Question 6 of 6: Blending Two Aggregates to Meet a Gradation Specification

Question 6: Blending Two Aggregates to Meet a Gradation Specification (12 + 8 = 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) — Feasible range of blend proportions (12 marks)

Given. Percent passing for aggregate A and aggregate B on nine sieves, together with lower and upper specification limits on each sieve, as tabulated in the question above.

Find. The minimum and maximum proportion of each aggregate, obtained algebraically rather than by trial and error.

Approach. Let $a$ be the decimal fraction of aggregate A in the blend, so $(1-a)$ is the fraction of aggregate B. On every sieve the blend gradation is the weighted average of the two, and each specification band converts that into a pair of linear inequalities in $a$. Solving them sieve by sieve produces a set of intervals; the feasible blend is their intersection, and the sieve that produces the largest lower limit and the sieve that produces the smallest upper limit are the governing sieves.

1. Write the blending equation. For any sieve, if $A_i$ and $B_i$ are the percentages passing for the two aggregates, the blend passing is $$P_i=a\,A_i+(1-a)\,B_i=B_i+a\,(A_i-B_i)$$ The specification requires $L_i \le P_i \le U_i$ on every sieve simultaneously.
2. Invert the inequality for one sieve. Provided $A_i \ne B_i$, substituting and rearranging gives the admissible interval directly: $$\frac{L_i-B_i}{A_i-B_i}\ \le\ a\ \le\ \frac{U_i-B_i}{A_i-B_i}\qquad (A_i>B_i)$$ Because $A_i \ge B_i$ on every sieve in this problem, the sense of the inequality never reverses. Where $A_i=B_i$ (the 19 mm sieve, both 100 per cent) the blend passes 100 per cent for any $a$, and the sieve imposes no constraint.
3. Apply the inversion sieve by sieve. Taking the 4.75 mm sieve as the worked example, $A=70$, $B=30$, so $A-B=40$, and the band 50 to 60 gives $$a_{\min}=\frac{50-30}{40}=0.500,\qquad a_{\max}=\frac{60-30}{40}=0.750$$ Repeating for every sieve produces the following table.

4. Intersect the intervals. The blend must satisfy every sieve at once, so $a$ must be at least the largest of the lower limits and at most the smallest of the upper limits: $$a_{\min}=\max_i\left(\frac{L_i-B_i}{A_i-B_i}\right)=0.500,\qquad a_{\max}=\min_i\left(\frac{U_i-B_i}{A_i-B_i}\right)=\frac{2}{3}=0.667$$ $$\boxed{0.500 \le a \le 0.667}$$ The lower limit is set jointly by the 4.75 mm, 2.36 mm, 0.60 mm and 0.15 mm sieves — all of which demand enough of the finer aggregate A — and the upper limit is set uniquely by the 0.30 mm sieve, which caps the fines.
5. State the answer as proportions of the two aggregates. Converting to percentages, $$\boxed{\text{Aggregate A: }50.0\ \text{to}\ 66.7\ \text{per cent};\quad \text{Aggregate B: }33.3\ \text{to}\ 50.0\ \text{per cent}}$$ The feasible window is 16.7 percentage points wide, which is comfortable: a plant can hold a blend ratio to about ±2 percentage points, so the specification is workable without exotic control.
6. Select a working blend and verify it. Choosing a value near the middle of the window gives the greatest tolerance to normal variation in the stockpiles. Taking $a=0.60$, that is 60 per cent aggregate A and 40 per cent aggregate B, the blend gradation is $P_i = 0.60A_i+0.40B_i$:

Every sieve is satisfied. In percent passing the tightest margin is on the 0.075 mm sieve (6 against a lower limit of 5), because the two aggregates differ by only 10 points there; the 0.30 mm and 0.15 mm sieves each sit 2 points inside a limit. Measured in blend proportion, which is what the plant controls, the nearest constraint is the 0.30 mm upper limit: $a = 0.60$ is 0.067 below $a_{\max}=0.667$ but 0.100 above $a_{\min}=0.500$. That is consistent with the algebra, which identified the 0.30 mm sieve alone as governing the upper end of the feasible range.

Part (b) — Plot on the semi-log gradation chart (8 marks)

The four required curves are plotted below on the semi-logarithmic chart supplied with the paper: percent passing on a linear vertical axis against sieve size on a logarithmic horizontal axis. The shaded band is the specification envelope, bounded by the dashed lower and upper limit curves; aggregate A and aggregate B are plotted as the two extreme gradations, and the selected 60 A / 40 B blend lies between them and inside the band on every sieve.

Three features of the plot are worth reading off. First, aggregate B plots below the envelope over the whole fine half of the chart — it passes 0 per cent on every sieve finer than 2.36 mm — so aggregate B alone can never satisfy the specification and a minimum proportion of A is unavoidable; that is the algebraic $a \ge 0.500$ made visible. Second, aggregate A plots above the envelope on every sieve from 9.5 mm down to 0.15 mm, so aggregate A alone is also inadmissible, being too fine; that is $a \le 0.667$. Third, the blend curve is smooth and continuous with no gap or hump, which confirms a well-graded dense mixture. Comparing it with the Fuller maximum density line for a 19 mm maximum size, $P = 100(d/19)^{0.45}$, which passes 73.2 per cent at 9.5 mm, 53.6 per cent at 4.75 mm, 39.1 per cent at 2.36 mm, 21.1 per cent at 0.60 mm and 8.3 per cent at 0.075 mm, the blend crosses the line: it is above it at 9.5 mm (81) and marginally at 4.75 mm (54), below it at 2.36 mm (34), above it again from 0.60 to 0.15 mm, and below it at 0.075 mm (6). The Superpave classification is therefore made at the primary control sieve, as defined in Question 5(b). The blend retains 19 per cent on the 9.5 mm sieve (the first sieve to retain more than 10 per cent), so its nominal maximum size is 12.5 mm, its maximum size 19.0 mm, and its primary control sieve 2.36 mm, where the control point is about 39 per cent (the maximum density line value; AASHTO M 323 tabulates 40 per cent for a 12.5 mm mix). The blend passes only 34 per cent there, so it is a coarse-graded dense mix — one that relies more on stone-on-stone contact, resists rutting well, and needs a little more compactive effort than a fine-graded surface mix.

To construct the plot by hand, mark the sieve openings on the logarithmic axis at their printed positions, plot each percent-passing value against its sieve, join the points with a smooth curve for each aggregate, hatch between the two specification-limit curves, and label the four curves in a legend. Practical drafting points: plot the specification limits first so the envelope is visible before any gradation is drawn; use the same symbol convention throughout; and always plot the blend from its computed percentages rather than sketching it by eye between A and B — on each sieve the 60 A / 40 B blend lies 60 per cent of the vertical distance from B towards A, not midway.