Percentage macronutrient intake in the United States by sex and age [19].
\r\n\t[2] J. V. Moloney, A. C. Newell. Nonlinear Optics. Westview Press, Oxford, 2004.
\r\n\t[3] M. Kauranen, A. V. Zayats. Nonlinear Plasmonics. Nature Photonics, vol. 6, 2012, pp. 737-748.
\r\n\t[4] P. Dombi, Z. Pápa, J. Vogelsang et al. Strong-field nano-optics. Reviews of Modern Physics, vol. 92, 2020, pp. 025003-1 – 025003-66.
\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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\r\n\t[8] Iam Choon Khoo. Nonlinear optics, active plasmonics and metamaterials with liquid crystals. Progress in Quantum Electronics, vol. 38, 2014, pp. 77- 117.
\r\n\t
Proteins are chains of amino acids which are involved in nearly every process in the body. Proteins function as enzymes, transcription factors, binding proteins, transmembrane transporters and channels, hormones, receptors, structural proteins, and signaling proteins [1]. However, the primary role of protein in the diet is to provide amino acids required for the synthesis of new proteins. We especially rely on dietary protein to provide the nine essential amino acids, which cannot be synthesized in the body. Protein intake greater than the dietary recommendations may prevent sarcopenia [2], help maintain energy balance [3], improve bone health [4, 5, 6, 7] and cardiovascular function [8, 9, 10], and aid in wound healing [11]. This chapter focuses on the role of dietary protein, and the associated health benefits, throughout the life cycle.
\nThe current dietary recommendations for protein intake include the estimated average requirement (EAR) [12] and the recommended dietary allowance [12]. For daily protein intake, the EAR for dietary protein is 0.66 g kg−1 day−1, and the RDA is 0.8 g kg−1 day−1 for all adults over 18 years of age. This can become confusing when trying to make recommendations for individuals at different stages of life. Even the Food and Nutrition Board recognizes a difference between what is recommended in the RDA and the level of protein intake needed for optimal health [12]. Therefore, there is a third recommendation for protein called the acceptable daily macronutrient range (ADMR) [13, 14]. The ADMR includes a recommendation for protein intakes ranging from 10 to 35% of daily energy (e.g., calorie intake), which makes the ADMR easier to use when developing dietary recommendations for protein [12].
\nA majority of the adult population in the United States exceeds the minimum recommendations for protein intake [15]. The current dietary protein intake in the United States is approximately 82 g d−1 for men and 67 g d−1 for women [16]. \nTable 1\n details the current protein intake as percent of energy intake in the United States based on sex and age. A majority of dietary protein comes from animal protein (46%), followed by plant protein (30%), dairy (16%), and mixed foods (8%) [16]. There is increasing evidence indicating that consuming dietary protein at levels above the current RDA (0.8 g dietary protein kg body weight−1 day−1) may be beneficial for children, adults, older adults, and physically active individuals [17]. For example, protein intake above the RDA may help reduce the risk of chronic diseases such as obesity, cardiovascular disease, type 2 diabetes, osteoporosis, and sarcopenia [13, 17]. However, high protein intake without a subsequent decrease in carbohydrates attenuates the beneficial effects of dietary protein [18].
\nAge | \nTotal | \nMen | \nWomen | \n
---|---|---|---|
Protein | \n|||
20–44 years | \n15.7 | \n16.1 | \n15.3 | \n
45–64 years | \n15.8 | \n16.0 | \n15.7 | \n
65–74 years | \n16.3 | \n16.6 | \n16.1 | \n
75 years and older | \n15.7 | \n16.1 | \n15.3 | \n
Percentage macronutrient intake in the United States by sex and age [19].
Adequate dietary protein intake is essential to support cellular integrity, growth, and physical function. Although protein malnutrition is not prevalent in the United States, there is little research on optimal protein requirements for health benefits in children. Current EARs are based on the factorial method and the nitrogen balance technique. The factorial method incorporates the estimated nitrogen (protein) requirement plus the rate of protein deposition and an estimate of the efficiency of protein utilization [20] which is derived from adult dietary protein needs [12]. By using the indicator amino acid oxidation method in a group of healthy children 6–11 years old, it was found that the mean and population-safe (upper 95% CI) protein requirements were 1.3 and 1.55 g kg−1 day−1, respectively. This is higher than the 2005 DRI for protein (0.76 and 0.95 g kg−1 day−1, respectively) [12]. A similar study using the nitrogen balance technique also found that protein requirements in children in this age range are above current recommendations at 1.2 g kg−1 day−1 [21]. These higher estimated protein requirements in children seem to be in line with current protein consumption patterns in different pediatric age groups. For instance, children 2–3 years old are currently daily consuming ~3.6 g/kg of ideal body weight, children 4–8 years old are currently consuming ~2.6 g kg−1 ideal body weight−1, and children 9–13 years old are consuming ~1.6 g kg−1 ideal body weight−1 [15]; however, the optimal protein intake for children is still under debate [22]. There are racial/ethnic differences in protein consumption in children (2–18 years old). For example, non-Hispanic black children eat about 5% below, non-Hispanic white children eat about 3% below, Hispanic children eat about 2% below, and Asian children eat less than 1% below the EAR for protein [15].
\nAlthough the currently established recommendations for protein intake in children may be lower than the requirements, the effect of diets higher in protein (e.g., 30% of total energy intake) in children is unclear [22]. Several studies have alluded to the potential benefit of higher protein intake dietary practices. For instance, diets higher in protein with a low glycemic index can be protective against obesity in children aged 5–18 years [23], and diets higher in protein can lead to smaller waist circumference, blood pressure, insulin, and serum cholesterol than lower-protein diets in children from the same age group. A recent cohort analysis found that protein intake in 8-year-olds is associated with higher fat-free mass [24], and an additional cohort analysis found that at ages 11, 15, and 22 years, protein intake is inversely associated with early adulthood BMI. However, protein intake at 2 years was positively associated with BMI and lean mass at age 22 [25], suggesting there are conflicting results regarding the benefits of increased dietary protein in children.
\nPregnancy is a period of rapid tissue growth during a short period of time. Maternal tissues, including breast, uterine, and adipose tissues, blood volume, and extracellular fluids, account for the largest amount of protein accretion during pregnancy at 60%. The remaining 40% of protein accretion occurs within the amniotic fluid, fetus, and placenta [26, 27]. In fact, protein needs to increase soon after conception to support tissue growth and development, maintenance of maternal homeostasis, and lactation preparation [27, 28, 29]. These alterations occur in an exponential way and only in response to adequate total energy intake. This means that protein deposition does not significantly change in the first trimester compared to pre-pregnancy, but increases during the second trimester and significantly increases to the highest levels of protein deposition in the third trimester. This variable period of growth makes it difficult to define recommendations regarding protein requirements. Thus, although current recommendations suggest constant protein intake throughout the duration of pregnancy, pregnancy may actually require an increase in protein intake throughout gestation to support adequate growth, although further research is needed. There are several benefits of protein intake during pregnancy including adequate maternal weight gain within recommendations, lower early pregnancy BMI, and decreased postpartum weight [30].
\nAlthough the benefits of increased protein intake during pregnancy are apparent as stated above, protein requirements during pregnancy are difficult to measure. This is due to the involved nature of some of the techniques used to measure protein requirements. Therefore, the current dietary protein recommendations during pregnancy are based on factorial estimates of recommendations for healthy, nonpregnant populations. Pregnancy protein needs have been derived from the EAR and RDA for healthy, nonpregnant populations and are set to 0.88 g kg−1 day−1 (EAR) and 1.1 g kg−1 day−1 (RDA) [12]. However, newer studies found protein needs to be 1.2 g kg−1 day−1 at 11–20 weeks, increasing to 1.52 g kg−1 day−1 at 30–38 weeks [31]. Both nonpregnant women of childbearing age (20–44 years) and pregnant women consume at or above the current recommendations of protein intake [32, 33]. One study [31] found that pregnant women consume the same amount of protein in early pregnancy (1.44 ± 0.30 g kg−1 day−1) as they do in late pregnancy (1.47 ± 0.53 g kg−1 day−1), not taking fluid retention and changes in body composition into account. These findings support others that have noted little overall change in dietary protein patterns from early to late pregnancy [33]. Collectively, these findings demonstrate that pregnant women meet the recommendations for dietary protein intake. Improvements may potentially be made to increase dietary protein requirements as pregnancy progresses.
\nAn important factor to consider when incorporating protein into the diet is how the source of dietary protein (e.g., protein derived from animal or plant sources) affects nutrient intake, nutrient adequacy, and diet quality [13, 34, 35]. Proteins with differing amino acid profiles exhibit varied digestion and absorption rates [36, 37, 38], and amino acid profiles depend directly on the quality and quantity of the dietary protein [37]. For example, the digestion and absorption rates of fast- (e.g., whey) versus slow (e.g., casein)-digesting proteins need to be taken into consideration when developing protein recommendations. One study provided young, healthy subjects with either a whey protein meal (30 g) or a casein meal (43 g) (both contained the same amount of leucine [one of the BCAAs]) and measured whole-body protein synthesis. Researchers determined that the subjects consuming the whey (fast) protein meal had a high, rapid increase in plasma amino acids, while subjects consuming the casein (slow) protein meal had a prolonged plateau of EAA [39]. In addition, the chemical structure and the presence of anti-nutritional compounds such as phytic acid within the protein source can influence digestion and amino acid availability [40]. Compared to animal sources, plant proteins are shown to have a lower anabolic impact on muscle; however, the reduced ability to elicit anabolic effects can be overcome by increasing protein intake and increasing the content of leucine [41].
\nWhether or not the amino acid source is derived from the whole protein or a mixture of free amino acids can also influence the rate of muscle protein synthesis [42]. For example, when older subjects were given either an EAA mixture (15 g) or a whey protein supplement (13.6 g) after an overnight fast, subjects consuming the EAA mixture had higher mixed muscle fractional synthetic rate [42], which is often associated with increases in muscle mass. The differing response could be due to the differing leucine content between the supplements (EAA, 2.8 g leucine, and whey, 1.8 g leucine) or because the EAA supplement was composed of individual amino acids while the whey protein supplement was intact protein. These subtle differences could influence the rate of appearance of the amino acids into blood circulation and thus the protein synthetic response.
\nAnother potential confounder of the protein synthetic response of various proteins is the form or texture of the protein itself, such as ground beef versus a beef steak [43]. When, older men consumed 135 g of protein as either ground beef or as a beef steak, the amino acids from the ground beef appeared more rapidly in the circulation than the amino acids from the beef steak. Whole-body protein balance was higher after consumption of the ground beef versus the beef steak. However, 6 h after the beef meals, muscle protein synthesis was not different [43]. Nonetheless, these data support that the form of the protein that is being consumed impacts digestion, absorption, and the rate of appearance of amino acids into circulation [35].
\nThe timing of dietary protein intake has received ample attention in the past several decades. Adults typically consume the majority of their protein intake at dinner (38 g) versus breakfast (13 g) [44]. However, recent research suggests that ingestion of more than 30 g of protein in a test meal does not further stimulate the effect of dietary protein on muscle protein synthesis [45]. This had led to discussion related to optimal timing of protein intake. For example, distributing protein intake throughout the day, timing of protein around nighttime eating, and protein eating at breakfast are all areas of increased interest. In general, research covering these topics is performed in young, healthy populations, or aging populations, and very few, if any, studies have been conducted in children and pregnant women.
\nBreakfast is often recognized as the most important meal of the day [46, 47, 48]. However, there is debate as to what defines the ideal breakfast meal [47], in addition to a lack of strong evidence to define which nutrients should be represented at breakfast [47]. A recent commentary published by the American Academy of Nutrition and Dietetics suggests that protein-containing foods (e.g., eggs, lean meat, and low-fat dairy products) should be included in breakfast meals [47]. Literature supports diets higher in protein aid in the treatment of chronic, metabolic diseases such as obesity, type 2 diabetes, and heart disease and have been shown to increase EE, improve satiety, regulate glycemic control, and improve body composition (reviewed in [13, 14, 34, 49]).
\nEating protein at night and immediately before bedtime has received substantial attention in the past decade. Although past common knowledge would claim that eating before bed precipitates negative effects on health and body composition, more recent studies show that there may be many metabolic, health, and body composition-related benefits [50]. Much of the previous research claiming the negative effects of nighttime eating was performed in shift workers [51], populations with night eating syndrome, who consume ≥50% of daily calories after dinner [52], and epidemiological data [53]. Although some of the negative effects of nighttime eating in these populations may include high BMI and abdominal obesity [54]; increased triglyceride concentration, dyslipidemia, and impaired glucose tolerance [55]; impaired kidney function [56]; and increased carbohydrate oxidation and decreased fat oxidation [57], many other factors need to be taken into consideration. For example, these populations are awake during abnormal hours and report sleep disturbances [58, 59]. In fact, the duration of sleep is inversely related to BMI [60, 61]. These populations also consume significantly more carbohydrate, protein, and fat throughout the day. Nonetheless, it is clear that eating large amounts of energy in the evening hours, in particular when the energy is carbohydrate- and fat-laden, may not be beneficial for health and body composition outcomes.
\nHowever, much more evidence has shown that eating a small protein snack (~200 kcal) before bed may elicit significant benefits. Improved muscle protein recovery, muscle mass, and strength gains mediated by enhanced overnight and next-morning muscle protein synthesis have been shown to be enhanced with 40 g of casein protein supplementation in elderly [62] and recreationally active men [63]. These effects are particularly enhanced when this dietary practice is added to the practice of resistance exercise [63]. In addition, reported hunger is lower and satiety is higher, and resting energy expenditure is higher the following morning after a small protein snack compared to a noncaloric placebo [50, 62]. Chronically (4 weeks) there are also reports of decreased blood pressure, decreased arterial stiffness [64], and a greater decrease in body fat in overweight and obese women when consuming nighttime protein [65, 66]. Importantly, these benefits are accompanied by no significant alterations in overnight or next-morning lipolysis, fat oxidation, substrate utilization, or any blood markers in obese men or resistance-trained young women [67].
\nCurrent research demonstrates that even distribution of protein intake throughout the day is more effective at stimulating a 24-h protein synthesis compared to an uneven distribution [68, 69]. This is supported by data from a longitudinal study on nutrition and aging, which found that even distribution of daily protein intake across meals is independently associated with greater muscle strength and higher muscle mass in older adult, but is not associated with loss in muscle mass [70] or mobility [71] over 2–3 years. However, there are some studies that fail to confirm the importance of spreading protein intake out over the course of the day [71, 72]. Additional studies have compared pulse feeding (72% of daily protein at lunch) versus protein being evenly distributed over four daily meals in hospitalized older patients for 6 weeks [73, 74]. These studies found that pulse feeding of protein increased postprandial amino acid bioavailability [75] and increased lean mass [74] compared to spreading protein intake throughout the day. Taken together, the optimal timing and distribution of protein intake still need to be determined.
\nObesity is a major public health concern [76] and is associated with the development of metabolic diseases such as cardiovascular disease, nonalcoholic fatty liver disease, and type 2 diabetes mellitus in both children and adults [77, 78]. Obesity is defined as having a body mass index (BMI) (weight in kilograms divided by height in centimeters squared) greater than or equal to 30.0. In 2015–2016, the prevalence of obesity (\nTable 2\n) in the United States was 39.6 for adults and 18.4% for youth [76]. Obesity also impacts racial and ethnic groups differently. For example, non-Hispanic black and Hispanic adults and youth have higher rates of obesity compared to non-Hispanic white and Asian populations [79].
\nAge group (years) | \nTotal (percent) | \nBoys or men (percent) | \nGirls or women (percent) | \n
---|---|---|---|
Youth, 2–19 | \n18.5 | \n19.1 | \n17.8 | \n
Young children, 2–5 | \n13.9 | \n14.3 | \n13.5 | \n
Youth, 6–11 | \n18.4\n$\n\n | \n20.4\n$\n\n | \n16.3 | \n
Adolescents, 12–19 | \n20.6\n$\n\n | \n20.2 | \n20.9\n$\n\n | \n
Adults, 20+ | \n39.6 | \n37.9 | \n41.1 | \n
Young adults, 20–39 | \n35.7 | \n34.8 | \n36.5 | \n
Middle-aged adults, 40–59 | \n42.8\n*\n\n | \n40.8\n*\n\n | \n44.7\n*\n\n | \n
Older adults, 60+ | \n41.0 | \n38.5 | \n43.1 | \n
Prevalence of obesity in the United States by age group and sex [76].
Significantly different from young children.
Significantly different from young adults.
A primary factor in controlling and preventing obesity and associated chronic diseases is through diet, for example, diets higher in protein [13, 14, 80, 81]. Diets higher in protein (>30% of energy intake) have been shown to improve body composition [82], improve glycemic response [81, 83, 84, 85], increase satiety [85, 86, 87], and increase postprandial energy metabolism [88, 89], which are all mediating factors of weight loss.
\nSarcopenia is the term for age-associated loss of muscle mass and function [35]. The loss of muscle function associated with sarcopenia is often referred to as dynapenia [90]. A loss or reduction in skeletal muscle function often leads to increased morbidity and mortality either directly, or indirectly, via the development of secondary diseases such as diabetes, obesity, and cardiovascular disease [91]. The causes of sarcopenia include poor nutrition, diminished responsiveness to anabolic hormones and/or nutrients, and a sedentary lifestyle.
\nThe loss in muscle mass observed with aging is often accompanied by an increase in fat mass [92], which can happen even in the absence of changes in BMI [35]. The loss in muscle mass results in a decrease in basal metabolic rate (BMR) or the amount of caloric energy we use while at rest [93]. The loss of muscle mass induces a 2–3% decrease in BMR per decade after the age of 20 and a 4% decline in BMR per decade after the age of 50 [93, 94]. Muscle loss and subsequent reduction in metabolic rate contribute to obesity that accompanies the aging process.
\nSeveral studies identify protein as a key nutrient for aging adults [2, 95]. Low protein intake is linked to a decrease in physical ability in aging adults [96]. However, protein intake greater than the dietary guidelines may prevent sarcopenia [96], help maintain BMR [3], improve bone health [4, 5, 6, 7], and improve cardiovascular function [8, 9, 10]. These benefits of increasing protein in the diet may improve function and quality of life in healthy older adults, as well as improve the ability of older patients to recover from disease and trauma [91].
\nCurrently, the dietary recommendations for protein intake are the same for all healthy adults above the age of 19. However, experts in the field of protein and aging recommend a protein intake between 1.2 and 2.0 g kg−1 day−1 or higher for elderly adults [91, 95, 97]. The RDA of 0.8 g kg−1 day−1 is well below these recommendations and reflects a value at the lowest end of the AMDR. It is estimated that 38% of adult men and 41% of adult women have dietary protein intakes below the RDA [16, 44].
\nBoth protein amount and source are important to consider when recommending protein intake to older adults [34, 35]. There are three important aspects to take into consideration when recommending a protein source: (1) the characteristics of the specific protein, such as the amount of essential amino acids (EAA); (2) the food matrix in which the protein is consumed, for example, as part of a beverage or a complete meal; and (3) the characteristics of the individuals consuming the food, including health status, physiological status, and energy balance [34]. In addition, the difference in digestibility and bioavailability of a protein can impact the quantity of protein that needs to be ingested to meet metabolic needs; this is especially important in older adults since gastric motility and nutrient absorption decrease with age. The speed of protein digestion and absorption of amino acids from the gut can influence whole-body protein building [36]. Proteins with differing amino acid profiles exhibit different digestion and absorption rates [36, 38, 98]. Amino acid availability depends directly on both the quality and quantity of the dietary protein [98].
\nOver the past 15 years, the gut microbiome has received increased attention regarding its role in impacting overall health [99]. Interestingly, it has been shown to influence diseases associated with metabolic health [100]. The intestinal mucosa houses nearly a trillion microorganisms, and the plasticity of this environment is highly reactive to changes in diet [101]. For instance, the gut becomes an active site for protein and amino acid metabolism prior to absorption. Following enzymatic denaturation by intestinal proteases, amino acids can become fermented into various metabolites which include short-chain fatty acids and ammonia [102]. The acute microbial response and long-term adaptation associated with dietary habits have become an important area of research.
\nAs gut assay methodologies improve, researchers have identified associations between microbial populations and their metabolite concentrations in response to dietary patterns. For instance, in vitro and human models demonstrate a potential negative link between animal protein intake and protein fermentation end products such as ammonia and trimethylamine-N-oxide [103, 104]. However, favorable outcomes associated with animal- and plant-based protein sources have been observed. For example, ingestion of both whey [105] and pea protein [106] has been shown to increase favorable gut bacterial species such as Bifidobacterium. In addition, supplementation with pea protein intake has been shown to increase the production of short-chain fatty acids, an important energy substrate utilized by enterocytes [106].
\nThere is sufficient evidence that protein intake higher than the current dietary recommendations is beneficial for most healthy individuals throughout the life cycle. However, benefits of dietary protein depend on the quality, the quantity, and the timing of protein intake. Although health benefits of dietary protein have been well-established for older adults, more research is needed to determine the health benefits of increased dietary protein intake through each state of life.
\nThis work was supported by a grant to J.I.B. and E.B. from the Arkansas Biosciences Institute.
\nThe authors have no conflicts of interest to declare.
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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