Composite Structures 113 (2014) 83–88

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Periodic and homogenized bending response of faceplates of filledweb-core sandwich beams

http://dx.doi.org/10.1016/j.compstruct.2014.03.0010263-8223/� 2014 Elsevier Ltd. All rights reserved.

⇑ Tel.: +358 50 511 3250.E-mail address: jani.romanoff@aalto.fi

Jani Romanoff ⇑Aalto University, School of Engineering, Department of Applied Mechanics, Tietotie 1C, 00076 Aalto, Finland

a r t i c l e i n f o

Article history:Available online 13 March 2014

Keywords:Web-core sandwichHomogenizationStress analysisExperiments

a b s t r a c t

The paper presents investigation on periodic and homogenized bending response of the faceplates offilled web-core sandwich beams. The investigation is carried out using analytical, numerical and exper-imental methods. It is shown that the interaction between the web and faceplates and the filling materialis complex phenomena. This affects the warping-induced deflection of the faceplates. The filling materialstiffens the beams and reduces the shear-induced warping-deflection and resulting normal stress in thefaceplates considerably. The homogenized solution is shown to agree with the periodic response well interms of deflections. However, the normal stresses in the faceplates are considerably underestimatedwith the homogenized solution.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The increasing demand for safe, sustainable and environmen-tally structures has increased the need to investigate new struc-tural solutions. Web-core sandwich panels have two facesseparated by unidirectional core. The fact that faceplates are farfrom neutral axis, makes these panels efficient in bending. Theunidirectional core makes the panels attractive to systems integra-tion. Typical materials in these panels are wood, GFRP, steel oraluminum and the production can be done through pultrution,lamination, adhesive bonding, friction-stir- or laser-welding. Thesepanels however suffer from high orthotrophy, especially in out-of-plane shear, which make them susceptible for high shear defor-mations, i.e. unit cell warping, opposite to the web plate direction;see Fig. 1.

Classical sandwich panels are based on the load carrying mech-anism where faces carry the global bending moment and the corethe shear forces; see for example the textbook in Ref. [1]. In emptyweb-core sandwich panels this load carrying becomes mixed, i.e.the faceplates carry also the shear forces; see Refs. [2–8]. If thevoids of the panels are filled stiff filling material the shear forceis carried almost totally by the foam. This means that the unit cellresponses is complex phenomena with foam, webs and faceplatesinteracting with each other and causing considerable stresses atunit cell level to the faceplates; see Refs. [6–13]. In practice thestructures are large and detailed modeling the unit cell and its’

response is not meaningful. Therefore, homogenization, i.e. averag-ing the response over unit cell, is often utilized. As the averagingprocess leads to averaging of the warping deflections and resultingstresses of the face and web plates, the question is raised on the er-ror introduced to the stress evaluation.

The aim of this paper is to investigate experimentally andnumerically the shear-induced normal stresses in web-core sand-wich beams. The organization of the paper is as follows. First, unitcell warping response is described analytically using basic Euler–Bernoulli beam theory to describe the behavior of the faceplates.Based on this description the instrumentation for the experimentalinvestigation presented in Ref. [14] is justified. Then, the homoge-nization method for the obtained response is presented. The resultsare complemented with FE analyses. Finally, a parametric investi-gation is carried out using FEA to investigate the scale effects.

2. Definitions

The sandwich plate is assumed to consist of small structuralelements representing the web and the faceplates, and the fillingmaterial. The core of the sandwich is considered to be betweenthe faceplates and it includes both the web plates and the voids be-tween them. The web plates are in the XZ-plane and have a thick-ness tw and a height hc see Fig. 1. The web plate spacing is denotedby s. The faceplates are in the xy-plane and have a thickness t. Thevoids between the face and web plates are filled by foam. Sub-scripts t, b, w and v are used for the top face, the bottom face theweb plates and the void, respectively. The plate has length L andbreadth B, total height h = tt + tb + hc and the neutral axes of the

Fig. 1. Filled web-core sandwich beam and the notations used.Fig. 2. (A) Deflection of empty unit cell due to constant shear force Q0, (B) addingfoam to the core sharing the same deflection at the unit cell ends, but not along unitcell length and (C) correcting the shape by introduction of interaction term qz,f fordistributed load.

84 J. Romanoff / Composite Structures 113 (2014) 83–88

faceplates have a distance of d = (tt + tb)/2 + hc. The sandwich platehas two coordinate systems, namely: global XYZ and local xyz. Theorigin of the global coordinate system is situated at the geometri-cal mid-plane of the sandwich plate. The origin of the local coordi-nate is located at the neutral axis of the structural element underconsideration. Notations F and M are used for the forces andmoments per unit breadth. The Young’s modulus, shear modulusand Poisson’s ratio are denoted with E, G and m, respectively. A unitcell is formed by the space limited by two web plates andfaceplates.

3. Theory

3.1. Unit cell warping-induced face plate deflection

As Fig. 1 shows the deflection of the faceplate oscillates withinunit cell. Here it is assumed that the faceplate acts as Euler–Ber-noulli beam, but the responses of the beam as a whole must beanalyzed using traditional or modified couple stress Timoshenkobeam theory; see Refs. [4–7,15]. Thus, the basic beam relationsfor slope, h, bending moment, M, shear force, Q, and distributedloading, q, are

hlðxÞ ¼ wð1Þl ðxÞMlðxÞ ¼ �Dwð2Þl ðxÞQ lðxÞ ¼ �Dwð3Þl ðxÞqlðxÞ ¼ Dwð4Þl ðxÞ

ð1Þ

where the local coordinate system is considered. Generally thedeflection at any point of the unit cell can be described using func-tions with constant, w0, even wl,even(x) and odd wl,odd(x) terms; i.e.superposition principle holds. Then the deflection can be written as:

wlðxÞ ¼ w0 þwl;oddðxÞ þwl;evenðxÞ ð2Þ

It is shown in Ref. [6] for empty beam, that the warping deflec-tion of the faceplate is (see also Fig. 2A),

wiQ ðxÞ ¼

Q Q s2d24Di

xs

ki1

sd�4

x2

s2 þ 3� �

þ 4 6Di

dkih

þ ki2

Di

Dw

!" #;

i ¼ t; b ð3Þ

where Q0 is the shear force associated with equal slopes of the faceplate at the ends of the unit cell. The total shear force is then givenas

Q ¼ Q 0 þ Q tf ð4Þ

where the Qtf is due to thick face plates effect or couple stress; seeRef. [15]. Further, the s and d are the distances between the mid-planes of consecutive web and face plates; Di is the bending rigidityof the face and web plates; and ki

h is the rotation stiffness of the con-nection between the face and web-plate. k1 and k2 are stiffnessparameters and given in detail in Ref. [6] and summarized inAppendix A. The deflection in Eq. (3) is presented by third orderodd-polynomial. This gives after three and four derivatives constantshear force, Q, and zero distributed load, q, for the faceplate. Usingthis deflection, the shear stiffness of the beam can be calculatedfrom:

DQempty ¼Q 0

cxz¼ Q 0

DwiQ=s¼ Q 0s

wiQ

s2

� ��wi

Q � s2

� � ð5Þ

Kolsters and Zenkert [4] showed that the shear stiffness of thefoam-filled sandwich panel can be approximated with very goodaccuracy by simply adding to this stiffness the stiffness contribu-tion of the foam as

DQ � DQ ;empty þ DQ ;foam ð6Þ

where the foam contribution is calculated according to basic sand-wich beam theory neglecting the elastic foundation effects; seeFig. 2B. In reality the unit cell warps both in vertical and horizontalplane, there is continuity requirement between the face, webs andthe core and the elastic foundation effects are present; see Fig. 2Cand Ref. [13]. This means that the warping deflection of unit cellwill have much more complex shape than given in Eq. (3). Excludingthe effect web-plate bending and connection rigidity, Grönroos [13],described this shape using combination of polynomials and hyper-bolic, odd, functions. As the hyperbolic functions can be approxi-mated with polynomials, the bending moment and distributedloading can be approximated locally as

MlðxÞ ¼ �Dið6a3xþ 20a5x3 þ . . .þ ðn� 1Þðn� 2Þan�1xn�3ÞqlðxÞ ¼ Dið120a5xþ . . .þ ðn� 1Þðn� 2Þðn� 3Þðn� 4Þan�1xn�5Þ;i ¼ t; b; n ¼ even ð7Þ

This means that in the empty panels the bending moment is lin-ear within the unit cell. Further, according to Eq. (3) distributedloading q is zero. For cases where foam interacts with the faceand web plates the higher order terms of the polynomial exist. This

Fig. 3. Instrumentation of the web-core sandwich beam.

J. Romanoff / Composite Structures 113 (2014) 83–88 85

is seen then both in bending moment, M, and distributed loading,q. In general the solution for unit cell deformation is analyticallyextremely difficult to obtain due to deforming of foam, web andface plates and web-to-face-intersection; see Fig. 2A.

3.2. Homogenized response of the unit cell

Homogenization means averaging unit cell response over theunit cell length and normalizing the result with unit cell length,s. In present case the integration is carried out over the unit celllength x = �s/2, . . .,s/2 and the mathematical facts of integrals ofeven and odd functions are used, i.e. [15]Z s=2

�s=2f ðxÞeven dx ¼ 2

Z s=2

0f ðxÞeven dx

Z s=2

�s=2f ðxÞodd dx ¼ 0

ð8Þ

Then the homogenization for constant w0 leads simply tow�0 ¼ w0 and for the odd and even terms we obtain [15]:

w�l;oddðxÞ ¼ 1s

R s=2�s=2 wl;oddðxÞdx¼ 0 w�l;evenðxÞ ¼ 2

s

R s=20 wl;evenðxÞdx

w�ð1Þl;odd ¼ h�l ¼ 2s

R s=20 wð1Þl;oddðxÞdx w�ð1Þl;even ¼ h�l ¼ 0

w�ð2Þl;odd ¼�M�lD ¼ 0 w�ð2Þl;even ¼�

M�lD ¼ 2

s

R s=20 wð2Þl;evenðxÞdx

w�ð3Þl;odd ¼�Q�lD ¼ 2

s

R s=20 wð3Þl;oddðxÞdx w�ð3Þl;even ¼�

Q�lD ¼ 0

w�ð4Þl;odd ¼q�

lD ¼ 0 w�ð4Þl;even ¼

q�l

D ¼ 2s

R s=20 wð4Þl;evenðxÞdx

ð9Þ

This result is important since it shows that the odd terms con-tribute only to the slope, and shear force carried out by the unitcell. Similarly, the even terms contribute only to the deflection,moment and distributed loading on the unit cell. Appendix B pre-sents this derivation for polynomials. This means that when theexperimental result is homogenized, the shear-induced bendingmoment gets zero value, being an odd-function.

4. Experiment

4.1. Test specimens

Details of the experiments can be found from Ref. [14]. Thebeams were made from steel and Divinycell foams, H-grade. Thesteel sandwich panels were manufactured by laser-welding andcut to beams having length L = 1080 mm and breadth B = 50 mm.The faceplates were measured to have thickness of tt = tb = 2.52 mm.The web plates had thickness and height of tw = 3.97 mm andhc = 40 mm respectively. The faceplates have Young’s modulus ofE = 221 GPa and for the web plates 200 GPa. The Poisson’s ratio is as-sumed to be m = 0.3. Three specimens were manufactured from H80and H200 each, while there was two empty specimens. The filledspecimens were manufactured by cutting the foam blocks to exactlyfill the voids. Then polyurethane-based Sikabond 545 adhesive wasapplied to foam blocks and activated by spraying water on the adhe-sive. While curing, the adhesive filled all gaps between the foamblocks and steel structure. The Divinycell H80 and H200 foams haveYoung’s modulus of 80–85 MPa and 200–210 MPa respectively,while the shear modulus is G = 23 MPa and 75 MPa respectively;see Ref. [16].

4.2. Instrumentation and test procedure

Within highly deforming beams, it is not practical to measurethe local deflection. This is due to flexibility of the beam, whichcauses high rigid body motion to the unit cells. Here, Eq. (7) is used

to instrument the unit cell in with four strain gauges to capture thenonlinear distribution of shear induced normal strains and furtherthe shear-induced bending moment, i.e.

r ¼ E�e ¼ Nt� 6

Mt2 () e ¼ 1

E�Ntþ 6

Mt2

� �ð10Þ

where E� is the effective Young’s modulus for plane stress. Eachspecimen was instrumented with 12 strain gauges (5 mm) locatedsymmetrically with respect to the mid-plane of the beam; seeFig. 3. The strain gauges were positioned at 2nd, 5th and 8th unitcells in order to guarantee that the strains were unaffected by localeffects arising from application of external loads. Eight straingauges were located at the constant shear region, x = 0, . . .,L/3 andx = 2L/3, . . .,L. The distance to web plates was 10 mm and 30 mm,so that the highly non-linear strains could be captured. Four straingauges were located at the constant bending moment region, x = L/3, . . .,2L/3. Deflection was measured from load introduction pointsat x = L/3 and x = 2L/3 with displacement gauges having measuringrange of ±20 mm, while the load was measured with force gaugehaving maximum measuring capacity of 10 kN. The test was carriedout three load cycles from by from 0 N to 400 N to 0 N.

5. Finite element analyses

The finite element analyses (FEA) were carried out to comple-ment the experimental findings. Linear elastic material modeland 3D, parabolic solid element mesh in Abaqus 6.8.1 was used.The rotation stiffness of welds was modeled with equivalent weldthickness of 2 mm. Thus, the minimum element size was 0.25 mmat the weld region and 0.625 mm elsewhere. In order, to decreasecomputational efforts in parametric investigations shell elementmodels were used to model the face and web-plates. In this casethe element size was set to 10 mm. The agreement between thistype of model and 3D-solid was within few percent. Details ofthe 3D solid FE-modeling can be found from Ref. [14].

6. Results

The deflection for empty and H200 beams is given in Fig. 4. Thecomparison of the normal stress at the top surface of top faceplateis presented in Fig. 5.

Fig. 4 clearly shows that the filling material decreases thedeflection; the filled beam has seven times smaller deflection thanthe empty beam. The difference is due to reduced shear deflections,which is seen as reduction of the waviness of the deflection line.The homogenized result is in good agreement with the periodic re-sponse. For shear-induced normal stress, the Fig. 5 confirms thereduction of stress levels as well. At the same time it is seen that

Fig. 4. Comparison deflection between empty and filled panels. F = 400 N. Solidlines are the FE-result, gray points the experimental and black dots the unit cellaverage of the FE-result.

Fig. 5. Comparison of the normal stress in the top surface of top faceplate.F = 400 N. Solid lines are the FE-result, gray points the experimental and black dotsthe unit cell average of the FE- and experimental result.

Fig. 6. Comparison of normal stress in the top surface of top faceplate. Empty beamwith unit cell size, L/s = 18, L/s = 9, L/s = 3. F = 400 N.

Fig. 7. Comparison of normal stress in the top surface of top faceplate. Beam filledwith isotropic foam E = 400 MPa and m = 0.3 and with unit cell size, L/s = 18, L/s = 9,L/s = 3 F = 400 N.

86 J. Romanoff / Composite Structures 113 (2014) 83–88

the normal stress becomes non-linear when filling material is used.When the average of the shear-induced response is taken, allresults fall into the same line in terms of stress; this line has theshape of the external bending moment. This confirms the shear-induced bending moment gets to zero when homogenization iscarried out. It is also seen that the homogenized response has con-siderably lower stress values than the periodic response.

6.1. Influence of length scales

As presented in Ref. [6] the unit cell response is coupled withthe global beam bending response. Therefore, numericalinvestigation is done by varying the unit cell size between L/s = 3, L/s = 9 and L/s = 18. The top surface normal stress of topfaceplate is presented in Fig. 6 for empty beams and in Fig. 7for filled beams.

Fig. 6 shows that for empty beams the shear-induced second-ary-bending moment have the same slope regardless of the unitcell size. This means that if the unit cell size doubles, i.e. L/s = 18changes to L/s = 9, then the moment and resulting stress doubles

as well, i.e. the error of homogenization to localization stress islinearly proportional to the unit cell size. Fig. 7 shows that for filledbeams this relation does not hold, but the similar periodic trend instress is visible. Both figures show that the averaged response con-verges to straight line as the external bending moment diagramindicates. Fig. 7 also shows that as the unit cell length is equal tothe s = L/3, the normal stress matches very closely the shape ofthe bending moment diagram of the external loading; the devia-tion is only seen at the unit cell ends.

7. Conclusions

The paper presented an experimental and numerical investiga-tion on the periodic and homogenized bending response of filledweb-core sandwich beams. First, experimental planning to mea-sure shear-induced bending stresses was presented. Then, theexperimental results from Ref. [14] were reviewed. Finite Element

J. Romanoff / Composite Structures 113 (2014) 83–88 87

Analyses were carried out to complement the experimentalfindings. Finally, homogenization was carried out to obtain under-standing of consequences of using filling material and homoge-nized beam models. At this stage also a parametric investigationwas carried out to gain understanding on the length scale effectsbetween the unit cell and the beam.

The investigation shows that the filling material efficientlyincreases the stiffness of the sandwich beams opposite to theweb-plate direction. This reduces the shear-induced warpingdeformation and resulting secondary faceplate bending stresseswithin the unit cell. This finding is in-line with that analyticallyand numerically observed in Refs. [4,13]. The investigation furthershows that the faceplate bending moment response within unitcell of empty beams in linear; while for filled beams it is highlynon-linear. Thus, the approximation made in Ref. [4] to estimatethe equivalent shear stiffness by direct summation of face, weband foam contributions directly is not exactly valid as linear bend-ing moment within the faceplates is assumed. Thus, the interactionis found to be significant and the method proposed by Grönroos in[13] is more correct since it models this non-linearity. However,this model does not consider elastic foundation effect of web-platenor it does the deformation at the connection between face andweb plates. This means that although the shear stiffness is mod-eled accurately the localization of stresses as proposed in Ref. [6]is not possible unless this interaction is properly modeled. As thisinvolves complex interaction between web, face and filling mate-rial bending and requires application of the modified couple stresstheory presented and validated in Refs. [15,17] this task is left forfuture work. Further investigations were carried out on the scaleeffect of the unit cell and issues related to homogenization of theresponses. It is shown that as the unit cell size is being decreased,the periodic stress and shear-induced bending moment responsedecrease as well. This is inline with findings from Ref. [6] for emptybeams, but the present investigation extends these conclusionsalso for filled panels. This means that, in order to capture the stressresponse of these panels correctly, the unit cell response must beaccounted for in the post-processing of the homogenized beam re-sults. In this development work the contributions made in Refs.[11,12] should be accounted for and modified for cases wherethe web plates are stiffer than the filling material. However, thisdevelopment is left for future work, as it requires considerableextension of the presented theory.

Appendix A. Shear stiffness for empty beam

The shear stiffness of the empty beam is derived in detail in Ref.[6]. The faceplate deflection is given as (Note. x-coordinate andzero deflection at the middle of the unit cell.)

wiQ ðxÞ ¼

QQ s2d24Di

xs

ki1

sd�4

x2

s2 þ 3� �

þ 4 6Di

dkih

þ ki2

Di

Dw

!" #ð11Þ

When the unit cell deforms under shear, the change of deflec-tion between the ends of the unit cell is:

DwiQ ¼ wi

Qs2

� ��wi

Q �s2

� �¼ Q Q s2d

12Diki

1sdþ 12

Di

dkih

þ 2ki2

Di

Dw

" #ð12Þ

Then, the shear stiffness is

DQ ¼Q Q

wQ=s¼ 1

sd12Di

ki1

sdþ 12 Di

dkih

þ 2ki2

DiDw

h i¼ 12Dw

s2 ki1

DwDiþ 12 d

sDw

dkih

þ 2ki2

ds

h i ð13Þ

where

kih ¼ Q Q s=hi

c

kt1 ¼ 1� kQ

kb1 ¼ kQ

kt2 ¼ 2� 3kQ

kb2 ¼ 3kQ � 1

ð14Þ

and

kQ ¼1þ 12 Dt

s1kth� 1

kbh

� �þ 6 Dt

Dw

ds

1þ 12 DtDw

ds þ

DtDb

ð15Þ

Appendix B. Unit cell averages for polynomials

Hyperbolic or trigonometric function can be written in polyno-mial series form. If the polynomial expansion is of n’th degree,where n is even number, then the polynomial expansion of deflec-tion with respect to mid-point of unit cell can be written as

wlðxÞ ¼ w0 þ a1x1 þ � � � þ an�1xn�1 þ anxn ð16Þ

where w0 is the unit cell mid-point deflection and it is assumed thatn is even. Then, the odd terms and their derivatives (level describedin superscript brackets) are written as:

wl;oddðxÞ ¼ a1x1 þ a3x3 þ . . .þ an�1xn�1

wð1Þl;oddðxÞ ¼ a1 þ 3a3x2 þ . . .þ ðn� 1Þan�1xn�2

wð2Þl;oddðxÞ ¼ 6a3xþ 20a5x3 þ . . .þ ðn� 1Þðn� 2Þan�1xn�3

wð3Þl;oddðxÞ ¼ 6a3 þ 60a5x2 þ . . .þ ðn� 1Þðn� 2Þðn� 3Þan�1xn�4

wð4Þl;oddðxÞ ¼ 120a5 þ . . .þ ðn� 1Þðn� 2Þðn� 3Þðn� 4Þan�1xn�5

ð17Þ

and the even terms as

wl;evenðxÞ ¼ a2x2 þ a4x4 þ . . .þ anxn

wð1Þl;evenðxÞ ¼ 2a2xþ 4a4x3 þ . . .þ nanxn�1

wð2Þl;evenðxÞ ¼ 2a2 þ 12a4x2 þ . . .þ nðn� 1Þanxn�2

wð3Þl;evenðxÞ ¼ 24a4xþ . . .þ nðn� 1Þðn� 2Þanxn�3

wð4Þl;evenðxÞ ¼ 24a4 þ . . .þ nðn� 1Þðn� 2Þðn� 3Þanxn�4

ð18Þ

For Eq. (17) we then obtain the average

w�l;oddðxÞ ¼1s

Z s=2

�s=2wl;oddðxÞdx ¼ 0

w�ð1Þl;oddðxÞ ¼ h�l ¼1s

Z s=2

�s=2wð1Þl;oddðxÞdx ¼ a1 þ a3

s2

� �2þ . . .þ an�1

s2

� �n�2

w�ð2Þl;oddðxÞ ¼ �M�

l

D¼ 1

s

Z s=2

�s=2wð2Þl;oddðxÞdx ¼ 0

w�ð3Þl;oddðxÞ ¼ �Q �lD¼ 1

s

Z s=2

�s=2wð3Þl;oddðxÞdx

¼ 6a3 þ 20a5s2

� �2þ . . .þ ðn� 1Þðn� 2Þan�1

s2

� �n�4

w�ð4Þl;oddðxÞ ¼ �q�lD¼ 1

s

Z s=2

�s=2wð4Þl;oddðxÞdx ¼ 0 ð19Þ

where Eq. (1) is utilized. Doing the same for the even terms, Eq.(18), gives

88 J. Romanoff / Composite Structures 113 (2014) 83–88

w�l;evenðxÞ¼1s

Z s=2

�s=2wl;evenðxÞdx¼1

3a3

s2

� �2þ���þ 1

ðnþ1Þans2

� �n

w�ð1Þl;evenðxÞ¼ h�l ¼1s

Z s=2

�s=2wð1Þl;evenðxÞdx¼0

w�ð2Þl;evenðxÞ¼�M�

l

D¼1

s

Z s=2

�s=2wð2Þl;evenðxÞdx¼2a2þ�� �þnan

s2

� �n�2

w�ð3Þl;evenðxÞ¼�Q �lD¼1

s

Z s=2

�s=2wð3Þl;evenðxÞdx¼0

w�ð4Þl;evenðxÞ¼�q�lD¼1

s

Z s=2

�s=2wð4Þl;evenðxÞdx¼24a4þ�� �þnðn�1Þðn�2Þan

s2

� �n�4

ð20Þ

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