D,8 • APPLICATIONS. THERMAL STRESSES 



range of interest, the transient thermal stresses in the cyUnder can be 

 calculated by use of Eq. 7-6 in the thermoelastic equations [12], relating 

 the stresses to instantaneous temperature distributions. This has been 

 done in [ii] with the following principal results for the long hollow cyhn- 

 der heated by Newtonian heat transfer through the inner boundary. 



At any instant t > the transient hoop stresses (j{r, t) have extreme 

 values at the boundaries, there being maximum compressive stress <Ta{t) = 

 <T{a, t) at the inner boundary and maximum tensile stress (rb{t) = a{h, t) 

 at the outer boundary. Similarly, axial stresses also have extreme values 



1.0 



H 

 P 



0.5 



CD 



0.5 



T 



1.0 



kt/pcd^ 



Fig. D,8. Comparison of temperature transients in 

 plane parallel slab and hollow cylinder. 



at any instant at the boundaries, r = a,b, where these extremes are equal 

 numerically to the local hoop stresses cra(t), ab{t) respectively. Radial 

 stresses are zero at the boundaries and rise to a maximum within the 

 shell, but this maximum is small compared with boundary stresses at 

 any instant. The thermoelastic equations [11] lead to rather simple ex- 

 pressions, as given below, for the boundary stresses o-„(0 and (Tb{t). These 

 boundary stresses in turn possess absolute maxima attained at some time 

 during the heating process. 



We define the dimensionless stress rj by 



V = 



(t(1_-jO 



(8-1) 



< 269 ) 



