right)-W_2{k}_h+c_{\alpha 2m}{L}_2\mathit{cos}{\alpha }_2$$E21

Since the force balance conditions for the upper and lower sliding blocks need to be met at the same time, a trial-and-error method can be used by first assuming a factor of safety ${\mathit{FS}}_a$. Subsequently, Eq. 18 can be used to calculate the active earth pressure acting on the upper sliding block $P_{a1}$, and then Eq. 20 can be used to calculate the active earth pressure acting on the lower sliding block $P_{a2}$. Finally, after the calculated value of $P_{a1}-P_{a2}$ becomes less than the error tolerance $\epsilon$, the slope stability factor of safety $\mathit{FS}$ is then set equal to ${\mathit{FS}}_a$. Based on this trial-and-error method [12], a calculation procedure was used to obtain the slope stability factor of safety $\mathit{FS}$.

5.2 Case study

The large-scale Tsaoling landslide induced by the 1999 Jiji earthquake was selected as a case study for the slope stability analysis. The back analyses of the slope stability were conducted under the following two conditions:

Condition 1: Considering the effects of shear banding and the horizontal and vertical ground vibration of the tectonic earthquake [12].

Condition 2: Only considering the effect of the horizontal ground vibration of the tectonic earthquake [14].

From Figure 13, points A to K can be roughly divided into five shear-band tilting slope steps: ABC, CDE, EFG, GHI, and IJK [12]. Each step in the shear-band tilting slope has a relatively gentle slope segment, AB, CD, EF, GH, and IJ, and a relatively steep slope segment, BC, DE, FG, HI, and JK. Using the sliding failure mechanism of multi-step shear-band tilting slopes, it was found that after sliding failure of the first step, each of the following steps would have a sliding failure in succession.

For the slope stability analysis of the Tsaoling landslide area, the coordinates, horizontal distance, elevation difference, and inclination angle of each line segment shown in Figure 13 were first recorded. Then, for the shear failure planes of the five non-shear banding zones AB, CD, EF, GH, and IJ shown in Figure 13, the unit weight for the corresponding sliding blocks was 24.52 $\mathrm{kN}/{\mathrm{m}}^3$, whereas for the five shear banding zones BC, DE, FG, HI, and JK it was 20.60 $\mathrm{kN}/{\mathrm{m}}^3$. From the distribution map of peak acceleration for the landslide area in Tsaoling, the peak ground acceleration (PGA) for the five step shear failure blocks in the Tsaoling landslide area can be obtained from PGA distribution map of the 921 Jiji earthquake reported by Earthquake Prediction Center, Central Weather Bureau, Taiwan [15]. The corresponding relationship between the PGA and seismic acceleration coefficients was provided by the Ministry of Economic Affairs [16]. The adopted horizontal seismic acceleration coefficient $k_h$ and vertical seismic acceleration coefficient $k_v$ corresponding to each PGA are given in Table 2.

Sliding block	PGA (g)	$k_h$	$k_v$
1st step	0.5	0.16	0.08
2nd step	0.8	0.24	0.12
3rd step	>0.8	0.24	0.12
4th step	>0.8	0.24	0.12
5th step	>0.8	0.24	0.12

Table 2.

The adopted $k_h$ and $k_v$ corresponding to each PGA for the five step sliding blocks [12].

In the slope stability analysis for the five stepped shear-band tilting slopes, the shear banding forces $P_s$, the horizontal ground vibration forces of the upper sliding block, the vertical ground vibration forces of the upper sliding block, the horizontal ground vibration forces of the lower sliding block, and the vertical ground vibration forces of the lower sliding block are given in Table 3.

Shear banding tilting slopes	Shear banding forces	Ground vibration forces
		Upper sliding block		Lower sliding block
	$P_s$ (kN)	$W_1{k}_h$ (kN)	$-W_1{k}_v$(kN)	$W_2{k}_h$(kN)	$-W_2{k}_v$(kN)
1st Step: ABC	44,485	14,235	−7118	11,204	−5601
2nd Step: CDE	51,652	24,793	−12,396	67,785	−33,892
3rd Step: EFG	30,151	14,472	−7236	37,706	−18,853
4th Step: GHI	69,198	33,215	−16,608	51,546	−25,773
5th Step: IJK	33,116	15,896	−7948	29,656	−14,828

Table 3.

The shear banding forces and ground vibration forces used in the case study for the shear-band tilting slopes [12].

The shear resistance strength parameters obtained through back-calculation under Condition 1 are given in Table 4.

	Upper sliding plane		Lower sliding plane 2
	Adhesion $c_{\alpha 1}$ (kPa)	Friction angle ${\delta }_1$ (degree)	Adhesion $c_{\alpha 2}$ (kPa)	Friction angle ${\delta }_2$ (degree)
1st Step: ABC	0	30.0	38.89	31.0
2nd Step: CDE	0	30.0	42.69	31.5
3rd Step: EFG	0	30.0	40.63	31.3
4th Step: GHI	0	32.0	123.36	36.0
5th Step: IJK	0	31.8	90.39	35.0

Table 4.

Shear resistance strength parameters obtained through back analyses under condition 1 [12].

The shear resistance strength parameters obtained through back-calculation under Condition 2 are given in Table 5.

	Upper sliding plane		Lower sliding plane 2
	Adhesion $c_{\alpha 1}$ (kPa)	Friction angle ${\delta }_1$ (degree)	Adhesion $c_{\alpha 2}$ (kPa)	Friction angle ${\delta }_2$ (degree)
1st Step: ABC	0	20.0	0	20.0
2nd Step: CDE	0	20.0	0	20.0
3rd Step: EFG	0	20.0	0	20.0
4th Step: GHI	0	20.0	0	20.0
5th Step: IJK	0	20.0	0	20.0

Table 5.

Shear resistance strength parameters obtained through back analyses under condition 2 [14].

The obtained safety factors using the shear resistance strength parameters of the sliding surface obtained from the back analysis under different conditions for the shear-band slope stability analysis are given in Table 6.

Shear banding tilting slope	FS
	The condition 1	The condition 2
1st Step: ABC	1.000 (100%)	0.120 (12.0%)
2nd Step: CDE	1.000 (100%)	0.134 (13.4%)
3rd Step: EFG	1.000 (100%)	0.136 (13.6%)
4th Step: GHI	1.000 (100%)	0.130 (13.0%)
5th Step: IJK	1.000 (100%)	0.131 (13.1%)

Table 6.

Shear-band slope stability analyses as obtained from the back-calculation results under conditions 1 and 2 [12].

For multi-step slope failure induced by the effects of shear banding and ground vibration of a tectonic earthquake, when the shear banding effect is ignored and only the horizontal ground vibration effect is considered, low shear strength parameters of the sliding plane are obtained by back analysis. When the shear-band slope stability analysis is carried out using the shear strength parameters obtained from the back analysis, only the back analysis results that consider the effects of the shear banding and the horizontal and vertical ground vibrations can obtain a factor of safety that meets the actual requirements.

For the Tsaoling landslide that occurred during the 921 Jiji earthquake, when the back analysis considered the effects of the shear banding and the horizontal and vertical ground vibration effects, Table 6 shows that the shear strength parameters obtained from the back analysis of the five-step shear-band slopes resulted in the calculated safety factors being consistent with the actual safety factor of 1.0; however, when the back analysis only considered the horizontal ground vibration effect, Table 6 shows that the shear strength parameters obtained from the back analysis of the five-step shear-band slopes resulted in the calculated safety factors being much lower than the actual safety factor of 1.0 [12].

6. Conclusions

1. The finite element simulation results of the plate under lateral compression show that only in the elastic–plastic strain softening model can the shear band be induced.

2. For the general shear failure plane proposed by Terzaghi, since the strain-softening model is replaced by the perfectly plastic model, the foundation soil will remain perfectly plastic when it should be strain softened, and then it will still maintain symmetry when it should be lost. Thus, the problem of overestimating the ultimate bearing capacity of foundations arises.

3. Both the strain softening effect and the tilting effect make the maximum lateral earth pressure of the retaining wall increase from the traditional Coulomb active earth pressure. The case study results show that the maximum lateral earth pressure of the retaining wall induced by strain softening can increase up to 49.2%, and the maximum lateral earth pressure of the retaining wall induced by the tilting effect can increase up to 99.4%. When the strain softening effect and the tilting effect are superimposed, the increase in the maximum lateral earth pressure of the retaining wall can be as high as 148.6%.

4. For tectonic earthquakes induced large-scale landslides, the case study results show that either through back- or forward-calculation, only when the shear banding effect (i.e., the primary effect) and the horizontal and vertical ground vibration effects (i.e., secondary effects) of tectonic earthquakes are considered, can the slope stability analysis results conform to the actual situation on site.

Acknowledgments

The previous work of this research is a part of the preliminary studies on the stability of existing earth structures as part of the Repair, Evaluation, Maintenance, and Rehabilitation Research Program (REMR). Financial support provided by the office of Chief of Engineers, U. S. Army is acknowledged with thanks. Close consultation with Dr. S. K. Saxena and Dr. J. F. Peters was of great benefit in determining the results of the computer analyses. The chance to use the computer facilities in the U. S. Army Engineer, Waterways Experiment Station to finish most of the computer work in this research is highly appreciated.