# Exploring the design space for nonlinear buckling of

composite thin-walled lenticular tubes under pure bending

Qilong Jia  Ning An<EMAIL_ADDRESS>Xiaofei Ma  Jinxiong Zhou
State Key Laboratory for Strength and Vibration of Mechanical Structures and
School of
Aerospace, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
Xi’an Institute of Space Radio Technology, Xi’an 710100, People’s Republic of
China

###### Abstract

This paper presents an automatic finite element simulation scheme accounting
for high geometric nonlinearity and the difference between linear and
nonlinear buckling of composite thin-walled lenticular tubes (CTLTs).
Parameterizing of cross-section shapes and generation of design space for
CTLTs with both circular and parabolic arcs were accomplished, and several key
factors were identified, in particular the contrary effect of lumbus length
and parabolic coefficient on the bending stiffness anisotropy. The first
quantitative comparison of triangular rollable and collapsible (TRAC) booms
and CTLTs is given in terms of bending performance in two directions, showing
that the optimal CTLT carefully selected from the design space demonstrates a
comparable or even better performance than the TRAC boom. This is of great
importance from both academic and engineering perspectives. Our efforts
enhance the understanding of nonlinear buckling and post-buckling behavior of
CTLTs, and provide guidelines for future design of CTLTs with desirable
performance.

###### keywords:

Nonlinear buckling; Composite thin-walled lenticular tubes (CTLTs); Finite
element method; Pure bending.

††journal: International Journal of Mechanical Sciences

##  1 Introduction

The composite thin-walled lenticular tube (CTLT), which is also known as the
collapsible tube mast (CTM), is a closed tube with lenticular cross-section
that can be commonly used for deploying large structures in space. The CTLT is
flattened and coiled for storage before and during launch, and capable of
deploying spontaneously for use once in orbit. This type of structure, since
was first introduced by NASA a few decades ago [1], has been developed and
employed as a basic component to deploy various large space structures, such
as solar arrays and antennas in many space missions. The first CTM ever used
in space was jointly developed by NASA and ESA for the ULYSSES mission [2].
The development of CTLT is also considered as a key technique of German
Aerospace Center DLR’s solar sailing technology [3, 4, 5, 6], and in 2009 an
agreement was made between ESA and DLR by which they started a three-step
project aiming to develop, prove, and demonstrate that CTLT can serve as a
safe and reliable component for long-lasting and deep space missions [7, 8].
NASA has also recently expressed an interest in small CTLTs as a candidate
solar sail boom for low-cost deep space exploration and science missions [9,
10, 11, 12, 13, 14]. As an example of commercial application, Oxford Space
Systems Ltd (OSS) in UK, is currently developing a 2.7-m-diameter wrapped-rib
antenna where 48 CTLTs are employed as wrapping ribs to deploy the metal mesh
reflector surface [15, 16, 17].

The CTLT may be subjected to various kinds of mechanical loads during
different stages of its working process such as flattening, coiling and
deploying, and as a slender thin-walled structure it could demonstrate a
complex and nonlinear behavior. A large number of fundamental research has
been carried out to study the flattening and wrapping process of CTLTs. Hu et
al. [18] performed a combined experimental, numerical, and analytical
investigation of the mechanical response of both compressive and tensile
flattening deformations of deployable CTLTs. Chen et al. [19] carried out some
experiments to test the large deformation behaviors of CTLTs in flattening and
wrapping process and developed three-dimensional finite element models to
predict the mechanical characteristics identified by experiments. Similar
studies were also reported by Bai et al. [20, 21, 22] for determining tensile,
compression and folding behaviors of CTLTs. The above-mentioned efforts have
been focused on the deformation and the associated failure during flattening
and folding, and localized buckling that occurs during flattening and folding
was observed as the dominant failure mode of deployable CTLTs [23, 24].
However, little attention was paid to study the instability or buckling
behavior of CTLTs. The only relevant work that we are aware of focused on the
linear buckling and nonlinear post-buckling response of the CTLT subjected to
uniaxial compression [25]. It was found that the critical buckling load
estimated directly by eigenvalue analysis is far greater than the realistic
experimental measurement. A closer critical buckling load was eventually
obtained by introducing proper initial imperfections and performing a post-
buckling analysis. However, the initial imperfections introduced in the post-
buckling analysis were determined from the experimental observations and this
may reduce the predictive significance of the model [26]. Therefore, reliable
simulation techniques need to be developed to accurately predict the critical
buckling load of CTLTs prior to conducting experiments [27].

A more accurate prediction of the critical buckling load can be obtained by
first performing a geometrically nonlinear response analysis and then
estimating the buckling load by a following eigenvalue analysis on the
deformed configuration. This method is referred to as the nonlinear buckling
analysis, and it has been well-developed and documented, and successfully
applied in the field of mechanical engineering in the past few decades. A
number of analytical solutions can be found for the nonlinear buckling
predictions of simple structures such as beam, truss, and shell problems with
ideal boundary conditions. Brendel et al. [28] formulated the eigenvalue
problem with the information at deformed states on the nonlinear pre-buckling
path and studied the effect of imperfections on the critial loads of several
cylindrical shells under uniform pressure and wind load. Kounadis et al. [29]
proposed a simplified approach for the nonlinear buckling analysis of frames
subjected to either bifurcational or limit-point instability. Wu et al. [30,
31] derived the design derivatives of the nonlinear critical load and proposed
an optimization procedure to maximize the load-carrying capacity of truss-
structures. For complex systems, analytical solutions are rarely obtainable
and numerical modelling is often required. Lindgaard et al. [32, 33, 34, 35]
developed a combination of numerical approach and optimization procedure to
maximize the critical nonlinear buckling load of laminated composite shell
structures and provided validation with several benchmark problems. Nguyen et
al. [36] formulated a finite element model using three-noded Timoshenko beam
elements to analyze the nonlinear buckling load of an inflatable beam made of
orthotropic technical textiles. Liang et al. [37, 38] proposed a reduced order
model that capable of predicting the nonlinear buckling behavior of variable
stiffness composite plates efficiently compared with typical full order finite
element model.

Structural instability arises as one of the biggest concerns for structural
design of thin-wall composite boom structures such as CTLTs in aerospace
engineering, because these structures are usually thin in thickness and have
long aspect ratio. TRAC (triangular rollable and collapsible) booms and CTLTs
have been considered promising (competitive) candidates for solutions of
various light-weight deployable structures [39, 40, 41]. Very recently a few
studies have been carried out to understand the nonlinear buckling behavior of
TRAC booms subjected to pure bending. Murphey et al. [42] analyzed the basic
structural mechanics including deployment stiffness, buckling strength, and
packaging constraints of TRAC booms using closed form analytical and finite
element approaches. Leclerc et al. [43] performed a study of the nonlinear
elastic buckling behavior of triangular rollable and collapsible (TRAC) booms
under pure bending and reported a good agreement between numerical predictions
and experimental measurements. Bessa et al. [44] constructed the design space
diagram for the TRAC booms and proposed a data-driven computational framework
combining Bayesian regression for optimizing the nonlinear critical buckling
load of TRAC booms. Despite the available study on instability of TRAC booms
and CTLTs, either numerical or experimental, reliable prediction of critical
loads with emphasis on variation of configuration and initial stress field in
the context of nonlinear buckling analysis remains elusive. A recent study on
TRAC booms reveals the subtle difference between classical linear buckling
analysis and nonlinear buckling, and the linear buckling may give biased
estimation on critical loads [45, 46]. But this has never been carried out for
CTLTs. Moreover, due to the unique cross-section shape of CTLT, it exhibits
strong bending stiffness anisotropy [47, 48, 49, 50], and thus a rational
design of CTLT is only achievable provided an accurate design space diagram is
available, and a trade-off is made by compromising bending performance in
different directions. Another benefit of generating and exploiting through a
design space is the possibility of unveiling some key design parameters which
might be missed otherwise. Last but not least, the obtained design space and
exploration permits parameter tailoring for optimal designs, and more
importantly, allows a possible quantitative comparison of the two
counterparts, TRAC booms and CTLTs.

In this paper, we develop a finite element method based numerical scheme that
predicts the nonlinear buckling and post-buckling behavior of CTLTs under pure
bending. We first study the mechanical response of a particular CTLT bent
along two perpendicular directions, and demonstrate the validity of the
developed analysis methods. A good agreement was obtained between the
predictions of critical buckling load extracted by the nonlinear buckling
analysis and that identified from the post-buckling response. We then perform
a systematic numerical study to construct the design space against nonlinear
buckling for a variety of CTLTs with equal weight but different cross-section
shapes. The cross-section shape is shown to be a convenient parameter for
controlling the pre-buckling stiffness and the nonlinear critical buckling
load of CTLTs. More specifically, we start by investigating the effect of the
size of web and lumbus on the bending resistance performance of conventional
CTLTs with circular arcs. We then introduce the concept of parabolic CTLTs by
replacing the conventional circular arcs with parabolas, and show how it is
possible, by just tuning the parabolic shape, to alter the range of attainable
mechanical responses and leverage the trade-off between the bending resistance
performance in the two directions. Finally, we made a comparison of the
bending performance between conventional circular CTLTs, parabolic CTLTs and
TRAC booms with equal weight. It is shown that the optimal CTLT carefully
selected from the design space demonstrates a comparable or even better
performance than TRAC boom loaded concurrently in two loading directions.

The paper is organized as follows. Section 2 describes the geometry design of
the traditional CTLTs with circular arcs. Section 3 introduces the numerical
analysis techniques that are used to investigate the nonlinear buckling and
post-buckling response of CTLTs. Section 4 presents the results obtained from
parametric studies, highlighting the effect of cross-section shape on the
nonlinear mechanical response of CTLTs. Finally, concluding remarks are
included in Section 5.

##  2 CTLT Geometry

Fig. 1 shows a schematic of the conventional CTLT with circular arcs. This
type of structure consists of joining two thin omega-shaped cylindrical shells
of thickness $t$. The cross-section of each omega-shaped shell consists of
four circular arcs of radius $R$ subtending an angle $60^{\circ}$, two
straight segments at the ends of width $w$ constituting the web, and one
straight segment called lumbus of length $L$ at center [51]. The two shells
are mirror-symmetric and the end flat parts are bonded together in the
unstressed configuration. The geometry of the CTLT is then fully characterized
by the longitudinal length $l$, thin-shell thickness $t$, and the cross-
section parameters: web width $w$, lumbus length $L$, and circular arc radius
$R$. In order to make a fair comparison between the performance of CTLTs with
different cross-section geometries and TRAC booms, the longitudinal length
$l$, thin-shell thickness $t$, and the flattening length of the cross-section
$s$ are set the same as those of TRAC booms reported in literature [43, 44],
i.e., $l=504$ mm, $t=0.071$ mm and $s=27.43$ mm. Finally, the flattening
length of the cross-section $s$, which can be expressed as
$s=2w+L+\frac{4}{3}{\pi}R$, is fixed constant, and therefore the mass of the
structure remains unchanged. The web width $w$ and the lumbus length $L$ are
selected as two independent variables that are used to adjust the cross-
section geometry of circular CTLTs.

Figure 1: Schematic of the CTLT with circular arcs. (a) A CTLT in the
deployed, i.e., unstressed configuration. (b) Cross-section of the CTLT with
geometric parameters indicated. Web width $w$ and lumbus length $L$ are
considered as the two independent variables, and circular arc radius $R$ is
found from the flattening length equation.

##  3 Finite element analyses

In this section, we will detail the finite element analyses that are used to
investigate the nonlinear buckling and post-buckling behavior of CTLTs
subjected to pure bending. First, we will introduce the finite element model
with the definition of boundary conditions and material properties (see
Section 3.1). Then, we will focus on the analyses to predict the critical
loads for the onset of buckling and the associated buckling modes (see Section
3.2). Finally, we will discuss the post-buckling response analysis (see
Section 3.3).

###  3.1 Finite element model

The finite element model is constructed utilizing the commercial software
Abaqus 2020 with python scripts allows for automated parametric studies. Four-
node general-purpose shell elements with reduced integration (Abaqus element
type S4R) were used and the accuracy of the mesh was ascertained through a
mesh refinement study, resulting in a relative mesh density of around 50
elements along the cross-section profile.

The nodes forming the two end cross-sections are kinematically coupled to two
reference points, in effect creating rigid cross-sections that match the end
conditions of the experimental setup [43]. All six degrees of freedom of the
reference point at end 1 are constrained, defining a clamped condition. At the
other end, a pure moment load is applied to the reference point 2. The
location of the two reference points is shown in Fig. 1(a). In the case of a
moment applied about $x-$axis $M_{X}$, the translational degrees of freedom
along $Y$ and $Z$ as well as the rotational degree of freedom around $X$ are
left free, while the other three degrees of freedom are all fixed, i.e., $u1=$
0, $u2=$ FREE, $u3=$ FREE, $ur1=$ FREE, $ur2=$ 0, $ur3=$ 0\. In the case of a
moment applied about $y-$axis $M_{Y}$, the same boundary conditions are used
with $X$ and $Y$ inverted.

In accordance with previously reported data for TRAC booms [43, 44], each CTLT
omega shell consists of four layers of unidirectional carbon fibers in an
epoxy resin and the layers are arranged in the stacking sequence
$[0^{\circ},90^{\circ}]_{S}$. The nominal orthotropic elastic material
properties of each layer are set as $E_{1}=128.0$ GPa, $E_{2}=6.5$ GPa,
$\nu_{12}=0.35$, $G_{12}=G_{13}=G_{23}=7.5$ GPa. In addition, a composite
shell section is defined to specify the material properties, thickness, and
orientation angle of each layer.

###  3.2 Instability analysis

In a nonlinear finite element formulation, the response of a structure is
calculated by iteratively solving the equation [52, 53, 54]:

$K_{T}\delta u=R$ (1)

where $K_{T}$ is the current tangent stiffness matrix where the loads are
applied, $\delta u$ is the incremental displacement vector, and $R$ is the
force residual for the iteration. For large deflection problems, the tangent
stiffness matrix $K_{T}$ consists of the sum of the initial stiffness matrix
$K_{0}$, the displacement stiffness matrix $K_{L}$, and the stress stiffness
matrix $K_{\sigma}$ as described by:

$K_{T}=K_{0}+K_{L}+K_{\sigma}$ (2)

The structure is stable only if the tangent stiffness matrix $K_{T}$ is
positive definite; in other words, the structure becomes unstable and buckling
is likely to occur when the tangent stiffness $K_{T}$ is singular. Therefore,
the onset of buckling can be predicted by looking for the loads for which the
model tangent stiffness matrix becomes singular, and so that the problem

$K_{T}\delta u=0$ (3)

has non-trivial incremental displacement vector $\delta u$ solutions.

####  3.2.1 Linear buckling analysis

In the linear buckling analysis, the pre-buckling displacement is assumed to
be small thus the displacement stiffness matrix $K_{L}=0$, and the tangent
stiffness matrix is approximated using only the initial stiffness matrix
$K_{0}$ and the stress stiffness matrix $K_{\sigma}$. By assuming the stress
stiffness matrix to be linearly proportional to an incremental load $Q$, Eq. 3
is reduced to an eigenvalue problem:

$(K_{0}+\lambda_{i}K_{\sigma})\phi_{i}=0$ (4)

where $K_{0}$ is the initial global stiffness matrix defined at the undeformed
configuration, $K_{\sigma}$ is the initial stress stiffness matrix caused by
the incremental load $Q$, $\lambda_{i}$ are the eigenvalues, i.e., critical
buckling load factors, and $\phi_{i}$ are the eigenvectors, i.e., buckling
modes. The eigenvalues and eigenvectors are ordered increasingly in magnitude,
such that $\lambda_{1}Q$ is the lowest linear critical buckling load and
$\phi_{1}$ is the corresponding linear buckling mode. This eigenvalue problem
is readily solved using Lanczos method by an eigenvalue buckling analysis step
(*Buckle) in Abaqus.

####  3.2.2 Nonlinear buckling analysis

In the nonlinear buckling analysis, the effect of the change in configuration
in the pre-buckling stage is considered and the full tangent stiffness matrix
from Eq. 2 must be utilized [55]. The nonlinear buckling analysis generally
consists of two analysis steps: (_i_) a nonlinear static analysis step
(*Static) and (_ii_) an eigenvalue buckling analysis step (*Buckle). In the
first step, a preload (”dead” load), $P$, is applied to the structure to
attain a deformed state, during which the large-displacement formulation is
used (NLGEOM=ON) to capture the geometric nonlinearity. Then the deformed
state of the model at the end of the first step is identified as the base
state for the second eigenvalue buckling step. Namely, the nonlinear critical
loads and buckling modes are calculated by applying a seperate incremental
load $Q$ to the deformed state of the structure with change of configurations
and attained stress-field incorporated. The eigenvalue problem is a bit
complicated as follows:

$(K_{0}+K_{L}+\lambda_{i}K_{\sigma})\phi_{i}=0$ (5)

where $K_{0}$ is the initial global stiffness matrix defined at the undeformed
configuration, $K_{L}$ is the global displacement stiffness matrix determined
at the deformed state by applying the preload $P$, and $K_{\sigma}$ is the
initial stress stiffness matrix caused by the incremental load $Q$.

While large deformation is included in the static analysis, the eigenvalue
buckling theory relies on there being little geometric change due to the
”live” buckling load, $\lambda_{i}Q$. Then the final nonlinear critical loads
predicted by the two-step nonlinear buckling analysis are given by
$P+\lambda_{i}Q$ and $\phi_{i}$ denote the corresponding nonlinear buckling
modes. Notice that the amount of preload would also have an effect on the
nonlinear buckling predictions of structures. To capture the nonlinear
buckling behavior of the thin-walled composite structures which are known for
their high sensitivity to geometric imperfections and having many buckling
modes with closely spaced eigenvalues, it often helps to apply enough preload
to deform the structure to just below the buckling load prior to performing
the eigenvalue extraction [56]. On the other hand, the structure should not be
preloaded above the buckling load; otherwise the adopted Lanczos algorithm in
Abaqus will issue an error message and terminate the analysis.

In the nonlinear buckling analysis, with the aim of finding the appropriate
preload which should be as large as possible while allowing the extraction of
eigenvalues, the following tricks could be helpful: (_i_) applying a
relatively large preload (greater than nonlinear buckling load) in the
nonlinear static analysis step, and thus this step is expected to be conducted
up to the predetermined load or fail to converge due to buckling; and (_ii_)
performing an eigenvalue analysis in the deformed state starting from the
gradually decreasing last available increment until the eigenvalues can be
properly extracted. The above procedure, illustrated as pseudocode in
Algorithm 1, is implemented into a Python script in Abaqus to run the
simulations automatically. In real practice, the preload in the nonlinear
static analysis is set equal to the linear critical buckling load obtained by
conducting a linear buckling analysis, because the linear buckling analysis
never underestimates the critical load of the CTLT as will be demonstrated in
the following sections.

Algorithm 1 Pseudocode for the nonlinear buckling analysis

Step 1. Run a nonlinear static analysis with a preload being applied and write
the model definition and deformed state at every increment to the files
required for restart.

Step 2. Perform an eigenvalue analysis from the last available increment until
the eigenvalues can be successfully extracted.

while True: do

Perform the eigenvalue analysis

if Eigenvalue analysis completes successfully then

break

end if

n = n - 1 (n is the maximum number of available increments obtained in Step 1)

end while

###  3.3 Post-buckling analysis

The nonlinear post-buckling response of the CTLT is investigated by
introducing an imperfection in the form of the two most critical buckling
modes, $v^{M}_{1}$ and $v^{M}_{2}$, obtained from the nonlinear buckling
analysis. Therefore, the mesh is perturbed by $v^{M}_{1}$ and $v^{M}_{2}$
scaled by a factor $\eta$, such that

${\delta}v^{M}=\eta(v^{M}_{1}+v^{M}_{2})$ (6)

where $\eta$ is chosen as 5% of thickness of the CTLT, i.e., $\eta=0.05t$.

##  4 Results and discussions

In this study numerical simulations are performed to explore the nonlinear
buckling behavior space of CTLTs characterized by different cross-section
geometries. First, the linear and nonlinear buckling and post-buckling
behavior of a particular CTLT is investigated (see Section 4.1). Then, the
behavior space of circular CTLTs is explored, highlighting the effect of the
size of web and lumbus on the pre-buckling stiffness and nonlinear critical
buckling load of conventional CTLTs (see Section 4.2). Next, the concept of
parabolic CTLTs is introduced and the design space of which is also probed,
demonstrating a wider tunability range of the mechanical response (see Section
4.3). Finally, the bending resistance performance of both circular and
parabolic CTLTs is compared to that of TRAC booms with equal weight, and a
discussion is presented (see Section 4.4).

###  4.1 Nonlinear buckling behavior of a particular circular CTLT

A beam develops compressive stresses on its inner surface when it is subjected
to a pure bending moment. For the CTLTs, the bending moment about $x-$axis
produces in-plane compressive stresses on the inner lumbus, while the bending
moment about $y-$axis produces in-plane compressive stresses on the inner web.
The compressive stress is the driving force for most buckling phenomena.
Moreover, the lumbus and web, as being parts of the CTLT, are thin-walled
structures and have a low bending stiffness. Therefore, the in-plane
compressive stress may lead to out-of-plane buckling within the lumbus or web
when the critical value is reached.

To demonstrate the typical response characteristics of the CTLTs under pure
bending, without loss of generality, in Fig. 2 we present the analysis results
of a particular CTLT characterized by $w=5.5$ mm and $L=3.5$ mm. Fig. 2a and
2b show the mechanical response of the CTLT when the bending moment is applied
about $x-$axis. We start by determining the critical loads for buckling of
this CTLT, only the lowest critical load is of interest here. The critical
load is determined from different perspectives. First, we estimate the
critical load by solving the eigenvalue problem through performing a liner
buckling analysis or a nonlinear buckling analysis as discussed previously.
The results are presented by two dashed lines perpendicular to the moment
axis. In Fig. 2a the red dashed line indicates the critical load obtained from
the linear buckling analysis, which we refer to as _linear critical load
(LCL)_ , and the blue dashed line indicates that obtained from the nonlinear
buckling analysis, which we refer to as _nonlinear critical load (NLCL)_.
Moreover, the critical buckling load can also be identified from the moment-
angle curve by performing a post-buckling analysis. For the post-buckling
analysis, the imperfections afore-mentioned should be introduced. It can be
seen from Fig. 2a that there exist two distinct regimes in the moment-angle
curve (as shown in blue solid line), i.e., a pre-buckling regime and a post-
buckling regime. The transition point on the moment-angle curve from pre-
buckling regime to post-buckling regime can be identified as the critical load
for buckling. From these results shown in Fig. 2a, we conclude that (_i_)
linear buckling analysis would overestimate the overall stability of the
structure in this case, i.e., $LCT>NLCT$, and (_ii_) the nonlinear critical
load ($NLCL$) estimated by the nonlinear buckling analysis agrees very well
with that identified from the post-buckling response. Furthermore, the
difference of linear and nonlinear buckling analysis is also manifested in
terms of buckling modes. As shown in Fig. 2b, linear buckling analysis
predicts a larger number of winkles within the lumbus than nonlinear buckling
analysis in this case. It should be pointed out the difference between linear
and nonlinear buckling highlighted herein is firstly pointed out by Leclerc et
al. [45]in the analysis of TRAC boom. The difference is elucidated once again
for CTLT bent about $x-$axis.

In sharp contrast, a different buckling behavior is observed when the pure
bending moment is loaded about $y-$axis. It can be seen in Fig. 2c that for
this case the critical loads predicted by linear and nonlinear buckling
analysis are basically the same, i.e., $LCL=NLCL$, and both are in good
agreement with that identified from the post-buckling moment-angle curve. This
difference is attributed to the different boundary conditions of the web and
lumbus in compression. More specifically, when the bending moment is applied
about $x-$axis the lumbus undergoing compressive stresses is constrained on
both sides by surrounding structures; when the bending moment is applied about
$y-$axis, only one side of the web in compression is constrained while the
inner edge remains free, which reduces the sensitivity of buckling load to
geometric nonlinearity. On the other hand, the post-buckling response under
the two loading conditions exhibits similar features, as they both contain two
distinct stages divided by $NLCL$, namely, a pre-buckling stage starting from
initiation to $NLCL$, and a post-buckling stage starting from $NLCL$ and
ending with buckling collapse. We also show, in Fig. 2a and 2c, that the pre-
buckling regime predicted by the post-buckling analysis is observed coincided
with the curve obtained by performing a nonlinear static analysis without
introducing any imperfections, which exhibits approximately a linear behavior.
Then the pre-buckling stiffness of CTLTs can be calculated as

$K=\frac{dM}{d\theta}$ (7)

where the derivative can be directly obtained through linear regression of the
data in pre-buckling regime.

Figure 2: Nonlinear buckling and post-buckling response of a particular CTLT
characterized by $w=5.5$ mm and $L=3.5$ mm when subjected to a pure bending
moment about (a-b) $x-$ axis and (c-d) $y-$axis.

###  4.2 Construction of design space for circular CTLTs

Having identified typical mechanical behavior of CTLTs under pure bending, we
then move on to construct the design space, aiming to probe the evolution of
pre-buckling stiffness $K$ and nonlinear critical buckling load $NLCL$. The
design space is constructed by varying the two independent cross-section
geometric parameters, which are the web width $w$ and lumbus length $L$ as
shown in Fig. 3a. More specifically, $w$ is varied from 1 to 10 mm and $L$ is
varied from 0 to 7 mm, and a number of diverse cross-section shapes are
determined as shown in Fig. 3b. Note that $L=0$ indicates a CTLT design
without lumbus. Fig. 3c and 3d present the evolution of the pre-buckling
stiffness $K_{x}$ and the nonlinear critical load $(NLCL)_{x}$ about $x-$axis
as a function of the web width $w$ and the lumbus length $L$. It is shown that
the pre-buckling stiffness $K_{x}$ strongly depends on the web width $w$ but
slightly on the lumbus length $L$, while the nonlinear critical load
$(NLCL)_{x}$ depends both highly on the web width $w$ and the lumbus length
$L$. Specifically, $K_{x}$ increases as $w$ decreases but tends to remain
unchanged as $L$ varies, while $(NLCL)_{x}$ increases as $w$ and/or $L$
decreases. Furthermore, the evolution of the pre-buckling stiffness $K_{y}$
and nonlinear critical load $(NLCL)_{y}$ about $y-$axis is presented in Fig.
3e and 3f. It is shown that the pre-buckling stiffness $K_{y}$ depends both
highly on the web width $w$ and the lumbus length $L$, while the nonlinear
critical load $(NLCL)_{y}$ depends solely on the web width $w$. Specifically,
$K_{y}$ increases with the increase of $w$ or $L$, while $(NLCL)_{y}$
increases with the decrease of $w$. In conclusion, there exists a common
region of the design space that maximize the pre-buckling stiffness about
$x-$axis and nonlinear critical buckling load about both $x$ and $y-$axes at
small web width, although a decrease in the pre-buckling stiffness about
$y-$axis is expected. Given the fact that the minimum pre-buckling stiffness
about $y-$axis ($\sim$20 Nm/rad) is yet much greater than the maximum pre-
buckling stiffness about $x-$axis ($\sim$6 Nm/rad), the smallest value of 1 mm
is determined to be the optimal size for the web.

Figure 3: The design space for nonlinear buckling of CTLTs with circular arcs.
(a) Schematic illustration of the cross-section geometry of the circular CTLT.
(b) A variety of cross-section shapes of circular CTLTs characterized by web
width $w\in[1,10]$ mm and lumbus length $L\in[0,7]$ mm. Heat map illustrating
the (c) pre-buckling stiffness $K_{x}$ and (d) nonlinear critical load
$(NLCL)_{x}$ as a function of the web width $w$ and lumbus length $L$ when the
moment is applied about $x-$axis. Heat map illustrating the (e) pre-buckling
stiffness $K_{y}$ (f) and nonlinear critical load $(NLCL)_{y}$ as a function
of the web width $w$ and lumbus length $L$ when the moment is applied about
$y-$axis.

Next, we take a closer look at the effect of the lumbus length $L$ on the
nonlinear buckling behavior of CTLTs by considering a group of samples with
$w$ fixed to be 1 mm and $L$ varies from 0 to 7 mm. Fig. 4a describes a trade-
off between the bending resistance performance about $x-$axis and that about
$y-$axis. Specifically, an increase in the lumbus length $L$ leads to a better
bending resistance performance about $x-$axis but in the meantime leads to a
worse bending resistance performance about $y-$axis. The effect of the lumbus
length is also reflected in terms of the nonlinear buckling modes as shown in
Fig. 4b. Firstly, for the cases the bending is applied about $x-$axis, two
distinct nonlinear buckling patterns were observed according to whether the
CTLT has lumbus or not. For the CTLT without lumbus, i.e., $L=0$, the buckling
mode is a diamond wave pattern, as often found in the buckling of thin-walled
cylindrical shells [57, 58, 59]. In contrast, for the CTLT with lumbus, i.e.,
$L>0$, the buckling mode is recognized as ”wrinkles” in the lumbus, and the
number of the wrinkles increases as the lumbus length $L$ increases from 1 to
7 mm. Secondly, for the cases bending is applied about $y-$axis, the buckling
mode is a wave pattern occurs in the compressed web, and the number of waves
also increases as the lumbus length increases from 0 to 7 mm. Finally,
considering the trade-off between the bending resistance performance about
$x-$ and $y-$axes, a value of 3 mm, which is in the middle of the range for
the lumbus length $L$, is selected as the optimal one. Note that for the
particular cross-sectional shape of CTLT of interest here, the lumbus length
has profound effect on the parameter selection and sectional shape design.
This effect, however, is omitted in current literature, and it is elucidated
for the first time with the aid of constructed design space diagram.

Figure 4: Effect of the lumbus length $L$ on (a) the post-buckling response
and (b) nonlinear buckling modes of CTLTs with $w=1$ mm when subjected to a
pure bending moment about $x-$axis (left) and $y-$axis (right).

###  4.3 Design space for parabolic CTLTs

So far, we have explored the nonlinear buckling behavior space of conventional
CTLTs with circular arcs, and reported the effect of web and lumbus sizes on
the bending performance of circular CTLTs. However, the constant curvature in
circular arcs may limit the potential use of conventional CTLTs. A new design
of CTLT with variable curvature arcs is achieved by replacing the circular
arcs with parabolic arcs, which is referred to as the parabolic CTLT [47]. The
cross-section geometry of the parabolic CTLT is illustrated in Fig. 5a. In
this section, we follow our previously developed numerical strategy for
parameterizing sectional shape and generating design space for circular CTLTs
and extend them to parabolic CTLTs. As shown in Fig. 5a, each parabolic arc
segment is defined by a parabola function:

$y=ax^{2}\;\;\;\;0\leq{x}\leq{x_{0}}$ (8)

where $a$ is the coefficient of the parabola. The arc length of a single
parabolic segment, $L_{P}$, is calculated using the following formula:

$L_{P}=\int_{0}^{x_{0}}\sqrt{(1+4a^{2}x^{2})}dx$ (9)

and then the total flattening length of the cross-section of the parabolic
CTLT is expressed as $s=2w+L+4L_{P}$. Recall that $s$ is set to 27.43 mm as
the same as its circular counterpart, and in this case the web width $w$ is
fixed to 1 mm. The parabolic CTLT thus has two parameters that can be varied
independently, i.e., the lumbus length $L$ and the coefficient of parabola
function $a$. Fig. 5b presents various cross-section geometries obtained by
varying the lumbus length $L$ from 0 to 7 mm and varying the coefficient $a$
from $10^{-2}$ to $10^{0}$. The coefficient $a$ determines how wide or narrow
the parabola is; the greater the coefficient $a$, the narrower the parabola.
Therefore, as shown in Fig. 5b, the cross-section becomes narrower in $x-$axis
and wider in $y-$axis as the coefficient $a$ increases for a given value of
lumbus length $L$. In theory, this will cause a decrease in the stiffness
about $x-$axis and in the meantime an increase in the stiffness about $y-$axis
[47, 48]. Heat maps in Fig. 5c and 5d present the effect of the coefficient
$a$ and the lumbus length $L$ on the bending performance of parabolic CTLTs
about $x-$axis. The pre-buckling stiffness $K_{x}$ and the nonlinear critical
load $(NLCL)_{x}$ both depend significantly on the coefficient $a$ and the
lumbus length $L$. Specifically, a greater coefficient $a$ and/or a smaller
lumbus length $L$ leads to an increase in both the pre-buckling stiffness
$K_{x}$ and the nonlinear critical load $(NLCL)_{x}$. Fig. 5e shows the
evolution of the pre-buckling stiffness about $y-$axis $K_{y}$ as a function
of the parabolic coefficient $a$ and the lumbus length $L$. In contrast with
the performance about $x-$axis, a greater coefficient $a$ and/or a smaller
lumbus length $L$ leads to a decrease in the pre-buckling stiffness about
$y-$axis $K_{y}$. In addition, as shown in Fig. 5f the parabolic coefficient
$a$ has greater effect on the nonlinear critical load $(NLCL)_{y}$ than the
lumbus length $L$, and indicatively, as $a$ increases from $10^{-2}$ to
$10^{0}$, $(NLCL)_{y}$ increases first, reaches a peak at $a\simeq 10^{-1}$,
and then decreases.

We also investigated in detail the effect of the parabolic coefficient $a$ on
the post-buckling response of parabolic CTLTs by considering a group of
samples characterized by $L=3$ mm and $a\in[10^{-2},10^{0}]$. An possible
compromise is made once again between the bending resistance performance about
$x-$axis and that about $y-$axis. It can be seen from Fig. 6 that a greater
value of parabolic coefficient $a$ leads to a better bending performance about
$x-$axis but a worse bending performance about $y-$axis. In addition, CTLTs
with circular arcs (black dash lines in Fig. 6) show a moderate performance in
both cases. These results indicate that a further tunability of the bending
performance of CTLTs can be achieved by altering the parabolic coefficient
$a$, and it could be useful in real-world engineering applications when
anisotropic stiffness property, e.g., a higher stiffness in one direction and
a lower stiffness in the other direction, is desirable.

Figure 5: The design space for nonlinear buckling of parabolic CTLTs. (a)
Schematic illustration of the cross-section geometry of the parabolic CTLT.
(b) A variety of cross-section shapes of parabolic CTLTs characterized by the
parabolic coefficient $a\in[10^{-2},10^{0}]$ and the lumbus length $L\in[0,7]$
mm. Heat map illustrating the (c) pre-buckling stiffness $K_{x}$ and (d)
nonlinear critical load $(NLCL)_{x}$ as a function of the parabolic
coefficient $a$ and the lumbus length $L$ when the moment is applied about
$x-$axis. Heat map illustrating the (e) pre-buckling stiffness $K_{y}$ and (f)
nonlinear critical load $(NLCL)_{y}$ as a function of the parabolic
coefficient $a$ and the lumbus length $L$ when the moment is applied about
$y-$axis. Figure 6: Effect of the parabolic coefficient $a$ on the post-
buckling response of parabolic CTLTs characterized by $L=3$ mm when subjected
to a pure bending moment about (a) $x-$axis and (b) $y-$axis.

###  4.4 Comparison with TRAC booms

Having explored the design space of circular and parabolic CTLTs, in this
subsection, we will perform a comparative study on the bending resistance
performance of circular and parabolic CTLTs to that of TRAC booms of equal
weight.

With the design space in hand, the variations of bending performance of both
circular and parabolic CTLTs are extracted readily from the heat maps in Fig.
3 and Fig. 5. The same physical quantity is taken from the nonlinear buckling
of TRAC booms reported very recently in [44]. A comparison of variation of the
nonlinear critical loads between TRAC booms, circular CTLTs and parabolic
CTLTs is given in Table 1. In summary, circular CTLTs has a narrower range of
variation $(NLCL)_{x}$ but a significantly wider range of variation of
$(NLCL)_{y}$ than TRAC booms for the case considered herein. By introducing
parabolas, the maximum value of $(NLCL)_{x}$ of parabolic CTLTs is beyond that
of TRAC booms. The maximum value of $(NLCL)_{y}$ of parabolic CTLTs is
slightly lower than that of circular CTLTs, but still much greater than that
of TRAC booms.

Table 1: Comparison of variation of nonlinear critical buckling moment $(NLCL)$ between TRAC booms, circular CTLTs and parabolic CTLTs | $(NLCL)_{x}$ | $(NLCL)_{y}$
---|---|---
TRAC booms [44] |  $0-0.77$ Nm |  $0-0.31$ Nm
Circular CTLTs |  $0-0.6$ Nm |  $0-2.0$ Nm
Parabolic CTLTs |  $0-0.82$ Nm |  $0-1.5$ Nm

In addition, as reported in [44], in the design space of TRAC booms there is a
common region that maximizes the nonlinear critical buckling load for both
loading directions at large flange angles and small web height, and this gives
rise to an optimal TRAC boom design characterized by a web height of 2 mm and
a flange angle of 300∘. In Fig. 7 the moment-angle curves of the optimal TRAC
boom were reproduced (blue lines) and used as the reference to evaluate the
performance of CTLTs. Notice that the results of TRAC boom that reproduced by
our approach are in excellent agreement with that reported in [44], which
validate the accuracy of the proposed model. A circular CTLT characterized by
$w=1$ mm and $L=3$ mm and a parabolic CTLT characterized by $a=10^{0}$ and
$L=3$ mm are chosen as the optimal candidates from the design space for
comparison. As shown in Fig. 7, the circular CTLT demonstrates a worse
performance than the others in $x-$axis but better in $y-$axis, while the
parabolic CTLT shows a comparable performance in both $x$ and $y-$axes to the
TRAC boom. These results show that the optimal CTLT with tailored parameters
derived from the design space demonstrates a comparable or even better
performance than the TRAC boom in either the two loading directions.

Figure 7: Comparison of bending resistance performance around (a) $x-$ axis
and (b) $y-$axis between the optimal TRAC boom (blue), circular CTLT (black),
and parabolic CTLT (red).

##  5 Conclusion

TRAC booms and CTLTs have been proposed as two promising candidates for
solutions for deploying various light-weight space structures. Since the thin-
walled tubes or booms are thin in thickness and have long aspect ratio,
structural instability arises as one of the biggest concerns for structural
designs. Recent study on nonlinear buckling analysis of TRAC booms reveals
that classical linear buckling may biasedly estimate the critical load of TRAC
booms under pure bending and necessitates the classification of nonlinear and
linear buckling. However, the recurrence of this difference needs to be
justified in CTLTs. Moreover, a design space is desperately needed, with which
a thorough exploration and tailoring of design parameters is admissible.

In the present paper, we carried out a comprehensive computational
investigation aiming at exploring the design space for nonlinear buckling
behavior of CTLTs with both circular and parabolic arcs. Our results reveal
the effect of cross-section geometric parameters such as the size of web,
lumbus, and the parabolic coefficient, on the bending resistance performance
of CTLTs. More specifically, we found that there is a common region at small
web width that maximizes the nonlinear critical buckling load of CTLTs
concurrently about both $x$ and $y-$axes under pure bending, whereas there is
a trade-off between the two loading directions in terms of pre-buckling
stiffness, namely, an increase in the web width and/or a descrease in the
parabolic coefficient would result in a decrease in the pre-buckling stiffness
about $x-$axis but an increase in the pre-buckling stiffness about $y-$axis.
More importantly, the contrary effects of the lumbus length on bending
performance of CTLTs in two directions are also unveiled. This profound effect
of the lumbus, nevertheless, has been ignored in the literature to our best
knowledge [51]. Moreover, we made a quantitative comparison on the bending
performance of TRAC booms and CTLTs under the same weight. The optimal
circular CTLT demonstrates a better bending resistance performance than the
optimal TRAC boom in $y-$ axis but a worse performance in $x-$axis, while the
optimal parabolic CTLT demonstrates a comparable or better performance in both
loading directions than TRAC boom.

We believe our efforts provide guidelines for engineers and scientists
attempting to design CTLTs with desirable bending resistance performance. We
also note that the nonlinear buckling simulation scheme proposed in this work
is readily extended for analyzing and optimizing the bending performance of
other kinds of thin-walled composite deployable structures [44, 60, 61,
62].The paradigm of parameterizing cross-section shape and then constructing a
design space in an automatic way can also be developed for optimal design of
other thin-walled slender structures.

## Acknowledgement

This research is supported by National Natural Science Foundation of China
(grant 11972277). Q. J. also acknowledges the support from Shanghai Rising-
Star Program (19QB1404000).

## Data availability

The Abaqus scripts used for the numerical analyses that support the findings
of this study are available from the corresponding author upon reasonable
request.

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