Unit No-2_Simple Stress Strain -4 ......

Simple Stress and Strain

1. Chapter No-2 Mechanics of Structure Simple Stress-Strain Created By Mr. ASHISH D. KALE Head of Civil Department K.K.Wagh Polytechnic,Nashik-03

2. Thermal stress-Strain

3. Thermal stress or Temperature Stress-Uniform Cross section Thermal stresses are the stresses induced in a body due to change in temperature, when the temperature of the body is raised or lowered and the body is restricted from expanding or contracting. Natural Deformation or free Deformation If a bar of Length ‘L’ having the coefficient of linear expansion ‘ α ’ is heated through a temperature rise of t ֯ c, It will expand naturally with out any stress by a small change in length ‘dL’ as shown in figure

4. When Thermal stress and strain is developed ? When free expansion or contraction is prevented by fixing the bar at the ends rigidly only then the temperature stress and strain is developed into the body or metal bar as shown in figure

5. Temperature Strain (e) if free expansion due to rise in temperature is prevented, then compressive temperature strain is set up, where as if free contraction due to fall in temperature is prevented, the tensile temperature strain is set up. Temperature Strain (e)= = e= Where , e= Temperature Strain, . Temperature Stress ( σ ) if free expansion due to rise in temperature is prevented, then compressive temperature stress is developed, where as if free contraction due to fall in temperature is prevented, then tensile temperature stress is developed We know, σ = but e= there for, σ = Where , E=Modulus of Elasticity,  

6. Force exerted by rigid support to prevent free deformation (P) Let ‘P’ be the force exerted by fixed rigid support We Know, P= σ A But σ = There for, P= A Where , E=Modulus of Elasticity,  

7. Temperature Stress in composite section Consider a composite member having two bars of different materials say,1 and 2 placed side by side with fixed connected at the ends and having length ‘L’ as shown in figure, where as P= A Let 1 = for bar material 1 = for bar material 2 Such that, 1 > Due to the rise in temperature by t ֯ c ,the natural expansion of material will be and that of material 2 will . At a certain instant, the composite bar will be under equilibrium and let the resultant elongation of the composite bar be ‘x’ as shown in figure Free expansion prevented by in bar 1= –x Compressive temperature strain induced in bar material 1 i.e. e 1 = = - ………………….(1) Slimily Tensile temperature strain induced in bar material 2 i.e. e 2 = = - ………………….(2)  

8. Temperature Stress in composite section Adding equation 1+2 e 1 + e 2 = - )+( - ) By solving above equation we get, + =( + ………………………(3) Compressive force on bar material 1= Tensile force on bar material 2 P 1 =P 2 A 1 = A 2 ………………………………………(4) Solving equation (3) and (4) simultaneously gives the unknown temperature stresses in two different material of composite section  

9. Field Examples of thermal stresses Gap between two rail is kept ,otherwise the temperature stresses will be developed due to free expansion get prevented. In bridge construction, there are expansion joints are provide so as to allow free expansion, other wise temperature stress will be developed due to the free expansion get prevented.

10. Numerical on Temperature Stress and strain-Uniform section A steel Bar of 30mm diameter is heated to 80 ֯ C and then clamped at the ends.it is then allowed to cool down to 30 ֯ During cooling only 1mm contraction was allowed .calculate temperature stress developed and reaction at the clamps. Take Length of bar=10m, =12x10 -6 / ֯ C,E=2X10 5 N/mm 2 Answer-: Step-1 Given Data: d=30mm,L=10m=10X10 3 mm, =12x10 -6 / ֯ C,E=2X10 5 N/mm 2 , δ =1mm,t 1 =80 ֯ C,t 2 =30 ֯ C,Hence t=t 1 -t 2 =80 ֯ - 30 ֯ = 50 ֯ C Step-2 Temperature Strain in the bar e = = = 5 X 10 -4 Step-3 Temperature Stress( σ ) =E X e=2x10 5 X 5 x = 100N/mm 2 (Tensile) Step-4 Due to contraction, nature of stress will be tensile, Let P be the reaction at the clamps We know, P= σ X A= 100 X X30 2 =70685.83N P=70685.83N (Tensile)