COURSE OUTLINE OF CE 425 – M3 FOUNDATION ENGINEERING 1. REVIEW OF THE BASIC FORMULAS OF SOIL MECHANICS 2. SHEAR STRENGTH OF SOIL 3. BEARING CAPACITY OF SOIL A) SHALLOW FOUNDATION B) DEEP FOUNDATION 4. LATERAL PRESSURES A) RETAINING WALLS B) SHEET PILES C) BRACED CUTS 5. SLOPE STABILITY REVIEW OF BASIC FORMULAS IN SOIL MECHANICS Assuming that the weight of the air is negligible, we can give the total weight of the sample as Void ratio (e) is defined as the ratio of the volume of voids to the volume of solids, or Porosity (n) is defined as the ratio of the volume of voids to the total volume, or The degree of saturation (S) is defined as the ratio of the volume of water to the volume of voids. It is commonly expressed as a percentage. The relationship between void ratio and porosity derived: Expressing porosity in terms of void ratio: The common terms used for weight relationships are: Moisture content (w) is also referred to as water content and is defined as the ratio of the weight of water to the weight of solids in a given volume of soil: Unit weight () is the weight of soil per unit volume. The unit weight, , expressed in terms of the weight of soil solids (Ws), moisture content (w), and the total volume (V) For soil engineers, unit weight is defined as the moist unit weight. Dry unit weight (d) is the weight per unit volume of soil, excluding water. The relationship of unit weight, dry unit weight, and moisture content Density equations: and where: = d = M= MS = density of soil (kg/m3) dry density of soil (kg/m3) total mass of the soil sample (kg) mass of soil solids in the sample (kg) The unit of total volume, V, is m3. In , if S = 1 The term relative density is commonly used to indicate the in situ denseness or looseness of granular soil. It is defined as Table Qualitative Description of Granular Soil Deposits Relative density in terms of porosity, n From relations of e and n Relative density in terms of unit weight dry, d Relative density in terms of density, d PLASTICITY AND STRUCTURE OF SOIL Parameters known as the Atterberg limits: Shrinkage limit – the moisture content, in percent, at which the transition from solid to semisolid state takes place. Plastic limit – the moisture content at the point of transition from semisolid to plastic state. Liquid limit – the moisture content at the point of transition from plastic to liquid state. A liquid limit device is used in determining the liquid limit. It consist of a brass cup and a hard rubber base. The brass cup can be dropped onto the base by a cam operated by a crank. Figure Liquid limit test: (c) soil pat before test; (d) soil pat after test The moisture content corresponding to N = 25, determined from The flow curve, gives the liquid limit of the soil. The slope of The line is defined as the flow index Flow index may be written as: where: IF = flow index w1 = moisture content of soil, in percent, corresponding to N1 blows w2 = moisture content corresponding to N2 blows Note: w2 and w1 are exchanged to yield a positive value even though the slope of the flow line is negative. So, equation of the flow line can be written in a general form as where: C = a constant Liquid limit in terms of an empirical equation as proposed By the U.S. Army Corps of Engineers (1949) Where: N = number of blows in the liquid limit device for a 12.7 mm (0.5 in.) groove closure wN = corresponding moisture content tan = 0.121 (but note that tan is not equal to 0.121 for all soils The plastic limit is defined as the moisture content in percent, at which the soil crumbles, when rolled into threads of 4.2 mm (1/8 in.) in diameter. The plastic limit is the lower limit of the plastic stage of soil. The plasticity index (PI) is the difference between the liquid Limit and the plastic limit of a soil, or Burmister (1949) classified the plasticity index in a qualitative manner as follows: - Defined as the moisture content, in percent, at which the volume of the soil mass stops to change as moisture is gradually lost from the soil, it shrinks. per continuing loss of moisture, a stage of equilibrium is reached, that is, more loss of moisture will result in no further volume change - Shrinkage limit tests (ASTM Test Designation D-427) are performed in the laboratory with a porcelain dish: Diameter = 44 mm (1.75 in) Height = 12.7 mm (1/2 inch) Inside of dish is coated with petroleum jelly and then filled completely with wet soil. Excess soil is struck off. The mass of the wet soil inside the dish is recorded then the pat of soil is oven-dried Volume of the oven-dried soil pat is determined by the displacement of mercury (hazardous) ASTM use the method of dipping the oven-dried soil in a melted pot of wax Cool wax-coated soil Volume is then determined by submerging the coated soil in water. - Shrinkage Limit can be determined by the expression: SL = wi(%) - w(%) where: wi = initial moisture content when the soil is placed in the shrinkage limit dish w = change in moisture content (that is, between the initial moisture content and the moisture content at the shrinkage limit) At the same time where: M1 = mass of the wet soil pat in the dish at the beginning of the test (g) M2 = mass of the dry soil pat (g) Also where: Vi = initial volume of the wet soil pat (that is, inside volume of the dish, cm3) Vf = volume of the oven-dried soil pat (cm3) w = density of water (g/cm3) Combining the three equations Another parameter that can be determined from the shrinkage limit test is Shrinkage ratio – the ratio of the volume change of soil as a percentage of the dry volume to the corresponding change in moisture content Expressed as where: V = change in volume M = corresponding change in the mass of moisture It can also be shown that Where: Gs = specific gravity of soil solids Liquidity Index – the relative consistency of a cohesive soil in the natural state and is expressed mathematically as a ratio where: w = in situ moisture content of the soil The in situ moisture content for a sensitive clay may be greater than the liquid limit, making it LI > 1 These soils, when remolded, can be transformed into a viscous form to flow like a liquid Soil deposits that are heavily overconsolidated may have a natural moisture content less than the plastic limit, expressed like LI < 0 Another index that is commonly used for engineering purposes is the consistency index (CI) which is defined as Where: w = in situ moisture content. If w is equal to the liquid limit, the consistency index is zero. Again if w = PI, then CI = 1 Activity, is the slope of the line correlating PI and % finer than 2 m. Activity is expressed as where: A = activity Activity is used as an index for identifying the swelling potential of clay soils Note: percentage of clay-size fraction = % finer than 2 m by weight Redefinition of Activity where: C’ is a constant for a given soil Figure: Simplified relationship between plasticity index and percentage of claysize fraction by weight (Seed, Woodward, and Lundgren, with permission from ASCE) - pertains to the relationship of the plasticity index to the liquid limit of a wide variety of natural soils Important feature of the chart: > The empirical A-line that is given by the equation PI = 0.73(LL – 20) An A-line separates the inorganic clays from the inorganic silts. Inorganic clay values lie above the A-line, and values for inorganic silts lie below the A-line… Figure: Plasticity chart * U-line is above the A-line. * U-line is approximately the upper limit of the relationship of the plasticity index to the liquid limit for any known soil. * Equation of the U-line can be given as CLASSIFICATION OF SOILS Textural Classification Soil texture (in general sense), refers to its surface appearance It is influenced by size of the individual particle present In the textural classification system, the soils are named after their principal components, such as sandy clay, silty clay, etc. • Sand size: 2.0 to 0.05 mm in diameter • Silt size: 0.05 to 0.002 mm in diameter • Clay size: smaller than 0.002 mm in diameter Example: The particle-size distribution of soil A shows 30% sand; 40% silt; 30% clay-size particles. Classify the soil using the USDA. Clay Silt Sand AASHTO Classification System It is developed in 1929 as the Public Road Administration classification system. 2. Plasticity: The term silty is applied when the fine fractions of the soil have a plasticity index of 10 or less. The term clayey is applied when the fine fractions have a plasticity index of 11 or more. 3. If cobbles and boulders (size larger than 75 mm) are encountered, they are excluded from the portion of the soil sample from which classification is made. However, the percentage of such material is recorded. Unified Soil Classification System - original form was proposed by Casagrande in 1942 for airfield construction works by Army Corps of Engineers during WW II - Revised in 1952 with the cooperation of the US Bureau of Reclamation - Widely used by the engineers at present (ASTM Test Designation D-2487) The system classifies soils into two broad categories: 1. Coarse-grained soils that are gravelly and sandy in nature with less than 50% passing through No. 200 sieve. - Group symbols start with prefixes of : G or S where: G - stands for gravel or gravelly soil S – for sand or sandy soil 2. Fine-grained soils are with 50% or more passing through the No. 200 sieve. - Group symbols start with prefixes of : M or C or O where: M - stands for inorganic silt C – inorganic clay O – for organic silts and clay and Pt - used for peat, muck, & other highly organic soil a Gravels with 5 to 12% fine require dual symbols: GW-GM, GW-GC, GP-M, GP-GC. b Sands with 5 to 12% fines require dual symbols: SW-SM, SW-SC, SP-SM, SP-SC. d If 4 ≤ PI ≤ 7 and plots in the hatched area in Figure 5.3, use dual symbol GC-GM or SC-SM. e If 4 ≤ PI ≤ 7 and plots in the hatched area in Figure 5.3, use dual symbol CL-ML. For proper classification according to this system some of all of the following information must be known: 1. Percent gravel – that is, the fraction passing the 76.2-mm (3”) sieve and retained on the No. 4 sieve (4.75-mm opening) 2. Percent of sand – that is, the fraction passing the No. 4 sieve (4.75-mm opening and retained on the No. 200 sieve (0.075-mm opening) 3. Percent of silt and clay – that is, the fraction finer than the No. 200 sieve (0.075-mm opening) 4. Uniformity coefficient (Cu) and the coefficient of gradation (Cc) 5. Liquid limit and plasticity index of the portion of soil passing the No. 40 sieve OR Fine fraction = F200 Coarse fraction = R200 Gravel fraction = R4 Sand fraction = R200 – R4 SOIL COMPACTION Figure Standard Proctor test equip ment: (a) mold For each test, the moisture content of the compacted soil is determined in the laboratory where with the known moisture content, the dry unit weight can be calculated as where: w(%) = percentage of moisture content Values of d determined from the above equation can be plotted against the corresponding moisture contents to obtain the maximum dry unit weight and the optimum moisture content for the soil. For a given moisture content w and degree of saturation S, the dry unit weight of compaction can be calculated by where: Gs = specific gravity of soil solids d = unit weight of water e = void ratio And or So, For a given moisture content, the theoretical maximum dry unit weight is obtained when no air is in the void spaces – that is, when the degree of saturation equals 100%. Therefore, the maximum dry unit weight at a given moisture content with zero air voids can be obtained by substituting S = 1 to the previous equation to where: zav = zero-air-void unit weight The compaction energy per unit volume used for the standard Proctor test can be given as In SI units In English units If the compaction effort per unit volume of soil is changed, the Moisture-unit weight curve also changes. Useful standard procedures for determining the field unit weight of compaction: 1. Sand cone method 2. Rubber balloon method 3. Nuclear method Sand cone device components: a) Glass or plastic jar with (Ottawa sand inside the jar) b) Metal cone (attached on top of jar) Dry weight of the soil that can be determined by the method: where: W2 = weight of moist soil excavated from the hole W3 = dry weight of the soil W1 = combined weight of jar, the cone, and the sand filling the jar. w = moisture content W4 = combined weight of the jar, the cone, and the remaining sand in the jar W5 = weight of sand to fill the hole and cone Relationships of the weights: W5 = W1 – W4 Volume of the excavated hole can be determined by: where: Wc = weight of sand to fill the cone only d(sand) = dry unit weight of Ottawa sand used Values of Wc and d(sand) are determined from the calibration done in the laboratory. Figure. Glass jar filled with Ottawa Figure. Field unit weight determined by sand Sand and with sand cone attached. Cone method. The dry unit weight of compaction made in the field can be by: The grain-size distribution of the backfill material is an important factor that controls the rate of densification. Brown has defined a quantity called the suitability number for rating the backfill as Where: D10, D20, and D50 = the diameters (in mm) through which, respectively, 10, 20, and 50 % of the material passes The smaller the value of SN, the more desirable the backfill material. The backfill rating system proposed by Brown: Dynamic compaction is a technique that involves the process of primarily dropping a heavy weight repeatedly on the ground at regular intervals. The weight of the hammer ranges from 80 to 360 kN (18 to 80 kip) Height of the hammer drop varies between 7.5 and 30.5 m (25 & 100 ft) The stress waves generated by the hammer drops aid in the densification. The three factors where degree of compaction at a given site is dependent upon: 1. Weight of hammer 2. Height of hammer drop 3. Spacing of locations at which the hammer is dropped. The significant depth of influence for compaction can be determined using: PERMEABILITY Soils are permeable due to the existence of interconnected voids through which water can flow from points of high energy to points of low energy. Importance of the study of permeability of soil: - Necessary for estimating the quantity of underground seepage under various hydraulic conditions; - For investigating problems involving the pumping of water for underground construction; and - For making stability analyses of earth dams and earth-retaining structures that are subject to seepage forces. Kinetic Energy - the ability of the fluid mass to do work by virtue of its velocity 1 W 2 1 2 KE = MV = V 2 g 2 KE Kinetic Head or Velocity Head = = W where: M = mass of the fluid V = velocity of flow W = weight of the fluid V2 2g Bernoulli’s equation applied to porous soil medium, the term containing the velocity head could be neglected because the seepage velocity is small. The expression becomes Where : i = hydraulic gradient L = distance between points A and B – that is, the length of flow over which the loss of head occurred 1856, the year Darcy published an equation for the discharge velocity of water through saturated soils from the expression Resulting into: where: v = discharge velocity, which is the quantity of water flowing in unit time through a unit gross cross-sectional area of soil at right angles to the direction of flow k = hydraulic conductivity (otherwise known as the coefficient of permeability) Combined, Finally, The empirical relationship of sand with a small uniformity coefficient for hydraulic conductivity as proposed by Hazen The equation is based from Hazen’s observations of loose, clean, filter sands. Presence of silt and clay may change the hydraulic conductivity of sand. Equations and relationships of characteristics and properties were made for different types of soils. Hydraulic conductivity and void ratio were also given relations expressed as A recent empirical relationship for k in conjunction with the above Relations by Chapius is In stratified soil, hydraulic conductivity in a given direction may change from layer to layer. An equivalent equation of hydraulic conductivity can be computed to simplify calculations. For horizontal direction of flow in every layer For stratified soil with vertical direction of flow in every layer In the field, the average hydraulic conductivity of a soil deposit in the direction of flow can be determined by performing pumping tests from wells. For an unconfined aquifer Integrating and solving for the permeability coefficient Which can also be written as For a confined aquifer The average hydraulic conductivity for a confined aquifer can also be determined by conducting a pumping test from a well with a perforated casing that penetrates the full depth of the aquifer and by observing the piezometric level in a number of observation wells at various radial distances. Pumping is continued at a uniform rate q until steady state is reached. (Figure is on the next slide) Because water can enter the test well only from the aquifer of thickness H, the steady state of discharge is expressed as For an unconfined aquifer Where: K = hydraulic conductivity Q = pumping discharge For a confined aquifer Where: K = hydraulic conductivity Q = pumping discharge H = depth of confined aquifer SEEPAGE In seepage, isotropic soils are considered. Isotropic soils exhibit properties with the same values when measured along the axes in all directions. A flow net is a graphical representation of the flow of water from upstream to downstream side. It shows the path of water as it travels through the soil. It consists two orthogonal families of curves: the flow lines and the equipotential lines. 1. A flow line is a line along which a water particle will travel from upstream to the downstream side in the permeable soil medium. 2. An equipotential line is a line along which the potential head at all point is equal. If piezometers are placed at different points along an equipotential line, the water level will rise to the same elevation in all of them. Figure 8.3b Completed flow net.. In any flow net, the strip between any two adjacent flow lines is called a flow channel. The rate of seepage through the flow channel is: h1, h2, h3, …hn – piezometric levels corresponding to equipotential lines flow elements (approximate squares) From Darcy’s law, the flow rate = kiA and can be written as And it shows, if flow elements are approximate squares, the drop in the piezometric levels between two adjacent equipotential lines is the same and is called the potential drop. and where: H = head difference between the upstream and downstream side Nd = number of potential drops From the above figure, for any flow channel, H = H1 – H2 and Nd = 6 If Number of flow channels is Nf, then total rate of flow through all the channels per unit length can be expressed as For rectangular flow elements: Condition: the width-to-length ratios for all elements are the same. Mathematical expression becomes and Figure 8.12a A weir. Figure 8.12a Uplift force under a hydraulic structure. The uplift force per unit length measured along the axis of the weir can be calculated by finding the area of the pressure diagram. hydraulic gradient Rate of seepage per unit length of the dam Solving for L using the rate of seepage, q = kiA (unit length of the dam) through the section bf equating q to q