Temperature Stresses

Definition

Temperature stress (or thermal stress) is the internal stress induced in a material when its natural tendency to expand or contract due to a temperature change is externally restricted. When a body is heated or cooled, it undergoes a change in dimension; if this movement is prevented, internal forces develop, leading to stress.

Main Content

1. Thermal Expansion and Contraction

• All materials expand when heated and contract when cooled, characterized by the coefficient of linear expansion ($\alpha$).
• If a body is allowed to expand or contract freely, no stress is developed, only a change in strain (length).

2. The Condition of Restriction

• Stress only arises when the material is "restrained." This happens when supports prevent the bar from changing its length.
• Example: A steel rail fixed rigidly between two concrete abutments on a hot summer day will experience high compressive stress because it wants to grow longer but cannot.

3. Mathematical Basis

• The free thermal deformation is given by $\delta_t = L \alpha \Delta T$, where $L$ is the original length, $\alpha$ is the coefficient of expansion, and $\Delta T$ is the temperature change.
• If the body is prevented from this movement, the stress ($\sigma$) is calculated by the formula: $\sigma = E \alpha \Delta T$, where $E$ is Young's Modulus.

[Visualizing Thermal Stress]
(Unrestricted Expansion)
|----L----|  --Heat-->  |-------L + δt-------|  (No Stress)

(Restricted Expansion)
|----L----|  --Heat-->  |--L--| (Blocked)       (Compressive Stress)
  Abutment              Abutment

Working / Process

1. Calculate Free Expansion

• Identify the material property ($\alpha$) and the total length of the component ($L$).
• Determine the change in temperature ($\Delta T = T_{final} - T_{initial}$).
• Calculate the expansion that would occur: $\delta_t = L \alpha \Delta T$.

2. Determine Equivalent Force

• Since the bar is blocked, the external supports exert an equivalent force to push the bar back to its original length.
• Use the relation: $Force = \frac{Area \times E \times \delta_t}{L}$.

3. Calculate Thermal Stress

• Apply the stress formula: Stress ($\sigma$) = Force / Area.
• Alternatively, simplify using $\sigma = E \alpha \Delta T$.
• Check if the result is positive (compressive stress due to heating) or negative (tensile stress due to cooling).

Advantages / Applications

• Shrink Fitting: Engineers intentionally use thermal stress to assemble parts. A metal ring is heated to expand, placed over a shaft, and allowed to cool. As it tries to contract, it grips the shaft tightly (Interference fit).
• Expansion Joints: Understanding these stresses allows engineers to design gaps in bridges and railway tracks to avoid buckling failure.
• Bimetallic Strips: Utilized in thermostats; two metals with different $\alpha$ values are bonded together. When heated, they bend due to the difference in thermal stress, triggering a switch.

Summary

Thermal stress occurs when a material's temperature-induced dimensional changes are obstructed. If expansion is blocked, compressive stress develops; if contraction is blocked, tensile stress occurs. Proper allowance for this behavior is critical in structural engineering to prevent mechanical failure.

Important terms to remember: - Coefficient of Thermal Expansion ($\alpha$): The rate at which a material expands per unit temperature. - Young’s Modulus ($E$): The measure of a material's stiffness. - Restraint: Any external condition that prevents thermal deformation.