Fineness modulus is an index that gives the mean size of the aggregates used in a concrete mix. It is an important parameter in concrete mix design as it helps to determine the proportions of various aggregate sizes to be used.

The fineness modulus is calculated by adding the cumulative percentages by mass of a sample of the fine aggregate retained on each of a specified series of sieves and dividing the sum by 100. The sieves used in the test are standard sieves with different sizes.

The calculation of fineness modulus involves the following steps:

1. Collect a representative sample of the fine aggregate, typically passing through a 4.75 mm sieve.

2. Weigh the sample and divide it into fractions by passing it through a series of sieves with sizes ranging from large to small. The sieves used are typically 4.75 mm, 2.36 mm, 1.18 mm, 600 microns, 300 microns, 150 microns, and 75 microns.

3. Determine the cumulative percentage passing through each sieve. This is done by weighing the aggregate retained on each sieve and dividing it by the total weight of the sample.

4. Calculate the cumulative percentage retained on each sieve by subtracting the cumulative percentage passing from 100.

5. Calculate the product of the cumulative percentage retained and the effective size of each sieve. The effective size is the mid-point of the sieve size range.

6. Sum up the products of the cumulative percentage retained and the effective size for all the sieves.

7. Divide the sum by 100 to get the fineness modulus.

The fineness modulus is a single value that represents the average size of the aggregate particles. A lower fineness modulus indicates a finer aggregate, while a higher fineness modulus indicates a coarser aggregate.

In conclusion, the fineness modulus is an index that gives the mean size of the aggregates used in a concrete mix. It is calculated by adding the cumulative percentages of the sample retained on each sieve and dividing the sum by 100.