First, we note that the stress range of 90 ksi exceeds the allowable stress for carbon steel f(1.25 Sh+0.25 SC)=30 ksi. Because of the design safety factors, and because Sa is a fatigue limit based on 7000 cycles of heat-up and cool-down, the stress range of 90 ksi does not mean that the pipe is in imminent danger of rupture.
Second, we note that the tee has plastically deformed since the stress of 90 ksi exceeds the material’s yield stress of 30 ksi. Here again, displacement limited plastic deformation does not mean that the pipe is in imminent danger of rupture. Neither does plastic deformation necessarily mean that when the line is cooled down it will have a permanent deformation. In the case of Figure 7-9, when the line will cool down the contraction of leg CD will re-straighten the vertical leg AB, even though the metal has been plastically deformed at the tee.
Third, we apply Markl’s fatigue failure relationship iSampl=245,000/N0.2 and note that, if supports at A and B are not removed, the tee would fail by through-wall fatigue crack after a number of heat-ups and cool-downs equal to N =(245,000/45,000)5=4780 cycles.
In light of these observations, the prudent thing to do, unless regulatory requirements dictate otherwise, is to
(a) Cut out supports A and B to bring the line to its intended design configuration. If the supports at A and B are cut while in hot service, the vertical leg will jump to the left.
The line was therefore shutdown and cooled down before cutting the supports.
(b) Remove the insulation and verify, visually and by touch, that the pipe has not buckled or wrinkled around the tee
(c) Inspect the tee’s base metal and three welds by a nondestructive technique (Chapter 16) to verify that the plastic deformation has not caused the material to crack by opening small pre-existing flaws.
7.9 FRACTURE MECHANICS APPROACH
In fracture mechanics, the growth of fatigue cracks can be represented by a crack growth curve that has three regions (regions I, II and III in Figure 7-10). In region I there is a threshold stress intensity factor, below which the fatigue crack does not form or propagate (equivalent to the endurance limit). The threshold stress intensity at a notch depends on the shape of the notch and can be approximated by [Barsom]
∆Kth=10 (r SY)0.2
∆Kth=threshold range of stress intensity factor, region I, ksi(in)0.5 r=notch radius, in
SY=yield stress, ksi
The threshold range is between 5 and 15 ksi(in)0.5 for steel and 3 to 6 ksi(in)0.5 for aluminum [Bannantine]. In region II, which corresponds to the propagation of a crack of depth “a” as a function of applied fatigue cycles “N”, the relationship between the crack growth rate da/dN and the range of applied stress intensity is
da/dN=crack growth per cycle, in/cycle m=material and environmental exponent C=material and environmental coefficient
∆K=range of applied stress intensity factor, ksi(in)0.5 (Chapter 21)
Figure 7-10 Fatigue Crack Growth Curve log(da/dN) vs. log(∆K)
The coefficient C depend on the material and the environment, for example in a non-corrosive environment [Fuchs], (C, m)=(3.0 10−10, 1) for austenitic stainless steel.
This equation can be modified to capture the shape of the fatigue crack growth curve in regions I and III, leading to
Kmax=maximum value of the stress intensity, ksi(in)0.5 KC=critical stress intensity, fracture toughness, ksi(in)0.5
In region III the crack has progressed to the point where the stress intensity K reaches a critical value KC, resulting in fracture of the remaining ligament in the cracked cross section (Chapter 21).
7.10 CORROSION FATIGUE
If the formation and propagation of a fatigue crack takes place in the presence of a corrosive fluid, then the existing crack which has been exposed to the corrosive fluid is corroded while the plane of metal just exposed during the last stress cycle is bare. The corroded region, with its passive oxide film, acts as the cathodic (or more noble) pole, while the recently exposed bare steel is anodic and corrodes (Chapter 20). The rate at which the fatigue crack progresses in this case is different than the growth rate in the absence of corrosion. Corrosion therefore influences the shape of the corrosion fatigue curve, da/dN vs. ∆K. Corrosion fatigue curves have been developed for a large number of metals and environments [ASM, Battelle].
7.11 SHAKEDOWN
Fatigue cracks tend to originate at sharp structural discontinuities or at existing crack flaws, where the local peak stress is large. This concentrated stress is often well above the material yield stress, creating a local plastic zone in the component. Fatigue design analysis rules generally require as a prerequisite that the material shakes-down to elastic behavior. What is meant is that for elastically based fatigue design rules to apply, the plastic zone must be small, surrounded by elastic material, so that plastic deformation remains confined; and as the cyclic deformation takes place, the strain will not continue to increase, it will not ratchet. In practice it is difficult to prove that shakedown occurs and simplified formulas have been introduced in the ASME Code to check for this condition of confined plasticity.
7.12 COLD SPRING
In principle, one way to reduce pipe expansion stresses and reaction loads on equipment nozzles in high temperature service is to cut short the pipe during construction, and pull or push it into alignment with the nozzle during erection, Figure 7-11. The pipe is said to be cold sprung. Then as the system heats up, the pipe will expand and relieve the cold spring imposed during construction, resulting in low stresses or reaction loads in service.
Ideally, if the pipe is cut short and cold sprung by the exact amount of hot expansion, then the reaction load would be zero in hot service. For example, in Figure 7-11, if the pipe will freely expand upwards an amount ∆L, it is cut short by ∆L during construction, forced into alignment with the nozzle by upward pull during erection, and bolted. Prior to service, while the line is still at ambient temperature, there is an initial downward force applied by the pipe to the nozzle.