3. Rocket Grain Thermal Loading

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3.1. Problem Description

Consider the circular port grain design shown in Figure 3.1 which has four materials: propellant, liner, insulation, and case. The grain dimensions are listed in Table 3.1. These are the stress-free dimensions at 145°F, which is the grain cure temperature. However, the propellant actually shrinks during cure, and its volume decreases due to the polymer cross linking that occurs. So the final stress-free dimensions of the propellant are different at the end of the cure cycle then at the beginning. This cure shrinkage effect is accounted for by adding 20°F to the propellant stress-free temperature. Table 3.2 lists the mechanical properties, including the stress-free temperature, for each material. Table 3.3 lists the propellant failure properties. These failure properties are lower three-sigma measured values to be used to compute the safety margins against grain failure.

Two failure modes are considered for this analysis: grain cracking and unbonding. Both of these failure modes can occur under cold storage thermal loading, and they can both result in catastrophic over-pressurization failure (i.e. explosion) during motor operation. Table 3.4 lists the grain failure modes considered here along with the required safety factor for each failure mode.

Figure 3.1. Solid Rocket Grain Geometry (Circular Port)

Table 3.1. Grain Dimensions

Radius	Value
R1	2.500
R2	7.663
R3	7.673
R4	7.923
R5	8.000

Table 3.2. Material Properties

Material	Designation (Reference)	Bulk Modulus (psi)	Tensile Modulus (psi)	Poisson Ratio	CTE (/°F)	Ref. Temp. (°F)	Density. (lb/in³)
Propellant	TP-H2808 (Ref. 1)	708e3	85	0.49998	5.0e-5	165	0.055
Liner	TL-H755A (Ref. 1)	350e3	126	0.49994	11.0e-5	145	0.050
Insulation	TI-R300 (Ref. 1)	500e3	7500	0.49750	6.0e-5	145	0.050
Steel	D6AC Steel (MIL-HDB-5J)	29e6	29e6	0.32	6.5e-6	145	0.286

Table 3.3. Propellant Failure Properties

Property	Value
Strain Endurance (%)	24
Normal Bond Strength (psi)	25

Table 3.4. Grain Failure Modes

Failure Mode	Critical Condition	Required Safety Factor
Cracking	Cold Storage at -50°F	1.5
Unbonding	Cold Storage at -50°F	1.5

3.2. Objectives

1. Use Concyl to compute the safety margins against grain cracking and unbonding.

2. Compare the results with the case fully supported axially and free axially.

3. Use the Concyl optimizer to compute the actual low temperature limit for cold storage.

4. Verify the Concyl results using FEA.

3.3. Key Assumptions

1. Linear, elastic, and isotropic materials

2. Linear kinematics

3. No temperature or pressure variation in axial direction

4. No radial variation in axial strain (generalized plane strain)

5. Propellant-to-liner interface is the weakest link in the bond system

3.4. Concyl Analysis

3.4.1. Cold Storage Thermal Loading

The two load cases listed in Table 3.5 were analyzed using Concyl. Load case 1 has the grain rigidly constrained axially (plane strain), and load case 2 has the grain unsupported axially (generalized plane strain). The unsupported condition was approximated by applying a uniform axial strain of aDT, where a is the CTE of the case material. This gives an axial thermal strain of -1.2675e-3. Listing 3.1 gives the Concyl input for load case 1, and Listing 3.2 gives the input for load case 2.

Table 3.5. Load Cases

Case	Condition	Temperature (°F)	Axial Constraint
1	Cold Storage	-50	Fully Constrained (ez = 0)
2	Cold Storage	-50	Unconstrained (ez = -aDT)

Table 3.6 shows the induced bore strain and bond stress for each load case along with the corresponding safety factors against cracking and unbonding. The safety factors are all greater than the required safety factors listed in Table 3.4 which means that the grain design does satisfy the design requirements for cold storage at -50°F. Comparison of the results from the load cases shows that the axial constraint condition (fixed or free) has negligible effect on the induced bore strain and minimal effect on the induced bond stress. Load case 1 with the grain fully constrained axially has 0.72% more induced strain and 1.6% more bond stress than load case 2. These results also show that the assumption of plane strain is conservative.

Table 3.7 lists the computed safety margins for each load case. This shows that the overall minimum safety margin against grain failure during cold storage is +0.14 and that the most critical failure mode is grain cracking from excessive bore hoop strain. The safety margins against grain unbonding are much higher as shown.

Table 3.6. Concyl Analysis Results

Load Case	Bore Hoop Strain (%) / Safety Factor	Normal Bond Stress (psi) / Safety Factor
1	14.0 / 1.71	7.39 / 3.38
2	13.9 / 1.72	7.27 / 3.43

Table 3.7. Safety Margin Summary

Load Case	Grain Cracking	Grain Unbonding
1	+0.14	+1.25
2	+0.14	+1.28

3.4.2. Cold Storage Limit (Concyl Optimizer)

Since the results in Table 3.7 show positive safety margins against grain failure at the required cold storage temperature of -50°F, the actual cold storage limit (for zero safety margin) is significantly lower than -50°F. Determining what the actual cold storage limit is for this grain design can be readily done using Concyl's powerful optimizer capability.

Figure 3.2 shows the optimizer settings dialog for this problem. The corresponding input file is listed in Listing 1.3. The desired hoop strain for zero safety margin is simply the actual strain endurance from Table 3.3 divided by the required grain cracking safety factor from Table 3.4. This gives 24/1.5 = 16% for the desired hoop strain. An initial search window of -100°F to 0°F was chosen which proved to be acceptable. Choosing the search window is typically a trial-and-error process.

Figure 3.2. Concyl Optimizer Settings

The results of the Concyl optimization analysis are listed in Table 3.8 for the two axial constraint conditions that correspond to load cases 1 and 2 in Section 3.4.1. The unconstrained condition (load case 2) gives a slightly lower cold storage limit than the constrained condition (load case 1) as shown. Based on these analysis results, a good engineering choice for the actual cold storage limit would be -75°F for this grain design.

Figure 3.3 shows a plot of the bore hoop stress variation with applied temperature for load case 1. Figure 3.4 shows a corresponding plot of the propellant normal bond stress as a function of temperature.

Table 3.8. Concyl Optimizer Results

Load Case	Cold Storage Limit	Critical Failure Mode
1	-79.9°F	Grain Cracking
2	-82.4°F	Grain Cracking

Figure 3.3. Bore Hoop Strain Versus Temperature (Plane Strain)

Figure 3.4. Normal Bond Stress Versus Temperature (Plane Strain)

3.5. Finite Element Analysis

3.5.1. Model Description

The small finite element model shown in Figure 3.5 was used to analyze the grain design with TEXLESP-2D. The model is a 2D axisymmetric model with 87 quadratic isoparametric elements. The elements are nine noded and can have a mixed formulation to properly simulate compressible behavior for materials that have a Poisson ratio near 0.5. This mixed formulation is called the Hermann formulation (Ref. 3) and adds a hydrostatic pressure dependent variable to the nine nodal displacement vectors for each element. The mixed element formulation was used for the propellant, liner, and insulation materials. Listing 1.4 gives the TEXLESP model input file.

Figure 3.5 shows the finite element model mesh, material boundaries, and applied boundary conditions. Since there is no axial variation in this problem, only one element is required in the axial direction. The model was given an arbitrary length of 0.1 inches in the axial direction. Axial displacement constraints were applied to the forward and aft edges as shown. Load case 1 has the aft edge fixed, and load case 2 has an applied axial displacement of -1.2675e-4 which corresponds to proper axial thermal strain from the Concyl analysis.

Figure 3.5. Finite Element Mesh and Material Boundaries

3.5.2. TEXLESP Results

Table 3.9 lists the TEXLESP grain analysis results. Note that these results are identical to the Concyl results (to three significant figures).

Table 3.9. TEXLESP Analysis Results

Load Case	Bore Hoop Strain (%) / Safety Factor	Normal Bond Stress (psi) / Safety Factor
1	14.0 / 1.71	7.39 / 3.38
2	13.9 / 1.72	7.27 / 3.43

3.6. Conclusions

This sample demonstrates how Concyl can be used to do structural analysis of solid rocket grains. The grain design considered here was shown to be structurally sound under the required cold storage thermal loading. Fully constrained and unconstrained axial boundary conditions were compared. Use of the Concyl optimizer was also demonstrated. All the Concyl results have excellent agreement with finite-element results. This verifies that Concyl is an accurate and efficient tool for analyzing complex thermal gradient loading in thick-walled pipes.

References

1. "Grain and Bond System Stress Analysis For The Patriot Service Life Evaluation Program", Toby.T.Norris, Thiokol Report No. U-88-2243.

2. "TEXLESP-2D Users Manual", Eric B. Becker, et al, NASA MSEC-RAPT-1564, 1988.

3. "Elasticity Equations for Incompressible and Nearly Incompressible Materials by a Variational Theorem", Hermann, L., AAA J. 3(10), 1965.

Listing 3.1. Concyl Input For Load Case 1

!Concyl Input File

!---------------------------------------------------

Sample 3, Load Case 1, Plane Strain

Cold Storage at -50F

!---------------------------------------------------

4 ! number of cylinders

4 ! number of materials

!---------------------------------------------------

! cylinder definitions

!---------------------------------------------------

2.5, 7.663, 7.673, 7.923, 8

1, 165

2, 145

3, 145

4, 145

!---------------------------------------------------

! material definitions

!---------------------------------------------------

1, Propellant, 85, 0.49998, 5e-005, 0.055

2, Liner, 126, 0.49994, 0.00011, 0.05

3, Insulation, 7500, 0.4975, 6e-005, 0.05

4, Case, 2.9e+007, 0.32, 6.5e-006, 0.283

!---------------------------------------------------

! load definitions

!---------------------------------------------------

0., 0. ! pressures

-50 ! uniform temperature

0 ! uniform axial strain

!---------------------------------------------------

! plot definitions

!---------------------------------------------------

ALL

Listing 3.2. Concyl Input For Load Case 2

!Concyl Input File

!---------------------------------------------------

Sample 3, Load Case 2, Applied Axial Strain

Cold Storage at -50F

!---------------------------------------------------

4 ! number of cylinders

4 ! number of materials

!---------------------------------------------------

! cylinder definitions

!---------------------------------------------------

2.5, 7.663, 7.673, 7.923, 8

1, 165

2, 145

3, 145

4, 145

!---------------------------------------------------

! material definitions

!---------------------------------------------------

1, Propellant, 85, 0.49998, 5e-005, 0.055

2, Liner, 126, 0.49994, 0.00011, 0.05

3, Insulation, 7500, 0.4975, 6e-005, 0.05

4, Case, 2.9e+007, 0.32, 6.5e-006, 0.283

!---------------------------------------------------

! load definitions

!---------------------------------------------------

0., 0. ! pressures

-50 ! uniform temperature

-0.0012675 ! uniform axial strain

!---------------------------------------------------

! plot definitions

!---------------------------------------------------

ALL

Listing 3.3. Concyl Input For Optimizer Run

!Concyl Input File

Sample 3, Case 2, Optimizer Run

4 ! number of cylinders

4 ! number of materials

!------------------------------------------

! cylinder definitions

!------------------------------------------

2.5, 7.663, 7.673, 7.923, 8

1, 165

2, 145

3, 145

4, 145

!------------------------------------------

! material definitions

!------------------------------------------

1, Propellant, 85, 0.49998, 5e-005, 0.055

2, Liner, 126, 0.49994, 0.00011, 0.05

3, Insulation, 7500, 0.4975, 6e-005, 0.05

4, Case, 2.9e+007, 0.32, 6.5e-006, 0.283

!------------------------------------------

! loading definitions

!------------------------------------------

0, 0 ! internal and external pressure

-50 ! uniform temperature

-1.2675e-3 ! uniform axial strain

!------------------------------------------

! plot file definitions

!------------------------------------------

ALL

!------------------------------------------

! optimizer input

!------------------------------------------

1 ! run type

7 ! parameter to vary

0 ! cylinder to vary

6 ! parameter to optimize

1 ! interface to optimize

-100, 0 ! search window

0.16 ! objective value

Listing 3.4. TEXLESP Input For Load Case 2

$ Concyl Sample 3 Test Case

$ CP Rocket Grain Analysis

$ English Units: in,lb,sec,psi

$ Load Case 2

$------------------------------------------------

$ Cold Storage Thermal Loading

$ With applied axial displacement

$ to simulate unrestrained condition

$ Temperature = -50

$------------------------------------------------

axisym

thermal,-50

$ material definition

setup,2,ij,175

iso,prop, 1, 85, 0.49998, 5.00e-5, 165

iso,liner, 2, 126, 0.49994, 1.11e-4, 145

iso,insul, 3, 7500, 0.49750, 6.00e-5, 145

iso,case, 4, 29e6, 0.32, 6.50e-6, 145

end,material

$ mesh definition

1,1, 151,3

2.500,7.663,7.663,2.500

0.000,0.000,0.100,0.100

151,1, 155,3

7.663,7.673,7.673,7.663

0.000,0.000,0.100,0.100

155,1, 165,3

7.673,7.923,7.923,7.673

0.000,0.000,0.100,0.100

165,1, 175,3

7.923,8.000,8.000,7.923

0.000,0.000,0.100,0.100

end,grid

$ element definitions

iloop,75

qqh,,1, 1,1

iend

iloop,2

qqh,,2, 151,1

iend

iloop,5

qqh,,3, 155,1

qq,,4, 165,1

iend

$ boundary conditions

iloop,87

bc,uy, 1,1, 1, 0.00

bc,uy, 1,1, 3, -1.2675e-4

iend

end,elements

$ linear solver

solve

$ post processing

stress

option,linear,17

end,stress

stop