A particular mechanical bracket has been subjected to constant-amplitude fatigue tests at the single level of Sr = 150 MPa, and the observed fatigue life was Nf = 10^6 cycles. Based on experience with similar devices, it is estimated that the parameter m of the S/N curve is in the range of

Question

A particular mechanical bracket has been subjected to constant-amplitude fatigue tests at the single level of [latex] S_{r} = 150 [/latex] MPa, and the observed fatigue life was [latex] N_{f}=10^{6} [/latex] cycles. Based on experience with similar devices, it is estimated that the parameter [latex] m [/latex] of the S/N curve is in the range of [latex] 3\leq m\leq 5 [/latex] . Based on this limited information, it is necessary to choose a level for the standard deviation [latex] \sigma _{X} [/latex] of the narrowband, mean-zero, Gaussian stress that will be applied to the bracket in actual service. Using the limiting values of [latex] m = 3 [/latex] and [latex] m = 5 [/latex] , find the level of [latex] \sigma _{X} [/latex] meeting each of two possible design situations: [latex] E[N(T)]=10^{6} [/latex] and [latex] E[N(T)]=10^{8} [/latex].

Accepted Answer

Beginning with [latex] m=3 [/latex] we find the value of [latex] K [/latex] in the S/N curve of Eq. 11.43 [latex]N_{f}(S_{r})=KS_{r}^{-m}[/latex] such that [latex] 10^{6}=K/(150)^{3} [/latex] , giving [latex] K=3.375 imes 10^{12} [/latex] . For [latex] E[N(T)]=10^{6} [/latex] , Eq. 11.50 [latex]E(\Delta D)=K^{-1}(2)^{3m/2}\lambda _{0}^{m/2}\Gamma \left(1+\frac{m}{2} ight) [/latex] then gives [latex] 10^{-6}=K^{-1}2^{4.5}(\sigma _{X})^{3}(3\pi ^{1/2}/4) [/latex] . Solving this gives the answer of [latex] \sigma _{X}=48.2 [/latex]MPa. Similarly, [latex] E[N(T)]=10^{8} [/latex] gives [latex] \sigma _{X}=10.4 [/latex]MPa.

For [latex] m=5 [/latex] , we proceed in the same way and find that [latex] K=7.594 imes 10^{16} [/latex] . For [latex] E[N(T)]=10^{6} [/latex] , Eq. 11.50 becomes [latex] 10^{-6}=K^{-1}2^{7.5}(\sigma _{X})^{5}(15\pi ^{1/2}/8) [/latex] , which gives [latex] \sigma _{X}=41.7 [/latex]MPa, and [latex] E[N(T)]=10^{8} [/latex] gives [latex] \sigma_{X}=16.6 [/latex]MPa.

Given the uncertainty about the value of [latex] m [/latex] , the safe choice is to use [latex] \sigma_{X}=41.7[/latex]MPa for a design condition of [latex] E[N(T)]=10^{6} [/latex] and [latex] \sigma_{X}=10.4 [/latex]MPa for a design condition of [latex] E[N(T)]=10^{8} [/latex] . That is, one should use the smaller stress for the given design condition, to avoid failure due to uncertainty about the value of [latex] m [/latex].

Note that one must be cautious in choosing conservative approximations for this type of problem. In particular, one cannot conclude that either choice of [latex] m [/latex] value is generally more conservative than the other. For the design condition of [latex] E[N(T)]=10^{6} [/latex] , it is more conservative to assume that [latex] m = 5 [/latex] , because this gives an allowable stress that is smaller than the value one would obtain by using [latex] m = 3 [/latex]. For the alternative design condition of [latex] E[N(T)]=10^{8} [/latex] , though, it is more conservative to assume that [latex] m=3 [/latex] , because this gives the smaller allowable stress value. In general, one must consider the range of possible m values when dealing with such limited experimental data rather than simply using the largest possible or smallest possible [latex] m [/latex]value.

Question 11.8: A particular mechanical bracket has been subjected to consta...

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