Six parameters for soil model.
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\r\n\t[3] M. Kauranen, A. V. Zayats. Nonlinear Plasmonics. Nature Photonics, vol. 6, 2012, pp. 737-748.
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\r\n\t[5] N. C. Panoiu, W. E. I. Sha, D.Y. Lei, G.-C. Li. Nonlinear optics in plasmonic nanostructures. Journal of Optics, 20, 2018, pp. 1-36.
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The common aetiologies of gangrene of the extremities are atherosclerosis and diabetes mellitus. An internist may encounter several other diseases which can cause gangrene of extremities. Systemic lupus erythematosus, progressive systemic sclerosis, Henoch-Schonlein purpura, anti-neutrophil cytoplasmic antibody(ANCA) associated vasculitis, Takayasu arteriitis, infective endocarditis, gangrene associated with procoagulant states due to malignancy, anticardiolipin antibody syndrome and disseminated intravascular coagulation are some of the important causes of gangrene of the extremities. Rare causes of gangrene include heparin induced thrombocytopenia (HIT), haemolytic uremic syndrome (HUS) and Human immunodeficiency virus (HIV) infection.
HIV infection involves all the systems of the body, and the cardiovascular system is no exception. HIV vasculopathy was first described as an entity in 1987 (Joshi et al., 1987). HIVassociated vasculopathymay present with arterial occlusive disease, aneurysmal disease, aortic dissection or spontaneous arteriovenous fistula (Mulaudzi, 2005; Nair, 1999, 2000, 2001; Pantula, 2009). Vasculitis is one of the less common but important consequences of HIV (Chetty, 2001). Aneursymal disease is thought to be due to vasculitis of vasa vasorum leading to inflammation with resultant transmural fibrosis or necrosis and the formation of true or false aneurysms (Mulauldzi et al., 2005). HIV-related thrombosis, which is segmental, shows a histological picture identical to that seen in aneurysms, with inflammatory changes confined to the vasa vasora with bland organising luminal thrombosis. This strongly suggests that aneurysm and thrombosis are different expressions of the same pathological process.
Peripheral arterial disease (PAD) is more prevalent in the HIV-infected population than in the general population (Periard, 2008). There is a six-fold increased risk for PAD in HIV-infected individuals as well as an earlier onset of the disease compared with HIV-negative patients (Periard et al., 2008). Broad spectrum of rheumatic syndromes are associated with HIV infection ranging from arthralgias to reactive arthritis, myopathy and fibromyalgia to more severe necrotizing vasculitis (Uppal & Achutan, 1997). Almost every pattern and type of vasculitis of small, medium and large vessels has been encountered in the HIV setting (Chetty, 2001). Widespread digital ischemic changes and gangrene of the hands and feet is an uncommon presentation in patients with HIV infection (Roh & Gertner, 1997). There are several reports on HIV associated occlusive vasculopathy from Africa. Reports from India on this subject are limited.
Data of 1311HIV reactive patients was obtained retrospectively from the medical records of department of medicine from the year 2000 to 2010. All HIV positive patients’ records were studied. Records of patients with gangrene were identified and studied.
HIV testing was done by HIV Vironostika (4th Generation ELISA, BioMerieux, France). All HIV reactive samples were then tested by HIV TRIDOT (J Mitra, India), a rapid assay that allows differentiation between HIV 1 and 2. The reactive samples were further confirmed by a Western Blot assay (HIV Blot 2.2, Gene labs Diagnostics, USA). CD4 counts were determined by using the FACS Count (Becton Dickinson, USA). HIV Viral loads were estimated by the AMPLICOR assay (Roche, USA).
In our experience over the last 10 years in a south Indian tertiary care hospital we have seen only 2 cases of HIV associated gangrene out of the 1311 cases HIV infected patients. We describe our experience of one of our patients of HIV infection with gangrene.
A woman aged 32 years presented to the outpatient department of our institute with complaints of pain and blackish discoloration of left foot since 1 month. She did not have diabetes mellitus or hypertension. She was non smoker and non alcoholic. There was no history of coronary artery disease or cerebrovascular diseases.
On examination she was emaciated. Pallor and oral thrush were noted. There was no lymphadenopathy. Vitals were normal. Cardiovascular, abdomen and respiratory systems were normal. Left foot was dry, gangrenous with blackish discoloration (Figure 1). Dorsalis pedis, posterior tibial pulses were not palpable. Sensations decreased on the foot. Hemoglobin was 10 gm/dL, Total leucocyte count was 10,000/cmm with neutrophils 66%, lymphocytes 28%, Eosinophils 4%, and monocytes 2%.Platelets were normal. Peripheralsmear examination showed normocytic, normochromic anemia. Erythrocyte sedimentation rate was 20mm at 1st hour. Liver function tests and renal function tests were normal. Lipid profile was normal. Anti nuclear factor (ANF), perinuclear and cytoplasmic anti neutrophil cytoplasmic antibodies (P and C-ANCA) were negative. Australian Antigen and HCV antibodies were negative.
She was evaluated and was found to have HIV-1 infection. CD4 count was 106 cells/µL. Patient’s spouse was tested reactive for HIV 1 and was receiving HAART. She had one male child who was 8 years old and had been tested non reactive for HIV infection. She was started on highly active antiretroviral therapy (HAART) and low dose prednisolone. Pain reduced and a line of demarcation developed.
HIV-associated vascular disease is a specific disease entity which differs from atherosclerotic disease in various aspects. HIV positive patients with vasculopathy are younger with an average age of 40 years in comparison to 55years in patients with atherosclerotic disease (Botes & Van Marle, 2007). There is also a lower incidence of the typical risk factors for atherosclerosis like smoking, hypertension, hypercholesterolemia and diabetes mellitus in these patients with HIV with peripheral vascular disease. CD4 T-cell
Gangrene involving left foot in a woman aged 32 years with HIV infection
count < 200 cells/μl was found as significant predictor of PAD in HIV (Periard et al., 2008).
Literature suggests that the incidence of symptomatic vasculitis is in the region of 0.4% to 1% of HIV-infected patients (Kaye, 1996; Kakrani et al., 2003). However, we have seen only2 cases of HIV associated gangrene over last 10 years. The reason for the low incidence in our experience is possibly due to the free antiretroviral treatment (ART) at the government funded ART centres in our state. Another reason for the low incidence of peripheral vascular disease is because we did not screen for any subclinical vascular disease in our patients. As observed in our cases CD4 counts in patients with HIV vasculopathy are below normal in more than 90% of patients and the CD4:CD8 ratio is usually reversed, indicative of advanced immunosuppression (Mulaudzi et al., 2005).
Infections, occlusive disease due to a hypercoagulable state, vasculitis are some of the mechanisms suggested for gangrene of the extremities in HIV. There have been reported findings of anti-phospholipid antibody syndrome, deficiencies of free protein S, protein C and anti-thrombin 3, but these have been sporadic reports. Chronic HIV infection, rather than its pharmacologic treatment, induces alterations of markers of endothelial dysfunction (Torriani et al 2008). However some opine that despite minimizing HIV plasma burden and subsequent associated inflammatory damage, antiretroviral medications may independently contribute to endothelial damage (Gaetano et al., 2003; Chai et al., 2005; Shankar et al., 2005).
HIV-infected patients tend to develop decreases in high density lipoprotein cholesterol (HDL-c) and low density lipoprotein cholesterol (LDL-c) levels, followed by an increase in plasma triglyceride levels, independent of any exposure to antiretroviral therapy (Grunfeld et al., 1992).
Viral infections have been implicated in the pathogenesis of systemic vasculitis, and many viruses, including HIV, are associated with vasculitis and occasionally gangrene. As in any immunocompromised state, opportunistic infections are likely in patients with HIV infection. Vasculitis and gangrene may be a manifestation of these opportunistic infections. Cytomegalovirus (CMV), Herpes zoster virus (HZV), toxoplasmosis, pneumocystis, salmonella, and Mycobacterium tuberculosis have all been associated with vasculitis in patients with HIV infection1. The two major mechanisms by which infection is thought to induce a vasculitis are direct microbial invasion, with resultant damage of the vessel wall, and immune mediated injury both humoral and cellular (Chetty, 2001).
Another mechanism for development of vasculitis in patients after start of antiretroviral therapy (ART) is immune-reconstitution inflammatory syndrome (IRIS). IRIS occurs within a few weeks to months after the start of HAART; patients most often present with clinical deterioration while the number of CD4 lymphocytes is increasing and the HIV viral load decreasing. IRIS could be implicated in the pathogenesis of the vascular complications (Venkataramana et al., 2006).
Various abnormalities predisposing to a hypercoagulable state have been detected in HIV patients, including antiphospholipid antibodies, lupus anticoagulant, increased Von Willebrand factor (vWF), deficiency in protein C and S, antithrombin and heparin co-factor. Viral-induced endothelial injury causes increased levels of Von Willebrand factor, total antigenic protein S, plasminogen activator inhibitor (PAI-1), endothelial-derived thrombomodulin and other procoagulant products of endothelial cell activation (Saif et al., 2001). Some of the rare causes which contribute to gangrene in HIV include thrombotic thrombocytopenic purpura (de Man et al., 1997).
HIV-associated arterial occlusive disease is recognised as a specific clinical entity.Median age of patients is between 30 - 40 years (Mulaudzi et al., 2005). Male preponderance has been observed in HIV associated vasculopathy (Mulaudzi et al., 2005). The vessels reported to be commonly affected are arteries of muscles and digits. In HIV related occlusive vascular disease two patterns of presentation have been reported(Robbs& Paruk, 2010). In the first type, there is no antecedent claudication and patients present with acute thrombosis. In this type angiographically, proximal vessels appear normal but there is segmental occlusion with poor distal runoff.The second group with occlusive disease appears to have premature atherosclerotic disease. The majority of the patients present with critical ischaemia with rest pain or gangrene.
In most patients the disease is confined to one limb, for reasons that are obscure. However there are case reports describing gangrene involving upper limbs in children with HIV (Despina et al., 2008). Cases of polyarteritis nodosa (PAN) like systemic necrotizing vasculitis in HIV infected individuals, with digital ischaemia are reported (Kakrani et al., 2003). As vasculopathies occur during the later stages of the HIV disease process and are a marker for advanced disease, a thorough search for opportunistic infections should be made in these patients. There are reports of increase in prevalence of Fournier’s gangrene in patients with HIV in their surgical ward during post HIV era (Elem& Ranjan, 1995). There are case reports from India on gangrene involving unusual sites like breast in HIV patients (Venkatramani et al., 2009.)
HIV associated gangrene may be associated with a known pathogen or trigger, or may occur in the absence of an obvious identifiable agent. To establish an opportunistic infection associated with the vascular pathology either a serological test, staining of smears on light microscopy, cultures, immunohistochemistry testing, and in situ hybridisation tests or viral markers may be done as are relevant based on the clinical presentation.
Work up for autoimmune diseases and procoagulant states- antinuclear antibodies, antiphospholipd antibodies, protein C, S and anti- thrombin III need to be planned in patients with HIV with gangrene.Patients with peripheral arterial disease can be easily and reliably identified by ankle brachial index (ABI) testing, and they presumably are at increased cardiovascular risk, assuming that ABI mortality correlations from the general population can be extrapolated to HIV-infected persons (Periard et al., 2008).
Doppler study of arterial and venous system is essential in all patients with ischemic changes. There is a pathognomonic sign with hypoechoic spotting within the arterial wall best described as string of pearls sign (Woolgar et al., 2002).
Diagnostic angiography in known HIV-positive patients with vasculopathy requires awareness of the manifestations of the disease (Scholtz, 2004). The uncommon sites and multiplicity of vascular involvement usually imply that additional images, with an increased volume of contrast media, will be required. Owing to the increased risk of thrombosis, care should be taken to adequately heparinize these patients during diagnostic or interventional procedures. In a series on vascular involvement in HIV patients, it was noted that the majority of the patients presented with occlusive disease, followed by aneurysms, usually atypical in location, multiple, clustered and with a predilection for the extracranial carotid arteries followed by the thoracoabdominal aorta and superficial femoral arteries(Sholtz, 2004).Angiographic appearances vary in the two types of HIV related occlusive vascular disease. In the acute thrombosis group, the angiographic picture is one in which the proximal vessels are normal, accompanied by aggressive peripheral thrombotic occlusion and very poor distal runoff.
A careful clinicopathological correlation is particularly important in skin biopsies showing a vasculitic reaction. There are five patterns of vasculitis described in HIV i.e. lymphocytic vasculitis(LyV), leucocytoclasticvasculitis(LCV) of small dermal vessels, neutrophilicvasculitis with vascular thrombosis and intradermal suppuration, vasculitis with palisaded neutrophilicneutrophilic and granulomatous dermatitis and large vessel vasculitis affecting subcutaneous vessels (Grayson, 2008).
The last three patterns usually infer systemic involvement. A neutrophilic (leucocytoclastic) vasculopathic reaction in the HIV positive individual may have many potential causes, including infection with HIV per se. Other viral causes of hypersensitivity vasculitis include CMV, hepatitis B virus, and Epstein–Barr virus. The dermal vessels in skin biopsies from cases of HSV or varicella zoster virus (VZV) infection frequently exhibit LCV. Recurrent varicella may even manifest with vasculitis in the absence of epidermal involvement. Drugs that may precipitate LCV include certain HAART agents, penicillin and the sulphonamides. The presence of acute LCV in association with vascular occlusion may signify either HIV-associated mixed cryoglobulinaemia (with small vessel occlusion by cryoglobulinprecipitates) or septic vasculitis (with dermal vascular occlusion by infected microthrombi). The latter may be associated with intradermal abscess formation and/or cutaneous infarction (Carlson, 2005).
HIV-associated vasculopathy is not cited in the World Health Organization (WHO) clinical staging system. As HIV associated vasculopathy and gangrene are pointers to an advanced stage of the disease it would be logical to offer antiretroviral therapy irrespective of the CD4 level. Recent publications have shown that HIV itself as a risk factor for peripheral artery disease and strongly recommended NRTI and NNRTI containing regimens (Ye, 2010).As a class, the protease inhibitors appear to have the greatest negative impact on total cholesterol and triglyceride levels; however, even within this class, certain agents e.g., atazanivir and darunavir do not have an adverse effect on lipids.
The treatment in patients with acute thrombosis type of occlusive disease depends on the clinical presentation. In those patients wherein the limb cannot be salvaged primary amputation is done. Where the limb is salvageable treatment options include endovascular procedures like thrombectomy and thrombolysis or bypass procedures. However, the limb salvage rate has been in the region of 27% (Mulaudzi et al., 2005). This is thought to be due to the fact that it is mainly a vasculitic process with superimposed thrombosis, and removing the thrombus does not, in effect, solve the problem, which has a very high re-thrombosis rate3.
Patients with HIV infection with widespread ischemic necrosis and gangrene may require treatment with corticosteroids (in the event of possible vasculitis), thrombolytic agents (for the thrombotic component), or both, unless there are contraindications to either (Roh & Gertner, 1997).
In those with chronic occlusive type of disease, in almost half of the patients, there may not be a possibility of reconstruction due to poor runoff or the limb may be beyond salvage. In others standard bypass procedures can be performed and only recently have endovascular procedures been attempted. In this group, the results are marginally better than in the patients with acute thrombosis type of occlusive disease. (Robbs & Paruk, 2010).
Patients should be optimized as per standard practice prior to surgical or endovascular intervention. Vascular surgical principles should be adhered to when managing patients with HIV-associated occlusive vasculopathy and management should be individualized. Various authors have identified a CD4 T-cell count of less than 200 cells/μl is an important risk factor for postoperative complications (Lin et al., 2004; Albaran, et al., 1998). A significant complication in all patients in whom surgery is attempted has been the high rate of superficial wound sepsis and graft sepsis. Some centers with vast experience with surgical procedures for occlusive vascular disease in HIV have reported a perioperative mortality of 6.95% (Van Marle et al., 2009). Primary amputation is usually done for very advanced arterial disease.
One of the unforeseen consequences of untreated long standing infection with HIV is the appearance of various rheumatic syndromes. Clinicians encountering a patient with gangrene should consider HIV as an important cause and screen for presence of HIV antibodies. Early detection of the HIV infection and appropriate and timely start of HAART can possibly prevent vascular complications. In every patient, workup for both traditional and specific risk factors including opportunistic infections and prothrombotic states should be done. Appropriate treatment for the ischemic limb as well as treatment of opportunistic infections and HAART when indicated may significantly improve the outcomes. Screening for subclinical vascular disease in asymptomatic patients with simple measures like ankle brachial index and appropriate timely treatment will go a long way in preventing the development of gangrene.
I sincerely thank my husband Dr. P. Narasingarao, Cardiac Surgeon at Kamineni Hospital, King Koti for the support and guidance during the preparation of the manuscript.
Slope instability is one of the most common forms of dam failure. Traditional slope stability analysis methods mainly depend on deterministic analysis, including limit equilibrium analysis and finite-element (FE) analysis. Equilibrium methods mainly include the ordinary method of slices, Bishop’s modified method, force equilibrium methods, Janbu’s generalized procedure of slices, Morgenstern and Price’s method, and Spencer’s method. All the equilibrium methods assume that the soil can be divided into slices, which is an artificial distinction. This assumption is the main characteristic that distinguishes different limit equilibrium methods. The main advantage of equilibrium methods is that they involve relatively simpler calculation, which leads to wide use [1, 2, 3, 4].
While the finite element method is another powerful approach for slope stability analysis, it can better reflect the stress–strain relationship of soils than the equilibrium methods. Slope failure in the finite-element model occurs naturally through the area in which the shear strength of the soil is insufficient to resist the shear stresses. There are several advantages of a FE approach to slope stability analysis over traditional limit equilibrium methods: (a) there is no assumption about the shape or location of the failure surface, (b) there are no slices and slice side forces, and (c) the FE method is able to monitor progressive failure up to and including overall shear failure [5, 6].
For a practical slope, not only the stress–strain relationship of soils but also the uncertainty of soil properties should be taken into consideration. However, traditional slope stability analysis methods always ignore the uncertainty and randomness of dam materials, which may overestimate the stability of dams. Attention was drawn to probabilistic slope stability analyses [7, 8]. Most probabilistic slope stability analyses continue to use classical slope stability analysis techniques which are mainly based on the equilibrium methods [9, 10, 11, 12]. An obvious deficiency of the traditional slope stability methods is that the shape of the failure surface is always fixed; therefore, the failure mechanism is not allowed to look for the most critical path through the soil. Besides, these traditional methods cannot take the importance of spatial correlation and local averaging of statistical geotechnical properties into consideration [13, 14, 15].
A more rigorous method, in which nonlinear finite-element methods are combined with random field generation techniques, called the random finite-element method (RFEM), was proposed by Griffiths and Fenton [16]. It can fully account for spatial correlation and averaging and is also a powerful slope stability analysis tool that does not require a priori assumptions relating to the shape or location of the failure mechanism.
In this chapter, a deterministic slope stability analysis based on strength reduction finite-element method is introduced first. After that, the slope is investigated using simple probabilistic methods, including first-order second-moment (FOSM) method, first-order reliability method (FORM), and Monte Carlo method. Further, RFEM is shown, in which spatial correlation and local averaging are illustrated in detail. Finally, the RFEM is applied to slope stability risk assessment, and the results can lead to higher probabilities of failure.
Deterministic slope stability analysis in this chapter is based on FE analysis. The program used is called SLOPE64 [6]. This program is for two-dimensional plane strain analysis. The soil is assumed to follow a linear elastic-perfectly plastic behavior characterized by the Mohr-Coulomb shear failure criterion. In the gravity load generation, the stiffness matrix generation, and the stress redistribution procedure, the program uses eight-node quadrilateral elements with simplified integration (four Gauss points per element). Initially, the soil is assumed to be elastic, and the model generates normal and tangential stresses at all Gauss points in the grid. These stresses are then compared with the Mohr-Coulomb failure criterion. If the stress at a particular Gauss point is within the Mohr-Coulomb failure envelope, it is assumed that the position remains elastic. If the stress is on or outside the failure envelope, it is considered that the point is yielding. The yield stresses are redistributed in the whole grids by the viscoplastic algorithm. Overall shear failure occurs when a sufficient number of Gauss points have yielded to allow a mechanism to develop [5, 6].
The soil model used in this program consists of six parameters, as shown in Table 1.
ϕ′ | Friction angle |
c′ | Cohesion |
ψ | Dilation angle |
E′ | Young’s modulus |
ν′ | Poisson’s ratio |
γ | Unit weight |
Six parameters for soil model.
The Mohr-Coulomb failure criterion used in this program can be written as follows:
where
The failure function F can be described as follows:
F < 0 stresses inside the failure envelope (elastic).
F = 0 stresses on the failure envelope (yielding).
F > 0 stresses outside the failure envelope (yielding and must be redistributed).
The FS of a soil slope is defined as the ratio between the strength of the soils and the actual load. It is exactly the same as that used in traditional limit equilibrium methods. The factored shear strength parameters cf′ and ϕf′ are therefore given by
In this program, in order to find the actual FS, it is necessary to start a systematic search for FS values that will cause the slope to fail. This is achieved by the program that repeatedly solves problems using a sequence of user-specified FS values.
Figure 1 shows a homogeneous slope with a foundation layer. The height of the slope (H) is 10 m, and the thickness of the foundation layer is H/2, 5 m. Soil parameters are shown in Table 2.
Homogeneous slope with a foundation layer.
ϕ′ | c′ | ψ | E′ | ν′ | γ |
---|---|---|---|---|---|
20° | 10 kPa | 0 | 10,000 kN/m2 | 0.3 | 20 kN/m3 |
Soil parameters.
Figure 2 shows the undeformed mesh of the homogeneous slope. The slope is inclined at an angle of 26.578° (2:1). The left boundary is fixed horizontally but is free along the vertical direction, and the base boundary is fixed in both directions. Gravity loads were applied to the mesh, and the trial FS gradually increased until convergence could not be achieved within the iteration limit. The deformed mesh and the nodal displacement vectors are shown in Figure 3(a) and (b), respectively. The critical FS is calculated to be 1.34.
Undeformed mesh of a homogeneous slope with a foundation layer.
(a) Deformed mesh of a homogeneous slope with a foundation layer; (b) nodal displacement vectors.
In this section, a homogeneous slope and an infinite slope are investigated using simple and classical probabilistic slope stability methods, including FOSM, FORM, and Monte Carlo method. These methods are illustrated one by one in detail followed by a simple example, respectively.
The FOSM method is a relatively simple method of including the effects of variability of input variables on a resulting dependent variable [17, 18]. It is basically a formalized methodology based on a first-order Taylor series expansion. This expansion is truncated after the linear term. The modified expansion is then used, along with the first two moments of the random variable(s), to determine the values of the first two moments of the dependent variable [19, 20, 21].
Consider a function f (X, Y) of two random variables X and Y. The Taylor series expansion of the function about the mean values (μX, μY) gives
where derivatives are evaluated at (μX, μY).
To a first order of accuracy, the expected value of the function is given by
and the variance by
Hence,
where E[X] and E[Y] are the expected values of X and Y, respectively; Var[X] and Var[Y] are the variances of X and Y, respectively; Cov[X,Y] is the covariance of X and Y, and Cov[X,Y] = E[(X-E[X])(Y-E[Y])].
If X and Y are uncorrelated,
In general, for a function of n uncorrelated random variables, the FOSM method tells us that
where the first derivatives are evaluated at the mean values (μX1, μX2,. .., μXn).
Here is another example on the homogeneous slope presented in Section 2.3; a probabilistic analysis using FOSM is investigated on this slope. The shear strength parameters are as follows:
According to Eqs. 4 and 7, the expect and variance of FS can be expressed as
Using a central difference estimate of the derivatives with perturbations of ±σ, then
where
Using program SLOPE64, FS calculated for each case is shown in Table 3.
ϕ′ | c′ | FS | ||
---|---|---|---|---|
μϕ′, μc′ | 20.0 | 10.0 | 1.34 | μFS = 1.34 |
μϕ′ + σϕ′, μc′ | 23.0 | 10.0 | 1.50 | ΔFSϕ′ = 0.3 |
μϕ′ - σϕ′, μc′ | 17.0 | 10.0 | 1.20 | |
μϕ′, μc′ + σc′ | 20.0 | 13.0 | 1.48 | ΔFSc′ = 0.28 |
μϕ′, μc′ - σc′ | 20.0 | 7.0 | 1.20 |
Factor of safety for five cases.
So, the variance of FS can be calculated by
Hence
Assume that the FS probability density function is normal distribution (as shown in Figure 4).
Normal distribution of FS.
Consider a “performance function” for this problem in which failure is defined when M < 0, the reliability index β in this case is given by
There are three different approaches calculating the reliability index β listed as follows.
For nonnegative loads and resistances (typical in geotechnical engineering), an alternative definition of the performance function could be
so that failure occurs when M < 0 as before.
Once more assuming R and Q are uncorrelated, the FOSM method gives
Hence
In the classical “resistance” versus “load” problem, the performance function can be defined as
Assuming R and Q are uncorrelated, the FOSM method gives
And
Hence
For nonnegative loads and resistances (typical in geotechnical engineering), an alternative definition of the performance function could be
so that failure occurs when M < 0 as before.
Once more assuming R and Q are uncorrelated, the FOSM method gives
Hence
Apparently, the reliability index β differs with the definition of the performance function, which is one of the major drawbacks of FOSM. Also, the method takes no account of the form of the probability density function describing the random variables, using only their mean and standard deviation, which ignores the effect of distribution of random variables to the results.
The major drawback to the FOSM method when used to compute probabilities relating to failure is that it can give different failure probabilities for the same problem [19, 22], which caused Hasofer and Lind to develop an improved approach, FORM [23].
As shown before, the reliability index β is given as
which measures how far the mean of the safety margin M is from zero (assumed to be the failure point) in units of number of standard deviations. The interesting point is on the probability that failure occurs, that is, M < 0. Therefore, a unique relationship between the reliability index (β) and the probability of failure (pf) is given by
where Φ is the standard normal cumulative distribution function. The point, line, or surface defined by M = 0 is called the failure surface.
The inconsistency of the FOSM method is due to that different definitions of margin M may have different mean estimates and different first derivatives. What the FOSM method does is calculating the distance from the average point to the failure surface in the gradient direction of the average point [18]. Hasofer and Lind solved the inconsistent problem by looking for the overall minimum distance between the average point and the failure surface, rather than just along the gradient direction [23].
In the general case, suppose that the safety margin M is a function of a sequence of random variables
which is the minimum distance between the failure surface (M = 0) and the mean point (E [X]) in units of number of standard deviations. For example, if M = f(X), then Eq. (24) simplifies to
Figure 5 gives an example for an infinite slope. In this example, H = 5 m, γ = 20 kN/m3, α = 30°, c′ and tanϕ′ are lognormal random variables with μc′ = 10 kPa, σc′ = 3 kPa (νc′ = 0.3) and ϕ′ = 30°, μtanϕ′ = 0.5774, σtanϕ′ = 0.1732 (νtanϕ′ = 0.3); the logarithmic normal distributions of c′ and tanϕ′ are shown in Figure 6.
Infinite slope.
Logarithmic normal distributions of (a) c′ and (b) tanϕ′.
FS for this slope can be expressed as follows [24]:
where H is the height of the slope, γ is the saturated unit weight, α is the slope angle to the horizontal direction, c′ is the effective cohesion, and ϕ′ is the effective friction angle.
Using Eq. (25), it can be calculated that
In practical applications, there are many different complex optimization algorithms, usually involving the gradient of M, which can find the point where the failure plane is perpendicular to the origin. The distance between these two points is β [25, 26]. Now, many spreadsheet programs include algorithms that allow users to specify only the minimum equations and constraints on the solution. Unfortunately, nonlinear failure surfaces can sometimes have multiple local minima, with respect to the mean point, which further complicates the problem. In this case, techniques such as simulated annealing may be necessary, but which still do not guarantee finding the global minimum. Monte Carlo simulation is an alternative means of computing failure probabilities which is simple in concept. Furthermore, it is not limited to first order and can be extended easily to very difficult failure problems with only a penalty in computing time to achieve a high level of accuracy [16].
The Monte Carlo method is a broad computational algorithm that relies on repeated random sampling to obtain numerical results. The basic concept is to use random numbers (sometimes pseudo-random numbers) to solve problems that might be deterministic in principle. This method was proposed in the 1940s and has been widely used in slope stability probability analysis [12, 27, 28, 29].
The idea of the Monte Carlo method is to randomly generate samples according to an input probability density function and evaluate the model response of each sample by a deterministic computational model. Consider the problem of determining the probability of failure of a system which has two random inputs, X1 and X2. The response of the system to these inputs is then defined as a function g (X1, X2). Obviously, the function g (X1, X2) is also random because the input variables are random. Assume that system failure will occur when g(X1, X2) > gcrit, where gcrit represents the critical state. In the space of (X1, X2) values, there will be some region in which g (X1, X2) > gcrit, and the problem boils down to assessing the probability that the particular (X1, X2) which actually occurs will fall into the failure region. So the probability pf can be defined as
Figure 7 shows the algorithm for Monte Carlo analysis of slope stability.
Algorithm for Monte Carlo analysis of slope stability.
Consider the same infinite slope given in Figure 5, c′ and tanϕ′ are lognormal random variables with μc′ = 10 kPa, σc′ = 3 kPa (νc′ = 0.3) and ϕ′ = 30°, μtanϕ′ = 0.5774, σtanϕ′ = 0.1732 (νtanϕ′ = 0.3), which are the same with the previous example. It can be calculated that
In this part, a new parameter spatial correlation and the local averaging method are illustrated first. After that, random finite-element method is presented. Finally, results from a full RFEM method are analyzed. Throughout this section, the probability of failure (pf) is compared with the traditional FS that would be obtained from charts or classical limit equilibrium methods.
In probabilistic slope stability study, the shear strength c and ϕ are assumed to be characterized statistically by a normal distribution or lognormal distribution defined by means μc and μtanϕ and standard deviations σc and σtanϕ. The probability of the strength that is less than a given value can be found from standard normal distribution table. When the variables are characterized by lognormal distribution, the lognormal can be transformed to normal as follows (take c for example):
The lognormal parameters μlnc and σlnc given μc and σc are obtained via the transformations:
in which the coefficient of variation of c, νc, is defined as
Unlike the former simulation, another parameter, the spatial correlation length θc or θlnc, will be considered in the following study. The spatial correlation length describes the significant correlation distance between spatially random values in the Gaussian field. Thus, a small value of θ refers to a ragged field, while a large value refers to a smooth field. In practice, the spatial correlation length can be estimated from a set of shear strength data (c and ϕ) taken over some spatial region simply by performing the statistical analyses on the data.
It has been suggested that typical νc values for undrained shear strength lie in the range of 0.1–0.5. The spatial correlation length, however, is less well documented and may well exhibit anisotropy, especially when soils are typically horizontally layered. To simplify in this chapter, the spatial correlation will be assumed to be isotropic.
The local average subdivision (LAS) method is a fast and accurate method that produces realizations of a discrete local average random process [30]. Consider a random process Z; Table 4 presents the local average procedure via the LAS method.
Stage 0 | ||||||||||||||||
Stage 1 | ||||||||||||||||
Stage 2 | ||||||||||||||||
Stage 3 | ||||||||||||||||
Stage 4 |
Procedure of the LAS method.
The algorithm proceeds as follows:
Generate a normally distributed random number
Subdivide
Their variances meet the requirements of local averaging theory.
The relationship between
The means of
Subdivide each cell in stage 1 into another two equal parts; the means and variances should be satisfied with the above three criteria, and another new requirement,
The above steps are repeated, and the cell is subdivided gradually until the size of the subunit reaches the expected requirement.
Using the RFEM approach to analyze a slope, each element is assigned a constant property, including the mean, standard deviation, and spatial correlation length of the shear strength, at each realization of the Monte Carlo process. The assigned property represents an average over the area of each finite element used to discretize the slope. If the point distribution is normal, local arithmetic averaging is used which results in a reduced variance but the mean is unaffected. In a lognormal distribution, however, local geometric averaging is used, and both the mean and the standard deviation are reduced by this form of averaging as is appropriate for situations in which low-strength regions dominate the effective strength. The reduction in both the mean and standard deviation is from
The mean of a lognormally random variable depends on both the mean and the variance of the underlying normal log variable:
Obviously, local averaging has a great influence on the form of a reduced mean and standard deviation. These adjustments are fully accounted for in the following RFEM analysis.
A powerful and general method of accounting for spatially random shear strength parameters and spatial correlation is the RFEM, which combines elastoplastic finite-element analysis with random field theory generated using the LAS method. Figure 8 shows a typical finite-element mesh for the test problem considered in this section. Most of the elements are square, and the elements adjacent to the slope are degenerated into triangles. Taking full account of element size in the local averaging process, the random field of shear strength values was generated and mapped onto the finite-element mesh. In a random field, the value assigned to each finite element is a random variable. The random variables can be correlated to one another by controlling the spatial correlation length θlnc as described previously. Figure 9a,b, and c shows the typical meshes corresponding to different spatial correlation lengths. Figure 9a shows a relatively low spatial correlation length of θ = 1, Figure 9b shows a medium spatial correlation length of θ = 5, and Figure 9c shows a relatively high spatial correlation length of θ = 10. In these figures, light regions represent weak- or low-strength soils, while dark regions represent strong- or high-strength soils. The shear strength distributions of these three cases come from the same lognormal distribution, and the only difference is the spatial correlation length. The slope stability analyses use the Tresca failure criterion which is an elastic-perfectly plastic stress–strain law. When the stresses exceed the yield stress, the program attempts to redistribute excess stresses to neighboring elements that still have reserves of strength. This process is iterative and will continue until the Tresca criterion and global equilibrium are satisfied at all points within the mesh under quite strict tolerances. Plastic stress redistribution is accomplished using a viscoplastic algorithm with eight-node quadrilateral elements and reduced integration in both the stiffness and stress redistribution parts of the algorithm [5, 6].
Undeformed mesh of a homogeneous slope with a foundation layer.
Deformed mesh at slope failure for three different spatial correlation lengths. (a) θ = 1; (b) θ = 5; (c) θ = 10.
Figure 9 shows three typical random field realizations and the associated failure mechanisms for slopes with θ = 1, 5, and 10. It can be concluded that spatial correlation length has a great influence on the failure surface morphology. When θ is low, the shear strength between neighbored elements varies severely; when θ is high, similar properties can be found between neighbored elements. In the RFEM approach, the failure mechanism is free to seek out the weakest path through the soil. Thus, the failure surface will tend to pass through elements with weaker shear strength.
In the following part, the two parameters of shear strength, c and ϕ, are defined as the random variable, respectively, to investigate the influence of spatial correlation length and coefficient of variance on the probability of failure.
Defining friction angle as a deterministic parameter, ϕ = 20°, and then fixing the mean of cohesion with μc = 10 kPa, Figure 10 shows the probability of failure pf as a function of the spatial correlation length θ for a range of coefficients of variation, with the mean cohesion fixed at μc = 10 kPa. Figure 11 shows the relationship between probability of failure pf and the coefficient of variation νc with two different spatial correlation lengths. It can be seen from Figure 10 that the probability of failure can be divided into two branches, with the probability of failure tending to unity or zero for higher and lower values of νc, respectively. Figure 11 demonstrates that when θ becomes large, the probability of failure is overestimated (conservative) when the coefficient of variation is relatively small and underestimated (unconservative) when the coefficient of variation is relatively high. The RFEM results show that the inclusion of spatial correlation and local averaging in this case will always lead to a smaller probability of failure.
Probability of failure versus spatial correlation length (the mean of cohesion is fixing at μc = 10 kPa).
Probability of failure versus coefficient of variance (the mean of cohesion is fixing at μc = 10 kPa).
Defining cohesion as a deterministic parameter, c = 10 kPa, and then fixing the mean of friction angle with μϕ = 20°, Figures 12 and 13 show the effect of the spatial correlation length θ and the coefficient of variation νϕ on the probability of failure for the test problem. It is obvious that Figures 12 and 13 show similar tendency with Figures 10 and 11. Comparing Figures 11 and 13, it can be concluded that the influence of spatial correlation length of ϕ on the probability of failure is less than that of c.
Probability of failure versus spatial correlation length (the mean of friction angle is fixing at μϕ = 20°).
Probability of failure versus coefficient of variance (the mean of friction angle is fixing at μϕ = 20°).
Defining cohesion c and friction angle ϕ as random parameters, and then fixing the mean of cohesion with μc = 10 kPa and the mean of friction angle with μϕ = 20°, Figure 14 shows the probability of failure versus spatial correlation length with different coefficients of variance of c and ϕ. Clearly, Figure 14 shows similar tendency with Figures 10 and 12. Figure 15 shows the probability of failure pf as a function of coefficient of variance νc for two different θ = 2 and 10, with the mean cohesion fixed at μc = 10 kPa, the mean of friction angle fixing at μϕ = 20° and νϕ fixing at 1. Similarly, Figure 16 shows the probability of failure pf as a function of coefficient of variance νϕ for two different θ = 2 and 10, with the mean cohesion fixed at μc = 10 kPa, the mean of friction angle fixing at μϕ = 20° and νc fixing at 1. Clearly, these two figures show a similar relationship with Figures 11 and 13. It is worth noting that defining ϕ as random has an apparent influence on the probability of failure versus coefficient of variance of cohesion. From Figure 15, pf is relatively higher than the case that only c is the only random parameter.
Probability of failure versus spatial correlation length (the mean of cohesion is fixing at μc = 10 kPa and the mean of friction angle is fixing at μϕ = 20°).
Probability of failure versus coefficient of variance of cohesion (the coefficient of variance of friction angle is fixing at νϕ = 1).
Probability of failure versus coefficient of variance of friction angle (the coefficient of variance of friction angle is fixing at νc = 1).
This chapter presents a deterministic slope stability analysis based on strength reduction finite-element method first. After that, three simple probabilistic methods, including FOSM, FORM, and Monte Carlo method, are introduced to perform a simple probabilistic slope stability analysis. Finally, the RFEM approach combining random field generation techniques and finite-element methods is shown and applied to slope stability risk assessment.
The elastoplastic finite-element slope stability method makes no a priori assumptions about the shape or location of the critical failure mechanism, offering significant benefits over traditional limit equilibrium methods on slope stability analysis.
FOSM, FORM, and Monte Carlo method are relatively basic and practical probabilistic analysis methods. Based on different algorithms, the uncertainty and randomness of the soil properties, especially the mean and standard deviation, can be taken into account from different views. However, there are some deficiencies, such as limit of accuracy and time-consuming on these methods.
The RFEM approach combines finite-element slope stability method and local averaging subdivision method, which can take full account of spatial correlation and local averaging. The influence of spatial correlation length and coefficient of variance on the probability of failure can be studied using a parametric approach. In the elastoplastic RFEM, the failure mechanism is free to seek out the weakest path through the soil, which leads to higher probabilities of failure than that conducted by finite-element local averaging alone.
In summary, simplified probabilistic analysis in which spatial variability is ignored can lead to unconservative estimates of the probability of failure, while the RFEM approach that considers spatial correlation and local averaging would be a practical method on slope stability risk assessment.
The author wishes to acknowledge the support from Professor D.V. Griffiths for his supervision during the period of the author’s visiting scholar at Colorado School of Mines. The author also acknowledges the support of the National Key Research and Development Program of China Grant No. 2018YFC1508602 and National Natural Science Foundation of China Grant No. 51539006.
Edited by Jan Oxholm Gordeladze, ISBN 978-953-51-3020-8, Print ISBN 978-953-51-3019-2, 336 pages,
\nPublisher: IntechOpen
\nChapters published March 22, 2017 under CC BY 3.0 license
\nDOI: 10.5772/61430
\nEdited Volume
This book serves as a comprehensive survey of the impact of vitamin K2 on cellular functions and organ systems, indicating that vitamin K2 plays an important role in the differentiation/preservation of various cell phenotypes and as a stimulator and/or mediator of interorgan cross talk. Vitamin K2 binds to the transcription factor SXR/PXR, thus acting like a hormone (very much in the same manner as vitamin A and vitamin D). Therefore, vitamin K2 affects a multitude of organ systems, and it is reckoned to be one positive factor in bringing about "longevity" to the human body, e.g., supporting the functions/health of different organ systems, as well as correcting the functioning or even "curing" ailments striking several organs in our body.
\\n\\nChapter 1 Introductory Chapter: Vitamin K2 by Jan Oxholm Gordeladze
\\n\\nChapter 2 Vitamin K, SXR, and GGCX by Kotaro Azuma and Satoshi Inoue
\\n\\nChapter 3 Vitamin K2 Rich Food Products by Muhammad Yasin, Masood Sadiq Butt and Aurang Zeb
\\n\\nChapter 4 Menaquinones, Bacteria, and Foods: Vitamin K2 in the Diet by Barbara Walther and Magali Chollet
\\n\\nChapter 5 The Impact of Vitamin K2 on Energy Metabolism by Mona Møller, Serena Tonstad, Tone Bathen and Jan Oxholm Gordeladze
\\n\\nChapter 6 Vitamin K2 and Bone Health by Niels Erik Frandsen and Jan Oxholm Gordeladze
\\n\\nChapter 7 Vitamin K2 and its Impact on Tooth Epigenetics by Jan Oxholm Gordeladze, Maria A. Landin, Gaute Floer Johnsen, Håvard Jostein Haugen and Harald Osmundsen
\\n\\nChapter 8 Anti-Inflammatory Actions of Vitamin K by Stephen J. Hodges, Andrew A. Pitsillides, Lars M. Ytrebø and Robin Soper
\\n\\nChapter 9 Vitamin K2: Implications for Cardiovascular Health in the Context of Plant-Based Diets, with Applications for Prostate Health by Michael S. Donaldson
\\n\\nChapter 11 Vitamin K2 Facilitating Inter-Organ Cross-Talk by Jan O. Gordeladze, Håvard J. Haugen, Gaute Floer Johnsen and Mona Møller
\\n\\nChapter 13 Medicinal Chemistry of Vitamin K Derivatives and Metabolites by Shinya Fujii and Hiroyuki Kagechika
\\n"}]'},components:[{type:"htmlEditorComponent",content:'This book serves as a comprehensive survey of the impact of vitamin K2 on cellular functions and organ systems, indicating that vitamin K2 plays an important role in the differentiation/preservation of various cell phenotypes and as a stimulator and/or mediator of interorgan cross talk. Vitamin K2 binds to the transcription factor SXR/PXR, thus acting like a hormone (very much in the same manner as vitamin A and vitamin D). Therefore, vitamin K2 affects a multitude of organ systems, and it is reckoned to be one positive factor in bringing about "longevity" to the human body, e.g., supporting the functions/health of different organ systems, as well as correcting the functioning or even "curing" ailments striking several organs in our body.
\n\nChapter 1 Introductory Chapter: Vitamin K2 by Jan Oxholm Gordeladze
\n\nChapter 2 Vitamin K, SXR, and GGCX by Kotaro Azuma and Satoshi Inoue
\n\nChapter 3 Vitamin K2 Rich Food Products by Muhammad Yasin, Masood Sadiq Butt and Aurang Zeb
\n\nChapter 4 Menaquinones, Bacteria, and Foods: Vitamin K2 in the Diet by Barbara Walther and Magali Chollet
\n\nChapter 5 The Impact of Vitamin K2 on Energy Metabolism by Mona Møller, Serena Tonstad, Tone Bathen and Jan Oxholm Gordeladze
\n\nChapter 6 Vitamin K2 and Bone Health by Niels Erik Frandsen and Jan Oxholm Gordeladze
\n\nChapter 7 Vitamin K2 and its Impact on Tooth Epigenetics by Jan Oxholm Gordeladze, Maria A. Landin, Gaute Floer Johnsen, Håvard Jostein Haugen and Harald Osmundsen
\n\nChapter 8 Anti-Inflammatory Actions of Vitamin K by Stephen J. Hodges, Andrew A. Pitsillides, Lars M. Ytrebø and Robin Soper
\n\nChapter 9 Vitamin K2: Implications for Cardiovascular Health in the Context of Plant-Based Diets, with Applications for Prostate Health by Michael S. Donaldson
\n\nChapter 11 Vitamin K2 Facilitating Inter-Organ Cross-Talk by Jan O. Gordeladze, Håvard J. Haugen, Gaute Floer Johnsen and Mona Møller
\n\nChapter 13 Medicinal Chemistry of Vitamin K Derivatives and Metabolites by Shinya Fujii and Hiroyuki Kagechika
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