Report Section 8 - Purdue University

Project Bellerophon 491

Author: Brandon White

A.5.0 Structures

A.5.1 Introduction Before igniting the first engine, a launch vehicle must prove that it can withstand the basic

structural forces during flight. Through iterative math models produced in MATLAB, we are

able to appropriately size all components of the launch vehicles to combat buckling, bending,

and shear forces. The propellant tanks, inter-stage skirts, inter-tank couplers, and nose cone are

designed in this manner. Whether it results in adding internal structural members (stringers and

support rings) or increasing the thickness of certain components (tanks, inter-stage skirts), our

launch vehicle’s structural success is justified in the following sections.

Prior to the final launch vehicle configuration, we had to be very flexible with our design. At a

moment’s notice we were able to create launch vehicles employing different materials for

different propellant types, without losing any confidence in mission success. As certain options

were eliminated, our analysis became more detailed and thorough. When the final mission

specifications were selected, we branched out to other investigations such as finite element

analysis and the design of the gondola that carries the launch vehicle to a 30 km altitude.

The design process, methods used, and research performed are presented in the subsequent

pages. We anticipate that the work accomplished by this group will be useful for anyone looking

to advance this design further.


Author: Chii Jyh Hiu

A.5.2 Design Methods A.5.2.1 Propellant Tanks A.5.2.1.1 Overview Our launch vehicle contains a number of propellant storage tanks, which comprise much of its

inert mass. The final launch vehicle configuration has a pressurant tank and an oxidizer tank in

the first stage of the rocket and a Liquid Injection Thrust Vector Control (LITVC) tank in the

second stage. The tanks are subjected to pressure loading as well as axial and bending loads in

flight and on the ground. In the preliminary design phase, liquid and cryogenic rocket fuels were

also considered, so we also designed fuel tanks. Collectively, we refer to the fuel and oxidizer

tanks as propellant tanks in this report.

The tanks are considered to be pressure-flight stabilized structures. This means that they are

designed to be strong enough to withstand ground loads while unpressurized, but are pressurized

while in flight, meaning that the launch vehicle may be transported and assembled while

unpressurised, resulting in safer and less complicated (and thus cheaper) ground logistics. At the

same time, treating the tanks as pressurized structures while in flight allows us to make the tanks

lighter and more structurally efficient.


Author: Sarah Shoemaker

A.5.2.1.2 Tank Material The primary materials that we are considering for our launch vehicle are aluminum, steel,

titanium, and composites. Research proved that these are the most common materials employed

in space flight. Our analysis involves finding different alloys of these materials and figuring out

which alloys give us the best results for cost, strength, weight, and manufacturability.

After compiling information on physical properties of the materials, we contacted manufacturing

companies to find out the costs associated with producing a tank. The data we obtained helps us

compare the materials suitably. These costs include: welding, riveting, and labor hours required

for production. These cost numbers are then placed into Matlab codes to give an overall cost for

the tank.

We looked at several different alloys for each of the materials. The strength of each of these

alloys is compiled into a database. These strengths are then compared to the costs output by the

code in order to figure out the best cost-to-strength ratio of the material. Along with the strength,

the manufacturability of the alloys is also considered. We looked at the time it would take to

weld or rivet the material, as well as the formability of each material. The alloys that we

analyzed were Aluminum 7075, Stainless Steel, Isotropic Carbon Fiber, and Titanium Ti-5Al-

2.55Sn. Table A.5.2.1.2.1 through Table A.5.2.1.2.4 show the database for each of the materials

considered for the tanks.

Table A.5.2.1.2.1 Materials Database for Aluminum 70751 Tanks

Variable Value Units Yield Stress 4.61*108 Pa Shear Stress 3.00*108 Pa Density 2.8*103 kg/m3 Young’s Modulus 6.79*1010 Pa Poisson’s Ratio 0.333 --



Table A.5.2.1.2.2 Materials Database for Stainless Steel3 Tanks


Table A.5.2.1.2.3 Materials Database for Isotropic Carbon Fiber4 Tanks


Table A.5.2.1.2.4 Materials Database for Titanium Ti-5Al-2.55Sn2 Tanks


Our propellant tanks are designed with Aluminum 7075. We chose this material because it has

been incorporated for many historical launch vehicle tank materials. Our analysis of strength-to-

weight ratios favors aluminum. The cost also favors aluminum due to the ease of

manufacturability. This material is used for all three stages on all three launch vehicle

configurations.



References 1 Setlak, Stanley J., “Aluminum Alloys; Cast,” Aerospace Structural Metals Handbook, Purdue University, Indiana, 2000. 2Setlak, Stanley J., “Titanium Alloys; Cast,” Aerospace Structural Metals Handbook, Purdue University, Indiana, 2000. 3Setlak, Stanley J., “Stainless Steel; Cast,” Aerospace Structural Metals Handbook, Purdue University, Indiana, 2000. 4Callister, W.D. Jr, Fundamentals of Materials Science and Engineering, 2nd Ed., Wiley & Sons, 2005



A.5.2.1.3 Preliminary Design Preliminary design for the propellant tanks and pressurant tanks is carried out by sizing the tanks

to contain the required amount of propellant and designing the tanks to withstand maximum in-

flight internal pressure. We use a safety factor of 1.25 to account for transient spikes in pressure

and to cover other failure modes. We also add 5% additional volume to the propellant tanks to

account for internal structure and dead space. We specify a minimum tank wall thickness of

0.75mm in the preliminary design algorithms for manufacturability and practicality.

A.5.2.1.3.1 Tanks The propellant tanks for liquid fuel/oxidizers are designed as cylindrical tanks with

hemispherical ends for the purposes of structural efficiency and ease of manufacture. The

hemispherical end configuration is stronger and lighter than using elliptical ends, but takes up

more space.1 Due to the small size of the launch vehicle, the space savings from using an

elliptical-end tank are negligible, so we incorporate the hemispherical end configuration instead.

Should a spherical tank be small enough to fit into the stage, we choose a spherical tank instead

of a cylindrical tank for structural efficiency. For cylindrical tanks, a maximum length to

diameter (L/D) ratio of 6.0 is chosen in preliminary design as a tradeoff between drag and

structural efficiency/dynamic stability. This value is later refined to 3.0 in final design based on

scaling from existing launch vehicle designs; provided more time to analyze the interaction

between size and drag/controllability, a more optimal aspect ratio range could have been

determined via simulation runs. However, since the final designs do not reach anywhere near the

maximum L/D ratio, we regard this exercise as not crucial to our current design and do not

pursue it any further.

Propellant tanks for solid propellants (and the solid components of hybrid rockets) are designed

as an open-ended cylinder with an elliptical cap. The same maximum L/D ratio is applied as with

the liquid propellant tank design. Spherical tanks are not appropriate for solid rocket fuel, so the

tanks were kept cylindrical.

The pressurant tank is designed as a spherical tank, as it is rated to a much higher internal

pressure (12 MPa) compared to the propellant tanks (typically ~ 2.0 MPa for liquid propellant



tanks and ~ 6.0 MPa for solid propellant tanks). The spherical tank configuration provides the

highest structural efficiency for a pressure vessel1 and is the ideal layout for a small, high-

pressure tank.

A.5.2.1.3.2 Inter-tank Couplers The inter-tank couplers connect the pressurant tank to the oxidizer tank, and the oxidizer tank to

the fuel tank. They are designed as cylindrical skin sections with longitudinal and hoop

stiffeners, and are designed to carry axial and shear load at maximum flight g-loading.



A.5.2.1.4 Stress Analysis

A.5.2.1.4.1 Tanks For the purpose of our analysis, we assume that the tanks carry only axial and bending loads, and

that the inter-tank couplers and inter-stage skirts carry only axial and shear loads. Tanks are

analyzed as thin-walled structures. We consider these assumptions to be a conservative and

reasonable approximation of the actual loads seen in the vehicle.

The oxidizer tank is manufactured from Aluminum 7075 spun in two halves, with a full-

thickness circumferential weld at the butt. This provides the optimal weld conditions for

strength, as the hoop stress in a cylindrical pressure vessel is twice the axial stress

(Eqs.(A.5.2.1.4.1) and (A.5.2.1.4.2)). Assuming a weld strength factor of 0.851 for a spot-

examined joint, this ensures that the tank wall thickness is designed entirely by the hoop stress

due to pressure, as the reserve factor for axial loading will consequently always be greater than

for hoop loading.

As mentioned above, the propellant tank is designed to the hoop stress seen due to pressure

loading due to internal pressure and hydrostatic pressure at maximum flight g-loading.

max_

tan _

/ 2ox ox oxox hoop

k ox

P D g ht

ρσ ⋅ + ⋅ ⋅= (A.5.2.1.4.1)

2max

_tan _

/ 22 4

ox ox ox ox oxox axial

k ox

P D g h P Dtρ πσ ⋅ + ⋅ ⋅ ⋅ ⋅

= −⋅

(A.5.2.1.4.2)

where σox_hoop is the hoop stress in the oxidizer tank (Pa), σox_axial is the axial stress in the oxidizer

tank (Pa), Pox is the internal pressure in the oxidizer tank (Pa), gmax is the maximum in-flight

acceleration (m/s2), h is the height of the fluid level (m), ttank_ox is the thickness of the tank wall

(m) and Dox.is the diameter of the oxidizer tank (m).

We then subject the model to further failure mode analyses, buckling and bending, and either add

structure or increase the wall thickness as needed to meet our strength requirements.



Tank buckling strength is calculated by using Baker’s buckling criteria3 (Eqs. A.5.2.1.4.3) for

unpressurized tanks, and using experimental data from Bruhn Figure C8.114 for pressurized

cylinders to determine the proportional increase in strength due to pressurization.

( )22

212 1s

crk E tP

Lπ

ν⋅ ⋅ ⎛ ⎞= ⋅⎜ ⎟− ⎝ ⎠

(A.5.2.1.4.3a)

( )2

22 1LzD t

ν⎛ ⎞⋅

= ⋅ −⎜ ⎟⋅⎝ ⎠ (A.5.2.1.4.3b)

0.750.85sk z= ⋅ (A.5.2.1.4.3c)

( )_ 1cr press cr crP P P= + Δ (A.5.2.1.4.3d)

where Pcr is the critical buckling stress of the structure (Pa), ks is the buckling coefficient, E is

the Young’s Modulus of the material (Pa), υ is the Poisson’s Ratio of the material, t is the

thickness of the inter-tank coupler (m), L is the length of the inter-tank coupler (m), D is

diameter of the tank (m), ΔPcr is the non-dimensionalized increase in critical buckling strength

(see Section A.5.2.1.6.2) and Pcr_press is the critical buckling stress of a pressurized tank, (Pa).

Tank bending strength is assessed using test data from Bruhn Figure C8.13a4 for unpressurized

cylinders and deriving the increase in tank bending allowable due to pressurization from Bruhn

Figure C8.144 for pressurized vessels.

Similar to the oxidizer tank, the pressurant tank is manufactured from spun Aluminum 7075 in

two hemispheres and joined together with a full thickness weld. Due to the higher criticality of

the tank, the weld of the pressurant is to be fully radiographically tested after manufacture.

Fortunately, as the pressurant feed tank is smaller than the oxidizer tank, this is easily achieved.



The pressurant tank is designed to withstand a wall stress calculated from Eq. (A.5.2.1.4.4).

max

_

/ 2press press presspress

tank press

P D g ht

ρσ

⋅ + ⋅ ⋅= (A.5.2.1.4.4)

where σpress is the stress in the pressurant tank (Pa), Ppress is the internal pressure in the pressurant

tank (Pa), gmax is the maximum in-flight acceleration in (m/s2), h is the height of the fluid level

(m), ttank_press is the thickness of the tank wall (m), and Dpress.is the diameter of the pressurant tank

(m).

The LITVC tank is found in the second stage of the rocket, and is designed as a spherical tank to

similar principles as the pressurant tank. We place the tank near the nozzle throat. If the need

arises, the LITVC tank could be redesigned as a toroidal tank, but this will require additional

work not covered in this report.

The LITVC tank is designed to withstand a wall stress calculated from Eq. (A.5.2.1.4.5).

max

_

/ 2LITVC LITVC LITVCLITVC

tank LITVC

P D g ht

ρσ ⋅ + ⋅ ⋅= (A.5.2.1.4.5)

where σLITVC is the stress in the LITVC tank (Pa), PLITVC is the internal pressure in the LITVC

tank (Pa), gmax is the maximum in-flight acceleration (m/s2), h is the height of the fluid level (m),

ttank_LITVC is the thickness of the tank wall (m) and DLITVC is the diameter of the LITVC tank (m).



A.5.2.1.4.2 Inter-tank Couplers The inter-tank couplers are designed to carry axial and shear load at maximum flight g-loading.

Fig. A.5.2.1.4.2.1: Inter-tank coupler showing internal supports, 1kg payload

(Chii Jyh Hiu) The inter-tank couplers are manufactured from rolled Aluminum sheet welded at the seams, with

equally spaced I-section hoops and z-section stringers riveted to the inside walls.

We design the inter-tank couplers to withstand axial loads by satisfying the Baker buckling

criteria3:

( )22

212 1s

crk E tP

Lπ

ν⋅ ⋅ ⎛ ⎞= ⋅⎜ ⎟− ⎝ ⎠

(A.5.2.1.4.6)

where Pcr is the critical buckling stress of the structure (Pa), ks is the buckling coefficient, E is

the Young’s Modulus of the material (Pa), υ is the Poisson’s Ratio of the material, t is the

thickness of the inter-tank coupler (m) and L is the length of the inter-tank coupler (m).

1.1527 m

1.1264 m



I-section hoops are added in evenly spaced increments until the inter-tank coupler meets or

exceeds the buckling criteria.

We also design the inter-tank couplers to withstand shear loads using the following relations for

shear stress:

1 12 26

r rr r skin

r r

y yB A t ry y

θ + −⎛ ⎞= + ⋅ ⋅ ⋅ + + +⎜ ⎟

⎝ ⎠ (A.5.2.1.4.7a)

2xx r rI B y= ⋅∑ (A.5.2.1.4.7b)

yr r r

xx

Sq B y

I⎛ ⎞

= − ⋅ ⋅⎜ ⎟⎝ ⎠

(A.5.2.1.4.7c)

rr

skin

qt

σ = (A.5.2.1.4.7d)

( )maxcrit r suσ σ σ= < (A.5.2.1.4.7e)

Fig. A.5.2.1.4.2.2: Inter-tank coupler stringer schematic for analysis

(Jesii Doyle)

where tskin is the skin thickness in m, θ is the angle between stringers (rad), yr is the vertical

distance from shear center to stringer r (m), Ar is the area of stringer r (m2), r is the stringer

t_skin

θ

θy

Sy



number, Ixx is the area moment of inertia (m4), qr is the shear flow through stringer r (N/m), σr is

the shear stress through stringer r (Pa), Sy is the shear force at shear center (N).

A.5.2.1.5 Effects of Propellant Type on Tank Requirements In preliminary design, we considered 4 major propellant types: Cryogenic (Liquid Oxygen

oxidizer + Liquid Hydrogen fuel), Storable (Hydrogen Peroxide oxidizer + Kerosene), Hybrid

(Hydrogen Peroxide oxidizer + HTPB fuel) and Solid (HTPBAPAN) propellant.

For the most part, the design requirements of the different propellant types is similar to the

blanket analysis covered in Section A.5.2.1.3 to A.5.2.1.4, but there are several nuances worth

mentioning here for anyone who wishes to replicate our preliminary design work.

The first challenge is that the MATLAB code for preliminary design tanks.m has to be versatile

enough to consider different propellant inputs and to perform different algorithms for different

cases as needed.

Cryogenic propellant presented a unique challenge, as the propellant tanks need thermal

insulation for the liquid oxygen and liquid hydrogen. We opt to use similar foam insulation to

that used on the Space Shuttle Main Tank5, with a 25.4 mm thick layer of foam insulation on the

fuel and oxidizer tanks, which adds a small amount of weight to the tanks. The low density of

hydrogen also necessitated very large tanks, which increases both the length and diameter of the

fuel tank. This is the prompt for us to implement the maximum L/D ratio of 3.0 discussed in

Section A.5.3.2.1.3.1. As the diameter and length of the tank increases, its structural efficiency

worsens, and we end up with larger inert mass fractions. In addition, it is found that despite the

low thermal conductivity of the insulating foam6, it is insufficient to keep the propellant cooled

for the rise time of a balloon launch, and thus limits us to ground launches (aircraft launches

were likewise limited by the large diameter and weight of the tanks).

Solid propellant tanks (Hybrid fuel and solid rockets) also require a separate branch in the code,

specifically that the tanks have to be cylindrical in shape. As the solid propellant tank is also the

combustion chamber, it develops high chamber pressures and also high internal temperature. The



solid propellant itself serves as a form of thermal insulation for the tank casing during burn, but

additional thermal insulation material may be required on the inner surface of the tank to prevent

the aluminum from melting. We assume that this extra weight is accounted for in the otherwise

nonexistent engine mass budgeted for the solid/hybrid motor, but further work will have to be

done in the area of thermal protection for more detailed design of the solid/hybrid motor. In

addition, the solid rocket motor used in the second stage requires a separate tank for LITVC,

which is accounted for in the MATLAB code.

References 1 Huzel, D.K., Huang D.H. Design of Liquid Propellant Rocket Engines, NASA SP-125, pp 329-352 2 Megyesevy, E.F., Pressure Vessel Handbook, 10th Edition, Pressure Vessel Publishing, pg 172

3 Baker, E.H., Kovalevsky, L., Rish, F.L., Structural Analysis of Shells, Robert E. Krieger Publishing Company, Huntington, NY, 1981, pgs. 229-240

4 Bruhn, E.F., Analysis and Design of Flight Vehicle Structures, Jacobs Publishing, 1973, Chapter C8 pgs. 347-353

5 National Aeronautics and Space Administration, “External Tank Thermal Protection System”, FS-2005-04-10-MSFC, Pub 8-40392, April 2005

6 Hart, G H, “Grounding The Space Shuttle, NASA’s Foam Insulation Problem”, www.insulation.org/articles/article.cfm?id=IO51204


Author: Steven Izzo

A.5.2.1.6 Pressure Vessel Buckling

A.5.2.1.6.1 Support Rings Buckling due to stress is a very important design consideration for launch vehicles.2 Research

on the stresses on similar launch vehicles, such as the Vanguard, confirms this.5 We design the

pressure vessels and solid engines not to buckle by integrating support rings inside the

cylindrical tanks. The ring requirements are found by comparing the applied buckling to a

critical buckling based on the geometry.

The first step in the buckling analysis is determining the load. We discover that the axial load

applied to a certain tank is equal to the mass of the entire vehicle above the tank multiplied by

the acceleration. The resulting equation for the applied axial load is Eq. (A.5.2.1.6.1.1).

gmP abovea ∗∗= l (A.5.2.1.6.1.1)

where mabove is the mass of the tank being analyzed with the propellant and mass of the launch

vehicle above the tank being analyzed (kg), l is the g loading, and g is the acceleration due to

gravity on the surface of the Earth (m/s2).

The launch vehicle has a large mass, and the acceleration could have a potential maximum of 30

g’s, so we are correct in our initial assessment of the importance of buckling support. The

dimensions and materials of the tank determine the critical buckling load for the launch vehicle.

The first method we incorporate is Euler’s buckling method1, using the following equations.

The moment of inertia is calculated using Eq. (A.5.2.1.6.1.2)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛=

44

224tddI π (A.5.2.1.6.1.2)

where d is the diameter of the tank (m), and t is the thickness (m).

The critical buckling load is then calculated using Eq. (A.5.2.1.1.6.1.3).


Author: Steven Izzo

2LEIP = (A.5.2.1.6.1.3)

where E is Young’s Modulus for the tank material (Pa), and L is the tank length (m).

For initial analysis, our code gathers the masses and dimensions of the tanks and other

components. Then the code calculates both the applied load and the critical buckling load. If the

applied load is the greater value, the code advises the user to resize the tank. With more research

and consideration, our code evolved to add inner support structures to the tank instead of

resizing.3 We determine that adding support rings inside the tank significantly aid in buckling

prevention and does not require the tank functions to be run again. Resizing the tanks would be

necessary if the tank thickness were changed. We made assume that such support rings would

not reduce the available volume in the tank because we oversize the tank initially. Without this

assumption, the tanks would have to be resized each time support rings are added. Therefore the

code determines the support needed to keep the tanks from buckling as opposed to just

determining whether it fails or not.

After extensive research we could not find a complete solution for designing the tanks with

support rings. From the research we did gather, the main source of support the rings provide is

by essentially separating the length of the tank into parts, Separation increases the critical

buckling load for the entire launch vehicle.2 The code starts with no rings and reiterates the

analysis, adding a ring each time, until the tank is structurally sound. This code could only find

the number of rings needed; a method to find ring dimensions was not known at the time. The

number of support rings we find is also an overestimation, as our design includes the effect of

internal pressure, which aids in buckling support. The code can be run for the worst cases, such

as if the pressure dropped but nearly all of the mass was still in the launch vehicle, or if

something happens that cannot be anticipated.

This code was run many times with many varying parameters. It was first tried for values

currently thought to be average, such as a first stage tank 4 meters long, 1 meter wide, with 2000

kg above it, and undergoing 2g’s acceleration. These values required no support rings. The code

was run several times to determine the conditions that did require support rings. All of the


Author: Steven Izzo

conditions were found to be outrageously extreme before support was necessary, such as a

thickness of 10-6 meters and diameter of 5 meters, or a mass of 100,000 kilograms, or an

acceleration of 5000 g’s. The research on launch vehicles did not match these findings, nearly

all vehicles required buckling support of some kind and vehicles did not have parameters similar

to the extremes we found5. Due to these tests, the method of stress analysis was revised.

After reviewing the code and methods gathered from the research, we chose to perform analysis

with methods other than Euler’s buckling method. Baker et. al. provided a critical buckling

stress specifically for thin shells.1

The curvature parameter is calculated from the below Eq. (A.5.2.1.6.1.4)

dtLZ

22 12 ν−= (A.5.2.1.6.1.4)

where L is the tank length (m), ν is Poisson’s Ratio for the tank material, d is the diameter (m),

and t is the thickness (m).

The buckling coefficient is calculated from the below Eq. (A.5.2.1.6.1.5) 75.85. ZK s = (A.5.2.1.6.1.5)

The critical buckling stress is calculated from the below Eq. (A.5.2.1.6.1.6)

( ) 22

22

112 LEtKP scr ν

π−

= (A.5.2.1.6.1.6)

where E is Young’s Modulus (Pa).

Using this analysis, while the current design test cases still did not need any support, for the

maximum value of 30 g’s and for only twice the mass or length, the larger tanks do require

support, so logically this method seemed correct. Bruhn’s text confirmed this method4 by

mentioning the very same equations.


Author: Steven Izzo

With the correct buckling analysis, we returned to the issue of sizing the support rings. For our

analysis we assume the rings take all of the stress and use the same method as the whole tank to

find the maximum load before the rings fail. A rectangular cross section was determined to be

the simplest method, as well as being the easiest to manufacture and weld. Starting very small,

the analysis reiterated until the minimum size ring was found that could support the load. The

largest size rings found for extreme test cases are only around one half of a millimeter square,

thus our initial assumption that the rings add minimal volume is confirmed. The code was then

ready to be incorporated into the costing codes and the rest of the structures codes.

References 1Baker, E.H., Kovalevsky, L., and Rish, F.L., Structural Analysis of Shells, Robert E. Krieger Publishing Company, Huntington, NY, 1981. 2Bedford, A., Fowler, W., and Liechti, K., Statics and Mechanics of Materials, Prentice Hall, Englewood Cliffs, NJ, 2002. 3Boddy, J., Mitchell, J., and Harris, L., “Systems Evaluation of Advanced Structures and Materials in Future Launch Vehicles,” AIAA Journal no. 1103-391, 1967. 4Bruhn, E.F., “Buckling Strength of Monocoque Cylinder,” Analysis and Design of Flight Vehicle Structures, S.R. Jacobs, 1973. 5Klemans, B., “The Vanguard Satellite Launching Vehicle,” The Martin Company, Engineering Report No.11022, April 1960



A.5.2.1.6.2 Effect of Pressure on Buckling and Bending Strength A pressurized vessel is stronger against buckling and bending than an unpressurized vessel. This

is because internal pressure stiffens the structure, provides resistance to axial loads, and

minimizes stress concentrations due to local geometric imperfections. In our analysis of the

propellant tanks, we take advantage of this phenomenon to optimize our structure and to avoid

having to add unnecessary internal supports.

For our analysis, we utilize curves based on empirical test data as presented in Bruhn Chapter

C81. Where possible, 90% probability curves with a confidence level of 95% are chosen, which

meets the required 90% probability requirements against catastrophic failure specified for our

mission.

A.5.2.1.6.2.1 Tank Buckling

Tank buckling is first modeled via Baker method 32, which gives us the nominal buckling

strength of an unpressurized tank.

( )22

212 1s

crk E tP

Lπ

ν⋅ ⋅ ⎛ ⎞= ⋅⎜ ⎟− ⎝ ⎠

(A.5.2.1.4.3a)

(See Section A.5.2.1.4.1 for more details)

We then employ empirical design curves from Bruhn Fig C8.111 to calculate the proportional

increase in buckling strength due to pressurizing the tank.

( )_ 1cr press cr crP P P= + Δ (A.5.2.1.4.3d)

where ΔPcr is the non-dimensionalized increase in critical buckling strength (see Section

A.5.2.1.6.2) and Pcr_press is the critical buckling stress of a pressurized tank, (Pa).

The non-dimensional increase in axial buckling strength ΔPcr is calculated by plotting data points

from Bruhn Fig C8.11 into Mathcad and using the linfit function to curvefit a function to

describe the relation in Matlab.



0.01 0.1 1 10 1000

0.05

0.1

0.15

0.2

0.25

Y

f z( )

X z,

( )

Fig. A.5.2.1.6.2.1 Non-dimensional increase in axial buckling strength versus tank internal pressure and dimensions

showing function fit curve

(Chii Jyh Hiu)

Using Mathcad, we obtain a high-correlation function fit of

273.0t2

DE

P098.0

t2D

EP

ln051.0P 4

2ktanktan

2ktanktan

cr +⎟⎠⎞

⎜⎝⎛⋅−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛⋅=Δ (A.5.2.1.6.2.1)

where ΔPcr is the non-dimensionalized increase in axial buckling strength, Ptank is the internal

pressure in the tank (Pa), E is the Young’s modulus of the material (Pa), Dtank is the tank

diameter (m) and t is the tank wall thickness (m).

The range of preliminary tank dimensions and pressures gives us between 15% - 20%

improvement in axial buckling stress for a pressurized cylindrical tank.

2ktanktan

t2D

EP

⎟⎠⎞

⎜⎝⎛

crPΔ



A.5.2.1.6.2.2 Tank Bending

Tank bending allowables are determined from Bruhn Fig C8.13a.1 The values from Bruhn are

determined from experimental results, and we plot data points and graph them in Microsoft

Excel with a log curvefit to derive an expression for allowable bending stress for a given L/D

ratio.

Fig. A.5.2.1.6.2.2 Allowable bending strength for an unpressurized cylindrical tank

(Chii Jyh Hiu)

From Fig. A.5.2.1.6.2.2, we obtain the following curvefits for unpressurized tank bending

allowable values:

y = 7.9915x-1.5949

y = 6.938x-1.5945

y = 6.3472x-1.6021

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

100 1000 10000

Fbcr

/E

D/2t

L/D = 2

L/D = 4

L/D = 8



For L/Dtank = 2, 1.5949

tan/ 7.99152

kDFbcr Et

−⎛ ⎞= ⋅⎜ ⎟⎝ ⎠

(A.5.2.1.6.2.2a)

For L/Dtank = 4,

1.5945tan/ 6.9382

kDFbcr Et

−⎛ ⎞= ⋅⎜ ⎟⎝ ⎠

(A.5.2.1.6.2.2b)

For L/Dtank = 8,

1.6021tan/ 6.34722

kDFbcr Et

−⎛ ⎞= ⋅⎜ ⎟⎝ ⎠

(A.5.2.1.6.2.2c)

where L is the tank length in m, Dtank is the tank diameter (m), Fbcr is the critical bending stress

(Pa), E is Young’s modulus (Pa), t is the tank wall thickness (m).

For L/D values between two lines, we choose the lower line (i.e. higher L/D ratio) for

conservativeness.

We then employ Bruhn Figure C8.14 to determine the increase in bending strength of a

pressurized tank. Data points from Bruhn are plotted into Microsoft Excel and a log curvefit is

applied.



Fig. A.5.2.1.6.2.3 Non-dimensional increase in pressurized tank bending stress

(Chii Jyh Hiu)

From Fig. A.5.2.1.6.2.3, we obtain an equation for the non-dimensional increase in bending

strength for a pressurized tank: 2

tan tan0.0604 ln 0.35832

k kb

P DCE t

⎛ ⎞⎛ ⎞Δ = ⋅ ⋅ +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ (A.5.2.1.6.2.3)

And the final bending allowable

*(1 )press bFbcr Fbcr C= + Δ (A.5.2.1.6.2.4)

where Fbcrpress is the bending stress allowable for a pressurized tank (Pa), Fbcr is the bending

stress allowable for an unpressurised tank (Pa), ΔCb is the non-dimensionalized increase in

bending strength, Ptank is the internal pressure in the tank (Pa), E is the Young’s modulus of the

material (Pa), Dtank is the tank diameter (m) and t is the tank wall thickness (m).

y = 0.0604ln(x) + 0.3583

0.1

1

0.01 0.1 1 10

ΔC

b (N

orm

aliz

ed)

P/E(D/2t)^2



For the preliminary design candidates, this resulted in a bending stress increase of 30-40% over

an unpressurized tank.

References 1 Bruhn, E.F., Analysis and Design of Flight Vehicle Structures, Jacobs Publishing, 1973, Chapter C8 pgs. 347-353 2 Baker, E.H., Kovalevsky, L., Rish, F.L., Structural Analysis of Shells, Robert E. Krieger Publishing Company, Huntington, NY, 1981, pgs. 229-240



A.5.2.1.7 Math Models We analyze the structural strength of the launch vehicle with MATLAB codes and function calls.

The master structures function is tanks.m (for preliminary designs) or tanskv2.m (for final

design).

A.5.2.1.7.1 Evolution of Final Code Preliminary designs are analyzed iteratively with the master MAT function mainloop.m. The

structures codes interact with the propellant codes to optimize vehicle inert mass and propellant

requirements. The required Δv and payload masses are entered into mainloop.m, which passes

the requirements into the propellant codes. The propellant codes then guess the Δv distribution

among the stages of the launch vehicle and calculate the required propellant masses and a target

inert mass fraction (IMF) which is passed to the structures code (tanks.m). tanks.m then designs

the launch vehicle structural components using the algorithm illustrated in Fig. A.5.2.1.7.1, and

determines whether it meets the target mass function. If the target IMF is not met, the loop is run

again with different variables until an optimal solution is reached. Due to the large number of

iterations in this process, we disable some of the less critical analyses in the structures codes and

use placeholder values and assumptions for mass instead.

For final design, the iterative loop is no longer required. We are able to use more sophisticated

algorithms to design and optimize the structure. Also, function calls that were disabled in

preliminary design (grayed out boxes) are re-enabled for final design calculations. The output

from the structures code represents the final design of the launch vehicle.



A.5.2.1.7.2 Algorithm Flowcharts

Fig. A.5.2.1.7.1 Algorithm flowchart of tanks.m/tanksv2.m

(Chii Jyh Hiu)

Propellant Type Propellant Mass

Tank Material Max g loading

Aerodynamic

Loads Calculate Dimensions to hold propellant volume

Spherical

Tank?

L/D > 3?

Inter-Tank Analysis

Add hoops and/or increment thickness Calculate stage mass

Tank Buckling & Bending

Inter-stage Skirt

Analysis

Add hoops and/or increment thickness

Add stringers/hoops

NO

YES

YES

NO

FAIL PASS

PASS

COST, Dimensions, Mass

PASS

FAIL

FAIL

PASS

Prop Code (in MAT)

---- Dashed Lines indicate path only used in MAT preliminary design.

Grayed-out boxes are

only used in final code. OUTPUT

Meet IMF

target?

PASS

FAIL

OUTPUT


Author: Jesii Doyle

A.5.2.2 Inter-Stage Skirts

A.5.2.2.1 Overview We design inter-stage skirts to act as both an aerodynamic fairing and structural support between

each stage of the launch vehicle. The inter-stage skirts consist of a 4.0mm thick skin, a variable

number of stringers, and a variable number of ring supports. The number of stringers, stringer

thickness and number of ring supports found in any inter-stage skirt is determined through

structural analysis of the skirt. The taper angle α of the inter-stage skirt remains fixed at 10° from

the vertical axis, and the top and bottom diameters are determined by the stage diameters above

and below the skirt. Figure A.5.2.2.1.1 shows a schematic of the inter-stage skirts. Figure

A.5.2.2.1.2 displays an example of a designed inter-stage skirt.

Fig. A.5.2.2.1.1: Inter-stage skirt stringer and ring support configuration

(Jesii Doyle)

α θ

Support Ring

Stringers


Author: Jesii Doyle

Fig. A.5.2.2.1.2: 5kg launch vehicle inter-stage skirt between stage 1 and stage 2

(Jesii Doyle)

The inter-stage skirts are designed to support the maximum applied axial force of the mass of the

stage above the inter-stage skirt multiplied by the maximum g-loading. Also, the inter-stage

skirts must withstand the maximum shear force that occurs during flight. The skin of the skirt

does not transfer any load, and simply acts as the aerodynamic fairing between stages. All static

and dynamic loads are transferred through the inter-stage skirt stringers and ring supports. The

inter-stage skirt located between the second and third stages of each launch vehicle also acts as

the mounting area for the avionics package.

Project Bel

A.5.2.2.2

We base

cone. Th

the stage

any supp

in the ske

In the ab

is the axi

the large

Applying

the wall

intends t

material

skirt’s co

length m

For certa

angle for

these situ

it so that

llerophon

2 Preliminary

the prelimi

he bottom of

e above it. W

port from str

etch.

Fig. A

ove figure, w

ial length of

end of the c

g our assum

thickness, a

o support th

and manufa

one angle is

must be long e

ain launch v

r the inter-sta

uations, we a

t the nozzle

y Design

inary design

f the cone co

We initially

ingers or sup

A.5.2.2.2.1. Geo

we illustrate

f the cone, r1

cone.

mptions and g

and the mate

he maximum

acturing. In

s prescribed

enough to ho

vehicle conf

age skirt wil

add a cylind

is housed by

Author: J

n of the inte

onnects to th

assume that

pport rings.

ometry for the

(Jessic

that the con

is the radiu

geometric co

erial to desig

m applied ax

n addition t

to be betwe

ouse the noz

figurations, w

ll not provid

der to the bot

y the skirt.

Jessica Schoenb

r-stage skirt

he stage belo

t the inter-st

In Fig. A.5

preliminary de

ca Schoenbaue

ne angle, α, i

us of the sma

onstraints, w

gn an inter-

xial load whi

to the const

een ten and

zzle.

we know th

e a skirt that

ttom of the c

Although pl

bauer

t on the sim

ow it and the

tage skirt is

5.2.2.2.1, the

esign of the inter)

is the taper a

all end of the

we vary the

stage skirt.

ile minimizi

traints ment

sixty degre

hat using the

t is long eno

cone that int

lacing the cy

mple geometr

e top of the

made of a t

e basic geom

ter-stage skir

angle from t

e cone, and r

quantities o

The inter-s

ing the cost

ioned above

ees1, and the

e ten degree

ugh to house

terfaces with

ylinder on th

ry of a trun

cone connec

thin wall wi

metry is prese

rt.

he vertical a

r2 is the radi

f the cone a

stage skirt d

associated t

e, the inter-

e inter-stage

e minimum

e the nozzle

h the stage b

he top of co

519

ncated

cts to

ithout

ented

axis, l

ius of

angle,

design

to the

-stage

skirt

taper

. For

below

one to


Author: Jessica Schoenbauer

interface with the stage above it reduces the mass (and hence the cost), we cannot place it in this

configuration because it may not be large enough in diameter to accommodate the nozzle. We

illustrate the two possible geometries for the inter-stage skirt and where they are in association

with the launch vehicle in the figure below.

Fig. A.5.2.2.2.2. Skirt configurations and their position in the launch vehicle.

(Jessica Schoenbauer)

The figure shows the two inter-stage skirts on a three stage launch vehicle. Skirt 1 connects the

first stage to the second stage. This inter-stage skirt could not house the nozzle by using the

minimum cone angle and therefore had the cylinder added to the bottom of it to allow for

housing of the nozzle. Skirt 2 follows the other configuration for the inter-stage skirt connecting

the second stage to the third stage. The bottom of the cone connects to the second stage while

the top of the cone connects to the third stage.

Skirt 1

Skirt 2



References 1 “NASA Space Vehicle Design Criteria (structures): Buckling of Thin-walled Truncated Cones,” NASA SP-8019, September 1968.

Project Bel

A.5.2.2.

Buckling

determin

ultimate

objects, a

Although

difficult

and num

are very

helpful.

Many di

explore t

The first

where D

(m), N is

the shell

The exac

Where C

llerophon

3 Effect of

g analysis of

ning its struc

and compre

above streng

h this value

and sometim

merical metho

useful to o

fferent meth

three of these

method1 for

is the bend

s the uniform

(m), and R i

ct solution fr

C is a constan

f Thickness

f a long and

ctural stabili

essive stress

gth considera

governs th

mes impossib

ods are nece

our analysis

hods have b

e methods, a

r analyzing b

ding rigidity

m axial comp

is the radius

rom this gene

nt of integrat

Autho

s on Buckli

slender obj

ity. General

.1 Therefor

ations.

he design of

ble to determ

essary to det

s because tr

been develop

all of which w

buckling beg

(N/m), w is

pressive forc

(m).

eral equation

tion.

or: Molly Kan

ing

ect, such as

lly, buckling

e, we see th

f the launch

mine.1 In m

termine the c

ying to dev

ped for this

were derived

gins with the

s the radial

ce (N), E is

n is then giv

ne

our launch

g occurs wit

hat buckling

h vehicle, ex

most instance

critical buck

velop more

application

d from Von

e general equ

deflection (m

Young’s mo

en by Eq. (A

vehicle, is v

th loads belo

governs the

xact solution

es, estimatio

kling values.

exact soluti

n. For our a

Karman’s eq

uation below

m), x is the

odulus (Pa),

A.5.2.2.3.2).

very importa

ow the valu

e design of

ns are extre

ons, assumpt

. These met

ions may no

analysis, we

quations.1

w.

(A.5.2.2

axial coord

h is thickne

(A.5.2.2

522

ant in

ues of

these

emely

tions,

thods

ot be

e will

2.3.1)

dinate

ess of

2.3.2)

Project Bel

Simplifyi

where μ i

The seco

The critic

Where C

shell (m)

The third

classic so

Where E

cylinder

In our d

ensure th

requirem

and shap

llerophon

ing, the criti

is Poisson’s

ond method2

cal stress val

Cc is approx

), R is the rad

d and final m

olution for ax

E is Young’s

(m).

esign, the le

he stability o

ments. The tr

e.

ical force can

ratio for the

2 for analysi

lue is found

ximately equ

dius of the c

method3 tha

xially compr

modulus (P

ength is dic

of the launch

rends seen f

Autho

n be found a

e material.

is of bucklin

via Eq. (A.5

ual to 0.6, E

ylinder (m),

at we incorp

ressed cylind

Pa), h is the

ctated by pro

h vehicle, co

from the thre

or: Molly Kan

as

ng assumes

5.2.2.3.4).

E is Young’s

and γ is rep

porate in our

ders. It simp

thickness of

opulsion sys

omponent th

ee different

ne

an unpressu

s modulus (P

presentative o

r analysis o

plifies to the

f the shell (m

stems, tanks

hickness is v

methods are

urized, long

Pa), t is the

of the R/t rat

of buckling i

e following f

m), and a is

s, and other

varied to rea

e also very d

(A.5.2.2

cylindrical

(A.5.2.2

thickness o

tio.

is considere

form

(A.5.2.2

the length o

r equipment.

ach the nece

different in v

523

2.3.3)

shell.

2.3.4)

of the

d the

2.3.5)

of the

. To

essary

value


Author: Molly Kane

Fig. 5.2.2.3.1: Method 1 Trends for Critical Pressure

(Molly Kane)

Method 1 shows an exponential rise in thickness for increased critical pressure.


(Molly Kane)

Method 2, however, shows a linear relationship, as does Method 3 below.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

4

5

6

7

8

9x 104

thickness (m)

Crit

ical

Pre

ssur

e (P

a)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

4

5

6

7x 106

thickness (m)

Crit

ical

Pre

ssur

e (P

a)


Author: Molly Kane


(Molly Kane)

Overall, the first method1 gives a low-end calculation to ensure the stability of our launch

vehicle. However, through this research it is seen that a more in-depth analysis of the launch

vehicle must be completed with piecewise steps and including the pressurized tank analysis.

This essentially divides the launch vehicle into many smaller elements and allows thicknesses to

be changed for each part, rather than for the entire structure.

References 1Wang, C.Y., Wang, C.M., Reddy, J.N., Exact Solutions for Buckling of Structural Members, CRC Press, Boca Raton, FL, 2005. 2Brush, D.O., Almroth, B.O., Buckling of Bars, Plates, and Shells, McGraw Hill, 1975, pgs. 161-165. 3Baker, E.H., Kovalevsky, L., Rish, F.L., Structural Analysis of Shells, Robert E. Krieger Publishing Company, Huntington, NY, 1981, pgs. 229-240.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

1

2

3

4

5

6

7

8

9

10x 108

thickness (m)

Crit

ical

Pre

ssur

e (P

a)



A.5.2.2.4 Buckling Analysis

We perform a buckling analysis to help drive the preliminary design of the skirt. We base the

analysis on the fact that the maximum axial load must not exceed the critical buckling allowable.

The critical buckling load for the skirt is given by Eq. (A.5.2.2.4.1).

( )2

22

13cos2ν

απγ−

=EtPcr (A.5.2.2.4.1)

where Pcr is the critical axial load on the cone (N), γ is the correlation factor to account for

differences between classical theory and predicted instability loads, E is Young’s modulus (Pa), t

is the wall thickness (m), α is the taper angle from the vertical axis, and ν is Poisson’s ratio.1

We employ γ equal to 0.33 as suggested by a NASA design manual document.1 This value gives

a lower bound on the experimental data obtained through testing of cones. We also limit α, the

taper angle from the vertical axis, from ten degrees to 60 degrees as recommended.1

We iterate through values of α and t that meet the required critical axial load. Through the

iterations, we choose the least expensive skirt based on material and manufacturing costs.

After we establish the geometry for the skirt, we determine the critical bending moment for the

skirt using Eq. (A.5.2.2.4.2).

( )2

21

2

13cos2ν

απγ−

=rEtM cr (A.5.2.2.4.2)

where Mcr is the critical bending load on the cone (N*m), γ is the correlation factor to account

for differences between classical theory and predicted instability loads, E is Young’s modulus

(Pa), t is the wall thickness (m), r1 is the radius of the small end of the cone (m), α is the taper

angle from the vertical axis, and ν is Poisson’s ratio.1

We apply γ equal to 0.41 as suggested by a NASA design manual document.1



We also find the critical torsional moment on the cone. We employ Eq. (A.5.2.2.4.3).

( )45

21

2

3

1128.52 ⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

−=

tr

ltEtTcr ν

γ (A.5.2.2.4.3)

where Tcr is the critical torsional load on the cone (N), γ is the correlation factor to account for

differences between classical theory and predicted instability loads, E is Young’s modulus (Pa), t

is the wall thickness (m), l is the axial length of the cone (m), ν is Poisson’s ratio, and r is given

by Eq. (A.5.2.2.4.4).1

2

121

1

221

1

22 1

211

211cos

rr

rr

rrrr

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++=

−

α (A.5.2.2.4.4)

where r2 is the radius of the large end of the cone (m), r1 is the radius of the small end of the cone

(m), and α is the taper angle from the vertical axis.1

We apply γ equal to 0.67 suggested by a NASA design manual document.1

References 1 “NASA Space Vehicle Design Criteria (structures): Buckling of Thin-walled Truncated Cones,” NASA SP-8019, September 1968.


Author: Jesii Doyle

A.5.2.2.5 Stringer Analysis Stringers are added to the structure to increase the inter-stage skirt’s ability to withstand shear

force. We choose rectangular cross-section stringers because that results in more simple

manufacturing and attachment to the skirt skin. Simplifying the manufacturing and attachment of

structural members decreases the overall cost of the stringers. Since we aim to minimize total

cost of the launch vehicle, low cost is a valuable feature of the inter-stage skirts. We assume that

the stringers must withstand both axial compression loading from the weight of the structure

above the skirt, and maximum shear loading that may occur during the mission. Figure

A.5.2.2.5.1 shows the general configuration of the inter-stage skirt.

Fig. A.5.2.2.5.1: General skirt geometry without internal structural support

(Jesii Doyle)

First, the stringers are designed to withstand the axial compression loading from the weight of

the structure above the inter-stage skirt. For this analysis, we assume that the rectangular cross-

section stringers carry the entire axial load, and that the load is distributed evenly through each

stringer. Therefore, the stringer is treated as a cantilever beam with a point force. Since the inter-

stage skirt is rigidly attached to the stages above and below it, we assume that the stringer can be

analyzed as if it is rigidly attached at the base. Figure A.5.2.2.5.2 shows the application of the

axial force on a stringer with respect to the inter-stage skirt angle α.

α

dupper stage

dlower stage

tskin


Author: Jesii Doyle

Fig. A.5.2.2.5.2: Axial force applied on stringer

(Jesii Doyle)

The applied stress on the stringer is determined by using Eq. (A.5.2.2.5.1).

IMc

=σ (A.5.2.2.5.1)

where M is the applied moment (N*m), c is the vertical distance from the center line (m), and I is

the area moment of inertia (m4).1

The applied force on one stringer is determined using Eq. (A.5.2.2.5.2).

nP

P applied= (A.5.2.2.5.2)

where Papplied is the total axial force multiplied by the maximum G-loading (N), and n is the

number of stringers.

We determine the applied moment by the following equation.

αα cos2

PhsinPLM A −= (A.5.2.2.5.3)

where P is the applied force on one stringer (N), L is the length of the stringer (m), h is the height

of the stringer cross-section (m), and α is the skirt taper angle.

The maximum applied stress occurs at the maximum applied moment and maximum vertical

distance from the center line. Since the load is considered constant through each stringer, the

maximum applied moment occurs at the furthest point from the applied load. That is, the

P

L b

h

α


Author: Jesii Doyle

maximum applied moment occurs at the base of the stringer. The maximum vertical distance

from the center line occurs at both the top and bottom faces of the stringer as it is shown in Fig.

A.5.2.2.5.2. Therefore, the maximum applied stress occurs at the top and bottom faces at the base

of the stringer, and can be calculated as shown in Eq. (A.5.2.2.5.4) below.

⎟⎠⎞

⎜⎝⎛ −=== αασ cos

2hsinL

bhP6

bhM6

2h

IM

22AA

max (A.5.2.2.5.4)

where all variables are as defined previously.

The maximum applied stress must be less than the yield stress of the stringer material multiplied

by the reserve factor for the inter-stage skirt to satisfy the catastrophic failure requirement.1

Therefore, the number of required stringers is determined by increasing the number of stringers

evenly until the requirement is obtained. If the number of required stringers is greater than the

maximum number of stringers that will fit in the smallest radius, the stringer thickness is

increased and the procedure for determining number of stringers is repeated. We iterate this

process until a valid solution is found.

This analysis of the stringers in axial compression results in a very large number of stringers

required. A large number of stringers results in a greater inter-stage skirt inert mass and a greater

manufacturing and attachment cost. To resolve this undesirable result, we add ring supports to

the inter-stage skirt structure. These ring supports also have a rectangular cross-section due to the

reduced manufacturing costs. The addition of the ring supports results in creating shorter

stringers, which can withstand greater axial loading. Figure A.5.2.2.5.3 displays the inter-stage

skirt stringer configuration with added ring supports.


Author: Jesii Doyle


(Jesii Doyle)

The maximum applied stress equation is now described by Eq. (A.5.2.2.5.5).

⎟⎟⎠

⎞⎜⎜⎝

⎛−= αασ cos

2hsin

nL

bhP6

ring2max (A.5.2.2.5.5)

where nring is the number of ring supports, and all other variables are as previously defined.

Next, we ensure that the stringers are able to withstand the maximum shear force applied to the

shear center of the inter-stage skirt. The shear stress in each stringer is determined by the

following equations.

skin

rr

rrxx

yr

2rrxx

r

1rskin

r

1rskinrr

tq

yBIS

q

yBI

yy2

6rt

yy2

6rtAB

=

−=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛++=

∑

−+

σ

θθ

(A.5.2.2.5.6a)

(A.5.2.2.5.6b)

(A.5.2.2.5.6c)

(A.5.2.2.5.6d)

where Ar is the cross-sectional area of each stringer (m2), tskin is the thickness of the inter-stage

skirt skin (m), r is the minimum skirt radius (m), θ is the angle between stringers, yr is the y-

distance from the center of the skirt to each stringer (m), and Sy is the applied shear force (N).2

α θ


Author: Jesii Doyle

The shear stress is analyzed at the minimum inter-stage skirt radius because that is where the

maximum shear stress occurs. The location of the maximum shear stress was determined by

calculating the shear stress in the stringers for varying radii from the minimum to maximum

inter-stage skirt radius. This maximum shear stress multiplied by the reserve factor must be less

than the ultimate shear strength of the stringer material to satisfy the catastrophic failure

requirement. If the previously determined number of stringers does not meet this requirement,

we must add additional stringers.

By accomplishing this analysis through Matlab code skirt_v3.m, we determine the final number

of stringers, ring supports, and stringer thicknesses for each inter-stage skirt for each launch

vehicle configuration. From these variables, the skirt mass and cost are also determined and

output through skirt_v3.m.

References 1 Bedford, A., Fowler, W., and Liechti, K.M., Statics and Mechanics of Materials, Pearson Education Inc., Upper Saddle River, New Jersey, 2003. 2 Megson, T.H.G., Aircraft Structures for Engineering Students, Vol. 3, Elsevier Butterworth-Heinemann, Burlington, MA, 1999, Ch. 10.2.


Author: Jesii Doyle

A.5.2.2.6 Math Models In the final inter-stage skirt analysis code, we incorporate stringer and ring internal supports. The

skin of the inter-stage skirt does not transfer any load, and simply acts as the aerodynamic fairing

between stages. All static and dynamic loads are transferred through the inter-stage skirt stringers

and ring supports. The inter-stage skirt stringers and ring supports are designed to support the

maximum applied axial force and the maximum shear force that occurs during launch.

The number of stringers and number of ring supports are designed so that the maximum applied

stress on one stringer multiplied by the reserve factor is less than the yield strength of the stringer

material. The maximum applied stress equation is described by Eq. (A.5.2.2.6.1).

⎟⎟⎠

⎞⎜⎜⎝

⎛−= αασ cos

2hsin

nL

bhP6

ring2max (A.5.2.2.6.1)

where σmax is the maximum applied stress (Pa), P is the force on one stringer (N), b is the width

of the stringer cross-section (m), h is the height of the stringer cross-section (m), L is the length

of the stringer (m), nring is the number of ring supports, and α is the taper angle from the vertical

axis.1

We check that the number of stringers is sufficient to overcome the maximum applied shear

force. The maximum calculated shear stress multiplied by the reserve factor must be less than the

shear stress allowable of the stringer material. The shear stress in each stringer is determined by

the following equation.

skin

rr

rrxx

yr

2rrxx

r

1rskin

r

1rskinrr

tq

yBIS

q

yBI

yy2

6rt

yy2

6rtAB

=

−=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛++=

∑

−+

σ

θθ

(A.5.2.2.6.2a)

(A.5.2.2.6.2b)

(A.5.2.2.6.2c)

(A.5.2.2.6.2d)

where Ar is the cross-sectional area of each stringer (m2), tskin is the thickness of the inter-stage

skirt skin (m), r is the minimum skirt radius (m), θ is the angle between stringers (rad), yr is the

y-distance from the center of the skirt to each stringer (m), and Sy is the applied shear force (N).2


Author: Jesii Doyle

References 1 Bedford, A., Fowler, W., and Liechti, K.M., Statics and Mechanics of Materials, Pearson Education Inc., Upper Saddle River, New Jersey, 2003. 2 Megson, T.H.G., Aircraft Structures for Engineering Students, Vol. 3, Elsevier Butterworth-Heinemann, Burlington, MA, 1999, Ch. 10.2.


Author: Jesii Doyle

A.5.2.2.7 Evolution of Final Code The structures code tanks.m incorporates the inter-stage skirt code into the launch vehicle. The

inter-stage skirt code consists of two parts: the inter-stage skirt analysis code

skirt_analysis_v3_str.m which accomplishes the structural analysis of a given skirt geometry

and the overall inter-stage skirt code skirt_v3.m which compiles the inert mass, length, and cost

values for the total inter-stage skirt package for each launch vehicle. The inputs to the inter-stage

code are the inert masses of all stages, the propellant masses of all stages, the diameters of each

stage, and the maximum forces applied to the launch vehicle. The outputs of the inter-stage skirt

code are the inert mass, vertical length, and cost of each skirt required for a particular launch

vehicle.

The overall inter-stage skirt code takes the necessary inputs from tanks.m, and sums all masses

for each stage. Then, we multiply the resultant masses by the gravitation velocity and the

maximum vertical G-forces to get the axial loading force. Also, the input shear force is

multiplied by the maximum normal G-forces to get the applied shear force. Next, the overall

inter-stage skirt code calls the inter-stage skirt analysis code for each skirt required for the launch

vehicle configuration. For example, if the launch vehicle has two stages only one inter-stage skirt

is required, whereas if the launch vehicle has three stages, two inter-stage skirts are required. The

inter-stage skirt analysis code determines the final outputs by minimizing individual skirt cost.

The overall code then takes the results from the analysis code, compiles the final inter-stage skirt

data, and outputs the information to the tanks.m code.

The first iteration of the inter-stage skirt analysis code does not include any internal supports.

Therefore, we only use the buckling analysis of a thin-walled tapered cone to determine the

required thickness of the inter-stage skirt skin. The use of only the buckling analysis was

determined to be appropriate because the only force currently considered is axial loading.

This iteration of the inter-stage skirt analysis code varies the taper angle of the truncated cone,

the skin thickness and the skin material, with given values for the upper and lower stage

diameters to minimize skirt cost. Figure A.5.2.2.7.1 shows the inter-stage skirt configuration

used for this iteration.


Author: Jesii Doyle

Fig. A.5.2.2.7.1: General skirt geometry without internal structural support

(Jesii Doyle)

We quickly conclude that this configuration is not feasible due to the launch vehicle stages inert

mass values. This skirt code results in an unreasonably large inter-stage skirt thickness due to the

large axial loading and lack of internal support structure. Also, we must take applied shear force

into consideration during the structural analysis of the skirt. A thin-walled structure with no

stringers will not support shear force.

Since the first iteration of the inter-stage skirt analysis code is infeasible, we revise the code to

include stringers as internal supports. The stringers have a rectangular cross-section and run

length-wise along the skirt skin. We assume that the stringers take the entire axial loading force.

Therefore the inter-stage skirt skin acts as a non-load-bearing fairing, and its thickness is set at

4.0mm to correspond to the common tank thickness. This iteration of the inter-stage skirt

analysis code varies the taper angle of the truncated cone, the number of stringers, the stringer

thickness and the stringer material to minimize skirt cost. Since the thickness of the inter-stage

skirt skin remains constant, the material that will result in minimum cost is aluminum. Therefore,

the skirt skin material is set as aluminum.

α

dlower stage

tskin

dupper stage


Author: Jesii Doyle

The second iteration of the inter-stage skirt analysis code results in a very large number of

stringers required to support the axial forces. A large number of stringers results in a greater skirt

inert mass and a greater manufacturing and attachment cost.

To resolve this undesirable result, we iterate the inter-stage skirt analysis code to add ring

supports to the inter-stage skirt structure. These ring supports also have a rectangular cross-

section. The addition of the ring supports results in creating shorter stringers, which can

withstand greater axial loading. This iteration of the inter-stage skirt analysis code varies the

taper angle of the truncated cone, the number of stringers, the stringer thickness, the stringer/ring

support material, and the number of ring supports to minimize skirt cost. Figure A.5.2.2.7.2

displays the inter-stage skirt configuration with added stringers ring supports.


(Jesii Doyle)

In this third iteration of the inter-stage skirt analysis code, we incorporate the applied shear force.

Up until this iteration, the applied shear force was ignored. We ensure that the number of

stringers required to overcome the axial loading are also able to withstand the maximum shear

force applied to the shear center of the inter-stage skirt. This maximum calculated shear stress

multiplied by the reserve factor must be less than the ultimate shear strength of the stringer

material. If the previously determined number of stringers does not meet this requirement, the

inter-stage skirt analysis code outputs a “no” value and more stringers must be added.

α θ


Author: Jesii Doyle

Since this iteration of the inter-stage skirt analysis code varies so many parameters, it produces a

very long run time for tanks.m. To overcome the long run time, we decide to set many

parameters constant. The stringer/ring support material is set to aluminum to reduce

manufacturing and assembly costs. The taper angle of the truncated cone is set to 10° because

that is the most common taper angle that resulted from the structural analysis. Also, the number

of stringers is set to a fraction of the total number of stringers that will fit in the minimum radius

of the skirt. Therefore, even though this number will vary according to the launch vehicle

geometry, it will not result in a time consuming for-loop in the code. We discover through

multiple runs of the inter-stage skirt analysis code that the approximate minimal number of

stringers required to overcome all necessary loading for all inter-stage skirts is 1/6th of the total

number of possible stringers.

The final iteration of the inter-stage skirt analysis code varies only two parameters. The stringer

thickness and the number of ring supports are iterated within the inter-stage skirt analysis code to

minimize skirt cost. The upper and lower stage diameters, skirt skin and internal support

material, taper angle of the truncated cone, and the number of stringers are held constant for each

inter-stage skirt configuration. This final iteration of the inter-stage skirt analysis code is called

skirt_analysis_v3_str.m and is incorporated in the skirt_v3.m overall code called by tanks.m.



A.5.2.2.8 Algorithm Flowcharts

We construct the final code for the inter-stage skirt analysis. The codes applies inputs of an axial

load, the radius of the stage below the skirt, the radius of the stage above the skirt, the length of

the nozzle, and the shear force. We then specify the cone angle to be equal to ten degrees and

the skin thickness to be four millimeters. Next, the number of ring supports is declared as 2.

The skirt length is then calculated and compared with the nozzle length. If the skirt length is less

than the nozzle length, then a cylinder is added to the bottom of the skirt so that the nozzle can

be housed within the skirt. We calculate the number of stringers, the load applied to each

stringer, and the length of each stringer next. We specify the stringer thickness to be equal to

three millimeters.

Then we determine the maximum stress in the skirt. If the skirt has a yield stress that is less than

the maximum stress, the code returns to the point where the stringer thickness is specified. The

code iterates through the stringer thickness until the yield stress is not exceeded or the stringer

thickness becomes greater than eight millimeters. In the situation that the stringer thickness is

exceeded, the code returns to the point where the number of ring supports is declared. We then

iterate through this process until the maximum stress does not exceed the yield stress.

After meeting the stress specification, we calculate the mass and the cost of manufacturing and

the materials for each skirt. We also determine the critical bending, pressure, and torsional

loads. Then, we choose the skirt that costs the least amount of money. Then we calculate the

shear stress in the skirt. If the shear stress is less than the yield shear stress, we return a pass. If

the shear stress is greater than the yield stress, we return a fail. The code then outputs the skirt

mass, length, cost, the number of stringers, a pass or fail, and the stringer thickness. In Fig.

A.5.2.2.8.1, the code algorithm is written out in a flow chart format so that the code may be

easily understood.



Fig. A.5.2.2.8.1. Flow chart of algorithm used for skirt analysis.


Inputs:Axial load

Radius of stage belowRadius of stage above

Nozzle lengthShear force

Start Skirt Code

Choose:Cone angle =10˚

Skin thickness = 0.004 m

Choose:No. of

Support rings

yesno

no

yes

Add cylinderTo bottom of

Skirt for nozzle

Outputs:Skirt massSkirt lengthSkirt cost

Number of stringersPass/Fail

Stringer thickness

yes(pass)

no(fail)



Stringer thickness

Calculate:No. of stringers

Load applied to stringerStringer length

ChooseStringer

thickness

CalculateMaximum

stress

Length>

nozzle

Yield stress

>Max

stress

Calculateskirt

length

Stringerthickness

<0.008 m

no

yes

IncrementStringer

thickness

CalculateSkirt mass

CalculateManufacturingand material

costs

Calculatecritical bending,pressure, and torsional loads

CalculateShear stress



ChooseStringer

thickness

CalculateMaximum

stress

Stringerthickness

<0.008 m

no

yes

no

yes

Yield stress

>Max

stress

IncrementStringer

thickness

CalculateSkirt mass


costs


Yieldstress

>Max shear

stress

End Skirt Code

Inputs:Axial load

Radius of stage belowRadius of stage above

Nozzle lengthShear force

Start Skirt Code

Choose:Cone angle =10˚

Skin thickness = 0.004 m

Choose:No. of

Support rings

yesno

no

yes

Add cylinderTo bottom of

Skirt for nozzle



Stringer thickness

yes(pass)

no(fail)



Stringer thickness



ChooseStringer

thickness

CalculateMaximum

stress

Length>

nozzle

Yield stress

>Max

stress

Calculateskirt

length

Stringerthickness

<0.008 m

no

yes

IncrementStringer

thickness

CalculateSkirt mass


costs


CalculateShear stress



ChooseStringer

thickness

CalculateMaximum

stress

Stringerthickness

<0.008 m

no

yes

no

yes

Yield stress

>Max

stress

IncrementStringer

thickness

CalculateSkirt mass


costs


Yieldstress

>Max shear

stress

End Skirt Code


Author: Vincent J. Teixeira

A.5.2.3 Nose Cone

A.5.2.3.1 Overview The final nose cone design revolves around a power-law body, with a blunted tip in order to

reduce the effects of heating throughout ascent. During original rocket design, drag losses

provided a severe limitation to the capabilities and mission parameters. Power-law bodies are

the optimum shape for minimum drag when it comes to the leading edge of a body of

revolution.1 A power-law body one whose revolving surface is governed by Eq. (A.5.2.3.1.1). mxr R

L⎛ ⎞= ⎜ ⎟⎝ ⎠

(A.5.2.3.1.1)

where x is the position measured along the axis of symmetry (m), L is the total axial length of the

power-law body (m), R is the radius of the body at the end-point (m), m is a pre-determined

power-law body coefficient, and r is the radius of power-law body at axial position x (m).

To start with, we choose a power-law coefficient (m = 0.7) that corresponded with the lowest

drag achieved during Auman and Wilks’ experiments.1 Further design steps require defining the

length of the nose cone, as well as thermal and structural analyses. The initial length of the nose

cone is set to three times the radius of the base. Using this as a starting point, we attempt to

reach a balance between elongating the nose cone to reduce drag and shortening the length in

order to reduce the nose cone heating. Since the nose cone will be subjected to a high heating

rate due to the velocity through the atmosphere, we conclude that further elongating the nose

cone increases overall cost due to the heightened thermal requirements. As Eq. (A.5.2.3.1.1)

shows, when x approaches zero (defined as the tip of the nose cone), the radius also approaches

zero (resulting in an increasingly sharp tip). Figure A.5.2.3.1.1 shows the general, two-

dimensional outline of a power-law body using m = 0.7.

Project Bel

Figure A

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body and blunted tip of the cone, which will be subjected to the greatest thermal loading. Since

we assume that the thermal loading lessens as we move further from the tip (as discussed in

A.5.2.3.3), it is possible to use aluminum for the lower third of the nose cone. Final

considerations for the nose cone involved necessary internal structures to support both the static

and expected dynamic loading during flight. While the nose cone is located near stagnation

point during nominal flight, calculated pressure loadings are low enough to negate the use of

excessive internal supports. Four internal aluminum stringers are placed symmetrically around

the nose cone in order to support the weight of the blunted titanium tip and are capable of

withstanding the expected dynamic loads within the reserve safety factor of 1.25.

Further work into the design of the nose cone should focus on the use of ablatives and current

software available for the thermal analysis of bodies subjected to the expected thermal loads.

Most contemporary space launches employ ablative shells over leading surfaces like the nose

cone in order to reduce the necessity of using expensive, difficult materials such as titanium.

Further research into the use of ablatives may open up alternatives for the materials used

throughout the nose cone. Unfortunately, we were advised that detailed analysis using Sandia

One-Dimensional Direct and Inverse Thermal (SODDIT) is outside the scope and deadlines of

this design phase, which leaves us limited to metallic alloys.

References 1Auman, L.M. and Wilks, B. “Supersonic and Hypersonic Minimum Drag for Bodies of Revolution.” AIAA 2003-

3417, Orlando, FL, June 2003.


Author: Molly Kane

A.5.2.3.2 Material Analysis For the selection of the nose cone material we began with research into historically used

materials in similar applications. The two most similar applications we have studied are the

Vanguard rocket and the design for the Purdue Hybrid Launch Vehicle.1,2

The most widely used materials employed for such applications include aluminum, titanium,

magnesium alloys, molybdenum, carbon-carbon composites, and hafnium diboride.3 Since our

project is optimized with respect to cost, many of the ceramic and composite materials are

eliminated from potential materials due to their extremely high manufacturing and raw material

costs.4 Ablative materials are not considered for this design. From here we figure out which

material is the least expensive that can still fulfill the requirements of the nose cone.

The tip of the nose cone must withstand the thermal and aerodynamic forces experienced when

attaining orbit. Two relatively attainable and low cost materials, aluminum and titanium, are

studied to determine their feasibility with the application.

Table A.5.2.3.2.1 Material Specifications of Aluminum Alloys3

Alloy Tmelt (K) ρ (g/cm^3) σcomp (Mpa) 2014 780.37 2.80 399.90 2024 774.82 2.77 399.90 2219 816.48 -- 386.11 6061 855.37 2.70 151.68 7075 749.82 2.80 275.79 2618 822.15 2.76 399.90 X7005 880.37 2.80 324.05 7049 -- 2.77 448.16 7175 749.82 2.80 -- Averages 801.01 2.77 348.19

The average melting temperatures of aluminum alloys is about 800 K. While inexpensive and

widely available, this temperature does not reach the constraints for the thermal loading that will

be seen by the nose cone while in is ascent to orbit.

Table A.5.2.3.2.2: Material Specifications of Titanium Alloys3


Author: Molly Kane

Alloy Tmelt (K) ρ (g/cm^3) σcomp (Mpa) Ti 1922.04 4.51 551.58 Ti-5Al-2.5Sn 1810.93 4.46 758.42 Ti-6Al-4V 1810.93 4.48 868.74 Ti-7Al-4Mo 1922.04 4.37 1068.69 Ti-8Al-1Mo-IV 1977.59 4.82 999.74 Ti-679 1977.59 4.70 999.74 Ti-8Mn 1772.04 -- 827.37 Ti-6-2-4-6 1922.04 4.56 1103.16 Averages 1889.40 4.56 897.18

Titanium alloys generally have a melting temperature around 1900 K. This value meets the

requirements of the similar rockets’ nose cones.1,2 While we are unable to complete a thorough

thermal analysis on our particular nose cone, we are able to design it from atmospheric

conditions during flight.

Ultimately we come to the conclusion that titanium is the cheapest and most easily accessible

material that could be considered for the nose cone of the rocket. This titanium material will

provide the tip of the nose cone with extra thermal protection and the remainder of the nose cone

is aluminum. This combination of metals allows us to reduce cost based on raw materials and

manufacturing of those materials.

References 1Klemans, B., The Vanguard Satellite Launching Vehicle, The Martin Company, Engineering Report No. 11022, April 1960. 2Tsohas, J., Droppers, L.J., Heister, S.D., “Sounding Rocket Technology Demonstration for Small Satellite Launch Vehicle Project”, 4th Responsive Space Conference, 2006. 3Brady, G.S., Clauser, H.R., Materials Hand Book, 13th Edition, McGraw-Hill, Inc, 1991. 4Jastrzebski, Z.D., The Nature and Properties of Engineering Materials, Second Edition. John Wiley & Sons, Inc., 1977.



A.5.2.3.3 Thermal Analysis Thermal analysis for the nose cone during ascent proves to be the limiting factor throughout the

design phase. An initial analysis of the power-law body as originally defined immediately

proves that the heating rate at the tip of the nose approaches infinity. This result implies infinite

heat transfer to the nose cone throughout flight. As an infinite heating rate is clearly

unacceptable, the first step in refinement requires blunting the tip of the nose cone in order to

bring the radius of curvature up. The heating rate of a leading body is dependent upon both the

physical shape of the object as well as the material properties. Heating rate is primarily

dependent upon the radius of curvature of the test body at a specific point as well as the specific

heat of the material used. The heating rate of a leading edge body can be theoretically

determined using Eq. (A.5.2.3.3.1).1

8 321.83 10 1

0.5pw w

n a

c Tq V

r h Vρ− ⎛ ⎞ ⎛ ⎞

= × −⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎝ ⎠&

(A.5.2.3.3.1)

where q is the heating rate per unit area (W/cm2), ρ is the density of the fluid (kg/m3), rn is the

radius of curvature of test body (m), V is the instantaneous velocity (m/s), cpw is the specific heat

of surface material (J/kg-K), and Tw is the instantaneous temperature at the surface (K).

We can see from Eq. (A.5.2.3.3.1) that the heating rate is dependent upon trajectory, material and

structural parameters. Since our design process does not entail changing the optimal trajectory

and therefore the velocity at any point in the launch, we are forced to focus on changes to both

the material and structural properties. Ideal design for meeting the thermal requirements entail

increasing the radius of curvature throughout the nose cone, especially at the stagnation point, as

well as employing a material with a higher specific heat. Eq. (A.5.2.3.3.1) clearly shows that as

the radius of curvature at a point decreases, it increases the instantaneous heat transfer, which

accumulates throughout the flight. Qualitative analysis alone is able to prove that the original

power-law body is unsuited to withstanding high velocity flight, which requires using a

simplified thermal analysis model with a blunted tip.

The initial heating rate equation requires a complicated iterative process as well as converting the

given heating rate from Eq. (A.5.2.3.3.1) to a heating rate per volume and then an overall



temperature. Initial steps to determine this heating rate requires a calculation of both the local

atmospheric enthalpy as well as the velocity contribution. The local, atmospheric enthalpy is

calculated using Eq. (A.5.2.3.3.2).

a ph C T= (A.5.2.3.3.2)

where ha is the local, atmospheric enthalpy (kJ/kg), Cp is the specific heat of air, defined as

1003.5 kJ/kg-K and T is the temperature at the desired altitude calculated using Standard

Atmosphere tables (K).

The velocity contribution is the 0.5V2 term of Eq. A.5.2.3.3.1, which contributes more to the

conditions on the surface of the nose cone due to our high velocity through high altitude/low-

density atmosphere. Figure A.5.2.3.3.1 shows the plot of the individual enthalpy terms, as well

as their combined value during the launch vehicle’s trajectory. This allows us to determine the

local conditions that will have an effect on the heating rate of the nose cone. Figure A.5.2.3.3.1

shows that since we are launching from a balloon at approximately 30km, the local atmospheric

enthalpy contributes very little to the overall enthalpy. As expected with a squared term, the

velocity contribution increases slowly at first and then rapidly as the velocity continues to

increase throughout ascent. While the velocity continues to increase until we reach the desired

velocity for our orbit, we only plotted our data through 65 km above Earth. At this altitude the

density of the air would be low enough that the air no longer operates under normal heating laws,

providing an upper limit for our calculations.



Fig. A.5.2.3.3.1: Enthalpy vs. time for proposed trajectory

(Vincent Teixeira)

The above research and analysis provides important insight into the factors that affect the heating

rate and overall temperature gain of the nose cone that we expect during ascent. However, we

are ultimately unable to both iterate and integrate the given function to provide an actual

temperature vs. time curve for ascent using various metallic alloys. Combining research from

Prof. Schneider1 and the tested components of the Vanguard rocket2, we decided to alter the tip

of the nose cone for a more favorable thermal survivability. Prof. Schneider simplifies the

heating rate calculation by assuming a blunt nosetip that serves as a massive heatsink.

Combining this with the Vanguard nose cone design, which used a solid titanium tip, we arrive at

the current design, which takes the original power-law body and replaces the sharp tip with a

solid blunt tip as shown earlier in Fig. A.5.2.3.1.2.

Ideally, the thickness of the nose cone skin would be determined by a similar thermal analysis in

order to provide the minimal mass necessary to protect both the interior of the nose cone and the

structural integrity of the nose cone itself. However, since we are unable to compute complete

solutions to the thermal loading of the body, we are unable to determine the minimum thickness

that our nose cone would need. Instead, we incorporate historical data from the Vanguard

rocket3 to define our thickness. We set the outer walls of our nose cone to be 1.75mm thick,

which is actually thicker than the 1.651mm (0.065in) nose cone used by the Vanguard rocket.2

As our thermal analysis shows, due to the high-altitude/low-density atmosphere of our ascent,

0 10 20 30 40 50 60 70 80 900

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (sec)

Ent

halp

y (k

J/kg

)

Local enthalpy, ha

Enthalpy due to velocityTotal enthalpy, ho



our thermal loading is expected to be less than that of the Vanguard rocket, which was ground-

launched.

A.5.2.3.4 Structural Analysis Once the nose cone is capable of handling the expected thermal loading, we begin to analyze the

structural capabilities of the nose. Of primary concern in this analysis is the stagnation pressure

on the blunt nose during ascent. Similar to the method used to determine the total enthalpy

during the ascent, local atmospheric pressure is calculated as a function of time during the ascent

using the Standard Atmospheric Tables while dynamic pressure is calculated using the absolute

velocity data provided by the Trajectory group. Stagnation pressure is therefore calculated using

Eq. (A.5.2.3.4.1).

20

12sP P Vρ= +

(A.5.2.3.4.1)

where P0 is the desired stagnation pressure (Pa), Ps is the local atmospheric pressure from the

Standard Atmosphere tables (Pa), ρ is the density of air at the current altitude (kg/m3) and V is

the absolute velocity of the launch vehicle (m/s).

Similar to the data gathered for enthalpy during ascent, the local atmospheric pressure

contribution is significantly smaller than that of the dynamic pressure, due mostly to the high-

altitude launch. Figure A.5.2.3.4.1 plots the stagnation pressure versus time for the launch

vehicle during ascent for the 5kg payload. As expected, the local atmospheric pressure drops off

quickly as the launch vehicle accelerates through the atmosphere. However, the dynamic

pressure curve initially starts at zero and increases quickly as a result of the rapidly accelerating

launch vehicle. Since the velocity term is squared, we expect the dynamic pressure to increase

rapidly and provide more of a contribution to the stagnation pressure before dropping off as a

result of the low-density atmosphere. Combining both values into a maximum stagnation

pressure allows us to determine the maximum axial loading for the nose cone.



Fig. A.5.2.3.4.1: Pressure vs. time for proposed trajectory

(Vincent Teixeira)

We initially assume that the solid titanium tip would be structurally capable of supporting the

stagnation pressure, which led to determining the need for axial strengthening throughout the

lower portion of the nose cone. In order to determine the compressive loading on any stringers

placed in the nose cone, we add the maximum expected stagnation pressure to the mass of the

solid titanium tip, at which point our reserve factor of safety of 1.25 is taken into account. Initial

tests assign the stringers to be made from aluminum in an effort to both reduce cost and mass. In

order to write a code that determines the necessary number of stringers to withstand the axial

loading, we arbitrarily set the stringer area. For this we choose to use stringers 3mm wide by

10mm deep, similar to those designed throughout the inter-stage skirts of the launch vehicle.

Using Eq. (A.5.2.3.4.2), we are able to calculate the required number of stringers to support both

the structural mass of the titanium tip as well as the stagnation pressure during ascent, assuming

that the titanium/aluminum wall does not carry any axial loading.

( )0tip tipc

s s

g m P An A

σ+

= (A.5.2.3.4.2)

where cσ is the yield stress of the stringers (Aluminum 7075), defined as 461 MPa for all

calculations, g is the assumed maximum G-loading during the flight, which we assume to be 6,

tipm is the mass of the solid titanium tip (kg), 0P is the maximum stagnation pressure calculated

0 20 40 60 80 100 120 140 1600

200

400

600

800

1000

1200

Time (sec)

Pre

ssur

e (P

a)

Atmospheric Pressure, Ps

Dynamic Pressure, qStagnation Pressure, P0



earlier (Pa), tipA is the area of the blunt tip (m2), sn is the required number of stringers, and sA is

the area of each individual stringer, arbitrarily set at 30mm2.

Using the 5kg payload as our test case, we find that the nose cone only requires 1.20 stringers to

support the required forces. For this calculation, we assume a maximum G-loading of 6,

concurrent with that provided by the Trajectory group and used throughout the structural analysis

of the entire launch vehicle. Since we clearly cannot have a fraction of a stringer, we decide to

include eight stringers in the nose cone, spaced evenly around the circumference in order to

support the necessary loading and meet the required factor of safety, set at 1.25 for structural

components. This stringer placement remains constant throughout all three launch vehicles in

order to provide added axial integrity to the nose cone.

Once the nose cone is capable of withstanding the expected thermal and structural loading, we

are able to finally calculate the required mass for the nose cone for each launch vehicle. Table

A.5.2.3.4.1 contains the mass of each nose cone.

Table A.5.2.3.4.1: Nose cone masses

Launch Vehicle Mass of Nose Cone (kg) 200g 1.7507 1kg 2.0435 5kg 1.7927



A.5.2.3.5 Math Model Flowchart

Fig. A.5.2.3.5.1: Nose Cone Math Model Flowchart

(Vincent Teixeira)

Nose Cone Mass

Input diameter of third stage

Input material properties

Define constants: Radius = D/2, Length = 3*R,

thickness = 1.75/1000, power coefficient = 0.7

Calculate volume/mass of titanium solid blunt tip modeled as 1/3 sphere

Calculate path integral for power-law curve from blunt tip (0.7*R) to bottom of cone (L)

Revolve path integral around axis of symmetry

Add stringer mass using pre-defined area of 30mm2, four stringers and length equal to path integral

Sum volumes, lengths and masses for each component to determine total mass and length

Mass and Length of nose cone



References 1Schneider, S.P., “Methods for Analysis of Preliminary Spacecraft Designs.” AAE 450 Spacecraft Design, Sept 2005. 2Klamans, B., “The Vanguard Satellite Launching Vehicle,” The Martin Company, Engineering Report No. 11022, April 1960.



A.5.2.4 Inertia Matrices

A.5.2.4.1 Elementary Theory We calculate inertia matrices for each launch vehicle. To find these matrices for most vehicles,

nine separate values must be obtained. These values are: three moments of inertia and six

products of inertia. Figure A.5.2.4.1.1 displays the typical inertia matrix that is required, along

with a visual representation of the coordinate system we used.

xx xy xz

xy yy yz

xz yz zz

I I II I II I I

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Fig. A.5.2.4.1.1: Inertia Matrix with Coordinate System

(Brandon White)

The coordinate system we chose for the launch vehicle is the following: the z-direction is along

the length of the vehicle, while the x-direction and y-direction are along two arbitrary radial

directions. For the particular configuration of our launch vehicle, a crucial assumption is made

that all products of inertia are equal to zero. This assumption is valid when the launch vehicle is

symmetric about the axis of rotation. Our launch vehicle is theoretically symmetric in the z-

direction, which is the axis of rotation. However, due to the fact that actual components of the

vehicle will not be symmetric about any axis, this creates inaccuracies in our calculations. In Fig.

A.5.2.4.1.2, the simplified inertia matrix we are using in final design is shown.

y

z

x



0 00 00 0

xx

yy

zz

II

I

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Fig. A.5.2.4.1.2: Simplified Inertia Matrix

(Brandon White)

A benefit to having a launch vehicle with a circular cross section is that the moments of inertia in

the x-direction and y-direction are going to be the same. The theory employed by the team is to

separate the entire launch vehicle into individual components, find the moments of inertia for

each component, and then sum the inertias together to get the total moments of inertia for the

launch vehicle.

The components of the launch vehicle that are used in the inertia calculations are summarized in

Table A.5.2.4.1.1. To approximate the moments of inertia of each component, the components

had to be configured as simple shapes.



Table A.5.2.4.1.1 Launch Vehicle Components and Approximate Shapes

Component Shape Nose Cone Hollow Cone Oxidizer Tanks Hollow Cylinders with End Plates Fuel Tanks Hollow Cylinders with End Plates Solid Fuel Casings Hollow Cylinders LITVC Point Mass Pressurant Tank Hollow Cylinders with End Plates Inter-Stage Skirts Truncated Hollow Cones Propellant Solid Cylinders Payload Point Mass Avionics 1st and 2nd Stage Truncated Hollow Cones Avionics 3rd Stage Point Mass Engine Nozzle Truncated Hollow Cone

A.5.2.4.2 Component Inertia

A.5.2.4.2.1 Cylinders For the components that are approximated as cylinders, moments of inertia are found using Eqs.

(A.5.2.4.2.1.1) through (A.5.2.4.2.1.5).

Hollow Cylinders:

( )22

21

iozz RRMI += (A.5.2.4.2.1.1)

where M is the total mass of the component (kg), Ro is the outer radius of the component (m),

and Ri is the inner radius of the component (m).

For the radial moments of inertia, the parallel axis theorem is employed to translate the moment

of inertia from the top of the component to the center of mass of the launch vehicle.

( ) 2222 33121 MxlRRMII ioyyxx +++==

(A.5.2.4.2.1.2)

where M is the total mass of the component (kg), Ro is the outer radius of the component (m), Ri

is the inner radius of the component (m), l is the approximate length of component (m), and x is

the distance from top of component to the launch vehicle center of mass (m).



The approximate length of each component is found because many times the propellant tanks are

designed to have hemispherical ends. The tanks are approximated, from Table A.5.2.4.1.1, as

hollow cylinders with end plates. The approximate length of the component is calculated using

Eq. (A.5.2.4.2.1.3). Figure A.5.2.4.2.1.1 visually explains this approximation. For solid rocket

motor casings, this approximation is not used because the casing is designed as a hollow cylinder

with square ends.

Fig. A.5.2.4.2.1.1: Hemispherical Tank Approximation

(Brandon White)

πRLl 4

+= (A.5.2.4.2.1.3)

where L is the total length of the cylindrical component (kg), and R is the radius of the

component (m).

Actual Approximate

L

R

2R/π

l



Solid Cylinders:

2

21 MRI zz =

(A.5.2.4.2.1.4)

where M is the total mass of the component (kg), R is the Radius of the component (m).

22

41

31 MxRlMII yyxx +⎟

⎠⎞

⎜⎝⎛ +==

(A.5.2.4.2.1.5)

where M is the total mass of the component (kg), R is the radius of the component (m), l is the

approximate length of component (m), and x is the distance from top of component to the launch

vehicle center of mass (m).

A.5.4.2.2.2 End Plates For cylinders that are designed with hemispherical ends, the approximate values for principal

moments of inertia are found for a square cylinder with cylindrical plates on each end. In the

axial direction, Eq. (A.5.2.4.2.2.1) is used. Equation (A.5.2.4.2.2.2) is the moment of inertia in

the radial direction.

22

21

21 MRMRI zz +=

(A.5.2.4.2.2.1)


2_

222

41

41

⎟⎠⎞

⎜⎝⎛+++== xMMRMxMRII yyxx

(A.5.2.4.2.2.2)

where M is the total mass of the end plate (kg), R is the radius of the end plate (m), x is the

distance from top of top end plate to the launch vehicle center of mass (m), and ⎟⎠⎞

⎜⎝⎛ _

x is the

distance from top of bottom end plate to the launch vehicle center of mass (m).

A5.4.2.2.3 Cones The principal moments of inertia for the nose cone are found by subtracting a small solid cone

from a larger solid cone. Figure A.5.2.4.2.3.1 shows this method.



Fig. A.5.2.4.2.3.1: Hollow Cone Approximation

(Brandon White)

For a solid cone, Eqs. (A.5.2.4.2.3.1) and (A.5.2.4.2.3.2) are employed.

2

103 MRI zz =

(A.5.2.4.2.3.1)


⎟⎠⎞

⎜⎝⎛ +== 22

203

53 RLMII yyxx

(A.5.2.4.2.3.2)

where M is the total mass of the component (kg), R is the radius of the component (m), L is the

vertical length of the component(m).

At first glance, this method appears very simple. However, the only properties of the nose cone

that are known are the vertical length, thickness, outer radius, mass and material. To use the

solid cone equations, both the larger and smaller solid cones had to be created with these

properties. The volumes of each solid cone are found and associated with a mass using the

material density. These are the masses used in Eqs. (A.5.2.4.2.3.1) and (A.5.2.4.2.3.2).

Knowing the thickness of the material provides enough information to calculate the vertical

length and radius of the smaller solid cone (Ex. The length of the smaller cone is the length of

the cone minus the thickness of the cone). The difference between the axial moments of inertia

for the two solid cones is the approximate axial moment of inertia for the hollow cone. For the

radial moment of inertia, the parallel axis theorem must be included after finding the difference

between the two solid cones. So, the radial moments of inertia are the difference of the two

calculations added to the product of the mass of the nose cone and the square of the distance

between the top of the nose cone and the vehicle center of mass.



A.5.2.4.2.4 Truncated Cones Certain components of the launch vehicle had to be approximated as truncated hollow cones. To

find the associated moments of inertia, the same method as the hollow cones is used with an

additional step. After finding a large hollow cone, a smaller hollow cone is subtracted off the

top, resulting in the truncated hollow cone approximation. Figure A.5.2.4.2.4.1 visually depicts

this.

Fig. A.5.2.4.2.4.1: Truncated Hollow Cone Approximation

(Brandon White)

A.5.2.4.2.5 Point Masses Components of the launch vehicle with completely unknown geometries had to be approximated

as point masses. The only inertia property known about the payload satellite, LITVC, and third

stage avionics is the mass. Without any other information, we are hamstrung into making the

decision to approximate them as point masses. Fortunately, all of these components are very

small in comparison to the rest of the launch vehicle. This approximation results in small

inaccuracies in final inertia values, but it must be announced that this approximation does cause

inaccuracies. Equations (A.5.2.4.2.5.1) and (A.5.2.4.2.5.2) are used for each point mass

component.

0=zzI (A.5.2.4.2.5.1)

2MxII yyxx == (A.5.2.4.2.5.2)

where M is the total mass of the component (kg), x is the distance from component to the launch

vehicle center of mass (m).



A.5.2.4.3 Evolution of Inertia Math Model The inertia model went through four central design phases, with each achieving more complexity

than the phase preceding it. Phase I, shown in Fig. A.5.2.4.3.1, was very basic. The model

included a single propellant tank in each stage, external skin to the propellant tanks, the

propellant itself, and the nose cone. As much as we would have liked to stop there, we knew the

final launch vehicle design would not be very close to this inertia design.

Fig. A.5.2.4.3.1: Phase I Launch Vehicle

(Brandon White)

Phase II of the math model design incorporated three crucial design features. The first of which

was that THERE IS NO SKIN. There is no external skin wrapped around the propellant tanks,

the outer walls of the launch vehicle (for the most part) are the propellant tank walls. Phase II

also includes both oxidizer and fuel tanks in each stage, rather than just one propellant tank. At

this juncture in design, the team was still considering cryogenic and storable propellants, which

required more than one tank per stage. Lastly, inter-stage skirts were added between stages.

Figure A.5.2.4.3.2 represents the conceptual launch vehicle design for Phase II.

PROP

PROP

PROP



Fig. A.5.2.4.3.2: Phase II Launch Vehicle

(Brandon White)

Phase III was designed in accordance to the final launch vehicle design. With the second and

third stages having solid rocket motors, the tanks were reverted back to only one tank in those

stages. For the first stage a hybrid motor was selected, which resulted in needing approximations

for a pressurant tank and oxidizer tank in addition to the solid propellant needed for the hybrid

motor. Engine nozzles were also included for each stage. Phase III also included the possibility

of having the inter-stage skirts being comprised of an angled section and a straight section.

Phase III (seen in Fig. A.5.2.4.3.3) marked the first time that the payload and avionics were

included in the inertia approximation. The payload was approximated as a solid cylinder inside

the nose cone, with dimensions scaled down appropriately to fit. We knew that the avionics

were going to be centrally based in the second stage, so they were approximated as a solid

cylinder inside the inter-stage skirt between the second and third stage.

O

O

O

F

F

F



Fig. A.5.2.4.3.3: Phase III Launch Vehicle

(Brandon White)

Phase IV marked the final inertia design that was actually used in final calculations. In this

configuration, LITVC is included as a point mass located at the base of the stage nozzle, the

payload and avionics in the third stage are point masses at the base of the nose cone, and

avionics in the first and second stages are truncated hollow cones that are wall mounted to the

inter-stage skirts. Figure A.5.2.4.3.4 displays the Phase IV launch vehicle configuration.

Fig. A.5.2.4.3.4: Phase IV Launch Vehicle

(Brandon White)

SOLID

SOLID

S

Skirt

Skirt

Pressurant Tank

Oxidizer Tank

Payload

SOLID

SOLID

S

Skirt

Skirt

Pressurant Tank

Oxidizer Tank

Payload, Avionics

Avionics

Avionics

LITVC



A.5.2.4.4 Inertia Requirements/Results The purpose of the inertia math model is to provide the Dynamics & Controls group with inertia

matrices at certain time instances during flight. The values that D&C requires are the matrices

before and after propellant burn of each stage. Figure A.5.2.4.4.1 provides these inertia matrices

for the 200g payload. Figure A.5.2.4.4.2 provides these inertia matrices for the 1kg payload.

Figure A.5.2.4.4.3 provides these inertia matrices for the 5kg payload. All values are in kg*m2.

Fig. A.5.2.4.4.1: Inertia Values for 200g Payload

(Brandon White)

Fig. A.5.2.4.4.2: Inertia Values for 1kg Payload

(Brandon White)

13567 0 00 13567 00 0 381

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

First Stage, Full

10814 0 00 10814 00 0 233

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦First Stage, Empty Second Stage, Full

1243 0 00 1243 00 0 25.6

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Second Stage, Empty Third Stage, Full Third Stage, Empty

684 0 00 684 00 0 12.5

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

25 0 00 25 00 0 0.60

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

6.75 0 00 6.75 00 0 0.14

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

25551 0 00 25551 00 0 751

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

First Stage, Full

19795 0 00 19795 00 0 446

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

First Stage, Empty Second Stage, Full

2766 0 00 2766 00 0 59

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦


1417 0 00 1417 00 0 28

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

19 0 00 19 00 0 0 .44

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

4.83 0 00 4.83 00 0 0.104

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦



Fig. A.5.2.4.4.2: Inertia Values for 5kg Payload

(Brandon White)

Figure A.5.2.4.4.3 shows an algorithmic flowchart of the inertia code which incorporates the

analyses discussed earlier.

Fig. A.5.2.4.4.3: Inertia Code Flowchart

(Brandon White)

108065 0 00 108065 00 0 3632

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

First Stage, Full

76990 0 00 76990 00 0 1909

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

First Stage, Empty Second Stage, Full

6663 0 00 6663 00 0 150

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦


3326 0 00 3326 00 0 68

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

20 0 00 20 00 0 0.46

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

8.97 0 00 8.97 00 0 0.11

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Collect Input Variables from tanks.m

Find Distances between Components and CM

Set Index n = 1 to represent the stage of the LV

n = n + 1

Calculate Ixx, Iyy, and Izz for Each Component n>3

no yes Sum Ixx, Iyy, and Izz for Entire LV

Set Index J = 1 to represent the particular phase in launch

to calculate Inertias

Sum Ixx, Iyy, and Izz for the Entire Stage

Set certain inertia values to 0 due to Phase J

J>6

no

J = J + 1

yes

Output Inertia Values for

Each Phase of Launch to

tanks.m

Inertia Code


Author: David Childers

A.5.2.5 Center of Mass

A.5.2.5.1 Algorithm Development We need to know the location of the center of mass for our launch vehicle to ensure that the

vehicle will have a stable flight throughout its trajectory. The first step is to examine how the

center of mass moves with respect to the propellant burn time that each stage undergoes.

Examining the movement is done by applying a linear relationship between values in project’s

launch vehicles and the size of the launch vehicle designed by John Tsohas and his partners.1

The stage lengths for our launch vehicles had previously been linearized against Vanguard by

comparing the payloads.2 The new lengths are then taken and employed in finding the burn

times, mass flow rates, and masses for the stages and propellants. Applying the known values

for burn time and mass flow rate of the propellant from Vanguard, we get an initial view of how

our center of mass changes during the launch vehicle’s flight as seen in Fig. A.5.2.5.1.1. The

figure shows how the center of mass moves during the rocket’s complete burn with the reference

being the bottom of the launch vehicle. The figure demonstrates that once a stage has

completely burned and is jettisoned, the center of mass “jumps” as a result of the loss of mass

that was coming from the previous stage.

Figure A.5.2.5.1.1. Center of mass location (y-axis) during propellant burn time (x-axis) for a 3-stage rocket.

(David Childers)



The initial steps of this type of process assume that there are not any increases or decreases in the

launch vehicle mass between each payload. We make this assumption because of the

inaccuracies that come from using a linearized relationship between values. We assume that

once a stage has finished its burn, the stage is jettisoned from the remainder of the launch

vehicle. Normally there is a period of time between the end of a stage burn and the point at

which the stage would typically be released. However, for our case, we ignore the period

between burn completion and jettison. Other assumptions are that the center of mass shown does

not account for the decrease of fuel and oxidizer in each tank and the fuel and oxidizer are

viewed as a single entity. We ignore the fact that the propellant is being burned to simplify the

calculations needed. For the same reason, we also view the two components as a single object.

Because this examination of the center of mass is just the first step, trying to implement several

complicated calculations and integrations becomes unnecessary until more concrete values are

known.

Looking at how the mass moved with time allows for an initial visualization of how the center of

mass is expected to move. The process of examining the center of mass location over the flight

time is refined to accommodate the needs of the dynamics and controls group. Knowing the

center of mass before and after a stage is applied to insure a controlled flight along the projected

trajectory. The new process employs finding the center of mass for each stage before and after

the burn.

The reference point that we apply for this measuring format is the top of the launch vehicle at the

tip of the nose cone. The reference point remains fixed at the location of the end of the nose

cone even after the cone is jettisoned at the end of the second stage burn. Within the structures

group, we assume that the nose cone is jettisoned. This assumption is not the same one

employed by D&C and trajectory. Within these groups, the nose cone is jettisoned after or

during the first stage. The affect of the difference in times should not have a great influence on

the final trajectory outcomes. The reason is the nose cone weighs only a few kilograms which

will only move the center of mass a few centimeters in either direction.



The reference location is exercised throughout the entire flight even after the nose cone is

ejected. In other words, the reference point remains at the location that the nose cone tip would

be if it was still attached to the launch vehicle. Many of the same assumptions are put into place

with the new procedure. The main assumptions are that the each mass is a point mass and the

launch vehicle is axisymmetric. Therefore, the location of each mass will be in the center of the

launch vehicle along the body axis.

Table A.5.2.5.1.1 Center of Mass location for the 200g payload

Variable Center of Mass From

Top of Launch Vehicle

Units

Stage1_Full 3.80 m

Stage1_Empty 2.59 m

Stage2_Full 1.85 m

Stage2_Empty 1.44 m

Stage3_Full 0.64 m

Stage3_Empty 0.73 m

Table A.5.2.5.1.2 Center of Mass location for the 1kg payload



Units

Stage1_Full 3.91 m

Stage1_Empty 2.65 m

Stage2_Full 1.84 m

Stage2_Empty 1.42 m

Stage3_Full 0.64 m

Stage3_Empty 0.75 m



Table A.5.2.5.1.3 Center of Mass location for the 5kg payload



Units

Stage1_Full 7.43 m

Stage1_Empty 5.33 m

Stage2_Full 2.42 m

Stage2_Empty 1.99 m

Stage3_Full 0.50 m

Stage3_Empty 0.69 m

Table A.5.2.5.1.1 through A.5.2.5.1.3 show the results for the center of mass locations for the

final cases of each payload. One key point to notice is that the center of mass moves back down

once the third stage is empty. This occurs because the engine in the final stage has more mass

than the payload which is located at the very top of the stage. Since the engine mass includes the

nozzle and the exact size and shape of the engine block itself is not a value that can easily be

determined, the engine mass is located along the length of the nozzle. The extent of the change

for the final stage also contributes to the loss of the nose cone mass. Since the cone is jettisoned

at the end of the second stage, any mass attributed with the cone is lost from the calculations of

the last stage. Since the center of mass moves in the same manner for each payload case, we

assume that the previously stated reasons are the likely sources of the outcomes rather than

believing that there is some underlying calculation error.

A.5.2.5.2 Math Model For each stage, the center of mass is found by calculating the sum of each individual mass

remaining multiplied by its location with respect to the top of the rocket. The location is found

by adding the center of mass of the component itself to the length remaining above it. For

example, the oxidizer tank in the first stage adds the top two stages, the nose cone, the pressurant

tank, and the tanks above it to the point found previously for just the oxidizer. The summation is

then divided by the total remaining mass. Once a stage has burned through all of its propellant,

the total mass accounts for loss of propellant by subtracting the propellant mass from the total



mass value when the stage is full. The center of mass CoM for a given rocket is calculated using

Eq. (A.5.2.5.2.1).

M

hmC

n

iii

oM

∑== 1 (A.5.2.5.2.1)

where mi is the mass (kg), hi is the center of mass with respect to the top of the launch vehicle of

each component (m), and M is the total remaining mass of the launch vehicle (kg).

For the cases that we solve for, M includes the current stage as well as the stages above it. Once

the current stage is empty, M is the gross liftoff mass minus the propellants of the stage that has

just been expended. The results are then passed onto the inertia matrices functions that find the

moments of inertia that are employed to insure the launch vehicle’s stability. Because we

assume the masses become point masses, the center of mass and center of gravity can be found

with the same equation and end up being at the same location.

A.5.2.5.3 Algorithm Flowchart The procedure that we employ to determine the center of mass is shown in Fig. A.5.2.5.3.1. The

center of mass algorithm is applied after the tank sizing and model analysis functions have been

ran.



Figure A.5.2.5.3.1. Flow chart for Center of Mass function which is called after the tanks and the model analysis

functions have been run.

(David Childers)

The mass.m function begins with inputting all of the masses and lengths of each component of

the launch vehicle. The inputs include fuel and oxidizer tanks and substances, avionics, engines,

nozzles, nose cone, and the interstage skirts. Since the tanks are either cylindrical or spherical in

shape, the centers of mass for the tanks are placed at the middle under the point mass system.

The point mass system allows for ease and simplicity in the calculations by turning the launch

vehicle into what can essentially be pictured as a cutout of the outline. The inter-stage skirts are

assumed to be trapezoidal in shape which places the mass at a point that is just below the center.

Engine masses are placed halfway along the nozzle length while the nose cone is conical in

shape leading the center of mass to be a third of the way up from the bottom of the cone.

Avionics bring about the most trouble in placing the equipment in locations that will be able to

acquire the size and also insure that the previously stated assumptions can still be implemented.

For this reason, we place the avionic components along the walls of the inter-stage skirts and

assume that we are able to cause the entirety of each stage’s package to circle along the wall,

creating symmetry in the mass distribution. This setup places the mass’s center in the same point

as the inter-stage skirt’s mass. The avionics of the third stage are placed as a point mass at the

bottom of the nose. The pressurant of the first stage and LITVC of the top two stages are treated

in the same manner as the other propellant components, and complete the main mass



contributors. The payload itself is assumed to be a point mass in the same manner as the third

stage avionics and is placed at the top of the stage just below the nose cone.

References 1Tsohas, J., “AAE 450 Spacecraft Design Spring 2008: Guest Lecture,” Space Launch Vehicle Design, URL: https://blackboard.purdue.edu/webct/urw/lc8056011.tp0/cobaltMainFrame.dowebct [sited 10 January 2008]. 2Wade, M., “The Vanguard Satellite Launching Vehicle,” URL: https://blackboard.purdue.edu/webct/urw/lc8056011.tp0/cobaltMainFrame.dowebct [sited 10 January 2008].



A.5.2.6 Other Component Analysis

A.5.2.6.1 Global Buckling of Complete Launch Vehicle Thin-walled structures are highly susceptible to buckling. The design of our launch vehicle is a

thin-walled body and therefore, it is necessary to perform a buckling analysis to determine a

critical buckling load. In previous sections, local buckling has been discussed. In this section,

we consider the overall global buckling of the launch vehicle. We determine the critical

buckling load of the launch vehicle by using an eigenanalysis.

The stability analyses, such as buckling, occur in two stages. These two stages are the pre-

buckle analysis and the buckling analysis.2 It is important to figure out the geometric stiffness

matrix so that in-plane stresses can be determined.2 We know it is necessary to find the in-plane

stress because the presence of in-plane stress causes the onset of buckling. However, the in-

plane stresses are usually not known in advance. Therefore, it is important to allow the degrees

of freedom to be such that the in-plane stresses can be evaluated. Then, we solve the eigenvalue

problem using the following equation:

[ ] [ ] { } 0E GK Kλ φ⎡ ⎤− =⎣ ⎦ (A.5.2.6.1.1)

where KE is the elastic stiffness, KG is the geometric stiffness, λ is the eigenvalue, and Φ is the

nodal displacements.2 We complete the buckling analysis by solving for the values of λ that

make the system unstable.

In performing the buckling analysis of our launch vehicle, we simplify the vehicle to represent a

column buckling problem. When the load on the column is applied through the center of gravity

of its cross section, the load is an axial load. In short columns loaded axially, it is likely that it

will fail due to the compression by the axial load before it will fail due to buckling. In long

columns loaded axially, the failure will occur as buckling. The critical load for buckling can be

calculated via Eq. (A.5.2.6.1.2).



2

2cEIP

Lπ

= (A.5.2.6.1.2)

where Pc is the critical buckling load (N), E is the modulus of elasticity (Pa), I is the axial

moment of inertia (m4), and L is the length of the launch vehicle (m).3 Now, we analyze the

above equation using the definition of the axial moment of inertia of a hollow cylinder in Eq.

(A.5.2.6.1.3).

( )4 4

64 O II D Dπ= − (A.5.2.6.1.3)

where DO is the outer diameter (m) and DI is the inner diameter (m).

We perform this analysis to allow the effect of the length and radius to be seen explicitly. We

see that in order to increase the critical buckling load, either the total length has to be decreased

or the radius has to be increased. Now that we understand the necessity of a global buckling

analysis and understand what parameters affect the critical buckling load, we construct the

algorithm employed in the global buckling analysis.

We model the three stage launch vehicle using fifteen elements and sixteen nodes. Each stage

contains five elements of equal length. The downfall to this approach is that the element length

is not consistent between various sized launch vehicles or stages. The number of elements and

nodes are fixed quantities. However, the number of elements used in the analysis is enough to

provide reliable values for the first couple modes. In Fig. A.5.2.6.1.1, we sketch a rough setup of

the global buckling finite element model for the launch vehicle.



Fig. A.5.2.6.1.1. Diagram of finite element model for the three stage launch vehicle applied for global buckling

analysis.


The figure shows that there are fifteen element and sixteen nodes. We designate elements one

through five to the first stage. The second stage has elements six through ten allocated to it, and

the third stage has elements eleven through fifteen designated to it. The figure illustrates that we

assign a different material to each stage to account for the possibility of a different material for

every stage plus the different properties such as the cross-section area and the moment of inertia.

The figure also demonstrates the boundary conditions and loading conditions applied to the

launch vehicle for the global buckling analysis. We fully fix the launch vehicle at node one

without any other degrees of freedom released on any of the other nodes in the model. We then

E1

E6

E11

E2

E3

E4

E5

E7E8E9E10

E12E13E14E15

N1

N2

N3

N4

N5

N6N7

N8

N9

N10

N11N12N13N14N15N16

Material 1

Material 2

Material 3

E1

E6

E11

E2

E3

E4

E5

E7E8E9E10

E12E13E14E15

N1

N2

N3

N4

N5

N6N7

N8

N9

N10

N11N12N13N14N15N16

E1

E6

E11

E2

E3

E4

E5

E7E8E9E10

E12E13E14E15

N1

N2

N3

N4

N5

N6N7

N8

N9

N10

N11N12N13N14N15N16

Material 1

Material 2

Material 3



apply a compressive axial load to the model at node sixteen. A similar model is created for the

two stage launch vehicle except the model employs a total of fourteen elements, which gives

seven elements per stage. Similar to the three stage launch vehicle, the two stage launch vehicle

also has a different material definition for every stage.

We did not construct the finite element model in any finite element modeling program. Instead,

we wrote a program called global_buck.m, using Matlab to generate a structure data file that we

employ in StaDyn4, the executable exercised by QED4, to complete the global buckling analysis.

The Matlab code is written so that a text file could be created which takes in several parameters.

The code is written so that the output from main_once.m can be applied as the input parameters

to the global buckling code. These input parameters include the number of stages in the launch

vehicle, the length of each stage, the diameter of each stage, the wall thickness of each stage, the

material of each stage, and the compressive axial load. The compressive axial load is the gross

lift off weight of the launch vehicle multiplied by the maximum expected gravity loading, which

is 6 G’s.

After these parameters are input into the code, the code employs the length of each stage to

compute the node locations. The code then defines the material properties for each stage and

computes the area of the cross-section and moment of inertia for each stage. After we calculate

these values, the values are sent to another code called editfiles2.m, which applies them to write

the structure data file. Once editfiles2.m writes the structure data file, it returns to the

global_buck.m, which executes a command line along with a command file to run the structure

data file for buckling using the StaDyn program. After StaDyn completes its analysis,

global_buck.m reads in the output file written by StaDyn with the results of the analysis. The

output of interest is the first lambda value. After the code reads in this value, it calculates a

maximum G loading that the structure can withstand before buckling. In Fig. A.5.2.6.1.2, the

process discussed above is written out in a flow chart format so that the code algorithm can be

followed more easily.



Fig. A.5.2.6.1.2. Flow chart of algorithm employed for global buckling analysis.


yes CalculateMax. G

durability

Inputs:No. of stagesStage lengths

Stage diametersWall thickness for stage

Material for stageCompressive load

32

Output:Maximum

withstandable G’s

Output:Launch Vehicle

failed

CalculateApplied load

No. ofstages

CalculateNode

locations

DetermineMaterial

properties

Calculatestage areas

Calculateinertias

Write StaDyninput deck

RunStaDyn

CalculateNode

locations

DetermineMaterial

properties


Calculateinertias


RunStaDyn

no Λ>

1.25

End GlobalBuckling Code

Start GlobalBuckling Code

yes CalculateMax. G

durability

Inputs:No. of stagesStage lengths

Stage diametersWall thickness for stage

Material for stageCompressive load

32

Output:Maximum

withstandable G’s

Output:Launch Vehicle

failed

CalculateApplied load

No. ofstages

CalculateNode

locations

DetermineMaterial

properties


Calculateinertias


RunStaDyn

CalculateNode

locations

DetermineMaterial

properties


Calculateinertias


RunStaDyn

no Λ>

1.25

End GlobalBuckling Code

Start GlobalBuckling Code



We run global_buck.m for the final design of the 200 g payload, 1 kg payload, and 5 kg payload

launch vehicles. The results from our analysis are summarized in Table A.5.2.6.1.1.

Table A.5.2.6.1.1 Maximum G’s the Launch Vehicle can

Withstand.

Launch Vehicle Max G durability Units 200 g payload 19.80 G’s 1 kg payload 28.42 G’s 5 kg payload 10.00 G’s

Dr. James Doyle, a professor of Aeronautics and Astronautics Engineering at Purdue University,

suggests using a knockdown factor of 0.60 to account for reductions in the strength due to

manufacturing and imperfections of the material.1 The values listed above apply a knockdown

factor of 0.50 to account for the topics brought to our attention and to allow some error for

applying a simplified column buckling approach. In addition to the knockdown factor, the

reserve factor of safety equal to 1.25 is also employed in reporting the results listed above.

Although we do not know if the knockdown factors applied are enough to allow for error using

the simplified approach, the maximum withstandable gravity loading predicted by the analysis

are much higher than the 6 G’s that we expect the launch vehicle to experience. Therefore, we

conclude that global buckling should not present any problems for our launch vehicle.

When we first looked at the results from our analysis, we were interested to see the difference in

each launch vehicle that would produce the differences in the maximum gravity loading

durability. In the following tables, the lengths and diameters of each stage are displayed to help

grasp the results of the analysis.



Table A.5.2.6.1.2 200 g Payload Launch Vehicle Length, Diameter, and Wall thickness of Each Stage.

Stage Length Diameter Wall thickness Units First 7.1478 1.3015 0.0037 m Second 2.5594 0.6741 0.0055 m Third 0.8945 0.2721 0.0022 m

Table A.5.2.6.1.3 1 kg Payload Launch Vehicle Length, Diameter, and Wall thickness of Each Stage.


Table A.5.2.6.1.4 5 kg Payload Launch Vehicle Length, Diameter, and Wall thickness of Each Stage.


Table A.5.2.6.1.5 Liftoff Mass for All Three Launch Vehicles.

Launch Vehicle Liftoff Mass Units 200 g payload 2583.83 kg 1 kg payload 1745.22 kg 5 kg payload 6294.8 kg

We look at these results and recall Eq. (A.5.2.6.1.3) to see what produced the differences in the

maximum gravity loading durability. We compare the 200 g payload launch vehicle and the 1 kg

payload launch vehicle and see that the 200g payload launch vehicle is larger in its geometry and

is more massive than the 1 kg payload launch vehicle by 1.5 times. The fact that the vehicle

geometry increased in a smaller proportion than the mass in the comparison of the two launch

vehicles, results in the higher gravity loading capability of the 1 kg payload launch vehicle over

the 200 g payload launch vehicle.



Now we compare the 1 kg payload launch vehicle and the 5 kg payload launch vehicle. As in the

case before, the ratio between the geometry of the two launch vehicles is less than half the ratio

between the masses of them. This, again, results in the higher gravity loading capability of the 1

kg payload launch vehicle over the 5 kg payload launch vehicle.

References 1James F. Doyle. Professor of Aeronautics and Astronautics Engineering. Purdue University. 2Doyle, James F., Guided Explorations on the Mechanics of Solids & Structures: strategies for learning and understanding. Purdue University, West Lafayette, IN. August 2007. 3Doyle, James F., Structural Dynamics and Stability: a modern course of analysis and applications. Purdue University, West Lafayette, IN. August 2007. 4ikayex Software Tools. QED: Static, Dynamic, Stability, and Nonlinear Analysis of Solids and Structures. Lafayette, IN. August 2007.



A.5.2.6.2 Stress Analysis on Balloon Launch Configurations In deciding that a balloon launch is the best launch option, how to carry the launch vehicle is a

main concern. There are two design configurations that we are analyzing in this project. The

first is a simple basket type gondola that simply holds the launch vehicle in a snug structure. The

vehicle then launches out of the gondola. The other design we are considering is a hook system,

where brackets attach to the launch vehicle on four sides. From the balloon would be some type

of hook system that would fall off when the launch vehicle was lifting off. After looking from

the structural point of view and at the stresses that are put on the launch vehicle, we decide to

design the basket type gondola system. The launch vehicle will simply sit in the gondola and

launch straight up. Figure A.5.6.2.1 shows the basket gondola and Fig. A.5.2.6.2.2 shows the

hook system.

Fig. A.5.2.6.2.1 Preliminary basket gondola design

(Sarah Shoemaker)

Launch Vehicle

Balloon

Basket Gondola



Fig. A.5.2.6.2.2 Preliminary hook system design

(Sarah Shoemaker)

In performing a balloon launch, there are several stresses that need to be considered in order to

make the launch successful. However, given the time frame of the project not all the stresses are

taken into account. The main stress that is analyzed is the stress on the gondola from the sitting

launch vehicle. This stress is used in designing the size and shape of the gondola.

The stress of the gondola is calculated using the area of the gondola base, the mass of the support

rails and support rings of the gondola, the mass of the launch vehicle, the mass of the avionics

support piece, and the mass of the avionics. All the masses are added up and multiplied by the

acceleration due to gravity to get the force exerted on the area of the gondola base. The area of

the gondola base is found with the inertia properties tool in CATIA. This area is used to find the

stress the gondola base is subjected to. This stress is then compared to the ultimate stress of the

material used for the base and if the stress on the base is less than the stress of the material then

the gondola is able to hold itself and the launch vehicle. The equation used for the force exerted

on the gondola base can be described by Eq. (A.5.6.2.1).

( )*rr r as aF m m m m g= + + + (A.5.6.2.1)where F is the force exerted on the area of the base of the gondola (N), mrr is the mass of the

support rails and support rings of the gondola (kg), mr is the mass of the launch vehicle (kg), mas

Balloon

Launch Vehicle

Hook System



is the mass of the avionics support piece (kg), ma is the mass of the avionics (kg) and g is the

gravitational acceleration (m/s2).

The stress the gondola base is experiencing can be described by Eq. (A.5.6.2.2).

/F Aσ = (A.5.6.2.2)

where σ is the stress the base of the gondola experiences (Pa), F is the force exerted on the

gondola base (N), and A is the area of the gondola base (m2). The strengths of the gondolas are

shown in Table A.5.6.2.1.

Table A.5.6.2.1 Strength of the Gondolas

Variable Value Units 200g Strength 56,301 Pa 1kg Strength 32,704 Pa 5kg Strength 75,479 Pa

The material being used for the gondola is aluminum. Aluminum was chosen because it is a

material that is cheap and light weight as well as easy to work with when it comes to welding

and riveting. Figure A.5.6.2.2 is the final gondola design drawn in CATIA.



Fig. A.5.6.2.2: CAD drawing of the balloon gondola

(Sarah Shoemaker)

The sizing of each of the gondolas for each of the payloads can be found in Tables A.5.6.2.2

through A.5.6.2.4.

Table A.5.6.2.2 Sizing of the Gondolas-200g Payload

Variable Value Units Mass 177.188 kg Length Width Ring Diam.

3.346 0.876 1.3015

m m m

Footnotes: All thicknesses of the beams and rails are 0.04 m.

Table A.5.6.2.3 Sizing of the Gondolas-1kg Payload


3.849 1.000 1.1264

m m m




Table A.5.6.2.2 Sizing of the Gondolas-200g Payload


5.133 1.380 1.8386

m m m


Originally the gondola was going to be in the shape of a triangle because triangles are one of the

strongest shapes. However, because of the nozzle of the launch vehicle needing to fit through

the base of the gondola, the shape had to be changed to fit the nozzle. A circular base was also

considered but because the launch vehicle needed to rest on the base a circle would not have

worked.

The final design of the gondola has four rails used in supporting the launch vehicle on its initial

assent. The reason for these rails is to help guide the launch vehicle so that it may launch in the

direction we decided. The gondola also has support rings riveted around the support rails. These

rings are needed to help support the rails from the stresses the rails will experience when the

launch vehicle takes off.

At the top of the gondola is a solid square with a hole cut out to fit the launch vehicle. The

platform is riveted to the support rails and is used for holding the avionics needed for controlling

and keeping track of the entire balloon-launch vehicle-gondola configuration. The avionics are

riveted to the platform. The base of the gondola is a square consisting of four beams welded

together. The support rails are welded to the four corners of the base. The shape of a square is

chosen because of its simplicity.

The gondola attaches to the balloon by its tethers. The tethers will be made out of a steel cable.

This cable has not had a stress analysis done on it. If an analysis were to be done, the tension in

the cable will need to be analyzed to make sure the cable will not snap under the pressure of the

tension force of the balloon and launch vehicle.



There are several stresses that are left out of this analysis due to the time constraint. One of these

is the stress on the connections from the gondola to the balloon. The connections are assumed to

be able to handle both the gondola and the stress from buoyancy of the balloon. If this analysis

was to be done there may have been some added mass to the top of the gondola to help with the

stress from the tension in the tethers from the balloon. Another stress that is not being analyzed

is the stress the balloon material is subjected to with the inflating of the balloon and the stress on

the material from the ascent considering pressure changes along the way.

Since we are assuming the launch vehicle launches through the balloon, there are stresses

involved in that as well. However, this stress analysis was not done again because of the time

constraint. If it were to be done, the stresses on the launch vehicle from bursting a balloon would

cause the launch vehicle to possibly go off course and therefore it would cause more stress on the

launch vehicle to control the course of the trajectory. Also there would be an added stress on the

nose cone when the launch vehicle was trying to burst the balloon. Another aspect not taken into

account is the amount of stress needed to burst the balloon.

Another stress analysis that is not being done is that of the swinging and rotating of the launch

vehicle-gondola combination. When something is attached to the bottom of a balloon it tends to

swing back and forth like a pendulum and in some cases the object will rotate while swinging.

This swinging and rotating will put added stress on the gondola joints where the tethers are

attached. It will also put added stress on the gondola support rails holding the launch vehicle

which will then put stress on the support rings. The swinging and rotating will also put stress on

the tether attachment to the balloon, adding stress to the balloon material.

When the launch vehicle takes off from the gondola, there will also be stresses on the gondola

which were not taken into account. The gondola was designed to minimize the stress on the

sides by making the support rails flush with the launch vehicle. This helps with when the launch

vehicle takes off, it is already touching the support rails and therefore not adding stresses to the

support rails. However what is not being taken into account is the stress on the support rails if

the launch vehicle were to take off at an angle due to the swinging and rotating. If the gondola

were to swing out to the side and the launch vehicle were to lift off at that point, the stresses on



the support rails the launch vehicle would be “resting on” would need to be analyzed. This

analysis would cause the mass of the gondola structure to increases in able to handle these

stresses.

The final stress on the gondola structure that is not being considered is the impact stress when

the gondola lands. We are assuming that after the launch vehicle lifts off and bursts the balloon,

the gondola will fall back to the ground. We have not done any analysis on how the impact of

this landing will affect the gondola structure because we are not planning on reusing the gondola.



A.5.2.6.3 Math Models The code that was used for the construction of the gondola is very simple. The code inputs the

gross lift off weight of the launch vehicle and adds it to the mass of the guide rails, support rings

and avionics bay. The code also inputs the area of the base of the gondola. Then the code uses

the total mass and multiplies it by the acceleration of gravity to obtain the force exerted on the

area of the gondola base. After the code generates the force, it then divides the force by the area

to get the stress exerted on the base of the gondola. This stress is compared to the maximum

strength of aluminum and if it is less than the aluminum strength then the gondola is able to hold

itself and the launch vehicle.

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The advantages we see in this model are that the tank could be designed independently of the

external body, and then the dimensions could be simply input for the body. Also with this

model, the tank could be completely supported without having to fasten it to the launch vehicle

directly. Riveting and bolting a tank directly could not be possible, as the tank would lose

pressure immediately, and welding is a more expensive process. With the model in Fig.

A.5.2.7.1, it is possible to hold the tank in place by riveting or bolting and not interfering with

the tank itself.

The design of the ring shown in the model is a simple rectangular cross section, which may have

been the least expensive option in the design of the ring because the simplicity means a much

smaller manufacturing cost.3 However, more complex structures were going to be looked at and

code was going be developed to determine the exact design of the support rings. This design

model was similar to other small payload launch vehicles such as the Vanguard.2 A portion of

the Vanguard’s tank mounting system is shown in Fig. A.5.2.7.2.

Fig. A.5.2.11.2: Tank mounting system for the Vanguard vehicle.

(The Martin Company2)

Later it was determined that this model should be changed. The more practical and efficient

method of launch vehicle design is to have the tanks and solid motor casings be the body of the

launch vehicle itself. Having two external structures of different materials is inefficient when it

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Author: Jesii Doyle

A.5.2.8 Bulkheads In our first attempt at designing a launch vehicle, we decide that our vehicle will have an outer

skin and that the fuel tanks will be sandwiched into the external structure. We use disk-shaped

bulkheads to separate the nozzle of the upper stage from the fuel tank of the lower stage. The

disk-shaped bulkhead has a hole in the center to pass electrical wires through to the stages below.

The structural analysis of the disk-shaped bulkhead results in a minimum thickness of the

bulkhead. First, we impose a deflection constraint on the bulkhead when the load is applied. The

nominal thickness is calculated by solving deflection Eqs. (A.5.2.8.3) or (A.5.2.8.7) with

maximum allowed vertical deflection. After the nominal thickness is determined, we calculate

the maximum applied stress on the bulkhead, multiply the stress by the reserve factor, and

compare this value to the yield stress of the bulkhead material. If the applied stress multiplied by

the reserve factor is less than the yield stress, the thickness is incremented. When the applied

stress multiplied by the reserve factor is greater than the yield stress, the minimum required

thickness has been found. Then, this process is repeated for each material candidate and the cost

of the bulkhead is output. These costs are then compared to find the lowest cost option.

In our initial design of the launch vehicle, two bulkhead configurations are considered. The

bulkheads themselves retain the same geometrical specifications for each configuration, but are

attached to the outer skin differently. The first bulkhead configuration is rigidly attached to the

outer skin, and the second bulkhead configuration is fixed on top of the tank below. Since the

first configuration is attached to the outer skin of the launch vehicle, the bulkhead will cause

more shear force to occur on that point of the launch vehicle skin, whereas the second

configuration does not impact the overall vehicle. This application of force on the overall vehicle

must be taken into consideration when we choose the final configuration.

First, we consider the structural analysis of the first bulkhead configuration. Figure A.5.2.8.1

displays this bulkhead configuration in the rocket, and Fig. A.5.2.8.2 gives the free body

diagram.


Author: Jesii Doyle

Fig. A.5.2.8.1: First bulkhead configuration and upper stage

(Jesii Doyle)

Fig. A.5.2.8.2: First bulkhead configuration free body diagram

(Jesii Doyle)

First, the applied load to the cross-section must be determined. The applied load to the bulkhead

is described in Eqs. (A.5.2.8.1) and (A.5.2.8.2).

nozzle

max

r2Pw

g*G*mP

π=

=

(A.5.2.8.1)

(A.5.2.8.2)

Bulkhead Inner Radius

w Nozzle Diameter

Stage Diameter

Tank

Engine

External Skin

Nozzle

Bulkhead


Author: Jesii Doyle

where m is the total mass of the stages above the bulkhead (kg), Gmax is the maximum G-loading,

g is the gravitational acceleration (m/s2), P is the applied force (N), rnozzle is the radius of the

nozzle in contact with the bulkhead (m), and w is the force per unit of circumference (N/m).1

The maximum displacement equation is shown in Eq. (A.5.2.8.3), solving for thickness.

( ) ( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎠⎞

⎜⎝⎛=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛=

⎥⎦⎤

⎢⎣⎡ −++=

⎟⎠⎞

⎜⎝⎛ −

−+⎟

⎠⎞

⎜⎝⎛−

=

⎥⎦

⎤⎢⎣

⎡ −⎟⎠⎞

⎜⎝⎛ −

−=

nozzle

2nozzlenozzle

2nozzle

nozzle

2nozzlenozzle

2

max

3

ralog21

ar

a4r

6L

1a

rr

alog1a

ra4

r3L

ba1

ab1

214C

ab

ba

41

balog

ab

211C

E)1(123L

4C6L*1C

ywat

31

υυ

υυ

υ

(A.5.2.8.3)

where t is the nominal thickness of the bulkhead (m), w is the force per unit of circumference

(N/m), a is the bulkhead outer radius (m), b is the bulkhead inner radius (m), ymax is the

maximum vertical displacement of the bulkhead’s free edge (m), υ is Poisson’s ratio, E is the

modulus of elasticity (Pa), and rnozzle is the radius of the nozzle in contact with the bulkhead (m).1

The calculated thickness is then input into the applied stress equation, which is shown in Eq.

(A.5.2.8.4).

( ).F.R*

traw*r2*6

2nozzlenozzle −

=π

σ (A.5.2.8.4)

where σ is the applied stress (Pa), rnozzle is the radius of the nozzle in contact with the bulkhead

(m), w is the applied force per unit of circumference (N/m), a is the bulkhead outer radius (m), t

is the thickness (m), and R.F. is the reserve factor.1

Once the minimum thickness is found for one type of bulkhead material, the process is repeated

for each possible material option. Finally, the weight and cost of each bulkhead is calculated.


Author: Jesii Doyle

Next, we consider the structural analysis of the second bulkhead configuration. Figure A.5.2.8.3

displays this bulkhead configuration in the rocket, and Fig. A.5.2.8.4 gives the free body

diagram.

Fig. A.5.2.8.3: Second bulkhead configuration, lower stage, and upper stage nozzle

(Jesii Doyle)

Fig. A.5.2.8.4: Second bulkhead configuration free body diagram

(Jesii Doyle)

First, the applied load to the cross-section of the second bulkhead configuration must be

determined. The applied load to the bulkhead is described in Eqs. (A.5.2.8.5) and (A.5.2.8.6).

nozzle

max

r2Pw

g*G*mP

π=

=

(A.5.2.8.5)

(A.5.2.8.6)

Tank

Engine

External Skin

Nozzle

Bulkhead

Bulkhead Inner Radius

Nozzle Dia.


Author: Jesii Doyle

where m is the total mass of the stages above the bulkhead (kg), Gmax is the maximum G-loading,

g is the gravitational acceleration (m/s2), P is the applied force (N), rnozzle is the radius of the

nozzle in contact with the bulkhead (m), and w is the force per unit of circumference (N/m).1

The maximum displacement equation is shown in Eq. (A.5.2.8.7) below, solving for thickness.

( )

( )

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−

−+⎟

⎠⎞

⎜⎝⎛+

=

⎟⎠⎞

⎜⎝⎛ −−=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛ −

−+⎟

⎠⎞

⎜⎝⎛−

=

⎭⎬⎫

⎩⎨⎧ −

⎥⎦

⎤⎢⎣

⎡+−⎟

⎠⎞

⎜⎝⎛ −

−=

2nozzle

nozzle

nozzle

2nozzle

nozzle

2nozzlenozzle

2

2

22

2vozzlenozzle

max

3

ar1

41

raln

21

a4r6L

1a

rr

alog1a

ra4

r3L

ab1

41

baln

21

ab9C

ab

ba1

217C

1ab

baln1

ab

a4b3C

ab

ba

41

balog

ab

211C

E1123L

b3Cr9L

b9Cr

7C1C

ywat

31

υυ

υυ

υ

υυ

υ

(A.5.2.8.7)

where t is the nominal thickness of the bulkhead (m), w is the force per unit of circumference

(N/m), a is the bulkhead outer radius (m), b is the bulkhead inner radius (m), ymax is the

maximum vertical displacement of the bulkhead’s free edge (m), υ is Poisson’s ratio, E is the

modulus of elasticity (Pa), and rnozzle is the radius of the nozzle in contact with the bulkhead (m).1

The calculated thickness is then input into the applied stress equation, which is shown in Eq.

(A.5.2.8.9).

( ) .F.R*t

brw*r2*62

nozzlenozzle −=

πσ (A.5.2.8.9)

where σ is the applied stress (Pa), rnozzle is the radius of the nozzle in contact with the bulkhead

(m), w is the applied force per unit of circumference (N/m), a is the bulkhead outer radius (m), t

is the thickness, and R.F. is the reserve factor.1


Author: Jesii Doyle

Once the minimum thickness is found for one type of bulkhead material, the process is repeated

for each possible material option. Finally, the weight and cost of each bulkhead is calculated for

each material option.

Subsequently, we decided to revise our launch vehicle configuration. The new launch vehicle

configuration does not have an overall external skin. Therefore, the inter-stage configuration

needs to be reconsidered. This new launch vehicle design results in the use of inter-stage skirts

between stages, and renders both bulkhead configurations obsolete.

References 1 Young, W.C., and Budynas, R.G., Roark’s Formulas for Stress and Strain (7th Edition), McGraw-Hill, 2002.



A.5.2.9 Avionics/Payload Mounts The avionics required to control and power the launch vehicle are being mounted on the inside

wall of the launch vehicle inter-stage skirt on the first and second stage. The avionics are placed

put into a ring that will be mounted by bolts to the inside of the inter-stage skirt wall. This ring

will be near the bottom of the inter-stage skirt to keep from getting in the way of the nozzle. The

size of this ring can be seen in Table A.5.2.9.1 through Table A.5.2.9.3.

Table A.5.2.9.1 Sizing of Avionics Mounting Ring-200g

Variable Value Units Stage1 Height 0.2 m

Stage1 Diameter 0.337 m Stage2 Height 0.2 m

Stage2 Diameter 0.144 m

Footnotes: Angle of ring is 10°. Thickness is 0.1m.

Table A.5.2.9.2 Sizing of Avionics Mounting Ring-1kg





Table A.5.2.9.3 Sizing of Avionics Mounting Ring-5kg





Figure A.5.2.9.1 below shows one of the rings the avionics will be placed in.



Fig. A.5.2.9.1 Avionics ring

(Jesii Doyle)

While we would have liked to perform a stress analysis on this avionics ring, time did not allow

us to do so. Consequently, we assume that whatever ring is used for the avionics will withstand

the g-forces and the bolts will hold. If there were to be a stress analysis done on the avionics

ring; the g-forces the ring would be subjected to, as well as the shear stress on the rivets, would

need to be considered and designed to make sure they hold.

As with the avionics, not much is being done with regards to the payload mounting. We assume

that we are mounting the small payload on a small aluminum beam structure. This beam

structure will be welded together and then riveted to the top of the third stage. The payload will

sit in the structure and will not move during the flight of the launch vehicle. The payload is

mounted in the third stage inside the nose cone.

There is no stress analysis on the g-forces on the support structure or the payload itself. If there

were to be a detailed stress analysis on the payload, the g-forces on the beams from the payload

would need to be considered. Also the shear stresses between the bulkheads and the mounting

beams holding the payload would need to be analyzed to make sure both the beams and the

bulkheads can withstand the stresses.


Author: Jesii Doyle

A.5.2.10 Internal Mounting Materials In the initial stage of structural design, the possible material options for the structure must be

evaluated. We establish some of the possible internal mounting materials by researching what

materials have been used historically, and what materials are currently being implemented in the

aerospace industry.

We consider the following material options in our analyses: aluminum 6061-T6, aluminum

7075-T6, titanium 6Al-4V, fiberglass, and a three-directional carbon fiber composite. The

common material not considered here is steel. Although steel is inexpensive it is not usually used

in the aerospace industry. Also, since the internal mounting structures are relatively small

compared to the entire launch vehicle, the cost reduction in steel is not significant. Table

A.5.2.10.1 shows the material properties of these materials.

Table A.5.2.10.1 Internal Mounting Material Properties

Material Ultimate Strength (MPa)

Yield Strength (MPa)

Density (g/cm3)

Al 6061-T61 262 241 2.70 Al 7075-T61 517 448 2.80 Ti 6Al-4V2 958 876 4.46 Fiberglass3 207 207 1.72 Carbon/Carbon4 310 159 1.90

The yield strength of the material is what will be most commonly used in our analysis. As

expected, titanium has the highest yield strength, and the carbon composite has the lowest.

Though the titanium material has the highest yield strength, it also has the highest density. Even

though less material will be structurally required if titanium is used, the total weight may not

necessarily be less. The most commonly used material options in the aerospace industry are the

aluminum alloys. We easily determine that these materials are very useful since they have

relatively high yield strength, and low density.

Since we are optimizing our launch vehicle by minimizing cost, the cost of the raw material for

each possible material option is necessary. Table A.5.2.10.2 shows the cost per weight and cost


Author: Jesii Doyle

per unit of strength. The cost per weight and cost per unit of strength show valuable ways of

comparing the material options. Table A.5.2.10.2 Internal Mounting Material Cost

Material Cost/Weight (USD/kg)

Cost/Strength (USD/MPa)

Al 6061-T61 4.41 0.02 Al 7075-T61 17.64 0.04 Ti 6Al-4V5 93.59 0.11 Fiberglass6 1121.98 5.60 Carbon/Carbon7 522.45 3.45

We can see that the lowest cost materials are the aluminum alloys. This is expected since these

are the two most common structural alloys used in the aerospace industry.

Subsequently, we decided that it is best for the launch vehicle design if there is a generic list of

material options for all structural components. Most of these internal mounting materials were

discarded because they would not be feasible for other structural components. Therefore, the

material options, properties and costs are revised in the final design of the launch vehicle.

References 1 Setlak, Stanley J., “Aluminum Alloys; Cast,” Aerospace Structural Metals Handbook, Purdue University, Indiana, 2000. 2Setlak, Stanley J., “Titanium Alloys; Cast,” Aerospace Structural Metals Handbook, Purdue University, Indiana, 2000. 3 “Properties,” Pultruded Products, Strongwell, January 2008. [http://www.strongwell.com/PDFfiles/Extren/EXTREN%20Properties.pdf. Accessed 1/16/2008.] 4 “Carbon matrix composite, Carbon/carbon, Three-directional orthogonal, Shape,” Engineered Materials Handbook, Vol. 1, Composites, ASM International, 1987. 5 “6Al-4V Titanium Sheet and Plate,” Titanium Joe, December 2007. [http://www.titaniumjoe.com/6al4v%20sheet.htm. Accessed 1/15/2008.] 6 “Suggested List Prices,” EXTREN: Fiberglass Structural Shapes & Plate, Strongwell, November 2006. [http://www.peabodyconcealment.com/ sitebuildercontent/sitebuilderfiles/StrongwellExtrenPrices.pdf. Accessed 1/15/2008.] 7 “Carbon Fiber – Structural,” Metals and Materials, The Robot Marketplace, December 2007. [http://www.robotmarketplace.com/marketplace_ carbonfiber_struc.html. Accessed 1/15/2008.]

Project Bel

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thickness

determin

where tcs

the case

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where fs

assume th

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References 1 Humble, Ronald W., Henry, Gary N., and Larson, Wiley J., Space Propulsion Analysis and Design, The McGraw-Hill Companies Inc., New York, 1995. 2DeVries, D. and Aadland, J. “Electonic Safe and Arm (ESA) Device for Solid Propellant Rocket Motor Initiation,” 37th AIAA/ASME/SAE/ASEE JPC Conference and Exhibit, Salt Lake City, Utah, 2001.



A.5.2.12 Composite Tanks We first examined carbon fiber composite tanks in the preliminary design phase. For preliminary

design, we assume a simplified material model consisting of an isotropic pre-preg carbon fiber

layup, which gives us conservative material strengths for a carbon fiber tank. Even with this

handicap, the performance of the carbon fiber tank is very attractive from a structural

perspective. The carbon fiber tank designed with these conservative assumptions would still be

half the weight of an aluminum tank of the same specifications, and we know that we could

achieve even better performance with further optimization.

However, a decision was made to abandon carbon fiber early on in the design process for several

reasons. First and foremost is the difficulty in accurately assessing carbon fiber material and

manufacturing costs. These figures are closely guarded trade secrets for aerospace companies.

What quotes we did obtain suggest that carbon fiber tanks are up to an order of magnitude more

expensive than metallic tanks. Since the main goal of the project is to minimize cost and not

maximize performance, we discard carbon fiber from our final design unless the required

performance absolutely could not be reached with metallic parts. In this section, we will discuss

the design of carbon fiber composite tanks in a little more detail than was covered in preliminary

design.

Carbon fiber by itself is not suitable as a primary tank material for rocket tanks. It has poor

corrosion and temperature resistance, and in resin-impregnated form, is too porous to contain

hydrogen gas. Fabrication of a purely carbon fiber tank is also problematic; as it requires a mold

for layup and is difficult to join in parts once cure. If formed in a single piece, it would require

that the mold be melted out of the exit port of the tank. For these reasons, a pure carbon fiber

tank is not practical or desirable. Instead, a carbon fiber winding over a metallic tank is more

useful in space applications.

Design of a carbon fiber wound tank follows slightly different principles than a metallic tank.

The first item of consideration is the wind angle. As the hoop stress in a cylindrical tank is twice

the axial stress (see Section A.5.2.1.4), the optimal winding angle is one which provides twice

the circumferential strength to the axial strength. Working out this value1



hoop stress, tan

/ 2sin windk

P DTt

α ⋅⋅ = (A.5.2.12.1a)

axial stress, ( )tan

/ 2cos2wind wind

k

P DT tant

α α⋅⋅ = ⋅

⋅ (A.5.2.12.1b)

2 2windtan α = (A.5.2.12.1c)

54.7windα = o (A.5.2.12.1d)

where T is the stress in the longitudinal fiber direction (Pa), D is the tank diameter (m), trank is the

tank wall thickness (m), αwind is the filament winding angle measured from the long axis of the

tank. Refer to Figure A.5.2.12.1.

Fig. A.5.2.12.1 Filament winding on tank

(Chii Jyh Hiu)

This wind angle αwind is both optimal for structural strength and necessary to avoid out of plane

deformation from pressure loading.

A fiber wound tank consists of unidirectional carbon fiber filaments wound around a metallic

tank. The stress analysis1 of such a compound tank is more complicated than for a single material

tank. Stresses in the material are now a function of the material properties.

T

T 1

tan αwind

axial direction

hoop direction

αwind



deformation, c mδ δ=( ) 2 2

,1 sincontact m contact c

m m c wind c

P P r P rE t E tα− ⋅

=⋅ ⋅ ⋅

(A.5.2.12.2a)

after solving for contact pressure,

metallic hoop stress, ( )contact mm

m

P P rt

σ− ⋅

= (A.5.2.12.2b)

composite hoop stress, contact cc

c

P rt

σ ⋅= (A.5.2.12.2c)

where δm and δc are the deformation in the metal and composite respectively (m), P is the

internal tank pressure, Pcontact is the contact pressure between the metal lining and the composite

sleeve, rm and rc are the radii of the metal and composite tank respectively (m), Em is the

Young’s modulus of the metal lining (Pa), Ec,1 is the Young’s modulus of the carbon fiber layup

in the principal fiber direction, αwind is the winding angle of the carbonf fiber measured from the

long axis of the tank, tm and tc are the thicknesses of the metal and composite respectively (m),

σm and σc are the hoop stresses seen in the metal and composite respectively (Pa). Refer to

Figure A.5.2.12.2.

Fig. A.5.2.12.2 Carbon fiber sleeve over metallic lining

(Chii Jyh Hiu)

A carbon fiber sleeve allows us to withstand higher loads than a metallic tank itself could

withstand. Due to the high elastic modulus of carbon fiber in the longitudinal direction (~145

metallic lining

carbon fiber sleeve

Pcontact

Pcontact

P



GPa2), a substantial improvement in metallic hoop strength can be obtained and significant

weight savings realized. We predict a weight savings of 2.5-3 times over a comparable metallic

tank, as opposed to a 2x savings from the preliminary design runs using isotropic layup

assumptions. Due to the appealing structural characteristics of carbon fiber, it may be prudent to

keep composite tanks in consideration for future designs, as the cost effectiveness of aerospace

carbon fiber continues to improve with industry adoption, so costs will only continue to fall.

References 1 Roylance, D, “Pressure Vessels”, Massachussets Institute of Technology, August 23, 2001 2. Callister, W.D. Jr, Fundamentals of Materials Science and Engineering, 2nd Ed., Wiley & Sons, 2005



A.5.2.13 Stage Separation Methods As the vehicle ascends, we need to get rid of the unwanted, used stages. There are a couple ways

this separation can occur. The way we have decided to release our stages is explosive bolts.

Explosive bolts are common throughout the history of lower earth orbit launch vehicles.

Using explosive bolts places added stress on the structure of the launch vehicle. These stresses

will occur at the points where the bolts are exploding. We are assuming that the bolts will

detonate and that the structure will be able to withstand the forces.

The location of the explosive bolts will be on the inter-stage skirt, in line with the end of the

nozzle. This is above the avionics and thus we are assuming the avionics will blow as well. This

location is chosen because we do not want the inter-stage skirt to interfere with the nozzle once

the previous stage is released. Also we did not want the separation to occur below the nozzle as

it would affect the flow coming out of the nozzle and throw the launch vehicle off course.

Figure A.5.2.13.1 shows this explosive bolt location.

Fig. A.5.2.13.1 Location of explosive bolts

(Sarah Shoemaker)

While we can not quantify the number of bolts necessary to separate the stages of the launch

vehicle, we are aware of their importance and how they affect the overall design. Further

analysis or research is required before deciding on a specific stage separation design.

Launch Vehicle

Explosive Bolt Plane Explosive Bolt Plane

Inter-Stage Skirt

Nozzle


Author: Steven Izzo

A.5.2.14 Final Stress Considerations The structures codes incorporate analysis of several stresses, including internal pressure, axial

buckling, buckling due to bending, and shear. All the analyses compare a critical stress value,

based on the geometry and materials of the launch vehicle, to an applied stress value, based on

the performance of the vehicle. The analysis models come from research on the large stresses

that occur on launch vehicles. The vehicle overall also incorporates a reserve safety factor of

1.25. Many other components of the vehicle, such as the tank size, are given an extra factor of

safety to incorporate unforeseen factors that could make applied stresses larger and critical

stresses smaller. Thus, these components are designed slightly stronger than necessary.

Once the initial design of the vehicle was complete, with the analysis of internal pressure, axial

buckling, buckling due to bending, and shear, we researched other potential stresses that can

occur on launch vehicles. We researched other potential stresses to ensure the following: our

codes were still valid, all other stresses were small in comparison to those accounted for, and the

vehicle would not fail with the addition of these stresses because of the safety factor.

The structures codes were always designed not to simply report success or failure, but to add

support to the vehicle until the vehicle was a success. These final stress considerations were

performed once the vehicle was completely designed, so these final stress analyses ensured that a

resize of the rocket was not necessary. The method was to run all the structures codes multiple

times over, adding applied stresses consistent with those found in research, and determining if

that added stress was within the safety factor of 1.25, meaning the vehicle did not have to be

resized.

Thrust vector control was the first source of extra stress. The structures codes were designed

with the assumption that the thrust was applied axially. When the thrust is angled for control

purposes, the thrust has a horizontal component and leads to bending stress.3 The structures

codes were ran for each payload, adding an additional bending stress resulting from the

horizontal component of the thrust at all angles (ranging from 0 to 90 degrees. Even at 90

degrees, the thrust pointing exactly horizontal, the added bending stress did not require extra

support. The number and size of the support structures remained the same.


Author: Steven Izzo

Another stress source is thrust misalignment. The thrust can be misaligned in any direction, and

can lead to extra bending and torsion.1 The thrust at various angles was input into the structures

codes again, this time as a source of shear stress. Again, for all angles, up to 90 degrees, the

thrust misalignment did not require extra support structure.

Spin stabilization is another source of stress. While the final spin rate does not cause any stress,

the rate at which the launch vehicle spins up becomes a source of stress. From Newton’s laws, a

change in angular momentum causes a moment,2 which causes torsion on the vehicle. The

structures code had to be run to find the limit on how fast spin-up could take place. The shear

stress was gradually increased to the point where it got large enough that a resize was necessary,

and from that value, the spin-up rate was calculated. The result was that the spin-up rate must

not exceed 170 rpm/sec.

The properties of the materials used on the launch vehicle change slightly with increasing or

decreasing temperature.3 The material properties in the structures codes were all assumed

constant at the room temperature values. Charts were found on the effect of temperature on

Young’s Modulus and Poisson’s Ratio. One such chart is shown in Fig. A.5.2.14.1.

Fig. A.5.2.14.1: Effect of Temperature of Young’s Modulus

(Engineering Toolbox4)


Author: Steven Izzo

If Young’s Modulus decreases, the critical stress decreases. The codes were run again for all

known values of the changing properties, and once again, even for the extreme values the launch

vehicle did not require extra support.

Thermal expansion is another source of stress commonly seen in launch vehicles. If there are

different materials adjacent to each other, they expand at different rates, and this can lead to

strain and stress.3 However, the launch vehicle was made of aluminum, and nearly all

components that need to be attached to each other were designed to be the same material.

Therefore, the effects of thermal expansion could be ignored.

Acoustics can lead to unseen stress as well. The vibrating air and expanding gasses leaving the

engines lead to increased vibrations.2 Basic vibration analysis was performed as part of the finite

element method. Although a complete analytical method was not performed at this time, the

acoustic vibrations did not require extra support. In general, acoustics are more of a concern for

liquid propelled launch vehicles as opposed to solid and hybrid, and for ground launches as

opposed to balloon launches.

All research has pointed out that a common source of failure in launch vehicles is on the local

level. Stress concentrations in places like joints, corners, and fastenings often lead to the failure

of launch vehicles in the past.3 This project was not designed on the local level, so a local stress

analysis was beyond the scope of this project, but it should noted that for an actual vehicle based

on this project to be built, a local stress analysis must be done first.

References 1. Klemans, B., “The Vanguard Satellite Launching Vehicle,” The Martin Company, Engineering Report No.11022, April 1960. 2. Pisacane, V., and Moore, R., Fundamentals of Space Systems, Oxford Press, New York, NY, 1994. 3. Sarafin, T., Spacecraft Structures and Mechanisms: From Concept to Launch. Microcosm, Inc., Torrance, CA, 1995. 4. “Young’s Modulus of Elasticity for Metals and Alloys,” The Engineering Toolbox, URL: http://www.engineeringtoolbox.com/young-modulus-d_773.html [cited 5 March 2008].



A.5.3 Closing Comments The Structures group was delegated the task of proving that our launch vehicles would not fail in

flight. By creating iterative math models, the propellant tanks, inter-stage skirts, and inter-tank

couplers are designed to withstand subjected forces due to buckling, bending, and shear. These

designs are found in an attempt to minimize GLOM without sacrificing structural integrity. We

add internal structural members like stringers or support rings to provide additional stability

when the primary component design is insufficient.

In addition to the previously mentioned elements of our launch vehicles, the nose cone and

vehicle gondola are also designed. While the gondola analysis employs the same methodology

as with the tanks and inter-stage skirts, the nose cone also had to survive thermal loads.

We are not able to complete all of the analyses we discuss to our satisfaction. If more time was

allotted, we would like to attempt further finite element analysis of the launch vehicle, and refine

our current math models to higher resolution. We are very pleased with the work accomplished

and look forward presenting our results to our adoring fans.


Written and Compiled by Brandon White

A.5.4 User’s Guide for Structures Codes Compiled and Edited by Brandon White The structures code is a conglomeration of 19 MATLAB scripts and functions. The following

document is meant to be a guide to anyone wishing to use the structures code. Each script or

function in the structures code folder is briefly described here, with a listing of inputs needed and

outputs generated.

Index bendtank.m C. Hiu page 614 buckle_main.m J. Schoenbauer page 615 editfiles2.m J. Schoenbauer page 616 editfiles_3.m J. Schoenbauer page 617 global_buck.m J. Schoenbauer page 618 editfiles_3.m J. Schoenbauer page 620 gond_strength.m S. Shoemaker page 622 InertiaFinal.m B. White page 623 intertank.m C. Hiu page 626 intertank_str.m J. Doyle page 627 mass.m D. Childers page 628 nose_cone_def.m V. Teixeira page 630 press_tank.m C. Hiu page 631 shear_calc.m J. Doyle page 632 skirt_analysis_v3_str.m J. Doyle page 633 skirt_v3.m J. Doyle page 635 tank_material_properties.m C. Hiu page 637 tanks.m C. Hiu page 639 tanksv2.m C. Hiu page 642


Written by Chii Jyh Hiu and Compiled by Brandon White

bendtank.m Written by Chii Jyh Hiu Revision 1.0 - 20 February 2008 Description:

bendtank.m is a function file that analyzes the propellant tanks in bending. It is called by tanks.m

or tanksv2.m.

Assumptions:

Tank bending allowable from Bruhn Figure C8.13a and bending allowable improvements for

pressurized tank from Bruhn Figure C8.14.

Input Section:

The call line of the function is: [Fbcr,Fbcr_press] = bendtank(E,D,t,L,P)

All of the variables that are passed into the m-file are described below: Variable Name Description E Tank material Young’s modulus (3-vector) [Pa] D Stage diameter(3-vector) [m] t Tank wall thickness (3-vector) [m] L Length of tank (3-vector) [m] P Tank internal pressure (3-vector) [Pa]

Output Section: Variable Name Description Fbcr Unpressurized tank bending allowable [Pa] Fbcr_press Pressurized tank bending allowable [Pa]


Written by Jessica Schoenbauer and Compiled by Brandon White

buckle_main.m Written by Jessica Schoenbauer Revision 1.0 – 25 March 2008 Description:

This is the main script to run the global buckling analysis on the launch vehicle. This code

formats inputs for global_buck.m and global_buck2.m and calls the codes. It also uses the

output, the buckling factor, from the global_buck.m and global_buck2.m codes to calculate the

maximum G’s that the launch vehicle can withstand.

Assumptions:

It is assumed that the launch vehicle can be simplified to represent a column-beam. It is assumed

that the number of stages, the stage lengths, the stage diameters, the wall thicknesses for each

stage, the material for each stage, the maximum acceleration in G’s that the rocket will

experience, and the gross lift off mass for the launch vehicle are all known quantities.

Important Notes:

It is necessary to have the StaDyn executable to run this code. It is also necessary to comment

out sections of the code. There is input for the 200g payload, the 1 kg payload, and the 5 kg

payload launch vehicles. Depending on the launch vehicle that is to be analyzed, the other two

vehicles should be commented out of the code.

Input Section:

The call line of the script is: buckle_main Output Section:

There is no “output” in the way a function does since this is a script. However, the resulting

maximum G’s that the launch vehicle can withstand is calculated and can be considered output.



editfiles2.m Written by Jessica Schoenbauer Revision 1.0 – 25 March 2008 Description:

This function is called by global_buck.m. It creates a structure data file that is used by StaDyn.

The first line of this function is: function [] = editfiles2(Node_yloc,Fy,MatProp)

Assumptions:

We assume a fixed number of nodes and elements for every launch vehicle.

Important Notes:

It is necessary to have the following file to run the program: template.3.

Input Section:

The call line of the function is: function [] = editfiles2(Node_yloc,Fy,MatProp);

All of the variables that are passed into the function are described below:

Variable Name Description Node_yloc A 16 vector containing the y locations for all the nodes. Fy The load applied to the launch vehicle in the analysis.

MatProp

A 3x8 matrix containing the material properties, material number, Young’s modulus, shear modulus, cross-section area, density, and cross-section moments of inertias. The 1st row is the 1st stage. The 2nd row is the 2nd stage. And the 3rd row is the 3rd stage.

Output Section:

It does not create any real output that is passed in Matlab, but it creates output that is written to

the structure data file which is used in StaDyn. The structure data file is called rocket.3.



editfiles_3.m Written by Jessica Schoenbauer Revision 1.0 – 25 March 2008 Description: This function is called by global_buck2.m. It creates a structure data file that is used by StaDyn. The first line of this function is: function [] = editfiles_3(Node_yloc,Fy,MatProp) Assumptions: We assume a fixed number of nodes and elements for every launch vehicle. Important Notes: Input Section: The call line of the function is: function [] = editfiles_3(Node_yloc,Fy,MatProp); All of the variables that are passed into the function are described below: Variable Name Description Node_yloc A 15 vector containing the y locations for all the nodes. Fy The load applied to the launch vehicle in the analysis.

MatProp

A 2x8 matrix containing the material properties, material number, Young’s modulus, shear modulus, cross-section area, density, and cross-section moments of inertias. The 1st row is the 1st stage. The 2nd row is the 2nd stage.

Output Section:

It does not create any real output that is passed in Matlab, but it creates output that is written to

the structure data file which is used in StaDyn. The structure data file is called rocket2.3.



global_buck.m Written by Jessica Schoenbauer Revision 1.0 – 25 March 2008 Description:

This function is called by buckle_main.m. It creates input for editfiles2.m and calls the function

using the inputs passed to it from buckle_main.m. The inputs that are created for editfiles2.m

consist of the node locations, the material properties, including the areas and inertias for each

stage cross-section, and the load.

The first line of this function is: function [pass,lambda] = global_buck(stage_len,stage_diam,stage_thk,stage_mat,load)

Assumptions:


Important Notes:

It is necessary to have the StaDyn executable to run this code. It is also necessary to have the

following file to run the program: runbuck, which is the file that contains the commands used by

StaDyn. StaDyn will also write an output file called stadyn.out. This file is read in to obtain the

output for the code.

Input Section:

The call line of the function is: function [pass,lambda] = global_buck(stage_len,stage_diam,stage_thk,stage_mat,load); All of the variables that are passed into the function are described below:

Variable Name Description stage_len A 3 vector containing the lengths of each stage, 1st to 3rd. stage_diam A 3 vector containing the stage diameters, 1st to 3rd. stage_thk A 3 vector containing the stage wall thicknesses, 1st to 3rd. Stage_mat A 3 vector containing the material for each stage, 1st to 3rd. load The maximum possible load applied to the launch vehicle.



Output Section:

A pass/fail parameter and the buckling load factor, lambda, are the outputs. The pass parameter

is 1 and the fail parameter is 2. The vehicle is considered to pass if the buckling load factor is

greater than the factor-of-safety, 1.25. It also outputs the buckling load factor output by StaDyn.



global_buck2.m Written by Jessica Schoenbauer Revision 1.0 – 25 March 2008 Description:

This function is called by buckle_main.m. It creates input for editfiles_3.m and calls the function

using the inputs passed to it from buckle_main.m. The inputs that are created for editfiles_3.m

consist of the node locations, the material properties, including the areas and inertias for each

stage cross-section, and the load.

The first line of this function is: function [pass,lambda] = global_buck2(stage_len,stage_diam,stage_thk,stage_mat,load)

Assumptions:


Important Notes:

It is necessary to have the StaDyn executable to run this code. It is also necessary to have the

following file to run the program: runbuck2.txt, which is the file that contains the commands

used by StaDyn. StaDyn will also write an output file called stadyn.out. This file is read in to

obtain the output for the code.

Input Section:

The call line of the function is: function [pass,lambda] = global_buck2(stage_len,stage_diam,stage_thk,stage_mat,load); All of the variables that are passed into the function are described below:

Variable Name Description stage_len A 2 vector containing the lengths of each stage, 1st to 2nd. stage_diam A 2 vector containing the stage diameters, 1st to 2nd. stage_thk A 2 vector containing the stage wall thicknesses, 1st to 2nd. Stage_mat A 2 vector containing the material for each stage, 1st to 2nd. load The maximum possible load applied to the launch vehicle.



Output Section:

A pass/fail parameter and the buckling load factor, lambda, are the outputs. The pass parameter

is 1 and the fail parameter is 2. The vehicle is considered to pass if the buckling load factor is

greater than the factor-of-safety, 1.25. It also outputs the buckling load factor output by StaDyn.


Written by Sarah Shoemaker and Compiled by Brandon White

gond_strength.m Written by Sarah Shoemaker Revision 1.1 - 19 March 2008

Description:

This code generates the strength the base of the gondola experiences.

Important Notes:

The gondola weight is acquired from the CATIA model.

Input Section:

The call line of the function is:

[stress] = gon_strength.m ( GLOW, area, gond_weight )


Variable Name Description GLOW Gross lift off weight of the launch vehicle [kg] area Area of the gondola base [m2]

gond_weight Mass of the guide rails, support rings, and avionics/avionics bay [kg]

Output Section:

Variable Name Description stress Stress on the gondola base [Pa]

Sample Output: ans = 7.5479e+004 The stress the gondola base experiences.



InertiaFinal.m Written by Brandon White Revision 1.0 - 1 March 2008 Description: InertiaFinal.m is a function embedded within tanks.m. The function calculates principal

moments of inertia for entire launch vehicle at various stages of flight. InertiaFinal.m uses inputs

from tanks.m and mass.m. InertiaFinal.m is a culmination of previous revisions under various

names (Inertia.m, Inertia1.m, Inertia2.m).

Assumptions:

All products of inertia are zero, launch vehicle is axisymmetric. LITVC is a point mass located

at the top of the second stage nozzle. Payload and avionics in the third stage are point masses at

the base of the nose cone. Nose cone is a perfect, right cone (not blunted). Avionics in the first

and second stages are wall mounted to the inter-stage skirts, utilizing a constant thickness of 10

cm. This thickness assumption for the avionics is extremely conservative and probably should be

updated.

Input Section:

The call line of the function is: [I1_full I1_empty I2_full I2_empty I3_full I3_empty] = InertiaFinal(Xcm, L_cone, M_cone, M_Ox, M_tank_Ox, L_Ox, t_Ox, M_Fuel, M_tank_Fuel, L_Fuel, t_Fuel, M_tank_press, D_tank_press, t_tank_press, D_tank_press, s_len, s_mass, D, M_Engine, L_Nozzle, payload, mass_avionics, LITVC_prop, LITVC_mt, Length_stage)




Variable Name Description

Xcm Launch Vehicle Center of mass at the 6 time steps, measured from nose cone [m]

L_cone Vertical length of the Nose Cone [m] M_cone Mass of the Nose Cone [kg] M_Ox Oxidizer mass for each stage [kg] M_tank_Ox Oxidizer tank mass for each stage [kg] L_Ox Length of Oxidizer tank for each stage [m] t_Ox Oxidizer tank thickness for each stage [m] M_Fuel Fuel mass for each stage [kg] M_tank_Fuel Fuel tank mass for each stage [kg] L_Fuel Length of Fuel tank for each stage [m] t_Fuel Oxidizer tank thickness for each stage [m] M_tank_press Mass of the Pressurant tank [kg] D_tank_press Diameter of Pressurant tank [m] t_tank_press Pressurant tank thickness [m]

D_tank_press Diameter of Pressurant tank [m], yes it is input twice. This is a mistake.

s_len Inter-stage skirt vertical length [m] s_mass Inter-stage skirt mass [kg] D Diameter for each stage [m] M_Engine Engine mass for each stage [kg] L_Nozzle Vertical length of the engine nozzle for each stage [m] payload Mass of payload [kg] mass_avionics Mass of the avionics [kg] LITVC_prop Mass of the LITVC [kg] LITVC_mt Mass of the LITVC tank [kg] Length_Stage Vertical length of each stage [m]

Output Section:

InertiaFinal.m outputs six 3-element arrays. These arrays represent the principle moments of

inertia at different phases of flight. The arrays are output in row format with the Ixx value being

the first element, the Iyy value being the second element, and the Izz value being the third element.




I1_full Principle moments of inertia with: All three stages, all full propellant [kg*m2]

I1_empty Principle moments of inertia with: All three stages, no propellant in the first stage [kg*m2]

I2_full Principle moments of inertia with: Second and Third stages, both with full propellant [kg*m2]

I2_empty Principle moments of inertia with: Second and Third stages, no propellant in the second stage [kg*m2]

I3_full Principle moments of inertia with: Third stage only, full propellant [kg*m2]

I3_empty Principle moments of inertia with: Third stage only, no propellant [kg*m2]



intertank.m Written by Chii Jyh Hiu Revision 1.0 - 17 February 2008 Description:

intertank.m is a function file that analyzes the inter-tank couplers for axial compressive strength.

It is called by tanks.m or tanksv2.m.

Assumptions:

Intertank buckling theory from Baker, E.H., Kovalevsky, L., Rish, F.L., Structural Analysis of

Shells, Robert E. Krieger Publishing Company, Huntington, NY, 1981, pgs. 229-240.

Important Notes:

Unlike most other MAT-derived functions, intertank.m accepts inputs as scalars

Input Section:

The call line of the function is: [mass_int, t, numhoop] = intertank(E,Sigma_y,Density,v,m_above,max_g,D,L)

All of the variables that are passed into the m-file are described below:

Variable Name Description E Inter-tank material Young’s modulus [Pa] Sigma_y Inter-tank material yield stress [Pa] Density Inter-tank material density [kg/m3] v Inter-tank material Poisson’s ratio m_above Mass of rocket above inter-tank [kg] max_g Maximum g-loading [g’s] D Stage diameter [m] L Length of intertank [m]

Output Section: Variable Name Description mass_int Inter-tank mass [kg] t Inter-tank wall mass [kg] numhoop Number of hoops


Written by Jesii Doyle and Compiled by Brandon White

intertank_str.m Written by Jesii Doyle Revision 1.0 - 27 February 2008 Description:

Intertank_str.m is a function file that determines the number of stringers needed in the inter-tank

coupler. It is called by tanks.m or tanksv2.m.

Input Section:

The call line of the function is: [nsi,mass_str _int] = intertank_str(D,Sy,t_skin,tank_material,L_intertank)

All of the variables that are passed into the m-file are described below: Variable Name Description D Inter-tank Diameter [m] Sy Inter-tank material yield stress [Pa] t_skin Inter-tank wall thickness [m] tank_material Inter-tank material L_intertank Length of inter-tank section [m]

Output Section: Variable Name Description nsi Number of stringers required mass_str_int Inter-tank stringer mass [kg]


Written by David Childers and Compiled by Brandon White

mass.m Written by David Childers Revision 2.0 - 5 March 2008 Description:

Calculates the center of mass for the launch vehicle.

Assumptions:

Point mass system. Mass is symmetric (located in center of launch vehicle). Locations are

approximated for avionics, engines, and LITVC because actual locations cannot be determined

based on the theoretical aspect of the project. Reference point is the top of the launch vehicle.

Stage numbering is from the bottom up.

Input Section:

The call line of the function is:

[CM_Full_3 CM_Emp_3 CM_Full_2 CM_Emp_2 CM_Full_1 CM_Emp_1]=mass(payload, M_Fuel,M_Ox,M_Engine,M_cone,M_tank_Fuel,M_tank_Ox,Mass_tank_press,LITVC_mt, s_mass,Mass_inert,D,D_tank_press,L_Nozzle,L_Ox,L_Fuel,Length_stage,s_len, L_cone,LITVC_prop,pressmass,Mass_intertank,mass_avionics)


Written by David Childers and Compiled by Brandon White

All of the variables that are passed into the function are described below: Variable Name Description Stages Number of stages payload Payload mass [kg] M_Fuel Fuel mass for each stage [kg] M_Ox Oxidizer mass for each stage [kg] M_engine Engine mass for each stage [kg] M_tank_Ox Oxidizer tank mass of each stage M_tank_Fuel Fuel tank mass of each stage Mass_inert Inert mass/stage M_cone Nose cone mass Mass_tank_press Pressurant tank mass [kg] LITVC_mt Mass of the LITVC tank [kg] s_mass Mass of the inter-stage skirts [kg] D Diameter of each stage [m] D_tank_press Pressurant tank diameter [m] L_Ox Oxidizer tank length/stage [m] L_Fuel Fuel tank length/stage [m] s_len Skirt length/stage [m] L_cone Nose cone length [m] Length_stage Total length of each stage [m] LITVC_prop Mass of the LITVC [kg] Mass_intertank Mass of material between the tanks [kg] mass_avionics Mass of the avionics [kg] Output Section: Variable Name Description CM_Full_3 Center of mass for full third stage [m] CM_Emp_3 Center of mass for empty third stage [m] CM_Full_2 Center of mass for full second stage [m] CM_Emp_2 Center of mass for empty second stage [m] CM_Full_1 Center of mass for full first stage [m] CM_Emp_1 Center of mass for empty first stage [m]


Written by Vince Teixeira and Compiled by Brandon White

nose_cone_def.m Written by Vince Teixeira Revision 1.0 - 24 February 2008 Description:

nose_cone_def.m is a function file that integrates the surface area of the nose cone using a

power-law body relationship.

Assumptions:

A one quarter sphere at the blunted tip at 30% of the nose cone length back from the tip.

Input Section:

The call line of the function is: [L_nose M_nose] = nose_cone_def(D_body)

All of the variables that are passed into the m-file are described below: Variable Name Description D_body Diameter of the final stage of the launch vehicle [m]

Output Section: Variable Name Description L_nose Length of the nose cone [m] M_nose Mass of the nose cone [kg]



press_tank.m Written by Chii Jyh Hiu Revision 1.1 - 20 February 2008 Description:

press_tank.m is a function file that analyzes the buckling strength of a pressurized cylindrical

tank. It is called by tanks.m or tanksv2.m.

Assumptions:

Tank buckling allowable from Baker and buckling allowable improvements for pressurized tank

from Bruhn Figure C8.11.

Input Section:

The call line of the function is: [Fcr_press,Fcr,DF]= press_tank(E,v,D,t,L,P)

All of the variables that are passed into the m-file are described below: Variable Name Description E Tank material Young’s modulus (3-vector)[Pa] v Poisson’s Ratio (3-vector) D Stage diameter (3-vector) [m] t Tank wall thickness (3-vector) [m] L Length of tank (3-vector) [m] P Tank internal pressure (3-vector) [Pa]

Output Section: Variable Name Description Fcr_press Unpressurized tank bending allowable [Pa] Fcr Pressurized tank bending allowable [Pa]



shear_calc.m Written by Jesii Doyle Revision 1.0 - 27 February 2008 Description: shear_calc.m is a function file that determines the maximum shear loading (applied through the shear center) allowable by the launch vehicle. It is called by tanks.m or tanksv2.m. Input Section:

The call line of the function is: [shear] = shear_calc(D,n_str,A_str,tskin,Sy,stages)

All of the variables that are passed into the m-file are described below: Variable Name Description D Stage diameter [m] n_str Number of Stringers in each stage A_str Cross sectional area of each stringer [m2] t_skin Tank wall thickness [m] Sy Stringer material yield stress [Pa] Stages Number of stages in the launch vehicle

Output Section: Variable Name Description shear Maximum Shear Force Capability of Launch Vehicle [N]



skirt_analysis_v3_str.m Written by Jesii Doyle Revision 2.0 - 24 March 2008 Description:

This function performs the analysis on the inter-stage skirt with stringers and ring supports, and

outputs the option with minimized cost. This code is called by skirt_v3.m.

Assumptions:

Constant taper angle of 10º

Constant skin thickness 4mm

Number of stringers is 1/6th of total possible stringers per radius

Important Notes:

This code is only called by the skirt_v3.m code, and should not need to be revised.

Input Section:

The call line of the function is: [mass,length,cost,ns,ring,yes_no,t_str] =

skirt_analysis_v3_str(P,r_bottom,r_top,l_noz,shear_f)


Variable Name Description P Vertical Load [N] r_bottom Radius of the bottom of the inter-stage skirt [m] r_top Raius of the top of the inter-stage skirt [m] l_noz Length of the nozzle [m] shear_f Applied shear force [N]



Output Section:

Variable Name Description mass Mass of the inter-stage skirt [kg] length Vertical length of the inter-stage skirt [m] cost Cost of the inter-stage skirt [USD] ns Number of stringers ring Number of support rings yes_no Shear stress check output t_str Stringer thickness [m]



skirt_v3.m Written by Jesii Doyle Revision 3.0 - 24 March 2008

Description:

This code calculates the mass, vertical length and cost of the inter-stage skirts between each

stage. This code is run within tanks.m and tanksv2.m.

Assumptions:

The inter-stage skirt shape is a truncated cone.

Only loads are maximum axial loads applied from mass above the inter-stage skirt and g-loading,

and maximum shear force.

Inter-stage skirt material: Aluminum

Stringer material: Aluminum

Input Section:

The call line of the function is: [s_mass,s_len,s_cost,ns,ring,yes_no_s,t_str] =

skirt_v3(L_Nozzle,M_Ox,M_Fuel,M_tank_Ox,M_tank_Fuel,M_Engine,D,payload_mass,m

ax_g,stages,m_press,M_tank_press,M_Cone,Sy)


Variable Name Description L_Nozzle Length of the nozzle for each stage [m] M_Ox Mass of the oxidizer for each stage [kg] M_Fuel Mass of the fuel for each stage [kg] M_tank_Ox Mass of the oxidizer tank for each stage [kg] M_tank_Fuel Mass of the fuel tank for each stage [kg] M_Engine Mass of the engine for each stage [kg] D Diameter for each stage [kg] payload_mass Mass of the payload [kg] max_g Max g-loading [kg] stages Total number of stages m_press Mass of the pressurant for each stage [kg] M_tank_press Mass of the pressurant tank for each stage [kg] M_cone Mass of the nose cone [kg] Sy Maximum shear force [N]



Output Section:

Variable Name Description s_mass Mass of inter-stage skirt for each stage [kg] s_len Length of inter-stage skirt for each stage [m] s_cost Inter-stage skirt cost for each stage [USD] ns Number of stringers in inter-stage skirt for each stage ring Number of support rings in inter-stage skirt for each stage yes_no_s Yes/No output t_str Stringer thickness [m]



tank_material_properties.m Written by Chii Jyh Hiu Revision 1.0 - 20 February 2008 Description:

tank_material_properties.m is a function file that returns material physical properties for use in

other calculations.

Assumptions:

Material strengths use B-basis, LT values where available. Tensile strengths are at yield, shear

stresses are at ultimate.

Important Notes:

Material properties for Carbon fiber are not authoritative. An isotropic carbon fiber layup was

assumed. As carbon fiber was abandoned early in the design process for cost reasons, there was

no incentive to refine the existing figures, which are kept for historic purposes.

Input Section:

The call line of the function is: [Sigma_y,Sigma_s,Density,E,Cost_kg,v] = tank_material_properties(tank_material)



tank_material Tank material - S: Steel, A: Aluminum, C: Composite, T: Titanium, X: n/a. (3-vector, string) [unitless]



Output Section: Variable Name Description Sigma_y Yield stress allowable [Pa] Sigma_s Shear stress allowable [Pa] Density Density [kg/m3] E Young’s modulus [Pa] Cost_kg Material raw cost [USD/kg] v Poisson’s ratio [unitless]



tanks.m Written by Chii Jyh Hiu Revision 1.7 - 20 February 2008 Description:

tanks.m calculates the required propellant and pressurant tank dimensions to meet propellant

storage and flight load requirements. It passes function calls to subsidiary functions to calculate

tank bending and buckling in-flight, as well as stresses on inter-tank couplers. It also calls cost

functions to calculate the manufacturing cost of the tanks. It is used in the Material Analysis

preliminary design level, where it is called by the mainloop.m master function and forms an

iterative loop with the various propulsion codes to optimize the inert mass fraction of the launch

vehicle.

Assumptions:

A Reserve Factor of 1.25 is applied across all stress analysis. We assume that hoop stresses in

the propellant tanks will always be greater than axial stresses. Max in-flight g-loading is applied

across all components. Inter-stage skirt function calls were disabled to speed iteration during

preliminary design runs.

Important Notes:

tanks.m expects 3-vector inputs for most variables. Exceptions are listed in the table below

Input Section:

The call line of the function is: [M_tank_press, t_tank_press, M_tank_Ox, M_tank_Fuel, t_Ox, t_Fuel, inert_mass_fraction_struct, Mass_inert, Length_stage, yes_no, COST_stage, Tot_Cost, D, L_Ox, L_Fuel] = tanks.m(Mat, Prop_Type, M_Ox, Ox_Vol, P_Ox, M_Fuel, Fuel_Vol, P_Fuel, D, L_Nozzle, M_Engine, g, inert_mass, m_press, vol_press, P_press, payload_mass, shear, bending)



All of the variables that are passed into the function are described below: Variable Name Description

Mat Tank material - S: Steel, A: Aluminum, C: Composite, T: Titanium, X: n/a. (3-vector, string) [unitless]

Prop_Type Propellant type - 1: Cryogenic, 2: Storable, 3: Hybrid, 4: Solid, 0: n/a (3-vector, string) [unitless]

M_Ox Mass of oxidizer (3-vector) [kg] Ox_Vol Volume of oxidizer (3-vector) [m3] P_Ox Oxidizer tank operating pressure (3-vector) [Pa] M_Fuel Mass of fuel (3-vector) [kg] Fuel_Vol Volume of fuel (3-vector) [m3] P_Fuel Fuel tank operating pressure (3-vector) [Pa] D Stage diameter (3-vector) [m] L_Nozzle Length of Nozzle and Engine (3-vector) [m] M_Engine Mass of Nozzle and Engine (3-vector) [kg] g Max in-flight acceleration (scalar) [g’s] inert_mass Target inert mass (3-vector) [kg] m_press Mass of pressurant (3-vector) [kg] vol_press Volume of pressurant (3-vector) [kg] P_press Pressurant tank max pressure (3-vector) [Pa] payload_mass Payload mass (scalar) [kg] shear Max shear (3-vector) [N] bending Max bending (3-vector) [Nm]



Output Section: Variable Name Description

M_tank_press Pressurant tank mass [kg]

t_tank_press Pressurant tank wall thickness [m]

M_tank_Ox Oxidizer tank mass [kg]

M_tank_Fuel Fuel tank mass [kg] t_Ox Oxidizer tank wall thickness [m] t_Fuel Fuel tank wall thickness [m] inert_mass_fraction_struct Inert mass fraction [kg] Mass_inert Inert mass [kg] Length_stage Length of stage [m] yes_no Success/Failure flag for target mass [unitless] COST_stage Cost of stage [USD] Tot_Cost Total Cost [USD] D Stage Diameter [m] L_Ox Length of oxidizer tank [m] L_Fuel Length of fuel tank [m]



tanksv2.m Written by Chii Jyh Hiu Revision 2.4 - 1 March 2008

Description:

tanksv2.m is a m-file that calculates the required propellant, pressurant and LITVC tank

dimensions to meet propellant storage and flight load requirements. It passes function calls to

subsidiary functions to calculate tank bending and buckling in-flight, as well as stresses on inter-

tank couplers and inter-stage skirts. It has calls to center of mass and inertia matrix functions. It

also calls cost functions to calculate the manufacturing cost of the tanks. It is used in final

analysis design, and calls input variables from the workspace that are calculated by running

mainonce.m and LITVC.m.

Assumptions:

A Reserve Factor of 1.25 is applied across all stress analysis. Max in-flight g-loading of 6gs is

applied across all components.

Input Section:



tank_material Tank material - S: Steel, A: Aluminum, C: Composite, T: Titanium, X: n/a. (3-vector, string) [unitless]

propellant_type Propellant type - 1: Cryogenic, 2: Storable, 3: Hybrid, 4: Solid, 0: n/a (3-vector, string) [unitless]

prop_mass Mass of propellant (3-vector) [kg] p_ox_tank Oxidizer tank operating pressure (3-vector) [Pa] p_fuel_tank Fuel tank operating pressure (3-vector) [Pa] diameter_final Stage diameter (3-vector) [m] nozzle_length Length of Nozzle and Engine (3-vector) [m] engine_mass Mass of Nozzle and Engine (3-vector) [kg] g Max in-flight acceleration (scalar) [g’s] payload_mass Payload mass (scalar) [kg]



Output Section: Variable Name Description M_tank_press Pressurant tank mass [kg] t_tank_press Pressurant tank wall thickness [kg] M_tank_Ox Oxidizer tank mass [kg] M_tank_Fuel Fuel tank mass [kg] t_Ox Oxidizer tank wall thickness [m] t_Fuel Fuel tank wall thickness [m] inert_mass_fraction_struct Inert mass fraction [kg] Mass_inert Inert mass [kg] Length_stage Length of stage [m] yes_no Success/Failure flag for target mass [unitless] COST_stage Cost of stage [USD] Tot_Cost Total Cost [USD] D Stage Diameter [m] L_Ox Length of oxidizer tank [m] L_Fuel Length of fuel tank [m] CM_Full Center of mass of fully fueled stage [m] CM_Empty Center of mass of empty stage [m] In_Full Mass moment of inertia of full nth stage [kgm2] In_Empty Mass moment of inertial of empty stage [kgm2]

Report Section 8 - Purdue University

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Transcript of Report Section 8 - Purdue University