thermal stress

thermal stresses are due to the change in temperature of the component and the component is mechanically restrained to expand or contract.

Conditions for existence of Thermal Stress:

1. There should be a temperature change in the component.
2. The thermal expansion or contraction should be restricted either completely or partially.

 

When the temperature of an object changes, its dimensions are changed which results in strain.

If α is the coefficient of thermal expansion (or contraction), the uniform thermal strain (ϵ_t) is given by: 

Where ΔT is the change in temperature.

So, change in length is given by,

Where L is the initial length.

coefficient of thermal expansion - The coefficient (α) is a property of the material and has a unit reciprocal of temperature change.

List of materials and their coefficient of thermal expansion.

 

Thermal stress under different stress condition

Case (a): When bar is free to expand:

Consider a bar of length L subjected to uniform temperature increase T, the bar being free to expand. Then the increase in the length of bar will be:

        

since the bar is free to expand, there will be no thermal stress in the bar.

Case (b): When bar is constrained:

consider a bar of length L, fixed at both ends between rigid supports.

This stress will be compressive when the change in temperature (T) is positive (i.e. increase) and tensile. When the change in temperature is negative (i.e. decrease).

Case (c): When Supports yield:

If the support yields by an amount λ. In this case, the total amount of expansion checked will be (ΔT - λ) .

 

Thus,

TEMPERATURE STRESSES IN BARS OF TAPERING SECTION

Consider a bar of uniformly tapering section, shown in Figure, rigidly fixed between two end supports. When the temperature is raised by ΔT,

Tapering bar with increasing cross-section

THERMAL STRESSES IN COMPOUNDS AND COMPOSITE BARS

Thermal stresses in compound bars (BARS in series):

Bars in series

 

TEMPERATURE STRESSES IN COMPOSITE BAR

When a composite bar, consisting of two materials having different coefficients of thermal expansion, is subjected to temperature change, opposite kinds of stresses (i.e. tensile and compressive) will be set up in the two materials.

Consider a composite bar of steel (S) and copper (C), subjected to increase in temperature.

https://gs-post-images.grdp.co/2020/8/11-img1596912533756-36.png-rs-high-webp.png

The free expansion  of copper due to temperature rise will be less in steel since  for copper is more than  of steel. In order to have common expansion , tensile stress will be developed in steel and compressive stress in copper.

Bars connected in parallel

 

Tensile force in steel = compressive force in copper

             

From compatibility equation

Final extension of copper = Final extension of steel     

                 

(Δ_s)_Th = Free expansion of steel, due to temperature rise

(Δ_c)_Th =Free expansion of copper, due to temperature rise

(Δ_s)_axial = Expansion of steel due to temperature stress (tensile)

(Δ_c)_axial  = Compression of copper due to Temperature stress (comp.)

Results,

Substituting the value of P