SOIL MECHANICS

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1 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

2 4.330 SOIL MECHANICS BERNOULLI S EQUATION IN SOIL u w g Z 0 0 (i.e. elocity of water in soil is negligible). Terefore: u Z w Slide of 37

3 A B CHANGE IN HEAD FROM POINTS A & B (H) A B A B Figure 5.. Das FGE (005). Were: u A w i Z A u L B w Z can be expressed in non-dimensional form i = Hydraulic Gradient L = Lengt of Flow between Points A & B B Slide 3 of 37

4 VELOCITY () VS. HYDRAULIC GRADIENT (i) General relationsip sown in Figure 5. Tree Zones:. Laminar Flow (I). Transition Flow (II) 3. Turbulent Flow (III) For most soils, flow is laminar. Terefore: i Figure 5.. Das FGE (005). Slide 4 of 37

5 DARCY S LAW (856) Were: = Discarge Velocity (i.e. quantity of water in unit time troug unit cross-sectional area at rigt angles to te direction of flow) = (i.e. coefficient of permeability) i = Hydraulic Gradient * Based on obserations of flow of water troug clean sands Slide 5 of 37

6 HYDRAULIC CONDUCTIVITY () w K Were: = Viscosity of Water K = Absolute Permeability (units of L ) Typical Values of per Soil Type Soil Type (cm/sec) (ft/min) Clean Grael Coarse Sand Fine Sand Silty Clay Clay < < after Table 5.. Das FGE (005) Slide 6 of 37

7 DISCHARGE AND SEEPAGE VELOCITIES q A A s Were: 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). Slide 7 of 37

8 Slide 8 of SOIL MECHANICS 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 ) ( ) ( ) ( ) ( Figure 5.3. Das FGE (005). DISCHARGE AND SEEPAGE VELOCITIES

9 EXAMPLE PROBLEM: FIND i AND q GIVEN: ft REQUIRED: Find Hydraulic Gradient (i) and Flow Rate (q) 5 ft CL (Imperious Layer) 5 ft SM ( = ft/min) CL (Imperious Layer) 75 ft Slide 9 of 37

10 EXAMPLE PROBLEM: FIND i AND q GIVEN: SOLUTION: Hydraulic Gradient (i): 5 ft 5 ft ft CL (Imperious Layer) i L i ft 75 ft cos SM ( = ft/min) CL (Imperious Layer) 75 ft Rate of Flow per Time (q): q q q ia ft (0.067)(5 ft)(cos min ft / min/ ft )( ft) Slide 0 of 37

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). Slide of 37

12 HYDRAULIC CONDUCTIVITY: LABORATORY TESTING Constant Head (ASTM D434) Q At A( i) t Figure 5.4. Das FGE (005). Were: Q = Quantity of water collected oer time t t = Duration of water collection Q A QL At t L Slide of 37

4.330 SOIL MECHANICS HYDRAULIC CONDUCTIVITY: LABORATORY TESTING Falling Head (No ASTM) d q A a L dt Were: A = Cross-sectional area of Soil a = Cross-sectional area of Standpipe

13 4.330 SOIL MECHANICS HYDRAULIC CONDUCTIVITY: LABORATORY TESTING Falling Head (No ASTM) d q A a L dt Were: A = Cross-sectional area of Soil a = Cross-sectional area of Standpipe Integrate from limits 0 to t t al A dt Figure 5.5. Das FGE (005). after rearranging aboe equation log e al A d after integration or.303 al At Integrate from limits to Log 0 Slide 3 of 37

14 HYDRAULIC CONDUCTIVITY: EMPIRICAL RELATIONSHIPS Uniform Sands - Hazen Formula (Hazen, 930): ( cm / sec) cd 0 Were: c = Constant between to.5 D 0 = Effectie Size (in mm) Sands Kozeny-Carman (Loudon 95 and Perloff and Baron 976): C e Were: C = Constant (to be determined) e = Void Ratio 3 e Sands Casagrande (Unpublised):.4e 0.85 Were: e = Void Ratio 0.85 e = 0.85 Normally Consolidated Clays (Samarasinge, Huang, and Drneic, 98): C n e e Were: C = Constant to be determined experimentally n = Constant to be determined experimentally e = Void Ratio Slide 4 of 37

15 4.330 SOIL MECHANICS HYDRAULIC CONDUCTIVITY: EMPIRICAL RELATIONSHIPS EXAMPLE NORMALLY CONSOLIDATED CLAYS GIVEN: Normally consolidated clay wit e and measurements from D Consolidation Test. Void Ratio (e) (cm/sec). 0.6 x x 0-7 REQUIRED: Find for same clay wit a oid ratio of.4. SOLUTION: Using (Samarasinge, Huang, and Drneic, 98) Equation: C C n e e n e e Substituting nown quantities 0.6cm / sec.5cm / sec..5 Example 5.6 Das PGE (005). n.5. Slide 5 of 37

16 e SOIL MECHANICS HYDRAULIC CONDUCTIVITY: EMPIRICAL RELATIONSHIPS EXAMPLE NORMALLY CONSOLIDATED CLAYS 0.6cm / sec.5cm / sec 0.6x0 C 7 C n e e cm / sec 0.58x cm / sec (0.58x0 7cm / sec).4..5 n.5. n 4.5.x0 7 cm / sec Example 5.6 Das PGE (005). Slide 6 of 37

17 STRATIFIED SOILS: EQUIVALENT HYDRAULIC CONDUCTIVITY HORIZONTAL DIRECTION Considering cross-section of Unit Lengt. Total flow troug cross-section can be written as: q q H H H... n H n Were: = Aerage Discarge Velocity = Discarge Velocity in Layer Figure 5.7. Das FGE (005). Slide 7 of 37

18 STRATIFIED SOILS: EQUIVALENT HYDRAULIC CONDUCTIVITY HORIZONTAL DIRECTION Substituting =i into q equation and using H to denote Horizontal Direction H ( eq) H i eq i ; i ;...; H n n i n Noting tat i eq =i =i = =i n H ( H... H ( eq) H H Hn n Were: H(eq) = Equialent in Horizontal Direction H H ) Figure 5.7. Das FGE (005). Slide 8 of 37

19 STRATIFIED SOILS: EQUIVALENT HYDRAULIC CONDUCTIVITY VERTICAL DIRECTION Total Head Loss = = Sum Head Loss in Eac Layer... V ( eq) H and V... Using Darcy s Law (=i) into equation and using V to denote Vertical Direction n n i... Vn Were: V(eq) = Equialent in Vertical Direction i n Figure 5.8. Das FGE (005). Slide 9 of 37

Field Measurements Taen: q, r, r,, q = Groundwater Flow into Well q also is rate of

20 4.330 SOIL MECHANICS FIELD PERMEABILITY TESTING BY PUMPING WELLS UNCONFINED PERMEABLE LAYER UNDERLAIN BY IMPERMEABLE LAYER Figure 5.9. Das FGE (005). Field Measurements Taen: q, r, r,, q = Groundwater Flow into Well q also is rate of discarge from pumping Equation: r q d r dr r field can be re-written as dr r q Soling Equation: d r.303q log0 r ( ) Slide 0 of 37

r q dr r r d rh dr can be re-written as H q d Figure 5.0. Das FGE (005).

21 FIELD PERMEABILITY TESTING BY PUMPING WELLS WELL PENETRATING CONFINED AQUIFER q = Groundwater Flow into Well q also is rate of discarge from pumping Equation: r q dr r r d rh dr can be re-written as H q d Figure 5.0. Das FGE (005). Field Measurements Taen: q, r, r,, Soling Equation: field q log0.77h ( r r ) Slide of 37

22 SOIL PERMEABILITY AND DRAINAGE SOIL MECHANICS after Casagrande and Fadum (940) and Terzagi et al. (996). Slide of 37

23 SOIL PERMEABILITY AND DRAINAGE COEFFICIENT OF PERMEABILITY CM/S (LOG SCALE) Drainage property Good drainage Poor drainage Practically imperious Application in eart dams and dies Perious sections of dams and dies Imperious sections of eart dams and dies 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., omogeneous clays below zone of weatering Imperious soils wic are modified by te effect of egetation and weatering; fissured, weatered 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 tis range.) Constant ead test in triaxial cell; reliable w it experience and no leas. Reliable; Little experience required Falling Head Per meameter; Range of unstable permeability;* muc experience necessary to correct interpretation Fairly reliable; considerable experience necessary (do in triaxial cell) Indirect determination of coefficient of permeability Computat ion: From te grain size distribution (e.g., Hazen s formula). Only applicable to clean, coesionless sands and graels Horizontal Capillarity Test: Very little experience necessary; especially useful for rapid testing of a large number of samples in te field w itout laboratory facilities. Computations: from consolidation tests; expensie laboratory equipment and considerable experience required *Due to migration of fines, cannels, and air in oids. From FHWA IF Ealuation of Soil and Roc Properties. Slide 3 of 37

Discarge Velocity in x Direction Figure 5.. Das FGE (005).

24 LAPLACE'S EQUATION OF CONTINUITY y z x Steady-State Flow around an imperious Seet Pile Wall Consider water flow at Point A: x = Discarge Velocity in x Direction Figure 5.. Das FGE (005). z = Discarge Velocity in z Direction Y Direction Out Of Plane Slide 4 of 37

25 LAPLACE'S EQUATION OF CONTINUITY Consider water flow at Point A (Soil Bloc at Pt A sown left) Rate of water flow into soil bloc in x direction: x dzdy Figure 5.. Das FGE (005). Rate of water flow into soil bloc in z direction: z dxdy Rate of water flow out of soil bloc in x,z directions: x z x x z z dxdzdy dz dxdy Slide 5 of 37

26 LAPLACE'S EQUATION OF CONTINUITY Consider water flow at Point A (Soil Bloc at Pt A sown left) Total Inflow = Total Outflow Figure 5.. Das FGE (005). x x x z dxdzdy z dz dxdy z dzdy dxdy 0 x x x z or z 0 z Slide 6 of 37

27 Slide 7 of SOIL MECHANICS 0 z x z i x i z x z z z z x x x x Consider water flow at Point A (Soil Bloc at Pt A sown left) Figure 5.. Das FGE (005). Using Darcy s Law (=i) LAPLACE'S EQUATION OF CONTINUITY

28 FLOW NETS: DEFINITION OF TERMS Flow Net: Grapical Construction used to calculate groundwater flow troug soil. Comprised of Flow Lines and Equipotential Lines. Flow Line: A line along wic a water particle moes troug a permeable soil medium. Flow Cannel: Strip between any two adjacent Flow Lines. Equipotential Lines: A line along wic te potential ead at all points is equal. NOTE: Flow Lines and Equipotential Lines must meet at rigt angles! Slide 8 of 37

29 FLOW NETS FLOW AROUND SHEET PILE WALL Figure 5.a. Das FGE (005). Slide 9 of 37

30 FLOW NETS FLOW AROUND SHEET PILE WALL Figure 5.b. Das FGE (005). Slide 30 of 37

31 FLOW NETS: BOUNDARY CONDITIONS. Te upstream and downstream surfaces of te permeable layer (i.e. lines ab and de in Figure b Das FGE (005)) are equipotential lines.. Because ab and de are equipotential lines, all te flow lines intersect tem at rigt angles. 3. Te boundary of te impreious layer (i.e. line fg in Figure b Das FGE (005)) is a flow line, as is te surface of te imperious seet pile (i.e. line acd in Figure b Das FGE (005)). 4. Te equipontential lines intersect acd and fg (Figure b Das FGE (005)) at rigt angles. Slide 3 of 37

32 FLOW NETS FLOW UNDER AN IMPERMEABLE DAM Figure 5.3. Das FGE (005). Slide 3 of 37

33 q q q SOIL MECHANICS FLOW NETS: DEFINITION OF TERMS Rate of Seepage Troug Flow Cannel (per unit lengt): Using Darcy s Law (q=a=ia) q n q l 3 l 3 4 l l Potential Drop H N d l 3 l 3... Figure 5.4. Das FGE (005). Were: H = Head Difference N d = Number of Potential Drops Slide 33 of 37

34 Terefore, flow troug one cannel is: q q N H N d If Number of Flow Cannels = N f, ten te total flow for all cannels per unit lengt is: HN d f SOIL MECHANICS FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE Figure 5.b. Das FGE (005). Slide 34 of 37

35 GIVEN: Flow Net in Figure 5.7. N f = 3 N d = 6 x = z =5x0-3 cm/sec DETERMINE: a. How ig water will rise in piezometers at points a, b, c, and d. b. Rate of seepage troug flow cannel II. c. Total rate of seepage SOIL MECHANICS FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE Figure 5.7. Das FGE (005). Slide 35 of 37

36 SOLUTION: Potential Drop = (5m.67m) 6 H N d 0.56m At Pt a: Water in standpipe = (5m x0.56m) = 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 SOIL MECHANICS FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE Figure 5.7. Das FGE (005). Slide 36 of 37

37 FLOW NETS FLOW AROUND SHEET PILE WALL EXAMPLE SOLUTION: q = 5x0-3 cm/sec = 5x0-5 m/sec H N d q = (5x0-5 m/sec)(0.56m) q =.8x0-5 m 3 /sec/m q HN N d f qn q = (.8x0-5 m 3 /sec/m) * 3 q = 8.4x0-5 m3/sec/m f Figure 5.7. Das FGE (005). Slide 37 of 37

ADVANCED SOIL MECHANICS

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 h 14.531 ADVANCED SOIL MECHANICS BERNOULLI S EQUATION IN SOIL u w g