Relative density

Geotechnical · FE Reference Handbook section

Geotechnical

2 formulas

10 exam-style examples

~49 min

All Geotechnical lectures

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

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
$%C = \gammad,field/\gammad,\max = 124.0/119.0 = 104.2%$
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 2

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
$%C = \gammad,field/\gammad,\max = 110.0/119.0 = 92.4%$
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 3

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
$%C = \gammad,field/\gammad,\max = 122.0/128.0 = 95.3%$
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 4

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
$%C = \gammad,field/\gammad,\max = 112.5/128.0 = 87.9%$
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 5

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
$%C = \gammad,field/\gammad,\max = 120.0/126.5 = 94.9%$
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 6

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
$%C = \gammad,field/\gammad,\max = 118.5/126.0 = 94.0%$
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 7

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
$%C = \gammad,field/\gammad,\max = 125.0/131.0 = 95.4%$
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 8

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
$%C = \gammad,field/\gammad,\max = 121.5/125.0 = 97.2%$
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 9

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
$%C = \gammad,field/\gammad,\max = 119.0/126.0 = 94.4%$
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

Example 10

Field compaction acceptance check — Relative density (10)

The standard Proctor maximum dry density is 127.0 pcf and the specification requires 96% compaction. A nuclear gauge reads a field dry density of 124.0 pcf. Does the lift pass, and by how much?

Given

γd,max = 127.0 pcf
$Required = 96%$
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 = 96%(127.0) = 121.9 pcf
Achieved
$%C = \gammad,field/\gammad,\max = 124.0/127.0 = 97.6%$
Compare — 97.6% ≥ 96% → PASSES
Margin — 2.1 pcf relative to the required density

Answer:

%C ≈ 97.6% → the lift passes (required 121.9 pcf)

Why the other options are there

102.4% (ratio inverted)
109.4% (moist density compared to dry maximum)

Reference: FE Reference Handbook — Geotechnical → Relative density