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A rectangular plate with modulus of elasticity , Poisson coefficient and expansion coefficient thermal has sides and and thickness and is inserted into a cavity rectangular slightly larger so that there are gaps of size as shown in the figure. The plate is subjected to a uniform temperature variation ΔT.

(i) What is the temperature variation ΔT that the plate must be subjected to in order to completely close the slack in x?

(ii) What is the temperature variation ΔT that the plate must be subjected to in order to completely close the slack in y? What is the corresponding stress ​​​x?

(iii) Determine the stress x and y on the plate for a temperature rise of ΔTc > ΔT. What are the main tensions 1 and​​​2 and the maximum shear stress max in this case?

User Kemo
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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.

User Thataustin
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