Research ArticleSeismic Behavior of Shear Connectors of Steel VierendeelSandwich Plate

Lan Jiang12 KejianMa13 Huagang Zhang 3 QinWu3 Hongna Lu3 and Qizhu Yang4

1Collage of Civil Engineering Hunan University Changsha HN 731 China2College of Electrical Engineering amp New Energy China Three Gorges University Yichang HB 717 China3Space Structures Research Center Guizhou University Guiyang GZ 851 China4College of Urban Construction Shaoyang University Shaoyang HN 739 China

Correspondence should be addressed to Huagang Zhang hgzhanggzueducn

Received 24 March 2019 Accepted 26 May 2019 Published 12 June 2019

Academic Editor Alberto Campagnolo

Copyright copy 2019 Lan Jiang et alThis is an open access article distributed under theCreative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As a new type of floor structure steel vierendeel sandwich plates are widely applied in large-span buildings with multiple storeysShear connectors are important stressed members of such plates To evaluate the seismic performance of the shear connectors afull-scale test piece in two different connection forms namely A and B is designed and tested under alternating load Test analysisof the two connection specimens covers the failure modes hysteresis curves andmain parameters (eg bearing capacity ductilitystiffness degradation and energy dissipation coefficient) The following results concerning type A connection are obtained Firstit exhibits good ductility and long yielding platform second elastoplasticity of steel is fully exerted with it third it absorbs anddissipates energy well with strong energy consumption and fourth when failure occurs cracks usually happen in the heat-affectedzone of the weld in the core zone The following conclusions about type B are drawn first it has large bearing capacity with highstiffness also when failure occurs the ribbed stiffeners crack and flexion deformity happen

1 Introduction

Steel vierendeel floor slab was invented by Ma Kejian et alin 2002 [1] In industrial and civil buildings the size of thecolumn net should be expanded and the thickness of thefloor structure and the structural weight should be reduceddue to the needs of production processes and functionalpurposes [2] Prestressed steel-concrete composite floor andcombined grid structures are commonly included in large-spanfloor coverings Steel vierendeel floor slab iswell adaptedto these requirements and is a structural form with broadapplication prospects [1] Steel vierendeel sandwich plate hasthe following advantages

(i) For buildings with large and small spans the nor-mal depth of steel vierendeel floor slab ranges fromspan30 to span25

(ii) The space between the top and bottom chords can bethreaded to save the layer height (Figure 1)

(iii) The floor slab is light because a hole exists betweenthe top and bottom chords

(iv) The construction speed is fast Spatial steel grid cas-sette structures can be constructed by steel vierendeelfloor slab and steel grid wall (Figure 1)

This structure is suitable for multi-storey large-spanindustrial and public buildings At present the steel vieren-deel sandwich plate structure is widely used in Chinaand has been applied to many pilot buildings (Figure 2)[3]

Steel vierendeel sandwich plate is mainly composed oftop chords bottom chords and web members (Figure 3)The top and bottom chords mainly comprise T-shaped steeland the web member is a square steel tube The threesections are typically connected by weld (Figure 3) and thesection size is usually not smaller than the width of thetop chord and the bottom chord flanges The connectionarea which is called web member shear connections joinsthe top chords and the bottom chords to a vertical barCommonly used connections in projects are types A andB (Figure 3) [4] The vertical shear connection is the mostcomplex component of the steel vierendeel sandwich plate

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 8047393 14 pageshttpsdoiorg10115520198047393

2 Mathematical Problems in Engineering

Steel vierendeel slab Steel grid wall Spatial steel grid cassettestructures

Concrete slab

Beam

Close-column

Steel vierendeel slab

Steel grid wall

Electrical and plumbing conduit

Figure 1 Configuration of spatial steel grid cassette structures

Steel vierendeel floor slab

A ssemble and construction

Multi-storey industrial building

Large-span office building

Steel grid wall

Figure 2 Application of steel vierendeel floor slab

[5 6] Chen Q et al [7] analyzed the static performanceof the shear connectors in steel vierendeel sandwich platethrough finite element method The design parameters ofthe shear connectors considerably influenced floor stiffnessBai Z Q et al [8] carried out a one-way static loadingtest of two shear connectors models and concluded thattype B connection displayed greater improvement in stiffnessthan type A In summary current research on the con-nection of steel vierendeel sandwich plate focuses on staticperformance

Steel vierendeel sandwich plate is a large-span floorstructure and its dynamic catastrophic mechanism is anurgent problem to be studied [4] Although many scholarsconducted research on the dynamic performance of steelstructure joints most of them focused on the bending con-nection of frame structure beams and columns thus far fewworks have investigated the shear connection of the web [9ndash11] Low frequency cyclic loading test is an effective methodused to evaluate the dynamic performance of a structure [12]This study aims to determine the dynamic performance ofthe shear connection of the vertical bar through low cyclicloading test

2 Connection Tests

21 Design of Test Specimens Vierendeel sandwich plate iscrossed constituted by Vierendeel trussThe bearing capacityof the plate is controlled by shear force due to the large cross-section of its vertical bar [13] A previous research testeda section of the open web with reference to literature toassess the performance of shear connectors [13] (Figure 4(a))During the test only the shearing action of the shear connec-tors under the horizontal force was considered Figure 4(b)shows the mechanical model and boundary conditions of theintersection test

Two full-scale test specimens were designed to examinethe mechanical properties of A- and B-type connections Thedimensions of the specimens are based on actual engineeringprojects The total height of the test piece (h) is 600 mmTheT-shaped section is used for the top and bottom chords andthe cross-section of the web member is a square tube Thedifferent parts are welded by manual arc welding Figure 5shows the dimensions of the fillet The type A connectionspecimen number is T1 and the type B connection number isT2The specimen design and structure are shown in Figure 5

Mathematical Problems in Engineering 3

Type A

Type B

Top chords

Bottom chords

Web memberCoverplate

Top chordsWeb memberCoverplate

Bottom chords

Ribbed stiffener

Figure 3 Types of connections

Bottom chord

Top chord

Web member

Range

5a

a

(a)

Web member

Top chord

Bottom chorda

hc

Pin support

Roller support

Cyclic loading

(b)

Figure 4 Mechanical model of the test (a) Object of study (b) Force diagram

and the component size parameters are shown in Table 1 Asshown in Figure 5 hc is the T-shaped centroid distance Thetest piece material used is Q235B steel

A displacement gauge is used to measure the displace-ment of the specimen during loading in the x directionfor the top chord The resistance strain gauge was used tomeasure the strain change of the web member connectionduring the test to determine the yield-destructive mechanismof the web connection Referring to the stress distributioncharacteristics of the shear connectors the strain gauge isplaced at a portion where the stress of the test piece is great[14] Each test piece has one displacement measuring pointand 24 strainmeasuring points Figure 6 shows themeasuringpoint arrangement

22 Test Set-Up and Loading A set of reaction force bracketsand loading systems were self-designed and fabricated on thebasis of experimental boundary conditions (Figure 4(b))The

reaction force bracket and loading system were bolted to thereaction floor Figure 7 shows the test loading device and itsarrangement The end A of specimens is box-shaped and theside has a round hole which is connected to the reactionframe through a pin to form a fixed hinge constraint The endB of specimens has no circular hole and is connected to anactuator through a pin The B-end C-end and D-end of thespecimens are restrained by four sets of limit bearings thatare independently designed with the reaction force bracket toform themovable hinge support constraint The limit bearingcomprises a bearing a screw rod and nuts The limit bearingand the reaction force are fixed by screws and bolts and thegap between the specimens can be adjusted An alternatingload is applied in the x direction at the B end of the specimenby an actuator The test protocols are determined accordingto the provisions of JGJ101-2015 [15] and the characteristics ofthis testThe reaction force data are collected through the loadcell inside the actuator [11] The actuator and power system

4 Mathematical Problems in Engineering

8

5

5

5

5

6

5

5

5

h 1h 1

h 2

a

h c

300

bf

h 1

tw

t f bf

h 1

tw

t f bc

b c

t c

tc

bc

b c

btbt

tc

t c

t tt t

Type A Type B

50

15

15

r=15

15

bg

b g t g

h 1

bc

h

Figure 5 Details of specimens (Units mm)

Table 1 Design parameters of specimens

Specimenssize T1 T2119886 (mm) 2400 2400ℎ1 (mm) 150 150ℎ2 (mm) 300 300ℎ119888 (mm) 530 530119887119891 (mm) 150 150119905119891 (mm) 8 8119905119908 (mm) 8 8119887119888 (mm) 150 150119905119888 (mm) 5 5119887119905 (mm) - 100119905119905 (mm) - 8119887119892 (mm) 250 250119905119892 (mm) 10 10

are purchased fromMTS Figure 8(a) shows the test protocolof specimen T1 and Figure 8(b) shows the test protocol ofspecimen T2 Every load cycle is 6 minutes long All the testdata are collected by static signal test and analysis system(model TZT3826E) while the sampling frequency is 10Hz

23 Material Properties The specimens are all made ofQ235B steel The same batch of steel plate-shaped test pieceswas simultaneously produced in the process of makingspecimens Figure 9 shows the specimen used for materialperformance test Test yield strength (119891119910) ultimate strength(119891119906) and elastic modulus (119864119904) were measured by homotaxialtensile test Table 2 shows the test piece parameters and testresults

3 Test Results and Discussion

31 Test Phenomenon and Failure Modes of Specimens Thetwo types of specimens appeared exhibited obvious elasticproperties at the beginning of loading

When the displacementwas loaded to 285mm specimenT1 showed remarkable sagging or convex deformation onthe joint that connected the side of the web plate and theweb of the chord (Figure 10(a)) When the displacement wasloaded to approximately 325 mm microcracks appeared inthe heat-affected zone of the joint between the chord weband the side of the web member (Figure 10(b)) The loadingprocess is accompanied by the sound of ldquoZhizhirdquo and thecrack is continuously increased with cyclic loading Part ofthe strain gauge failed to fall off When the displacementwas loaded to 365 mm the cracks at the four joints were

Mathematical Problems in Engineering 5

A1

A2

A3A4

A5A6A7

A8 A9A10A11 A12

A13A14 A15

A16A17A18

A19

A20

A21

A22

A23

A24

Pin support

Load-end

Top chord

Bottom chord

Top chord

A

A A-A

B B

B-B

Displacement sensor

Resistance strain gauge

Displacement sensor

geResistance strain gaug

xz

xy

yz

Coverplate

Web member

Web

Flange

Web

Flange Flange

Web

(a)

B1B2B3

B4B5B6

B7

B8

B9

B10

B11B12 B13

B14B15B16B17B18

B19 B20B21B22B23

B24

Load-end

Top chord

Bottom chord

Top chord

Pin support A-A

B-B

A

A

B B

Coverplate

Web member

Ribbed stiffener

Displacement sensor

Resistance strain gauge

xz

xy

yz

Web

Flange

Web

Flange

Web

Flange

Ribbed stiffener

(b)

Figure 6 Arrangement of displacement sensors and strain gauges (a) Specimen T1 (b) Specimen T2

Table 2 Mechanical property index of steel

Square tube T-section steel Cover plateS-1 S-2 S-3

119867 (mm) 50 50 50119862 (mm) 17 17 171198710 (mm) 84 84 841198860 (mm) 5 8 101198630 (mm) 30 30 301198870 (mm) 20 20 20119891y (MPa) 232 261 255119891u (MPa) 393 410 402119864s (MPa) 203e5 211e5 209e5

successively expanded into U-shaped cracks (Figure 10(c))and the bearing capacity of the members reaches a maximum

of 523 kN When displacement was loaded to the maximumset displacement of 425 mm except for the increase of thecrack at the joint of the chord and the web the microcrackappeared in the heat-affected zone where the cover and theweb were connected and the bearing capacity descendedIn general when damage occurred the bearing capacity ofthe specimen T1 did not significantly decrease Thus plasticdeformational development was not completed

When the displacement was loaded to 185 mm micro-cracks appeared at the joint between the stiffener rib andthe web of the chord in specimen T2 (Figure 11(a)) Withthe cyclic loading cracks appeared at the four joints andthe specimen performed the elastoplastic stage When thedisplacement was loaded to 205 mm the crack on the web ofthe chord began to expand obliquely and increased (Figures11(a) and 11(b)) When the displacement was loaded to 225mm the bearing capacity reached a maximum of 1625 kN

6 Mathematical Problems in Engineering

Strong floor

Reaction frame system

Actuator

Specimen

Hinge support

Movable hinge support

Movable hinge support

Movable hinge support

PinPin Actuator

Bearing

Bearing

Reaction frame

x

yz

A

C

B

D

Nuts

Screw rod

(a)

T1 T2

Reaction frame system

Actuator

Specimen Specimen

Reaction frame

Strong floor

Reaction frame system

Reaction frame

Actuator

Strong floor

(b)

Figure 7 Schematic of the test set-up (a) Self-design of loading device (b) Test photos

When the displacement was loaded to 245 mm the jointbetween the stiffener and the web began to buckle Thebearing capacity of the specimen entered the descendingphase and the crack on the web of the chord expandedrapidly When the displacement was loaded to a maximumof 265 mm the whole stiffening plate exhibited an ldquoSrdquo-type buckling and the buckling deformation is shown inFigure 11(d) After destroying the specimen the bearingcapacity is obviously reduced indicating that the plasticdeformation development is perfect

32 Hysteretic Curves The hysteresis curve is a comprehen-sive reflection of the seismic performance of the structure andthe main basis for structural seismic elastoplastic dynamic

response analysis [16] According to the literature [17ndash19] theinitial geometric defects and residual stress only affect theinitial maximum load reduction of the member and posesminimal effect on the subsequent hysteresis performanceof the member Therefore the initial geometric defects andresidual stress of themember are not considered in this paperThe hysteresis performance of the web connection node ismainly contributed by the node domain and the two partsof the rodTherefore the total reaction force P of the loadingpoint is characterized by the horizontal displacement Δ of therod loading pointThe total reaction force P of the node is theload applied by the actuator and the horizontal displacementof the loading point is the horizontal displacement of theactuator The load P-Δ hysteresis curves for specimen T1 and

Mathematical Problems in Engineering 7

0 60 120 180 240

0

20

40

60Numbers of cycles 5533333333

Time (min)

1

minus20

minus40

minus60

Top

Chor

d Ti

pD

ispla

cem

ent (

Δ[m

m])

(a)

0 50 100 150 200 250 300

0

10

20

30 555Numbers of cycles 5533333333

Time (min)

1

minus10

minus20

minus30

Top

Chor

d Ti

pD

ispla

cem

ent (

Δ[m

m])

(b)

Figure 8 Test protocols (a) Specimen T1 (b) Specimen T2

H C L0 C H

a0

D0 b 0

(a)

S-1

S-2

S-3

(b)

Figure 9 Test specimen details (a) Size indication (b) Specimen of material tests

Bottom chords

Web members

(a)

Top chords

Web members

(b)

Bottom chords

Web members

(c)

Web members

Cover plates

(d)

Figure 10 Test phenomenon and failure modes of specimen T1

8 Mathematical Problems in Engineering

Top chords

Stiffener

(a)

Bottom chords

Stiffener

(b)

Bottom chords

Stiffener

(c)

Top chords

Bottom chords

Web members

Stiffener

(d)

Figure 11 Test phenomenon and failure modes of specimen T2

0 20 40 60

0

20

40

60

P (k

N)

peak points

minus20

minus20

minus40

minus40minus60

minus60

Δ (mm)

(a)

P (k

N)

0 10 20 30

0

50

100

150

200peak points

minus30minus200

minus150

minus100

minus50

minus20 minus10

Δ (mm)

(b)

Figure 12 Experimental hysteresis curves for specimens (a) Specimen T1 (b) Specimen T2

T2 are shown inFigures 12(a) and 12(b) respectively Figure 12shows the following

A The hysteresis loops of the two connected specimensare similar in shape all of which are spindle-shaped and nopinching phenomenon are observed

B At the initial stage of loading the specimen is in anelastic working status so that the development of the P-Δcurve is basically linear The loading and unloading curvesare basically coincident that is the residual deformation issubstantially zero Residual deformation increases with loaddue to the drumming or yielding of steel in the node domainand the hysteresis loop area gradually increases therebyforming a full shuttle shape The reason is that the yieldingextent of the pipewall in the node zone is gradually increasedand the stiffness is also decreased as deformation increased

C In contrast with specimen T2 the hysteresis curve ofspecimen T1 is fuller and has stronger energy consumption

33 Skeleton Curve and Bearing Capacity The skeleton curveis the trajectory of the maximum peak of P-Δ curve at theloading end under each cyclic load In this study the skeletoncurve of the specimens can be obtained by connecting theload peak points of various hysteretic curves of the specimensuccessively The peak points used to obtain the skeletoncurve are shown in Figure 12 Figures 13(a) and 13(b) show

the skeleton curve of specimens T1 and T2 respectively Thebuckling displacement and yield load of the structure areobtained from the skeleton curve using the energy method[20] Table 3 shows the skeleton characteristic values of eachspecimen

When the loading for specimen T1 starts the skeletoncurve is a straight line and is basically in an elastic workingstatus As the load increases to approximately 065 of theultimate load the skeleton curve is gradually bended towardthe displacement axis and the slope of the curve is graduallydecreased After the peak load the load does not dropsignificantly When the loading for specimen T2 starts theskeleton curve is a straight line and is basically in an elasticworking status As the load is increased to approximately085 of the ultimate load the skeleton curve begins tobend toward the displacement axis and the slope of thecurve is decreased After the peak load the load is droppedsignificantly

Table 3 shows the experimental values of the bearingcapacity of the two types of joint specimens Under thecondition that the basic parameters of the structure are con-sistent the bearing capacity of specimen T2 is approximatelythree times that of specimen T1 The connection method ofthe vertical rod poses significant influence on the bearingcapacity of the specimen

Mathematical Problems in Engineering 9

Table 3 The measured value of bearing capacity displacement ductility coefficient and final damage

Specimen Loading direction Yield point Ultimate point Damage point 120583 119863PykN Δ 119910mm P119906kN Δ 119906mm P119889kN Δ 119889mm

T1 +119909 3260 1170 5237 3730 4960 4229 319 066-119909 2847 1232 5220 3726 4961 4224 302

T2 +119909 9588 950 15671 2041 11005 2420 215 061-119909 9233 948 16267 2241 13597 2439 236

0 20 40 60

0

20

40

60

P (k

N)

Damage pointUltimate pointYield point

Δ (mm)

minus20

minus20

minus40

minus40minus60

minus60

(a)

P (k

N)

0 10 20 30

0

140

70

210

Damage pointUltimate pointYield point

Δ (mm)

minus70

minus140

minus210minus20 minus10minus30

(b)

Figure 13 Skeleton curves (a) Specimen T1 (b) Specimen T2

34 Stiffness and Ductility The stiffness of the connectedspecimen is related to the displacement and the number ofcycles As the alternating load and displacement amplitudeincrease plastic deformationdevelops and stiffness graduallydegenerates

In this study the stiffness of the specimen is expressed bythe secant stiffness that corresponds to the displacement inthe same stage [15]119870119894 is the stiffness of the specimen in i loadcycle which can be calculated by

119870119894 =1003816100381610038161003816+1198651198941003816100381610038161003816 +1003816100381610038161003816minus11986511989410038161003816100381610038161003816100381610038161003816+Δ 119894

1003816100381610038161003816 +1003816100381610038161003816minusΔ1198941003816100381610038161003816 (1)

where 119865119894 is the peak load of the specimen in i load cycles Δ 119894is the displacement of the specimen in i load cycles

Figure 14(a) shows the stiffness degradation curve ofspecimen T1 and Figure 14(b) shows the stiffness degra-dation curve of specimen T2 Figure 14(a) shows that asΔ increases the stiffness of specimen T1 deteriorates andstiffness degradation is severe when control displacement islarge Figure 14(b) shows that when the control displacementis between minus20 mm and +20 mm the stiffness curve of spec-imen T2 is relatively flat When the control displacement isgreater than +20mm or less than minus20mm stiffness is rapidlydegraded The stiffness of specimen T2 is approximately 3times that of specimen T1

The load and displacement values of yield points limitpoints and damage points in Figure 13 are listed in Table 3

The yield point is calculated by the energy method [20] Thepeak points of the curve in Figure 13 are defined as limitpoints The corresponding points on the skeleton curves aredefined as damage points while the load decreases to 85 ofthe limit load [15]

The ductility coefficient 120583 is a measure of structureductility [21] which is expressed as the ratio of the ultimatedisplacement of the structure to the yield displacement

120583 =Δ 119906Δ119910 (2)

where Δ 119906 is the ultimate displacement and Δ119910 is the yielddisplacement Table 3 shows the main calculation resultsTable 3 shows that the ductility 120583 of T1 specimen is between30 and 32 and the ductility 120583 of T2 specimen is between 21and 24This finding shows that the two connection methodsexhibit good ductility and the ductility of type A connectionis better than that of type B connection

The damage of the connected specimen response can beobtained through the stiffness degradation value in each loadcycle

119863119894 =Δ1198701198941198700 (3)

where119863119894 is the damage of the specimen in each load cycle1198700is the stiffness of the joint specimen and Δ119870119894 is the stiffness

10 Mathematical Problems in Engineering

0 20 40 60

Stiff

ness

(kN

m)

Δ (mm)minus20minus40minus60

1000

1500

2000

2500

3000

3500

(a)

Stiff

ness

(kN

m)

0 10 20 302000

4000

6000

8000

10000

12000

14000

Δ (mm)minus30 minus20 minus10

(b)

Figure 14 Stiffness degeneration curve (a) Specimen T1 (b) Specimen T2

degradation value of the joint in each load cycle which canbe calculated by

Δ119870119894 = 119870119894 minus 119870119894minus1 (4)

where 119870119894 and 119870119894minus1 are the stiffness of the specimen in i andi-1 load cycles respectively The final damage of the specimencan be obtained by summing 119863119894

119863 =119899

sum119894=1

119863119894 (5)

where n is the total number of cycles when the specimen isloaded to failure

Table 3 shows the cumulative damage of the specimenunder the action of alternating load indicating that the finaldamage D of specimen T1 is slightly greater than that ofspecimen T2 The elastoplastic properties of the material arefully exerted

35 Energy Dissipation Performance Energy dissipationcapacity and damping are important indicators formeasuringstructural dynamic performance The energy dissipationcapacity is measured by the area surrounded by the hysteresiscurve of load deformation and is usually evaluated by theenergy dissipation coefficient E andor the equivalent viscousdamping coefficient 120577119890119902 The calculation is presented asfollows

E =119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(6)

120577119890119902 =12120587sdot119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(7)

where 119878(119860119861119862+119862119863119860) is the area surrounded by the hysteresiscurve in Figure 15 119878(119874119861119864+119874119863119865) is the sum of the area of thetriangle OBE and the triangle ODF in Figure 15 The energyconsumption of the yield point peak point and damage pointis shown in Table 4 Table 4 shows that the energy dissipationcoefficient and the equivalent viscosity damping coefficientof each specimen increase with displacement The energyconsumption of specimen T1 is better than that of specimenT2

X

FB

EC

D

AF o

Figure 15 Load-displacement hysteretic loop

36 Strain Analysis The strain measuring points in Figure 6were monitored during the test Figure 16 shows the distribu-tion of specimen T1 strain along with loading displacementchanges Figure 17 shows the distribution of specimen T2strain with loading displacement changes

Figure 16 shows that the type A connection is connectedto the end portion of the web in the upper chord and the webside Stress concentration occurs Strain is larger when theend portion is close The main strain direction on the sidewall is radial with the intersection of the edge of the chordweb and the side wall of the vertical rod as the origin Theend of the upper chord is connected to the sidewall of theweb where the flange strain is large in the edge and smallin the middle The end of the upper chord is connected tothe sidewall of the web and the end of the web is close tothe flange where the strain at the measuring point is smallerand the strain away from the flange is larger In general whenthe loading displacement reaches Δ119910 most of the measuredstrain at the sidewall reaches yield strain and the strain atthe end of the upper chord is still in the elastic phase When

Mathematical Problems in Engineering 11

Table 4 Energy dissipation coefficient and equivalent viscous damping coefficient

Specimen Yield point Ultimate point Damage pointE 120577119890119902 E 120577119890119902 E 120577119890119902

T1 0347 0055 1173 0187 1465 0233T2 0198 0032 0404 0064 0859 0137

0

4000

8000

A4A5A6

A13A14A15

0 20 40 60Δ (mm)

minus20minus40minus60minus8000

minus4000

Stra

in(times

10minus6)

(a)

0

1000

2000

A22A23A24

A19A20A21

minus2000

minus1000

0 20 40 60Δ (mm)

minus20minus40minus60

Stra

in(times

10minus6)

(b)

0

1000

2000

3000

A7A8A9

A1A2A3

0 20 40 60Δ (mm)

minus20minus40minus60minus2000

minus1000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

4000

A10A11A12

A16A17A18

0 20 40 60Δ (mm)

minus20minus40minus60

minus2000

minus3000

minus1000

Stra

in(times

10minus6)

(d)

Figure 16 Strain of specimen T1 (a) The part of web near the weld (b) Top chord (c) The part of web near the top chord (d) The part ofweb near the bottom chord

the loading displacement reaches Δ 119906 most of the measuredstrain of the sidewall reaches the peak value and the strainat the end of the chord gradually reaches yield strain Whenthe loading displacement reachesΔ119889 the stress redistributionoccurs due to the large crack at the joint between the sidewalland the chord web The strain at the measuring point showsa downward trend and the strain at the end of the chordcontinues to increase

The strain on the stiffening plate in the loading processamong the measuring points of the type B connection is thelargest followed by the end of the chord and the side wallstrain of the web (Figure 17) In general when the loadingdisplacement reaches Δ119910 the strain at the point where thestiffener is connected to the web of the chord initially reachesyield strain and the strain at the joint of the bar and the

stiffener bar also rapidly reach the yield strain However theother points still remain in the elastic stageWhen the loadingdisplacement reaches Δ 119906 the strain at the point where thestiffener is connected to the web of the chord reaches thepeak and the strain at the other points continues to increaseWhen the loading displacement reaches Δ119889 the strain atthe point where the stiffener is connected to the web ofthe chord is gradually decreased due to the overall bucklingdeformation of the stiffener and the chord web

4 Conclusions

The main conclusions are summarized on the basis ofcyclic pseudo-static seismic testing of web member shearconnections as follows

12 Mathematical Problems in Engineering

0 10 20 30

0

1000

2000

3000

B4B5B6

B1B2B3

Δ (mm)minus10minus20minus30

minus4000

minus3000

minus2000

minus1000

Stra

in(times

10minus6)

(a)

0

2000

4000

6000

B10B11B12

B7B8B9

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(b)

0

2000

4000

B16B17B18

B13B14B15

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

B22B23B24

B19B20B21

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus1000

minus3000

Stra

in(times

10minus6)

(d)

Figure 17 Strain of specimen T2 (a) Flank of web member (b) Top chord (c)The part of ribbed stiffener near the top chord (d)The part ofribbed stiffener near the bottom chord

(1) The force performance of the vertical rod connectionis achieved by adopting the self-designed loadingsystem which can truly reflect the mechanical perfor-mance and deformation characteristics of the verticalrod shear connection under the action of alternatingload

(2) The ductility of the material in the heat-affected zonedeteriorates due to the stress concentration at thetype A connection between the chord and the sidewall of the vertical rod as well as the heat inputof the welding Added with alternating load theplastic strain in the heat-affected zone is accumulatedand the ductility is reduced Therefore a U-shapedfracture crack is formed on the outside of the heat-affected zone of the chord web and the side wall ofthe vertical rod for specimen T1

(3) In the type B connection the strain at the connectionbetween the stiffener and the chord web is the largestthereby yielding first As the loading displacement

increases the joint between the chord web and thestiffening plate cracks and develops eventually caus-ing the stiffening plate to be buckled and undergostructural damage

(4) When the main design parameters are the same thestiffness of type B connection is approximately threetimes that of the type A connection and the bearingcapacity is approximately three times that of the typeA connection

(5) The yielding platform is long and the stiffness isdegraded slowly so that the energy can be absorbedand dissipated well due to the sufficient plasticitydevelopment of the Type A connection materialEnergy consumption performance is remarkably bet-ter than that of the type B connectionIn summary the two types of vertical rod connec-tions investigated in this study exhibit good seismicperformance and can meet the engineering designrequirements

Mathematical Problems in Engineering 13

Notation119886 Specimen length1198860 Thickness of specimen ends for material

test1198870 Working width of specimen ends for

material test119887c Web member width119887f Chord width119887g Cover-plate width119887t Width of ribbed stiffener119862 Transition length of specimen ends for

material test119863 Final damage of the specimen119863i Damage of specimen in each load cycle1198630 Width of specimen ends for material test119864 Energy dissipation coefficient119864s Youngrsquos modulus119891u Ultimate strength of steel119891y Yield strength of steel119867 Length of specimen ends for material testℎ Overall height of specimenℎ1 Chord heightℎ2 Net distance between top chord and

bottom chordℎc The distance between the top and bottom

chord centroid1198700 Stiffness of specimen119870i Stiffness of the specimen in i load cycles119870i-1 Stiffness of the specimen in i-1 load cycles1198710 Working length of specimen ends for

material test119899 Total cycling number when the specimen

is loaded to failure119875 Counter-force of loading point119875d Failure load of specimen119875u Ultimate load of specimen119875y Yield load of specimen119878(119860119861119862+119862119863119860) The area surrounded by the hysteresis

curve119878(119874119861119864+119874119863119865) The area of the triangle OBE and the

triangle ODF119905c Pip thickness of web member119905f Chord flange thickness119905g Cover-plate thickness119905t Thickness of ribbed stiffener119905w Chord web thicknessΔ Displacement of loading pointΔ d Damage displacement of specimenΔ u Ultimate displacement of specimenΔ y Yield displacement of specimenΔ119870i Stiffness degradation value of the

specimen in each load cycle120577119890119902 Equivalent viscous damping coefficient120583 Ductility coefficient

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank the National Natural ScienceFoundation of China (51568012) for financially supportingthis research

References

[1] K J Ma H G Zhang and T Zheng ldquoTheory and practiceof new type architectural space gridding structuresrdquo ChinaCommunications Press 2006

[2] J S Xu and L H Xue ldquoModern prestressed concrete floorstructurerdquoChina Architecture amp Building Press 2013

[3] Z Chen G Wu D Feng and K Ma ldquoNumerical study of thestatic and dynamic characteristics of reinforced concrete cas-sette structures for high-rise buildingsrdquo The Structural Designof Tall and Special Buildings vol 28 no 3 pp 1ndash24 2019

[4] L Jiang K J Ma and H G Zhang ldquoThe dynamic property andcomfort degree study on the steel-concrete composite vieren-deel sandwich platerdquo Earthquake Engineering and EngineeringDynamics vol 37 no 6 pp 122ndash131 2017

[5] H Luan K Ma Y Qin X Li and Z Chen ldquoInvestigationon structural behavior of an innovative orthogonal-diagonalsteel open-web sandwich floor systemrdquo Advances in StructuralEngineering vol 19 no 2 pp 353ndash371 2016

[6] H Luan K Ma Y Qin Z Chen and Y Wei ldquoInvestigation ofthe structural behavior of an innovative steel open-web floorsystemrdquo International Journal of Steel Structures vol 17 no 4pp 1365ndash1378 2017

[7] Q Chen J C Xiao K J Ma and Z C Huang ldquoEffects of shearconnector joints on static behavior in steel open-web sandwichplatesrdquo Journal of Guangxi University vol 43 no 1 pp 41ndash492018 (Chinese)

[8] Z Q Bai X D Liu D Xing Y H Wei and K J Ma ldquoExperi-mental research on mechanics characteristics for shear block ofsteel tube with t-section steel ribsrdquo Industrial Construction vol48 no 1 pp 159ndash164 2018

[9] C C Chen S W Chen M D Chung and M C LinldquoCyclic behaviour of unreinforced and rib-reinforced momentconnectionsrdquo Journal of Constructional Steel Research vol 61no 1 pp 1ndash21 2005

[10] C-C Chen C-C Lin and C-H Lin ldquoDuctile moment con-nections used in steel column-tree moment-resisting framesrdquoJournal of Constructional Steel Research vol 62 no 8 pp 793ndash801 2006

[11] H S Park D C Lee B K Oh et al ldquoPerformance-basedmulti-objective optimal seismic retrofit method for a steel moment-resisting frame considering the life-cycle costrdquo MathematicalProblems in Engineering vol 2014 Article ID 305737 14 pages2014

[12] D Sosa D Arevalo E DMora et al ldquoExperimental and analyt-ical study of slender reinforced concrete shear wall under cyclicin-plane lateral loadrdquo Mathematical Problems in Engineeringvol 2017 Article ID 4020563 14 pages 2017

[13] L Hu Theoretical Analysis and Experimental Study on SemiAssembled Composite Open-Web Sandwich Plate Outsourcing

14 Mathematical Problems in Engineering

U Type Steel Plate [Unpublished PhD Dissertation] HunanUniversity 2017

[14] L Jiang K JMaH G Zhang and L Li ldquoThe designmethod ofsteel Vierendeel beam with shear connectionrdquo Xirsquoan Universityof Architecture amp Technology (Natural Science Edition) vol 50no 3 pp 354ndash360 2018

[15] JGJ 101-2015 Specification of Testing Methods for EarthquakeResistant Building Chinese Building Industry Press 2015

[16] Z L Li C T Zhang W L Fan et al ldquoExperimental research onseismic behavior of a new type of thin-walled square steel tubebeam-column connectionrdquo Engineering Mechanics vol 30 no2 pp 72ndash81 2013

[17] G Ballio and F M Mazzolani Theory and Design of SteelStructures Chapman and Hall London UK 1983

[18] S Banno I H P Mamaghani T Usami and E Mizuno ldquoCyclicelastoplastic large deflection analysis of thin steel platesrdquo Journalof Engineering Mechanics vol 124 no 4 pp 363ndash370 1998

[19] I H P Mamaghani T Usami and E Mizuno ldquoInelastic largedeflection analysis of structural steel members under cyclicloadingrdquo Engineering Structures vol 18 no 9 pp 659ndash6681996

[20] G L Bai J Q Zhao N J Du B Xie andH Gao ldquoExperimentalstudy on seismic behavior of SRC abnormal interior joints anddesign advicerdquo Journal of Building Structures vol 39 no 7 pp33ndash45 2018

[21] W B Kratzig and Y S Petryna ldquoQuasistatic seismic damageindicators for rc structures from dissipating energies in tangen-tial subspacesrdquoMathematical Problems in Engineering vol 2014Article ID 615792 11 pages 2014

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Mathematical Problems in Engineering 3

Type A

Type B

Top chords

Bottom chords

Web memberCoverplate

Top chordsWeb memberCoverplate

Bottom chords

Ribbed stiffener

Figure 3 Types of connections

Bottom chord

Top chord

Web member

Range

5a

a

(a)

Web member

Top chord

Bottom chorda

hc

Pin support

Roller support

Cyclic loading

(b)

Figure 4 Mechanical model of the test (a) Object of study (b) Force diagram

and the component size parameters are shown in Table 1 Asshown in Figure 5 hc is the T-shaped centroid distance Thetest piece material used is Q235B steel

A displacement gauge is used to measure the displace-ment of the specimen during loading in the x directionfor the top chord The resistance strain gauge was used tomeasure the strain change of the web member connectionduring the test to determine the yield-destructive mechanismof the web connection Referring to the stress distributioncharacteristics of the shear connectors the strain gauge isplaced at a portion where the stress of the test piece is great[14] Each test piece has one displacement measuring pointand 24 strainmeasuring points Figure 6 shows themeasuringpoint arrangement

22 Test Set-Up and Loading A set of reaction force bracketsand loading systems were self-designed and fabricated on thebasis of experimental boundary conditions (Figure 4(b))The

reaction force bracket and loading system were bolted to thereaction floor Figure 7 shows the test loading device and itsarrangement The end A of specimens is box-shaped and theside has a round hole which is connected to the reactionframe through a pin to form a fixed hinge constraint The endB of specimens has no circular hole and is connected to anactuator through a pin The B-end C-end and D-end of thespecimens are restrained by four sets of limit bearings thatare independently designed with the reaction force bracket toform themovable hinge support constraint The limit bearingcomprises a bearing a screw rod and nuts The limit bearingand the reaction force are fixed by screws and bolts and thegap between the specimens can be adjusted An alternatingload is applied in the x direction at the B end of the specimenby an actuator The test protocols are determined accordingto the provisions of JGJ101-2015 [15] and the characteristics ofthis testThe reaction force data are collected through the loadcell inside the actuator [11] The actuator and power system

4 Mathematical Problems in Engineering

8

5

5

5

5

6

5

5

5

h 1h 1

h 2

a

h c

300

bf

h 1

tw

t f bf

h 1

tw

t f bc

b c

t c

tc

bc

b c

btbt

tc

t c

t tt t

Type A Type B

50

15

15

r=15

15

bg

b g t g

h 1

bc

h

Figure 5 Details of specimens (Units mm)

Table 1 Design parameters of specimens

Specimenssize T1 T2119886 (mm) 2400 2400ℎ1 (mm) 150 150ℎ2 (mm) 300 300ℎ119888 (mm) 530 530119887119891 (mm) 150 150119905119891 (mm) 8 8119905119908 (mm) 8 8119887119888 (mm) 150 150119905119888 (mm) 5 5119887119905 (mm) - 100119905119905 (mm) - 8119887119892 (mm) 250 250119905119892 (mm) 10 10

are purchased fromMTS Figure 8(a) shows the test protocolof specimen T1 and Figure 8(b) shows the test protocol ofspecimen T2 Every load cycle is 6 minutes long All the testdata are collected by static signal test and analysis system(model TZT3826E) while the sampling frequency is 10Hz

23 Material Properties The specimens are all made ofQ235B steel The same batch of steel plate-shaped test pieceswas simultaneously produced in the process of makingspecimens Figure 9 shows the specimen used for materialperformance test Test yield strength (119891119910) ultimate strength(119891119906) and elastic modulus (119864119904) were measured by homotaxialtensile test Table 2 shows the test piece parameters and testresults

3 Test Results and Discussion

31 Test Phenomenon and Failure Modes of Specimens Thetwo types of specimens appeared exhibited obvious elasticproperties at the beginning of loading

When the displacementwas loaded to 285mm specimenT1 showed remarkable sagging or convex deformation onthe joint that connected the side of the web plate and theweb of the chord (Figure 10(a)) When the displacement wasloaded to approximately 325 mm microcracks appeared inthe heat-affected zone of the joint between the chord weband the side of the web member (Figure 10(b)) The loadingprocess is accompanied by the sound of ldquoZhizhirdquo and thecrack is continuously increased with cyclic loading Part ofthe strain gauge failed to fall off When the displacementwas loaded to 365 mm the cracks at the four joints were

Mathematical Problems in Engineering 5

A1

A2

A3A4

A5A6A7

A8 A9A10A11 A12

A13A14 A15

A16A17A18

A19

A20

A21

A22

A23

A24

Pin support

Load-end

Top chord

Bottom chord

Top chord

A

A A-A

B B

B-B

Displacement sensor

Resistance strain gauge

Displacement sensor

geResistance strain gaug

xz

xy

yz

Coverplate

Web member

Web

Flange

Web

Flange Flange

Web

(a)

B1B2B3

B4B5B6

B7

B8

B9

B10

B11B12 B13

B14B15B16B17B18

B19 B20B21B22B23

B24

Load-end

Top chord

Bottom chord

Top chord

Pin support A-A

B-B

A

A

B B

Coverplate

Web member

Ribbed stiffener

Displacement sensor

Resistance strain gauge

xz

xy

yz

Web

Flange

Web

Flange

Web

Flange

Ribbed stiffener

(b)

Figure 6 Arrangement of displacement sensors and strain gauges (a) Specimen T1 (b) Specimen T2

Table 2 Mechanical property index of steel

Square tube T-section steel Cover plateS-1 S-2 S-3

119867 (mm) 50 50 50119862 (mm) 17 17 171198710 (mm) 84 84 841198860 (mm) 5 8 101198630 (mm) 30 30 301198870 (mm) 20 20 20119891y (MPa) 232 261 255119891u (MPa) 393 410 402119864s (MPa) 203e5 211e5 209e5

successively expanded into U-shaped cracks (Figure 10(c))and the bearing capacity of the members reaches a maximum

of 523 kN When displacement was loaded to the maximumset displacement of 425 mm except for the increase of thecrack at the joint of the chord and the web the microcrackappeared in the heat-affected zone where the cover and theweb were connected and the bearing capacity descendedIn general when damage occurred the bearing capacity ofthe specimen T1 did not significantly decrease Thus plasticdeformational development was not completed

When the displacement was loaded to 185 mm micro-cracks appeared at the joint between the stiffener rib andthe web of the chord in specimen T2 (Figure 11(a)) Withthe cyclic loading cracks appeared at the four joints andthe specimen performed the elastoplastic stage When thedisplacement was loaded to 205 mm the crack on the web ofthe chord began to expand obliquely and increased (Figures11(a) and 11(b)) When the displacement was loaded to 225mm the bearing capacity reached a maximum of 1625 kN

6 Mathematical Problems in Engineering

Strong floor

Reaction frame system

Actuator

Specimen

Hinge support

Movable hinge support

Movable hinge support

Movable hinge support

PinPin Actuator

Bearing

Bearing

Reaction frame

x

yz

A

C

B

D

Nuts

Screw rod

(a)

T1 T2

Reaction frame system

Actuator

Specimen Specimen

Reaction frame

Strong floor

Reaction frame system

Reaction frame

Actuator

Strong floor

(b)

Figure 7 Schematic of the test set-up (a) Self-design of loading device (b) Test photos

When the displacement was loaded to 245 mm the jointbetween the stiffener and the web began to buckle Thebearing capacity of the specimen entered the descendingphase and the crack on the web of the chord expandedrapidly When the displacement was loaded to a maximumof 265 mm the whole stiffening plate exhibited an ldquoSrdquo-type buckling and the buckling deformation is shown inFigure 11(d) After destroying the specimen the bearingcapacity is obviously reduced indicating that the plasticdeformation development is perfect

32 Hysteretic Curves The hysteresis curve is a comprehen-sive reflection of the seismic performance of the structure andthe main basis for structural seismic elastoplastic dynamic

response analysis [16] According to the literature [17ndash19] theinitial geometric defects and residual stress only affect theinitial maximum load reduction of the member and posesminimal effect on the subsequent hysteresis performanceof the member Therefore the initial geometric defects andresidual stress of themember are not considered in this paperThe hysteresis performance of the web connection node ismainly contributed by the node domain and the two partsof the rodTherefore the total reaction force P of the loadingpoint is characterized by the horizontal displacement Δ of therod loading pointThe total reaction force P of the node is theload applied by the actuator and the horizontal displacementof the loading point is the horizontal displacement of theactuator The load P-Δ hysteresis curves for specimen T1 and

Mathematical Problems in Engineering 7

0 60 120 180 240

0

20

40

60Numbers of cycles 5533333333

Time (min)

1

minus20

minus40

minus60

Top

Chor

d Ti

pD

ispla

cem

ent (

Δ[m

m])

(a)

0 50 100 150 200 250 300

0

10

20

30 555Numbers of cycles 5533333333

Time (min)

1

minus10

minus20

minus30

Top

Chor

d Ti

pD

ispla

cem

ent (

Δ[m

m])

(b)

Figure 8 Test protocols (a) Specimen T1 (b) Specimen T2

H C L0 C H

a0

D0 b 0

(a)

S-1

S-2

S-3

(b)

Figure 9 Test specimen details (a) Size indication (b) Specimen of material tests

Bottom chords

Web members

(a)

Top chords

Web members

(b)

Bottom chords

Web members

(c)

Web members

Cover plates

(d)

Figure 10 Test phenomenon and failure modes of specimen T1

8 Mathematical Problems in Engineering

Top chords

Stiffener

(a)

Bottom chords

Stiffener

(b)

Bottom chords

Stiffener

(c)

Top chords

Bottom chords

Web members

Stiffener

(d)

Figure 11 Test phenomenon and failure modes of specimen T2

0 20 40 60

0

20

40

60

P (k

N)

peak points

minus20

minus20

minus40

minus40minus60

minus60

Δ (mm)

(a)

P (k

N)

0 10 20 30

0

50

100

150

200peak points

minus30minus200

minus150

minus100

minus50

minus20 minus10

Δ (mm)

(b)

Figure 12 Experimental hysteresis curves for specimens (a) Specimen T1 (b) Specimen T2

T2 are shown inFigures 12(a) and 12(b) respectively Figure 12shows the following

A The hysteresis loops of the two connected specimensare similar in shape all of which are spindle-shaped and nopinching phenomenon are observed

B At the initial stage of loading the specimen is in anelastic working status so that the development of the P-Δcurve is basically linear The loading and unloading curvesare basically coincident that is the residual deformation issubstantially zero Residual deformation increases with loaddue to the drumming or yielding of steel in the node domainand the hysteresis loop area gradually increases therebyforming a full shuttle shape The reason is that the yieldingextent of the pipewall in the node zone is gradually increasedand the stiffness is also decreased as deformation increased

C In contrast with specimen T2 the hysteresis curve ofspecimen T1 is fuller and has stronger energy consumption

33 Skeleton Curve and Bearing Capacity The skeleton curveis the trajectory of the maximum peak of P-Δ curve at theloading end under each cyclic load In this study the skeletoncurve of the specimens can be obtained by connecting theload peak points of various hysteretic curves of the specimensuccessively The peak points used to obtain the skeletoncurve are shown in Figure 12 Figures 13(a) and 13(b) show

the skeleton curve of specimens T1 and T2 respectively Thebuckling displacement and yield load of the structure areobtained from the skeleton curve using the energy method[20] Table 3 shows the skeleton characteristic values of eachspecimen

When the loading for specimen T1 starts the skeletoncurve is a straight line and is basically in an elastic workingstatus As the load increases to approximately 065 of theultimate load the skeleton curve is gradually bended towardthe displacement axis and the slope of the curve is graduallydecreased After the peak load the load does not dropsignificantly When the loading for specimen T2 starts theskeleton curve is a straight line and is basically in an elasticworking status As the load is increased to approximately085 of the ultimate load the skeleton curve begins tobend toward the displacement axis and the slope of thecurve is decreased After the peak load the load is droppedsignificantly

Table 3 shows the experimental values of the bearingcapacity of the two types of joint specimens Under thecondition that the basic parameters of the structure are con-sistent the bearing capacity of specimen T2 is approximatelythree times that of specimen T1 The connection method ofthe vertical rod poses significant influence on the bearingcapacity of the specimen

Mathematical Problems in Engineering 9

Table 3 The measured value of bearing capacity displacement ductility coefficient and final damage

Specimen Loading direction Yield point Ultimate point Damage point 120583 119863PykN Δ 119910mm P119906kN Δ 119906mm P119889kN Δ 119889mm

T1 +119909 3260 1170 5237 3730 4960 4229 319 066-119909 2847 1232 5220 3726 4961 4224 302

T2 +119909 9588 950 15671 2041 11005 2420 215 061-119909 9233 948 16267 2241 13597 2439 236

0 20 40 60

0

20

40

60

P (k

N)

Damage pointUltimate pointYield point

Δ (mm)

minus20

minus20

minus40

minus40minus60

minus60

(a)

P (k

N)

0 10 20 30

0

140

70

210

Damage pointUltimate pointYield point

Δ (mm)

minus70

minus140

minus210minus20 minus10minus30

(b)

Figure 13 Skeleton curves (a) Specimen T1 (b) Specimen T2

34 Stiffness and Ductility The stiffness of the connectedspecimen is related to the displacement and the number ofcycles As the alternating load and displacement amplitudeincrease plastic deformationdevelops and stiffness graduallydegenerates

In this study the stiffness of the specimen is expressed bythe secant stiffness that corresponds to the displacement inthe same stage [15]119870119894 is the stiffness of the specimen in i loadcycle which can be calculated by

119870119894 =1003816100381610038161003816+1198651198941003816100381610038161003816 +1003816100381610038161003816minus11986511989410038161003816100381610038161003816100381610038161003816+Δ 119894

1003816100381610038161003816 +1003816100381610038161003816minusΔ1198941003816100381610038161003816 (1)

where 119865119894 is the peak load of the specimen in i load cycles Δ 119894is the displacement of the specimen in i load cycles

Figure 14(a) shows the stiffness degradation curve ofspecimen T1 and Figure 14(b) shows the stiffness degra-dation curve of specimen T2 Figure 14(a) shows that asΔ increases the stiffness of specimen T1 deteriorates andstiffness degradation is severe when control displacement islarge Figure 14(b) shows that when the control displacementis between minus20 mm and +20 mm the stiffness curve of spec-imen T2 is relatively flat When the control displacement isgreater than +20mm or less than minus20mm stiffness is rapidlydegraded The stiffness of specimen T2 is approximately 3times that of specimen T1

The load and displacement values of yield points limitpoints and damage points in Figure 13 are listed in Table 3

The yield point is calculated by the energy method [20] Thepeak points of the curve in Figure 13 are defined as limitpoints The corresponding points on the skeleton curves aredefined as damage points while the load decreases to 85 ofthe limit load [15]

The ductility coefficient 120583 is a measure of structureductility [21] which is expressed as the ratio of the ultimatedisplacement of the structure to the yield displacement

120583 =Δ 119906Δ119910 (2)

where Δ 119906 is the ultimate displacement and Δ119910 is the yielddisplacement Table 3 shows the main calculation resultsTable 3 shows that the ductility 120583 of T1 specimen is between30 and 32 and the ductility 120583 of T2 specimen is between 21and 24This finding shows that the two connection methodsexhibit good ductility and the ductility of type A connectionis better than that of type B connection

The damage of the connected specimen response can beobtained through the stiffness degradation value in each loadcycle

119863119894 =Δ1198701198941198700 (3)

where119863119894 is the damage of the specimen in each load cycle1198700is the stiffness of the joint specimen and Δ119870119894 is the stiffness

10 Mathematical Problems in Engineering

0 20 40 60

Stiff

ness

(kN

m)

Δ (mm)minus20minus40minus60

1000

1500

2000

2500

3000

3500

(a)

Stiff

ness

(kN

m)

0 10 20 302000

4000

6000

8000

10000

12000

14000

Δ (mm)minus30 minus20 minus10

(b)

Figure 14 Stiffness degeneration curve (a) Specimen T1 (b) Specimen T2

degradation value of the joint in each load cycle which canbe calculated by

Δ119870119894 = 119870119894 minus 119870119894minus1 (4)

where 119870119894 and 119870119894minus1 are the stiffness of the specimen in i andi-1 load cycles respectively The final damage of the specimencan be obtained by summing 119863119894

119863 =119899

sum119894=1

119863119894 (5)

where n is the total number of cycles when the specimen isloaded to failure

Table 3 shows the cumulative damage of the specimenunder the action of alternating load indicating that the finaldamage D of specimen T1 is slightly greater than that ofspecimen T2 The elastoplastic properties of the material arefully exerted

35 Energy Dissipation Performance Energy dissipationcapacity and damping are important indicators formeasuringstructural dynamic performance The energy dissipationcapacity is measured by the area surrounded by the hysteresiscurve of load deformation and is usually evaluated by theenergy dissipation coefficient E andor the equivalent viscousdamping coefficient 120577119890119902 The calculation is presented asfollows

E =119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(6)

120577119890119902 =12120587sdot119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(7)

where 119878(119860119861119862+119862119863119860) is the area surrounded by the hysteresiscurve in Figure 15 119878(119874119861119864+119874119863119865) is the sum of the area of thetriangle OBE and the triangle ODF in Figure 15 The energyconsumption of the yield point peak point and damage pointis shown in Table 4 Table 4 shows that the energy dissipationcoefficient and the equivalent viscosity damping coefficientof each specimen increase with displacement The energyconsumption of specimen T1 is better than that of specimenT2

X

FB

EC

D

AF o

Figure 15 Load-displacement hysteretic loop

36 Strain Analysis The strain measuring points in Figure 6were monitored during the test Figure 16 shows the distribu-tion of specimen T1 strain along with loading displacementchanges Figure 17 shows the distribution of specimen T2strain with loading displacement changes

Figure 16 shows that the type A connection is connectedto the end portion of the web in the upper chord and the webside Stress concentration occurs Strain is larger when theend portion is close The main strain direction on the sidewall is radial with the intersection of the edge of the chordweb and the side wall of the vertical rod as the origin Theend of the upper chord is connected to the sidewall of theweb where the flange strain is large in the edge and smallin the middle The end of the upper chord is connected tothe sidewall of the web and the end of the web is close tothe flange where the strain at the measuring point is smallerand the strain away from the flange is larger In general whenthe loading displacement reaches Δ119910 most of the measuredstrain at the sidewall reaches yield strain and the strain atthe end of the upper chord is still in the elastic phase When

Mathematical Problems in Engineering 11

Table 4 Energy dissipation coefficient and equivalent viscous damping coefficient

Specimen Yield point Ultimate point Damage pointE 120577119890119902 E 120577119890119902 E 120577119890119902

T1 0347 0055 1173 0187 1465 0233T2 0198 0032 0404 0064 0859 0137

0

4000

8000

A4A5A6

A13A14A15

0 20 40 60Δ (mm)

minus20minus40minus60minus8000

minus4000

Stra

in(times

10minus6)

(a)

0

1000

2000

A22A23A24

A19A20A21

minus2000

minus1000

0 20 40 60Δ (mm)

minus20minus40minus60

Stra

in(times

10minus6)

(b)

0

1000

2000

3000

A7A8A9

A1A2A3

0 20 40 60Δ (mm)

minus20minus40minus60minus2000

minus1000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

4000

A10A11A12

A16A17A18

0 20 40 60Δ (mm)

minus20minus40minus60

minus2000

minus3000

minus1000

Stra

in(times

10minus6)

(d)

Figure 16 Strain of specimen T1 (a) The part of web near the weld (b) Top chord (c) The part of web near the top chord (d) The part ofweb near the bottom chord

the loading displacement reaches Δ 119906 most of the measuredstrain of the sidewall reaches the peak value and the strainat the end of the chord gradually reaches yield strain Whenthe loading displacement reachesΔ119889 the stress redistributionoccurs due to the large crack at the joint between the sidewalland the chord web The strain at the measuring point showsa downward trend and the strain at the end of the chordcontinues to increase

The strain on the stiffening plate in the loading processamong the measuring points of the type B connection is thelargest followed by the end of the chord and the side wallstrain of the web (Figure 17) In general when the loadingdisplacement reaches Δ119910 the strain at the point where thestiffener is connected to the web of the chord initially reachesyield strain and the strain at the joint of the bar and the

stiffener bar also rapidly reach the yield strain However theother points still remain in the elastic stageWhen the loadingdisplacement reaches Δ 119906 the strain at the point where thestiffener is connected to the web of the chord reaches thepeak and the strain at the other points continues to increaseWhen the loading displacement reaches Δ119889 the strain atthe point where the stiffener is connected to the web ofthe chord is gradually decreased due to the overall bucklingdeformation of the stiffener and the chord web

4 Conclusions

The main conclusions are summarized on the basis ofcyclic pseudo-static seismic testing of web member shearconnections as follows

12 Mathematical Problems in Engineering

0 10 20 30

0

1000

2000

3000

B4B5B6

B1B2B3

Δ (mm)minus10minus20minus30

minus4000

minus3000

minus2000

minus1000

Stra

in(times

10minus6)

(a)

0

2000

4000

6000

B10B11B12

B7B8B9

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(b)

0

2000

4000

B16B17B18

B13B14B15

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

B22B23B24

B19B20B21

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus1000

minus3000

Stra

in(times

10minus6)

(d)

Figure 17 Strain of specimen T2 (a) Flank of web member (b) Top chord (c)The part of ribbed stiffener near the top chord (d)The part ofribbed stiffener near the bottom chord

(1) The force performance of the vertical rod connectionis achieved by adopting the self-designed loadingsystem which can truly reflect the mechanical perfor-mance and deformation characteristics of the verticalrod shear connection under the action of alternatingload

(2) The ductility of the material in the heat-affected zonedeteriorates due to the stress concentration at thetype A connection between the chord and the sidewall of the vertical rod as well as the heat inputof the welding Added with alternating load theplastic strain in the heat-affected zone is accumulatedand the ductility is reduced Therefore a U-shapedfracture crack is formed on the outside of the heat-affected zone of the chord web and the side wall ofthe vertical rod for specimen T1

(3) In the type B connection the strain at the connectionbetween the stiffener and the chord web is the largestthereby yielding first As the loading displacement

increases the joint between the chord web and thestiffening plate cracks and develops eventually caus-ing the stiffening plate to be buckled and undergostructural damage

(4) When the main design parameters are the same thestiffness of type B connection is approximately threetimes that of the type A connection and the bearingcapacity is approximately three times that of the typeA connection

(5) The yielding platform is long and the stiffness isdegraded slowly so that the energy can be absorbedand dissipated well due to the sufficient plasticitydevelopment of the Type A connection materialEnergy consumption performance is remarkably bet-ter than that of the type B connectionIn summary the two types of vertical rod connec-tions investigated in this study exhibit good seismicperformance and can meet the engineering designrequirements

Mathematical Problems in Engineering 13

Notation119886 Specimen length1198860 Thickness of specimen ends for material

test1198870 Working width of specimen ends for

material test119887c Web member width119887f Chord width119887g Cover-plate width119887t Width of ribbed stiffener119862 Transition length of specimen ends for

material test119863 Final damage of the specimen119863i Damage of specimen in each load cycle1198630 Width of specimen ends for material test119864 Energy dissipation coefficient119864s Youngrsquos modulus119891u Ultimate strength of steel119891y Yield strength of steel119867 Length of specimen ends for material testℎ Overall height of specimenℎ1 Chord heightℎ2 Net distance between top chord and

bottom chordℎc The distance between the top and bottom

chord centroid1198700 Stiffness of specimen119870i Stiffness of the specimen in i load cycles119870i-1 Stiffness of the specimen in i-1 load cycles1198710 Working length of specimen ends for

material test119899 Total cycling number when the specimen

is loaded to failure119875 Counter-force of loading point119875d Failure load of specimen119875u Ultimate load of specimen119875y Yield load of specimen119878(119860119861119862+119862119863119860) The area surrounded by the hysteresis

curve119878(119874119861119864+119874119863119865) The area of the triangle OBE and the

triangle ODF119905c Pip thickness of web member119905f Chord flange thickness119905g Cover-plate thickness119905t Thickness of ribbed stiffener119905w Chord web thicknessΔ Displacement of loading pointΔ d Damage displacement of specimenΔ u Ultimate displacement of specimenΔ y Yield displacement of specimenΔ119870i Stiffness degradation value of the

specimen in each load cycle120577119890119902 Equivalent viscous damping coefficient120583 Ductility coefficient

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank the National Natural ScienceFoundation of China (51568012) for financially supportingthis research

References

[1] K J Ma H G Zhang and T Zheng ldquoTheory and practiceof new type architectural space gridding structuresrdquo ChinaCommunications Press 2006

[2] J S Xu and L H Xue ldquoModern prestressed concrete floorstructurerdquoChina Architecture amp Building Press 2013

[3] Z Chen G Wu D Feng and K Ma ldquoNumerical study of thestatic and dynamic characteristics of reinforced concrete cas-sette structures for high-rise buildingsrdquo The Structural Designof Tall and Special Buildings vol 28 no 3 pp 1ndash24 2019

[4] L Jiang K J Ma and H G Zhang ldquoThe dynamic property andcomfort degree study on the steel-concrete composite vieren-deel sandwich platerdquo Earthquake Engineering and EngineeringDynamics vol 37 no 6 pp 122ndash131 2017

[5] H Luan K Ma Y Qin X Li and Z Chen ldquoInvestigationon structural behavior of an innovative orthogonal-diagonalsteel open-web sandwich floor systemrdquo Advances in StructuralEngineering vol 19 no 2 pp 353ndash371 2016

[6] H Luan K Ma Y Qin Z Chen and Y Wei ldquoInvestigation ofthe structural behavior of an innovative steel open-web floorsystemrdquo International Journal of Steel Structures vol 17 no 4pp 1365ndash1378 2017

[7] Q Chen J C Xiao K J Ma and Z C Huang ldquoEffects of shearconnector joints on static behavior in steel open-web sandwichplatesrdquo Journal of Guangxi University vol 43 no 1 pp 41ndash492018 (Chinese)

[8] Z Q Bai X D Liu D Xing Y H Wei and K J Ma ldquoExperi-mental research on mechanics characteristics for shear block ofsteel tube with t-section steel ribsrdquo Industrial Construction vol48 no 1 pp 159ndash164 2018

[9] C C Chen S W Chen M D Chung and M C LinldquoCyclic behaviour of unreinforced and rib-reinforced momentconnectionsrdquo Journal of Constructional Steel Research vol 61no 1 pp 1ndash21 2005

[10] C-C Chen C-C Lin and C-H Lin ldquoDuctile moment con-nections used in steel column-tree moment-resisting framesrdquoJournal of Constructional Steel Research vol 62 no 8 pp 793ndash801 2006

[11] H S Park D C Lee B K Oh et al ldquoPerformance-basedmulti-objective optimal seismic retrofit method for a steel moment-resisting frame considering the life-cycle costrdquo MathematicalProblems in Engineering vol 2014 Article ID 305737 14 pages2014

[12] D Sosa D Arevalo E DMora et al ldquoExperimental and analyt-ical study of slender reinforced concrete shear wall under cyclicin-plane lateral loadrdquo Mathematical Problems in Engineeringvol 2017 Article ID 4020563 14 pages 2017

[13] L Hu Theoretical Analysis and Experimental Study on SemiAssembled Composite Open-Web Sandwich Plate Outsourcing

14 Mathematical Problems in Engineering

U Type Steel Plate [Unpublished PhD Dissertation] HunanUniversity 2017

[14] L Jiang K JMaH G Zhang and L Li ldquoThe designmethod ofsteel Vierendeel beam with shear connectionrdquo Xirsquoan Universityof Architecture amp Technology (Natural Science Edition) vol 50no 3 pp 354ndash360 2018

[15] JGJ 101-2015 Specification of Testing Methods for EarthquakeResistant Building Chinese Building Industry Press 2015

[16] Z L Li C T Zhang W L Fan et al ldquoExperimental research onseismic behavior of a new type of thin-walled square steel tubebeam-column connectionrdquo Engineering Mechanics vol 30 no2 pp 72ndash81 2013

[17] G Ballio and F M Mazzolani Theory and Design of SteelStructures Chapman and Hall London UK 1983

[18] S Banno I H P Mamaghani T Usami and E Mizuno ldquoCyclicelastoplastic large deflection analysis of thin steel platesrdquo Journalof Engineering Mechanics vol 124 no 4 pp 363ndash370 1998

[19] I H P Mamaghani T Usami and E Mizuno ldquoInelastic largedeflection analysis of structural steel members under cyclicloadingrdquo Engineering Structures vol 18 no 9 pp 659ndash6681996

[20] G L Bai J Q Zhao N J Du B Xie andH Gao ldquoExperimentalstudy on seismic behavior of SRC abnormal interior joints anddesign advicerdquo Journal of Building Structures vol 39 no 7 pp33ndash45 2018

[21] W B Kratzig and Y S Petryna ldquoQuasistatic seismic damageindicators for rc structures from dissipating energies in tangen-tial subspacesrdquoMathematical Problems in Engineering vol 2014Article ID 615792 11 pages 2014

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Submit your manuscripts atwwwhindawicom

4 Mathematical Problems in Engineering

8

5

5

5

5

6

5

5

5

h 1h 1

h 2

a

h c

300

bf

h 1

tw

t f bf

h 1

tw

t f bc

b c

t c

tc

bc

b c

btbt

tc

t c

t tt t

Type A Type B

50

15

15

r=15

15

bg

b g t g

h 1

bc

h

Figure 5 Details of specimens (Units mm)

Table 1 Design parameters of specimens

Specimenssize T1 T2119886 (mm) 2400 2400ℎ1 (mm) 150 150ℎ2 (mm) 300 300ℎ119888 (mm) 530 530119887119891 (mm) 150 150119905119891 (mm) 8 8119905119908 (mm) 8 8119887119888 (mm) 150 150119905119888 (mm) 5 5119887119905 (mm) - 100119905119905 (mm) - 8119887119892 (mm) 250 250119905119892 (mm) 10 10

are purchased fromMTS Figure 8(a) shows the test protocolof specimen T1 and Figure 8(b) shows the test protocol ofspecimen T2 Every load cycle is 6 minutes long All the testdata are collected by static signal test and analysis system(model TZT3826E) while the sampling frequency is 10Hz

23 Material Properties The specimens are all made ofQ235B steel The same batch of steel plate-shaped test pieceswas simultaneously produced in the process of makingspecimens Figure 9 shows the specimen used for materialperformance test Test yield strength (119891119910) ultimate strength(119891119906) and elastic modulus (119864119904) were measured by homotaxialtensile test Table 2 shows the test piece parameters and testresults

3 Test Results and Discussion

31 Test Phenomenon and Failure Modes of Specimens Thetwo types of specimens appeared exhibited obvious elasticproperties at the beginning of loading

When the displacementwas loaded to 285mm specimenT1 showed remarkable sagging or convex deformation onthe joint that connected the side of the web plate and theweb of the chord (Figure 10(a)) When the displacement wasloaded to approximately 325 mm microcracks appeared inthe heat-affected zone of the joint between the chord weband the side of the web member (Figure 10(b)) The loadingprocess is accompanied by the sound of ldquoZhizhirdquo and thecrack is continuously increased with cyclic loading Part ofthe strain gauge failed to fall off When the displacementwas loaded to 365 mm the cracks at the four joints were

Mathematical Problems in Engineering 5

A1

A2

A3A4

A5A6A7

A8 A9A10A11 A12

A13A14 A15

A16A17A18

A19

A20

A21

A22

A23

A24

Pin support

Load-end

Top chord

Bottom chord

Top chord

A

A A-A

B B

B-B

Displacement sensor

Resistance strain gauge

Displacement sensor

geResistance strain gaug

xz

xy

yz

Coverplate

Web member

Web

Flange

Web

Flange Flange

Web

(a)

B1B2B3

B4B5B6

B7

B8

B9

B10

B11B12 B13

B14B15B16B17B18

B19 B20B21B22B23

B24

Load-end

Top chord

Bottom chord

Top chord

Pin support A-A

B-B

A

A

B B

Coverplate

Web member

Ribbed stiffener

Displacement sensor

Resistance strain gauge

xz

xy

yz

Web

Flange

Web

Flange

Web

Flange

Ribbed stiffener

(b)

Figure 6 Arrangement of displacement sensors and strain gauges (a) Specimen T1 (b) Specimen T2

Table 2 Mechanical property index of steel

Square tube T-section steel Cover plateS-1 S-2 S-3

119867 (mm) 50 50 50119862 (mm) 17 17 171198710 (mm) 84 84 841198860 (mm) 5 8 101198630 (mm) 30 30 301198870 (mm) 20 20 20119891y (MPa) 232 261 255119891u (MPa) 393 410 402119864s (MPa) 203e5 211e5 209e5

successively expanded into U-shaped cracks (Figure 10(c))and the bearing capacity of the members reaches a maximum

of 523 kN When displacement was loaded to the maximumset displacement of 425 mm except for the increase of thecrack at the joint of the chord and the web the microcrackappeared in the heat-affected zone where the cover and theweb were connected and the bearing capacity descendedIn general when damage occurred the bearing capacity ofthe specimen T1 did not significantly decrease Thus plasticdeformational development was not completed

When the displacement was loaded to 185 mm micro-cracks appeared at the joint between the stiffener rib andthe web of the chord in specimen T2 (Figure 11(a)) Withthe cyclic loading cracks appeared at the four joints andthe specimen performed the elastoplastic stage When thedisplacement was loaded to 205 mm the crack on the web ofthe chord began to expand obliquely and increased (Figures11(a) and 11(b)) When the displacement was loaded to 225mm the bearing capacity reached a maximum of 1625 kN

6 Mathematical Problems in Engineering

Strong floor

Reaction frame system

Actuator

Specimen

Hinge support

Movable hinge support

Movable hinge support

Movable hinge support

PinPin Actuator

Bearing

Bearing

Reaction frame

x

yz

A

C

B

D

Nuts

Screw rod

(a)

T1 T2

Reaction frame system

Actuator

Specimen Specimen

Reaction frame

Strong floor

Reaction frame system

Reaction frame

Actuator

Strong floor

(b)

Figure 7 Schematic of the test set-up (a) Self-design of loading device (b) Test photos

When the displacement was loaded to 245 mm the jointbetween the stiffener and the web began to buckle Thebearing capacity of the specimen entered the descendingphase and the crack on the web of the chord expandedrapidly When the displacement was loaded to a maximumof 265 mm the whole stiffening plate exhibited an ldquoSrdquo-type buckling and the buckling deformation is shown inFigure 11(d) After destroying the specimen the bearingcapacity is obviously reduced indicating that the plasticdeformation development is perfect

32 Hysteretic Curves The hysteresis curve is a comprehen-sive reflection of the seismic performance of the structure andthe main basis for structural seismic elastoplastic dynamic

response analysis [16] According to the literature [17ndash19] theinitial geometric defects and residual stress only affect theinitial maximum load reduction of the member and posesminimal effect on the subsequent hysteresis performanceof the member Therefore the initial geometric defects andresidual stress of themember are not considered in this paperThe hysteresis performance of the web connection node ismainly contributed by the node domain and the two partsof the rodTherefore the total reaction force P of the loadingpoint is characterized by the horizontal displacement Δ of therod loading pointThe total reaction force P of the node is theload applied by the actuator and the horizontal displacementof the loading point is the horizontal displacement of theactuator The load P-Δ hysteresis curves for specimen T1 and

Mathematical Problems in Engineering 7

0 60 120 180 240

0

20

40

60Numbers of cycles 5533333333

Time (min)

1

minus20

minus40

minus60

Top

Chor

d Ti

pD

ispla

cem

ent (

Δ[m

m])

(a)

0 50 100 150 200 250 300

0

10

20

30 555Numbers of cycles 5533333333

Time (min)

1

minus10

minus20

minus30

Top

Chor

d Ti

pD

ispla

cem

ent (

Δ[m

m])

(b)

Figure 8 Test protocols (a) Specimen T1 (b) Specimen T2

H C L0 C H

a0

D0 b 0

(a)

S-1

S-2

S-3

(b)

Figure 9 Test specimen details (a) Size indication (b) Specimen of material tests

Bottom chords

Web members

(a)

Top chords

Web members

(b)

Bottom chords

Web members

(c)

Web members

Cover plates

(d)

Figure 10 Test phenomenon and failure modes of specimen T1

8 Mathematical Problems in Engineering

Top chords

Stiffener

(a)

Bottom chords

Stiffener

(b)

Bottom chords

Stiffener

(c)

Top chords

Bottom chords

Web members

Stiffener

(d)

Figure 11 Test phenomenon and failure modes of specimen T2

0 20 40 60

0

20

40

60

P (k

N)

peak points

minus20

minus20

minus40

minus40minus60

minus60

Δ (mm)

(a)

P (k

N)

0 10 20 30

0

50

100

150

200peak points

minus30minus200

minus150

minus100

minus50

minus20 minus10

Δ (mm)

(b)

Figure 12 Experimental hysteresis curves for specimens (a) Specimen T1 (b) Specimen T2

T2 are shown inFigures 12(a) and 12(b) respectively Figure 12shows the following

A The hysteresis loops of the two connected specimensare similar in shape all of which are spindle-shaped and nopinching phenomenon are observed

B At the initial stage of loading the specimen is in anelastic working status so that the development of the P-Δcurve is basically linear The loading and unloading curvesare basically coincident that is the residual deformation issubstantially zero Residual deformation increases with loaddue to the drumming or yielding of steel in the node domainand the hysteresis loop area gradually increases therebyforming a full shuttle shape The reason is that the yieldingextent of the pipewall in the node zone is gradually increasedand the stiffness is also decreased as deformation increased

C In contrast with specimen T2 the hysteresis curve ofspecimen T1 is fuller and has stronger energy consumption

33 Skeleton Curve and Bearing Capacity The skeleton curveis the trajectory of the maximum peak of P-Δ curve at theloading end under each cyclic load In this study the skeletoncurve of the specimens can be obtained by connecting theload peak points of various hysteretic curves of the specimensuccessively The peak points used to obtain the skeletoncurve are shown in Figure 12 Figures 13(a) and 13(b) show

the skeleton curve of specimens T1 and T2 respectively Thebuckling displacement and yield load of the structure areobtained from the skeleton curve using the energy method[20] Table 3 shows the skeleton characteristic values of eachspecimen

When the loading for specimen T1 starts the skeletoncurve is a straight line and is basically in an elastic workingstatus As the load increases to approximately 065 of theultimate load the skeleton curve is gradually bended towardthe displacement axis and the slope of the curve is graduallydecreased After the peak load the load does not dropsignificantly When the loading for specimen T2 starts theskeleton curve is a straight line and is basically in an elasticworking status As the load is increased to approximately085 of the ultimate load the skeleton curve begins tobend toward the displacement axis and the slope of thecurve is decreased After the peak load the load is droppedsignificantly

Table 3 shows the experimental values of the bearingcapacity of the two types of joint specimens Under thecondition that the basic parameters of the structure are con-sistent the bearing capacity of specimen T2 is approximatelythree times that of specimen T1 The connection method ofthe vertical rod poses significant influence on the bearingcapacity of the specimen

Mathematical Problems in Engineering 9

Table 3 The measured value of bearing capacity displacement ductility coefficient and final damage

Specimen Loading direction Yield point Ultimate point Damage point 120583 119863PykN Δ 119910mm P119906kN Δ 119906mm P119889kN Δ 119889mm

T1 +119909 3260 1170 5237 3730 4960 4229 319 066-119909 2847 1232 5220 3726 4961 4224 302

T2 +119909 9588 950 15671 2041 11005 2420 215 061-119909 9233 948 16267 2241 13597 2439 236

0 20 40 60

0

20

40

60

P (k

N)

Damage pointUltimate pointYield point

Δ (mm)

minus20

minus20

minus40

minus40minus60

minus60

(a)

P (k

N)

0 10 20 30

0

140

70

210

Damage pointUltimate pointYield point

Δ (mm)

minus70

minus140

minus210minus20 minus10minus30

(b)

Figure 13 Skeleton curves (a) Specimen T1 (b) Specimen T2

34 Stiffness and Ductility The stiffness of the connectedspecimen is related to the displacement and the number ofcycles As the alternating load and displacement amplitudeincrease plastic deformationdevelops and stiffness graduallydegenerates

In this study the stiffness of the specimen is expressed bythe secant stiffness that corresponds to the displacement inthe same stage [15]119870119894 is the stiffness of the specimen in i loadcycle which can be calculated by

119870119894 =1003816100381610038161003816+1198651198941003816100381610038161003816 +1003816100381610038161003816minus11986511989410038161003816100381610038161003816100381610038161003816+Δ 119894

1003816100381610038161003816 +1003816100381610038161003816minusΔ1198941003816100381610038161003816 (1)

where 119865119894 is the peak load of the specimen in i load cycles Δ 119894is the displacement of the specimen in i load cycles

Figure 14(a) shows the stiffness degradation curve ofspecimen T1 and Figure 14(b) shows the stiffness degra-dation curve of specimen T2 Figure 14(a) shows that asΔ increases the stiffness of specimen T1 deteriorates andstiffness degradation is severe when control displacement islarge Figure 14(b) shows that when the control displacementis between minus20 mm and +20 mm the stiffness curve of spec-imen T2 is relatively flat When the control displacement isgreater than +20mm or less than minus20mm stiffness is rapidlydegraded The stiffness of specimen T2 is approximately 3times that of specimen T1

The load and displacement values of yield points limitpoints and damage points in Figure 13 are listed in Table 3

The yield point is calculated by the energy method [20] Thepeak points of the curve in Figure 13 are defined as limitpoints The corresponding points on the skeleton curves aredefined as damage points while the load decreases to 85 ofthe limit load [15]

The ductility coefficient 120583 is a measure of structureductility [21] which is expressed as the ratio of the ultimatedisplacement of the structure to the yield displacement

120583 =Δ 119906Δ119910 (2)

where Δ 119906 is the ultimate displacement and Δ119910 is the yielddisplacement Table 3 shows the main calculation resultsTable 3 shows that the ductility 120583 of T1 specimen is between30 and 32 and the ductility 120583 of T2 specimen is between 21and 24This finding shows that the two connection methodsexhibit good ductility and the ductility of type A connectionis better than that of type B connection

The damage of the connected specimen response can beobtained through the stiffness degradation value in each loadcycle

119863119894 =Δ1198701198941198700 (3)

where119863119894 is the damage of the specimen in each load cycle1198700is the stiffness of the joint specimen and Δ119870119894 is the stiffness

10 Mathematical Problems in Engineering

0 20 40 60

Stiff

ness

(kN

m)

Δ (mm)minus20minus40minus60

1000

1500

2000

2500

3000

3500

(a)

Stiff

ness

(kN

m)

0 10 20 302000

4000

6000

8000

10000

12000

14000

Δ (mm)minus30 minus20 minus10

(b)

Figure 14 Stiffness degeneration curve (a) Specimen T1 (b) Specimen T2

degradation value of the joint in each load cycle which canbe calculated by

Δ119870119894 = 119870119894 minus 119870119894minus1 (4)

where 119870119894 and 119870119894minus1 are the stiffness of the specimen in i andi-1 load cycles respectively The final damage of the specimencan be obtained by summing 119863119894

119863 =119899

sum119894=1

119863119894 (5)

where n is the total number of cycles when the specimen isloaded to failure

Table 3 shows the cumulative damage of the specimenunder the action of alternating load indicating that the finaldamage D of specimen T1 is slightly greater than that ofspecimen T2 The elastoplastic properties of the material arefully exerted

35 Energy Dissipation Performance Energy dissipationcapacity and damping are important indicators formeasuringstructural dynamic performance The energy dissipationcapacity is measured by the area surrounded by the hysteresiscurve of load deformation and is usually evaluated by theenergy dissipation coefficient E andor the equivalent viscousdamping coefficient 120577119890119902 The calculation is presented asfollows

E =119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(6)

120577119890119902 =12120587sdot119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(7)

where 119878(119860119861119862+119862119863119860) is the area surrounded by the hysteresiscurve in Figure 15 119878(119874119861119864+119874119863119865) is the sum of the area of thetriangle OBE and the triangle ODF in Figure 15 The energyconsumption of the yield point peak point and damage pointis shown in Table 4 Table 4 shows that the energy dissipationcoefficient and the equivalent viscosity damping coefficientof each specimen increase with displacement The energyconsumption of specimen T1 is better than that of specimenT2

X

FB

EC

D

AF o

Figure 15 Load-displacement hysteretic loop

36 Strain Analysis The strain measuring points in Figure 6were monitored during the test Figure 16 shows the distribu-tion of specimen T1 strain along with loading displacementchanges Figure 17 shows the distribution of specimen T2strain with loading displacement changes

Figure 16 shows that the type A connection is connectedto the end portion of the web in the upper chord and the webside Stress concentration occurs Strain is larger when theend portion is close The main strain direction on the sidewall is radial with the intersection of the edge of the chordweb and the side wall of the vertical rod as the origin Theend of the upper chord is connected to the sidewall of theweb where the flange strain is large in the edge and smallin the middle The end of the upper chord is connected tothe sidewall of the web and the end of the web is close tothe flange where the strain at the measuring point is smallerand the strain away from the flange is larger In general whenthe loading displacement reaches Δ119910 most of the measuredstrain at the sidewall reaches yield strain and the strain atthe end of the upper chord is still in the elastic phase When

Mathematical Problems in Engineering 11

Table 4 Energy dissipation coefficient and equivalent viscous damping coefficient

Specimen Yield point Ultimate point Damage pointE 120577119890119902 E 120577119890119902 E 120577119890119902

T1 0347 0055 1173 0187 1465 0233T2 0198 0032 0404 0064 0859 0137

0

4000

8000

A4A5A6

A13A14A15

0 20 40 60Δ (mm)

minus20minus40minus60minus8000

minus4000

Stra

in(times

10minus6)

(a)

0

1000

2000

A22A23A24

A19A20A21

minus2000

minus1000

0 20 40 60Δ (mm)

minus20minus40minus60

Stra

in(times

10minus6)

(b)

0

1000

2000

3000

A7A8A9

A1A2A3

0 20 40 60Δ (mm)

minus20minus40minus60minus2000

minus1000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

4000

A10A11A12

A16A17A18

0 20 40 60Δ (mm)

minus20minus40minus60

minus2000

minus3000

minus1000

Stra

in(times

10minus6)

(d)

Figure 16 Strain of specimen T1 (a) The part of web near the weld (b) Top chord (c) The part of web near the top chord (d) The part ofweb near the bottom chord

the loading displacement reaches Δ 119906 most of the measuredstrain of the sidewall reaches the peak value and the strainat the end of the chord gradually reaches yield strain Whenthe loading displacement reachesΔ119889 the stress redistributionoccurs due to the large crack at the joint between the sidewalland the chord web The strain at the measuring point showsa downward trend and the strain at the end of the chordcontinues to increase

The strain on the stiffening plate in the loading processamong the measuring points of the type B connection is thelargest followed by the end of the chord and the side wallstrain of the web (Figure 17) In general when the loadingdisplacement reaches Δ119910 the strain at the point where thestiffener is connected to the web of the chord initially reachesyield strain and the strain at the joint of the bar and the

stiffener bar also rapidly reach the yield strain However theother points still remain in the elastic stageWhen the loadingdisplacement reaches Δ 119906 the strain at the point where thestiffener is connected to the web of the chord reaches thepeak and the strain at the other points continues to increaseWhen the loading displacement reaches Δ119889 the strain atthe point where the stiffener is connected to the web ofthe chord is gradually decreased due to the overall bucklingdeformation of the stiffener and the chord web

4 Conclusions

The main conclusions are summarized on the basis ofcyclic pseudo-static seismic testing of web member shearconnections as follows

12 Mathematical Problems in Engineering

0 10 20 30

0

1000

2000

3000

B4B5B6

B1B2B3

Δ (mm)minus10minus20minus30

minus4000

minus3000

minus2000

minus1000

Stra

in(times

10minus6)

(a)

0

2000

4000

6000

B10B11B12

B7B8B9

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(b)

0

2000

4000

B16B17B18

B13B14B15

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

B22B23B24

B19B20B21

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus1000

minus3000

Stra

in(times

10minus6)

(d)

Figure 17 Strain of specimen T2 (a) Flank of web member (b) Top chord (c)The part of ribbed stiffener near the top chord (d)The part ofribbed stiffener near the bottom chord

(1) The force performance of the vertical rod connectionis achieved by adopting the self-designed loadingsystem which can truly reflect the mechanical perfor-mance and deformation characteristics of the verticalrod shear connection under the action of alternatingload

(2) The ductility of the material in the heat-affected zonedeteriorates due to the stress concentration at thetype A connection between the chord and the sidewall of the vertical rod as well as the heat inputof the welding Added with alternating load theplastic strain in the heat-affected zone is accumulatedand the ductility is reduced Therefore a U-shapedfracture crack is formed on the outside of the heat-affected zone of the chord web and the side wall ofthe vertical rod for specimen T1

(3) In the type B connection the strain at the connectionbetween the stiffener and the chord web is the largestthereby yielding first As the loading displacement

increases the joint between the chord web and thestiffening plate cracks and develops eventually caus-ing the stiffening plate to be buckled and undergostructural damage

(4) When the main design parameters are the same thestiffness of type B connection is approximately threetimes that of the type A connection and the bearingcapacity is approximately three times that of the typeA connection

(5) The yielding platform is long and the stiffness isdegraded slowly so that the energy can be absorbedand dissipated well due to the sufficient plasticitydevelopment of the Type A connection materialEnergy consumption performance is remarkably bet-ter than that of the type B connectionIn summary the two types of vertical rod connec-tions investigated in this study exhibit good seismicperformance and can meet the engineering designrequirements

Mathematical Problems in Engineering 13

Notation119886 Specimen length1198860 Thickness of specimen ends for material

test1198870 Working width of specimen ends for

material test119887c Web member width119887f Chord width119887g Cover-plate width119887t Width of ribbed stiffener119862 Transition length of specimen ends for

material test119863 Final damage of the specimen119863i Damage of specimen in each load cycle1198630 Width of specimen ends for material test119864 Energy dissipation coefficient119864s Youngrsquos modulus119891u Ultimate strength of steel119891y Yield strength of steel119867 Length of specimen ends for material testℎ Overall height of specimenℎ1 Chord heightℎ2 Net distance between top chord and

bottom chordℎc The distance between the top and bottom

chord centroid1198700 Stiffness of specimen119870i Stiffness of the specimen in i load cycles119870i-1 Stiffness of the specimen in i-1 load cycles1198710 Working length of specimen ends for

material test119899 Total cycling number when the specimen

is loaded to failure119875 Counter-force of loading point119875d Failure load of specimen119875u Ultimate load of specimen119875y Yield load of specimen119878(119860119861119862+119862119863119860) The area surrounded by the hysteresis

curve119878(119874119861119864+119874119863119865) The area of the triangle OBE and the

triangle ODF119905c Pip thickness of web member119905f Chord flange thickness119905g Cover-plate thickness119905t Thickness of ribbed stiffener119905w Chord web thicknessΔ Displacement of loading pointΔ d Damage displacement of specimenΔ u Ultimate displacement of specimenΔ y Yield displacement of specimenΔ119870i Stiffness degradation value of the

specimen in each load cycle120577119890119902 Equivalent viscous damping coefficient120583 Ductility coefficient

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank the National Natural ScienceFoundation of China (51568012) for financially supportingthis research

References

[1] K J Ma H G Zhang and T Zheng ldquoTheory and practiceof new type architectural space gridding structuresrdquo ChinaCommunications Press 2006

[2] J S Xu and L H Xue ldquoModern prestressed concrete floorstructurerdquoChina Architecture amp Building Press 2013

[3] Z Chen G Wu D Feng and K Ma ldquoNumerical study of thestatic and dynamic characteristics of reinforced concrete cas-sette structures for high-rise buildingsrdquo The Structural Designof Tall and Special Buildings vol 28 no 3 pp 1ndash24 2019

[4] L Jiang K J Ma and H G Zhang ldquoThe dynamic property andcomfort degree study on the steel-concrete composite vieren-deel sandwich platerdquo Earthquake Engineering and EngineeringDynamics vol 37 no 6 pp 122ndash131 2017

[5] H Luan K Ma Y Qin X Li and Z Chen ldquoInvestigationon structural behavior of an innovative orthogonal-diagonalsteel open-web sandwich floor systemrdquo Advances in StructuralEngineering vol 19 no 2 pp 353ndash371 2016

[6] H Luan K Ma Y Qin Z Chen and Y Wei ldquoInvestigation ofthe structural behavior of an innovative steel open-web floorsystemrdquo International Journal of Steel Structures vol 17 no 4pp 1365ndash1378 2017

[7] Q Chen J C Xiao K J Ma and Z C Huang ldquoEffects of shearconnector joints on static behavior in steel open-web sandwichplatesrdquo Journal of Guangxi University vol 43 no 1 pp 41ndash492018 (Chinese)

[8] Z Q Bai X D Liu D Xing Y H Wei and K J Ma ldquoExperi-mental research on mechanics characteristics for shear block ofsteel tube with t-section steel ribsrdquo Industrial Construction vol48 no 1 pp 159ndash164 2018

[9] C C Chen S W Chen M D Chung and M C LinldquoCyclic behaviour of unreinforced and rib-reinforced momentconnectionsrdquo Journal of Constructional Steel Research vol 61no 1 pp 1ndash21 2005

[10] C-C Chen C-C Lin and C-H Lin ldquoDuctile moment con-nections used in steel column-tree moment-resisting framesrdquoJournal of Constructional Steel Research vol 62 no 8 pp 793ndash801 2006

[11] H S Park D C Lee B K Oh et al ldquoPerformance-basedmulti-objective optimal seismic retrofit method for a steel moment-resisting frame considering the life-cycle costrdquo MathematicalProblems in Engineering vol 2014 Article ID 305737 14 pages2014

[12] D Sosa D Arevalo E DMora et al ldquoExperimental and analyt-ical study of slender reinforced concrete shear wall under cyclicin-plane lateral loadrdquo Mathematical Problems in Engineeringvol 2017 Article ID 4020563 14 pages 2017

[13] L Hu Theoretical Analysis and Experimental Study on SemiAssembled Composite Open-Web Sandwich Plate Outsourcing

14 Mathematical Problems in Engineering

U Type Steel Plate [Unpublished PhD Dissertation] HunanUniversity 2017

[14] L Jiang K JMaH G Zhang and L Li ldquoThe designmethod ofsteel Vierendeel beam with shear connectionrdquo Xirsquoan Universityof Architecture amp Technology (Natural Science Edition) vol 50no 3 pp 354ndash360 2018

[15] JGJ 101-2015 Specification of Testing Methods for EarthquakeResistant Building Chinese Building Industry Press 2015

[16] Z L Li C T Zhang W L Fan et al ldquoExperimental research onseismic behavior of a new type of thin-walled square steel tubebeam-column connectionrdquo Engineering Mechanics vol 30 no2 pp 72ndash81 2013

[17] G Ballio and F M Mazzolani Theory and Design of SteelStructures Chapman and Hall London UK 1983

[18] S Banno I H P Mamaghani T Usami and E Mizuno ldquoCyclicelastoplastic large deflection analysis of thin steel platesrdquo Journalof Engineering Mechanics vol 124 no 4 pp 363ndash370 1998

[19] I H P Mamaghani T Usami and E Mizuno ldquoInelastic largedeflection analysis of structural steel members under cyclicloadingrdquo Engineering Structures vol 18 no 9 pp 659ndash6681996

[20] G L Bai J Q Zhao N J Du B Xie andH Gao ldquoExperimentalstudy on seismic behavior of SRC abnormal interior joints anddesign advicerdquo Journal of Building Structures vol 39 no 7 pp33ndash45 2018

[21] W B Kratzig and Y S Petryna ldquoQuasistatic seismic damageindicators for rc structures from dissipating energies in tangen-tial subspacesrdquoMathematical Problems in Engineering vol 2014Article ID 615792 11 pages 2014

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Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 5

A1

A2

A3A4

A5A6A7

A8 A9A10A11 A12

A13A14 A15

A16A17A18

A19

A20

A21

A22

A23

A24

Pin support

Load-end

Top chord

Bottom chord

Top chord

A

A A-A

B B

B-B

Displacement sensor

Resistance strain gauge

Displacement sensor

geResistance strain gaug

xz

xy

yz

Coverplate

Web member

Web

Flange

Web

Flange Flange

Web

(a)

B1B2B3

B4B5B6

B7

B8

B9

B10

B11B12 B13

B14B15B16B17B18

B19 B20B21B22B23

B24

Load-end

Top chord

Bottom chord

Top chord

Pin support A-A

B-B

A

A

B B

Coverplate

Web member

Ribbed stiffener

Displacement sensor

Resistance strain gauge

xz

xy

yz

Web

Flange

Web

Flange

Web

Flange

Ribbed stiffener

(b)

Figure 6 Arrangement of displacement sensors and strain gauges (a) Specimen T1 (b) Specimen T2

Table 2 Mechanical property index of steel

Square tube T-section steel Cover plateS-1 S-2 S-3

119867 (mm) 50 50 50119862 (mm) 17 17 171198710 (mm) 84 84 841198860 (mm) 5 8 101198630 (mm) 30 30 301198870 (mm) 20 20 20119891y (MPa) 232 261 255119891u (MPa) 393 410 402119864s (MPa) 203e5 211e5 209e5

successively expanded into U-shaped cracks (Figure 10(c))and the bearing capacity of the members reaches a maximum

of 523 kN When displacement was loaded to the maximumset displacement of 425 mm except for the increase of thecrack at the joint of the chord and the web the microcrackappeared in the heat-affected zone where the cover and theweb were connected and the bearing capacity descendedIn general when damage occurred the bearing capacity ofthe specimen T1 did not significantly decrease Thus plasticdeformational development was not completed

When the displacement was loaded to 185 mm micro-cracks appeared at the joint between the stiffener rib andthe web of the chord in specimen T2 (Figure 11(a)) Withthe cyclic loading cracks appeared at the four joints andthe specimen performed the elastoplastic stage When thedisplacement was loaded to 205 mm the crack on the web ofthe chord began to expand obliquely and increased (Figures11(a) and 11(b)) When the displacement was loaded to 225mm the bearing capacity reached a maximum of 1625 kN

6 Mathematical Problems in Engineering

Strong floor

Reaction frame system

Actuator

Specimen

Hinge support

Movable hinge support

Movable hinge support

Movable hinge support

PinPin Actuator

Bearing

Bearing

Reaction frame

x

yz

A

C

B

D

Nuts

Screw rod

(a)

T1 T2

Reaction frame system

Actuator

Specimen Specimen

Reaction frame

Strong floor

Reaction frame system

Reaction frame

Actuator

Strong floor

(b)

Figure 7 Schematic of the test set-up (a) Self-design of loading device (b) Test photos

When the displacement was loaded to 245 mm the jointbetween the stiffener and the web began to buckle Thebearing capacity of the specimen entered the descendingphase and the crack on the web of the chord expandedrapidly When the displacement was loaded to a maximumof 265 mm the whole stiffening plate exhibited an ldquoSrdquo-type buckling and the buckling deformation is shown inFigure 11(d) After destroying the specimen the bearingcapacity is obviously reduced indicating that the plasticdeformation development is perfect

32 Hysteretic Curves The hysteresis curve is a comprehen-sive reflection of the seismic performance of the structure andthe main basis for structural seismic elastoplastic dynamic

response analysis [16] According to the literature [17ndash19] theinitial geometric defects and residual stress only affect theinitial maximum load reduction of the member and posesminimal effect on the subsequent hysteresis performanceof the member Therefore the initial geometric defects andresidual stress of themember are not considered in this paperThe hysteresis performance of the web connection node ismainly contributed by the node domain and the two partsof the rodTherefore the total reaction force P of the loadingpoint is characterized by the horizontal displacement Δ of therod loading pointThe total reaction force P of the node is theload applied by the actuator and the horizontal displacementof the loading point is the horizontal displacement of theactuator The load P-Δ hysteresis curves for specimen T1 and

Mathematical Problems in Engineering 7

0 60 120 180 240

0

20

40

60Numbers of cycles 5533333333

Time (min)

1

minus20

minus40

minus60

Top

Chor

d Ti

pD

ispla

cem

ent (

Δ[m

m])

(a)

0 50 100 150 200 250 300

0

10

20

30 555Numbers of cycles 5533333333

Time (min)

1

minus10

minus20

minus30

Top

Chor

d Ti

pD

ispla

cem

ent (

Δ[m

m])

(b)

Figure 8 Test protocols (a) Specimen T1 (b) Specimen T2

H C L0 C H

a0

D0 b 0

(a)

S-1

S-2

S-3

(b)

Figure 9 Test specimen details (a) Size indication (b) Specimen of material tests

Bottom chords

Web members

(a)

Top chords

Web members

(b)

Bottom chords

Web members

(c)

Web members

Cover plates

(d)

Figure 10 Test phenomenon and failure modes of specimen T1

8 Mathematical Problems in Engineering

Top chords

Stiffener

(a)

Bottom chords

Stiffener

(b)

Bottom chords

Stiffener

(c)

Top chords

Bottom chords

Web members

Stiffener

(d)

Figure 11 Test phenomenon and failure modes of specimen T2

0 20 40 60

0

20

40

60

P (k

N)

peak points

minus20

minus20

minus40

minus40minus60

minus60

Δ (mm)

(a)

P (k

N)

0 10 20 30

0

50

100

150

200peak points

minus30minus200

minus150

minus100

minus50

minus20 minus10

Δ (mm)

(b)

Figure 12 Experimental hysteresis curves for specimens (a) Specimen T1 (b) Specimen T2

T2 are shown inFigures 12(a) and 12(b) respectively Figure 12shows the following

A The hysteresis loops of the two connected specimensare similar in shape all of which are spindle-shaped and nopinching phenomenon are observed

B At the initial stage of loading the specimen is in anelastic working status so that the development of the P-Δcurve is basically linear The loading and unloading curvesare basically coincident that is the residual deformation issubstantially zero Residual deformation increases with loaddue to the drumming or yielding of steel in the node domainand the hysteresis loop area gradually increases therebyforming a full shuttle shape The reason is that the yieldingextent of the pipewall in the node zone is gradually increasedand the stiffness is also decreased as deformation increased

C In contrast with specimen T2 the hysteresis curve ofspecimen T1 is fuller and has stronger energy consumption

33 Skeleton Curve and Bearing Capacity The skeleton curveis the trajectory of the maximum peak of P-Δ curve at theloading end under each cyclic load In this study the skeletoncurve of the specimens can be obtained by connecting theload peak points of various hysteretic curves of the specimensuccessively The peak points used to obtain the skeletoncurve are shown in Figure 12 Figures 13(a) and 13(b) show

the skeleton curve of specimens T1 and T2 respectively Thebuckling displacement and yield load of the structure areobtained from the skeleton curve using the energy method[20] Table 3 shows the skeleton characteristic values of eachspecimen

When the loading for specimen T1 starts the skeletoncurve is a straight line and is basically in an elastic workingstatus As the load increases to approximately 065 of theultimate load the skeleton curve is gradually bended towardthe displacement axis and the slope of the curve is graduallydecreased After the peak load the load does not dropsignificantly When the loading for specimen T2 starts theskeleton curve is a straight line and is basically in an elasticworking status As the load is increased to approximately085 of the ultimate load the skeleton curve begins tobend toward the displacement axis and the slope of thecurve is decreased After the peak load the load is droppedsignificantly

Table 3 shows the experimental values of the bearingcapacity of the two types of joint specimens Under thecondition that the basic parameters of the structure are con-sistent the bearing capacity of specimen T2 is approximatelythree times that of specimen T1 The connection method ofthe vertical rod poses significant influence on the bearingcapacity of the specimen

Mathematical Problems in Engineering 9

Table 3 The measured value of bearing capacity displacement ductility coefficient and final damage

Specimen Loading direction Yield point Ultimate point Damage point 120583 119863PykN Δ 119910mm P119906kN Δ 119906mm P119889kN Δ 119889mm

T1 +119909 3260 1170 5237 3730 4960 4229 319 066-119909 2847 1232 5220 3726 4961 4224 302

T2 +119909 9588 950 15671 2041 11005 2420 215 061-119909 9233 948 16267 2241 13597 2439 236

0 20 40 60

0

20

40

60

P (k

N)

Damage pointUltimate pointYield point

Δ (mm)

minus20

minus20

minus40

minus40minus60

minus60

(a)

P (k

N)

0 10 20 30

0

140

70

210

Damage pointUltimate pointYield point

Δ (mm)

minus70

minus140

minus210minus20 minus10minus30

(b)

Figure 13 Skeleton curves (a) Specimen T1 (b) Specimen T2

34 Stiffness and Ductility The stiffness of the connectedspecimen is related to the displacement and the number ofcycles As the alternating load and displacement amplitudeincrease plastic deformationdevelops and stiffness graduallydegenerates

In this study the stiffness of the specimen is expressed bythe secant stiffness that corresponds to the displacement inthe same stage [15]119870119894 is the stiffness of the specimen in i loadcycle which can be calculated by

119870119894 =1003816100381610038161003816+1198651198941003816100381610038161003816 +1003816100381610038161003816minus11986511989410038161003816100381610038161003816100381610038161003816+Δ 119894

1003816100381610038161003816 +1003816100381610038161003816minusΔ1198941003816100381610038161003816 (1)

where 119865119894 is the peak load of the specimen in i load cycles Δ 119894is the displacement of the specimen in i load cycles

Figure 14(a) shows the stiffness degradation curve ofspecimen T1 and Figure 14(b) shows the stiffness degra-dation curve of specimen T2 Figure 14(a) shows that asΔ increases the stiffness of specimen T1 deteriorates andstiffness degradation is severe when control displacement islarge Figure 14(b) shows that when the control displacementis between minus20 mm and +20 mm the stiffness curve of spec-imen T2 is relatively flat When the control displacement isgreater than +20mm or less than minus20mm stiffness is rapidlydegraded The stiffness of specimen T2 is approximately 3times that of specimen T1

The load and displacement values of yield points limitpoints and damage points in Figure 13 are listed in Table 3

The yield point is calculated by the energy method [20] Thepeak points of the curve in Figure 13 are defined as limitpoints The corresponding points on the skeleton curves aredefined as damage points while the load decreases to 85 ofthe limit load [15]

The ductility coefficient 120583 is a measure of structureductility [21] which is expressed as the ratio of the ultimatedisplacement of the structure to the yield displacement

120583 =Δ 119906Δ119910 (2)

where Δ 119906 is the ultimate displacement and Δ119910 is the yielddisplacement Table 3 shows the main calculation resultsTable 3 shows that the ductility 120583 of T1 specimen is between30 and 32 and the ductility 120583 of T2 specimen is between 21and 24This finding shows that the two connection methodsexhibit good ductility and the ductility of type A connectionis better than that of type B connection

The damage of the connected specimen response can beobtained through the stiffness degradation value in each loadcycle

119863119894 =Δ1198701198941198700 (3)

where119863119894 is the damage of the specimen in each load cycle1198700is the stiffness of the joint specimen and Δ119870119894 is the stiffness

10 Mathematical Problems in Engineering

0 20 40 60

Stiff

ness

(kN

m)

Δ (mm)minus20minus40minus60

1000

1500

2000

2500

3000

3500

(a)

Stiff

ness

(kN

m)

0 10 20 302000

4000

6000

8000

10000

12000

14000

Δ (mm)minus30 minus20 minus10

(b)

Figure 14 Stiffness degeneration curve (a) Specimen T1 (b) Specimen T2

degradation value of the joint in each load cycle which canbe calculated by

Δ119870119894 = 119870119894 minus 119870119894minus1 (4)

where 119870119894 and 119870119894minus1 are the stiffness of the specimen in i andi-1 load cycles respectively The final damage of the specimencan be obtained by summing 119863119894

119863 =119899

sum119894=1

119863119894 (5)

where n is the total number of cycles when the specimen isloaded to failure

Table 3 shows the cumulative damage of the specimenunder the action of alternating load indicating that the finaldamage D of specimen T1 is slightly greater than that ofspecimen T2 The elastoplastic properties of the material arefully exerted

35 Energy Dissipation Performance Energy dissipationcapacity and damping are important indicators formeasuringstructural dynamic performance The energy dissipationcapacity is measured by the area surrounded by the hysteresiscurve of load deformation and is usually evaluated by theenergy dissipation coefficient E andor the equivalent viscousdamping coefficient 120577119890119902 The calculation is presented asfollows

E =119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(6)

120577119890119902 =12120587sdot119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(7)

where 119878(119860119861119862+119862119863119860) is the area surrounded by the hysteresiscurve in Figure 15 119878(119874119861119864+119874119863119865) is the sum of the area of thetriangle OBE and the triangle ODF in Figure 15 The energyconsumption of the yield point peak point and damage pointis shown in Table 4 Table 4 shows that the energy dissipationcoefficient and the equivalent viscosity damping coefficientof each specimen increase with displacement The energyconsumption of specimen T1 is better than that of specimenT2

X

FB

EC

D

AF o

Figure 15 Load-displacement hysteretic loop

36 Strain Analysis The strain measuring points in Figure 6were monitored during the test Figure 16 shows the distribu-tion of specimen T1 strain along with loading displacementchanges Figure 17 shows the distribution of specimen T2strain with loading displacement changes

Figure 16 shows that the type A connection is connectedto the end portion of the web in the upper chord and the webside Stress concentration occurs Strain is larger when theend portion is close The main strain direction on the sidewall is radial with the intersection of the edge of the chordweb and the side wall of the vertical rod as the origin Theend of the upper chord is connected to the sidewall of theweb where the flange strain is large in the edge and smallin the middle The end of the upper chord is connected tothe sidewall of the web and the end of the web is close tothe flange where the strain at the measuring point is smallerand the strain away from the flange is larger In general whenthe loading displacement reaches Δ119910 most of the measuredstrain at the sidewall reaches yield strain and the strain atthe end of the upper chord is still in the elastic phase When

Mathematical Problems in Engineering 11

Table 4 Energy dissipation coefficient and equivalent viscous damping coefficient

Specimen Yield point Ultimate point Damage pointE 120577119890119902 E 120577119890119902 E 120577119890119902

T1 0347 0055 1173 0187 1465 0233T2 0198 0032 0404 0064 0859 0137

0

4000

8000

A4A5A6

A13A14A15

0 20 40 60Δ (mm)

minus20minus40minus60minus8000

minus4000

Stra

in(times

10minus6)

(a)

0

1000

2000

A22A23A24

A19A20A21

minus2000

minus1000

0 20 40 60Δ (mm)

minus20minus40minus60

Stra

in(times

10minus6)

(b)

0

1000

2000

3000

A7A8A9

A1A2A3

0 20 40 60Δ (mm)

minus20minus40minus60minus2000

minus1000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

4000

A10A11A12

A16A17A18

0 20 40 60Δ (mm)

minus20minus40minus60

minus2000

minus3000

minus1000

Stra

in(times

10minus6)

(d)

Figure 16 Strain of specimen T1 (a) The part of web near the weld (b) Top chord (c) The part of web near the top chord (d) The part ofweb near the bottom chord

the loading displacement reaches Δ 119906 most of the measuredstrain of the sidewall reaches the peak value and the strainat the end of the chord gradually reaches yield strain Whenthe loading displacement reachesΔ119889 the stress redistributionoccurs due to the large crack at the joint between the sidewalland the chord web The strain at the measuring point showsa downward trend and the strain at the end of the chordcontinues to increase

The strain on the stiffening plate in the loading processamong the measuring points of the type B connection is thelargest followed by the end of the chord and the side wallstrain of the web (Figure 17) In general when the loadingdisplacement reaches Δ119910 the strain at the point where thestiffener is connected to the web of the chord initially reachesyield strain and the strain at the joint of the bar and the

stiffener bar also rapidly reach the yield strain However theother points still remain in the elastic stageWhen the loadingdisplacement reaches Δ 119906 the strain at the point where thestiffener is connected to the web of the chord reaches thepeak and the strain at the other points continues to increaseWhen the loading displacement reaches Δ119889 the strain atthe point where the stiffener is connected to the web ofthe chord is gradually decreased due to the overall bucklingdeformation of the stiffener and the chord web

4 Conclusions

The main conclusions are summarized on the basis ofcyclic pseudo-static seismic testing of web member shearconnections as follows

12 Mathematical Problems in Engineering

0 10 20 30

0

1000

2000

3000

B4B5B6

B1B2B3

Δ (mm)minus10minus20minus30

minus4000

minus3000

minus2000

minus1000

Stra

in(times

10minus6)

(a)

0

2000

4000

6000

B10B11B12

B7B8B9

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(b)

0

2000

4000

B16B17B18

B13B14B15

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

B22B23B24

B19B20B21

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus1000

minus3000

Stra

in(times

10minus6)

(d)

Figure 17 Strain of specimen T2 (a) Flank of web member (b) Top chord (c)The part of ribbed stiffener near the top chord (d)The part ofribbed stiffener near the bottom chord

(1) The force performance of the vertical rod connectionis achieved by adopting the self-designed loadingsystem which can truly reflect the mechanical perfor-mance and deformation characteristics of the verticalrod shear connection under the action of alternatingload

(2) The ductility of the material in the heat-affected zonedeteriorates due to the stress concentration at thetype A connection between the chord and the sidewall of the vertical rod as well as the heat inputof the welding Added with alternating load theplastic strain in the heat-affected zone is accumulatedand the ductility is reduced Therefore a U-shapedfracture crack is formed on the outside of the heat-affected zone of the chord web and the side wall ofthe vertical rod for specimen T1

(3) In the type B connection the strain at the connectionbetween the stiffener and the chord web is the largestthereby yielding first As the loading displacement

increases the joint between the chord web and thestiffening plate cracks and develops eventually caus-ing the stiffening plate to be buckled and undergostructural damage

(4) When the main design parameters are the same thestiffness of type B connection is approximately threetimes that of the type A connection and the bearingcapacity is approximately three times that of the typeA connection

(5) The yielding platform is long and the stiffness isdegraded slowly so that the energy can be absorbedand dissipated well due to the sufficient plasticitydevelopment of the Type A connection materialEnergy consumption performance is remarkably bet-ter than that of the type B connectionIn summary the two types of vertical rod connec-tions investigated in this study exhibit good seismicperformance and can meet the engineering designrequirements

Mathematical Problems in Engineering 13

Notation119886 Specimen length1198860 Thickness of specimen ends for material

test1198870 Working width of specimen ends for

material test119887c Web member width119887f Chord width119887g Cover-plate width119887t Width of ribbed stiffener119862 Transition length of specimen ends for

material test119863 Final damage of the specimen119863i Damage of specimen in each load cycle1198630 Width of specimen ends for material test119864 Energy dissipation coefficient119864s Youngrsquos modulus119891u Ultimate strength of steel119891y Yield strength of steel119867 Length of specimen ends for material testℎ Overall height of specimenℎ1 Chord heightℎ2 Net distance between top chord and

bottom chordℎc The distance between the top and bottom

chord centroid1198700 Stiffness of specimen119870i Stiffness of the specimen in i load cycles119870i-1 Stiffness of the specimen in i-1 load cycles1198710 Working length of specimen ends for

material test119899 Total cycling number when the specimen

is loaded to failure119875 Counter-force of loading point119875d Failure load of specimen119875u Ultimate load of specimen119875y Yield load of specimen119878(119860119861119862+119862119863119860) The area surrounded by the hysteresis

curve119878(119874119861119864+119874119863119865) The area of the triangle OBE and the

triangle ODF119905c Pip thickness of web member119905f Chord flange thickness119905g Cover-plate thickness119905t Thickness of ribbed stiffener119905w Chord web thicknessΔ Displacement of loading pointΔ d Damage displacement of specimenΔ u Ultimate displacement of specimenΔ y Yield displacement of specimenΔ119870i Stiffness degradation value of the

specimen in each load cycle120577119890119902 Equivalent viscous damping coefficient120583 Ductility coefficient

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank the National Natural ScienceFoundation of China (51568012) for financially supportingthis research

References

[1] K J Ma H G Zhang and T Zheng ldquoTheory and practiceof new type architectural space gridding structuresrdquo ChinaCommunications Press 2006

[2] J S Xu and L H Xue ldquoModern prestressed concrete floorstructurerdquoChina Architecture amp Building Press 2013

[3] Z Chen G Wu D Feng and K Ma ldquoNumerical study of thestatic and dynamic characteristics of reinforced concrete cas-sette structures for high-rise buildingsrdquo The Structural Designof Tall and Special Buildings vol 28 no 3 pp 1ndash24 2019

[4] L Jiang K J Ma and H G Zhang ldquoThe dynamic property andcomfort degree study on the steel-concrete composite vieren-deel sandwich platerdquo Earthquake Engineering and EngineeringDynamics vol 37 no 6 pp 122ndash131 2017

[5] H Luan K Ma Y Qin X Li and Z Chen ldquoInvestigationon structural behavior of an innovative orthogonal-diagonalsteel open-web sandwich floor systemrdquo Advances in StructuralEngineering vol 19 no 2 pp 353ndash371 2016

[6] H Luan K Ma Y Qin Z Chen and Y Wei ldquoInvestigation ofthe structural behavior of an innovative steel open-web floorsystemrdquo International Journal of Steel Structures vol 17 no 4pp 1365ndash1378 2017

[7] Q Chen J C Xiao K J Ma and Z C Huang ldquoEffects of shearconnector joints on static behavior in steel open-web sandwichplatesrdquo Journal of Guangxi University vol 43 no 1 pp 41ndash492018 (Chinese)

[8] Z Q Bai X D Liu D Xing Y H Wei and K J Ma ldquoExperi-mental research on mechanics characteristics for shear block ofsteel tube with t-section steel ribsrdquo Industrial Construction vol48 no 1 pp 159ndash164 2018

[9] C C Chen S W Chen M D Chung and M C LinldquoCyclic behaviour of unreinforced and rib-reinforced momentconnectionsrdquo Journal of Constructional Steel Research vol 61no 1 pp 1ndash21 2005

[10] C-C Chen C-C Lin and C-H Lin ldquoDuctile moment con-nections used in steel column-tree moment-resisting framesrdquoJournal of Constructional Steel Research vol 62 no 8 pp 793ndash801 2006

[11] H S Park D C Lee B K Oh et al ldquoPerformance-basedmulti-objective optimal seismic retrofit method for a steel moment-resisting frame considering the life-cycle costrdquo MathematicalProblems in Engineering vol 2014 Article ID 305737 14 pages2014

[12] D Sosa D Arevalo E DMora et al ldquoExperimental and analyt-ical study of slender reinforced concrete shear wall under cyclicin-plane lateral loadrdquo Mathematical Problems in Engineeringvol 2017 Article ID 4020563 14 pages 2017

[13] L Hu Theoretical Analysis and Experimental Study on SemiAssembled Composite Open-Web Sandwich Plate Outsourcing

14 Mathematical Problems in Engineering

U Type Steel Plate [Unpublished PhD Dissertation] HunanUniversity 2017

[14] L Jiang K JMaH G Zhang and L Li ldquoThe designmethod ofsteel Vierendeel beam with shear connectionrdquo Xirsquoan Universityof Architecture amp Technology (Natural Science Edition) vol 50no 3 pp 354ndash360 2018

[15] JGJ 101-2015 Specification of Testing Methods for EarthquakeResistant Building Chinese Building Industry Press 2015

[16] Z L Li C T Zhang W L Fan et al ldquoExperimental research onseismic behavior of a new type of thin-walled square steel tubebeam-column connectionrdquo Engineering Mechanics vol 30 no2 pp 72ndash81 2013

[17] G Ballio and F M Mazzolani Theory and Design of SteelStructures Chapman and Hall London UK 1983

[18] S Banno I H P Mamaghani T Usami and E Mizuno ldquoCyclicelastoplastic large deflection analysis of thin steel platesrdquo Journalof Engineering Mechanics vol 124 no 4 pp 363ndash370 1998

[19] I H P Mamaghani T Usami and E Mizuno ldquoInelastic largedeflection analysis of structural steel members under cyclicloadingrdquo Engineering Structures vol 18 no 9 pp 659ndash6681996

[20] G L Bai J Q Zhao N J Du B Xie andH Gao ldquoExperimentalstudy on seismic behavior of SRC abnormal interior joints anddesign advicerdquo Journal of Building Structures vol 39 no 7 pp33ndash45 2018

[21] W B Kratzig and Y S Petryna ldquoQuasistatic seismic damageindicators for rc structures from dissipating energies in tangen-tial subspacesrdquoMathematical Problems in Engineering vol 2014Article ID 615792 11 pages 2014

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Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

Strong floor

Reaction frame system

Actuator

Specimen

Hinge support

Movable hinge support

Movable hinge support

Movable hinge support

PinPin Actuator

Bearing

Bearing

Reaction frame

x

yz

A

C

B

D

Nuts

Screw rod

(a)

T1 T2

Reaction frame system

Actuator

Specimen Specimen

Reaction frame

Strong floor

Reaction frame system

Reaction frame

Actuator

Strong floor

(b)

Figure 7 Schematic of the test set-up (a) Self-design of loading device (b) Test photos

When the displacement was loaded to 245 mm the jointbetween the stiffener and the web began to buckle Thebearing capacity of the specimen entered the descendingphase and the crack on the web of the chord expandedrapidly When the displacement was loaded to a maximumof 265 mm the whole stiffening plate exhibited an ldquoSrdquo-type buckling and the buckling deformation is shown inFigure 11(d) After destroying the specimen the bearingcapacity is obviously reduced indicating that the plasticdeformation development is perfect

32 Hysteretic Curves The hysteresis curve is a comprehen-sive reflection of the seismic performance of the structure andthe main basis for structural seismic elastoplastic dynamic

response analysis [16] According to the literature [17ndash19] theinitial geometric defects and residual stress only affect theinitial maximum load reduction of the member and posesminimal effect on the subsequent hysteresis performanceof the member Therefore the initial geometric defects andresidual stress of themember are not considered in this paperThe hysteresis performance of the web connection node ismainly contributed by the node domain and the two partsof the rodTherefore the total reaction force P of the loadingpoint is characterized by the horizontal displacement Δ of therod loading pointThe total reaction force P of the node is theload applied by the actuator and the horizontal displacementof the loading point is the horizontal displacement of theactuator The load P-Δ hysteresis curves for specimen T1 and

Mathematical Problems in Engineering 7

0 60 120 180 240

0

20

40

60Numbers of cycles 5533333333

Time (min)

1

minus20

minus40

minus60

Top

Chor

d Ti

pD

ispla

cem

ent (

Δ[m

m])

(a)

0 50 100 150 200 250 300

0

10

20

30 555Numbers of cycles 5533333333

Time (min)

1

minus10

minus20

minus30

Top

Chor

d Ti

pD

ispla

cem

ent (

Δ[m

m])

(b)

Figure 8 Test protocols (a) Specimen T1 (b) Specimen T2

H C L0 C H

a0

D0 b 0

(a)

S-1

S-2

S-3

(b)

Figure 9 Test specimen details (a) Size indication (b) Specimen of material tests

Bottom chords

Web members

(a)

Top chords

Web members

(b)

Bottom chords

Web members

(c)

Web members

Cover plates

(d)

Figure 10 Test phenomenon and failure modes of specimen T1

8 Mathematical Problems in Engineering

Top chords

Stiffener

(a)

Bottom chords

Stiffener

(b)

Bottom chords

Stiffener

(c)

Top chords

Bottom chords

Web members

Stiffener

(d)

Figure 11 Test phenomenon and failure modes of specimen T2

0 20 40 60

0

20

40

60

P (k

N)

peak points

minus20

minus20

minus40

minus40minus60

minus60

Δ (mm)

(a)

P (k

N)

0 10 20 30

0

50

100

150

200peak points

minus30minus200

minus150

minus100

minus50

minus20 minus10

Δ (mm)

(b)

Figure 12 Experimental hysteresis curves for specimens (a) Specimen T1 (b) Specimen T2

T2 are shown inFigures 12(a) and 12(b) respectively Figure 12shows the following

A The hysteresis loops of the two connected specimensare similar in shape all of which are spindle-shaped and nopinching phenomenon are observed

B At the initial stage of loading the specimen is in anelastic working status so that the development of the P-Δcurve is basically linear The loading and unloading curvesare basically coincident that is the residual deformation issubstantially zero Residual deformation increases with loaddue to the drumming or yielding of steel in the node domainand the hysteresis loop area gradually increases therebyforming a full shuttle shape The reason is that the yieldingextent of the pipewall in the node zone is gradually increasedand the stiffness is also decreased as deformation increased

C In contrast with specimen T2 the hysteresis curve ofspecimen T1 is fuller and has stronger energy consumption

33 Skeleton Curve and Bearing Capacity The skeleton curveis the trajectory of the maximum peak of P-Δ curve at theloading end under each cyclic load In this study the skeletoncurve of the specimens can be obtained by connecting theload peak points of various hysteretic curves of the specimensuccessively The peak points used to obtain the skeletoncurve are shown in Figure 12 Figures 13(a) and 13(b) show

the skeleton curve of specimens T1 and T2 respectively Thebuckling displacement and yield load of the structure areobtained from the skeleton curve using the energy method[20] Table 3 shows the skeleton characteristic values of eachspecimen

When the loading for specimen T1 starts the skeletoncurve is a straight line and is basically in an elastic workingstatus As the load increases to approximately 065 of theultimate load the skeleton curve is gradually bended towardthe displacement axis and the slope of the curve is graduallydecreased After the peak load the load does not dropsignificantly When the loading for specimen T2 starts theskeleton curve is a straight line and is basically in an elasticworking status As the load is increased to approximately085 of the ultimate load the skeleton curve begins tobend toward the displacement axis and the slope of thecurve is decreased After the peak load the load is droppedsignificantly

Table 3 shows the experimental values of the bearingcapacity of the two types of joint specimens Under thecondition that the basic parameters of the structure are con-sistent the bearing capacity of specimen T2 is approximatelythree times that of specimen T1 The connection method ofthe vertical rod poses significant influence on the bearingcapacity of the specimen

Mathematical Problems in Engineering 9

Table 3 The measured value of bearing capacity displacement ductility coefficient and final damage

Specimen Loading direction Yield point Ultimate point Damage point 120583 119863PykN Δ 119910mm P119906kN Δ 119906mm P119889kN Δ 119889mm

T1 +119909 3260 1170 5237 3730 4960 4229 319 066-119909 2847 1232 5220 3726 4961 4224 302

T2 +119909 9588 950 15671 2041 11005 2420 215 061-119909 9233 948 16267 2241 13597 2439 236

0 20 40 60

0

20

40

60

P (k

N)

Damage pointUltimate pointYield point

Δ (mm)

minus20

minus20

minus40

minus40minus60

minus60

(a)

P (k

N)

0 10 20 30

0

140

70

210

Damage pointUltimate pointYield point

Δ (mm)

minus70

minus140

minus210minus20 minus10minus30

(b)

Figure 13 Skeleton curves (a) Specimen T1 (b) Specimen T2

34 Stiffness and Ductility The stiffness of the connectedspecimen is related to the displacement and the number ofcycles As the alternating load and displacement amplitudeincrease plastic deformationdevelops and stiffness graduallydegenerates

In this study the stiffness of the specimen is expressed bythe secant stiffness that corresponds to the displacement inthe same stage [15]119870119894 is the stiffness of the specimen in i loadcycle which can be calculated by

119870119894 =1003816100381610038161003816+1198651198941003816100381610038161003816 +1003816100381610038161003816minus11986511989410038161003816100381610038161003816100381610038161003816+Δ 119894

1003816100381610038161003816 +1003816100381610038161003816minusΔ1198941003816100381610038161003816 (1)

where 119865119894 is the peak load of the specimen in i load cycles Δ 119894is the displacement of the specimen in i load cycles

Figure 14(a) shows the stiffness degradation curve ofspecimen T1 and Figure 14(b) shows the stiffness degra-dation curve of specimen T2 Figure 14(a) shows that asΔ increases the stiffness of specimen T1 deteriorates andstiffness degradation is severe when control displacement islarge Figure 14(b) shows that when the control displacementis between minus20 mm and +20 mm the stiffness curve of spec-imen T2 is relatively flat When the control displacement isgreater than +20mm or less than minus20mm stiffness is rapidlydegraded The stiffness of specimen T2 is approximately 3times that of specimen T1

The load and displacement values of yield points limitpoints and damage points in Figure 13 are listed in Table 3

The yield point is calculated by the energy method [20] Thepeak points of the curve in Figure 13 are defined as limitpoints The corresponding points on the skeleton curves aredefined as damage points while the load decreases to 85 ofthe limit load [15]

The ductility coefficient 120583 is a measure of structureductility [21] which is expressed as the ratio of the ultimatedisplacement of the structure to the yield displacement

120583 =Δ 119906Δ119910 (2)

where Δ 119906 is the ultimate displacement and Δ119910 is the yielddisplacement Table 3 shows the main calculation resultsTable 3 shows that the ductility 120583 of T1 specimen is between30 and 32 and the ductility 120583 of T2 specimen is between 21and 24This finding shows that the two connection methodsexhibit good ductility and the ductility of type A connectionis better than that of type B connection

The damage of the connected specimen response can beobtained through the stiffness degradation value in each loadcycle

119863119894 =Δ1198701198941198700 (3)

where119863119894 is the damage of the specimen in each load cycle1198700is the stiffness of the joint specimen and Δ119870119894 is the stiffness

10 Mathematical Problems in Engineering

0 20 40 60

Stiff

ness

(kN

m)

Δ (mm)minus20minus40minus60

1000

1500

2000

2500

3000

3500

(a)

Stiff

ness

(kN

m)

0 10 20 302000

4000

6000

8000

10000

12000

14000

Δ (mm)minus30 minus20 minus10

(b)

Figure 14 Stiffness degeneration curve (a) Specimen T1 (b) Specimen T2

degradation value of the joint in each load cycle which canbe calculated by

Δ119870119894 = 119870119894 minus 119870119894minus1 (4)

where 119870119894 and 119870119894minus1 are the stiffness of the specimen in i andi-1 load cycles respectively The final damage of the specimencan be obtained by summing 119863119894

119863 =119899

sum119894=1

119863119894 (5)

where n is the total number of cycles when the specimen isloaded to failure

Table 3 shows the cumulative damage of the specimenunder the action of alternating load indicating that the finaldamage D of specimen T1 is slightly greater than that ofspecimen T2 The elastoplastic properties of the material arefully exerted

35 Energy Dissipation Performance Energy dissipationcapacity and damping are important indicators formeasuringstructural dynamic performance The energy dissipationcapacity is measured by the area surrounded by the hysteresiscurve of load deformation and is usually evaluated by theenergy dissipation coefficient E andor the equivalent viscousdamping coefficient 120577119890119902 The calculation is presented asfollows

E =119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(6)

120577119890119902 =12120587sdot119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(7)

where 119878(119860119861119862+119862119863119860) is the area surrounded by the hysteresiscurve in Figure 15 119878(119874119861119864+119874119863119865) is the sum of the area of thetriangle OBE and the triangle ODF in Figure 15 The energyconsumption of the yield point peak point and damage pointis shown in Table 4 Table 4 shows that the energy dissipationcoefficient and the equivalent viscosity damping coefficientof each specimen increase with displacement The energyconsumption of specimen T1 is better than that of specimenT2

X

FB

EC

D

AF o

Figure 15 Load-displacement hysteretic loop

36 Strain Analysis The strain measuring points in Figure 6were monitored during the test Figure 16 shows the distribu-tion of specimen T1 strain along with loading displacementchanges Figure 17 shows the distribution of specimen T2strain with loading displacement changes

Figure 16 shows that the type A connection is connectedto the end portion of the web in the upper chord and the webside Stress concentration occurs Strain is larger when theend portion is close The main strain direction on the sidewall is radial with the intersection of the edge of the chordweb and the side wall of the vertical rod as the origin Theend of the upper chord is connected to the sidewall of theweb where the flange strain is large in the edge and smallin the middle The end of the upper chord is connected tothe sidewall of the web and the end of the web is close tothe flange where the strain at the measuring point is smallerand the strain away from the flange is larger In general whenthe loading displacement reaches Δ119910 most of the measuredstrain at the sidewall reaches yield strain and the strain atthe end of the upper chord is still in the elastic phase When

Mathematical Problems in Engineering 11

Table 4 Energy dissipation coefficient and equivalent viscous damping coefficient

Specimen Yield point Ultimate point Damage pointE 120577119890119902 E 120577119890119902 E 120577119890119902

T1 0347 0055 1173 0187 1465 0233T2 0198 0032 0404 0064 0859 0137

0

4000

8000

A4A5A6

A13A14A15

0 20 40 60Δ (mm)

minus20minus40minus60minus8000

minus4000

Stra

in(times

10minus6)

(a)

0

1000

2000

A22A23A24

A19A20A21

minus2000

minus1000

0 20 40 60Δ (mm)

minus20minus40minus60

Stra

in(times

10minus6)

(b)

0

1000

2000

3000

A7A8A9

A1A2A3

0 20 40 60Δ (mm)

minus20minus40minus60minus2000

minus1000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

4000

A10A11A12

A16A17A18

0 20 40 60Δ (mm)

minus20minus40minus60

minus2000

minus3000

minus1000

Stra

in(times

10minus6)

(d)

Figure 16 Strain of specimen T1 (a) The part of web near the weld (b) Top chord (c) The part of web near the top chord (d) The part ofweb near the bottom chord

the loading displacement reaches Δ 119906 most of the measuredstrain of the sidewall reaches the peak value and the strainat the end of the chord gradually reaches yield strain Whenthe loading displacement reachesΔ119889 the stress redistributionoccurs due to the large crack at the joint between the sidewalland the chord web The strain at the measuring point showsa downward trend and the strain at the end of the chordcontinues to increase

The strain on the stiffening plate in the loading processamong the measuring points of the type B connection is thelargest followed by the end of the chord and the side wallstrain of the web (Figure 17) In general when the loadingdisplacement reaches Δ119910 the strain at the point where thestiffener is connected to the web of the chord initially reachesyield strain and the strain at the joint of the bar and the

stiffener bar also rapidly reach the yield strain However theother points still remain in the elastic stageWhen the loadingdisplacement reaches Δ 119906 the strain at the point where thestiffener is connected to the web of the chord reaches thepeak and the strain at the other points continues to increaseWhen the loading displacement reaches Δ119889 the strain atthe point where the stiffener is connected to the web ofthe chord is gradually decreased due to the overall bucklingdeformation of the stiffener and the chord web

4 Conclusions

The main conclusions are summarized on the basis ofcyclic pseudo-static seismic testing of web member shearconnections as follows

12 Mathematical Problems in Engineering

0 10 20 30

0

1000

2000

3000

B4B5B6

B1B2B3

Δ (mm)minus10minus20minus30

minus4000

minus3000

minus2000

minus1000

Stra

in(times

10minus6)

(a)

0

2000

4000

6000

B10B11B12

B7B8B9

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(b)

0

2000

4000

B16B17B18

B13B14B15

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

B22B23B24

B19B20B21

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus1000

minus3000

Stra

in(times

10minus6)

(d)

Figure 17 Strain of specimen T2 (a) Flank of web member (b) Top chord (c)The part of ribbed stiffener near the top chord (d)The part ofribbed stiffener near the bottom chord

(1) The force performance of the vertical rod connectionis achieved by adopting the self-designed loadingsystem which can truly reflect the mechanical perfor-mance and deformation characteristics of the verticalrod shear connection under the action of alternatingload

(2) The ductility of the material in the heat-affected zonedeteriorates due to the stress concentration at thetype A connection between the chord and the sidewall of the vertical rod as well as the heat inputof the welding Added with alternating load theplastic strain in the heat-affected zone is accumulatedand the ductility is reduced Therefore a U-shapedfracture crack is formed on the outside of the heat-affected zone of the chord web and the side wall ofthe vertical rod for specimen T1

(3) In the type B connection the strain at the connectionbetween the stiffener and the chord web is the largestthereby yielding first As the loading displacement

increases the joint between the chord web and thestiffening plate cracks and develops eventually caus-ing the stiffening plate to be buckled and undergostructural damage

(4) When the main design parameters are the same thestiffness of type B connection is approximately threetimes that of the type A connection and the bearingcapacity is approximately three times that of the typeA connection

(5) The yielding platform is long and the stiffness isdegraded slowly so that the energy can be absorbedand dissipated well due to the sufficient plasticitydevelopment of the Type A connection materialEnergy consumption performance is remarkably bet-ter than that of the type B connectionIn summary the two types of vertical rod connec-tions investigated in this study exhibit good seismicperformance and can meet the engineering designrequirements

Mathematical Problems in Engineering 13

Notation119886 Specimen length1198860 Thickness of specimen ends for material

test1198870 Working width of specimen ends for

material test119887c Web member width119887f Chord width119887g Cover-plate width119887t Width of ribbed stiffener119862 Transition length of specimen ends for

material test119863 Final damage of the specimen119863i Damage of specimen in each load cycle1198630 Width of specimen ends for material test119864 Energy dissipation coefficient119864s Youngrsquos modulus119891u Ultimate strength of steel119891y Yield strength of steel119867 Length of specimen ends for material testℎ Overall height of specimenℎ1 Chord heightℎ2 Net distance between top chord and

bottom chordℎc The distance between the top and bottom

chord centroid1198700 Stiffness of specimen119870i Stiffness of the specimen in i load cycles119870i-1 Stiffness of the specimen in i-1 load cycles1198710 Working length of specimen ends for

material test119899 Total cycling number when the specimen

is loaded to failure119875 Counter-force of loading point119875d Failure load of specimen119875u Ultimate load of specimen119875y Yield load of specimen119878(119860119861119862+119862119863119860) The area surrounded by the hysteresis

curve119878(119874119861119864+119874119863119865) The area of the triangle OBE and the

triangle ODF119905c Pip thickness of web member119905f Chord flange thickness119905g Cover-plate thickness119905t Thickness of ribbed stiffener119905w Chord web thicknessΔ Displacement of loading pointΔ d Damage displacement of specimenΔ u Ultimate displacement of specimenΔ y Yield displacement of specimenΔ119870i Stiffness degradation value of the

specimen in each load cycle120577119890119902 Equivalent viscous damping coefficient120583 Ductility coefficient

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank the National Natural ScienceFoundation of China (51568012) for financially supportingthis research

References

[1] K J Ma H G Zhang and T Zheng ldquoTheory and practiceof new type architectural space gridding structuresrdquo ChinaCommunications Press 2006

[2] J S Xu and L H Xue ldquoModern prestressed concrete floorstructurerdquoChina Architecture amp Building Press 2013

[3] Z Chen G Wu D Feng and K Ma ldquoNumerical study of thestatic and dynamic characteristics of reinforced concrete cas-sette structures for high-rise buildingsrdquo The Structural Designof Tall and Special Buildings vol 28 no 3 pp 1ndash24 2019

[4] L Jiang K J Ma and H G Zhang ldquoThe dynamic property andcomfort degree study on the steel-concrete composite vieren-deel sandwich platerdquo Earthquake Engineering and EngineeringDynamics vol 37 no 6 pp 122ndash131 2017

[5] H Luan K Ma Y Qin X Li and Z Chen ldquoInvestigationon structural behavior of an innovative orthogonal-diagonalsteel open-web sandwich floor systemrdquo Advances in StructuralEngineering vol 19 no 2 pp 353ndash371 2016

[6] H Luan K Ma Y Qin Z Chen and Y Wei ldquoInvestigation ofthe structural behavior of an innovative steel open-web floorsystemrdquo International Journal of Steel Structures vol 17 no 4pp 1365ndash1378 2017

[7] Q Chen J C Xiao K J Ma and Z C Huang ldquoEffects of shearconnector joints on static behavior in steel open-web sandwichplatesrdquo Journal of Guangxi University vol 43 no 1 pp 41ndash492018 (Chinese)

[8] Z Q Bai X D Liu D Xing Y H Wei and K J Ma ldquoExperi-mental research on mechanics characteristics for shear block ofsteel tube with t-section steel ribsrdquo Industrial Construction vol48 no 1 pp 159ndash164 2018

[9] C C Chen S W Chen M D Chung and M C LinldquoCyclic behaviour of unreinforced and rib-reinforced momentconnectionsrdquo Journal of Constructional Steel Research vol 61no 1 pp 1ndash21 2005

[10] C-C Chen C-C Lin and C-H Lin ldquoDuctile moment con-nections used in steel column-tree moment-resisting framesrdquoJournal of Constructional Steel Research vol 62 no 8 pp 793ndash801 2006

[11] H S Park D C Lee B K Oh et al ldquoPerformance-basedmulti-objective optimal seismic retrofit method for a steel moment-resisting frame considering the life-cycle costrdquo MathematicalProblems in Engineering vol 2014 Article ID 305737 14 pages2014

[12] D Sosa D Arevalo E DMora et al ldquoExperimental and analyt-ical study of slender reinforced concrete shear wall under cyclicin-plane lateral loadrdquo Mathematical Problems in Engineeringvol 2017 Article ID 4020563 14 pages 2017

[13] L Hu Theoretical Analysis and Experimental Study on SemiAssembled Composite Open-Web Sandwich Plate Outsourcing

14 Mathematical Problems in Engineering

U Type Steel Plate [Unpublished PhD Dissertation] HunanUniversity 2017

[14] L Jiang K JMaH G Zhang and L Li ldquoThe designmethod ofsteel Vierendeel beam with shear connectionrdquo Xirsquoan Universityof Architecture amp Technology (Natural Science Edition) vol 50no 3 pp 354ndash360 2018

[15] JGJ 101-2015 Specification of Testing Methods for EarthquakeResistant Building Chinese Building Industry Press 2015

[16] Z L Li C T Zhang W L Fan et al ldquoExperimental research onseismic behavior of a new type of thin-walled square steel tubebeam-column connectionrdquo Engineering Mechanics vol 30 no2 pp 72ndash81 2013

[17] G Ballio and F M Mazzolani Theory and Design of SteelStructures Chapman and Hall London UK 1983

[18] S Banno I H P Mamaghani T Usami and E Mizuno ldquoCyclicelastoplastic large deflection analysis of thin steel platesrdquo Journalof Engineering Mechanics vol 124 no 4 pp 363ndash370 1998

[19] I H P Mamaghani T Usami and E Mizuno ldquoInelastic largedeflection analysis of structural steel members under cyclicloadingrdquo Engineering Structures vol 18 no 9 pp 659ndash6681996

[20] G L Bai J Q Zhao N J Du B Xie andH Gao ldquoExperimentalstudy on seismic behavior of SRC abnormal interior joints anddesign advicerdquo Journal of Building Structures vol 39 no 7 pp33ndash45 2018

[21] W B Kratzig and Y S Petryna ldquoQuasistatic seismic damageindicators for rc structures from dissipating energies in tangen-tial subspacesrdquoMathematical Problems in Engineering vol 2014Article ID 615792 11 pages 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

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Dierential EquationsInternational Journal of

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AnalysisInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

0 60 120 180 240

0

20

40

60Numbers of cycles 5533333333

Time (min)

1

minus20

minus40

minus60

Top

Chor

d Ti

pD

ispla

cem

ent (

Δ[m

m])

(a)

0 50 100 150 200 250 300

0

10

20

30 555Numbers of cycles 5533333333

Time (min)

1

minus10

minus20

minus30

Top

Chor

d Ti

pD

ispla

cem

ent (

Δ[m

m])

(b)

Figure 8 Test protocols (a) Specimen T1 (b) Specimen T2

H C L0 C H

a0

D0 b 0

(a)

S-1

S-2

S-3

(b)

Figure 9 Test specimen details (a) Size indication (b) Specimen of material tests

Bottom chords

Web members

(a)

Top chords

Web members

(b)

Bottom chords

Web members

(c)

Web members

Cover plates

(d)

Figure 10 Test phenomenon and failure modes of specimen T1

8 Mathematical Problems in Engineering

Top chords

Stiffener

(a)

Bottom chords

Stiffener

(b)

Bottom chords

Stiffener

(c)

Top chords

Bottom chords

Web members

Stiffener

(d)

Figure 11 Test phenomenon and failure modes of specimen T2

0 20 40 60

0

20

40

60

P (k

N)

peak points

minus20

minus20

minus40

minus40minus60

minus60

Δ (mm)

(a)

P (k

N)

0 10 20 30

0

50

100

150

200peak points

minus30minus200

minus150

minus100

minus50

minus20 minus10

Δ (mm)

(b)

Figure 12 Experimental hysteresis curves for specimens (a) Specimen T1 (b) Specimen T2

T2 are shown inFigures 12(a) and 12(b) respectively Figure 12shows the following

A The hysteresis loops of the two connected specimensare similar in shape all of which are spindle-shaped and nopinching phenomenon are observed

B At the initial stage of loading the specimen is in anelastic working status so that the development of the P-Δcurve is basically linear The loading and unloading curvesare basically coincident that is the residual deformation issubstantially zero Residual deformation increases with loaddue to the drumming or yielding of steel in the node domainand the hysteresis loop area gradually increases therebyforming a full shuttle shape The reason is that the yieldingextent of the pipewall in the node zone is gradually increasedand the stiffness is also decreased as deformation increased

C In contrast with specimen T2 the hysteresis curve ofspecimen T1 is fuller and has stronger energy consumption

33 Skeleton Curve and Bearing Capacity The skeleton curveis the trajectory of the maximum peak of P-Δ curve at theloading end under each cyclic load In this study the skeletoncurve of the specimens can be obtained by connecting theload peak points of various hysteretic curves of the specimensuccessively The peak points used to obtain the skeletoncurve are shown in Figure 12 Figures 13(a) and 13(b) show

the skeleton curve of specimens T1 and T2 respectively Thebuckling displacement and yield load of the structure areobtained from the skeleton curve using the energy method[20] Table 3 shows the skeleton characteristic values of eachspecimen

When the loading for specimen T1 starts the skeletoncurve is a straight line and is basically in an elastic workingstatus As the load increases to approximately 065 of theultimate load the skeleton curve is gradually bended towardthe displacement axis and the slope of the curve is graduallydecreased After the peak load the load does not dropsignificantly When the loading for specimen T2 starts theskeleton curve is a straight line and is basically in an elasticworking status As the load is increased to approximately085 of the ultimate load the skeleton curve begins tobend toward the displacement axis and the slope of thecurve is decreased After the peak load the load is droppedsignificantly

Table 3 shows the experimental values of the bearingcapacity of the two types of joint specimens Under thecondition that the basic parameters of the structure are con-sistent the bearing capacity of specimen T2 is approximatelythree times that of specimen T1 The connection method ofthe vertical rod poses significant influence on the bearingcapacity of the specimen

Mathematical Problems in Engineering 9

Table 3 The measured value of bearing capacity displacement ductility coefficient and final damage

Specimen Loading direction Yield point Ultimate point Damage point 120583 119863PykN Δ 119910mm P119906kN Δ 119906mm P119889kN Δ 119889mm

T1 +119909 3260 1170 5237 3730 4960 4229 319 066-119909 2847 1232 5220 3726 4961 4224 302

T2 +119909 9588 950 15671 2041 11005 2420 215 061-119909 9233 948 16267 2241 13597 2439 236

0 20 40 60

0

20

40

60

P (k

N)

Damage pointUltimate pointYield point

Δ (mm)

minus20

minus20

minus40

minus40minus60

minus60

(a)

P (k

N)

0 10 20 30

0

140

70

210

Damage pointUltimate pointYield point

Δ (mm)

minus70

minus140

minus210minus20 minus10minus30

(b)

Figure 13 Skeleton curves (a) Specimen T1 (b) Specimen T2

34 Stiffness and Ductility The stiffness of the connectedspecimen is related to the displacement and the number ofcycles As the alternating load and displacement amplitudeincrease plastic deformationdevelops and stiffness graduallydegenerates

In this study the stiffness of the specimen is expressed bythe secant stiffness that corresponds to the displacement inthe same stage [15]119870119894 is the stiffness of the specimen in i loadcycle which can be calculated by

119870119894 =1003816100381610038161003816+1198651198941003816100381610038161003816 +1003816100381610038161003816minus11986511989410038161003816100381610038161003816100381610038161003816+Δ 119894

1003816100381610038161003816 +1003816100381610038161003816minusΔ1198941003816100381610038161003816 (1)

where 119865119894 is the peak load of the specimen in i load cycles Δ 119894is the displacement of the specimen in i load cycles

Figure 14(a) shows the stiffness degradation curve ofspecimen T1 and Figure 14(b) shows the stiffness degra-dation curve of specimen T2 Figure 14(a) shows that asΔ increases the stiffness of specimen T1 deteriorates andstiffness degradation is severe when control displacement islarge Figure 14(b) shows that when the control displacementis between minus20 mm and +20 mm the stiffness curve of spec-imen T2 is relatively flat When the control displacement isgreater than +20mm or less than minus20mm stiffness is rapidlydegraded The stiffness of specimen T2 is approximately 3times that of specimen T1

The load and displacement values of yield points limitpoints and damage points in Figure 13 are listed in Table 3

The yield point is calculated by the energy method [20] Thepeak points of the curve in Figure 13 are defined as limitpoints The corresponding points on the skeleton curves aredefined as damage points while the load decreases to 85 ofthe limit load [15]

The ductility coefficient 120583 is a measure of structureductility [21] which is expressed as the ratio of the ultimatedisplacement of the structure to the yield displacement

120583 =Δ 119906Δ119910 (2)

where Δ 119906 is the ultimate displacement and Δ119910 is the yielddisplacement Table 3 shows the main calculation resultsTable 3 shows that the ductility 120583 of T1 specimen is between30 and 32 and the ductility 120583 of T2 specimen is between 21and 24This finding shows that the two connection methodsexhibit good ductility and the ductility of type A connectionis better than that of type B connection

The damage of the connected specimen response can beobtained through the stiffness degradation value in each loadcycle

119863119894 =Δ1198701198941198700 (3)

where119863119894 is the damage of the specimen in each load cycle1198700is the stiffness of the joint specimen and Δ119870119894 is the stiffness

10 Mathematical Problems in Engineering

0 20 40 60

Stiff

ness

(kN

m)

Δ (mm)minus20minus40minus60

1000

1500

2000

2500

3000

3500

(a)

Stiff

ness

(kN

m)

0 10 20 302000

4000

6000

8000

10000

12000

14000

Δ (mm)minus30 minus20 minus10

(b)

Figure 14 Stiffness degeneration curve (a) Specimen T1 (b) Specimen T2

degradation value of the joint in each load cycle which canbe calculated by

Δ119870119894 = 119870119894 minus 119870119894minus1 (4)

where 119870119894 and 119870119894minus1 are the stiffness of the specimen in i andi-1 load cycles respectively The final damage of the specimencan be obtained by summing 119863119894

119863 =119899

sum119894=1

119863119894 (5)

where n is the total number of cycles when the specimen isloaded to failure

Table 3 shows the cumulative damage of the specimenunder the action of alternating load indicating that the finaldamage D of specimen T1 is slightly greater than that ofspecimen T2 The elastoplastic properties of the material arefully exerted

35 Energy Dissipation Performance Energy dissipationcapacity and damping are important indicators formeasuringstructural dynamic performance The energy dissipationcapacity is measured by the area surrounded by the hysteresiscurve of load deformation and is usually evaluated by theenergy dissipation coefficient E andor the equivalent viscousdamping coefficient 120577119890119902 The calculation is presented asfollows

E =119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(6)

120577119890119902 =12120587sdot119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(7)

where 119878(119860119861119862+119862119863119860) is the area surrounded by the hysteresiscurve in Figure 15 119878(119874119861119864+119874119863119865) is the sum of the area of thetriangle OBE and the triangle ODF in Figure 15 The energyconsumption of the yield point peak point and damage pointis shown in Table 4 Table 4 shows that the energy dissipationcoefficient and the equivalent viscosity damping coefficientof each specimen increase with displacement The energyconsumption of specimen T1 is better than that of specimenT2

X

FB

EC

D

AF o

Figure 15 Load-displacement hysteretic loop

36 Strain Analysis The strain measuring points in Figure 6were monitored during the test Figure 16 shows the distribu-tion of specimen T1 strain along with loading displacementchanges Figure 17 shows the distribution of specimen T2strain with loading displacement changes

Figure 16 shows that the type A connection is connectedto the end portion of the web in the upper chord and the webside Stress concentration occurs Strain is larger when theend portion is close The main strain direction on the sidewall is radial with the intersection of the edge of the chordweb and the side wall of the vertical rod as the origin Theend of the upper chord is connected to the sidewall of theweb where the flange strain is large in the edge and smallin the middle The end of the upper chord is connected tothe sidewall of the web and the end of the web is close tothe flange where the strain at the measuring point is smallerand the strain away from the flange is larger In general whenthe loading displacement reaches Δ119910 most of the measuredstrain at the sidewall reaches yield strain and the strain atthe end of the upper chord is still in the elastic phase When

Mathematical Problems in Engineering 11

Table 4 Energy dissipation coefficient and equivalent viscous damping coefficient

Specimen Yield point Ultimate point Damage pointE 120577119890119902 E 120577119890119902 E 120577119890119902

T1 0347 0055 1173 0187 1465 0233T2 0198 0032 0404 0064 0859 0137

0

4000

8000

A4A5A6

A13A14A15

0 20 40 60Δ (mm)

minus20minus40minus60minus8000

minus4000

Stra

in(times

10minus6)

(a)

0

1000

2000

A22A23A24

A19A20A21

minus2000

minus1000

0 20 40 60Δ (mm)

minus20minus40minus60

Stra

in(times

10minus6)

(b)

0

1000

2000

3000

A7A8A9

A1A2A3

0 20 40 60Δ (mm)

minus20minus40minus60minus2000

minus1000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

4000

A10A11A12

A16A17A18

0 20 40 60Δ (mm)

minus20minus40minus60

minus2000

minus3000

minus1000

Stra

in(times

10minus6)

(d)

Figure 16 Strain of specimen T1 (a) The part of web near the weld (b) Top chord (c) The part of web near the top chord (d) The part ofweb near the bottom chord

the loading displacement reaches Δ 119906 most of the measuredstrain of the sidewall reaches the peak value and the strainat the end of the chord gradually reaches yield strain Whenthe loading displacement reachesΔ119889 the stress redistributionoccurs due to the large crack at the joint between the sidewalland the chord web The strain at the measuring point showsa downward trend and the strain at the end of the chordcontinues to increase

The strain on the stiffening plate in the loading processamong the measuring points of the type B connection is thelargest followed by the end of the chord and the side wallstrain of the web (Figure 17) In general when the loadingdisplacement reaches Δ119910 the strain at the point where thestiffener is connected to the web of the chord initially reachesyield strain and the strain at the joint of the bar and the

stiffener bar also rapidly reach the yield strain However theother points still remain in the elastic stageWhen the loadingdisplacement reaches Δ 119906 the strain at the point where thestiffener is connected to the web of the chord reaches thepeak and the strain at the other points continues to increaseWhen the loading displacement reaches Δ119889 the strain atthe point where the stiffener is connected to the web ofthe chord is gradually decreased due to the overall bucklingdeformation of the stiffener and the chord web

4 Conclusions

The main conclusions are summarized on the basis ofcyclic pseudo-static seismic testing of web member shearconnections as follows

12 Mathematical Problems in Engineering

0 10 20 30

0

1000

2000

3000

B4B5B6

B1B2B3

Δ (mm)minus10minus20minus30

minus4000

minus3000

minus2000

minus1000

Stra

in(times

10minus6)

(a)

0

2000

4000

6000

B10B11B12

B7B8B9

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(b)

0

2000

4000

B16B17B18

B13B14B15

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

B22B23B24

B19B20B21

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus1000

minus3000

Stra

in(times

10minus6)

(d)

Figure 17 Strain of specimen T2 (a) Flank of web member (b) Top chord (c)The part of ribbed stiffener near the top chord (d)The part ofribbed stiffener near the bottom chord

(1) The force performance of the vertical rod connectionis achieved by adopting the self-designed loadingsystem which can truly reflect the mechanical perfor-mance and deformation characteristics of the verticalrod shear connection under the action of alternatingload

(2) The ductility of the material in the heat-affected zonedeteriorates due to the stress concentration at thetype A connection between the chord and the sidewall of the vertical rod as well as the heat inputof the welding Added with alternating load theplastic strain in the heat-affected zone is accumulatedand the ductility is reduced Therefore a U-shapedfracture crack is formed on the outside of the heat-affected zone of the chord web and the side wall ofthe vertical rod for specimen T1

(3) In the type B connection the strain at the connectionbetween the stiffener and the chord web is the largestthereby yielding first As the loading displacement

increases the joint between the chord web and thestiffening plate cracks and develops eventually caus-ing the stiffening plate to be buckled and undergostructural damage

(4) When the main design parameters are the same thestiffness of type B connection is approximately threetimes that of the type A connection and the bearingcapacity is approximately three times that of the typeA connection

(5) The yielding platform is long and the stiffness isdegraded slowly so that the energy can be absorbedand dissipated well due to the sufficient plasticitydevelopment of the Type A connection materialEnergy consumption performance is remarkably bet-ter than that of the type B connectionIn summary the two types of vertical rod connec-tions investigated in this study exhibit good seismicperformance and can meet the engineering designrequirements

Mathematical Problems in Engineering 13

Notation119886 Specimen length1198860 Thickness of specimen ends for material

test1198870 Working width of specimen ends for

material test119887c Web member width119887f Chord width119887g Cover-plate width119887t Width of ribbed stiffener119862 Transition length of specimen ends for

material test119863 Final damage of the specimen119863i Damage of specimen in each load cycle1198630 Width of specimen ends for material test119864 Energy dissipation coefficient119864s Youngrsquos modulus119891u Ultimate strength of steel119891y Yield strength of steel119867 Length of specimen ends for material testℎ Overall height of specimenℎ1 Chord heightℎ2 Net distance between top chord and

bottom chordℎc The distance between the top and bottom

chord centroid1198700 Stiffness of specimen119870i Stiffness of the specimen in i load cycles119870i-1 Stiffness of the specimen in i-1 load cycles1198710 Working length of specimen ends for

material test119899 Total cycling number when the specimen

is loaded to failure119875 Counter-force of loading point119875d Failure load of specimen119875u Ultimate load of specimen119875y Yield load of specimen119878(119860119861119862+119862119863119860) The area surrounded by the hysteresis

curve119878(119874119861119864+119874119863119865) The area of the triangle OBE and the

triangle ODF119905c Pip thickness of web member119905f Chord flange thickness119905g Cover-plate thickness119905t Thickness of ribbed stiffener119905w Chord web thicknessΔ Displacement of loading pointΔ d Damage displacement of specimenΔ u Ultimate displacement of specimenΔ y Yield displacement of specimenΔ119870i Stiffness degradation value of the

specimen in each load cycle120577119890119902 Equivalent viscous damping coefficient120583 Ductility coefficient

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank the National Natural ScienceFoundation of China (51568012) for financially supportingthis research

References

[1] K J Ma H G Zhang and T Zheng ldquoTheory and practiceof new type architectural space gridding structuresrdquo ChinaCommunications Press 2006

[2] J S Xu and L H Xue ldquoModern prestressed concrete floorstructurerdquoChina Architecture amp Building Press 2013

[3] Z Chen G Wu D Feng and K Ma ldquoNumerical study of thestatic and dynamic characteristics of reinforced concrete cas-sette structures for high-rise buildingsrdquo The Structural Designof Tall and Special Buildings vol 28 no 3 pp 1ndash24 2019

[4] L Jiang K J Ma and H G Zhang ldquoThe dynamic property andcomfort degree study on the steel-concrete composite vieren-deel sandwich platerdquo Earthquake Engineering and EngineeringDynamics vol 37 no 6 pp 122ndash131 2017

[5] H Luan K Ma Y Qin X Li and Z Chen ldquoInvestigationon structural behavior of an innovative orthogonal-diagonalsteel open-web sandwich floor systemrdquo Advances in StructuralEngineering vol 19 no 2 pp 353ndash371 2016

[6] H Luan K Ma Y Qin Z Chen and Y Wei ldquoInvestigation ofthe structural behavior of an innovative steel open-web floorsystemrdquo International Journal of Steel Structures vol 17 no 4pp 1365ndash1378 2017

[7] Q Chen J C Xiao K J Ma and Z C Huang ldquoEffects of shearconnector joints on static behavior in steel open-web sandwichplatesrdquo Journal of Guangxi University vol 43 no 1 pp 41ndash492018 (Chinese)

[8] Z Q Bai X D Liu D Xing Y H Wei and K J Ma ldquoExperi-mental research on mechanics characteristics for shear block ofsteel tube with t-section steel ribsrdquo Industrial Construction vol48 no 1 pp 159ndash164 2018

[9] C C Chen S W Chen M D Chung and M C LinldquoCyclic behaviour of unreinforced and rib-reinforced momentconnectionsrdquo Journal of Constructional Steel Research vol 61no 1 pp 1ndash21 2005

[10] C-C Chen C-C Lin and C-H Lin ldquoDuctile moment con-nections used in steel column-tree moment-resisting framesrdquoJournal of Constructional Steel Research vol 62 no 8 pp 793ndash801 2006

[11] H S Park D C Lee B K Oh et al ldquoPerformance-basedmulti-objective optimal seismic retrofit method for a steel moment-resisting frame considering the life-cycle costrdquo MathematicalProblems in Engineering vol 2014 Article ID 305737 14 pages2014

[12] D Sosa D Arevalo E DMora et al ldquoExperimental and analyt-ical study of slender reinforced concrete shear wall under cyclicin-plane lateral loadrdquo Mathematical Problems in Engineeringvol 2017 Article ID 4020563 14 pages 2017

[13] L Hu Theoretical Analysis and Experimental Study on SemiAssembled Composite Open-Web Sandwich Plate Outsourcing

14 Mathematical Problems in Engineering

U Type Steel Plate [Unpublished PhD Dissertation] HunanUniversity 2017

[14] L Jiang K JMaH G Zhang and L Li ldquoThe designmethod ofsteel Vierendeel beam with shear connectionrdquo Xirsquoan Universityof Architecture amp Technology (Natural Science Edition) vol 50no 3 pp 354ndash360 2018

[15] JGJ 101-2015 Specification of Testing Methods for EarthquakeResistant Building Chinese Building Industry Press 2015

[16] Z L Li C T Zhang W L Fan et al ldquoExperimental research onseismic behavior of a new type of thin-walled square steel tubebeam-column connectionrdquo Engineering Mechanics vol 30 no2 pp 72ndash81 2013

[17] G Ballio and F M Mazzolani Theory and Design of SteelStructures Chapman and Hall London UK 1983

[18] S Banno I H P Mamaghani T Usami and E Mizuno ldquoCyclicelastoplastic large deflection analysis of thin steel platesrdquo Journalof Engineering Mechanics vol 124 no 4 pp 363ndash370 1998

[19] I H P Mamaghani T Usami and E Mizuno ldquoInelastic largedeflection analysis of structural steel members under cyclicloadingrdquo Engineering Structures vol 18 no 9 pp 659ndash6681996

[20] G L Bai J Q Zhao N J Du B Xie andH Gao ldquoExperimentalstudy on seismic behavior of SRC abnormal interior joints anddesign advicerdquo Journal of Building Structures vol 39 no 7 pp33ndash45 2018

[21] W B Kratzig and Y S Petryna ldquoQuasistatic seismic damageindicators for rc structures from dissipating energies in tangen-tial subspacesrdquoMathematical Problems in Engineering vol 2014Article ID 615792 11 pages 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

Top chords

Stiffener

(a)

Bottom chords

Stiffener

(b)

Bottom chords

Stiffener

(c)

Top chords

Bottom chords

Web members

Stiffener

(d)

Figure 11 Test phenomenon and failure modes of specimen T2

0 20 40 60

0

20

40

60

P (k

N)

peak points

minus20

minus20

minus40

minus40minus60

minus60

Δ (mm)

(a)

P (k

N)

0 10 20 30

0

50

100

150

200peak points

minus30minus200

minus150

minus100

minus50

minus20 minus10

Δ (mm)

(b)

Figure 12 Experimental hysteresis curves for specimens (a) Specimen T1 (b) Specimen T2

T2 are shown inFigures 12(a) and 12(b) respectively Figure 12shows the following

A The hysteresis loops of the two connected specimensare similar in shape all of which are spindle-shaped and nopinching phenomenon are observed

B At the initial stage of loading the specimen is in anelastic working status so that the development of the P-Δcurve is basically linear The loading and unloading curvesare basically coincident that is the residual deformation issubstantially zero Residual deformation increases with loaddue to the drumming or yielding of steel in the node domainand the hysteresis loop area gradually increases therebyforming a full shuttle shape The reason is that the yieldingextent of the pipewall in the node zone is gradually increasedand the stiffness is also decreased as deformation increased

C In contrast with specimen T2 the hysteresis curve ofspecimen T1 is fuller and has stronger energy consumption

33 Skeleton Curve and Bearing Capacity The skeleton curveis the trajectory of the maximum peak of P-Δ curve at theloading end under each cyclic load In this study the skeletoncurve of the specimens can be obtained by connecting theload peak points of various hysteretic curves of the specimensuccessively The peak points used to obtain the skeletoncurve are shown in Figure 12 Figures 13(a) and 13(b) show

the skeleton curve of specimens T1 and T2 respectively Thebuckling displacement and yield load of the structure areobtained from the skeleton curve using the energy method[20] Table 3 shows the skeleton characteristic values of eachspecimen

When the loading for specimen T1 starts the skeletoncurve is a straight line and is basically in an elastic workingstatus As the load increases to approximately 065 of theultimate load the skeleton curve is gradually bended towardthe displacement axis and the slope of the curve is graduallydecreased After the peak load the load does not dropsignificantly When the loading for specimen T2 starts theskeleton curve is a straight line and is basically in an elasticworking status As the load is increased to approximately085 of the ultimate load the skeleton curve begins tobend toward the displacement axis and the slope of thecurve is decreased After the peak load the load is droppedsignificantly

Table 3 shows the experimental values of the bearingcapacity of the two types of joint specimens Under thecondition that the basic parameters of the structure are con-sistent the bearing capacity of specimen T2 is approximatelythree times that of specimen T1 The connection method ofthe vertical rod poses significant influence on the bearingcapacity of the specimen

Mathematical Problems in Engineering 9

Table 3 The measured value of bearing capacity displacement ductility coefficient and final damage

Specimen Loading direction Yield point Ultimate point Damage point 120583 119863PykN Δ 119910mm P119906kN Δ 119906mm P119889kN Δ 119889mm

T1 +119909 3260 1170 5237 3730 4960 4229 319 066-119909 2847 1232 5220 3726 4961 4224 302

T2 +119909 9588 950 15671 2041 11005 2420 215 061-119909 9233 948 16267 2241 13597 2439 236

0 20 40 60

0

20

40

60

P (k

N)

Damage pointUltimate pointYield point

Δ (mm)

minus20

minus20

minus40

minus40minus60

minus60

(a)

P (k

N)

0 10 20 30

0

140

70

210

Damage pointUltimate pointYield point

Δ (mm)

minus70

minus140

minus210minus20 minus10minus30

(b)

Figure 13 Skeleton curves (a) Specimen T1 (b) Specimen T2

34 Stiffness and Ductility The stiffness of the connectedspecimen is related to the displacement and the number ofcycles As the alternating load and displacement amplitudeincrease plastic deformationdevelops and stiffness graduallydegenerates

In this study the stiffness of the specimen is expressed bythe secant stiffness that corresponds to the displacement inthe same stage [15]119870119894 is the stiffness of the specimen in i loadcycle which can be calculated by

119870119894 =1003816100381610038161003816+1198651198941003816100381610038161003816 +1003816100381610038161003816minus11986511989410038161003816100381610038161003816100381610038161003816+Δ 119894

1003816100381610038161003816 +1003816100381610038161003816minusΔ1198941003816100381610038161003816 (1)

where 119865119894 is the peak load of the specimen in i load cycles Δ 119894is the displacement of the specimen in i load cycles

Figure 14(a) shows the stiffness degradation curve ofspecimen T1 and Figure 14(b) shows the stiffness degra-dation curve of specimen T2 Figure 14(a) shows that asΔ increases the stiffness of specimen T1 deteriorates andstiffness degradation is severe when control displacement islarge Figure 14(b) shows that when the control displacementis between minus20 mm and +20 mm the stiffness curve of spec-imen T2 is relatively flat When the control displacement isgreater than +20mm or less than minus20mm stiffness is rapidlydegraded The stiffness of specimen T2 is approximately 3times that of specimen T1

The load and displacement values of yield points limitpoints and damage points in Figure 13 are listed in Table 3

The yield point is calculated by the energy method [20] Thepeak points of the curve in Figure 13 are defined as limitpoints The corresponding points on the skeleton curves aredefined as damage points while the load decreases to 85 ofthe limit load [15]

The ductility coefficient 120583 is a measure of structureductility [21] which is expressed as the ratio of the ultimatedisplacement of the structure to the yield displacement

120583 =Δ 119906Δ119910 (2)

where Δ 119906 is the ultimate displacement and Δ119910 is the yielddisplacement Table 3 shows the main calculation resultsTable 3 shows that the ductility 120583 of T1 specimen is between30 and 32 and the ductility 120583 of T2 specimen is between 21and 24This finding shows that the two connection methodsexhibit good ductility and the ductility of type A connectionis better than that of type B connection

The damage of the connected specimen response can beobtained through the stiffness degradation value in each loadcycle

119863119894 =Δ1198701198941198700 (3)

where119863119894 is the damage of the specimen in each load cycle1198700is the stiffness of the joint specimen and Δ119870119894 is the stiffness

10 Mathematical Problems in Engineering

0 20 40 60

Stiff

ness

(kN

m)

Δ (mm)minus20minus40minus60

1000

1500

2000

2500

3000

3500

(a)

Stiff

ness

(kN

m)

0 10 20 302000

4000

6000

8000

10000

12000

14000

Δ (mm)minus30 minus20 minus10

(b)

Figure 14 Stiffness degeneration curve (a) Specimen T1 (b) Specimen T2

degradation value of the joint in each load cycle which canbe calculated by

Δ119870119894 = 119870119894 minus 119870119894minus1 (4)

where 119870119894 and 119870119894minus1 are the stiffness of the specimen in i andi-1 load cycles respectively The final damage of the specimencan be obtained by summing 119863119894

119863 =119899

sum119894=1

119863119894 (5)

where n is the total number of cycles when the specimen isloaded to failure

Table 3 shows the cumulative damage of the specimenunder the action of alternating load indicating that the finaldamage D of specimen T1 is slightly greater than that ofspecimen T2 The elastoplastic properties of the material arefully exerted

35 Energy Dissipation Performance Energy dissipationcapacity and damping are important indicators formeasuringstructural dynamic performance The energy dissipationcapacity is measured by the area surrounded by the hysteresiscurve of load deformation and is usually evaluated by theenergy dissipation coefficient E andor the equivalent viscousdamping coefficient 120577119890119902 The calculation is presented asfollows

E =119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(6)

120577119890119902 =12120587sdot119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(7)

where 119878(119860119861119862+119862119863119860) is the area surrounded by the hysteresiscurve in Figure 15 119878(119874119861119864+119874119863119865) is the sum of the area of thetriangle OBE and the triangle ODF in Figure 15 The energyconsumption of the yield point peak point and damage pointis shown in Table 4 Table 4 shows that the energy dissipationcoefficient and the equivalent viscosity damping coefficientof each specimen increase with displacement The energyconsumption of specimen T1 is better than that of specimenT2

X

FB

EC

D

AF o

Figure 15 Load-displacement hysteretic loop

36 Strain Analysis The strain measuring points in Figure 6were monitored during the test Figure 16 shows the distribu-tion of specimen T1 strain along with loading displacementchanges Figure 17 shows the distribution of specimen T2strain with loading displacement changes

Figure 16 shows that the type A connection is connectedto the end portion of the web in the upper chord and the webside Stress concentration occurs Strain is larger when theend portion is close The main strain direction on the sidewall is radial with the intersection of the edge of the chordweb and the side wall of the vertical rod as the origin Theend of the upper chord is connected to the sidewall of theweb where the flange strain is large in the edge and smallin the middle The end of the upper chord is connected tothe sidewall of the web and the end of the web is close tothe flange where the strain at the measuring point is smallerand the strain away from the flange is larger In general whenthe loading displacement reaches Δ119910 most of the measuredstrain at the sidewall reaches yield strain and the strain atthe end of the upper chord is still in the elastic phase When

Mathematical Problems in Engineering 11

Table 4 Energy dissipation coefficient and equivalent viscous damping coefficient

Specimen Yield point Ultimate point Damage pointE 120577119890119902 E 120577119890119902 E 120577119890119902

T1 0347 0055 1173 0187 1465 0233T2 0198 0032 0404 0064 0859 0137

0

4000

8000

A4A5A6

A13A14A15

0 20 40 60Δ (mm)

minus20minus40minus60minus8000

minus4000

Stra

in(times

10minus6)

(a)

0

1000

2000

A22A23A24

A19A20A21

minus2000

minus1000

0 20 40 60Δ (mm)

minus20minus40minus60

Stra

in(times

10minus6)

(b)

0

1000

2000

3000

A7A8A9

A1A2A3

0 20 40 60Δ (mm)

minus20minus40minus60minus2000

minus1000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

4000

A10A11A12

A16A17A18

0 20 40 60Δ (mm)

minus20minus40minus60

minus2000

minus3000

minus1000

Stra

in(times

10minus6)

(d)

Figure 16 Strain of specimen T1 (a) The part of web near the weld (b) Top chord (c) The part of web near the top chord (d) The part ofweb near the bottom chord

the loading displacement reaches Δ 119906 most of the measuredstrain of the sidewall reaches the peak value and the strainat the end of the chord gradually reaches yield strain Whenthe loading displacement reachesΔ119889 the stress redistributionoccurs due to the large crack at the joint between the sidewalland the chord web The strain at the measuring point showsa downward trend and the strain at the end of the chordcontinues to increase

The strain on the stiffening plate in the loading processamong the measuring points of the type B connection is thelargest followed by the end of the chord and the side wallstrain of the web (Figure 17) In general when the loadingdisplacement reaches Δ119910 the strain at the point where thestiffener is connected to the web of the chord initially reachesyield strain and the strain at the joint of the bar and the

stiffener bar also rapidly reach the yield strain However theother points still remain in the elastic stageWhen the loadingdisplacement reaches Δ 119906 the strain at the point where thestiffener is connected to the web of the chord reaches thepeak and the strain at the other points continues to increaseWhen the loading displacement reaches Δ119889 the strain atthe point where the stiffener is connected to the web ofthe chord is gradually decreased due to the overall bucklingdeformation of the stiffener and the chord web

4 Conclusions

The main conclusions are summarized on the basis ofcyclic pseudo-static seismic testing of web member shearconnections as follows

12 Mathematical Problems in Engineering

0 10 20 30

0

1000

2000

3000

B4B5B6

B1B2B3

Δ (mm)minus10minus20minus30

minus4000

minus3000

minus2000

minus1000

Stra

in(times

10minus6)

(a)

0

2000

4000

6000

B10B11B12

B7B8B9

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(b)

0

2000

4000

B16B17B18

B13B14B15

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

B22B23B24

B19B20B21

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus1000

minus3000

Stra

in(times

10minus6)

(d)

Figure 17 Strain of specimen T2 (a) Flank of web member (b) Top chord (c)The part of ribbed stiffener near the top chord (d)The part ofribbed stiffener near the bottom chord

(1) The force performance of the vertical rod connectionis achieved by adopting the self-designed loadingsystem which can truly reflect the mechanical perfor-mance and deformation characteristics of the verticalrod shear connection under the action of alternatingload

(2) The ductility of the material in the heat-affected zonedeteriorates due to the stress concentration at thetype A connection between the chord and the sidewall of the vertical rod as well as the heat inputof the welding Added with alternating load theplastic strain in the heat-affected zone is accumulatedand the ductility is reduced Therefore a U-shapedfracture crack is formed on the outside of the heat-affected zone of the chord web and the side wall ofthe vertical rod for specimen T1

(3) In the type B connection the strain at the connectionbetween the stiffener and the chord web is the largestthereby yielding first As the loading displacement

increases the joint between the chord web and thestiffening plate cracks and develops eventually caus-ing the stiffening plate to be buckled and undergostructural damage

(4) When the main design parameters are the same thestiffness of type B connection is approximately threetimes that of the type A connection and the bearingcapacity is approximately three times that of the typeA connection

(5) The yielding platform is long and the stiffness isdegraded slowly so that the energy can be absorbedand dissipated well due to the sufficient plasticitydevelopment of the Type A connection materialEnergy consumption performance is remarkably bet-ter than that of the type B connectionIn summary the two types of vertical rod connec-tions investigated in this study exhibit good seismicperformance and can meet the engineering designrequirements

Mathematical Problems in Engineering 13

Notation119886 Specimen length1198860 Thickness of specimen ends for material

test1198870 Working width of specimen ends for

material test119887c Web member width119887f Chord width119887g Cover-plate width119887t Width of ribbed stiffener119862 Transition length of specimen ends for

material test119863 Final damage of the specimen119863i Damage of specimen in each load cycle1198630 Width of specimen ends for material test119864 Energy dissipation coefficient119864s Youngrsquos modulus119891u Ultimate strength of steel119891y Yield strength of steel119867 Length of specimen ends for material testℎ Overall height of specimenℎ1 Chord heightℎ2 Net distance between top chord and

bottom chordℎc The distance between the top and bottom

chord centroid1198700 Stiffness of specimen119870i Stiffness of the specimen in i load cycles119870i-1 Stiffness of the specimen in i-1 load cycles1198710 Working length of specimen ends for

material test119899 Total cycling number when the specimen

is loaded to failure119875 Counter-force of loading point119875d Failure load of specimen119875u Ultimate load of specimen119875y Yield load of specimen119878(119860119861119862+119862119863119860) The area surrounded by the hysteresis

curve119878(119874119861119864+119874119863119865) The area of the triangle OBE and the

triangle ODF119905c Pip thickness of web member119905f Chord flange thickness119905g Cover-plate thickness119905t Thickness of ribbed stiffener119905w Chord web thicknessΔ Displacement of loading pointΔ d Damage displacement of specimenΔ u Ultimate displacement of specimenΔ y Yield displacement of specimenΔ119870i Stiffness degradation value of the

specimen in each load cycle120577119890119902 Equivalent viscous damping coefficient120583 Ductility coefficient

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank the National Natural ScienceFoundation of China (51568012) for financially supportingthis research

References

[1] K J Ma H G Zhang and T Zheng ldquoTheory and practiceof new type architectural space gridding structuresrdquo ChinaCommunications Press 2006

[2] J S Xu and L H Xue ldquoModern prestressed concrete floorstructurerdquoChina Architecture amp Building Press 2013

[3] Z Chen G Wu D Feng and K Ma ldquoNumerical study of thestatic and dynamic characteristics of reinforced concrete cas-sette structures for high-rise buildingsrdquo The Structural Designof Tall and Special Buildings vol 28 no 3 pp 1ndash24 2019

[4] L Jiang K J Ma and H G Zhang ldquoThe dynamic property andcomfort degree study on the steel-concrete composite vieren-deel sandwich platerdquo Earthquake Engineering and EngineeringDynamics vol 37 no 6 pp 122ndash131 2017

[5] H Luan K Ma Y Qin X Li and Z Chen ldquoInvestigationon structural behavior of an innovative orthogonal-diagonalsteel open-web sandwich floor systemrdquo Advances in StructuralEngineering vol 19 no 2 pp 353ndash371 2016

[6] H Luan K Ma Y Qin Z Chen and Y Wei ldquoInvestigation ofthe structural behavior of an innovative steel open-web floorsystemrdquo International Journal of Steel Structures vol 17 no 4pp 1365ndash1378 2017

[7] Q Chen J C Xiao K J Ma and Z C Huang ldquoEffects of shearconnector joints on static behavior in steel open-web sandwichplatesrdquo Journal of Guangxi University vol 43 no 1 pp 41ndash492018 (Chinese)

[8] Z Q Bai X D Liu D Xing Y H Wei and K J Ma ldquoExperi-mental research on mechanics characteristics for shear block ofsteel tube with t-section steel ribsrdquo Industrial Construction vol48 no 1 pp 159ndash164 2018

[9] C C Chen S W Chen M D Chung and M C LinldquoCyclic behaviour of unreinforced and rib-reinforced momentconnectionsrdquo Journal of Constructional Steel Research vol 61no 1 pp 1ndash21 2005

[10] C-C Chen C-C Lin and C-H Lin ldquoDuctile moment con-nections used in steel column-tree moment-resisting framesrdquoJournal of Constructional Steel Research vol 62 no 8 pp 793ndash801 2006

[11] H S Park D C Lee B K Oh et al ldquoPerformance-basedmulti-objective optimal seismic retrofit method for a steel moment-resisting frame considering the life-cycle costrdquo MathematicalProblems in Engineering vol 2014 Article ID 305737 14 pages2014

[12] D Sosa D Arevalo E DMora et al ldquoExperimental and analyt-ical study of slender reinforced concrete shear wall under cyclicin-plane lateral loadrdquo Mathematical Problems in Engineeringvol 2017 Article ID 4020563 14 pages 2017

[13] L Hu Theoretical Analysis and Experimental Study on SemiAssembled Composite Open-Web Sandwich Plate Outsourcing

14 Mathematical Problems in Engineering

U Type Steel Plate [Unpublished PhD Dissertation] HunanUniversity 2017

[14] L Jiang K JMaH G Zhang and L Li ldquoThe designmethod ofsteel Vierendeel beam with shear connectionrdquo Xirsquoan Universityof Architecture amp Technology (Natural Science Edition) vol 50no 3 pp 354ndash360 2018

[15] JGJ 101-2015 Specification of Testing Methods for EarthquakeResistant Building Chinese Building Industry Press 2015

[16] Z L Li C T Zhang W L Fan et al ldquoExperimental research onseismic behavior of a new type of thin-walled square steel tubebeam-column connectionrdquo Engineering Mechanics vol 30 no2 pp 72ndash81 2013

[17] G Ballio and F M Mazzolani Theory and Design of SteelStructures Chapman and Hall London UK 1983

[18] S Banno I H P Mamaghani T Usami and E Mizuno ldquoCyclicelastoplastic large deflection analysis of thin steel platesrdquo Journalof Engineering Mechanics vol 124 no 4 pp 363ndash370 1998

[19] I H P Mamaghani T Usami and E Mizuno ldquoInelastic largedeflection analysis of structural steel members under cyclicloadingrdquo Engineering Structures vol 18 no 9 pp 659ndash6681996

[20] G L Bai J Q Zhao N J Du B Xie andH Gao ldquoExperimentalstudy on seismic behavior of SRC abnormal interior joints anddesign advicerdquo Journal of Building Structures vol 39 no 7 pp33ndash45 2018

[21] W B Kratzig and Y S Petryna ldquoQuasistatic seismic damageindicators for rc structures from dissipating energies in tangen-tial subspacesrdquoMathematical Problems in Engineering vol 2014Article ID 615792 11 pages 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

Table 3 The measured value of bearing capacity displacement ductility coefficient and final damage

Specimen Loading direction Yield point Ultimate point Damage point 120583 119863PykN Δ 119910mm P119906kN Δ 119906mm P119889kN Δ 119889mm

T1 +119909 3260 1170 5237 3730 4960 4229 319 066-119909 2847 1232 5220 3726 4961 4224 302

T2 +119909 9588 950 15671 2041 11005 2420 215 061-119909 9233 948 16267 2241 13597 2439 236

0 20 40 60

0

20

40

60

P (k

N)

Damage pointUltimate pointYield point

Δ (mm)

minus20

minus20

minus40

minus40minus60

minus60

(a)

P (k

N)

0 10 20 30

0

140

70

210

Damage pointUltimate pointYield point

Δ (mm)

minus70

minus140

minus210minus20 minus10minus30

(b)

Figure 13 Skeleton curves (a) Specimen T1 (b) Specimen T2

34 Stiffness and Ductility The stiffness of the connectedspecimen is related to the displacement and the number ofcycles As the alternating load and displacement amplitudeincrease plastic deformationdevelops and stiffness graduallydegenerates

In this study the stiffness of the specimen is expressed bythe secant stiffness that corresponds to the displacement inthe same stage [15]119870119894 is the stiffness of the specimen in i loadcycle which can be calculated by

119870119894 =1003816100381610038161003816+1198651198941003816100381610038161003816 +1003816100381610038161003816minus11986511989410038161003816100381610038161003816100381610038161003816+Δ 119894

1003816100381610038161003816 +1003816100381610038161003816minusΔ1198941003816100381610038161003816 (1)

where 119865119894 is the peak load of the specimen in i load cycles Δ 119894is the displacement of the specimen in i load cycles

Figure 14(a) shows the stiffness degradation curve ofspecimen T1 and Figure 14(b) shows the stiffness degra-dation curve of specimen T2 Figure 14(a) shows that asΔ increases the stiffness of specimen T1 deteriorates andstiffness degradation is severe when control displacement islarge Figure 14(b) shows that when the control displacementis between minus20 mm and +20 mm the stiffness curve of spec-imen T2 is relatively flat When the control displacement isgreater than +20mm or less than minus20mm stiffness is rapidlydegraded The stiffness of specimen T2 is approximately 3times that of specimen T1

The load and displacement values of yield points limitpoints and damage points in Figure 13 are listed in Table 3

The yield point is calculated by the energy method [20] Thepeak points of the curve in Figure 13 are defined as limitpoints The corresponding points on the skeleton curves aredefined as damage points while the load decreases to 85 ofthe limit load [15]

The ductility coefficient 120583 is a measure of structureductility [21] which is expressed as the ratio of the ultimatedisplacement of the structure to the yield displacement

120583 =Δ 119906Δ119910 (2)

where Δ 119906 is the ultimate displacement and Δ119910 is the yielddisplacement Table 3 shows the main calculation resultsTable 3 shows that the ductility 120583 of T1 specimen is between30 and 32 and the ductility 120583 of T2 specimen is between 21and 24This finding shows that the two connection methodsexhibit good ductility and the ductility of type A connectionis better than that of type B connection

The damage of the connected specimen response can beobtained through the stiffness degradation value in each loadcycle

119863119894 =Δ1198701198941198700 (3)

where119863119894 is the damage of the specimen in each load cycle1198700is the stiffness of the joint specimen and Δ119870119894 is the stiffness

10 Mathematical Problems in Engineering

0 20 40 60

Stiff

ness

(kN

m)

Δ (mm)minus20minus40minus60

1000

1500

2000

2500

3000

3500

(a)

Stiff

ness

(kN

m)

0 10 20 302000

4000

6000

8000

10000

12000

14000

Δ (mm)minus30 minus20 minus10

(b)

Figure 14 Stiffness degeneration curve (a) Specimen T1 (b) Specimen T2

degradation value of the joint in each load cycle which canbe calculated by

Δ119870119894 = 119870119894 minus 119870119894minus1 (4)

where 119870119894 and 119870119894minus1 are the stiffness of the specimen in i andi-1 load cycles respectively The final damage of the specimencan be obtained by summing 119863119894

119863 =119899

sum119894=1

119863119894 (5)

where n is the total number of cycles when the specimen isloaded to failure

Table 3 shows the cumulative damage of the specimenunder the action of alternating load indicating that the finaldamage D of specimen T1 is slightly greater than that ofspecimen T2 The elastoplastic properties of the material arefully exerted

35 Energy Dissipation Performance Energy dissipationcapacity and damping are important indicators formeasuringstructural dynamic performance The energy dissipationcapacity is measured by the area surrounded by the hysteresiscurve of load deformation and is usually evaluated by theenergy dissipation coefficient E andor the equivalent viscousdamping coefficient 120577119890119902 The calculation is presented asfollows

E =119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(6)

120577119890119902 =12120587sdot119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(7)

where 119878(119860119861119862+119862119863119860) is the area surrounded by the hysteresiscurve in Figure 15 119878(119874119861119864+119874119863119865) is the sum of the area of thetriangle OBE and the triangle ODF in Figure 15 The energyconsumption of the yield point peak point and damage pointis shown in Table 4 Table 4 shows that the energy dissipationcoefficient and the equivalent viscosity damping coefficientof each specimen increase with displacement The energyconsumption of specimen T1 is better than that of specimenT2

X

FB

EC

D

AF o

Figure 15 Load-displacement hysteretic loop

36 Strain Analysis The strain measuring points in Figure 6were monitored during the test Figure 16 shows the distribu-tion of specimen T1 strain along with loading displacementchanges Figure 17 shows the distribution of specimen T2strain with loading displacement changes

Figure 16 shows that the type A connection is connectedto the end portion of the web in the upper chord and the webside Stress concentration occurs Strain is larger when theend portion is close The main strain direction on the sidewall is radial with the intersection of the edge of the chordweb and the side wall of the vertical rod as the origin Theend of the upper chord is connected to the sidewall of theweb where the flange strain is large in the edge and smallin the middle The end of the upper chord is connected tothe sidewall of the web and the end of the web is close tothe flange where the strain at the measuring point is smallerand the strain away from the flange is larger In general whenthe loading displacement reaches Δ119910 most of the measuredstrain at the sidewall reaches yield strain and the strain atthe end of the upper chord is still in the elastic phase When

Mathematical Problems in Engineering 11

Table 4 Energy dissipation coefficient and equivalent viscous damping coefficient

Specimen Yield point Ultimate point Damage pointE 120577119890119902 E 120577119890119902 E 120577119890119902

T1 0347 0055 1173 0187 1465 0233T2 0198 0032 0404 0064 0859 0137

0

4000

8000

A4A5A6

A13A14A15

0 20 40 60Δ (mm)

minus20minus40minus60minus8000

minus4000

Stra

in(times

10minus6)

(a)

0

1000

2000

A22A23A24

A19A20A21

minus2000

minus1000

0 20 40 60Δ (mm)

minus20minus40minus60

Stra

in(times

10minus6)

(b)

0

1000

2000

3000

A7A8A9

A1A2A3

0 20 40 60Δ (mm)

minus20minus40minus60minus2000

minus1000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

4000

A10A11A12

A16A17A18

0 20 40 60Δ (mm)

minus20minus40minus60

minus2000

minus3000

minus1000

Stra

in(times

10minus6)

(d)

Figure 16 Strain of specimen T1 (a) The part of web near the weld (b) Top chord (c) The part of web near the top chord (d) The part ofweb near the bottom chord

the loading displacement reaches Δ 119906 most of the measuredstrain of the sidewall reaches the peak value and the strainat the end of the chord gradually reaches yield strain Whenthe loading displacement reachesΔ119889 the stress redistributionoccurs due to the large crack at the joint between the sidewalland the chord web The strain at the measuring point showsa downward trend and the strain at the end of the chordcontinues to increase

The strain on the stiffening plate in the loading processamong the measuring points of the type B connection is thelargest followed by the end of the chord and the side wallstrain of the web (Figure 17) In general when the loadingdisplacement reaches Δ119910 the strain at the point where thestiffener is connected to the web of the chord initially reachesyield strain and the strain at the joint of the bar and the

stiffener bar also rapidly reach the yield strain However theother points still remain in the elastic stageWhen the loadingdisplacement reaches Δ 119906 the strain at the point where thestiffener is connected to the web of the chord reaches thepeak and the strain at the other points continues to increaseWhen the loading displacement reaches Δ119889 the strain atthe point where the stiffener is connected to the web ofthe chord is gradually decreased due to the overall bucklingdeformation of the stiffener and the chord web

4 Conclusions

The main conclusions are summarized on the basis ofcyclic pseudo-static seismic testing of web member shearconnections as follows

12 Mathematical Problems in Engineering

0 10 20 30

0

1000

2000

3000

B4B5B6

B1B2B3

Δ (mm)minus10minus20minus30

minus4000

minus3000

minus2000

minus1000

Stra

in(times

10minus6)

(a)

0

2000

4000

6000

B10B11B12

B7B8B9

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(b)

0

2000

4000

B16B17B18

B13B14B15

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

B22B23B24

B19B20B21

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus1000

minus3000

Stra

in(times

10minus6)

(d)

Figure 17 Strain of specimen T2 (a) Flank of web member (b) Top chord (c)The part of ribbed stiffener near the top chord (d)The part ofribbed stiffener near the bottom chord

(1) The force performance of the vertical rod connectionis achieved by adopting the self-designed loadingsystem which can truly reflect the mechanical perfor-mance and deformation characteristics of the verticalrod shear connection under the action of alternatingload

(2) The ductility of the material in the heat-affected zonedeteriorates due to the stress concentration at thetype A connection between the chord and the sidewall of the vertical rod as well as the heat inputof the welding Added with alternating load theplastic strain in the heat-affected zone is accumulatedand the ductility is reduced Therefore a U-shapedfracture crack is formed on the outside of the heat-affected zone of the chord web and the side wall ofthe vertical rod for specimen T1

(3) In the type B connection the strain at the connectionbetween the stiffener and the chord web is the largestthereby yielding first As the loading displacement

increases the joint between the chord web and thestiffening plate cracks and develops eventually caus-ing the stiffening plate to be buckled and undergostructural damage

(4) When the main design parameters are the same thestiffness of type B connection is approximately threetimes that of the type A connection and the bearingcapacity is approximately three times that of the typeA connection

(5) The yielding platform is long and the stiffness isdegraded slowly so that the energy can be absorbedand dissipated well due to the sufficient plasticitydevelopment of the Type A connection materialEnergy consumption performance is remarkably bet-ter than that of the type B connectionIn summary the two types of vertical rod connec-tions investigated in this study exhibit good seismicperformance and can meet the engineering designrequirements

Mathematical Problems in Engineering 13

Notation119886 Specimen length1198860 Thickness of specimen ends for material

test1198870 Working width of specimen ends for

material test119887c Web member width119887f Chord width119887g Cover-plate width119887t Width of ribbed stiffener119862 Transition length of specimen ends for

material test119863 Final damage of the specimen119863i Damage of specimen in each load cycle1198630 Width of specimen ends for material test119864 Energy dissipation coefficient119864s Youngrsquos modulus119891u Ultimate strength of steel119891y Yield strength of steel119867 Length of specimen ends for material testℎ Overall height of specimenℎ1 Chord heightℎ2 Net distance between top chord and

bottom chordℎc The distance between the top and bottom

chord centroid1198700 Stiffness of specimen119870i Stiffness of the specimen in i load cycles119870i-1 Stiffness of the specimen in i-1 load cycles1198710 Working length of specimen ends for

material test119899 Total cycling number when the specimen

is loaded to failure119875 Counter-force of loading point119875d Failure load of specimen119875u Ultimate load of specimen119875y Yield load of specimen119878(119860119861119862+119862119863119860) The area surrounded by the hysteresis

curve119878(119874119861119864+119874119863119865) The area of the triangle OBE and the

triangle ODF119905c Pip thickness of web member119905f Chord flange thickness119905g Cover-plate thickness119905t Thickness of ribbed stiffener119905w Chord web thicknessΔ Displacement of loading pointΔ d Damage displacement of specimenΔ u Ultimate displacement of specimenΔ y Yield displacement of specimenΔ119870i Stiffness degradation value of the

specimen in each load cycle120577119890119902 Equivalent viscous damping coefficient120583 Ductility coefficient

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank the National Natural ScienceFoundation of China (51568012) for financially supportingthis research

References

[1] K J Ma H G Zhang and T Zheng ldquoTheory and practiceof new type architectural space gridding structuresrdquo ChinaCommunications Press 2006

[2] J S Xu and L H Xue ldquoModern prestressed concrete floorstructurerdquoChina Architecture amp Building Press 2013

[3] Z Chen G Wu D Feng and K Ma ldquoNumerical study of thestatic and dynamic characteristics of reinforced concrete cas-sette structures for high-rise buildingsrdquo The Structural Designof Tall and Special Buildings vol 28 no 3 pp 1ndash24 2019

[4] L Jiang K J Ma and H G Zhang ldquoThe dynamic property andcomfort degree study on the steel-concrete composite vieren-deel sandwich platerdquo Earthquake Engineering and EngineeringDynamics vol 37 no 6 pp 122ndash131 2017

[5] H Luan K Ma Y Qin X Li and Z Chen ldquoInvestigationon structural behavior of an innovative orthogonal-diagonalsteel open-web sandwich floor systemrdquo Advances in StructuralEngineering vol 19 no 2 pp 353ndash371 2016

[6] H Luan K Ma Y Qin Z Chen and Y Wei ldquoInvestigation ofthe structural behavior of an innovative steel open-web floorsystemrdquo International Journal of Steel Structures vol 17 no 4pp 1365ndash1378 2017

[7] Q Chen J C Xiao K J Ma and Z C Huang ldquoEffects of shearconnector joints on static behavior in steel open-web sandwichplatesrdquo Journal of Guangxi University vol 43 no 1 pp 41ndash492018 (Chinese)

[8] Z Q Bai X D Liu D Xing Y H Wei and K J Ma ldquoExperi-mental research on mechanics characteristics for shear block ofsteel tube with t-section steel ribsrdquo Industrial Construction vol48 no 1 pp 159ndash164 2018

[9] C C Chen S W Chen M D Chung and M C LinldquoCyclic behaviour of unreinforced and rib-reinforced momentconnectionsrdquo Journal of Constructional Steel Research vol 61no 1 pp 1ndash21 2005

[10] C-C Chen C-C Lin and C-H Lin ldquoDuctile moment con-nections used in steel column-tree moment-resisting framesrdquoJournal of Constructional Steel Research vol 62 no 8 pp 793ndash801 2006

[11] H S Park D C Lee B K Oh et al ldquoPerformance-basedmulti-objective optimal seismic retrofit method for a steel moment-resisting frame considering the life-cycle costrdquo MathematicalProblems in Engineering vol 2014 Article ID 305737 14 pages2014

[12] D Sosa D Arevalo E DMora et al ldquoExperimental and analyt-ical study of slender reinforced concrete shear wall under cyclicin-plane lateral loadrdquo Mathematical Problems in Engineeringvol 2017 Article ID 4020563 14 pages 2017

[13] L Hu Theoretical Analysis and Experimental Study on SemiAssembled Composite Open-Web Sandwich Plate Outsourcing

14 Mathematical Problems in Engineering

U Type Steel Plate [Unpublished PhD Dissertation] HunanUniversity 2017

[14] L Jiang K JMaH G Zhang and L Li ldquoThe designmethod ofsteel Vierendeel beam with shear connectionrdquo Xirsquoan Universityof Architecture amp Technology (Natural Science Edition) vol 50no 3 pp 354ndash360 2018

[15] JGJ 101-2015 Specification of Testing Methods for EarthquakeResistant Building Chinese Building Industry Press 2015

[16] Z L Li C T Zhang W L Fan et al ldquoExperimental research onseismic behavior of a new type of thin-walled square steel tubebeam-column connectionrdquo Engineering Mechanics vol 30 no2 pp 72ndash81 2013

[17] G Ballio and F M Mazzolani Theory and Design of SteelStructures Chapman and Hall London UK 1983

[18] S Banno I H P Mamaghani T Usami and E Mizuno ldquoCyclicelastoplastic large deflection analysis of thin steel platesrdquo Journalof Engineering Mechanics vol 124 no 4 pp 363ndash370 1998

[19] I H P Mamaghani T Usami and E Mizuno ldquoInelastic largedeflection analysis of structural steel members under cyclicloadingrdquo Engineering Structures vol 18 no 9 pp 659ndash6681996

[20] G L Bai J Q Zhao N J Du B Xie andH Gao ldquoExperimentalstudy on seismic behavior of SRC abnormal interior joints anddesign advicerdquo Journal of Building Structures vol 39 no 7 pp33ndash45 2018

[21] W B Kratzig and Y S Petryna ldquoQuasistatic seismic damageindicators for rc structures from dissipating energies in tangen-tial subspacesrdquoMathematical Problems in Engineering vol 2014Article ID 615792 11 pages 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

0 20 40 60

Stiff

ness

(kN

m)

Δ (mm)minus20minus40minus60

1000

1500

2000

2500

3000

3500

(a)

Stiff

ness

(kN

m)

0 10 20 302000

4000

6000

8000

10000

12000

14000

Δ (mm)minus30 minus20 minus10

(b)

Figure 14 Stiffness degeneration curve (a) Specimen T1 (b) Specimen T2

degradation value of the joint in each load cycle which canbe calculated by

Δ119870119894 = 119870119894 minus 119870119894minus1 (4)

where 119870119894 and 119870119894minus1 are the stiffness of the specimen in i andi-1 load cycles respectively The final damage of the specimencan be obtained by summing 119863119894

119863 =119899

sum119894=1

119863119894 (5)

where n is the total number of cycles when the specimen isloaded to failure

Table 3 shows the cumulative damage of the specimenunder the action of alternating load indicating that the finaldamage D of specimen T1 is slightly greater than that ofspecimen T2 The elastoplastic properties of the material arefully exerted

35 Energy Dissipation Performance Energy dissipationcapacity and damping are important indicators formeasuringstructural dynamic performance The energy dissipationcapacity is measured by the area surrounded by the hysteresiscurve of load deformation and is usually evaluated by theenergy dissipation coefficient E andor the equivalent viscousdamping coefficient 120577119890119902 The calculation is presented asfollows

E =119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(6)

120577119890119902 =12120587sdot119878(119860119861119862+119862119863119860)119878(119874119861119864+119874119863119865)

(7)

where 119878(119860119861119862+119862119863119860) is the area surrounded by the hysteresiscurve in Figure 15 119878(119874119861119864+119874119863119865) is the sum of the area of thetriangle OBE and the triangle ODF in Figure 15 The energyconsumption of the yield point peak point and damage pointis shown in Table 4 Table 4 shows that the energy dissipationcoefficient and the equivalent viscosity damping coefficientof each specimen increase with displacement The energyconsumption of specimen T1 is better than that of specimenT2

X

FB

EC

D

AF o

Figure 15 Load-displacement hysteretic loop

36 Strain Analysis The strain measuring points in Figure 6were monitored during the test Figure 16 shows the distribu-tion of specimen T1 strain along with loading displacementchanges Figure 17 shows the distribution of specimen T2strain with loading displacement changes

Figure 16 shows that the type A connection is connectedto the end portion of the web in the upper chord and the webside Stress concentration occurs Strain is larger when theend portion is close The main strain direction on the sidewall is radial with the intersection of the edge of the chordweb and the side wall of the vertical rod as the origin Theend of the upper chord is connected to the sidewall of theweb where the flange strain is large in the edge and smallin the middle The end of the upper chord is connected tothe sidewall of the web and the end of the web is close tothe flange where the strain at the measuring point is smallerand the strain away from the flange is larger In general whenthe loading displacement reaches Δ119910 most of the measuredstrain at the sidewall reaches yield strain and the strain atthe end of the upper chord is still in the elastic phase When

Mathematical Problems in Engineering 11

Table 4 Energy dissipation coefficient and equivalent viscous damping coefficient

Specimen Yield point Ultimate point Damage pointE 120577119890119902 E 120577119890119902 E 120577119890119902

T1 0347 0055 1173 0187 1465 0233T2 0198 0032 0404 0064 0859 0137

0

4000

8000

A4A5A6

A13A14A15

0 20 40 60Δ (mm)

minus20minus40minus60minus8000

minus4000

Stra

in(times

10minus6)

(a)

0

1000

2000

A22A23A24

A19A20A21

minus2000

minus1000

0 20 40 60Δ (mm)

minus20minus40minus60

Stra

in(times

10minus6)

(b)

0

1000

2000

3000

A7A8A9

A1A2A3

0 20 40 60Δ (mm)

minus20minus40minus60minus2000

minus1000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

4000

A10A11A12

A16A17A18

0 20 40 60Δ (mm)

minus20minus40minus60

minus2000

minus3000

minus1000

Stra

in(times

10minus6)

(d)

Figure 16 Strain of specimen T1 (a) The part of web near the weld (b) Top chord (c) The part of web near the top chord (d) The part ofweb near the bottom chord

the loading displacement reaches Δ 119906 most of the measuredstrain of the sidewall reaches the peak value and the strainat the end of the chord gradually reaches yield strain Whenthe loading displacement reachesΔ119889 the stress redistributionoccurs due to the large crack at the joint between the sidewalland the chord web The strain at the measuring point showsa downward trend and the strain at the end of the chordcontinues to increase

The strain on the stiffening plate in the loading processamong the measuring points of the type B connection is thelargest followed by the end of the chord and the side wallstrain of the web (Figure 17) In general when the loadingdisplacement reaches Δ119910 the strain at the point where thestiffener is connected to the web of the chord initially reachesyield strain and the strain at the joint of the bar and the

stiffener bar also rapidly reach the yield strain However theother points still remain in the elastic stageWhen the loadingdisplacement reaches Δ 119906 the strain at the point where thestiffener is connected to the web of the chord reaches thepeak and the strain at the other points continues to increaseWhen the loading displacement reaches Δ119889 the strain atthe point where the stiffener is connected to the web ofthe chord is gradually decreased due to the overall bucklingdeformation of the stiffener and the chord web

4 Conclusions

The main conclusions are summarized on the basis ofcyclic pseudo-static seismic testing of web member shearconnections as follows

12 Mathematical Problems in Engineering

0 10 20 30

0

1000

2000

3000

B4B5B6

B1B2B3

Δ (mm)minus10minus20minus30

minus4000

minus3000

minus2000

minus1000

Stra

in(times

10minus6)

(a)

0

2000

4000

6000

B10B11B12

B7B8B9

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(b)

0

2000

4000

B16B17B18

B13B14B15

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

B22B23B24

B19B20B21

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus1000

minus3000

Stra

in(times

10minus6)

(d)

Figure 17 Strain of specimen T2 (a) Flank of web member (b) Top chord (c)The part of ribbed stiffener near the top chord (d)The part ofribbed stiffener near the bottom chord

(1) The force performance of the vertical rod connectionis achieved by adopting the self-designed loadingsystem which can truly reflect the mechanical perfor-mance and deformation characteristics of the verticalrod shear connection under the action of alternatingload

(2) The ductility of the material in the heat-affected zonedeteriorates due to the stress concentration at thetype A connection between the chord and the sidewall of the vertical rod as well as the heat inputof the welding Added with alternating load theplastic strain in the heat-affected zone is accumulatedand the ductility is reduced Therefore a U-shapedfracture crack is formed on the outside of the heat-affected zone of the chord web and the side wall ofthe vertical rod for specimen T1

(3) In the type B connection the strain at the connectionbetween the stiffener and the chord web is the largestthereby yielding first As the loading displacement

increases the joint between the chord web and thestiffening plate cracks and develops eventually caus-ing the stiffening plate to be buckled and undergostructural damage

(4) When the main design parameters are the same thestiffness of type B connection is approximately threetimes that of the type A connection and the bearingcapacity is approximately three times that of the typeA connection

(5) The yielding platform is long and the stiffness isdegraded slowly so that the energy can be absorbedand dissipated well due to the sufficient plasticitydevelopment of the Type A connection materialEnergy consumption performance is remarkably bet-ter than that of the type B connectionIn summary the two types of vertical rod connec-tions investigated in this study exhibit good seismicperformance and can meet the engineering designrequirements

Mathematical Problems in Engineering 13

Notation119886 Specimen length1198860 Thickness of specimen ends for material

test1198870 Working width of specimen ends for

material test119887c Web member width119887f Chord width119887g Cover-plate width119887t Width of ribbed stiffener119862 Transition length of specimen ends for

material test119863 Final damage of the specimen119863i Damage of specimen in each load cycle1198630 Width of specimen ends for material test119864 Energy dissipation coefficient119864s Youngrsquos modulus119891u Ultimate strength of steel119891y Yield strength of steel119867 Length of specimen ends for material testℎ Overall height of specimenℎ1 Chord heightℎ2 Net distance between top chord and

bottom chordℎc The distance between the top and bottom

chord centroid1198700 Stiffness of specimen119870i Stiffness of the specimen in i load cycles119870i-1 Stiffness of the specimen in i-1 load cycles1198710 Working length of specimen ends for

material test119899 Total cycling number when the specimen

is loaded to failure119875 Counter-force of loading point119875d Failure load of specimen119875u Ultimate load of specimen119875y Yield load of specimen119878(119860119861119862+119862119863119860) The area surrounded by the hysteresis

curve119878(119874119861119864+119874119863119865) The area of the triangle OBE and the

triangle ODF119905c Pip thickness of web member119905f Chord flange thickness119905g Cover-plate thickness119905t Thickness of ribbed stiffener119905w Chord web thicknessΔ Displacement of loading pointΔ d Damage displacement of specimenΔ u Ultimate displacement of specimenΔ y Yield displacement of specimenΔ119870i Stiffness degradation value of the

specimen in each load cycle120577119890119902 Equivalent viscous damping coefficient120583 Ductility coefficient

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank the National Natural ScienceFoundation of China (51568012) for financially supportingthis research

References

[1] K J Ma H G Zhang and T Zheng ldquoTheory and practiceof new type architectural space gridding structuresrdquo ChinaCommunications Press 2006

[2] J S Xu and L H Xue ldquoModern prestressed concrete floorstructurerdquoChina Architecture amp Building Press 2013

[3] Z Chen G Wu D Feng and K Ma ldquoNumerical study of thestatic and dynamic characteristics of reinforced concrete cas-sette structures for high-rise buildingsrdquo The Structural Designof Tall and Special Buildings vol 28 no 3 pp 1ndash24 2019

[4] L Jiang K J Ma and H G Zhang ldquoThe dynamic property andcomfort degree study on the steel-concrete composite vieren-deel sandwich platerdquo Earthquake Engineering and EngineeringDynamics vol 37 no 6 pp 122ndash131 2017

[5] H Luan K Ma Y Qin X Li and Z Chen ldquoInvestigationon structural behavior of an innovative orthogonal-diagonalsteel open-web sandwich floor systemrdquo Advances in StructuralEngineering vol 19 no 2 pp 353ndash371 2016

[6] H Luan K Ma Y Qin Z Chen and Y Wei ldquoInvestigation ofthe structural behavior of an innovative steel open-web floorsystemrdquo International Journal of Steel Structures vol 17 no 4pp 1365ndash1378 2017

[7] Q Chen J C Xiao K J Ma and Z C Huang ldquoEffects of shearconnector joints on static behavior in steel open-web sandwichplatesrdquo Journal of Guangxi University vol 43 no 1 pp 41ndash492018 (Chinese)

[8] Z Q Bai X D Liu D Xing Y H Wei and K J Ma ldquoExperi-mental research on mechanics characteristics for shear block ofsteel tube with t-section steel ribsrdquo Industrial Construction vol48 no 1 pp 159ndash164 2018

[9] C C Chen S W Chen M D Chung and M C LinldquoCyclic behaviour of unreinforced and rib-reinforced momentconnectionsrdquo Journal of Constructional Steel Research vol 61no 1 pp 1ndash21 2005

[10] C-C Chen C-C Lin and C-H Lin ldquoDuctile moment con-nections used in steel column-tree moment-resisting framesrdquoJournal of Constructional Steel Research vol 62 no 8 pp 793ndash801 2006

[11] H S Park D C Lee B K Oh et al ldquoPerformance-basedmulti-objective optimal seismic retrofit method for a steel moment-resisting frame considering the life-cycle costrdquo MathematicalProblems in Engineering vol 2014 Article ID 305737 14 pages2014

[12] D Sosa D Arevalo E DMora et al ldquoExperimental and analyt-ical study of slender reinforced concrete shear wall under cyclicin-plane lateral loadrdquo Mathematical Problems in Engineeringvol 2017 Article ID 4020563 14 pages 2017

[13] L Hu Theoretical Analysis and Experimental Study on SemiAssembled Composite Open-Web Sandwich Plate Outsourcing

14 Mathematical Problems in Engineering

U Type Steel Plate [Unpublished PhD Dissertation] HunanUniversity 2017

[14] L Jiang K JMaH G Zhang and L Li ldquoThe designmethod ofsteel Vierendeel beam with shear connectionrdquo Xirsquoan Universityof Architecture amp Technology (Natural Science Edition) vol 50no 3 pp 354ndash360 2018

[15] JGJ 101-2015 Specification of Testing Methods for EarthquakeResistant Building Chinese Building Industry Press 2015

[16] Z L Li C T Zhang W L Fan et al ldquoExperimental research onseismic behavior of a new type of thin-walled square steel tubebeam-column connectionrdquo Engineering Mechanics vol 30 no2 pp 72ndash81 2013

[17] G Ballio and F M Mazzolani Theory and Design of SteelStructures Chapman and Hall London UK 1983

[18] S Banno I H P Mamaghani T Usami and E Mizuno ldquoCyclicelastoplastic large deflection analysis of thin steel platesrdquo Journalof Engineering Mechanics vol 124 no 4 pp 363ndash370 1998

[19] I H P Mamaghani T Usami and E Mizuno ldquoInelastic largedeflection analysis of structural steel members under cyclicloadingrdquo Engineering Structures vol 18 no 9 pp 659ndash6681996

[20] G L Bai J Q Zhao N J Du B Xie andH Gao ldquoExperimentalstudy on seismic behavior of SRC abnormal interior joints anddesign advicerdquo Journal of Building Structures vol 39 no 7 pp33ndash45 2018

[21] W B Kratzig and Y S Petryna ldquoQuasistatic seismic damageindicators for rc structures from dissipating energies in tangen-tial subspacesrdquoMathematical Problems in Engineering vol 2014Article ID 615792 11 pages 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

Table 4 Energy dissipation coefficient and equivalent viscous damping coefficient

Specimen Yield point Ultimate point Damage pointE 120577119890119902 E 120577119890119902 E 120577119890119902

T1 0347 0055 1173 0187 1465 0233T2 0198 0032 0404 0064 0859 0137

0

4000

8000

A4A5A6

A13A14A15

0 20 40 60Δ (mm)

minus20minus40minus60minus8000

minus4000

Stra

in(times

10minus6)

(a)

0

1000

2000

A22A23A24

A19A20A21

minus2000

minus1000

0 20 40 60Δ (mm)

minus20minus40minus60

Stra

in(times

10minus6)

(b)

0

1000

2000

3000

A7A8A9

A1A2A3

0 20 40 60Δ (mm)

minus20minus40minus60minus2000

minus1000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

4000

A10A11A12

A16A17A18

0 20 40 60Δ (mm)

minus20minus40minus60

minus2000

minus3000

minus1000

Stra

in(times

10minus6)

(d)

Figure 16 Strain of specimen T1 (a) The part of web near the weld (b) Top chord (c) The part of web near the top chord (d) The part ofweb near the bottom chord

the loading displacement reaches Δ 119906 most of the measuredstrain of the sidewall reaches the peak value and the strainat the end of the chord gradually reaches yield strain Whenthe loading displacement reachesΔ119889 the stress redistributionoccurs due to the large crack at the joint between the sidewalland the chord web The strain at the measuring point showsa downward trend and the strain at the end of the chordcontinues to increase

The strain on the stiffening plate in the loading processamong the measuring points of the type B connection is thelargest followed by the end of the chord and the side wallstrain of the web (Figure 17) In general when the loadingdisplacement reaches Δ119910 the strain at the point where thestiffener is connected to the web of the chord initially reachesyield strain and the strain at the joint of the bar and the

stiffener bar also rapidly reach the yield strain However theother points still remain in the elastic stageWhen the loadingdisplacement reaches Δ 119906 the strain at the point where thestiffener is connected to the web of the chord reaches thepeak and the strain at the other points continues to increaseWhen the loading displacement reaches Δ119889 the strain atthe point where the stiffener is connected to the web ofthe chord is gradually decreased due to the overall bucklingdeformation of the stiffener and the chord web

4 Conclusions

The main conclusions are summarized on the basis ofcyclic pseudo-static seismic testing of web member shearconnections as follows

12 Mathematical Problems in Engineering

0 10 20 30

0

1000

2000

3000

B4B5B6

B1B2B3

Δ (mm)minus10minus20minus30

minus4000

minus3000

minus2000

minus1000

Stra

in(times

10minus6)

(a)

0

2000

4000

6000

B10B11B12

B7B8B9

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(b)

0

2000

4000

B16B17B18

B13B14B15

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

B22B23B24

B19B20B21

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus1000

minus3000

Stra

in(times

10minus6)

(d)

Figure 17 Strain of specimen T2 (a) Flank of web member (b) Top chord (c)The part of ribbed stiffener near the top chord (d)The part ofribbed stiffener near the bottom chord

(1) The force performance of the vertical rod connectionis achieved by adopting the self-designed loadingsystem which can truly reflect the mechanical perfor-mance and deformation characteristics of the verticalrod shear connection under the action of alternatingload

(2) The ductility of the material in the heat-affected zonedeteriorates due to the stress concentration at thetype A connection between the chord and the sidewall of the vertical rod as well as the heat inputof the welding Added with alternating load theplastic strain in the heat-affected zone is accumulatedand the ductility is reduced Therefore a U-shapedfracture crack is formed on the outside of the heat-affected zone of the chord web and the side wall ofthe vertical rod for specimen T1

(3) In the type B connection the strain at the connectionbetween the stiffener and the chord web is the largestthereby yielding first As the loading displacement

increases the joint between the chord web and thestiffening plate cracks and develops eventually caus-ing the stiffening plate to be buckled and undergostructural damage

(4) When the main design parameters are the same thestiffness of type B connection is approximately threetimes that of the type A connection and the bearingcapacity is approximately three times that of the typeA connection

(5) The yielding platform is long and the stiffness isdegraded slowly so that the energy can be absorbedand dissipated well due to the sufficient plasticitydevelopment of the Type A connection materialEnergy consumption performance is remarkably bet-ter than that of the type B connectionIn summary the two types of vertical rod connec-tions investigated in this study exhibit good seismicperformance and can meet the engineering designrequirements

Mathematical Problems in Engineering 13

Notation119886 Specimen length1198860 Thickness of specimen ends for material

test1198870 Working width of specimen ends for

material test119887c Web member width119887f Chord width119887g Cover-plate width119887t Width of ribbed stiffener119862 Transition length of specimen ends for

material test119863 Final damage of the specimen119863i Damage of specimen in each load cycle1198630 Width of specimen ends for material test119864 Energy dissipation coefficient119864s Youngrsquos modulus119891u Ultimate strength of steel119891y Yield strength of steel119867 Length of specimen ends for material testℎ Overall height of specimenℎ1 Chord heightℎ2 Net distance between top chord and

bottom chordℎc The distance between the top and bottom

chord centroid1198700 Stiffness of specimen119870i Stiffness of the specimen in i load cycles119870i-1 Stiffness of the specimen in i-1 load cycles1198710 Working length of specimen ends for

material test119899 Total cycling number when the specimen

is loaded to failure119875 Counter-force of loading point119875d Failure load of specimen119875u Ultimate load of specimen119875y Yield load of specimen119878(119860119861119862+119862119863119860) The area surrounded by the hysteresis

curve119878(119874119861119864+119874119863119865) The area of the triangle OBE and the

triangle ODF119905c Pip thickness of web member119905f Chord flange thickness119905g Cover-plate thickness119905t Thickness of ribbed stiffener119905w Chord web thicknessΔ Displacement of loading pointΔ d Damage displacement of specimenΔ u Ultimate displacement of specimenΔ y Yield displacement of specimenΔ119870i Stiffness degradation value of the

specimen in each load cycle120577119890119902 Equivalent viscous damping coefficient120583 Ductility coefficient

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank the National Natural ScienceFoundation of China (51568012) for financially supportingthis research

References

[1] K J Ma H G Zhang and T Zheng ldquoTheory and practiceof new type architectural space gridding structuresrdquo ChinaCommunications Press 2006

[2] J S Xu and L H Xue ldquoModern prestressed concrete floorstructurerdquoChina Architecture amp Building Press 2013

[3] Z Chen G Wu D Feng and K Ma ldquoNumerical study of thestatic and dynamic characteristics of reinforced concrete cas-sette structures for high-rise buildingsrdquo The Structural Designof Tall and Special Buildings vol 28 no 3 pp 1ndash24 2019

[4] L Jiang K J Ma and H G Zhang ldquoThe dynamic property andcomfort degree study on the steel-concrete composite vieren-deel sandwich platerdquo Earthquake Engineering and EngineeringDynamics vol 37 no 6 pp 122ndash131 2017

[5] H Luan K Ma Y Qin X Li and Z Chen ldquoInvestigationon structural behavior of an innovative orthogonal-diagonalsteel open-web sandwich floor systemrdquo Advances in StructuralEngineering vol 19 no 2 pp 353ndash371 2016

[6] H Luan K Ma Y Qin Z Chen and Y Wei ldquoInvestigation ofthe structural behavior of an innovative steel open-web floorsystemrdquo International Journal of Steel Structures vol 17 no 4pp 1365ndash1378 2017

[7] Q Chen J C Xiao K J Ma and Z C Huang ldquoEffects of shearconnector joints on static behavior in steel open-web sandwichplatesrdquo Journal of Guangxi University vol 43 no 1 pp 41ndash492018 (Chinese)

[8] Z Q Bai X D Liu D Xing Y H Wei and K J Ma ldquoExperi-mental research on mechanics characteristics for shear block ofsteel tube with t-section steel ribsrdquo Industrial Construction vol48 no 1 pp 159ndash164 2018

[9] C C Chen S W Chen M D Chung and M C LinldquoCyclic behaviour of unreinforced and rib-reinforced momentconnectionsrdquo Journal of Constructional Steel Research vol 61no 1 pp 1ndash21 2005

[10] C-C Chen C-C Lin and C-H Lin ldquoDuctile moment con-nections used in steel column-tree moment-resisting framesrdquoJournal of Constructional Steel Research vol 62 no 8 pp 793ndash801 2006

[11] H S Park D C Lee B K Oh et al ldquoPerformance-basedmulti-objective optimal seismic retrofit method for a steel moment-resisting frame considering the life-cycle costrdquo MathematicalProblems in Engineering vol 2014 Article ID 305737 14 pages2014

[12] D Sosa D Arevalo E DMora et al ldquoExperimental and analyt-ical study of slender reinforced concrete shear wall under cyclicin-plane lateral loadrdquo Mathematical Problems in Engineeringvol 2017 Article ID 4020563 14 pages 2017

[13] L Hu Theoretical Analysis and Experimental Study on SemiAssembled Composite Open-Web Sandwich Plate Outsourcing

14 Mathematical Problems in Engineering

U Type Steel Plate [Unpublished PhD Dissertation] HunanUniversity 2017

[14] L Jiang K JMaH G Zhang and L Li ldquoThe designmethod ofsteel Vierendeel beam with shear connectionrdquo Xirsquoan Universityof Architecture amp Technology (Natural Science Edition) vol 50no 3 pp 354ndash360 2018

[15] JGJ 101-2015 Specification of Testing Methods for EarthquakeResistant Building Chinese Building Industry Press 2015

[16] Z L Li C T Zhang W L Fan et al ldquoExperimental research onseismic behavior of a new type of thin-walled square steel tubebeam-column connectionrdquo Engineering Mechanics vol 30 no2 pp 72ndash81 2013

[17] G Ballio and F M Mazzolani Theory and Design of SteelStructures Chapman and Hall London UK 1983

[18] S Banno I H P Mamaghani T Usami and E Mizuno ldquoCyclicelastoplastic large deflection analysis of thin steel platesrdquo Journalof Engineering Mechanics vol 124 no 4 pp 363ndash370 1998

[19] I H P Mamaghani T Usami and E Mizuno ldquoInelastic largedeflection analysis of structural steel members under cyclicloadingrdquo Engineering Structures vol 18 no 9 pp 659ndash6681996

[20] G L Bai J Q Zhao N J Du B Xie andH Gao ldquoExperimentalstudy on seismic behavior of SRC abnormal interior joints anddesign advicerdquo Journal of Building Structures vol 39 no 7 pp33ndash45 2018

[21] W B Kratzig and Y S Petryna ldquoQuasistatic seismic damageindicators for rc structures from dissipating energies in tangen-tial subspacesrdquoMathematical Problems in Engineering vol 2014Article ID 615792 11 pages 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Mathematical Problems in Engineering

0 10 20 30

0

1000

2000

3000

B4B5B6

B1B2B3

Δ (mm)minus10minus20minus30

minus4000

minus3000

minus2000

minus1000

Stra

in(times

10minus6)

(a)

0

2000

4000

6000

B10B11B12

B7B8B9

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(b)

0

2000

4000

B16B17B18

B13B14B15

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus4000

minus6000

Stra

in(times

10minus6)

(c)

0

1000

2000

3000

B22B23B24

B19B20B21

0 10 20 30Δ (mm)

minus10minus20minus30

minus2000

minus1000

minus3000

Stra

in(times

10minus6)

(d)

Figure 17 Strain of specimen T2 (a) Flank of web member (b) Top chord (c)The part of ribbed stiffener near the top chord (d)The part ofribbed stiffener near the bottom chord

(1) The force performance of the vertical rod connectionis achieved by adopting the self-designed loadingsystem which can truly reflect the mechanical perfor-mance and deformation characteristics of the verticalrod shear connection under the action of alternatingload

(2) The ductility of the material in the heat-affected zonedeteriorates due to the stress concentration at thetype A connection between the chord and the sidewall of the vertical rod as well as the heat inputof the welding Added with alternating load theplastic strain in the heat-affected zone is accumulatedand the ductility is reduced Therefore a U-shapedfracture crack is formed on the outside of the heat-affected zone of the chord web and the side wall ofthe vertical rod for specimen T1

(3) In the type B connection the strain at the connectionbetween the stiffener and the chord web is the largestthereby yielding first As the loading displacement

increases the joint between the chord web and thestiffening plate cracks and develops eventually caus-ing the stiffening plate to be buckled and undergostructural damage

(4) When the main design parameters are the same thestiffness of type B connection is approximately threetimes that of the type A connection and the bearingcapacity is approximately three times that of the typeA connection

(5) The yielding platform is long and the stiffness isdegraded slowly so that the energy can be absorbedand dissipated well due to the sufficient plasticitydevelopment of the Type A connection materialEnergy consumption performance is remarkably bet-ter than that of the type B connectionIn summary the two types of vertical rod connec-tions investigated in this study exhibit good seismicperformance and can meet the engineering designrequirements

Mathematical Problems in Engineering 13

Notation119886 Specimen length1198860 Thickness of specimen ends for material

test1198870 Working width of specimen ends for

material test119887c Web member width119887f Chord width119887g Cover-plate width119887t Width of ribbed stiffener119862 Transition length of specimen ends for

material test119863 Final damage of the specimen119863i Damage of specimen in each load cycle1198630 Width of specimen ends for material test119864 Energy dissipation coefficient119864s Youngrsquos modulus119891u Ultimate strength of steel119891y Yield strength of steel119867 Length of specimen ends for material testℎ Overall height of specimenℎ1 Chord heightℎ2 Net distance between top chord and

bottom chordℎc The distance between the top and bottom

chord centroid1198700 Stiffness of specimen119870i Stiffness of the specimen in i load cycles119870i-1 Stiffness of the specimen in i-1 load cycles1198710 Working length of specimen ends for

material test119899 Total cycling number when the specimen

is loaded to failure119875 Counter-force of loading point119875d Failure load of specimen119875u Ultimate load of specimen119875y Yield load of specimen119878(119860119861119862+119862119863119860) The area surrounded by the hysteresis

curve119878(119874119861119864+119874119863119865) The area of the triangle OBE and the

triangle ODF119905c Pip thickness of web member119905f Chord flange thickness119905g Cover-plate thickness119905t Thickness of ribbed stiffener119905w Chord web thicknessΔ Displacement of loading pointΔ d Damage displacement of specimenΔ u Ultimate displacement of specimenΔ y Yield displacement of specimenΔ119870i Stiffness degradation value of the

specimen in each load cycle120577119890119902 Equivalent viscous damping coefficient120583 Ductility coefficient

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank the National Natural ScienceFoundation of China (51568012) for financially supportingthis research

References

[1] K J Ma H G Zhang and T Zheng ldquoTheory and practiceof new type architectural space gridding structuresrdquo ChinaCommunications Press 2006

[2] J S Xu and L H Xue ldquoModern prestressed concrete floorstructurerdquoChina Architecture amp Building Press 2013

[3] Z Chen G Wu D Feng and K Ma ldquoNumerical study of thestatic and dynamic characteristics of reinforced concrete cas-sette structures for high-rise buildingsrdquo The Structural Designof Tall and Special Buildings vol 28 no 3 pp 1ndash24 2019

[4] L Jiang K J Ma and H G Zhang ldquoThe dynamic property andcomfort degree study on the steel-concrete composite vieren-deel sandwich platerdquo Earthquake Engineering and EngineeringDynamics vol 37 no 6 pp 122ndash131 2017

[5] H Luan K Ma Y Qin X Li and Z Chen ldquoInvestigationon structural behavior of an innovative orthogonal-diagonalsteel open-web sandwich floor systemrdquo Advances in StructuralEngineering vol 19 no 2 pp 353ndash371 2016

[6] H Luan K Ma Y Qin Z Chen and Y Wei ldquoInvestigation ofthe structural behavior of an innovative steel open-web floorsystemrdquo International Journal of Steel Structures vol 17 no 4pp 1365ndash1378 2017

[7] Q Chen J C Xiao K J Ma and Z C Huang ldquoEffects of shearconnector joints on static behavior in steel open-web sandwichplatesrdquo Journal of Guangxi University vol 43 no 1 pp 41ndash492018 (Chinese)

[8] Z Q Bai X D Liu D Xing Y H Wei and K J Ma ldquoExperi-mental research on mechanics characteristics for shear block ofsteel tube with t-section steel ribsrdquo Industrial Construction vol48 no 1 pp 159ndash164 2018

[9] C C Chen S W Chen M D Chung and M C LinldquoCyclic behaviour of unreinforced and rib-reinforced momentconnectionsrdquo Journal of Constructional Steel Research vol 61no 1 pp 1ndash21 2005

[10] C-C Chen C-C Lin and C-H Lin ldquoDuctile moment con-nections used in steel column-tree moment-resisting framesrdquoJournal of Constructional Steel Research vol 62 no 8 pp 793ndash801 2006

[11] H S Park D C Lee B K Oh et al ldquoPerformance-basedmulti-objective optimal seismic retrofit method for a steel moment-resisting frame considering the life-cycle costrdquo MathematicalProblems in Engineering vol 2014 Article ID 305737 14 pages2014

[12] D Sosa D Arevalo E DMora et al ldquoExperimental and analyt-ical study of slender reinforced concrete shear wall under cyclicin-plane lateral loadrdquo Mathematical Problems in Engineeringvol 2017 Article ID 4020563 14 pages 2017

[13] L Hu Theoretical Analysis and Experimental Study on SemiAssembled Composite Open-Web Sandwich Plate Outsourcing

14 Mathematical Problems in Engineering

U Type Steel Plate [Unpublished PhD Dissertation] HunanUniversity 2017

[14] L Jiang K JMaH G Zhang and L Li ldquoThe designmethod ofsteel Vierendeel beam with shear connectionrdquo Xirsquoan Universityof Architecture amp Technology (Natural Science Edition) vol 50no 3 pp 354ndash360 2018

[15] JGJ 101-2015 Specification of Testing Methods for EarthquakeResistant Building Chinese Building Industry Press 2015

[16] Z L Li C T Zhang W L Fan et al ldquoExperimental research onseismic behavior of a new type of thin-walled square steel tubebeam-column connectionrdquo Engineering Mechanics vol 30 no2 pp 72ndash81 2013

[17] G Ballio and F M Mazzolani Theory and Design of SteelStructures Chapman and Hall London UK 1983

[18] S Banno I H P Mamaghani T Usami and E Mizuno ldquoCyclicelastoplastic large deflection analysis of thin steel platesrdquo Journalof Engineering Mechanics vol 124 no 4 pp 363ndash370 1998

[19] I H P Mamaghani T Usami and E Mizuno ldquoInelastic largedeflection analysis of structural steel members under cyclicloadingrdquo Engineering Structures vol 18 no 9 pp 659ndash6681996

[20] G L Bai J Q Zhao N J Du B Xie andH Gao ldquoExperimentalstudy on seismic behavior of SRC abnormal interior joints anddesign advicerdquo Journal of Building Structures vol 39 no 7 pp33ndash45 2018

[21] W B Kratzig and Y S Petryna ldquoQuasistatic seismic damageindicators for rc structures from dissipating energies in tangen-tial subspacesrdquoMathematical Problems in Engineering vol 2014Article ID 615792 11 pages 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 13

Notation119886 Specimen length1198860 Thickness of specimen ends for material

test1198870 Working width of specimen ends for

material test119887c Web member width119887f Chord width119887g Cover-plate width119887t Width of ribbed stiffener119862 Transition length of specimen ends for

material test119863 Final damage of the specimen119863i Damage of specimen in each load cycle1198630 Width of specimen ends for material test119864 Energy dissipation coefficient119864s Youngrsquos modulus119891u Ultimate strength of steel119891y Yield strength of steel119867 Length of specimen ends for material testℎ Overall height of specimenℎ1 Chord heightℎ2 Net distance between top chord and

bottom chordℎc The distance between the top and bottom

chord centroid1198700 Stiffness of specimen119870i Stiffness of the specimen in i load cycles119870i-1 Stiffness of the specimen in i-1 load cycles1198710 Working length of specimen ends for

material test119899 Total cycling number when the specimen

is loaded to failure119875 Counter-force of loading point119875d Failure load of specimen119875u Ultimate load of specimen119875y Yield load of specimen119878(119860119861119862+119862119863119860) The area surrounded by the hysteresis

curve119878(119874119861119864+119874119863119865) The area of the triangle OBE and the

triangle ODF119905c Pip thickness of web member119905f Chord flange thickness119905g Cover-plate thickness119905t Thickness of ribbed stiffener119905w Chord web thicknessΔ Displacement of loading pointΔ d Damage displacement of specimenΔ u Ultimate displacement of specimenΔ y Yield displacement of specimenΔ119870i Stiffness degradation value of the

specimen in each load cycle120577119890119902 Equivalent viscous damping coefficient120583 Ductility coefficient

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank the National Natural ScienceFoundation of China (51568012) for financially supportingthis research

References

[1] K J Ma H G Zhang and T Zheng ldquoTheory and practiceof new type architectural space gridding structuresrdquo ChinaCommunications Press 2006

[2] J S Xu and L H Xue ldquoModern prestressed concrete floorstructurerdquoChina Architecture amp Building Press 2013

[3] Z Chen G Wu D Feng and K Ma ldquoNumerical study of thestatic and dynamic characteristics of reinforced concrete cas-sette structures for high-rise buildingsrdquo The Structural Designof Tall and Special Buildings vol 28 no 3 pp 1ndash24 2019

[4] L Jiang K J Ma and H G Zhang ldquoThe dynamic property andcomfort degree study on the steel-concrete composite vieren-deel sandwich platerdquo Earthquake Engineering and EngineeringDynamics vol 37 no 6 pp 122ndash131 2017

[5] H Luan K Ma Y Qin X Li and Z Chen ldquoInvestigationon structural behavior of an innovative orthogonal-diagonalsteel open-web sandwich floor systemrdquo Advances in StructuralEngineering vol 19 no 2 pp 353ndash371 2016

[6] H Luan K Ma Y Qin Z Chen and Y Wei ldquoInvestigation ofthe structural behavior of an innovative steel open-web floorsystemrdquo International Journal of Steel Structures vol 17 no 4pp 1365ndash1378 2017

[7] Q Chen J C Xiao K J Ma and Z C Huang ldquoEffects of shearconnector joints on static behavior in steel open-web sandwichplatesrdquo Journal of Guangxi University vol 43 no 1 pp 41ndash492018 (Chinese)

[8] Z Q Bai X D Liu D Xing Y H Wei and K J Ma ldquoExperi-mental research on mechanics characteristics for shear block ofsteel tube with t-section steel ribsrdquo Industrial Construction vol48 no 1 pp 159ndash164 2018

[9] C C Chen S W Chen M D Chung and M C LinldquoCyclic behaviour of unreinforced and rib-reinforced momentconnectionsrdquo Journal of Constructional Steel Research vol 61no 1 pp 1ndash21 2005

[10] C-C Chen C-C Lin and C-H Lin ldquoDuctile moment con-nections used in steel column-tree moment-resisting framesrdquoJournal of Constructional Steel Research vol 62 no 8 pp 793ndash801 2006

[11] H S Park D C Lee B K Oh et al ldquoPerformance-basedmulti-objective optimal seismic retrofit method for a steel moment-resisting frame considering the life-cycle costrdquo MathematicalProblems in Engineering vol 2014 Article ID 305737 14 pages2014

[12] D Sosa D Arevalo E DMora et al ldquoExperimental and analyt-ical study of slender reinforced concrete shear wall under cyclicin-plane lateral loadrdquo Mathematical Problems in Engineeringvol 2017 Article ID 4020563 14 pages 2017

[13] L Hu Theoretical Analysis and Experimental Study on SemiAssembled Composite Open-Web Sandwich Plate Outsourcing

14 Mathematical Problems in Engineering

U Type Steel Plate [Unpublished PhD Dissertation] HunanUniversity 2017

[14] L Jiang K JMaH G Zhang and L Li ldquoThe designmethod ofsteel Vierendeel beam with shear connectionrdquo Xirsquoan Universityof Architecture amp Technology (Natural Science Edition) vol 50no 3 pp 354ndash360 2018

[15] JGJ 101-2015 Specification of Testing Methods for EarthquakeResistant Building Chinese Building Industry Press 2015

[16] Z L Li C T Zhang W L Fan et al ldquoExperimental research onseismic behavior of a new type of thin-walled square steel tubebeam-column connectionrdquo Engineering Mechanics vol 30 no2 pp 72ndash81 2013

[17] G Ballio and F M Mazzolani Theory and Design of SteelStructures Chapman and Hall London UK 1983

[18] S Banno I H P Mamaghani T Usami and E Mizuno ldquoCyclicelastoplastic large deflection analysis of thin steel platesrdquo Journalof Engineering Mechanics vol 124 no 4 pp 363ndash370 1998

[19] I H P Mamaghani T Usami and E Mizuno ldquoInelastic largedeflection analysis of structural steel members under cyclicloadingrdquo Engineering Structures vol 18 no 9 pp 659ndash6681996

[20] G L Bai J Q Zhao N J Du B Xie andH Gao ldquoExperimentalstudy on seismic behavior of SRC abnormal interior joints anddesign advicerdquo Journal of Building Structures vol 39 no 7 pp33ndash45 2018

[21] W B Kratzig and Y S Petryna ldquoQuasistatic seismic damageindicators for rc structures from dissipating energies in tangen-tial subspacesrdquoMathematical Problems in Engineering vol 2014Article ID 615792 11 pages 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

14 Mathematical Problems in Engineering

U Type Steel Plate [Unpublished PhD Dissertation] HunanUniversity 2017

[14] L Jiang K JMaH G Zhang and L Li ldquoThe designmethod ofsteel Vierendeel beam with shear connectionrdquo Xirsquoan Universityof Architecture amp Technology (Natural Science Edition) vol 50no 3 pp 354ndash360 2018

[15] JGJ 101-2015 Specification of Testing Methods for EarthquakeResistant Building Chinese Building Industry Press 2015

[16] Z L Li C T Zhang W L Fan et al ldquoExperimental research onseismic behavior of a new type of thin-walled square steel tubebeam-column connectionrdquo Engineering Mechanics vol 30 no2 pp 72ndash81 2013

[17] G Ballio and F M Mazzolani Theory and Design of SteelStructures Chapman and Hall London UK 1983

[18] S Banno I H P Mamaghani T Usami and E Mizuno ldquoCyclicelastoplastic large deflection analysis of thin steel platesrdquo Journalof Engineering Mechanics vol 124 no 4 pp 363ndash370 1998

[19] I H P Mamaghani T Usami and E Mizuno ldquoInelastic largedeflection analysis of structural steel members under cyclicloadingrdquo Engineering Structures vol 18 no 9 pp 659ndash6681996

[20] G L Bai J Q Zhao N J Du B Xie andH Gao ldquoExperimentalstudy on seismic behavior of SRC abnormal interior joints anddesign advicerdquo Journal of Building Structures vol 39 no 7 pp33ndash45 2018

[21] W B Kratzig and Y S Petryna ldquoQuasistatic seismic damageindicators for rc structures from dissipating energies in tangen-tial subspacesrdquoMathematical Problems in Engineering vol 2014Article ID 615792 11 pages 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom