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Bending Performance of Concrete Sandwich Walls with Actual Boundary Conditions

Bending Performance of Concrete Sandwich Walls with Actual Boundary Conditions applied sciences Article BendingPerformanceof Concrete Sandwich WallswithActual Boundary Conditions 1 1,2 1,2, 1 DaweiYan , Haiying Wan ,AnyingChen * andBingWang SchoolofCivilEngineering,Hefei UniversityofTechnology,Hefei230009,China AnhuiKeyLaboratoryof CivilEngineeringStructuresandMaterials,HefeiUniversityofTechnology, Hefei230009,China * Correspondence: anyingchen@hfut.edu.cn Abstract: Concrete sandwich walls are commonly used as the exterior wall panels of a structure, in which the wall suffers out‑of‑plane bending under strong wind conditions. This paper aims to investigate the bending performance of concrete sandwich walls under actual boundary conditions throughexperimental and analytical methods. In total, four concrete sandwich wallsweretested to detect the influence of openings and loading direction. Typical failure patterns were characterized anddiscussed. Theload‑displacementcurvesoffourtestspecimenswereanalyzed. Itwasindicated that the bearing capacity of the walls under negative bending conditions was higher than that un‑ der positive bending conditions, owing to the additional constraints provided by the steel beams. Straindistributionsofwallspecimenswerealsodiscussedinordertoobtainthecompositeactionof the sandwich walls between the upper and lower layers of concrete. In addition, the finite element model (FEM) was developed by ABAQUS to provide insights into the bending performance of the sandwichwalls. Throughcomparisonwiththetestresults,theFEMwasverifiedwithagoodlevelof accuracy. Subsequently,thedegreeofcompositeactionofthesandwichwallswasassessedinterms of both the moment of inertia and bearing capacity. From the experimental and numerical results, itdemonstratedthatthebearingcapacityofconcretesandwichedwallundernegativedirectionwas higher than that under positive direction owing to the constraints of steel beam. The derived com‑ posite action degree could be employed to evaluate the out‑plane bending stiffness and strength of sandwiched concrete wall. Both the experimental and analytical results in this paper are beneficial forthe designof sandwich wallsunder bending conditions. Citation: Yan,D.; Wan,H.;Chen, A.; Wang,B.BendingPerformanceof Keywords: concrete sandwich wall; bending performance; experimental program; actual boundary ConcreteSandwichWallswith constraints;composite action degree ActualBoundaryConditions. Appl. Sci. 2023, 13,1229. https://doi.org/ 10.3390/app13031229 AcademicEditor: Laurent 1. Introduction Daudeville Exteriorwallpanels are important parts of a structure, playing an indispensable role Received: 29December2022 inshelteringagainstenvironmentalconditionssuchaswindandrain. Inthemeantime,it Revised: 12 January2023 isnecessaryforthewallpaneltohavesatisfactorythermalandsoundinsulation,whichis Accepted: 13January 2023 beneficial in providing a comfortable living environment for human beings. Moreover, a Published: 17January2023 low self‑weight is also expected for sandwich walls, especially in steel structures. There‑ fore, various lightweight and thermal insulation materials have been employed for use in the sandwich layer within the wall panel. Moreover, many other structural components alsoadoptedsandwichedconfigurations[ 1–3]. However,thesematerialshavealowbear‑ Copyright: © 2023 by the authors. ing capacity. This also leads to more challenges for sandwich walls under bending condi‑ Licensee MDPI, Basel, Switzerland. tions. This article is an open access article Manyscholarshavecarriedoutexperimentalandnumericalprogramsinordertoan‑ distributed under the terms and alyzetheperformanceofconcretesandwichwallsundervariousloadingconditions[4–14]. conditions of the Creative Commons Amongthem,Xuetal.[5],RaoandPoluraju[8]andXueetal.[9]performedexperimental Attribution (CC BY) license ( https:// creativecommons.org/licenses/by/ investigations on concrete sandwich wall panels in order to analyze their seismic perfor‑ 4.0/). mance. LeeandPessiki[6]andYuetal.[11]analyzedthethermalbehavioroftheconcrete Appl. Sci.2023,13, 1229. https://doi.org/10.3390/app13031229 https://www.mdpi.com/journal/applsci Appl. Sci.2023,13, 1229 2 of17 sandwich walls based on experimental and numerical approaches. Kumar et al., [4] re‑ ported the test results of FRP‑reinforced concrete sandwich walls under concentric axial loading. Kontoleon et al., [10] discussed the vulnerability of concrete sandwich walls to fire, and Garhwal et al. [ 13] studied the corrosion performance of the concrete sandwich wall panels. In addition, Jensen et al. [7] developed an analytical model to predict the me‑ chanicalbehaviorofconcretesandwichwallsundervariousloadings. Itcanbeconcluded that sandwich wall panels have attracted the attention of many scholars, and are widely employedinengineeringpractice. Untilnow,greatattentionhasbeenpaidtothebendingperformanceofvarioussand‑ wich walls [15–22]. For example, Mercedes et al. [19] presented an experimental and nu‑ mericalinvestigationofsandwichpanelswithavegetal‑fabric‑reinforcedcementitiousma‑ trixlayer. Gallettietal.[ 20]proposedthedesignofasandwichpanelunderbendingcondi‑ tions. McCann [21] examined the behavior of recycled glass bead sandwich panels under bending conditions. Liu et al. [22] reported the bending performance of curved sandwich panels. From these studies, it was revealed that the sandwich wall panels exhibited satis‑ factory bending performance. Among these studies, the sandwich panels were subjected tobendingloadswithpinnedsupports. Thesandwichpanelsweregenerallyemployedas exteriorwallsandwereconnectedtothemainstructure,usuallybasedonangledsteelcon‑ nectors,inwhichtheactualconstraintofthesandwichwallpanelusedasanexteriorwall was significantly different from the pinned supports. Therefore, it is necessary to detect thebendingperformance ofsandwich wallsunder actual boundary conditions. Against the background, current investigations mainly explored the bending perfor‑ mance of the concrete sandwiched wall under pinned supports. Little attention has been paidonthebendingbehaviorofconcretesandwichedwallunderactualsupportconstraints. This paper attempts to discuss the out‑of‑plane bending performance of sandwich walls connected to the top and bottom beams, in which the sandwich wall analyzed in this pa‑ per did not bear vertical loads. In total, four sandwich wall specimens with extruded polystyrene foam (XPS) layers were prefabricated. Different loading directions and open‑ ingdetailsweredesignedforthevariousspecimens. Out‑of‑planebendingtestswereper‑ formed to detect the bending behavior, and typical failure modes were discussed. Load– deflection curves were also characterized in some key points. Strain distributions of the sandwich walls along their thicknesses were analyzed to reveal the composite action be‑ tweentheconcreteandXPSlayers. Subsequently,finiteelementmodeling(FEM)wasalso established, of which the accuracy of the FEM was validated by the test results. Stresses ofthesteeltrussandreinforcementwireswerealsodiscussed. Furthermore,thedegreeof compositeactionbetweenthetwolayersofconcretewasassessedintermsofthemoment of inertia and bearing capacity. The experimental and analytical performance could pro‑ videareferenceforthebendingperformanceofsandwichwallsconnectedtothesteelbeams. 2. ExperimentalProgram 2.1. TestSpecimens Two groups of wall test specimens were prefabricated with various opening configu‑ rations. Each group consisted of two test specimens with the same constructional details. Therewerefourtestspecimensoverall,namelyWP,WN,WPO,andWNO,inwhichPand Nrepresentpositiveloadingandnegativeloading,respectively;Oreferredtothetestspec‑ imenwithanopening. Thetotalheightandwidthofwallspecimensweresetat2920mm and 2760, respectively. Extruded polystyrene (XPS) was employed as the insulation ma‑ terial, and two layers of concrete were clamped on both sides of XPS, forming a concrete sandwichwallpanel,asshowninFigure1. Thethicknessofthetwolayersofconcreteand XPS was 50 mm. To avoid any cracking of the concrete sandwich walls, the thickness of the concrete layer was chosen at 50 mm, with a nominal compression strength of 30 MPa. The steel bar truss was embedded in the concrete layer, in which the reinforcements with a diameter of 6 mm were placed at average spaces of 150 mm. In addition, the steel bar Appl. Sci. 2023, 13, x FOR PEER REVIEW 3 of 18 concrete and XPS was 50 mm. To avoid any cracking of the concrete sandwich walls, the Appl. Sci.2023,13, 1229 3 of17 thickness of the concrete layer was chosen at 50 mm, with a nominal compression strength of 30 MPa. The steel bar truss was embedded in the concrete layer, in which the reinforce- ments with a diameter of 6 mm were placed at average spaces of 150 mm. In addition, the steel bar truss deck, composed of short reinforcements with diameters of 6 mm, was used trussdeck,composedofshortreinforcementswithdiametersof6mm,wasusedtofirmly to firmly connect the two layers of concrete without passing through the XPS layer. connectthetwolayersof concretewithout passing through the XPS layer. (a) XPS (b) Figure1. Details of test specimens. (a) Specimens WP and WN. (b) Specimens WPO andWNO. 150 2640 130 150 2640 130 Appl. Sci.2023,13, 1229 4 of17 In the construction process, the bottom layer of concrete was first poured with the reinforcements embedded in it. Subsequently, the XPS layer was placed on top of the bottomlayerofconcrete. Finally,the upper layerof concrete waspoured. 2.2. MaterialProperties The density and thermal conductivity of XPS used in this paper were 30 kg/m and 0.03 W/(m.k), respectively. Key reinforcements were extracted to conduct material tests in order to accurately capture the necessary properties from the material according to the tensiletestsmethodforreinforcementsshowninGB/T2975[23]. Testresultsfromtherein‑ forcementcomponentsareshowninTable1. Materialtestswerealsoperformedtoacquire theconcretematerialproperties, asdepicted in Table2. Table1. Materialpropertiesof reinforcement. Diameterof 2 2 2 f (N/mm ) f (N/mm ) E(N/mm ) Elongation (%) y u Reinforcement(mm) 6 428.7 563.5 1.98 × 10 20.6 5 407.3 452.3 2.01 × 10 18.9 8 442.9 484.6 2.11 × 10 19.5 Note: f ,f andEaretheyieldstrength, ultimatestrength,andelasticmodulusofreinforcements. y u Table2. Materialpropertiesof concrete. 2 2 Size (mm) CuringDays E (N/mm ) f (N/mm ) c cu C11 150 × 150 × 300 28 3.08 × 10 C12 150 × 150 × 300 28 2.99 × 10 C13 150 × 150 × 300 28 3.06 × 10 Mean 3.04 × 10 C21 150 × 150 × 150 28 33.25 C22 150 × 150 × 150 28 31.92 C23 150 × 150 × 150 28 32.12 Mean 32.43 Note: E andf aretheelasticmodulus andcompressivestrengthofconcrete. c cu 2.3. LoadingProcedure To investigate the performance of the wall specimens under both positive and nega‑ tivebendingconditions,specimensWPandWPOweretestedunderpositivebendingcon‑ ditions,whereasspecimensWNandWNOweresubjectedtonegativebendingconditions. Test specimens were connected to the steel beam using top and bottom angle connectors, asshowninFigure 2. Figure3displaystheschematicviewofthetestsetup. Itisrelativelydifficulttosimu‑ late uniform loads on the wall surface. A simplified loading method was employed using rigidbeamstoallocatetheconcentrateforceoneightpointsonthetopofsandwichedcon‑ crete wall, as shown in Figure 4. Moreover, the exterior sandwiched concrete wall was generally connected to the top and bottom steel beams through steel angles and bolts in engineeringpractice. Therefore,thesimilarconnectionmethodwasalsoemployedinthis paper,asshowninFigure 4a. Beforetheformalloading,preloadingwasperformedtocheckwhethertheequipment wasfunctional. Duringtheformalloading,astep‑by‑steploadingprocedurewasadopted usingahydraulicjack. Theincrementsinloadingforceofeachstepweresetat10kN.Test specimensweresubjected to the loadingof each step for about 15 min until failure. Appl. Sci. 2023, 13, x FOR PEER REVIEW 5 of 18 Appl. Sci.2023,13, 1229 5 of17 Figure2. Connectiondetails (Unit: mm). Figure 2. Connection details (Unit: mm). Figure 3 displays the schematic view of the test setup. It is relatively difficult to sim- ulate uniform loads on the wall surface. A simplified loading method was employed using rigid beams to allocate the concentrate force on eight points on the top of sandwiched concrete wall, as shown in Figure 4. Moreover, the exterior sandwiched concrete wall was generally connected to the top and bottom steel beams through steel angles and bolts in engineering practice. Therefore, the similar connection method was also employed in this paper, as shown in Figure 4a. Before the formal loading, preloading was performed to check whether the equip- ment was functional. During the formal loading, a step-by-step loading procedure was adopted using a hydraulic jack. The increments in loading force of each step were set at 10 kN. Test specimens were subjected to the loading of each step for about 15 min until failure. Appl. Sci. 202Appl. 3, 13Sci. , x F2023 OR, 13 PE, 1229 ER REVIEW 6 of17 6 of 18 Appl. Sci. 2023, 13, x FOR PEER REVIEW 6 of 18 (a) (b) (a) (b) Figure 3. Test setup. (a) Loading in a positive direction. (b) Loading in a negative direction. Figure3. Testsetup. (a)Loading in a positivedirection. (b) Loading in a negativedirection. Figure 3. Test setup. (a) Loading in a positive direction. (b) Loading in a negative direction. (a) (b) Figure 4. Layout of loading points (Unit: mm). (a) Loading schematic. (b) Loading points. (a) Figure4. Layout of loading points(Unit: mm). (a) Loading schematic. ((b b))Loading points. 3. Test Results Figure 4. Layout of loading points (Unit: mm). (a) Loading schematic. (b) Loading points. 3.1. Failure Modes 3. Test Results In total, 29 regions, each with a width of 100 mm, were divided along its span, iden- tified as #1–29. Specimens WP and WN exhibited similar failure patterns. Specimen WP 3.1. Failure Modes was set as the example used to describe the failure modes. No obvious failure modes and In total, 29 regions, each with a width of 100 mm, were divided along its span, iden- bending deformations were noticed in the specimen before cracking. When the loading tified as #1–29. Specimens WP and WN exhibited similar failure patterns. Specimen WP force reached 35 kN, cracking was initiated in section #16 and its depth was about 2/3 was t is m ee ts a ts h a tt h o ef e th xe a w map lll e th u ics ke nd e stso . W de hs ec nr tih be e l o th ad e ifn acir le uarse e d m to o d 50 e s k.N N , a on o ob thv eiro cu ra sc f ka a il p u pr ee a rm ed o des and in region #12. As the load increased to 78 kN, slight bending deformation was noticed. bending deformations were noticed in the specimen before cracking. When the loading With the further increase in load, more and more cracks were noticed and penetrated the force reached 35 kN, cracking was initiated in section #16 and its depth was about 2/3 thickness of the wall. When the load equaled 140 kN, slippage between the concrete layer times that of the wall thickness. When the load increased to 50 kN, another crack appeared in region #12. As the load increased to 78 kN, slight bending deformation was noticed. With the further increase in load, more and more cracks were noticed and penetrated the thickness of the wall. When the load equaled 140 kN, slippage between the concrete layer Appl. Sci.2023,13, 1229 7 of17 3. TestResults 3.1. FailureModes Intotal,29regions,eachwithawidthof100mm,weredividedalongitsspan,identi‑ fiedas#1–29. SpecimensWPandWNexhibitedsimilarfailurepatterns. SpecimenWPwas setastheexampleusedtodescribethefailuremodes. Noobviousfailuremodesandbend‑ ing deformations were noticed in the specimen before cracking. When the loading force Appl. Sci. 2023, 13, x FOR PEER REVIEW 7 of 18 reached35kN,crackingwasinitiatedinsection#16anditsdepthwasabout2/3timesthat ofthewallthickness. Whentheloadincreasedto50kN,anothercrackappearedinregion #12. Astheloadincreasedto78kN,slightbendingdeformationwasnoticed. Withthefur‑ ther increase in load, more and more cracks were noticed and penetrated the thickness of and XPS was detected, and the crack along the span was noticed. When the load increased thewall. Whentheloadequaled140kN,slippagebetweentheconcretelayerandXPSwas detected, and the crack along the span was noticed. When the load increased to 150 kN, to 150 kN, the width of crack along the span grew to 5 mm and the XPS fractured. After- the width of crack along the span grew to 5 mm and the XPS fractured. Afterwards, the wards, the deflection of the wall specimen increased sharply when the load reached to 160 deflection of the wall specimen increased sharply when the load reached to 160 kN and kN and the loading procedure was terminated. Typical failure modes of specimen WP are the loading procedure was terminated. Typical failure modes of specimen WP are shown shown in Figure 5. inFigure 5. (a) (b) (c) (d) Figure 5. Typical failure modes of specimen WP. (a) First crack at the load of 35 kN. (b) Bending Figure 5. Typical failure modes of specimen WP. (a) First crack at the load of 35 kN. (b) Bending deformation at the end of the test. (c) Bending deformation in the test. (d) Cracks at the bottom of deformation at the end of the test. (c) Bending deformation in the test. (d) Cracks at the bottom of the wall. thewall. Moreover, specimens WPO and WNO exhibited similar failure modes. Before the load increased to 33 kN, no test phenomena were detected among the specimens. Subse- quently, a series of cracks emerged and spread to the middle of the wall thickness. When the load increased to 52 kN, obvious bending deformation was noticed in the test speci- mens. A penetration crack along the wall thickness was observed when the load reached 100 kN. At the same time, some cracks also appeared around the openings. As the load increased to 135 kN, slippage between the XPS and concrete was also noticed. With the load further increasing to 145 kN, the deflection increased significantly, whereas the load was kept almost constant. The test also ended at this time. After the loading, it demon- strated that the sandwich wall remained almost flat, and the connection angle exhibited obvious deformation. Typical failure modes are displayed in Figure 6. Appl. Sci.2023,13, 1229 8 of17 Moreover, specimens WPO and WNO exhibited similar failure modes. Before the load increased to 33 kN, no test phenomena were detected among the specimens. Subse‑ quently, a series of cracks emerged and spread to the middle of the wall thickness. When the load increased to 52 kN, obvious bending deformation was noticed in the test speci‑ mens. A penetration crack along the wall thickness was observed when the load reached 100kN.Atthesametime,somecracksalsoappearedaroundtheopenings. Astheloadin‑ creasedto135kN,slippagebetweentheXPSandconcretewasalsonoticed. Withtheload further increasing to 145 kN, the deflection increased significantly, whereas the load was kept almost constant. The test also ended at this time. After the loading, it demonstrated Appl. Sci. 2023, 13, x FOR PEER REVIEW 8 of 18 that the sandwich wall remained almost flat, and the connection angle exhibited obvious deformation. Typical failure modesare displayedin Figure6. (a) (b) (c) (d) Figure 6. Typical failure modes of specimen WPO. (a) Cracks at a load of 50 kN. (b) Cracks around Figure 6. Typical failure modes of specimen WPO. (a) Cracks at a load of 50 kN. (b) Cracks around the opening. (c) Deformation of the connection angle. (d) Residual deformation after the loading. theopening. (c)Deformation of the connection angle. (d)Residual deformation after the loading. In conclusion, the bottom concrete first cracked under bending. Subsequently, the In conclusion, the bottom concrete first cracked under bending. Subsequently, the XPS layer also exhibited fracture and slippage between the XPS and concrete, which was XPS layer also exhibited fracture and slippage between the XPS and concrete, which was detected both under positive and negative bending conditions. However, the crack was detected both under positive and negative bending conditions. However, the crack was mainly located on the 1/3 and middle span, and did not penetrate the wall along its thick‑ mainly located on the 1/3 and middle span, and did not penetrate the wall along its thick- ness,whichindicatedthatthetwolayersofconcretecouldworktogetherthroughthesteel ness,truss which connection. indicated Inaddition, that the no twolocal layer crushing s of concre wasfound te coduring uld work the to test. gether Meanwhile, througthe h the steel additional constraints provided by the actual boundary conditions increased the stiffness truss connection. In addition, no local crushing was found during the test. Meanwhile, the andstrengthofthesandwichedwallto some extent. additional constraints provided by the actual boundary conditions increased the stiffness and strength of the sandwiched wall to some extent. 3.2. Load–Deflection Curves Several key points were selected to analyze their deflection during the test. The de- flections of the key points were measured by the LVDTs, and the load was recorded by the pressure sensor. For specimens WP and WN, the deflections of five points were rec- orded in the test. For specimens WPO and WNO, the deflections of six points were rec- orded, as can be seen in Figure 7. Appl. Sci.2023,13, 1229 9 of17 3.2. Load–DeflectionCurves Several key points were selected to analyze their deflection during the test. The de‑ flectionsofthekeypointsweremeasuredbytheLVDTs,andtheloadwasrecordedbythe Appl. Sci. 2023, 13, x FOR PEER REVIEW 9 of 18 pressure sensor. For specimens WP and WN, the deflections of five points were recorded in the test. For specimens WPO and WNO, the deflections of six points were recorded, as canbe seeninFigure 7. (a) (b) Figure 7. Selected deflection points. (a) Deflection points of specimens WP and WN. (b) Deflection Figure 7. Selected deflection points. ( a) Deflection points of specimens WP and WN. ( b) Deflection points of specimens WPO and WNO. pointsofspecimens WPO and WNO. Theload–deflectioncurvesoftestspecimensarepresentedinFigure 8. Forspecimens The load–deflection curves of test specimens are presented in Figure 8. For specimens WPandWN,similarload–deflectioncurvesweredisplayedatthefivepoints. Overall,the WP and WN, similar load–deflection curves were displayed at the five points. Overall, the resultsshowedthatthedeflectionincreasedwiththedecreaseinthedistancebetweenthe results showed that the deflection increased with the decrease in the distance between the points and middle span. In comparison, the deflection of specimen WP was larger than points and middle span. In comparison, the deflection of specimen WP was larger than that of specimen WN, owing to the fact that the steel beam provided additional vertical that of specimen WN, owing to the fact that the steel beam provided additional vertical constraints for the wall specimens when subjected to negative bending conditions. More‑ over,theload–deflectioncurveofspecimenWPreachedthefirstinflectionpointwhenthe constraints for the wall specimens when subjected to negative bending conditions. More- uniformload(q)equaledto3kN/m duetotheconcretecracking,whereasthatofspecimen over, the load–deflection curve of specimen WP reached the first inflection point when WN reached the first inflection point when q reached 9 kN/m . The load–deflection curve the uniform load (q) equaled to 3 kN/m due to the concrete cracking, whereas that of of specimen WP reached the second inflection point when q was 13 kN/m , and that of specimen WN reached the first inflection point when q reached 9 kN/m . The load–deflec- specimenreachedthesecondinflectionpointwhen qwas13.5kN/m . Thisalsoindicated tion curve of specimen WP reached the second inflection point when q was 13 kN/m , and thatthecrackingload of specimenWN washigher than that of specimen WP. that of specimen reached the second inflection point when q was 13.5 kN/m . This also Figure 8 depicts the load–displacement curves of specimens WPO and WNO. It also demonstrated that the deflection of the points increased with the decrease in the distance indicated that the cracking load of specimen WN was higher than that of specimen WP. between the point and the middle span. However, the maximum deflection of specimen Figure 8 depicts the load–displacement curves of specimens WPO and WNO. It also WPOwasalmostequaltothatofspecimenWNO.Incomparison,theinflectionontheload– demonstrated that the deflection of the points increased with the decrease in the distance displacement curve of specimen WPO appeared when q equaled to 3 kN/m , indicating between the point and the middle span. However, the maximum deflection of specimen thattheconcretestartedcracking,whilethatofspecimenWNOemergedwhenqwasabout WPO was almost equal to that of specimen WNO. In comparison, the inflection on the 7.5kN/m . Afterwards,theload–displacementcurvesofspecimensWPOandWNOboth 2 2 load–displacement curve of specimen WPO appeared when q equaled to 3 kN/m , indi- exhibited their second points of inflection when q increased to 12.5 kN/m . Therefore, it couldbedeterminedthatthecrackingloadofspecimenWPOwaslowerthanthatofWNO. cating that the concrete started cracking, while that of specimen WNO emerged when q It also showed that 2the deflections on S5 and S6 were significantly larger than those of S1 was about 7.5 kN/m . Afterwards, the load–displacement curves of specimens WPO and and S2. This could be explained by the fact that the center of the eight loading points was WNO both exhibited their second points of inflection when q increased to 12.5 kN/m . closertoS5andS6. Therefore, it could be determined that the cracking load of specimen WPO was lower than that of WNO. It also showed that the deflections on S5 and S6 were significantly larger than those of S1 and S2. This could be explained by the fact that the center of the eight loading points was closer to S5 and S6. Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 18 Appl. Sci.2023,13, 1229 10 of17 Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 18 10 10 S1 S1 10 10 S2 S2 S1 S1 S3 S3 S2 S2 S4 S4 S3 S3 S5 5 S5 S4 S4 S5 S5 0 20 40 60 80 100 0 20 40 60 80 D (mm) 0 20 40 60 80 10 0 D (mm) 0 20 40 60 80 (a) (b) D (mm) D (mm) 25 25 (a) (b) 25 25 20 20 20 20 S1 S2 S1 S1 S3 S2 S2 S1 S4 S3 S3 S2 S5 S4 5 S4 S3 S6 S5 S5 S4 5 S6 S6 S5 0 S6 0 20 40 60 80 0 20 40 60 80 D (mm) 0 20 40 60 80 D (mm) 0 20 40 60 80 (c D ) (mm) (d) D (mm) (c) (d) Figure 8. Load–deflection curves. (a) Specimen WP. (b) Specimen WN. (c) Specimen WPO. (d) Specimen WNO. Figure 8. Load–deflection curves. (a) Specimen WP. (b) Specimen WN. (c) Specimen WPO. (d) Figure8. Load–deflectioncurves. ( a)SpecimenWP.(b)SpecimenWN.(c)SpecimenWPO.(d)Spec‑ Specimen WNO. imenWNO. 3.3. Strain Responses 3.3. Strain Responses Figure 9 shows the positions of strain gauges distributed throughout specimen WP 3.3. StrainResponses in the 1/4 and middle span. As can be seen in Figure 10, the strain remained almost at zero Figure 9 shows the positions of strain gauges distributed throughout specimen WP Figure 9 shows the positions of strain gauges distributed throughout specimen WP before q reached 12 kN/m . Afterwards, strains #5 and #6 decreased significantly, indicat- in the 1/4 and middle span. As can be seen in Figure 10, the strain remained almost at zero in the 1/4 and middle span. As can be seen in Figure 10, the strain remained almost at ing that the upper concrete 2was in compression, whereas strains #1 and #2 of the lower before q reached 12 kN/m . Afterwards, strains #5 and #6 decreased significantly, indicat- zero before q reached 12 kN/m . Afterwards, strains #5 and #6 decreased significantly, concrete layer remained at zero due to the fact that the concrete in the middle span exhib- ing that the upper concrete was in compression, whereas strains #1 and #2 of the lower indicating that the upper concrete was in compression, whereas strains #1 and #2 of the ited serious cracking and the strain gauges were broken. Similarly, strains #11 and #12 concrete layer remained at zero due to the fact that the concrete in the middle span exhib- lower concrete layer remained at zero due to the fact that the concrete in the middle span also decreased, whereas strains #7 and #8 increased sharply when q was higher than 12 ited serious cracking and the strain gauges were broken. Similarly, strains #11 and #12 exhibited serious cracking and the strain gauges were broken. Similarly, strains #11 and kN/m . This revealed that the upper and lower layers of concrete were in states of com- also decreased, whereas strains #7 and #8 increased sharply when q was higher than 12 #12 also decreased, whereas strains #7 and #8 increased sharply when q was higher than pression 2 and tension, respectively. kN/m . This revealed that the upper and lower layers of concrete were in states of com- 12 kN/m . This revealed that the upper and lower layers of concrete were in states of pression and tension, respectively. compressionandtension,respectively. Figure9. Strain distributions. p (kN/m ) p (kN/m ) p (kN/m ) p (kN/m ) p (kN/m ) p (kN/m ) p (kN/m ) p (kN/m ) Appl. Sci. 2023, 13, x FOR PEER REVIEW 11 of 18 Figure 9. Strain distributions. -200 Appl. Sci. 2023, 13, x FOR PEER REVIEW 11 of 18 -400 0 -200 Appl. Sci.2023,13, 1229 11 of17 -600 5 -400 Figure 9. Strain distributions. -600 -800 04 8 12 16 04 8 12 16 p(kN/m ) p(kN/m ) (a) (b) Figure 10. Strains of specimen WP. (a) Middle span. (b) 1/4 span. -200 4. Finite Element Modeling To gain insights into the stress state and work mechanism of the sandwich wall under -400 0 bending conditions, the numerical model was developed by ABAQUS software in this -200 -600 5 section, which was also beneficial for proposing design equation of the sandwiched wall. -400 -600 -800 4.1. Material Models 0 4 8 12 16 0 4 8 12 16 p(kN/m ) For the reinforcements embedded in the concrete slabp(k , N a b /m ili )near model was generally employed to simulate the behavior of reinforcements, as shown in Figure 11a. (a) (b) For the concrete in the sandwich wall, the stress–strain relationship demonstrated in Figure 10. Strains of specimen WP. (a) Middle span. (b) 1/4 span. Figure10. Strains of specimen WP.(a)Middle span. (b)1/4 span. GB 50010 [24] was adopted, and the concrete damage plastic model in ABAQUS was also employed to simulate the behavior of concrete, which is shown in Figure 11b. In the con- 4. Finite Element Modeling 4. FiniteElementModeling crete damaged plasticity model, the dilation angle, eccentricity, stress ratio between the To To gain gain iinsights nsights iinto nto tthe he sstress tress ststate ate anand d wwork ork mmechanism echanism ofof thethe sansandwich dwich waw ll all unun‑ der concrete under biaxial compression and uniaxial compression, K and viscosity parameter der benbending ding conconditions, ditions, thethe nunumerical merical momodel del ww asas dedev veleloped oped bby y AABAQUS BAQUS ssoftw oftwaare re in in this this were taken as 30, 0.1, 1.16, 0.667, and 0.0005, respectively. Moreover, the damage variables section,whichwasalsobeneficialfor proposing design equation of the sandwiched wall. section, which was also beneficial for proposing design equation of the sandwiched wall. in GB 50010 [24] were also assigned for the concrete in the sandwich wall. XPS exhibited relatively low stiffness and strength compared with the concrete and provided little con- 4.1. MaterialModels 4.1. Material Models tribution on the strength and stiffness of the sandwich wall, which is tentatively ignored Forthereinforcementsembeddedintheconcreteslab,abilinearmodelwasgenerally For the reinforcements embedded in the concrete slab, a bilinear model was generally in the FEM. employedtosimulate the behaviorof reinforcements, as shown in Figure 11a. employed to simulate the behavior of reinforcements, as shown in Figure 11a. For the concrete in the sandwich wall, the stress–strain relationship demonstrated in GB 50010 [24] was adopted, and the concrete damage plastic model in ABAQUS was also σ to employed to simulate the behavior of concrete, which is shown in Figure 11b. In the con- crete damaged plasticity model, the dilation angle, eccentricity, stress ratio between the f B concrete under biaxial compression and uniaxial compression, K and viscosity parameter were taken as 30, 0.1, 1.16, 0.667, and 0.0005, respectively. Moreover, the damage variables W =1 t (1-d )E t 0 in GB 50010 [24] were also assigned for the W co =0 ncrete in the sandwich wall. XPS exhibited (1-d )E relatively low stiffness and stren c gt 0h c(1 o- m d p )(1 a-rde)d E with the concrete and provided little con- c t 0 D W =1 W =0 F c tribution on the strength and stiffness of the sandwich wall, which is tentatively ignored in the FEM. ε y ε Uniaxial stress-strain curve (a) (b) to Figure 11. b Material models of concrete and reinforcements. (a) Reinforcements. (b) Concrete. Figure11. Materialmodels of concrete and reinforcements. (a) Reinforcements. (b)Concrete. 4.2. Element and Mesh Size For the concrete in the sandwich wall, the stress–strain relationship demonstrated in GB 50010 [24] was adopted, and the concrete damage plastic model in ABAQUS was also W =1 t (1-d )E t 0 W =0 employed to simulate the behavior of concrete, which is shown in Figure 11b. In the con‑ (1-d )E (1-d )(1-d )E c 0 c t 0 crete damaged plasticity model, the dilation angle, eccentricity, stress ratio between the D W =1 ε W =0 concrete under biaxial compression and uniaxial compression, K and viscosity parameter weretakenas30,0.1,1.16,0.667,and0.0005,respectively. Moreover,thedamagevariables o 0 in GB 50010 [24] were also assigned for the concrete in the sandwich wall. XPS exhibited ε y relatively low stiffness and strength compared with the concrete and provided little con‑ tribution on the strength and stiffness of the sandwich wall, which is tentatively ignored Uniaxial stress-strain curve intheFEM. (a) (b) Figure 11. Material models of concrete and reinforcements. (a) Reinforcements. (b) Concrete.  ( ) 2 (04) e (me) ε (με) Appl. Sci. 2023, 13, x FOR PEER REVIEW 12 of 18 Appl. Sci.2023,13, 1229 12 of17 4.2. Element and Mesh Size The concrete and XPS layers were all modeled using solid elements (C3D8R). Truss 4.2. ElementandMesh Size elements were assigned for reinforcement. Both XPS and reinforcements were embedded The concrete and XPS layers were all modeled using solid elements (C3D8R). Truss in the concrete without considering separation. The mesh sizes of XPS, concrete and rein- elements were assigned for reinforcement. Both XPS and reinforcements were embedded in the concrete without considering separation. The mesh sizes of XPS, concrete and re‑ forcements were set as 50 mm, whereas that of the sandbag was taken as 100 mm. A de- inforcements were set as 50 mm, whereas that of the sandbag was taken as 100 mm. A tailed FEM model is displayed in Figure 12. The static analysis step was employed to ap- detailed FEM model is displayed in Figure 12. The static analysis step was employed to ply the out-bending loads and the complete integral algorithm was used. The actual run applytheout‑bendingloadsandthecompleteintegralalgorithmwasused. Theactualrun time of each model was about 40 min. The sandwiched wall was tied to the supports and time of each model was about 40 min. The sandwiched wall was tied to the supports and all degree of freedoms of the supports were constrained. alldegreeoffreedomsof the supports wereconstrained. (a) (b) Figure 12. Finite element model of sandwich wall with reasonable mesh sizes. (a) Concrete layer. Figure 12. Finite element model of sandwich wall with reasonable mesh sizes. (a) Concrete layer. (b) Reinforcements. (b)Reinforcements. 4.3. AnalysisofFEMResults 4.3. Analysis of FEM Results The comparison results between the FEM and test results are demonstrated in The comparison results between the FEM and test results are demonstrated in Figure Figure 13. It can be concluded that the load–displacement curve of FEM coincided well 13. It can be concluded that the load–displacement curve of FEM coincided well with the with the test results, indicating that the FEM was able to reasonably predict the bending test results, indicating that the FEM was able to reasonably predict the bending perfor- performanceoftheconcrete sandwichwall. ] mance of the concrete sandwich wall. Figure14presentsthestressdistributionsofreinforcementsinspecimenWP.Itshowed thatthereinforcementwiremainlyappearedtoyieldinthemiddlespananditssupports. 20 20 The maximum stress was about 500 MPa. On the other hand, the steel truss parallel to thespandirectionexhibitedyieldingmainlyinthemiddlespan,andthemaximumstress 16 16 of the steel truss was 464 MPa, lower than that of the reinforcement wire. However, the steeltrussperpendiculartothespandirectionshowedrelativelylowerstressthanthetruss which was parallel to the span direction. Figure 15 displays the stress distributions of re‑ 12 12 inforcements of specimen WPO. It could be noticed that only a small portion of reinforce‑ mentsinthewireachievedtheiryieldstress,mainlylocatedwiththemiddleand1/3span 8 8 near the opening. Comparatively, the reinforcements in steel truss parallel to the span Test Test direction showed higher stress levels with the maximum stress value of 439 MPa mainly 4 4 FEM FEM distributed on the web reinforcements. Moreover, the strengthen reinforcements around the opening showed relatively high stress levels, indicating that the stress concentration 0 20 40 appeared 60 around 80 the 10opening. 0 It could be concluded that the reinforcements in the con‑ 0 20 40 60 80 100 D (mcrete m) sandwich walls did not all reach their yield strength, indicating D (mm) that the two layers of concrete could not work together fully in the concrete sandwich wall. Therefore, it is (a) (b) necessary to discuss the composite action between the two layers of the concrete for the concretesandwichwall. p ( kN/m ) p ( kN/m ) Appl. Sci. 2023, 13, x FOR PEER REVIEW 12 of 18 4.2. Element and Mesh Size The concrete and XPS layers were all modeled using solid elements (C3D8R). Truss elements were assigned for reinforcement. Both XPS and reinforcements were embedded in the concrete without considering separation. The mesh sizes of XPS, concrete and rein- forcements were set as 50 mm, whereas that of the sandbag was taken as 100 mm. A de- tailed FEM model is displayed in Figure 12. The static analysis step was employed to ap- ply the out-bending loads and the complete integral algorithm was used. The actual run time of each model was about 40 min. The sandwiched wall was tied to the supports and all degree of freedoms of the supports were constrained. (a) (b) Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 18 Figure 12. Finite element model of sandwich wall with reasonable mesh sizes. (a) Concrete layer. (b) Reinforcements. 4.3. Analysis of FEM Results The comparison results between the FEM and test results are demonstrated in Figure 20 20 13. It can be concluded that the load–displacement curve of FEM coincided well with the Appl. Sci.2023,13, 1229 13 of17 test results, indicating that the FEM was able to reasonably predict the bending perfor- 16 16 mance of the concrete sandwich wall. 12 12 20 20 8 8 Test Test FEM FEM 4 4 0 12 12 0 20 40 60 80 100 0 20 40 60 80 100 D (mm) D (mm) 8 8 (c) (d) Test Test Figure 13. Comparison between FEM and test. (a) WP. (b) WN. (c) WPO. (d) WPN. FEM FEM Figure 14 presents the stress distributions of reinforcements in specimen WP. It Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 18 0 0 0 20 40 60 80 100 0 20 40 60 80 100 showed that the reinforcement wire mainly appeared to yield in the middle span and its D (mm) D (mm) supports. The maximum stress was about 500 MPa. On the other hand, the steel truss (a) (b) parallel to the span direction exhibited yielding mainly in the middle span, and the max- 24 24 imum stress of the steel truss was 464 MPa, lower than that of the reinforcement wire. However, the steel truss perpendicular to the span direction showed relatively lower stress than the truss which was parallel to the span direction. Figure 15 displays the stress 16 16 distributions of reinforcements of specimen WPO. It could be noticed that only a small portion of reinforcements in the wire achieved their yield stress, mainly located with the 12 12 middle and 1/3 span near the opening. Comparatively, the reinforcements in steel truss parallel to the span direction showed higher stress levels with the maximum stress value 8 8 Test Test of 439 MPa mainly distributed on the web reinforcements. Moreover, the strengthen rein- FEM FEM forcements around the opening showed r 4elatively high stress levels, indicating that the stress concentration appeared around the opening. It could be concluded that the rein- 0 0 forcements in the concrete sandwich walls did not all reach their yield strength, indicating 0 20 40 60 80 100 0 20 40 60 80 100 D (mm) that the two layers of concrete could not work together fully D i (n m m th ) e concrete sandwich (w c) all. Therefore, it is necessary to discuss the composite action (d b )e tween the two layers of the concrete for the concrete sandwich wall. Figure 13. Comparison between FEM and test. (a) WP. (b) WN. (c) WPO. (d) WPN. Figure13. Comparison betweenFEMand test. (a)WP. (b) WN. (c) WPO. (d)WPN. Figure 14 presents the stress distributions of reinforcements in specimen WP. It showed that the reinforcement wire mainly appeared to yield in the middle span and its supports. The maximum stress was about 500 MPa. On the other hand, the steel truss parallel to the span direction exhibited yielding mainly in the middle span, and the max- imum stress of the steel truss was 464 MPa, lower than that of the reinforcement wire. However, the steel truss perpendicular to the span direction showed relatively lower stress than the truss which was parallel to the span direction. Figure 15 displays the stress distributions of reinforcements of specimen WPO. It could be noticed that only a small portion of reinforcements in the wire achieved their yield stress, mainly located with the Appl. Sci. 2023, 13, x FOR PEER REVIEW 14 of 18 middle and 1/3 span near the opening. Comparatively, the reinforcements in steel truss parallel to the span direction showed higher stress levels with the maximum stress value (a) (b) of 439 MPa mainly distributed on the web reinforcements. Moreover, the strengthen rein- forcements around the opening showed relatively high stress levels, indicating that the stress concentration appeared around the opening. It could be concluded that the rein- forcements in the concrete sandwich walls did not all reach their yield strength, indicating that the two layers of concrete could not work together fully in the concrete sandwich wall. Therefore, it is necessary to discuss the composite action between the two layers of the concrete for the concrete sandwich wall. (c) Figure 14. Stress distributions of specimen WP. (a) Stress of reinforcement wire. (b) Stress of steel Figure 14. Stress distributions of specimen WP. (a) Stress of reinforcement wire. (b) Stress of steel truss parallel to the span. (c) Mises stress contour plots of steel truss perpendicular to the span. trussparalleltothespan. (c)Misesstresscontourplotsofsteeltrussperpendiculartothespan. The The non-English term ‘平均’ means ‘Average’. non‑Englishterm‘ 平均’ means ‘Average’. (a) (b) (a) (b) (c) (d) Figure 15. Stress distributions of specimen WPO. (a) Stress of reinforcement wire. (b) Stress of steel truss parallel to the span. (c) Stress of steel truss perpendicular to the span. (d) Stress of strengthened steel truss. The non-English term ‘平均’ means ‘Average’. 5. Analytical Model of Composite Action The degree of composite action (K) of the sandwich wall is an important factor that can reveal the interactive working degree of the components. When K equals to 0, it means that the two layers of concrete worked separately and the moment of inertia of the wall should be the summation of two layers of concrete. When K equals to 1, it means that the two layers of concrete works together, the moment of inertia of the wall should be calcu- lated based on the whole section. Currently, there are two methods assessing the p ( kN/m ) p ( kN/m ) p ( kN/m ) p ( kN/m ) 2 p ( kN/m ) p ( kN/m ) Appl. Sci. 2023, 13, x FOR PEER REVIEW 14 of 18 (c) Figure 14. Stress distributions of specimen WP. (a) Stress of reinforcement wire. (b) Stress of steel Appl. Sci.2023,13, 1229 14 of17 truss parallel to the span. (c) Mises stress contour plots of steel truss perpendicular to the span. The non-English term ‘平均’ means ‘Average’. (a) (b) (c) (d) Figure 15. Stress distributions of specimen WPO. (a) Stress of reinforcement wire. (b) Stress of Figure15. StressdistributionsofspecimenWPO.(a)Stressofreinforcementwire. (b)Stressofsteel steel truss parallel to the span. (c) Stress of steel truss perpendicular to the span. (d) Stress of trussparalleltothespan. (c)Stressofsteeltrussperpendiculartothespan. (d)Stressofstrengthened strengthened steel truss. The non-English term ‘平均’ means ‘Average’. steeltruss. The non‑English term ‘平均’means ‘Average’. 5. Analytical Model of Composite Action 5. AnalyticalModelof CompositeAction The degree of composite action (K) of the sandwich wall is an important factor that Thedegreeofcompositeaction(K)ofthesandwichwallisanimportantfactorthatcan can reveal the interactive working degree of the components. When K equals to 0, it means revealtheinteractiveworkingdegreeofthecomponents. WhenKequalsto0,itmeansthat that the two layers of concrete worked separately and the moment of inertia of the wall thetwolayersofconcreteworkedseparatelyandthemomentofinertiaofthewallshould should be the summation of two layers of concrete. When K equals to 1, it means that the bethesummationoftwolayersofconcrete. WhenKequalsto1,itmeansthatthetwolayers two layers of concrete works together, the moment of inertia of the wall should be calcu- of concrete works together, the moment of inertia of the wall should be calculated based lated based on the whole section. Currently, there are two methods assessing the on the whole section. Currently, there are two methods assessing the composite action of sandwich walls. One is related to the level of stiffness [ 25], and the other is related to the bearingcapacity[26],both of whichare expressed as: I − I 0 nc K = (1) I − I c nc F − F 0 nc K = (2) F − F 0 nc in which K and K denote the degree of composite action related to stiffness and bearing capacity,respectively. I andI representthemomentofinertiaofthesandwichwall,with nc c the degrees of composite action being 0 and 1, respectively. I is the moment of inertia of thesandwichwallwhenreachingthecrackingload. F andF representtheultimatebear‑ nc c ing capacity of the sandwich wall with composite action degrees of 0 and 1, respectively. F representstheultimate bearingcapacity of the sandwich wall. 0 Appl. Sci. 2023, 13, x FOR PEER REVIEW 15 of 18 composite action of sandwich walls. One is related to the level of stiffness [25], and the other is related to the bearing capacity [26], both of which are expressed as: I I 0 nc K = (1) I I c nc F F 0 nc K  (2) F F 0 nc in which Ks and Kb denote the degree of composite action related to stiffness and bearing capacity, respectively. Inc and Ic represent the moment of inertia of the sandwich wall, with the degrees of composite action being 0 and 1, respectively. I0 is the moment of inertia of Appl. Sci.2023,13, 1229 15 of17 the sandwich wall when reaching the cracking load. Fnc and Fc represent the ultimate bear- ing capacity of the sandwich wall with composite action degrees of 0 and 1, respectively. F0 represents the ultimate bearing capacity of the sandwich wall. Generally, the degree of composite action for the sandwich wall gradually decreased Generally,thedegreeofcompositeactionforthesandwichwallgraduallydecreased with the increase in load. Figure 16 presents the strain distribution of the sandwich wall with the increase in load. Figure 16 presents the strain distribution of the sandwich wall with K values of 0 and 1. with K valuesof0and 1. (a) (b) Figure 16. Strain distributions of the sandwich wall. (a) K = 0. (b) K = 1. Figure16. Strain distributionsof the sandwich wall. (a)K = 0. (b) K = 1. For the sandwich wall before cracking, the moment of inertia could be expressed as For the sandwich wall before cracking, the moment of inertia could be expressed [27]: as[27]: A = bh + (n − 1)A (3) 0 s (3) A bh  (n 1)A 0 s 0.5bh + (n − 1)A h s 0 x = (4) bh + (n − 1)A 2 s 0.5bh  n  1 Ah   s 0 [ ] x  (4) 0 3 2 I = x + (h − x ) + (n − 1)A (h − x ) (5) 0 0 s 0 0 bh  n 01 A   in which A is the area of reinforcements under tension, and b and h represent the width b 3 2 andthicknessofthe sandwichwall.   I  x  h x  n  1 A h x (5)       0 0 0 s 0 0 Aftercracking,the moment ofinertia could be expressed as:   2 2 in which As is the area of reinforcements under tension, and b and hh represent the width x = ( n µ + 2nµ − nµ)h (6) cr 0 and thickness of the sandwich wall. After cracking, the moment of i1 nertia could be expressed as: I = bx + nA (h − x ) (7) cr cr s 0 cr 2 2 (6) x  ( n m  2nm nm )h inwhich µ = A /bh and n = E /E . cr s 0 s 0 0 Theultimatebearingcapacitycouldbeeasilycalculatedaccordingtothestraindistri‑ bution. Tables3and4listthedegreeofcompositeactionfortestspecimensbasedontheir 1 2 (7) I  bx nA h x strengthandstiffness,respectively.   cr cr s 0 cr Table3. Degreeof composite action related to strength. m  A /bh s 0 in which and n  E /E . s 0 Specimen F (kN) F (kN) F (kN) k nc c exp WP 37.60 289.98 121 33.05% WN 37.60 289.98 131 37.01% WPO 33.40 269.68 138 44.26% WNO 33.40 269.68 141 45.53% Table4. Degreeof composite action related to stiffness. 4 4 4 Specimen I (mm ) I (mm ) I (mm ) k nc c exp 6 7 7 WP 1.73 × 10 10.37 × 10 6.15 × 10 58.65% 6 7 7 WN 59.61% 1.73 × 10 10.37 × 10 6.25 × 10 6 7 7 WPO 1.56 × 10 9.72 × 10 6.37 × 10 65.03% 6 7 7 WNO 1.56 × 10 9.72 × 10 6.54 × 10 66.83% It could be observed that the degree of composite action of the test specimens based on their bearing capacities ranged from 33.05% to 45.53%, and, based on the moment of inertia, varied from 58.65% to 66.83%. It could also be seen that the degree of composite actionofthesandwichwallsundernegativedirectionwasslightlyhigherthanthatunder Appl. Sci.2023,13, 1229 16 of17 positivedirection,duetothefactthatthesteelbeamprovidedadditionalconstraintsunder negative bending conditions. The composite action degree of the sandwich wall without an opening was also slightly lower than that with an opening, owing to the fact that the two area layers of concrete decreased, whereas the connection force of the steel bar truss remainedconstant. Therefore,itisadvisedtojudgethedegreeofcompositeactionrelated to the bearing capacity of the sandwich concrete wall with an XPS layer of 30%, and with thedegreeatthemomentofinertiaat50%. Oncethedegreesofcompositeactionhavebeen determined, the bearing capacity and moment of inertia could be calculated and used in designingthistypeofsandwich concrete wall. 6. Conclusions Inthispaper,theconcretesandwichwallspecimensconsideringtheactualboundary conditions under positive and negative bending were analyzed using experimental and analyticalmethods. The following conclusionscan be drawn: 1. Typicalfailuremodesofthesandwichwallspecimensobservedfromthetestwerethe cracking of concrete, fracture of XPS and bending deformation. In comparison, the specimens under positive and negative bending conditions exhibited similar failure modes, and the deflection under positive bending conditions was larger than that under negative bending conditions, owing to the fact that the constraints provided bythesteelbeamincreasedthestiffnessofthewallundernegativedirectiontosome extent. 2. The FEM considering the concrete damaged model was developed and compared withtheexperimentalresults. ItdemonstratedthattheFEMcouldaccuratelycapture theload‑deflection curvesof testspecimens. 3. Through the analyses of reinforcements embedded in the wall, it could be observed that the reinforcement wire of the specimen all reached their yield stress in the area occurring maximum deflection in the test. In addition, the reinforcements near the openingdemonstratedthe stress concentration effect. 4. Thecompositeactiondegreebetweenthetwolayersofconcretewasassessedinterms of the stiffness and bearing capacity. The composite action degrees related to the stiffness and bearing capacity were 58.65–66.83% and 33.05–45.53%, respectively. It issuggestedthatthemomentofinertiaofthesandwichwallpanelcouldbecalculated bytakingthecompositeactiondegreeof50%,andthebearingcapacityoftheconcrete sandwich wallcouldbe observedby taking the composite action degree of 30%. Author Contributions: Conceptualization, D.Y. and H.W.; methodology, A.C.; software, B.W.; vali‑ dation,D.Y.;formalanalysis,A.C.;investigation,D.Y.;resources,A.C.;datacuration,B.W.;writing— originaldraftpreparation,D.Y.;writing—reviewandediting,A.C.;visualization,H.W.;supervision, A.C.;projectadministration,A.C.;fundingacquisition,A.C.Allauthorshavereadandagreedtothe publishedversionof the manuscript. Funding: ThisresearchwasfundedbytheUniversitySynergyInnovationProgramofAnhuiProvince [GXXT‑2019‑005]. InstitutionalReviewBoard Statement: Notapplicable. InformedConsent Statement: Not applicable. DataAvailabilityStatement: Dataare contained within the article. Acknowledgments: ThisworkissupportedbytheUniversitySynergyInnovationProgramofAnhui Province(GXXT‑2019‑005). Thefinancial support is highly appreciated. ConflictsofInterest: Theauthors declare no conflict of interest. Appl. Sci.2023,13, 1229 17 of17 References 1. Sahoo, S.; Panda, S.; Singh, V.K. Experimental and numerical investigation of static and free vibration responses of woven glass/epoxylaminated composite plate. Proc. Inst. Mech. Eng. PartL. J. Mater. Des. Appl. 2017,231,463–478. [CrossRef] 2. Nguyen,T.;Dung,M.;Phung,V.;Phan,H.C.;TaDuc,T.;NguyenThiCam,N.;NguyenThi,D.BendingofSymmetricSandwich FGMBeamswith Shear Connectors. Math. Probl. Eng. TheoryMethods Appl. 2021,2021Pt 30, 7596300.1–7596300.15. [CrossRef] 3. Nguyen, C.T.; Do Van, T.; Cong, P.H.; Zenkour Ashraf, M.; Doan, D.H.; Minh, P.V. Finite element modeling of the bending and vibrationbehaviorofthree‑layercompositeplateswith a crack in the core layer. 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MDPI and/or the editor(s) disclaim responsibility for any injury to people orpropertyresulting fromanyideas, methods, instructionsor products referred to in the content. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Sciences Multidisciplinary Digital Publishing Institute

Bending Performance of Concrete Sandwich Walls with Actual Boundary Conditions

Yan, Dawei; Wan, Haiying; Chen, Anying; Wang, Bing
Applied Sciences , Volume 13 (3) – Jan 17, 2023
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Abstract

applied sciences Article BendingPerformanceof Concrete Sandwich WallswithActual Boundary Conditions 1 1,2 1,2, 1 DaweiYan , Haiying Wan ,AnyingChen * andBingWang SchoolofCivilEngineering,Hefei UniversityofTechnology,Hefei230009,China AnhuiKeyLaboratoryof CivilEngineeringStructuresandMaterials,HefeiUniversityofTechnology, Hefei230009,China * Correspondence: anyingchen@hfut.edu.cn Abstract: Concrete sandwich walls are commonly used as the exterior wall panels of a structure, in which the wall suffers out‑of‑plane bending under strong wind conditions. This paper aims to investigate the bending performance of concrete sandwich walls under actual boundary conditions throughexperimental and analytical methods. In total, four concrete sandwich wallsweretested to detect the influence of openings and loading direction. Typical failure patterns were characterized anddiscussed. Theload‑displacementcurvesoffourtestspecimenswereanalyzed. Itwasindicated that the bearing capacity of the walls under negative bending conditions was higher than that un‑ der positive bending conditions, owing to the additional constraints provided by the steel beams. Straindistributionsofwallspecimenswerealsodiscussedinordertoobtainthecompositeactionof the sandwich walls between the upper and lower layers of concrete. In addition, the finite element model (FEM) was developed by ABAQUS to provide insights into the bending performance of the sandwichwalls. Throughcomparisonwiththetestresults,theFEMwasverifiedwithagoodlevelof accuracy. Subsequently,thedegreeofcompositeactionofthesandwichwallswasassessedinterms of both the moment of inertia and bearing capacity. From the experimental and numerical results, itdemonstratedthatthebearingcapacityofconcretesandwichedwallundernegativedirectionwas higher than that under positive direction owing to the constraints of steel beam. The derived com‑ posite action degree could be employed to evaluate the out‑plane bending stiffness and strength of sandwiched concrete wall. Both the experimental and analytical results in this paper are beneficial forthe designof sandwich wallsunder bending conditions. Citation: Yan,D.; Wan,H.;Chen, A.; Wang,B.BendingPerformanceof Keywords: concrete sandwich wall; bending performance; experimental program; actual boundary ConcreteSandwichWallswith constraints;composite action degree ActualBoundaryConditions. Appl. Sci. 2023, 13,1229. https://doi.org/ 10.3390/app13031229 AcademicEditor: Laurent 1. Introduction Daudeville Exteriorwallpanels are important parts of a structure, playing an indispensable role Received: 29December2022 inshelteringagainstenvironmentalconditionssuchaswindandrain. Inthemeantime,it Revised: 12 January2023 isnecessaryforthewallpaneltohavesatisfactorythermalandsoundinsulation,whichis Accepted: 13January 2023 beneficial in providing a comfortable living environment for human beings. Moreover, a Published: 17January2023 low self‑weight is also expected for sandwich walls, especially in steel structures. There‑ fore, various lightweight and thermal insulation materials have been employed for use in the sandwich layer within the wall panel. Moreover, many other structural components alsoadoptedsandwichedconfigurations[ 1–3]. However,thesematerialshavealowbear‑ Copyright: © 2023 by the authors. ing capacity. This also leads to more challenges for sandwich walls under bending condi‑ Licensee MDPI, Basel, Switzerland. tions. This article is an open access article Manyscholarshavecarriedoutexperimentalandnumericalprogramsinordertoan‑ distributed under the terms and alyzetheperformanceofconcretesandwichwallsundervariousloadingconditions[4–14]. conditions of the Creative Commons Amongthem,Xuetal.[5],RaoandPoluraju[8]andXueetal.[9]performedexperimental Attribution (CC BY) license ( https:// creativecommons.org/licenses/by/ investigations on concrete sandwich wall panels in order to analyze their seismic perfor‑ 4.0/). mance. LeeandPessiki[6]andYuetal.[11]analyzedthethermalbehavioroftheconcrete Appl. Sci.2023,13, 1229. https://doi.org/10.3390/app13031229 https://www.mdpi.com/journal/applsci Appl. Sci.2023,13, 1229 2 of17 sandwich walls based on experimental and numerical approaches. Kumar et al., [4] re‑ ported the test results of FRP‑reinforced concrete sandwich walls under concentric axial loading. Kontoleon et al., [10] discussed the vulnerability of concrete sandwich walls to fire, and Garhwal et al. [ 13] studied the corrosion performance of the concrete sandwich wall panels. In addition, Jensen et al. [7] developed an analytical model to predict the me‑ chanicalbehaviorofconcretesandwichwallsundervariousloadings. Itcanbeconcluded that sandwich wall panels have attracted the attention of many scholars, and are widely employedinengineeringpractice. Untilnow,greatattentionhasbeenpaidtothebendingperformanceofvarioussand‑ wich walls [15–22]. For example, Mercedes et al. [19] presented an experimental and nu‑ mericalinvestigationofsandwichpanelswithavegetal‑fabric‑reinforcedcementitiousma‑ trixlayer. Gallettietal.[ 20]proposedthedesignofasandwichpanelunderbendingcondi‑ tions. McCann [21] examined the behavior of recycled glass bead sandwich panels under bending conditions. Liu et al. [22] reported the bending performance of curved sandwich panels. From these studies, it was revealed that the sandwich wall panels exhibited satis‑ factory bending performance. Among these studies, the sandwich panels were subjected tobendingloadswithpinnedsupports. Thesandwichpanelsweregenerallyemployedas exteriorwallsandwereconnectedtothemainstructure,usuallybasedonangledsteelcon‑ nectors,inwhichtheactualconstraintofthesandwichwallpanelusedasanexteriorwall was significantly different from the pinned supports. Therefore, it is necessary to detect thebendingperformance ofsandwich wallsunder actual boundary conditions. Against the background, current investigations mainly explored the bending perfor‑ mance of the concrete sandwiched wall under pinned supports. Little attention has been paidonthebendingbehaviorofconcretesandwichedwallunderactualsupportconstraints. This paper attempts to discuss the out‑of‑plane bending performance of sandwich walls connected to the top and bottom beams, in which the sandwich wall analyzed in this pa‑ per did not bear vertical loads. In total, four sandwich wall specimens with extruded polystyrene foam (XPS) layers were prefabricated. Different loading directions and open‑ ingdetailsweredesignedforthevariousspecimens. Out‑of‑planebendingtestswereper‑ formed to detect the bending behavior, and typical failure modes were discussed. Load– deflection curves were also characterized in some key points. Strain distributions of the sandwich walls along their thicknesses were analyzed to reveal the composite action be‑ tweentheconcreteandXPSlayers. Subsequently,finiteelementmodeling(FEM)wasalso established, of which the accuracy of the FEM was validated by the test results. Stresses ofthesteeltrussandreinforcementwireswerealsodiscussed. Furthermore,thedegreeof compositeactionbetweenthetwolayersofconcretewasassessedintermsofthemoment of inertia and bearing capacity. The experimental and analytical performance could pro‑ videareferenceforthebendingperformanceofsandwichwallsconnectedtothesteelbeams. 2. ExperimentalProgram 2.1. TestSpecimens Two groups of wall test specimens were prefabricated with various opening configu‑ rations. Each group consisted of two test specimens with the same constructional details. Therewerefourtestspecimensoverall,namelyWP,WN,WPO,andWNO,inwhichPand Nrepresentpositiveloadingandnegativeloading,respectively;Oreferredtothetestspec‑ imenwithanopening. Thetotalheightandwidthofwallspecimensweresetat2920mm and 2760, respectively. Extruded polystyrene (XPS) was employed as the insulation ma‑ terial, and two layers of concrete were clamped on both sides of XPS, forming a concrete sandwichwallpanel,asshowninFigure1. Thethicknessofthetwolayersofconcreteand XPS was 50 mm. To avoid any cracking of the concrete sandwich walls, the thickness of the concrete layer was chosen at 50 mm, with a nominal compression strength of 30 MPa. The steel bar truss was embedded in the concrete layer, in which the reinforcements with a diameter of 6 mm were placed at average spaces of 150 mm. In addition, the steel bar Appl. Sci. 2023, 13, x FOR PEER REVIEW 3 of 18 concrete and XPS was 50 mm. To avoid any cracking of the concrete sandwich walls, the Appl. Sci.2023,13, 1229 3 of17 thickness of the concrete layer was chosen at 50 mm, with a nominal compression strength of 30 MPa. The steel bar truss was embedded in the concrete layer, in which the reinforce- ments with a diameter of 6 mm were placed at average spaces of 150 mm. In addition, the steel bar truss deck, composed of short reinforcements with diameters of 6 mm, was used trussdeck,composedofshortreinforcementswithdiametersof6mm,wasusedtofirmly to firmly connect the two layers of concrete without passing through the XPS layer. connectthetwolayersof concretewithout passing through the XPS layer. (a) XPS (b) Figure1. Details of test specimens. (a) Specimens WP and WN. (b) Specimens WPO andWNO. 150 2640 130 150 2640 130 Appl. Sci.2023,13, 1229 4 of17 In the construction process, the bottom layer of concrete was first poured with the reinforcements embedded in it. Subsequently, the XPS layer was placed on top of the bottomlayerofconcrete. Finally,the upper layerof concrete waspoured. 2.2. MaterialProperties The density and thermal conductivity of XPS used in this paper were 30 kg/m and 0.03 W/(m.k), respectively. Key reinforcements were extracted to conduct material tests in order to accurately capture the necessary properties from the material according to the tensiletestsmethodforreinforcementsshowninGB/T2975[23]. Testresultsfromtherein‑ forcementcomponentsareshowninTable1. Materialtestswerealsoperformedtoacquire theconcretematerialproperties, asdepicted in Table2. Table1. Materialpropertiesof reinforcement. Diameterof 2 2 2 f (N/mm ) f (N/mm ) E(N/mm ) Elongation (%) y u Reinforcement(mm) 6 428.7 563.5 1.98 × 10 20.6 5 407.3 452.3 2.01 × 10 18.9 8 442.9 484.6 2.11 × 10 19.5 Note: f ,f andEaretheyieldstrength, ultimatestrength,andelasticmodulusofreinforcements. y u Table2. Materialpropertiesof concrete. 2 2 Size (mm) CuringDays E (N/mm ) f (N/mm ) c cu C11 150 × 150 × 300 28 3.08 × 10 C12 150 × 150 × 300 28 2.99 × 10 C13 150 × 150 × 300 28 3.06 × 10 Mean 3.04 × 10 C21 150 × 150 × 150 28 33.25 C22 150 × 150 × 150 28 31.92 C23 150 × 150 × 150 28 32.12 Mean 32.43 Note: E andf aretheelasticmodulus andcompressivestrengthofconcrete. c cu 2.3. LoadingProcedure To investigate the performance of the wall specimens under both positive and nega‑ tivebendingconditions,specimensWPandWPOweretestedunderpositivebendingcon‑ ditions,whereasspecimensWNandWNOweresubjectedtonegativebendingconditions. Test specimens were connected to the steel beam using top and bottom angle connectors, asshowninFigure 2. Figure3displaystheschematicviewofthetestsetup. Itisrelativelydifficulttosimu‑ late uniform loads on the wall surface. A simplified loading method was employed using rigidbeamstoallocatetheconcentrateforceoneightpointsonthetopofsandwichedcon‑ crete wall, as shown in Figure 4. Moreover, the exterior sandwiched concrete wall was generally connected to the top and bottom steel beams through steel angles and bolts in engineeringpractice. Therefore,thesimilarconnectionmethodwasalsoemployedinthis paper,asshowninFigure 4a. Beforetheformalloading,preloadingwasperformedtocheckwhethertheequipment wasfunctional. Duringtheformalloading,astep‑by‑steploadingprocedurewasadopted usingahydraulicjack. Theincrementsinloadingforceofeachstepweresetat10kN.Test specimensweresubjected to the loadingof each step for about 15 min until failure. Appl. Sci. 2023, 13, x FOR PEER REVIEW 5 of 18 Appl. Sci.2023,13, 1229 5 of17 Figure2. Connectiondetails (Unit: mm). Figure 2. Connection details (Unit: mm). Figure 3 displays the schematic view of the test setup. It is relatively difficult to sim- ulate uniform loads on the wall surface. A simplified loading method was employed using rigid beams to allocate the concentrate force on eight points on the top of sandwiched concrete wall, as shown in Figure 4. Moreover, the exterior sandwiched concrete wall was generally connected to the top and bottom steel beams through steel angles and bolts in engineering practice. Therefore, the similar connection method was also employed in this paper, as shown in Figure 4a. Before the formal loading, preloading was performed to check whether the equip- ment was functional. During the formal loading, a step-by-step loading procedure was adopted using a hydraulic jack. The increments in loading force of each step were set at 10 kN. Test specimens were subjected to the loading of each step for about 15 min until failure. Appl. Sci. 202Appl. 3, 13Sci. , x F2023 OR, 13 PE, 1229 ER REVIEW 6 of17 6 of 18 Appl. Sci. 2023, 13, x FOR PEER REVIEW 6 of 18 (a) (b) (a) (b) Figure 3. Test setup. (a) Loading in a positive direction. (b) Loading in a negative direction. Figure3. Testsetup. (a)Loading in a positivedirection. (b) Loading in a negativedirection. Figure 3. Test setup. (a) Loading in a positive direction. (b) Loading in a negative direction. (a) (b) Figure 4. Layout of loading points (Unit: mm). (a) Loading schematic. (b) Loading points. (a) Figure4. Layout of loading points(Unit: mm). (a) Loading schematic. ((b b))Loading points. 3. Test Results Figure 4. Layout of loading points (Unit: mm). (a) Loading schematic. (b) Loading points. 3.1. Failure Modes 3. Test Results In total, 29 regions, each with a width of 100 mm, were divided along its span, iden- tified as #1–29. Specimens WP and WN exhibited similar failure patterns. Specimen WP 3.1. Failure Modes was set as the example used to describe the failure modes. No obvious failure modes and In total, 29 regions, each with a width of 100 mm, were divided along its span, iden- bending deformations were noticed in the specimen before cracking. When the loading tified as #1–29. Specimens WP and WN exhibited similar failure patterns. Specimen WP force reached 35 kN, cracking was initiated in section #16 and its depth was about 2/3 was t is m ee ts a ts h a tt h o ef e th xe a w map lll e th u ics ke nd e stso . W de hs ec nr tih be e l o th ad e ifn acir le uarse e d m to o d 50 e s k.N N , a on o ob thv eiro cu ra sc f ka a il p u pr ee a rm ed o des and in region #12. As the load increased to 78 kN, slight bending deformation was noticed. bending deformations were noticed in the specimen before cracking. When the loading With the further increase in load, more and more cracks were noticed and penetrated the force reached 35 kN, cracking was initiated in section #16 and its depth was about 2/3 thickness of the wall. When the load equaled 140 kN, slippage between the concrete layer times that of the wall thickness. When the load increased to 50 kN, another crack appeared in region #12. As the load increased to 78 kN, slight bending deformation was noticed. With the further increase in load, more and more cracks were noticed and penetrated the thickness of the wall. When the load equaled 140 kN, slippage between the concrete layer Appl. Sci.2023,13, 1229 7 of17 3. TestResults 3.1. FailureModes Intotal,29regions,eachwithawidthof100mm,weredividedalongitsspan,identi‑ fiedas#1–29. SpecimensWPandWNexhibitedsimilarfailurepatterns. SpecimenWPwas setastheexampleusedtodescribethefailuremodes. Noobviousfailuremodesandbend‑ ing deformations were noticed in the specimen before cracking. When the loading force Appl. Sci. 2023, 13, x FOR PEER REVIEW 7 of 18 reached35kN,crackingwasinitiatedinsection#16anditsdepthwasabout2/3timesthat ofthewallthickness. Whentheloadincreasedto50kN,anothercrackappearedinregion #12. Astheloadincreasedto78kN,slightbendingdeformationwasnoticed. Withthefur‑ ther increase in load, more and more cracks were noticed and penetrated the thickness of and XPS was detected, and the crack along the span was noticed. When the load increased thewall. Whentheloadequaled140kN,slippagebetweentheconcretelayerandXPSwas detected, and the crack along the span was noticed. When the load increased to 150 kN, to 150 kN, the width of crack along the span grew to 5 mm and the XPS fractured. After- the width of crack along the span grew to 5 mm and the XPS fractured. Afterwards, the wards, the deflection of the wall specimen increased sharply when the load reached to 160 deflection of the wall specimen increased sharply when the load reached to 160 kN and kN and the loading procedure was terminated. Typical failure modes of specimen WP are the loading procedure was terminated. Typical failure modes of specimen WP are shown shown in Figure 5. inFigure 5. (a) (b) (c) (d) Figure 5. Typical failure modes of specimen WP. (a) First crack at the load of 35 kN. (b) Bending Figure 5. Typical failure modes of specimen WP. (a) First crack at the load of 35 kN. (b) Bending deformation at the end of the test. (c) Bending deformation in the test. (d) Cracks at the bottom of deformation at the end of the test. (c) Bending deformation in the test. (d) Cracks at the bottom of the wall. thewall. Moreover, specimens WPO and WNO exhibited similar failure modes. Before the load increased to 33 kN, no test phenomena were detected among the specimens. Subse- quently, a series of cracks emerged and spread to the middle of the wall thickness. When the load increased to 52 kN, obvious bending deformation was noticed in the test speci- mens. A penetration crack along the wall thickness was observed when the load reached 100 kN. At the same time, some cracks also appeared around the openings. As the load increased to 135 kN, slippage between the XPS and concrete was also noticed. With the load further increasing to 145 kN, the deflection increased significantly, whereas the load was kept almost constant. The test also ended at this time. After the loading, it demon- strated that the sandwich wall remained almost flat, and the connection angle exhibited obvious deformation. Typical failure modes are displayed in Figure 6. Appl. Sci.2023,13, 1229 8 of17 Moreover, specimens WPO and WNO exhibited similar failure modes. Before the load increased to 33 kN, no test phenomena were detected among the specimens. Subse‑ quently, a series of cracks emerged and spread to the middle of the wall thickness. When the load increased to 52 kN, obvious bending deformation was noticed in the test speci‑ mens. A penetration crack along the wall thickness was observed when the load reached 100kN.Atthesametime,somecracksalsoappearedaroundtheopenings. Astheloadin‑ creasedto135kN,slippagebetweentheXPSandconcretewasalsonoticed. Withtheload further increasing to 145 kN, the deflection increased significantly, whereas the load was kept almost constant. The test also ended at this time. After the loading, it demonstrated Appl. Sci. 2023, 13, x FOR PEER REVIEW 8 of 18 that the sandwich wall remained almost flat, and the connection angle exhibited obvious deformation. Typical failure modesare displayedin Figure6. (a) (b) (c) (d) Figure 6. Typical failure modes of specimen WPO. (a) Cracks at a load of 50 kN. (b) Cracks around Figure 6. Typical failure modes of specimen WPO. (a) Cracks at a load of 50 kN. (b) Cracks around the opening. (c) Deformation of the connection angle. (d) Residual deformation after the loading. theopening. (c)Deformation of the connection angle. (d)Residual deformation after the loading. In conclusion, the bottom concrete first cracked under bending. Subsequently, the In conclusion, the bottom concrete first cracked under bending. Subsequently, the XPS layer also exhibited fracture and slippage between the XPS and concrete, which was XPS layer also exhibited fracture and slippage between the XPS and concrete, which was detected both under positive and negative bending conditions. However, the crack was detected both under positive and negative bending conditions. However, the crack was mainly located on the 1/3 and middle span, and did not penetrate the wall along its thick‑ mainly located on the 1/3 and middle span, and did not penetrate the wall along its thick- ness,whichindicatedthatthetwolayersofconcretecouldworktogetherthroughthesteel ness,truss which connection. indicated Inaddition, that the no twolocal layer crushing s of concre wasfound te coduring uld work the to test. gether Meanwhile, througthe h the steel additional constraints provided by the actual boundary conditions increased the stiffness truss connection. In addition, no local crushing was found during the test. Meanwhile, the andstrengthofthesandwichedwallto some extent. additional constraints provided by the actual boundary conditions increased the stiffness and strength of the sandwiched wall to some extent. 3.2. Load–Deflection Curves Several key points were selected to analyze their deflection during the test. The de- flections of the key points were measured by the LVDTs, and the load was recorded by the pressure sensor. For specimens WP and WN, the deflections of five points were rec- orded in the test. For specimens WPO and WNO, the deflections of six points were rec- orded, as can be seen in Figure 7. Appl. Sci.2023,13, 1229 9 of17 3.2. Load–DeflectionCurves Several key points were selected to analyze their deflection during the test. The de‑ flectionsofthekeypointsweremeasuredbytheLVDTs,andtheloadwasrecordedbythe Appl. Sci. 2023, 13, x FOR PEER REVIEW 9 of 18 pressure sensor. For specimens WP and WN, the deflections of five points were recorded in the test. For specimens WPO and WNO, the deflections of six points were recorded, as canbe seeninFigure 7. (a) (b) Figure 7. Selected deflection points. (a) Deflection points of specimens WP and WN. (b) Deflection Figure 7. Selected deflection points. ( a) Deflection points of specimens WP and WN. ( b) Deflection points of specimens WPO and WNO. pointsofspecimens WPO and WNO. Theload–deflectioncurvesoftestspecimensarepresentedinFigure 8. Forspecimens The load–deflection curves of test specimens are presented in Figure 8. For specimens WPandWN,similarload–deflectioncurvesweredisplayedatthefivepoints. Overall,the WP and WN, similar load–deflection curves were displayed at the five points. Overall, the resultsshowedthatthedeflectionincreasedwiththedecreaseinthedistancebetweenthe results showed that the deflection increased with the decrease in the distance between the points and middle span. In comparison, the deflection of specimen WP was larger than points and middle span. In comparison, the deflection of specimen WP was larger than that of specimen WN, owing to the fact that the steel beam provided additional vertical that of specimen WN, owing to the fact that the steel beam provided additional vertical constraints for the wall specimens when subjected to negative bending conditions. More‑ over,theload–deflectioncurveofspecimenWPreachedthefirstinflectionpointwhenthe constraints for the wall specimens when subjected to negative bending conditions. More- uniformload(q)equaledto3kN/m duetotheconcretecracking,whereasthatofspecimen over, the load–deflection curve of specimen WP reached the first inflection point when WN reached the first inflection point when q reached 9 kN/m . The load–deflection curve the uniform load (q) equaled to 3 kN/m due to the concrete cracking, whereas that of of specimen WP reached the second inflection point when q was 13 kN/m , and that of specimen WN reached the first inflection point when q reached 9 kN/m . The load–deflec- specimenreachedthesecondinflectionpointwhen qwas13.5kN/m . Thisalsoindicated tion curve of specimen WP reached the second inflection point when q was 13 kN/m , and thatthecrackingload of specimenWN washigher than that of specimen WP. that of specimen reached the second inflection point when q was 13.5 kN/m . This also Figure 8 depicts the load–displacement curves of specimens WPO and WNO. It also demonstrated that the deflection of the points increased with the decrease in the distance indicated that the cracking load of specimen WN was higher than that of specimen WP. between the point and the middle span. However, the maximum deflection of specimen Figure 8 depicts the load–displacement curves of specimens WPO and WNO. It also WPOwasalmostequaltothatofspecimenWNO.Incomparison,theinflectionontheload– demonstrated that the deflection of the points increased with the decrease in the distance displacement curve of specimen WPO appeared when q equaled to 3 kN/m , indicating between the point and the middle span. However, the maximum deflection of specimen thattheconcretestartedcracking,whilethatofspecimenWNOemergedwhenqwasabout WPO was almost equal to that of specimen WNO. In comparison, the inflection on the 7.5kN/m . Afterwards,theload–displacementcurvesofspecimensWPOandWNOboth 2 2 load–displacement curve of specimen WPO appeared when q equaled to 3 kN/m , indi- exhibited their second points of inflection when q increased to 12.5 kN/m . Therefore, it couldbedeterminedthatthecrackingloadofspecimenWPOwaslowerthanthatofWNO. cating that the concrete started cracking, while that of specimen WNO emerged when q It also showed that 2the deflections on S5 and S6 were significantly larger than those of S1 was about 7.5 kN/m . Afterwards, the load–displacement curves of specimens WPO and and S2. This could be explained by the fact that the center of the eight loading points was WNO both exhibited their second points of inflection when q increased to 12.5 kN/m . closertoS5andS6. Therefore, it could be determined that the cracking load of specimen WPO was lower than that of WNO. It also showed that the deflections on S5 and S6 were significantly larger than those of S1 and S2. This could be explained by the fact that the center of the eight loading points was closer to S5 and S6. Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 18 Appl. Sci.2023,13, 1229 10 of17 Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 18 10 10 S1 S1 10 10 S2 S2 S1 S1 S3 S3 S2 S2 S4 S4 S3 S3 S5 5 S5 S4 S4 S5 S5 0 20 40 60 80 100 0 20 40 60 80 D (mm) 0 20 40 60 80 10 0 D (mm) 0 20 40 60 80 (a) (b) D (mm) D (mm) 25 25 (a) (b) 25 25 20 20 20 20 S1 S2 S1 S1 S3 S2 S2 S1 S4 S3 S3 S2 S5 S4 5 S4 S3 S6 S5 S5 S4 5 S6 S6 S5 0 S6 0 20 40 60 80 0 20 40 60 80 D (mm) 0 20 40 60 80 D (mm) 0 20 40 60 80 (c D ) (mm) (d) D (mm) (c) (d) Figure 8. Load–deflection curves. (a) Specimen WP. (b) Specimen WN. (c) Specimen WPO. (d) Specimen WNO. Figure 8. Load–deflection curves. (a) Specimen WP. (b) Specimen WN. (c) Specimen WPO. (d) Figure8. Load–deflectioncurves. ( a)SpecimenWP.(b)SpecimenWN.(c)SpecimenWPO.(d)Spec‑ Specimen WNO. imenWNO. 3.3. Strain Responses 3.3. Strain Responses Figure 9 shows the positions of strain gauges distributed throughout specimen WP 3.3. StrainResponses in the 1/4 and middle span. As can be seen in Figure 10, the strain remained almost at zero Figure 9 shows the positions of strain gauges distributed throughout specimen WP Figure 9 shows the positions of strain gauges distributed throughout specimen WP before q reached 12 kN/m . Afterwards, strains #5 and #6 decreased significantly, indicat- in the 1/4 and middle span. As can be seen in Figure 10, the strain remained almost at zero in the 1/4 and middle span. As can be seen in Figure 10, the strain remained almost at ing that the upper concrete 2was in compression, whereas strains #1 and #2 of the lower before q reached 12 kN/m . Afterwards, strains #5 and #6 decreased significantly, indicat- zero before q reached 12 kN/m . Afterwards, strains #5 and #6 decreased significantly, concrete layer remained at zero due to the fact that the concrete in the middle span exhib- ing that the upper concrete was in compression, whereas strains #1 and #2 of the lower indicating that the upper concrete was in compression, whereas strains #1 and #2 of the ited serious cracking and the strain gauges were broken. Similarly, strains #11 and #12 concrete layer remained at zero due to the fact that the concrete in the middle span exhib- lower concrete layer remained at zero due to the fact that the concrete in the middle span also decreased, whereas strains #7 and #8 increased sharply when q was higher than 12 ited serious cracking and the strain gauges were broken. Similarly, strains #11 and #12 exhibited serious cracking and the strain gauges were broken. Similarly, strains #11 and kN/m . This revealed that the upper and lower layers of concrete were in states of com- also decreased, whereas strains #7 and #8 increased sharply when q was higher than 12 #12 also decreased, whereas strains #7 and #8 increased sharply when q was higher than pression 2 and tension, respectively. kN/m . This revealed that the upper and lower layers of concrete were in states of com- 12 kN/m . This revealed that the upper and lower layers of concrete were in states of pression and tension, respectively. compressionandtension,respectively. Figure9. Strain distributions. p (kN/m ) p (kN/m ) p (kN/m ) p (kN/m ) p (kN/m ) p (kN/m ) p (kN/m ) p (kN/m ) Appl. Sci. 2023, 13, x FOR PEER REVIEW 11 of 18 Figure 9. Strain distributions. -200 Appl. Sci. 2023, 13, x FOR PEER REVIEW 11 of 18 -400 0 -200 Appl. Sci.2023,13, 1229 11 of17 -600 5 -400 Figure 9. Strain distributions. -600 -800 04 8 12 16 04 8 12 16 p(kN/m ) p(kN/m ) (a) (b) Figure 10. Strains of specimen WP. (a) Middle span. (b) 1/4 span. -200 4. Finite Element Modeling To gain insights into the stress state and work mechanism of the sandwich wall under -400 0 bending conditions, the numerical model was developed by ABAQUS software in this -200 -600 5 section, which was also beneficial for proposing design equation of the sandwiched wall. -400 -600 -800 4.1. Material Models 0 4 8 12 16 0 4 8 12 16 p(kN/m ) For the reinforcements embedded in the concrete slabp(k , N a b /m ili )near model was generally employed to simulate the behavior of reinforcements, as shown in Figure 11a. (a) (b) For the concrete in the sandwich wall, the stress–strain relationship demonstrated in Figure 10. Strains of specimen WP. (a) Middle span. (b) 1/4 span. Figure10. Strains of specimen WP.(a)Middle span. (b)1/4 span. GB 50010 [24] was adopted, and the concrete damage plastic model in ABAQUS was also employed to simulate the behavior of concrete, which is shown in Figure 11b. In the con- 4. Finite Element Modeling 4. FiniteElementModeling crete damaged plasticity model, the dilation angle, eccentricity, stress ratio between the To To gain gain iinsights nsights iinto nto tthe he sstress tress ststate ate anand d wwork ork mmechanism echanism ofof thethe sansandwich dwich waw ll all unun‑ der concrete under biaxial compression and uniaxial compression, K and viscosity parameter der benbending ding conconditions, ditions, thethe nunumerical merical momodel del ww asas dedev veleloped oped bby y AABAQUS BAQUS ssoftw oftwaare re in in this this were taken as 30, 0.1, 1.16, 0.667, and 0.0005, respectively. Moreover, the damage variables section,whichwasalsobeneficialfor proposing design equation of the sandwiched wall. section, which was also beneficial for proposing design equation of the sandwiched wall. in GB 50010 [24] were also assigned for the concrete in the sandwich wall. XPS exhibited relatively low stiffness and strength compared with the concrete and provided little con- 4.1. MaterialModels 4.1. Material Models tribution on the strength and stiffness of the sandwich wall, which is tentatively ignored Forthereinforcementsembeddedintheconcreteslab,abilinearmodelwasgenerally For the reinforcements embedded in the concrete slab, a bilinear model was generally in the FEM. employedtosimulate the behaviorof reinforcements, as shown in Figure 11a. employed to simulate the behavior of reinforcements, as shown in Figure 11a. For the concrete in the sandwich wall, the stress–strain relationship demonstrated in GB 50010 [24] was adopted, and the concrete damage plastic model in ABAQUS was also σ to employed to simulate the behavior of concrete, which is shown in Figure 11b. In the con- crete damaged plasticity model, the dilation angle, eccentricity, stress ratio between the f B concrete under biaxial compression and uniaxial compression, K and viscosity parameter were taken as 30, 0.1, 1.16, 0.667, and 0.0005, respectively. Moreover, the damage variables W =1 t (1-d )E t 0 in GB 50010 [24] were also assigned for the W co =0 ncrete in the sandwich wall. XPS exhibited (1-d )E relatively low stiffness and stren c gt 0h c(1 o- m d p )(1 a-rde)d E with the concrete and provided little con- c t 0 D W =1 W =0 F c tribution on the strength and stiffness of the sandwich wall, which is tentatively ignored in the FEM. ε y ε Uniaxial stress-strain curve (a) (b) to Figure 11. b Material models of concrete and reinforcements. (a) Reinforcements. (b) Concrete. Figure11. Materialmodels of concrete and reinforcements. (a) Reinforcements. (b)Concrete. 4.2. Element and Mesh Size For the concrete in the sandwich wall, the stress–strain relationship demonstrated in GB 50010 [24] was adopted, and the concrete damage plastic model in ABAQUS was also W =1 t (1-d )E t 0 W =0 employed to simulate the behavior of concrete, which is shown in Figure 11b. In the con‑ (1-d )E (1-d )(1-d )E c 0 c t 0 crete damaged plasticity model, the dilation angle, eccentricity, stress ratio between the D W =1 ε W =0 concrete under biaxial compression and uniaxial compression, K and viscosity parameter weretakenas30,0.1,1.16,0.667,and0.0005,respectively. Moreover,thedamagevariables o 0 in GB 50010 [24] were also assigned for the concrete in the sandwich wall. XPS exhibited ε y relatively low stiffness and strength compared with the concrete and provided little con‑ tribution on the strength and stiffness of the sandwich wall, which is tentatively ignored Uniaxial stress-strain curve intheFEM. (a) (b) Figure 11. Material models of concrete and reinforcements. (a) Reinforcements. (b) Concrete.  ( ) 2 (04) e (me) ε (με) Appl. Sci. 2023, 13, x FOR PEER REVIEW 12 of 18 Appl. Sci.2023,13, 1229 12 of17 4.2. Element and Mesh Size The concrete and XPS layers were all modeled using solid elements (C3D8R). Truss 4.2. ElementandMesh Size elements were assigned for reinforcement. Both XPS and reinforcements were embedded The concrete and XPS layers were all modeled using solid elements (C3D8R). Truss in the concrete without considering separation. The mesh sizes of XPS, concrete and rein- elements were assigned for reinforcement. Both XPS and reinforcements were embedded in the concrete without considering separation. The mesh sizes of XPS, concrete and re‑ forcements were set as 50 mm, whereas that of the sandbag was taken as 100 mm. A de- inforcements were set as 50 mm, whereas that of the sandbag was taken as 100 mm. A tailed FEM model is displayed in Figure 12. The static analysis step was employed to ap- detailed FEM model is displayed in Figure 12. The static analysis step was employed to ply the out-bending loads and the complete integral algorithm was used. The actual run applytheout‑bendingloadsandthecompleteintegralalgorithmwasused. Theactualrun time of each model was about 40 min. The sandwiched wall was tied to the supports and time of each model was about 40 min. The sandwiched wall was tied to the supports and all degree of freedoms of the supports were constrained. alldegreeoffreedomsof the supports wereconstrained. (a) (b) Figure 12. Finite element model of sandwich wall with reasonable mesh sizes. (a) Concrete layer. Figure 12. Finite element model of sandwich wall with reasonable mesh sizes. (a) Concrete layer. (b) Reinforcements. (b)Reinforcements. 4.3. AnalysisofFEMResults 4.3. Analysis of FEM Results The comparison results between the FEM and test results are demonstrated in The comparison results between the FEM and test results are demonstrated in Figure Figure 13. It can be concluded that the load–displacement curve of FEM coincided well 13. It can be concluded that the load–displacement curve of FEM coincided well with the with the test results, indicating that the FEM was able to reasonably predict the bending test results, indicating that the FEM was able to reasonably predict the bending perfor- performanceoftheconcrete sandwichwall. ] mance of the concrete sandwich wall. Figure14presentsthestressdistributionsofreinforcementsinspecimenWP.Itshowed thatthereinforcementwiremainlyappearedtoyieldinthemiddlespananditssupports. 20 20 The maximum stress was about 500 MPa. On the other hand, the steel truss parallel to thespandirectionexhibitedyieldingmainlyinthemiddlespan,andthemaximumstress 16 16 of the steel truss was 464 MPa, lower than that of the reinforcement wire. However, the steeltrussperpendiculartothespandirectionshowedrelativelylowerstressthanthetruss which was parallel to the span direction. Figure 15 displays the stress distributions of re‑ 12 12 inforcements of specimen WPO. It could be noticed that only a small portion of reinforce‑ mentsinthewireachievedtheiryieldstress,mainlylocatedwiththemiddleand1/3span 8 8 near the opening. Comparatively, the reinforcements in steel truss parallel to the span Test Test direction showed higher stress levels with the maximum stress value of 439 MPa mainly 4 4 FEM FEM distributed on the web reinforcements. Moreover, the strengthen reinforcements around the opening showed relatively high stress levels, indicating that the stress concentration 0 20 40 appeared 60 around 80 the 10opening. 0 It could be concluded that the reinforcements in the con‑ 0 20 40 60 80 100 D (mcrete m) sandwich walls did not all reach their yield strength, indicating D (mm) that the two layers of concrete could not work together fully in the concrete sandwich wall. Therefore, it is (a) (b) necessary to discuss the composite action between the two layers of the concrete for the concretesandwichwall. p ( kN/m ) p ( kN/m ) Appl. Sci. 2023, 13, x FOR PEER REVIEW 12 of 18 4.2. Element and Mesh Size The concrete and XPS layers were all modeled using solid elements (C3D8R). Truss elements were assigned for reinforcement. Both XPS and reinforcements were embedded in the concrete without considering separation. The mesh sizes of XPS, concrete and rein- forcements were set as 50 mm, whereas that of the sandbag was taken as 100 mm. A de- tailed FEM model is displayed in Figure 12. The static analysis step was employed to ap- ply the out-bending loads and the complete integral algorithm was used. The actual run time of each model was about 40 min. The sandwiched wall was tied to the supports and all degree of freedoms of the supports were constrained. (a) (b) Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 18 Figure 12. Finite element model of sandwich wall with reasonable mesh sizes. (a) Concrete layer. (b) Reinforcements. 4.3. Analysis of FEM Results The comparison results between the FEM and test results are demonstrated in Figure 20 20 13. It can be concluded that the load–displacement curve of FEM coincided well with the Appl. Sci.2023,13, 1229 13 of17 test results, indicating that the FEM was able to reasonably predict the bending perfor- 16 16 mance of the concrete sandwich wall. 12 12 20 20 8 8 Test Test FEM FEM 4 4 0 12 12 0 20 40 60 80 100 0 20 40 60 80 100 D (mm) D (mm) 8 8 (c) (d) Test Test Figure 13. Comparison between FEM and test. (a) WP. (b) WN. (c) WPO. (d) WPN. FEM FEM Figure 14 presents the stress distributions of reinforcements in specimen WP. It Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 18 0 0 0 20 40 60 80 100 0 20 40 60 80 100 showed that the reinforcement wire mainly appeared to yield in the middle span and its D (mm) D (mm) supports. The maximum stress was about 500 MPa. On the other hand, the steel truss (a) (b) parallel to the span direction exhibited yielding mainly in the middle span, and the max- 24 24 imum stress of the steel truss was 464 MPa, lower than that of the reinforcement wire. However, the steel truss perpendicular to the span direction showed relatively lower stress than the truss which was parallel to the span direction. Figure 15 displays the stress 16 16 distributions of reinforcements of specimen WPO. It could be noticed that only a small portion of reinforcements in the wire achieved their yield stress, mainly located with the 12 12 middle and 1/3 span near the opening. Comparatively, the reinforcements in steel truss parallel to the span direction showed higher stress levels with the maximum stress value 8 8 Test Test of 439 MPa mainly distributed on the web reinforcements. Moreover, the strengthen rein- FEM FEM forcements around the opening showed r 4elatively high stress levels, indicating that the stress concentration appeared around the opening. It could be concluded that the rein- 0 0 forcements in the concrete sandwich walls did not all reach their yield strength, indicating 0 20 40 60 80 100 0 20 40 60 80 100 D (mm) that the two layers of concrete could not work together fully D i (n m m th ) e concrete sandwich (w c) all. Therefore, it is necessary to discuss the composite action (d b )e tween the two layers of the concrete for the concrete sandwich wall. Figure 13. Comparison between FEM and test. (a) WP. (b) WN. (c) WPO. (d) WPN. Figure13. Comparison betweenFEMand test. (a)WP. (b) WN. (c) WPO. (d)WPN. Figure 14 presents the stress distributions of reinforcements in specimen WP. It showed that the reinforcement wire mainly appeared to yield in the middle span and its supports. The maximum stress was about 500 MPa. On the other hand, the steel truss parallel to the span direction exhibited yielding mainly in the middle span, and the max- imum stress of the steel truss was 464 MPa, lower than that of the reinforcement wire. However, the steel truss perpendicular to the span direction showed relatively lower stress than the truss which was parallel to the span direction. Figure 15 displays the stress distributions of reinforcements of specimen WPO. It could be noticed that only a small portion of reinforcements in the wire achieved their yield stress, mainly located with the Appl. Sci. 2023, 13, x FOR PEER REVIEW 14 of 18 middle and 1/3 span near the opening. Comparatively, the reinforcements in steel truss parallel to the span direction showed higher stress levels with the maximum stress value (a) (b) of 439 MPa mainly distributed on the web reinforcements. Moreover, the strengthen rein- forcements around the opening showed relatively high stress levels, indicating that the stress concentration appeared around the opening. It could be concluded that the rein- forcements in the concrete sandwich walls did not all reach their yield strength, indicating that the two layers of concrete could not work together fully in the concrete sandwich wall. Therefore, it is necessary to discuss the composite action between the two layers of the concrete for the concrete sandwich wall. (c) Figure 14. Stress distributions of specimen WP. (a) Stress of reinforcement wire. (b) Stress of steel Figure 14. Stress distributions of specimen WP. (a) Stress of reinforcement wire. (b) Stress of steel truss parallel to the span. (c) Mises stress contour plots of steel truss perpendicular to the span. trussparalleltothespan. (c)Misesstresscontourplotsofsteeltrussperpendiculartothespan. The The non-English term ‘平均’ means ‘Average’. non‑Englishterm‘ 平均’ means ‘Average’. (a) (b) (a) (b) (c) (d) Figure 15. Stress distributions of specimen WPO. (a) Stress of reinforcement wire. (b) Stress of steel truss parallel to the span. (c) Stress of steel truss perpendicular to the span. (d) Stress of strengthened steel truss. The non-English term ‘平均’ means ‘Average’. 5. Analytical Model of Composite Action The degree of composite action (K) of the sandwich wall is an important factor that can reveal the interactive working degree of the components. When K equals to 0, it means that the two layers of concrete worked separately and the moment of inertia of the wall should be the summation of two layers of concrete. When K equals to 1, it means that the two layers of concrete works together, the moment of inertia of the wall should be calcu- lated based on the whole section. Currently, there are two methods assessing the p ( kN/m ) p ( kN/m ) p ( kN/m ) p ( kN/m ) 2 p ( kN/m ) p ( kN/m ) Appl. Sci. 2023, 13, x FOR PEER REVIEW 14 of 18 (c) Figure 14. Stress distributions of specimen WP. (a) Stress of reinforcement wire. (b) Stress of steel Appl. Sci.2023,13, 1229 14 of17 truss parallel to the span. (c) Mises stress contour plots of steel truss perpendicular to the span. The non-English term ‘平均’ means ‘Average’. (a) (b) (c) (d) Figure 15. Stress distributions of specimen WPO. (a) Stress of reinforcement wire. (b) Stress of Figure15. StressdistributionsofspecimenWPO.(a)Stressofreinforcementwire. (b)Stressofsteel steel truss parallel to the span. (c) Stress of steel truss perpendicular to the span. (d) Stress of trussparalleltothespan. (c)Stressofsteeltrussperpendiculartothespan. (d)Stressofstrengthened strengthened steel truss. The non-English term ‘平均’ means ‘Average’. steeltruss. The non‑English term ‘平均’means ‘Average’. 5. Analytical Model of Composite Action 5. AnalyticalModelof CompositeAction The degree of composite action (K) of the sandwich wall is an important factor that Thedegreeofcompositeaction(K)ofthesandwichwallisanimportantfactorthatcan can reveal the interactive working degree of the components. When K equals to 0, it means revealtheinteractiveworkingdegreeofthecomponents. WhenKequalsto0,itmeansthat that the two layers of concrete worked separately and the moment of inertia of the wall thetwolayersofconcreteworkedseparatelyandthemomentofinertiaofthewallshould should be the summation of two layers of concrete. When K equals to 1, it means that the bethesummationoftwolayersofconcrete. WhenKequalsto1,itmeansthatthetwolayers two layers of concrete works together, the moment of inertia of the wall should be calcu- of concrete works together, the moment of inertia of the wall should be calculated based lated based on the whole section. Currently, there are two methods assessing the on the whole section. Currently, there are two methods assessing the composite action of sandwich walls. One is related to the level of stiffness [ 25], and the other is related to the bearingcapacity[26],both of whichare expressed as: I − I 0 nc K = (1) I − I c nc F − F 0 nc K = (2) F − F 0 nc in which K and K denote the degree of composite action related to stiffness and bearing capacity,respectively. I andI representthemomentofinertiaofthesandwichwall,with nc c the degrees of composite action being 0 and 1, respectively. I is the moment of inertia of thesandwichwallwhenreachingthecrackingload. F andF representtheultimatebear‑ nc c ing capacity of the sandwich wall with composite action degrees of 0 and 1, respectively. F representstheultimate bearingcapacity of the sandwich wall. 0 Appl. Sci. 2023, 13, x FOR PEER REVIEW 15 of 18 composite action of sandwich walls. One is related to the level of stiffness [25], and the other is related to the bearing capacity [26], both of which are expressed as: I I 0 nc K = (1) I I c nc F F 0 nc K  (2) F F 0 nc in which Ks and Kb denote the degree of composite action related to stiffness and bearing capacity, respectively. Inc and Ic represent the moment of inertia of the sandwich wall, with the degrees of composite action being 0 and 1, respectively. I0 is the moment of inertia of Appl. Sci.2023,13, 1229 15 of17 the sandwich wall when reaching the cracking load. Fnc and Fc represent the ultimate bear- ing capacity of the sandwich wall with composite action degrees of 0 and 1, respectively. F0 represents the ultimate bearing capacity of the sandwich wall. Generally, the degree of composite action for the sandwich wall gradually decreased Generally,thedegreeofcompositeactionforthesandwichwallgraduallydecreased with the increase in load. Figure 16 presents the strain distribution of the sandwich wall with the increase in load. Figure 16 presents the strain distribution of the sandwich wall with K values of 0 and 1. with K valuesof0and 1. (a) (b) Figure 16. Strain distributions of the sandwich wall. (a) K = 0. (b) K = 1. Figure16. Strain distributionsof the sandwich wall. (a)K = 0. (b) K = 1. For the sandwich wall before cracking, the moment of inertia could be expressed as For the sandwich wall before cracking, the moment of inertia could be expressed [27]: as[27]: A = bh + (n − 1)A (3) 0 s (3) A bh  (n 1)A 0 s 0.5bh + (n − 1)A h s 0 x = (4) bh + (n − 1)A 2 s 0.5bh  n  1 Ah   s 0 [ ] x  (4) 0 3 2 I = x + (h − x ) + (n − 1)A (h − x ) (5) 0 0 s 0 0 bh  n 01 A   in which A is the area of reinforcements under tension, and b and h represent the width b 3 2 andthicknessofthe sandwichwall.   I  x  h x  n  1 A h x (5)       0 0 0 s 0 0 Aftercracking,the moment ofinertia could be expressed as:   2 2 in which As is the area of reinforcements under tension, and b and hh represent the width x = ( n µ + 2nµ − nµ)h (6) cr 0 and thickness of the sandwich wall. After cracking, the moment of i1 nertia could be expressed as: I = bx + nA (h − x ) (7) cr cr s 0 cr 2 2 (6) x  ( n m  2nm nm )h inwhich µ = A /bh and n = E /E . cr s 0 s 0 0 Theultimatebearingcapacitycouldbeeasilycalculatedaccordingtothestraindistri‑ bution. Tables3and4listthedegreeofcompositeactionfortestspecimensbasedontheir 1 2 (7) I  bx nA h x strengthandstiffness,respectively.   cr cr s 0 cr Table3. Degreeof composite action related to strength. m  A /bh s 0 in which and n  E /E . s 0 Specimen F (kN) F (kN) F (kN) k nc c exp WP 37.60 289.98 121 33.05% WN 37.60 289.98 131 37.01% WPO 33.40 269.68 138 44.26% WNO 33.40 269.68 141 45.53% Table4. Degreeof composite action related to stiffness. 4 4 4 Specimen I (mm ) I (mm ) I (mm ) k nc c exp 6 7 7 WP 1.73 × 10 10.37 × 10 6.15 × 10 58.65% 6 7 7 WN 59.61% 1.73 × 10 10.37 × 10 6.25 × 10 6 7 7 WPO 1.56 × 10 9.72 × 10 6.37 × 10 65.03% 6 7 7 WNO 1.56 × 10 9.72 × 10 6.54 × 10 66.83% It could be observed that the degree of composite action of the test specimens based on their bearing capacities ranged from 33.05% to 45.53%, and, based on the moment of inertia, varied from 58.65% to 66.83%. It could also be seen that the degree of composite actionofthesandwichwallsundernegativedirectionwasslightlyhigherthanthatunder Appl. Sci.2023,13, 1229 16 of17 positivedirection,duetothefactthatthesteelbeamprovidedadditionalconstraintsunder negative bending conditions. The composite action degree of the sandwich wall without an opening was also slightly lower than that with an opening, owing to the fact that the two area layers of concrete decreased, whereas the connection force of the steel bar truss remainedconstant. Therefore,itisadvisedtojudgethedegreeofcompositeactionrelated to the bearing capacity of the sandwich concrete wall with an XPS layer of 30%, and with thedegreeatthemomentofinertiaat50%. Oncethedegreesofcompositeactionhavebeen determined, the bearing capacity and moment of inertia could be calculated and used in designingthistypeofsandwich concrete wall. 6. Conclusions Inthispaper,theconcretesandwichwallspecimensconsideringtheactualboundary conditions under positive and negative bending were analyzed using experimental and analyticalmethods. The following conclusionscan be drawn: 1. Typicalfailuremodesofthesandwichwallspecimensobservedfromthetestwerethe cracking of concrete, fracture of XPS and bending deformation. In comparison, the specimens under positive and negative bending conditions exhibited similar failure modes, and the deflection under positive bending conditions was larger than that under negative bending conditions, owing to the fact that the constraints provided bythesteelbeamincreasedthestiffnessofthewallundernegativedirectiontosome extent. 2. The FEM considering the concrete damaged model was developed and compared withtheexperimentalresults. ItdemonstratedthattheFEMcouldaccuratelycapture theload‑deflection curvesof testspecimens. 3. Through the analyses of reinforcements embedded in the wall, it could be observed that the reinforcement wire of the specimen all reached their yield stress in the area occurring maximum deflection in the test. In addition, the reinforcements near the openingdemonstratedthe stress concentration effect. 4. Thecompositeactiondegreebetweenthetwolayersofconcretewasassessedinterms of the stiffness and bearing capacity. The composite action degrees related to the stiffness and bearing capacity were 58.65–66.83% and 33.05–45.53%, respectively. It issuggestedthatthemomentofinertiaofthesandwichwallpanelcouldbecalculated bytakingthecompositeactiondegreeof50%,andthebearingcapacityoftheconcrete sandwich wallcouldbe observedby taking the composite action degree of 30%. Author Contributions: Conceptualization, D.Y. and H.W.; methodology, A.C.; software, B.W.; vali‑ dation,D.Y.;formalanalysis,A.C.;investigation,D.Y.;resources,A.C.;datacuration,B.W.;writing— originaldraftpreparation,D.Y.;writing—reviewandediting,A.C.;visualization,H.W.;supervision, A.C.;projectadministration,A.C.;fundingacquisition,A.C.Allauthorshavereadandagreedtothe publishedversionof the manuscript. Funding: ThisresearchwasfundedbytheUniversitySynergyInnovationProgramofAnhuiProvince [GXXT‑2019‑005]. InstitutionalReviewBoard Statement: Notapplicable. InformedConsent Statement: Not applicable. DataAvailabilityStatement: Dataare contained within the article. Acknowledgments: ThisworkissupportedbytheUniversitySynergyInnovationProgramofAnhui Province(GXXT‑2019‑005). Thefinancial support is highly appreciated. ConflictsofInterest: Theauthors declare no conflict of interest. Appl. Sci.2023,13, 1229 17 of17 References 1. Sahoo, S.; Panda, S.; Singh, V.K. Experimental and numerical investigation of static and free vibration responses of woven glass/epoxylaminated composite plate. Proc. Inst. Mech. Eng. PartL. J. Mater. Des. Appl. 2017,231,463–478. [CrossRef] 2. Nguyen,T.;Dung,M.;Phung,V.;Phan,H.C.;TaDuc,T.;NguyenThiCam,N.;NguyenThi,D.BendingofSymmetricSandwich FGMBeamswith Shear Connectors. Math. Probl. Eng. TheoryMethods Appl. 2021,2021Pt 30, 7596300.1–7596300.15. [CrossRef] 3. Nguyen, C.T.; Do Van, T.; Cong, P.H.; Zenkour Ashraf, M.; Doan, D.H.; Minh, P.V. Finite element modeling of the bending and vibrationbehaviorofthree‑layercompositeplateswith a crack in the core layer. 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Journal

Applied Sciences – Multidisciplinary Digital Publishing Institute

Published: Jan 17, 2023

Keywords: concrete sandwich wall; bending performance; experimental program; actual boundary constraints; composite action degree

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