Aggregate Compaction — Quantity and Cost: 18-Env-A3 Geotechnical and Hydrogeological Engineering, December 2015, Q3

Question 3: Aggregate Compaction — Quantity and Cost (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Given data
Quantity	Symbol	Value
Target compacted dry unit weight	$\gamma_{d,\text{target}}$	20 kN/m³
Compacted fill volume needed	$V_{\text{compacted}}$	50 m³
Loose (as-sold) moist unit weight	$\gamma_{\text{loose}}$	17 kN/m³
Loose degree of saturation	$S$	30%
Specific gravity of solids	$G_s$	2.65
Purchase price	—	\$3120 / metric tonne

Find. (a) total weight and volume of loose material to purchase; (b) void ratio of the compacted fill and the purchase cost.

Approach. The mass of solids is conserved between the loose (borrow) and compacted (fill) states — compaction only reduces the void volume, not the solids — so the dry weight of solids required for the target compacted volume fixes the dry weight (and hence the loose volume and moist purchase weight) that must be hauled in.

Dry weight of solids required in the finished fill. $$W_s=\gamma_{d,\text{target}}\times V_{\text{compacted}}=20\times50=\boxed{1000\ \text{kN}}.$$
Characterize the loose material's phase relations. Combining $\gamma_{\text{loose}}=\dfrac{G_s\gamma_w(1+w)}{1+e}$ with $S\,e=w\,G_s$ (two equations, two unknowns $e,w$) and solving simultaneously: $$e_{\text{loose}}=0.640,\qquad w_{\text{loose}}=\frac{S\,e_{\text{loose}}}{G_s}=\frac{0.30\times0.640}{2.65}=\boxed{7.25\%},$$ $$\gamma_{d,\text{loose}}=\frac{\gamma_{\text{loose}}}{1+w_{\text{loose}}}=\frac{17}{1.0725}=\boxed{15.85\ \text{kN/m}^3}.$$
(a) Loose volume and moist weight to purchase. The same 1000 kN of solids, at the loose-state dry unit weight, occupies $$V_{\text{loose}}=\frac{W_s}{\gamma_{d,\text{loose}}}=\frac{1000}{15.85}=\boxed{63.1\ \text{m}^3},$$ and since the yard sells by moist (as-delivered) weight, $$W_{\text{moist}}=\gamma_{\text{loose}}\times V_{\text{loose}}=17\times63.1=\boxed{1072.5\ \text{kN}}.$$
Convert to purchase units and price. With 1 metric tonne-force $=\gamma_w=9.81\ \text{kN}$: $$\text{mass}=\frac{1072.5}{9.81}=\boxed{109.3\ \text{tonnes}},\qquad \text{cost}=109.3\times\$3120=\boxed{\$341{,}100}.$$
(b) Void ratio of the compacted material at the target $\gamma_{d,\text{target}}=20\ \text{kN/m}^3$: $$e_{\text{compacted}}=\frac{G_s\gamma_w}{\gamma_{d,\text{target}}}-1=\frac{2.65\times9.81}{20}-1=1.300-1=\boxed{0.300}.$$

Quantity	Value
(a) Loose volume to purchase, $V_{\text{loose}}$	63.1 m³
(a) Moist weight to purchase, $W_{\text{moist}}$	1072.5 kN (109.3 tonnes)
(b) Void ratio of compacted material, $e_{\text{compacted}}$	0.300
(b) Purchase cost	≈ \$341,100

Concept & Theory

Topic. Borrow-to-fill earthwork quantities and compaction phase relations.

Key relations. $\gamma=G_s\gamma_w(1+w)/(1+e)$; $S\,e=w\,G_s$; $e=G_s\gamma_w/\gamma_d-1$.

Why this works. Compaction rearranges soil particles to reduce the void ratio without adding or removing solids, so equating the dry weight of solids between the loose borrow state and the target compacted state is the correct conservation law for sizing a cut-to-fill quantity; every other loose-state property follows from the two independent phase-relation equations in $e$ and $w$.

Common pitfall. Pricing the DRY tonnage instead of the MOIST (as-delivered) weight the yard actually sells, and using 1000 kg/tonne instead of $\gamma_w=9.81\ \text{kN/tonne}$ when converting a weight in kN to metric tonnes-force.

Source. Das, Principles of Geotechnical Engineering, ch. on compaction.

Question 3 of 6: Aggregate Compaction — Quantity and Cost

Question 3: Aggregate Compaction — Quantity and Cost (20 marks)