ADVANCED SOIL MECHANICS

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Jeremy Summers
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1 BERNOULLI S EQUATION h Where: u w g Z h = Total Head u = Pressure = Velocity g = Acceleration due to Graity w = Unit Weight of Water

2 h ADVANCED SOIL MECHANICS BERNOULLI S EQUATION IN SOIL u w g Z 0 0 (i.e. elocity of water in soil is negligible). Therefore: u h Z w

3 h h A h B CHANGE IN HEAD FROM POINTS A & B (H) h h h A B h h A h B Figure 5.1. Das FGE (005). h Where: u A w i Z A u h L B w Z h can be expressed in non-dimensional form i = Hydraulic Gradient L = Length of Flow between Points A & B B

4 VELOCITY () VS. HYDRAULIC GRADIENT (i) General relationship shown in Figure 5. Three Zones: 1. Laminar Flow (I). Transition Flow (II) 3. Turbulent Flow (III) For most soils, flow is laminar. Therefore: i Figure 5.. Das FGE (005).

5 DARCY S LAW (1856) Where: = Discharge Velocity (i.e. quantity of water in unit time through unit cross-sectional area at right angles to the direction of flow) k = (i.e. coefficient of permeability) i = Hydraulic Gradient * Based on obserations of flow of water through clean sands

6 HYDRAULIC CONDUCTIVITY (k) k w K Where: = Viscosity of Water K = Absolute Permeability (units of L ) Typical Values of k per Soil Type Soil Type k (cm/sec) k (ft/min) Clean Grael Coarse Sand Fine Sand Silty Clay Clay < < after Table 5.1. Das FGE (005)

7 DISCHARGE AND SEEPAGE VELOCITIES q A A s Where: q = Flow Rate (quantity of water/unit time) A = Total Cross-sectional Area A = Area of Voids s = Seepage Velocity Figure 5.3. Das FGE (005).

8 n e e V V V V V V V A L A A A A A A A A q s s s s s s s s s 1 1 ) ( ) ( ) ( ) ( Figure 5.3. Das FGE (005). DISCHARGE AND SEEPAGE VELOCITIES

9 EXAMPLE PROBLEM: FIND i AND q GIVEN: 1 ft REQUIRED: Find Hydraulic Gradient (i) and Flow Rate (q) 5 ft CL (Imperious Layer) 1 15 ft SM (k = ft/min) CL (Imperious Layer) 175 ft

10 EXAMPLE PROBLEM: FIND i AND q GIVEN: SOLUTION: Hydraulic Gradient (i): 5 ft 15 ft 1 ft CL (Imperious Layer) 1 i h L i 1 ft 175 ft cos SM (k = ft/min) CL (Imperious Layer) 175 ft Rate of Flow per Time (q): q q q kia ft (0.067)(15 ft)(cos1 min ft / min/ ft )(1 ft)

11 HYDRAULIC CONDUCTIVITY: LABORATORY TESTING Constant Head (ASTM D434) Falling Head (no ASTM) Figure 5.4. Das FGE (005). Figure 5.5. Das FGE (005).

12 HYDRAULIC CONDUCTIVITY: LABORATORY TESTING Constant Head (ASTM D434) Q At A( ki) t Figure 5.4. Das FGE (005). Where: Q = Quantity of water collected oer time t t = Duration of water collection Q k A k QL Aht h t L

HYDRAULIC CONDUCTIVITY: LABORATORY TESTING Falling Head (No ASTM) h dh q k A a L dt Where: A = Cross-sectional area of Soil a = Cross-sectional area of Standpipe Integrate from

13 HYDRAULIC CONDUCTIVITY: LABORATORY TESTING Falling Head (No ASTM) h dh q k A a L dt Where: A = Cross-sectional area of Soil a = Cross-sectional area of Standpipe Integrate from limits 0 to t t al Ak dt Figure 5.5. Das FGE (005). after rearranging aboe equation h log e 1 h al Ak k dh h after integration or.303 al At Integrate from limits h 1 to h Log 10 h h 1

14 HYDRAULIC CONDUCTIVITY: EMPIRICAL RELATIONSHIPS Uniform Sands - Hazen Formula (Hazen, 1930): k( cm / sec) cd 10 Where: c = Constant between 1 to 1.5 D 10 = Effectie Size (in mm) Sands Kozeny-Carman (Loudon 195 and Perloff and Baron 1976): k C 1 e 1 Where: C = Constant (to be determined) e = Void Ratio 3 e Sands Casagrande (Unpublished): k 1.4e k0.85 Where: e = Void Ratio k 0.85 e = 0.85 Normally Consolidated Clays (Samarasinghe, Huang, and Drneich, 198): k C n e 1 e Where: C = Constant to be determined experimentally n = Constant to be determined experimentally e = Void Ratio

15 HYDRAULIC CONDUCTIVITY: EMPIRICAL RELATIONSHIPS EXAMPLE NORMALLY CONSOLIDATED CLAYS GIVEN: Normally consolidated clay with e and k measurements from 1D Consolidation Test. k Void Ratio (e) (cm/sec) x x 10-7 REQUIRED: Find k for same clay with a oid ratio of 1.4. SOLUTION: Using (Samarasinghe, Huang, and Drneich, 198) Equation: k 1 k C C n e 1 1 e 1 n e 1 e Substituting known quantities 0.6cm / sec 1.5cm / sec Example 5.6 Das PGE (005). n.5.

16 k e ADVANCED SOIL MECHANICS HYDRAULIC CONDUCTIVITY: EMPIRICAL RELATIONSHIPS EXAMPLE NORMALLY CONSOLIDATED CLAYS 0.6cm / sec 1.5cm / sec 0.6x10 C k 1 7 C n e1 1 e cm / sec 0.581x cm / sec (0.581x10 7cm / sec) n.5. n x10 7 cm / sec Example 5.6 Das PGE (005).

17 STRATIFIED SOILS: EQUIVALENT HYDRAULIC CONDUCTIVITY HORIZONTAL DIRECTION Considering cross-section of Unit Length 1. Total flow through cross-section can be written as: q q 1 H 1 1 H 1 1 H... n 1 H n Where: = Aerage Discharge Velocity 1 = Discharge Velocity in Layer 1 Figure 5.7. Das FGE (005).

18 STRATIFIED SOILS: EQUIVALENT HYDRAULIC CONDUCTIVITY HORIZONTAL DIRECTION Substituting =ki into q equation and using H to denote Horizontal Direction 1 k H ( eq) k H1 1 i eq i ; k i ;...; H n k n i n Noting that i eq =i 1 =i = =i n k 1 H ( k H k... k H ( eq) H1 1 H Hn n Where: k H(eq) = Equialent in Horizontal Direction H H ) Figure 5.7. Das FGE (005).

19 STRATIFIED SOILS: EQUIVALENT HYDRAULIC CONDUCTIVITY VERTICAL DIRECTION Total Head Loss = h h = Sum Head Loss in Each Layer... k h V ( eq) h 1 1 h H h and V 1... Using Darcy s Law (=ki) into equation and using V to denote Vertical Direction h n n k i... 1 k Vn Where: k V(eq) = Equialent in Vertical Direction i n Figure 5.8. Das FGE (005).

Field Measurements Taken: q, r 1, r, h 1, h q = Groundwater Flow into Well q also is rate

20 FIELD PERMEABILITY TESTING BY PUMPING WELLS UNCONFINED PERMEABLE LAYER UNDERLAIN BY IMPERMEABLE LAYER Figure 5.9. Das FGE (005). Field Measurements Taken: q, r 1, r, h 1, h q = Groundwater Flow into Well q also is rate of discharge from pumping Equation: r 1 q k dh rh dr r k field can be re-written as dr r h q k Soling Equation: h 1 1 hdh r1.303q log10 r ( h h )

dr 1 r r h dh rh dr can be re-written as h kh q dh Figure 5.10. Das FGE (005).

21 FIELD PERMEABILITY TESTING BY PUMPING WELLS WELL PENETRATING CONFINED AQUIFER q = Groundwater Flow into Well q also is rate of discharge from pumping Equation: r q 1 k dr 1 r r h dh rh dr can be re-written as h kh q dh Figure Das FGE (005). Field Measurements Taken: q, r 1, r, h 1, h Soling Equation: k field q log10.77h ( h 1 r1 r h )

22 SOIL PERMEABILITY AND DRAINAGE ADVANCED SOIL MECHANICS after Casagrande and Fadum (1940) and Terzagi et al. (1996).

23 SOIL PERMEABILITY AND DRAINAGE COEFFICIENT OF PERMEABILITY CM/S (LOG SCALE) Drainage property Good drainage Poor drainage Practically imperious Application in earth dams and dikes Perious sections of dams and dikes Imperious sections of earth dams and dikes Type of soil Clean grael Clean sands, Clean sand and grael mixtures Very fine sands, organic and inorganic silts, mixtures of sand, silt, and clay glacial till, stratified clay deposits, etc. Imperious soils e.g., homogeneous clays below zone of weathering Imperious soils which are modified by the effect of egetation and weathering; fissured, weathered clays; fractured OC clays Direct determination of coefficient of permeability Direct testing of soil in its original position (e.g., well points). If properly conducted, reliable; considerable experience required. Constant Head Permeameter; little experience required. (Note: Considerable experience also required in this range.) Constant head test in triaxial cell; reliable w ith experience and no leaks. Reliable; Little experience required Falling Head Per meameter; Range of unstable permeability;* much experience necessary to correct interpretation Fairly reliable; considerable experience necessary (do in triaxial cell) Indirect determination of coefficient of permeability Computat ion: From the grain size distribution (e.g., Hazen s formula). Only applicable to clean, cohesionless sands and graels Horizontal Capillarity Test: Very little experience necessary; especially useful for rapid testing of a large number of samples in the field w ithout laboratory facilities. Computations: from consolidation tests; expensie laboratory equipment and considerable experience required *Due to migration of fines, channels, and air in oids. From FHWA IF Ealuation of Soil and Rock Properties.

= Discharge Velocity in x Direction Figure 5.11. Das FGE (005).

24 LAPLACE'S EQUATION OF CONTINUITY y z x Steady-State Flow around an imperious Sheet Pile Wall Consider water flow at Point A: x = Discharge Velocity in x Direction Figure Das FGE (005). z = Discharge Velocity in z Direction Y Direction Out Of Plane

25 LAPLACE'S EQUATION OF CONTINUITY Consider water flow at Point A (Soil Block at Pt A shown left) Rate of water flow into soil block in x direction: x dzdy Figure Das FGE (005). Rate of water flow into soil block in z direction: z dxdy Rate of water flow out of soil block in x,z directions: x z x x z z dxdzdy dz dxdy

26 0 0 z x or dxdy dzdy dxdy dz z dzdy dx x z x z x z z x x Consider water flow at Point A (Soil Block at Pt A shown left) Figure Das FGE (005). Total Inflow = Total Outflow LAPLACE'S EQUATION OF CONTINUITY

Pt A shown left) Figure 5.11. Das FGE (005).

27 0 z h k x h k z h k i k x h k i k z x z z z z x x x x Consider water flow at Point A (Soil Block at Pt A shown left) Figure Das FGE (005). Using Darcy s Law (=ki) LAPLACE'S EQUATION OF CONTINUITY

28 FLOW NETS: DEFINITION OF TERMS Flow Net: Graphical Construction used to calculate groundwater flow through soil. Comprised of Flow Lines and Equipotential Lines. Flow Line: A line along which a water particle moes through a permeable soil medium. Flow Channel: Strip between any two adjacent Flow Lines. Equipotential Lines: A line along which the potential head at all points is equal. NOTE: Flow Lines and Equipotential Lines must meet at right angles!

29 FLOW NETS FLOW AROUND SHEET PILE WALL Figure 5.1a. Das FGE (005).

30 FLOW NETS FLOW AROUND SHEET PILE WALL Figure 5.1b. Das FGE (005).

31 FLOW NETS: BOUNDARY CONDITIONS 1. The upstream and downstream surfaces of the permeable layer (i.e. lines ab and de in Figure 1b Das FGE (005)) are equipotential lines.. Because ab and de are equipotential lines, all the flow lines intersect them at right angles. 3. The boundary of the imperious layer (i.e. line fg in Figure 1b Das FGE (005)) is a flow line, as is the surface of the imperious sheet pile (i.e. line acd in Figure 1b Das FGE (005)). 4. The equipontential lines intersect acd and fg (Figure 1b Das FGE (005)) at right angles.

32 FLOW NETS FLOW UNDER AN IMPERMEABLE DAM Figure Das FGE (005).

33 q q q ADVANCED SOIL MECHANICS FLOW NETS: DEFINITION OF TERMS Rate of Seepage Through Flow Channel (per unit length): Using Darcy s Law (q=a=kia) 3 q n q k h h 1 l 1 k h h 3 l k h h 3 4 l 1 h 1 h h h 3 h 3 h 4... H N d l Potential Drop l 3 l 3... Figure Das FGE (005). Where: H = Head Difference N d = Number of Potential Drops

34 Therefore, flow through one channel is: q k q k HN N H N d If Number of Flow Channels = N f, then the total flow for all channels per unit length is: d f ADVANCED SOIL MECHANICS FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE Figure 5.1b. Das FGE (005).

35 GIVEN: Flow Net in Figure N f = 3 N d = 6 k x =k z =5x10-3 cm/sec DETERMINE: a. How high water will rise in piezometers at points a, b, c, and d. b. Rate of seepage through flow channel II. c. Total rate of seepage ADVANCED SOIL MECHANICS FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE Figure Das FGE (005).

36 SOLUTION: Potential Drop = (5m 1.67m) 6 H N d 0.56m At Pt a: Water in standpipe = (5m 1x0.56m) = 4.44m At Pt b: Water in standpipe = (5m x0.56m) = 3.88m At Pts c and d: Water in standpipe = (5m 5x0.56m) =.0m ADVANCED SOIL MECHANICS FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE Figure Das FGE (005).

37 FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE SOLUTION: q k k = 5x10-3 cm/sec k = 5x10-5 m/sec H N d q = (5x10-5 m/sec)(0.56m) q =.8x10-5 m 3 /sec/m q k HN N d f qn q = (.8x10-5 m 3 /sec/m) * 3 q = 8.4x10-5 m3/sec/m f Figure Das FGE (005).

38 FLOW NETS: RULES FOR CREATING FLOW NETS (FROM UTEXAS) 1. Head drops between adjacent equipotential lines must be constant (or, in those rare cases where this is not desirable, clearly stated, just as in topographic contour maps)!. Equipotential lines must match known boundary conditions. 3. Flow lines can neer cross. Equi. Flow Line

39 FLOW NETS: STEPS FOR DRAWING (LADD, MIT) 1. Draw problem in ink.. Draw in known equipotential and flow boundary lines. 3. Sketch or 3 flow lines. 4. Draw corresponding equipotential lines (check for squares and intersections) 5. Keep adjusting (and adjusting, and adjusting ) Equi. Flow Line

40 FLOW NETS: RULES FOR CREATING FLOW NETS (FROM UTEXAS) 4. Refraction of flow lines must account for differences in hydraulic conductiity. 5. For isotropic media. Equi. a) Flow lines must intersect equipotential lines at right angles. b) The flow line-equipotential polygons should approach curilinear squares, as shown in the Figure to the right. Flow Line

41 FLOW NETS: RULES FOR CREATING FLOW NETS (FROM UTEXAS) 6. The quantity of flow between any two adjacent flow lines must be equal. Equi. 7. The quantity of flow between any two stream lines is always constant. Flow Line

42 FLOW NETS: DRAWING PROCEDURE (AFTER HARR (196, P. 3) 1. Draw the boundaries of the flow region to scale so that all equipotential lines and flow lines that are drawn can be terminated on these boundaries.. Sketch lightly three or four flow lines, keeping in mind that they are only a few of the infinite number of cures that must proide a smooth transition between the boundary flow lines. As an aid in spacing of these lines, it should be noted that the distance between adjacent flow lines increases in the direction of the larger radius of curature. 3. Sketch the equipotential lines, bearing in mind that they must intersect all flow lines, including the boundary streamlines, at right angles and that the enclosed figures must be (curilinear) squares.

43 FLOW NETS: DRAWING PROCEDURE (FROM HARR (196, P. 3) 4. Adjust the locations of the flow lines and the equipotential lines to satisfy the requirements of step 3. This is a trail-and-error process with the amount of correction being dependent upon the position of the initial flow lines. The speed with which a successful flow net can be drawn is highly contingent on the experience and judgment of the indiidual. A beginner will find the suggestions in Casagrande (1940) to be of assistance. 5. As a final check on the accuracy of the flow net, draw the diagonals of the squares. These should also form smooth cures that intersect each other at right angles.

44 FLOW NETS: EXAMPLES Wrong Wrong Correct! Unconfined groundwater flow nets on a slope

45 FLOW NETS: EXAMPLES Cross-sectional flow net of a homogeneous and isotropic aquifer (Hubbert, 1940).

46 FLOW NETS: EXAMPLES Contour map of the piezometric surface near Saannah, Georgia, 1957, showing closed contours resulting from heay local groundwater pumping (from Bedient, after USGS Water-Supply Paper 1611).

47 FLOW NETS: DAM EXAMPLES

SOIL MECHANICS

4.330 SOIL MECHANICS BERNOULLI S EQUATION Were: u w g Z = Total Head u = Pressure = Velocity g = Acceleration due to Graity w = Unit Weigt of Water Slide of 37 4.330 SOIL MECHANICS BERNOULLI S EQUATION