The dilemma for a designer in attempting to implement the “best” failure prevention strategy often seems to rest upon the choice between the reliabilityapproach and the de- sign safety factor approach. The spotty availability of pertinent statistical data often leads a designer to simply ignore the data that areavailable. Thus the basic principle of utilizing as much quantitative information as possible when making design decisions is violated.
Another viewpoint sometimes taken is that when using the probabilistic design ap- proach (reliability approach), the design safety factor should be abandoned. If, indeed, probabilistic descriptions were available for strength, load, environment, manufacturing, inspection, and maintenance practices, there would be no need for a design safety factor.
However, such complete information is rarely, if ever, available to a designer.
Rather than choosing between the two approaches, a more productive viewpoint might be to combine the best attributesof each in making design decisions. Thus, where well- defined probabalistic data are available for describing strength, loading, manufacturing practice, or other design parameters, these quantitative probablistic data might be incor- porated piecewise in the design safety factor approach. As more precise probabilistic data are incorporated, the semiquantitative rating numbers (RNs)tend to be driven toward more negative values, since more precise information in these cases would prompt a perceived need to lowerthe design safety factor. The magnitude of the design safety factor calculated from (2-86) would, in such cases, be driven toward lower values, ultimately approaching the lower limiting value prescribed by (2-87). Because of using the available probabilistic information, this approach promotes more precision in making design decisions about ma- terials and dimensions, and, at the same time, preserves the designer’s ability to account for uncertainties and variabilities in parameters that are unsupported by statistical data.
The approach has much to recommend it.
37An excellent example is provided by C. R. Mischke in ref. 30, pp. 1–10.
Example 2.15 Change in Selection of Design Safety Factor as a Result of Higher- Reliability Strength Data
Since the determination of a design safety factor of 1.25 in Example 2.11, an exten- sive body of experimental data has been discovered for fatigue strength of the material. This data bank indicates that the fatigue endurance limit is normally distributed. Since the data are extensive, accurate estimates of the mean and standard deviation may be deter- mined with a high level of confidence. How would this newly discovered body of statistical data influence the value of the design safety factor determined in Example 2.11?
Solution
The only change in determining from the rating factors would involve reevaluation of rating number 3, which was chosen as RN in Example 2.11. The new, much more accurate, data might suggest a change in value to RN . Thus, from (2.85) for this case
and since
Using the statistically significant new strength data therefore results in a design safety factor reduction of about 8 percent. It should be cautioned that an 8 percent reduction in de- sign safety factor does not necessarily correspond to an 8 percent increase in the calculated design-allowable stress. The reason for this is that the fatigue endurance limit value used in the calculation of design-allowable stress would (because the probabilistic approach is being used to define fatigue strength) now depend upon the choice of an appropriate fatigue strength reliability level for the application.
nd = 1.15 t 6 -6
t = 0 + 0 - 3 - 3 + 3 + 0 + 0 - 4 = -7 -1 -3
nd
S¿f
nd
Problems 85
2-1. In the context of machine design, explain what is meant by the terms failureandfailure mode.
2-2. Distinguish the difference between high-cycle fatigueand low-cycle fatigue, giving the characteristics of each.
2-3. Describe the usual consequences of surface fatigue.
2-4. Compare and contrast ductile ruptureandbrittle fracture.
2-5. Carefully define the terms creep,creep rupture, and stress rupture, citing the similarities that relate these three failure modes and the differences that distinguish them from one another.
2-6. Give a definition for fretting, and distinguish among the related failure phenomena of fretting fatigue,fretting wear, and fretting corrosion.
2-7. Give a definition of wear failureand list the major subcat- egories of wear.
2-8. Give a definition of corrosion failure, and list the major subcategories of corrosion.
2-9. Describe what is meant by a synergisticfailure mode, give three examples, and for each example describe how the syner- gistic interaction proceeds.
2-10. Taking a passenger automobile as an example of an en- gineering system, list all failure modes that you think might be significant, and indicate where in the auto you think each fail- ure mode might be active.
2-11. For each of the following applications, list three of the more likely failure modes, describing why each might be ex- pected: (a) high-performance automotive racing engine, (b) pressure vessel for commercial power plant, (c) domestic wash- ing machine, (d) rotary lawn mower, (e) manure spreader, (f) 15-inch oscillating fan.
2-12. In a tension test of a steel specimen having a 6-mm-by- 25-mm rectangular net cross section, a gage length of 20 cm was used. Test data include the following observations: (1) load at the onset of yielding was 37.8 kN, (2) ultimate load was 65.4 kN, (3) rupture load was 52 kN, (4) total deformation in the gage length at 18 kN load was 112 . Determine the following:
a. Nominal yield strength b. Nominal ultimate strength c. Modulus of elasticity
mm
Problems
2-13. A tension test on a 0.505-inch diameter specimen of cir- cular cross section was performed, and the following data were recorded for the test:
a. Plot an engineering stress-strain curve for this material.
b. Determine the nominal yield strength.
c. Determine the nominal ultimate strength.
d. Determine the approximate modulus of elasticity.
e. Using the available data and the stress-strain curve, make your best guess as to what type of material the spec- imen was manufactured from.
f. Estimate the axially applied tensile load that would cor- respond to yielding of a 2-inch diameter bar of the same material.
g. Estimate the axially applied tensile load that would be required to produce ductile rupture of the 2-inch bar.
h. Estimate the axial spring rate of the 2-inch bar if it is 2 feet long.
2-14. An axially loaded straight bar of circular cross section will fail to perform its design function if the applied static axial load produces permanent changes in length after the load is removed. The bar is 12.5 mm in diameter, has a length of 180 cm, and is made of Inconel 601.38The axial load required for this application is 25 kN. The operating environment is room-temperature air.
a. What is the probable governing failure mode?
b. Would you predict that failure does take place? Explain your logic.
2-15. A 1.25-inch diameter round bar of material was found in the stock room, but it was not clear whether the material was alu- minum, magnesium, or titanium. When a 10-inch length of this bar was tensile-tested in the laboratory, the force-deflection curve obtained was as shown in Figure P2.15. It is being proposed that a vertical deflection-critical tensile support rod made of
this material, having a 1.128-inch diameter and 7-foot length, be used to support a static axial tensile load of 8000 pounds. A total deflection of no more than 0.040 inch can be tolerated.
a. Using your best engineering judgment, and recording your supporting calculations, what type of material do you believe this to be?
b. Would you approve the use of this material for the pro- posed application? Clearly show your analysis supporting your answer.
2.16. A 304 stainless-steel alloy, annealed, is to be used in a de- flection-critical application to make the support rod for a test package that must be suspended near the bottom of a deep cylindrical cavity. The solid support rod is to have a diameter of 20 mm and a precisely machined length of 5 m. It is to be ver- tically oriented and fixed at the top. The 30 kN test package is to be attached at the bottom, placing the vertical rod in axial tension. During the test, the rod will experience a temperature increase of 80C. If the total deflection at the end of the rod must be limited to a maximum of 8 mm, would you approve the design?
2.17. A cylindrical 2024-T3 aluminum bar with a diameter of 25 mm and length of 250 mm is vertically oriented with a static- axial load of 100 kN attached at the bottom.
a. Neglect stress concentrations and determine the maxi- mum normal stress in the bar and identify where it occurs.
b. Determine the elongation of the bar.
c. Assume the temperature of the bar is nominally 20°C when the axial load is applied. Determine the temperature change that would be required to bring the bar back to its original 250 mm length.
2-18. A portion of a tracking radar unit to be used in an an- timissile missile defense system is sketched in Figure P2.18.
TABLE P2.13 Tension Test Data
Load, lb Elongation,in
1000 0.0003
2000 0.0007
3000 0.0009
4000 0.0012
5000 0.0014
6000 0.002
7000 0.004
8000 0.085
9000 0.150
10,000 0.250
11,000 (maximum load) 0.520
38See Chapter 3 for material properties data.
Force,F, lb
Deflection,, in.
0.01 0.02 0.03 0.04
0 2000 4000 6000 8000 10,000 12,000 14,000 16,000 18,000
Figure P2.15
Force-deflection curve for unknown material.
Problems 87
The radar dish that receives the signals is labeled D and is at- tached by frame members A, B, C, and E to the tracking struc- ture S. Tracking structure Smay be moved angularly in two planes of motion (azimuthal and elevational) so that the dish D can be aimed at an intruder missile and locked on the target to follow its trajectory.
Due to the presence of electronic equipment inside the box formed by frame members A, B, C, and E, the approximate temperature of member E may sometimes reach while the temperature of member B is about . At other times, members B and E will be about the same temperature. If the temperature difference between members B and E is , and joint resistance to bending is negligible, by how many feet would the line of sight of the radar tracking unit miss the in- truder missile if it is 40,000 feet away, and
a. the members are made of steel?
b. the members are made of aluminum?
c. the members are made of magnesium?
2-19. Referring to Figure P2.19, it is absolutely essential that the assembly slab be precisely level before use. At room tem- perature, the free unloaded length of the aluminum support bar is 80 inches, the free unloaded length of the nickel-steel support bar is 40 inches, and a line through A-B is absolutely level be- fore attaching slab W. If slab W is then attached, and the tem- perature of the entire system is slowly and uniformly increased to above room temperature, determine the magnitude and direction of the vertical adjustment of support “C” that would be required to return slab surface A-B to a level position.
(For material properties, see Chapter 3.) 150°F
50°F 150°F
200°F
2-20. Referring to the pinned mechanism with a lateral spring at point B, shown in Figure 2.5, do the following:
a. Repeat the derivation leading to (2-23), using the con- cepts of upsetting momentandresisting moment, to find an expression for critical load.
b. Use an energy method to again find an expression for critical load in the mechanism of Figure 2.5, by equating change in the potential energyof vertical force to strain energy storedin the spring. (Hint:Use the first two terms of the series expansion for cos to approximate cos .) c. Compare results of part (a) with results of part (b).
2-21. Verify the value of effective length 2Lfor a column fixed at one end and free at the other [see Figure 2.7(b)] by writ- ing and solving the proper differential equation for this case, then comparing the result with text equation (2-35).
2-22. A solid cylindrical steel bar is 50 mm in diameter and 4 m long. If both ends are pinned, estimate the axial load required to cause the bar to buckle.
2-23. If the same amount of material used in the steel bar of problem 2-22 had been formed into a hollow cylindrical bar of the same length and supported at the ends in the same way, what would the critical buckling load be if the tube wall thickness were (a) 6 mm, (b) 3 mm, and (c) 1.5 mm? What conclusion do you draw from these results?
2-24. If the solid cylindrical bar of problem 2-22 were fixedat both ends, estimate the axial load required to cause the bar to buckle.
2-25. A steel pipe 4 inches in outside diameter, and having 0.226-inch wall thickness, is used to support a tank of water weighing 10,000 pounds when full. The pipe is set vertically in a heavy, rigid concrete base, as shown in Figure P2.25. The pipe material is AISI 1060 cold-drawn steel with 90,000 psi and 70,000 psi. A safety factor of 2 on loadis desired.
a. Derive a design equation for the maximum safe height Habove the ground level that should be used for this ap- plication. (Use the approximation I L pD3t>8.2
Syp
Su Le
a a
Pa 20 in.
S
Line of sight
Azimuth control bearing Elevation
control bearing
= elevation angle = azimuth angle
D B
C E
A
15 in.
(Ambient = 70°F) (Adjustment)
A B
C
40 in.
(unloaded) 80 in.
(unloaded)
Nickel steel 0.500 in. D.
round Aluminum
0.625 in. D.
round
Assembly slab W = 3000 lb
48 in.
12 in.
12 in.
Figure P2.18
Sketch of radar tracking unit.
Figure P2.19
Assembly slab configuration.
b. Compute a numerical value for .
c. Would compressive yielding be a problem in this de- sign? Justify your answer.
2-26. Instead of using a steel pipe for supporting the tank of problem 2-25, it is being proposed to use a W625 wide- flange beam for the support, and a plastic line to carry the wa- ter. (See Appendix Table A.3 for beam properties.) Compute the maximum safe height above ground level that this beam could support and compare the result with the height
145 inches, as determined in problem 2-25(b).
2-27. A steel pipe is to be used to support a water tank using a configuration similar to the one shown in Figure P2.25. It is be- ing proposed that the height Hbe chosen so that failure of the supporting pipe by yieldingand by bucklingwould be equally likely. Derive an equation for calculating the height, , that would satisfy the suggested proposal.
Heq
1Hmax2pipe
1Hmax2beam
1Hmax2pipe 2-28. A steel pipe made of AISI 1020 cold-drawn material (see Table 3.3) is to have an outside diameter of D15 cm, and is to support a tank of liquid fertilizer weighing 31 kN when full, at a height of 11 meters above ground level, as shown in Figure P 2.28. The pipe is set vertically in a heavy, rigid concrete base.
A safety factor of n2.5on loadis desired.
a. Using the approximation that , derive a design equation, using symbols only, for the minimum pipe wall thickness that should be used for this application.
Write the equation explicitly for tas a function of H,W,n, andD, defining all symbols used.
b. Compute a numerical value for thickness t.
c. Would compressive yielding be a problem in this de- sign? Justify your answer.
2-29. A connecting link for the cutter head of a rotary mining machine is shown in Figure P2.29. The material is to be AISI
I L 1pD3t2>8 Tank
10,000 lb
D= 4.0 in.
t= 0.226 in.
H
Figure P2.25
Water tank supported by a steel pipe.
Figure P2.28
Liquid fertilizer tank and support.
D= 15 cm
t H= 11 m
W= 31 kN
Section A-A 0.500 in.
1.00"
0.500 in.
Very low friction between pins and connecting link 1.00 in.
Very-high-precision fits will permit no deflection or angular displacement of
's in this plane.
a a
c
c A
A
CL
CL CL 20.0 in.
Figure P2.29
Connecting link for rotary mining machine. (Not to scale.)
Problems 89
1020 steel, annealed. The maximum axial load that will be ap- plied in service is 10,000 pounds (compression) along the centerline, as indicated in Figure P2.29. If a safety factor of at least 1.8 is desired, determine whether the link would be ac- ceptable as shown.
2-30. A steel wire of 2.5-mm diameter is subjected to torsion.
The material has a tensile strength of Syp690 MPa and the wire is 3 m long. Determine the torque at which it will fail and identify the failure mode.
2-31. A sheet-steel cantilevered bracket of rectangular cross section 0.125 inch by 4.0 inches is fixed at one end with the 4.0- inch dimension vertical. The bracket, which is 14 inches long, must support a vertical load, P, at the free end.
a. What is the maximum load that should be placed on the bracket if a safety factor of 2 is desired? The steel has a yield strength of 45,000 psi.
b. Identify the governing failure mode.
2-32. A hollow tube is to be subjected to torsion. Derive an equation that gives the length of this tube for which failure is equally likely by yielding or by elastic instability.
2-33. A steel cantilever beam 1.5 m long with a rectangular cross section 25 mm wide by 75 mm deep is made of steel that has a yield strength of Syp276 MPa. Neglecting the weight of the beam, from what height, h, would a 60 N weight have to be dropped on the free end of the beam to produce yielding.
Neglect stress concentrations.
2-34. A utility cart used to transport hardware from a warehouse to a loading dock travels along smooth, level rails. At the end of the line the cart runs into a cylindrical steel bumper bar of 3.0- inch diameter and 10-inch length, as shown in Figure P 2.34.
Assuming a perfectly “square” contact, frictionless wheels, and negligibly small bar mass, do the following:
a. Use the energy method to derive an expression for max- imum stress in the bar.
b. Calculate the numerical value of the compressive stress induced in the bar if the weight of the loaded cart is 1100 lb and it strikes the bumper bar at a velocity of 5 miles per hour.
Pmax
reduction in stress level that would be experienced by the beam of Example 2.7 if it were supported by a spring with k390 lb\in at each of the simple supports, instead of being rigidly supported.
2-36. A tow truck weighing 22 kN is equipped with a 25-mm nominal diameter tow rope that has a metallic cross-sectional area of 260 mm2, an elastic modulus of 83 GPa, and an ultimate strength of Su1380 MPa. A 7-m long tow rope is attached to a wrecked vehicle and the driver tries to jerk the wrecked vehi- cle out of a ditch. If the tow truck is traveling at 8 km/hr when the slack in the rope is taken up, and the wrecked vehicle does not move, would you expect the rope to break?
2-37. An automobile that weighs 14.3 kN is traveling toward a large tree in such a way that the bumper contacts the tree at the bumper’s midspan between supports that are 1.25 m apart.
If the bumper is made of steel with a rectangular cross section 1.3 cm thick by 13.0 cm deep, and it may be regarded as sim- ply supported, how fast would the automobile need to be trav- eling to just reach the 1725 MPa yield strength of the bumper material?
2-38. a. If there is zero clearance between the bearing and the journal (at point B in Figure P2.38), find the maximum stress in the steel connecting rod A–B, due to impact, when the 200-psi pressure is suddenly applied.
b. Find the stress in the same connecting rod due to im- pact if the bearing at Bhas a 0.005-inch clearance space between bearing and journal and the 200-psi pressure is suddenly applied. Compare the results with part (a) and draw conclusions.
c.m.
v= 5 mph W= 1100 lb
d= 3.0"
A= 7.07 in.2 E= 30 106 psi Syp= 130,000 psi L= 10 in.
Figure P2.34
Utility cart and bumper bar.
Piston diameter = 3.0 in.
p = 200 psi
A = 1 in.2 20°
A
B
C
7"
Figure P2.38
Sketch of a connecting rod in an internal combustion engine.
2-35. If the impact factor, the bracketed expression in (2-57) and (2-58), is generalized, it may be deduced that for any elastic structure the impact factor is given by [1
]. Using this concept, estimate the 21 + 12h>ymax-static2
2-39. Carefully define the terms creep, creep rupture, and stress rupture, citing the similarities that relate these three fail- ure modes and the differences that distinguish them from one another.