Mechanical Properties of Materials

      Solve each of the following problems as indicated using your choice of computational software or hand calculations. Where MATLAB is used, be sure to publish your solution in pdf format. If other computational software is used, be sure to print your solution to pdf format. Hand solutions should be scanned to pdf. Combine your solutions into one pdf file for upload. 1. For the notched member of Fig. 10.11, answer the following: a. Determine your own value of kt, and then estimate kf. (Note that the half-circular notch shape gives w1 = w2 – 2ρ). b. For 106 cycles, use your kf value with the smooth specimen fatigue strength from the S-N curve to estimate the notched specimen strength. How good is the agreement with the test data? 2. A double-edge-notched plate is made of the SAE 1015 steel of Table 9.1 and is subjected to an axial force P, as in Figure A.11 (b). Dimensions are w1 = 36.0 mm, w2 = 54.0 mm, ρ = 6.30 mm, and t = 5.00 mm. The fatigue limit for completely reversed (σm = 0) loading of unnotched material can be estimated from Fig. 9.24. a. Determine kt, and then estimate kf. b. What completely reversed force amplitude Pa can be applied to the notched member for 106 cycles with a safety factor of 2.5 in stress? 3. A shaft with a step-down in diameter has dimensions, as defined in Fig. A.12 (b), of d1 = 50, d2 = 60, and ρ = 1.25 mm. The shaft is subjected to bending and is made of a quenched and tempered low-alloy steel having an ultimate tensile strength of 950 MPa. The fatigue limit for completely reversed (σm = 0) loading of unnotched material can be estimated from Fig. 9.24. a. Determine kt, and then estimate kf. b. What completely reversed bending moment amplitude Ma can be applied to the notched shaft for 106 cycles? A safety factor of 2.0 in stress is required. 4. A plate with a round hole is made of the 2024-T4 aluminum of Table 9.1, and it has dimensions d = 6.0, w = 30, and t = 5.0 mm, as defined in Fig. A.11 (a). The fatigue limit for completely reversed (σm = 0) loading of unnotched material can be estimated from Fig. 9.25. a. Determine kt and estimate kffor loading in tension. b. What completely reversed axial force amplitude Pa can be applied for 5 x 108 cycles without fatigue failure? A safety factor of 2.5 in stress is required. 5. A circular rod made of the SAE4142 (450HB) steel of Table 9.1 is loaded in bending and has a step change in diameter. The dimensions , as defined in Fig. A.12 (b), are d1 = 20, d2 = 24, and ρ = 2.0 mm, and the fillet radius is machined. For a safety factor of 1.5 on stress, estimate the largest completely reversed bending moment amplitude that can be applied for 106 cycles. 6. A circumferentially grooved round bar made of 7075-T6 aluminum (Table 3.2) is subjected to 5 x 108 cycles of completely reversed bending. The bar has dimensions, as defined in Fig. A.12 (c), of d1 = 15, d2 = 21.5, and ρ = 0.75 mm. The notch radius ρ has a ground surface finish, which is estimated to reduce the polished specimen fatigue limit of the material (see Fig. 9.25) by a factor ms = 0.8. A safety factor of 1.5 in stress is required. a. What completely reversed moment amplitude Ma can be applied in the actual service of this bar? b. If a completely reversed moment amplitude of Ma = 13 N·m will be applied in service, is the design adequate? If not, what new value of notch radius ρ will allow the bar to just meet the required safety factor? 7. For the notched member of Fig. 10.11, estimate the repeatedly applied zero-to-maximum nominal stress Smax that corresponds to a life of Nf = 105 cycles for both of the following equations: a. Using Equation 10.21: 𝑆𝑆𝑎𝑎𝑎𝑎 = 𝜎𝜎𝑎𝑎𝑎𝑎 𝑘𝑘𝑓𝑓 = 𝑆𝑆𝑎𝑎 1 − 𝑆𝑆𝑚𝑚 𝜎𝜎𝑢𝑢 (Eq. 10.21) b. Using Equation 10.27: 𝑆𝑆𝑎𝑎𝑎𝑎 = �𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚𝑆𝑆𝑎𝑎, 𝑆𝑆𝑎𝑎𝑎𝑎 = 𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚�1 − 𝑅𝑅 2 , (a, b) (Eq. 10.27) 8. Consider the S-N curve of Fig. 9.27 for the bar of 2024-T4 aluminum with notch radius ρ = 1.59 mm. Note that nominal stress S is defined as in Fig. A.12 (c). If a similar bar is subjected to a mean bending moment Mm = 5.0 N·m, estimate the moment amplitude Ma that will cause failure in 105 cycles for both of the following equations: a. Using Equation 10.21 (given above). b. Using Equation 10.27 (given above). 9. The aluminum alloy 2024-T4 of Table 9.1 is to be used in the form of a plate with a central hole under axial loading, as in Fig. A.11 (a). The dimensions are w = 60, d = 9.0, and t = 20 mm. For a life of Nf = 107 cycles, estimate the mean force Pm if the force amplitude is Pa = 55 kN: a. Using Equation 10.21 (given above). b. Using Equation 10.27 (given above).