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When a rod is clamped at both ends and temperature changes by Delta T, it cannot expand or contract. The prevented thermal strain (alpha * Delta T) produces stress:

$sigma_{thermal}$ = Y * alpha * Delta T $F_{thermal}$ = Y * A * alpha * Delta T

Key insight: thermal stress is independent of length. Both the free expansion (alphaDelta TL) and the strain (alpha*Delta T) are determined by the ratio, where L cancels.

Practical significance: For a steel rail (Y = 200 GPa, alpha = 12 * 10^{-6} per K), a 30 K rise produces sigma = 72 MPa — about 30% of yield stress. This is why railway tracks need expansion gaps, bridges have roller supports, and concrete slabs have expansion joints.

Related applications: (1) Bimetallic strips — two metals with different alpha bonded together; differential expansion causes bending (used in thermostats). (2) Shrink fitting — a heated ring (expanded) is placed on a shaft and allowed to cool, creating enormous clamping force. (3) Pre-stressed concrete — steel rods under tension are embedded in concrete; when released, they compress the concrete, compensating for its weak tensile strength.