A 250 -m-long bridge is improperly designed so that it...

A 250 -m-long bridge is improperly designed so that it cannot expand with temperature. It is made of concrete with $\alpha=12 \times$ $10^{-6}\left(^{\circ} \mathrm{C}\right)^{-1}$ .( a) Assuming the maximum change in temperature at the site is expected to be $20^{\circ} \mathrm{C}$ , find the change in length the span would undergo if it were free to expand. (b) Show that the stress on an object with Young's modulus $Y$ when raised by $\Delta T$ with its ends firmly fixed is given by $\alpha Y \Delta T$ (c) If the maximum stress the bridge can withstand without crumbling is $2.0 \times 10^{7} \mathrm{Pa}$ , will it crumble because of this temperature increase? Young's modulus for concrete is about $2.0 \times 10^{10} \mathrm{Pa}$ .

Key Concepts

Thermal Stress Analysis

Thermal stress analysis involves assessing the stresses induced in a material when its thermal expansion is constrained. When a material is prevented from expanding freely, the thermal strain is converted into mechanical stress. This stress is given by the product of the material’s Young's modulus and the thermal strain, leading to the formula stress = ?Y?T, which is pivotal in determining whether a structure can safely withstand temperature variations.

Young's Modulus

Young's modulus (Y) is a measure of a material's stiffness, representing the ratio of stress (force per unit area) to strain (relative deformation) in the elastic region of the stress-strain curve. It is a fundamental property used in understanding how materials respond to applied loads and deformations.

Coefficient of Thermal Expansion

The coefficient of thermal expansion (?) is a material property that quantifies how much a material's length changes per unit length per degree change in temperature. It is critical in calculating thermal expansion in structures and must be considered during design to prevent issues from temperature variations.

Free Thermal Expansion

Free thermal expansion refers to the natural change in dimensions (such as length) of a material when it is heated or cooled without any external constraints. The change in length can be calculated using the formula ?L = ?L?T, which predicts the expansion or contraction based on the material's coefficient of thermal expansion, original length, and temperature change.

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