91Ó°ÊÓ

The modulus of elasticity, often known as Young's modulus, is a fundamental material property that measures a material's ability to deform elastically when forces are applied. In simpler terms, it tells us how stiff or stretchy a material is. For steel, the modulus of elasticity is given as 200 GPa, which is equivalent to 200 billion pascals. This high value shows that steel is a very stiff material.
In our problem, this property is crucial because it allows us to calculate the stress in the steel rails when they undergo temperature changes. The relationship between stress and strain in materials undergoing elastic deformation is described by the equation:\[E = \frac{\sigma}{\varepsilon}\]where \(E\) is the modulus of elasticity, \(\sigma\) is the stress, and \(\varepsilon\) is the strain. When analyzing thermal stress, since the strain is zero due to the constraint, we can directly calculate stress from the modulus of elasticity and thermal strain.

The coefficient of thermal expansion is a material property that quantifies how much a material will expand when the temperature changes. This is denoted by \(\alpha\) and is measured in terms of length change per degree of temperature change, usually units like \/^{\circ} \text{C}^{-1}\. For steel in our exercise, the coefficient of thermal expansion is \(11.7 \times 10^{-6} /^{\circ} \text{C}\).
This property comes into play significantly in thermal stress calculations. As the temperature increases, materials want to expand. The amount of expansion can be calculated using:\[\varepsilon = \alpha \Delta T\]where \(\Delta T\) is the change in temperature. For a freely expanding material, this strain (\(\varepsilon\)) shows how much longer the material gets.
In scenarios where the material cannot expand freely due to constraints, like welded continuous steel rails, this thermal strain converts into stress, creating what's known as "thermal stress."

Normal stress is the internal force per area within a material arising from externally applied forces or other actions like temperature changes. In our exercise, the normal stress develops in the steel rails due to temperature changes. If the rails are prevented from expanding by welding, the stress is derived from thermal strain.
To calculate the stress when the rails are welded:- Identify that strain is zero, as they can't expand.- Use the modulus of elasticity and thermal strain formula: \[\sigma = E \cdot \alpha \cdot \Delta T\]This product equals stress (\(\sigma\)), which is pressure. Substituting the values:- \(E = 200 \times 10^9\) Pa- \(\alpha = 11.7 \times 10^{-6} /^{\circ} \text{C}\)- \(\Delta T = 42^{\circ} \text{C}\)Results in stress calculation:\[\sigma = 200 \times 10^9 \times 11.7 \times 10^{-6} \times 42 = 98.28 \text{MPa}\]Thus, if the rails are continuous, they experience a stress of about 98.28 MPa due to the change in temperature.