9.1 Weight and volume relationships

Achievable FE Civil

9. Soil mechanics

Our FE Civil course is currently in development and is a work-in-progress.

Weight and volume relationships

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This chapter covers the following:

Weight and volume relationships
Soil compaction and classification parameters
Permeability and seepage

Weight and volume relationships

In soil mechanics, weight-volume relationships help you describe and predict soil behavior. They’re based on the three-phase system, which treats soil as a mixture of:

solids
water
air

A three-phase soil diagram illustrating the relationships between weights (or masses) and volumes of air, water, and solids. — Three-Phase diagram

Three-phase system

Definitions

Soil is idealized as a system consisting of three components: solids, water, and air. Engineering relationships are derived by relating the weights and volumes of these components.

We’ll use the following symbols:

$V$ = total volume
$V_{s}$ = volume of solids
$V_{w}$ = volume of water
$V_{a}$ = volume of air
$W$ = total weight
$W_{s}$ = weight of solids
$W_{w}$ = weight of water

The total volume is the sum of the three phase volumes:

$V = V_{s} + V_{w} + V_{a}$

The volume of voids is the part of the soil volume not occupied by solids (so it includes both air and water):

$V_{v} = V_{w} + V_{a}$

Water content

Water content is the ratio of water weight to solids weight:

$w = \frac{W _{w}}{W _{s}} \times 100%$

Total/bulk unit weight

Total (bulk) unit weight is total weight per total volume:

$γ = \frac{W}{V}$

Dry unit weight

Dry unit weight uses only the weight of solids (but still divides by the total volume):

$γ_{d} = \frac{W _{s}}{V}$

Saturated unit weight

Saturated unit weight applies when the voids are completely filled with water (no air):

$γ_{s a t} = \frac{W _{s} + W _{w}}{V}$

Void ratio

Void ratio compares void volume to solids volume:

$e = \frac{V _{v}}{V _{s}}$

Porosity

Porosity is the fraction of the total volume that is void space:

$n = \frac{V _{v}}{V} \times 100%$

Degree of saturation

Degree of saturation is the fraction of the void space that is filled with water:

$S = \frac{V _{w}}{V _{v}} \times 100%$

Specific gravity of solids

Specific gravity of solids compares the density of soil solids to the density of water:

$G_{s} = \frac{ρ _{s}}{ρ _{w}} = \frac{W _{s} / V _{s}}{ρ _{w}}$

Quick conversion formula

These relationships are commonly used to convert between properties:

$γ = \frac{( 1 + w ) G _{s} γ _{w}}{1 + e}$

$γ_{d} = \frac{G _{s} γ _{w}}{1 + e}$

$e = \frac{G _{s} γ _{w}}{γ _{d}} - 1$

$S e = G_{s} w$

$n = \frac{e}{1 + e}$

$e = \frac{n - 1}{n}$

Example problem

Given:

$V = 0.003 m^{3}$
$W_{s} = 40 N$
$w = 25%$
$G_{s} = 2.7$

Find:

Volume of voids
Porosity
Bulk unit weight
Dry unit weight

Solution

1. Weight of water:

Convert the water content to a decimal and multiply by the solids weight.

$W_{w} = w \cdot W_{s} = 0.25 \times 40 = 10 N$

2. Total weight:

Add the solids and water weights.

$W = W_{s} + W_{w} = 40 + 10 = 50 N$

3. Bulk unit weight:

Divide total weight by total volume.

$γ = \frac{W}{V} = \frac{50}{0.003} = 16, 667 N/m^{3}$

4. Dry unit weight:

Divide solids weight by total volume.

$γ_{d} = \frac{W _{s}}{V} = \frac{40}{0.003} = 13, 333 N/m^{3}$

5. Volume of solids:

Use the relationship between solids weight, specific gravity, and the unit weight of water.

$V_{s} = \frac{W _{s}}{G _{s} \cdot γ _{w}}$

Assume $γ_{w} = 9, 810 N/m^{3}$ :

$V_{s} = \frac{40}{2.7 \times 9810} \approx 0.00151 m^{3}$

6. Volume of voids:

Subtract the solids volume from the total volume.

$V_{v} = V - V_{s} = 0.003 - 0.00151 = 0.00149 m^{3}$

7. Porosity:

Compute void volume as a fraction of total volume.

$n = \frac{V _{v}}{V} \times 100% = \frac{0.00149}{0.003} \times 100% \approx 49.7%$

Soil compaction and classification parameters

Relative density

Relative density compares the in-place void ratio (or dry unit weight) of a granular soil to its loosest and densest possible states.

$D_{r} = [\frac{( e _{max} - e )}{( e _{max} - e _{min} )}] \times 100$

$D_{r} = [\frac{( γ _{D field} - γ _{D min} )}{( γ _{D max} - γ _{D min} )}] [\frac{γ _{D max}}{γ _{D field}}] \times 100$

Relative compaction (%)

Relative compaction compares the field dry unit weight to the maximum dry unit weight from a compaction test.

$RC = (\frac{γ _{D field}}{γ _{D max}}) \times 100$

Plasticity index

Plasticity index is the range of water contents over which a fine-grained soil behaves plastically.

$P I = LL - P L$

$LL$ = liquid limit
$P L$ = plastic limit

Coefficient of uniformity

The coefficient of uniformity describes how spread out the particle sizes are.

$C_{U} = \frac{D _{60}}{D _{10}}$

Coefficient of concavity (or curvature)

The coefficient of concavity (curvature) describes the shape of the gradation curve.

$C_{C} = \frac{( D _{30} ) ^{2}}{D _{10} \times D _{60}}$

$D_{n}$ is the particle size (diameter in $mm$ ) at which $n$ percent of the particles are finer. The “mean particle size” ( $D_{50}$ ) is the particle diameter at which $50%$ of the particles are finer.

Definitions

$D_{n}$ is the particle diameter at which $n$ of the soil particles are finer by weight.

A grain-size distribution curve showing cumulative percent finer versus particle diameter for soil classification. — Particle size diameter example

Permeability and seepage

Hydraulic conductivity (coefficient of permeability)

Hydraulic conductivity $k$ measures how easily water flows through soil.

Constant head test:

$k = \frac{Q}{i A t _{e}}$

$i = \frac{d h}{d L}$

where, $Q = total quantity of water$

Falling head test:

$k = 2.303 (\frac{a L}{A t _{e}}) lo g_{10} (\frac{h _{1}}{h _{2}})$

Where:

$A$ = cross-sectional area of test specimen perpendicular to flow
$a$ = cross-sectional area of reservoir tube
$t_{e}$ = elapsed time
$h_{1}$ = head at time $t = 0$
$h_{2}$ = head at time $t = t_{e}$
$L$ = length of soil column

Discharge velocity and seepage velocity

Definitions

The actual velocity at which water moves through the void spaces of soil.

Seepage velocity is the rate of movement of an element of water through a soil.

$v = ki$

$v_{s} = \frac{ki}{n}$

where, $v = discharge velocity$ and $v_{s} = seepage velocity$

Flow nets

Definitions

A graphical representation of flow lines and equipotential lines used to analyze seepage through soils.

A flow net is a combination of flow lines and equipotential lines.

A flow line is a line along which a water particle travels.
An equipotential line connects points of equal total head.

Key properties used when constructing and interpreting a flow net:

There are no flow along equipotential lines, which are 90 degrees to flow lines.
The total head along an equipotential line is equal at all points.
Flow lines cannot cross other flow lines and equipotential lines cannot cross other equipotential lines.
Equipotential lines intersect the flow lines at right angles.
Flow net must be constructed so that each element is a curvilinear square (sides may be curved but a circle must be inscribed within it that touches all its 4 sides).

The total flow rate though a flow net, is solved for by:

$Q = k Δ h \frac{N _{f}}{N _{d}} L$

Where:

$Q$ = total flow rate
$N_{f}$ = number of flow channels in a flow net
$N_{d}$ = number of potential drops
$Δ h$ = head change from upstream to downstream
$k$ = coefficient of permeability
$L$ = length of structure (i.e. bank-to-bank)

Factor of safety against seepage liquefaction

$F S_{s} = \frac{i _{c}}{i _{e}}$

$i_{c} = \frac{( γ _{sat} - γ _{w} )}{γ _{w}}$

where, $i_{e} = seepage exit gradient$

Weight and volume relationships

Three-phase system: soil = solids + water + air
Key formulas:
- $V = V_{s} + V_{w} + V_{a}$ (total volume)
- $V_{v} = V_{w} + V_{a}$ (void volume)
Water content: $w = \frac{W _{w}}{W _{s}} \times 100%$
Unit weights:
- Bulk: $γ = \frac{W}{V}$
- Dry: $γ_{d} = \frac{W _{s}}{V}$
- Saturated: $γ_{s a t} = \frac{W _{s} + W _{w}}{V}$
Void ratio: $e = \frac{V _{v}}{V _{s}}$
Porosity: $n = \frac{V _{v}}{V} \times 100%$
Degree of saturation: $S = \frac{V _{w}}{V _{v}} \times 100%$
Specific gravity: $G_{s} = \frac{ρ _{s}}{ρ _{w}}$
Conversion formulas:
- $γ = \frac{( 1 + w ) G _{s} γ _{w}}{1 + e}$
- $γ_{d} = \frac{G _{s} γ _{w}}{1 + e}$
- $e = \frac{G _{s} γ _{w}}{γ _{d}} - 1$
- $S e = G_{s} w$
- $n = \frac{e}{1 + e}$

Soil compaction and classification parameters

Relative density: compares in-place void ratio/dry unit weight to loosest/densest states
- $D_{r} = [\frac{( e _{ma x} - e )}{( e _{ma x} - e _{min} )}] \times 100$
Relative compaction: field dry unit weight vs. maximum dry unit weight
- $RC = (\frac{γ _{D f i e l d}}{γ _{D ma x}}) \times 100$
Plasticity index: $P I = LL - P L$
- Range of water contents for plastic behavior
Coefficient of uniformity: $C_{U} = \frac{D _{60}}{D _{10}}$
Coefficient of concavity: $C_{C} = \frac{( D _{30} ) ^{2}}{D _{10} \times D _{60}}$
- $D_{n}$ = particle diameter at which $n %$ is finer

Permeability and seepage

Hydraulic conductivity ( $k$ ): ease of water flow through soil
- Constant head: $k = \frac{Q}{i A t _{e}}$
- Falling head: $k = 2.303 (\frac{a L}{A t _{e}}) lo g_{10} (\frac{h _{1}}{h _{2}})$
Discharge velocity: $v = ki$
Seepage velocity: $v_{s} = \frac{ki}{n}$
Flow nets:
- Graphical tool: flow lines (water paths) + equipotential lines (equal head)
- Properties: lines intersect at 90°, no crossing of same type, curvilinear squares
- Flow rate: $Q = k Δ h \frac{N _{f}}{N _{d}} L$
Factor of safety against seepage liquefaction:
- $F S_{s} = \frac{i _{c}}{i _{e}}$
- $i_{c} = \frac{( γ _{s a t} - γ _{w} )}{γ _{w}}$ (critical gradient)

Next | 9.2 Consolidation and stress

Weight and volume relationships

This chapter covers the following:

Weight and volume relationships
Soil compaction and classification parameters
Permeability and seepage

Weight and volume relationships

In soil mechanics, weight-volume relationships help you describe and predict soil behavior. They’re based on the three-phase system, which treats soil as a mixture of:

solids
water
air

Three-phase system

Definitions

Soil is idealized as a system consisting of three components: solids, water, and air. Engineering relationships are derived by relating the weights and volumes of these components.

We’ll use the following symbols:

$V$ = total volume
$V_{s}$ = volume of solids
$V_{w}$ = volume of water
$V_{a}$ = volume of air
$W$ = total weight
$W_{s}$ = weight of solids
$W_{w}$ = weight of water

The total volume is the sum of the three phase volumes:

$V = V_{s} + V_{w} + V_{a}$

The volume of voids is the part of the soil volume not occupied by solids (so it includes both air and water):

$V_{v} = V_{w} + V_{a}$

Water content

Water content is the ratio of water weight to solids weight:

$w = \frac{W _{w}}{W _{s}} \times 100%$

Total/bulk unit weight

Total (bulk) unit weight is total weight per total volume:

$γ = \frac{W}{V}$

Dry unit weight

Dry unit weight uses only the weight of solids (but still divides by the total volume):

$γ_{d} = \frac{W _{s}}{V}$

Saturated unit weight

Saturated unit weight applies when the voids are completely filled with water (no air):

$γ_{s a t} = \frac{W _{s} + W _{w}}{V}$

Void ratio

Void ratio compares void volume to solids volume:

$e = \frac{V _{v}}{V _{s}}$

Porosity

Porosity is the fraction of the total volume that is void space:

$n = \frac{V _{v}}{V} \times 100%$

Degree of saturation

Degree of saturation is the fraction of the void space that is filled with water:

$S = \frac{V _{w}}{V _{v}} \times 100%$

Specific gravity of solids

Specific gravity of solids compares the density of soil solids to the density of water:

$G_{s} = \frac{ρ _{s}}{ρ _{w}} = \frac{W _{s} / V _{s}}{ρ _{w}}$

Quick conversion formula

These relationships are commonly used to convert between properties:

$γ = \frac{( 1 + w ) G _{s} γ _{w}}{1 + e}$

$γ_{d} = \frac{G _{s} γ _{w}}{1 + e}$

$e = \frac{G _{s} γ _{w}}{γ _{d}} - 1$

$S e = G_{s} w$

$n = \frac{e}{1 + e}$

$e = \frac{n - 1}{n}$

Example problem

Given:

$V = 0.003 m^{3}$
$W_{s} = 40 N$
$w = 25%$
$G_{s} = 2.7$

Find:

Volume of voids
Porosity
Bulk unit weight
Dry unit weight

Solution

1. Weight of water:

Convert the water content to a decimal and multiply by the solids weight.

$W_{w} = w \cdot W_{s} = 0.25 \times 40 = 10 N$

2. Total weight:

Add the solids and water weights.

$W = W_{s} + W_{w} = 40 + 10 = 50 N$

3. Bulk unit weight:

Divide total weight by total volume.

$γ = \frac{W}{V} = \frac{50}{0.003} = 16, 667 N/m^{3}$

4. Dry unit weight:

Divide solids weight by total volume.

$γ_{d} = \frac{W _{s}}{V} = \frac{40}{0.003} = 13, 333 N/m^{3}$

5. Volume of solids:

Use the relationship between solids weight, specific gravity, and the unit weight of water.

$V_{s} = \frac{W _{s}}{G _{s} \cdot γ _{w}}$

Assume $γ_{w} = 9, 810 N/m^{3}$ :

$V_{s} = \frac{40}{2.7 \times 9810} \approx 0.00151 m^{3}$

6. Volume of voids:

Subtract the solids volume from the total volume.

$V_{v} = V - V_{s} = 0.003 - 0.00151 = 0.00149 m^{3}$

7. Porosity:

Compute void volume as a fraction of total volume.

$n = \frac{V _{v}}{V} \times 100% = \frac{0.00149}{0.003} \times 100% \approx 49.7%$

Soil compaction and classification parameters

Relative density

Relative density compares the in-place void ratio (or dry unit weight) of a granular soil to its loosest and densest possible states.

$D_{r} = [\frac{( e _{max} - e )}{( e _{max} - e _{min} )}] \times 100$

$D_{r} = [\frac{( γ _{D field} - γ _{D min} )}{( γ _{D max} - γ _{D min} )}] [\frac{γ _{D max}}{γ _{D field}}] \times 100$

Relative compaction (%)

Relative compaction compares the field dry unit weight to the maximum dry unit weight from a compaction test.

$RC = (\frac{γ _{D field}}{γ _{D max}}) \times 100$

Plasticity index

Plasticity index is the range of water contents over which a fine-grained soil behaves plastically.

$P I = LL - P L$

$LL$ = liquid limit
$P L$ = plastic limit

Coefficient of uniformity

The coefficient of uniformity describes how spread out the particle sizes are.

$C_{U} = \frac{D _{60}}{D _{10}}$

Coefficient of concavity (or curvature)

The coefficient of concavity (curvature) describes the shape of the gradation curve.

$C_{C} = \frac{( D _{30} ) ^{2}}{D _{10} \times D _{60}}$

Definitions

$D_{n}$ is the particle diameter at which $n$ of the soil particles are finer by weight.

Permeability and seepage

Hydraulic conductivity (coefficient of permeability)

Hydraulic conductivity $k$ measures how easily water flows through soil.

Constant head test:

$k = \frac{Q}{i A t _{e}}$

$i = \frac{d h}{d L}$

where, $Q = total quantity of water$

Falling head test:

$k = 2.303 (\frac{a L}{A t _{e}}) lo g_{10} (\frac{h _{1}}{h _{2}})$

Where:

$A$ = cross-sectional area of test specimen perpendicular to flow
$a$ = cross-sectional area of reservoir tube
$t_{e}$ = elapsed time
$h_{1}$ = head at time $t = 0$
$h_{2}$ = head at time $t = t_{e}$
$L$ = length of soil column

Discharge velocity and seepage velocity

Definitions

The actual velocity at which water moves through the void spaces of soil.

Seepage velocity is the rate of movement of an element of water through a soil.

$v = ki$

$v_{s} = \frac{ki}{n}$

where, $v = discharge velocity$ and $v_{s} = seepage velocity$

Flow nets

Definitions

A graphical representation of flow lines and equipotential lines used to analyze seepage through soils.

A flow net is a combination of flow lines and equipotential lines.

A flow line is a line along which a water particle travels.
An equipotential line connects points of equal total head.

Key properties used when constructing and interpreting a flow net:

There are no flow along equipotential lines, which are 90 degrees to flow lines.
The total head along an equipotential line is equal at all points.
Flow lines cannot cross other flow lines and equipotential lines cannot cross other equipotential lines.
Equipotential lines intersect the flow lines at right angles.
Flow net must be constructed so that each element is a curvilinear square (sides may be curved but a circle must be inscribed within it that touches all its 4 sides).

The total flow rate though a flow net, is solved for by:

$Q = k Δ h \frac{N _{f}}{N _{d}} L$

Where:

$Q$ = total flow rate
$N_{f}$ = number of flow channels in a flow net
$N_{d}$ = number of potential drops
$Δ h$ = head change from upstream to downstream
$k$ = coefficient of permeability
$L$ = length of structure (i.e. bank-to-bank)

Factor of safety against seepage liquefaction

$F S_{s} = \frac{i _{c}}{i _{e}}$

$i_{c} = \frac{( γ _{sat} - γ _{w} )}{γ _{w}}$

where, $i_{e} = seepage exit gradient$

Achievable FE Civil

9. Soil mechanics

Our FE Civil course is currently in development and is a work-in-progress.

Weight and volume relationships

8 min read

Font

Discuss

Feedback

This chapter covers the following:

Weight and volume relationships
Soil compaction and classification parameters
Permeability and seepage

Weight and volume relationships

In soil mechanics, weight-volume relationships help you describe and predict soil behavior. They’re based on the three-phase system, which treats soil as a mixture of:

solids
water
air

Three-phase system

Definitions

Soil is idealized as a system consisting of three components: solids, water, and air. Engineering relationships are derived by relating the weights and volumes of these components.

We’ll use the following symbols:

$V$ = total volume
$V_{s}$ = volume of solids
$V_{w}$ = volume of water
$V_{a}$ = volume of air
$W$ = total weight
$W_{s}$ = weight of solids
$W_{w}$ = weight of water

The total volume is the sum of the three phase volumes:

$V = V_{s} + V_{w} + V_{a}$

The volume of voids is the part of the soil volume not occupied by solids (so it includes both air and water):

$V_{v} = V_{w} + V_{a}$

Water content

Water content is the ratio of water weight to solids weight:

$w = \frac{W _{w}}{W _{s}} \times 100%$

Total/bulk unit weight

Total (bulk) unit weight is total weight per total volume:

$γ = \frac{W}{V}$

Dry unit weight

Dry unit weight uses only the weight of solids (but still divides by the total volume):

$γ_{d} = \frac{W _{s}}{V}$

Saturated unit weight

Saturated unit weight applies when the voids are completely filled with water (no air):

$γ_{s a t} = \frac{W _{s} + W _{w}}{V}$

Void ratio

Void ratio compares void volume to solids volume:

$e = \frac{V _{v}}{V _{s}}$

Porosity

Porosity is the fraction of the total volume that is void space:

$n = \frac{V _{v}}{V} \times 100%$

Degree of saturation

Degree of saturation is the fraction of the void space that is filled with water:

$S = \frac{V _{w}}{V _{v}} \times 100%$

Specific gravity of solids

Specific gravity of solids compares the density of soil solids to the density of water:

$G_{s} = \frac{ρ _{s}}{ρ _{w}} = \frac{W _{s} / V _{s}}{ρ _{w}}$

Quick conversion formula

These relationships are commonly used to convert between properties:

$γ = \frac{( 1 + w ) G _{s} γ _{w}}{1 + e}$

$γ_{d} = \frac{G _{s} γ _{w}}{1 + e}$

$e = \frac{G _{s} γ _{w}}{γ _{d}} - 1$

$S e = G_{s} w$

$n = \frac{e}{1 + e}$

$e = \frac{n - 1}{n}$

Example problem

Given:

$V = 0.003 m^{3}$
$W_{s} = 40 N$
$w = 25%$
$G_{s} = 2.7$

Find:

Volume of voids
Porosity
Bulk unit weight
Dry unit weight

Solution

1. Weight of water:

Convert the water content to a decimal and multiply by the solids weight.

$W_{w} = w \cdot W_{s} = 0.25 \times 40 = 10 N$

2. Total weight:

Add the solids and water weights.

$W = W_{s} + W_{w} = 40 + 10 = 50 N$

3. Bulk unit weight:

Divide total weight by total volume.

$γ = \frac{W}{V} = \frac{50}{0.003} = 16, 667 N/m^{3}$

4. Dry unit weight:

Divide solids weight by total volume.

$γ_{d} = \frac{W _{s}}{V} = \frac{40}{0.003} = 13, 333 N/m^{3}$

5. Volume of solids:

Use the relationship between solids weight, specific gravity, and the unit weight of water.

$V_{s} = \frac{W _{s}}{G _{s} \cdot γ _{w}}$

Assume $γ_{w} = 9, 810 N/m^{3}$ :

$V_{s} = \frac{40}{2.7 \times 9810} \approx 0.00151 m^{3}$

6. Volume of voids:

Subtract the solids volume from the total volume.

$V_{v} = V - V_{s} = 0.003 - 0.00151 = 0.00149 m^{3}$

7. Porosity:

Compute void volume as a fraction of total volume.

$n = \frac{V _{v}}{V} \times 100% = \frac{0.00149}{0.003} \times 100% \approx 49.7%$

Soil compaction and classification parameters

Relative density

Relative density compares the in-place void ratio (or dry unit weight) of a granular soil to its loosest and densest possible states.

$D_{r} = [\frac{( e _{max} - e )}{( e _{max} - e _{min} )}] \times 100$

$D_{r} = [\frac{( γ _{D field} - γ _{D min} )}{( γ _{D max} - γ _{D min} )}] [\frac{γ _{D max}}{γ _{D field}}] \times 100$

Relative compaction (%)

Relative compaction compares the field dry unit weight to the maximum dry unit weight from a compaction test.

$RC = (\frac{γ _{D field}}{γ _{D max}}) \times 100$

Plasticity index

Plasticity index is the range of water contents over which a fine-grained soil behaves plastically.

$P I = LL - P L$

$LL$ = liquid limit
$P L$ = plastic limit

Coefficient of uniformity

The coefficient of uniformity describes how spread out the particle sizes are.

$C_{U} = \frac{D _{60}}{D _{10}}$

Coefficient of concavity (or curvature)

The coefficient of concavity (curvature) describes the shape of the gradation curve.

$C_{C} = \frac{( D _{30} ) ^{2}}{D _{10} \times D _{60}}$

Definitions

$D_{n}$ is the particle diameter at which $n$ of the soil particles are finer by weight.

Permeability and seepage

Hydraulic conductivity (coefficient of permeability)

Hydraulic conductivity $k$ measures how easily water flows through soil.

Constant head test:

$k = \frac{Q}{i A t _{e}}$

$i = \frac{d h}{d L}$

where, $Q = total quantity of water$

Falling head test:

$k = 2.303 (\frac{a L}{A t _{e}}) lo g_{10} (\frac{h _{1}}{h _{2}})$

Where:

$A$ = cross-sectional area of test specimen perpendicular to flow
$a$ = cross-sectional area of reservoir tube
$t_{e}$ = elapsed time
$h_{1}$ = head at time $t = 0$
$h_{2}$ = head at time $t = t_{e}$
$L$ = length of soil column

Discharge velocity and seepage velocity

Definitions

The actual velocity at which water moves through the void spaces of soil.

Seepage velocity is the rate of movement of an element of water through a soil.

$v = ki$

$v_{s} = \frac{ki}{n}$

where, $v = discharge velocity$ and $v_{s} = seepage velocity$

Flow nets

Definitions

A graphical representation of flow lines and equipotential lines used to analyze seepage through soils.

A flow net is a combination of flow lines and equipotential lines.

A flow line is a line along which a water particle travels.
An equipotential line connects points of equal total head.

Key properties used when constructing and interpreting a flow net:

There are no flow along equipotential lines, which are 90 degrees to flow lines.
The total head along an equipotential line is equal at all points.
Flow lines cannot cross other flow lines and equipotential lines cannot cross other equipotential lines.
Equipotential lines intersect the flow lines at right angles.
Flow net must be constructed so that each element is a curvilinear square (sides may be curved but a circle must be inscribed within it that touches all its 4 sides).

The total flow rate though a flow net, is solved for by:

$Q = k Δ h \frac{N _{f}}{N _{d}} L$

Where:

$Q$ = total flow rate
$N_{f}$ = number of flow channels in a flow net
$N_{d}$ = number of potential drops
$Δ h$ = head change from upstream to downstream
$k$ = coefficient of permeability
$L$ = length of structure (i.e. bank-to-bank)

Factor of safety against seepage liquefaction

$F S_{s} = \frac{i _{c}}{i _{e}}$

$i_{c} = \frac{( γ _{sat} - γ _{w} )}{γ _{w}}$

where, $i_{e} = seepage exit gradient$

Weight and volume relationships

Three-phase system: soil = solids + water + air
Key formulas:
- $V = V_{s} + V_{w} + V_{a}$ (total volume)
- $V_{v} = V_{w} + V_{a}$ (void volume)
Water content: $w = \frac{W _{w}}{W _{s}} \times 100%$
Unit weights:
- Bulk: $γ = \frac{W}{V}$
- Dry: $γ_{d} = \frac{W _{s}}{V}$
- Saturated: $γ_{s a t} = \frac{W _{s} + W _{w}}{V}$
Void ratio: $e = \frac{V _{v}}{V _{s}}$
Porosity: $n = \frac{V _{v}}{V} \times 100%$
Degree of saturation: $S = \frac{V _{w}}{V _{v}} \times 100%$
Specific gravity: $G_{s} = \frac{ρ _{s}}{ρ _{w}}$
Conversion formulas:
- $γ = \frac{( 1 + w ) G _{s} γ _{w}}{1 + e}$
- $γ_{d} = \frac{G _{s} γ _{w}}{1 + e}$
- $e = \frac{G _{s} γ _{w}}{γ _{d}} - 1$
- $S e = G_{s} w$
- $n = \frac{e}{1 + e}$

Soil compaction and classification parameters

Relative density: compares in-place void ratio/dry unit weight to loosest/densest states
- $D_{r} = [\frac{( e _{ma x} - e )}{( e _{ma x} - e _{min} )}] \times 100$
Relative compaction: field dry unit weight vs. maximum dry unit weight
- $RC = (\frac{γ _{D f i e l d}}{γ _{D ma x}}) \times 100$
Plasticity index: $P I = LL - P L$
- Range of water contents for plastic behavior
Coefficient of uniformity: $C_{U} = \frac{D _{60}}{D _{10}}$
Coefficient of concavity: $C_{C} = \frac{( D _{30} ) ^{2}}{D _{10} \times D _{60}}$
- $D_{n}$ = particle diameter at which $n %$ is finer

Permeability and seepage

Hydraulic conductivity ( $k$ ): ease of water flow through soil
- Constant head: $k = \frac{Q}{i A t _{e}}$
- Falling head: $k = 2.303 (\frac{a L}{A t _{e}}) lo g_{10} (\frac{h _{1}}{h _{2}})$
Discharge velocity: $v = ki$
Seepage velocity: $v_{s} = \frac{ki}{n}$
Flow nets:
- Graphical tool: flow lines (water paths) + equipotential lines (equal head)
- Properties: lines intersect at 90°, no crossing of same type, curvilinear squares
- Flow rate: $Q = k Δ h \frac{N _{f}}{N _{d}} L$
Factor of safety against seepage liquefaction:
- $F S_{s} = \frac{i _{c}}{i _{e}}$
- $i_{c} = \frac{( γ _{s a t} - γ _{w} )}{γ _{w}}$ (critical gradient)

Next | 9.2 Consolidation and stress

Weight and volume relationships

This chapter covers the following:

Weight and volume relationships
Soil compaction and classification parameters
Permeability and seepage

Weight and volume relationships

In soil mechanics, weight-volume relationships help you describe and predict soil behavior. They’re based on the three-phase system, which treats soil as a mixture of:

solids
water
air

Three-phase system

Definitions

Soil is idealized as a system consisting of three components: solids, water, and air. Engineering relationships are derived by relating the weights and volumes of these components.

We’ll use the following symbols:

$V$ = total volume
$V_{s}$ = volume of solids
$V_{w}$ = volume of water
$V_{a}$ = volume of air
$W$ = total weight
$W_{s}$ = weight of solids
$W_{w}$ = weight of water

The total volume is the sum of the three phase volumes:

$V = V_{s} + V_{w} + V_{a}$

The volume of voids is the part of the soil volume not occupied by solids (so it includes both air and water):

$V_{v} = V_{w} + V_{a}$

Water content

Water content is the ratio of water weight to solids weight:

$w = \frac{W _{w}}{W _{s}} \times 100%$

Total/bulk unit weight

Total (bulk) unit weight is total weight per total volume:

$γ = \frac{W}{V}$

Dry unit weight

Dry unit weight uses only the weight of solids (but still divides by the total volume):

$γ_{d} = \frac{W _{s}}{V}$

Saturated unit weight

Saturated unit weight applies when the voids are completely filled with water (no air):

$γ_{s a t} = \frac{W _{s} + W _{w}}{V}$

Void ratio

Void ratio compares void volume to solids volume:

$e = \frac{V _{v}}{V _{s}}$

Porosity

Porosity is the fraction of the total volume that is void space:

$n = \frac{V _{v}}{V} \times 100%$

Degree of saturation

Degree of saturation is the fraction of the void space that is filled with water:

$S = \frac{V _{w}}{V _{v}} \times 100%$

Specific gravity of solids

Specific gravity of solids compares the density of soil solids to the density of water:

$G_{s} = \frac{ρ _{s}}{ρ _{w}} = \frac{W _{s} / V _{s}}{ρ _{w}}$

Quick conversion formula

These relationships are commonly used to convert between properties:

$γ = \frac{( 1 + w ) G _{s} γ _{w}}{1 + e}$

$γ_{d} = \frac{G _{s} γ _{w}}{1 + e}$

$e = \frac{G _{s} γ _{w}}{γ _{d}} - 1$

$S e = G_{s} w$

$n = \frac{e}{1 + e}$

$e = \frac{n - 1}{n}$

Example problem

Given:

$V = 0.003 m^{3}$
$W_{s} = 40 N$
$w = 25%$
$G_{s} = 2.7$

Find:

Volume of voids
Porosity
Bulk unit weight
Dry unit weight

Solution

1. Weight of water:

Convert the water content to a decimal and multiply by the solids weight.

$W_{w} = w \cdot W_{s} = 0.25 \times 40 = 10 N$

2. Total weight:

Add the solids and water weights.

$W = W_{s} + W_{w} = 40 + 10 = 50 N$

3. Bulk unit weight:

Divide total weight by total volume.

$γ = \frac{W}{V} = \frac{50}{0.003} = 16, 667 N/m^{3}$

4. Dry unit weight:

Divide solids weight by total volume.

$γ_{d} = \frac{W _{s}}{V} = \frac{40}{0.003} = 13, 333 N/m^{3}$

5. Volume of solids:

Use the relationship between solids weight, specific gravity, and the unit weight of water.

$V_{s} = \frac{W _{s}}{G _{s} \cdot γ _{w}}$

Assume $γ_{w} = 9, 810 N/m^{3}$ :

$V_{s} = \frac{40}{2.7 \times 9810} \approx 0.00151 m^{3}$

6. Volume of voids:

Subtract the solids volume from the total volume.

$V_{v} = V - V_{s} = 0.003 - 0.00151 = 0.00149 m^{3}$

7. Porosity:

Compute void volume as a fraction of total volume.

$n = \frac{V _{v}}{V} \times 100% = \frac{0.00149}{0.003} \times 100% \approx 49.7%$

Soil compaction and classification parameters

Relative density

Relative density compares the in-place void ratio (or dry unit weight) of a granular soil to its loosest and densest possible states.

$D_{r} = [\frac{( e _{max} - e )}{( e _{max} - e _{min} )}] \times 100$

$D_{r} = [\frac{( γ _{D field} - γ _{D min} )}{( γ _{D max} - γ _{D min} )}] [\frac{γ _{D max}}{γ _{D field}}] \times 100$

Relative compaction (%)

Relative compaction compares the field dry unit weight to the maximum dry unit weight from a compaction test.

$RC = (\frac{γ _{D field}}{γ _{D max}}) \times 100$

Plasticity index

Plasticity index is the range of water contents over which a fine-grained soil behaves plastically.

$P I = LL - P L$

$LL$ = liquid limit
$P L$ = plastic limit

Coefficient of uniformity

The coefficient of uniformity describes how spread out the particle sizes are.

$C_{U} = \frac{D _{60}}{D _{10}}$

Coefficient of concavity (or curvature)

The coefficient of concavity (curvature) describes the shape of the gradation curve.

$C_{C} = \frac{( D _{30} ) ^{2}}{D _{10} \times D _{60}}$

Definitions

$D_{n}$ is the particle diameter at which $n$ of the soil particles are finer by weight.

Permeability and seepage

Hydraulic conductivity (coefficient of permeability)

Hydraulic conductivity $k$ measures how easily water flows through soil.

Constant head test:

$k = \frac{Q}{i A t _{e}}$

$i = \frac{d h}{d L}$

where, $Q = total quantity of water$

Falling head test:

$k = 2.303 (\frac{a L}{A t _{e}}) lo g_{10} (\frac{h _{1}}{h _{2}})$

Where:

$A$ = cross-sectional area of test specimen perpendicular to flow
$a$ = cross-sectional area of reservoir tube
$t_{e}$ = elapsed time
$h_{1}$ = head at time $t = 0$
$h_{2}$ = head at time $t = t_{e}$
$L$ = length of soil column

Discharge velocity and seepage velocity

Definitions

The actual velocity at which water moves through the void spaces of soil.

Seepage velocity is the rate of movement of an element of water through a soil.

$v = ki$

$v_{s} = \frac{ki}{n}$

where, $v = discharge velocity$ and $v_{s} = seepage velocity$

Flow nets

Definitions

A graphical representation of flow lines and equipotential lines used to analyze seepage through soils.

A flow net is a combination of flow lines and equipotential lines.

A flow line is a line along which a water particle travels.
An equipotential line connects points of equal total head.

Key properties used when constructing and interpreting a flow net:

There are no flow along equipotential lines, which are 90 degrees to flow lines.
The total head along an equipotential line is equal at all points.
Flow lines cannot cross other flow lines and equipotential lines cannot cross other equipotential lines.
Equipotential lines intersect the flow lines at right angles.
Flow net must be constructed so that each element is a curvilinear square (sides may be curved but a circle must be inscribed within it that touches all its 4 sides).

The total flow rate though a flow net, is solved for by:

$Q = k Δ h \frac{N _{f}}{N _{d}} L$

Where:

$Q$ = total flow rate
$N_{f}$ = number of flow channels in a flow net
$N_{d}$ = number of potential drops
$Δ h$ = head change from upstream to downstream
$k$ = coefficient of permeability
$L$ = length of structure (i.e. bank-to-bank)

Factor of safety against seepage liquefaction

$F S_{s} = \frac{i _{c}}{i _{e}}$

$i_{c} = \frac{( γ _{sat} - γ _{w} )}{γ _{w}}$

where, $i_{e} = seepage exit gradient$