A steel bar of length L and cross-sectional area A is fitted between rigid supports.
Write an expression for the stress in the bar when the temperature is raised by an amount ΔT. Write your answer in terms of α,E and ΔT.
Is the stress compressive or tensile?
compressive
tensile
sigma = -E*alpha*DeltaT
Compressive
To find the expression for the stress in the bar when the temperature is raised by ΔT, we need to consider the thermal expansion of the bar.
The change in length of the bar, ΔL, due to the change in temperature can be expressed as:
ΔL = α * L * ΔT
where α is the coefficient of thermal expansion of the material, L is the original length of the bar, and ΔT is the change in temperature.
To calculate the stress in the bar, we need to consider Hooke's Law, which states that stress is directly proportional to strain. In this case, the strain is defined as the change in length divided by the original length:
ε = ΔL / L
Using the equation for ΔL, we can rewrite the strain equation as:
ε = (α * L * ΔT) / L
Simplifying further:
ε = α * ΔT
Now, we can use Hooke's Law to find the stress (σ). The stress is defined as the force acting on the cross-sectional area of the bar:
σ = E * ε
where E is the Young's modulus of the material.
Substituting the value of ε from above:
σ = E * (α * ΔT)
Therefore, the expression for the stress in the bar when the temperature is raised by an amount ΔT is:
σ = E * α * ΔT
Now, to determine whether the stress is compressive or tensile, we need to know the properties of the material. If the coefficient of thermal expansion (α) is positive, which means the material expands with an increase in temperature, the stress will be tensile. If α is negative, which means the material contracts with an increase in temperature, the stress will be compressive.
Therefore, the stress in the bar when the temperature is raised is either compressive or tensile depending on the sign of the coefficient of thermal expansion (α).