The problem states that an element is subjected to biaxial normal tensile strains of 0.0030 and 0.0020. We need to determine the normal strain in the plane of maximum shear strain.

To solve this problem, we can use the concept of principal strains and the relationship between principal strains and normal strains in the plane of maximum shear strain.

Principal strains are the strains that occur in the directions of maximum and minimum normal stress. The normal strains in the plane of maximum shear strain are equal to half the difference between the principal strains.

Let's solve the problem step by step:

1. Given data:
- Normal tensile strain in the first direction (ε1) = 0.0030
- Normal tensile strain in the second direction (ε2) = 0.0020

2. Determine the principal strains:
The principal strains (ε1' and ε2') can be calculated using the formulas:
ε1' = (ε1 + ε2) / 2 + √((ε1 - ε2) / 2)^2 + γ^2
ε2' = (ε1 + ε2) / 2 - √((ε1 - ε2) / 2)^2 + γ^2

Where γ is the maximum shear strain.

Substituting the given values:
ε1' = (0.0030 + 0.0020) / 2 + √((0.0030 - 0.0020) / 2)^2 + γ^2
ε2' = (0.0030 + 0.0020) / 2 - √((0.0030 - 0.0020) / 2)^2 + γ^2

3. Simplify the equations:
ε1' = 0.0025 + √(0.0000025 + γ^2)
ε2' = 0.0025 - √(0.0000025 + γ^2)

4. Determine the normal strain in the plane of maximum shear strain:
The normal strain in the plane of maximum shear strain (εn) is equal to half the difference between the principal strains.
εn = (ε1' - ε2') / 2

Substituting the simplified equations:
εn = (0.0025 + √(0.0000025 + γ^2) - 0.0025 + √(0.0000025 + γ^2)) / 2
= √(0.0000025 + γ^2) / 2

5. Solve for γ:
To find the maximum shear strain γ, we can use the given normal tensile strains and the equation:
γ = √(ε1^2 + ε2^2 - ε1*ε2)

Substituting the given values:
γ = √(0.0030^2 + 0.0020^2 - 0.0030*0.0020)
= √(0.000009 + 0.000004 - 0.000006)
= √0.000007
= 0.0026

6. Substitute γ into the equation for εn:
εn