Asphalt Concrete Volumetrics and Marshall Mix Design: 16-Civ-B11 Structural Materials, Undated paper, Q4

Question 4: Asphalt Concrete Volumetrics and Marshall Mix Design (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.

Part (a) — The three volumetric quantities defined (9 marks)

i. Air voids, VTM (voids in the total mix). The air voids are the small pockets of air distributed between the coated aggregate particles in the compacted mixture, expressed as a percentage of the total volume of the compacted specimen: $$VTM=100\,\frac{G_{mm}-G_{mb}}{G_{mm}} .$$ The measurement compares the bulk relative density of the compacted specimen with the theoretical maximum relative density of the same mixture with no air in it, so the difference between them is precisely the air. Air voids are the single most important volumetric property. Too few (below about 3 per cent) and the mix has no room to densify further under traffic, so the binder is squeezed to the surface, the mixture flushes and it ruts and shoves. Too many (above about 5 per cent as designed, or 8 per cent in place) and the interconnected voids admit air and water, so the binder oxidises, the mix ravels and moisture damage and stripping follow. The design air-void content of 4 per cent used in this question is the industry standard compromise, chosen because a mix compacted to 7 or 8 per cent air on the road will densify to about 4 per cent under a few years of traffic.

ii. Voids in the mineral aggregate, VMA. The VMA is the total intergranular void space between the aggregate particles in the compacted mixture — that is, the air voids plus the volume occupied by the effective (non-absorbed) binder — expressed as a percentage of the bulk volume of the specimen: $$VMA=100-\frac{G_{mb}\,P_s}{G_{sb}} .$$ It is calculated on the aggregate's bulk relative density, which means that binder absorbed into the pores of the stone is counted as part of the aggregate and not as part of the void space, which is exactly the intention. The VMA is the room available for binder, and the specification sets a minimum for it (as a function of the nominal maximum aggregate size) because a mix with too little VMA cannot hold a durable binder film no matter how much binder is added — adding binder to such a mix merely drives the air voids to zero. A VMA that is too high, on the other hand, gives a mix with an unstable, binder-rich structure. The minimum VMA is therefore the specification's way of controlling the film thickness and hence the durability of the mixture.

iii. Voids filled with asphalt, VFA. The VFA is the percentage of the intergranular void space (the VMA) that is occupied by the effective binder rather than by air: $$VFA=100\,\frac{VMA-VTM}{VMA} .$$ It is not independent of the other two — it is a ratio derived from them — but it is a useful design control because it expresses the balance between them. A low VFA means a thin binder film and a dry, permeable, easily ravelled mix; a high VFA means the voids are nearly full of binder, leaving nothing for the mix to densify into, and the mixture will flush and rut. The Asphalt Institute limits (65 to 78 per cent for medium traffic, narrowing to 65 to 75 per cent for heavy traffic) are set to keep the mixture in the band that is durable without being unstable, and the VFA criterion frequently governs the design of fine-graded mixes where the air voids and VMA criteria are both satisfied.

Part (b) — Marshall design asphalt content (11 marks)

Given.

Quantity	Value
Asphalt cement	AC-30, specific gravity G_b = 1.031
Aggregate	9.5 mm nominal maximum particle size, bulk specific gravity G_sb = 2.696
Theoretical maximum specific gravity	G_mm = 2.470, measured at P_b = 5.0 per cent
Design air-void content	4 per cent
Traffic level	Medium (Asphalt Institute MS-2 criteria)
Trial mixes (P_b / G_mb / stability, kN / flow, 0.25 mm)	4.0 / 2.360 / 6.3 / 9 · 4.5 / 2.378 / 6.7 / 10 · 5.0 / 2.395 / 5.4 / 12 · 5.5 / 2.405 / 5.1 / 15 · 6.0 / 2.415 / 4.7 / 22

Find. The design asphalt content that gives 4 per cent air voids, and a demonstration that the mixture at that binder content satisfies every Asphalt Institute medium-traffic criterion, including the minimum VMA for a 9.5 mm nominal maximum size.

Approach. Back-calculate the effective specific gravity of the aggregate from the one measured maximum specific gravity, use it to compute $G_{mm}$ at every trial binder content, then obtain VTM, VMA and VFA at each point, plot the six design properties against binder content, interpolate the binder content at 4 per cent air voids, and check the interpolated stability, flow, VMA and VFA against the medium-traffic limits.

Back-calculate the effective specific gravity of the aggregate. The maximum specific gravity was measured at only one binder content, so the aggregate property that lets it be transferred to the others must be extracted first: $$G_{se}=\frac{100-P_b}{\dfrac{100}{G_{mm}}-\dfrac{P_b}{G_b}}=\frac{100-5.0}{\dfrac{100}{2.470}-\dfrac{5.0}{1.031}}=\frac{95}{40.486-4.850}$$ $$\boxed{\,G_{se}=2.666\,}$$
Check the absorbed asphalt content, and deal with what it reveals. The absorbed binder follows from the difference between the effective and bulk relative densities of the aggregate: $$P_{ba}=100\,\frac{G_{se}-G_{sb}}{G_{sb}\,G_{se}}\,G_b=100\times\frac{2.666-2.696}{2.696\times 2.666}\times 1.031=-0.43 .$$ The effective specific gravity has come out below the bulk specific gravity, which is physically impossible — the effective value is measured on a volume that excludes the aggregate pores filled with binder and must therefore be the larger of the two — and the absorbed binder content is formally negative. The data as printed are internally inconsistent by a small margin; the correct engineering response is to report zero absorption and carry the effective binder content equal to the total binder content: $$\boxed{\,P_{ba}=0,\qquad P_{be}=P_b\ \text{at every trial point}\,}$$ This does not affect the design, because VTM, VMA and VFA are all computed from $G_{mb}$, $G_{mm}$ and $G_{sb}$ directly and none of them uses $P_{ba}$.
Compute the theoretical maximum specific gravity at each binder content. With $G_{se}$ fixed, the maximum specific gravity of the same aggregate at any other binder content is $$G_{mm}=\frac{100}{\dfrac{100-P_b}{G_{se}}+\dfrac{P_b}{G_b}} .$$ At $P_b=4.0$ per cent this gives $G_{mm}=100/(96/2.666+4/1.031)=2.5068$, and repeating at each trial point gives 2.5068, 2.4883, 2.4700, 2.4520 and 2.4342 for 4.0 to 6.0 per cent respectively. The value at 5.0 per cent reproduces the measured 2.470, which confirms the back-calculation.

Compute VTM, VMA and VFA at each binder content. Applying the three definitions of part (a) in turn, $$VTM=100\,\frac{G_{mm}-G_{mb}}{G_{mm}},\qquad VMA=100-\frac{G_{mb}(100-P_b)}{G_{sb}},\qquad VFA=100\,\frac{VMA-VTM}{VMA},$$ and taking the 4.5 per cent mix as the worked instance, $VTM=100(2.4883-2.378)/2.4883=4.43$, $VMA=100-2.378(95.5)/2.696=15.76$ and $VFA=100(15.76-4.43)/15.76=71.9$. The complete design table is:

P_b (%)	G_mb	G_mm	VTM (%)	VMA (%)	VFA (%)	Stability (kN)	Flow (0.25 mm)
4.0	2.360	2.5068	5.86	15.96	63.3	6.3	9
4.5	2.378	2.4883	4.43	15.76	71.9	6.7	10
5.0	2.395	2.4700	3.04	15.61	80.5	5.4	12
5.5	2.405	2.4520	1.92	15.70	87.8	5.1	15
6.0	2.415	2.4342	0.79	15.80	95.0	4.7	22

The VMA column shows the characteristic shallow minimum at 5.0 per cent binder, which is the expected shape and a useful check on the arithmetic.

Interpolate the design asphalt content at 4 per cent air voids. The air-void curve falls monotonically and crosses 4 per cent between the 4.5 and 5.0 per cent trial points: $$P_b=4.5+0.5\times\frac{4.43-4.00}{4.43-3.04}=4.5+0.5\times 0.3086$$ $$\boxed{\,P_b=4.65\ \text{per cent by weight of total mix}\,}$$
Read the remaining properties at the design binder content. Interpolating each design curve at $P_b=4.65$ per cent gives $G_{mb}=2.383$, stability 6.30 kN, flow 10.6 units of 0.25 mm, $VMA=15.72$ per cent and $VFA=74.6$ per cent.

Check every Asphalt Institute medium-traffic criterion. The minimum VMA is taken at the design air-void content of 4 per cent for a 9.5 mm nominal maximum aggregate size, which MS-2 sets at 15.0 per cent. Comparing each design value with its limit:

Property at P_b = 4.65 %	Value	Medium-traffic criterion	Verdict
Marshall stability	6.30 kN	minimum 5.34 kN (1 200 lb)	Pass
Flow	10.6 (0.25 mm)	8 to 18	Pass
Air voids VTM	4.0 per cent	3 to 5 per cent	Pass
VMA	15.7 per cent	minimum 15.0 per cent (9.5 mm NMPS at 4 % air)	Pass
VFA	74.6 per cent	65 to 78 per cent	Pass

$$\boxed{\ \text{Design asphalt content}=4.65\ \text{per cent; all five medium-traffic criteria satisfied}\ }$$ The design is comfortable rather than marginal on four of the five criteria; VFA at 74.6 per cent is the closest to a limit, sitting 3.4 points below the 78 per cent maximum, so any increase in binder content would put it at risk first. Note also that the stability falls steeply beyond 4.5 per cent binder — at 5.0 per cent it is only 5.4 kN against the 5.34 kN minimum — which is a second, independent reason not to drift richer than the 4.65 per cent design value.

Figure 4.1 — the six Marshall design property curves. The design asphalt content is set where the air-void curve crosses 4 per cent, and the remaining five properties are then read at that binder content and checked against the Asphalt Institute medium-traffic limits.

Check: the printed data give an effective aggregate specific gravity of 2.666 against a stated bulk specific gravity of 2.696, so the absorbed asphalt content computes as −0.43 per cent. Absorption cannot be negative; the discrepancy is roughly 1 per cent in $G_{se}$ and is within the combined uncertainty of a $G_{sb}$ determination and a single Rice test, so it is almost certainly a rounding or transcription artefact in the question rather than a real result. The answer above reports P_ba = 0 and P_be = P_b, states the inconsistency, and proceeds — correctly, because the design quantities VTM, VMA and VFA are computed from G_mb, G_mm and G_sb and are entirely unaffected by the absorbed-binder term. A candidate who instead used G_sb in place of G_se to build the G_mm column would obtain air voids about 1.1 points higher throughout and a design binder content near 5.3 per cent, which is the error this check is designed to catch.

Quantity	Value
Effective specific gravity of aggregate, G_se	2.666
Absorbed asphalt, P_ba	0 (computes as −0.43 per cent; reported as zero — see callout)
Design asphalt content	4.65 per cent by weight of total mix
G_mb at design	2.383
Air voids VTM at design	4.0 per cent
VMA at design (minimum required 15.0 per cent)	15.7 per cent
VFA at design (65 to 78 per cent)	74.6 per cent
Marshall stability at design (minimum 5.34 kN)	6.30 kN
Flow at design (8 to 18)	10.6 units of 0.25 mm
Overall	Mix satisfies all Asphalt Institute medium-traffic criteria

Topic. Asphalt mixture volumetrics and the Marshall design procedure: turning three measured relative densities into a binder content.

Key relations. $G_{se}=(100-P_b)\big/\left(100/G_{mm}-P_b/G_b\right)$; $G_{mm}=100\big/\left[(100-P_b)/G_{se}+P_b/G_b\right]$; $P_{ba}=100\,G_b(G_{se}-G_{sb})/(G_{sb}G_{se})$; $P_{be}=P_b-P_{ba}P_s/100$; $VTM=100(G_{mm}-G_{mb})/G_{mm}$; $VMA=100-G_{mb}P_s/G_{sb}$; $VFA=100(VMA-VTM)/VMA$. Asphalt Institute MS-2 medium-traffic criteria: stability ≥ 5.34 kN, flow 8 to 18, air voids 3 to 5 per cent, VFA 65 to 78 per cent, and a minimum VMA that depends on nominal maximum aggregate size (15.0 per cent at 9.5 mm, 14.0 per cent at 12.5 mm, 13.0 per cent at 19 mm) evaluated at the design air-void content.

Why this works. Everything in this calculation rests on the distinction between three ways of measuring the volume of the aggregate. The bulk specific gravity counts the whole particle including its permeable pores; the apparent specific gravity excludes them entirely; the effective specific gravity counts the particle including only those pores that the binder does not penetrate, and so lies between the other two. That is why $G_{se}$ is the right density for computing how much binder is available to fill mix voids, while $G_{sb}$ is the right density for computing the space between particles. Using the wrong one is the classic source of a mis-designed mix. The Marshall procedure itself works because the five design properties respond to binder content in opposite directions — air voids fall, VFA rises, stability peaks and falls, flow rises steadily — so the acceptable band is bounded on both sides and picking the binder content at 4 per cent air voids lands near the middle of it.

Common pitfall. The commonest error by far is to use $G_{sb}$ where $G_{se}$ belongs when building the $G_{mm}$ column, which shifts every air-void value by roughly a point and the design binder content by half a point. The second is to compute the VMA from the effective binder content or from $G_{se}$; the VMA definition uses $G_{sb}$ precisely so that absorbed binder is counted with the stone. The third is forgetting that the minimum VMA depends on nominal maximum aggregate size and is read at the design air-void content, so that quoting the 19 mm figure of 13 per cent for a 9.5 mm mix would wrongly pass a deficient mixture.

Source. Asphalt Institute MS-2 Asphalt Mix Design Methods, 7th ed., Ch. 4 and 5 (volumetric relations and the Marshall method, including the design criteria tables); Mamlouk & Zaniewski, Materials for Civil and Construction Engineers, 4th ed., Ch. 9 (Asphalt Binders and Asphalt Mixtures); ASTM D6926 (specimen preparation) and D6927 (stability and flow).

Question 4 of 5: Asphalt Concrete Volumetrics and Marshall Mix Design

Question 4: Asphalt Concrete Volumetrics and Marshall Mix Design (20 marks)

Part (a) — The three volumetric quantities defined (9 marks)

Part (b) — Marshall design asphalt content (11 marks)