Structures 61 (2024) 105963

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Structures
journal homepage: www.elsevier.com/locate/structures

Investigation of the mechanical performance of the sandwich cylindrical

shell with a truss core under eccentric loading
Mohammad J. Zarei a, Shahabeddin Hatami a, *, Mojtaba Gorji Azandariani b, c,
Mohammad Gholami a
a
Department of Civil Engineering, Yasouj University, Yasouj, Iran
b
Structural Engineering Division, Faculty of Civil Engineering, Semnan University, Semnan, Iran
c
Centre for Infrastructure Engineering, Western Sydney University, Penrith, Australia

A R T I C L E I N F O A B S T R A C T

Keywords: Sandwich structures, a class of composites, leverage the connection of diverse materials to achieve superior
Sandwich structure mechanical properties, such as elevated stiffness, a favorable strength-to-weight ratio, impressive energy ab­
Cylindrical shell sorption capability, and effective thermal insulation. This study delves into the mechanical response of a cy­
Truss core
lindrical sandwich shell with a truss core when subjected to eccentric compression loading through simulation.
Eccentric compression load
The investigation encompasses an analysis of failure modes, ultimate load-bearing capacities, and specimen
Debonding
Composite structure deformations. A comprehensive examination is undertaken to elucidate the impacts of adhesive stiffness and
Numerical method eccentricity load on critical failure loads. Consistency is observed in comparisons between finite element
Buckling mode simulation outcomes across distinct buckling modes and analytical critical values. Noteworthy influences on
Ultimate deformation buckling modes are identified, including element shape and mesh size. Local buckling emerges as the predom­
inant failure mode, attributed to a high diameter-to-thickness ratio (D/t). The study underscores the significant
enhancement in buckling load achieved through a robust adhesive bond. Moreover, the discernible effects of load
eccentricity on the ultimate load-bearing capacity of cylindrical sandwich shells are highlighted, providing
valuable insights for manufacturing engineers seeking to optimize structural performance in related applications.
Also, by considering the type of meshing and its effects on the critical load, which was rarely considered in
previous studies, this research enables achieving a more accurate modeling of the sandwich structure and
analyzing the results as best as possible in this field. In conclusion, this work not only contributes to the current
understanding of mechanical responses in sandwich structures but also sets the stage for future developments in
this evolving field.

1. Introduction Conceivable applications for sandwich structures in civil engineering

include covering flat or curved roofs, interior or exterior walls, columns,
Composite structures which possess a low-density core enclosed and members of space trusses (3D truss). Face sheets are usually made of
between two thin, stiff face sheets are known as sandwich structures, sheet metal or laminated composite. The cores of the sandwich structure
which are composed of two main components: the face sheets and the include truss core, foam, honeycomb, corrugated, hybrid, and folded
core construction. In most cases, the face layers are adhesively bonded to core [3–10]. In an experimental study [11], continuous flax fibers with
the core layer on either side [1,2]. The adhesive used to bond these two short polyester fibers/polylactic acid matrix sandwich composites are
entities must have enough strength to withstand the stresses established fabricated via vacuum-assisted material extrusion. Compared with ma­
between the face and the core. An I-beam inspires a sandwich structure’s terial extrusion, vacuum-assisted material extrusion yields higher
basic design principle. Given sandwich structures, the face sheets bending and short-beam shear strengths. The facing layers of sandwich
resemble the flanges of the I-beam, and the core represents the web of composites are moderately thinner than the core and rarely go beyond
the I-beam that connects both the flanges. The face sheets are resistant to several millimeters, whereas the core thickness is often over 50 mm.
transverse and in-plane loads, and the core withstands shear forces. However, exceptional dimensions can be found [12]. Sandwich

* Corresponding author.
E-mail addresses: hatami@yu.ac.ir (S. Hatami), mgorji@semnan.ac.ir, gorji1365@yahoo.com (M. Gorji Azandariani).

https://doi.org/10.1016/j.istruc.2024.105963
Received 16 November 2023; Received in revised form 13 January 2024; Accepted 23 January 2024
Available online 2 February 2024
2352-0124/© 2024 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
M.J. Zarei et al. Structures 61 (2024) 105963

structures primarily use aluminum alloy as their facings material [13]. modes of the structure to be Euler buckling, local buckling of the face
Other facing materials are fiber-reinforced polymer [14–16], and sheet, global buckling of the cylinder, core member buckling, and face
epoxy-carbon prepreg [17,18]. The core is constructed of honeycomb yielding. The primary cause of failure was identified as being local
[19–21], metal [22,23], and polymers [24,25]. The weakest point of the buckling in cylindrical shells, which is consistent with experimental
sandwich structure when loads are inserted place is the core/facing results. Alphonse et al. [40] reviewed the influence of the mechanical
bond. This interface typically uses adhesives to connect the core and the behavior of sandwich structures constructed of two metal face sheets
face sheets [12]. In an investigation by Cao et al. [26], fused deposition and a core material; the properties were examined by varying the loads
modeling printed polylactic acid was used as core material and sand­ on these structures. Cao et al. [41] designed and tested creative sand­
wiched between two unidirectional glass fiber reinforced polymer skins wich cylindrical shells with a pyramidal truss core. The axial compres­
to form a sandwich composite by compression-molding process, which sive testing results established the structures failed at the cylinder shells
provides a good manufacturing strategy for skin/core interphase yielding and obeyed the post-plastic local buckling. Wu et al. [42]
modification. Cao [27] also showed the interleaving of the heating explored the axial compression performance of cylindrical shells with a
element with multiwalled carbon nanotub sheets can help improve the low-density truss core. Through theoretical analysis, they identified four
bonding quality in the resistance welding of thermoplastic composites. different failure modes: Euler buckling, local buckling, face panel
The sandwich structure’s ability to execute multiple functions mainly crushing, and axisymmetry buckling of the sandwich cylinders. The
depends on the structure’s configuration and the sandwich materials mechanism map of the failure modes provided guidance on the creation
selected. Notable features of these structures include impressive energy of the cylinder specimens. Wang and Fu [43] analyzed the acoustic
absorption, superior ballistic resistance performance, high response of functionally graded material (FGM) sandwich cylindrical
stiffness-to-weight ratios, and excellent thermal and acoustic insulation. shell submerged in convected fluids. An evaluation was performed on
The beneficial aspects of sandwich structures apply to many engineering two varieties of FGM sandwich cylindrical shells, one with an FGM face
projects, such as aerospace industries [28], marine manufacturing [29, shell and a homogeneous ceramic or metal core, and the other with an
30], and integrated thermal protection systems [31,32]. If the geometry FGM core and a homogeneous face shell.
of the face sheet is cylindrical, the sandwich structure has a high resis­
tance under mechanical loads because of the high buckling load.
1.2. Necessity and novelty of research
Investigating the mechanical characteristics of the sandwich cylindrical
shell with a truss core can give us a better comprehension of performing
Composite structures, specifically sandwich cylinders with truss
this system, failure modes, and how to fixing any imperfections.
cores, have long been a focal point in engineering due to their impressive
mechanical properties. The intrinsic strength of these structures, stem­
1.1. Literature review
ming from the interplay between low-density cores and rigid face sheets,
has made them a cornerstone in various applications. While extensive
Fan et al. [33] manufactured a CFRC cylinder sandwich using
studies have probed their behavior under axial pressure, our research
Kagome truss cores and validated the mechanical behaviors of the cyl­
seeks to bridge a critical knowledge gap by delving into the response of
inder by the axial compression test. Analysis of the results shows sand­
these structures when subjected to eccentric compression loads. The
wich composite cylindrical shells are more robust and stiffer than
motivation behind this study arises from a notable gap in existing
composite cylindrical shells with a grid-stiffened configuration. Xiong
literature. While axial loading scenarios have been extensively explored,
et al. [34] conducted a combined experiment and finite element model
the eccentric compression behavior of sandwich cylinders with truss
analysis to examine the effects of core deboning on the failure modes of a
cores remains relatively uncharted territory. To investigate the ultimate
curved sandwich structure with a pyramidal truss core under bending
load carrying capacity and mode shapes of these structures when com­
load. The authors determined the maximum load capacity of the struc­
pressed eccentrically, as shown in Fig. 1, numerical research was con­
ture with varying thicknesses of the face sheet. The primary cause of
ducted by the finite element modeling with ABAQUS software.
failure in the experimental results was core deboning, which was in
Understanding how these structures perform under such conditions is
contrast to the simulation results. Xiong et al. [35] produced
imperative, especially considering potential real-world applications
sandwich-walled cylindrical shells with a pyramidal aluminum truss
where loads may not always act uniformly. Moreover, the eccentric
core. Axial compression tests were conducted to assess the failure modes
compression scenario introduces a unique set of challenges that differ
of the structures and were compared with an analytical failure map that
from conventional axial loading. Eccentricity induces asymmetrical
considered shell buckling, Euler buckling, face-crushing, and local
stress distribution, leading to potential points of weakness in the struc­
buckling between reinforcements. The results of the experiment
ture. Investigating the buckling resistance of sandwich cylinders in these
matched the analytically expected. The authors concluded that local
conditions becomes crucial for ensuring their structural integrity in
buckling and face crushing modes could both be present, which are the
practical applications.
primary sources of structural collapse.
Xiong et al. [36] studied composite cylindrical shells’ mechanical
behavior and failure modes with truss cores under axial compression
load. The analysis investigated four distinct failure modes and theoret­
ical models were constructed to gauge the failure strength of each failure
mode. Analytical estimates were used during the experimental part to
construct sandwich cylindrical shells, thus examining several failure
scenarios. The measured collapse loads were comparable to the calcu­
lated predictions. Li et al. [37] designed and manufactured a distinct
sandwich composite cylindrical shell with a truss core strengthened by
carbon fibers. Tests revealed the manner in which the cylinder was
subjected to uniaxial compression. Yang et al. [38] used an FE model to
investigate the effect of imperfections, positions, and shapes on the
modal parameters of the sandwich cylindrical panels through the
ABAQUS software. Wang et al. [39] examined the eigenvalue buckling
of the sandwich cylindrical shell with a truss core under axial Fig. 1. Schematic of the sandwich cylindrical truss core under eccentric
compression using the FEM analysis. The authors identified the failure compression load.

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M.J. Zarei et al. Structures 61 (2024) 105963

Our research introduces a novel dimension to the study of truss-core and magnesium, are increasingly regarded as alternatives to steel. In the
sandwich cylinders by scrutinizing their buckling resistance under current study, aluminum metal is considered the core and cylindrical
eccentric compressive loads. The innovation lies in the comprehensive shell material. The sandwich parts are merged to make a single part for
exploration of eccentric compression-induced failure modes and the linear buckling analysis. The mutual contact among parent parts is
critical load-bearing capacity of the sandwich cylinders. The present mainly defined as cohesive surface contact, simulated by ABAQUS/Ex­
study analyzes sandwich cylindrical shells’ buckling modes and critical plicit analysis. In the simulated model of the sandwich cylindrical shell,
failure load under eccentric compression. Then, providing some adhe­ the truss core unit cell (Fig. 2(a) and (b)) is constructed with the spec­
sive stiffness between the core and the cylindrical shells is investigated ifications given in Table 1.
to achieve the total strength of the structure close to the entire bond The lattice truss core comprises 60 × 16 two-layer truss cells to
state. This deliberate inclusion allows us to not only estimate the accommodate the cylinder shells (Fig. 3(a)), and a sandwich cylindrical
buckling load and mode shapes using Eigenvalue buckling predictions shell comprising two cylinder shells with radii of 200 mm and 250 mm,
but also scrutinize the influence of adhesive stiffness on the critical a length of 3000 mm, and a thickness of 0.5 mm, and the lattice truss
failure load. The research is structured to provide a holistic under­ core is showed in Fig. 3(b). The material properties are given in Table 2.
standing of how the interplay between eccentric compression, adhesive Fig. 4 shows that the sandwich structure is represented by limiting all
stiffness, and the inherent characteristics of truss-core sandwich cylin­ degrees of freedom of the reference points to the two endpoints,
ders collectively influences their mechanical behavior. By examining excluding the translational degrees of freedom on the upward axial line
these interdependencies, aim to offer insights that can guide structural and the rotational degrees of freedom at the moment planes (UR1) on
design and optimization, contributing to advancements in the field of both sides. Loading of the FE model is achieved through displacement
composite engineering. control, and a concentrated force is implemented on the RP-1, allowing
for the change to generate various eccentricities. The initial neutral
1.3. Objectives and aims of study surface in response to any combination of axial compression and
bending moment can be determined by referring to Fig. 4. The lower
This research focuses on advancing the comprehension of truss-core part will experience compression as the load process is ongoing, while
sandwich cylinders under eccentric compressive loads. The primary the upper part will experience tension. Stress in the upper region may
objectives include a comprehensive exploration of the structures’ shift from tension to compression, yet remain within the elastic stage.
response to non-uniform loading conditions, determining their ultimate Jin et al. [45] reported that a metal material’s tensile and compressive
load-carrying capacity, analyzing mode shapes during eccentric properties could be the same during an elastic stage.
compression, investigating the impact of adhesive stiffness on critical Multiple convergence studies that include linear buckling analysis
failure loads, employing advanced finite element method simulations for are performed to show the influence of inclined meshes on the efficiency
numerical research, and contributing to the understanding of failure of the buckling loads and mode shapes of the sandwich cylindrical
modes. A variety of numerical models have been produced by this structure under eccentric compression load with different eccentricities.
software to accommodate different structures and its capacity to account These curves show the subtle but necessary relationship between mesh
for nonlinear problems has been established. In order to fully under­ and stress field orientation, helpful for finite element analysts in struc­
stand the sandwich structure’s behavior under an eccentric compression tural engineering. Fig. 5 Illustrates two types of element shape meshes.
load, first, the buckling load and modes shape is estimated using This paper investigates sandwich cylindrical shells with merged parts
Eigenvalue buckling prediction. Then, the effect of the adhesive stiffness in linear buckling analysis with radii of 200 and 250 mm with a similar
between the core and cylindrical shells on the critical failure load was thickness of 0.5 and length equal to exactly 3000 mm, with an element
study and discussed using cohesive surface contact with a traction- size of 20 mm under eccentric compression load. The sandwich cylin­
separation law. This innovative approach is poised to contribute drical shell specimens (D) are described in Table 3.
significantly to the ongoing evolution of composite structures, particu­
larly in applications where eccentric compression loads are prevalent. 3. Results and discussion

2. Numerical method In this paper, will analyze the load combinations encountered in
practice. It is suggested to consider combine of two or more loadings that
The following section presents the simulation model using ABAQUS could lead to buckling, assuming that the sum of the various critical load
finite element software [44]. Because of the modeling of eccentric ratios is equal to one. It has been illustrated by both theoretical [46] and
compression load (combined in-plane compression and bending loads),
Shell elements (S3R and S4R) are used to model the cylinder shell.
Table 1
Typical geometries used in truss cores are Kagome, tetrahedrons, pyra­
Geometry parameters of unit truss core.
mids, and X-shaped. Nevertheless, the last few years have seen the
r l
development of several derivative designs, for example, the hourglass ω α

truss. The tow-node linear beam element, B31, is adopted to model the 0.3 mm 32 mm 55⸰ 55⸰
pyramidal truss core. Lightweight metals, such as aluminum, titanium,

Fig. 2. Pyramidal Truss core cell: (a) Truss core cell geometry, (b) simulated 2-layer pyramidal truss core cell.

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M.J. Zarei et al. Structures 61 (2024) 105963

and bending is:

Rc + Rb = 1 (2)

where the compressive and bending critical load is represented by the

quantities Rc and Rb , respectively:
Rc = P/Pcr (3)

Rb = M/Mcr (4)

where the compressive and bending loads denoted as P and M, respec­

tively. The permissible axial load (Pcr ) can be established from the
axial stress given in Eq. (1) and the permissible bending moment (Mcr )
can be identified directly from Eq. (6):
Fig. 3. Truss core sandwich cylindrical shells: (a) lattice truss core, (b) sand­
wich cylindrical shells. 2Eπt2
Pcr = 2πrtσ cr = √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ (5)
3(1 − ν2 )
Table 2
Material properties of AlSi10Mg used in core and cylinder shells. Eπ rt2
Mcr = πr2 tσ cr = √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ (6)
3(1 − ν2 )
Property value

Modulus of elasticity 70 GPa Jin et al. [45] conducted parametric analyses using the validated FE
Yield strength (Rp 0.2%) 240 MPa models. Three parameters have a noteworthy influence on the behaviour
Ultimate strength 360 MPa of thin cylinders under combined axial compression and bending loads.
5
Density 2.67 × 10− N/mm3
An increased slenderness, Larger D/t ratio, and eccentricity ratio (e/D)
Poisson’s Ratio 0.3
lead to a decrease in ultimate strength. Through this supposition, the
traditional mechanical equations are still valid before buckling failure
experimental [47] results. This assumption can be conservative,
particularly in combined bending and torsion, and combined compres­
sion and torsion. Using alternative methods to estimate the influence of
combined loads can achieve more precise and less overestimated
buckling loads.
Analytical examinations which investigated the failure of cylindrical
thin-walled shells initially concentrated on axial compression loads.
Gere and Tymoshenko [47] concluded that the critical stress for a long
cylindrical shell can be determined using Eq. (1):
Et
σcr = √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
̅ (1)
r 3(1 − ν2 )

where σcr is the critical stress, r is the mean shell radius, t is the wall
thickness, E is the young modulus, and ν is the Poisson’s ratio. The Fig. 5. The element shape of the mesh technique: (a) Quad element shape
equation which is suggested for when combining with axial compression (S4R), (b) Tri element shape (S3R).

Fig. 4. Loading and boundary conditions of the sandwich cylindrical shell under eccentric compression load.

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M.J. Zarei et al. Structures 61 (2024) 105963

Table 3
Sandwich specimens with an internal radius of 200 mm and external radius of 250 mm.
Parameter study specimen

D.1 D.2 D.3 D.4 D.5 D.6 D.7 D.8 D.9 D.10 D.11 D.12

Element Shape Quad Tri Quad Tri Quad Tri Quad Tri Quad Tri Quad Tri
Eccentric (e-mm) 25 25 50 50 125 125 200 200 300 300 1000 1000

occurs. The cross-section appears to be circular within the pre-buckling shape results computed by research literature. A mesh composed of quad
range, and the strain at which buckling occurs is impacted significantly elements shows a pattern of square indentations that resembles a
by the D/t ratio but is not impacted much by slenderness and chequerboard. Meshes triangular element shape show off an intriguing
eccentricity. spiral buckling mode that appears to match the angle of the spiral mesh.
The critical load value of the cylindrical face sheets in this study This result illustrates that the cylindrical shell responds to changes in
(with radii of 200 mm and 250 mm) obtained from Eq. (5) is the same element shape when under axial compressive load. Fig. 6(b) presents the
value as 66.548 kN (for the total of two face sheets is approximately buckling mode of the cylindrical shell in response to pure bending in an
133 kN). The critical moment obtained from Eq. (6) for the inner and eigenvalue analysis using quad and tri S4R elements, displaying nearly
outer cylindrical shells is 6.654 × 103 kN.mm and 8.317 × 103 kN.mm, identical results regardless of the element shape.
respectively (for the total of two face sheets is approximately Now, this paper presents a technique for creating finely structured
14.971 ×103 kN.mm. The values obtained from the above equations are finite element meshes for evaluating the stress and buckling of a sand­
without considering the effect of core resistance. The flexural stiffness of wich cylindrical shell structure with a truss core. A study was conducted
the lattice truss core is low, therefore, the critical failure load of the to evaluate the eigenvalue buckling convergence with eccentric
sandwich cylindrical shells can be expected to be slightly higher than the compression load, which encompassed both buckling shapes and ca­
critical load determined during the examination of cylindrical shells. pacities. The results showed that a regular orthogonal finite element
Before beginning the analysis of the sandwich structure, it is useful to mesh composed of quadrilateral shell elements showed different be­
analyse the behaviour of the components of the sandwich structure haviors depending on its orientation in relation to the principal axes of
separately (specially the face sheets that are resistant to transverse and stress in the structure. The buckling modes of the cylindrical sandwich
in-plane loads) to better understand how these types of structures shells with merge parts (D.1 to D.12) are depicted in Fig. 7, and the
behave. Fig. 6(a) illustrates the first buckling mode of a long cylindrical critical buckling load is listed in Tables 4 to 6.
shell when subjected to an axial load with a scale of 50, and two types of The buckling analysis of the sandwich structure subjected to an
element shape mesh (tri and quad), which are compared with the eccentric compression load has produced the following results. Different
research result [48,49]. As can be seen, the first buckling mode obtained buckling modes indicate the structure’s sensitivity to the element shape
from the eigenvalue analysis by ABAQUS agrees well with the mode mesh (tri and quad). The buckling modes are much less sensitive to the

Fig. 6. The first buckling mode of the long cylindrical shells compared with the research literature result: (a) under axial compressive load, (b) under pure
bending load.

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M.J. Zarei et al. Structures 61 (2024) 105963

Fig. 7. The first buckling modes of the sandwich cylindrical shell in the modeled specimens under axial compressive and bending loads with the scale of 60.

Table 4 Table 6
Critical buckling load (kN) of Sandwich cylindrical shell (mesh size = 20 mm). Critical buckling load (kN) of Sandwich cylindrical shell (mesh size =10 mm).
Eccentricity (mm) Element Shape Eccentricity (mm) Adhesive Bond Stiffness (N/mm3)

Specimen Quadrilateral Specimen Triangle K= 0 K= 100 K= 3 × 105

25 D.1 105.615 D.2 123.209 50 86.411 101.038 109.092

50 D.3 86.392 D.4 103.366 125 61.185 64.089 75.808
125 D.5 62.587 D.6 75.569 250 37.506 43.805 45.569
200 D.7 47.859 D.8 57.811 1000 12.408 14.428 15.751
300 D.9 37.989 D.10 42.306
1000 D.11 13.608 D.12 16.264
eccentricity of the applied load. (see Fig. 7). The critical load of tri
element shape is approximately 10% more than the quad element at the
Table 5 same eccentricity (e). The tri elements have fewer degrees of freedom as
Critical buckling load (kN) of Sandwich cylindrical shell (Eccentricity = compared to the quad elements (Since the number of nodes in tri ele­
50 mm). ments is lesser than quad elements); hence, tri elements are stiffer than
Mesh Size Element Shape
quad elements. For example, in the meshes of structures with an element
(mm) size of 20 mm, the tri and quad element shapes have 25275 and 30627
Triangle Quadrilateral
nodes, respectively. So the shell elements S3R and S4R (three-dimen­
Number of Buckling Load Number of Buckling Load sional) have a total of 151650 and 183762 degrees of freedom respec­
DoF (kN) DoF (kN) tively, given that each node is composed of three translations and three
60 26766 254.481 30612 61.160 rotations, which makes up a total of six degrees of freedom. The struc­
50 32550 196.512 37980 79.083 ture has a buckling load of 42.306 kN at a 300 mm eccentricity with the
40 41532 182.166 51348 80.709
tri element shape (see Fig. 8), in which the buckling load is reduced by
30 63252 135.713 77976 85.355
20 151650 103.366 183762 86.392 approximately 70%. Therefore, as the eccentricity of the load (e) in­
15 241932 99.309 288696 89.809 creases, the buckling load decreases significantly, which can be
10 518598 92.693 602526 90.435 considered and controlled in the analyses of the sandwich structure.
Fig. 9 shows the mesh convergence of the sandwich structure under
eccentric compression load with a 50 mm eccentricity. The tri element
shape exhibits a slower convergence than the quad element shape and
The least precise forecast of the critical buckling load. Nevertheless, the

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M.J. Zarei et al. Structures 61 (2024) 105963

50 mm, the critical load is reduced by approximately 22% relative to the

pure compression condition (e= 0, and buckling load of 133 kN).
Fig. 10 demonstrates the buckling modes for cylindrical shells at
various mesh resolutions and element shapes when the eccentricity is
50 mm. It is seen that the critical buckling modes are influenced by the
element shape and not as much by the size of the mesh. Tri-element
meshes demonstrate remarkable buckling modes that seem to be in
line with the angle of the spiral mesh. This is due to the spiraling of the
mesh acting as a slight imperfection which, while not enough to affect
the critical load, is significant enough to cause the buckling mode to be
more beneficial in alignment with the spiral axis. The triangular element
shape has a slower convergence rate when compared to the quadrilateral
element shape. The smaller the mesh size, the more precise the buckling
mode and critical load of the sandwich structure will be.
In the nonlinear analysis, to better understand the influence of the
failure modes, stresses, and presence of truss core on the behavior of the
sandwich cylindrical shells, we first simulated concentric cylindrical
Fig. 8. The Force-eccentric displacement curves of sandwich cylindrical shells shells without lattice truss core by Explicit dynamic analysis with quad
with quad and tri element shapes. element shape and 10 mm mesh size. Then, the weakest joint stiffness, in
which the shells and core are individually loaded until the adhesive
reaches a suitable stiffness and the structure continues to function.
Plastic buckling will not be considered in this paper since its occurrence
is usually due to a low value of yield stress and a low value of ratio. So
local buckling is the dominant failure mode. The nonlinear analysis
undertook an investigation of the load-displacement curves of sandwich
structures with a range of load eccentricities and the effect on the critical
buckling load. The behavior of the adhesive bond and its modeling in the
software are described below.
Cohesive surface contact can serve as a model of a bonded connec­
tion, with the potential for damage and bond failure. Cohesive behavior
of contact can be described as a surface interaction property, repre­
senting the separation at interfaces and restrict to surface regions those
that are initially connected or may join upon meeting. The constitutive
response of the cohesive surface is modeled using the general traction-
separation model, which assumes an initial elastic behavior that is
then followed by the initiation and evolution of damage. The elastic
behavior results from an elastic constitutive matrix which shows the
relation between the normal and shear stresses and the normal and shear
Fig. 9. Mesh convergence of the sandwich cylindrical shell at the separations at the interface. We can express the elastic behavior as:
50 mm eccentric. ⎧ ⎫ ⎡ ⎤⎧ ⎫
⎨ tn ⎬ knn kns knt ⎨ δn ⎬
⎣
t = ts = kns kss kst ⎦ δs = Kδ (7)
regular mesh with a quad element shape can only initially obtain a ⎩ ⎭
tt knt kst ktt
⎩ ⎭
δt
higher precision with a lesser number of elements. By using a piecewise
linear approximation for the displacement field of the finite element The nominal traction stress vector, t, for three-dimensional problems
model, an over-constraint situation is created, thus leading to an is composed of three components: the normal traction (tn ) in the local 3-
excessively stiff system. To make up for this extra rigidity, the essential direction, and the two shear traction components (ts , tt ) in the local 1-
buckling load will be higher than it should be and will consequently and 2-directions, respectively. The behavior of adaptability is then
converge on the accurate value from above as the mesh resolution shown by the following separations, designated by δ. By default, no
grows. An even more mysterious outcome, and more complicated to association will be made between the normal and tangential stiffness
explain, is why meshes with quad element shapes converge from below, components. The terms Knn , Kss , and Ktt must be characterized to
achieving outstanding accuracy with fewer elements initially. Never­ comprehend the uncoupled traction-separation behavior. It is possible to
theless, utilizing tri element shapes from above causes meshes to use damage modeling to simulate the relationship between two cohesive
converge, resulting in a less critical load because of the finer mesh. The surfaces deteriorating and then eventually failing. Damage initiation
result suggests that the optimal mesh angle for this seemingly effortless and damage evolution are the basis for the failure mechanism. However,
load condition might be inclined. As a result, the orientation of the when a damage initiation criterion is met, destruction can occur ac­
components could be a vital factor in the finite element modelling of cording to a user-defined damage evolution law. Fig. 11 depicts a linear
issues in structural mechanics, since a unique misfortunate orientating traction-separation response with an associated failure mechanism. The
might be the point at which the convergence of the inclined mesh will be deterioration procedure begins when the contact stresses or separations
at its worst [43,44]. Eventually, the tri element shape needs finer mesh meet particular damage initiation criteria. The maximum stress criterion
than the quad element shape to achieve great convergence. The buckling is employed, which is articulated:
load in the tri element shape (92.693 kN) has a load difference of { }
〈tn 〉 ts tt
approximately 6% compared to the square element shape (87.435 kN) at max o
; o ; o =1 (8)
tn ts tt
a 10 mm mesh size. Comparing these results with the buckling analysis
of sandwich structures (D.3 and D.4) with the 20 mm element size shows
where the peak values of contact stress t on , tos , and t ot refer to when the
that a significant difference in buckling load has not been in the tri and
separation is purely in the contact normal or either of the two shear
quad element shapes with increasing mesh sizes. At an eccentricity of

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M.J. Zarei et al. Structures 61 (2024) 105963

Fig. 10. The first buckling modes of the sandwich cylindrical shell under eccentric compression load at the 50 mm eccentricity and different mesh sizes with a scale
of 60.

fracture behavior is studied by a cohesive law that relates tractions and

separations across cohesive surfaces [50]. We will analyze the cohesive
stiffness in competition with cylindrical shells/core bond stresses. When
the contact stress values exceed the cohesive strength, damage initiation
occurs at the desired joint. Further loading causes the cohesive stiffness
to deteriorate, leading to nodes rupturing and, eventually, the struc­
ture’s complete failure. Cohesive failures produced signatures on the
load-displacement indentation curves, which evaluated the failure load
according to the input cohesive stiffness values. Finally, the simulation
results are examined. The effect of joint stiffness will be examined in
debonding among the lattice truss core and cylinder shells and its effect
Fig. 11. Linear traction-separation response. on the failure load of sandwich cylindrical shells with a quad element
shape and 10 mm mesh size. According to the past research, the simu­
directions. The Macaulay brackets showed by the symbol 〈〉 signify a lated model is established under eccentric compressive load with
purely compressive displacement, which is a contact penetration, or a cohesive surface properties, including different cohesive stiffness (K),
purely compressive stress state that will not cause damage. traction at damage initiation of 10 MPa, and effective separation at the
The damage evolution regulation explains how the cohesive stiffness complete failure of 0.02 mm. In the weakest joint stiffness, the cylinder
diminishes when the initiation criterion is fulfilled. The damage evolu­ shells and core are individually loaded (separate state), in which case
tion is substantially determined by two variables. The initial variable is the structure has the least critical failure load. As the adhesive bond
determining the effective separation at the complete failure, δfm , becomes somewhat stiffer, the sandwich structure can approach its full
compared to the effective separation at damage initiation, δom , or the strength in the surrendered state by bearing more load and improving
energy dissipated through failure, Gc . The fracture energy is equivalent the critical failure load.
to the area beneath the tension-separation graph (as seen in Fig. 11). Fig. 12 illustrates that the critical load decreases as the eccentricity of
This research uses effective separation at complete collapse and soft­ the load increases. As the adhesive stiffness decreases in the joints be­
ening linearly. Models that proposed different softening behavior and tween the core and the cylindrical shells, the structure parts have
diminishment of stiffness can create significant convergence issues. The separately withstood loading, which lessens the probability of the crit­
ical failure load. Furthermore, the inability of the structure to resist the

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M.J. Zarei et al. Structures 61 (2024) 105963

Fig. 12. The load-displacement curves of sandwich structures under eccentric compression load with different eccentricities.

eccentric compression load, in which the structure suffers from buckling = 125 mm, the critical force was diminished by 50% compared to the
and sudden failure. Therefore, the critical load at different cohesive axial load case. Consequently, the eccentricity of the load in these
stiffness could be varied between two separate and continuous sandwich structures significantly influences the critical buckling load.
structure approaches. Moreover, it displays a considerable load of ec­ Fig. 14 shows the Load-displacement curve of cylindrical shells and
centricity influence on the critical load corresponding to the local the sandwich structure by two weak and strong adhesive strengths with
buckling of the sandwich structure. buckling/ultimate deformations at the eccentricity of 125 mm. Local
Fig. 13 displays the effect of an eccentric compressive load on the buckling is the dominant failure mode. Cylindrical shells, without a core,
structure’s load-bearing capacity when adhesive stiffness is 3e5 N/mm3. illustrated the minor critical load of 60 kN. Despite the core in weak
An increase in the load’s eccentricity leads to a decrease in the critical adhesive bonding, the critical load has reached 63 kN. However, it has
load. When the eccentricity of the load was arranged at the midpoint of e an individual approach that does not significantly improve the critical
load. The compressive capacity of cylindrical sandwich shells with a
strong adhesive is about 75 kN, and this enhancement is perceptible.
Consequently, adhesive stiffness has a more substantial effect on the
critical load under the eccentric compression load. Furthermore,
elevating the adhesive stiffness enhances the critical moment. As shown
in Fig. 14, the Sandwich cylindrical shells with high adhesive stiffness
can reach the highest buckling load by restricting the buckling defor­
mation than the weak adhesive bond and the cylindrical shells without
core presence. Thus, increasing the adhesive stiffness bond allows the
structure to reach its compressive load against a near-continuous axial
compressive load and improves the critical failure load value. The ad­
hesives which are more stiffened experience more pronounced buckling
deformations, and the buckling load increases considerably when elastic
buckling deformations are inhibited by the adhesive.
Once again, these structures are examined under axial compressive
load with the eccentricity of 250 mm in Fig. 15. The sandwich cylin­
drical shells with high adhesive stiffness can improve the critical load
compared to weak adhesive bonds and single cylindrical shells. This
leads to the maximum buckling load of 46 kN, which is constrained by
Fig. 13. Load-displacement curves of sandwich structure with adhesive stiff­ the adhesive’s ability to cause elastic buckling deformation.
ness of 3e5 N/mm3 at the different load eccentric.

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M.J. Zarei et al. Structures 61 (2024) 105963

Fig. 14. The load-displacement curve of structures under compressive load with an eccentricity of 125 mm and the initial buckling and ultimate deformations.

Fig. 15. The load-displacement curves of structures under compressive load at an eccentricity of 250 mm with the buckling and ultimate deformations.

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M.J. Zarei et al. Structures 61 (2024) 105963

Furthermore, in the presence of the truss core, the buckling load 2. In nonlinear dynamic analysis, to better understand the influence
improved by approximately 20% compared to single cylindrical shells of the failure modes, stresses, and presence of truss core on the behavior
with possessing a capacity of 36 kN under load. Accordingly, the sand­ of the sandwich cylindrical shells, we first simulated concentric cylin­
wich structure can be an excellent choice to increase the load-bearing drical shells without lattice truss core. Then, the weakest joint stiffness,
capability of columns in buildings. in which the shells and core are individually loaded until the adhesive
reaches a suitable stiffness and the structure continues to function. This
4. Future research and limitations paper will not take plastic buckling into consideration as it is generally
caused by a low yield stress and a low D/t ratio value. Thus, local
In future research, a comprehensive understanding of cylindrical buckling is the most prominent mode of failure. An investigation of the
sandwich shells with truss cores can be achieved through a synergistic load-displacement curves of sandwich structures with distinctive load
combination of experimental and numerical analyses. Integrating eccentricities and their relation to the critical buckling load is con­
physical experiments will validate and refine the presented numerical ducted. A rise in the load’s eccentricity is associated with a decrease in
models, particularly under the influence of manufacturing errors and the critical load. Setting the eccentricity of the load to a midpoint of e
different eccentric loads. Investigations should delve into the impact of = 125 mm resulted in 50% reduction in the critical force relative to the
manufacturing errors on the critical failure load, considering variations axial load case. Therefore, the load eccentricity in these structures
in skin thickness and adhesive bonding quality. Parametric studies can significantly influences the critical buckling load. As a result, by
systematically evaluate the influence of design parameters, while providing appropriate cohesive stiffness between the core and the cyl­
exploring dynamic loading conditions will assess the structure’s inder shells, this structure may achieve total resistance to the axial
response to dynamic forces. The exploration of multi-material configu­ compressive load close to its persistent condition. Furthermore, it has
rations and optimization techniques can further enhance structural the potential to increase the critical failure load value.
performance. Despite valuable insights, this study has limitations, The outstanding strengths of this study include a high number of
including simplifications in modeling, limited consideration of samples for achieving precise results, accurate correlation of values
manufacturing errors, and a focus on a single loading condition. Future obtained from modeling with analytical and engineering relationships,
research should address these limitations, prioritizing experimental comparison with similar samples, and a comprehensive study for
validation, considering environmental factors, and exploring scale-up selecting the mesh type and dimensions. Additionally, this research
considerations. This approach will contribute to a more nuanced un­ points out weaknesses such as the limited dimensions and thickness in
derstanding of cylindrical sandwich shells with truss cores, ensuring the parameter studies, which can be addressed in future investigations.
applicability to diverse engineering scenarios. The focus of this study on
eccentric compression loading provides valuable insights into specific Declaration of Competing Interest
failure modes. However, future research should consider additional
loading conditions, such as lateral loads or dynamic forces, to compre­ The authors declare that they have no known competing financial
hensively understand the structural response under diverse scenarios. interests or personal relationships that could have appeared to influence
Exploring how the structure behaves under different loading conditions the work reported in this paper.
will provide a more holistic understanding of its capabilities and limi­
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