Consider temperature rise of a composite bar consisting of two members; one of steel and other of brass rigidly fastened to each other. If allowed to expand freely;
expansion of brass bar: AB = Lαb∆t
expansion of steel bar: AC = Lαs∆t
Since coefficient of thermal expansion of brass is greater than that of steel, expansion of brass will be more.But the bars are fastened together and accordingly both will expand to the same final position represented by DD with net expansion of composite system AD equal to dl. To attain this position, brass bar is pushed back and the steel bar is pulled.
Obviously compressive stress will be induced in brass bar and tensile stress will be developed in steel bar. Under equilibrium state;
compressive force in brass = tensile force is steel
σcAc = σs As
Corresponding to brass rod:
Reduction in elongation,
DB = AB – AD = Lαbσt – δL
Strain eb = (Lαb∆t – δL)/L = αb∆t – e;
Where e = δL/L, is the actual strain of the composite system Corresponding to steel rod:
extra elongation, CD = AD – AC = δL – Lαs ∆t
Strain es = (δL – Lαs ∆t)/L = e – αb∆t
Adding eb and es , we get
eb + es = (δb – δs ) ∆t,
It is also called as compatibility condition.
It may be pointed out that the nature of the stresses in the bars will get reversed if there is reduction in the temperature of the composite system.
