The 2014-T6 aluminum rod has a diameter of 0.5 in. and is...

The 2014-T6 aluminum rod has a diameter of 0.5 in. and is lightly attached to the rigid supports at $A$ and $B$ when $T_{1}=70^{\circ} \mathrm{F}$. Determine the force $P$ that must be applied to the collar so that, when $T=0^{\circ} \mathrm{F}$, the reaction at $B$ is zero.

Key Concepts

Thermal Expansion

Thermal expansion is the phenomenon where materials change their dimensions in response to temperature changes. It is quantified by the coefficient of thermal expansion, which links the temperature change to the proportional change in length of a material, forming the basis for analyzing free thermal deformations.

Thermal Stress

Thermal stress arises when a material’s natural tendency to expand or contract due to temperature changes is restrained by supports or connections. This constraint leads to the development of internal stresses proportional to the material’s modulus of elasticity and the thermal strain, which are critical in ensuring structural integrity.

Compatibility of Deformations

Compatibility of deformations is a principle in structural mechanics that ensures the displacements due to thermal effects and applied mechanical loads satisfy the physical constraints imposed by the structure’s supports and connections. It involves equating the free thermal deformation with the deformation imposed by external forces and boundary conditions.

Statics Equilibrium

Statics equilibrium is a fundamental concept in structural analysis that requires the sum of forces and moments acting on a structure to be zero. This principle is used to determine unknown forces, such as the applied force needed to balance thermal-induced deformations, ensuring that all reactions and loads are in equilibrium.

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