The 40 -ft-long A-36 steel rails on a train track are laid...

The 40 -ft-long $\mathrm{A}-36$ steel rails on a train track are laid with a small gap betwocn them to allow for thermal expansion. Determine the required gap $\delta$ so that the rails just touch one another when the temperature is increased from $T_{1}=-20^{\circ} \mathrm{F}$ to $T_{2}=90^{\circ} \mathrm{F}$. Using this gap, what would be the axial force in the rails if the temperature rises to $T_{3}=110^{\circ} \mathrm{F} ?$ The cross-sectional area of each rail is 5.10 in $^{2}$

Key Concepts

Linear Thermal Expansion

This concept refers to the increase in a material's length due to a rise in temperature. It is quantified by the formula ?L = L???T, where ?L is the change in length, L? is the original length, ? is the coefficient of thermal expansion, and ?T is the change in temperature. It is a fundamental idea in thermal mechanics and materials science, affecting how structures expand or contract with temperature fluctuations.

Thermal Stress

Thermal stress develops when a material is prevented from freely expanding or contracting due to temperature changes. When a structure such as a rail is restrained (for instance, by adjoining components or boundary conditions), the thermal expansion creates internal forces (stresses) that can lead to potential failure. The stress is calculated using the relationship ? = E??T, where E is the Young’s modulus of the material.

Expansion Gaps in Structures

Expansion gaps are deliberate spaces introduced in structural systems to accommodate changes in dimensions due to temperature variations. In applications like rail tracks, these gaps prevent buckling or excessive forces by allowing the components to expand without imposing significant stress on the system. This proactive design is critical in ensuring safety and longevity of structures subjected to wide temperature ranges.

Elasticity and Material Properties

Elasticity describes how materials deform and return to their original shape under applied forces, as governed by Hooke’s Law. Key material properties such as Young’s modulus and the coefficient of thermal expansion determine how a material responds to thermal loading and mechanical stress. Understanding these properties is vital when predicting the behavior of materials under temperature changes, ensuring that structures are designed to withstand induced stresses.

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