To address the given questions, we'll consider the thermal expansion of the rectangular plate and the resulting stresses.
(i) To completely close the slack in the x-direction, we need to determine the temperature variation ΔT. This can be found using the equation for thermal expansion:
ΔL = αLΔT
Where:
ΔL = change in length
α = coefficient of thermal expansion
L = original length
ΔT = temperature variation
In this case, we are interested in closing the slack in the x-direction, so ΔL in the x-direction should be equal to the gap size in the x-direction (δx).
Therefore:
δx = αxLxΔT
Simplifying for ΔT:
ΔT = δx / (αxLx)
(ii) To completely close the slack in the y-direction, we follow a similar approach. The gap size in the y-direction (δy) can be closed by a temperature variation ΔT determined by:
ΔT = δy / (αyLy)
To find the corresponding stress in the x-direction, we can use Hooke's Law for stress and strain:
σx = E(εx - νεy)
Where:
σx = stress in the x-direction
E = modulus of elasticity
εx = strain in the x-direction
ν = Poisson's ratio
εy = strain in the y-direction
(iii) For a temperature rise of ΔTc > ΔT, we need to determine the stresses σx and σy. The strains εx and εy can be found using the equations for thermal expansion:
εx = αxΔTc
εy = αyΔTc
Substituting these values into the stress equation, we have:
σx = E(αxΔTc - ναyΔTc)
σy = E(αyΔTc - ναxΔTc)
The principal stresses σ1 and σ2 can be calculated using:
σ1 = (σx + σy) / 2 + sqrt(((σx - σy) / 2)^2 + τxy^2)
σ2 = (σx + σy) / 2 - sqrt(((σx - σy) / 2)^2 + τxy^2)
Where:
τxy = max shear stress
The maximum shear stress can be determined using:
max = sqrt(((σx - σy) / 2)^2 + τxy^2)
These equations allow us to determine the stress components and the principal stresses for the given conditions.