Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2018 — 18-Geol-A6 Soil Mechanics. Three-hour, closed-book exam; a Casio or Sharp approved calculator, a compass and a ruler are permitted. Six questions constitute the complete 100-mark paper: Questions 1–5 (10+15+20+20+20 marks) are compulsory; Question 6 offers eight 5-mark optional items of which the source asks for three.
Each reading is cross-checked below by an independent physical-consistency test (Q2: the 12%-point saturation comes out to $\approx100\%$, matching its plotted position essentially on the $S=100\%$ curve; Q4: the two digitized curves in Figure Q4-2 are treated as the overconsolidated-clay sample (upper, steep-kneed curve, per the question's own description) and an unused second dataset (lower curve, not needed for a single-sample problem); Q5: the resulting pore pressures give positive effective stress at every point and a linear uplift diagram, the expected signature of a consistent equipotential count). Assumed $G_s=2.70$ (not given) for Q2(c)'s phase relations — typical for a silty/clayey compacted fill and independently supported by the $S\approx100\%$ check above.
Given. Standard and Modified Proctor test points plotted on $\gamma_d$ (dry unit weight, kN/m$^3$) vs. $w$ (water content, %), together with three zero-air-void ($S=90,95,100\%$) reference curves (Figure Q2-1); test point of interest at $w=12\%$.
Find. (a) axis labels; (b) qualitative shape, OMC/$\gamma_{d,max}$ for both curves, and the line of optimums; (c) $e$, $S$, $\gamma_t$, $\theta$, $n$, $\rho_d$ at $w=12\%$.
Approach. Read the two curve peaks directly off the plot; for part (c) use the standard phase-relation identities with $G_s=2.70$ (assumed — not given; see the check note banner above).
Fig. Q2-1 — Standard and Modified Proctor curves with $S=90/95/100\%$ zero-air-void curves ($G_s=2.70$); the 12%-water-content test point is ringed.
(a) Axes. Horizontal axis: water content, $w$ (%). Vertical axis: dry unit weight, $\gamma_d$ (kN/m$^3$).
(b)(i) Compaction curves. Both curves rise to a single peak (maximum $\gamma_d$) and then fall as $w$ increases further — below the peak, added water lubricates particles and improves packing (dry side); beyond the peak, added water displaces solids without expelling all of the entrained air (wet side). The Modified curve (higher compactive energy) sits above and to the left of the Standard curve: more compaction energy raises $\gamma_{d,max}$ and lowers the OMC needed to reach it.
(b)(ii) OMC / $\gamma_{d,max}$. Reading each curve's peak from the plot: Modified Proctor peaks at $w\approx10\%$, $\gamma_{d,max}\approx20.4\text{ kN/m}^3$; Standard Proctor peaks at $w\approx15\%$, $\gamma_{d,max}\approx18.5\text{ kN/m}^3$.
(b)(iii) Line of optimums. The line joining the two peaks, $(10\%,20.4)\to(15\%,18.5\text{ kN/m}^3)$, slope $\approx-0.38\text{ kN/m}^3$ per %; it runs roughly parallel to the zero-air-void curves at a saturation somewhat below $100\%$ (typically $S\approx85$–$95\%$ at optimum), and is the locus optimum points would trace out for any other compactive effort on this soil.
(c) Phase relations at $w=12\%$. From the Modified Proctor curve, $\gamma_d=20.0\text{ kN/m}^3$ at $w=12\%$. With $G_s=2.70$, $\gamma_w=9.81\text{ kN/m}^3$:
i) Void ratio: $$e=\frac{G_s\gamma_w}{\gamma_d}-1=\frac{2.70\times9.81}{20.0}-1=\boxed{0.324}$$
ii) Degree of saturation: $$S=\frac{wG_s}{e}=\frac{0.12\times2.70}{0.324}=0.999\approx\boxed{100\%}$$
(the point plots essentially on the $S=100\%$ curve — confirms both the $\gamma_d$ reading and the assumed $G_s$).
iii) Total unit weight: $$\gamma_t=\gamma_d(1+w)=20.0(1.12)=\boxed{22.4\text{ kN/m}^3}$$
iv) Volumetric water content: $$\theta=\frac{w\gamma_d}{\gamma_w}=\frac{0.12\times20.0}{9.81}=\boxed{24.5\%}$$
v) Porosity: $$n=\frac{e}{1+e}=\frac{0.324}{1.324}=\boxed{24.5\%}$$
vi) Dry density: $$\rho_d=\frac{\gamma_d}{g}=\frac{20.0}{9.81}=\boxed{2.04\text{ Mg/m}^3}$$
(cross-check: $\theta=Sn=0.999\times0.245=0.245$ — matches iv) directly.)
