Structural Integrity Assessment on Solid Propellant Rocket Motors B. Nageswara Rao K L University Presentation in Pravartana 2016: Symposium on Mechanics at IIT Kanpur during February 12-14, 2016
In a solid rocket motor, the combustion reaction generates a large amount of thermal/potential energy that is converted to kinetic energy by expansion through a nozzle, whereby the required lift-off thrust is created.
For the solid rocket motor to perform successfully in its mission, it is necessary for it to retain its structural integrity under a wide variety of mechanical loads, that are imposed on it during storage and operational phases.
This lecture deals with solid rocket motor propellant grain structural integrity analysis, including materials characterization, structural analysis, and failure criteria for margin/factor of safety estimation.
Figure-1.1: Free- standing grains and case-bonded grains
Figure-1.2: Cross-section of a typical case-bonded solid rocket motor. (A) Chamber; (B) Head end dome; (C) Nozzle; (D) Igniter; (E) Nozzle convergent portion; (F) Nozzle divergent portion; (G) Port; (H) Inhibitor; (I) Nozzle throat insert; (J) Lining; (K) Insulation; (L) Propellant; (M) Nozzle exit plane; (N) SITVC system; (O) Segment joint.
Figure-1.3: Evaluation of Structural Integrity.
For selection of grain configuration, the main factors taken into account are: • Volume available for the propellant grain • Grain length to diameter ratio • Grain diameter to web thickness ratio • Thrust versus time curve, which gives a good idea of what should be the Burning area versus web burned curve (see Figure-2.1) • Volumetric loading fraction which can be estimated from required total impulse and actual specific impulse of available propellants • Critical loads such as thermal cycles, pressure rise at ignition, acceleration, internal flow Figure-2.1: Progressive, regressive and neutral burning
Figure-2.3 Typical solid propellant grain geometries.
Cylindrical configurations Axisymmetric configurations Three-dimensional geometries
Propellants • CTPB (carboxy-terminated polybutadiene); • HTPB (hydroxy-terminated polybutadiene); • PBAN (polybutadiene acrylonitrile), • PS (polysulfide); • PVC (polyvinyl chloride). Loads • pressure load • Thermal load • Gravity load
Figure-2.4: Basic geometric parameters of a right- circular cylinder geometry.
Structural Integrity Analysis Materials Characterization Structural Analysis Failure Criteria
Modeling of Structural Response with the Development of Computational Methods Observation of Response Phenomena Development of Computational Models Development / Assembly of Software / Hardware to implement the Computational Models Post-processing and Interpretation of Results Use of Computational Models in the Analysis and Design of Structures
Based on the nature of the final matrix equations, finite element methods are often referred to as: displacement method force method mixed method Commercial codes (viz., MARC, NASTRAN, NISA, ANSYS, etc.) and user’s guides are currently available to solve structural problems. Need experience selection of suitable elements for modeling specification of boundary conditions for the intended structural analysis under the specified loading conditions interpretation of finite element analysis results
Materials Characterization Figure-6.1: Tensile Specimens.
Figure-6.2: Uniaxial stress-strain behavior at constant strain-rate. t = + P K log P property change due to ageing 0 t 0
Figure-6.5: Master stress relaxation modulus curve with reduced time
Figure-6.3: Failure boundary Figure-6.4: Variation of strain envelope for HTPB propellant from with temperature reduced fracture data of uniaxial tensile strain rate specimens.
Structural Response under various loads Effect of Thermal Shrinkage The Effects of gravity Pressure rise at firing
STAR GRAIN CONFIGURATION FOR IGNITER MOTOR
GRAIN CONFIGURATION OF A TYPICAL MOTOR
HEAD END GRAIN CONFIGURATION
MID SEGMENT GRAIN CONFIGURATION
NOZZLE END SEGMENT GRAIN CONFIGURATION
* To idealize the grain configuration, the following elements are required (i) Axi-symmetric element(2-D model) (ii) Plane- strain element(2-D model) (iii) 3-D Brick element * TYPES OF LOADS ( i ) Pressure load ( ii ) Thermal load ( iii ) Gravity load
MATERIALS IN SRM. ( i ) Casing (isotropic/orthotropic) ( ii ) Liner (Insulation material) ( iii ) Solid Propellant material Young’s Modulus and Poisson’s ratio for the solid propellant material will be specified from Master stress relaxation modulus(MSRM) curve and the Bulk- modulus(K).
Master stress relaxation modulus (MSRM) curve of a HTPB-based propellant grain
* The relaxation modulus curve is represented by means of Prony series in the form ∞ − t = + E ( t ) E ∑ A exp ∞ rel k (1) τ 2 a = k 1 T k Where E ∞ is the equilibrium modulus, t is the time τ k ’ S are relaxation times − c ( T T ) 1 R = log ( a ) e T (2) + c ( T T ) - 2 R
c 1, c 2 are material constants T R – Reference temperature. T- Temperature * For the specified time, the modulus of the propellant material is = E [ sL ( E ( t ))] = rel s 1 / 2 t (3) 1 E = − ν 1 (4) 2 3 K
• Hybrid - stress - displacement formulation : [ ] { } ε { } (5) = B q (1) Strain – displacement relation [ ] { } σ β (2) Stress function { } (6) = P #The elements in [ P ] matrix are functions of co-ordinates. - #These functions are obtained from equilibrium quations(without body forces) and Compatibility equations. #{ β }’s are unknown constants of element which will be expressed in terms of element displacement{q}
[ ] [ ] [ ] [ ] − = T 1 (3)Element stiffness matrix: k G H G (7) [ ] [ ] [ ][ ] dv Where = T H P C P ∫ (8) v And [C] is a compliance matrix Matrix (9) [ ] [ ] [ ] dv = T G P B ∫ v { } [ ] [ ] { } − = 1 β (10) H G q - : (4) Load vector computations are as per standard procedures available in FEAST-C (5) Assembly of element stiffness matrices and load vector are as per FEAST-C Architecture
* Solution of displacements by solving the following system of linear equations [ ] { } { } G = δ K F (11) G Through, - (i) Frontal solver (or) Cholesky solver(Band solver) available in FEAST-C.
(10) From the displacements for each element,{ β } are computed as { } [ ] [ ] { } − = 1 β H G q (12) Using { β }: Stresses in the element : [ ] { } σ β { } = P { } [ ] { } Strains in the element : ε = σ C
TYPES OF ELEMENTS 4 Node iso-parametric axi-symmetric element 4 Node iso-parametric plane strain element 8 Node 3-D brick element calculation of relaxation modulus at particular time from prony series
Validation of axi-symmetric,Plane strain and 3 D brick element with closed form solution and MARC software PROBLEM DESCRIPTION • Three layered infinitely long thick cylindrical shell of propellant grain, insulation and casing under Case (a) pressure load Case (b) Thermal load Case (c) Gravity load • Inside layer is propellant grain, middle layer is insulation and outer layer is casing
GEOMETRICAL DETAILS • Grain inner radius = 50 cm • Grain outer radius = 138.9 cm • Insulation thickness = 0.5 cm • Casing thickness = 0.78 cm LOAD DETAILS Case (a) Internal pressure = 50ksc Case (b) Thermal load of –38.0 o C Case ( c) Gravity load of 1 g acting downward
MATERIAL PROPERTIES Material Young’s Poisso Density Coefficient modulus n’s of thermal Kg/cm 3 ratio expansion (KSC) (/ o C) Casing 190000 0.3 0.0078 0.000011 Insulation 20 0.499 0.00178 0.0003 Propellant Case (a) 50 0.499 0.00178 0.0001 Case (b) 20 Case (c) 20
Comparison of results with closed form solution for Pressure load Location ANALYT Axi- Plane 3 D Brick ICAL symmetric Strain Element [ref 1] Radial disp. 2.6086 2.609 2.606 2.631 at inner port 2.6086 2.558 (cm) (MARC) (MARC) Hoop strain 5.21 5.149 5.095 5.18 at inner port 5.1810 5.506 (%) (MARC) (MARC)
Comparison of results with closed form solution for Thermal load Location ANALY Axi- Plane 3 D TICAL symmetri Strain Brick c Elemen [ref 1] t Maximum radial 0.4545 0.4617 0.462 0.4564 Stress at the 0.4545 0.4662 interface of (MARC) (MARC) propellant and insulation (ksc) Hoop strain at 3.35 3.365 3.32 3.326 inner port (%) 3.327 3.26 (MARC) (MARC)
Comparison of results with closed form solution for gravity load Location ANALYT Axi- 3 D Brick ICAL symmetric Element [ref 1] Maximum slump 0.7875 0.7895 0.7859 displacement for 0.7892 vertical storage,w (MARC) (cm)
1.60 S lump Vs Time Axi-symmetric model 1.20 3-D model max in cm 0.80 lump W S 0.40 0.00 -8.00 -4.00 0.00 4.00 8.00 t in seconds (log base 10 scale) Time march analysis for slump estimation of S200 Mid-Segement
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