Partially Restrained Tapered Bar:
Let us consider the bar of Figure, but now let it be free to extend by an amount ΔL′(ΔL′ < ΔL) . By following similar arguments as in the previous section, now the compressive load P developed in the bar will be given by
P = πE d1 d2 (ΔL - ΔL′)/4L
= πE d1 d2 (L α ΔT - ΔL′)/4L
and hence,
σ = E d1 d2 (L α ΔT - ΔL′)/L( d1 - (d1 - d2 )/L x)2
and the maximum stress,
σmax = E d1 (L α ΔT - ΔL′)/Ld2