In this scenario, we have a steel bar that is fixed at both ends and subjected to a non-uniform increase in temperature. The temperature change along the length of the bar is given by the expression tx = t× x³/L³, where tx is the temperature at a distance x from one end of the bar, t is the overall temperature change, and L is the length of the bar.


To calculate the compressive stress in the bar, we need to consider the thermal expansion and Young's modulus of the material.


The thermal expansion coefficient, denoted as α, represents how much a material expands or contracts with a change in temperature. It is given by the formula α = (1/L) * (dL/dt), where L is the original length of the bar and (dL/dt) is the rate of change of length with respect to temperature.


Young's modulus, denoted as E, is a measure of the stiffness of a material. It represents the ratio of stress to strain under elastic deformation. The stress-strain relationship is given by the formula σ = E * ε, where σ is the stress, E is Young's modulus, and ε is the strain.


To calculate the compressive stress in the bar, we can use the formula σ = α * E * ΔT, where σ is the compressive stress, α is the thermal expansion coefficient, E is Young's modulus, and ΔT is the temperature change.

Substituting the given expression for temperature change, tx = t× x³/L³, we can rewrite the formula as σ = (t× x³/L³) * E * α.


The compressive stress in the steel bar fixed at both ends and subjected to a non-uniform increase in temperature can be calculated using the formula σ = (t× x³/L³) * E * α. By substituting the values of the temperature change, Young's modulus, and thermal expansion coefficient, we can determine the compressive stress at any point along the length of the bar.