Fatigue verification of high loaded bolts of a rocket combustion chamber.

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1 Fatigue veriication o high loaded bolts o a rocket combustion chamber. Marcus Lehmann 1 & Dieter Hummel 1 1 Airbus Deence and Space, Munich Zusammenassung Rocket engines withstand intense thermal and structural loads. Either do the bolted interaces between the engine components. Thereore particular emphasis is placed on quality assurance and veriication by incoming inspection or asteners. As part o the incoming inspection a atigue test is perormed to ensure high bolt durability covering the dynamic loads during engine operation. The diiculty is that a signiicant dierence between test loads and light loads exists which lead to a nonlinear relation between test results and expected operational lie. To link both load cases a multi parametrical model is chosen and need to be adapted to both cases which is realized by a sensitivity analysis. While the parameters are scattered about the uniying parameter set also the lie expectation varies or both load cases. Thereore a robustness analysis is inally perormed to project the result variety under light conditions onto the test result scattering. Keywords: Bolt analysis, Robustness veriication Kontakt: Airbus DS, München, Space Propulsion, Structure Mechanics & Design, marcus.lehmann@airbus.com 1 Introduction Bolts or screws connect constructional parts with each other. The threaded bolt shat and its evenly shaped counterpart, as nut or threaded blind hole, transmit orces by a shapeclosed connection. In case o overloading the bolt will ail and thereby loose its orce transmitting capability. A bolt can be overloaded by overstressing which will lead to ductile ailure, preerably at the irst thread in contact. Another overloading mechanism is known as atigue that becomes critical ater a certain number o load cycles. The second is outlined in this paper. 2 Basics on bolt analysis 2.1 Pretension During bolt mounting into a blind hole or a nut, pretension is to be generated. Continuous torque tightening increases the bolt orce and the lange orce in the same rate while the value o deormation depends on the stiness o both components. Due to the tension load the bolt is strained by the law o elasticity l B = F B /K B with the bolt orce F B and the bolt s rigidity K B that leads to the absolute bolt deormation l B. With the same orce F B but with a dierent lange stiness K F the lange parts are compressed about 11. Weimarer Optimierungs- und Stochastiktage November

2 l F = F B /K F. Here l F denotes the deormation o the lange area in an imaginary cylinder between the bolt head and the nut. K F is the corresponding lange stiness. The mounted and pre-stressed interace is loaded by the operational orce F L. I F L is oriented in tension direction the bolt will be additionally stressed while the lange compression decreases. Hence the operational load is taken by both components in dependence rom their stiness. The ratio between the orce raction taken by lange decompression F LF and the part covered by the bolt F LB is deined by the orce ratio Φ: Φ = K B K B + K F = F LB F L. (1) The bolt and lange behaviour due to pretension and operational load are illustrated in igure 1. stresses the bolt. This eect can be seen in the relations o equ. (1). Due to this advantageous behaviour the bolts stress range per cycle is decreased which crucially increases bolt lie. 2.2 Stress distribution Loaded by an axial orce F ax the nominal stress σ nom within the bolt shat equals to: σ nom = F ax A st, (2) with A st as stress area. Notch eects at the thread ground lead to a local stress concentration σ max = K σ nom. The stress concentration actor K depends beside others on the depth o the thread and the radius o the thread ground. To estimate the magnitude o K tables are presented in engineering literature, e.g. Young and Budynas [2002]. As result o the stress concentration at the thread grounds a stress distribution equivalent to igure 2 occurs. Figure 1: Load-deormation-curve o a classical bolt connection. The eect o the orce ratio becomes substantial or dynamic loading domains. The high durability o bolted joints is to be attributed to Φ and the act that an operational load is partly taken by the relie o the prestressed langes. The higher the lange stiness is compared to the stiness o the bolt shat, the lower the actual impact o operational loads Figure 2: Stress distribution along threaded bolt. Stress concentration at thread grounds. When the locally increased stress reaches the yield limit σ y, local plastic deormations will occur. For the present study the Neuber rule is used to approximate the magnitude o plastic deormation the Neuber rule is used. Neuber expects a hyperbola in the stressstrain-ield where the product o stress and 11. Weimarer Optimierungs- und Stochastiktage November

3 strain stays constant σε = σ 2 max/e. When the Neuber hyperbola its the endpoint o the linear extrapolated stress-strain line σ max,el, it will cross the yield curve at the point σ max,neuber. This point approximates the stress-strain relation ater yielding. See igure 3 or visualisation. As yield curve a bilinear approximation is used. It is deined by the yield limit (R p0.2 /E, R p0.2 ) and at ultimate conditions (A, R M ). are chosen in accordance to the bolt material. A better adjustable orm o (3) is used with the parameters C 1 to C 4 that can be it to actual material behaviour: ε = (N ) = C 1 R m σ m E N C 2 + D C 3 u N C 4. (4) Aligned values or C 1 to C 4 can be ound or dierent materials in Lemaitre and Chaboche [1990]. Varying the constants C 1 to C 4 o (4) inluence the ε N Curve as shown in igure 4. The actual sensitivity o the model towards these Coin Manson parameters is analysed in section 4. Figure 3: Plastic stress-strain state obtained by Neuber approximation. 2.3 Fatigue damage The bolt lie prediction is realized by the Coin Manson approach. With the universal slope proposed by Lemaitre and Chaboche [1990]: ε univ. = (N ) = 3.5 R m σ m E N Du 0.6 N 0.6 (3) the total strain range ε is related to the number o cycles until ailure N. R m being the ultimate strength, D u the ductility o the material and σ m the mean stress o the load cycle. Herein the values in the exponents are itted to a wide range o dierent materials or universal validity. To reach our needs these constants are considered as material speciic and Figure 4: Coin Mansons atigue curve. Variety o the constants C 2 and C 4 are shown by the diuse bluish areas. 11. Weimarer Optimierungs- und Stochastiktage November

4 3 Bolt validation procedure and uncertainties To accept the bolts or light application a ew bolts per batch are submitted to several dierent test procedures. To check towards atigue ailure a cycling test is perormed. It is shown that the load conditions during the test dier to those experienced during rocket launch. The objective o this investigation is to correlate the results o the atigue test into the circumstances during light. Finally it is to show that required cycles during light can be validated by a certain number o test cycles. 3.1 Validation test conditions For atigue testing the bolt is inserted to the testing device with contact at thread and bolt head. No lange material is considered. Loads applied through the device are ully covered by the bolt itsel. The ull range o alternating testing loads is applied to the bolt. The diagram in igure 5a displays the load-deormation curve o this behaviour. The large load range o 2F a combined with the stress concentration actor at thread ground leads to local cyclic plastiication as earlier shown in igure 2. According to the Neuber approximation this opens the stressstrain hysteresis, stretches the stress range and reduces the bolt lie signiicantly. 3.2 Flight conditions The considered bolts connect the combustion chamber to the injector. During mounting a high pretension F p is applied to avoid interace sliding. The dynamic interace loads F L occur in a moderate level which leads to relatively low alternating bolt orce F a compared to the pretension orce F p. The ratio can be seen in the load-deormation curve in igure 5b. With high lange stiness, which is given in this case, the dynamic loads on-top to pretension are Figure 5: Load-deormation-curve under a) test conditions and b) light conditions. mostly covered by lange relie. The actual bolt load F a alternates in a much smaller stress range compared to the test case. That leads to a solely elastic dynamic behaviour with a much smaller strain range. According to Coin Manson and as seen in igure 4, a small εlight results in signiicant longer bolt lie than under test conditions 3.3 Correlation o test results to light conditions To compare test result with light live expectations the above mentioned inluences need to be considered. Slight uncertainties o yield 11. Weimarer Optimierungs- und Stochastiktage November

5 stress R p0,2 and strain at rupture A lead to contrary changes in the calculation o the strain range via the Neuber Approximation approach and spreads the resulting live expectation. Additional uncertainties occur by varying the Coin Manson coeicients C 1 to C 4. Also the stress concentration actor o the threat K is not a deinite value but depends on geometrical width ratio and the edge radius which is not deinitely detectable. It is treated as variable during ollowing investigations. In the ollowing lie expectations will be calculated by considering a certain set o the mentioned variables. Each parameter set gives two results: N,test lie expectation under test conditions and N,light lie expectation during light. Finally it is to show that the light requirement is reached in any case. Which means or the model in any possible combination o input variables. Parameter combinations that lead to lower lie expectation need to be excluded by the choice o test conditions. Table 1: Model sensitivity under test and light conditions towards input parameters. Parameter unit Test Flight A m [%] 0 1 E [ MP a / MP a ] 0 0 K [%] 1 4 R p0.2 [ MP a / MP a ] 9 36 R m [ MP a / MP a ] C 1 [%] 47 3 C 2 [%] 17 5 C 3 [%] 1 0 C 4 [%] 6 0 parameters in this case. Table 1 lists the sensitivities or both cases. Considering the inluencing parameters as normal distributed, the model is ed with speciic parameter sets. The distribution o the parameters is evaluated rom test data, or in the case o the Coin Manson variables rom literature. For each input set an output o the two lie expectations is obtained - one or test and one or light. 4 Robustness under light conditions The entirety o variables that are inluencing the bolt s lie expectation is analysed. Ater perorming a sensitivity analysis with optis- Lang it occurred that the inluence o the parameters varies rom light to test case. The test case shows the highest sensitivity to the Coin Manson variables C 1 and C 2 that are responsible or the high cycle domain o the slope. For the light case these variables have a minor impact. Despite to its small strain range ε light the sensitivity is mostly strength driven. R p0.2 and R m are the most inluencing Figure 6: Criteria or atigue test. With the optislang sotware a robustness analysis was executed. As result o the lie evaluation the plot in igure 6 is drawn. It shows the lie results o 5000 parameter sets. For each set the expected light lie is prognosed or an calculated test lie. As requirement or light acceptance the bolts have to withstand 11. Weimarer Optimierungs- und Stochastiktage November

6 the speciied loads, even with the worst possible combination o material parameters. To speak in colours regarding igure 6 all points below the light requirements must not be accepted. That means all test results in the red area will lead to bolt rejection rom light worthiness. I one o the ew tested bolts per incoming batch shows an unacceptable atigue durability the whole batch will not be mounted. The actual test requirement is inally deined at a higher number o cycles during test to meet an additional saety actor. Test results in the orange area can achieve acceptance level by perorming additional analysis. The big plus o the indierent orange area is the early recognition o any disadvantegeous changes o production methods. I processes change the inal product may be aected in a negative way. With the demanding test requirement changes can early be detected and counteractions can be prepared. The bolts that meet the test requirement, illustrated by the green area in the plot, are accepted or light without urther analysis. Literature Jean Lemaitre and Jean-Louis Chaboche. Mechanics o solid materials. Cambridge university press, Warren Clarence Young and Richard Gordon Budynas. Roark s ormulas or stress and strain, volume 7. McGraw-Hill New York, The acceptance regarding bolt lie could inally be veriied. With the possibility o taking all parameters into account within a single analysis, the understanding o its sensitivities was improved. Having the bandwidth o each parameter in mind the spread o the bolt lie expectation was shown. In the anthill plot this lie expectation was projected on the durability under testing conditions. With the relations between light and test a new test criteria was ound that disqualiies unacceptable bolts beore they go to light. 11. Weimarer Optimierungs- und Stochastiktage November

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