Analysis of the Stiffness Ratio at the Interim Layer of Frame-Supported Multi-Ribbed Lightweight Walls Under Low-Reversed Cyclic Loading

applied sciences

Article Analysis of the Stiffness Ratio at the Interim Layer of Frame-Supported Multi-Ribbed Lightweight Walls under Low-Reversed Cyclic Loading

Suizi Jia 1,*, Wanlin Cao 1, Yuchen Zhang 2 and Quan Yuan 3

Received: 23 October 2015; Accepted: 11 January 2016; Published: 15 January 2016 Academic Editor: César Vasques

1 College of Architecture and Civil Engineering, Beijing University of Technology, No.100, Pingleyuan, Chaoyang District, Beijing 100124, China; [email protected] 2 Academy of Railway Sciences, Scientiﬁc & Technological Information Research Institute, No. 2 Daliushu Road, Haidian District, Beijing 100081, China; [email protected] 3 School of Civil Engineering, Beijing Jiaotong University, No.3 Shangyuancun, Haidian District, Beijing 100044, China; [email protected] * Correspondence: [email protected]; Tel.: +86-10-6739-6617; Fax: +86-10-6739-6617

Abstract: By conducting low-reversed cyclic loading tests, this paper explores the load-bearing performance of frame-supported multi-ribbed lightweight wall structures. The finite-element software (OpenSees) is used to simulate the process with the shear wall width (at the frame-supported layer) and different hole opening approaches (for multi-ribbed lightweight walls) as variables. The conclusions reflect the influence of the stiffness ratio at the interim layer on the load-bearing performance of the structure. On this basis, the paper identifies the preferable numerical range for engineering design, which provides a solid foundation for the theoretical advancement of the structures in this study.

Keywords: frame-supported structure; multi-ribbed lightweight wall; OpenSees; stiffness ratio; interim layer

1. Introduction The wide application of frame-supported structures serves the increasing demand for modern buildings that are designed and constructed with multiple functions for various purposes. However, the uneven distribution of stiffness and mass in the vertical direction may induce a sudden change in structural stiffness under seismic influence, which undermines the overall structure [1–3]. Framework-masonry structures (FM structures), with a design scheme where the upper part is relatively stronger than the lower part, have poor seismic resistance based on their behavior in previous earthquakes [4]. The Olive View Medical Center in San Fernando, for instance, suffered from crashed ground columns and buckled reinforced steel in the main building [5,6]. Another example occurred in Skopje, Yugoslavia. A five-storey building on October Street (literally translated) was heavily damaged at the ground floor which consisted of stores with no partition wall, while the upper floors—primarily partitioned residences—were only slightly damaged. [7]. These findings were confirmed in China, where FM structure stands were some of the most extensively damaged structures in both Wenchuan and Yushu earthquakes (in Sichuan and Qinghai Province, respectively) [8]. Comparatively, the masonry structure of framework seismic walls presents a good choice in terms of city reconstruction and business decentralization. The China Academy of Building Research conducted an experimental study on the seismic performance of a 1/2-scale-seven-floor brick house with a seismic wall framework. The conclusions indicate that, as the weak layer emerges, elastic-plastic deformation takes place in

Appl. Sci. 2016, 6, 21; doi:10.3390/app6010021 www.mdpi.com/journal/applsci Appl. Sci. 2016, 6, 21 2 of 19 a collective manner, which means that severe damage is likely to occur. Therefore, the study suggests that the stiffness ratio at the interim layer falls somewhere between 1.2 and 2.0 [9]. Xi’an University of Architecture and Technology conducted quasi-dynamic response tests to using 1/2-scale models, where a two-layer framework seismic wall supports a four-layer brick house. The results show the peak values for the seismic shear force and for the moment occur at the interim layer, which remains vulnerable to external force. Therefore, to enhance the lateral stiffness and load-bearing performance in the horizontal direction, structural columns are added to the middle of the interim layer for further support. In the meantime, the stiffness ratio between the third layer and the second layer is between 1.2 and 1.4, whereas that between the second and the first should not exceed 1.4 [4]. The frame-work shear wall structure delivers strong performance in seismic resistance. Although a ground shear wall with increased wall thickness can cope with any sudden change in stiffness at the interim layer, its overuse undermines the layout design and the daily function of the building. At the same time, frame-supported columns tend to be underused as they bear only limited shear force during earthquakes [10]. Both Japan and South Korea have placed frame-supported shear wall structures under quasi-dynamic and vibration tests. The majority of the structural damage is noticed in the frame, whereas the upper structure remains in the elastic stage. In other words, the overall specimen is unevenly damaged [11,12]. The Southeast University of China (Jiangsu Province) scaled a reinforced framework structure with the bottom layer of a large bay supported by specially shaped columns to 1/6 of the prototype and performed seismic simulations on a vibration test rig. The damage could be categorized as structural damage of the beam hinge. As the plastic hinge at the beam end develops, the wall corners crack; therefore, stronger corner design is highly recommended [13]. Several researchers and scholars have studied the seismic design of frame-supported structures, which tend to display an uneven distribution of stiffness and mass. The results show that the inter-layer displacement and ductility increase with decreasing stiffness and strength of the first layer. Therefore, the use of a multiple-defense design in the upper structure is recommended to mitigate the accumulation of damage and withstand collapse in an aftershock [14,15]. On this basis, this paper suggests multi-ribbed lightweight wall for the construction of buildings with large spaces at the bottom, which helps to create a new structural system—frame-supported multi-ribbed lightweight wall structure (FSMRL wall structure). The paper presents low-reversed cyclic loading tests of differential-span FSMRL wall structures with their scales reduced by half of the prototype. The finite-element software (OpenSees) [16] is used to simulate the load-bearing performance of the specimens with variable stiffness ratios at the interim layer. Thanks to this application, the paper identifies the numerical range of the stiffness ratio for engineering design.

2. Multi-Ribbed Lightweight Wall Structure The multi-ribbed lightweight wall structure (or MRL wall structure), with a layer-by-layer embedded design, is composed of cast-in-situ concealed frames, layer slabs and prefabricated MRL walls [17,18], which can be further divided into environmentally friendly embedded lightweight blocks and densely arranged ribbed beams and ribbed columns. The frame, constructed of reinforced concrete edge columns, connected columns and concealed beams, restrains the displacement of the wall. These parts make up the major load-bearing members of the structure, namely the MRL wall (see Figure1). Compared with structures of similar functions, MRL wall structures offer the following advantages:

(1) Outstanding structural integrity and working performance

The manufacturing technique promises consistent deformation between the MRL wall, concealed frame and layer slab, which then work together to bear the exerted force. From an overall perspective, the MRL wall can be viewed as a composite material structure, with the lightweight blocks acting as the matrix and the ribbed grid as the reinforced ﬁber. With smaller mechanical difference between the grid and blocks, the structure works as a new type of load-bearing member, standing out for its Appl. Sci. 2016, 6, 21 3 of 19

strengthAppl. andSci. 2016 overall, 6, 21 integrity. It needs to be pointed out that the blocks and the ribbed grid—mutually restraint and coordinately functioned—contribute to greater stiffness, which means block cracking only occursblock within cracking a certain only occurs range within under a cyclic certain loading. range un Theder cyclic cracks loading. then tend The cracks to close then up tend under to close reversed loading,up under in other reversed words loading, the blocks—though in other words the cracked—still blocks—though bear cracked—still the exerted bear force the and exerted participate force in and participate in lateral force resistance [19]. lateral force resistance [19].

(a) (b)

Figure 1. Sketch map of a multi-ribbed lightweight wall structure; (a) Multi-ribbed lightweight wall Figure 1. Sketch map of a multi-ribbed lightweight wall structure; (a) Multi-ribbed lightweight wall structure; (b) Multi-ribbed lightweight wall. structure; (b) Multi-ribbed lightweight wall. (2) Multiple seismic lines of defense (2) MultipleFrom seismicthe perspective lines of of defense the applied material, the embedded block is composed of a brittle material with strong stiffness, whereas the multi-ribbed grid and the concealed frame use reinforced From the perspective of the applied material, the embedded block is composed of a brittle material concrete, which has high ductility. Overall, the structure has average stiffness, with a load-bearing with strongperformance stiffness, greater whereas than the the framework multi-ribbed structure, grid and butthe less concealed than the shear frame wall use structure. reinforced Under concrete, whichhorizontal has high loading, ductility. the Overall, lightweight the structureblocks with has lo averagew elastic stiffness,moduli are with the afi load-bearingrst to crack. However, performance greaterdue than to the the restraint framework of the structure,ribbed grid, but the lessseismic than en theergy shear is dissipated wall structure. through friction Under between horizontal blocks, loading, the lightweightforming the blocksfirst seismic with line low of elastic defense. moduli Thanks are to the the criss-crossed first to crack. ribbed However, beams and due ribbed to the columns, restraint of the ribbedbetter plane grid, theinteraction seismic (i.e., energy densely is dissipatedarranged ribbed through beams friction and ribbed between columns blocks, form forming cross grids, the first seismicand line increase of defense. the in-plane Thanks stability to the of criss-crossedthe MRL wall)—together ribbed beams with and the ribbedsupport columns,of the lightweight better plane interactionblocks—results (i.e., densely in stronger arranged structural ribbed beamsstiffness and in ribbedcomparison columns with form that crossof the grids, concealed and increaseframe. the Therefore, the structure is prone to collapse before its counterpart does, forming the second seismic in-plane stability of the MRL wall)—together with the support of the lightweight blocks—results in line of defense. In the final stage of force bearing, the cooperation between the concealed frame and stronger structural stiffness in comparison with that of the concealed frame. Therefore, the structure the ribbed columns withstands the external force applied, preventing the occurrence of structural is pronecollapse to collapse in major beforeseismic itsincidence—hence counterpart does, the third forming seismic the line second of defense. seismic line of defense. In the final stagePrevious of force studies bearing, and the analysis cooperation show that between the failure the concealed process-the frame graded and process the ribbed of energy columns withstandsconsumption—for the external instance—starts force applied, with preventing the lightweight the occurrence blocks, then of structural moves to collapse the grids in and major finally seismic incidence—henceto the concealed the frame, third all seismic of which line make of defense. up the multiple seismic lines of defense [20]. Previous(3) Easy adjustment studies and of structural analysis stiffness show that the failure process-the graded process of energy consumption—for instance—starts with the lightweight blocks, then moves to the grids and finally to The adjustment of the stiffness in lateral force resistance can be achieved by changing the block the concealed frame, all of which make up the multiple seismic lines of defense [20]. material, the grid layout and the dimensions of the ribbed beams and ribbed columns. As the MRL (3) wallEasy structure adjustment can easily of structural cooperate stiffness with other structural system(s), it promises easy adjustment of stiffness distribution and force-bearing capacity [21]. The(4) adjustmentLow dead weight, of the limited stiffness construction in lateral duration, force resistance and tangible can bebenefits achieved on the by energetic, changing social, the block material,environmental, the grid layout and and economic the dimensions fronts of the ribbed beams and ribbed columns. As the MRL wall structure can easily cooperate with other structural system(s), it promises easy adjustment of The dead weight of the MRL wall structure is approximately 7.2 kN/m2, equivalents to 65% of stiffness distribution and force-bearing capacity [21]. that of a brick masonry structure, 68% of that of a frame structure, and 72% of that of a shear-wall structure. Engineering practice has found that the MRL wall structure cuts the civil cost by 4%–6% (4) Low dead weight, limited construction duration, and tangible benefits on the energetic, social, compared with masonry concrete structures, 10%–12% compared with framework structures, and environmental, and economic fronts more than 15% compared with shear wall structures. At the same time, the thinner wall offers a The6%–8% dead increase weight in ofusable the MRLfloorage. wall These structure figures is co approximatelynfirm that this new 7.2 kN/mstructural2, equivalents system has great to 65% of potential for industrialization [22]. that of a brick masonry structure, 68% of that of a frame structure, and 72% of that of a shear-wall structure. Engineering practice has found that the MRL wall structure cuts the civil cost by 4%–6% 3/19 compared with masonry concrete structures, 10%–12% compared with framework structures, and Appl. Sci. 2016, 6, 21 4 of 19 more than 15% compared with shear wall structures. At the same time, the thinner wall offers a 6%–8% increase in usable floorage. These figures confirm that this new structural system has great potential for industrialization [22]. Appl. Sci. 2016, 6, 21 (5) Residence pilot projects (5) Appl.Residence Sci. 2016, 6 ,pilot 21 projects As 1,000,000 m2 residence pilot projects have been constructed in Hebei, Shanxi, Henan and As 1,000,000 m2 residence pilot projects have been constructed in Hebei, Shanxi, Henan and many other(5) Residence provinces pilot [23 projects], the MRL wall structure has achieved every aspect of social production many other provinces [23], the MRL wall structure has achieved every aspect of social production and practiceAs (see 1,000,000 Figure m22 ),residence and our pilot efforts projects toward have thisbeen end constructed have yielded in Hebei, both Shanxi, economic Henan andand social and practice (see Figure 2), and our efforts toward this end have yielded both economic and social results.many In reference other provinces to the [23], Code the for MRL Masonry wall structure Structure has achieved Design every (GB50003-2011), aspect of social the production Load Code for results.and practice In reference (see Figure to the 2), Code and forour Masonryefforts toward Structure this end Design have (GB50003-2011), yielded both economic the Load and Code social for Building Structures (GB50009-2012), the Code for the Design of Concrete Structures (GB50010-2010), Buildingresults. Structures In reference (GB50009-2012), to the Code for the Masonry Code forStructure the Design Design of Concrete(GB50003-2011), Structures the Load(GB50010-2010), Code for and theandBuilding Code the Code for Structures Seismicfor Seismic (GB50009-2012), Design Design of of Buildings Buildings the Code (GBfor(GB th 50011-2010),e Design of Concrete the the Ministry Ministry Structures of Housing of (GB50010-2010), Housing and andUrban Urban Development,Development,and the Code P.R. P.R. China,for China,Seismic has has Design issued issued of the Buildingsthe industrial industri (GBal 50011-2010), standard—the the DirectiveMinistry Directive of Rules Housing Rules for forMulti-Ribbedand Multi-Ribbed Urban LightweightLightweightDevelopment, Wall Wall Structures—which P.R. Structures—which China, has issued became thebecame industri effective effectiveal standard—the from from 1 June 1 June Directive 2014 2014 (JGJ/T275-2013). Rules(JGJ/T275-2013). for Multi-Ribbed Details Details about the designaboutLightweight the and design construction Wall and Structures—which construction are provided are provided became in Figure effective in 3Figure[24]. from 3 [24]. 1 June 2014 (JGJ/T275-2013). Details about the design and construction are provided in Figure 3 [24].

FigureFigure 2.FigureResidence 2. Residence2. Residence pilot pilot pilot projects projects projects using using using the the thesystem—frame-supported system—fsystem—frame-supported multi-ribbed multi-ribbed multi-ribbed lightweight lightweight lightweight wall wall wall structurestructurestructure (FSMRL (FSMRL (FSMRL for for short) for short) short) wall wall wall structure. structure. structure.

(a) (a)

(b) (b) Figure 3.FigureDetails 3. Details of the of design the design and and construction constructionof of FSMRLFSMRL wall wall structures. structures. (a) Slab (a) joints; Slab joints; (b) Structure (b) Structure Figureconstruction 3. Details process. of the design and construction of FSMRL wall structures. (a) Slab joints; (b) Structure construction process. construction process. 3. Low-Reversed Cyclic Loading Test on FSMRL Wall Structure 3. Low-Reversed Cyclic Loading Test on FSMRL Wall Structure 3. Low-ReversedIn accordance Cyclic with Loading a frame-supported Test on FSMRL brick maso Wallnry Structure structure [4], the specimen is designed to In accordancewithstandIn accordance an earthquake with with a frame-supported a frame-supported with an intensity bricklevelbrick of masonrymaso eightnry when structure structure the site [4],soil [4 the],is type the specimen specimenII. Built is on designed theories is designed to to withstandwithstand an earthquake an earthquake with with an an intensity intensity level level ofof eighteight when when the the site site soil soil is type is type II. Built II. Built on theories on theories 4/19 4/19 Appl. Sci. 2016, 6, 21 5 of 19

Appl. Sci. 2016, 6, 21 applied in the ﬁeld, the specimen is scaled to 1/2 of the prototype (see Figure4), and the preparation is performedapplied by in anchoringthe field, the the specimen rib reinforcement is scaled to 1/2 of theof the MRL prototype wall in (see the Figu upperre 4), half and to the the preparation edge columns at eachis performed end and theby headanchoring beams, the whereasrib reinforcement the wall of is the laid MRL on thewall frame-supported in the upper half beamto the (FSedge beam) usingcolumns the bed at mortar each end method and the [24 ,head25]. Tablesbeams,1 –whereas3 provide the the wall key is parameterslaid on the frame-supported of reinforcement, beam concrete (FS beam) using the bed mortar method [24,25]. Tables 1–3 provide the key parameters of and block material. The applicationt aims to unveil the inﬂuence of the stiffness ratio (at the interim reinforcement, concrete and block material. The applicationt aims to unveil the influence of the layer)stiffness on the load-bearingratio (at the interim performance layer) ofon thethe overallload-bearing structure. performance Additionally, of the the overall authors structure. conducted a heightAdditionally, cut to the the bottom authors layerconducted to avoid a height any cut premature to the bottom cracking layer to avoid of the any frame-supported premature cracking columns of (FS columns).the frame-supported columns (FS columns).

(a)

(b)

(e)

Figure 4. Structural diagram of the FSMRL wall. (a) FSMRL-1 Differential-span FSMRL wall without holes; Figure 4. Structural diagram of the FSMRL wall. (a) FSMRL-1 Differential-span FSMRL wall without (b) FSMRL-2 Differential-span FSMRL wall with holes; (c) The reinforcement for the outer frame; (d) holes; (b) FSMRL-2 Differential-span FSMRL wall with holes; (c) The reinforcement for the outer The reinforcement for the ribbed beam and ribbed column; (e) Reinforcement details of the cross section. frame; (d) The reinforcement for the ribbed beam and ribbed column; (e) Reinforcement details of the cross section. 5/19 Appl. Sci. 2016, 6, 21 6 of 19

Appl. Sci. 2016, 6, 21 Table 1. Concrete test results.

Table 1. Concrete test results. Ribbed Beam and Edge Column and Major Member Frame at the Bottom RibbedRibbed Column Beam and ConnectedEdge Column Column and Major Member Frame at the Bottom Cubic compression strength/MPa 34.13Ribbed 30.02 Column Connected 32.27 Column Cubic Elasticcompression modulus/MPa strength/MPa 3.11 34.13ˆ 10 4 2.98 30.02ˆ 104 3.09 ˆ 32.27104 Elastic modulus/MPa 3.11 × 104 2.98 × 104 3.09 × 104 Table 2. Steel bar test results. Table 2. Steel bar test results. Speciﬁcation φ6 φ8 φ12 φ14 φ20 Specification Ф6 Ф8 Ф12 Ф14 Ф20 2 Area/mmArea/mm2 28.328.3 50.350.3 113.1 113.1 153.9 153.9 314.2 314.2 Yield strength/MPa 430.55 613.06 367.42 441.76 525.33 StrengthYield strength/MPa limit/MPa 533.14430.55 644.90613.06 367.42 527.54 441.76 576.03 525.33 739.02 ElasticStrength modulus/MPa limit/MPa 2.14533.14ˆ 105 2.11644.90ˆ 105 2.03527.54ˆ 10 5 2.12576.03ˆ 10 5 739.022.08 ˆ 105 Elastic modulus/MPa 2.14 × 105 2.11 × 105 2.03 × 105 2.12 × 105 2.08 × 105 Table 3. Block test results. Table 3. Block test results. Dry Density (kN/m3) Compression Strength/MPa Tensile Strength/MPa Elastic Modulus/MPa Dry Density (kN/m3) Compression Strength/MPa Tensile Strength/MPa Elastic Modulus/MPa 7.327.32 4.15 4.15 0.42 0.42 2.252.25 ×ˆ 10103 3

See Figure5 5 for for thethe modelmodel preparationpreparation of of the the specimens. specimens.

(a)

(b)

Figure 5. The fabricationfabrication ofof the the specimens. specimens. ( a()a The) The preparation preparation of of the the MRL MRL wall; wall; (b) ( Theb) The making making of the of specimensthe specimens in stages. in stages.

The test setup shown in Figure 6 was used. The horizontal loading was exerted repetitively by The test setup shown in Figure6 was used. The horizontal loading was exerted repetitively by the vertical oil-pumped jacks starting with a fixed force of 1000 kN. The prototype—a six-layered the vertical oil-pumped jacks starting with a ﬁxed force of 1000 kN. The prototype—a six-layered structure—is put under an external force of 1000 kN on a layer basis, with the top four floors structure—is put under an external force of 1000 kN on a layer basis, with the top four ﬂoors loaded to the two layers in the bottom. With its scale reduced by half, the model bears a loading of loaded to the two layers in the bottom. With its scale reduced by half, the model bears a loading of 1000 kN—1000 × 4 × 1/4 at the vertical direction). It needs to be noted that the low-reversed cyclic 1000 kN—1000 ˆ 4 ˆ 1/4 at the vertical direction). It needs to be noted that the low-reversed cyclic loading test can be viewed as a quasi-static loading test where the loading rates can be slow. loading test can be viewed as a quasi-static loading test where the loading rates can be slow. One One single cyclic takes 10–15 min, which is then followed by a 5–10 min break to view the results. single cyclic takes 10–15 min, which is then followed by a 5–10 min break to view the results. When When the displacement sensor read a stable value, the horizontal loading was increased by a margin of 30 kN repeatedly until yielding of the steel reinforcing bars in each specimen. Then, a new type of loading pattern was applied, as the experimental parameters changed from force to displacement

6/19 Appl. Sci. 2016, 6, 21 7 of 19 the displacement sensor read a stable value, the horizontal loading was increased by a margin of Appl. Sci. 2016, 6, 21 30Appl. kN Sci. repeatedly 2016, 6, 21 until yielding of the steel reinforcing bars in each specimen. Then, a new type of loading pattern was applied, as the experimental parameters changed from force to displacement with with a fixed increase of 5 mm each time, and the operation was repeated immediately. See Figure 7 a ﬁxed increase of 5 mm each time, and the operation was repeated immediately. See Figure7 for the for the horizontal loading curves for the two specimens. horizontal loading curves for the two specimens.

Figure 6. The loading device. FigureFigure 6.6. The loading device.

Figure 7. Low-reversed cyclic loading curve. (a) The horizontal-loading curve for FSMRL-1; (b) The Figure 7.7. Low-reversedLow-reversed cycliccyclic loadingloading curve.curve. (a)) TheThe horizontal-loadinghorizontal-loading curvecurve forfor FSMRL-1;FSMRL-1; ((b)) TheThe horizontal-loading curve for FSMRL-2. horizontal-loading curvecurve forfor FSMRL-2.FSMRL-2.

The damage of the FSMRL wall structure goes through three stages, namely, the stage of preliminary development (with the force control at 0.40 to 0.55 times the yield limit—Py), the stage of preliminary development (with the force control at 0.40 to 0.55 times the yield limit—Py), the stage of rapid development (with the force control at 0.55 to 1.10 times the Py) and the stage of destruction rapid development (with the force control at 0.55 to 1.10 times the Py) and the stage of destruction (displacement control). While the previous two stages are service states, and the destruction stage is the ultimate state.

7/19 Appl. Sci. 2016, 6, 21 8 of 19

The damage of the FSMRL wall structure goes through three stages, namely, the stage of preliminary development (with the force control at 0.40 to 0.55 times the yield limit—Py), the stage of rapid development (with the force control at 0.55 to 1.10 times the Py) and the stage of destruction (displacement control). While the previous two stages are service states, and the destruction stage is the ultimate state. At the stage of preliminary development, cracks are ﬁrst found at the blocks, which then tend Appl. Sci. 2016, 6, 21 to close under reversed loading. At this point, the blocks, the ribbed beams, and the ribbed columns work cooperatively.At the stage In theof preliminary lower part development, of the structure, cracks minorare first crackingfound at the occurs blocks, at which the FS then columns tend to and the FS beams.close As under the loading reversed continues, loading. At thethis crackspoint, th ate theblocks, above-mentioned the ribbed beams, locationsand the ribbed increase columns in quantity; during thework stage cooperatively. of rapid In development the lower part thatof the follows, structure, theminor existing cracking cracks occurs extendat the FS viacolumns the frameand grids the FS beams. As the loading continues, the cracks at the above-mentioned locations increase in accompaniedquantity; by during exfoliation the stage (of of rapid the blocks) development as plastic that follows, hinges the areexisting formed cracks at extend the jointsvia the offrame the ribbed beams andgrids the accompanied ribbed columns. by exfoliation In the (of lower the blocks) structure, as plastic cracks hinges appear are formed equidistantly at the joints of on the the ribbed FS columns, and cracksbeams increase and the in ribbed quantity columns. in In the the FS lower beams. structure, As thecracks destruction appear equidistantly occurs, on the the experiment FS columns, reaches an enduringand cracks stage. increase As the in displacementquantity in the FS amplitude beams. As the increases, destruction the occurs, load-bearing the experiment capacity reaches of an the walls decrease.enduring More cracks stage. canAs the be noticeddisplacement at the amplitude block-grid increases, interface, the load-bearing and exfoliation capacity occurs of the on walls a large-scale, decrease. More cracks can be noticed at the block-grid interface, and exfoliation occurs on a large- which resultsscale, which in the results destruction in the destruction of some of blocks. some bloc Finally,ks. Finally, concrete concrete crushing crushing at at thethe FS column column bases, yieldingbases, of the yielding longitudinal of the longitudinal steels, and steels, inter-layer and inter-layer sliding sliding occur occur (See (See Figure Figure8). 8).

(a)

(b)

(c)

Figure 8.FigureThree 8. stages Three ofstages accumulative of accumulative damage. damage. (a) Preliminary(a) Preliminary development development damageddamaged stage; stage; (b ()b ) Rapid Rapid development damaged stage; (c) Destruction stage. development damaged stage; (c) Destruction stage.

8/19 Appl. Sci. 2016, 6, 21 Appl. Sci. 2016, 6, 21 9 of 19 4. Theoretical Determination of the Interim Stiffness Ratio for Frame-Supported Multi-Ribbed Lightweight Walls 4. Theoretical Determination of the Interim Stiffness Ratio for Frame-Supported Multi-Ribbed Lightweight Walls 4.1. The Regulations of Stiffness Ratio at the Interim Layer 4.1. TheThe Regulations Technical Specification of Stiffness Ratio for atthe the Concrete Interim LayerStructure of Tall Building (JGJ3-2010) regulations in the appendixThe Technical E [26], Specification where the interim for the structural Concrete Structuremember is of defined Tall Building as the (JGJ3-2010) component regulations set due to layoutin the appendixor structural E [26 change(s)], where theamong interim the upper structural and memberlower floor, is defined with some as the typical component examples set duebeing to interimlayout or beam, structural interim change(s) slab, interim among truss, the etc. upper The and located lower floor floor, of withsuch somea component typical examplesis referred being to as interim layer beam, interim slab, interim truss, etc. The located floor of such a component is referred to as interimOn layerthis basis, the paper focuses on the up-and-down stiffness ratio at the interim layer of the FSMRLOn wall this basis,structure the (hereinafter paper focuses refers on theto as up-and-down stiffness ratio stiffness at the interim ratio at layer) the interim and discusses layer of the optimalFSMRL wallnumerical structure range (hereinafter accordingly. refers The to stiffness as stiffness ratio ratio at the at interim the interim layer layer) γe can and be calculated discusses theby usingoptimal Equation numerical (1). range accordingly. The stiffness ratio at the interim layer γe can be calculated by using Equation (1). Δ H γ= 11 e Δ∆1 H1 (1) γe “ 22H (1) ∆2 H2

4.2. Stiffness Stiffness Calculation fo forr the FSMRL Wall Structure (1) ForFor MRL MRL walls, walls, the the elastic elastic stiffnes stiffnesss in in the the lateral lateral force force resistance resistance (or (or KK)) can can be calculated using EquationsEquations (2)–(5) (2)–(5) (shown (shown in in section section 6) 6) [27] [27 ] Based onon previousprevious wall-related wall-related experiments, experiments, the the equations equations incorporate incorporate bending bending deformation deformation and andstructural structural destruction destruction and takeand take into accountinto account the inﬂuence the influence of concerned of concerned variables, variables, such such as the as axial the axialcompressive compressive ratio, ratio, minor minor cracks, cracks and, the and construction the construction procedure. procedure. 0.3ημ+ (2 0.4) K = 0.3ηccNp2µN ` 0.4q K1 “1 3 μ (2) HHH3 + µH (2) 12E IGA` 12EcceqIeq G cc A eqeq E = ＋ Eq AeqAA cq q (3) Aeq “ Ac `Ec Aq (3) Ec bAh= / beqeq“ A eqeq{h (4)(4)

1 33 IeqI “= bbheqh (5)(5) eq12 eq For MRL walls with holes, the calculation of the elastic stiffness in the lateral force resistance can For MRL walls with holes, the calculation of the elastic stiffness in the lateral force resistance can be conducted using Equation (6) (see Figure9). be conducted using Equation (6) (see Figure 9). 1 K1 “= (6) K1 1 1 11`+ (6) KIII KI ` KII KΙΙΙKK Ι+ ΙΙ

Figure 9. SketchSketch of of the the stiffness calculation for MRL walls with holes.

9/19 Appl. Sci. 2016, 6, 21 10 of 19 Appl. Sci. 2016, 6, 21

Note:Note: KKII,, KKIIII andand KKIIIIII standstand for for the the piece-by-piece piece-by-piece calculation calculation results results of the specimen stiffnessstiffness inin laterallateral force force resistance, resistance, whereas whereas the layer the heightlayer Hheight represents H represents the height ofthe the height embrasure of the or windowsill.embrasure or windowsill. (2) Stiffness calculation for reinforced concrete frame and seismic wall (2) Stiffness calculation for reinforced concrete frame and seismic wall The authors introduced the D-value method which can be expressed by Equation (7). The authors introduced the D-value method which can be expressed by Equation (7). 12i K “ α 12ci (7) K2 =αc h2 c (7) 2 c h2 (3) Calculation of layer stiffness (3) Calculation of layer stiffness TheThe calculatedcalculated valuevalue forfor thethe layerlayer stiffnessstiffness isis derivedderived fromfrom thethe linearlinear superpositionsuperposition ofof thethe frameframe stiffnessstiffness andand thethe MRLMRL wallwall stiffnessstiffness (see (see Equation Equation (8)). (8)).

KK=+ K K “ K112` K2 (8)(8) Engineering practice normally reservesÿ a largeÿ space in the lower layer of the FSMRL wall Engineering practice normally reserves a large space in the lower layer of the FSMRL wall structure. However, to avoid the early occurrence of FS column failure, this paper shortens the structure. However, to avoid the early occurrence of FS column failure, this paper shortens the columns and introduces reinforcement measures. The height cut is conducted proportionally; the columns and introduces reinforcement measures. The height cut is conducted proportionally; the layer layer height—which is 2.6 m in practice—is 1.3 m in this paper. The paper makes use of the height—which is 2.6 m in practice—is 1.3 m in this paper. The paper makes use of the ﬁnite-element finite-element software OpenSees ([16]) to simulate the influence of height changes in the FS layer on software OpenSees ([16]) to simulate the inﬂuence of height changes in the FS layer on the load-bearing the load-bearing performance of the overall structure. The height of the FS layer is set at 1.3 m, 1.4 m, performance of the overall structure. The height of the FS layer is set at 1.3 m, 1.4 m, 1.5 m, 1.6 m, 1.5 m, 1.6 m, or 1.7 m to simulate an actual structure with a layer height of 2.6 m, 2.8 m, 3.0 m, 3.2 m, or or 1.7 m to simulate an actual structure with a layer height of 2.6 m, 2.8 m, 3.0 m, 3.2 m, or 3.4 m 3.4 m respectively. The model provides a comparative diagram for skeleton curves, which are respectively. The model provides a comparative diagram for skeleton curves, which are displayed as displayed as follows (see Figure 10). follows (see Figure 10).

FigureFigure 10.10. ComparativeComparative diagramdiagram forfor thethe skeletonskeleton curvescurves ofof thethe models.models.

UsingUsing EquationEquation (2)(2) toto EquationEquation (8),(8), thethe stiffnessstiffness ratioratio (at(at thethe interiminterim layer)layer) cancan bebe determineddetermined forfor specimensspecimens withwith differentdifferent frameframe heights.heights. TableTable4 4shows shows the the relation relation between between the the stiffness stiffness ratio ratio and and the the structure’sstructure’s crack,crack, yield,yield, andand peakpeak loadingloading (see(see FigureFigure 11 11).).

TableTable 4.4. StiffnessStiffness ratioratio atat thethe interiminterim layerlayer withwith differentdifferent frame frame supported supported (FS) (FS) layer layer heights. heights.

HeightHeight of FS LayerLayer (mm) (mm) 1300 1400 1500 1600 1700 Stiffness ratio at the interim layer 1.66 2.02 2.42 2.88 3.39 Stiffness ratio at the interim layer 1.66 2.02 2.42 2.88 3.39

10/19 Appl. Sci. 2016, 6, 21 11 of 19 Appl.Appl. Sci.Sci. 20162016,, 66,, 2121

FigureFigure 11.11. RelationRelation between between the the interim interim stiffnessstiffness ratioratio andand load-bearingload-bearing performance.performance.

FigureFigure 1111 showsshows thatthat thethe overalloverall curvescurvescurves forfor thethethe crack,crack,crack, yield,yield,yield, andand peakpeak loadingloadingloading decreasedecrease withwith increasingincreasing stiffness stiffness ratio. ratio. AA A closercloser closer looklook look showsshows shows thth that,at,at, comparatively,comparatively, comparatively, thethe the crack-loadingcrack-loading crack-loading curvecurve curve followsfollows follows aa amoremore more steadysteady steady variationvariation variation pattern,pattern, pattern, whereaswhereas whereas thethe the decredecre decreaseasease ofof of thethe the otherother other twotwo variablesvariables is is steeper, steeper, whichwhich indicatesindicates thatthatthat the thethe stiffness stiffnessstiffness ratio ratioratio has hashas a strongeraa strongerstronger influence influenceinfluence on theonon structuralthethe structuralstructural strength strengthstrength at the atat yielding thethe yieldingyielding stage andstagestage under andand underunder peak loading. peakpeak loading.loading. StiffnessStiffness adjustmentadjustment ofof of thethe the FSFS FS layerlayer layer andand and holehole hole openopen openinginging ofof thethe of theMRLMRL MRL wallwall wall areare performedperformed are performed inin anan effort ineffort an efforttoto obtainobtain to obtain differentdifferent different stiffnessstiffness stiffness ratios,ratios, ratios, eithereither either greategreate greaterrr thanthan than oror orsmallersmaller smaller thanthan than 1.1. 1. TheThe The authorsauthors authors againagain useuse OpenSeesOpenSees toto simulatesimulate thethe loadload bearingbearing processprocess toto identifyiidentifydentify thethe preferablepreferable stiffnessstiffness ratioratio forfor thethe interiminterim layerlayer ofof thethe FSMRLFSMRL wallwall structurestructure [[16,28].[16,28].16,28]. In In the the si simulationsimulationmulation of of the the specimen’s specimen’s physical physical performance, performance, thethe material-specializedmaterial-specializedmaterial-specialized modelmodel Concrete02ConcConcrete02rete02 (provided(provided byby OpenSeesOpenSees [[16])—which[16])—which16])—which takestakes intointo accountaccountaccount thethe concrete’sconcrete’s strengthstrength andand linearlinearlinear tensiletensiletensile emollescence—isemollescence—isemollescence—is appliedapplied inin thethe studystudystudy ofofof thethethe constitutiveconstitutiveconstitutive relationrelation ofof thethethe concreteconcreteconcrete (see(see(see FigureFiguFigurere 1212).12).). TheTheThe uniaxialuniaxialuniaxial materialmaterialmaterial modelmodelmodel (Hysteretic(Hysteretic(Hysteretic Material)Material)Material) isis chosenchosenchosen forfor thethe simulationsimulation ofof rebar,rebar, becausebecausebecause thethethe modelmodelmodel doesdoesdoes notnotnot eliminateeliminateeliminate thethethe stiffnessstiffnessstiffness degradationdegradationdegradation uponupon thethe removalremoval ofofof thethethe external externalexternal force, force,force, and andand it itit is is fullyis fullyfully able ableable to reflecttoto reflectreflect the the pinchingthe pinchingpinching effect, effect,effect, (see (see Figure(see FigureFigure 13). The13).13). beam-columnTheThe beam-columnbeam-column unit makesunitunit makesmakes use of use theuse non-linearofof thethe non-lineanon-linea fiber modelrr fiberfiber that modelmodel OpenSees thatthat OpenSeesOpenSees provides in providesprovides its simulation inin itsits (seesimulationsimulation Figure (see(see14). FigureFigure In accordance 14).14). InIn accordanceaccordance with the equivalent withwith ththee equivalentequivalent principle ofprincipleprinciple compressive ofof compressivecompressive stiffness andstiffnessstiffness bending andand stiffness,bendingbending stiffness,stiffness, the MRL thethe wall MRLMRL is analyzed wallwall isis analyzedanalyzed as a reinforced asas aa reinforcedreinforced concrete concreteconcrete plate, where plate,plate, the wherewhere ShellMITC4 thethe ShellMITC4ShellMITC4 unit (model unitunit provided(model(model providedprovided by OpenSees) byby OpenSees)OpenSees) is defined isis defined bydefined a multi-layered byby aa multi-layeredmulti-layered shell element shellshell elementelement [29,30]. [29,30].[29,30].

FigureFigure 12.12. Stress-strainStress-strain relationrelation ofof concrete.concrete.

11/1911/19 Appl.Appl.Appl. Sci. Sci.Sci. 2016 2016,, ,6 66,, ,21 2121 12 of 19

Figure 13. Skeleton curve of the hysteretic material. FigureFigure 13.13. SkeletonSkeleton curvecurve ofof thethe hysteretichysteretic materialmaterial.. .

FigureFigureFigure 14. 14.14. The The composite composite of ofof beam beambeam and andand column columncolumn element. element.element.

The test data collected were compared with the calculation results of the hysteretic curve and TheTheThe testtest datadata collected collectedcollected were werewere compared comparedcompared with withwith the thethe calculation calculationcalculation results resultsresults of of theof thethe hysteretic hysteretichysteretic curve curvecurve and andand the the skeleton curve to verify the model, refer to Figure 15 for comparison of hysteretic curves and theskeletonthe skeletonskeleton curve curvecurve to verify toto verifyverify the model, thethe model,model, refer to referrefer Figure toto Fi Fi15guregure for comparison1515 forfor comparisoncomparison of hysteretic ofof hysteretichysteretic curves and curvescurves skeleton andand skeleton curves of the calculated values and the test data for FSMRL-2. skeletoncurvesskeleton of curves curves the calculated of of the the calculated calculated values and va values thelues testand and data the the test fortest FSMRL-2.data data for for FSMRL-2. FSMRL-2.

(((aa))) (((bb))) FigureFigure 15.15. ComparativeComparative diagramsdiagrams forfor hysteretichysteretic andand skeletonskeleton curvescurves (calculated(calculated valuesvalues andand testtest data).data). FigureFigure 15.15. Comparative diagrams for hysteretic and skeleton curves (calculated values and test data). ((aa)) hysteretichysteretic curve;curve; ((bb)) skeletonskeleton curve.curve. ((aa)) hysteretichysteretic curve;curve; ((bb)) skeletonskeleton curve.curve.

FromFrom FigureFigure 15,15, itit cancan bebe concludedconcluded thatthat thethe calculatedcalculated valuesvalues ofof bothboth hysteretichysteretic curvescurves andand From Figure 15, it can be concluded that the calculated values of both hysteretic curves and skeletonskeleton curvescurves areare consistentconsistent wiwiththth thethethe correspondingcorrespondingcorresponding datadatadata collectedcollectedcollected fromfromfrom thethethe test.test.test. TheTheThe hysteretichysteretichysteretic skeleton curves are consistent with the corresponding data collected from the test. The hysteretic loops loopsloopsloops tend tendtend to toto expand expandexpand horizontally horizontallyhorizontally,, , which whichwhich indicates indicatesindicates a aa fairly fairlyfairly good goodgood capacity capacitycapacity for forfor energy energyenergy dissipation. dissipation.dissipation. Due DueDue tototo block blockblock and andand concrete concreteconcrete exfoliation exfoliationexfoliation and andand the thethe yielding yieldingyielding of ofof the thethe reinforced reinforcedreinforced steel, steel,steel, the thethe hysteretic hysteretichysteretic loops loopsloops starts startsstarts to toto

12/1912/19 Appl. Sci. 2016, 6, 21 13 of 19 tend to expand horizontally, which indicates a fairly good capacity for energy dissipation. Due to blockAppl. and Sci. concrete 2016, 6, 21 exfoliation and the yielding of the reinforced steel, the hysteretic loops starts to “pinch”Appl. in Sci. the 2016 middle, 6, 21 in the later stages of loading, whereas sliding occurs between the upper and the “pinch” in the middle in the later stages of loading, whereas sliding occurs between the upper and lower layers. The skeleton curves are consistent in terms of the trends, and the loading simulation was the“pinch” lower in layers. the middle The skeleton in the curveslater stages are consistent of loading, in whereasterms of thesliding trends, occurs and betweenthe loading the simulation upper and proven accurate from the cracking, yielding, and peaking of the damage to the specimens. wasthe lowerproven layers. accurate The from skeleton the cracking,curves are yielding consistent, and in peakingterms of ofthe the trends, damage and to the the loading specimens. simulation was proven accurate from the cracking, yielding, and peaking of the damage to the specimens. 4.3. Analysis4.3. Analysis of the of Load-Bearingthe Load-Bearing Performance Performance whenwhen the Stiffness Stiffness Ratio Ratio Exceeds Exceeds 1 1 As4.3. Table AnalysisAs Table1 shows, of 1 the shows, Load-Bearing the heightthe height increasePerformance increase of thewhenof the FS the FS layer Stiffness layer results results Ratio in Exceedsin a a stiffness stiffness 1 decrease,decrease, indicating the deteriorationthe deteriorationAs Table of the 1 shows,of load-bearing the load-bearing the height performance increase performance of the of of theFS the layer overall overall results structure. in a stiffness In In practice, practice, decrease, because because indicating a large a large spacespacethe is generally deterioration is generally reserved reservedof the load-bearing for for the the lower lower performance layer, layer, thethe layer of the height height overall tends tends structure. to tobe beconsiderable. In considerable. practice, because In this In regard, thisa large regard, shearshearspace wall wall is structures generally structures canreserved can be be introduced—when for introduced—when the lower layer, the necessary—to layer height lowertends lower tothe be the stiffness considerable. stiffness ratio ratio at In the this at interim the regard, interim layer, which helps to improve the load-bearing performance of the structure at large. Here, the paper layer,shear which wall helps structures to improve can be the introduced—when load-bearing performance necessary—to of lower the structure the stiffness at large.ratio at Here, the interim the paper takes the FSMRL wall structures with an FS layer height of 1700 mm as the study object, both sides takeslayer, the FSMRL which helps wall structuresto improve withthe load-bearing an FS layer perf heightormance of 1700 of the mm structure as the at study large. object, Here, the both paper sides of of which are enhanced by shear walls of different widths, namely 0 mm for MX-1, 200 mm for MX-2, whichtakes are enhancedthe FSMRL by wall shear structures walls ofwith different an FS layer widths, height namely of 1700 0 mmmm foras the MX-1, study 200 object, mm both for MX-2, sides 300 300of which mm for are MX-3, enhanced 400 mmby shear for MX-4, walls 500of differ mm entfor widths,MX-5, 600 namely mm 0for mm MX-6, for MX-1, 700 mm 200 for mm MX-7, for MX-2, and mm for MX-3, 400 mm for MX-4, 500 mm for MX-5, 600 mm for MX-6, 700 mm for MX-7, and 800 mm 800300 mm forfor MX-3,MX-8. 400 With mm a foruniformed MX-4, 500 thickness mm for MXof -5,100 600 mm, mm the for shearMX-6, walls 700 mm applied for MX-7, for thisand for MX-8.experiment—both800 mm With for a uniformedMX-8. horizontal With thickness a uniformedand vertical—use of 100 thickness mm, φ the 6of reinforcement shear 100 mm, walls the applied with shear an forwalls interval this applied experiment—both of 100 for mmthis horizontal(Seeexperiment—both Figure and vertical—use 16). The horizontal detailedϕ6 reinforcement and vertical—use is with shown anφ6 intervalinreinforcement Figure of 4. 100The mm withcalculated (Seean interval Figure results 16 ofof). stiffness100 The mm detailed reinforcementratio(See areFigure summarized is 16). shown The indetailed in Figure Table reinforcement5.4. The calculated is shown results in Figure of stiffness 4. The ratio calculated are summarized results of stiffness in Table 5. ratio are summarized in Table 5.

FigureFigure 16.16. The study study model. model. Figure 16. The study model. TableTable 5. Stiffness 5. Stiffness ratio ratio at at the the interim interim layer (with (with shear shear wall wall width width as variable). as variable). TableModel 5. Stiffness ratio at MX-1MX-2MX-3MX-4MX-5MX-6 the interim layer (with shear wall width as variable). MX-7 MX-8 StiffnessModel ratio Modelat the interim layer MX-1 MX-1MX-2MX-3MX-4MX-5MX-63.39 MX-2 3.26 MX-3 3.00 2.62MX-4 2.19 MX-5 1.78 MX-6 MX-71.42 MX-7 MX-81.13 MX-8 StiffnessStiffness ratio at ratio the at interim the interim layer layer 3.393.39 3.26 3.26 3.00 3.00 2.62 2.62 2.19 2.19 1.78 1.78 1.42 1.421.13 1.13

Figure 17. Force exerted on the FSMRL wall structures (with shear wall width as variable). FigureFigure 17. Force 17. Force exerted exerted on on the the FSMRL FSMRL wall wall structuresstructures (with (with shear shear wall wall width width as variable). as variable). 13/19 13/19 Appl. Sci. 2016, 6, 21 14 of 19 Appl. Sci. 2016, 6, 21

The simulation of the load-bearing performance is conducted using OpenSees. To streamline the calculation process, the authors conduct monotone loading to obtain loading-displacementloading-displacement curves, whichwhich areare displayeddisplayed inin FigureFigure 1717.. Based on the force-displacement curves, the relationrelation between the yield loading and peak loading exerted and the stiffness ratioratio atat thethe interiminterim layerlayer werewere identiﬁedidentified (See(See FigureFigure 18 18).). It can be concluded from Figure 18 that the increa increasese in the interim stiffness ratio triggers a decline inin thethe yieldyield andand peakpeak loading.loading. When the stiffness ratio exceeds 2.5, the decreasedecrease ofof thethe peakpeak loadingloading increasesincreases in in speed, speed, whereas whereas the the loading loading exerted exerted at th ate the yield yield of the of specimen the specimen drops drops dramatically dramatically after afterthe ratio the exceeds ratio exceeds 3. The 3. paper The paperrecommends recommends a not-le ass-than not-less-than 2.5 interim 2.5 interimstiffness stiffness ratio for ratio the FSMRL for the FSMRLwall structure, wall structure, because because the yield the yieldstrength strength need needss to be to bemaintained maintained at ata acertain certain level level withoutwithout compromising thethe overalloverall structural reliabilityreliability (The(The higherhigher the interim stiffness ratio grows, the faster the frame-supported-layerframe-supported-layer stiffness stiffness decreases. decreases. In theIn meantime,the meantime, the drop the of drop the yield of the loading yield inﬂuences loading theinfluences overall the structural overall reliability).structural reliability).

Figure 18. Relation between the interim stiffness ratio and load-bearing performance.

4.4. Analysis of the Load-Bearing Performance when the Stiffness Ratio FallsFalls BelowBelow 11 Due to the shear wall added for this experiment and the hole opening of a large amount at the upper structure (for (for functional functional purpos purposes),es), the the stiffness stiffness ratio ratio falls falls below below 1. Therefore, 1. Therefore, it is itpossible is possible that thatthe upper the upper structure structure may mayencounter encounter premature premature failure failure caused caused by the by extremely the extremely low lowstiffness stiffness in the in thelateral lateral force force resistance. resistance. To this To thisend, end,the paper the paper adjusts adjusts the location the location and the and dimensions the dimensions of the of hole the holeopening opening in an in attempt an attempt to obtain to obtain a host a host of ofstiffness stiffness ratios ratios below below 1. 1. The The simulation simulation is performedperformed using OpenSees, which provides the variation pattern for the inﬂuenceinfluence of the stiffness ratio on the overall structure. The FSMRL wall structures with 100 mm thick and 800 mm wide shear walls at both sides (whose reinforcement is displayed in Figure 16) use a 1700 mm FS layer height, and the upper structure varies in MX-9 to MX-16 from a frame with no blocks embedded to MRL walls of different widths (see Figure 19 and Table6). The detailed reinforcement and dimensions for the opening are shown in Figure4. The interim stiffness ratios for the different structural layouts are calculated, and the results are listed in Table7.

Table 6. FSMRL wall structure with different hole opening.

Model Number MX-9 MX-10 MX-11 MX-12 MX-13 MX-14 MX-15 MX-16 The Opening Rate

Slab 1(a) 1 2/3 2/3 2/3 1/3(b) 1/3 1/3 1/3 Slab 2 1 1 2/3 2/3 2/3 1/3 1/3 1/3 Figure 19. PartialSlab 3FSMRL wall structure 1with different 1 hole 1 opening. 3/4 (a) MX-9 3/4 (a frame 3/4 with 1/2 no blocks 1/4 at the upper structure) (b) MX-16 (blocks embedded to slab 1, 2, and 3 with different opening rates at the upper structure).

14/19 Appl. Sci. 2016, 6, 21

The simulation of the load-bearing performance is conducted using OpenSees. To streamline the calculation process, the authors conduct monotone loading to obtain loading-displacement curves, which are displayed in Figure 17. Based on the force-displacement curves, the relation between the yield loading and peak loading exerted and the stiffness ratio at the interim layer were identified (See Figure 18). It can be concluded from Figure 18 that the increase in the interim stiffness ratio triggers a decline in the yield and peak loading. When the stiffness ratio exceeds 2.5, the decrease of the peak loading increases in speed, whereas the loading exerted at the yield of the specimen drops dramatically after the ratio exceeds 3. The paper recommends a not-less-than 2.5 interim stiffness ratio for the FSMRL wall structure, because the yield strength needs to be maintained at a certain level without compromising the overall structural reliability (The higher the interim stiffness ratio grows, the faster the frame-supported-layer stiffness decreases. In the meantime, the drop of the yield loading influences the overall structural reliability).

Figure 18. Relation between the interim stiffness ratio and load-bearing performance.

4.4. Analysis of the Load-Bearing Performance when the Stiffness Ratio Falls Below 1 Due to the shear wall added for this experiment and the hole opening of a large amount at the upper structure (for functional purposes), the stiffness ratio falls below 1. Therefore, it is possible that the upper structure may encounter premature failure caused by the extremely low stiffness in the lateral force resistance. To this end, the paper adjusts the location and the dimensions of the hole opening in an attempt to obtain a host of stiffness ratios below 1. The simulation is performed usingAppl. Sci. OpenSees,2016, 6, 21 which provides the variation pattern for the influence of the stiffness ratio on15of the 19 overall structure.

Appl. Sci. 2016, 6, 21 Appl. Sci. 2016, 6, 21 The FSMRL wall structures with 100 mm thick and 800 mm wide shear walls at both sides The FSMRL wall structures with 100 mm thick and 800 mm wide shear walls at both sides (whose reinforcement is displayed in Figure 16) use a 1700 mm FS layer height, and the upper (whose reinforcement is displayed in Figure 16) use a 1700 mm FS layer height, and the upper structure varies in MX-9 to MX-16 from a frame with no blocks embedded to MRL walls of different structure varies in MX-9 to MX-16 from a frame with no blocks embedded to MRL walls of different widths (see Figure 19 and Table 6). The detailed reinforcement and dimensions for the opening are widths (see Figure 19 and Table 6). The detailed reinforcement and dimensions for the opening are shown in Figure 4. The interim stiffness ratios for the different structural layouts are calculated, and shown in Figure 4. The interim stiffness ratios for the different structural layouts are calculated, and the results are listed in Table 7. the results are listed in Table 7. Table 6. FSMRL wall structure with different hole opening. Table(a )6. FSMRL wall structure with different hole opening.(b) Model Number Model Number MX-9 MX-10 MX-11 MX-12 MX-13 MX-14 MX-15 MX-16 FigureFigure 19. 19.The Partial PartialOpening FSMRL FSMRL Rate wall wall structure structureMX-9 with withMX-10 different different MX-11 hole MX-12 opening. MX-13 (a) (a) MX-9 MX-9 MX-14 (a (a frame frameMX-15 with withMX-16 no no blocks blocks The Opening Rate atat the the upper upper structure) structure)Slab 1 (b) (b) MX-16 MX-16 (blocks 1(blocks embedded embedded 2/3 2/3 to to slab slab2/3 1, 1, 2, 2, and and1/3 3 3 with with1/3 different different 1/3 opening opening1/3 rates rates at at Slab 1 1 2/3 2/3 2/3 1/3 1/3 1/3 1/3 the upper structure).Slab 2 1 1 2/3 2/3 2/3 1/3 1/3 1/3 the upper structure).SlabSlab 23 1 1 1 1 2/3 1 2/3 3/4 2/33/4 1/33/4 1/31/2 1/31/4 Slab 3 1 1 1 3/4 3/4 3/4 1/2 1/4

Table 7.7. Stiffness ratioratio atat thethe interiminterim14/19 layerlayer (with(with holehole asas variable).variable). Table 7. Stiffness ratio at the interim layer (with hole as variable). Model Number MX-9 MX-10 MX-11 MX-12 MX-13 MX-14 MX-15 MX-16 ModelModel Number Number MX-9MX-9 MX-10MX-10 MX-11 MX-11 MX-12 MX-12 MX-13 MX-13 MX-14 MX-14 MX-15 MX-15 MX-16 MX-16 Interim stiffness ratio 0. 11 0. 21 0. 32 0. 42 0. 52 0. 62 0. 72 0. 82 InterimInterim stiffness stiffness ratio ratio 0.11 0. 11 0.21 0. 21 0.32 0. 32 0. 0.42 42 0. 0.5252 0. 62 0.62 0. 72 0.72 0. 82 0.82 Figure 20 depicts the load-bearing performances of the specimens. Figure 20 depicts the load-bearing performances of the specimens.

Figure 20. Force exerted on the FSMRL wall structures (with hole as variable). Figure 20. ForceForce exerted on the FSMRL wall stru structuresctures (with hole as variable). Based on the force-displacement curves listed above, the relation between the yield and peak Based on the force-displacement curves listed above, the relation between the yield and peak loadingBased and on the the interim force-displacement stiffness ratio are curves iden listedtified above,for each the specimen relation (see between Figure the 21). yield and peak loading and the interim stiffness ratio are identiﬁedidentified forfor eacheach specimenspecimen (see(see FigureFigure 2121).).

Figure 21. Relation between interim stiffness ratio and load bearing performance. Figure 21. Relation between interim stiffness ratio and load bearing performance. Figure 21. Relation between interim stiffness ratio and load bearing performance. Figure 21 shows that when the stiffness ratio falls below 0.4, the yield and peak loading limit Figure 21 shows that when the stiffness ratio falls below 0.4, the yield and peak loading limit drop at an alarming rate due to the MRL wall’s relatively low stiffness level in lateral force resistance. drop at an alarming rate due to the MRL wall’s relatively low stiffness level in lateral force resistance.

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Appl. Sci. 2016, 6, 21 Figure 21 shows that when the stiffness ratio falls below 0.4, the yield and peak loading limit drop atThe an premature alarming rate failure due undermines to the MRL wall’sthe overall relatively bearin lowg capacity; stiffness leveltherefore, in lateral an interim force resistance. stiffness ratio The prematureof at least 0.4 failure must undermines be guaranteed the for overall the structure. bearing capacity; therefore, an interim stiffness ratio of at least 0.4 must be guaranteed for the structure. 4.5. Suggested Numerical Range for the Interim Stiffness Ratio in the Engineering Design Stage 4.5. Suggested Numerical Range for the Interim Stiffness Ratio in the Engineering Design Stage This paper integrates the findings of Figure 18 and Figure 21 on the influence of the interim This paper integrates the ﬁndings of Figures 18 and 21 on the inﬂuence of the interim stiffness stiffness ratio on the specimen’s load-bearing performance, and presents Figure 22 as a ratio on the specimen’s load-bearing performance, and presents Figure 22 as a conclusive summary. conclusive summary.

FigureFigure 22.22. Relation between interim stiffnessstiffness ratioratio andand loadload bearingbearing performance.performance.

TheThe followingfollowing cancan bebe concludedconcluded fromfrom FigureFigure 22 22:: (1) When the stiffness ratio is approximately 1, the yield and peak loading reach a maximum, which (1) When the stiffness ratio is approximately 1, the yield and peak loading reach a maximum, which can be interpreted as fairly good load-bearing performance with even distribution of stiffness can be interpreted as fairly good load-bearing performance with even distribution of stiffness between the upper and the lower structure. between the upper and the lower structure. (2) Based on the variation pattern of the yield and peak loading, the paper takes into consideration (2) Based on the variation pattern of the yield and peak loading, the paper takes into consideration the seismic performance required and identifies the numerical range for the interim stiffness the seismic performance required and identiﬁes the numerical range for the interim stiffness ratio, which falls between 0.4 and 2.5. The maximum displacements for MX-12, MX-13, MX-14, ratio, which falls between 0.4 and 2.5. The maximum displacements for MX-12, MX-13, MX-14, MX-15, and MX-16 are 1/300, 1/433, 1/555, 1/820, and 1/900, respectively, but the inter-layer MX-15, and MX-16 are 1/300, 1/433, 1/555, 1/820, and 1/900, respectively, but the inter-layer displacement angle does not exceed 1/800. On this basis, the paper suggests a numerical range displacement angle does not exceed 1/800. On this basis, the paper suggests a numerical range of 0.7 to 2.5 for engineering design. of 0.7 to 2.5 for engineering design.

5.5. Summary Summary and and Conclusions Conclusions TheThe paper studied the influence inﬂuence of of the the interim interim sti stiffnessffness ratio ratio on on the the load-b load-bearingearing performance performance of ofthe the FSMRL FSMRL wall wall structure structure and and presented presented evenly-dis evenly-distributedtributed figures ﬁgures by by adding adding shear walls withwith differentdifferent widthswidths to to the the FS FS layer layer or byor usingby using various various hole openinghole opening approaches; approaches; the load-bearing the load-bearing process isprocess then simulated is then simulated using OpenSees using software,OpenSees which softwa providesre, which relationship provides curvesrelationship for the curves stiffness for ratio the andstiffness the yield ratio and/orand the peak yield loading. and/or peak loading. (1) The paper recommends a not-less-than 2.5 interim stiffness ratio for the FSMRL wall structure, (1) The paper recommends a not-less-than 2.5 interim stiffness ratio for the FSMRL wall structure, because the yield strength needs to be maintained at a certain level without compromising the because the yield strength needs to be maintained at a certain level without compromising the overall structural reliability. At the same time, an interim stiffness ratio of at least 0.4 must be overall structural reliability. At the same time, an interim stiffness ratio of at least 0.4 must be guaranteed for the structure, because if the stiffness ratio falls below 0.4, the yield and peak guaranteed for the structure, because if the stiffness ratio falls below 0.4, the yield and peak loading limit drop at an alarming rate. loading limit drop at an alarming rate. (2) The increase of hole opening ratio—the area of the hole to the area of the slab—brings up the (2) The increase of hole opening ratio—the area of the hole to the area of the slab—brings up the maximum displacement angle, based on the limit of the inter-layer displacement angle, maximum displacement angle, based on the limit of the inter-layer displacement angle, which which does not exceed 1/800, the range is adjusted to 0.7 to 2.5 for reasonable engineering design does not exceed 1/800, the range is adjusted to 0.7 to 2.5 for reasonable engineering design in practice. in practice. Notation

γe: equivalent stiffness ratio at the interim layer H1: height of the interim layer (frame-supported layer)

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Notation

γe: equivalent stiffness ratio at the interim layer H1: height of the interim layer (frame-supported layer) ∆1: elative displacement at the interim layer H2: eight of the upper storey (MRL wall layer) ∆2: relative displacement of the upper storey H: wall height; Ieq: inertia moment of equivalent wall section H: height of wall section; Ae: area of equivalent wall section beq: width of equivalent wall section; Gc: shear modulus for concrete µN: axial compressive ratio (when µN falls below 0.3, µN is 0.3; likewise if µN exceeds 0.6, then µN is 0.6) Ac: sum of the cross-section area of concrete used for ribbed columns, connected columns and edge members (flange concrete excluded) Aq: sum of the cross-section area for embedded blocks µ: non-uniform coefficient for the shear stress distribution at the cross-section (for a rectangle cross-section, µ = 1.2) ηc: coefficient that reflects the restraint of the ribbed beams and the ribbed columns on the embedded blocks, which can be set to 1.05 (or ηc = 1.05) Ec: elastic modulus of concrete; Eq: elastic modulus of embedded block αc: modified coefficient for the lateral-resistant stiffness; ic: frame stiffness or the stiffness of seismic wall $fpc: concrete compressive strength at 28 days (compression is negative) $epsc0: concrete strain at maximum strength $fpcu: concrete crushing strength $epsU: concrete strain at crushing strength $λ: ratio between unloading slope at $epsU and initial slope (The initial slope for this model is 2 ˆ $fpc/$epsc0) $s1p $e1p: stress and strain (or force & deformation) at first point of the envelope in the positive direction $s2p $e2p: stress and strain (or force & deformation) at second point of the envelope in the positive direction $s3p $e3p: stress and strain (or force & deformation) at third point of the envelope in the positive direction (optional) $s1n $e1n: stress and strain (or force & deformation) at first point of the envelope in the negative direction $s2n $e2n: stress and strain (or force & deformation) at second point of the envelope in the negative direction $s3n $e3n: stress and strain (or force & deformation) at third point of the envelope in the negative direction (optional) K0: initial elastic stiffness µ: exponent of non-linear hardening component $β: power used to determine the degraded unloading stiffness based on ductility, µ´$β (optional, default = 0.0) In cases $s3p > $s2p and abs ($s3n) > abs ($s2n), the envelope of the hysteretic material after $e3p or $e3n follows the slope defined by the 2nd and 3rd points of the envelope. In cases $s3p <= $s2p and abs ($s3n) <= abs($s2n), the envelope of the hysteretic material after $e3p or $e3n is a flat line with a constant stress (or force) equal to $s3p or $s3n Acknowledgments: This research was conducted with the financial support of the National Natural Science Foundation of China (NO. 51508009) and the Project was supported by the Beijing Postdoctoral Research Foundation (2015ZZ-29). National Science and technology support program of China (NO. 2015BAL03B01). Author Contributions: Suizi Jia performed the experiments, analyzed the data, and wrote the manuscript; Suizi Jia, Wanlin Cao and Quan Yuan designed the experiments; Suizi Jia and Yuchen Zhang modified the final paper. Conflicts of Interest: The authors declare no conflict of interest.

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