Two steel rods and copper rod of equal length l0 and equal...

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

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

Key Concepts

Constrained Thermal Expansion in Composite Systems

When materials with different thermal expansion properties are joined together in a single structure, each material's attempt to expand by its own amount is limited by the constraint of the overall system geometry. This constrained expansion results in one material experiencing less expansion than its free thermal expansion would predict, while the more expanding material might extend beyond its free expansion if the system is forced to maintain continuity, leading to a redistribution of forces among the different components.

Thermal Stress

Thermal stress arises in structures when parts of a system are prevented from expanding or contracting freely due to constraints. In composite structures where materials with differing thermal expansion coefficients are rigidly joined, the mismatch in the natural expansion tendencies produces internal forces. These internal forces, or thermal stresses, attempt to equalize the deformation, potentially leading to tension or compression in various components.

Thermal Expansion

Thermal expansion refers to the tendency of materials to change in size—typically to expand—when their temperature increases. This property is quantified using the coefficient of thermal expansion, which indicates how much a unit length of material will change per degree of temperature change. In engineering problems, the differential expansion of various materials under the same temperature change is a key factor in understanding induced stresses and deformations.

Coefficient of Thermal Expansion

The coefficient of thermal expansion is a material-specific constant that describes the fractional change in length per unit temperature change. Different materials typically have different coefficients; for instance, metals such as copper and steel expand at different rates when heated. This difference is crucial in the design of composite systems where materials joined together are subjected to temperature variations, as it leads to internal forces if the natural expansions are constrained.

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