The casing is subjected to the following:

Size: px

Start display at page:

Download "The casing is subjected to the following:"

Eric Hubbard
5 years ago
Views:

1 16.50 Leture 13 Subjet: Roket asing design; Strutural modeling Thus far all our modeling has dealt with the fluid mehanis and thermodynamis of rokets. This is appropriate beause it is these features that set rokets apart from most other devies. On the other hand it is not possible to understand the harateristis and limitations of rokets as systems without at least a rudimentary understanding of their strutural harateristis whih determine their mass, durability et. To aid suh understanding we must develop some simple Strutural odels. The first step is to understand the Loads that the roket's struture must withstand. Let us begin with a Solid Propellant Roket 1) Loads We model the roket as a sphere- ylinder full of fuel, ated on by the thrust F and the payload reation (a+g)( pay) where a is the aeleration and g the gravitational aeleration. m pay Propellent L A Casing A Nozzle Image by IT OpenCourseWare. The asing is subjeted to the following: a) internal pressure, p b) shear loads from the propellant grain, whih is bonded to the 1

2 ) ompressive loads To find the fores in the asing due to the pressure, we make free- body diagrams. First onsider setion A- A. Setion A- A From this free body diagram, the irumferential stress σ y is! y = p t There is also an axial load generated by the ombination of the internal pressure, the ompressive load from the payload and the ompressive stress in the propellant. Construting another free- body diagram, [Pay+(Case+prop x/l]a Ap x A p σp σx x [ pay + ( + prop ) L ](a+g) + σx πt = p A + σ p A p 1 σ x = πt {p A + σ p A p - [ pay + ( + prop ) x L ] (a+g)} This stress is most positive (tension) at x=0:

3 ( p (! ) A +! p A ) p pa ase p =! x ax = " t " t 4t where we have assumed hydrostati grain equilibrium, i.e.,! = p. At the other end, x= L, w e have either least tension or possibly ompression of the asing wall; this last possibility would imply bukling problems. Putting x= L, paas e " ( 0 " (! ) = nozzle )(a + g) x min # t and realling that a + g = F / 0 = F p A t / 0, p (! ) = [A # (1 nozzle x min F # )A t ] " t 0 So, unless A is less than about A / there will still be tension at the asing s base. ) State of Stress t F, With this representation of the loads we an now dedue the State of Stress in the asing wall, subjet to some simplifying assumptions. We assume t <<, so the stress may be desribed as a state of plane stress in x- y oordinates, where x is axial and y tangential. In these oordinates imagine a small triangular element. The stresses on the element in general are tensions σx, σy and shear τxy = τ yx : : p x α τ xy σ x τ xy σ y τ x', y' σ x' y Image by IT OpenCourseWare. Now onsider th e stresses on a plane perpendiular to a line making the angle α with the x axis. In treatments of stress and strain it is shown that σx'= σ x α + σy sin α + τxy sinα os α τx' y' = - (os σ x - σ y ) sinα osα + τ xy (osα - sin α) It is then shown that the maximum (or minimum) of σx' ours at an angle α suh that τ xy tan α = σx- σ y and that these stresses are: 3

4 σ x + σ y! x "! σ1, = ± ( y ) +# xy In our τxy = 0, so σmax is simply the larger of σ x or σy. and usually σy> σ x : y = p p σ t ; ( σ x ) ax = 4t = 1 σ y So the asing must be designed to withstand the hoop stresses: p t = " (a safety fator) 3) Case ass Again assuming onstant t<< and hemispherial aps, = (! +!L)t ρ se = π 3 L p a (1+ ) ρ σ y The mass of the propellant is 3 L 1 1 prop = ( π π ) ε ρprop = 3 L π ( ) ε ρ prop where ε is the fration of volume filled by propellant.! ult L p 3(1+ ) = 3 L (1+ )# " "! prop prop Typially, for p 50 atm= 750 psi, σy 150,000 psi (about 100kg/mm ρ= in European notation), and =.3 ln lb3 (steel), ρ prop =.06 ln lb3 L pr op L and for L = 3,.068. pr op Is this a reasonable estimate? For the inuteman, from Sutton & Ross = 557 lb prop = 45,831 = prop It seems we have been somewhat onservative in our design parameters. 4) What is left out of our estimate? 4

5 a) Attahments b) End bells have 1/ the stress - make 1/ as thik? ) Shear loads on propellant grain 5

6 Is this a reasonable estimate? For the inuteman, from Sutton & Ross m = 557 lb m prop = 45,831 } =.056 4) What is left out of our estimate? a) Attahments b) End bells have 1/ the stress - make 1/ as thik? ) Shear loads on propellant grain It seems we have been somewhat onservative in our design parameters. 6

7 IT OpenCourseWare Introdution to Propulsion Systems Spring 01 For information about iting these materials or our Terms of Use, visit:

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS CHAPER MECHANICS OF MAERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John. DeWolf orsion Leture Notes: J. Walt Oler exas eh University 006 he MGraw-Hill Companies, In. All rights reserved. Contents