CHAPTER 8 RELIABILITY BASED DESIGN OF A TYPICAL ROCKET MOTOR CASE CONTAINING SUFACE CRACK

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Nigel Fletcher
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1 144 CHAPTER 8 RELIABILITY BASED DESIGN OF A TYPICAL ROCKET MOTOR CASE CONTAINING SUFACE CRACK The Reliability-Based Design approach adopts a probability based design framework to ensure high reliability and safety. Reliability-Based Design (RBD) has been a valuable evolution of deterministic design. This is a more rational approach that quantifies the reliability or risk of failure in probabilistic terms and includes these terms directly in design. This chapter studies about the RBD of a high strength Maraging steel rocket motor case subjected to internal pressure using the proposed probabilistic failure assessment methodology. The MTPFC is used to predict the failure of the rocket motor case. Generally, launch vehicle is propelled by forces termed thrusts, which provide the desired component of acceleration. Solid propellant rockets are examples of a pure reaction system in which the propulsive forces are produced by the ejection of mass (propellants) initially contained in the system. Self-contained systems of this type are called rocket motors and can be operated in space as well as in atmosphere, since they do not require an external propulsive fluid. A solid propellant rocket is the simplest form of chemical propulsion. The fuel and oxidizer are both incorporated in a single solid called the propellant grain, located inside a container called the combustion chamber. A schematic illustration of this type of motor is shown in Figure 8.1. Many metallic materials, such as steels, are useful for motor

145 cases because they tend to have a high modulus of elasticity and high yield strength. Expansion of the propellant grain is a less severe problem with metallic cases.

2 145 cases because they tend to have a high modulus of elasticity and high yield strength. Expansion of the propellant grain is a less severe problem with metallic cases. The principal parts of a solid propellant rocket motor are the grain, the casing, the insulation, the nozzle and the igniter. The rocket motor case shown in Figure 8.1 is considered for this study. M A Chamber I Nozzle throat insert B Head end dome J Lining C Nozzle K Insulation D Igniter L Propellant E Nozzle convergent portion M Nozzle exit plane F Nozzle divergent portion N SITVC system G Port O Segment joint H Inhibitor Figure 8.1 Cross-Section of a Typical Rocket Motor Case 8.1 DETERMINISTIC DESIGN The design of maraging steel rocket motor case using traditional deterministic design procedure is presented. Cylindrical portion of the rocket motor case experiences high stresses under internal pressure and hence it governs the design. The specifications of the solid rocket motor case considered are given below.

3 146 Material: M250 grade maraging steel Ultimate tensile strength, ult= 1744 MPa (Weld) 0.2% proof stress, ys= 1701 MPa (Weld) % elongation = 8 Youngs modulus, E =186.4 GPa Poisson s ratio, 0. 3 Maximum expected operating pressure (MEOP) = 7.2 MPa Proof pressure = 8.1 MPa Design pressure = 9 MPa Diameter of the cylindrical shell, D i (= 2R i ) = 3 m Minimum detectable surface crack size = length (2c) x depth (a) = 3 mm x 1 mm For the given design specifications, several steps are involved in the thickness evaluation of the cylindrical shell portion of the rocket motor case. Initially, the thickness will be estimated from the burst pressure formula for the unflawed cylindrical vessel. For the minimum detectable flaw size, an optimum thickness will be arrived for the design pressure through fracture analysis by suitably modifying the estimated thickness of the unflawed cylindrical vessel Design of Rocket Motor Case without Crack The thickness of the rocket motor case is computed deterministically as mm for the design pressure 9 MPa. Faupel s burst pressure prediction Equation (8.1) is used to assess the failure with the assumption that the rocket motor case does not contain flaw. 2 ys t R i exp pm ys ult 1 (8.1)

4 exp mm Design of Rocket Motor Case with crack It is a well-known fact that the load carrying capacity of the motor case reduces in the presence of crack like defects. Damage tolerant and failsafe approaches have been employed in the design of rocket motor cases. The residual strength of a component is assessed with an assumed pre-existing crack. In this study, high strength rocket motor case with an assumed preexisting crack of size 3 mm x 1 mm is considered. The thickness computed in Section may not be adequate for the minimum detectable crack size of 3 mm x 1 mm and the specified design pressure of 9 MPa. Assumption of the minimum detectable crack size in the design does not mean that such a crack actually does exist in each pressure vessel. It means that a crack of this size could exist in any realized pressure vessel without knowledge of its existence. The fracture mechanics approach assumes the existence of the minimum detectable crack size in each pressure vessel and determines the service stress/environment conditions that will not permit the crack to grow to failure during the service life of each particular mission. The modified two parameter fracture criterion described in Chapter 5 is used here for the evaluation of failure pressure (p bf ) of rocket motor case containing an axial surface crack.

5 148 q f Kmax K F 1 m 1 m f (8.2) u u The three fracture parameters of M250 grade maraging steel in the Equation (8.2) are:k F =235.7 MPa m; m 0.6 ; and q = For the minimum detectable crack size of 3 mm x 1 mm and the specified design pressure of 9 MPa, the shell wall thickness using deterministic fracture mechanics approach is found as 8.1 mm and is reported in Table 8.1. Similarly, for different crack sizes thickness values are computed and are presented in Table 8.1. Table 8.1 Failure Pressure of Maraging Steel Rocket Motor Case Containing an Axial Surface Crack Shell wall Failure pressure, p bf (MPa) thickness Surface cracks (2c mm x a mm) t (mm) No-flaw 3 x 1 5 x 2 7 x RELIABILITY BASED DESIGN In this section, reliability-based design of rocket motor case using probabilistic failure assessment methodology explained in Chapter 3 is elaborated and analysed. Table 8.2 shows the statistical parameters of random input variables. The failure pressure (P bf ) of rocket motor case is predicted

6 149 for a particular thickness value by performing probabilistic failure analysis and are summarized in Table 8.3. Monte-Carlo simulation method is used to perform probabilistic fracture analysis using modified two parameter fracture criterion as per the procedure explained in Chapter 3. Table 8.2 Statistical Properties of Random input Variables for Rocket motor case Statistical properties Material Diameter (D) mm Crack depth (a) mm Crack length (2c) mm Ultimate strength ult) MPa Yield strength ys) MPa Fracture parameter (K F ) Mean (µ) M250 maraging steel rocket motor cases 2% 10% 10% 7% 7% 14% COV Probability distribution Log Log Reference (Sang-Min Lee et al 2006) Obtained Table 8.3 shows the statistical parameters of predicted failure pressure. The COV of the operating pressure is assumed as 10% based on the reference (Lin et al 2004) for reliability computation. The reliability of the motor case is calculated using Equation (3.23) by considering predicted failure pressure using proposed methodology as strength (S) variable and given design pressure as stress (L) variable and is presented in Table 8.3. It is observed from Table 8.3 that thickness of the motor case is obtained as 11.5 against high reliability of 99.99% for the specified crack size of crack length 2c =3mm and depth (a) = 1mm.

7 150 Table 8.3 Reliability against Thickness for M250 Maraging Steel Rocket Motor Case (2c =3mm and a = 1mm) Thickness (t) mm Predicted failure pressure, p bf (MPa) Design pressure, p (MPa) mean Standard mean Standard Reliability (%) From Table 8.3 it is observed that thickness obtained by deterministic approach when subjected to uncertainty yields a reliability of 79.61% which is unsafe for aerospace structures like motor cases. In RBD the target reliability is 99.99%. Table 8.4 shows the comparison of thickness obtained through different approaches for M250 grade Maraging steel rocket motor case for the specific defect of length 2c=3 mm and depth of a=1 mm. Table 8.4 Comparison of Thickness Obtained Through Different Approaches (2c =3mm and a = 1mm) Design pressure MPa Unflawed Recommended thickness (t) mm Flawed deterministic Flawed Probabilistic at 99.99% reliability The same procedure is repeated for various defect sizes (say crack length and depth) and it is shown in Table 8.5. From the Table shown below it is clear observed that when defect size increases, thickness value also

8 151 increases for the specified reliability of 99.99% for various operating/design pressures. Table 8.5 Recommended Thickness for Various Crack Dimensions and Design Pressure for the Specified Reliability R= 99.99% Design/operating pressure value MPa 9.0 Recommended thickness (t), mm a 1mm 2mm 3mm 2c 3mm mm mm mm mm mm mm mm mm RESULTS AND DISCUSSION The Reliability-based Design of a high strength Maraging steel rocket motor case containing surface crack is considered for this study. The following observations are made from the study. (i) In this study, reliability-based design of rocket motor cases is explored. The thickness of the rocket motor cases is predicted for the specified reliability using the probabilistic failure assessment methodology. It is found from Table 8.1 that by the deterministic

9 152 design procedure, for the design pressure of 9 MPa the recommended thickness, for unflawed cylindrical pressure vessel is 6.72 mm and for flawed cylindrical pressure vessel the recommended thickness is around 8.1 mm. (ii) Table 8.3 depicts that thickness obtained through probabilistic approach is high when compared with deterministic design method. Figure 8.2 Effect of defect size on design of rocket motor case

10 153 The effect of operating pressure on thickness is presented in Table 8.5 by varying the operating pressure. The above Figure 8.2 shows that when the operating pressure increases, the required thickness to meet the target reliability of 99.99% also increases and it is found that the rate of increase in thickness increases with the increase in flaw size and depth. The effect of reliability on thickness is also studied for the design pressure of 7.2 MPa. From Table 8.6 it is observed that as reliability increases, the thickness of the motor case also increases. Table 8.6 Effect of reliability on the thickness for various flaw sizes Obtained Thickness (t) mm Reliability % 2c=3mm; a=1mm 2c=5mm; a=2mm 2c=7mm; a=3mm SUMMARY This section encompasses a reliable design of pressure vessel by incorporating uncertainties in the design. A high strength maraging steel rocket motor case containing surface crack is considered for this study. The minimum thickness is computed using deterministic design and probabilistic design approach using modified two parameter fracture criterion. It is concluded that relationship between thickness and reliability has been studied. The need of probabilistic approach in the damage tolerant design of structures such as rocket motor cases has been ensured. The effect of reliability on the thickness for various flaw sizes is also studied.

CHAPTER 9 INFLUENCE OF RESIDUAL STRESSES ON THE FAILURE PRESSURE OF CYLINDRICAL VESSELS

150 CHAPTER 9 INFLUENCE OF RESIDUAL STRESSES ON THE FAILURE PRESSURE OF CYLINDRICAL VESSELS In the design of pressure vessels, evaluation of failure pressure that a cylindrical pressure vessel can withstand