A rectangular beam made of S-590 alloy has depth 2c = 50 mm and thickness b = 20 mm. It is loaded with a moment of M = 1.50 kN·m at a temperature of 725°C. At this temperature, and for stresses in the range 100 to 400 MPa, constants for Eq. 15.71 are m = 10.74 and B = 3.91 × 10^−29, where t is in

Question

A rectangular beam made of S-590 alloy has depth 2c = 50 mm and thickness b = 20 mm. It is loaded with a moment of M = 1.50 kN·m at a temperature of 725°C. At this temperature, and for stresses in the range 100 to 400 MPa, constants for Eq. 15.71

[latex]\varepsilon=B \sigma^m t[/latex] (15.71)

are m = 10.74 and [latex]B = 3.91 × 10^{−29}[/latex], where t is in units of hours. (From data in [Grant 50].) Estimate the stress in the beam, the creep strain after 16,000 hours, and the life to creep rupture.

Accepted Answer

The stress at the edge of the beam is obtained directly from Eq. 15.73.

[latex]M=\frac{2 m b c^2 \sigma_c}{1+2 m}=\frac{2 m b c^2}{1+2 m}\left(\frac{\varepsilon_c}{B t} ight)^{1 / m}[/latex] (15.73)

Retaining units of mm for b and c, and substituting [latex]M = 1.50 × 10^{6}[/latex] N·mm, we obtain

[latex]\sigma_c=\frac{M(1+2 m)}{2 m b c^2}=125.6 MPa[/latex]

After 16,000 hours, the creep strain at the edge of the beam is

[latex]\varepsilon_c=B \sigma_c^m t=3.91 imes 10^{-29}(125.6 MPa )^{10.74}(16,000 h )=0.0218[/latex]

From constants C, [latex]b_{1} , ext{ and } b_{2}[/latex] in Table 15.3, the Larson–Miller parameter and the estimated life to rupture from Eq. 15.20 are

[latex]\begin{gathered}\log t_r=\frac{P_{L M}}{T}-C \quad \quad (a)\ P_{L M}=b_0+b_1 x+b_2 x^2+b_3 x^3 \quad \quad (b)\end{gathered}[/latex] (15.20)

[latex]P_{L M}=38,405-8206 \log \sigma=38,405-8206 \log (125.6 MPa )=21,181 K \cdot \log ( h )[/latex]

[latex]\log t_r=\frac{P_{L M}}{T}-C=\frac{21,181}{(725+273) K }-17=4.224, \quad t_r=16,730 ext { hours }[/latex]

It is assumed in this calculation that the steady-state creep strain is large compared with both the elastic and the transient creep strain, so that Eq. 15.71 applies on an approximate basis.

Question 15.8: A rectangular beam made of S-590 alloy has depth 2c = 50 mm ...

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