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Chapter 4

Q. 4.5

An initially unstressed short steel cylinder, internal radius 0.2 m and external radius 0.3 m, is subjected to a temperature distribution of the form T = a + b  log_{e} r to ensure constant heat flow through the cylinder walls. With this form of distribution the radial and circumferential stresses at any radius r , where the temperature is T. are given by

\sigma _{r}=A-\frac{B}{r^{2}} -\frac{\alpha ET}{2(1-\nu )}

\sigma _{H}=A+\frac{B}{r^{2}} -\frac{\alpha ET}{2(1-\nu )}-\frac{ E\alpha b}{2(1-\nu )}

If the temperatures at the inside and outside surfaces are maintained at 200°C and 100°C respectively, determine the maximum circumferential stress set up in the cylinder walls. For steel, E = 207 GN/m², ν = 0.3 and \alpha = 11 × 10^{-6} per °C


Verified Solution

T=a+b\log _{e}r
200=a+b\log _{e}0.2=a+b(0.6931-2.3026)
∴     200=a-1.6095 b                  (1)
also            100=a+b\log _{e}0.3=a+b(1.0986-2.3026)
100=a-1.204   b                       (2)
(2)-(1),                100=-0.4055   b
Also                \frac{E\alpha }{2(1-\nu )} =\frac{207\times 10^{9}\times 11\times 10^{-6}}{2(1-0.29)}
=1.6\times 10^{6}

Therefore substituting in the given expression for radial stress

\sigma _{r}=A-\frac{B}{r^{2}} -1.6\times 10^{6}T

At r = 0.3, \sigma _{r} = 0 and T = 100

0=A-\frac{B}{0.09}-1.6 \times 10^{6} \times 100      (3)

At r = 0.2, \sigma _{r} = 0 and T = 200

0=A-\frac{B}{0.04}-1.6\times 10^{6}\times 200    (4)

(4)-(3)     0=B(11.1-25)-1.6\times 10^{8}
B=-11.5\times 10^{6}

and from (4),

A=25B+3.2\times 10^{8}
=(-2.88+3.2)10^{8}=0.32\times 10^{8}

substituting in the given expression for hoop stress,

\sigma _{H}=0.32\times 10^{8}-\frac{11.5\times 10^{6}}{r^{2}} -1.6\times 10^{6}T+1.6\times 10^{6}\times 247

At  r = 0.2, \sigma _{H}=(0.32-2.88-3.2+3.96)10^{8}=-180    MN/m^{2}

At r = 0.3,    \sigma _{H}=(0.32-1.28-1.6+3.96)10^{8}=+140   MN/m^{2}

The maximum tensile circumferential stress therefore occurs at the outside radius and has a value of 140 MN/m². The maximum compressive stress is 180 MN/m² at the inside radius