Thermal stress

Thermal stress is created when a change in size or volume is constrained due to a change in temperature.

Due to change in the temperature the material expand or contract. The amount δ_T, is given by

δT=αL(Tf−Ti)=αLΔTδT=αL(Tf−Ti)=αLΔT

where α is the coefficient of thermal expansion in m/m°C, L is the length in meter, T_i and T_f are the initial and final temperatures, respectively in °C. For steel, α= 11.25 × 10^-6 m/m°C.

If temperature deformation is permitted to occur freely, no load or stress will be induced in the structure. In some cases where temperature deformation is not permitted, an internal stress is created. The internal stress created is termed as
thermal stress.

Case 1- when beam is fixed at both end

For a homogeneous rod mounted between unyielding supports as shown, the thermal stress is computed as:

deformation due to temperature changes;

δT=αLΔTδT=αLΔT

deformation due to equivalent axial stress;
δP=PLAE=σLEδP=PLAE=σLE

δT=δPδT=δP

αLΔT=σLEαLΔT=σLE

σ=EαΔTσ=EαΔT

where σ is the thermal stress in MPa, E is the modulus of elasticity of the rod in MPa.

Case 2- if beam is partialy fix at both end

If the wall yields a distance of x as shown, the following calculations will be made:

δT=x+δPδT=x+δP

αLΔT=x+σLEαLΔT=x+σLE

where σ represents the thermal stress.

Take note that as the temperature rises above the normal, the rod will be in compression, and if the temperature drops below the normal, the rod is in tension.
 Case 3- if beam end free at one end

Due to change in temperature of  material only strain is appear but no stress will appear. 

Because we know that stress appear if two equal and opposite forces are acting on the body.