Relative density

Geotechnical · FE Reference Handbook section

Geotechnical

2 formulas

10 exam-style examples

~49 min

All Geotechnical lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Relative density within Geotechnical. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what relative density describes physically and when it applies.
State every one of the 2 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: unit weight in pcf with depth in ft gives stress in psf, not psi.

Lecture

Why this section exists. Relative density is the part of Geotechnical that lets you connect a layered soil profile beneath a footing or wall to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as a phase-diagram quantity, an effective stress, or a bearing capacity. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. unit weight in pcf with depth in ft gives stress in psf, not psi. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Truck-mounted drill rig taking a soil boring beside a bridge, with sample jars in the foreground. — Photo 1. Where this shows up in practice: relative density.
Capstone Studio instructional photograph

Geotechnical — Relative density: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes a layered soil profile beneath a footing or wall. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 2 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

Dr	Quantity produced by "Dr = [(emax – e)/(emax – emin)] × 100" — read its definition and unit from the handbook line directly above the equation.

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Traffic density from flow and speed

A freeway lane carries 1,800 veh/h at an average speed of 90 km/h. Find the density and the average spacing.

Given

q = 1,800 veh/h
u = 90 km/h

Find

Density k and spacing

Start with the thinking

q = ku is the fundamental relation.
Spacing is the reciprocal of density.

Step-by-step solution

Density
$k = q / u = 1, 800/90$
Evaluate
$k = 20.0 v e h / k m / l an e$
Spacing
$s = 1, 000/ k = 1, 000/20.0$
Result
$k = 20 v e h / k m, s p a c in g = 50.0 m$

Answer: k = 20 veh/km/lane, spacing = 50 m

Why the other options are there

162,000 (q multiplied by u)
k = 0.05 (units not converted)

Reference: FE Reference Handbook — Transportation — Traffic flow relationships

Example 2

Field compaction acceptance check — Relative density

The standard Proctor maximum dry density is 119.0 pcf and the specification requires 95% compaction. A nuclear gauge reads a field dry density of 124.0 pcf. Does the lift pass, and by how much?

Given

γd,max = 119.0 pcf
Required = 95%
γd,field = 124.0 pcf

Find

Required density, percent compaction achieved, pass/fail

Start with the thinking

Percent compaction always compares dry densities, never moist ones.
Compare the achieved percentage against the specification.

Step-by-step solution

Required density — γd,req = 95%(119.0) = 113.1 pcf
Achieved
Compare — 104.2% ≥ 95% → PASSES
Margin — 11.0 pcf relative to the required density

Answer: %C ≈ 104.2% → the lift passes (required 113.1 pcf)

Why the other options are there

96.0% (ratio inverted)
116.7% (moist density compared to dry maximum)

Reference: FE Reference Handbook — Geotechnical → Relative density

Example 3

Field compaction acceptance check — Relative density (2)

The standard Proctor maximum dry density is 119.0 pcf and the specification requires 94% compaction. A nuclear gauge reads a field dry density of 110.0 pcf. Does the lift pass, and by how much?

Given

γd,max = 119.0 pcf
Required = 94%
γd,field = 110.0 pcf

Find

Required density, percent compaction achieved, pass/fail

Start with the thinking

Percent compaction always compares dry densities, never moist ones.
Compare the achieved percentage against the specification.

Step-by-step solution

Required density — γd,req = 94%(119.0) = 111.9 pcf
Achieved
Compare — 92.4% < 94% → FAILS
Margin — -1.9 pcf relative to the required density

Answer: %C ≈ 92.4% → the lift fails (required 111.9 pcf)

Why the other options are there

108.2% (ratio inverted)
103.5% (moist density compared to dry maximum)

Reference: FE Reference Handbook — Geotechnical → Relative density

Example 4

Field compaction acceptance check — Relative density (3)

The standard Proctor maximum dry density is 128.0 pcf and the specification requires 95% compaction. A nuclear gauge reads a field dry density of 122.0 pcf. Does the lift pass, and by how much?

Given

γd,max = 128.0 pcf
Required = 95%
γd,field = 122.0 pcf

Find

Required density, percent compaction achieved, pass/fail

Start with the thinking

Percent compaction always compares dry densities, never moist ones.
Compare the achieved percentage against the specification.

Step-by-step solution

Required density — γd,req = 95%(128.0) = 121.6 pcf
Achieved
Compare — 95.3% ≥ 95% → PASSES
Margin — 0.4 pcf relative to the required density

Answer: %C ≈ 95.3% → the lift passes (required 121.6 pcf)

Why the other options are there

104.9% (ratio inverted)
106.8% (moist density compared to dry maximum)

Reference: FE Reference Handbook — Geotechnical → Relative density

Example 5

Field compaction acceptance check — Relative density (4)

The standard Proctor maximum dry density is 128.0 pcf and the specification requires 97% compaction. A nuclear gauge reads a field dry density of 112.5 pcf. Does the lift pass, and by how much?

Given

γd,max = 128.0 pcf
Required = 97%
γd,field = 112.5 pcf

Find

Required density, percent compaction achieved, pass/fail

Start with the thinking

Percent compaction always compares dry densities, never moist ones.
Compare the achieved percentage against the specification.

Step-by-step solution

Required density — γd,req = 97%(128.0) = 124.2 pcf
Achieved
Compare — 87.9% < 97% → FAILS
Margin — -11.7 pcf relative to the required density

Answer: %C ≈ 87.9% → the lift fails (required 124.2 pcf)

Why the other options are there

113.8% (ratio inverted)
98.4% (moist density compared to dry maximum)

Reference: FE Reference Handbook — Geotechnical → Relative density

Example 6

Field compaction acceptance check — Relative density (5)

The standard Proctor maximum dry density is 126.5 pcf and the specification requires 95% compaction. A nuclear gauge reads a field dry density of 120.0 pcf. Does the lift pass, and by how much?

Given

γd,max = 126.5 pcf
Required = 95%
γd,field = 120.0 pcf

Find

Required density, percent compaction achieved, pass/fail

Start with the thinking

Percent compaction always compares dry densities, never moist ones.
Compare the achieved percentage against the specification.

Step-by-step solution

Required density — γd,req = 95%(126.5) = 120.2 pcf
Achieved
Compare — 94.9% < 95% → FAILS
Margin — -0.2 pcf relative to the required density

Answer: %C ≈ 94.9% → the lift fails (required 120.2 pcf)

Why the other options are there

105.4% (ratio inverted)
106.2% (moist density compared to dry maximum)

Reference: FE Reference Handbook — Geotechnical → Relative density

Example 7

Field compaction acceptance check — Relative density (6)

The standard Proctor maximum dry density is 126.0 pcf and the specification requires 98% compaction. A nuclear gauge reads a field dry density of 118.5 pcf. Does the lift pass, and by how much?

Given

γd,max = 126.0 pcf
Required = 98%
γd,field = 118.5 pcf

Find

Required density, percent compaction achieved, pass/fail

Start with the thinking

Percent compaction always compares dry densities, never moist ones.
Compare the achieved percentage against the specification.

Step-by-step solution

Required density — γd,req = 98%(126.0) = 123.5 pcf
Achieved
Compare — 94.0% < 98% → FAILS
Margin — -5.0 pcf relative to the required density

Answer: %C ≈ 94.0% → the lift fails (required 123.5 pcf)

Why the other options are there

106.3% (ratio inverted)
105.3% (moist density compared to dry maximum)

Reference: FE Reference Handbook — Geotechnical → Relative density

Example 8

Field compaction acceptance check — Relative density (7)

The standard Proctor maximum dry density is 131.0 pcf and the specification requires 94% compaction. A nuclear gauge reads a field dry density of 125.0 pcf. Does the lift pass, and by how much?

Given

γd,max = 131.0 pcf
Required = 94%
γd,field = 125.0 pcf

Find

Required density, percent compaction achieved, pass/fail

Start with the thinking

Percent compaction always compares dry densities, never moist ones.
Compare the achieved percentage against the specification.

Step-by-step solution

Required density — γd,req = 94%(131.0) = 123.1 pcf
Achieved
Compare — 95.4% ≥ 94% → PASSES
Margin — 1.9 pcf relative to the required density

Answer: %C ≈ 95.4% → the lift passes (required 123.1 pcf)

Why the other options are there

104.8% (ratio inverted)
106.9% (moist density compared to dry maximum)

Reference: FE Reference Handbook — Geotechnical → Relative density

Example 9

Field compaction acceptance check — Relative density (8)

The standard Proctor maximum dry density is 125.0 pcf and the specification requires 97% compaction. A nuclear gauge reads a field dry density of 121.5 pcf. Does the lift pass, and by how much?

Given

γd,max = 125.0 pcf
Required = 97%
γd,field = 121.5 pcf

Find

Required density, percent compaction achieved, pass/fail

Start with the thinking

Percent compaction always compares dry densities, never moist ones.
Compare the achieved percentage against the specification.

Step-by-step solution

Required density — γd,req = 97%(125.0) = 121.3 pcf
Achieved
Compare — 97.2% ≥ 97% → PASSES
Margin — 0.3 pcf relative to the required density

Answer: %C ≈ 97.2% → the lift passes (required 121.3 pcf)

Why the other options are there

102.9% (ratio inverted)
108.9% (moist density compared to dry maximum)

Reference: FE Reference Handbook — Geotechnical → Relative density

Example 10

Field compaction acceptance check — Relative density (9)

The standard Proctor maximum dry density is 126.0 pcf and the specification requires 93% compaction. A nuclear gauge reads a field dry density of 119.0 pcf. Does the lift pass, and by how much?

Given

γd,max = 126.0 pcf
Required = 93%
γd,field = 119.0 pcf

Find

Required density, percent compaction achieved, pass/fail

Start with the thinking

Percent compaction always compares dry densities, never moist ones.
Compare the achieved percentage against the specification.

Step-by-step solution

Required density — γd,req = 93%(126.0) = 117.2 pcf
Achieved
Compare — 94.4% ≥ 93% → PASSES
Margin — 1.8 pcf relative to the required density

Answer: %C ≈ 94.4% → the lift passes (required 117.2 pcf)

Why the other options are there

105.9% (ratio inverted)
105.8% (moist density compared to dry maximum)

Reference: FE Reference Handbook — Geotechnical → Relative density

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given a layered soil profile beneath a footing or wall, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Relative density contains 2 relations; you must be able to find this page in under 15 seconds.
Exam style: a phase-diagram quantity, an effective stress, or a bearing capacity.
Unit rule: unit weight in pcf with depth in ft gives stress in psf, not psi.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

unit weight in pcf with depth in ft gives stress in psf, not psi
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.