Article Simplified Calculation Model and Experimental Study of Latticed Concrete-Gypsum Composite Panels NanJiang1,2andShaochunMa1,* Received:21August2015;Accepted:19October2015;Published:27October2015 AcademicEditor:JorgedeBrito 1 SchoolofCivilEngineering,TianjinUniversity,Tianjin300072,China;[email protected] 2 KeyLaboratoryofCoastalCivilEngineeringStructureandSafety(TianjinUniversity), MinistryofEducation,Tianjin300072,China * Correspondence:[email protected];Tel.:+86-136-3969-1265 Abstract: In order to address the performance complexity of the various constituent materials of (dense-column)latticedconcrete-gypsumcompositepanelsandthedifficultyinthedetermination of the various elastic constants, this paper presented a detailed structural analysis of the (dense-column) latticed concrete-gypsum composite panel and proposed a feasible technical solutiontosimplifiedcalculation. Inconformitywithmechanicalrules,atypicalpanelelementwas selectedanddividedintotwohomogenouscompositesub-elementsandasecondaryhomogenous element, respectively for solution, thus establishing an equivalence of the composite panel to a simple homogenous panel and obtaining the effective formulas for calculating the various elastic constants. Finally, the calculation results and the experimental results were compared, which revealedthatthecalculationmethodwascorrectandreliableandcouldmeetthecalculationneeds of practical engineering and provide a theoretical basis for simplified calculation for studies on compositepanelelementsandstructuresaswellasareferenceforcalculationsofotherpanels. Keywords: lattice;composite;equivalentconstants;simplifiedcalculation;experiment;method 1. Introduction Gypsum is one of the three major cementing materials that are widely used in the fields of building materials, food, precision casting, models and molds, medicine, paper, and paint fillers. As early as 2000~3000 BC, humans attempted using gypsum as a cementing material to build the famousEgyptianpyramids,ancientRomanarchitecture,MogaoCavesatDunhuang,etc. SinceLouis XIV, known as France’s Sun King, issued a decree in 1667, calcined gypsum (building gypsum) started to be widely used in the building industry. Building gypsum is primarily comprised of beta-calcium sulfate hemihydrate (β-CaSO ‚1/2H O), whose content should be not lower than 4 2 60.0%[1]. Buildinggypsumexpandsby0.05%~0.15%afterhardening, withacompressivestrength of5~10MPa,aflexuralstrengthof2.5MPa,andadensityof13.0~14.5kN/m3[2]. Intheconstruction industry, gypsum is generally used for the cementing material, decorative gypsum boards, reliefs, romancolumns, etc.; non-load-bearingpartitionssuchashollowgypsumboardsanddense-column composite panels with low load-bearing capacity; and load-bearing, anti-seismic walls with high load-bearingcapacitysuchasthe(dense-column)latticedconcrete-gypsumcompositepanelstudied in this paper. The (dense-column) latticed concrete-gypsum composite panel is a composite of environmentally-friendly gypsum, fiberglass, concrete, and steel bars that has excellent seismic performance, is lighter than ordinary shear walls, and contains non-toxic, harmless, pollution-free gypsum that can adjust indoor humidity automatically with good thermal and sound insulation Materials2015,8,7199–7216;doi:10.3390/ma8105375 www.mdpi.com/journal/materials Materials2015,8,7199–7216 and fire resistance [3]. The development and wide use of (dense-column) latticed concrete-gypsum compositepanelshavepromotedtheinnovationinnewwallmaterialsandthedevelopmentofgreen, energy-savingwalls. Thiskindofcompositepanelmakesfulluseoftheadvantagesofitsconstitute materials. The tensile fiber increases the gypsum board’s ability to withstand forces. The gypsum board performs the (dense-column) latticed concrete-gypsum composite panel’s maintenance and thermal and sound insulation functions while serving as a stressed member [4]. The excellent materialpropertiesofconcreteenableittobearheavyloads.Asindicatedabove,thecompositepanel reflects an effective composite of the functional and mechanical aspects of the various constituent materials. ScholarsinAustralia,China,India,theUnitedStates,andItalyhaveconductedextensive experimental studies on composite panels [5–9]. However, few achievements have been made in simplifying calculations that are difficult to operate and implement. A great number of gypsum partitions that exist in the gypsum board of a (dense-column) latticed concrete-gypsum composite panel partition the hollow gypsum board into several horizontal and vertical gypsum cavities. After concrete pouring, a lattice of invisible beam-invisible column forms, making simplified calculation difficult and challenging. The greatest difficulty confronting the simplified calculation of composite panels lies in the determination of basic elastic constants due to the complexity of the composite[10],directlyleadingtothedifficultyinengineeringsoftwaremodeling,agreatnumberof computingelements,agreatamountofcalculationwork,andafailuretomakeeffectivecalculation. The(dense-column)latticedconcretegypsumcompositepanelinthisstudyprimarilyconsisted oftheporousgypsumboardwithcavitiesandtheinvisiblebeam-invisiblecolumnlatticeofconcrete. The composite panel with this structure not only has a high bearing capacity, but also facilitates the arrangement of steel bars and the processing of nodes. In order to find a feasible simplified calculation model for the (dense-column) latticed concrete-gypsum composite panel, a technical solution was established. A typical element of this composite panel was selected to establish an effectivecalculationmodel.Throughaseriesoftheoreticaldeviations,afeasiblesimplifiedcalculation model was obtained. Finally, the reliability of the simplified calculation model was validated by comparingtheseismicexperimentalresultsandthecalculationresultsofthecompositepanel. 2. TechnicalSolutiontoSimplifiedCalculation As shown in Figure 1, (1) is a gypsum partition; (2) is a cavity formed vertically by gypsum partitions;(3)isacavityformedhorizontallybygypsumpartitions;(4)isaninvisibleconcretebeam formedatthehorizontalcavityafterconcretepouring;and(5)isaninvisibleconcretecolumnformed at the vertical cavity after concrete pouring. The invisible beams and invisible columns form an entire beam-column lattice structure. The porous gypsum board primarily consists of two gypsum side panels and n gypsum partitions (lengthˆwidthˆheight=94mmˆ20mmˆ160mm). Gypsum partitions are reasonably arranged between the two gypsum side panels to form many horizontal gypsum cavities (length ˆ width = 220 mm ˆ 94 mm) and vertical gypsum cavities(lengthˆwidth=230mmˆ94mm)intheentiregypsumboard. Thesecavitiescanbeused asthetemplatesforconcretepouring. The above introduction revealed that a (dense-column) latticed concrete gypsum composite panel has a complex structure, and the performance of its constituent materials differs greatly, which determines the special complexity of its mechanical properties and makes it difficult to calculate. Therefore,itishardtoobtaintheaccuratemechanicalresultsofcompositepanels. Simple homogenizationhasbeenincompetent,andanewwayisrequired.Theaimofthisstudyistoenable the calculation results to meet engineering requirements, simplify the calculation method as much as possible and significantly reduce the amount of calculations and time required for calculations. The specific technical solution is shown in Figure 2. From the perspective of the mechanics of the composite, a typical homogenous composite element of the (dense-column) latticed concrete gypsumcompositepanelwasselected,asshowninFigure2b. Thehomogenouscompositeelement was then divided into two homogeneous composite sub-elements, as shown in Figure 2c,d. These 7200 Materials2015,8,7199–7216 two homogenous composite sub-elements were rendered homogenously equivalent, respectively. Then the two equivalent homogenous composite sub-elements were used to constitute a secondary homogenous element, as shown in Figure 2e. The secondary homogeneous element was used to Materials 2015, 8, page–page  substitute the original homogenous composite element, and the basic dimensions and mechanical propMemrateteicreihaslason 2fi0c1ta5h,l  e8,p cproaogmpe–eppraotgiseei st eopf atnheel wcoemrepomsiatien tpaainneeld  wunerceh amnagiendtabineefdo reunacnhdanagfteedr ebqeufoivrea leanndt traefatetrm  ent, equivalent treatment, thus realizing the transformation from a complex (dense‐column) latticed  thusrealizingthetransformationfromacomplex(dense-column)latticedconcretegypsumcomposite mcoencchraenteic gayl ppsruomp ecrotimesp oosfi tet hpea nceol mtop ao ssiimte pplea hnoeml owgeerneo umsa pinantaeiln, eadn du gnrcehaatlnyg reedd ubceinfogr teh ea dndif fiacuftletry   paneltoasimplehomogenouspanel,andgreatlyreducingthedifficultyincalculation. Thevaluable eiqnu ciavlacluelnatt iotrne.a Ttmhee nvta, luthaubsl er reeasluizlitns go bthtaei nterdan fsifnoarlmlya wtioilnl  bfero omf  gar ecaotm ppralecxti c(adl esnigsen‐icfoiclaunmcen )t ol atthtiec eind‐  resultsobtainedfinallywillbeofgreatpracticalsignificancetothein-depthstudiesonthemechanical cdoenpctrhe tset ugydpiessu omn  ctohme pmoescithea pnaicnaell  ptoro ap seirmtipesle o hf o(mdeongseen‐cooulsu pmann)e ll,a attnicde dgr ceoantlcyr ereted ugcyipnsgu tmhe c doimffipcouslittye   propiepnra tcniaeelslc ueollfeamt(idoeennn.t ssT eah-necd ov lsautlrumuacnbtu)lerl aerset.ts iucletds ocbotnacinreedte fginyapllsyu wmillc obme opfo gsrieteatp parnaectlicealel msigennitfsicaanndce sttor uthcetu irne‐s. depth studies on the mechanical properties of (dense‐column) latticed concrete gypsum composite  panel elements and structures.    Figure 1. Perspective of the gypsum board.  Figure1.Perspectiveofthegypsumboard.  Figure 1. Perspective of the gypsum board.    Figure 2. Technical solution to the calculation of elastic constants: (a) the (dense‐column) latticed    concrete gypsum composite panel; (b) a typical homogenous composite element; (c) the homogenous  FiguFrceiogmu2.rpeo Ts2ei.t ceTh enscuihcbna‐ielclaselom lseuonltuito tiIno; nt( odt)o t htthheee c cahalolccmuulolaagtteiioonnon uoosf f ceeolalmasstpitcoi csciotceno snstsautnbat‐nse:lt es(ma:)e (tanh)te  tIh(Id;e ean(ndsdee‐ nc(osele)u -mcao nlsu)e cmlaotnntdi)caelradyt  ticed choonmcroegtee ngoyupss ucmom cpoomspitoe.s ite panel; (b) a typical homogenous composite element; (c) the homogenous  concretegypsumcompositepanel;(b)atypicalhomogenouscompositeelement;(c)thehomogenous composite sub‐element I; (d) the homogenous composite sub‐element II; and (e) a secondary  composite sub-element I; (d) the homogenous composite sub-element II; and (e) a secondary 3. Thheoomroegtiecnaolu Ds ecorimvpatoisoitnes.  of Elastic Constants  homogenouscomposite. 33.. 1T.h Beaosrice tAicsaslu mDpetriiovnast ions of Elastic Constants  3. TheoreticalDerivationsofElasticConstants (1) The gypsum board, fiberglass, and concrete were all isotropic materials and linearly elastic;  3.1. Basic Assumptions  (2) the fiberglass was evenly distributed in the gypsum board, and the various materials of the  3.1. BasicAssumptions comp(1o)s Tithe ep gaynpels ubmon bdoedar fdir, mfiblye;r galnads s(,3 a) nthde c eofnfcercetste o wf tehree  raelsl iidsoutarlo sptirce smseast earniadl ss tarnadin lsi nine atrhley  vealarsiotiuc;s   ((m21))a ttTehrhei eaflisgb yoerfp gtshluaesm sc owbmoapsao resdvit,eefin plbyae nrdegils lwatrseisbr,ue atiegndnd oicrneo dtnh c[e1r 1eg]ty.e pwsuemre baolalridso, tarnodp itchem vaateriroiualss manatderliianlse aorfl ytheel astic; (2) thcoemfipbWoesirittghel  agpsraesnaetwl i nba‐ospnledavneeden  sfltiyrifmfndleyiss; star [ni1bd2u ](,t3 fe)l dothoriens e cftafhence ctsog noysfp ttrhsauein mr ethsbeid oouauartld‐ os,tfr‐aepsnlasdense t ahdneedfo vsrtamrraaiiontiuso snin om tfh aceot emvraipraoilossuitsoe  f the mpaanteerlisa. lTsh oefr tehfoer ceo, omnplyo sthitee  ipna‐pnlealn we earceti iognn noereedds [ 1to1 ]b. e considered for a composite panel. In other words,  With great in‐plane stiffness [12], floors can constrain the out‐of‐plane deformation of composite  panels. Therefore, only the in‐plane action needs to b3e  considered for a composite panel. In other words,  7201 3 Materials2015,8,7199–7216 compositepanelbondedfirmly; and(3)theeffectsoftheresidualstressesandstrainsinthevarious materialsofthecompositepanelwereignored[11]. Withgreatin-planestiffness[12],floorscanconstraintheout-of-planedeformationofcomposite panels. Therefore, only the in-plane action needs to be considered for a composite panel. In other words, a composite panel can be treated as a plane. In the following, formula derivation was Materials 2015, 8, page–page  conductedforthesimplifiedcalculationofcompositepanels. a composite panel can be treated as a plane. In the following, formula derivation was conducted for  3.2. CalculationofElasticConstants the simplified calculation of composite panels.  E , E , G , and G represent the elastic modulus and shear modulus of the concrete and c g c g 3.2. Calculation of Elastic Constants  gypsum board, respectively (obtained through material tests). The equivalent elastic modulus of the homoEgc,e Engo, uGsc, acondm Gpgo rseitperessuenbt- ethleem eleanstticI minodthuelusX anodr sYhedairr emcotidounluiss oEf Itxheo croEncIyre,tae nadndi tgsyepqsuumiv alent shearbomarodd, urleusspeacntidvePlyo is(osobnta’isnerdat itohrionutghhe  mXYateprilaaln eteastrse). GThe aenqduivvalen,tr eeslpaestcitci vmeloyd.uSluims iolaf rltyh,et hose Ixy Ixy ofthheohmoomgoengoeunso cuosmcpoomsipteo ssiutbe‐esluebm-eenlet mI einn tthaer eX Eor Y, Edirec,tGion is, EaIxn odr vEIy, a,nrde sitpse ecqtiuvivelayl,enatn sdhetharo seof IIx IIy IIxy IIxy the smecoodnudluasr yanhdo mPooigsseonno’su rsaetiloe mine tnhte aXreY Epla,nEe ,arGe G,Ixya nanddv vIxy,, rreessppeeccttiivveellyy. .SDimuielatrolys, ythmomse eotfr yt,h1e /4 of x y xy xy thetwhoomhoogmenooguesn cooumspcoosmitep ossuibte‐esleumbe-enlte maree nEtsIIxw, EeIrIye, aGnIIaxyl,y zanedd ,vaInIxyd, trhesepeencttiivreelys,e caonndd athroyseh oomf othgee nous secondary homogenous element are Ex, Ey, Gxy, and vxy, respectively. Due to symmetry, 1/4 of the two  elementwasanalyzed,asdetailedbelow. homogenous composite sub‐elements were analyzed, and the entire secondary homogenous element  was analyzed, as detailed below.  3.2.1. HomogenousCompositeSub-ElementI 3F.i2g.1u. rHeo3maosgheonwouss1 C/o4mopfothsietet hSurebe‐E-dleimmeennts Ii onalhomogenouscompositesub-elementI.Foreaseof derivatioFni,gtuhree f3oar scheodwias g1r/4a mofs thoef tthhreehe‐odmimoegnesnioonuasl hcoommopgoesnioteuss ucobm-epleomsiteen stuibn‐etlheemXen–tZ I.a Fnodr eXa–seY opf lanes weredgeriivveanti,onre, sthpee fcotrivcee ldyi.agFriagmurse of3 tbh,ec haormeotgheenocuaslc cuolmatpioosnitem soudb‐eelleamnednte iqnu thivea Xle–nZt amndo Xd–eYl pfolarnseos lving EIx; wFiegrue rgeiv3edn,,e reasrpeectthiveelcya. lFciugluarteio 3nb,cm aored ethlea cnadlcuelqautioivna mleondteml aondde elqfuoirvasloenlvt imngodGelI xfoyr; sFoilgvuinrge E3Ifx;, g are the cFailgcuurlea t3iodn,e maroed tehlea cnadlcueqlautiiovna lmenotdmel oadnedl efqourisvoallvenint gmEodIye,l afnodr sFoilgvuinrge G3hIxy;i sFtighuercea 3lcf,ugl aatrieo nthme odel forsocalvlciunlgatvion m. odel and equivalent model for solving EIy, and Figure 3h is the calculation model for  Ixy solving vIxy.    Figure 3. Models of Homogenous Composite Sub‐element I: (a) 1/4 of the three‐dimensional  Figure 3. Models of Homogenous Composite Sub-element I: (a) 1/4 of the three-dimensional homohgomenooguensocuosm cpomospiotesisteu bsu-ebl‐eemlemenetntI ;I;( b(b))t thhee ccaallccuullaattiioonn mmooddele flofro sroslvoilnvgin EgIx;E (c) ;e(qcu)ievqauleinvta mleondteml odel Ix for solving EIx; (d) the calculation model for solving GIxy; (e) equivalent model for solving GIxy;   for solving E ; (d) the calculation model for solving G ; (e) equivalent model for solving G ; Ix Ixy Ixy (f) the calculation model for solving EIy; (g) equivalent model for solving EIy; and (h) the calculation  (f)thecalculationmodelforsolvingE ;(g)equivalentmodelforsolvingE ;and(h)thecalculation model for solving vIxy.  Iy Iy modelforsolvingv . Ixy (1) Elastic modulus EIx: As shown in Figure 3b,c, the planar calculation model of the homogenous  composite sub‐element I was divided into the three calculation domains, A1, B1, and C1. B1 and C1 are  the gypsum boards, and A1 is concrete. In the7 X20 d2irection, the sum of the normal stresses of the  4 Materials2015,8,7199–7216 (1) Elastic modulus E : As shown in Figure 3b,c, the planar calculation model of the Ix homogenous composite sub-element I was divided into the three calculation domains, A , B , and 1 1 C . B andC arethegypsumboards,andA isconcrete. IntheXdirection,thesumofthenormal 1 1 1 1 stressesofthecalculationmodelofthecompositepanelandthatoftheequivalentmodelshouldbe equalunderthesameloading[13]. σ “σ ,σ h B{2“σ h b{2`σ h pB{2´b{2q (1) Ix C1x Ix 1 A1x 1 B1x 1 where σ , σ , σ , and σ arethe stresses ofconcrete, gypsum board (namely domain B and A1x B1x C1x Ix 1 C ), and the equivalent model in the X direction, respectively. ε is the strain of the equivalent 1 Ix elementIintheXdirection. Thestraininthehomogenouscompositesub-elementIintheXdirection ε shouldbethesamebeforeandafterequivalenttreatment. Ix σ {E “σ {E “σ {E “ε (2) Ix Ix A1x c B1x g Ix Letb{B“λ. Equation(2)wassubstitutedintoEquation(1)toobtain E “λE `p1´λqE (3) Ix c g (2) Elastic modulus E : As shown in Figure 3f,g, σ , σ , and σ are the stresses of concrete, Iy cy gy Iy thegypsumboard, andtheequivalentmodelintheYdirection, respectively, and ε isthestrainof Iy theequivalentelementIintheYdirection. Similarly, σ BL{4“σ bl{4`σ pBL´blq{4 (4) Iy c g σ {E “σ {E “σ {E “ε (5) Iy Iy c c g g Iy Letl{L“β. Equation(5)andλweresubstitutedintoEquation(4)toobtain E “λβE `p1´λβqE (6) Iy c g (3) Shear modulus G : τ , τ and τ are the shear stresses of concrete, the gypsum board Ixy cx gx Ix (namelydomainB andC ), andtheequivalentmodelintheX direction. AsshowninFigure3d,e, 1 1 thesimilarmethodwasusedtoobtain G “λβG `p1´λβqG (7) Ixy c g (4)Poisson’sratiov :Theplanarcalculationmodelofthehomogenouscompositesub-element Ixy IwasdividedintothethreecalculationdomainsA , B , andC . B andC arethegypsumboards, 2 2 2 2 2 andA isconcrete.Whereσ ,σ ,andσ arethestressesofconcreteandgypsumboard(namely 2 A2x B2x C2x domain B and C ) in the X direction. As shown in Figure 3h, the strain difference between the 2 2 combinationofA andB andC intheXdirectionis0.5lpν ´ν qε . 2 2 2 g c Iy Accordingtotheconditionsforequilibriumofforces, pB´bqσ “bσ (8) C2x B2x Since ε “ν ε “ν ε ´σ {E (9) Ix Ixy Iy g Iy C2x g Thedisplacementsorstrainsontheleftandrightsideswererequiredtosatisfythecoordination: σ L{2E `σ l{2E `σ pL´lq{2E “pν ´ν qε l{2 (10) C2x g B2x c B2x g g c Iy 7203 Materials2015,8,7199–7216 LetE {E “α. λandβweresubstitutedtoobtain g c ν “ν ´βλpν ´ν q{rλ`p1´λqp1´β`αβqs (11) Ixy g g c 3.2.2. HomogenousCompositeSub-ElementII (1)ElasticmodulusE : AsshowninFigure4d,e,similarly, IIx Materials 2015, 8, page–page  σ h B{2“σ h b{2`σ h pB´bq{2,σ {E “σ {E “σ {E “ε (12) IIx 2 Dx 2 Ex 2 IIx IIx Dx c Ex g IIx σ h B/2σ hb/2σ h (Bb)/2, σ /E σ /E σ /E ε   (12) IIx 2 Dx 2 Ex 2 IIx IIx Dx c Ex g IIx   Figure 4. Models of Homogenous Composite Sub‐element II” (a) 1/4 of the three‐dimensional  Figure 4. Models of Homogenous Composite Sub-element II: (a) 1/4 of the three-dimensional homogenous composite sub‐element II; (b) the calculation model for solving EIIy; (c) equivalent model  homogenous composite sub-element II; (b) the calculation model for solving E ; (c) equivalent for solving EIIy; (d) the calculation model for solving EIIx; (e) equivalent model for solvIiInyg EIIx; (f) the  model for solving E ; (d) the calculation model for solving E ; (e) equivalent model for solving calculation modeIlI fyor solving GIIxy; and (g) equivalent model for soIIlxving GIIxy.   E ; (f)thecalculationmodelforsolvingG ;and(g)equivalentmodelforsolvingG . IIx IIxy IIxy σIIy is the stress of the equivalent model in the Y direction. σDx, σEx, and σIIx are the stresses of  σconcrisetteh, egysptrseusms obfoathrde, eaqnud itvhael eeqnutimvaolednetl minodthele inY tdheir Xec dtiiorenc.tiσon. ε,IIσx an,da εnIIdy aσre thea rsetrathines sotfr tehses esof IIy Dx Ex IIx equivalent model in the X and Y directions.  concrete,gypsumboard,andtheequivalentmodelintheXdirection. ε andε arethestrainsof IIx IIy Therefore,  theequivalentmodelintheXandYdirections. Therefore, E λE (1λ)E   (13) IIx c g E “λE `p1´λqE (13) IIx c g (2) Elastic modulus EIIy: As shown in Figure 4b,c, similarly,  (2)ElasticmσodBuLlu/s4EIIyσ: AbsL/sh4owσninBFigburLe4/b4,,c ,σsim/ilEarly, σ /E σ /E ε   (14) IIy cy gy IIy IIy cy c gy g IIy σ BL{4“σ bL{4`σ pB´bqL{4,σ {E “σ {E “σ {E “ε (14) TherefIoIyre,  cy gy IIy IIy cy c gy g IIy Therefore, EIIy λEc(1λ)Eg  (15) E “λE `p1´λqE (15) (3) Shear modulus GIIxy: τcx, τgx anIdIy τIIx are cthe shear strgesses of concrete (namely domain D),   (t3h)e Sghyepasrumm obdouarldu s(nGamel:y τdo,mτain aEn),d aτnd thaer eeqthueivsahleenatr mstordeesls eins othfec oXn dcirreetceti(onna. mAes lsyhdowomn ainin   D), IIxy cx gx IIx Figure 4f,g, the same method was used to obtain  the gypsum board (namely domain E), and the equivalent model in the X direction. As shown in Figure4f,g,thesamemethodwasusedtoGobtainλG (1λ)G   (16) IIxy c g (4) Poisson’s ratio vIIxy: As showGn IinIx Fyi“guλreG 4cd`, wp1he´n λthqeG cgomposite panel is stretched evenly in (16) the X direction, its contraction strain is equal to the sum of the contraction strains of gypsum and  concrete in the Z direction, namely  ε  ε B/2[b/2 (Bb)/2]ε   (17) IIz IIxz IIx c g IIx Similarly, λ was substituted to obtain     λ （1-λ）   (18) IIxy IIxz IIyz c g   7204 6 Materials2015,8,7199–7216 (4) Poisson’s ratio v : As shown in Figure 4d, when the composite panel is stretched evenly IIxy intheXdirection,itscontractionstrainisequaltothesumofthecontractionstrainsofgypsumand concreteintheZdirection,namely ε “ν ε B{2“rν b{2`ν pB´bq{2sε (17) IIz IIxz IIx c g IIx Similarly,λwassubstitutedtoobtain ν “ν “ν “λν `p1´λqν (18) IIxy IIxz IIyz c g Materials 2015, 8, page–page  3.2.3. SecondaryHomogenousElement 3.2.3. Secondary Homogenous Element  As shown in Figure 5a, the secondary homogenous element consisted of the homogenous composite sAusb -sehloewmne nints FIigaunred 5IaI,. tσhe, sσeco,nτda,rya nhdomτogaennodusa reeletmheentn ocornmsiasltesdt roefs sthees ahnomdosgheenaoruss tresses x y x y oftheseccoomnpdoasirtye shuobm‐eloegmeennotsu Is aenlde ImI. eσnx,t σ,yr,e τsx,p aencdt iτvy ealnyd. are the normal stresses and shear stresses of the  secondary homogenous element, respectively.    Figure  5.  Models  of  Secondary  Homogenous  Element:  (a)  the  three‐dimensional  secondary  Figure 5. Models of Secondary Homogenous Element: (a) the three-dimensional secondary homogenous element; (b) the calculation model for solving Ey; (c) equivalent model for solving Ey;   homoge(ndo) uthse ecalelcmuleantito;n( mb)odthele focra slcoulvliantgi oEnx; amnod d(ee)l efqourivsaolelvnit nmgodEeyl; fo(cr )soelqvuinigv aElxe. ntmodelforsolvingEy; (d)thecalculationmodelforsolvingEx;and(e)equivalentmodelforsolvingEx. (1) Elastic modulus Ey: As shown in Figure 5b,c, the sums of the normal strains of the two models  under the same loading should be equal.  (1)ElasticmodulusE :AsshowninFigure5b,c,thesumsofthenormalstrainsofthetwomodels y underthesameloadingshouldbeequal.εy(h1h2)h1εIyh2εIIy  (19) The stresses in the Y direction are equal, namely  ε ph `h q“h ε `h ε (19) y 1 2 1 Iy 2 IIy E ε E ε E ε σ   (20) Iy Iy IIy IIy y y y ThestressesintheYdirectionareequal,namely Let h /h ζ, which was substituted into Equation (19) to obtain  1 2 EIyEεIy“(1EIζI)yEεIIEy “/E(Eyεy“ζEσy)  (21) (20) y Iy IIy Iy IIy Leth1{h(22) “Elaζs,tiwc mhoicdhulwusa Esx:s Aubs ssthiotuwtne din iFnigtourEe q5ud,aet,i tohne m(1e9t)hotod ofobrt caailnculating EIIx was used.  Therefore,  E “p1`ζqE E {pE `ζE q (21) y E =(ζEIy IIEy )/Iy(1ζ) IIy (22) x Ix IIx (2)Elas(t3i)c Smheoadr umloudsuEluxs: GAxsy: sThhoe wcanlcuinlaFtioignu mreod5edl ,aen,dth eequmiveatlhenotd mfoodrecl aalrceu slhaotiwnng iEn IFIixguwreass 5uf,sge. d. TheTrheef omreet,hod for calculating GIIxy was used.  Therefore,  E =pζE `E q{p1`ζq (22) x Ix IIx G =(ζG G )/(1ζ)  (23) xy Ixy IIxy (4) Poisson’s ratio vxy: The method for calculating vIIxy was used to obtain  7205  ( ζ )/(1ζ)  (24) xy Ixy IIxy 7 Materials2015,8,7199–7216 (3) Shear modulus G : The calculation model and equivalent model are shown in Figure 5f,g. xy ThemethodforcalculatingG wasused. IIxy Therefore, G =pζG `G q{p1`ζq (23) xy Ixy IIxy (4)Poisson’sratiov : Themethodforcalculatingv wasusedtoobtain xy IIxy ν “pν ζ`ν q{p1`ζq (24) xy Ixy IIxy 4. Experiment Materials 2015, 8, page–page  4.1. SampleDesignandLoadingSystem 4. Experiment  Three samples (1520 mm ˆ 1520 mm ˆ 120 mm) were designed for the seismic experiment on Materials 2015, 8, page–page  the compo4s.1i.t eSampaplne eDlesiingnt ahnids Lsotaudidngy .SyTshteemy  were numbered Q-1, Q-2, and Q-3, respectively. The plan and elevati4o. nExopferiemaecnht sample are shown in Figures 6 and 7 respectively. The steel bars arranged Three samples (1520 mm × 1520 mm × 120 mm) were designed for the seismic experiment on the  in the comcopmopsiotseitep paanneell iwn ethries sHtuRdyB.4 T0h0eyd wefeorer mnuemdbesrteede lQ‐b1a, rQs‐2w, ainthd Qa‐3d, iraemspeectteivreolyf. T1h4e mplman, aansd shown in 4.1. Sample Design and Loading System  Figure 8. Telheveateioxnp oefr iemacehn staamlprlee saurel tsshoowfnth ine Fmigautreers i6a lasndo f7,t hreespceoctmiveploy.s Tithee pstaenele blaarsr earsrahnogwedn ins uthbes equently. Three samples (1520 mm × 1520 mm × 120 mm) were designed for the seismic experiment on the  composite panel were HRB400 deformed steel bars with a diameter of 14 mm, as shown in Figure 8.  The compressive strength and elastic modulus of the gypsum board were 5.52 MPa and 4350 MPa, composite panel in this study. They were numbered Q‐1, Q‐2, and Q‐3, respectively. The plan and  The  experimental  results  of  the  materials  of  the  composite  panel  are  shown  subsequently.   respectivelye.leTvahtieonc oofm eapchre ssasmivpele asrter eshnogwthn ina nFdiguerelas s6t iacndm 7,o rdesupleuctsiveolfy. tThhee sctoeenl cbraerst earrwanegreed 2in4 .t6he3  MPa and The compressive strength and elastic modulus of the gypsum board were 5.52 MPa and 4350 MPa,  2.72ˆ104rMecsopPmeacpt,oivsreietleys p.p aTenhceetli  vwcoeemlryep .HreRTssBhi4ve0e0 t sdetrenefonsrigmltehe dsa tnsrtdee enel lgbatashtrisc  womfitohtd haue dluiHas moRef Bteth4r e0o f0c 1o4sn tcmereemtle,  abwsa esrrhseo w2w4n.a 6is3n  FM6i5gP4ua.r 0ea 0n8.d M   Pa (D8), The  experimental  results  of  the  materials  of  the  composite  panel  are  shown  subsequently.   669.45MP2a.7(2D ×1 140)4, ManPda, 6re7s6p.e9c2tivMelPy.a T(hDe 2te0n)s,irlee ssptreencgtitvh eolfy th[1e 4H].RB400 steel bars was 654.00 MPa (D8),   The compressive strength and elastic modulus of the gypsum board were 5.52 MPa and 4350 MPa,  669.45 MPa (D14), and 676.92 MPa (D20), respectively [14].  respectively. The compressive strength and elastic modulus of the concrete were 24.63 MPa and   2.72 × 104 MPa, respectively. The tensile strength of the HRB400 steel bars was 654.00 MPa (D8),   669.45 MPa (D14), and 676.92 MPa (D20), respectively [14].    Figure 6. Plane of the composite panel.  Figure6.Planeofthecompositepanel.   Figure 6. Plane of the composite panel.    Figure 7. Elevation of the composite panel.    Figure 7. Elevation of the composite panel.  Figure7.Elevationofthecompositepanel. 7206 8 8 Materials2015,8,7199–7216 Materials 2015, 8, page–page  Materials 2015, 8, page–page    Figure 8. Reinforcement of the composite panel.  Figure8.Reinforcementofthecompositepanel. The loading devices at the experimental site are shown in Figure 9 and consisted of a horizontal  Thloealdoiandg idnegvdicee vaincde saa vtetrhtiecaelx lpoaedriimnge dnetavlicsei t[e15a–r1e7]s.h Tohwe nhoinrizFoingtualr ec y9claicn dloacdo nwsaisst eimdpoofseadh boyr iaz ontal loading10d00e vkiNce haynddraualivc ejartcikc awlhlooFsaiegd ubirnaec g8k.  dRweeaivnsif coseercceu[m1r5eedn–t1  to7of]  tt.hheeT  chroeemacphtooiosrintiez  wpoaannltlea lo.l nc ythcel irciglhota dsidwe.a Tshiem vperotsiceadl by a load was imposed by two 500 kN hydraulic jacks, which were secured to the sliding supports above.  1000kN hydraulic jack whose back was secured to the reaction wall on the right side. The vertical The Tslhide ilnoga dsiunpgp doertvsi cwese raet  stheceu erxepde rtoim thene traela scittieo anr est seheol wbena min  aFbigouvree.  9In a nordd ceorn tsoi satcecdu oraf tae lhyo sriimzounltaatle   loadwasimposedbytwo500kNhydraulicjacks,whichweresecuredtotheslidingsupportsabove. ltohaed iancgti odne voicf ev aenrtdic aa l vleoratdicianlg l ooand itnhge  dceovmicpeo s[1it5e– 1p7a]n. eTlh uen hdoerri zeoanrttahlq cuyackleic  elovaendt sw ians  timhep opsreodce bssy  oaf   Theslidingsupportsweresecuredtothereactionsteelbeamabove. Inordertoaccuratelysimulate 1h0o0r0i zkoNn thaly dcyracluicli cl ojaacdkin wg,h tohsee  vbearctkic wala jsa cskesc uwreedre t oe ntahbel ereda ctoti omno wvea lsl yonnc hthroe nroiguhstl ys iwdeit. hT hthee v searmticpalle   the action of vertical loading on the composite panel under earthquake events in the process of lwoaidth w thaes  iamidp oosf ead s bliyd itnwgo  c5a0r0,  wkNhi hchy dernasuulrice dja tchkast,  wthhei cvhe rwtiecrael  lsoeacdur aeldw taoy tsh aec stleidd inong  sthuep pceonrttse ra boof vthe.e   horizoTntohtpea  lsolcfi dythicnelgi sc asmluoppapldeo.ir nAtsg  rw,igtehirdee  dsveisceturritrbiecudat iltvoje at hbckeea srmewa wcetiaroesn p eslnatecaeebld lb ebedeatmwto eaebmno ovthvee.e  Itnws yoonr 5dc0eh0rr  kotoNn a ohcuycusdlrryaatuewlliycit  jshaimctkhus elaantseda  mple withthttheheea  aicdcotimoofnpa oossfil tievd eiprntaigncaeclla  lrion,a wdoihrndigce hro nteo n tshgueen rceeodrmatteph oaustnittiehf oeprmavne erclto iumcanpldrleeors aseidvareat hlswqtruaeayskss eoa encvt eethdnetso  tnoinpt h tsheuecr fepanrcoteec erosfos  ftohtfeh  etop ofthehscaoomrmizppoolnest.iatlAe  cpyraicnglieicdl  lionda itdshtienr iglbo, uatdhtieivn evge bprterioaccamel sjswa.c Aakss  dpwisleaprcleae cedenmbabeelntewtd me teoent emrth owveaets ws pyolnac5che0rd0o naktoN uthshely ys idwdreiat hou flt ihtcheej a ssacamkmsppallene  dthe compowtsoiit tahec qtphuaein raeei ldthi one fd oairs sdpllieadrcinetomg gecnaertn ,o ewfr thahiteceh su aenmnispfuolerre mwd ittchhoa tmth teph lreoe avsdes.ri tvAice adsli talrole ainsdsd aioclnwataothyr sew atacostp epdlsa uocnerdf ta hacete  tchoeefn ettehnrde o ocfof t tmhheep  osite tfoipxe odf  tbheea msa matp tlhe.e A b oritgtoidm d oisft rtihbeu tsiavme pbleea mto  wdeatse pctl awcehde tbheetrw teheen  stahme tpwleo  i5n0 0it sk Nen htiyredtrya umliocv jaecdk sin a nthde   panelintheloadingprocess. Adisplacementmeterwasplacedatthesideofthesampletoacquire tehxep ceorimmpeonstiatle p proacneesls  iinn  oorrddeerr t oto e lgimeninearatete t huen difiosprmla cecommenptr eesrsriovre g esntreersast eodn b yth teh et ofipx esdu brfeaacme  dofu rtihneg   thedisplacementofthesamplewiththeload.Adialindicatorwasplacedattheendofthefixedbeam csoumbspeoqsuiteen pt adnaetla  ipnr tohcee slsoiandgi.n Rge psirsotcaenscse.  Ast rdaiisnp glaacuegmese nwt emree tienrs twalales dp laatc tehde  akte tyh ep ossiditeio onfs t ohfe  tshaem spteleel   at the bottom of the sample to detect whether the sample in its entirety moved in the experimental tboa arc, qiunivries itbhlee  dciospnclarceetem beneat mof,  tihnev sisaimblpel ec ownicthre tthe ec loolaudm. nA,  dainadl i noduitcsaidtoer  gwyapss pulmac ebdo aartd th oef e tnhde  opf atnheel   procesfssixaiemndp oblreed atoem rm taeota teshuleirm eb tiohntetao stmtera toihnfe st hidnei  stshpaelm avcpaerleimo tueosn  dmteeatterercroti arwlgsh eoefnt hteherera  stthaemed psbalyem. tFpholeer  eifinxx aiemtsd peblneet, iatrhmeety 4d 5mu° orstivrneagdin si gnua btuhsgeee q uent dataperoxonpc etehrsiesm isneangmt.apRll epe rcsooiscutelasdsn  bicnee  ousrstderdear it ntoo gm ealeiumagsiunersaet weth teher ese hdieinsapsr tldaaecllefeomdrmeanattt eitohrrneo ork fge teyhnepe proaastneietdilo .b Tnyhs teho edf aftithxaee cdos bltleeeaecmtlebd da bur,yri intnhgve  isible concrestdueibsbspeelqaacumeem,niten ndvta itmsai ebptlereor,cc deosinasilcn rigne.dt eRicecasotioslruta, mnancned,  svatranardiinoou gusa trusegisdeisset awgneycrepe s situnrmastianbl lgeoadau ragdte tsoh, feat nkhdee ytph peao nsseietnliossonarsms  oapft tltaehceth osetdme etelo a sure the strbatahinre, s jianicnvkisst hiwbeleerv eca osrtnioocrrueedste im nb aae atcemormi, aiplnsuvtoiesfri bftlohere  creosaanlmc‐rtiepmtleee  .ccooFlluolemrctneio,x naa.mn dp oleu,tstihdee 4g5y˝pssutrma inbogaradu goef tohne tphaenesla mple couldsbaemupslee dtot moemaseuarseu trhee tshtreaisnhs eina rthdee vfaorrimouast mioanteorifatlhs eofp thaen sealm. Tphlee. Fdoart eaxcaomllpelcet, ethde b4y5°t shtreadini sgpaulagcee ment on the sample could be used to measure the shear deformation of the panel. The data collected by the  meter,dialindicator,andvariousresistancestraingauges,andthesensorsattachedtothejackswere displacement meter, dial indicator, and various resistance strain gauges, and the sensors attached to  storedinacomputerforreal-timecollection. the jacks were stored in a computer for real‐time collection.    Figure 9. Loading devices at the experimental site figure.  9   Figure 9. Loading devices at the experimental site figure.  Figure9.Loadingdevicesattheexperimentalsitefigure. 9 7207 Materials2015,8,7199–7216 Materials 2015, 8, page–page  TThhee quqausai‐ssi-tsattaicti tcestte mstetmhoetdh dodefindeedfi nine dthein SptehceificSapteiocnifi ocfa tTioesntinogf MTeestthinogds Mfore tEhaordtshqfuoarkeE Raretshisqtuaankte  BRueilsdisintagn t(JBGuJi1l0d1in‐9g6()J GwJa1s0 1p-9e6rf)owrmasedp e[r1f8o]r. mFeirdst[, 1a8 ].coFnirsstta,nat cvoenrtsitcaanl tlvoaedrt icwaalsl oiamdpwosaesdi,m apnods ethde,na nad  htohreinzoantahlo rcizyocnlitca llocyadcl icwlaosa dimwpaosseimd.p oWsehde.nW  thheen  cthome pcoomsitpeo spitaenepla nwelasw aisn inthteh eelealsatsitci cssttaaggee, ,  aa loloaadd‐c-coonntrtorollleledd loloaaddiningg ssyysstetemm wwaass aaddoopptetedd. .WWhhenen ththe ecocommppoosistiete ppaanneel lwwaass inin ththee eelalasstitcic‐p-plalasstitcic  sstataggee, ,aa loloaadd‐ -aanndd ddisispplalacceemmeennt‐t-ccoonnttrroollleledd llooaaddiinngg ssyysstteemm wwaass uusseedd uunnttiill tthhee ssaammppllee ffrraaccttuurreedd..  44.2.2. .MMaainin EExxppereirmimenentatal lRReseusultlst s 44.2.2.1.1. .EExxppeerirmimeenntatal lPProrocceessss aanndd FFrraacctuturree PPhheennoommeennaa  TThhee eexxppeerrimimeennt twwaass ccoonndduucctetedd toto sstutuddyy ththee bbeeaarriningg aanndd ddeefoforrmmaatitoionn ccaappaabbiliiltiiteiess, ,hhyyssteterreetitcic  ccuurrvveess aanndd sskkeeleletotonn ccuurvrveess, ,eenneergrgyy ddisisssipipaatitoionn pproroppeertriteiess (s(suucchh aas sddaammppiningg, ,dduuctcitliiltiyty, ,aanndd sstitfiffnfneessss),) , mmaateterriaial lddeefoforrmmaatitoionn (s(suucchh aass ggyyppssuumm, ,ccoonnccrreetete, ,aanndd ssteteeel lbbaarrss),) ,aanndd frfraacctuturree cchhaarraaccteterrisistitcicss oof fththee  tytyppicicaal lssaammppleless oof fththee (d(deennssee‐c-coolulummnn) )lalatttitciceedd ccoonnccrreetete ggyyppssuumm ccoommppoossitiete ppaanneel.l .TThhee eexxppeerrimimeennt toonn  QQ‐1-1 wwaass ddeessccrribibeedd aass ffoolllolowwss. .WWhhenen ththe esasmampplel ewwasa sppuulllelded oovveer raa ddisitsatanncece oof f00.7.722 mmmm, ,4455° ˝fifinnee, , inincclilnineedd crcarcakcsk sapappepaeraerde dbebloewlo wthet hineitiinali thiaolrihzoornitzaoln ctraalckcr aatc kthea tfotohte off otohte coofmthpeosciotem ppaonseilt.e Apsa tnheel . loAasdt hwealso aindcwreaasseindc, rbeoatshe din,cblointhedin acnlidn ehdoarinzdonhtoarli zcoranctkasl cwraecrke sown etrheeo innctrheeaisnec. rWeahsee.nW thhee snatmhepslea mwpasle  pwulalsedp uovlleerd ao dviesrtaancdei sotfa 3n.c2e8 omfm3.,2 a8 lomt mof, daelnoste o4f5°d ienncslein4e5d˝ cirnacclkins ewdhcicrahc dkisffwerheidc hind leifnfegrtehd aipnpeleanrgedth  ina ptpheea mreidddinlet haenmd ildowdleera pnadrtlos wofe rthpea rctosmofptohseitceo pmapnoesl.i tSeopmaen eolf. Sthoem ceraocfkths einc rtahcek smiindtdhlee mpaidrtd olef pthaer t coofmtphoesciotem ppaonseitl einptaenrseelcitnetde,r sceacutseidn,gc athues ignygptshuemg ybposaurdm tboo paeredl tsoligpheetllys. lAigsh ttlhye. lAosadth aeplpolaiedda tpop tlhieed  ctoomtpheosciotem ppaonseilt ewpaasn inelcrweaasseidn,c trheea soeudt,stidhee sotuetesli dbaersst oeef lthbea rpsaonfelt hyeiepldaende,l ayniedl dtheed ,parnindcitphael pinrcinlicniepda l cirnacclkin gerdadcuraacllky gtoroakd ushalalpyet.o Wokhesnh athpee .saWmphleen wtahse psualmlepdl eovwera sa dpiusltlaendceo ovfe r11a.5d0i mstamn,c iencolfin1e1d.5 c0ramckms , foinrcmliende da clraarcgkes pfoiercme eodf anelat rigne tphiee cmeiodfdnlee tpianrtt hoef mthied dcolemppaorstitoef pthaeneclo, mthpeo gsiytepspuamne bl,otahred gpyepesluedm  sberoiaorudslpye, ealnedd stherei ocuosnlcyr,eatne datt htheec ofonoctr eotfe tahtet hcoemfopootsoitfet hpeanceolm wpaoss ictreapckaende lawndas ccrruaschkeedd, aans dshcoruwsnh eind , Faigsushreo w10na.i nThFeig euxrpee1r0ima.eTnht eweaxsp eenridmeden wthweans tehned ceodmwphoesniteth peacnoeml fpraocstiuterepda nseerliforuacstlyu.r edseriously. (a) Q‐1  (b) Q‐2 (c) Q‐3  Figure 10. Fracture of samples (a) Q‐1; (b) Q‐2; (c) Q‐3.   Figure10.Fractureofsamples(a)Q-1;(b)Q-2;(c)Q-3. The cyclic experimental process revealed that the three composite panel samples underwent four  Thecyclicexperimentalprocessrevealedthatthethreecompositepanelsamplesunderwentfour stages: the elastic stage when the load was initially applied to the panel, the elastic‐plastic stage when  stages:theelasticstagewhentheloadwasinitiallyappliedtothepanel,theelastic-plasticstagewhen the panel yielded, and the fracture stage when the displacement and deformation of the panel was  the panel yielded, and the fracture stage when the displacement and deformation of the panel was great while the load applied decreased gradually. The sample was approximately in a square shape  greatwhiletheloadapplieddecreasedgradually. Thesamplewasapproximatelyinasquareshape and exhibited shear fracture. The final fracture of the samples shown in Figure 10 were primarily  and exhibited shear fracture. The final fracture of the samples shown in Figure 10 were primarily reflected  by  the  cracking  and  crushing  of  the  concrete  at  the  foot  of  the  panel  samples   reflectedbythecrackingandcrushingoftheconcreteatthefootofthepanelsamples(Q-1andQ-3) (Q‐1 and Q‐3) or the occurring of the 45° principal inclined crack at the foot of the sample (Q‐2).  ortheoccurringofthe45˝principalinclinedcrackatthefootofthesample(Q-2). 4.2.2. Hysteretic Characteristics  4.2.2. HystereticCharacteristics Hysteretic curves are the lateral load‐displacement curves for the composite panel under cyclic  Hystereticcurvesarethelateralload-displacementcurvesforthecompositepanelundercyclic loading. Figure 11 shows the hysteretic and skeleton curves for the panel samples Q‐1, Q‐2, and Q‐3  loading. Figure 11 shows the hysteretic and skeleton curves for the panel samples Q-1, Q-2, and obtained through experimental measurement. It can be seen that before the composite panel cracked,  the load‐displacement curves generally showed a linear change. The hysteresis loops initially took a  7208 10