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Undrained stability of an active planar trapdoor in non-homogeneous clays with a linear increase of strength with depth
Computers and Geotechnics 81 (2017) 284–293
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Technical Communication
Undrained stability of an active planar trapdoor in non-homogeneous clays with a linear increase of strength with depth Suraparb Keawsawasvong, M.Eng., Ph.D. (Research Fellow), Boonchai Ukritchon ⇑, Sc.D. (Associate Professor) Geotechnical Research Unit, Department of Civil Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok 10330, Thailand
a r t i c l e
i n f o
Article history: Received 17 June 2016 Received in revised form 6 August 2016 Accepted 28 August 2016
Keywords: Limit analysis Numerical analysis Stability Trapdoor Active failure
a b s t r a c t Finite element limit analysis was employed to determine the upper and lower bound solutions of the active failure of a planar trapdoor in non-homogeneous clays that have a linear increase of strength with depth. Influences of cover ratio, dimensionless strength gradient and trapdoor roughness on predicted failure mechanisms and stability factors were determined. In all cases, the exact stability factors were accurately bracketed by computed bound solutions within 1%. Accurate closed-form equations to predict the exact estimates of stability factors, trapdoor pressure and factor of safety using the new proposed factors for the cohesion and strength gradient are presented. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Active and passive failures of a trapdoor are classical stability problems in geotechnical engineering, in addition to the bearing capacity of shallow foundations, retaining walls and slopes. The active and passive collapses of a trapdoor corresponds to a determination of the limit loads that act on the downward and upward movements of the trapdoor, respectively. The original study of stress distribution on a trapdoor in sand was experimentally investigated by Terzaghi [1]. In addition to being an important benchmark solution in theoretical geomechanics, the stability of a trapdoor is of considerable practical interest in many applications. The passive failure of a trapdoor corresponds to the pullout problem of a vertical plate anchor in soils, and has been studied widely [2,3]. On the other hand, the solution of the active trapdoor can be applied to analyze the collapse of an underground roof in tunnels and mining works [4], the gravitational flow of a granular material through hoppers [5,6], and the stability of buried pipes subjected to the loss of ground support [7]. A relatively large number of studies on the stability of an active trapdoor have been investigated previously and include physical model tests [1,5,8–13], numerical analysis by the limit equilibrium method (LEM) [14], finite element analysis [10,15] and limit analysis [9,16,17]. Some researchers have considered the planar ⇑ Corresponding author. E-mail addresses: [email protected] (S. Keawsawasvong), boonchai.uk@ gmail.com (B. Ukritchon). http://dx.doi.org/10.1016/j.compgeo.2016.08.027 0266-352X/Ó 2016 Elsevier Ltd. All rights reserved.
trapdoor problem in a purely and homogeneously cohesive soil [16–19]. However, no study on the undrained active failure of a planar trapdoor in non-homogenous clay with a linear increase of undrained shear strength has been reported, and so is the subject of this note. The problem definition of an active trapdoor under the plane strain condition is shown in Fig. 1. A depth of a non-homogeneous clay layer (H) rests on a rigid boundary and a rigid trapdoor of width (B). The clay behaves as a rigid-perfectly plastic Tresca material with the associated flow rule. The non-homogeneous clay has an undrained shear strength that increases linearly with the depth from the ground surface (su0) with a strength gradient (q). Despite the strength non-homogeneity, the clay is assumed to have a constant unit weight (c). The ground surface is fully loaded by a uniform surcharge (rs). To produce the active failure, the trapdoor is assumed to move downwards as a rigid body that is resisted by the externally applied uniform trapdoor pressure (rt). Such a failure happens due to the actions of the surcharge (rs) and the soil unit weight (c) and is resisted by the trapdoor pressure (rt) and the shear resistance of clay (su0, q). This study considers either purely rough or smooth cases, where the former corresponds to a perfectly rough surface at both the trapdoor and the bottom boundary, and the latter corresponds to a perfectly smooth surface of these. It is expected that the true boundary condition should lie somewhere between these two cases. Following previous studies of an undrained collapse of an active trapdoor in cohesive soils [18,19], the solution of this problem is
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285
Fig. 1. Active trapdoor under a plane strain condition.
represented by the dimensionless stability factor (N), as shown in Eq. (1);
N¼
rs þ cH rt su0
¼f
H qH ; ;a B su0
ð1Þ
where H/B is the cover ratio, qH/su0 is the dimensionless strength gradient, a is the adhesion factor at the interfaces of rigid trapdoor and bottom boundary = sua/su, and sua and su are the undrained shear strength at the interfaces and the surrounding soil, respectively. Note that the negative sign in front of the trapdoor pressure (rt) denotes that it is the resistance of the problem, while the surcharge (rs) and the soil unit weight (c) are the driving stresses that cause the active failure. The rough and smooth cases correspond to a = 1 and 0, respectively. With respect to previous studies on the active failure of a trapdoor in a homogeneous clay [16–19], the most accurate solutions
were derived by employing finite element limit analysis (FELA) to solve this problem [18,19]. Sloan et al. [18] presented a significant improvement on N bound solutions as compared to previous studies [16,17], especially for deep trapdoors and the influence of interface roughness on the N solutions that became significant when H/B > 5. Their bound solutions had an accuracy of 7–11% for all H/B ratios. Subsequently, Martin [19] proposed the stress and velocity fields of new slip-line solutions for a shallow trapdoor (H/B = 0.5–2) and validated the results using FELA with an adaptive mesh refinement and confirmed that the slip-line solutions were actually the exact ones only for H/B < 1.3. This paper extends these previous studies on planar trapdoors [18,19] in homogeneous clays to investigate the influence of the linear strength non-homogeneity of clay on the undrained bound N solutions of active failure, which has not been addressed before. The study employed the state-of-art FELA software, OptumG2 [20], to accurately determine the undrained active collapse of a trapdoor
Fig. 2. Numerical model of the undrained stability of an active planar trapdoor in OptumG2.
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Fig. 3. Comparison of the final adaptive meshes of an active planar trapdoor with different cover ratios (H/B) and rough (a = 1) or smooth (a = 0) surfaces and qH/su0 = 0.
under the plane strain condition. Three parametric studies were performed; namely (i) the cover ratio (H/B) from shallow to deep trapdoors over a range of 0.5–10, (ii) a dimensionless strength gradient (qH/su0) between 0 and 4 and (iii) an adhesion factor (a) of 1 or 0 (either rough or smooth interfaces at the trapdoor and bottom support, respectively). The latter case should cover the true boundary condition of the trapdoor in practice. The effects of these three parameters on the predicted failure mechanisms are discussed and compared. A closed-form equation is proposed for the exact estimate of the true solutions of the active failure of a trapdoor based on a nonlinear regression analysis. New trapdoor stability factors, with respect to cohesion and strength gradient, were deduced from the proposed equation. Expressions that utilize these factors are given for predicting the trapdoor pressure and factor of safety (FS) in a general strength profile that cover homogenous (su0 – 0, q = 0) and non-homogenous strengths (su0 – 0, q – 0). The latter includes the special case of a linear strength but zero strength at the ground surface (su0 = 0, q – 0). Thus, the proposed closedform expressions serve as a convenient and accurate tool to determine the solutions for an active trapdoor failure in practice.
2. Method of analysis At present, the emerging numerical technique of FELA has become a powerful approach for analyzing various complex stability problems in geotechnical engineering, as demonstrated by Sloan [21]. The technique combines the powerful capabilities of finite element discretization for handling complicated soil stratifications, loadings and boundary conditions with the plastic bound theorem to bracket the exact limit load by upper and lower bound solutions. The underlying bound theorems assume a rigid-perfectly plastic material with an associated flow rule. Details of limit
analysis and FELA are omitted here but can be found in Chen and Liu [22] and Sloan [21], respectively. The plane strain numerical model for simulating an active failure of a trapdoor in OptumG2 is shown in Fig. 2. A symmetrical condition at the centerline of the problem is utilized to prepare the model. As a result, only one half of the domain is considered in the analysis. The trapdoor is modelled as a rigid plate element, while the clay behaves as a Tresca material with an associated flow rule that is governed by su0 and q. The interface elements are defined at the trapdoor and along the rigid bottom boundary of the problem with the adhesion factor (a). To produce the solutions for either rough or smooth trapdoors, the adhesion factor is set as either unity or zero, respectively. Due to possible stress and velocity discontinuities at the corner of the trapdoor, a small segment of vertical line is employed at that point and defined as interface elements, where their adhesion factor is fully rough (i.e. a = 1). All interface elements allows velocity and stress discontinuities to happen in the upper bound and lower bound calculations, respectively. It should be noted that the small vertical line of interface is not sensitive to the accuracy of lower bound solutions since stress discontinuities are modelled at all inter-element edges. In contrast, this vertical interface is very crucial in order to produce accurate upper bound solutions since velocity discontinuities are not permitted to happen along all inter-element edges but they are expected to happen at the corner of trapdoor. Thus, this interface is used to introduce a line of velocity discontinuity at this corner. The left (centerline) and right boundaries of the model are fixed horizontally, while the bottom boundary is fixed in both directions. A uniform surcharge pressure (rs) is applied downwards at the ground surface of the model, while the uniform trapdoor pressure (rt) is applied upwards on the trapdoor. The left and right dimension of the domain is chosen to be large enough that the plastic zone is contained within the domain and does not intersect the
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Fig. 4. Comparison of the incremental displacement vectors of an active planar trapdoor with different cover ratios (H/B) and rough (a = 1) or smooth (a = 0) surfaces and qH/su0 = 0.
right boundary. Based on various numerical simulations, the optimal sizes of the domain along the horizontal direction corresponded to 2H, as shown in Fig. 2, while the size along the vertical direction is an input parameter of trapdoor depth (H). Thus, the computed upper and lower bound loads are unaffected by the selected domain size. The following summarizes the upper and lower bound FELA of the plane strain condition in OptumG2 for analyzing the active failure of a planar trapdoor in homogeneous and non-homogenous clays. In the upper bound analysis, the soil mass is discretized into six-noded triangular elements. The unknown velocity components at each node employ a quadratic interpolation within each element and are continuous between adjacent elements. Except for the interface elements, velocity discontinuities are not permitted to happen along all shared edges of adjacent elements, which is similar to the conventional displacement-based finite element method. A rigid plate element is used at the bottom to simulate the trapdoor. Velocity discontinuities are permitted to happen at the interface elements that are defined at the soil-trapdoor interface, the small vertical line extending from the trapdoor corner, and the rigid bottom support of the problem. These velocity discontinuities are implemented by collapsing patches of regular triangular elements to zero thickness [23,24]. The upper bound calculation is formulated as a second-order conic programming (SOCP) [25,26]. Constraints on the kinematically admissible velocity conditions include the velocity boundary conditions and the compatibility equations with the associated flow rule at triangular and interfaces elements. The objective function of the upper bound
SOCP is defined to minimize the unit weight of soil that is obtained through invoking the principle of virtual work by equating the rate of work done by external loads (c, rs and rt) with the internal energy dissipation at the triangular and interface elements. In the lower bound analysis, three-noded triangular elements were employed to discretize the soil mass. Unlike a conventional displacement-based finite element method, each element has its own unique node resulting in possible stress discontinuities to occur along all the shared edges of adjacent elements. Like the upper bound model, the interface elements are defined at the soil-trapdoor interface, the small vertical line extending from the trapdoor corner and the bottom support. The lower bound calculation was formulated as a SOCP [25,27], which satisfies the statically admissible stress conditions, including the equilibrium equations at each triangular element and along all stress discontinuities, stress boundary conditions, and no stress violation of the yield criterion in the soil mass. The objective function of the lower bound SOCP was defined to directly maximize the unit weight of the soil that has an inter-relationship with unknown nodal stresses through the vertical equilibrium equation. It should be noted that stress discontinuity is the key feature in lower bound calculations and used to improve the accuracy of the lower bound collapse load, as extensively discussed by Sloan [21]. Therefore, it is necessary to employ stress discontinuities defined by all inter-element edges in a lower bound mesh consisting of three-noded triangular elements. Six-noded triangular elements have never been implemented in any previous formulation of lower bound calculation since this employment does not produce
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Fig. 5. Comparison of the incremental displacement vectors of an active planar trapdoor with different strength gradients (qH/su0) and rough (a = 1) or smooth (a = 0) surfaces and H/B = 2.
a rigorous lower bound solution. Employing these elements results in unknown stresses to vary quadratically over the element, and hence it is not possible to ensure that the stress field satisfies the equilibrium equations everywhere within the element. Such elements can only be employed in upper bound calculations so as to obtain a rigorous upper bound solution (see [20,21,27]). Both the upper and lower bound SOCP in the plane problem were solved to obtain rigorous bound solutions using the general purpose optimization solver, SONIC [28], which is based on the interior-point method [29,30]. Plane strain FELA using SOCP in OptumG2 is comparable to that of previous formulations [26,27], and full details of the numerical FELA formulation in OptumG2 can be found in [20]. A powerful feature in OptumG2, the automatically adaptive mesh refinement, was also employed for both the upper and lower bound analyses to determine the tight upper and lower bound solutions. In this study, the shear dissipation control to the bounds gap in OptumG2 was based on the formulation proposed by Ciria et al. [31] and used for the adaptive mesh refinement. Five adaptive steps were used for all analyses following the suggested value in the software manual, such that this value is adequate enough to obtain an accurate solution. In the course of five adaptive
iterations, an initial mesh with 5000 elements was automatically adapted and increased to a final mesh with 10,000 elements. As mentioned by Sloan et al. [18], there are a variety of ways to determine the stability factor (N). For example, a fixed value of c and rt is setup and the surcharge is optimized. In this study, for both the rough and smooth trapdoors, a fixed value of rs and rt was setup and c, the body force of the problem, was optimized directly in both the SOCP upper and lower bound analyses to give the minimum and maximum values, respectively. Thus, the trapdoor stability factor (N) and the corresponding dimensionless input parameters, H/B, a (1 or 0) and qH/su0, were calculated according to Eq. (1). 3. Results and comparisons The comparisons of predicted failure mechanisms of the active trapdoor that are affected by different cover ratios (H/B) for a homogeneous strength profile (qH/su0 = 0) are shown in Figs. 3 and 4. The results of the failure mechanism include the final adaptive meshes and the incremental displacement vectors, respectively. These results clearly show the influence of the cover ratio on the predicted failure mechanism from shallow (H/B = 0.5) to
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Fig. 6. Comparison of the incremental shear strain contours of an active planar trapdoor with different strength gradients (qH/su0) and rough (a = 1) or smooth (a = 0) surfaces and H/B = 2.
deep (H/B = 10) trapdoors. The failure of a shallow trapdoor corresponds to a vertical slip mechanism, as first proposed by Davis [16] and later improved by Gunn [17] using a three-parameter rigid block mechanism. The failure width of this mechanism at the ground surface is localized and does not extend beyond the corner of the trapdoor. Note that the adaptive mesh for the case H/B = 0.5 of the present study is quite similar to that of Martin [19] whose results are omitted. Once the H/B ratio increases (H/B = 2), the vertical slip mode changes to a curvilinear one that is accompanied by the radial shear zone around the trapdoor corner and a rigid triangular zone that is located in front of the trapdoor. For intermediate and deep trapdoors (H/B = 6 and 10, respectively), there is more lateral spreading of the plastic zones in both the radial and curvilinear shear ones, resulting in a significant failure width at the ground surface. Such results are attributed from an influence of geometric nonlinearity of the cover ratio (H/B) that induces more mobilized shear stress along a longer curvilinear surface for a more resisting force to against failure and requires larger plastic zone for a more driving force (cH) to cause failure. The results of the smooth cases are also compared with those of the rough cases in these figures. For shallow trapdoors (H/B = 0.5 and 2), the differences in the failure mechanism between the smooth and rough cases are quite small, but the effect of a smooth interface on the failure mechanism becomes significant for intermediate and deep trapdoors (H/B = 6 and 10, respectively). The
smooth cases reveal a curvilinear shear zone that starts from the ground surface and ends at a certain position on the bottom smooth boundary instead of being at the corner of the trapdoor in the rough case. In addition, a relative sliding of soils along the bottom boundary is expected to occur only in the smooth cases. Such results are attributed from the influence of shear tractions along rough surfaces of trapdoor and bottom support that contribute some shear resistances to against failure. Since the failure of shallow trapdoor is caused by a vertical slip plane directly passing through its corner, an effect of these shear tractions on rough surfaces contributes to a lesser degree in shear resistance, as compared to deep trapdoors that are associated with curvilinear shear surface. For smooth cases, there is no shear traction along the bottom support, and hence sliding can take place along the bottom interface. On the other hand, the presence of shear traction along the bottom interface induces the curvilinear shear surface to pass at the corner of trapdoor instead of sliding along the bottom interface. The influence of qH/su0 on the failure mechanism of the rough cases for shallow (H/B = 2) and deep (H/B = 10) trapdoors are compared in Figs. 5 and 6 for qH/su0 values of 0, 1 and 4. The results of the failure mechanism include the incremental displacement vectors and the incremental shear strain contours, respectively. The effects of qH/su0 on the failure mechanism of smooth and rough trapdoors are quite similar. Increases in qH/su0 result in a
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(a)
(a) 35
Rough Cases
8
30
(σs+γ H-σt)/su0
(σs+γ H-σt)/su0
ρH/su = 4
25
6
4
=2 =1 = 0.75 = 0.5 = 0.25 =0
20 15 10
2
5 0
0 0
2
4
6
8
0
10
2
4
6
H/B
(b)
(b)
Smooth Cases
8
8
10
H/B 35 30 25
(σs+γ H-σt)/su0
(σs+γ H-σt)/su0
6
4
H/B = 10 =8 =6 =4 =2 =1 = 0.5 = 0.1
20 15 10
2
5 0
0 0
2
4
6
8
10
0
1
2
ρH/su0
H/B Fig. 7. Relationship between the stability factor (N) and cover ratio (H/B) for (a) rough and (b) smooth interfaces.
decreased size of the shear bands and curvilinear shear zones around the corner of the trapdoor for smooth and rough trapdoors. As a result, the width of the failure mechanism extending to the ground surface also decreases accordingly. It should be noted that the triangular rigid zone above the trapdoor becomes more curvilinear and shifted higher towards the ground surface as the qH/su0 factor increases. For small value of qH/su0 6 1, there is no difference in failure mechanism between smooth and rough trapdoors. However, for a large value of qH/su0 = 4, some difference can be observed, where the size of shear bands of smooth trapdoors depicted by incremental shear strain (see Fig. 6) is slightly larger than that of rough trapdoors. Such results are attributed from the influence of linear strength gradient (q) that generates more shear resistance in shear bands and plastic zone in addition to the effect of shear resistance due to homogeneous strength (su0). Therefore, an increase in qH/su0 gives rise to a higher shear resistance of soils, and thus resulting in a decreased size of shear bands and plastic zone. The validation of the present study of the trapdoor in homogeneous clay is shown for the rough and smooth cases in Fig. 7. These results compared the trapdoor stability factors (N) between the computed bound solutions and the existing solutions by FELA for
3
4
Fig. 8. Undrained stability of an active planar trapdoor as (a) a function of H/B with different dimensionless strength gradients (qH/su0) and (b) a function of the dimensionless strength gradient (qH/su0) with different cover ratios (H/B).
Table 1 Optimal value of statistical constants. a1
a2
a3
a4
a5
a6
a7
a8
2.5245
0.1139
0.2867
1.1527
0.0131
0.1510
1.1996
1.2324
the rough [18,19] and smooth [18] cases. The prior solutions of Davis [16] and Gunn [17] are omitted in these comparisons since they have been verified previously [18,19]. Solutions of Sloan et al. [18] and Martin [19] represent the most accurate and upto-date solutions for this comparison. Generally, the predicted trends in the trapdoor stability factor as a function of the cover ratio agree very well with that of the existing solutions [18,19]. Obviously, there is an excellent agreement in the rough surface cases between the present study and Martin [19] for a shallow trapdoor, where H/B = 0.5–2. For H/B = 2–10, the results of present study indicate a significant improvement over the previous FELA solutions [18] of about 1–8%. The influence of H/B and qH/su0 on the trapdoor stability factors are illustrated in Fig. 8. In all the cases of these parametric studies
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(a)
(a)
10
R2
= 99.91%
8 Nc
6 Nρ
Nc and Nρ
Nc, Proposed equaon
8
6
4
4 2
2 0 0
0 0
2
4
6
8
Nc, FELA
(b)
2
4
10
6
8
10
H/B
(b) 2.7
10
R2 = 99.98%
2.4
2.1
Nc /Nρ
Nρ, Proposed equaon
8
6
1.8
4 1.5
2
1.2
0
2
0
2
4
6
8
10
(σs+γH-σt)/su0, Proposed equaon
Nρ, FELA 35 30
6
8
10
H/B
0
(c)
4
Fig. 10. Relationship between the stability factors Nc and Np and the cover ratio (H/B) showing a comparison between the data from (a) the FELA and Eq. (2) and (b) the smooth and rough case from Eq. (2).
an increase in the H/B factor. In addition, the trapdoor stability factor in a given rough case is generally larger than that in the corresponding smooth case, where the differences in the N factor among the two cases can be seen for the intermediate and/or deep trapdoors (H/B > 6) only. Interestingly, an increasing linear relationship between the trapdoor stability factor and qH/su0 can be observed for all H/B ratios.
R2 = 99.97%
25 20
4. Closed-form equation
15 10 5 0 0
5
10
15
20
25
30
35
(σs+γ H-σt)/su0, FELA Fig. 9. Comparison of the undrained stability of an active planar trapdoor between the proposed Eq. (2) and FELA for the stability factors (a) Nc, (b) Np and (c) N.
and both interface roughness (rough and smooth), the exact N factor can be accurately bracketed by the computed upper and lower bound solutions within 1%. As expected for both the rough and smooth cases, an increase in the N factor is nonlinearly related to
The trial-and-error method of curve fitting was performed to determine the mathematical equation that best matched with the FELA computed data. The average bound solutions were chosen in the curve fitting method since the differences in the bound solutions were very small for all cases and the exact estimate of the N factor can be accurately predicted using the average data. A closedform equation, consisting of a linear function of adhesion factor (a) and qH/su0, and a complicated nonlinear function of H/B for the exact estimate of the N factor for an undrained stability of active trapdoor in both homogeneous and non-homogenous clays was proposed in Eq. (2);
N ¼ ða1 þ a2 aÞ
1þ
H a7 B a a3 HB 7
qH H a8
þ ða4 þ a5 aÞ
where a1–a8 are constant coefficients.
su0
B
1 þ a6 ðHBÞ
a8
ð2Þ
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Note that the proposed equation requires eight constant coefficients. By multiplying su0 to both sides in Eq. (2) and rearranging terms, a general solution of the exact estimate of rt for an undrained active collapse in homogenous and non-homogenous clays can be obtained, as shown in Eq. (3);
rt ¼ rs þ cH Nc su0 Nq ðqHÞ
ð3Þ
where
Nc ¼ ða1 þ a2 aÞ
1þ
H a7 B a a3 HB 7
; Nq ¼ ða4 þ a5 aÞ
1þ
H a8 B a a6 HB 8
It is worth noting two special cases of the trapdoor pressure can be predicted from Eq. (3), namely (i) a homogenous clay (q = 0) and (ii) a linear increase of strength with zero strength at the ground surface (su0 = 0), as in Eqs. (4) and (5), respectively:
caseðiÞ q ¼ 0 caseðiiÞ su0 ¼ 0
rt ¼ rs þ cH Nc su0 rt ¼ rs þ cH Nq ðqHÞ
ð4Þ ð5Þ
According to Eqs. (3)–(5), Nc and Nq may be interpreted such that Nc represents the stability factor of the trapdoor with respect to the soil cohesion and Nq represents its strength gradient, where they can be completely separated to the cases of q = 0 and su0 = 0, respectively. These two factors are a linear function of the adhesion factor (a) of the soil-trapdoor interface and a nonlinear function of the cover ratio (H/B). The least square method [32,33] was employed to determine the optimal solution of the eight constant coefficients (a1–a8). Additional upper and lower bound analyses for the cases of su0 = 0 (q – 0), H/B = 0.5–10 and a = 1 and 0 were analyzed and used to fit the Nq factor. The curve fitting method tried to fit all predictions of Nc, Nq and N with the average bound solutions. The optimal solution of all the coefficients is summarized in Table 1 and comparisons of Nc, Nq and N between the predicted and average solutions are shown in Fig. 9. The predicted values of these factors corresponded very well with the average solutions, with coefficients of determination (R2) for Nc, Nq and N of 99.91%, 99.98% and 99.97%, respectively. Thus, the proposed closed-form equation is sufficiently accurate to provide an exact estimate of the active trapdoor pressure in practice. The variations in the proposed Nc and Nq factors as a function of H/B is shown in Fig. 10, along with the average solutions of these factors for comparison. Increases in the H/B ratio resulted in higher values of these two stability factors. In addition, there was no difference in each factor between the smooth and rough cases when H/B 6 4. The influence of the adhesion factor on Nc can be seen for H/B > 5, but the influence of Nq was still small even for a deep trapdoor (H/B = 10). Lastly, the proposed Nc/Nq ratio is shown in Fig. 10 (b), where it ranged from 2.4–1.3 and decreased nonlinearly with an increasing H/B ratio. In practice, geotechnical engineers wish to determine the factor of safety (FS) of the active trapdoor when the input parameters of the problem are known, including the trapdoor dimensions (H and B), soil parameters (c, su0i and qi) and loadings (rs and rt). For such practical problems, Eq. (3) together with the concept of the mobilized shear strength can be employed to determine the FS. It is important to distinguish between the input strength parameters (su0i and qi) and the mobilized ones (su0 and q), where the latter is used in Eq. (3). Thus, the mobilized strength parameters based on this concept are given as: su0 = su0i/FS and q = qi/FS. Substituting these expressions into Eq. (3), the FS of the active trapdoor can be determined directly from all input parameters from Eq. (6).
FS ¼
su0i Nc þ qi HN q rs þ cH rt
ð6Þ
It is important to interpret a physical meaning of Eq. (6). The FS is defined as the ratio of the total shear resistance to the net driving pressure of the problem. The former is contributed from the effects of soil cohesion (su0iNc) and the linear strength gradient (qiHNq), while the latter is calculated from the sum of a uniform surcharge on the ground surface (rs) and an overburden pressure (cH) less the trapdoor pressure (rt) that is the resistance of the problem. Thus, Eq. (6) represents a general solution of the FS for the active trapdoor for homogenous (qi = 0) and non-homogeneous (su0i – 0, qi – 0) clays, where the latter includes the special case of zero strength at the ground surface (su0i = 0, qi – 0). 5. Conclusions This note presents a new plasticity solution of the active collapse of a planar trapdoor in non-homogeneous clays that have a linear increase of strength with depth. Three parametric studies on the dimensionless stability factors (N) were performed for the cover ratio (H/B), adhesion factor (a) at the interfaces for a rigid trapdoor and bottom boundary, and the dimensionless strength gradient (qH/su0). In all cases, the exact stability factors were accurately bracketed by the computed bound solutions within 1%. For homogeneous clay, the present study succeeded in providing a significant improvement on the bound solutions of the active trapdoor that had a H/B ratio in the range of 2–10. New bound solutions were presented for the non-homogeneous cases for H/B = 0.5–10. An accurate closed-form equation was proposed from nonlinear regression analysis of the average bound solutions to predict an exact estimate of the stability factor. Two new stability factors (Nc and Nq) were deduced from the proposed equation and accounted for the influences of the cohesion and strength gradient on the trapdoor pressure. These two factors were a linear function of the adhesion factor (a) of the soil-trapdoor interface and a nonlinear function of the H/B ratio. The proposed equation was generalized to determine exact estimates of rt and FS for a practical problem with an arbitrary linear increase of undrained strength profile (i.e., su0i = any, qi = any) that includes two special cases of homogenous strength (su0i – 0, qi = 0) and linear strength with zero strength at the ground surface (su0i = 0, qi – 0). References [1] Terzaghi K. Stress distribution in dry and saturated sand above a yielding trapdoor. In: Proc., 1st international conference on soil mechanics and foundation engineering, Cambridge, Mass.. [2] Merifield RS, Sloan SW, Yu HS. Stability of plate anchors in undrained clay. Géotechnique 2001;51(2):141–53. [3] Merifield RS, Sloan SW. The ultimate pullout capacity of anchors in frictional soils. Can Geotech J 2006;43:852–68. [4] Suchowerska AM, Merifield RS, Carter JP, Clausen J. Prediction of underground cavity roof collapse using the Hoek-Brown failure criterion. Comput Geotech 2012;44:93–103. [5] Costa Y, Zornberg J, Bueno B, Costa C. Failure mechanisms in sand over a deep active trapdoor. J Geotech Geoenviron Eng 2009. http://dx.doi.org/10.1061/ (ASCE)GT.1943-5606.0000134: 1741-1753. [6] Jenike A. Steady gravity flow of frictional-cohesive solids in converging channels. J Appl Mech 1964;31(1):5–11. [7] Enstad G. On the theory of arching in mass flow hoppers. Chem Eng Sci 1975;30(10):1273–83. [8] Ladanyi B, Hoyaux B. A study of the trap-door problem in a granular mass. Can Geotech J 1969;6(1):1–15. [9] Vardoulakis I, Graf B, Gudehus G. Trap-door problem with dry sand: a statical approach based upon model kinematics. Int J Anal Meth Geomech 1981;5:57–78. [10] Tanaka T, Sakai T. Progressive failure and scale effect of trap-door problems with granular materials. Soils Found 1993;33(1):11–22. [11] Santichaianaint K. Centrifuge modeling and analysis of active trapdoor in sand Ph.D. thesis. Dept. of Civil, Environmental and Architectural Engineering, University of Colorado at Boulder; 2002. [12] Iglesia G, Einstein H, Whitman R. Investigation of soil arching with centrifuge tests. J Geotech Geoenviron Eng 2013. http://dx.doi.org/10.1061/(ASCE) GT.1943-5606.0000998: 04013005.
S. Keawsawasvong, B. Ukritchon / Computers and Geotechnics 81 (2017) 284–293 [13] Iglesia GR, Einstein HH, Whitman RV. Validation of centrifuge model scaling for soil systems via trapdoor tests. J Geotech Geoenviron Eng 2011. http://dx. doi.org/10.1061/(ASCE)GT.1943-5606.0000517: 1075–1089. [14] Terzaghi K. Theoretical soil mechanics. New York: Wiley; 1943. [15] Koutsabeloulis NC, Griffiths DV. Numerical modelling of the trap door problem. Géotechnique 1989;39(1):77–89. [16] Davis EH. Theories of plasticity and the failure of soil masses. In: Lee IK, editor. Soil mechanics: selected topics. London: Butterworths; 1968. [17] Gunn MJ. Limit analysis of undrained stability problems using a very small computer. In: Proceedings of the symposium on computer applications in geotechnical problems in highway engineering, Cambridge. [18] Sloan SW, Assadi A, Purushothaman N. Undrained stability of a trapdoor. Géotechnique 1990;40(1):45–62. [19] Martin CM. Undrained collapse of a shallow plane-strain trapdoor. Géotechnique 2009;59(10):855–63. [20] Krabbenhoft K, Lyamin A, Krabbenhoft J, Optum computational engineering (OptumG2); 2015,. [21] Sloan SW. Geotechnical stability analysis. Géotechnique 2013;63(7):531–72. [22] Chen WF, Liu XL. Limit analysis in soil mechanics. Amsterdam: Elsevier; 1990. [23] Krabbenhoft K, Lyamin AV, Hjiaj M, Sloan SW. A new discontinuous upper bound limit analysis formulation. Int J Numer Meth Eng 2005;63:1069–88. [24] Lyamin AV, Krabbenhoft K, Abbo AJ, Sloan SW. General approach to modelling discontinuities in limit analysis. In: Proceedings of IACMAG, Turin, Italy.
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[25] Krabbenhoft K, Lyamin AV, Sloan SW. Formulation and solution of some plasticity problems as conic programs. Int J Solids Struct 2007;44:1533–49. [26] Makrodimopoulos A, Martin CM. Lower bound limit analysis of cohesivefrictional materials using second-order cone programming. Int J Numer Meth Eng 2006;66:604–34. [27] Makrodimopoulos A, Martin CM. Upper bound limit analysis using simplex strain elements and second-order cone programming. Int J Anal Meth Geomech 2007;31:835–65. [28] Krabbenhoft K, Lyamin AV, Sloan SW, SONIC: a conic programming code for plasticity problems: version 1.0/LA: SOCP for limit analysis authors, Edition illustrated Publisher University of Newcastle – Centre of Full Employment & Equity; 2006. [29] Wright MH. The interior-point revolution in optimization: history, recent developments, and lasting consequences. Bull Am Math Soc 2005;42:39–56. [30] Krabbenhoft K, Lyamin AV, Sloan SW, Wriggers P. An interior-point method for elastoplasticity. Int J Numer Meth Eng 2007;69:592–626. [31] Ciria H, Peraire J, Bonet J. Mesh adaptive computation of upper and lower bounds in limit analysis. Int J Numer Meth Eng 2008;75:899–944. [32] Sauer T. Numerical analysis. UK: Pearson Education Limited; 2014. [33] Walpole RE, Myers RH, Myers SL, Ye K. Probability & statistics for engineering and scientists. 7th ed. New Jersey, USA: Prentice Hall; 2002.
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Technical Communication
Undrained stability of an active planar trapdoor in non-homogeneous clays with a linear increase of strength with depth Suraparb Keawsawasvong, M.Eng., Ph.D. (Research Fellow), Boonchai Ukritchon ⇑, Sc.D. (Associate Professor) Geotechnical Research Unit, Department of Civil Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok 10330, Thailand
a r t i c l e
i n f o
Article history: Received 17 June 2016 Received in revised form 6 August 2016 Accepted 28 August 2016
Keywords: Limit analysis Numerical analysis Stability Trapdoor Active failure
a b s t r a c t Finite element limit analysis was employed to determine the upper and lower bound solutions of the active failure of a planar trapdoor in non-homogeneous clays that have a linear increase of strength with depth. Influences of cover ratio, dimensionless strength gradient and trapdoor roughness on predicted failure mechanisms and stability factors were determined. In all cases, the exact stability factors were accurately bracketed by computed bound solutions within 1%. Accurate closed-form equations to predict the exact estimates of stability factors, trapdoor pressure and factor of safety using the new proposed factors for the cohesion and strength gradient are presented. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Active and passive failures of a trapdoor are classical stability problems in geotechnical engineering, in addition to the bearing capacity of shallow foundations, retaining walls and slopes. The active and passive collapses of a trapdoor corresponds to a determination of the limit loads that act on the downward and upward movements of the trapdoor, respectively. The original study of stress distribution on a trapdoor in sand was experimentally investigated by Terzaghi [1]. In addition to being an important benchmark solution in theoretical geomechanics, the stability of a trapdoor is of considerable practical interest in many applications. The passive failure of a trapdoor corresponds to the pullout problem of a vertical plate anchor in soils, and has been studied widely [2,3]. On the other hand, the solution of the active trapdoor can be applied to analyze the collapse of an underground roof in tunnels and mining works [4], the gravitational flow of a granular material through hoppers [5,6], and the stability of buried pipes subjected to the loss of ground support [7]. A relatively large number of studies on the stability of an active trapdoor have been investigated previously and include physical model tests [1,5,8–13], numerical analysis by the limit equilibrium method (LEM) [14], finite element analysis [10,15] and limit analysis [9,16,17]. Some researchers have considered the planar ⇑ Corresponding author. E-mail addresses: [email protected] (S. Keawsawasvong), boonchai.uk@ gmail.com (B. Ukritchon). http://dx.doi.org/10.1016/j.compgeo.2016.08.027 0266-352X/Ó 2016 Elsevier Ltd. All rights reserved.
trapdoor problem in a purely and homogeneously cohesive soil [16–19]. However, no study on the undrained active failure of a planar trapdoor in non-homogenous clay with a linear increase of undrained shear strength has been reported, and so is the subject of this note. The problem definition of an active trapdoor under the plane strain condition is shown in Fig. 1. A depth of a non-homogeneous clay layer (H) rests on a rigid boundary and a rigid trapdoor of width (B). The clay behaves as a rigid-perfectly plastic Tresca material with the associated flow rule. The non-homogeneous clay has an undrained shear strength that increases linearly with the depth from the ground surface (su0) with a strength gradient (q). Despite the strength non-homogeneity, the clay is assumed to have a constant unit weight (c). The ground surface is fully loaded by a uniform surcharge (rs). To produce the active failure, the trapdoor is assumed to move downwards as a rigid body that is resisted by the externally applied uniform trapdoor pressure (rt). Such a failure happens due to the actions of the surcharge (rs) and the soil unit weight (c) and is resisted by the trapdoor pressure (rt) and the shear resistance of clay (su0, q). This study considers either purely rough or smooth cases, where the former corresponds to a perfectly rough surface at both the trapdoor and the bottom boundary, and the latter corresponds to a perfectly smooth surface of these. It is expected that the true boundary condition should lie somewhere between these two cases. Following previous studies of an undrained collapse of an active trapdoor in cohesive soils [18,19], the solution of this problem is
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285
Fig. 1. Active trapdoor under a plane strain condition.
represented by the dimensionless stability factor (N), as shown in Eq. (1);
N¼
rs þ cH rt su0
¼f
H qH ; ;a B su0
ð1Þ
where H/B is the cover ratio, qH/su0 is the dimensionless strength gradient, a is the adhesion factor at the interfaces of rigid trapdoor and bottom boundary = sua/su, and sua and su are the undrained shear strength at the interfaces and the surrounding soil, respectively. Note that the negative sign in front of the trapdoor pressure (rt) denotes that it is the resistance of the problem, while the surcharge (rs) and the soil unit weight (c) are the driving stresses that cause the active failure. The rough and smooth cases correspond to a = 1 and 0, respectively. With respect to previous studies on the active failure of a trapdoor in a homogeneous clay [16–19], the most accurate solutions
were derived by employing finite element limit analysis (FELA) to solve this problem [18,19]. Sloan et al. [18] presented a significant improvement on N bound solutions as compared to previous studies [16,17], especially for deep trapdoors and the influence of interface roughness on the N solutions that became significant when H/B > 5. Their bound solutions had an accuracy of 7–11% for all H/B ratios. Subsequently, Martin [19] proposed the stress and velocity fields of new slip-line solutions for a shallow trapdoor (H/B = 0.5–2) and validated the results using FELA with an adaptive mesh refinement and confirmed that the slip-line solutions were actually the exact ones only for H/B < 1.3. This paper extends these previous studies on planar trapdoors [18,19] in homogeneous clays to investigate the influence of the linear strength non-homogeneity of clay on the undrained bound N solutions of active failure, which has not been addressed before. The study employed the state-of-art FELA software, OptumG2 [20], to accurately determine the undrained active collapse of a trapdoor
Fig. 2. Numerical model of the undrained stability of an active planar trapdoor in OptumG2.
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Fig. 3. Comparison of the final adaptive meshes of an active planar trapdoor with different cover ratios (H/B) and rough (a = 1) or smooth (a = 0) surfaces and qH/su0 = 0.
under the plane strain condition. Three parametric studies were performed; namely (i) the cover ratio (H/B) from shallow to deep trapdoors over a range of 0.5–10, (ii) a dimensionless strength gradient (qH/su0) between 0 and 4 and (iii) an adhesion factor (a) of 1 or 0 (either rough or smooth interfaces at the trapdoor and bottom support, respectively). The latter case should cover the true boundary condition of the trapdoor in practice. The effects of these three parameters on the predicted failure mechanisms are discussed and compared. A closed-form equation is proposed for the exact estimate of the true solutions of the active failure of a trapdoor based on a nonlinear regression analysis. New trapdoor stability factors, with respect to cohesion and strength gradient, were deduced from the proposed equation. Expressions that utilize these factors are given for predicting the trapdoor pressure and factor of safety (FS) in a general strength profile that cover homogenous (su0 – 0, q = 0) and non-homogenous strengths (su0 – 0, q – 0). The latter includes the special case of a linear strength but zero strength at the ground surface (su0 = 0, q – 0). Thus, the proposed closedform expressions serve as a convenient and accurate tool to determine the solutions for an active trapdoor failure in practice.
2. Method of analysis At present, the emerging numerical technique of FELA has become a powerful approach for analyzing various complex stability problems in geotechnical engineering, as demonstrated by Sloan [21]. The technique combines the powerful capabilities of finite element discretization for handling complicated soil stratifications, loadings and boundary conditions with the plastic bound theorem to bracket the exact limit load by upper and lower bound solutions. The underlying bound theorems assume a rigid-perfectly plastic material with an associated flow rule. Details of limit
analysis and FELA are omitted here but can be found in Chen and Liu [22] and Sloan [21], respectively. The plane strain numerical model for simulating an active failure of a trapdoor in OptumG2 is shown in Fig. 2. A symmetrical condition at the centerline of the problem is utilized to prepare the model. As a result, only one half of the domain is considered in the analysis. The trapdoor is modelled as a rigid plate element, while the clay behaves as a Tresca material with an associated flow rule that is governed by su0 and q. The interface elements are defined at the trapdoor and along the rigid bottom boundary of the problem with the adhesion factor (a). To produce the solutions for either rough or smooth trapdoors, the adhesion factor is set as either unity or zero, respectively. Due to possible stress and velocity discontinuities at the corner of the trapdoor, a small segment of vertical line is employed at that point and defined as interface elements, where their adhesion factor is fully rough (i.e. a = 1). All interface elements allows velocity and stress discontinuities to happen in the upper bound and lower bound calculations, respectively. It should be noted that the small vertical line of interface is not sensitive to the accuracy of lower bound solutions since stress discontinuities are modelled at all inter-element edges. In contrast, this vertical interface is very crucial in order to produce accurate upper bound solutions since velocity discontinuities are not permitted to happen along all inter-element edges but they are expected to happen at the corner of trapdoor. Thus, this interface is used to introduce a line of velocity discontinuity at this corner. The left (centerline) and right boundaries of the model are fixed horizontally, while the bottom boundary is fixed in both directions. A uniform surcharge pressure (rs) is applied downwards at the ground surface of the model, while the uniform trapdoor pressure (rt) is applied upwards on the trapdoor. The left and right dimension of the domain is chosen to be large enough that the plastic zone is contained within the domain and does not intersect the
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Fig. 4. Comparison of the incremental displacement vectors of an active planar trapdoor with different cover ratios (H/B) and rough (a = 1) or smooth (a = 0) surfaces and qH/su0 = 0.
right boundary. Based on various numerical simulations, the optimal sizes of the domain along the horizontal direction corresponded to 2H, as shown in Fig. 2, while the size along the vertical direction is an input parameter of trapdoor depth (H). Thus, the computed upper and lower bound loads are unaffected by the selected domain size. The following summarizes the upper and lower bound FELA of the plane strain condition in OptumG2 for analyzing the active failure of a planar trapdoor in homogeneous and non-homogenous clays. In the upper bound analysis, the soil mass is discretized into six-noded triangular elements. The unknown velocity components at each node employ a quadratic interpolation within each element and are continuous between adjacent elements. Except for the interface elements, velocity discontinuities are not permitted to happen along all shared edges of adjacent elements, which is similar to the conventional displacement-based finite element method. A rigid plate element is used at the bottom to simulate the trapdoor. Velocity discontinuities are permitted to happen at the interface elements that are defined at the soil-trapdoor interface, the small vertical line extending from the trapdoor corner, and the rigid bottom support of the problem. These velocity discontinuities are implemented by collapsing patches of regular triangular elements to zero thickness [23,24]. The upper bound calculation is formulated as a second-order conic programming (SOCP) [25,26]. Constraints on the kinematically admissible velocity conditions include the velocity boundary conditions and the compatibility equations with the associated flow rule at triangular and interfaces elements. The objective function of the upper bound
SOCP is defined to minimize the unit weight of soil that is obtained through invoking the principle of virtual work by equating the rate of work done by external loads (c, rs and rt) with the internal energy dissipation at the triangular and interface elements. In the lower bound analysis, three-noded triangular elements were employed to discretize the soil mass. Unlike a conventional displacement-based finite element method, each element has its own unique node resulting in possible stress discontinuities to occur along all the shared edges of adjacent elements. Like the upper bound model, the interface elements are defined at the soil-trapdoor interface, the small vertical line extending from the trapdoor corner and the bottom support. The lower bound calculation was formulated as a SOCP [25,27], which satisfies the statically admissible stress conditions, including the equilibrium equations at each triangular element and along all stress discontinuities, stress boundary conditions, and no stress violation of the yield criterion in the soil mass. The objective function of the lower bound SOCP was defined to directly maximize the unit weight of the soil that has an inter-relationship with unknown nodal stresses through the vertical equilibrium equation. It should be noted that stress discontinuity is the key feature in lower bound calculations and used to improve the accuracy of the lower bound collapse load, as extensively discussed by Sloan [21]. Therefore, it is necessary to employ stress discontinuities defined by all inter-element edges in a lower bound mesh consisting of three-noded triangular elements. Six-noded triangular elements have never been implemented in any previous formulation of lower bound calculation since this employment does not produce
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Fig. 5. Comparison of the incremental displacement vectors of an active planar trapdoor with different strength gradients (qH/su0) and rough (a = 1) or smooth (a = 0) surfaces and H/B = 2.
a rigorous lower bound solution. Employing these elements results in unknown stresses to vary quadratically over the element, and hence it is not possible to ensure that the stress field satisfies the equilibrium equations everywhere within the element. Such elements can only be employed in upper bound calculations so as to obtain a rigorous upper bound solution (see [20,21,27]). Both the upper and lower bound SOCP in the plane problem were solved to obtain rigorous bound solutions using the general purpose optimization solver, SONIC [28], which is based on the interior-point method [29,30]. Plane strain FELA using SOCP in OptumG2 is comparable to that of previous formulations [26,27], and full details of the numerical FELA formulation in OptumG2 can be found in [20]. A powerful feature in OptumG2, the automatically adaptive mesh refinement, was also employed for both the upper and lower bound analyses to determine the tight upper and lower bound solutions. In this study, the shear dissipation control to the bounds gap in OptumG2 was based on the formulation proposed by Ciria et al. [31] and used for the adaptive mesh refinement. Five adaptive steps were used for all analyses following the suggested value in the software manual, such that this value is adequate enough to obtain an accurate solution. In the course of five adaptive
iterations, an initial mesh with 5000 elements was automatically adapted and increased to a final mesh with 10,000 elements. As mentioned by Sloan et al. [18], there are a variety of ways to determine the stability factor (N). For example, a fixed value of c and rt is setup and the surcharge is optimized. In this study, for both the rough and smooth trapdoors, a fixed value of rs and rt was setup and c, the body force of the problem, was optimized directly in both the SOCP upper and lower bound analyses to give the minimum and maximum values, respectively. Thus, the trapdoor stability factor (N) and the corresponding dimensionless input parameters, H/B, a (1 or 0) and qH/su0, were calculated according to Eq. (1). 3. Results and comparisons The comparisons of predicted failure mechanisms of the active trapdoor that are affected by different cover ratios (H/B) for a homogeneous strength profile (qH/su0 = 0) are shown in Figs. 3 and 4. The results of the failure mechanism include the final adaptive meshes and the incremental displacement vectors, respectively. These results clearly show the influence of the cover ratio on the predicted failure mechanism from shallow (H/B = 0.5) to
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Fig. 6. Comparison of the incremental shear strain contours of an active planar trapdoor with different strength gradients (qH/su0) and rough (a = 1) or smooth (a = 0) surfaces and H/B = 2.
deep (H/B = 10) trapdoors. The failure of a shallow trapdoor corresponds to a vertical slip mechanism, as first proposed by Davis [16] and later improved by Gunn [17] using a three-parameter rigid block mechanism. The failure width of this mechanism at the ground surface is localized and does not extend beyond the corner of the trapdoor. Note that the adaptive mesh for the case H/B = 0.5 of the present study is quite similar to that of Martin [19] whose results are omitted. Once the H/B ratio increases (H/B = 2), the vertical slip mode changes to a curvilinear one that is accompanied by the radial shear zone around the trapdoor corner and a rigid triangular zone that is located in front of the trapdoor. For intermediate and deep trapdoors (H/B = 6 and 10, respectively), there is more lateral spreading of the plastic zones in both the radial and curvilinear shear ones, resulting in a significant failure width at the ground surface. Such results are attributed from an influence of geometric nonlinearity of the cover ratio (H/B) that induces more mobilized shear stress along a longer curvilinear surface for a more resisting force to against failure and requires larger plastic zone for a more driving force (cH) to cause failure. The results of the smooth cases are also compared with those of the rough cases in these figures. For shallow trapdoors (H/B = 0.5 and 2), the differences in the failure mechanism between the smooth and rough cases are quite small, but the effect of a smooth interface on the failure mechanism becomes significant for intermediate and deep trapdoors (H/B = 6 and 10, respectively). The
smooth cases reveal a curvilinear shear zone that starts from the ground surface and ends at a certain position on the bottom smooth boundary instead of being at the corner of the trapdoor in the rough case. In addition, a relative sliding of soils along the bottom boundary is expected to occur only in the smooth cases. Such results are attributed from the influence of shear tractions along rough surfaces of trapdoor and bottom support that contribute some shear resistances to against failure. Since the failure of shallow trapdoor is caused by a vertical slip plane directly passing through its corner, an effect of these shear tractions on rough surfaces contributes to a lesser degree in shear resistance, as compared to deep trapdoors that are associated with curvilinear shear surface. For smooth cases, there is no shear traction along the bottom support, and hence sliding can take place along the bottom interface. On the other hand, the presence of shear traction along the bottom interface induces the curvilinear shear surface to pass at the corner of trapdoor instead of sliding along the bottom interface. The influence of qH/su0 on the failure mechanism of the rough cases for shallow (H/B = 2) and deep (H/B = 10) trapdoors are compared in Figs. 5 and 6 for qH/su0 values of 0, 1 and 4. The results of the failure mechanism include the incremental displacement vectors and the incremental shear strain contours, respectively. The effects of qH/su0 on the failure mechanism of smooth and rough trapdoors are quite similar. Increases in qH/su0 result in a
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(a)
(a) 35
Rough Cases
8
30
(σs+γ H-σt)/su0
(σs+γ H-σt)/su0
ρH/su = 4
25
6
4
=2 =1 = 0.75 = 0.5 = 0.25 =0
20 15 10
2
5 0
0 0
2
4
6
8
0
10
2
4
6
H/B
(b)
(b)
Smooth Cases
8
8
10
H/B 35 30 25
(σs+γ H-σt)/su0
(σs+γ H-σt)/su0
6
4
H/B = 10 =8 =6 =4 =2 =1 = 0.5 = 0.1
20 15 10
2
5 0
0 0
2
4
6
8
10
0
1
2
ρH/su0
H/B Fig. 7. Relationship between the stability factor (N) and cover ratio (H/B) for (a) rough and (b) smooth interfaces.
decreased size of the shear bands and curvilinear shear zones around the corner of the trapdoor for smooth and rough trapdoors. As a result, the width of the failure mechanism extending to the ground surface also decreases accordingly. It should be noted that the triangular rigid zone above the trapdoor becomes more curvilinear and shifted higher towards the ground surface as the qH/su0 factor increases. For small value of qH/su0 6 1, there is no difference in failure mechanism between smooth and rough trapdoors. However, for a large value of qH/su0 = 4, some difference can be observed, where the size of shear bands of smooth trapdoors depicted by incremental shear strain (see Fig. 6) is slightly larger than that of rough trapdoors. Such results are attributed from the influence of linear strength gradient (q) that generates more shear resistance in shear bands and plastic zone in addition to the effect of shear resistance due to homogeneous strength (su0). Therefore, an increase in qH/su0 gives rise to a higher shear resistance of soils, and thus resulting in a decreased size of shear bands and plastic zone. The validation of the present study of the trapdoor in homogeneous clay is shown for the rough and smooth cases in Fig. 7. These results compared the trapdoor stability factors (N) between the computed bound solutions and the existing solutions by FELA for
3
4
Fig. 8. Undrained stability of an active planar trapdoor as (a) a function of H/B with different dimensionless strength gradients (qH/su0) and (b) a function of the dimensionless strength gradient (qH/su0) with different cover ratios (H/B).
Table 1 Optimal value of statistical constants. a1
a2
a3
a4
a5
a6
a7
a8
2.5245
0.1139
0.2867
1.1527
0.0131
0.1510
1.1996
1.2324
the rough [18,19] and smooth [18] cases. The prior solutions of Davis [16] and Gunn [17] are omitted in these comparisons since they have been verified previously [18,19]. Solutions of Sloan et al. [18] and Martin [19] represent the most accurate and upto-date solutions for this comparison. Generally, the predicted trends in the trapdoor stability factor as a function of the cover ratio agree very well with that of the existing solutions [18,19]. Obviously, there is an excellent agreement in the rough surface cases between the present study and Martin [19] for a shallow trapdoor, where H/B = 0.5–2. For H/B = 2–10, the results of present study indicate a significant improvement over the previous FELA solutions [18] of about 1–8%. The influence of H/B and qH/su0 on the trapdoor stability factors are illustrated in Fig. 8. In all the cases of these parametric studies
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(a)
(a)
10
R2
= 99.91%
8 Nc
6 Nρ
Nc and Nρ
Nc, Proposed equaon
8
6
4
4 2
2 0 0
0 0
2
4
6
8
Nc, FELA
(b)
2
4
10
6
8
10
H/B
(b) 2.7
10
R2 = 99.98%
2.4
2.1
Nc /Nρ
Nρ, Proposed equaon
8
6
1.8
4 1.5
2
1.2
0
2
0
2
4
6
8
10
(σs+γH-σt)/su0, Proposed equaon
Nρ, FELA 35 30
6
8
10
H/B
0
(c)
4
Fig. 10. Relationship between the stability factors Nc and Np and the cover ratio (H/B) showing a comparison between the data from (a) the FELA and Eq. (2) and (b) the smooth and rough case from Eq. (2).
an increase in the H/B factor. In addition, the trapdoor stability factor in a given rough case is generally larger than that in the corresponding smooth case, where the differences in the N factor among the two cases can be seen for the intermediate and/or deep trapdoors (H/B > 6) only. Interestingly, an increasing linear relationship between the trapdoor stability factor and qH/su0 can be observed for all H/B ratios.
R2 = 99.97%
25 20
4. Closed-form equation
15 10 5 0 0
5
10
15
20
25
30
35
(σs+γ H-σt)/su0, FELA Fig. 9. Comparison of the undrained stability of an active planar trapdoor between the proposed Eq. (2) and FELA for the stability factors (a) Nc, (b) Np and (c) N.
and both interface roughness (rough and smooth), the exact N factor can be accurately bracketed by the computed upper and lower bound solutions within 1%. As expected for both the rough and smooth cases, an increase in the N factor is nonlinearly related to
The trial-and-error method of curve fitting was performed to determine the mathematical equation that best matched with the FELA computed data. The average bound solutions were chosen in the curve fitting method since the differences in the bound solutions were very small for all cases and the exact estimate of the N factor can be accurately predicted using the average data. A closedform equation, consisting of a linear function of adhesion factor (a) and qH/su0, and a complicated nonlinear function of H/B for the exact estimate of the N factor for an undrained stability of active trapdoor in both homogeneous and non-homogenous clays was proposed in Eq. (2);
N ¼ ða1 þ a2 aÞ
1þ
H a7 B a a3 HB 7
qH H a8
þ ða4 þ a5 aÞ
where a1–a8 are constant coefficients.
su0
B
1 þ a6 ðHBÞ
a8
ð2Þ
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S. Keawsawasvong, B. Ukritchon / Computers and Geotechnics 81 (2017) 284–293
Note that the proposed equation requires eight constant coefficients. By multiplying su0 to both sides in Eq. (2) and rearranging terms, a general solution of the exact estimate of rt for an undrained active collapse in homogenous and non-homogenous clays can be obtained, as shown in Eq. (3);
rt ¼ rs þ cH Nc su0 Nq ðqHÞ
ð3Þ
where
Nc ¼ ða1 þ a2 aÞ
1þ
H a7 B a a3 HB 7
; Nq ¼ ða4 þ a5 aÞ
1þ
H a8 B a a6 HB 8
It is worth noting two special cases of the trapdoor pressure can be predicted from Eq. (3), namely (i) a homogenous clay (q = 0) and (ii) a linear increase of strength with zero strength at the ground surface (su0 = 0), as in Eqs. (4) and (5), respectively:
caseðiÞ q ¼ 0 caseðiiÞ su0 ¼ 0
rt ¼ rs þ cH Nc su0 rt ¼ rs þ cH Nq ðqHÞ
ð4Þ ð5Þ
According to Eqs. (3)–(5), Nc and Nq may be interpreted such that Nc represents the stability factor of the trapdoor with respect to the soil cohesion and Nq represents its strength gradient, where they can be completely separated to the cases of q = 0 and su0 = 0, respectively. These two factors are a linear function of the adhesion factor (a) of the soil-trapdoor interface and a nonlinear function of the cover ratio (H/B). The least square method [32,33] was employed to determine the optimal solution of the eight constant coefficients (a1–a8). Additional upper and lower bound analyses for the cases of su0 = 0 (q – 0), H/B = 0.5–10 and a = 1 and 0 were analyzed and used to fit the Nq factor. The curve fitting method tried to fit all predictions of Nc, Nq and N with the average bound solutions. The optimal solution of all the coefficients is summarized in Table 1 and comparisons of Nc, Nq and N between the predicted and average solutions are shown in Fig. 9. The predicted values of these factors corresponded very well with the average solutions, with coefficients of determination (R2) for Nc, Nq and N of 99.91%, 99.98% and 99.97%, respectively. Thus, the proposed closed-form equation is sufficiently accurate to provide an exact estimate of the active trapdoor pressure in practice. The variations in the proposed Nc and Nq factors as a function of H/B is shown in Fig. 10, along with the average solutions of these factors for comparison. Increases in the H/B ratio resulted in higher values of these two stability factors. In addition, there was no difference in each factor between the smooth and rough cases when H/B 6 4. The influence of the adhesion factor on Nc can be seen for H/B > 5, but the influence of Nq was still small even for a deep trapdoor (H/B = 10). Lastly, the proposed Nc/Nq ratio is shown in Fig. 10 (b), where it ranged from 2.4–1.3 and decreased nonlinearly with an increasing H/B ratio. In practice, geotechnical engineers wish to determine the factor of safety (FS) of the active trapdoor when the input parameters of the problem are known, including the trapdoor dimensions (H and B), soil parameters (c, su0i and qi) and loadings (rs and rt). For such practical problems, Eq. (3) together with the concept of the mobilized shear strength can be employed to determine the FS. It is important to distinguish between the input strength parameters (su0i and qi) and the mobilized ones (su0 and q), where the latter is used in Eq. (3). Thus, the mobilized strength parameters based on this concept are given as: su0 = su0i/FS and q = qi/FS. Substituting these expressions into Eq. (3), the FS of the active trapdoor can be determined directly from all input parameters from Eq. (6).
FS ¼
su0i Nc þ qi HN q rs þ cH rt
ð6Þ
It is important to interpret a physical meaning of Eq. (6). The FS is defined as the ratio of the total shear resistance to the net driving pressure of the problem. The former is contributed from the effects of soil cohesion (su0iNc) and the linear strength gradient (qiHNq), while the latter is calculated from the sum of a uniform surcharge on the ground surface (rs) and an overburden pressure (cH) less the trapdoor pressure (rt) that is the resistance of the problem. Thus, Eq. (6) represents a general solution of the FS for the active trapdoor for homogenous (qi = 0) and non-homogeneous (su0i – 0, qi – 0) clays, where the latter includes the special case of zero strength at the ground surface (su0i = 0, qi – 0). 5. Conclusions This note presents a new plasticity solution of the active collapse of a planar trapdoor in non-homogeneous clays that have a linear increase of strength with depth. Three parametric studies on the dimensionless stability factors (N) were performed for the cover ratio (H/B), adhesion factor (a) at the interfaces for a rigid trapdoor and bottom boundary, and the dimensionless strength gradient (qH/su0). In all cases, the exact stability factors were accurately bracketed by the computed bound solutions within 1%. For homogeneous clay, the present study succeeded in providing a significant improvement on the bound solutions of the active trapdoor that had a H/B ratio in the range of 2–10. New bound solutions were presented for the non-homogeneous cases for H/B = 0.5–10. An accurate closed-form equation was proposed from nonlinear regression analysis of the average bound solutions to predict an exact estimate of the stability factor. Two new stability factors (Nc and Nq) were deduced from the proposed equation and accounted for the influences of the cohesion and strength gradient on the trapdoor pressure. These two factors were a linear function of the adhesion factor (a) of the soil-trapdoor interface and a nonlinear function of the H/B ratio. The proposed equation was generalized to determine exact estimates of rt and FS for a practical problem with an arbitrary linear increase of undrained strength profile (i.e., su0i = any, qi = any) that includes two special cases of homogenous strength (su0i – 0, qi = 0) and linear strength with zero strength at the ground surface (su0i = 0, qi – 0). References [1] Terzaghi K. Stress distribution in dry and saturated sand above a yielding trapdoor. In: Proc., 1st international conference on soil mechanics and foundation engineering, Cambridge, Mass.. [2] Merifield RS, Sloan SW, Yu HS. Stability of plate anchors in undrained clay. Géotechnique 2001;51(2):141–53. [3] Merifield RS, Sloan SW. The ultimate pullout capacity of anchors in frictional soils. Can Geotech J 2006;43:852–68. [4] Suchowerska AM, Merifield RS, Carter JP, Clausen J. Prediction of underground cavity roof collapse using the Hoek-Brown failure criterion. Comput Geotech 2012;44:93–103. [5] Costa Y, Zornberg J, Bueno B, Costa C. Failure mechanisms in sand over a deep active trapdoor. J Geotech Geoenviron Eng 2009. http://dx.doi.org/10.1061/ (ASCE)GT.1943-5606.0000134: 1741-1753. [6] Jenike A. Steady gravity flow of frictional-cohesive solids in converging channels. J Appl Mech 1964;31(1):5–11. [7] Enstad G. On the theory of arching in mass flow hoppers. Chem Eng Sci 1975;30(10):1273–83. [8] Ladanyi B, Hoyaux B. A study of the trap-door problem in a granular mass. Can Geotech J 1969;6(1):1–15. [9] Vardoulakis I, Graf B, Gudehus G. Trap-door problem with dry sand: a statical approach based upon model kinematics. Int J Anal Meth Geomech 1981;5:57–78. [10] Tanaka T, Sakai T. Progressive failure and scale effect of trap-door problems with granular materials. Soils Found 1993;33(1):11–22. [11] Santichaianaint K. Centrifuge modeling and analysis of active trapdoor in sand Ph.D. thesis. Dept. of Civil, Environmental and Architectural Engineering, University of Colorado at Boulder; 2002. [12] Iglesia G, Einstein H, Whitman R. Investigation of soil arching with centrifuge tests. J Geotech Geoenviron Eng 2013. http://dx.doi.org/10.1061/(ASCE) GT.1943-5606.0000998: 04013005.
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