Module 2 - Weight and Volume Relationships of Soil PDF

Geotechnical Engineering 1 (Soil Mechanics) Module 2: Weight and Volume Relationships of Soil Objectives: After studying these topics students be able to: 1. Understand the significance of weight and volume relationships of soils. 2. Know the importance and application of weight and volume relations...

Geotechnical Engineering 1 (Soil Mechanics) Module 2: Weight and Volume Relationships of Soil Objectives: After studying these topics students be able to: 1. Understand the significance of weight and volume relationships of soils. 2. Know the importance and application of weight and volume relationships of soils in determination of the properties of soils. Content: A. Introduction The soil mass consists of solids and voids. The voids may be partially or wholly filled with water or air. Although the solids and voids in a sample of soil do not occupy separate volumes. Va Air Wa Vw Water Ww Vs Solid Particles Ws Vv V Where: V = total volume of soil Va = volume of air Vw = volume of water Vs = volume of solids Vv = volume of voids W = total weight of soil Wa = weight of air Ww = weight of water Ws = weight of solids W e = void ratio n = porosity S = degree of saturation  = moisture content or water content  = unit weight of soil d = dry unit weight of soil Gs = specific gravity of soil w = unit weight of water sat = saturated unit weight of soil sub = submerged unit weight of soil Note: w = 9. 81 kN/m3 = 9, 810 N/m3 = 1, 000kg/m3 = 1 g/cc = 62. 4 lb/ft3 B. Basic Formulas 1. Void ratio (e) is defined as the ratio of the volume of voids to the volume of solids. e= vv Vs If n is given, e = n 1−n 2. Porosity (n) is defined as the ratio of the volume of voids to the volume. n= Vv V If e is given, n = MBV Geotechnical Engineering 1 e 1+e Page | 1 Where: Air void ratio = e−  Gs 1+e 3. Degree of saturation (S) is defined as the ratio of the volume of water to the volume of voids. Vw x 100% Vv If , Gs and e are given, S = S=  Gs e For fully saturated soil where S = 100%, e =  Gs 4. Moisture content or water content () is defined as the ratio of the weight of water to the weight of solids in a given volume of soil. Ww Ws  = x 100% 5. Unit weight or Bulk unit weight () is the weight of soil per unit volume. W V  = 6. Dry unit weight (d) is defined as weight of soil solids per unit volume. Ws d = V 7. Density of soil (ρ) is defined as the mass of soil per unit volume. m V  = 8. Dry density of soil (ρd) is defined as mass of soil solids per unit volume. d = ms V 9. Specific gravity (Gs) is defined as the unit weight soil per unit weight of water. Gs = s w 10. Effective unit weight or Buoyant unit weight (’) is the weight of a saturated soil, surrounded by water, per unit volume of soil or is the weight of soil solids in a submerged soil per unit volume.  ′ = sat − w ′ = ′ = (Gs − 1)w 1+e [Gs – 1 − e (1 − S)]w 1+e 11. Specific volume (V’) is the volume of soil per unit volume of solids. V′ = V Vs V′ = 1 + e 12. Saturated unit weight (sat) is the weight of a saturated soil per unit volume. sat = MBV Geotechnical Engineering 1 Wsat V Page | 2 C. Various Forms of Weight and Volume Relationships 1. Moist unit weight () Va Air Wa Vw Water Ww Vs Solid Particles Ws Vv V Given Relationship , Gs, e ( 1 +  )Gs w S, Gs, e ( Gs + Se )w W 1+e 1+e , Gs, S ( 1 +  )Gs w  Gs 1+ S , Gs, n Gs w ( 1 − n)( 1 +  ) S, Gs, n Gs w ( 1 − n ) + nSw 2. Dry unit weight (d) Vv = Va Wa Air V W Vs Given ,  Ws G s, e Relationship  1+  Gs w Gs, n Gs w ( 1 − n) Gs, , S Gs w  Gs 1+ S e S w ( 1 + e ) S, e,  sat,, e MBV Geotechnical Engineering 1 Solid Particles 1+e sat – e w 1+e Page | 3 sat − nw sat,, n ( sat − w )Gs sat, Gs Gs − 1 3. Saturated unit weight (sat) Vv = Vw Ww Water W V Vs Solid Particles Given Ws G s, e Relationship ( Gs + e )w Gs, n [(1 − n)Gs + n]w Gs, sat ( 1 + sat )Gs w (1 + satGs ) sat, e (e)( 1 + sat )(w ) (sat )( 1 + e) 1+e (1 + sat )nw sat , n sat e w d, e d + d, n d + n w d, S (1 − 1+e 1 )  + w Gs d d ( 1 + sat ) d, sat Typical Values of Unit Weight for Soils sat ( kN/m3 ) 20 – 22 18 – 20 18 – 20 16 – 22 Soil Type Gravel Sand Silt Clay d ( kN/m3 ) 15 – 17 13 – 16 14 – 18 14 – 21 Relative density is commonly used to indicate the in situ denseness or looseness of granular soil or is an index that quantifies the degree of packing between the loosest and densest possible state of coarse-grained soils as determined by experiments. Dr = emax − e emax − emin Where: Dr= relative density, usually given as a percentage e = in situ void ratio of the soil MBV Geotechnical Engineering 1 Page | 4 emax = void ratio of the soil in the loosest state emax =  Gs − min  w min −1 w emin = void ratio of the soil in the densest state emin = w max − 1 w nmax nmin emin = 1− nmax 1− nmin ( 1− nmin )( nmax − n ) ( nmax − nmin )( 1−n ) d − d(min) d(max) emax = Dr =  Gs − min  Dr = [ d(max) − d(min) Dr = ][ d ] 1 1 −  d(min) d 1 1 − d(min) d(max) Where: d = in situ dry unit weight (at a void ratio of e) d(max) = dry unit weight in the densest condition (at a void ratio e min) d(min) = dry unit weight in the loosest condition (at a void ratio of emax) Qualitative Description of Granular Soil Relative Density % 0-15 15-50 50-70 70-85 85-100 MBV Geotechnical Engineering 1 Description of Soil Deposit Very loose Loose Medium Dense Very dense Page | 5 Problems 1. Prove the following forms of weight and volume relationships: a. γ = (1+ω)Gs γw 1+e , Given: ω, Gs , , e b. γ = Gs γw (1 − n)(1 + ω), Given: ω, Gs , n c. γd = Gs γw , Given: Gs , e 1+e d. γd = Gs γw (1 − n), Given: Gs , n e. γd = γsat − f. γsat = eγw , Given: 1+e (Gs +e)γw 1+e γsat , e , Given: Gs, e g. γsat = [(1 − n)Gs + n]γw , Given: Gs , n h. γsat = (e)(1+ωsat )γw (ωsat)(1+e) , Given: ωsat , e 2. The moist unit weight of a soil is 19.2 kN/m3. Given that Gs = 2. 69 and ω =9.8%, determine a. b. c. d. Dry unit weight Void ratio Porosity Degree of saturation 3. The unit weight of a soil is 96 lb/ft3. The moisture content of this soil is 17% when the degree of saturation is 60%. Determine a. Void ratio b. Specific gravity of solids c. Saturated unit weight 4. For a moist soil, the following are given: V = 0.25 ft 3, W = 30.75 lb, ω = 9.8%, and Gs = 2.66. Determine a. b. c. d. e. f. Unit weight Dry unit weight Void ratio Porosity Degree of saturation Volume occupied by water 5. For a sandy soil, emax = 0.72, emin = 0.46, and Gs = 2.68. What is the moist unit weight of compaction (kN/m3) in the field if Dr = 78% and ω = 9%? References: 1. 2. 3. 4. 5. 6. Geotechnical Engineering (Revised Third Edition) by C. Venkatramaiah, 2012 Principles of Geotechnical Engineering (Seventh Edition) by Braja M. Das, 2010 Soil Mechanics and Foundations (Third Edition) by Muni Budhu, 2011 Soil Mechanics 7th Edition, R.F. Craig, 2004 Basic Fundamentals of Geotechnical Engineering by Venancio L. Besavilla Jr., 1998 Fundamentals of Geotechnical Engineering by Diego Inocencio T. Gillesania, 2006 MBV Geotechnical Engineering 1 Page | 6