When the temperature is at 30^∘ C, the A-36 steel pipe fits snugly between the two fuel tanks. W

step 1 So this question is the same as the previous question. But this time, instead of fixed support a

When the temperature is at 30^∘ C, the A-36 steel pipe fits...

When the temperature is at $30^{\circ} \mathrm{C}$, the $\mathrm{A}-36$ steel pipe fits snugly between the two fuel tanks. When fuel flows through the pipe, the temperatures at ends $A$ and $B$ rise to $130^{\circ} \mathrm{C}$ and $80^{\circ} \mathrm{C},$ respectively. If the temperature drop along the pipe is linear, determine the average normal stress developed in the pipe. Assume the walls of each tank act as a spring, each having a stiffness of $k=900 \mathrm{MN} / \mathrm{m}$

Key Concepts

Thermal Expansion

Thermal expansion is the tendency of matter to change its dimensions in response to a change in temperature. For solids, the coefficient of thermal expansion quantifies this response, and when expansion is constrained, internal stresses develop as the material is unable to freely expand or contract.

Thermal Stress

Thermal stress arises when temperature-induced expansion or contraction in a material is impeded by external constraints. The stresses are computed based on the difference between the unconstrained expansion and the actual deformation permitted by the supports or connections of the structure.

Linear Temperature Distribution

A linear temperature distribution assumes that the temperature changes at a constant rate along a specific dimension. In the context of the problem, this assumption simplifies the determination of thermal strain by allowing a direct relationship between distance along the pipe and the temperature change, and ultimately facilitating the calculation of the induced stresses.

Mechanical Constraints and Spring Stiffness Modeling

Mechanical constraints, such as those provided by the walls of the tanks modeled as springs with defined stiffness, resist the free expansion of the pipe. The concept of modeling these supports as springs helps in quantifying the force necessary to produce a given displacement, which is crucial in linking the thermal expansion to the resulting stresses in the structure.

Material Mechanical Properties

Material mechanical properties, including Young’s modulus, play a critical role in determining how a material reacts to stress. Young's modulus relates the stress to the strain under elastic conditions, and it is an essential parameter when evaluating the effect of thermal expansion on the overall stress within the material.

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