3) Aft bolted connection analysis: (See Figure 1.0)

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1 Given: Both static and dynamic (fatigue) failure criteria will be used. A mimum factor of safety =2 will be adhered to. For fatigue analysis the ASME elliptic model with Von Mises equivalent stress will be used Assumptions: Chamber pressure = 500 psi Maximum chamber pressure fluctuations = 25% of mean chamber pressure [] Thermal effects will itially be neglected because of short burn times and sulative properties of fuel gra (Temperature factor = ) Loadg is purely axial. (Size factor =) Fd: ) Pressure vessel analysis: (See Figure.0) Combustion chamber material is 606 T6 alumum OD = 2.35 Length = 8.00 THK= 0.40 Only use th wall assumption if radius to thickness ratio = 20: [2] For static analysis fd maximum tensile and shear stressed plane and out of plane. *Calculate safety factor usg given material properties For dynamic analysis use Von Mises equivalent stress with ASME elliptic Sigma alternatg = 25% sigma mean [] Fd corrected endurance limit usg correction coefficients as applicable to [2]. 2) Forward bolted connection analysis: (See Figure.0) Material # = SS 36L annealed plate (0.205 ches thick) Material #2 = 606 T6 alumum (0.875 ches thick) Assume reusable connection (75% of proof load) Use static and dynamic situations Use both bolt yieldg and jot separation failure criteria Fatigue analysis Von Mises equivalent stress, ASME elliptic, S.F. =2 Assume bolts extend from nuts 2 threads No shanks on bolts 3) Aft bolted connection analysis: (See Figure.0) Material #2 = 606 T6 alumum (0.875 ches thick) Material #2 = 606 T6 alumum (0.875 ches thick) Assume reusable connection (75% of proof load) Use static and dynamic situations Use both bolt yieldg and jot separation failure criteria Fatigue analysis Von Mises equivalent stress, ASME elliptic, S.F. =2 Assume bolts extend from nuts 2 threads No shanks on bolts 4) Flange to chamber weld analysis: (See Figure.0) Comg soon! Practicg TIG weldg and learng more about weld quality. References: [] NASA /TP [2] Mechanical Engeerg Design, Shigley & Meschke 5th ed.

2 Section : Analysis of Forward Bolted Connection. The forward bolted connection of the test motor is comprised of an annealed staless steel plate connected to an alumum plate by 8 M6 x.0 class 2.9 bolts. The values for the bolts are taken from Shigley and shown below. Proof Strength: S pb := psi Ultimate Strength: S ub := psi [2] pg. 343 Table 8-6 Yield Strength: S yb := psi Endurance Limit: S eb := psi [2] pg. 353 Table 8-2 Threaded Area: := [2] pg.327 Table 8- Length of Bolt: L b := 0.63 (6 mm) Diameter of Bolt: d b := Modulus of Elasticity: E b := psi (6mm) The nut and washer specs are given as: Thickness of Nut: H b := 0.20 (5.2 mm) [2] pg.767 Table A-28 Thickness of Washers: W t := 0.02 (0.5mm) Diameter of Washer: W d := 0.5 Washer Modulus of Elasticity: E w := psi The Material properties for the SS 36L annealed staless steel plate are taken from Matweb.com Thickness of Plate: SS t := Modulus of Elastisity: E ss := psi 2

3 The material properties for the 606 T6 alumum plate are also taken from Matweb.com. Thickness of Plate: Modulus of Elasticity: Al t := E al := psi Section 2: Static Analysis of Bolts. The maximum applied force to the groupg of bolts will occur when the maximum chamber pressure has been applied. For analysis a pressure of 625 psi will be used. This pressure acts over the the capped area shown figure FB. Figure FB The applied force is thus: F max := ( 625psi) π () Section 2.: Stiffness of the Bolt F max = To exame how the bolted connection will react when an external load is applied the stiffness of the bolt, washer, and two plates must be examed. The bolts used have no shank and thus the siffness of the bolt is simply k b := E b L b (2) [2] pg. 337 eq. 8-0 k b =

4 Section 2.2: Stiffness of the Members To determe the stiffness of the washer and two plates the frustrum method outled by Shigley will be used. With this method the diamond shaped area shown the figure below is considered. Figure FB2 For the washers and each of the plates a separate stiffness value k is determed and the total stiffenss is determed the same manner as sprgs series. The equation for the dividual frustrum is: k mem = ln 0.577π E d (.5 t D d) ( D d) (.5t D d) ( D d) (3) [2] pg. 340 eq. 8-4 Where d is the nomal diameter of the bolt, D is the shortest width of the frustrum, t is the thickness of the frustrum and E is Youngs Modulus for the material beg evaluated. The frustrum starts at D =.5d and grows at an angle of 30 degrees until it is at a maximum at a thickness equal to /2 the total grip. The grip is the total thickness of the clamped material. In this case the grip is equal to 0.55 ch (4mm). By spectg figure FB2 and followg the geometry set out above the followg values for the variables equation (3) can be determed for each of the materials. By sertg these values to equation (3) the stiffness of each of the materials and thus the stiffness of the combed members can be calculated. ) Washer: d := D := t := E w = psi k wash :=

5 2) Steel Plate (): D := t := E ss = psi k ss := ) Steel Plate (2): D := 0.5 t := 0.49 k ss2 := ) Alumum Plate: D := t := k al := E al = 0 7 psi To fd the equivalent stiffness of the members: = k mem k wash k ss k ss2 k al k wash k wash k ss k ss2 k al k wash = k mem := Section 2.3: Bolt Strength and Jot Separation. For static analysis and jot separation a constant C, called the Jot Constant must be defed. It's value can be used to determe jot separation and stress on the bolt. C := k b k b k mem (4) [2] pg.347 eq. 8-2 C =

6 Tensile Stress the Bolt: The proof strength of a bolt is the limitg factor for the allowable stress of the bolt. A load factor n, can be calculated that tells whether the bolt stress is less than the proof strength. A value of n greater than dicates that the bolt stress is less than the proof stength. n p = S p A t C P (5) [2] pg. 347 eq Here, P is the applied tensile load on one of the eight connections, and Fi is the preload of the bolt. For a reusable connection Shigley suggests a preload equal to 75% of the proof load which is defed as the product of the threaded area At, and the proof strength. However, after a complete analysis the factor of safety fatigue was below the imposed mimum of 2 and a preload of 50% was decided upon. Therefore, := 0.50 S pb (6) [2] pg. 349 eqs. 8-25,26 P := psi Puttg the above values to equation (6) a value for n may be determed. n p := S pb C P n p = The bolts are loaded below their proof strengths. Even the extreme case where the entire load is taken by one bolt we get an n value greater than unity. Jot Separation: If the ternal pressure of the motor is great enough to cause separation between the Alumum and Staless steel plates with the connection the entire load would be place on the bolts. The followg equation returns a factor of safety agast jot separation. n s := P ( C) (7) [2] pg. 348 eq n s = Aga, we see that the connection is safe agast jot separation and the external load will be shared between the bolts and the connected members. 6

7 Section 3: Fatigue Analysis of Bolts. For fatigue analysis of the forward bolted section we first must fd the mean and alternatg stress the bolt. To fd this stress we use the equation: σ b = C P A t A t (8) [2] pg. 347 eq. 8-22b The applied tensile load is to have a mean value of 500 psi and an alternatg value of 25 psi. Aga, this load will be taken up evenly by all eight bolts, 500 P m := 8 25 P a := 8 Because all of the other variables equation (8) are dependent of the applied load the value for the mean and alternatg stress the bolts can be determed. σ bm := C P m σ ba := C P a σ bm = σ ba = Usg the ASME Elliptic criteria for fatigue safety the factor of safety for each of the bolts can be determed. ASME Elliptic is of the form: sovlg for n: n σ ba S e 2 2 n σ bm = S y (9) [2] pg. 707 Table 8- n f := S yb S eb σ ba Syb σ bm Seb n f =

8 Section 4: Conclusion of Forward Bolted Analysis The eight M6 x.0 class 2.9 bolts used the forward connection were evaluated for tensile strength, separation, and fatigue. For each, the mimum required safety factor of two was exceeded. It is recomended that the bolt preload is 50% of the proof load to assure a satisfactory factor of safety for fatigue. A summary of important values is given below. Maximum Applied Force F max = Bolt Preload: = Mean Stress on Bolt Alternatg Stress on Bolt σ bm = σ ba = Stiffness of Bolt k b = Stiffness of Members Jot Constant Static Safety of Bolt k mem = C = n p = Static Safety of Jot Separation n s = Fatigue Factor of Safety of Bolt n f =

9 Section 5: Aft Bolted Connection. The analysis of the aft bolted connection of the test motor will be analyzed for static, fatigue, and jot separation the same manner as the forward bolted connection. The aft connection differs from the forward that the steel plate has been substituted with another alumum plate. The applied loadg and fasteners rema the same. Analysis for the aft bolted connection will beg with the determation of a new jot constant C which corporates the changes material. This new value will then be simply serted to equations 5,7, and 8 from the sections above to determe the appropriate factors of safety. Section 6: Stiffness of Aft Members. Usg the same method outled section 2.2 the stiffness of the aft members will be determed. Equation 3 is repeated here for reference. k mem = ln 0.577π E al d (.5 t D d) ( D d) (.5t D d) ( D d) (3) [2] pg. 340 eq. 8-4 ) Washer: d := D := t := E w = psi k wash := ) Alumum: D := t := E al = 0 7 psi k al := Aga the stiffnesses of the material are added series resultg an equivalent stiffness. k wash k al k al k wash = k aft.mem :=

10 Section 7: Determation of Jot Constant and Design Analysis. Now that the new stiffness has been determed a jot constant for the aft connection can be determed. Equation 4, repeated here, is used to determe the jot constant. C := k b k b k aft.mem (4) [2] pg.347 eq. 8-2 C = 0.39 With the new jot constant static failure, jot separation and fatigue failure can be evaluated. For static loadg, the load factor n p, can be calculated that tells whether the bolt stress is less than the proof strength. A value of n greater than dicates that the bolt stress is less than the proof stength. n p := S pb C P n p = We see here that the bolt is statically safe agast failure. The followg equation returns a factor of safety agast jot separation. n s := P ( C) n s = Aga, we see that the connection is safe agast jot separation and the external load will be shared between the bolts and the connected members. As section 3 the fatigue analysis of the bolts used the connection equation 8, repeated here, gives the stress actg on the bolt as a function of the jot constant, the applied mean and alternatg loads, and the pre-load on the bolt. The mean and alternatg stresses are analyzed usg the ASME Elliptic criteria to determe a factor of safety agast fatigue failure. Aga the lower pre-load of 50% proof load will be used. σ b = C P A t A t (8) [2] pg. 347 eq. 8-22b 0

11 σ bm := C P m σ ba := C P a σ bm = σ ba = n f := S yb S eb σ ba Syb σ bm Seb n f = The factor of safety for fatigue is satisfactory. Section 8: Conclusion of Aft Connection Analysis. The eight M6 x.0 class 2.9 bolts used the aft connection were evaluated for tensile strength, separation, and fatigue. For each, the mimum required safety factor of two was exceeded. It is recomended that the bolt preload is 50% of the proof load to assure a satisfactory factor of safety for fatigue. A summary of important values is given below. Maximum Applied Force F max = Bolt Preload: = Mean Stress on Bolt Alternatg Stress on Bolt σ bm = σ ba = Stiffness of Bolt k b = Stiffness of Members Jot Constant Static Safety of Bolt k mem = C = 0.39 n p = Static Safety of Jot Separation n s = Fatigue Factor of Safety of Bolt n f = 2.265

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