chapter 6 structural stress and buckling analyses...

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CHAPTER 6

STRUCTURAL STRESS AND BUCKLING ANALYSES OF

MICRO-SATELLITE

6.1 LINEAR STATIC ANALYSIS

Linear static analysis represents the most basic type of analysis.

The term "linear" means that the computed displacement or stress is linearly

related to the applied force. The term “static” means that the forces do not

vary with time or, that the time variation is insignificant and can therefore be

safely ignored. The static analysis equation is:

[K]{u} = {f}

Where ‘K’ is the system stiffness matrix, ‘f’ is the applied force vector, and

‘u’ is the displacement vector. Once the displacements are computed, the

solver uses these to compute element forces, stresses, reaction forces, and

strains. The applied forces may be used independently or combined with each

other. The loads can also be applied in multiple loading subcases, in which

each subcase represents a particular loading or boundary condition. Multiple

loading subcases provide a means of solution efficiency, whereby the solution

time for subsequent subcases is a small fraction of the solution time for the

first.

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6.2 OBJECTIVE

The reliability of satellite structural components is greatly

increased, and their cost and weight reduced by the systematic and rigorous

application of stress analysis principles as an integral part of the design

process. The structural stress analysis performed guides the design of the

satellite and sizing of the components and provide a high degree of

confidence before launch. Structural stress analysis is performed in order to

ensure that a structure will fulfill its intended function in a given loads

environment. It is important to anticipate all the possible failure modes and

design it against them. For a space structure, the common failures to be

considered are as follows:

Ultimate failure, rupture and collapse due to the stresses

exceeding the ultimate strength of the material.

Detrimental yielding that undermines structural integrity or

performance due to stresses exceeding the yield strength of the

material.

Instability (buckling) under a combination of loads,

deformation and part geometry such that the structure faces

collapse before buckling strength of the material is reached.

“Excessive” elastic static or dynamic deformations causing

loss of function, preload or alignment, interference, and

undesirable vibration noise.

6.3 FAILURE MODES OF SANDWICH PANELS

The micro-satellite structures like bottom deck, middle deck, top

deck, cross webs, vertical webs and solar panels are constructed of

honeycomb sandwich panels where the skin is made of aluminium alloy and

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the core is made of aluminium honeycomb. The following are the common

failure modes seen in the sandwich panels.

6.3.1 Strength

The skin and core materials should be able to withstand the tensile,

compressive and shear stresses induced by the design load. The skin to core

adhesive must be capable of transferring the shear stresses between skin and

core. The Figure 6.1 shows the skin compressive failure.

Figure 6.1 Skin compressive failures

6.3.2 Stiffness

The sandwich panel should have sufficient bending and shear

stiffness to prevent excessive deflection. The Figure 6.2 shows the excessive

deflection.

Figure 6.2 Excessive deflection

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6.3.3 Panel buckling

The core thickness and shear modulus must be adequate to prevent

the panel from buckling under end compression loads. The Figure 6.3 shows

the panel buckling mode of failure.

Figure 6.3 Panel buckling

6.3.4 Shear crimping

The core thickness and shear modulus must be adequate to prevent

the core from prematurely failing in shear under end compression loads. The

Figure 6.4 shows the shear crimping in sandwich panel.

Figure 6.4 Shear crimping

6.3.5 Skin wrinkling

The compressive modulus of the facing skin and the core

compression strength must both be high enough to prevent a skin wrinkling

failure. The Figure 6.5 shows the failure due to skin wrinkling.

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Figure 6.5 Skin wrinkling

6.3.6 Intra cell buckling

For a given skin material, the core cell size must be small enough to

prevent intra cell buckling. The Figure 6.6 shows the intra cell buckling

happening in sandwich panel.

Figure 6.6 Intra cell buckling

6.3.7 Local compression

The core compressive strength must be adequate to resist local

loads on the panel surface. The Figure 6.7 shows the failure by local

compression.

Figure 6.7 Local compression

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6.4 SUMMARY OF SATELLITE LOADS

The most common loads encountered on a satellite in the space

applications are given in Table 6.1.

Table 6.1 Loads acting on the satellite

Loading (Event) Significance

Inertia loads (Launch and landing) Loads that drive the design ofprimary structure

Vibration (Flight and Orbitoperations)

Structurally transmitted, Causingfatigue/fracture

Vibro-acoustic (Launch) Acoustically transmitted,especially for low mass/area parts

Thermally induced (Flight and orbitoperations)

Dictates allowable temperaturesand gradients, compatibility ofmaterials

Pressurization and flow induced(flight and orbit operations)

For pressure vessels, pipe lines,housings

Mechanical/Thermal(Fabrication/Assembly))

Material Residual stresses,Fastener/seal preloads,misalignment

Mechanical/Thermal(Verificationtesting)

May limit useful life of material

Mechanical/Inertial(Ground handlingand Transportation)

Important for the design ofmechanical ground supportequipment (MGSE) and spacecraftinterface with MGSE, may limituseful life of material.

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Structural loads are specified at maximum expected level and

referred to as the design or limit loads. Usually, two or more of these loads act

simultaneously and their combined effect needs to be considered. It was noted

that the loads environment applied to the structure during the verification

testing may be more significant than the loads experienced during flight.

Many structural failures have occurred during the action of the combined

loads on the satellite structures. Therefore, these loads must be considered

very carefully in the strength and fatigue calculations. The impact of the

satellite with the orbital debris was not included in the possible loads a

structure may encounter. The micro satellite considered is subjected to both

static and dynamic loads and are given below.

Longitudinal acceleration (Static+ Dynamic): 7g compression/2.5g tension

Lateral acceleration (Static + Dynamic) : 6g

Load factor : 1.25

The load cases derived for the analysis are given in Table 6.2.

Table 6.2 Load cases for stress analysis

Load case number

(LC)

Lateral X Lateral Y Longitudinal Z

LC 1 7.25g 0 -8.25g

LC 2 0 7.25g -8.25g

LC 3 7.25g 0 3.75g

LC 4 0 7.25g 3.75g

LC 5 5.75g 5.75g -8.25g

LC 6 5.75g 5.75g 3.75g

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A finite element model of the micro-satellite structure with all the

subsystems was developed in MSC PATRAN and analysis was done using

MSC NASTRAN inorder to predict deformations, internal forces and stresses.

It was based on an idealization of the actual structure using simplified

assumptions on geometry, loads and boundary conditions. Structural stress

analysis should define and address all the loads acting on the primary and

secondary structures. The following results were noted from the analysis.

Deformation of the micro-satellite structure

Von Mises stress distribution

Forces acting on the Multi Point Constraint

6.5 FAILURE CHECK MODES OF MICRO-SATELLITE

STRUCTURE

Adequacy of the structure to withstand the calculated forces and

stresses is checked by calculating the margin of safety (MS) values which is

defined as

Failure is predicted for MS < 0 where SF is the safety factor.

Failure stress or force is determined by means of failure theories. Some of the

most commonly used ones are summarized below

(a) Maximum Normal Stress theory is used to predict ultimate failure with

MS = Ftu/(SF x max) - 1. Here Ftu is the ultimate tensile stress of the material

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and max is the maximum normal stress due to the external loading. In general,

this theory is more applicable to brittle materials.

(b) Maximum Shear Stress theory is used to conservatively predict ultimate

failure for ductile materials. MS = Fsu/(SF x max) - 1, where Fsu is the

ultimate shear stress of the material and max is the predicted maximum shear

stress. This theory can also be used to predict the onset of yield by replacing

Fsu by Fsy, the shear yield strength of the material.

(c) Distortion Energy theory is used for predicting the initiation of yield in a

structure for the micro-satellite considered and gives more accurate results

than the maximum shear stress theory. The margin of safety is given by

MS = Fty/(SF x VM) - 1, where Fty is the tensile yield stress, and VM is the

Von Mises stress. A similar criterion used for the prediction of failure in

laminated composite materials by the Tsai-Hill Theory. A typical SF value

used for the ultimate failure of the spacecraft structures is 1.25. In addition a

yield SF typically equal to 1.25 is selected to prevent structural damage or

detrimental yielding during structural testing or flight. The tensile yield

strength of the materials is listed in the Table 6.3. Figures 6.8 to 6.13 show the

Von mises stress distribution in the satellite structures. The Table 6.4 shows

the margin of safety values calculated by using Distortion theory for the

maximum Von mises stress values identified in the micro-satellite structures

for the derived load cases.

Table 6.3 Tensile yield strength of the materials

Material Tensile strength in MPa

Aluminium Face sheet 280

Aluminum Honey comb 0.816

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Figure 6.8 Von mises Stress distribution in bottom deck for LC2

Figure 6.9 Von mises Stress distribution in middle deck for LC2

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Figure 6.10 Von mises Stress distribution in top deck for LC6

Figure 6.11 Von mises Stress distribution in cross webs for LC2

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Figure 6.12 Von mises Stress distribution in vertical webs for LC2

Figure 6.13 Von mises Stress distribution in solar panels for LC5

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Table 6.4 Margin of safety of the satellite structures

Structure Load case Von mises stress(MPa)

Margin ofSafety

Bottom deckLayer 1 LC 2 71.6 2.13Layer 2 LC 5 0.571 0.14Layer 3 LC 2 71.5 2.13

Middle deckLayer 1 LC 2 41.3 4.42Layer 2 LC 2 0.630 0.03Layer 3 LC 2 41.2 4.42

Top deckLayer 1 LC 6 36.7 5.10Layer 2 LC 5 0.341 0.909Layer 3 LC 6 36.6 5.10

Cross webs ( Between Bottom deck and Middle deck)Layer 1 LC 2 138 0.623Layer 2 LC 6 0.0958 5.79Layer 3 LC 2 138 0.623

Vertical webs ( Between Middle deck and Top deck)Layer 1 LC 2 24.8 8.03Layer 2 LC 2 0.0903 6.00Layer 3 LC 2 24.6 8.03

Solar panelsLayer 1 LC 5 21.8 9.27Layer 2 LC 2 0.0457 13.24Layer 3 LC 5 21.8 9.27

Interface ringLayer 1 LC 2 34.4 5.51Layer 2 LC 2 35.2 5.36

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6.6 MULTI POINT CONSTRAINT FORCE ON THE JOINTS

Multi Point Constraints (MPC) establishes geometric relationships

that have to be met by the displacements of certain nodes of the structure.

This is useful for modelling very stiff elements, without numerical difficulties

that would imply to model them as truly flexible bodies of high stiffness.

MSC NASTRAN has included rigid elements namely RBAR, RBE, etc that

generate internally these relationships from the geometric data provided by

the user. These forces are often essential for the correct dimensioning of the

structure. This concept is used in the case of screwed joints, very stiff

elements, etc. The M6 inserts were used to integrate the bottom deck of the

satellite with the interface ring at 12 points. All the structural inserts except

the edge inserts and the interface ring are M4 inserts and all subsystem

integration with primary structure are M4 inserts. The M6 inserts can

withstand a load of 600 N and M4 inserts can withstand the load of 400 N.

The Table 6.5 shows the maximum MPC force taken from MSC PATRAN/

MSC NASTRAN for the 6 different load cases shown in Table 6.2.

Table 6.5 MPC forces of the joints

StructuralComponent

Insert type Load CaseMaximum

MPC Force (N)

Bottom deck M4 LC 2 345

Middle deck M4 LC 2 320

Top deck M4 LC 1 85.6

Cross webs M4 LC 2 384

Vertical webs M4 LC 2 153

Solar panels M4 LC 5 242

Interface ring M6 LC 5 292

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The stress values of the micro-satellite structures obtained from the

Von Mises stress distribution and the MPC forces obtained at the joint

locations are well within the safe limits. The static analysis results show that a

high value of margin of safety has been obtained for certain structures, but it

was proved that for small satellite, dynamic design consideration is the main

criteria on the structure capability under applied load. Since the thickness of

the aluminum face sheet is already small the reduction of the thickness of the

aluminum face sheet has only negligible effect on the weight reduction of the

satellite. So the analytical results may be used for the design.

6.7 SANDWICH INSTABILITY

The margins of safety for the Sandwich Instabilities of the micro-

satellite structures are calculated as shown below.

6.7.1 Face wrinkling of Bottom deck

For 0.25 mm face skin and 40 mm core sandwich

Density of the core dcore = 32 kg/m3

Density of the face skin dface = 2700 kg/m3

Elastic modulus of face skin Ef = 70 GPa

Elastic modulus of core Ec E fd face

d core

Elastic modulus of core Ec = 829.6 MPa

Thickness of face skin tf = 0.25 mm

Thickness of core tc = 40 mm

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Stress acting in core c = E ft cE f

t fEc2/1

33.0

= 195 MPa

Maximum stress acting in face skin a = 71.6 MPa

(From FE analysis)

Margin of Safety MS = 1ac

Margin of Safety MS = 1.72

6.7.2 Face wrinkling of Middle deck






d core




Stress acting in core c = E ft cE f

t fEc2/1

33.0

= 275.7 MPa

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Maximum stress acting in face skin a = 41.3 MPa

(From FE analysis)



6.7.3 Face wrinkling of Cross webs






d core




Stress acting in the core c = E ft cE f

t fEc2/1

33.0

= 284 MPa

Maximum stress acting in the face skin a= 138 MPa

(From FE analysis)

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6.7.4 Intracellular buckling / dimpling of Bottom and Middle deck

For core with 0.25 mm face skin



Poisson’s ratio = 0.3

Cell size of honeycomb core S = 4.76 mm

Stress acting in the core c =2

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St fE f

= 416 MPa

Maximum stress acting in the face skin

of bottom deck and middle deck a = 71.6 MPa

(from FE analysis)



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6.7.5 Intracellular buckling / dimpling of Cross webs, Vertical webs,

Top deck and solar panels for core with 0.20 mm face skin



Poisson’s ratio = 0.3

Cell size of honeycomb core S = 4.76 mm

Stress acting in the core c =2

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St fE f

= 266 MPa

Maximum stress acting in the face skin

of bottom deck and middle deck a = 138 MPa

(from FE analysis)

Margin of Safety MS = 1a

c


The margin of safety values obtained for the instabilities like face

wrinkling, intra cellular buckling/dimpling of micro-satellite sandwich

structures like bottom deck, middle deck, top deck, cross webs, vertical webs

and solar panels are positive, and it is clear that there is no possible modes of

failure of sandwich panels for the loads considered.

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6.8 SUBSYSTEM LOADS ON INSERT

The margins of safety for the subsystem loads on the insert are

calculated as shown below.

6.8.1 Calculation for largest subsystem Bus Electronics loads on M4

insert

Out of plane acceleration acting on the subsystem (LongG) = 30g

In plane acceleration acting on the subsystem (LatG) = 20g

Center of Gravity of the subsystem bus Electronics C.G = 0.06524 m

Mass of the subsystem = 3. 068 kg

Pitch Diameter of inserts PCdia = 0.209 m

Lateral force acting on the subsystem LatF = Lat G*Mass

= 601.94 N

Longitudinal force acting on the subsystem LongF = LongG*Mass

= 902.91 N

Moment = Lat F*C.G

= 39.27 Nm

Number of inserts resisting Lateral load N = 4

Longitudinal non-uniformity factor (Longnuf) = 1

Load per insert in Longitudinal load (LongL) =4

LongF Longnuf

= 225.72 N

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Load per insert in Lateral load (LatL) =NPC

Momentdia *

= 46.97 N

Total pull out load on insert Lsi = LongL+ LatL

= 272.69 N

Inplane load on insert Lso =N

LatF

= 150.48 N

Maximum inplane load of M4 insert Pci = 2000 N

Maximum outplane load of M4 insert Pco = 1200 N

Margin of Safety MS =22

1

PL

PL

co

so

ci

si

- 1


6.8.2 Calculation for subsystem Power Electronics loads on M4

insert



Center of Gravity of the subsystem Power Electronics C.G = 0.0548 m

Mass of the subsystem = 2.084 kg


Lateral force acting on the subsystem (LatF) = Lat G*Mass

= 408.88 N

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Longitudinal force acting on the subsystem (LongF) = LongG*Mass

= 613.32 N

Moment = Lat F*C.G

= 22.40 Nm




LongF Longnuf

= 153.33 N

Load per insert in Lateral load LatL =NPC

Momentdia *

= 30.60 N


= 183.93 N


LatF

= 102.22 N




1

PL

PL

co

so

ci

si

- 1


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6.8.3 Calculation for subsystem Power Distribution loads on M4

insert



Center of Gravity of the subsystem Power Distribution C.G = 0.03225 m

Mass of the subsystem = 1.178 kg


Lateral force acting on the subsystem (LatF) = Lat G*Mass

= 231.12 N

Longitudinal force acting on the subsystem (LongF) = LongG*Mass

= 348.68 N

Moment = Lat F*C.G

= 7.45 Nm




LongF Longnuf

= 86.67 N

Load per insert in Lateral load (LatL) =NPC

Momentdia *

= 9.09 N


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= 95.75 N


LatF

= 57.78 N




1

PL

PL

co

so

ci

si

- 1


The margins of safety calculations of subsystem loads of bus

Electronics, Power Electronics and Power Distribution which have mass

values greater than 1 kg on M4 inserts show positive values and it is evident

that the inserts used can withstand the loads of the subsystem packages

accommodated in the micro-satellite structure. Similar calculations are made

for all the subsystems which are comparatively small and have less mass than

the subsystems for which the calculations are shown, the margins of safety

values obtained are still higher.

6.9 BUCKLING ANALYSIS

In linear static analysis, a structure is normally considered to be in a

state of stable equilibrium. As the applied load is removed, the structure is

assumed to return to its original position. However, under certain

combinations of loadings, the structure may become unstable. When this

loading is reached, the structure continues to deflect without an increase in the

magnitude of the loading. In this case, the structure has actually buckled or

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has become unstable; hence, the term “instability” is often used

interchangeably with the term “buckling”.

The principal sources of external load tending to buckle the satellite

are the direct pressure of photons from the sun (solar pressure) and the

pressure exerted by impact with the molecules of the very thin atmosphere

through which the satellite must orbit (dynamic pressure).The loads for

buckling analysis are as same as that of the static analysis. The Table 6.6

shows the derived load cases for the buckling analysis.

Table 6.6 Load cases for buckling analysis

Load case number

(LC)Lateral X Lateral Y Longitudinal Z

LC 1 7.25g 0 -8.25g

LC 2 0 7.25g -8.25g

LC 3 7.25g 0 3.75g

LC 4 0 7.25g 3.75g

LC 5 5.75g 5.75g -8.25g

LC 6 5.75g 5.75g 3.75g

6.9.1 Buckling load factor (BLF)

The Buckling Load Factor (BLF) is an indicator of the factor of

safety against buckling or the ratio of the buckling loads to the currently

applied loads. The BLF values are directly calculated from MSC

PATRAN/MSC NASTRAN. A knock down factor of 0.6 is used for

incorporating the material non-homogeneity, material imperfections, etc.The

Table 6.7 shows the buckling status for the BLF values calculated.

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Table 6.7 Interpretation of the Buckling Load Factor (BLF)

BLF value Buckling status Remarks

0<BLF<1 Buckling predicted The applied loads exceed the estimatedcritical loads. Buckling will occur.

BLF=1 Buckling predicted The applied loads are exactly equal tothe critical loads. Buckling is expected.

-1<BLF<1 Buckling possible Buckling is predicted if the loaddirections are reversed.

BLF=-1 Buckling possible Buckling is predicted if the loaddirections are reversed.

BLF>1 Buckling notpredicted

The applied loads are less than theestimated critical loads.

BLF<-1 Buckling notpredicted

The applied loads are less than theestimated critical loads even if theloading directions are reversed.

6.9.2 Verification for Global stability of micro-satellite

The buckling analysis is carried out for all the derived load cases

and the Table 6.8 shows the minimum buckling load factors obtained for all

the load cases. The Figure 6.14 shows the buckling plot obtained for mode 11

of LC 1 and Figure 6.15 shows the buckling plot obtained for mode 9 of LC 2.

Table 6.8 Buckling Load Factor values (BLF)

Load case number Minimum Buckling Load Factor

LC 1 -1.86

LC 2 5.16

LC 3 1.90

LC 4 -5.22

LC 5 -2.36

LC 6 2.41

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The minimum buckling load factor values obtained for the micro-

satellite for all the derived load cases are BLF>1 and BLF<-1.From Table 6.8

it is seen that the structure would not undergo any buckling instability as BLF

values are either less than -1 or greater than +1.

Figure 6.14 Buckling plot obtained for mode 11 of LC 1

Figure 6.15 Buckling plot obtained for mode 9 of LC 2

chapter 6 structural stress and buckling analyses...

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