Probing the Stability of Ladder-Type Coilable Space Structures

Fabien Royer

∗

and Sergio Pellegrino

†

California Institute of Technology, Pasadena, California 91125

https://doi.org/10.2514/1.J060820

This paper analyzes the buckling and postbuckling behavior of ultralight ladder-type coilable structures, called

strips, composed of thin-shell longeronsconnectedby thinrods. Basedon recent research on the stability of cylindrical

and spherical shells, the stability of strip structures loaded by normal pressure is studied by applying controlled

perturbations throughlocalized probing. A plot of these disturbancesfor increasingpressureis the stability landscape

for the structure, which gives insight into the structure

’

s buckling, postbuckling, and sensitivity to disturbances. The

probing technique is generalized to higher-order bifurcations along the postbuckling path, and low-energy escape

paths into buckling that cannot be predicted by a classical eigenvalue formulation are identified. It is shown that the

stability landscape for a pressure-loaded strip is similar to the landscape for classical shells, such as the axially loaded

cylinder and the pressure-loaded sphere. Similarlyto classical shells, the stability landscape for the strip shows that an

early transition into buckling can be triggered by small disturbances; however, while classical shell structures buckle

catastrophically, strip structures feature a large stable postbuckling range.

Nomenclature

= total area of strip

= area of longeron webs and battens

= batten cross-sectional width

= longeron cross-sectional web width

= batten cross-sectional height

= strip length

= pressure applied on area

= nonlinear buckling pressure

−

lin

= linear buckling pressure

= minimum postbuckling pressure

max

= maximum postbuckling pressure

= longeron cross-sectional radius

= batten spacing

= longeron flange thickness

= strip width

= probe location along strip axis

= longeron cross-sectional opening angle

I. Introduction

HIN-SHELL structures have been used extensively for aero-

space applications as they enable lightweight vehicles. Since the

early 1920s, discrepancies between shell buckling experiments and

theoretical buckling predictions based on linear bifurcation analysis

based on perfect shell geometries were observed. The experimental

buckling loads were lower than the analytical predictions, and the

discrepancywas later linked to the presence ofinitial imperfections in

the shell geometry [1

–

3]. Considerable efforts were made to find safe

lower bounds for the buckling load of these structures, which led to

the NASA space vehicle design criteria for the buckling of thin-

walled circular cylinders (NASA SP-8007) [4].

Today, these empirical buckling criteria are still used but are seen

as very conservative, and have some inherent limitations. To address

these shortcomings, the NASA

’

s Shell Buckling Knockdown Factor

(SBKF) Project was established in 2007 to develop less-conservative,

robust shell buckling design factors by testing shells with known

imperfections, as well as nonuniformities in loading and boundary

conditions [5]. The introduction of precisely engineered imperfec-

tions in spherical shells showed that buckling could be accurately

predicted if the initial geometry is known accurately [6]. However, in

many applications, measuring the shape of the structure before use

can be both expensive and difficult. The traditional buckling and

postbuckling prediction method uses a linear combination of the first

buckling modes as imperfection [7,8] and showed increased accuracy

compared with the classical linear bifurcation approach. The impor-

tance of local deformations at the onset of buckling was linked to

localization effects that cannot be described as a combination of

eigenmodes [9].

In particular, postbuckling paths exhibiting localization are found

in cylindrical and spherical shells. In most cases, these paths are

broken away from the fundamental path but approach it asymptoti-

cally, and can be reached before the first eigenvalue is attained if a

small amount of disturbing energy is input into the structure [10,11].

For these early buckling routes, the structure exhibits a single dimple

localized deformation and sits on a ridge of total potential energy

separating the prebuckling energy well and a lower energy, localized

postbuckling well. This mode of deformation is thus called mountain

pass point and it has been shown that the single dimple corresponds in

fact to the cylindrical shell lowest, mountain pass point [10], i.e. the

postbuckling solution that can be reached with a minimal energy

barrier. An experimental procedure to determine the fundamental

path meta-stability was proposed in 2013 [12] and has been used

experimentally [13]. Comparisons with earlier work showed that

the onset of meta-stability often referred to as

“

shock sensitivity

”

[12] gives an accurate lower bound for experimental buckling

loads [14,15].

The objective of the present paper is to apply these recent break-

throughs in understanding cylindrical and spherical shell buckling to

more complex thin shell structures made of composite materials. In

particular, the present authors are currently investigating structural

architectures for ultralight, coilable space structures for large,

deployable, flat spacecraft for the Caltech Space Solar Power (SSPP)

project [16]. In the deployed configuration, each spacecraft measures

up to

60 m

×

60 m

in size and is composed of ultralight ladder-type

coilable strips of equal width, arranged to form a square, and each

strip supports photovoltaic and power transmission elements. This

structure is described in a previous paper [17] and is shown in Fig. 1.

Scaled laboratory prototypes of this structural concept have been

built and tested [18,19].

Ladder-type structures consist of two triangular rollable and col-

lapsible (TRAC) [20] longerons, connected transversely by rods

(battens), and will be referred to as a

strip

in this paper. In the

Presented as Paper 2020-1437 at the AIAA Scitech 2020 Forum, Orlando,

FL, January 6

–

10, 2020; received 17 April 2021; revision received 9 Sep-

tember 2021; accepted for publication 8 November 2021; published online 3

Institute of Aeronautics and Astronautics, Inc., with permission. All requests

for copying and permission to reprint should be submitted to CCC at www.

also AIAA Rights and Permissions www.aiaa.org/randp.

*Graduate Student; Currently at MIT AeroAstro, 125 Massachusetts Ave,

Cambridge, MA 02139; fabienr@mit.edu. Member AIAA.

†

Joyce and Kent Kresa Professor of Aerospace and Professor of Civil

Engineering, Graduate Aerospace Laboratories, 1200 E California Blvd.

MC 105-05; sergiop@caltech.edu. Fellow AIAA.

2000

AIAA J

OURNAL

Vol. 60, No. 4, April 2022

Downloaded by CALIFORNIA INST OF TECHNOLOGY on August 4, 2022 | http://arc.aiaa.org | DOI: 10.2514/1.J060820

proposed SSPP architecture, the strips are simply supported at the

ends with boundary conditions that do not allow any tension to be

applied. The battens are rectangular-cross-section carbon fiber rods,

and the longerons are thin composite shells. The present study

considers a simplified version of the strips under development for

SSPP, forming rectangles instead of trapezoids and with simpler

boundary conditions.

In orbit, the main loading is solar pressure, modeled as a static

pressure. This is the main loading condition that will be considered in

the present study. Dynamic load conditions can be modeled using an

equivalent static pressure.

The purpose of the present study is to understand and quantify the

buckling and postbuckling behavior of a strip, and explore the

sensitivity of its transition to buckling, considering the effects of

disturbances and imperfections. Recent work studied the longeron

’

deployed behavior and showed that localized buckles form under

transverse bending, and that the postbuckling regime immediately

after the first bifurcation restabilizes quickly. The studies focused on

replicating experimental results using nonlinear buckling and post-

buckling simulations [21

–

24] and further work used nonlinear buck-

ling load computations coupled with modern machine learning

techniques to design optimized longeron geometries [25].

The present paper takes a different approach. The stability of the

prebuckling (fundamental) path is assessed and insights into early

transitions into the postbuckling regime are gained by using localized

probing to apply a perturbation to the structure. The probing tech-

nique is then generalized to higher-order bifurcations arising from the

postbuckling path. Low-energy escape paths into buckling that can-

not be predicted by a classical eigenvalue problem are identified.

The paper is structured as follows. Section II highlights some

experimental observations on the buckling of a cantilevered strip,

to gain initial insights, and Sec. III reviews the concept of stability

landscape for thin shell buckling, which is used in the rest of the

paper. Section IV presents the numerical computation of stability

landscapes for a specific strip structure with boundary conditions

similar to those used for coilable strips, and Sec. Vextends the use of

these landscapes to the postbuckling regime. In Sec. VI the effect of

the strip length on the stability landscape is investigated. Section VII

generalizes the probing to the entire structure, unveiling the existence

of low-energy escape paths into buckling, similar to the low-energy

paths observed for the cylinder and the sphere. Finally, Sec. VIII

presents a preliminary analysis of the effect of geometrical imper-

fections on the strip stability landscape.

Terminology:

Brief descriptions of the key terminology used in the paper are

provided here.

•

Linear buckling eigenvalue/load

: buckling eigenvalue estimate

obtained by solving the buckling eigenvalue problem for the unde-

formed structure.

•

Nonlinear buckling eigenvalue/load

: buckling eigenvalue esti-

mate obtained by iteratively solving the buckling eigenvalue problem

as the structure is loaded. A first buckling eigenvalue prediction is

performed without any preload. This first estimate gives the linear

buckling eigenvalue. The structure is then preloaded under the linear

buckling eigenvalue and a second buckling eigenvalue prediction is

performed. This step is repeated until the preload converges to the

buckling eigenvalue estimate, to give the nonlinear buckling eigen-

value. The associated mode shape is denoted as the nonlinear buck-

ling mode.

•

Minimum postbuckling load

: load at which a specific post-

buckling path restabilizes. This is the lowest load reached on the

postbuckling path, unless no stable paths exist in the postbuckling

regime. Note that the structure can reach lower load values if there

exist bifurcations on the stable postbuckling path. In this case, one

will associate one minimal postbuckling load with each bifurcation.

•

Maximum postbuckling load

: ultimate load that can be sus-

tained by a structure before it loses its overall stiffness.

II. Experimental Observations on Strip Buckling

A 0.8-m-long strip prototype with three battens spaced at 0.2 m

was built and tested in the cantilever configuration shown in Fig. 2b.

The longeron end cross sections at one end of the structure were built

into a stiff plate and the cross sections at the other end were attached

to a stiff composite rod. These boundary conditions did not allow the

end cross sections to deform. Two load cells mounted on an acrylic

beam were attached to a translation stage and were placed in direct

contact with the two longerons

’

ends (using steel balls). When

actuated, the linear stage displaced the two longeron ends and the

reaction force at each end was measured. The end displacements of

the longerons were also measured, using laser displacement gauges.

The results of this experiment are shown in Fig. 2a and the buckles

that were observed are shown in Fig. 2b. It can be seen that longeron 2

is 10% softer than longeron 1, due to a slightly smaller cross section

introduced by the manufacturing process. The two longerons exhibit

different buckling behaviors. A local buckle at the location of the first

batten connection appears (buckle 1 in Fig. 2b) in longeron 2 for a tip

deflection of 1.5 mm. As the tip deflection is increased, a gradual

softening of longeron 2 is observed, which physically corresponds to

the growth of buckle 1. No other buckles appear in longeron 2. For

longeron 1, the linear regime extends to a tip deflection of 2.5 mm,

and then the first buckle (buckle 2) appears close to the cantilever

base. Unlike longeron 2, the behavior of longeron 1 after the appear-

ance of buckle 2 is linear. A second buckle (buckle 3) appears 0.5 m

away from the cantilever base, for a tip deflection of 8.5 mm. The

postbuckling regime of longeron 1 after the formation of buckle 3 is

also linear. Finally, unstable buckling of longeron 1 occurs for a tip

deflection of 17.5 mm (buckle 4). At this point buckle 3 disappears.

This ultimate buckling is of a different type and consists of a doubly

curvedS-shaped flange deformation, which creates a localized region

of very high curvature and leads to the formation of a crack at this

location. Because the two longerons are connected at the tip by a

carbon fiber tube, the drop in reaction force in longeron 1 is accom-

panied by a sudden increase of reaction force in longeron 2.

Based on this experiment, the following observations can be made

regarding the behavior of the tested structure.

Fig. 1 Overview of ladder-type structure for SSPP.

ROYER AND PELLEGRINO

2001

Downloaded by CALIFORNIA INST OF TECHNOLOGY on August 4, 2022 | http://arc.aiaa.org | DOI: 10.2514/1.J060820