1 _L 

NASA Technical Memorandum 101636 

Stability and Dynamic Analysis of a Slender 

Column with Curved Longitudinal Stiffeners 

Mark S. Lake 

AUGUST 1989 

&IALTSXS (BlSA-Tfi-1Q1636) OF A SLEIDEB STABILITP COLUUII AYD UITB DPlAflIC CgBVED n89-27275 

LOECITUDIBU, STIPFEHBBS ~UAS~. #s Langley 

Research Center) 67 p CSCL 20s Unclas 63/39 0224569 

National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton, Virginia 23665 
TABLE OF CONTENTS 

Page . 

TABLE OF CONTENTS ............................. i 

LIST OF TABLES ................................. iii 

LIST OF FIGURES ............................... iv 

LIST OF SYMBOLS ................................. vi 

CHAPTER 

1. INTRODUCTION ............................ 1 

1.1 ReviewofResearch ....................... 2 

1.2 Definition of Problem and Objectives ............ 4 

2. ANALYSIS ................................ 

2.1 Cross-sectional Geometric Properties ........... 

2.1.1 Effect of Stiffener Kinematic Attachment on 

Geometric Properties ................. 

2.1.2 Calculation of Geometric Properties ........ 

2.1.3 Optimization Considerations for a Stiffened 

Column .......................... 

2.2 Analysis of Stiffener Local Buckling ............ 

2.2.1 Functional Dependence of Stiffener Local 

2.2.2 Derivation of Approximate Explicit Relationship 

Buckling on Geometric Design Parameters ... 

for Stiffener Local Buckling. ............ 

2.3 Column Design Algorithm ................... 

10 

10 

11 

14 

17 

19 

20 

24 

29 

3. EXAMPLES AND DISCUSSION ................. 33 

i 
3.1 Generation of Design Curves for Example Problems. . 33 

3.2 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . 39 

4. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 

REFERENCES . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 47 

APPENDIX A 

Derivation of Stiffener Cross-Sectional Inertias . . . . . . . . . . . . . 49 

APPENDIX B 

- DEsign of Diffened alumn Code (DESTCO) . . . . . . . . . . . . . 51 

APPENDIX C 

Vibration Frequencies of the Stiffened Column . , . . . . . . . . . . . 57 
LIST OF TABLES 

Table 

Number 7 Title Page 

Number 

2.1 Mamx of Normalized Stiffener Local Buckling Stresses From 

VIPASA Analysis ........................... 26 

3.1 Nominal Dimensions for Example Central Columns ........... 34 

3.2 Design Data for Space Station Freedom Column. ............ 34 

3.3 Design Data for Precision Segmented Reflector Column. ........ 34 

c 

... 

u1 
Figure 

Number 

1.1 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6 

3.1 

3.2 

3.3 

3.4 

3.5 

3.6 

3.7 

A. 1 

B. 1 

c. 1 

LIST OF FIGURES 

Page 

- Title Number 

Section of a cylindrical column with curved deployable stiffeners . . . . . 2 

Symmetry of stiffened column cross section. . . . . . . . . . . . . . . 14 

Stiffened column cross-sectional geometric parameters. . . . . . . . . . 15 

Stiffener local buckling mode due to uniform axial compression. . . . . . 20 

Typical plot of curved stiffener as represented in VIPASA . . . . . . . . 25 

Plot of normalized stiffener local buckling stresses from VIPASA 

analysis.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 

Comparison of normalized stresses from VIPASA and explicit 

expression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 

Stiffener thickness variation with arc angle for minimum mass designs . . 35 

Minimum mass design curve for Space Station Freedom column . . . . . 36 

Minimum mass design curve for Precision Segmented Reflector column. . 36 

Verification of Space Station Freedom column design curve . . . . . . . 38 

Design curve for Space Station Freedom column including 10% 

mechanism mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 

Comparison of approximate and exact design curves for Space Station 

Freedom column including 10% mechanism mass . . . . . . . . . . . . 41 

Family of approximate general stiffened column design curves . . . . . . 42 

Cross-sectional parameters for stiffener moments of inertia. . . . . . . . 50 

Flow chart of DESTCO program . . . . . . . . . . . . . . . . . . . . 51 

General Stiffened column design curves with vibration frequency increase . 59 

iv 
LIST OF SYMBOLS 

c1 

N 

cross-sectional area 

coefficient expressions defined for use in equations for moment of 

inertia 

constants used in bilinear equation for stiffener local buckling stress 

distance between discrete stiffener attachment locations 

width of flat plate 

constant in column vibration frequency equation 

percent increase in stiffened column mass from that of the central 

column 

percent increase in stiffened column buckling load from that of the 

central column 

Young's modulus 

summation index 

general cross-sectional moments of inertia 

equivalent cross-sectional moments of inertia of stiffened column 

cross-sectional moments of inertia of central column and stiffener 

column length 

stiffener deployment mechanism mass as a percentage of the mass of 

the central column 

mass density 

number of data points used in least-squares approximation of stress 

function 

V Poisson's ratio 

V 
r 

0, 

t 

8 

a, P 

0 

critical buckling load of column 

arc angle of stiffener 

deployment angle of stiffener 

deployment angle of stiffener which maximizes cross-sectional 

moment of inertia 

radius of curved shell 

radius of gyration of cross section 

inner and outer radii of central column, and outer radius of stiffener 

average radius of stiffener 

local buckling stress of stiffener 

Euler buckling stress 

thickness of curved shell 

thickness of central column 

thickness of stiffener 

orientation angle of moment of inertia reference axis 

surrogate angles defined for use in equations for moment of inertia 

vibration frequency of stiffened column 

vibration frequency of colunin without stiffeners 

vi 
CHAPTER 1 

INTRODUCTION 

A significant percentage of future manned and unmanned space flight missions will 

involve very large spacecraft whose components will be of such size and/or number as to 

require on-orbit assembly and integration. Although the goals of these missions may be 

quite different, the design of these spacecraft gives rise to at least one significant common 

technological requirement. This requirement is to develop the ability to construct and 

operate large, lightweight structures in space. 

Most of these structures share the characteristic that their design is primarily driven by 

stiffness as opposed to strength considerations. This along with the constraint of low 

packaged volume allowances tends to drive the use of very long slender truss members. 

However, many of these structures are also required to withstand significant loads due to 

thermal gradients, spacecraft operations, and attitude control maneuvers. Consequently, 

stability of these slender members becomes a design concern. As a compromise, concepts 

have been developed for truss members which exhibit high buckling loads and, yet, 

package very efficiently. 

One of these concepts consists of a cylindrical central column having three deployable 

longitudinal stiffeners each of which are sections of a right circular cylinder with an inner 

diameter equal to the outer diameter of the central column as shown in Fig. 1.1. While 

packaged, the stiffeners lie flush around the outside of the central column, and when 

deployed, the stiffeners are rotated outward to an angle which maximizes the moment of 

inertia of the cross section. This thesis is the summary of a study of the deployable curved 

stiffener concept. Results presented herein demonsrate the stability characteristics of 

columns employing these stiffeners, and outline a procedure for designing these columns 

for application to large space structures. 

1 
* Isometric view shows only a 

rCentral column 

Stiffener 

Deployment angle 

Deployed 

Figure 1.1. Section of a cylindrical column with curved deployable stiffeners. 

1.1 Review of Research 

Technology development programs have been in existence for more than ten years to 

address the problem of constructing and operating large, lightweight structures in space. 

From these programs, concepts have been developed for large truss systems to support a 

variety of large aperture antennas and platforms (1,2) and a permanently manned Space 

Station (3). Typically, these truss systems share the design goals of high stiffness, low 

weight, and low packaged volume (4, 5). Constrained optimization studies have been 

conducted to determine geometric proportions of minimum mass structures using constant 

diameter cylindrical columns (6,7, 8). Results from these studies support the use of truss 

members having the maximum diameters allowed for packaging, and lengths which are set 

by local buckling or vibration requirements. 

The issue of improving member buckling loads without reducing member lengths has 

given rise to a number of design concepts for columns which retain efficient packaged 

volumes. References 8 and 9 describe a nestable tapered column concept which allows 

effectively large diameter columns to be assembled from very compactly packaged tapered 

2 

ORIGINAL PAGE 

BLACK AND WHITE PilOTOGFIAPH 
, 

column halves. Results of buckling tests and analyses of nestable tapered columns are 

presented in reference 10 and verify that greatly improved buckling loads can be achieved 

for columns of a given length without sacrificing the component packaging efficiency. 

Other studies have addressed the use of built-up columns for applications requiring 

very long spans (4, 11). These columns are assembled from efficiently packaged smaller 

components using a wide variety of assembly geomemes, and they exhibit large buckling 

loads in proportion to their mass and packaged volume. 

The disadvantage to the use of built-up or nestable tapered columns is the requirement 

for on-orbit component assembly. The increase in truss construction time required because 

of component assembly will probably be acceptable for missions involving manual 

assembly of smaller truss structures or automated assembly of larger structures. However, 

there is certainly a class of missions whose assembly requirements will preclude the use of 

structural components which must be assembled on-orbit. These missions include large 

manually assembled structures and any deployable structures. For these missions it might 

be practical to investigate the use of slender, constant diameter columns with deployable 

stiffeners. 

This column concept was introduced in reference 12, and has the characteristics of 

achieving a substantial increase in column buckling load for a moderate increase in mass 

and no change in the packaged dimensions (outer diameter and length) of the column. The 

utility of the concept is very broad in that it may be substituted with essentially any column 

in any large space truss structure without affecting the packaging of the structure. The 

disadvantages are the added complexity and cost of implementing deployable stiffener 

mechanisms. Furthermore, the geometry of the structure indicates that it will exhibit 

stability modes in addition to simple Euler column buckling that must be accounted for 

during design. 

The study presented in reference 12 considered only column geomemes which 

behaved as simple Euler columns. The objectives of that study were to evaluate stiffener 

attachment schemes and determine the finite element modeling complexity necessary to 

predict Euler buckling behavior in the column. The results showed that certain stiffener 

3 
attachment schemes greatly improved column buckling load whereas others showed less of 

an improvement. Also, it was determined that modeling this type of structure with flat plate 

finite elements requires the use of a very fine mesh to ensure solution accuracy. 

The present study focuses on the analysis of stiffener local buckling and derivation of 

an explicit empirical expression to determine this behavior. This expression is incorporated 

into a column design routine which determines minimum mass geometric proportions of the 

stiffeners such that Euler buckling remains the fundamental stability mode of the column. 

Classical work by Donne11 (1 3) and Timoshenko (14) resulted in explicit expressions for 

local buckling of long slender curved panels for a variety of loadings and boundary 

conditions. However, it will be demonstrated that none of these solutions are applicable to 

the present problem. 

1.2 Definition of Problem and Objectives 

The stiffened column concept requires hinge mechanisms to attach the stiffeners to the 

central column. These mechanisms are perceived to be more complex than basic hinges 

because they must allow free rotation of the stiffener during deployment and yet firmly lock 

the stiffener in its deployed position to provide maximum resistance to stiffener local 

buckling. Furthermore, the efficient analysis of large space structures constructed from 

many of these columns requires the normal operating axial stiffness of the columns to be 

linear. This requirement may be satisfied in one of two ways. First, it may be possible to 

design a stiffener hinge mechanism which not only freezes rotational motion upon 

deployment, but also freezes axial motion, and thus provides an ideal (zero free play) shear 

load transfer between the central column and the stiffener. Or, second, it may be desirable 

to isolate the stiffeners from the central column by mounting the hinges to cylindrical 

collars which slip over the outside of the central column. This approach allows a transfer 

of only lateral load and no shear load between the central column and the stiffeners and thus 

removes the stiffeners from the normal operating load path which is primarily axial tension 

or compression of the central column. 

The study presented in reference 12 identified three stiffener kinematic attachment 

schemes which represent the behavior of possible stiffener hinge designs. The first 

4 
kinematic attachment scheme allows the stiffeners to move independently of the central 

column and each other in the axial direction while all move together in the lateral direction. 

This scheme represents a design which uses hinges with no shear load transfer capability 

but rigid restraint against stiffener rotation. The second kinematic attachment scheme 

allows shear load transfer between individual stiffeners but not between the stiffeners and 

the central column while all components move together in the lateral direction. This scheme 

represents a design which uses hinges that become rigid in both rotation and shear upon 

deployment, but the stiffeners and hinges are attached to isolation collars which slide along 

the central column. The third kinematic attachment scheme allows direct shear and 

rotational load transfer between the stiffeners and the central column. This scheme 

represents a design which also uses hinges that become rigid in both rotation and shear 

upon deployment, but in this case the hinges are attached directly to the central column. 

All three kinematic attachment schemes result in column designs which exhibit linear 

axial stiffness within the normal operating tension and compression load range. For the 

first two attachment schemes this axial stiffness is simply the axial stiffness of the central 

column, and for the third it is the stiffness of the central column plus that of the stiffeners. 

The difference in these kinematic attachments is the way in which they allow the stiffeners 

to influence the stability of the central column. It will be shown that stiffened columns may 

exhibit either linear or nonlinear stability behavior. The nature of the fundamental or lowest 

stability mode will depend upon the type of kinematic attachment used for the stiffener and 

the geometric proportions of the stiffener and central column. 

It is apparent that any of these kinematic attachments would result in columns having 

one stability mode which could be predicted using simple linear Euler theory for a simply 

supported column (Eq. 1.1). This should be obvious for a column using the third 

kinematic attachment scheme because it would be a simple prismatic column directly 

obeying the assumptions of Euler theory, The first and second kinematic attachment 

schemes can also be understood to give rise to simple linear Euler behavior if one realizes 

the stiffener contributions in these cases can be viewed as distributed lateral elastic 

restraints. Results were presented in reference 12 which verified Euler column behavior 

5 
for test specimens constructed with each of the three kinematic attachment schemes 

described. It should be noted that the effective cross-sectional moment of inertia, I, in Eq. 

1.1 depends on the stiffener kinematic attachment scheme employed. 

In addition to the simple linear Euler stability modes described, a column designed 

with rigidly fixed stiffeners (third kinematic attachment scheme) would also have linear 

stiffener local buckling modes. These modes are similar to those predicted for stiffened 

panels under uniform axial compression (13, and they would be characterized by lateral 

flexing of the stiffeners with sinusoidal amplitude variation along the column length and 

essentially no lateral motion of the central column. A number of classical linear solution 

procedures exist for calculating the local buckling stress of stiffeners with simple crosssections. Furthermore, many linear numerical procedures exist for calculating the local 

buckling stress of stiffeners with complex cross-sections. Therefore, it is a relatively 

straightforward task to analyze the critical loads of stiffened columns exhibiting linear 

stability behavior. 

Brush and Almroth (16) have outlined general situations in which nonlinear analysis 

is needed to capture stability behavior. One of these situations is represented by stiffened 

columns employing either the first or second stiffener kinematic attachment scheme. These 

kinematic attachments allow only lateral load to be transferred between the stiffeners and 

the central column. Since the central column is nominally straight and loaded through 

uniform axial compression, the only way for the stiffeners to become loaded is through a 

lateral motion of the central column. This lateral motion can be determined with linear 

analysis and is equivalent to Euler column buckling. However, since this lateral motion 

only occurs as a result of passing a linear bifurcation point, predicting local stiffener 

instability beyond this point using common numerical routines such as finite element 

analysis may require nonlinear solution procedures. 

The present study will only address linear stability analyses of the stiffened column. 

These analyses will include Euler column buckling and linear local buckling of a uniformly 

compressed stiffener. Therefore, the results are most applicable to columns designed with 

hinge mechanisms which become rigid upon deployment (the third stiffener kinematic 

attachment scheme). 

6 
The primary stability design goal of this type of stiffened column is simply to ensure 

that its fundamental buckling mode is an Euler column mode, and thus, the value of the 

corresponding buckling load can be predicted using Eq. 1.1. Providing this guarantee 

requires the ability to analyze the possible stiffener local buckling modes and relate the 

corresponding critical loads to the design parameters of the column. 

As stated previously, the problem of stiffener local buckling is represented by a 

slender curved panel under uniform axial compression. Donne11 (1 3) and Timoshenko (14) 

have generated solutions for the stability of curved panels under uniform compression for a 

variety of boundary conditions, and in some cases have been successful in deriving exact 

explicit expressions for the critical stress. The boundary conditions applicable to this 

problem are a clamped connection along one of the stiffener's generators, free along the 

other, and pinned at both ends. The exact solution for this combination of boundary 

conditions and applied loading has not been generated to date, however, because of the 

character of the governing differential equations it can be shown that this solution would 

have the fonn of a transcendental eighth-order system of equations. A solution of this form 

would offer little insight to the relationship between the critical stress and the geometric 

parameters of the column. 

Since no exact explicit expression exists to characterize this behavior, a reasonable 

alternative is to generate an approximate explicit expression based on empirical results. 

These results may be generated in a number of ways. First, it is certainly possible to 

construct a family of test specimens which represent the expected range of design 

parameters, and base the empirical expression on experimental data accumulated from these 

specimens. Second, it is possible to derive the transcendental eighth-order system of 

equations representing the exact solution to the governing differential equations, and 

generate a family of analytical data from iterative numerical solution of this system. Third, 

it is possible to employ existing numerical structural analysis techniques (e.g. finite element 

method or the finite smp method) to generate numerical results. 

The obvious disadvantage in obtaining results through experimental study is the cost 

and time necessary to construct and test an adequate series of specimens. Therefore, 

numerical analysis techniques may bC better suited for this task. Such techniques have 

been demonstrated to be very accurate for linear stability calculations. Probably the most 

7 
widely used numerical structural analysis technique is the finite element method. Results 

presented in reference 12 indicate that applying this technique to the modeling of a slender 

stiffened column would require a very fine mesh and result in excessive solution times to 

ensure solution accuracy. This is very undesirable for an analytical study requiring the 

generation of a set of parametric solutions. 

Alternatively, references 17 and 18 describe an efficient and accurate solution 

technique using a linked-plate representation of the structure. This technique is used to 

model prismatic branched plate or shell structures acted upon by longitudinally invariant 

loads as assemblies of thin flat plate smps rigidly connected along their longitudinal edges. 

The two-dimensional plate equations describing each plate strip are reduced to onedimensional equations through the assumption of sinusoidal longitudinal deformations, and 

a generalized stiffness mamx representing the structure is assembled from equilibrium and 

compatibility considerations. Reference 18 describes the structural analysis code VIPASA 

which generates this linked-plate representation of the structure and uses it in the solution 

of linear stability and vibration problems. 

There are two fundamental advantages to using the VIPASA routine for analysis of 

prismatic branched plate and shell structures with longitudinally invariant applied loadings. 

First, the solution accuracy is guaranteed for structures which obey the kinematic 

constraints imposed. This is because it is a numerical solution to the linear differential 

equations governing the structure. Second, by reducing the 2-D plate theory to a 1-D 

theory, the size of the problem is greatly reduced and, consequently, the solution times are 

very small compared to other routines such as those based on finite element analysis. 

Because of these considerations, all stability calculations for the present study are 

performed using the VIPASA analysis routine imbedded in the design optimization code, 

PASCO (19, 20). 

The present study focuses on the derivation of an approximate empirical expression 

for buckling of a curved stiffener, clamped along one edge and free along the other. The 

solution is sought for bounded ranges of the stiffener geometric parameters. The bounds 

for these ranges are derived from considerations set forth by typical large space structure 

design studies. The empirical expression for stiffener local buckling is used in formulating 

design rules for stiffened columns which exhibit Euler buckling as a fundamental stability 

8 
mode. Additionally, studies are conducted on the cross-sectional moment of inertia of the 

stiffened column to develop rules governing mass minimization of the design. Finally, 

these results are assembled in a computer design program which is used to generate 

stiffened column designs for two current large space structure applications. 

9 
CHAPTER 2 

ANALYSIS 

This chapter is a summary of analyses performed to derive rules governing the 

minimum mass design of a slender cylindrical column stiffened with curved deployable 

stiffeners. These rules are determined from two sources: 1) consideration of equivalent 

cross-sectional properties, and 2) consideration of stiffener stability. The first section of 

the chapter deals with calculation of equivalent cross-sectional properties. These properties 

determine column mass, packaged size, and equivalent flexural stiffness, and a knowledge 

of their functional dependence on the cross-sectional geometric parameters leads to 

minimum mass design rules. The second section of the chapter outlines analyses 

performed to determine an approximate explicit expression governing stiffener local 

buckling. This expression is used to define a design condition that guarantees Euler 

buckling as the fundamental stability mode of the column. The third section of the chapter 

outlines the development and use of a stiffened column design procedure. Finally, a 

computer program which automates the design procedure is discussed as well as the steps 

for employing it in a design exercise. 

2.1 Cross-sectional Geometric Properties 

As previously stated, the ultimate goal in the design of a stiffened column is insuring 

the fundamental stability mode of that column to be Euler buckling. Given that 

consequence, the column can be characterized by equivalent properties such as the crosssectional area and moment of inertia. Calculating these properties involves evaluating 

double integrals over the column cross section. Derivations for these equivalent properties 

will be presented later. First, the general nature of the cross section will be examined in 

detail in order to simplify the equivalent representation of the column. 

10 
2.1.1 Effect of Stiffener Kinematic Attachment on 

Geometric Properties 

As stated in the introduction, the equivalent cross-sectional properties of the stiffened 

column depend not only on the cross-sectional shape but also on the stiffener kinematic 

attachment scheme. Stiffened columns employing any of the three attachment schemes 

identified in the introduction can be represented by one of two distinct equivalent members. 

One of these equivalent members represents the first stiffener attachment scheme (no shear 

load transfer between stiffeners or the stiffeners and the central column). The other 

equivalent member represents either the second or the third stiffener attachment scheme 

(ideal shear load transfer between stiffeners, and either ideal or no shear load transfer 

between the stiffeners and the central column). 

A stiffened column using either the second or third stiffener attachment scheme is 

represented by a single member having the equivalent moment of inertia of the entire cross 

section about axes centered at its centroid. This is seen clearly for the third attachment 

scheme in which the stiffeners are rigidly attached and represent simple extensions of the 

central column. This also applies to the second attachment scheme in which the stiffener 

assembly behaves as one member and the central column behaves as another so that the 

equivalent moment of inertia is the sum of the inertias of the two components with respect 

to their respective centroids. However, due to symmetry, the two components have the 

same centroid location. Thus, summing their moments of inertia with respect to their 

respective centroids is equivalent to calculating the moment of inertia of the entire section. 

A stiffened column using the first stiffener attachment scheme can be thought of as an 

assembly of members which assume the same lateral deflection without any shear load 

transfer between them. The equivalent cross-sectional moment of inertia of this assembly 

is simply the sum of the inertias of the individual elements with respect to their respective 

centroids, and consequently, the flexural stiffness of the column is the Young's modulus 

times this equivalent moment of inertia. Before concentrating on stiffened columns 

employing the third stiffener attachment scheme, a short discussion is presented to explain 

the consequences of employing the first stiffener attachment scheme. 

The flexural stiffness of a column employing the first kinematic stiffener attachment 

11 
scheme has some interesting characteristics which can be determined by employing the 

procedure just established for calculating its equivalent cross-sectional moment of inertia. 

The total moment of inertia is simply the sum of the inertias of the individual components 

about their respective centroids. Expressions for the moments of inertia of two of the 

stiffeners can be derived from the the moments of inertia of the remaining stiffener by 

applying axis rotation transformations known to be as follows (21): 

2 2 

2 2 

I,' = I, cos 8 + I, sin 8- IXysin28 

I,' = I, cos 8 + I, sin 8+ I,, sin2 8 

I,, = z( I, - 1, ) sin28+ I,, cos2 e I] 

where I,, I,, and I,, are the cross-sectional moments of inertia in the original coordinate 

system, 8 is the angle of rotation of the coordinate system, and I,', ZyB, and Zxy' are the 

cross-sectional moments of inertia in the rotated coordinate system. 

It should be noted that, relative to a given stiffener, one of the others is rotated by 

2fl3 and the remaining one is rotated by -2fl3. Therefore, applying the transformation and 

summing the resulting inertias gives: 

where I,,, I,,, and Ixyr, are the equivalent cross-sectional moments of inertia, ICC is the 

moment of inertia of the central column relative to its centroid, and I, and I, are the 

principal moments of inertia of one of the stiffeners relative to its centroid. 

12 
Two very interesting results are shown in Eq. 2.2. First the equivalent product of 

inertia, Ixye, of the cross section is equal to zero despite the fact that the individual 

stiffeners may have a non-zero product of inertia. Second, the equivalent moments of 

inertia about the x and y axes are identical. These two results taken together dictate that the 

equivalent moment of inertia of the cross section has only one value, independent of the 

orientation angle of the reference axis. This is certainly significant and desirable from the 

standpoint that it precludes the column from having a preferred direction of Euler buckling. 

Another significant characteristic is that the same expression will result (Eq. 2.2) 

regardless of the value of stiffener deployment angle (see Fig. 1.1). This can be 

understood by realizing two points. First, the constraint that the stiffeners differ in 

orientation by 2d3 exists independently of deployment angle. Second, the reference axes 

can always be oriented parallel to the principal axes of one of the stiffeners so I,, and Isy 

will have the same value for any deployment angle. Therefore, not only is the equivalent 

cross-sectional moment of inertia for a given deployment angle independent of the 

orientation angle of the reference axis, but the value of the moment of inertia is also 

independent of deployment angle. Thus, the cross section has the same equivalent flexural 

stiffness regardless of whether or not the stiffeners are deployed. 

These results contribute valuable information for the design of a stiffened column. It 

is obvious that designing a column with hinges having no shear load transfer capability 

would be pointless. The same improvement in equivalent cross-sectional moment of inertia 

could be achieved with less added mass by simply increasing the thickness of the central 

column. Thus, only the columns with stiffeners attached through hinges having effectively 

rigid shear load transfer capability are further analyzed. These correspond to the second 

and third kinematic att;ichmcnt schemes. 

The independence of cross-sectional moment of inertia with reference axis orientation 

can also be demonstrated for columns employing stiffener hinges with rigid shear load 

transfer capability. A rigorous mathematical proof of this point would involve rather 

lengthy expressions and, therefore, an equally conclusive proof by deduction is performed. 

The basis of this proof is the fact that the moment of inertia of a cross section made up of 

three infinitesimal area elements located at the vertices of an equilateral triangle is 

independent of the orientation of the reference axis. 

13 
Each stiffener in the stiffened column cross section can be broken up into a large 

number of differential area elements with each element having corresponding elements on 

both of the other stiffeners as indicated in Fig. 2.1. Due to symmetry, at any arbitrary 

deployment angle, these three differential elements are all located the same radial distance 

from the centroid of the cross section and the same distance from each other. Therefore, 

the three differential area elements are located at the vertices of an equilateral mangle whose 

centroid is coincident with the centroid of the cross section. Consequently, the moment of 

inertia of these three area elements is independent of the reference axis orientation for any 

arbitrary deployment angle. Furthermore, the moment of inertia of the stiffener assembly is 

simply the sum of the inertias of a set of differential triangles subdividing the stiffener 

assembly cross section, and therefore this moment of inertia is also independent of the 

reference axis orientation. Finally, the total moment of inertia of the cross section is the 

sum of the inertia of the stiffener assembly and that of the central column, and thus, this 

quantity is also independent of reference axis orientation angle. 

Yt 

Equilateral [r \ yyJJce;erEnts 

triangle 1 / 

Centroid 

,J/ --- 

Figure 2.1. Symmetry of stiffened column cross section. 

2.1.2 Calculation of Geometric Properties 

The previous result establishes that it is only necessary to calculate one equivalent 

cross-sectional moment of inertia to completely characterize the flexural stiffness of the 

14 
stiffened column. The following derivation of this expression is performed assuming 

arbitrary values for all cross-sectional geometric parameters. Referring to Fig. 2.2, these 

parameters include central column inner radius (ri) and outer radius (ro), stiffener outer 

radius (rp), arc angle (N, and deployment angle (@). 

Figure 2.2. Stiffened column cross-sectional geometric parameters. 

It should be apparent that Eq. 2.2, which were derived for a column using stiffener 

attachment hinges with no shear load transfer capability, are also applicable to the 

derivation of the equivalent cross-sectional moment of inertia for columns using hinges 

with a rigid shear connection. The only modification necessary is that for the later case, I, 

and I,, denote the x and y moments of inertia of the stiffener relative to the centroidal 

coordinate axes of the entire cross section. Given these definitions, it is possible to derive 

the following expressions for these component moments of inertia. The derivation of these 

expressions is given in Appendix A. 

(2.3) 

I, = a1 sin2# + a2 sin#( sina - sin (a - y,) 

+ a3 ( ~2 + (sida - sin2(a - v)) /4 ) 

15 
where: 

Additionally, the moment of inertia of the central column is derived in reference 2 1, 

and the expression follows. 

Substituting Eq. 2.3 and 2.5 into Eq. 2.2 gives the expression for the cross-sectional 

moment of inertia of a stiffened column. After simplification using trigonometric identities, 

this expression becomes: 

x44 

4 I =-( ro- ri ) +3al (1 -cos@) 

A similar procedure may be followed to obtain the expression for cross-sectional area 

of the stiffened column. This derivation is very straightforward and the result is given in 

Eq. 2.7 without presentation of the details. 

A =2r2+(x-T)r:-xri 2p 2 

(2.7) 

16 
2.1.3 Optimization Considerations for a Stiffened Column 

As mentioned in the introduction, the primary design goal of the stiffened column is 

to produce columns whose fundamental stability mode is Euler buckling, and, 

consequently, whose buckling load will be proportional to the cross-sectional moment of 

inertia presented in Eq. 2.6. Therefore, mass optimization of a column having a certain 

length involves simply minimizing the cross-sectional area of the column for a given crosssectional moment of inertia. It is reasonable to assert that minimizing the cross-sectional 

area for a given moment of inertia is equivalent to the conjugate task of maximizing the 

moment of inertia for a given area. Performing this conjugate optimization will prove more 

useful for many applications; because a common set of design constraints is a maximum 

mass allowance and maximum packaged size allowance (outer radius of stiffeners, rp). 

The first issue to be addressed in obtaining the maximum moment of inertia for a 

given area is calculation of the optimum stiffener deployment angle. Eq. 2.7 demonstrates 

a straightforward observation that the cross-sectional area is not dependent on the stiffener 

deployment angle. Furthermore the trigonometric functions in Eq. 2.6 having @ as an 

argument can only go through at most one half cycle for the whole range of allowed values 

of @, (0s @ ,< n). Therefore, there exists a unique value for the deployment angle which 

will produce a maximum cross-sectional moment of inertia for given values of the other 

cross-sectional geometry parameters (ri, r,, rp, and w}. This value can be determined by 

differentiating Eq. 2.6 with respect to @, setting the resulting expression equal to zero, and 

solving for 41. Differentiating Eq. 2.6 with respect to qj gives: 

2 d!. = 3al sin@ + 6~2 sin(F) [I/. COS ($ COS (" @ 

Reducing this expression using trigonometric identities and the definitions in Eq. 2.4, and 

equating it to zero, gives the following: 

0 = a, sin# + a2 sin (TI - cos a -- I Y) 

17 
Since the surrogate angle a is defined in terms of @ (see Eq. 2.4), it is necessary to 

further employ trigonometric identities to solve the above expression for 4. After this is 

done the following expression is derived for the deployment angle,@, which maximizes the 

cross-sectional moment of inertia for given values of the stiffener radii and arc angle. 

where: 

Eq. 2.6, 2.7, and 2.8 represent three constraint conditions on the design of a 

stiffened column. The next section will address stiffener stability analyses which will lead 

to one more design condition. That will bring the total number of conditions to four, with 

the total number of unknown parameters equal to seven (Ti, ro, rp, @, w, I, and A). 

Obviously, most truss applications will also specify the column buckling load and, thus, 

through Eq. 1.1, specify the moment of inertia, I. With this additional constraint, the total 

number of conditions come to five for the seven specified parameters. 

Although the system is still indeterminate at this point, it would certainly be possible 

to locate a minimum column mass. However, performing this optimization will invariably 

produce a design in which rp + 00, causing the cross-sectional area and, consequently, the 

mass to become infinitesimal. The implication of this is simple; without imposing 

constraints on the magnitudes of the column radii, mass optimization produces no useful 

results. Furthermore, this argument can be extended to show that the only way of 

producing useful designs from a mass optimization routine is to make the routine 

determinate by suppling two independent constraints on the column radii. 

The example of mass minimization with no constraints on column radii was just 

18 
shown to result in designs in which rP 3 -, Similarly, applying only one constraint to the 

column radii leaves the problem indeterminate and will result in useless minimum mass 

designs. For example, packaging requirements or on-orbit handling requirements for most 

truss applications will impose maximum allowances on the outer radius, rP. This additional 

constraint eliminates the possibility of obtaining infinite radii from a mass minimization 

routine. However, performing the mass minimization will invariably produce a design in 

which ri = r,, or, in other words, one with all of the material in the stiffeners and no central 

column. This result can be easily deduced by realizing that the most efficient location for 

the material is in the stiffeners, because they are at a greater distance from the centroid than 

the central column. Likewise, it should be apparent that applying a maximum constraint to 

any of the radii will give the same mimimum mass design result, ri = I,. Therefore, in 

order to produce useful minimum mass designs it is necessary to supply two independent 

constraints on the column radii. Thus, the design procedure consists of seven independent 

constraint equations and seven independent unknown parameters. 

This statement should be alarming because it implies that it is possible to locate the 

minimum of a parameter in a fully determined system. However, mass minimization 

considerations form the basis of the derivation of the remaining design condition dealing 

with stiffener stability. Therefore, the resulting determinate system will, implicitly, 

describe the minimum mass design. 

2.2 Analysis of Stiffener Local Buckling 

The discussion in the introduction defined the need for a relationship between the 

geometric design parameters of the stiffened column and the critical stress for stiffener local 

buckling. The stiffener is represented as a long slender curved panel of constant radius, 

and the loading condition that produces stiffener local buckling is uniform axial 

compression (see Fig. 2.3). The applicable boundary conditions for this problem were 

determined to be a clamped connection along one of the stiffener's generators and free 

along the other. Furthermore, it was established that an explicit empirical expression 

would be more useful for impletnmtation in a design routine than the complex 

transcendental eight-order system of equations that would be derived from an exact solution 

of the governing differential equation$. This section presents a preliminary discussion of 

explicit expressions derived for related problems to give insight into the functional 

If 

19 
dependence of stiffener local buckling on the geometric design parameters. Finally, the 

section concludes with the derivation of an explicit empirical expression relating stiffener 

local buckling stress to the geometric design parameters. 

Clamped edge -\ 

--/ 

Pinned end Pinned end 

Free edge Local buckling mode 

Figure 2.3. Stiffener local buckling mode due to uniform axial compression. 

2.2.1 Functional Dependence of Stiffener Local Buckling on 

Geometric Design Parameters 

The six geometric design parameters being considered for the stiffened column (I, ri, 

ro, rp, 9, and yi) were defined in the previous section. However, the analysis of stiffener 

local buckling only depends on four of these (l, ro, rp, and @, because the central column 

geometry and orientation of the stiffener with respect to the central column have no effect 

on stiffener local buckling. This result leads to the assumption that the stiffener hinge line 

can be represented as a clamped connection for the stiffener stability analysis. An 

equivalent, and more convenient set of geometric parameters for use in studying stiffener 

local buckling are I, r,, r,, and yi. Where r, and r, are the mean radius and thickness of the 

stiffener and are defined as: 

, ts = rp - r, rp + ro 

2 r, = - (2.10) 

In reference 14, Timoshenko and Gere derived expressions for the buckling stress of 

flat and curved panels, pinned at both ends and acted on by uniform axial compression, for 

a variety of boundary conditions on both generator edges. These expressions provide 

explicit relationships between the critical stress and the panel geometric parameters (rs, ts, I, 

20 
and wfor a curved panel and b, t, and I for a flat panel). Although, reference 14 does not 

address the solution of curved panel stability for the boundary conditions of the present 

problem, the explicit relationships derived for similar problems provide insight as to the 

form of an explicit expression for the present problem. 

Results from reference 14 indicate that long, slender flat panels (6 e< I) restrained by 

clamped boundaries along one or both of the generator edges tend to buckle as infinitely 

long panels. In other words, the buckling mode will include a large number of sine waves 

along the length of the panel, and, consequently, the value of the buckling stress is 

independent of panel length, 1. Since all practical stiffened column designs will be very 

slender and thus, the stiffeners will be characterized by rsw <<I, it seems reasonable that 

the stiffener buckling stress will also be independent of column length. 

Also, results from reference 14 show that the buckling stress of curved panels tends 

to be a polynomial function of the non-dimensional quotient, tJrs, regardless of the panel 

size or boundary conditions. Therefore, it follows that the explicit expression sought 

herein for the local buckling of curved stiffeners will, most probably, only be a function of 

two independent geometry parameters, firs and ty. 

Another observation from the work presented in reference 14 is that there are a 

number of different exact expressions for critical stress of a curved panel derived for a 

given set of boundary conditions. These different expressions only apply within distinct 

ranges of the geometry parameters, tJrs and y. Therefore, it is very important to determine 

the ranges of these parameters applicable to a particular problem so that the proper 

expression may be applied. This observation also has significance in the derivation of an 

approximate empirical expression for stiffener local buckling. Since this expression is to 

be derived from a matrix of parametric data, it is important to fiist define the ranges for tJrs 

and w by determining bounding values of these parameters that would be expected in 

design applications. 

It is easy to establish an absolute range for w, the stiffener arc angle, as 0 5 w52M3 

(see Fig. 2.2). However, it should also be noted that there exists a practical lower limit to 

the arc angle, such that below this limit the structural improvement offered by the stiffeners 

21 
is offset by the cost, in complexity and mass, of incorporating the hinged attachments. The 

preliminary nature of this design study makes it difficult to assess these costs and set a 

definite value for this lower limit. However, a lower limit of d3 will be assumed for 

generating parametric data leading to an explicit expression for stiffener critical stress, thus, 

I,Y will be bounded as shown in Eq. 2.1 1, 

zqfs- 2a 

3 3 (2.11) 

It is more difficult to set limits on the range of the parameter tJrr It must be recalled 

here that the explicit expression for stiffener local buckling is sought as a means of 

establishing a stiffened column design condition which would result in a minimum mass 

design having the same values for both stiffener local buckling stress and Euler buckling 

stress. Therefore, the values of tJrs that would characterize this design boundary for all 

perceived stiffened column applications can only be determined by understanding this 

design condition. 

This lack of a priori knowledge about the possible range for tJrs necessitates the use 

of an iterative approach to determine this range. This iterative approach involves making an 

initial guess for the applicable range of firs, and generating parametric data throughout this 

range from which a design condition could be derived and, consequently, the range for tJrs 

could be refined. This approach can be applied on subsequent iterations until the bounding 

values of this range converge. Furthermore, it is possible to use the explicit expression for 

critical stress of a related problem, a slender flat panel clamped along one generator and free 

along the other, to determine an initial guess for the range of tJr,. The design condition 

derived from this range of empirical data can then be used, as stated, to verify or modify 

the range of tJrs. 

Timoshenko and Gere (14) derived the explicit relationship for the local buckling 

stress of a slender flat panel, clamped along one generator edge and free along the other, 

and this expression is given in Eq. 2.12. It should be apparent that the critical stress for a 

curved stiffener is greater than the critical stress of a flat stiffener having the same width 

and length because of the higher lateral bending stiffness of the curved stiffener. 

Preliminary numerical stability analyses performed on curved stiffeners indicates that, in 

22 
most cases, the critical stress of a curved stiffener is roughly two orders of magnitude 

greater than that of the flat stiffener. So, a reasonable approximation to the local buckling 

stress of a curved stiffener is given by Eq. 2.13, where the flat panel width, b, has been 

replaced by the equivalent curved panel width, rsv/. 

os = 1.328 E ($ (2.12) 

(2.13) 

The other equation needed to construct the aforementioned minimum mass design 

condition is the expression for the Euler buckling stress, o,, of the stiffened column. This 

expression can be determined from EQ. 1.1 by recognizing that the equivalent crosssectional moment of inertia is equal to the cross-sectional area times the radius of gyration, 

p, squared (Eq. 2.14). The resulting expression is given in Eq. 2.15. 

I = p2A - 

2 

oe=n E [;[ 

(2.14) 

(2.15) 

As mentioned previously, a practical stiffened column design would have Euler 

buckling as its fundamental stability mode. The minimum mass design condition is derived 

by equating the Euler buckling stress with the stiffener local buckling stress. Applying the 

expressions in Q. 2.13 and 2.15 gives the following approximate minimum mass design 

condition. 

1. 

It can be shown that the radius of &ration of the stiffened column is typically within 

twenty percent of the value of the dadfhs of the stiffener, so assuming that p s r, and 

simplifying the above expression gives the following approximate minimum mass design 

condition. 

23 
(2.16) 

It is now possible to define an approximate range for the non-dimensional parameter 

tJrs by specifying the minimum and maximum values for the parameters y, and rsjl. Eq. 

2.1 1 established a range for the stiffener arc angle, y, and an appropriate range for rJ1 may 

be specified by considering the dimensions of slender columns currently being considered 

for large space structure applications. A lower bound for rs/l is represented by the 

candidate truss components for the Space Station Freedom discussed in reference 22. The 

diagonal members of this truss have an rsjl z .0035. An upper bound for rJ1 is represented 

by the candidate truss components for the Precision Segmented Reflector testbed structure 

discussed in reference 23. The members of this truss have an average rsll s .016. 

Therefore, by applying the minimum and maximum bounds of rJ1 and y, respectively, to 

Eq. 2.15, an approximate range for fJrs is defined to be: 

.001 &_<.01 

rS 

(2.17) 

2.2.2 Derivation of Approximate Explicit Relationship for 

Stiffener Local Buckling 

As indicated, stiffener local buckling should only be a function of two independent 

parameters, tJrS and I,Y, and Eq. 2.11 and 2.17 define approximate ranges for these 

parameters that are applicable to the present problem. Therefore, it is now possible to 

conduct a parametric study to determine an approximate explicit relationship for stiffener 

local buckling based on local buckling analysis of a large set of stiffeners having 

dimensions falling within the bounds set in Eq. 2.1 1 and 2.17. 

In the introduction, it was explained that the present study will employ the VIPASA 

structural analysis routine to generate an efficient linked-plate representation of the structure 

for numerical solution of the stability problem. As explained, this method of structural 

representation assumes lengthwise sinusoidal deformations of the structure, and thus 

requires that only the cross section of the structure be discretized. Fig. 2.4 shows a typical 

curved stiffener as represented in VIPASA. The constant radius curvature of the stiffener 

24 
is approximated with eight discrete flat plate strips. The dashed line in Fig. 2.4 shows the 

lateral deformation corresponding to a typical stiffener local buckling mode. As stated, the 

magnitude of this deformation varies sinusoidally along the length of the stiffener with 

essentially the same wavelength as that for an infinitely long stiffener. 

Free 

edge 

Typical local buckling 

mode 

Discrete jlat plate strip 

Undeformed stiffener 

longitudinal edge longitudinal 0 

Figure 2.4. Typical plot of curved stiffener as represented in VIPASA. 

Preliminary parametric analyses were performed, using the VIPASA model, to verify 

that the local buckling stress of the curved stiffener is only a function of w and tJ/rs. 

Although this data is not presented, indeed, this assumption was found to be true 

throughout the parametric ranges defined in Eq. 2.11 and 2.17. Therefore, subsequent 

parametric analysis are performed using a value of one inch for the stiffener radius of 

curvature, rs, and thus variations in the parameter firs are achieved by varying only the 

thickness, ts. 

A matrix of values for stiffener local buckling stress can be generated by selecting a 

set of discrete values for the parameters tyand fJrs, and performing a VIPASA stability 

analysis on the resulting family of stiffeners. Table 2.1 presents the values of these 

stresses normalized to the Young's modulus of the stiffener for a family of thirty designs. 

By reviewing this data, it is apparent that the local buckling stress monotonically increases 

with increasing fJ/rs, and decreasing v. Furthermore, the corresponding rates of increase 

of the local buckling stress are approximately constant for both parameters. 

25 
Table 2.1. Matrix of Normalized Stiffener Local Buckling Stresses From 

VIPASA Analysis. 

0.001 

48.9 

32.0 

24.2 

16.7 

12.7 

w 0.002 

150. 

121. 

87.4 

65.4 

50.6 

111 13 

511: 112 

A /2 

7A 112 

2A /3 

Os 6 

-x 10 , fort$s= E 

0.004 

3 17. 

309. 

297. 

245. 

195. 

0.006 

500. 

477. 

463. 

449. 

411. 

0.008 

697. 

649. 

631. 

623. 

603. 

0.0 10 

909. 

837. 

803. 

797. 

774. 

This result indicates that a reasonable approximate expression for the normalized local 

buckling stress might be a bilinear equation in wand tslrs. The general form for this 

bilinear equation is given in Eq. 2.18, where the coefficients bl, b2, b,, and b4 are 

constants. 

lo00 

800 

600 

6, 

-- x 106 

E 

400 

200 

0 

................................................................................ // +' ............ c, 2' ............ //+ \ %+ ....... +\ .......... 3 

4 .L.' .. + 9 + Data point from r--- VIPASA analysis / p--- ................................................................... .................. I ............... *...>.-. * ................................. / / - -.A 

/ 

/ / / 

/ 

-N - - -*/- -* - - I. ~ 

.. ..""""""""............., ! ............... L ................ / ................ L .......... 1. ................. 

Figure 2.5. Plot of normalized stiffener local buckling stresses from VIPASA analysis. 

26 
The data from Table 2.1 is plotted in Fig. 2.5 in the form of a carpet plot, and best-fit 

interpolation lines corresponding to constant values of w and rJrs are dashed in to 

demonstrate the near linear behavior of the local buckling stress. An important observation 

from this plot is that the slopes of the lines for the constant values of ware very nearly the 

same. Also, the slopes of the lines for constant values of firs are very nearly the same with 

the exception of the line for firs = 0.001. Neglecting the case of rJrS = 0.001 for the 

moment, it follows that the local buckling stress should be adequately represented with the 

planar Eq. 2.19 which is derived from Eq. 2.18 with b4 = 0. 

(2.19) 

Considering now the case of tJrS = 0.001, it is noted that these data points 

correspond to very low local buckling stresses and would not be accurately predicted with a 

planar expression of the form in Eq. 2.19. Results of design studies to be presented in the 

following chapter will demonstrate that practical stiffened column designs will exhibit 

buckling stresses above these values. Therefore, a planar expression for local buckling 

stress will prove sufficiently accurate for the applications being considered. 

The method of least-squares (see reference 24) is selected for calculating the 

coefficients bl, b2, and b3 in Eq. 2.19 given the set of discrete data in Table 2.1. 

Application of this method results in a system of three linear algebraic equations in bl, b2, 

and b3. This system is given in Eq. 2.20 in which the summation index i denotes the iLh 

data point in Table 2.1, and N is the total number of data points. 

N EW 

i=l 

i=l i= I 

N 

i=l i=l 

1=l E (%Ii 

N 

i=l 

(2.20) 

27 
Because Eq. 2.20 is only a third-order system, the solutions for bl, bZ, and bj are 

most easily derived by Gauss elimination and back substitution. Implementing this 

procedure in a computer program produces the appropriate values of bl, b2, and b3. After 

substitution of these values into Eq. 2.19, the approximate, explicit expression for stiffener 

local buckling stress becomes: 

os = E (6.0 x - 8.74 x w+ 8.97 X Io-2 (&)) 

rS 

(2.21) 

800 

600 

200 

0 

-2o() L .................................................................................................................. ~ ...................................... 

Figure 2.6. Comparison of normalized stresses from VIPASA and explicit expression. 

Fig. 2.6 presents a comparison of the stresses predicted using Eq. 2.21 and those 

from the VIPASA analysis, summarized in Table 2.1. The difference between Fig. 2.6 and 

Fig. 2.5 is that the dashed lines in Fig. 2.6 are the best-fit planar representation of the 

stresses derived from Eq. 2.21. As anticipated, the agreement between the approximate 

stresses and the VIPASA results is quite good with the exception of the very low stress 

region of the graph where fs/rs < .002 and w 2 n/2. In this region Eq. 2.21 predicts 

ncgntive values for the local buckling stress which is obviously inadmissible. Neglecting 

the three data points for fJr, = .001 and w2 n/2, the rms error of the remaining values 

predicted using Eq. 2.21 is only 7.2%. Therefore, it is reasonable to apply Eq. 2.21 to 

predict the local buckling stress of a curved stiffener unless fJrs < .002 and w-> x/2. 

28 
Further studies of local buckling for stiffeners with firs > .OZ show that application 

of Eq. 2.21 to this region results in local buckling stress values which are consistently low 

when compared to values from VIPASA analysis. This implies that applying Eq. 2.21 to 

the design of stiffened columns with stiffeners having fs/rs > .OZ would result in a 

conservative design, and, thus, one which is guaranteed to exhibit Euler buckling as a 

fundamental stability mode. 

It was stated previously that a minimum mass design is one which exhibits the same 

critical stress for both Euler column buckling and stiffener local buckling. However, to 

avoid stability mode interaction and ensure Euler buckling for the fundamental stability 

mode, stiffened column designs will be sought for which the stiffener local buckling stress 

is greater than the Euler buckling stress. Thus, equating these stresses provides a limiting 

condition for stiffened column design. This design condition is given in Eq. 2.22 and is 

derived from the stress expressions in Eq. 2.15 and 2.21. 

a2 (;r S(6.O x ZO-5 - 8.74 x ZP5 w+ 8.97 x ZO-2 (k)) (2.22) 

It was pointed out in the first section of this chapter that, with the addition of this 

design condition, the procedure for generating a minimum mass stiffened column design is 

characterized by seven parameters; (Ti, To, rp, 9, y, I, and A), or, equivalently (ri, rs, c,, #, 

y, I, and A), and is governed by five constraint Eq.; 1.1, 2.6, 2.7, 2.8, and 2.22. It was 

also demonstrated that it is necessary to specify two additional independent conditions on 

the radii of the stiffened column in order to make the problem determinate, and solvable for 

a useful design. The following section will address the selection of these additional 

conditions for general design problems, and then will define a set of conditions and the 

resulting design routine to be used in exqple design problems to demonstrate the potential 

mass savings offered by the stiffened column concept. 

2.3 Column Design Algorithm 

A main goal of the present study is to demonstrate the potential mass savings offered 

by the stiffened column concept. This swings will be quantified by comparing design 

curves of mass versus buckling load for minimum mass stiffened column designs with the 

29 
corresponding design curves for the central column with varying thickness. These curves 

will allow the comparison of two designs which can be used interchangeably for a given 

application because they have the same outer radius, length, and buckling load. To 

generate these design curves a design algorithm must be established which solves the 

determinate system consisting of Eq. 1.1, 2.6, 2.7, 2.8, and 2.22 and two additional 

conditions on the column radii. 

In general, when developing a stiffened column design routine there are a variety of 

independent conditions on the column radii that can be chosen, and the most useful ones 

depend on the particular application. For example, it was noted previously that often a 

maximum value for rp will be given from packaging or on-orbit handling considerations. 

Also, manufacturing and handling requirements impose minimum values on wall thickness 

for both the central column (ro-ri) and the stiffeners (rp-ro or, equivalently, ts). Alternately, 

it may be desireable to implement curved deployable stiffeners on a fixed central column 

design. In which case, both ri and ro would be defined, and, thus provide the additional 

design conditions needed. 

In the present study, the later of these additional radius constraints are employed so 

that mass penalty as a function of increased buckling load can be studied for a fixed central 

column design. This approach is taken because it is well suited for demonstrating the 

potential mass savings offered by the stiffened column. A family of minimum mass 

stiffener designs will be generated for the given central column which have optimum values 

of t,, y, and @, and, thus, the minimum total column mass, for a range of values of the 

buckling load. The resulting mass versus buckling load curve will show the minimum 

percent increase of mass necessary to achieve a particular percent increase in buckling load. 

As mentioned, the mass savings of the stiffened column will be quantified by comparing 

this curve to the corresponding mass versus buckling load curve derived for a simple 

circular column having the same outer radius and buckling load. 

By selecting values for ri and ro, and defining the buckling load, Pcr, it should be 

obvious that the determinate system describing the minimum mass stiffened column 

reduces to a five equation (eqs. 1.1, 2.6, 2.7, 2.8, and 2.22 ) and five unknown (t,, @, y, 

I, and A) system. Furthermore, it can be seen that EQ. 1.1 and 2.7 and the variables, I and 

A uncouple, and leave a three by three system to be solved. However, reviewing the 

30 
remaining equations, it is seen that this system is highly coupled and nonlinear, thus 

difficult to solve. An alternate approach to solving this system is to select the value for vat 

the outset, and allow the buckling load, Pcr, to become a solution variable. This approach 

results in a relatively simple iterative solution procedure for the nonlinear system, which 

will be shown to converge quite quickly to the desired system solution. 

The first step in developing this solution procedure is to rearrange Eq. 2.22 by 

solving for tr The resulting expression is given in Eq. 2.23. 

+ 9.74 x IO4 v- 6.69 x (2.23) 

Eq. 2.23 appears to be an explicit expression for f,, however r, and p are both 

implicit functions of t, through the remaining system equations and Eq. 2.10 and 2.14 

(recall that w has been defined at the outset). Therefore, Eq. 2.23 is in the correct form for 

application of fixed-point iteration to determine the solution for f, (see reference 24). This 

iteration procedure is quite simple. An initial estimate for t, is made and substituted into the 

right hand side of Eq. 2.23, then the equation is solved to determine the new estimate for 

f,, and this procedure is repeated until t, converges. At each iteration step, the new values 

for r, and p are calculated from Eq. 2.6,2.7,2.8,2.10, and 2.14 which uncouple if solved 

in the order: 2.8, 2.6,2.7 then 2.10 and 2.14. 

Rather than making an initial estimate for f, and solving the system of equations for 

initial estimates of r, and p, it is easier to begin the iteration process by making initial 

estimates for r, and p. These initial values are assumed to be the outer radius of the central 

column, To. Although this initial estimate implies that t, = 0 (from Eq. 2.10) it will be 

shown that the fixed-point iteration procedure will, nonetheless, converge rapidly to the 

desired solution. 

By determining a converged solution for t,, converged solutions for r,, @, p, I, and A 

are also implicitly determined using this process. Finally, the value for buckling load, Pcr, 

can be determined from the value of I, and Eq. 1.1. Therefore, the result of this iteration 

procedure is the minimum mass design and the corresponding buckling load of a stiffened 

31 
column having a stiffener arc angle = y. 

A complete curve of minimum column mass versus buckling load can be generated by 

determining minimum mass design for the complete range of y. It will be shown that as w 

is increased, so will the minimum mass and buckling load of the column. Finally using 

Eq. 1.1, 2.5 and 2.7 with I,, = w = 0, it is possible to calculate a set of masses and 

buckling loads of the central column for a range of thicknesses. 

Appendix B presents a flow chart and a listing of the FORTRAN computer program 

DESTCO that was written to perform the fixed-point iterative solution of the above system 

of equations, and generate the mass versus buckling load design curve data for both the 

minimum mass stiffened column and the central column. The program was written and 

executed using a FORTRAN compiler on a desk-top personal computer. 

32 
CHAPTER 3 

EXAMPLES AND DISCUSSION 

This chapter is a summary of results from stiffened column design studies conducted 

using the design program DESTCO. These studies will serve to illustrate the procedures 

involved in generating minimum mass stiffened column designs, as well as verify that the 

resulting designs exhibit Euler buckling as their fundamental stability mode. The first 

section of the chapter presents two stiffened column design examples which are applicable 

to typical large space structure missions currently envisioned and analyses to verify the 

DESTCO results. The final section presents a discussion of the results, and additional 

practical considerations for implementation of stiffened column designs. 

3.1 Generation of Design Curves for Example Problems 

In the last chapter it was explained that the design routine embedded in the DESTCO 

program requires that the central column size be completely specified (ri, ro, and 2). 

Therefore, application of this routine results in the set of minimum mass curved stiffeners 

for the given central column and the corresponding values for percent increase in column 

buckling load and mass. 

Two example central column designs have been selected for illustrative purposes in 

generating minimum mass stiffener desip. The first is from the Space Station Freedom 

five-meter truss structure (see reference 22), and the second is from the Precision 

Segmented Reflector two-meter truss structure (see reference 23). The nominal dimensions 

assumed for these two central columns an given in Table 3.1. 

33 
Table 3.1. Nominal Dimensions for Example Central Columns. 

Example Column 

I I I I I 

ri (in) ro (in) 1 (in) 

Precision Segmented 

Reflector 

I SpaceSlationFreedom I 0.9675 I 1.0325 I 196.8 I 

0.4700 0.5300 78.7 

I 

w 1, (in) 

A I3 DO14 

5A 112 .0016 

A I2 .0017 

7n 112 .0019 

2~ 13 .002 1 

The dimensions in Table 3.1 are used as input to DESTCO to generate the set of 

minimum mass stiffener designs for the range of stiffener arc angles given in Eq. 2.1 1. 

Tables 3.2 and 3.3 present these results for the Space Station Freedom column and the 

Precision Segmented Reflector column, respectively. 

.0244 

.0249 

.0256 

.02& 

.0275 

Table 3.2. Design Data for Space Station Freedom Column. 

~~ 

.191 

.192 

.193 

.194 

.195 

w 

A /3 

5~ 112 

A 12 

7A I12 

2n I3 

.0019 

.0022 

.0025 

.0028 

.003 1 

.0018 

.0021 

.0024 

.0027 

.0030 

115. 

120. 

125. 

129. 

.212 

.217 

.225 

.235 

.248 

A (in21 

.415 

.417 

.420 

.424 

A29 

Table 3.3. Design Data for Precision Segmented Reflector Column. 

ts lr s 

DO26 

.0029 

.0032 

.003 6 

.0039 

110. 

115. 

120. 

125. 

129. 

34 
It should be recalled that the empirical expression for stiffener local buckling derived 

in the last chapter (eq. 2.21) is only applicable for the range .OOZ I fJrs I .OZ with the 

exception of .002 I tJrs when y/ 2 x/2. Initial approximate stiffener local buckling 

analyses determined these bounds as appropriate for typical stiffened column design 

problems to be considered. Fig. 3.1 shows the actual variation in rJr, for the minimum 

mass designs of the two examples considered. It is seen that, indeed, these designs all lie 

within the aforementioned bounds on firs, and thus, the analyses imbedded in the 

DESTCO program are applicable to these examples. 

0.004 

0.003 

0.002 ts 

rs 

- 

- 

Precision Segmented - 

Reflector column - - Space Station 

Freedom column - 

t 

0 n/3 

Stiffener arc angle, y/ 

2d3 

Figure 3.1. Stiffener thickness variation with arc angle for minimum mass designs. 

It is important that the designs generated using DESTCO have values for tJrs which 

are within the acceptable limits. If not, one could either derive a new stiffener local 

buckling expression, applicable for the new range, or one could use the original results and 

accept the lower solution accuracy. Recall that the error in employing the DESTCO 

program outside of the specified range on tsjrs, should result in only near-minimum mass 

designs, but the fundamental stability de of these designs should still be Euler buckling. 

35 
10 

8 

6 

Percent Increase 

in Column Mass 

4 

2 

Data from Table 3.2 

I 

0 10 20 30 40 50 

Percent Increase in 

Column Buckling Load 

Figure 3.2. Minimum mass design curve for Space Station Freedom column. 

10 

8 

6 

Percent Increase 

in Column Mass 

4 

2 

0 10 20 30 40 50 

Percent Increase in 

Column Buckling Load 

Figure 3.3. Minimum mass design curve for Precision Segmented Reflector column. 

36 
The minimum mass designs presented in Tables 3.2 and 3.3 represent the stiffener 

thickness, for a given stiffener arc angle, that results in equal Euler buckling and stiffener 

local buckling stresses (see Eq. 2.23). Designs with larger thicknesses will have Euler 

buckling as their fundamental stability mode, and designs with smaller thicknesses will 

have stiffener local buckling as their fundamental stability mode. Additionally, for designs 

which behave as Euler columns, the percent increases in buckling load and mass are equal 

to the percent increases in cross-sectional inertia and area, respectively. Therefore, the data 

in Tables 3.2 and 3.3 can be used to generate a portion of a stiffened column design curve 

showing minimum percent mass increase versus percent buckling load increase. 

Fig. 3.2 and 3.3 show the complete minimum mass design curves for the Space 

Station Freedom column and the Precision Segmented Reflector column, respectively. The 

two dashed lines superimposed over these curves represent stiffened columns having 

stiffener arc angles of n/3 and 2x13 and are generated using Eq. 2.6 and 2.7 with 

successively increasing values of stiffener thickness. The portion of the design curve 

between the origin and point B represents designs with increasing values of stiffener 

thickness and arc angle, for which the Euler buckling stress equals the stiffener local 

buckling stress. The portion of the curve between points A and B was generated from data 

in Tables 3.2 and 3.3, and is extrapolated to the origin to show an approximation of the 

curve in this region. The portion beyond point B, represents designs having 1y=2x/3 (the 

imposed upper limit on stiffener arc angle) and increasing stiffener thicknesses. As 

mentioned, designs in this region have Euler buckling stresses which are less than their 

stiffener local buckling stresses. 

The lighter weight solid lines in Fig. 3.2 and 3.3 show the mass penalty necessary to 

get the given buckling load increase by simply increasing the thickness of the central 

column. These lines are generated using Eq. 2.5 and 2.7. The difference between the 

thickened central column line and the minimum mass design curve is the mass savings 

attributed to the stiffened column concept. The slope of the thickened central column line is 

approximately one. This follows from the fact that, for a thin circular cross section, both 

the inertia and the area are proportional to the thickness, and thus, a given increase in 

thickness will cause the same percent increase in both area and inertia. Conversely, the 

slope of the minimum mass design curve beyond point B is approximately 1/4, and thus, a 

37 
given percent increase in buckling load in this region, will require only 1/4 of that 

percentage in increased mass. 

The design curves in Fig. 3.2 and 3.3 can be verified through further analysis. As 

mentioned, the portion of the curves between points A and B should represent the 

boundary between designs with Euler buckling as their fundamental stability mode and 

designs with stiffener local buckling as their fundamental stability mode. Fig. 3.4 is an 

exploded view of this portion of the design curve for the Space Station Freedom column. 

The square data points along the design curve correspond to designs which were analysed 

using VIPASA and verified to have Euler buckling as a fundamental stability mode. 

Additionally, models of these designs, with slightly smaller values for stiffener thickness, 

were analysed and determined to exhibit stiffener local buckling as their fundamental 

stability mode. These data points are indicated with stars in Fig. 3.4. These results 

indicate that, indeed, the design curves presented represent the minimum mass designs and 

the boundary between designs with Euler buckling as their fundamental stability mode and 

those with stiffener local buckling as their fundamental stability mode. 

5 

4 

3 

Percent Increase 

in Column Mass 

2 

1 .-' , **/ 0 m Local sliffcner 1 */ 

s, ' I.... I....) ... .l....l 

0 5 10 15 20 25 

Percent Increase in 

Column Buckling Load 

Figure 3.4. Verification of Space Station Freedom column design curve. 

38 
3.2 Discussion of Results 

It should be reiterated that the design curves presented in Fig. 3.2 and 3.3 present the 

percent increase in only the structural mass of the stiffened columns, with no allowance for 

the added mass of the hinge and deployment mechanisms attaching the stiffeners to the 

central column. This added mechanism mass can be estimated as a percentage of the mass 

of the central column, and can be considered a constant for the entire range of stiffener 

designs. Fig. 3.5 shows the minimum mass design curve for the Space Station Freedom 

column when the mass of the hinge and deployment mechanisms is assumed to be 10 

percent of the central column mass. This curve is constructed by simply adding 10 percent 

to the minimum mass design curve in Fig. 3.2. 

Percent Increase 

in Column Mass 

0 10 20 30 40 50 

Percent Increase in 

Column Buckling Load 

Figure 3.5. Design curve for Space Station Freedom column including 10% 

mechanisim mass. 

It is apparent from this figure that the added mechanism mass significantly changes 

the character of the minimum mass design curve. When mechanism mass is considered, 

small to moderate increases in buckling load can be achieved with less mass penalty by 

simply increasing the wall thickness of the central column. In this example, buckling load 

increases up to about 15 percent would favor the thickened central column over the central 

39 
column with deployable stiffeners. Also, this cross over point occurs fairly close to point 

B which is the start of the portion of the design curve on which 1y=2n/3. This implies that 

the addition of an appreciable mechanism mass causes the region of the design curve 

between points A and B in Fig. 3.2 and 3.3 to become essentially unimportant. Thus the 

consideration of stiffener local buckling is unnecessary. Therefore the practical column 

design problem is reduced to the consideration of only two design regions: thickened 

central column for low buckling load increases and stiffened column with 1y=2~/3 for 

higher buckling load increases. 

The upper and lower portions of the design curve in Fig. 3.5 are generated using Eq. 

2.5, 2.6 and 2.7, and by linearly increasing the value of stiffener thickness (upper portion 

of curve) or central column thickness (lower portion of curve). The near linear shape these 

segments of the curve suggest that Eq. 2.5, 2.6 and 2.7 may be linearized with respect to 

these two thicknesses and still give accurate results. Performing this linearization is exactly 

equivalent to invoking the following small thickness assumptions. 

(Ti + r,) = (r, + rp) = 2r 

tcc = r, - ri << r (3.1) 

t, = rp - r, << r 

Additionally, a further simplification can be made by observing a result presented in 

reference 12. In this reference, it was demonstrated that the cross-sectional inertia of the 

stiffened column does not vary more than five percent for deployment angles within 10 to 

15 degrees of the optimum value. Therefore, it is possible to assume the deployment 

angle, I$ = 2n/3 and still ensure a reasonable approximate result. Applying this assumption 

and the small thickness assumptions in Eq. 3.1 to Eq. 2.5,2.6 and 2.7 gives the following 

linearized equations governing the design of a thickened central column and a stiffened 

column with 1y=2n/3. 

I,, m3tcc 

I z m3(tcc + 4tJ 

40 
A r2m(tcc + t,) (3.4) 

These equations illustrate the conclusions previously drawn. First, an increase in 

thickness of the central column will cause the same percent increase in both area and inertia. 

Second, an increase in thickness of the stiffeners will give rise to a percent increase in 

inertia which is four times the percent increase in area 

Eq. 3.2,3.3, and 3.4 can be used to generate an approximate stiffened column design 

curve. Fig. 3.6 presents this approximate design curve superimposed over the exact curve 

from Fig. 3.5. In this example, the approximate curve is in reasonable agreement with the 

exact curve. Furthermore, the approximate curve gives consistently conservative results 

(slightly higher mass increase for a given buckling load increase). Therefore, it is 

reasonable to simply use the approximate expressions given in Eq. 3.2, 3.3, and 3.4 to 

generate frst iteration design curves. 

Percent Increase 

in Column Mass 

20 

15 

10 

5 

Approximate curve 

Exact curve 

- 

--- 

0 10 20 30 40 50 

Pcrcent Increase in 

Column Buckling Load 

Figure 3.6. Comparison of approximate and exact design curves for Space Station 

Freedom column including 10% mechanism mass. 

41 
It should be noticed that the approximate percent mass increase and buckling load 

increase are only linear functions of the percent increase in tcc and f,. Consequently, a 

general expression can be easily derived from Eq. 3.3 and 3.4 which relates the percent 

increase in stiffened column mass (AM) to the percent increase in buckling load (AP,,,) nnd 

the deployment mechanism mass as a percentage of the central column mass (Adm). This 

expression is given in Eq. 3.5 and applies to any stiffened column design having w = 2M3. 

AM = .25APc, + M,,, (3.5) 

A family of approximate stiffened column design curves can be generated from Eq. 

3.5 using a set of values for the deployment mechanism mass. Fig. 3.7 presents one such 

family of curves. These curves provide a first iteration estimate of the percent mass 

increase necessary to achieve a given percent buckling load increase for any stiffened 

col L) mn application. 

150 r 

100 - 

Percent Increase / 

in Column Mass 

50 

0 50 100 150 200 250 

Pcrccnt Increase in 

Column Buckling Load 

Figure 3.7. Family of approximate general stiffened column design curves. 

42 
As an illustrative example, if one desires to increase the buckling load of a circular 

column by 200 percent using deployable stiffeners, and one assumes that the mass of the 

deployment mechanism would be 50 percent of the mass of the central column, then Eq. 

3.5 and, similarly, Fig. 3.7 shows that this would require approximately a 100 percent 

increase in total column mass. This compares to a 200 percent increase in total mass if the 

increased buckling load is achieved through simply thickening the central column. 

Finally, for further evaluating the usefulness of stiffened columns, Appendix C 

presents the effect of deployable stiffeners on column fundamental vibration frequency. 

The results from these calculations indicate that there is an increase in column vibration 

frequency with the addition of deployable stiffeners just as there is an increase in column 

buckling load. However, the percent increase in vibration frequency is typically an order 

of magnitude less than the percent increase in buckling load, thus deployable stiffeners 

appear not to be well suited for an application only requiring an increase in vibration 

frequency. 

43 
CHAPTER 4 

CONCLUSIONS 

This paper has presented the results of a stability design study of a slender column 

with curved longitudinal stiffeners for large space structure applications. These stiffeners 

are attached to the central column through deployment mechanisms which allow the 

stiffeners to package flush around the outside of the central column. The stiffened column 

concept is attractive because it allows the buckling load of a slender column to be increased 

at a fairly low cost in increased mass, and with no change to the packaged size of the 

column. Furthermore, the use of deployable stiffeners eliminates the need for extra onorbit assembly time which is characteristic of other high-buckling-load, low-packagedvolume column concepts. 

The symmetry of the stiffened column cross section gives rise to a zero product of 

inertia and, consequently, a constant value for moment of inertia regardless of reference 

axis orientation. Therefore, the stiffened column is perfectly suited for stability 

applications because it does not have a "preferred" direction of buckling. 

The kinematic attachment constraints offered by the stiffener deployment mechanisms 

are very important in determining the stability modes of the stiffened column. Stiffener 

attachment mechanisms which do not allow shear load to be transferred may require nonlinear numerical analysis to determine stiffener buckling modes. For the purposes of this 

study, the stiffener attachment is assumed to provide perfect shear load transfer, and thus, 

the stiffener buckling analysis performed is strictly linear. 

An approximate explicit expression describing stiffener local buckling stress was 

derived from numerical analysis results using a one dimensional "linked-plate" 

approximation to the stiffener differential equations. This approximate expression was 

44 
demonstrated to be accurate to within ten percent for the appropriate range in geometric 

parameters. Also, equations were derived which relate the set of geomemc parameters to 

column mass and buckling load. The resulting indeterminate system was studied to 

determine mass minimization considerations. The need for additional geometric constraints 

was defined, and one set was selected for construction of a determinate minimum mass 

stiffened column design routine. 

The resulting column design routine was used to construct design curves for two 

current large space structure applications: the Space Station Freedom column and the 

Precision Segmented Reflector column. These curves are comprised of two segments: one 

representing designs for which Euler buckling stress equals stiffener local buckling stress 

and the other representing designs for which the stiffener arc angle is fixed at 2x13 and 

Euler buckling stress is less than stiffener local buckling stress. The designs on the first 

segment of the design curve were verified through further numerical analysis of the entire 

stiffened column cross section using the "linked-plate" structural representation. 

These examples demonstrate that stiffener local buckling is only a concern for 

stiffened columns having approximately 20 percent or less increase in buckling load over 

that of the central column. Also, this lower design range proves to be of no practical 

interest after allowing for the increased mass necessary for the stiffener deployment 

mechanisms. Therefore, stiffener local buckling proves to not be a concern for practical 

stiffened column designs. Consequently, the stiffened column design problem is reduced 

to determining the necessary stiffener thickness and corresponding optimum deployment 

angle to achieve a given percent increase in cross-sectional inertia. 

Furthermore, the equations for cross-sectional inertia and area of the stiffened column 

can be linearized in terms of the stiffener and central column thicknesses by invoking small 

thickness approximations. These approximate equations result in a very simple 

approximate design equation which relates the percent increase in column mass to the 

percent increase in column buckling load and the stiffener deployment mechanism mass. 

Neglecting deployment mechanism mass, it turns out that a given percent increase in area of 

the stiffened column (through increased stiffened thickness) will give rise to four times that 

percent increase in inertia. This is compared to the mass penalty associated with increasing 

the buckling load of the central column by simply increasing its thickness. In this case, a 

45 
given percent increase in area of the central column will cause only the same percent 

increase in inertia. 

The potential mass savings offered by the stiffened column are clear. In the ideal case 

where the deployment mechanisms are assumed to add negligible mass, it offers a given 

buckling load improvement at one fourth the added mass of a circular column with the same 

outer radius. In practical cases where the deployment mechanisms are known to add a 

certain amount of mass, the mass savings of the stiffened column are small for small 

buckling load increases, but approach the ideal rate for higher buckling load increases. 

Finally, results from calculations of column fundamental vibration frequency indicate 

that there is an increase in column vibration frequency with the addition of deployable 

stiffeners just as there is an increase in column buckling load. However, the percent 

increase in vibration frequency is typically an order of magnitude less than the percent 

increase in buckling load, thus deployable stiffeners appear not to be well suited for an 

application only requiring an increase in vibration frequency. 

46 
REFERENCES 

1. Large Space Antenna Systems Technology - 1982, Parts 1 and 2. NASA CP2269, November 30-December 3,1982. 

2. Large Space Antenna Systems Technology - 1984, Parts I and 2. NASA CP2368, December 4-6,1984. 

3. Mikulas, Martin M., Jr.; Croomes, Scott D.; Schneider, William; Bush, Harold G.; 

Nagy, Kornell; Pelischek, Timothy; Lake, Mark S.; and Wesselski, Clarence: 

Space Station Truss Structures and Construction Considerations. NASA TM86338, January 1985. 

4. Mikulas, Martin M., Jr.; Bush, Harold G.; and Card, Michael F.: Structural 

Stiflness, Strength and Dynamic Characteristics of Large Tetrahedral Space Truss 

Structures. NASA TMX-74001, March 1977. 

5. Mikulas, Martin M., Jr.; and Bush, Harold G.: Advances in Structural Concepts. 

Published in Large Space Antenna Systems Technology - 1982, Part I, NASA 

CP-2269 November 30-December 3,1982, pp. 257-283. 

Reinert, R. P.: Weight Optimization of Ultra Large Space Structures. Presented at 

the 38th Annual Conference of the Society of Allied Weight Engineers, Inc., May 

1979, SAW paper No. 1301. 

Heard, Walter L., Jr.; Bush, Harold G.; Walz, Joseph E.; and Rehder, John J.: 

Structural Sizing Considerations for Large Space Platforms. AIAA Journal of 

Spacecraft and Rockets, Vol. 18, No. 6, Nov.-Dec., 1981, pp. 556-564. 

6. 

7. 

8. Bush, Harold G.; Heard, Walter L., Jr.; Walz, Joseph E.; and Rehder, John J.: 

Deployable and Erectable Concepts for Large Spacecraft. Presented at the 39th 

Annual Conference of the Society of Allied Weight Engineers, Inc. May 1980, 

SAW paper No. 1374. 

Bush, Harold G.; Heard, Walter L., Jr.: General Description of Nestable Column 

Structural and Assembly Technology. NASA TM-83255, December 198 1. 

Heard, W. L., Jr.; Bush, H. G.; and Agranoff, Nancy: Buckling Tests of 

Structural Elements Applicable to Large Space Trusses. NASA TM-78628, 

October 1978. 

9. 

10. 

1 1. Mikulas, Martin M., Jr,: Structural Efficiency of Long Lightly Loaded Truss and 

Isogrid Columns for Space Applications. NASA TM-78687, July 1978. 

47 
12. 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

20. 

21. 

22. 

23. 

24. 

Lake, Mark S.; and Wu, K. Chauncey: Preliminary Investigation of Stability of a 

Fin-Stiflened Slender Strut. NASA TM 4034, April 1988. 

Donnell, Lloyd Hamilton: Beams, Plates, and Shells. McGraw-Hill Book Co., 

1976. 

Timoshenko, Stephen P.; and Gere, James M.: Theory of Elastic Stability, Second 

ed. McGraw-Hill Book Co., 1961. 

Murray, N. W.: Introduction to the Theory of Thin- Walled Structures. Clarendon 

Press, 1984. 

Brush, Don 0.; and Almroth, Bo 0.: Buckling of Bars, Plates, and Shells. 

McGraw-Hill Book Co., 1975. 

Wittrick, W. H.: General Sinusoidal Stiffness Matrices for Buckling and Vibration 

Analyses of Thin Flat- Walled Structures International Journal of Mechanical 

Sciences 1968, Vol. 10, pp. 949-966. 

Wittrick, W. H.; and Williams, F. W.: Buckling and Vibration of Anisotropic or 

Isotropic Plate Assemblies Under Combined Loadings. International Journal of 

Mechanical Sciences 1974, Vol. 16, pp. 209-239. 

Stroud, W. Jefferson; and Anderson, Melvin S.: PASCO: Structural Panel 

Analysis and Sizing Code, Capability and Analytical Foundations. NASA TM80181, November 1981. 

Anderson, Melvin S.; Stroud, W. Jefferson; Durling, Barbara J.; and Hennessy, 

Katherine W.: PASCO: Structural Panel Analysis and Sizing Code, User's 

Manual. NASA TM-80182, November 198 1. 

Roark, Raymond J.; and Young, Warren C.: Formulas for Stress and Strain, Fifth 

ed. McGraw-Hill Book Co., 1982. 

Mikulas, Martin M., Jr.; and Bush, Harold G.: Design, Construction and 

Utilization of a Space Station Assembled from 5-Meter Erectable Struts. NASA 

TM-89043, October 1986. 

Collins, Timothy J.; and Fichter, W. B.: Support Trusses for Large Precision 

Segmented Reflectors: Preliminary Design and Analysis. NASA TM- 10 1560, 

March 1989. 

Burden, Richard L.; and Faires, J. Douglas: Numerical Analysis, Third ed. PWSKent Publishing Co., 1985. 

48 
APPENDIX A 

DERIVATION OF STIFFENER CROSS-SECTIONAL INERTIAS 

The cross-sectional moment of inertia components of the curved stiffener can be 

calculated for arbitrary stiffener dimensions and deployment angle by performing the 

appropriate double integrals. The general expressions for these integrals are given in Eq. 

A.l. 

Before evaluating these integrals, it is necessary to define a convenient coordinate 

system and appropriate limits for the integrals. Due to the circular curvature of the stiffener 

cross section, it is convenient to use a polar coordiante system centered at the stiffener's 

center of curvature. Fig. A.l shows a stiffener of arc angle vat the deployment angle @, 

with the column cross section oriented such that the stiffener's center of deployment 

rotation lies along the global y axis. This center of deployment rotation lies along the 

middle surface through the stiffener's thickness and, for this study, is defined to be a 

distance away from the edge equal to half the stiffener thickness ( (rp-r0)/2 ). Integration in 

polar coordinates will be conducted relative to the local coordinate system (x', y') which is 

centered at the global coordinates (x,,y,). Eq. A.2 gives the expressions for these global 

coordinates. 

49 
Figure A. 1. Cross-sectional parameters for stiffener moments of inertia. 

As seen in Fig. A. 1, a differential area element can be defined at any angle, 0, and 

radius, r, relative to the local coordinate system. Also, the global coordinates of this are3 

element are easily defined as in Eq. A.3. 

Now it is possible to express the explicit polar form of Eq. A.l. These expressions 

are given in Eq. A.4. Performing these integrations will result in Eq. 2.3 and Eq. 2.4 

given in the main text. 

(r,, + rp[l - cos($))/2 + rsin(8) rdrd6 

(A.4) 

isx = 1.: 

50 
APPENDIX B 

ILE;SIGN OF aIFFENED ULUMN CODE (DESTCO) 

Select value of stiffener 

arc height 

The computer program DESTCO was developed for the design of stiffened columns 

using design rules developed in the present study. The program was written and executed 

using a FORTRAN 77 compiler on a desk-top personal computer. Fig. B.l presents a 

flow chart which explains program execution, and following is a program listing. 

- 

w Interactive data input 

c I 

L 

Calculate deployment angle, inertia, 1 

area, and radius of gyration 

Figure B. 1. Flow chart of DESTCO program. 

SUBROUTINE INERTIA 

Calculatc deployment angle 

and corresponding maximum 

inertia for cross section 

51 

1 

Stare minimum mass 

dcsign information 

I 

I 

I L A Generate comparison mass and buckling load data I 
c ..................................................................... 

C 

C DESTCO 

C 

C THIS IS A PROGRAM TO GENERATE THE DESIGN CURVE FOR A LATERC ALLY STIFFENED SLENDER COLUMN. THE CROSS SECTION INCLUDES A 

C CIRCULAR TUBE WITH THREE CYLINDRICAL OPEN SECTIONS FUNC CTIONING AS DEPLOYABLE STIFFENERS WHICH INCREASE THE BENDING . C STIFFNESS OF THE SECTION. 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

INPUT QUANTITIES INCLUDE THE CORE TUBE DIMENSIONS (INNER RADIUS, OUTER RADIUS, AND LENGTH), A MAXIMUM VALUE OF THE BUCKLING LOAD IMPROVEMENT RATIO TO BE STUDIED, AND AN INTEGER 

NUMBER OF EQUALLY SPACED STIFFENER ARC HEIGHTS TO BE ANALYZED. 

THE PROGRAM BEGINS BY CALCULATING ALLOWED STIFFENER THICKNESS 

FOR A SERIES OF STIFFENER ARC HEIGHTS (IE, 30, 60, 90, AND 120 

DEGREES) USING A RECURSIVE APPLICATION OF A DESIGN EQUATION 

DERIVED FROM THE STABILITY OF AN INFINITELY LONG CURVED PANEL 

CLAMPED ALONG ONE EDGE AND FREE ALONG THE OTHER. THIS PROCEDURE REQUIRES CALCULATING THE DEPLOYED INERTIA OF THE 

CROSS-SECTION AND THE CORRESPONDING OPTIMUM VALUE OF THE FIN 

DEPLOYMENT ANGLE FOR INITIAL APPROXIMATIONS OF THE CROSS-SECTIONAL GEOMETRY. SUBSEQUENT APPROXIMATIONS ARE THEN DERIVED 

FROM THE PREVIOUS VALUES. 

FINALLY, THE SET OF INERTIAS AND AREAS ARE NORMALIZED TO THE 

INERTIA AND AREA OF CORE TUBE SO THAT A DATA FILE CAN BE GENERATED FOR A PLOT OF NORMALIZED WEIGHT INCREASE VS. NORMALIZED 

BUCKLING LOAD IMPROVEMENT. 

INPUT VARIABLES ARE AS FOLLOWS: 

RI = INNER RADIUS OF TUBE 

RO = OUTER RADIUS OF TUBE (INNER OF STIFFENER) 

L = LENGTH OF COLUMN 

PMAX = RATIO OF MAXIMUM DESIGN BUCKLING LOAD 

NARC = NUMBER OF STIFFENER ARC HEIGHTS 

TO THAT OF THE CORE TUBE 

TO BE ANALYZED 

OTHER INTERNAL VARIABLES ARE AS FOLLOWS: 

PHI = DEPLOYMENT ANGLE OF STIFFENERS 

PSI = ARC HEIGHT OF STIFFENERS 

RP = OUTER RADIUS OF STIFFENERS 

............................................................... 

REAL IX, ICORE, INORM, L 

CHARACTER"1 TAB 

DIMENSION T (10) , PDESIGN (lo) , WDESIGN (lo) , P (5,s) , W (5,5) 

TAB=Z'09' 

IOUT=l 

IDAT=2 

WRITE (*,I) 

WRITE (*,2) 

OPEN(IOUT,FILE='STIFF. STRUT DESIGN.OUT',STATUS='NEW') 

.. 

52 
OPEN(IDAT,FILE='STIFF. STRUT DESIGN.PLT',STATUS='NEW') 

1 FORMAT (5X,28HDATA WILL BE SENT TO PRINTER) 

2 FORMAT (5X,58HPLOT DATA WILL BE ECHOED TO FILE "STIFF. STRUT DE 

5 CONTINUE 

XSIGN. PLT") 

c 

C READ IN DATA AND ECHO TO OUTPUT FILE 

C 

10 FORMAT (SX, 32HINPUT INNER RADIUS OF TUBE (IN) : ) 

20 FORMAT (5X,32HINPUT OUTER RADIUS OF TUBE (IN):) 

30 FORMAT (SX,28HINPUT LENGTH OF COLUMN (IN) :) 

40 FORMAT (5X,48HINPUT MAXIMUM STRUT-TO-CORE BUCKLING LOAD RATIO:) 

50 FORMAT (5X,48HINPUT NUMBER OF ARC HEIGHTS TO BE STUDIED (<lo):) 

60 FORMAT (10X165H************************************************ 

I*****************) 

70 FORMAT (lox, 1H*, 63X,lH*) 

80 FORMAT (10X165H* GENERAL DESIGN OF A LATERALLY STIFFENED SL 

90 FORMAT (lox, 1H*, 63X, 1H*) 

1ENDER COLUMN *) 

100 FORMAT (lOX,65H****************X"******************************** 

..................... 

110 FORMAT (9X,24HINNER RADIUS OF TUBE = ,F9.4,5H (IN)) 

120 FORMAT (9X,24HOUTER RADIUS OF TUBE = ,F9.4,5H (IN)) 

130 FORMAT (9X,24HLENGTH OF TUBE = ,F9.4,5H (IN)) 

140 FORMAT (9X,24HMAX. BUCKLING RATIO = ,F9.6) 

150 FORMAT (9X,24HNUMBER OF ARC HEIGHTS = ,141 

WRITE (*,lo) 

READ (*,*) RI 

WRITE (*,20) 

READ (*,*) RO 

WRITE (*,30) 

READ (*,*) L 

WRITE (*,40) 

READ (*,*) PMAX 

WRITE (*,SO) 

READ (*,*) NARC 

WRITE (IOUT, 60) 

WRITE (IOUT,70) 

WRITE (IOUT, 80) 

WRITE (IOUT, 90) 

WRITE (IOUT, 100) 

WRITE (IOUT,110) RI 

WRITE (IOUT,120) RO 

WRITE (IOUT,130) L 

WRITE (IOUT, 140) PMAX 

WRITE (IOUT, 150) NARC 

PI=ACOS (-1.0) 

C 

C DETERMINE THE INERTIA AND AREA OF THE CORE TUBE 

C 

ICORE=PI* ( (RO**4) - (RI**4) ) /4. 

ACORE=PI*((RO**2)-(RI**2)) 

PSIINC=PI/ (3" (NARC-1) ) 

PDESIGN(1)=1.0 

WDESIGN ( 1 ) =1.0 

C 

C DETERMINE SET OF ALLOWABLE THICKNESSES USING A RECURSIVE 

53 
C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

APPROACH. THE FIRST STEP DETERMINES THE ALLOWED THICKNESS 

BASED ON A FIRST GUESS AT THE RADIUS OF THE STIFFENER AND THE 

RADIUS OF GYRATION OF THE STRUT EQUAL TO THE OUTER RADIUS OF 

THE CORE. THE SECOND AND SUBSEQUENT STEPS REFINE THE ESTIMATE 

OF THE ALLOWED THICKNESS USING VALUES FOR THE STIFFFENER RADIUS 

AND RADIUS OF GYRATION BASED ON RESULTS FROM THE PREVIOUS GUESS. 

ITERATION IS HALTED WHEN THE THICKNESS CONVERGES TO .5 PERCENT 

(. 005) . AT EACH ITERATION STEP, SUBPROGRAM "INERTIA" IS CALLED 

TO CALCULATE THE OPTIMUM DEPLOYMENT ANGLE FOR THE GIVEN DESIGN. 

THE DESIGN EQUATION COMES FROM EQUATING THE EULER STRESS 

OF THE COLUMN TO THE LOCAL BUCKLING STRESS OF AN INFINITELY 

LONG CURVED STIFFENER, CLAMPED ALONG ONE EDGE AND FREE ALONG 

THE OTHER. 

DO 400 I=l,NARC 

ICOUNT=O 

RHO=RO 

RFIN=RO 

T (I) =O . 0 

200 CONTINUE 

PSI=PSIINC* (1-1)t(P1/3.) 

C 

C BEGIN ITERATION TO DETERMINE ALLOWED THICKNESS FOR THE 

C GIVEN ARC HEIGHT. 

C 

ICOUNT=ICOUNT+l 

TMl=T (I) 

T(I)=RFIN* (110.* ((RHO/L)**2)t.O00974*PSI-.O00669) 

RP=ROtT (I ) 

CALL INERTIA (RI , RO, RP , IX, PHI, PSI ) 

C 

C 

C 

C 

C 

C 

C 

CALCULATE STRUT AREA AND NEW RADIUS OF GYRATION 

ATUBE=PI* ( (RP**2) - (RO**2) ) * (PSI*3) / (PI*2) +ACORE 

RHO=SQRT (IX/ATUBE) 

RFIN=RO+T (I) /2 

TEST FOR CONVERGENCE AND CHECK THE MAXIMUM ITERATION 

COUNTER. 

IF (ICOUNT.EQ.1) GO TO 200 

TEST=ABS (TEST) 

IF (ICOUNT.GE.20) GO TO 250 

IF (TEST.GT.0.005) GO TO 200 

TEST= (T (I) /TM~) -1.0 

250 CONTINUE 

J=I+1 

WDESIGN (J) =ATUBE/ACORE 

PSIDEG=PSI*180/PI 

PDESIGN ( J) =IX/ICORE 

PHIDEG=PHI*~~~/PI 

C 

C PRINTOUT INERTIA AND AREA 

C 

260 FORMAT (/12X,21HARC HEIGHT OF FIN = ,F8.3,10H (DEG) 1 

270 FORMAT (/13X, ZOHTHICKNESS OF FIN = , E12.5,SH (IN) ) 

54 
280 FORMAT 

290 FORMAT 

300 FORMAT 

XDEG) ) 

WRITE 

WRITE 

WRITE 

WRITE 

WRITE 

400 CONTINUE 

C 

(5X,28HOPTIMUM DEPLOYMENT ANGLE = ,F8.3,10H ( 

(27X,6HIX = ,E12.5,7H(IN**4) 

(25X,8HAREA = ,E12.5,7H(IN**Z) /) 

IOUT, 260) PSIDEG 

IOUT, 270) T (I) 

IOUT, 280) PHIDEG 

IOUT,290) IX 

IOUT, 300) ATUBE 

C ENTER LOOP TO DETERMINE A SERIES OF STRUT DESIGNS 

C FOR PSI=O, 30, 60, 90, 120 DEGREES. THIS DATA CAN 

C BE USED TO PLOT FOR COMPARISON WITH THE DESIGN 

C CURVES ALREADY GENERATED. BASE THE MAXIMUM THICKNESS 

C ON AN APPROXIMATION TO A BUCKLING LOAD = (PMAX*PCORE) 

C 

TINCs (PMAX-1. ) *ICORE/ (52.67* (RO**3) 

TFIN=TINC 

PSIINC=PI/6. 

DO 500 I=1,5 

P (I,l)=l. 0 

w (I, 1)=1.0 

500 CONTINUE 

DO 520 J=2,5 

RP=RO+TFIN 

P (I, J)=PI* ( (RP**4) -(RI**4) ) / (4"ICORE) 

W (1, J)=PI* ( (RP**2) -(RI**2) ) /ACORE 

DO 510 I=2,5 

PSI=PSIINC* (1-1) 

CALL INERTIA (RI,RO,RP, IX,PHI,PSI) 

ATUBE=PI* ( (RP**2) - (RO**2) ) * (PSI*3) / (PI*2) 

P (I, J) =IX/ICORE 

X + ACORE 

W (I, J) =ATUBE/ACORE 

510 CONTINUE 

520 CONTINUE 

TFIN=TFINtTINC 

C 

C WRITE NORMALIZED INERTIAS AND AREAS TO OUTPUT 

C DATA FILE. 

c 

DO 550 

WRITE 

530 FORMAT 

540 FORMAT 

550 CONTINUE 

N=NARCt 1 

DO 560 

WRITE 

560 CONTINUE 

STOP 

END 

I=l, 5 

(10 (F12.6,Al) 1 

(F12.6,Al, F12.6) 

IDAT,530) (W(K,I) ,TAB,P(K,I) ,TAB,K=1,5) 

I=1,N 

IDAT, 5 4 0 ) WDESIGN ( I ) , TAB, PDES IGN ( I ) 

c. .................................................................... C 

C SUBROUTINE INERTIA CALCULATES THE DEPLOYMENT ANGLE, PHI 

55 
C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

WHICH GIVES THE IMUM INERTIA FOR THE GIVEN RADII 

(RI,RO,RP). USING THIS VALUE OF PHI, IT CALCULATES THE 

AREA MOMENT OF INERTIA OF THE SECTION 

INPUT VARIABLES ARE AS FOLLOWS: 

RI = INNER RADIUS OF TUBE 

RO = OUTER RADIUS OF TUBE 

RP = OUTER RADIUS OF STIFFENERS 

PSI = ARC HEIGHT OF STIFFENERS 

(INNER RADIUS OF STIFFENERS) 

OUTPUT VARIABLES ARE AS FOLLOWS: 

IX = MAXIMUM MOMENT OF INERTIA FOR CROSS-SECTION 

PHI = DEPLOYMENT ANGLE FOR MAXIMUM INERTIA 

c..................................................................... 

REAL IX 

PI=ACOS (-1.0) 

Al=(PSI/8.0) * (RP**z-R0**2) * ( (RPtRO) ""2) 

A2=(RP**3-R0**3) * (RPtRO) /3.0 

A3=3.0*PSI*(RP**4-RO**4)/8.OtPI*(RO**4-R1**4) /4.O 

CALCULATE OPTIMUM DEPLOYMENT ANGLE 

BETA=(PI/2.)-(PS1/2.)tZ.*ATAN( (RP-R0)/(2.*(RP+RO))) 

PHI=ATAN ( (A2*SIN (PSI/2. ) *COS (BETA) 

IF (PHI.LT.0.) THEN 

PHI=PHItPI 

END IF 

/ 

x (A2*SIN (PSI/2. ) *SIN (BETA) -AI) 

DETERMINE INERTIA OF CROSS SECTION 

THETAE=PI/2.otPHI+2.o*ATAN((RP-RO)/(~.o*(RP+Ro))) 

IX=3.*A1* (l.O-COS(PHI))t6.*A2*SIN(PHI/2.)*SIN(PSI/~.~ 

RETURN 

END 

X*COs((PHItPSI-2.*THETAE)/Z.)tA3 

56 
APPENDIX C 

VIBRATION FREQUENCIES OF THE STIFFENED COLUMN 

Local vibration of truss members is a concern in the design of large space structures 

because of the desire to avoid interaction between local vibration modes and global 

vibration modes as well as active attitude control systems. Therefore, it is necessary to 

determine the effect deployable stiffeners have on the vibration frequency of the column in 

order to throughly evaluate their usefulness for application to large space structures. 

The linearized expressions for column cross-sectional area and inertia given in Eq. 

3.3 and 3.4 can be used to derive an approximation for the fundamental vibration frequency 

of the column. From this, an expression can be derived for percent increase in vibration 

frequency as a function of increase in mass and buckling load. Thus the general stiffened 

column design curves given in Fig. 3.7 can be amended to include information on vibration 

frequency increase. 

The vibration frequency of interest corresponds to the half-sine wave lateral vibration 

of a pin-pinned column. The formula for this frequency is given in Eq. C-1 (21). 

where p is defined as the mass density of the material. 

For the purpose of determining percent increase in vibration frequency, it is sufficient 

to consider the leading parenthetic phrase in Eq. C.l to be a constant. Thus the vibration 

frequency is simply proportional to the square-root of the ratio of moment of inertia to area. 

Inserting Eq. 3.3 and 3.4 into Eq. C.l and accounting for the additional deployment 

57 
mechani mma s, the following -quation 

~ ~ 

arise to describe the unstiffened (initial) vibration 

frequency (0,) and the stiffened vibration frequency (0). 

oo=cdE =c 

where: 

The percent increase in vibration frequency from that of the ustiffened column is then 

simply: 

=7-I + Mm + tJtcc 

Eq. C.4 can be expressed in terms of the percent increase in column buckling load by 

noticing APcr = 4ts/tcc, as shown in Chapter 3. This resulting expression is given in Eq. 

c.5. 

-1 I +Mu 

I + M, + .25Mcr 

Setting do to any constant value in Eq. C.5 results in an equation which is linear in 

Per. Furthermore it is possible to superimpose these lines of constant do over the general 

stiffened column design curves presented in Fig. 3.7. Fig. C. 1 presents the resulting plot. 

58 
150 

100 

Percent 

Increase in 

Column Mass 

50 

0 50 100 150 200 250 

Petcent Increase in 

Column Buckling Load 

Figure C. 1. General stiffened column design curves with vibration frequency increase. 

Notice that the line corresponding to the thickened central column is also the line for 

which there is zero percent increase in vibration frequency. This result follows from Eq. 

C.2 in which it is shown that the vibration frequency of a thin walled circular column is 

independent of wall thickness. As expected, there is an increase in vibration frequency 

with the addition of deployable stiffeners just as there is an increase in column buckling 

load. However, it is important to note that the magnitude of increase in vibration frequency 

is typically much less than that for buckling load (e.g. approximately one order of 

magnitude less for a mechanism mass equal to 50%). 

59 
Report Documentation Page 

1. Report No. 

NASA TM-101636 

2. Government Accession No. 

4. Title and Subtitle 

Stability and Dynamic Analysis of a Slender Column 

with Curved Longitudinal Stiffeners 

17. Key Words (Suggested by Authods)) 

Stability 

Curved Stiffener 

7. Authorls) 

18. Distribution Statement 

Unclassified - Unlimited 

Mark S. Lake 

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of pages 

9. Performing Organization Name and Address 

NASA Langley Research Center 

Hampton, VA 23665-5225 

22. Price 

12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

15. Supplementary Notes 

3. Recipient's Catalog No. 

5. Report Date 

August 1989 

6 Performing Organization Code 

8. Performing Organization Report No. 

10. Work Unit No. 

506-43-31-02 

11. Contract or Grant No. 

13 Type of Report and Period Covered 

Technical Memorandum 

14 Sponsoring bgency Code 

Slender Column 

Mass Subject Category - 39 

1 A04 I 66 Unclassified 1 Unclassified 

I I I 

iASA FORM 1626 OCT 86