Two steel rods and a copper rod of equal length ln and equal...

Two steel rods and a copper rod of equal length $l_{n}$ and equal cross-sectional area are joined rigidly as shown, All the rods are in a state of zero tension at 0 " $C$. The temperature of system increases upto $30^{\circ} \mathrm{C}$, then (a) tensile force on either steel rod is half of copper rod (b) the net expansion in copper rod is less than the thermal expansion of the copper rod (c) the expansion in either steel rod is larger than the thermal expansion in steel rod (d) all of the above

Two steel rods and a copper rod of equal length $l_{n}$ and equal cross-sectional area are joined rigidly as shown, All the rods are in a state of zero tension at 0 " $C$. The temperature of system increases upto $30^{\circ} \mathrm{C}$, then
(a) tensile force on either steel rod is half of copper rod
(b) the net expansion in copper rod is less than the thermal expansion of the copper rod
(c) the expansion in either steel rod is larger than the thermal expansion in steel rod
(d) all of the above

Key Concepts

Compatibility Conditions in Composite Structures

When different materials are rigidly joined in a structure, the overall deformation must satisfy compatibility conditions, meaning that the total deformation or expansion must be the same across the joints. This requirement leads to a redistribution of the free thermal strains, and-balancing the induced thermal stresses based on differences in material properties such as the coefficient of thermal expansion and elasticity.

Thermal Stress

Thermal stress arises when a material is prevented from expanding or contracting freely due to external constraints or connected components with different expansion properties. This stress is a mechanical force generated internally when the differential thermal expansions are forced to be compatible with each other, potentially causing tension or compression within the material.

Statically Indeterminate Structures

A statically indeterminate structure is one in which the static equilibrium equations are insufficient to determine all internal forces and deformations. In such structures, additional considerations such as compatibility conditions and material constitutive relationships (like stress-strain relationships) are necessary. When joining different rods with different thermal properties, the structure becomes statically indeterminate, and the additional constraints cause the induced thermal stresses to depend on both the material properties and the imposed constraints.

Thermal Expansion

Thermal expansion describes the tendency of a material to change its dimensions (usually length) in response to a temperature change. The expansion is typically proportional to both the original length and the temperature change, governed by the coefficient of thermal expansion. This concept is crucial in understanding how materials behave when heated or cooled, especially when their expansion is mechanically constrained.

Coefficient of Thermal Expansion

The coefficient of thermal expansion is a material-specific parameter that quantifies how much a material expands or contracts per unit length for a unit change in temperature. This property is vital when comparing different materials, as it helps predict the amount of free expansion each material would experience under the same thermal conditions if there were no mechanical constraints.

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