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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 expands or contract. The amount Î´T is given by
δT=αL(TfTi)=αLΔTδT=αL(Tf−Ti)=αLΔT
where Î± is the coefficient of thermal expansion in m/m°C, L is the length in meter, Ti and Tf are the initial and final temperatures, respectively in °C. For steel, Î±= 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, internal stress is created. The internal stress created is termed thermal stress.
Case 1- when the beam is fixed at both ends
For a homogeneous rod mounted between unyielding supports as shown, the thermal stress is computed as:
homogeneous rod mounted between unyielding supports

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 Ïƒ is the thermal stress in MPa, E is the modulus of elasticity of the rod in MPa.
Case 2- if the beam is partially fixed at both ends
If the wall yields a distance of x as shown, the following calculations will be made:
Rod in yielding support

δT=x+δPδT=x+δP
αLΔT=x+σLEαLΔT=x+σLE
where Ïƒ 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 appears if two equal and opposite forces are acting on the body. 
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