Available via license: CC BY 4.0
Content may be subject to copyright.
Citation: Pradhan, E.M.; Lange, J.
Warping Torsion in Sandwich Panels:
Analyzing the Structural Behavior
through Experimental and Numerical
Studies. Materials 2024,17, 460. https:
//doi.org/10.3390/ma17020460
Academic Editor: Vit Smilauer
Received: 18 December 2023
Revised: 10 January 2024
Accepted: 16 January 2024
Published: 18 January 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
materials
Article
Warping Torsion in Sandwich Panels: Analyzing the Structural
Behavior through Experimental and Numerical Studies
Eric Man Pradhan * and Jörg Lange
Institute for Steel Construction and Materials Mechanics, Technical University of Darmstadt,
64287 Darmstadt, Germany
*Correspondence: pradhan@stahlbau.tu-darmstadt.de
Abstract: Recently, there has been a growing interest in the use of sandwich panels that, beyond
handling well-known bending stress, can withstand torsional stresses. This is particularly relevant
for wall applications when the panels are equipped with photovoltaic or supplemental curtain wall
modules. This research article presents a detailed exploration of the structural behavior of eccentri-
cally loaded sandwich panels, with a specific focus on warping torsion. Experimental and numerical
studies were conducted on polyisocyanurate (PU) core sandwich panels, commonly employed in
building envelopes. These studies involved various dimensions and material properties, while
omitting longitudinal joints. The experimental study provided essential insights and validated the nu-
merical model in ANSYS. Enabling parametric variation, the numerical analysis extends the analysis
beyond the experimental scope. Results revealed a high degree of correlation between experimental,
numerical, and analytical solutions, regarding the rotation, as well as the normal and shear stress of
the panel. Confirming the general applicability of warping torsion in sandwich panels with certain
limitations, the study contributes valuable data for applications and design of eccentrically loaded
sandwich panels, laying the foundation for potential engineering calculation methods.
Keywords: sandwich panels; eccentrically loaded; warping torsion; structural behavior; rotation;
stress analysis; parametric studies; experimental investigations; numerical model; finite element
1. Introduction
1.1. Motivation and Objective
Sandwich panels, consisting of two thin steel face sheets with a shear-deformable
polyisocyanurate (PU) or mineral wool core in between have become a well-established so-
lution for cost-effective building envelopes, especially within the industrial building sector
in Europe. They offer the benefits of a prefabricated lightweight component, delivering
excellent thermal insulation and sealing performance. In Germany alone, around 20 mil-
lion square meters of sandwich panels, employed for cladding and roofing applications,
were manufactured and installed annually. Figure 1illustrates a typical cross-section with
notations indicating the dimensions and material properties.
Materials2024,17,x.hps://doi.org/10.3390/xxxxxwww.mdpi.com/journal/materials
Article
WarpingTorsioninSandwichPanels:AnalyzingtheStructural
BehaviorthroughExperimentalandNumericalStudies
EricManPradhan*andJörgLange
InstituteforSteelConstructionandMaterialsMechanics,TechnicalUniversityofDarmstadt,
64287Darmstadt,Germany;info@stahlbau.tu-darmstadt.de
*Correspondence:pradhan@stahlbau.tu-darmstadt.de
Abstract:Recently,therehasbeenagrowinginterestintheuseofsandwichpanelsthat,beyond
handlingwell-knownbendingstress,canwithstandtorsionalstresses.Thisisparticularlyrelevant
forwallapplicationswhenthepanelsareequippedwithphotovoltaicorsupplementalcurtainwall
modules.Thisresearcharticlepresentsadetailedexplorationofthestructuralbehaviorofeccentri-
callyloadedsandwichpanels,withaspecificfocusonwarpingtorsion.Experimentalandnumerical
studieswereconductedonpolyisocyanurate(PU)coresandwichpanels,commonlyemployedin
buildingenvelopes.Thesestudiesinvolvedvariousdimensionsandmaterialproperties,whileomit-
tinglongitudinaljoints.Theexperimentalstudyprovidedessentialinsightsandvalidatedthenu-
mericalmodelinANSYS.Enablingparametricvariation,thenumericalanalysisextendstheanalysis
beyondtheexperimentalscope.Resultsrevealedahighdegreeofcorrelationbetweenexperimental,
numerical,andanalyticalsolutions,regardingtherotation,aswellasthenormalandshearstressof
thepanel.Confirmingthegeneralapplicabilityofwarpingtorsioninsandwichpanelswithcertain
limitations,thestudycontributesvaluabledataforapplicationsanddesignofeccentricallyloaded
sandwichpanels,layingthefoundationforpotentialengineeringcalculationmethods.
Keywords:sandwichpanels;eccentricallyloaded;warpingtorsion;structuralbehavior;rotation;
stressanalysis;parametricstudies;experimentalinvestigations;numericalmodel;finiteelement
1.Introduction
1.1.MotivationandObjective
Sandwichpanels,consistingoftwothinsteelfacesheetswithashear-deformable
polyisocyanurate(PU)ormineralwoolcoreinbetweenhavebecomeawell-established
solutionforcost-effectivebuildingenvelopes,especiallywithintheindustrialbuilding
sectorinEurope.Theyofferthebenefitsofaprefabricatedlightweightcomponent,deliv-
eringexcellentthermalinsulationandsealingperformance.InGermanyalone,around20
millionsquaremetersofsandwichpanels,employedforcladdingandroofingapplica-
tions,weremanufacturedandinstalledannually.Figure1illustratesatypicalcross-sec-
tionwithnotationsindicatingthedimensionsandmaterialproperties.
Figure1.Definitionofthesandwichpanelcross-section(withoutjointgeometry).
b
d
C
t
1
t
2
E
C
,G
C
E
F
,G
F
E
F
,G
F
y,η
z,ζ
D
topface sheet (steel)
bottom face sheet (steel)
core (PU)
e
e
1
e
2
Citation:Pradhan,E.M.;Lange,J.
Warpin gTorsioninSandwich
Panels:AnalyzingtheStructural
BehaviorthroughExperimentaland
NumericalStudies.Materials2024,
17,x.hps://doi.org/10.3390/xxxxx
AcademicEditor:VitSmilauer
Received:18December2023
Revised:10January2024
Accepted:16January2024
Published:18January2024
Copyright:©2024bytheauthors.
Submiedforpossibleopenaccess
publicationunderthetermsand
conditionsoftheCreativeCommons
Aribution(CCBY)license
(hps://creativecommons.org/license
s/by/4.0/).
Figure 1. Definition of the sandwich panel cross-section (without joint geometry).
Materials 2024,17, 460. https://doi.org/10.3390/ma17020460 https://www.mdpi.com/journal/materials
Materials 2024,17, 460 2 of 20
In recent developments, sandwich wall panels are subjected to eccentric loads in
addition to the well-known bending loads. For instance, sandwich panels equipped with
photovoltaic or supplemental curtain wall modules have gained popularity [
1
,
2
]. The
latter represents a novel approach that offers cost-effective and visually appealing façade
solutions for both new constructions and building renovations. Façade modules of diverse
architectural design can be fastened to the exterior face of the sandwich wall panels using
rails (e.g., omega profiles), as shown in Figure 2. In this context, sandwich panels are
the primary load-bearing component of the façade. In cases where horizontally oriented
sandwich panels are used, the eccentric loads, arising from the dead weight of the additional
façade modules (G), induces a torsional moment (M
T
) and resulting stresses. In summary,
it is evident that with the increase in eccentricity or weight of the attached modules, the
magnitude of the torsional load experienced by the sandwich panel increases too.
Materials2024,17,xFORPEERREVIEW2of20
Inrecentdevelopments,sandwichwallpanelsaresubjectedtoeccentricloadsinad-
ditiontothewell-knownbendingloads.Forinstance,sandwichpanelsequippedwith
photovoltaicorsupplementalcurtainwallmoduleshavegainedpopularity[1,2].Thelat-
terrepresentsanovelapproachthatofferscost-effectiveandvisuallyappealingfaçade
solutionsforbothnewconstructionsandbuildingrenovations.Façademodulesofdiverse
architecturaldesigncanbefastenedtotheexteriorfaceofthesandwichwallpanelsusing
rails(e.g.,omegaprofiles),asshowninFigure2.Inthiscontext,sandwichpanelsarethe
primaryload-bearingcomponentofthefaçade.Incaseswherehorizontallyorientedsand-
wichpanelsareused,theeccentricloads,arisingfromthedeadweightoftheadditional
façademodules(G),inducesatorsionalmoment(MT)andresultingstresses.Insummary,
itisevidentthatwiththeincreaseineccentricityorweightoftheaachedmodules,the
magnitudeofthetorsionalloadexperiencedbythesandwichpanelincreasestoo.
Figure2.Sandwichwallpanelwithsupplementaryfaçademodules(schematicsketch).
Previouscalculationmodelsofeccentricallyloadedsandwichpanels[3–10]havepri-
marilyreliedonSaint-Venant’storsiontheory.However,recentresearch[11–17]hasun-
veiledthepresenceoftorsion-inducednormalstressesinthefacesheets.Thefindingsin-
dicatethatadditionalmechanicaleffects,suchaswarpingtorsion,mustbegivensignifi-
cantconsideration.Subsequently,aresearchprojectatTUDarmstadthasbeeninitiated
withtheaimofdevelopinganewengineeringcalculationapproachtodescribethestruc-
turalbehaviorofeccentricallyloadedsandwichpanelsfromafundamentalstatic-mechan-
icalperspective.Toachievethisgoal,comprehensiveexperimental,numerical,andana-
lyticalstudiesareconductedonsandwichpanelsofvaryingdimensionsandmaterial
properties.Thesestudiesaimtothoroughlyassessthetheoryofwarpingtorsion,taking
intoaccountitslimitations.Thisresearcharticlepresentsanddiscussestheresultsofthe
recentexperimentalandnumericalparameterstudies,withaspecificfocusontheanalysis
oftherotationandstressanalysisinrelationtowarpingtorsion.
1.2.StateoftheResearchandStateoftheArt
Forsandwichstructures,thetopicoftorsionwasfirstexploredinthemid-1950sand
iscontinuouslyrelevantuntiltoday.In1956,Seide[3]derivedthetorsionalstiffnessfor
rectangularsandwichpanelsbasedonSaint-Venant’stheory.Inthiscontext,thetorsional
stiffnessofthefacesheets,thecore,andtheirinterdependentinfluenceareconsidered.A
similarapproachwasadoptedbyCheng[4–6]althoughwiththesimplificationthatthe
coreshearstressesparalleltothefacesheetplanearenegligible.Thegeneralsolutions
presentedbySeideandCheng,respectively,provedimpracticalforconstructionapplica-
tionsduetothemathematicalcomplexityoftheexpressions,involvinginfiniteseries.A
differentapproachwastakenbyStammandWie[7]in1974.Followingthemethodology
supplementary façade modules
sandwich wallpanel omega profiles
G
M
T
Figure 2. Sandwich wall panel with supplementary façade modules (schematic sketch).
Previous calculation models of eccentrically loaded sandwich panels [
3
–
10
] have
primarily relied on Saint-Venant’s torsion theory. However, recent research [
11
–
17
] has
unveiled the presence of torsion-induced normal stresses in the face sheets. The findings in-
dicate that additional mechanical effects, such as warping torsion, must be given significant
consideration. Subsequently, a research project at TU Darmstadt has been initiated with
the aim of developing a new engineering calculation approach to describe the structural
behavior of eccentrically loaded sandwich panels from a fundamental static-mechanical
perspective. To achieve this goal, comprehensive experimental, numerical, and analytical
studies are conducted on sandwich panels of varying dimensions and material proper-
ties. These studies aim to thoroughly assess the theory of warping torsion, taking into
account its limitations. This research article presents and discusses the results of the recent
experimental and numerical parameter studies, with a specific focus on the analysis of the
rotation and stress analysis in relation to warping torsion.
1.2. State of the Research and State of the Art
For sandwich structures, the topic of torsion was first explored in the mid-1950s and
is continuously relevant until today. In 1956, Seide [
3
] derived the torsional stiffness for
rectangular sandwich panels based on Saint-Venant’s theory. In this context, the torsional
stiffness of the face sheets, the core, and their interdependent influence are considered. A
similar approach was adopted by Cheng [
4
–
6
] although with the simplification that the
core shear stresses parallel to the face sheet plane are negligible. The general solutions
presented by Seide and Cheng, respectively, proved impractical for construction applica-
tions due to the mathematical complexity of the expressions, involving infinite series. A
different approach was taken by Stamm and Witte [
7
] in 1974. Following the methodology
commonly used for determining the torsional stiffness of thin-walled, closed cross-sections,
Materials 2024,17, 460 3 of 20
they formulated a differential equation and found a more practicable solution. In their
calculation model, they conceptually divided the core of the sandwich panel into an infinite
number of vertical lamellae, each of infinitesimal width, which can deform independently
of one another. This approach was justified by their assumption that core shear stresses in
the face sheet plane can be neglected.
Torsion in sandwich panels gained increased practical relevance in the building indus-
try during the 1980s, when the use of sandwich panels as cladding components became
widespread. When a larger opening, such as a window, is introduced in a sandwich panel,
it results in an eccentric load to the adjacent panel connected through a joint. This, in
return, leads to torsional stress, unless an additional substructure is employed. In this
context, Höglund [
8
] devised a formula for the torsional stiffness of sandwich panels while
calculating sandwich constructions with variably positioned windows. The formula was
validated through four experimental tests. In these calculations, Höglund utilized the
second Bredt formula, which is typically applied to thin-walled closed cross-sections. An
idealized cross-section for the sandwich panel was assumed, excluding 33% of the inner
core area from consideration. In 2005, Böttcher [
18
] subsequently developed a simplified
engineering model for an assembly of panels.
While the extensive number of theoretical publications might suggest that this is a
well-explored subject, the above findings were questioned by Rädel [
10
,
11
] in the 2010s.
During her research on sandwich panels with openings, she conducted two-panel tests that
revealed the existence of normal stresses resulting from torsional loads. This discovery
fundamentally contradicted the Saint-Venant-based torsion theories for sandwich panels
mentioned earlier. According to conventional definitions, no torsion-induced normal
stresses can hereby occur. Moreover, she demonstrated that the calculated torsional stiffness
significantly exceeded the experimentally determined values by a factor of 1.4 to 1.6 (in
comparison to Stamm and Witte’s approach) or even 6 to 10 (in comparison to Seide’s
simplified approach).
In response to these findings, Pozorski and Wojciechowski [
12
,
13
,
16
] addressed this
issue by incorporating the consideration of warping torsion. They derived a first approach
of an analytical beam model for sandwich panels subjected to torsion, similar to the
formulas commonly used in steel construction for straight bars with open cross-sections.
Experimental and numerical studies have verified this in principle for the analyzed cases,
while high attention was paid to the boundary conditions, including 1D-, 2D-, and 3D-
scenarios. In addition, they noted that the more sophisticated general solution in Seide’s
torsional stiffness calculation matches that of Stamm and Witte.
For additionally incorporating local effects caused by the soft core, Elmalich and
Rabinovitch [
19
] in 2014 and Wurf et al. [
17
] in 2022 developed individual analytical
models based on the extended high-order sandwich panel theory. Wurf et al. demonstrated
favorable agreement between their calculated results and a 3D numerical solution in ANSYS
for a selected example, utilizing the approach from Pozorski and Wojciechowski [
13
] as
a reference. In detail, the results of their new approach aligned more closely with their
finite element (FE) model than those shown in [
13
]. Nevertheless, the question remains
whether the additional computational effort is justified for these minor deviations, in terms
of practical applications.
From an industrial standpoint, the need to understand and account for eccentric
loads in sandwich panels has grown in importance. In recent years, a rising number of
manufacturers have sought European Technical Approvals (ETAs) for their sandwich panel
systems, particularly those incorporating additional features such as façades or photovoltaic
modules. In typical applications involving sandwich panels, the impact of torsion has
historically been downplayed due to its usually negligible effect, even when dealing
with minor load eccentricities. However, the increasing demand for improved thermal
insulation has highlighted the influence of eccentricity on torsional loading, particularly as
core thicknesses continue to increase.
Materials 2024,17, 460 4 of 20
Nonetheless, the current European harmonized standard for sandwich panels, EN
14509:2013 [
20
], does not address torsion. On a regulatory level, the CIB 378 [
21
] briefly
mentions torsional load calculations concerning sandwich panels with openings but does
not extensively consider warping torsion. Current efforts are underway to develop a new
ECCS recommendation on sandwich panels [
22
], which will cover the effects of point and
line loads, including torsion.
1.3. Fundamentals
In the example illustrated in Figure 2, the sandwich panel is subjected to eccen-
tric loads due to the additional façade modules, and it can simultaneously also experi-
ence out-of-plane bending loads from wind or temperature. Numerous well-established
publications [
7
,
9
,
23
] and the standard EN 14509:2013 offer practical calculation formulas
for determining stiffnesses, deformations, and stresses under bending loads based on the
First-Order Shear Deformation Theory. A distinctive feature, compared to the theory of
shear-stiff beams, is that the deformations arise from both bending and shear components.
In these references, sandwich panels are typically treated as beams for modelling as well
as analysis, despite their plate-like appearance. In the case of torsion, this simplification
is also to be initially assumed in this article. This serves as a crucial step in gaining an
understanding of the structural behavior. Consequently, the term “torsion” is employed
here in analogy to beams when addressing eccentrically loaded sandwich panels.
Torsion in a beam occurs when rotations, denoted as
ϑ
(x), appear around its longitudi-
nal axis x, and these rotations vary along its length, indicating that the beam undergoes
rotation with increasing torsion
ϑ′
(x) > 0. In general, torsion can be divided into Saint-
Venant’s torsion (type I, free torsion) where shear stresses
τxy
are the primary concern, and
warping torsion (type II, Vlasow torsion), in which additionally normal stresses, known
as warping normal stresses
σw
, emerge. In the field of structural engineering, the analysis
typically assumes the Bernoulli and Wagner hypotheses. This means that the beams’ axis
remains straight, rotations
ϑ
(x) occur solely around the longitudinal axis, and no cross-
sectional deformations take place in the loaded state. Furthermore, influences of secondary
shear deformations are neglected. Additionally, in the case of Saint-Venant’s torsion, it is
presumed that warping
ω
or deformations out of the cross-sectional plane occur without
constraint. The general differential equation for torsion is well-established. It can be ex-
pressed by the relationship between total torsional moment M
T
and the derivations of the
beam’s rotation, while introducing the decay factor
λ
or the structural parameter
ε
; see
Equations (1)–(3).
MT(x)
EIW=−λ2ϑ′(x)+ϑ′′′ (x)(1)
MT(x)=MT,I(x)+MT,II (x)(2)
λ=sGIT
EIW=ε
L(3)
A multitude of solutions for this differential equation, considering various boundary
conditions and load applications, can be found in [
24
]. However, this article primarily
focuses on a single span beam experiencing singular torsional moments M
T,i
applied to an
arbitrary location
xi=α·L
, while m
T
equals zero, as illustrated in Figure 3. Recognizing
the significance of boundary conditions [
13
,
25
], this article introduces both warping (k
w
)
and torsional springs (kd,x) to the static system.
The rotation
ϑ(ξ=x/L)
can be determined following the solution of the system
of equations as presented in Figure 4, using Equation (4), which includes the auxiliary
parameters denoted as X see Equation (5).
Materials 2024,17, 460 5 of 20
Materials2024,17,xFORPEERREVIEW5of20
Figure3.Staticsystemofabeamsubjectedtoasingulartorsionalmoment.
Therotationϑ
ξ = x/Lcanbedeterminedfollowingthesolutionofthesystemof
equationsaspresentedinFigure4,usingEquation(4),whichincludestheauxiliarypa-
rametersdenotedasX
seeEquation(5).
ϑξ=ϑ
ξ=ϑ
j,0+ϑ’
j,0 sinhεξ+M
T,j,0εξ-sinhεξ+M
W,j,01-coshεξ(4)
ϑ
=ϑ M
T=MT
GIT⋅L
ε k
d,x =kd,x
GIT⋅L
ε
ϑ
’=ϑ’⋅L
ε M
W=MW
GIT k
w =kw
GITε
L
(5)
Figure4.Systemofequationsforatorsionalloadedbeamwithwarpingandtorsionalsprings.
Beforetheseuniversalequationsandmethodscanbeemployedforsandwichpanels,
itisnecessarytoestablishthepanelsstiffness.Inlinewiththecurrentstateofresearch
[11,13,16,17],thefollowingstiffnessvaluesareconsideredappropriate,asproposedby
StammandWie[7].TheyintroducedthetorsionalstiffnessdenotedasGITasfollows.
GIT=4 GFe2b t1t2
t1+t21 − tanh(Ωb/2)
Ωb/2 (6)
Ω=Gxz,C
GF
t1+ t2
dC·t1t2(7)
ApplyingthewarpingtorsiontheoryonasandwichpanelinanalogytoanI-beam,
thefollowingEquation(8)isderived.
EIW=EF b3
12 e12t1+e22t2(8)
M
T,i
x
i
=α
i
Lβ
i
L
k
w,2
k
w,1
k
d,x,2
k
d,x,1
1 2
0-1 000 0 0
1,0
0
00 0 0 1+
1,0
0
00100 0 0 0
00 0 0 0 - - +0
1-1- -1
2,0
0
01-0
2,0
0
0-0 0
00 1 0 0 0 -1 0
=
Figure 3. Static system of a beam subjected to a singular torsional moment.
ϑ(ξ)=ϑ(ξ)=ϑj,0 +ϑ′j,0sinh(ε ξ )+MT,j,0 [ε ξ −sinh(ε ξ)] +MW,j,0 [1−cosh(ε ξ)] (4)
ϑ=ϑMT=MT
GIT·L
εkd,x =kd,x
GIT·L
ε
ϑ′=ϑ′·L
εMW=MW
GITkw=kw
GITε
L(5)
Materials2024,17,xFORPEERREVIEW5of20
Figure3.Staticsystemofabeamsubjectedtoasingulartorsionalmoment.
Therotationϑ
ξ = x/Lcanbedeterminedfollowingthesolutionofthesystemof
equationsaspresentedinFigure4,usingEquation(4),whichincludestheauxiliarypa-
rametersdenotedasX
seeEquation(5).
ϑξ=ϑ
ξ=ϑ
j,0+ϑ’
j,0 sinhεξ+M
T,j,0εξ-sinhεξ+M
W,j,01-coshεξ(4)
ϑ
=ϑ M
T=MT
GIT⋅L
ε k
d,x =kd,x
GIT⋅L
ε
ϑ
’=ϑ’⋅L
ε M
W=MW
GIT k
w =kw
GITε
L
(5)
Figure4.Systemofequationsforatorsionalloadedbeamwithwarpingandtorsionalsprings.
Beforetheseuniversalequationsandmethodscanbeemployedforsandwichpanels,
itisnecessarytoestablishthepanelsstiffness.Inlinewiththecurrentstateofresearch
[11,13,16,17],thefollowingstiffnessvaluesareconsideredappropriate,asproposedby
StammandWie[7].TheyintroducedthetorsionalstiffnessdenotedasGITasfollows.
GIT=4 GFe2b t1t2
t1+t21 − tanh(Ωb/2)
Ωb/2 (6)
Ω=Gxz,C
GF
t1+ t2
dC·t1t2(7)
ApplyingthewarpingtorsiontheoryonasandwichpanelinanalogytoanI-beam,
thefollowingEquation(8)isderived.
EIW=EF b3
12 e12t1+e22t2(8)
M
T,i
x
i
=α
i
Lβ
i
L
k
w,2
k
w,1
k
d,x,2
k
d,x,1
1 2
0-1 000 0 0
1,0
0
00 0 0 1+
1,0
0
00100 0 0 0
00 0 0 0 - - +0
1-1- -1
2,0
0
01-0
2,0
0
0-0 0
00 1 0 0 0 -1 0
=
Figure 4. System of equations for a torsional loaded beam with warping and torsional springs.
Before these universal equations and methods can be employed for sandwich pan-
els, it is necessary to establish the panels stiffness. In line with the current state of
research [
11
,
13
,
16
,
17
], the following stiffness values are considered appropriate, as pro-
posed by Stamm and Witte [
7
]. They introduced the torsional stiffness denoted as
GIT
as follows.
GIT=4 GFe2bt1t2
t1+t21−tanh(Ωb/2)
Ωb/2 (6)
Ω=Gxz,C
GF
t1+t2
dC·t1t2(7)
Applying the warping torsion theory on a sandwich panel in analogy to an I-beam,
the following Equation (8) is derived.
EIW=EFb3
12 e12t1+e22t2(8)
The torsion-induced stresses in the core and face sheets of the sandwich panel can be
computed using the following equations for both xz- and xy-stress components. Here, it
should be noticed that, in contrast to the shear stress distribution of a torsion-loaded I-beam,
Materials 2024,17, 460 6 of 20
the primary shear stress in the “flanges” (here: face sheets) is not uniformly distributed
along the panel width but is hyperbolic with a peak at mid-width.
τxz,I,C =Ωsinh(Ωy)
cosh(Ωb/2)−sinh(Ωb/2)
Ωb/2
MT,I
2be (9)
τxy, I,F =cosh(Ωb/2)−cosh(Ωy)
cosh(Ωb/2)−sinh(Ωb/2)
Ωb/2
MT,I
2bet1,2 (10)
τxy,II,F =MT,II Sw
IWt1,2 (11)
σw,F =MW
IWt1,2 ωM(12)
2. Parametric Analysis Methods for Eccentrically Loaded Sandwich Panels
2.1. General
To comprehensively assess warping torsion in sandwich panels, a series of experi-
mental and numerical parameter studies was conducted. PU foam core sandwich panels
with quasi-planar face sheets, commonly employed in European building envelopes, are
the subjects of these studies. Various dimensions and material properties were taken into
account, while longitudinal joints were neglected. The primary focus of these studies is
directed towards the structural response, particularly in terms of rotation, as well as normal
and shear stresses in the face sheets and core.
The experimental study supplied crucial data on actual sandwich panels used in
construction and permits the validation of the numerical model in ANSYS. In contrast, the
numerical study, based on a calibrated model, provides a valuable advantage by enhancing
the experimental scope and offering deeper insights into aspects that may not be readily
determined through direct measurements.
2.2. Experimental Study
2.2.1. Test Setup and Measurements
As part of the experimental studies, a newly developed eccentric 6-point bending test
was conducted on several PU sandwich panels from two different manufacturers. This
test setup is primarily derived from the standardized test used to assess the load-bearing
capacity of sandwich panels in accordance with EN 14509:2013, Annex A. The significant
modifications to the conventional test are elaborated upon in the following and can also be
referenced in Figure 5.
•
In addition to the bending loads, a torsional moment is induced by the eccentric
application (e = 350 mm) of four-point loads.
•
The point loads are applied to the top face sheet using a steel plate with the dimensions
100 mm ×150 mm ×10 mm and an elastomer layer underneath.
•The pin support on both sides is designed based on a fork support.
•
The maximum load was set at 50% of the experimentally determined ultimate load for
the centrically loaded reference specimen, ensuring that the tested materials remain
within the linear-elastic range. This reference specimen was exclusively subjected to
bending following EN 14509:2013, A.5.
Materials 2024,17, 460 7 of 20
Materials2024,17,xFORPEERREVIEW7of20
(a)(b)
Figure5.
Eccentric6-pointbendingtestsetup:(
a
)fullview;(
b
)lateralperspective.
Figure6illustratesthetestsetupschematicallyinthetopview.Here,thearrange-
mentofthebasicmeasurementscarriedoutforalltestspecimensareshown.Intotal,nine
straingauges(S1–S9)wereappliedonboththetopandtheboomfacesheets,positioned
atmidspan(x
1
)andbetweenthethirdandfourthloadintroductionpoints(x
2
).Addition-
ally,fourdisplacementtransducers(D1–D4)werepositionedatx
1
andx
2
toassessthe
verticaldisplacementoftheboomfacesheet.Supplementarymeasurementswerealso
taken.However,theyarenotexhaustivelydiscussedinthisarticle.
Figure6.
Tes tsetupintopviewandarrangementofthebasicmeasurement.
x,ξ
y,η
b
L
s
/2
L
s
/2
load introduction (F
z
):F1–F4
strain gauges (ε
x
):S1–S9
displacement transducers (u
z
):D1–D4
x
2
=
S3|S6
D2
S2|S5
S1|S4
F1
S9
S8
S7
D1
D4
D3
F2 F3 F4
topface layerX
bottom face layerX
Legend
x
1
x
2
L
s
/8 L
s
/4 L
s
/4 L
s
/4 L
s
/8
L
s
=L-100mm
x
1
=L
s
/2
e
e=350mm
L
s
/2+1,100mm|TA-serie
L
s
/2+1,475mm|TJ-serie
Figure 5. Eccentric 6-point bending test setup: (a) full view; (b) lateral perspective.
Figure 6illustrates the test setup schematically in the top view. Here, the arrangement
of the basic measurements carried out for all test specimens are shown. In total, nine strain
gauges (S1–S9) were applied on both the top and the bottom face sheets, positioned at
midspan (x
1
) and between the third and fourth load introduction points (x
2
). Additionally,
four displacement transducers (D1–D4) were positioned at x1and x2to assess the vertical
displacement of the bottom face sheet. Supplementary measurements were also taken.
However, they are not exhaustively discussed in this article.
Materials2024,17,xFORPEERREVIEW7of20
(a)(b)
Figure5.
Eccentric6-pointbendingtestsetup:(
a
)fullview;(
b
)lateralperspective.
Figure6illustratesthetestsetupschematicallyinthetopview.Here,thearrange-
mentofthebasicmeasurementscarriedoutforalltestspecimensareshown.Intotal,nine
straingauges(S1–S9)wereappliedonboththetopandtheboomfacesheets,positioned
atmidspan(x
1
)andbetweenthethirdandfourthloadintroductionpoints(x
2
).Addition-
ally,fourdisplacementtransducers(D1–D4)werepositionedatx
1
andx
2
toassessthe
verticaldisplacementoftheboomfacesheet.Supplementarymeasurementswerealso
taken.However,theyarenotexhaustivelydiscussedinthisarticle.
Figure6.
Tes tsetupintopviewandarrangementofthebasicmeasurement.
x,ξ
y,η
b
L
s
/2
L
s
/2
load introduction (F
z
):F1–F4
strain gauges (ε
x
):S1–S9
displacement transducers (u
z
):D1–D4
x
2
=
S3|S6
D2
S2|S5
S1|S4
F1
S9
S8
S7
D1
D4
D3
F2 F3 F4
topface layerX
bottom face layerX
Legend
x
1
x
2
L
s
/8 L
s
/4 L
s
/4 L
s
/4 L
s
/8
L
s
=L-100mm
x
1
=L
s
/2
e
e=350mm
L
s
/2+1,100mm|TA-serie
L
s
/2+1,475mm|TJ-serie
Figure 6. Test setup in top view and arrangement of the basic measurement.
Materials 2024,17, 460 8 of 20
2.2.2. Fork Support
In the context of the eccentric 6-point bending test, the utilization of a fork support on
both sides proves beneficial for investigating the structural response of sandwich panels
under torsion. However, achieving an ideal fork support in practical applications, especially
for sandwich panels, composed of materials with significantly different stiffnesses, is a
challenging endeavor. The primary aim was to develop a support system that fulfills
the following requirements: ideally preventing rotations caused by a torsional moment
(M
T
), allowing warping to a certain extent, and remaining hinged concerning bending
moments (My).
As depicted in Figure 7, the newly devised bearing construction represents an op-
timized iteration of the one utilized in a previous study [
14
]. The sandwich panels are
positioned upon a flat steel plate (I) that is welded to a cylindrical steel element (II) sup-
ported by a roller bearing (III), enabling it to rotate with minimal friction. The roller
bearings are securely attached to the substructure through a fixed support (IV). To restrict
deformations of the cylindrical steel, three semi-circular plain bearings are symmetrically
arranged beneath it. At the top of the sandwich panel, a square hollow section (V) is
positioned and secured to the flat steel using threaded rods (VI) to prevent rotation around
the longitudinal axis of the panel.
Materials2024,17,xFORPEERREVIEW8of20
2.2.2.ForkSupport
Inthecontextoftheeccentric6-pointbendingtest,theutilizationofaforksupport
onbothsidesprovesbeneficialforinvestigatingthestructuralresponseofsandwichpan-
elsundertorsion.However,achievinganidealforksupportinpracticalapplications,es-
peciallyforsandwichpanels,composedofmaterialswithsignificantlydifferentstiff-
nesses,isachallengingendeavor.Theprimaryaimwastodevelopasupportsystemthat
fulfillsthefollowingrequirements:ideallypreventingrotationscausedbyatorsionalmo-
ment(M
T
),allowingwarpingtoacertainextent,andremaininghingedconcerningbend-
ingmoments(M
y
).
AsdepictedinFigure7,thenewlydevisedbearingconstructionrepresentsanopti-
mizediterationoftheoneutilizedinapreviousstudy[14].Thesandwichpanelsarepo-
sitioneduponaflatsteelplate(I)thatisweldedtoacylindricalsteelelement(II)supported
byarollerbearing(III),enablingittorotatewithminimalfriction.Therollerbearingsare
securelyaachedtothesubstructurethroughafixedsupport(IV).Torestrictdefor-
mationsofthecylindricalsteel,threesemi-circularplainbearingsaresymmetricallyar-
rangedbeneathit.Atthetopofthesandwichpanel,asquarehollowsection(V)isposi-
tionedandsecuredtotheflatsteelusingthreadedrods(VI)topreventrotationaround
thelongitudinalaxisofthepanel.
Figure7.
Newlydevelopedbearingconstruction:(
a
)sideview;(
b
)frontview.
2.2.3.TestProgram
Duringtheexperimentalstudy,atotalof20distinctconfigurationsweretested.In
thiscontext,aconfigurationisdefinedasthespecificparametersetoftheinvestigated
sandwichpanels,seeFigure8.TheanalyzedparametersincludethetotallengthL,the
corethicknessD,andthefacesheetthicknesst
1
andt
2
,respectively.Eachconfiguration
wasassessedwithuptothreespecimens,resultinginatotalnumberof52tests.Thepa-
rameterrangesfortheseconfigurationsarelistedinTable1.
Figure 7. Newly developed bearing construction: (a) side view; (b) front view.
2.2.3. Test Program
During the experimental study, a total of 20 distinct configurations were tested. In
this context, a configuration is defined as the specific parameter set of the investigated
sandwich panels, see Figure 8. The analyzed parameters include the total length L, the core
thickness D, and the face sheet thickness t
1
and t
2
, respectively. Each configuration was
assessed with up to three specimens, resulting in a total number of 52 tests. The parameter
ranges for these configurations are listed in Table 1.
Materials 2024,17, 460 9 of 20
Materials2024,17,xFORPEERREVIEW9of20
Figure8.Definitionoftheconfiguration.
Tab le1.Assessedparameterrangefortheeccentric6-pointbendingtests.
ParameterSymbolValue s
TotallengthL4000–6000mm
CorethicknessD40–220mm
Sheetthicknesst1,t20.40–0.63mm
Toensurecomparability,auniformtotalwidthofb=900mmwasmaintainedacross
allthetestedsandwichpanels.Infact,thegeometryanddesignofthelongitudinaljoints
significantlyvariesdependingonthemanufacturerandthecorethickness.Therefore,
thesejointsweresymmetricallyremovedfrombothsidesbeforethetests.Thewidthin-
dicatedherereferstotheremainingwidth(b=900mm).
2.2.4.MaterialTests
ThematerialpropertiesofthePUcoresandwichpanelsunderexaminationwere
testedatroomtemperatureinaccordancewithEN14509:2013,AnnexA.Table2presents
asummaryofthevaluerangesfortheshearandYoung’smodulusofthespecimen.
Tab le2.Assessedparameterrangefortheeccentric6-pointbendingtests.
ParameterSymbolMeanVal uesinMPa
ShearmodulusGC,xz3.0–4.4
Young’smodulus(compression)ECc3.3–5.2
Young’smodulus(tension)ECt3.3–4.7
2.3.NumericalStudy
Forthenumericalinvestigation,aparametric3DFEmodelofthetestsdescribedin
Section2.2.wasdevelopedinANSYSWorkbench2022R2[26],asshowninFigure9.In
thismodel,thefacesheetswereidealizedasplanesurfacesusing4-nodedshellelements
(SHELL281),whilethecorewasmodelledwith8-nodedsolidelements(SOLID185).Sim-
ilarsolidelementswereemployedforthebearingstructure,exceptforthethreadedrods,
forwhichBEAM188elementswerechosen.
TA100_6_6.3−5_1
Torsion
manufacturer
Dinmm
t
1
in10
−1
mm
t
2
in10
−1
mm
no.
Linm
Figure 8. Definition of the configuration.
Table 1. Assessed parameter range for the eccentric 6-point bending tests.
Parameter Symbol Values
Total length L 4000–6000 mm
Core thickness D 40–220 mm
Sheet thickness t1, t20.40–0.63 mm
To ensure comparability, a uniform total width of b = 900 mm was maintained across
all the tested sandwich panels. In fact, the geometry and design of the longitudinal joints
significantly varies depending on the manufacturer and the core thickness. Therefore, these
joints were symmetrically removed from both sides before the tests. The width indicated
here refers to the remaining width (b = 900 mm).
2.2.4. Material Tests
The material properties of the PU core sandwich panels under examination were
tested at room temperature in accordance with EN 14509:2013, Annex A. Table 2presents a
summary of the value ranges for the shear and Young’s modulus of the specimen.
Table 2. Assessed parameter range for the eccentric 6-point bending tests.
Parameter Symbol Mean Values in MPa
Shear modulus GC,xz 3.0–4.4
Young’s modulus (compression) ECc 3.3–5.2
Young’s modulus (tension) ECt 3.3–4.7
2.3. Numerical Study
For the numerical investigation, a parametric 3D FE model of the tests described in
Section 2.2. was developed in ANSYS Workbench 2022 R2 [
26
], as shown in Figure 9. In
this model, the face sheets were idealized as plane surfaces using 4-noded shell elements
(SHELL281), while the core was modelled with 8-noded solid elements (SOLID185). Similar
solid elements were employed for the bearing structure, except for the threaded rods, for
which BEAM188 elements were chosen.
The steel face sheets of the sandwich panel and all other steel components were
characterized using an isotropic linear elastic material model with E
F
= 210,000 MPa and
µF
= 0.3. The PU foam sandwich core was defined as an anisotropic linear elastic material
with shear and Young’s moduli set equally in all directions, and a Poisson’s ratio
µC
of 0.25,
according to [13,18,23]. The mesh size was specified at 25 mm on average.
Materials 2024,17, 460 10 of 20
Materials2024,17,xFORPEERREVIEW10of20
Figure9.
NumericalmodelinANSYSWorkbench2022R2,exemplaryfortheconfiguration
TA160_5-6.3-5:(
a
)intheundeformedstate;(
b
)inthedeformedstate(F=3kN).
Thesteelfacesheetsofthesandwichpanelandallothersteelcomponentswerechar-
acterizedusinganisotropiclinearelasticmaterialmodelwithE
F
=210,000MPaandµ
F
=
0.3.ThePUfoamsandwichcorewasdefinedasananisotropiclinearelasticmaterialwith
shearandYou ng’smodulisetequallyinalldirections,andaPoisson’sratioµ
C
of0.25,
accordingto[13,18,23].Themeshsizewasspecifiedat25mmonaverage.
Boundaryconditionswereestablishedfortherollerbearingsbydefiningageneral
“Body-Ground”connection,fixingtherelevanttranslationsandrotations.Sincetheface
sheetsandcoreweretreatedasonecomponentintheANSYStoolDesignModelerthere
wasnoneedtoexplicitlydefinecontactwithinthesandwichpanel.Incontrast,contact
elementsbetweenthesandwichpanelandthebearingstructureweremanuallydefined
asCONTA174
and
TARGE170,respectively.Forboththeupperandlowerinterfaces,three
typesofcontactwereconsidered:bonded,rough,andfrictional.Thecontacttypesdiffer
intheforcestheytransfer.Bondedcontacttransmitscompressive,tensile,andshear
forces,whereasthenon-linearcontacts(roughandfrictional)excludetensileforcesbut
transmitshearandcompressiveforces.Inthefrictionaltype,shearforcesaretransmied
basedonthefrictionvalue.Forthenon-linearcontacttypes,the“LargeDeflection”seing
isconfiguredasrecommended,consideringtheeffectsfromsecond-ordertheory[26].
Thenumericalstudyisdividedintotwodistinctparts,withtechnicaldataconcerning
materialandgeometricalbeingcategorizedaccordingly.
Inthevalidationofthenumericalmodel,africtionalcontacttypewasselectedand
theexperimentallydeterminedmaterialpropertiesanddimensionswereemployed,
asdetailedinTables1and2.
Inthesubsequentextendednumericalinvestigations,aroughcontacttypewascho-
sen.Table 3providesinformationregardingtheconfigurationofthereferencemodels
fortheextendednumericalstudyandtherangeoftheanalyzedparametersforpo-
tentialpracticalusecases.Thevaluesspecifiedfortheelasticandshearmoduliapply
uniformlyinallthreespatialdirectionsunlessotherwisespecified.
Figure 9. Numerical model in ANSYS Workbench 2022 R2, exemplary for the configuration
TA160_5-6.3-5: (a) in the undeformed state; (b) in the deformed state (F = 3 kN).
Boundary conditions were established for the roller bearings by defining a general
“Body-Ground” connection, fixing the relevant translations and rotations. Since the face
sheets and core were treated as one component in the ANSYS tool DesignModeler there
was no need to explicitly define contact within the sandwich panel. In contrast, contact
elements between the sandwich panel and the bearing structure were manually defined as
CONTA174 and TARGE170, respectively. For both the upper and lower interfaces, three
types of contact were considered: bonded, rough, and frictional. The contact types differ in
the forces they transfer. Bonded contact transmits compressive, tensile, and shear forces,
whereas the non-linear contacts (rough and frictional) exclude tensile forces but transmit
shear and compressive forces. In the frictional type, shear forces are transmitted based
on the friction value. For the non-linear contact types, the “Large Deflection” setting is
configured as recommended, considering the effects from second-order theory [26].
The numerical study is divided into two distinct parts, with technical data concerning
material and geometrical being categorized accordingly.
•
In the validation of the numerical model, a frictional contact type was selected and the
experimentally determined material properties and dimensions were employed, as
detailed in Tables 1and 2.
•
In the subsequent extended numerical investigations, a rough contact type was chosen.
Table 3provides information regarding the configuration of the reference models for
the extended numerical study and the range of the analyzed parameters for poten-
tial practical use cases. The values specified for the elastic and shear moduli apply
uniformly in all three spatial directions unless otherwise specified.
Furthermore, two load situations (LS) were considered for the simulation with respect
to the extended numerical studies. They involved uniformly distributed surface loads
with deformable behavior acting on a defined area (indicated in parentheses). Primarily,
in LS I, four vertical eccentric surface loads F were applied (150
×
100 mm), similar to the
eccentric 6-point bending test. Secondly, in LS II, a superposition of “pure” bending and
“pure” torsion loads was introduced. For the bending part, four vertical centric surface
loads p (150 mm
×
b) were applied, and for the torsional part, opposite-directed horizontal
surface loads (L
×
b) in the plane of the face sheets were introduced, see Figure 10. Both
load situations were applied in a single time step each.
Materials 2024,17, 460 11 of 20
Table 3. Reference values and parameter range for the extended numerical study.
Parameter Symbol Reference Parameter Range
shear modulus 1GC4 MPa 2–6 MPa
Young’s modulus EC4 MPa 4 MPa
Poisson’s ratio µC0.25 0.25
core thickness D 100 mm 40–250 mm
total width b 900 mm 450–1350 mm
total length L 5000 mm 3000–8000 mm
sheet thickness 1t1, t20.5 mm 0.4–0.6 mm
total load 1F 3 kN 1 kN–5 kN
1
In the present article, results from the extended numerical study are shown only with G
C
= 4 MPa,
t1= t2= 0.5 mm, and F = 3 kN.
Materials2024,17,xFORPEERREVIEW11of20
Tab le3.Referencevaluesandparameterrangefortheextendednumericalstudy.
ParameterSymbolReferenceParameterRange
shearmodulus1GC4MPa2–6MPa
Young’smodulusEC4MPa4MPa
Poisson’sratioµC0.250.25
corethicknessD100mm40–250mm
totalwidthb900mm450–1350mm
totallengthL5000mm3000–8000mm
sheetthickness1t1,t20.5mm0.4–0.6mm
totalload1F3kN1kN–5kN
1Inthepresentarticle,resultsfromtheextendednumericalstudyareshownonlywithGC=4MPa,
t1=t2=0.5mm,andF=3kN.
Furthermore,twoloadsituations(LS)wereconsideredforthesimulationwithre-
specttotheextendednumericalstudies.Theyinvolveduniformlydistributedsurface
loadswithdeformablebehavioractingonadefinedarea(indicatedinparentheses).Pri-
marily,inLSI,fourverticaleccentricsurfaceloadsFwereapplied(150×100mm),similar
totheeccentric6-pointbendingtest.Secondly,inLSII,asuperpositionof“pure”bending
and“pure”torsionloadswasintroduced.Forthebendingpart,fourverticalcentricsur-
faceloadsp(150mm×b)wereapplied,andforthetorsionalpart,opposite-directedhor-
izontalsurfaceloads(L×b)intheplaneofthefacesheetswereintroduced,seeFigure10.
Bothloadsituationswereappliedinasingletimestepeach.
Figure10.LoadsituationsI(6-pointbendingtests)andII(superpositionofbendingandtorsion)
illustratedonthecross-sectionofasandwichpanel.Inparentheses,theareaonwhichtheindividual
surfaceloadsareapplied.
2.4.AnalyticalStudy
Inthisarticle,theanalyticalstudyservesthespecificpurposeofestablishingrelation-
shipsbetweentheoutcomesoftheexperimentalandnumericalmethods,particularly
withregardtowarpingtorsion.Forthewarpingtorsiontheorycalculations,formulasde-
scribedinSection1.3wereappliedwithandwithouttorsionalandwarpingsprings.
Intheconsiderationofsprings,atest-basedbest-fitmethodwasemployed,usingthe
MethodofLeastSquaresandadoptingthebasicapproachoutlinedin[27].Thisapproach
aimedtodeterminethespringvalueskd,xandkwforeachtestedorsimulatedconfiguration
ofsandwichpanels.Ifnotexplicitlymentionedotherwise,thiscasewasappliedinthis
articlewhenreferringtotheanalyticalapproach.Inthealternativecaseofneglectingat
leastonespringtype,kw→0orkd,x→∞ wasapplied,respectively.
Thesuperpositionprinciplewasappliedconcerningbothbendingandtorsion,as
wellasthefour-pointload,inaccordancewiththeinitialassumptionthattestingandsim-
ulationoccurredwithinthelinearelasticrangeofthesandwichpanels.Forthebending
behavior,therelevantequationsforthedisplacementandstressanalysisofthefacesheets
andthecoreofthesandwichpanelswereutilizedfrom[7].
LSILSII
4×F
4×p
h
h
(150mm× 100mm)
(L×b)
(150mm×b)
Figure 10. Load situations I (6-point bending tests) and II (superposition of bending and torsion)
illustrated on the cross-section of a sandwich panel. In parentheses, the area on which the individual
surface loads are applied.
2.4. Analytical Study
In this article, the analytical study serves the specific purpose of establishing relation-
ships between the outcomes of the experimental and numerical methods, particularly with
regard to warping torsion. For the warping torsion theory calculations, formulas described
in Section 1.3 were applied with and without torsional and warping springs.
In the consideration of springs, a test-based best-fit method was employed, using the
Method of Least Squares and adopting the basic approach outlined in [
27
]. This approach
aimed to determine the spring values k
d,x
and k
w
for each tested or simulated configuration
of sandwich panels. If not explicitly mentioned otherwise, this case was applied in this
article when referring to the analytical approach. In the alternative case of neglecting at
least one spring type, kw→0 or kd,x →∞was applied, respectively.
The superposition principle was applied concerning both bending and torsion, as well
as the four-point load, in accordance with the initial assumption that testing and simulation
occurred within the linear elastic range of the sandwich panels. For the bending behavior,
the relevant equations for the displacement and stress analysis of the face sheets and the
core of the sandwich panels were utilized from [7].
3. Results from Experimental and Numerical Studies Considering Warping Torsion
3.1. General
First, representative experimental results are shown in conjunction with the numerical
and analytical solutions. Second, the results of the extended numerical study beyond the
scope of the experiments are provided.
3.2. Experimental Results and Validation of the Numerical Model
Figure 11 shows the experimentally determined rotation–load (a) and the strain–load
(b) relationships for each tested configuration. In both diagrams, a clear, nearly linear
association is observed between the applied load and the rotation or strain at midspan. The
rotation
ϑ
is calculated under the assumption of an idealized, linearized deformation of
Materials 2024,17, 460 12 of 20
the face sheets across the width of the panel. The value u
z
represents the displacement
transducers, and c signifies the distance between these measurement points.
ϑ(x1)=uz, D1 −uz,D2
c(13)
Materials2024,17,xFORPEERREVIEW12of20
3.ResultsfromExperimentalandNumericalStudiesConsideringWarpi ngTorsion
3.1.General
First,representativeexperimentalresultsareshowninconjunctionwiththenumeri-
calandanalyticalsolutions.Second,theresultsoftheextendednumericalstudybeyond
thescopeoftheexperimentsareprovided.
3.2.ExperimentalResultsandValidationoftheNumericalModel
Figure11showstheexperimentallydeterminedrotation–load(a)andthestrain–load
(b)relationshipsforeachtestedconfiguration.Inbothdiagrams,aclear,nearlylinearas-
sociationisobservedbetweentheappliedloadandtherotationorstrainatmidspan.The
rotationϑiscalculatedundertheassumptionofanidealized,linearizeddeformationof
thefacesheetsacrossthewidthofthepanel.Thevalueuzrepresentsthedisplacement
transducers,andcsignifiesthedistancebetweenthesemeasurementpoints.
ϑ x u, u,
c(13)
ThemeasuredvalueofthestraingaugeS3,locatedatx=L/2,y=−b/2+50mm,z=
−D/2isexpressedinµm/m.Inthefollowing,theresultingnormalstressisprovidedin
MPabymultiplyingthesestrainvalueswiththemodulusofelasticityofE=210,000MPa.
(a)(b)
Figure11.Loaddiagrams:(a)rotation–loadrelation;(b)strain–loadrelation.
InthefollowingfourdiagramsinFigures12and13,thefocusisonconfigurations
TA160_5_6.3-5andTA40_5_6.3-5withanappliedloadofF=3kN.Figure12displaysthe
deflectioncurvesoftheboomfacesheetatpointsx1(midspan)andx2(betweenthethird
andfourthloadintroduction).Thedatapointsobtainedfromdisplacementtransducers
D1toD4closelyalignwiththeredandbluecurves,indicatinggoodagreement.
0,00
0,01
0,02
0,03
0,04
0246810
rotation
ϑinrad
load
FinkN
rotation‐load relation (exp.)
x=L/2,z=D/2
0.03
0.02
0.01
0.00
TA160_6_6.3-5TJ220_6_4-4 TA160_5_6.3-5
TA160_4_6.3-5
TA120_6_5-4
TA120_5_5-4
TA120_4_5-4
TA100_6_6.3-5
TA100_5_6.3-5
TA100_4_6.3-5
TA40_5_6.3-5
TA40_4_6.3-5
TJ220_5_4-4
TJ220_4_4-4
TJ150_6_4-4
TJ150_5_4-4
TJ150_4_4-4
TJ120_6_4-4
TJ120_5_4-4
TJ120_4_4-4
0.04
-800
-600
-400
-200
0
0246810
strain
ε
x
inµm/m
load
FinkN
strain‐load relation (exp.)
x=L/2,y=−b/2+50mm,z=−D/2
TA160_6_6.3-5TJ220_6_4-4 TA160_5_6.3-5
TA160_4_6.3-5
TA120_6_5-4
TA120_5_5-4
TA120_4_5-4
TA100_6_6.3-5
TA100_5_6.3-5
TA100_4_6.3-5
TA40_5_6.3-5
TA40_4_6.3-5
TJ220_5_4-4
TJ220_4_4-4
TJ150_6_4-4
TJ150_5_4-4
TJ150_4_4-4
TJ120_6_4-4
TJ120_5_4-4
TJ120_4_4-4
Figure 11. Load diagrams: (a) rotation–load relation; (b) strain–load relation.
The measured value of the strain gauge S3, located at x = L/2, y =
−
b/2 + 50 mm,
z =
−
D/2 is expressed in
µ
m/m. In the following, the resulting normal stress is provided in
MPa by multiplying these strain values with the modulus of elasticity of E = 210,000 MPa.
In the following four diagrams in Figures 12 and 13, the focus is on configurations
TA160_5_6.3-5 and TA40_5_6.3-5 with an applied load of F = 3 kN. Figure 12 displays the
deflection curves of the bottom face sheet at points x
1
(midspan) and x
2
(between the third
and fourth load introduction). The data points obtained from displacement transducers D1
to D4 closely align with the red and blue curves, indicating good agreement.
Materials2024,17,xFORPEERREVIEW13of20
(a)(b)
Figure12.Rotationofthepanelanddeflectionoftheboomfacesheetacrossthepanelwidthatx1
andx2,shownasexemplaryfortwoconfigurations:(a)TA160_5_6.3-5;(b)TA40_5_6.3-5.
Figure13displaysthedistributionofnormalstressinthetopfacesheetacrossthe
widthofthepanelatpointsx1andx2.InFigure13a,anearlylinearcurveisevident,while
inFigure13b,itexhibitsnon-linearity.Inbothcases,thenumericalcurvescloselymatch
theexperimentaldatapoints,whichisnotthecasefortheanalyticalapproachofpart(b).
Similarresultshavebeenobtainedfortheboomfacesheetbutarenotfurtherdiscussed.
(a)(b)
Figure13.Normalstressofthetopfacesheetacrossthepanelwidthatx1andx2,shownasexem-
plaryfortwoconfigurations:(a)TA160_5_6.3-5;(b)TA40_5_6.3-5.
Tovalidateandquantifythepresentedqualitativealignmentbetweentheexperi-
mentalandnumericalapproach,scaerplotsincludingcoefficientofdeterminationR2are
introduced,includingall20testedandsimulatedsandwichpanelconfigurationsconcern-
ingtherotationandthenormalstressatmidspan,seeFigure14.Forbothmagnitudes,the
R2valueishigh(>0.95),whichprovesthereisastrongcorrelationbetweentheapproaches
acrossthewiderangeofconsideredparametersandsupportsthequalitativecompliance
shownabove.Atthesametime,thenumericalmodelisvalidated.Inaddition,theresult
oftheanalyticalcalculationshowsacomparablyhighcorrelationwiththeothertwoap-
proaches.
0
2
4
6
8
10
12
-450 -350 -250 -150 -50 50 150 250 350 450
u
z
inmm
yinmm
vertical displacement u
z
x=x
1
andx
2
,z=D/2
TA160_5_6.3−5_1(exp.)
TA160_5_6.3−5_2(exp.)
TA160_5_6.3−5_3(exp.)
numerical
analytical
x
1
x
2
configuration:
TA160_5_6.3−5
0
20
40
60
80
100
120
140
-450 -350 -250 -150 -50 50 150 250 350 450
u
z
inmm
yinmm
vertical displacement u
z
x=x
1
andx
2
,z=D/2
TA40_5_6.3−5_1(exp.)
TA40_5_6.3−5_2(exp.)
TA40_5_6.3−5_3(exp.)
numerical
analytical
x
1
x
2
configuration:
TA40_5_6.3−5
x
1
=L/2
-35
-30
-25
-20
-15
-10
-5
0
-450 -350 -250 -150 -50 50 150 250 350 450
σ
x
inMPa
yinmm
normalstressσ
x
x=x
1
andx
2
,z=−D/2
TA160_5_6.3−5_1(exp.)
TA160_5_6.3−5_2(exp.)
TA160_5_6.3−5_3(exp.)
numerical
analytical
x
1
x
2
configuration:
TA160_5_6.3−5
-120
-100
-80
-60
-40
-20
0
-450 -350 -250 -150 -50 50 150 250 350 450
σ
x
inMPa
yinmm
normalstressσ
x
x=x
1
andx
2
,z=−D/2
TA40_5_6.3−5_1(exp.)
TA40_5_6.3−5_2(exp.)
TA40_5_6.3−5_3(exp.)
numerical
analytical
x
1
x
2
configuration:
TA40_5_6.3−5
Figure 12. Rotation of the panel and deflection of the bottom face sheet across the panel width at x
1
and x2, shown as exemplary for two configurations: (a) TA160_5_6.3-5; (b) TA40_5_6.3-5.
Materials 2024,17, 460 13 of 20
Materials2024,17,xFORPEERREVIEW13of20
(a)(b)
Figure12.Rotationofthepanelanddeflectionoftheboomfacesheetacrossthepanelwidthatx1
andx2,shownasexemplaryfortwoconfigurations:(a)TA160_5_6.3-5;(b)TA40_5_6.3-5.
Figure13displaysthedistributionofnormalstressinthetopfacesheetacrossthe
widthofthepanelatpointsx1andx2.InFigure13a,anearlylinearcurveisevident,while
inFigure13b,itexhibitsnon-linearity.Inbothcases,thenumericalcurvescloselymatch
theexperimentaldatapoints,whichisnotthecasefortheanalyticalapproachofpart(b).
Similarresultshavebeenobtainedfortheboomfacesheetbutarenotfurtherdiscussed.
(a)(b)
Figure13.Normalstressofthetopfacesheetacrossthepanelwidthatx1andx2,shownasexem-
plaryfortwoconfigurations:(a)TA160_5_6.3-5;(b)TA40_5_6.3-5.
Tovalidateandquantifythepresentedqualitativealignmentbetweentheexperi-
mentalandnumericalapproach,scaerplotsincludingcoefficientofdeterminationR2are
introduced,includingall20testedandsimulatedsandwichpanelconfigurationsconcern-
ingtherotationandthenormalstressatmidspan,seeFigure14.Forbothmagnitudes,the
R2valueishigh(>0.95),whichprovesthereisastrongcorrelationbetweentheapproaches
acrossthewiderangeofconsideredparametersandsupportsthequalitativecompliance
shownabove.Atthesametime,thenumericalmodelisvalidated.Inaddition,theresult
oftheanalyticalcalculationshowsacomparablyhighcorrelationwiththeothertwoap-
proaches.
0
2
4
6
8
10
12
-450 -350 -250 -150 -50 50 150 250 350 450
u
z
inmm
yinmm
vertical displacement u
z
x=x
1
andx
2
,z=D/2
TA160_5_6.3−5_1(exp.)
TA160_5_6.3−5_2(exp.)
TA160_5_6.3−5_3(exp.)
numerical
analytical
x
1
x
2
configuration:
TA160_5_6.3−5
0
20
40
60
80
100
120
140
-450 -350 -250 -150 -50 50 150 250 350 450
u
z
inmm
yinmm
vertical displacement u
z
x=x
1
andx
2
,z=D/2
TA40_5_6.3−5_1(exp.)
TA40_5_6.3−5_2(exp.)
TA40_5_6.3−5_3(exp.)
numerical
analytical
x
1
x
2
configuration:
TA40_5_6.3−5
x
1
=L/2
-35
-30
-25
-20
-15
-10
-5
0
-450 -350 -250 -150 -50 50 150 250 350 450
σ
x
inMPa
yinmm
normalstressσ
x
x=x
1
andx
2
,z=−D/2
TA160_5_6.3−5_1(exp.)
TA160_5_6.3−5_2(exp.)
TA160_5_6.3−5_3(exp.)
numerical
analytical
x
1
x
2
configuration:
TA160_5_6.3−5
-120
-100
-80
-60
-40
-20
0
-450 -350 -250 -150 -50 50 150 250 350 450
σ
x
inMPa
yinmm
normalstressσ
x
x=x
1
andx
2
,z=−D/2
TA40_5_6.3−5_1(exp.)
TA40_5_6.3−5_2(exp.)
TA40_5_6.3−5_3(exp.)
numerical
analytical
x
1
x
2
configuration:
TA40_5_6.3−5
Figure 13. Normal stress of the top face sheet across the panel width at x
1
and x
2
, shown as exemplary
for two configurations: (a) TA160_5_6.3-5; (b) TA40_5_6.3-5.
Figure 13 displays the distribution of normal stress in the top face sheet across the
width of the panel at points x
1
and x
2
. In Figure 13a, a nearly linear curve is evident, while
in Figure 13b, it exhibits non-linearity. In both cases, the numerical curves closely match
the experimental data points, which is not the case for the analytical approach of part (b).
Similar results have been obtained for the bottom face sheet but are not further discussed.
To validate and quantify the presented qualitative alignment between the experimental
and numerical approach, scatter plots including coefficient of determination R
2
are intro-
duced, including all 20 tested and simulated sandwich panel configurations concerning the
rotation and the normal stress at midspan, see Figure 14. For both magnitudes, the R
2
value
is high (>0.95), which proves there is a strong correlation between the approaches across
the wide range of considered parameters and supports the qualitative compliance shown
above. At the same time, the numerical model is validated. In addition, the result of the
analytical calculation shows a comparably high correlation with the other two approaches.
Materials2024,17,xFORPEERREVIEW14of20
(a)(b)
Figure14.Scaerplotsoftheexperimental,numerical,andanalyticalresults:(a)rotationϑ(x=L/2,
z=D/2);(b)normalstressσx(x=L/2,y=−b/2+50mm,z=−D/2).
3.3.FurtherNumericalStudies
Intheexperimentalstudy,practicableconstraintsmaylimittheabilitytomeasureall
pointsofinterestindetail.Consequently,specificpointsareselectivelyanalyzedandas-
sessedinthenumericalapproach.Furthermore,significantinsightscanbeextractedfrom
thenumericalparametricstudytobreakdownthefundamentalstructuralbehaviorof
eccentricallyloadedsandwichpanelsintospecificparametersorparameterratios.This
processfacilitatesidentifyingpotentialexpressionsforengineeringcalculationmethods.
3.3.1.ShearStressAnalysis
Tohighlighttheshearstressdistribution,thenumericalresultsofthetworepresenta-
tiveconfigurationsTA160_5_6.3-5andTA40_5_6.3-5areshowninrelationtotheanalyti-
calsolutionsinFigure15.Ontherightside(b),thestressesinthemiddleofthecorewere
calculatedbysuperimposingtheverticalforceandtorsion.Fortheverticalforce,auni-
formlydistributedcoreshearstressacrossthepanelwidthwasassumed,neglectingthe
effectsoflocalloadintroduction.
(a)(b)
Figure15.Shearstressdistributionacrossthepanelwidth:(a)shearstressofthefacesheets(x=x2,
z=−D/2);(b)shearstressofthecore(x=x2,z=0).
0,000
0,005
0,010
0,015
0,020
0,025
0,030
0,000 0,005 0,010 0,015 0,020 0,025 0,030
experimental
ϑinrad
numerical |analytical
ϑinrad
rotation ϑ
x=L/2,z=D/2
R²=0.99
R²=0.98
0.030
0.025
0.020
0.010
0.005
0.000
0.000
0.005 0.010 0.015 0.020 0.025 0.030
0.015
alltested configurations
numerical
analytical
-120
-100
-80
-60
-40
-20
0
-120-100-80-60-40-200
experimental
σxinMPa
numerical |analytical
σxinMPa
normalstressσ
x
x=L/2,y=−b/2+50mm,z=−D/2
R²=0.99
R²=0.96
numerical
analytical
alltested configurations
0
2
4
6
8
10
12
-450 -350 -250 -150 -50 50 150 250 350 450
τ
xy,F
inMpa
yinmm
shear stressτ
xy,F
x=x
2
,z=−D/2
TA40_5_6.3−5
TA160_5_6.3-5
numerical
analytical
-0,01
0,00
0,01
0,02
0,03
0,04
0,05
-450 -350 -250 -150 -50 50 150 250 350 450
yinmm
shear stressτ
xz,c
x=x
2
,z=0
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
τ
xz,C
inMPa
TA40_5_6.3−5
TA160_5_6.3−5
numerical
analytical
Figure 14. Scatter plots of the experimental, numerical, and analytical results: (a) rotation
ϑ
(x = L/2, z = D/2); (b) normal stress σx(x = L/2, y = −b/2 + 50 mm, z = −D/2).
Materials 2024,17, 460 14 of 20
3.3. Further Numerical Studies
In the experimental study, practicable constraints may limit the ability to measure
all points of interest in detail. Consequently, specific points are selectively analyzed and
assessed in the numerical approach. Furthermore, significant insights can be extracted
from the numerical parametric study to break down the fundamental structural behavior
of eccentrically loaded sandwich panels into specific parameters or parameter ratios. This
process facilitates identifying potential expressions for engineering calculation methods.
3.3.1. Shear Stress Analysis
To highlight the shear stress distribution, the numerical results of the two representa-
tive configurations TA160_5_6.3-5 and TA40_5_6.3-5 are shown in relation to the analytical
solutions in Figure 15. On the right side (b), the stresses in the middle of the core were cal-
culated by superimposing the vertical force and torsion. For the vertical force, a uniformly
distributed core shear stress across the panel width was assumed, neglecting the effects of
local load introduction.
Materials2024,17,xFORPEERREVIEW14of20
(a)(b)
Figure14.Scaerplotsoftheexperimental,numerical,andanalyticalresults:(a)rotationϑ(x=L/2,
z=D/2);(b)normalstressσx(x=L/2,y=−b/2+50mm,z=−D/2).
3.3.FurtherNumericalStudies
Intheexperimentalstudy,practicableconstraintsmaylimittheabilitytomeasureall
pointsofinterestindetail.Consequently,specificpointsareselectivelyanalyzedandas-
sessedinthenumericalapproach.Furthermore,significantinsightscanbeextractedfrom
thenumericalparametricstudytobreakdownthefundamentalstructuralbehaviorof
eccentricallyloadedsandwichpanelsintospecificparametersorparameterratios.This
processfacilitatesidentifyingpotentialexpressionsforengineeringcalculationmethods.
3.3.1.ShearStressAnalysis
Tohighlighttheshearstressdistribution,thenumericalresultsofthetworepresenta-
tiveconfigurationsTA160_5_6.3-5andTA40_5_6.3-5areshowninrelationtotheanalyti-
calsolutionsinFigure15.Ontherightside(b),thestressesinthemiddleofthecorewere
calculatedbysuperimposingtheverticalforceandtorsion.Fortheverticalforce,auni-
formlydistributedcoreshearstressacrossthepanelwidthwasassumed,neglectingthe
effectsoflocalloadintroduction.
(a)(b)
Figure15.Shearstressdistributionacrossthepanelwidth:(a)shearstressofthefacesheets(x=x2,
z=−D/2);(b)shearstressofthecore(x=x2,z=0).
0,000
0,005
0,010
0,015
0,020
0,025
0,030
0,000 0,005 0,010 0,015 0,020 0,025 0,030
experimental
ϑinrad
numerical |analytical
ϑinrad
rotation ϑ
x=L/2,z=D/2
R²=0.99
R²=0.98
0.030
0.025
0.020
0.010
0.005
0.000
0.000
0.005 0.010 0.015 0.020 0.025 0.030
0.015
alltested configurations
numerical
analytical
-120
-100
-80
-60
-40
-20
0
-120-100-80-60-40-200
experimental
σxinMPa
numerical |analytical
σxinMPa
normalstressσ
x
x=L/2,y=−b/2+50mm,z=−D/2
R²=0.99
R²=0.96
numerical
analytical
alltested configurations
0
2
4
6
8
10
12
-450 -350 -250 -150 -50 50 150 250 350 450
τ
xy,F
inMpa
yinmm
shear stressτ
xy,F
x=x
2
,z=−D/2
TA40_5_6.3−5
TA160_5_6.3-5
numerical
analytical
-0,01
0,00
0,01
0,02
0,03
0,04
0,05
-450 -350 -250 -150 -50 50 150 250 350 450
yinmm
shear stressτ
xz,c
x=x
2
,z=0
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
τ
xz,C
inMPa
TA40_5_6.3−5
TA160_5_6.3−5
numerical
analytical
Figure 15. Shear stress distribution across the panel width: (a) shear stress of the face sheets
(x = x2,z=−D/2); (b) shear stress of the core (x = x2, z = 0).
3.3.2. Extended Numerical Parametric Studies
Through the extended numerical parametric study based on the reference model
(see Table 3), incorporating 48 configurations, important relations between the outcome
variables (rotation, stress) and the geometric input variables were found and compared to
the analytical approach based on warping torsion.
While conducting experiments, achieving a “pure” torsional loading on sandwich
panels is not practicable. However, for the subsequent extensive numerical investigation,
a more effective understanding of the structural response to pure torsion can be attained
by decomposing the bending and torsion component from the applied eccentric four-
point-load, as described in load situation LS II. Figure 16 illustrates that this approach is
acceptable for the present investigation. The deviation of the results
σx
(x
1
), u
z
(x
1
),
τxy,f
(x
2
) obtained from the validated numerical model applying LS I and LS II is low (< 6%),
presumably resulting from the concentrated load introduction, see [28].
Materials 2024,17, 460 15 of 20
Materials2024,17,xFORPEERREVIEW15of20
3.3.2.ExtendedNumericalParametricStudies
Throughtheextendednumericalparametricstudybasedonthereferencemodel(see
Table3),incorporating48configurations,importantrelationsbetweentheoutcomevaria-
bles(rotation,stress)andthegeometricinputvariableswerefoundandcomparedtothe
analyticalapproachbasedonwarpingtorsion.
Whileconductingexperiments,achievinga“pure”torsionalloadingonsandwich
panelsisnotpracticable.However,forthesubsequentextensivenumericalinvestigation,
amoreeffectiveunderstandingofthestructuralresponsetopuretorsioncanbeaained
bydecomposingthebendingandtorsioncomponentfromtheappliedeccentricfour-
point-load,asdescribedinloadsituationLSII.Figure16illustratesthatthisapproachis
acceptableforthepresentinvestigation.Thedeviationoftheresultsσx(x1),uz(x1),τxy,f(x2)
obtainedfromthevalidatednumericalmodelapplyingLSIandLSIIislow(<6%),pre-
sumablyresultingfromtheconcentratedloadintroduction,see[28].
Figure16.Deviationofresults(σx,uz,τxy)oftheloadsituationsIandII.
UsingthevalidatedFEmodel,theparameterwindowoftheexperimentalparameter
study(seeTab les1and2)iscomprehensivelyexpandedregardingthegeometryofthe
panels(seeTable3).Thenumericalsolutionwascategorizedusingthethreeanalytical
approachesconsideringcomplete(blue),none(violet),andpartialwarpingrestraint
(green).Throughatest-basedbest-fitmethodmentionedinSection2.4,thetorsionspring
constant,herethevaluekdx=0.79wasdeterminedforthedescribedsix-pointbendingtest
setup.However,fiingnotonlykdxbutalsokw,andbothvaluesseparatelyforeachtested
sandwichpanelconfiguration,leadstothegreendatapoints.
PloingtherotationovertheratioD/Lrevealsahyperbolicrelationshipbetween
thesequantities(seeFigure17).Thenumericalresultsalignbetweenbothanalyticallyde-
termineddatapointswithandwithoutwarpingrestraintacrosstheentirespectrum.This
alsoappliestotheconfiguration-dependentbest-fitapproach.
-6%
-4%
-2%
0%
2%
4%
6%
-450 -350 -250 -150 -50 50 150 250 350 450
deviation in%
yinmm
deviation of results
load situation Ivs II
normalstressσ
x
(x
1
)
displacement u
z
(x
1
)
shear stressτ
xy,F
(x
2
)
TA_160_5_6.3−5
TA_40_5_6.3−5
Figure 16. Deviation of results (σx, uz,τxy) of the load situations I and II.
Using the validated FE model, the parameter window of the experimental parameter
study (see Tables 1and 2) is comprehensively expanded regarding the geometry of the
panels (see Table 3). The numerical solution was categorized using the three analytical
approaches considering complete (blue), none (violet), and partial warping restraint (green).
Through a test-based best-fit method mentioned in Section 2.4, the torsion spring constant,
here the value k
dx
= 0.79 was determined for the described six-point bending test setup.
However, fitting not only k
dx
but also k
w
, and both values separately for each tested
sandwich panel configuration, leads to the green data points.
Plotting the rotation over the ratio D/L reveals a hyperbolic relationship between
these quantities (see Figure 17). The numerical results align between both analytically
determined data points with and without warping restraint across the entire spectrum.
This also applies to the configuration-dependent best-fit approach.
Materials2024,17,xFORPEERREVIEW16of20
Figure17.Rotationdependentongeometricratios.
SimilarobservationsaremaderegardingtheshearstressesinFigure18,bothinthe
facesheets(a)andinthecore(b).Intheleftcase(a),thereisahyperbolicrelationship
betweenshearstressandtheproductofthesheetthicknesst1,thecorethicknessD,and
thesquarerootofthecomponentlengthL.Intherightcase(b),thereisahyperbolicrela-
tionshipbetweenthetorsionalinducedcoreshearstressandthequotientofDandthe
squarerootofL.Itisnoteworthythatthegreendatapointsfollowtheredonesdespite
differentinitialparameters.
(a)(b)
Figure18.Shearstressdependentongeometricratios:(a)shearstressofthefacesheets(x=x2,y=
0,z=−D/2);(b)torsion-inducedshearstressofthecore(x=x2,y=−b/2,z=0).
4.DiscussiononExperimentalandNumericalResultsconsideringWarpin gTorsion
4.1.General
Throughoutthecomprehensivestudies,aconsistencyintheresultsobtainedfrom
theexperimental,numerical,andanalyticalmethodswasobserved.Indetail,thefindings
willbediscussedwithregardtotheassumptions,calculationmodels,andpossiblelimi-
tationsofthemethods.
0,00
0,01
0,02
0,03
0,04
0,05
0,00 0,02 0,04 0,06 0,08 0,10
ϑinrad
D/L
rotation ϑ
x=x
1
,z=D/2
0.05
0.04
0.03
0.01
0.02
0.02 0.04 0.06 0.08 0.10
0.00
0.00
numerical
analytical
(→0, =0.79)
analytical
(→∞, =0.79)
analytical
( , )
0
4
8
12
16
20
0 2000 4000 6000 8000 10000 12000
τ
xy,F
inMPa
shear stressτ
xy,F
x=x
2
,y=0,z=−D/2
t
1
D
numerical
analytical
(→0, =0.79)
analytical
(→∞, =0.79)
analytical
( , )
10,000 12,000
0,000
0,005
0,010
0,015
0,020
0,025
0,030
012345
τ
xz,C
inMPa
shear stressτ
xz,C
x=x
2
,y=b/2,z=0
0.030
0.025
0.020
0.010
0.015
0.000
D/
numerical
analytical
(→0, =0.79)
analytical
(→∞, =0.79)
analytical
( , )
0.050
Figure 17. Rotation dependent on geometric ratios.
Similar observations are made regarding the shear stresses in Figure 18, both in the
face sheets (a) and in the core (b). In the left case (a), there is a hyperbolic relationship
between shear stress and the product of the sheet thickness t
1
, the core thickness D, and
the square root of the component length L. In the right case (b), there is a hyperbolic
relationship between the torsional induced core shear stress and the quotient of D and the
Materials 2024,17, 460 16 of 20
square root of L. It is noteworthy that the green data points follow the red ones despite
different initial parameters.
Materials2024,17,xFORPEERREVIEW16of20
Figure17.Rotationdependentongeometricratios.
SimilarobservationsaremaderegardingtheshearstressesinFigure18,bothinthe
facesheets(a)andinthecore(b).Intheleftcase(a),thereisahyperbolicrelationship
betweenshearstressandtheproductofthesheetthicknesst1,thecorethicknessD,and
thesquarerootofthecomponentlengthL.Intherightcase(b),thereisahyperbolicrela-
tionshipbetweenthetorsionalinducedcoreshearstressandthequotientofDandthe
squarerootofL.Itisnoteworthythatthegreendatapointsfollowtheredonesdespite
differentinitialparameters.
(a)(b)
Figure18.Shearstressdependentongeometricratios:(a)shearstressofthefacesheets(x=x2,y=
0,z=−D/2);(b)torsion-inducedshearstressofthecore(x=x2,y=−b/2,z=0).
4.DiscussiononExperimentalandNumericalResultsconsideringWarpin gTorsion
4.1.General
Throughoutthecomprehensivestudies,aconsistencyintheresultsobtainedfrom
theexperimental,numerical,andanalyticalmethodswasobserved.Indetail,thefindings
willbediscussedwithregardtotheassumptions,calculationmodels,andpossiblelimi-
tationsofthemethods.
0,00
0,01
0,02
0,03
0,04
0,05
0,00 0,02 0,04 0,06 0,08 0,10
ϑinrad
D/L
rotation ϑ
x=x
1
,z=D/2
0.05
0.04
0.03
0.01
0.02
0.02 0.04 0.06 0.08 0.10
0.00
0.00
numerical
analytical
(→0, =0.79)
analytical
(→∞, =0.79)
analytical
( , )
0
4
8
12
16
20
0 2000 4000 6000 8000 10000 12000
τ
xy,F
inMPa
shear stressτ
xy,F
x=x
2
,y=0,z=−D/2
t
1
D
numerical
analytical
(→0, =0.79)
analytical
(→∞, =0.79)
analytical
( , )
10,000 12,000
0,000
0,005
0,010
0,015
0,020
0,025
0,030
012345
τ
xz,C
inMPa
shear stressτ
xz,C
x=x
2
,y=b/2,z=0
0.030
0.025
0.020
0.010
0.015
0.000
D/
numerical
analytical
(→0, =0.79)
analytical
(→∞, =0.79)
analytical
( , )
0.050
Figure 18. Shear stress dependent on geometric ratios: (a) shear stress of the face sheets (x = x
2
, y = 0,
z = −D/2); (b) torsion-induced shear stress of the core (x = x2,y=−b/2, z = 0).
4. Discussion on Experimental and Numerical Results considering Warping Torsion
4.1. General
Throughout the comprehensive studies, a consistency in the results obtained from the
experimental, numerical, and analytical methods was observed. In detail, the findings will
be discussed with regard to the assumptions, calculation models, and possible limitations
of the methods.
4.2. Rotation of the Sandwich Panel
The rotation of the sandwich panels was calculated using vertical displacement, as-
suming a linear distribution across the panel width. This assumption was validated as the
deflection exhibited linearity across the width for both examined configurations. The dis-
placement curves TA160_5_6.3-5 and TA40_5_6.3-5 are nearly identical for both considered
locations, x
1
and x
2
, across all three methods. Correspondingly, the derived rotation values
from this analysis are also in agreement as demonstrated across the entire scope of tested
configurations in the scatter plot presented in Figure 14a (R
2
> 0.98). Residuals at high
rotation values can be attributed to fluctuations of the material properties or inaccurate
measurements. In the extended numerical study, a hyperbolic relationship between the
rotation and the ratio D/L was identified (see Figure 17). In this context, the numerical
solution aligns closely with the analytical results, considering full, none, or partial warping
restraint. Notably, the configuration-dependent test-based best-fit method exhibits the most
accurate alignment with the numerical solution.
4.3. Normal Stresses in the Face Sheets
According to classical warping torsion theory, linear stress distributions in the flanges
(in this case: face sheets) are anticipated. Unlike an eccentrically loaded homogeneous
plate, the maximum normal stresses are expected to be localized on the load-averted side.
Both statements are generally confirmed for the presented configuration TA160_5_6.3-
5 across all three methods, particularly for x
1
, as illustrated in Figure 13a. However, for
TA40_5_6.3-5, featuring the lowest examined core thickness of D = 40 mm, qualitative and
quantitative deviations are apparent; see Figure 13b. While the numerical solution aligns
with the non-linear distribution obtained from the experimental data points, the analytical
approach, by definition, leads to a linear curve. Notably, considering that non-linearity
is more prominent on the load-averted side (
−
b/2
≤
y < 0), the linear approach appears
Materials 2024,17, 460 17 of 20
to remain suitable for the load-facing side (0 < y
≤
b/2). Influences resulting from the
boundary conditions or the geometrically non-linear load effects may contribute to favoring
this non-linearity. Considerations of these factors were integrated into the FE model but
were not comprehensively addressed in the analytical model of warping torsion.
In the finite element model, frictional contact definitions between the face sheet and the
bearing structure capture the normal stress gradient more accurately than other considered
contact definitions. The friction value, calibrated to 0.1, accounts for the effective area
activated for friction (considering the presence of lined face sheets) and the characteristics
of the material surfaces. The newly designed fork support may exhibit clearance due to
the compressibility of the core and stiffness in lateral bending. Analytically, this effect was
considered only through the introduction of the torsional and warping spring for both
support conditions using the test-based best-fit method.
In the ANSYS setting “Large Deflection”, effects from second-order theory were
considered. However, this aspect seems contradictory to the experimentally determined
displacement–load and strain–load relationship (see Figure 11), assumed to be nearly linear
for the applied load range.
Conversely, for core thicknesses above D = 40 mm, the significance of the contact
definition and the second-order theory (“Large Deflection”) gradually decreases for the
tested configurations. Preliminary numerical studies on sandwich panels with thicker
core thicknesses, neglecting these assumptions, have already indicated this. This trend is
evident in the scatter plot (Figure 14b), which also highlights the high correlation between
the different approaches (R
2
> 0.98). It is presumed that in sandwich panels with low core
thicknesses, a secondary structural mechanism occurs in addition to the warping torsion.
4.4. Shear Stresses in the Face Sheets and in the Core
The shear stress analysis is conducted based on the analytical and the validated
numerical model. For both depicted shear stresses (see Figure 15), observed in the top
face sheets and in the middle of the core, the numerical and analytical solutions exhibit
perfect alignment for the configuration TA160_5_6.3-5. Deviations, however, occur for
TA40_5_6.3-5. Regarding the shear stress
τxy,F
, a slight asymmetry is evident, with both
lines coinciding on the load-averted side, excluding the fact that the numerical solution does
not approach zero in the edge. In contrast, for the shear stress
τxz,C
, both solutions align on
the load-faced side but diverge on the load-averted side. Presumably, both phenomena are
linked to the effects of the local load introduction. It is noteworthy that the value from the
torsional induced shear stress in the core is comparable to the value from the vertical load
component. Nevertheless, the analytical formula for determining the shear stress can be
confirmed in principle.
Similar to the rotation analysis, the extended numerical study reveals hyperbolic
relationships between the considered magnitudes and geometric ratios. For the shear stress
(see Figure 18) in the face sheets, the abscissa is the product of the core thickness, the square
root of the total length and the sheet thickness. Meanwhile, the abscissa for the core shear
stress is represented by the quotient of core thickness and the square root of the total length.
Once again, the test-based best-fit approach most accurately aligns with the numerical
solution, capturing effects beyond the purely analytical approach.
4.5. Superposition Principle of Bending and Torsion in Sandwich Panels
The presented displacement and stress curves confirm that bending and torsion can
be superimposed to represent eccentric loads for sandwich panels, despite their plate-like
appearance. Furthermore, the small deviations between the load situations I (six-point
bending test) and II (superposition of bending and torsion), as introduced in Figure 10 and
illustrated in Figure 16, indicate additionally the influence of the local load introduction
concerning the transversal direction is recognizable but small.
Materials 2024,17, 460 18 of 20
5. Conclusions, Contribution, and Future Work
In this research article, the structural behavior of eccentrically loaded sandwich panels
with a PU foam core was investigated through a combination of experimental and numerical
studies and was compared to the analytical approach based on warping torsion. For this
purpose, a newly developed eccentric six-point single-span bending test was introduced
with the primary goal of gathering essential data on rotation and normal stress values for
various sandwich panels with different geometries and material properties. Subsequently,
a numerical model was developed and validated, enabling an expanded parameter study
and a detailed examination regarding shear stresses. The key findings and contributions
from this study are summarized as follows:
•
A high correlation was observed between experimental, numerical, and analytical
results for 20 different parameter configurations, particularly concerning the rotation
and normal stress values at the midspan of the sandwich panels.
•
The qualitative curve of the stress and displacement distribution across the width of
the panel are in general aligned for all three methodologies, in particular for the tested
sandwich panels with a core thickness higher than D = 40 mm.
•
The shear stress of the face sheets and the core of the sandwich panel from the analytical
approach on warping torsion agrees in general with the numerical results.
•
The superposition principle concerning bending and torsion loadings is confirmed for
the tested configurations. Only minor deviations were observed in comparison to the
eccentric loading.
•
The applicability of warping torsion for PU sandwich panels has been demonstrated
within a considerable practical parameter range, considering various material prop-
erties and dimensions. A test-based best-fit method based on warping torsion was
applied for the analytical approach. However, it is important to acknowledge the
limitations of this approach regarding non-linear stress distributions.
•
From the extensive parametric study based on a validated numerical model hyperbolic
relations were found between the rotation values and the ratio D/L, face shear stress
and the ratio t1·D/√L, as well as the core shear stress and the ratio D ·√L.
•
The magnitude of stresses may not necessarily be negligible in terms of design consid-
erations and can become significant.
For future research, it is recommended to address the constraints of the warping torsion
theory, particularly in terms of geometric and stiffness ratios, which are evident in the
non-linear stress distribution. The presented numerical approach is well-suited to conduct
further detailed studies with the aim of finding a practicable engineering calculation
model. While it may involve a more complex mathematical process, it is suggested to
explore alternative analytical methods, such as the sandwich plate theory [
7
], for calculating
eccentrically loaded sandwich panels. Mechanically complex, but not considered in this
article, influences from secondary shear deformation could be relevant [11].
Nevertheless, the primary objective should remain focused on developing a simplified
analytical or semi-analytical approach based on the data obtained from the experimental
and numerical studies. This approach should concentrate on the design of safe and eco-
nomically efficient building envelopes, while exploring the potential to expand the scope
of applications for sandwich panels.
Author Contributions: Writing—original draft, E.M.P.; Writing—review & editing, J.L. All authors
have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The data presented in this study are available on request from the
corresponding author.
Materials 2024,17, 460 19 of 20
Conflicts of Interest: The authors declare no conflicts of interest.
References
1.
Lange, J.; Berner, K. Sandwichelemente im Hochbau. In Stahlbaukalender 2020: Neue Normung im Hochbau, Leichtbau; Kuhlmann,
U., Ed.; Ernst & Sohn: Berlin, Germany, 2020; pp. 905–971. ISBN 9783433032909.
2.
Berner, K.; Misiek, T.; Raabe, O. Sandwichbauteile mit vorgehängter Fassade—Bemessungskonzepte. Stahlbau 2018,87, 427–437.
[CrossRef]
3. Seide, P. On the Torsion of Rectangular Sandwich Plates. J. Appl. Mech. 1956,23, 191–194. [CrossRef]
4.
Cheng, S. Torsion of Rectangular Sandwich Plate. Report no. 1871. Available online: https://ir.library.oregonstate.edu/concern/
defaults/s7526h30x (accessed on 7 June 2023).
5.
Cheng, S. Supplement to Torsion of Sandwich Panels of Trapezoidal, Triangular, and Rectangular Cross Sections. Report
no. 1874-A. Available online: https://ir.library.oregonstate.edu/concern/defaults/z890rz25w (accessed on 28 September 2023).
6. Cheng, S. A Formula for the Torsional Stiffness of Rectangular Sandwich Plates. J. Appl. Mech. 1961,28, 307–310. [CrossRef]
7.
Stamm, K.; Witte, H. Sandwichkonstruktionen: Berechnung, Fertigung, Ausführung; Springer: Wien, Austria, 1974;
ISBN 9783709183359.
8.
Höglund, T. Load Bearing Strength of Sandwich Panel Walls with Window Openings. In Proceedings of IABSE Colloquium; IABSE
Reports; IABSE: Stockholm, Sweden, 1986; pp. 349–356.
9. Zenkert, D. The Handbook of Sandwich Construction; EMAS Publishing: Cradley Heath, UK, 1997; ISBN 9780947817961.
10.
Rädel, F.; Lange, J. Eccentrically Loaded Sandwich Elements. In Proceedings of the 6th European Conference on Steel and
Composite Structures—Eurosteel 2011, Budapest, Hungary, 31 August–2 September 2011; ISBN 978-92-9147-103-4.
11.
Rädel, F. Untersuchungen zur Tragfähigkeit von Sandwichelementen mit Öffnungen. Ph.D. Thesis, Technical University of
Darmstadt, Darmstadt, Germany, 2013.
12.
Pozorski, Z. Numerical Analysis of Sandwich Panels Subjected to Torsion. In Selected Topics in Contemporary Mathematical Modeling,
Proceedings of the Conference on Mathematical Modeling in Physics and Engineering, Czestochowa, Poland, 18–21 September 2017;
Grzybowski, A.Z., Ed.; Publishing Office of Czestochowa University of Technology: Czestochowa, Poland, 2017; pp. 109–122.
ISBN 9788371936685.
13.
Pozorski, Z.; Wojciechowski, S. The Influence of Symmetrical Boundary Conditions on the Structural Behaviour of Sandwich
Panels Subjected to Torsion. Symmetry 2020,12, 2093. [CrossRef]
14.
Pradhan, E.M.; Lange, J. Das Tragverhalten exzentrisch beanspruchter Sandwichelemente. In Proceedings of the 23rd DASt-
Forschungskolloquium 2022, Dortmund, Germany, 22–23 February 2022; pp. 61–64.
15.
Pradhan, E.M.; Lange, J. Parametric Studies of Eccentrically Loaded Sandwich Panels Regarding Geometric Ratios. In Contempo-
rary Problems: Book of Abstracts, Proceedings of the XXVIII LSCE—Lightweight Structures in Civil Engineering, Poznan, Poland, 1–2
December 2022; Publishing House of Poznan University of Technology: Poznan, Poland, 2022; ISBN 978-83-7775-675-1.
16.
Wojciechowski, S. Analiza Do´swiadczalna, Numeryczna i Teoretyczna Skr˛ecanych Statycznie Paneli Warstwowych o Sztywnych
Okładzinach i Podatnym Rdzeniu: Experimental, Numerical and Theoretical Analysis of Statically Twisted Sandwich Panels with
Rigid Facings and Deformable Core. Ph.D. Thesis, Poznan University of Technology, Poznan, Poland, 2022.
17.
Wurf, A.; Rabinovitch, O.; Frostig, Y. High-order torsion analysis of sandwich beams of rectangular cross-section. J. Sandw. Struct.
Mater. 2022,25, 284–307. [CrossRef]
18.
Böttcher, M. Wand-Sandwichelemente mit Öffnungen. Ph.D. Thesis, Technical University of Darmstadt, Darmstadt, Germany,
2005.
19. Elmalich, D.; Rabinovitch, O. Twist in soft-core sandwich plates. J. Sandw. Struct. Mater. 2014,16, 577–613. [CrossRef]
20.
EN 14509:2013; Self-Supporting Double Skin Metal Faced Insulating Panels—Factory Made Products—Specifications. CEN:
Brussels, Belgium, 2013.
21.
ECCS. Preliminary European Recommendations for the Design of Sandwich Panels with Openings: A State of the Art Report, 1st ed.; ECCS:
Brussels, Belgium, 2014; no. 134.
22.
ECCS. Preliminary European Recommendations for the Design of Sandwich with Point and Line Loads; 2nd draft; ECCS: Brussels,
Belgium.
23. Davies, J.M. Lightweight Sandwich Construction; Blackwell Science: Oxford, UK; Malden, MA, USA, 2001; ISBN 9780470779965.
24. Cywi´nski, C. Drillträger-Formeln für die wichtigsten Belastungsfälle. Stahlbau 1983,52, 245–252.
25.
Studzi´nski, R.; Pozorski, Z.; Garstecki, A. Structural behavior of sandwich panels with asymmetrical boundary conditions. J.
Constr. Steel Res. 2015,104, 227–234. [CrossRef]
26. Ansys Mechanical. Mechanical User’s Guide; Release 2022 R2.; Ansys Mechanical: Canonsburg, PA, USA, 2022.
Materials 2024,17, 460 20 of 20
27.
Kemmer, G.; Keller, S. Nonlinear least-squares data fitting in Excel spreadsheets. Nat. Protoc. 2010,5, 267–281. [CrossRef]
[PubMed]
28.
Pozorska, J.; Pozorski, Z.; Janik, Ł. Numerical Simulations of Structural Behavior of Sandwich Panels Subjected to Concentrated
Static Loads. J. Appl. Math. Comput. Mech. 2017,16, 113–121. [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.
