William J. Likos, Ph.D. Department of Civil and Environmental Engineering University of Wisconsin-Madison GLE/CEE 330 Lecture Notes Soil Mechanics Phase Relations: (aka Weight-Volume Relationships)

Soil is a three-phase system S = Solids W = Water A = Air Relative amount of each phase will affect behavior “block diagram” (portions courtesy of HKUST)

M t = M a + M w + M s 0 V t = V a + V w + V s V v = V a + V w (void volume) GsGsGsGs We will use specific gravity (G s ) To go from mass to volume or from volume to mass

Unit Weight,  Units: 1) Total Unit Weight,  t or  ~80 pcf to 130 pcf ~12.5 to 20.5 kN/m 3 2) Dry Unit Weight,  d ~40 pcf to 120 pcf ~6.3 to 18.8 kN/m 3 3) Unit Weight of Solids,  s 4) Unit Weight of Water,  w 5) Buoyant Unit Weight,  b

Volumetric Ratios 1) Void Ratio, 0 < e < ∞ 2) Porosity, n (often expressed as %) 3) Degree of Saturation, S or S r If S < 100%, then “unsaturated” If S = 100%, then “saturated”

(Coduto, 1999) Some Typical Values

(Lambe and Whitman, 1979) Compaction to Reduce e and n Loose Dense Compact High n High e Low n Low e VvVvVvVv VsVsVsVs VtVtVtVt VsVsVsVs VtVtVtVt VvVvVvVv

Simple cubic (SC), e = 0.91 Cubic-tetrahedral (CT), e = 0.65 Engineering Implications Loose soils (high e) will contract when sheared and are relatively weak Dense soils (low e) will expand (dilate) when sheared and are relatively strong contraction dilation

Water Content 1) Gravimetric Water Content, w 2) Volumetric Water Content,  1) Mass of Moist Soil, M m 2) Mass of Dry Soil, M d 105 C 3) M w = M m - M d 4) w = (M m – M d )/M d

Specific Gravity, G s Property of soil solids Property of soil solids Measure of unit weight relative to unit weight of water Measure of unit weight relative to unit weight of water So we can go from weight to volume if G x is known G water = 1 G iron = 7.5 G s = specific gravity of soil solids G s for most soil minerals is between 2.6 and 2.8 G s ≈ 2.65 for quartz sand (often assumed)

Some Useful Relationships Useful b/c W t and w are easy to measure Divide by V t Useful to construct compaction curve (Chap 6)

Some Useful Relationships

Void ratio and porosity are directly related

Saturated Unit Weight (when all voids filled with water, S = 100%)

Dry Unit Weight “Sewing” Equation (very useful) Proof: May be used to plot void ratio (or porosity) profile In homogeneous saturated deposits (assume G s, S)

Relative Density, Dr How dense is a soil relative to its maximum and minimum possible density?

Solving W-V Problems Usually given w, G s,  (these are easy to measure!) Usually given w, G s,  (these are easy to measure!) Might need to assume G s (2.65 for sand, 2.70 for clay) Might need to assume G s (2.65 for sand, 2.70 for clay) If saturated (below GWT), typically assume S = 100% If saturated (below GWT), typically assume S = 100% USE PHASE DIAGRAMS USE PHASE DIAGRAMS Might need to assume ONE of the following (but only one): Might need to assume ONE of the following (but only one): V t = 1.0 ft3 (or 1.0 m3) W t = 100 lb (or 100 kN) W s = 100 lb (or 100 kN)