4.3 Gradation Calculations, Reporting & Combined Wash-Sieve Analysis

Determining aggregate gradation requires converting raw laboratory scale masses into standardized individual and cumulative percentage distributions. While modern laboratory software often automates these computations, certified technicians must master the underlying mathematical principles to detect calculation errors, verify software output, and successfully pass the calculation-intensive sections of the ACI written examination.

Furthermore, because almost all natural and manufactured aggregates contain fine dust of fracture or clay, testing laboratories routinely conduct a combined test: running ASTM C117 (Wash Test) first, followed immediately by ASTM C136 (Dry Sieve Analysis) on the washed, dried residue. Integrating data from both standards into a unified gradation report involves a critical rule regarding the mathematical denominator that frequently trips up exam candidates.

Fundamental Gradation Formulas

A complete gradation table reports aggregate distribution across four fundamental parameters: individual mass retained, individual percent retained, cumulative percent retained, and cumulative percent passing.

1. Individual Percent Retained

The proportion of aggregate particles resting on a specific sieve relative to the total original sample mass:

$\text{Individual } \% \text{ Retained} = \left[ \frac{m_i}{M_{\text{dry}}} \right] \times 100$

Where:

• $m_i$ = mass of aggregate retained on the individual sieve $i$, in grams (g).
• $M_{\text{dry}}$ = total original oven-dry mass of the sample before any washing or sieving, in grams (g).

2. Cumulative Percent Retained

The total proportion of aggregate particles that are coarser than the opening of the current sieve. It represents the progressive accumulation of all material held back by the current sieve and all sieves above it:

$\text{Cumulative } \% \text{ Retained}_i = \sum_{k=1}^{i} (\text{Individual } \% \text{ Retained}_k)$

Alternatively, it can be computed directly from cumulative retained mass:

$\text{Cumulative } \% \text{ Retained}_i = \left[ \frac{\sum_{k=1}^i m_k}{M_{\text{dry}}} \right] \times 100$

3. Cumulative Percent Passing

The proportion of the total sample that is finer than the opening of the current sieve and has passed through its apertures:

$\text{Cumulative } \% \text{ Passing}_i = 100 - \text{Cumulative } \% \text{ Retained}_i$

[!NOTE] ASTM Reporting Precision: ASTM C136 Section 9.1 requires percentages to be calculated to the nearest 0.1% on the basis of the total mass of the initial dry sample. Section 10.2 then requires them to be reported to the nearest whole number, except if the percentage passing the 75-µm (No. 200) sieve is less than 10%, in which case it is reported to the nearest 0.1%. Note the threshold: at 10% or more, even the minus-No. 200 value is reported as a whole number. ASTM C117 Section 11.1.1 uses the mirror-image wording for the wash test — report to the nearest 0.1%, except report to the nearest whole number if the result is 10% or more — and Section 11.1.2 additionally requires a statement of which procedure (A or B) was used. Fineness modulus is reported to the nearest 0.01 (Section 10.3).

The Combined Wash and Sieve Analysis Protocol

When an aggregate sample is tested for full grading compliance, running ASTM C136 alone produces flawed results because fine particles adhere electrostatically to coarser grains. The standardized sequence requires performing ASTM C117 prior to ASTM C136.

+--------------------------------------------------------------------------+
|             COMBINED ASTM C117 + ASTM C136 TESTING PROTOCOL              |
+--------------------------------------------------------------------------+
                                      |
                                      v
                 +-----------------------------------------+
                 |     INITIAL DRY SPECIMEN (ASTM C117)    |
                 |      Oven-dry at 110 ± 5°C to constant  |
                 |       Record Original Dry Mass = B      |
                 +-----------------------------------------+
                                      |
                                      v
                 +-----------------------------------------+
                 |           WASHING & DECANTING           |
                 |      Agitate in water over No. 200      |
                 |        Rinse until decant is clear      |
                 +-----------------------------------------+
                                      |
                                      v
                 +-----------------------------------------+
                 |           DRY WASHED RESIDUE            |
                 |      Oven-dry at 110 ± 5°C to constant  |
                 |        Record Washed Dry Mass = C       |
                 |          Wash Loss = (B - C)            |
                 +-----------------------------------------+
                                      |
                                      v
                 +-----------------------------------------+
                 |         DRY SIEVING (ASTM C136)         |
                 |     Sieve Washed Mass C through stack   |
                 |     Weigh individual fractions & pan    |
                 |       Verify 0.3% mass check on C       |
                 +-----------------------------------------+
                                      |
                                      v
                 +-----------------------------------------+
                 |        FINAL GRADATION REPORTING        |
                 |    CRITICAL: Universal Denominator is B |
                 |  Total Passing No. 200 = (B - C) + Pan  |
                 +-----------------------------------------+

The Golden Denominator Rule

On the ACI Level 1 exam, the single most common mathematical error in combined analysis is selecting the wrong denominator:

• $B$ = Original dry sample mass before washing (from ASTM C117)
• $C$ = Oven-dry sample mass after washing (from ASTM C117)
• $B - C$ = Mass of micro-fines washed out through the No. 200 sieve
• The dried material placed into the ASTM C136 sieve shaker is the washed mass $C$.
• THE GOLDEN RULE: The original dry mass before washing ($B$) MUST be used as the denominator for EVERY individual and cumulative percentage calculation across the entire sieve stack!

[!WARNING] The Washed Denominator Trap: If a technician inadvertently uses the washed dry mass ($C$) as the denominator when calculating sieve percentages, every calculated percent retained will be artificially inflated, causing cumulative passing percentages to plunge and falsely indicating that an in-spec aggregate fails project gradation limits!

Step-by-Step Worked Tabular Example

To illustrate the exact mathematical mechanics, consider a complete concrete sand acceptance test conducted under combined ASTM C117 and ASTM C136.

Laboratory Raw Data

• Original dry sample mass before washing ($B$): $500.0\text{ g}$
• Oven-dry sample mass after washing ($C$): $486.0\text{ g}$
• Wash loss finer than No. 200 ($B - C$): $500.0\text{ g} - 486.0\text{ g} = 14.0\text{ g}$
• Mass of washed specimen charged to sieves: $486.0\text{ g}$

Following 12 minutes of mechanical shaking and hand-sieving sufficiency verification, the individual masses retained were weighed:

• 3/8 in. (9.5 mm): $0.0\text{ g}$
• No. 4 (4.75 mm): $12.5\text{ g}$
• No. 8 (2.36 mm): $65.0\text{ g}$
• No. 16 (1.18 mm): $87.5\text{ g}$
• No. 30 (600 µm): $112.5\text{ g}$
• No. 50 (300 µm): $115.0\text{ g}$
• No. 100 (150 µm): $68.5\text{ g}$
• No. 200 (75 µm): $21.0\text{ g}$
• Sieve Pan: $3.5\text{ g}$

Step 1: Pre-Gradation 0.3% Mass Balance Check

Before computing percentages, verify that mass was conserved during sieving of the washed sample ($C = 486.0\text{ g}$): $M_{\text{after}} = 0.0 + 12.5 + 65.0 + 87.5 + 112.5 + 115.0 + 68.5 + 21.0 + 3.5 = 485.5\text{ g}$ $\Delta M = \left[ \frac{|486.0 - 485.5|}{486.0} \right] \times 100 = \left[ \frac{0.5}{486.0} \right] \times 100 = 0.103\%$ Because $0.103% \le 0.300%$, the sieve operation is valid!

Step 2: Full Tabular Calculations

All percentage calculations utilize the original dry mass $B = 500.0\text{ g}$ as the divisor:

Explaining the Total Passing No. 200 Synthesis

Notice that the material passing the No. 200 sieve originates from two separate phases:

1. Washing phase: $14.0\text{ g}$ washed through the No. 200 sieve during ASTM C117 ($2.8%$).
2. Dry sieving phase: $3.5\text{ g}$ passed through the No. 200 sieve into the pan during ASTM C136 ($0.7%$).

• Total minus No. 200 material = $14.0\text{ g} + 3.5\text{ g} = 17.5\text{ g}$.
• Reported Total Percent Passing No. 200 = $(17.5 / 500.0) \times 100 = \mathbf{3.5%}$.
• Why the two routes differ by 0.1%: subtracting the cumulative retained gives $100.0 - 96.4 = 3.6%$, or $18.0\text{ g}$, while the measured wash loss plus dry pan is $17.5\text{ g}$. The $0.5\text{ g}$ gap is not a rounding artifact — it is exactly the sieving loss found in the Step 1 mass check ($486.0 - 485.5 = 0.5\text{ g}$). Subtraction silently pushes every gram of unaccounted-for material into the minus-No. 200 fraction. The measured value is the one to report: $17.5\text{ g} / 500.0\text{ g} = \mathbf{3.5%}$. Whenever the two routes disagree by more than rounding, look for lost mass before you look at your arithmetic.

ASTM C33 Gradation Envelopes: Fine & Coarse Aggregates

ASTM C33 / C33M (Standard Specification for Concrete Aggregates) establishes the legal grading envelopes that aggregates must satisfy to produce workable, high-strength concrete.

1. Fine Aggregate Grading Requirements (ASTM C33 Section 6.1)

Additional Fine Aggregate Rules in ASTM C33

• Maximum Retained Between Consecutive Sieves: Not more than 45% of the fine aggregate shall be retained between any two consecutive standard sieves (e.g., between No. 16 and No. 30, or between No. 30 and No. 50). This prevents severe "gap-grading" within the sand fraction.
• Sands near the No. 50 / No. 100 minimums (ASTM C33 Note 2): C33 does not publish a reduced grading band for air-entrained concrete. What it says is that concrete made with fine aggregate grading near the minimums for percent passing the 300-µm (No. 50) and 150-µm (No. 100) sieves sometimes has difficulty with workability, pumping, or excessive bleeding, and that the accepted remedies are entrained air, additional cement, or an approved mineral admixture to supply the deficient fines. (AASHTO M 6 does carry a numeric reduction footnote; do not attribute it to ASTM C33.)
• Alternative acceptance (ASTM C33 Section 6.3): fine aggregate failing the grading limits still meets the specification if the supplier demonstrates to the purchaser that concrete of the specified class made with it has relevant properties at least equal to concrete made with a reference sand from a source with an acceptable performance record.

2. Coarse Aggregate Standard Sizes (ASTM C33 Table 2)

ASTM C33 designates coarse aggregate gradations by standard numerical size numbers. Three sizes dominate commercial ready-mix concrete production:

+--------------------------------------------------------------------------+
|                   COMMON ASTM C33 COARSE AGGREGATE SIZES                 |
+--------------------------------------------------------------------------+
|                                                                          |
|   SIZE 57:  Nominal 25.0 mm to 4.75 mm (1 in. to No. 4)                  |
|             1-1/2": 100% | 1": 95-100% | 1/2": 25-60% | No. 4: 0-10%     |
|             Standard structural concrete for slabs, beams, columns       |
|                                                                          |
|   SIZE 67:  Nominal 19.0 mm to 4.75 mm (3/4 in. to No. 4)                |
|             1": 100% | 3/4": 90-100% | 3/8": 20-55% | No. 4: 0-10%       |
|             Congested rebar, pump mixes, residential flatwork            |
|                                                                          |
|   SIZE 8:   Nominal 9.5 mm to 2.36 mm (3/8 in. to No. 8)                 |
|             1/2": 100% | 3/8": 85-100% | No. 4: 10-30% | No. 8: 0-10%    |
|             Pea gravel, self-consolidating concrete, thin toppings       |
+--------------------------------------------------------------------------+

• Size No. 57: Nominal range 25.0 mm to 4.75 mm (1 in. to No. 4). Widely used for standard commercial foundations, bridge piers, and thick slabs. Requires 100% passing 1-1/2 in., 95–100% passing 1 in., 25–60% passing 1/2 in., 0–10% passing No. 4, and 0–5% passing No. 8.
• Size No. 67: Nominal range 19.0 mm to 4.75 mm (3/4 in. to No. 4). The most common commercial mix size in the United States, optimized for heavily reinforced walls, columns, and concrete pumping. Requires 100% passing 1 in., 90–100% passing 3/4 in., 20–55% passing 3/8 in., 0–10% passing No. 4, and 0–5% passing No. 8.
• Size No. 8: Nominal range 9.5 mm to 2.36 mm (3/8 in. to No. 8) ("pea gravel"). Used in highly congested prestressed members, architectural concrete, thin architectural toppings, and as the coarse component in Self-Consolidating Concrete (SCC).

Practical Concrete Consequences of Out-of-Spec Gradations

When aggregate gradations violate ASTM C33 boundaries, fresh and hardened concrete properties suffer immediate degradation:

1. Deficient Fine Sand (Low Passing No. 50 & No. 100)

• Excessive Bleeding: Without sufficient fine sand particles ($< 300\text{ µm}$) to choke capillary channels, free water rushes upward to the flatwork surface.
• Sand Streaking and Segregation: Heavy coarse particles settle rapidly, leaving water channels and sandy streaks along vertical formwork.
• Harsh Finishability: The fresh concrete tears under troweling and cannot produce a smooth, closed architectural surface.

2. Excessive Fine Sand (High Passing No. 50 & No. 100)

• High Water Demand: Sand surface area escalates dramatically, requiring massive additions of mixing water (or high-range water reducers) to lubricate particles.
• Reduced Compressive Strength: Higher water demand increases the $w/cm$ ratio, causing severe strength loss.
• Shrinkage and Crazing: High paste volume promotes plastic shrinkage cracking, drying shrinkage cracking, and surface map-cracking (crazing).

3. Gap-Graded or Segregated Coarse Aggregate

• Pump Line Blockages: Mixes lacking intermediate aggregate fractions (such as 3/8-in. or No. 4 particles) experience "grout lock," where paste squeezes through coarse stone voids under pumping pressure, creating rock jams inside the boom line.
• Honeycombing: Coarse stones bridge across closely spaced reinforcing bars, leaving open structural voids (honeycomb) that require expensive structural chipping and pressure grouting.