Soil Phase Relations for the FE Civil Exam

NCEES FE Civil spec: Geotechnical Engineering, B. Phase relations. FE Reference Handbook: Civil Engineering chapter, Geotechnical section (Phase Relationships).

What the FE Civil Spec Covers

Phase relations describe soil as three phases: solids, water and air. FE problems give you two or three measured quantities (for example moist unit weight, water content and specific gravity of solids) and ask for another one such as void ratio, degree of saturation or dry unit weight. These are some of the most reliable points on the geotechnical portion of the exam because they are pure algebra.

Key Concepts

Water content $w$ is weight of water over weight of solids.
Void ratio $e$ is volume of voids over volume of solids.
Porosity $n$ is volume of voids over total volume.
Degree of saturation $S$ is volume of water over volume of voids (0 for dry, 1 for saturated).
Specific gravity of solids $G_{s}$ is the unit weight of the solid particles over that of water, typically 2.60 to 2.80.

The fastest method is to assume a convenient quantity, usually the volume of solids equal to 1, fill in a phase diagram, and read off whatever is asked. The relationships below are the shortcuts that result.

Core Formulas

w = \frac{W _{w}}{W _{s}} e = \frac{V _{v}}{V _{s}} n = \frac{V _{v}}{V} = \frac{e}{1 + e}

S e = w G_{s}

γ = \frac{( G _{s} + S e ) γ _{w}}{1 + e} γ_{d} = \frac{G _{s} γ _{w}}{1 + e} = \frac{γ}{1 + w}

γ_{s a t} = \frac{( G _{s} + e ) γ _{w}}{1 + e} γ^{'} = γ_{s a t} - γ_{w}

D_{r} = \frac{e _{ma x} - e}{e _{ma x} - e _{min}}

Relative density $D_{r}$ applies to granular soils and compares the in-place void ratio with the loosest and densest states.

Solving strategy

Write down what is given and what is asked, with symbols.
If no single formula links them, set $V_{s} = 1$ (or $W_{s} = 1$ ) and build the phase diagram.
Fill in weights from $G_{s}$ and $w$ , and volumes from $e$ and $S$ .
Compute the requested ratio and sanity check it against typical ranges.

Quantity	Typical range
Specific gravity of solids, Gs	2.60 to 2.80 (2.65 for quartz sand)
Void ratio, sands	about 0.4 to 0.9
Void ratio, soft clays	about 0.8 to 1.5 or more
Saturated unit weight	about 18 to 22 kN/m³ (115 to 140 lb/ft³)

A computed degree of saturation above 1.0, or a dry unit weight above the saturated unit weight, means an input was entered wrong. Catching that takes seconds and saves a wrong answer. Also watch units: if the problem gives masses in grams and volumes in cubic centimeters, density of water is 1.0 g/cm³, and you convert to unit weight only at the end.

Where It Is in the FE Reference Handbook

Search the Civil Engineering chapter, Geotechnical section, under Phase Relationships. It includes a phase diagram and definitions for each unit weight, void ratio, porosity and relative density. The Se = wGs identity appears there only as the degree of saturation formula S = wGs/e, so know how to rearrange it.

Worked Example

A soil sample has a moist unit weight of 19.0 kN/m³, a water content of 15 percent and $G_{s} = 2.70$ . Find the dry unit weight, void ratio, porosity, degree of saturation and the saturated unit weight. Use $γ_{w} = 9.81$ kN/m³.

Step 1: dry unit weight.

γ_{d} = \frac{γ}{1 + w} = \frac{19.0}{1.15} = 16.52 kN/m^{3}

Step 2: void ratio.

e = \frac{G _{s} γ _{w}}{γ _{d}} - 1 = \frac{( 2.70 ) ( 9.81 )}{16.52} - 1 = 0.603

Step 3: porosity.

n = \frac{e}{1 + e} = \frac{0.603}{1.603} = 0.376

Step 4: degree of saturation.

S = \frac{w G _{s}}{e} = \frac{( 0.15 ) ( 2.70 )}{0.603} = 0.672 (67%)

Step 5: saturated unit weight.

γ_{s a t} = \frac{( 2.70 + 0.603 ) ( 9.81 )}{1.603} = 20.21 kN/m^{3}

Check: the saturated unit weight must exceed the moist unit weight (20.21 versus 19.0), which it does. The submerged unit weight would be 20.21 minus 9.81 = 10.40 kN/m³.

Common Mistakes

Entering water content as 15 instead of 0.15 in the formulas.
Dividing water weight by total weight instead of dry solids weight.
Using the moist unit weight where the dry unit weight belongs in the void ratio formula.
Mixing up void ratio and porosity, especially when a problem gives porosity and asks for void ratio.
Using the saturated unit weight below the water table in an effective stress problem but forgetting to subtract pore pressure.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

A wet soil sample contains 20 g of water and 100 g of dry solids. What is its gravimetric water content? See the worked solution or practice the Soil Mechanics deck
A soil has porosity n = 0.40. What is its approximate void ratio e? See the worked solution or practice the Soil Mechanics deck
A soil has moist unit weight γ = 19 kN/m³ and gravimetric water content w = 10%. What is its approximate dry unit weight? See the worked solution or practice the Soil Mechanics deck
A sand has maximum void ratio 0.90, minimum void ratio 0.45, and an in-situ void ratio of 0.60. Its relative density is most nearly: See the worked solution or practice the Soil Mechanics deck
A soil has water content w = 18%, specific gravity of solids G_s = 2.70, and void ratio e = 0.65. Its degree of saturation is most nearly: See the worked solution or practice the Soil Mechanics deck
A saturated soil has G_s = 2.68 and void ratio e = 0.70. Taking γ_w = 9.81 kN/m³, its saturated unit weight is most nearly: See the worked solution or practice the Soil Mechanics deck

Related Calculators and Examples

Frequently Asked Questions

What is the difference between void ratio and porosity?

Void ratio is the volume of voids divided by the volume of solids, so it can exceed 1. Porosity is the volume of voids divided by the total volume, so it is always less than 1. They convert with n = e/(1 + e) and e = n/(1 - n).

Is water content measured by weight or by volume in geotechnical work?

By weight. Geotechnical water content is the weight of water divided by the weight of dry solids, so it can exceed 100 percent in very soft clays. Volumetric water content is a different quantity used in hydrology.

What value of the unit weight of water should I use on the FE?

Use 9.81 kN/m³ in SI units and 62.4 lb/ft³ in US units unless the problem states otherwise.

How do I find void ratio if only porosity is given?

Use e = n/(1 - n). For example, a porosity of 0.40 gives a void ratio of 0.40/0.60 = 0.67. Going the other way, n = e/(1 + e).

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