GATE Civil Engineering (CE) 2023 Shift 2 Solved Paper

A 2D thin plate with modulus of elasticity, 𝐸 = 1.0 N/m², and Poisson’s ratio, 𝜇 = 0.5, is in plane stress condition. The displacement field in the plate is given by 𝑢 = 𝐶𝑥²𝑦 and 𝑣 = 0, where 𝑢 and 𝑣 are displacements (in m) along the 𝑋 and 𝑌 directions, respectively, and 𝐶 is a constant (in m⁻²). The distances x and y along X and Y, respectively, are in m. The stress in the 𝑋 direction is 𝜎_𝑋𝑋 = 40𝑥𝑦 N/m², and the shear stress is 𝜏_𝑋𝑌 = 𝛼𝑥² N/m². What is the value of 𝛼 (in N/m⁴, in integer)? ________