Explanation:

In order to determine the principal strains and their directions on a plane surface, at least three strain readings using strain gauges are required.

Principal Strains:
Principal strains are the maximum and minimum strains experienced by a material in a given direction. They are represented by ε1 and ε2. The principal strains are perpendicular to each other.

Strain Gauges:
Strain gauges are electrical devices that measure the strain or deformation in an object. They consist of a metallic foil pattern that changes its electrical resistance when it is stretched or compressed.

Calculating Principal Strains:
To determine the principal strains and their directions on a plane surface, strain gauges are placed at different orientations on the surface. The strain gauges measure the strains in their respective directions.

Minimum Three Strain Readings:
To calculate the principal strains, at least three strain readings are needed. These readings should be taken at different orientations on the plane surface.

Steps to Determine Principal Strains:
1. Place three strain gauges at different orientations on the plane surface.
2. Apply a known force or load to the surface.
3. Measure the strain in each strain gauge using the strain gauge's electrical resistance change.
4. Use the measured strains to calculate the principal strains using mathematical formulas.

Why Three Strain Readings are Required:
Three strain readings are required because there are two principal strains (maximum and minimum) and their directions to be determined. By using three strain readings, the values of the principal strains can be calculated, and the directions can be determined based on the orientation of the strain gauges.

Additional Information:
If four or more strain gauges are used, the excess readings can be redundant and may not provide additional information for determining the principal strains and their directions accurately. Therefore, three strain readings are sufficient for this purpose.

Hence, the correct answer is option 'C' - 3 strain readings are needed to determine the principal strains and their directions on a plane surface using strain gauges.