tio

Keywords:A. Layered structuresB. Stress transferC. Analytical modelingLaminate mechanics

s thconf th

inner interlayer. In order to obtain a fast and easy to use tool, something that is necessary for an analytical

tep ofodelin

than the sum of its components as standalone structural elements.Composite structures can be broadly subdivided into mixed, lami-nated, sandwich, and layered structures.

Mixed (or hybrid) structures are composite structures whosecomponents are made of materials different from one another.Mixed structures use the in-plane shear stress transfer to makethe components behave in the most advantageous way. Examplesare steelconcrete beams [57], which make the concrete carry

series of sharp teeth (which dig into the wood), connection

and concentrated loads, and to dissipate the energy released bycracking, so attenuating crack propagation. Moreover, a laminatedstructure may have non-structural advantages as regards thermalinsulation, sound attenuation, and radiation absorption. On theother hand, lamination implies a certain reduction in stiffnessand strength, with respect to the equivalent monolithic structure(i.e., the monolith with thickness equal to the total thickness ofthe laminated). Thus, laminated structures use the interface in-plane shear stresses to minimize those reductions. Examples arepolymeric laminates [9] and laminated glass [1016]. The poly-meric interlayer of laminated glass, in particular, prevents the

Composites: Part B 47 (2013) 365378

Contents lists available at

te

evE-mail address: paofor@iuav.itto the composite plate with discontinuous connection.A composite structure can be dened as a structure made up of

distinct components connected to each other. The connectiontransfers in-plane shear stresses between the components, whichprovide the composite structure with greater stiffness and strength

enhancements (ribs, embossments, indentations), welded steel(spiral) bars.

Laminated structures are composite structures manufacturedby connecting two or more layers together, with thin interlayers.Lamination allows the composite structure to tolerate impactsplate with continuous connection [3,4]. This last step is devotedmembers. The rst two steps were devoted to the composite beam[1,2]; the third and fourth steps were devoted to the composite

ments, named shear connectors. Examples of shear connectorsare welded (headed) studs, short length of steel channels, bolts,1. Introduction

This paper represents the fth sscope is the analytical and exact m1359-8368/$ - see front matter 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compositesb.2012.11.004model to be chosen over nite elements and empirical formulas, modeling process is developed withinthe framework of two-dimensional elasticity. In so doing, the model also represents a means for attainingfull comprehension of the mechanical phenomena that are involved, something that neither three-dimensional elasticity nor nite elements and empirical formulas can attain. The transition from threeto two-dimensional behavior is obtained by relating the normal stress in the direction transverse tothe plate to the distortion in the interlayer. The two-dimensional behavior is governed using kinematicand force assumptions that do not impose appreciable constraints on the stressstrain state and struc-tural behavior. Starting from these assumptions, the paper develops the relationships between displace-ments and interface stresses, for both continuous and discontinuous connection. The latter relationships,which are used in this model, and the former relationships, which were used in a previously presentedmodel, are discussed and compared to each other. The subsequent sections of the paper describe themodel and present some real case applications of discontinuously-connected layered plate.

2012 Elsevier Ltd. All rights reserved.

a research line whoseg of composite exural

prevailing compressive stress and the steel prevailing tensilestress, and timberconcrete structures [8], which make the con-crete carry prevailing compressive stress. The connection betweenthe components is normally obtained by using discontinuous ele-Available online 19 November 2012distribution of discontinuous connectors is replaced by a ctitious continuous medium (interlayer).Accordingly, the plate is modeled as an equivalent three-layered plate: Two outer layers and a connectingLayered plate with discontinuous connec

Paolo ForaboschiUniversit IUAV di Venezia, Convento delle Terese, Dorsoduro 2206, 30123 Venice, Italy

a r t i c l e i n f o

Article history:Received 18 October 2012Received in revised form 31 October 2012Accepted 4 November 2012

a b s t r a c t

The subject of this paper icontinuous way, i.e. via disthe mechanical behavior o

Composi

journal homepage: www.elsll rights reserved.n: Exact mathematical model

e plate composed of two identical layers connected to each other in a dis-tinuous elements (connectors). This paper presents a model that describesis plate by a system of exact, analytical (explicit) equations. The discrete

SciVerse ScienceDirect

s: Part B

ier .com/locate /composi tesb

included [1719,25,34,45]. Moreover, this research is intimatelyconnected with the topic of composite structures with imperfect

s: Papropagation of a crack to the other glass ply, eliminates fallingglass shards, and guarantees substantial residual strength aftercracking [16].

Sandwich structures are composite structures fabricated byattaching two relatively stiff skins (facings or face-sheets) to alow-to-moderate stiffness core. The design criterion of the cross-section is to place a stiff material where the normal stresses andtheir bending lever arms are greater, i.e. outward from the cen-troid, and a soft material where the normal stresses and theirbending lever arms are lower, i.e. inward to the centroid. In otherwords, an interlayer is interposed between two layers with the

Nomenclature

B side of the layered plate parallel to y-axisE elastic modulus of the layers (in tension and compres-

sion)Es elastic modulus of the connector (discontinuous con-

necting element)Et elastic modulus of the interlayer, which is assumed to

be nilG shear elastic modulus of the layersGt shear elastic modulus of the interlayerIs modulus of inertia of the individual connector (discon-

tinuous connecting element)h thickness of each layerL side of the layered plate parallel to x-axis, with LP Bk half the thickness of the interlayerM mp/LN np/Bp lateral load, applied on the upper surface of the upper

layer (F/L2)tu, tv x and y components of the in-plane shear stress trans-

mitted through the upper interface, between the inter-layer and the upper layer

366 P. Foraboschi / Compositepurpose of increasing both the resultant forces in each layer andtheir lever arm, without increasing the maximum stress acting inthe layers. Considering that the soft material is much lighter thanthe stiff material and that the stiff material has also adequatestrength, sandwich structures can achieve higher stiffness-to-den-sity and strength-to-density ratios than other comparable struc-tures, although the stiffness of the interlayer may be drasticallylower than that of a layer. Examples of skin materials are rein-forced polymers, metals, and timber [1725]. The core can be com-posed of a continuous materialexamples are polymers, foams,balsa wood [17,18,2025]or a micro-structured periodic mate-rialexamples are honeycombs, corrugated, lattice, ceramic tiles[19,2629].

Layered structures are composite structures manufactured byconnecting two or more equal layers together, without interposingany space interlayer. The connection can be either discontinuous orcontinuous. Examples of discontinuous connections are studs,nails, dowels, pins, spikes, screws, dovetails, comb joints, ngers[3044]. Examples of continuous connections are polymers (chem-ical bonding), welds, post-vulcanization, interlocking devices(cohesion and mobilized friction mechanisms between the roughinterfaces) [9,1820,22,3133]. The difference between a layeredand a mixed structure is that the components of the latter are dif-ferent from one another, contrary to the former. The difference be-tween a layered and a laminated structure is that the latter isassembled together through the interposition of one or more spacelayers (thin, but not negligible), contrary to the former. The same istrue for a sandwich structure, with the difference that the inter-layer may be thick.interface, which provides important Refs. [4654].

2. Standard modeling assumptions

Analytical modeling of composite plates can be developed in theThis paper focuses on the layered plate with discontinuous con-nection (two layers). Laminated plates with discontinuous inter-layer [58,16] and sandwich plates with discontinuous core are

u, v, w components of displacements in the x, y, z directions,respectively

ut, vt x and y components of the relative in-plane displace-ment of the upper interface with respect to the to theplate middle surface (which is immovable)

tvm, vtm maximum value of tv and vtTu shear force transmitted by an individual connecting ele-

ment in the x directionwMax maximum deection of the plate (at the center)z0, z0 0 out of plane axes with origin O0 and O0 0 on the middle

plane of the upper and lower layers, respectivelyDx, Dy x-axis and y-axis spacing of the connectorsm Poissons ratio of the layersryi ry stresses at the upper surface of the lower layerrym ry stresses at the lower surface of the lower layerrMax maximum stress in the layered platerz normal stress in the direction transverse to the platehab angle of rotation of the segment abf h + 2k = lever arm of the internal couple of the in-plane

forces

rt B 47 (2013) 365378framework of either three-dimensional or two-dimensionalelasticity.

Exact three-dimensional elasticity solutions were constructed in1969 for the composite laminates in cylindrical bending [55] and in1970 for rectangular bidirectional composites and sandwich plates[56]. These analytical models represent important achievementsnot only in sandwich theory but in mechanics. Unfortunately, thesemodels are neither easy nor expressive, and it is neither straightfor-ward nor fast to obtain the solutions, something that conversely isnecessary for an analytical model to be chosen over nite elementmodels or empirical formulas. Moreover, the three-dimensional ap-proach does not capture the dening attribute of layered structures,because it does not distinguish between essential behaviors andside effects. Thus, three-dimensional analytical modeling does noteven represent a means for attaining scientic understanding ofthe mechanical behavior of the layered plate.

In the two-dimensional elasticity framework, conversely, mod-eling process can be developed so as to obtain a simple analyticalformulation whose application not only is very easy, but even dras-tically less time consuming than to generate any nite elementmesh or to apply any empirical formula [3,4]. Furthermore, thetwo-dimensional approach reproduces directly and explicitly theprocess of deformation and mechanical failure of the layered plateand the connection between these processes and their underlyingmechanisms, which provides fully understanding of the phenom-ena involved. For these reasons, this research was developed with-in the framework of the two-dimensional elasticity.

Two-dimensional elasticity differs from three-dimensional elas-ticity mainly in the normal stress in the direction transverse to the

plate, rz, which the latter considers while the former ignores. In or-der to replace the three-dimensional model, thus, a two-dimen-sional model has to reproduce the effects of the rz.

The layered plate is here simulated by the three-layered model(Fig. 1): Two identical layers of stiff material that reproduce thebehavior of the outer components and a more compliant innerinterlayer that reproduces the connection between the layers. Thismodel is a thorough and straightforward schematization of anysymmetric composite plate. Therefore, not only does it simulatethe layered plate, but it also simulates laminated and sandwichplates. In fact, the layers can reproduce the behavior of any outerface sheets and the interlayer can reproduce both continuous anddiscontinuous connections.

The development of the three-layered model within the two-dimensional elasticity framework calls for the following modelingassumptions, which are about (1) the plate, (2) the interfaces, (3)the layers, (4) and the interlayer.

(1) The structural behavior is elastic.(2) The displacement functions have no discontinuity at the

interfaces.(3) Each layer behaves according to the Kirchhoff-Love plate

hypotheses.(4) The interlayer does not transmit any stresses through the

cross-sections and its thickness does not change during

P. Foraboschi / Composites: Pabending.

The rst assumption is supported by the fact that inelasticityusually provides the ultimate limit states of layered plates withhigher capacity/demand ratios than the serviceability limit states[511,1620,2427,3841,5557]. With particular regard to theconnection between the layers, if it is relatively compliant, inelas-ticity would be possible only beyond the ultimate limit state of theplate, while if it is less compliant, failure is due to delamination,displaying little inelasticity [11,21,30,35,38,39,42,43,5860]. Thus,these specic layered plates are dimensioned to satisfy the service-ability deection, which entails linear behavior, while inelasticityis pointless for their dimensioning and assessment.

Fig. 1. Diagram of the layered plate. The interlayer may reproduce the behavior of adiscontinuous connection (shown in the gure) or a continuous connection. The

gure accounts for the elastic properties of the components. The out-of-plane axes,z0 , z0 0 , and z have origins O0 , O0 0 , and O, which lie on the middle plane of the upperlayer, lower layer, and interlayer, respectively.The second assumption coincides with the denition of inter-layer [4851,60,61].

The third assumption, which consists of modeling the layersalong the lines of classical plate theory, reects that a layer sub-stantially has a bending deection and falls within the categoryof thin plates with small deection [56].

As regards small deection, results from nite element modelsshowed that the layered plates that fulll the deection limits pre-scribed by the serviceability limit states of almost all the structuralapplications do not exhibit any detectable geometric non-linearityeven at ultimate. Thus, the single layer may be extremely thin,since the composite action keeps the deection down. As regardsignoring the shear strain, three-dimensional elasticity solutionsthat take into account shear deformation [55,56] showed that thethickness-to-span ratio beyond which the shear strain noticeablyaffects the behavior is very high, to the extent that the layer isnot a plate but a three-dimensional solid [4].

The fourth assumption reproduces every discontinuous connec-tion directly (the subject of this paper) and, indirectly, the vastmajority of the continuous connections.

A discontinuous connection does not exhibit continuous cross-sections; thus, no rx and ry stress is transmitted through the inter-layer. Moreover, each individual connector does not exhibit anyelongation or shortening; thus, the thickness of the interlayer can-not change.

The vast majority of the continuous connection have low elasticmodulus or/and small thickness. Thus, the normal stresses actingon the interlayer cross-sections are negligible and/or have negligi-ble lever arms. Moreover, the transverse shear stresses that main-tain the rotational equilibrium are negligible and/or act on thincross-sections as well. Hence, the interlayer bending momentsand shear forces are negligible, i.e., the stresses acting onto theinterlayer cross-sections have no effects [1625,47,51,56].

Moreover, almost all the continuous connections change thethickness only marginally due to the normal stress in the directiontransverse to the plate, rz. Changing the interlayer thickness wouldimply losing the anti-asymmetry of the behavior, which would en-tail that a layer has to bear a greater fraction of the external loadthen the other layer. Thus, design avoids this undesirable condition.

The fourth assumption does not impose any constraint to thedistortion due to compatibility between interlayer and layers(deection derivative).

Ultimately, the assumptions do not impose constraints on thestressstrain state of the layered plate. Thus, a model based onthese assumptions does not need to be veried by experimentsor nite element models. Conversely, such a model allows niteelement models and empirical formulas to be veried.

3. Modeling of the connection through the interlayer

The assumptions, in particular the fourth one, allow one to sim-ulate the mechanical behavior of the connection between the lay-ers by using an interlayer.

This section refers to an innitesimal element cut out of theinterlayer by two pairs of planes at an innitesimal distance fromeach other, parallel to the xz and yz planes; i.e., innitesimal paral-lelepiped of sides dx, dy, 2k (Fig. 2), where k is the semi-thicknessof the interlayer.

Let xa and ya denote the x and y coordinates of the vertex a; uat itsin-plane translation ut, and wa its deection w (Fig. 3). The transla-tions ut of the vertexes b, c, d can be obtained by using continuityand anti-symmetry of displacements with respect to the middleplane. The deections w of the same vertexes can be obtained by

rt B 47 (2013) 365378 367using the fourth assumption.

ubt uxa;k uat ; wb wa 1

hac @w@x

5

hab utk 6

3.1. Continuous connection between the layers

This Section is devoted to simulating the continuous connection

s: Part B 47 (2013) 365378368 P. Foraboschi / Compositeuct uat @utxa

@xdx; wc wa @wxa

@xdx 2

udt uxa dx;k uct ; wd wc; 3The strains in the face abdcmay be described by using the deriv-

ative of the angle hab, along with the angles hac plus hab of its sidesac and ab. Conversely, the curvature is meaningless, since the shearstrain is dominant. From (1)(3):

@hab@x

1k @ut@x

4

between the layers by using an interlayer. Novelty does not stemfrom the results, which were already presented and used [1,3,4],but from the generalized mechanical framework within the inter-layer model is obtained.

According to (4),@hab@x

induces ex and hence rx in the face abdc(Figs. 4 and 5):

ex 1 @ut z; rx Et @ut z 7

Fig. 2. Innitesimal element cut out of the interlayer. Shadowed area: face abdc,which the developments refer to. The origin O of the axis z lies on the middle planeof the interlayer.

Fig. 3. Innitesimal face abdc of the element cut out of the interlayer (Fig. 2).Undeformed face: dashed lines. Deformed face: solid bold lines with shadowedarea. The gure shows the in-plane translation ut, the deection w of the vertexes,and the angles of rotation of the sides, including the signs.Fig. 4. Innitesimal face abdc of the solid element cut out of the interlayer. Prolesof the normal stress acting on the x cross-sections. Shadowed area: upper semi-k @x k @x

where Et is the elastic modulus of the interlayer. Tensile strains andtensile stresses are positive.

Eq. (7) explain the rst statement of the forth assumption. Theelasticity moduli of almost all the cores and interlayers, Et, are verysmall. Thus, the normal stresses rx and ry given by (7) are small. Ifmoreover the interlayer is thin, z is small; consequently the normalstresses are particularly small and their effects are insignicant.That is why rx and ry are ignored (Figs. 4 and 5).

According to (5), the interlayer is subjected to the distortion@w@x

.This distortion results from the anti-symmetric rotations hac andhbd of the two sides ac and bd (Fig. 3). Due to these rotations, thestraight lines normal to the middle surface can exhibit (1) eitheranti-symmetric bending (2) or relative deection.

(1) Anti-symmetric bending induces variations of the transversenormal stresses, rz, which are transmitted between theinterlayer and layers through the interfaces. Thus, anti-symmetric bending implies that any straight line normalto the middle surface, in particular of the two sides aband cd, remain neither straight nor normal to the middlesurface after deformation (i.e., double-curvature bendingshape; Fig. 6).

(2) Relative deection induces shear stresses sxz and syz on thecross-sections of the interlayer. Thus, relative deectionimplies that any straight line normal to the middle surface,in particular of the two sides ab and cd, do not remain nor-mal to the middle surface after deformation, whereas theyremain straight (as shown in Fig. 3).thickness of the interlayer. The gure shows the elongation of the z-ber after

deformation, i.e., from the length dx to dx0 1 1k @ut@x

z

dx.

metric bending in the interlayer, which consists of variable rz(Fig. 6). The variation of rz over the sides ac (and over bd) is equiv-alent to a bending moment (couple).

The variation of rz is governed by the anti-symmetric rotationsof the sides ac and bd. Under the working hypothesis that the shearstresses acting on the cross-sections of the interlayer have uniformdistribution (Fig. 6):

dwdx

limDx!0

Drz Dy Dx2 k3 Ei Dy Dx3

limDx!0

Drz k3 Ei Dx

k3 Ei

drzdx

8

Eq. (8) proves that rz is an exogenous variable, i.e., it can bedetermined outside the model of the layered plate, by using anindependent equation.

Eq. (8) is valid for Et/k ratios lower than the critical value, other-wise the distortion induces relative deection instead of anti-sym-metric bending. For these higher ratios, For these higher ratios ofEz/k the distortion is not induced by the rz stresses and their vari-ations, but by the shear stresses.

Ultimately, Eq. (8) explains why the in-plane shear stressestransmitted by a continuous connection through the interfacesdo not depend on the deection derivative: The distortion is not

(a) (b)

P. Foraboschi / Composites: Part B 47 (2013) 365378 369The interlayer performs the distortion@w@x

following the less stiffbehavior between anti-symmetric bending and relative deection.The stiffness of each behavior depends on the Et/k ratio. If this ratiois negligible to very low, the stiffness of the anti-symmetric bend-

Fig. 5. Innitesimal element cut out of the interlayer: proles of the normal stressacting on the x and y cross sections. The model ignores these stresses.ing is lower than the stiffness of the relative deection, and viceversa. Although the critical ratio is very low, the vast majority ofthe practical applications can be analyzed under the assumption

that the Et/k ratio is nil. Thus, the distortion@w@x

induces anti-sym-

Fig. 6. Face abdc of the innitesimal element cut out of the interlayer. Shadowedarea: upper semi-thickness of the interlayer. Stresses rz acting on the sides ac andbd (interfaces). The variation of rz over the sides ac is equivalent to a bendingmoment (couple); the same for the side bd. The gure represents these two bendingmoments, together with the balancing shear stresses.(c)Fig. 7. Shear distortion and shear stresses in the interlayer. Positive signs of thevectors are illustrated (note that tu and tv are applied to the interlayer). (a) Faceabdc. Undeformed state: solid bold lines. Deformed state due to shear distortion:

dashed lines. Angle hab and hcd of the sides ab and cd. (b) Shear stresses acting on theinnitesimal face abdc. (c) Shear stresses acting on the innitesimal element cut outof the interlayer.

shear distortion. It follows that, the distortiondwdx

plays no role in

the composite behavior of the plate.According to (6), the angle hab induces shear strain czx and cxz

together with shear stress szx and sxz in the interlayer, which areconstant over the thickness k (Fig. 7):

czx utk

; szx Gt utk 9

According to Eqs. (7)(9), hence, the interlayer is free from nor-mal stresses, bending moments, and transverse shear forces, whileit transmits in-plane shear stresses through the interfaces togetherwith the balancing shear stresses on the cross-sections (Figs. 8 and9). The x and y components of the in-plane shear stresses transmit-ted through the upper interface, tu and tv, are:

tu Gt utk ; tv Gt v tk

10

The sign of an interface in-plane shear stress is due to the factthat Eqs. (10) consider the shear stress acting on the lower edgeof the upper layer (Figs. 79). Thus, tu and tv acting on the inter-layer have opposite sign with respect to (10). The signs of tu andtv acting onto the cross-sections of the interlayer follow from therotational equilibrium of the innitesimal element (Figs. 79).

Physical interpretation of Eqs. (8) and (10) can be derived fromthe fact that Et 0 means that the boundaries of the interlayer are

Fig. 9. Perspective of the shear and normal stresses acting on the innitesimalelement cut out of the interlayer. Positive signs of the vectors are illustrated. Themodel ignores the shear stresses acting on the cross-sections and all the normalstresses, as well as their effects.

Fig. 10. Diagram of the discontinuous connection that does not transfer anybending moment to the layers: The behavior of the interlayer is reproduced by aspatial truss.

Fig. 11. Mesh of the truss of Fig. 10. The in-plane force transmitted through thehinge depends on the elongation of the diagonal pin-joined truss (solid bold line).

370 P. Foraboschi / Composites: Part B 47 (2013) 365378utterly compliant with the two anti-symmetric angles hac and hbdimposed by the layers to the interlayer. Conversely, the angleshab and hcd, which are not anti-symmetric, imply the transmissionof signicant shear stress through the interfaces.

3.2. Discontinuous connection between the layers

The research developed the subject in a comprehensive manner.To reach this, the research did not consider how the discontinuousconnection was made but it considered only the actions that thediscontinuous connection exchanged with the layers. Using thisapproach, all the practical applications [3045] can be grouped

Fig. 8. Shear and normal stresses acting on the innitesimal face abdc of the

element cut out of the interlayer. Positive signs of the vectors are illustrated. Themodel considers only the shear stress tu (and tv acting on the face abfe, which is notshown).into two general categories, according to whether or not the con-nection transfers bending moments, together with in-plane forces.

The behavior of the discontinuous connection that does nottransmit any bending moment through the interfaces can be repro-duced by transverse and diagonal straight bars hinged at bothends, to form a spatial pin-joined truss, with the pins along theinterfaces (Fig. 10). The truss exchanges transverse and in-planhe forces at the pins. The in-plane forces depend on the in-planedisplacements of the hinges but do not appreciably depend onthe vertical displacement of the hinges (Fig. 11). Accordingly, thiscategory of discontinuous connection has the same behavior asthe continuous connection, except for the fact that the former isThe elongation depends on ut and Dw, as well as on k and Dx. Since Dw/ut 1 andk/Dx 1, the in-plane forces depend primarily on the in-plane displacements whilethey depend moderately on the variation of the deection.

a discrete system. Thus, the behavior of this truss can be closelyapproximated by smearing its stiffness along x and y and thenusing (10).

A discontinuous connection that transmits bending momentsthrough the interfaces can be simulated by transverse straight barsbuilt-in at both ends, at certain x-axis and y-axis spacing (Fig. 12).Each end of the bars is xed into the layer at the distance from thelayer edge necessary to rigidly restrain the end against relativerotation with respect to the layer. This distance, which dependson the layers and interlayer, is the semi-thickness k.

The bars exchange bending moments (couples) and in-planeforces at the ends. The bending moment at an end depends on therotation of the end and is balanced by the in-plane force at boththe ends of the bar. Thus, the in-plane forces substantially dependon both the in-plane displacements and the rotation of the bar ends

Gt Dx Dy k3

16

stresses tu and tv acting onto the cross-sections of the interlayerhave the sign that follows from the rotational equilibrium of theinnitesimal element (Fig. 7).

3.4. Continuous and discontinuous connection relationships

The interlayer that reproduces the continuous connection,which is described by Eq. (10), is completely different from theinterlayer that reproduces the discontinuous connection, which isdescribed by Eq. (15). It could be useful to substantiate this asser-tion with examples.

Consider an interlayer made of the same material as the layers.In this case the entire cross-section of the layered plate remainsplane during deformation. At any abscissa, therefore, the cross-sec-tion of the interlayer rotates through the same angle as the twocross-sections of the layers:

utk @w

@x;

v tk @w

@y17

P. Foraboschi / Composites: Pa(deection derivative). This means that this category of discontinu-ous connection drastically differs from the continuous connection,regardless of being discrete. More specically, the discontinuousconnection exhibits the same double-curvature bending shape that,in the continuous connection, is exhibited by every straight line nor-mal to the middle surface (Figs. 12 and 13). However, the interfaceshear transfer of the latter and the former are different from one an-other. Thus, thebehaviorof thebars cannotbe simulatedbyapplyingEq. (10) to the smeared model. All things considered, henceforwardonly this case is referred to as discontinuous connection.

This research dened the interlayer that models the discontin-uous connection and derived the analytical model of the layeredplate with this interlayer.

Let Es and Is denote the modulus of elasticity and the moment ofinertia of the individual (transverse) bar. The in-plane translationsut of the upper and lower ends of the bar produces the strain-planeshear force Tuu (Fig. 12):

Tuu 12 Es Isk3

ut 11

where the sign of Tuu results from the signs of ut and from the factthat Eq. (9) considers the Tuu vector applied to the lower edge ofthe upper layer. Hence, Tuu adopts the same sign convention as tu.

The rotations hp = hw of, respectively, the upper and lower endsof the bar produce the in-plane shear force Tuh (Fig. 13):

Tuh 12 Es Isk2

hp 12

with the same sign convention; in particular hp and hw are positive ifclockwise.

In nal analysis, the relationship between the in-plane shearforce Tu applied to the lower edge of the upper layer and the dis-placements is:

Fig. 12. Diagram of the discontinuous connection that transfers bending moment tothe layers: the behavior of the interlayer is reproduced by a whole of transversebars. Periodic reiteration of a bar at spacing Dx and Dy, each one having both endsrigidly xed against rotation (built-in into the layers). Dotted line: deformed bar.

The distance of the built-in end to the edge of the layer is the semi-thickness k.Accordingly, the actual interface is the solid line that bounds the shadowed area,while the dashed line shows the edges of the layers only.The sign is due to the fact that Eqs. (15) consider the shearstress acting on the lower surface of the upper layer. The shearTu 12 Es Isk2

utk hp

13

3.3. Interlayer that simulates the discontinuous connection

The in-plane shear force Tu can be smeared into the in-planeshear stress tu. In so doing, the discontinuous connection is simu-lated by a ctitious continuous interlayer. Let Dx and Dy denotethe x-axis and y-axis spacing of the connectors, respectively, i.e.the spacing between the bars (Fig. 1).

tu TuDx Dy 12 Es IsDx Dy k2

utk hac

14

The x and y components of the in-plane shear stresses transmit-ted through the upper interface, tu and tv, can be nally worked outby using Eq. (5):

tu Gt utk Gt @w@x

; tv Gt v tk Gt @w@y

15

where Gt represents the equivalent shear elastic modulus:

12 Es Is

Fig. 13. Element of length dx cut out of the layered plate. Deformation of the bar(pointed line) due to the rotation of the layers, which causes the upper and lowerends of the bar to rotate by an angle of hp = hw. The rotations of the ends of the barengender the in-plane shear force Tuh.

rt B 47 (2013) 365378 371Substitution of (17) into the equation that pertains to this situ-ation, i.e. into (10), leads to:

it is sufcient to measure the relative displacement Ut betweenthe layers due to a shear action Tu obtained by a force Tu appliedto one layer and an opposite force Tu applied to the other layer(considering also the balancing couples, since the two forces Tuare not coaxial).

Let K denote the shear stiffness of the individual connection. Bydenition: K = Tu/Ut. Considering that the bar is xed against rota-tion at both the ends:

k 12 Es Is

K3

r22

4. Discontinuously-connected layered plate: system denition

The major conclusion which can be drawn from Section 3 is thatthe model presented in [4] covers the continuous connection only,while the layered plate with discontinuous connection has to bemodeled by the interlayer described by Eq. (15). The researchwas then focused towards developing the model of the layeredplate with discontinuous connection by using said interlayer.

The reference structure used was the three-layered plate withcross-section formed by two layers, each one of thickness h, plusan interlayer of thickness 2k, restrained at the boundary, and sub-jected to a lateral load p (Fig. 1). The assumptions are stated in Sec-tion 2; the interlayer is governed by Eq. (15).

The nomenclature adopted throughout this paper is the same asthat adopted in [4]. In particular tu and tv denote the x and y com-ponents of the in-plane shear stresses transmitted between theinterlayer and layers through the interfaces. Positive directions oftu and tv applied to the lower surface of the upper layer are those

s: Part B 47 (2013) 365378tu Gt @w@x

; tv Gt @w@y

18

which reproduce the real behavior in this limiting condition.Conversely, substitution of (17) into (15) leads to tu = 0 and

tv = 0. If the interlayer is modeled using Eq. (15) in lieu of Eq.(10), thus, no in-plane shear stress is transferred through the inter-face in this limiting condition, while the real behavior exhibits themaximum shears stress transfer.

Consider a continuum connection that provides the layers withno in-plane shear stresses. The cross-section of the interlayer ro-tates through the angles:

utk @w

@x h2 k ;

v tk @w

@y h2 k 19

Substitution of (19) into (10) leads to:

tu Gt @w@x

h2 k ; tv Gt

@w@y

h2 k 20

For k that approaches innite, tu and tv of Eq. (20) approach zero,which reproduce the real behavior in this limiting condition.

Substitution of (19) into (15) leads to:

tu Gt 1 h2 k

@w@x

; tv Gt 1 h2 k

@w@y

21

For k that approaches innite, tu and tv of Eq. (21) approaches

Gt @w@x

and Gt @w@y

, respectively, while the real behavior is that

the interfaces transfer no in-plane shears stresses.In (21), moreover, tu and tv result to be nil for k = h/2, indepen-

dently on Gt, which is another erroneous prediction. According to(20), in addition, for both k and Gt that approach zero in such a

way that their ratioGtk

remains the same, tu and tv do not vary.

According to (21), on the contrary, tu and tv vary (i.e.,h Gt2 k

@w@x

andh Gt2 k

@w@y

do not vary, but Gt @w@x

and Gt @w@y

vary with Gt).

The real behavior in this limiting condition is that that tu and tvare almost constant. Again, this behavior is captured by (10) whileit is missed by (15).

To sum up, this Section has explained why [4] has used Eq. (10)and not Eq. (15). Actually, those numerical models that use Eq. (15)for continuously-connected layered plates misestimate the stiff-ness of the connection.

3.5. Thickness of the interlayer that reproduces the discontinuousconnection

Eqs. (15) replace the discontinuous connection with the inter-layer, as long as Eq. (16) and the semi-thickness k are substitutedinto Eq. (15). The input data of Eqs. (15) and (16) can be taken di-rectly from the layered plate, except for k, which does not repre-sent a real geometric dimension but a ctitious thickness. In fact,each transverse bar is built-in at both ends (Figs. 12 and 13) andthe rigid restraint against relative rotation calls for an adequateinsertion depth of the built-in end into the layer. The insertiondepth necessary to x the bar to the layer depends on the stiffnessof the bar and layer. This insertion is reproduced by the semi-thick-ness of the interlayer k. Accordingly, k is a mechanical parameterrather than a geometric dimension. This entails that k has to bedetermined experimentally by testing a sample of layered plate.

372 P. Foraboschi / CompositeThe sample may consist of a portionDx Dy of the plate (a connec-tor together with the relevant parts of the layers). The test has todetermine the shear stiffness of the connection. To achieve this,Fig. 14. Element cut out of the layered plate by two pairs of planes parallel to the xzand yz planes, at an innitesimal distance from each other. Layer out-of-plane shearforces acting on the x and y cross-sections; layer bending moments and layer

normal (axial) forces acting on the x + dx and y + dy cross-sections. The gure showsthe positive direction of the vectors. The gures illustrate as well the transverseload p.

of the axes that they are parallel to (Fig. 7). Moreover, u, v, wdenote the components of the displacement in the x, y, z directions(i.e., u and v are the in-plane displacements, w is the deection);the positive direction of each displacement component is that ofthe relevant axis. Furthermore, ut and vt denote the x and y compo-nents of the in-plane displacement of the upper interface with re-spect to the middle surface of the plate (i.e., ut and vt are u and v forz = k). The positive directions of ut and vt are those of the axis thatthey are parallel to. Anti-symmetric behavior implies that the mid-dle surface of the plate is immovable. Thus, ut and vt are absolutedisplacements.

(Figs. 7 and 1417):

@NX@x

@VXY@y

tu 23

The derivatives of Nxy and Vxy are available in [4], while tu is gi-ven by Eq. (15):

a h2

2 @

3w@x3

a h @2ut@x2

m a h2

2 @

3w@x@y2

m a h @2v t

@x@y

G h2 @3w

@x@y2 G h @

2ut@y2

G h @2v t

@x@y Gt utk

Gt @w@x

0 24

where:

Fig. 15. Innitesimal element cut out of the layered plate, with the layer twistingmoments and layer in-plane shear forces, acting on the x + dx and y + dy cross-sections, in the positive direction.

P. Foraboschi / Composites: PaFig. 16. Layer internal actions upon the x and y cross-sections as well as upon theupper interface, of an innitesimal element cut out of the upper layer.The material properties of the layers are the elastic modulus,the shear (elastic) modulus, and Poissons ratio, E, G, and m,respectively.

The mathematical model developments consider (Figs. 1417)both an innitesimal element cut out of the upper layer and aninnitesimal element cut out of the layered plate (two pairs ofcut planes at an innitesimal distance from each other, parallelto the xz and yz planes respectively). Presented here are the devel-opments for the x direction; the expressions for the y direction canbe obtained by inverting the index x with y and vice-versa in the xexpressions.

5. Model of the discontinuously-connected layered plate

The mechanical behavior of the discontinuously-connected lay-ered plate is described by the 10 internal actions Nx, Ny, Mx, My, Vx,Vy (Fig. 14), Vxy, Mxy (Fig. 15), tu (Fig. 7), tv (by virtue of sxy = syx,Mxy =Myx and Vxy = Vyx), and by the three displacements w, ut, vt(Figs. 3 and 17). These 13 unknowns are functions of x and y. To ob-tain the model, rst the relationships between the 10 internal ac-tions and the three displacements were found, and then 3equations involving the displacements were dened.

The reduction of the 13 unknowns to Vx, Vy, tu, tv, w, ut, vt doesnot involve the interlayer. Thus, the same reduction carried out in[4] also holds true here. Hence, the relationships available in [4],together with (15), represented the starting point of the develop-ment of the model. Thus, all that was needed to obtain the modelwas to nd the remaining relationships between the internal ac-tions and the displacements, and the equations involving thedisplacements.

5.1. Layer in-plane translational equilibrium

The translational equilibrium along the x-axis implies that

Fig. 17. Innitesimal rectangular element dx. Positive signs of the displacementsand the differential internal actions.

rt B 47 (2013) 365378 373a E1 m2 25

f G @w 2 V 0 28

t3 2 2 3 29

s: Paa h @x @y a h

5.4. Governing equations of the discontinuously-connected layeredplate

The collection of (29), (24), and the form of (24) written for the yaxis gives a system of three linear differential equations with con-stant coefcients. The coefcients are known, since they combinegeometric and mechanical characteristics of the plate. Thus, thissystem of three differential equations completely describes themechanical behavior of the layered plate with discontinuous con-nection spread onto an equivalent continuous interlayer.

@4w@x4

2 @4w

@x2@y2 @

4w@y4

6 f Gtk a h3

@ut@x

@v t@y

6 f Gta h3

@2w@x2

@2w

@y2

! 6 pa h3

a h2 @

3w@x3

m a h2 G h

@

3w@x@y2

a @2ut@x2

G @2ut@y2

Gt utk hGth @w@x

G m a @2v t

@x@y 0

a h2 @

3w@y3

m a h2 G h

@

3w@x2@y

G m a @2ut

@x@y2 2

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

30t@x x

5.3. Layer lateral translational equilibrium

The out-of-plane shear forces do not depend on w, ut and vt. Inorder to reduce the unknowns to w, ut and vt, hence, Vx and Vy haveto be eliminated by determining Vx from (28) and Vy from the anal-ogous equation, then substituting them into the lateral transla-tional equilibrium equation.

@4w@x4

2 @4w

@x2@y2 @

4w@y4

6 f Gtk a h3

@ut@x

@v t@y

6 f G @2w @2w !

6 p5.2. Global rotational equilibrium

The rotational equilibrium of an innitesimal element cut out ofthe layered plate around the y-axis is the sum of Vx, Nx, Vxy multi-plied by the lever arm of these vectors, plus Mx and Mxy (Figs. 1417). The rotational contribution DR given by Nx and Vxy has the fol-lowing nite form:

DR @Nx@x

f @Vxy@y

f @Nx@x

@Vxy@y

f 26

where f h 2 k. Substitution of (23) and (15) into (26) trans-forms (26) into:

DR f tu f Gt utk f Gt @w@x

27

The rotational equilibrium of the innitesimal element aroundthe y-axis provides the following nite form equation, which re-sults from taking the moments of all the forces acting on the ele-ment with respect to the y-axis and using (27) as well as someexpressions obtained in [4]:

a h36

@3w@x3

m @3w

@y3

! a h

3 1 m6

@3w

@x@y2 f Gt utk

374 P. Foraboschi / Compositea @ v t@y2

G @ v t@x2

Gt v tk hGth @w@y

0>>:6. Analytical solution of the exact equations

The response of the three-layered plate to a doubly sinusoidalload has double trigonometric distribution along the x and y axes,with the same period as the load. Thus, the analytical solution ofEq. (30) can be obtained by expanding the load function p andthe unknown displacement functions w, ut, vt in double trigono-metric series with periods equal to the sides of the plate and tosubmultiples of them, i.e. double trigonometric series, having argu-ments Mx and Ny

M m pL

31

N n pB

32

where m and n are odd, integer numbers; moreover, L and B are themaximum values of the x and y coordinates that describe the geom-etry of the plate (0 6 x 6 L; 0 6 y 6 B). This analytical solution is en-tirely composed of explicit terms and it approaches the exactsolution as precisely as desired. Thus, this analytical solution canbe referred to as exact and closed-form.

This section accounts for the solution of the rectangular layeredplate of sides L and Bwith LP B, subjected to lateral uniformly dis-tributed static loading p, being simply-supported at the edges. Thesolutions for different geometries, loads, and restraints can be ob-tained in the same way. Considering the boundary conditions, andthe conditions of symmetry and anti-symmetry, the uniform loadand the unknown displacement functions can be expressed as:

px; y X1m1

X1n1

16 pm n p2 sinM x sinN x 33

wx; y X1m1

X1n1

wmn sinM x sinN y 34

utx; y X1m1

X1n1

ut-mn cosM x sinN y 35

v tx; y X1m1

X1n1

v t-mn sinM x cosN y 36

The orders m and n of each term of the series (33)(36) are in-cluded into M and N dened by (31) and (32). Upon using the sub-stitution (33)(36) into Eq. (30), the system of three differential Eq.(30) can be converted into m n systems of linear algebraic equa-tions, which can be solved individually. The solution of each mnsystem consists of the unknown coefcient wmn, ut-mn, vt-mn ofthe series, i.e., the solution of the mechanical problem.

M4 2 M2 N2 N4 6 f Gta h3

M2 6 f Gta h3

N2

wmn

6 f Gtk a h3

M ut-mn 6 f Gtk a h3

N v t-mn 96 pm n a p2 h3

a h2

M3 m a h2

G h

M N2 GthM

wmn

a M2 G N2 Gtk h

ut-mn G m a M N v t-mn 0

a h2

N3 m a h2

G h

M2 N Gth N

wmn

G m a M N ut-mn a N2 G M2 Gt

v t-mn 0

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

rt B 47 (2013) 365378k h37

Arranging Eq. (37) into the matrix form:

W33 f/g31 fcg31 38with

W11 M4 2 M2 N2 N4 6 f Gta h3

M2 6 f Gta h3

N2 39

W12 6 f Gtk a h3

M 40

W13 6 f Gt3 N 41

of the displacements converge rapidly to the exact solutionw, ut, vt.Effectively, a solution with m and n from 1 to 7 is almost exact.

7. Displacement and stress results

In order to obtain the entire structural response, it is sufcientto calculate the maximum of: deection, layer normal stress, andinterface shear stress.

The maximum deection is: wMax = w(x = L/2; y = B/2). Consid-ering (48):

wMax X1 X1

wmn 1mn2

2 54

epor

P. Foraboschi / Composites: Part B 47 (2013) 365378 375k a h

W21 a h2 M3 m a h

2 G h

M N2 Gt

hM 42

W22 a M2 G N2 Gtk h 43

W23 G m a M N 44

W31 a h2 N3 m a h

2 G h

M2 N Gt

h N 45

W32 W23 G m a M N 46

W33 a N2 G M2 Gtk h 47

/11 wmn 48

/21 ut-mn 49

/31 v t-mn 50

c11 96 p

m n a p2 h351

c21 0 52

c31 0 53The inversion of the matrix [W] provides the three-element vec-

tor {/}, which is the analytical solution for the simple term that theload has been divided into. Hence, wmn, ut-mn, vt-mn obtained fromany odd m and n are the exact, closed-form (explicit) solution forthe simple mn term that the load p has been broken up into.

The series obtained by recombining the terms that the functionshave been divided into, is the exact solution for the load thatapproximates p. The automatic calculus immediately reaches thesolution, even for high values of m and n. However, the seriesexpansion of the load converges fast to p; so, the series expansion

Table 1Application of the model to layered plates with discontinuous connection. The table r

L (mm) 2000 2500 4000B (mm) 2000 1800 2500h (mm) 2.250 6.500 40.002k (mm) 6.000 13.000 80.00Gt (N/mm2) 42.00 42.25 500.00K (N/mm3) 14.00 6.50 12.50E (N/mm2) 20,7000 20,7000 58,000m 0.30 0.34 0.15p (kN/m2) 0.75 2.40 3.50wMax (mm) 3.1721 1.1373 0.0874r (N/mm2) 9.8785 7.0379 0.5161Maxtum (N/mm2) 0.0557 0.0653 0.0226tvm (N/mm2) 0.0557 0.0763 0.0280m1n1

The maximum stress function is ry at the lower surface of thelower layer, denoted rym (since LP B). The stress function rymcan be obtained by taking into account that (i) ry at the upper sur-face of the lower layer, denoted by ryi, differs from rym for the bi-triangular stress prole due toMy, (ii) ryi can be obtained by insert-ing @ut=@x and @v t=@y into the elastic relationship between strainand plane-stress, (iii) My can be obtained by the relationship be-tween bending moment My and deection w. All considered:

rym a h @2w@y2

a h m @2w@x2

a @v t@y

m a @ut@x

55

The derivatives of ut and vt are negative since the displacementsut and vt given by (30) are those of the upper interface, while(55) considers the lower interface. The maximum of the stressfunction (55), denoted by rMax, occurs at x = L/2; y = B/2, i.e.,rMax = rym(x = L/2; y = B/2). Substituting (48)(50):

rMax a X1m1

X1n1

N2 m M2 h wmn N v t-mn

m M ut-mn 1mn2

2 56At any point of the interface |vt|P |ut| and j dwdy jP j

dwdx

j, sinceLP B. Therefore, tvP tu. The interface in-plane shear stress tv,

interface in-plane displacement vt, and cross-section angledwdy

reach the maximum at x = L/2 and y = B.

tvm Gt 1k v t x L2

; y B

Gt @w@y

x L2

; y B

57

The y-component of the shear force Tv transmitted through anindividual connector reaches the maximum, denoted by Tvm:

Tvm tvm Dx Dy Gt 1k v t x L2

; y B

@w@y

x L2

; y B

Dx Dy 58The maximum shear force transmitted through an individual

connector can be obtained by calculating the maximum of the vec-tor composition Tu + Tv.

ts as well the values of the ratios K = Gt/k used in [4].

4250 4500 5000 52502750 3500 3500 3500100.00 30.00 75.00 50.0045.00 20.00 75.00 30.00326.25 28.50 90.00 117.7514.50 2.85 2.40 7.8516,000 180,000 32,000 12,50000.25 0.33 0.15 0.208.50 6.50 10.00 5.500.2186 2.0411 0.7189 0.46620.5555 7.8465 1.3503 2.2586

0.0434 0.0905 0.0602 0.05820.0530 0.1050 0.0720 0.0713

The ultimate load can be obtained by setting up a proportionbetween the actual-to-ultimate stress ratio and the actual-to-ulti-mate load ratio.

8. Application of the model: Case studies

The assumptions of this model synthesize the mechanicalbehavior of the layered plate without imposing considerable con-

Fig. 18. Discontinuously-connected layered plate having: L = 4750 mm; B = 4750 mm; h = 55 mm; 2k = 25 mm; E = 7500 N/mm2; m = 0.22; Gt = 6.50 N/mm2; K = 0.52 N/mm2;p = 6.00 kN/m2. The maximum deection resulted to be: wMax = 11.8977 mm. The maximum in-plane interface shear stresses resulted to be tum = tvm = 0.0818 N/mm2. Strainand stress proles across the layered cross-section at the center (where they reach the maximum): Strains at the edges of the layers and interlayer; stresses at the edges of thelayers. Shadowed areas: tensile strains and stresses. The geometric measures are in mm.

; h =

376 P. Foraboschi / Composites: Part B 47 (2013) 365378Fig. 19. Discontinuously-connected layered plate having: L = 6750 mm; B = 6750 mm

p = 8.60 kN/m2. The maximum deection resulted to be: wMax = 20.0140 mm. The maximand stress proles across the layer cross-section at the center (where they reach the maxlayers. Shadowed areas: tensile strains and stresses. The geometric measures are in mm85 mm; 2k = 45 mm; E = 9400 N/mm2; m = 0.23; Gt = 4.50 N/mm2; K = 0.20 N/mm2;

um in-plane interface shear stresses resulted to be tum = tvm = 0.0901 N/mm2. Strainimum): Strains at the edges of the layers and interlayer; stresses at the edges of the.

prole of the layered cross-section [62,63].This analysis has considered exactly the same cases that were

considered in [4], to allow direct comparisons to be made between

s: Pathe results. The comparisons of the results of Table 1 with the anal-ogous results obtained in [4] point out the differences in behaviordue to the same interlayer that models, respectively, a discontinu-ous and continuous connection. To make the comparison immedi-ate and explicit, Table 1 also reports the stiffness K that would haveto be used if the connection were continuous rather than discon-tinuous, i.e. the value of K that was used in [4]. Not only do thesecases highlight the differences in behavior between layered platewith discontinuous and continuous connections, but also theyunderline the errors that can be made if the interlayer is not sim-ulated with the correct model.

The comparisons show that, all the other parameters beingequal (geometry and elastic parameters), the discontinuously-con-nected layered plate has signicant extra-stiffness and appreciableextra-strength with respect to the continuously-connected layeredplate. As regards the cases of Table 1, the maximum deection andnormal stress reached by the layered plate with continuous con-nection are 1.43.4 times greater and 1.21.6 times greater,respectively, than the values reached by the layered plate with dis-continuous connection. This means that the models of the layeredplate with continuous soft connection that erroneously considerdwdx

anddwdy

as shear distortion are affected by a substantial system-

atic error, as large as is shown by the above comparisons and evengreater if the interlayer is very thick and/or soft.

9. Conclusions

This research focuses on the layered plate with discontinuousconnection between the layers. In order to develop the subject ina comprehensive manner, the research groups the practical appli-cations into two general categories. The rst being that the discon-tinuous connection transmits bending moments through theinterfaces (together with in-plane shear actions), and the secondthat it does not. In so doing, the research has been developed con-sidering only the actions transferred by the connection, but with-out considering the conguration of the specic connection andof the individual connectors.

The research proves that the model [4], which is devoted to thelayered plate with continuous connection, applies also to the lay-ered plates with discontinuous connection that does not transmitany bending moments through the interfaces, but only in-planestraints. Thus, this model does not need experiment or nite ele-ment verications; moreover it replaces nite element modelsand empirical formulas in the majority of the cases.

However, nite element remains the best method for analyzinglayered plates that do not comply with the assumptions (Section 2)or that entail complicated series expansions (due to the shape,load, and boundary conditions). Yet, nite element models sufferfrom high values of the layer-to-interlayer elastic moduli andthicknesses ratios, which impinge on the results [57]. Thus, niteelement results have to be checked and nite element models haveto be adjusted against exact results [15,18,20,36,38,41].

In order to estimate the capacity of a nite element model tosimulate the layered plate with discontinuous connection and totune the free-parameters, exact solutions are provided in this sec-tion (Table 1; Figs. 18 and 19). Table 1 also provides a benchmarkfor empirical formulas. Figs. 18 and 19 also show the zigzag strain

P. Foraboschi / Compositeshear actions. Conversely, the model [4] does not apply to the lay-ered plates with discontinuous connection that transmits bendingmoments together with in-plane shear actions.Throughout the research, hence, only the latter category is re-ferred to as discontinuous connection.

Considering that no analytical model was available in the liter-ature for the layered plate with discontinuous connection, this re-search has lled this gap. This paper presents the main results ofthis research, which are the continuous interlayer that reproducesthe behavior of the discontinuous connection and the model of thelayered plate with discontinuous connection. The mechanicalbehavior of the composite plate is simulated by using the three-layered plate model having the above-mentioned interlayer, whichreproduces the structural response with a system of three exact,analytical and explicit (closed-form) equations.

Only basic mechanical assumptions are made (in particular Kir-chhoff-Love plate hypotheses), which do not impose constraints onthe stressstrain state and behavior of the layered plate. Thus, themodel does not need to be veried by experiments or nite ele-ment models. Conversely, such a model allows nite element mod-els and empirical formulas to be veried.

The modeling assumptions allow for the reduction of themechanical problem into the two-dimensional framework. Transi-tion from three to two-dimensional behavior has been accom-plished by reproducing the effects of the normal stresses in thedirection transverse to the plate. In so doing, the model does notlose any prediction accuracy but is faster and easier to use, com-pared to the three-dimensional model. Moreover, three-dimen-sional differs from two-dimensional modeling process inbehaviors that the plate is not explicitly designed to have (side-ef-fects). Thus, three-dimensional modeling process does not capturethe dening attributes of composite plates, while this model pro-vides scientic understanding of the mechanical phenomenainvolved.

The input data of the model are the geometric and elasticparameters of the layers and interlayer, plus the ctitious semi-thickness k of the interlayer. The semi-thickness is the depth nec-essary to restraint the discontinuous connecting element againstrelative rotation with respect to the layer (i.e., to x a connectorinto the layer, so as to obtain the built-in end). Hence, the thickness2k of the interlayer is a mechanical rather than a geometricdimension. Consequently, k has to be determined experimentally,by testing the connection. However, any test that can measurethe stiffness of a connection provides adequate results. All thingsconsidered, the thickness of the interlayer is also an easily obtain-able and objective input data.

An analytical model does not dismiss nite element models asunnecessary. Actually, nite element models remain a viablemeans for structural analysis of layered plates with discontinuousconnection that do not comply with the modeling assumptions(Section 2) or whose shapes, loads, and boundary conditions entailcomplicated series expansions.

However, the way that the discontinuous connection is numer-ically modeled will greatly inuence the performance of the wholelayered plate and may drastically impinge on the results. Thus, anite element model has to be checked and calibrated against ex-act results in advance. In order to make this task easier and to pro-vide a benchmark, this paper presents exact solutions.

The applications of this model to practical cases highlight thatthe layered plate with discontinuous connection reaches signi-cantly lower deections and appreciably lower stresses than thelayered plate with continuous soft connection [4], all the otherparameters being equal (in particular, the discontinuous and con-tinuous connection are smeared into the same continuous inter-layer). This is due to the stiffness of the connection against theangles of rotation imposed by the layers to the interlayer (in-plane

rt B 47 (2013) 365378 377axis of rotation). For an individual connector, this stiffness is con-siderable. Accordingly, the interlayer that simulates the discontin-uous connection has to include proper shear stiffness against the

[31] De Matteis G, Landolfo R. Structural behaviour of sandwich panel shear walls:

378 P. Foraboschi / Composites: Part B 47 (2013) 365378deection derivatives (Eq. (15)). For a continuous connection pro-vided by a compliant core (soft interlayer), conversely, this stiff-ness is almost nil. Accordingly, the interlayer that simulates thecontinuous connection has to be utterly compliant to the anglesimposed by the layers to the interlayer [4].

Those models of the layered plates with soft continuous con-nection that use the interlayer devoted to the discontinuous con-nection (i.e., that consider an interlayer shear distortion) areaffected by a bias error of up to 250% for deection predictionsand of up to 60% for normal stress predictions.

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Layered plate with discontinuous connection: Exact mathematical model1 Introduction2 Standard modeling assumptions3 Modeling of the connection through the interlayer3.1 Continuous connection between the layers3.2 Discontinuous connection between the layers3.3 Interlayer that simulates the discontinuous connection3.4 Continuous and discontinuous connection relationships3.5 Thickness of the interlayer that reproduces the discontinuous connection

4 Discontinuously-connected layered plate: system definition5 Model of the discontinuously-connected layered plate5.1 Layer in-plane translational equilibrium5.2 Global rotational equilibrium5.3 Layer lateral translational equilibrium5.4 Governing equations of the discontinuously-connected layered plate

6 Analytical solution of the exact equations7 Displacement and stress results8 Application of the model: Case studies9 ConclusionsReferences