Combined Steady and Variable Stresses, Gerber Method for Combination of Stresses, Goodman Method for Combination of Stresses, Soderberg Method for Combination of Stresses.
- 1. Title Mechanical Manufacturing Submitted to: Sir DR.Saleem Submitted by: Roll no# 18-MT-21 Punjab Tianjin University Of Technology Lahore
- 2. Combined Steady and Variable Stress The failure points from fatigue tests made with different steels and combinations of mean and variable stresses are plotted in Fig. as functions of variable stress (σv ) and mean stress (σm). The most significant observation is that, in general, the failure point is little related to the mean stress when it is compressive but is very much a function of the mean stress when it is tensile. In practice, this means that fatigue failures are rare when the mean stress is compressive (or negative). Therefore, the greater emphasis must be given to the combination of a variable stress and a steady (or mean) tensile stress There are several ways in which problems involving this combination of stresses may be solved. : • Gerber method • Goodman method • Soderberg method Gerber Method for Combination of Stresses
- 3. The relationship between variable stress (σv ) and mean stress (σm) for axial and bending loading for ductile materials are shown in Fig. 6.15. The point σe represents the fatigue strength corresponding to the case of complete reversal (σm = 0) and the point σu represents the static ultimate strength corresponding to σv = 0. A parabolic curve drawn between the endurance limit (σe ) and ultimate tensile strength (σu) was proposed by Gerber in 1874. Generally, the test data for ductile material fall closer to Gerber parabola as shown in Fig. 6.15, but because of scatter in the test points, a straight-line relationship (i.e. Goodman line and Soderberg line) is usually preferred in designing machine parts. Liquid refrigerant absorbs heat as it vaporizes inside the evaporator coil of a refrigerator. The heat is released when a compressor turns the refrigerant back to liquid According to Gerber, variable stress, Where F.S. = Factor of safety, σm = Mean stress (tensile or compressive),
- 4. σu = Ultimate stress (tensile or compressive), and σe = Endurance limit for reversal loading. Considering the fatigue stress concentration factor (Kf), the equation may be written as • Goodman Method for Combination of Stresses A straight line connecting the endurance limit (σe ) and the ultimate strength (σu), as shown by line AB in Fig. 6.16, follows the suggestion of Goodman. A Goodman line is used when the design is based on ultimate strength and may be used for ductile or brittle materials. In Fig. 6.16, line AB connecting σe and σu is called Goodman's failure stress line. If a suitable factor of safety (F.S.) is applied to endurance limit and ultimate strength, a safe stress line CD may be drawn parallel to the line AB. Let us consider a design point P on the line CD. Now from similar triangles COD and PQD, PQ/QD CO/OD = OD-OQ/OD − = 1 – OQ/OD (Q QD = OD – O Q)
- 5. This expression does not include the effect of stress concentration. It may be noted that for ductile materials, the stress concentration may be ignored under steady loads. Since many machine and structural parts that are subjected to fatigue loads contain regions of high stress concentration, therefore equation (i) must be altered to include this effect. In such cases, the fatigue stress concentration factor (Kf) is used to multiply the variable stress (σv). The equation (i) may now be written as F.S. = Factor of safety, σm = Mean stress, σu= Ultimate stress, σv= Variable stress, σe= Endurance limit for reversed loading, and Kf= Fatigue stress concentration factor. Considering the load factor, surface finish factor and size factor, the equation (ii) may be written as Kb= Load factor for reversed bending load , Ksur= Surface finish factor, and Ksz= Size factor.
- 6. The equation (iii) is applicable to ductile materials subjected to reversed bending loads (tensile or compressive). For brittle materials, the theoretical stress concentration factor (Kt ) should be applied to the mean stress and fatigue stress concentration factor (Kf ) to the variable stress. Thus for brittle materials, the equation (iii) may be written as When a machine component is subjected to a load other than reversed bending, then the endurance limit for that type of loading should be taken into consideration. Thus for reversed axial loading (tensile or compressive), the equations (iii) and (iv) may be written as where suffix ‘s’denotes for shear. For reversed torsional or shear loading, the values of ultimate shear strength (τu) and endurance shear strength (τe) may be taken as follows: τu = 0.8 σu; and τe = 0.8 σe Soderberg Method for Combination of Stresses: A straight line connecting the endurance limit (σe) and the yield strength (σy), as shown by the line AB in Fig., follows the suggestion of Soderberg line. This line is used when the design is based on yield strength.
- 7. In this central heating system, a furnace burns fuel to heat water in a boiler. A pump forces the hot water through pipes that connect to radiators in each room. Water from the boiler also heats the hot water cylinder. Cooled water returns to the boiler. Proceeding in the same way as discussed in Art 6.20, the line AB connecting σe and σy, as shown in Fig. 6.17, is called Soderberg's failure stress line. If a suitable factor of safety (F.S.) is applied to the endurance limit and yield strength, a safe stress line CD may be drawn parallel to the line AB. Let us consider a design point P on the line CD. Now from similar triangles COD and PQD,
- 8. For machine parts subjected to fatigue loading, the fatigue stress concentration factor (Kf) should be applied to only variable stress (σv). Thus the equations (i) may be written as Considering the load factor, surface finish factor and size factor, the equation (ii) may be written as Since σeb = σe × Kb and Kb = 1 for reversed bending load, therefore σeb = σe may be substituted in the above equation. Example 1: An automobile engine part rotates, and in each rotation stress varies from Smax=20,000 psi to Smin=1,000 psi. The material has Su= 80,000 psi, Syp = 60,000 psi, Se=28,000 psi. Assume K=Kf=1. Find Nfs, with (i) Soderberg’s, (ii) Goodman’s and (iii) modified Goodman’s equations.