MechanicalBehaviorandCalculationMethodforRCFifteen-Pile ... · 2020. 9. 18. · A ﬁfteen-pile cap of mixed passenger and freight railway contains two-ﬂoor caps. e plane dimension

Research ArticleMechanical Behavior and Calculation Method for RC Fifteen-PileCap of Mixed Passenger and Freight Railway Bridge

Hongmeng Huang 1 Lu Cui2 and Wei Lu1

1School of Civil Engineering Northwest Minzu University Lanzhou 730030 China2Yantai Academy of Urban Construction and Design Co Ltd Yantai 264003 China

Correspondence should be addressed to Hongmeng Huang huanghongmeng_1987163com

Received 19 July 2020 Revised 13 August 2020 Accepted 21 August 2020 Published 18 September 2020

Academic Editor Xueping Fan

Copyright copy 2020 Hongmeng Huang et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

e thickness reinforcement and concrete strength grade of railway caps in China are generally determined according to theforce yet the method for calculating the force is unclear To date there is no desirable calculation method for analyzing the capsBased on the fifteen-pile thick cap of mixed passenger and freight railway the influencing factors on cap bearing capacity wereanalyzed using finite element method (FEM) e variations of load-bearing capacity and mechanical behavior of thick cap werecharacterized by introducing rigid angle α Results indicated that ultimate load-bearing value of the cap increased linearly with theincrease of concrete strength grade and an increasing load-bearing capacity of the reinforcement distributed in the pile diameterrange was larger than that of the uniform reinforcement when the reinforcement ratio was 015 it increased by 93 e capshowed punching failure when αlt 45deg e reaction force at each pile top under vertical load was not equal thereby the cap wasnot absolutely rigid e principal compressive stress trajectories in the concrete were distributed in the range of connecting thepile and the outer edge of the pier and the effective tensile stresses in the reinforcement were mainly distributed in the diameterrange of pile and pile connection which is in accord with the stress mode of the ordinary spatial truss model Based on this aspatial truss model applicable to the design of railway caps is proposed and a method for calculating reaction force at pile top andformulas for calculating the bearing capacity of strut and tie were presented e feasibility of the proposed method was alsoverified by comparison with FEM results

1 Introduction

e cap is an important component for transferring theupper load and its internal forces are complex which isespecially the case for the forces of thick caps of group pilesfoundation Many researchers have studied the factors af-fecting the bearing capacity and failure forms of caps Guo[1] and Souza et al [2] respectively introduced punching-span ratio and shear span-depth ratio to distinguish thedifferent failure forms of the caps Huang et al [3] analyzedthe impact exerted on bearing capacity of pier and pile afterincreasing pile cap height by using the standard formula andnumerical simulation method Bloodworth et al [4] studiedthe effects of shear span cap width and reinforcement ratioon the shear behavior of the cap under full-width wallloading observed strut-and-tie model (STM) and thereby

proposed an improvedmodified strut-and-tie method whichgives more accurate predictions for the analysis of a four-pilecap

STM is considered a commonly used method for theanalysis and design of reinforced concrete structures [5]which was first introduced as an alternative approach for thedesign of pile caps and D-region members into the ACI 318-02 [6] Based on ACI 318 [7] and BS 8110-1 [8] Chetchotisakand Teerawong [6] introduced the strength reduction factoras a safety index to ensure the safety of caps designed by STMmethod Abdul-Razzaq and Farhood [9] designed andmanufactured 12 RC cap specimens with different numberof piles comparatively studied the difference of bearingcapacity and failure mechanism of the specimens designedaccording to the traditional section design method and theSTM and pointed out the shortcomings of cap design in ACI

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 8833256 13 pageshttpsdoiorg10115520208833256

318 Existing research on the application of STM mainlyfocused on the few pile caps although STM was modifiedand improved by model tests and FEM and a higher pre-cision STM was proposed [2 10ndash16] In the literature thecompatibility and tensile contributions of concrete materials[2] modifying the node and tie parameters [14] the sizeeffect (decreasing strength while increasing member size)[15] or the constitutive laws and strain compatibility ofcracked reinforced concrete [10] were considered

Most of the aforementioned research is based on few pilecaps in the building structure CANCSA S6-14 [17] JTG3362-2018 [18] and AASHTO LRFD-2017 [19] all putforward STM applicable to the calculation of pile caps andmade specific provisions on the calculation parameters ofstrut and tie However at present there are few studiesdealing with railway caps In order to meet the requirementsof bearing capacity and stiffness thick cap and group pilesfoundation are mostly adopted in railway bridge Me-chanical characteristics of railway bridge caps are differentfrom those of few pile caps in building structure ethickness reinforcement and concrete strength grade ofrailway caps in China are generally determined according tothe force the pile cap is usually designed by controlling therigid angle and setting steel bars mesh at the bottom [20]Although this method has proved to be reliable in long-termengineering construction practices it is unclear how theforce was calculated To date there is no better calculationmethod for analyzing the caps In this study finite elementmethod (FEM) was used to study the influencing factors ofbearing capacity and mechanical behavior of caps based onthe fifteen-pile thick cap of mixed passenger and freightrailway For more accurate mechanical characteristics of thecap ANSYS software was used to establish the analyticalmodel of the prototype cap Moreover by taking into ac-count factors such as concrete strength grade pile spacingcap thickness reinforcement ratio reinforcement distrib-uted form load form and their variations this paper an-alyzed the influence of each factor on bearing capacityfailure mechanism inside the cap stress trajectories in theconcrete stresses distribution in the reinforcement and lawof the reaction force at pile top is study was seeking todetermine whether STM method is suitable for the calcu-lation of railway cap and proposed the calculation method ofcap bearing capacity

2 Establishment of Finite ElementAnalysis Model

A fifteen-pile cap of mixed passenger and freight railwaycontains two-floor caps e plane dimension of the first-floor and second-floor cap is 186mtimes 122m and134mtimes 70m respectively the thickness of each floor cap is3m and the concrete strength grade is C30 e cap isdesigned according to the reinforcement on six sides withsteel bars being evenly distributed orthogonally at its bot-tom e steel bar grade is HRB335 with a diameter of25mm and a spacing of 10 cm and the reinforcement ratio is017 e cap is connected with the bored pile with adiameter of 150 cm e concrete strength grade of pile and

pier is C30 and C35 respectively e main dimensions ofthe cap and number of piles are shown in Figure 1

According to the structural characteristics and calcula-tion requirements of railway caps the stress-strain rela-tionship of concrete under uniaxial compression[Figure 2(a)] was referring to the curve specified in GB50010-2010 (2015 edition) [21] Concrete was simulatedusing SOLID65 3D solid element in ANSYS software andthe failure criterion was selected based on the five-parametermodel of WillamndashWarnke criterion

SOLID65 can be used to calculate the cracking andcrushing of concrete with the special cracking and crushingcapabilities Combined with the material characteristics ofreinforced concrete the shear transfer coefficients are in-troduced and the uniaxial tensile strength ft and uniaxialcompressive strength fc of concrete are added to simulate thecracking and crushing of concrete In this paper sheartransfer coefficient for an open crack βt 05 shear transfercoefficient for a closed crack βc 095 uniaxial tensilestrength ft and uniaxial compressive strength fc were de-termined according to the concrete strength grade

Constitutive relation of steel bar was simplified asuniaxial stress and the ideal elastic-plastic model wasadopted to ignore the stress strengthening stage[Figure 2(b)] When constructing the finite element modelthe steel bars were diffused in the concrete elementsaccording to the volume ratio of reinforcement and thecoordinate system and the concrete element was regarded asa continuous uniform material without taking into con-sideration the bond and slip between steel bar and concrete

e pile-soil interaction model adopted the linear elastichypothesis and used spring to simulate soil stiffness [22]at is to say it took into consideration the lateral resistanceof soil on the pile side and its distribution the vertical re-sistance of soil and the compression of pile itself econstraint of soil on pile foundation was equivalent to aseries of discontinuous springs which were simulated byspring element COMBIN14 in ANSYS software Springstiffness coefficient was calculated by the ldquomrdquo method basedon the linear elastic hypothesis Figure 3 shows the finiteelement model

3 Analysis of Influencing Factors on Cap

In order to analyze the effects of some factors such asconcrete strength grade pile spacing cap thickness bottomreinforcement ratio reinforcement distributed form andload form on the mechanical behavior of the cap rigid angleα was introduced which is defined as the angle of the ex-tension of the outer edge of the pier root to the outer edge ofthe pile top e change of pile spacing or cap thickness wasconsidered as the change of α e parameters of eachinfluencing factor were changed according to Table 1 andother unchanged parameters can be seen in Figure 1 InTable 1 the concentrated reinforcement was within the pilediameter range e combined load containing horizontalforce vertical force and bendingmoment was considered byapplying the seismic force at the pier bottom to the cap topthrough the seismic calculation of the bridge structure

2 Advances in Civil Engineering

31 Load-Bearing Capacity Analysis e relationships ofvertical ultimate load at cap top with concrete strengthgrade pile spacing cap thickness and reinforcement ratioare shown in Figure 4 Vertical ultimate load at cap top withdifferent reinforcement ratios and reinforcement distributedforms is shown in Table 2

As can be seen from Figure 4 and Table 2 the ultimateload increased linearly with the increase of concretestrength grade e ultimate load had its maximum whenthe longitudinal pile spacing was 48m (α 34deg) and itdecreased as spacing gt48m e ultimate load showed anincreasing trend with the increase of cap thickness and itsincreasing tendency began to slow down as cap thicknessgt30m e ultimate load increased with the increase ofthe reinforcement ratio and its increasing tendency beganto slow down as reinforcement ratio gt015 When

reinforcement ratio of the cap was 012 and 015 theload-bearing capacity of the reinforcement distributedin the pile diameter range increased by 93 and42 compared to that of the uniform reinforcementrespectively

32 Failure Mechanism When α 26deg or α 27deg [seeFigure 5(a)] cracks first appeared around the pile and theyexpanded upward as load increased showing the fastestspeed around the middle pile At cap bottom radial cracksappeared along the line between the middle pile and otherpiles and developed continuously Since the inside edge pileof No 2 to No 4 and No 12 to No 14 [see Figure 1(a)]shared the same vertical line with the edge of the second-floor cap cracks developed vertically

1860

130 4 times 400 130

130

2 times

480

130

1220

700

1340No 1

No 2 No 3 No 4

No 5

No 6 No 7 No 8 No 9 No 10

No 11No 12 No 13 No 14

No 15

Unit cm

(a)

Unit cm

300

300

Pier

hei

ght h

Pile

leng

th l

21deg

α = 34deg

700

1220

55 330 330 55

Pier

Second-floor cap

First-floor cap

Piles

150 150 150

(b)

Figure 1 Main layout and section dimensions of pile and cap (a) Plane dimensions and pilesrsquo number (b) Elevation dimensions

Advances in Civil Engineering 3

When αlt 45deg [see Figures 5(b) and 5(c)] cracks firstappeared at the inner edge of the edge pile and extendedobliquely upward As cracks extended to a certain heightthey began to appear around the middle pile and spreadobliquely to the top of the middle span ey then

converged with the oblique cracks of No 2 to No 4 andNo 12 to No 14 edge piles and spread obliquely towardscap top Finally they connected to the oblique cracks ofother piles at cap top declaring cap damage by showingpunching failure

σ

ε

fc

05 fc

0 εc εcu

fc = ultimate axial compressivestrength of concreteεc = peak compressive strain ofconcrete corresponding to fcεcu = the compressive strain of concretewhen the stress in the descending branchof the stress-strain curve is equal to 05 fc

(a)

σ

ε

fs

εs0

fs = tensile strength of steel barεs = strain of steel bar corresponding to fs

(b)

Figure 2 Stress-strain curve of materials (a) Concrete under uniaxial compression (b) Steel bar under monotonic tension

Equivalentspring element

First-floor cap

Second-floor cap

Pier

Figure 3 Finite element analysis model of pile cap

Table 1 Parameter variation of influencing factors

Influencing factors Parameter variationConcrete strength grade C25 C30 C35 C40 C45Longitudinal pile spacing 42m (α 26deg) 48m (α 34deg) 54m (α 41deg) 60m (α 47deg) 66m (α 52deg)First-floor cap thickness 20m (α 46deg) 25m (α 39deg) 30m (α 34deg) 35m (α 30deg) 40m (α 27deg)

Reinforcement ratio Uniform reinforcement 006 012 015 018 020 025Concentrated reinforcement 12 015

Load form Vertical load combined load


When αge 45deg [see Figure 5(d)] cracks first appearedaround the edge pile began to appear around the middle pilewith increasing load and developed obliquely upwardMeanwhile bending cracks appeared at the midspan betweenpiles at cap bottom and expanded upward After intersectingwith oblique cracks on the edge pile cracks continued toexpand upward When cracks extended to cap top cap wasdeclared to be damaged showing bending failure

33 Reaction Force at Pile Top Since the structure and loadwere symmetrical the reaction forces at No 1 to No 3 andNo 6 to No 8 pile top within 14 cap were extracted therein

No 1 pile was corner pile No 8 pile was middle pile and therest were edge piles [see Figure 1(a)]

Reaction forces at pile top in this study were consistentwith results of spatial truss model test and numerical sim-ulation presented by He et al [23] As can be seen fromFigure 6 and Table 3 the reaction force at each pile top undervertical load was not equal the middle pile top had thehighest value followed by edge pile top and corner pile topAs α increased the proportion of reaction force decreased atcorner pile No 2 edge pile and No 3 edge pile while thatproportion increased at the middle pile No 6 edge pile andNo 7 edge pile Under ultimate load the maximum pilereaction force at α 52deg and α 27deg was 197- and 14-fold

Ulti

mat

e loa

d (1

06 N)

200

220

240

260

280

300

320

340

360

380

400

C30 C35 C40 C45C25Concrete strength grade

(a)

Ulti

mat

e loa

d (1

06 N)

180

200

220

240

260

280

300

48 54 60 6642Longitudinal pile spacing (m)

(b)

Ulti

mat

e loa

d (1

06 N)

220

240

260

280

300

320

25 30 35 4020First-floor cap thickness (m)

(c)

Ulti

mat

e loa

d (1

06 N)

240

250

260

270

280

290

300

009 012 015 018 021 024 027006Bottom reinforcement ratio ()

(d)

Figure 4 Influencing curve of vertical ultimate load at cap top with parameter variation of different influencing factors (a) Concretestrength grade (b) Pile spacing (c) Cap thickness (d) Reinforcement ratio

Table 2 Vertical ultimate load at cap top with different reinforcement ratios and reinforcement distributed forms

Reinforcement distributed form Uniform reinforcement Concentratedreinforcement

Reinforcement ratio () 006 012 015 018 020 025 012 015Ultimate load (106N) 2472 2593 2754 2836 2868 2902 2834 2871


greater than the minimum pile reaction force respec-tively e reaction force at pile top was closer when αwassmaller but the cap was not absolutely rigid When α waslarger the punching was more serious Under the actionof load the capacity of load transferred to corner pile wasweakened with the generation of damage inside the capAs a result corner pile could not continue to bear theload and the load was transferred to the middle pilewhich caused middle pile failure in advance ereforemiddle pile should be considered strengthening in capdesign

34 Stress Trajectory Figures 7(a) and 7(c) show the dis-tribution of elastic principal compressive stresses in theconcrete under vertical load and combined load (hori-zontal force vertical force and bending moment) re-spectively e principal compressive stresses were mainlytransmitted along the line between the pile and the outeredge of pier while the stress trajectories in other areas wereless distributed Concrete in this area provided capstructure safety e first-floor cap top and the pile topexperienced a state of three-way compression Figures 7(b)and 7(d) show the distribution of elastic principal tensile

Cracks

(a)

Cracks

(b)

Cracks

(c)

Cracks

(d)

Figure 5 Cracks distribution varying with α (a) α 26deg (b) α 34deg (c) α 41deg (d) α 52deg


stresses in the concrete under vertical load and thecombined load respectively Concrete at a certain height atcap bottom was under tension which was borne by steelbars

35 Steel Bar Stress Distribution Figure 8 shows the stressdistribution curve of horizontal steel bars at cap bottomemaximum stress of steel bars was distributed between pileand pile connection After cracking of concrete at capbottom the effective range of reinforcement effect wasroughly within the pile diameter range and the stresses of

steel bars at other positions were relatively small e steelbars within the pile diameter range between pile and pileconnection served as tie bars

4 Calculation Method for Bearing Capacity ofTie and Strut

e failure mechanism elastic principal stress trajectories inthe concrete and stresses distribution in the reinforcementwere summarized as follows According to the developmentof cracks inside the cap cracked reinforced concrete carriedload principally by compressive stresses in the concrete and

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)

No1 pile No2 pileNo3 pile

No6 pileNo7 pileNo8 pile

4

5

6

7

8

9

100 150 200 250 30050Load value (106 N)

(a)Re

actio

n fo

rce p

ropo

rtio

n at

pile

top

()



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(b)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(c)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)


No6 pileNo7 pile No8 pile

4

5

6

7

8

9

100 150 200 250 300 35050Load value (106 N)

(d)

Figure 6 Curve of the reaction force proportion at pile top with vertical load when α changes (a) α 26deg (b) α 52deg (c) α 46deg (d) α 27deg


Table 3 Reaction force at each pile top under ultimate load (including self-weight loads) (unit 106N)

Calculated parameters Ultimate loade reaction force at pile top

No 1 pile No 2 pile No 3 pile No 6 pile No 7 pile No 8 pile

Cap thickness

20m (α 46deg) 2464 110 169 186 162 213 22725m (α 39deg) 2728 136 180 207 179 225 24230m (α 34deg) 3064 148 180 194 174 208 21835m (α 30deg) 3122 166 213 230 202 244 25640m (α 27deg) 3334 188 224 241 214 255 264

Pile spacing

42m (α 26deg) 2961 156 204 223 188 226 24754m (α 41deg) 2843 145 184 202 20 237 25260m (α 47deg) 2608 131 165 178 19 226 23866m (α 52deg) 2222 109 136 145 169 202 214

Note When cap thickness 30m the corresponding pile spacing 48m To avoid repetition results for pile spacing 48m are not presented in Table 3

Principalcompressive

stresses region

A = ndash0739E + 07 B = ndash0649E + 07 C = ndash0559E + 07 D = ndash0470E + 07 E = ndash0380E + 07 F = ndash0290E + 07 G = ndash0200E + 07 H = ndash0110E + 07 I = ndash196672

(a)

Principaltensile

stresses region

A = ndash0171E + 07 B = ndash0133E + 07 C = ndash946447 D = ndash566984 E = ndash187521 F = 191941 G = 571404 H = 950867 I = 0133E + 07

(b)

Figure 7 Continued


tensile stresses in the reinforcement e mechanicalcharacteristic of railway cap conforms to the simplifiedcalculationmodel of the spatial truss in which concrete lyingbetween pier outer edge and piles inside cap serves as strutsthe steel bars at cap bottom within the pile diameter rangeare tie bars the junction surface at cap top serves as the topnode and the junction points between struts and tie barsserve as the bottom node From isosurface of elastic principalcompressive stress trajectories in the concrete it can be seenthat concrete struts intersected at a surface but not a pointunder vertical load which is similar to the composite spatial

truss model Under combined load with a large proportionof bending moment mechanical characteristics of capconform to the ordinary spatial truss model e simplifiedspatial truss model under different loads is shown inFigure 9

According to STM and the calculation method of strutand tie in CANCSA S6-14 JTG 3362-2018 and AASHTOLRFD-2017 based on the mechanical behavior and failuremechanism of railway cap and code for design of concretestructures of railway bridge and culvert using the allowablestress method the calculation method for the design of


stresses region


(c)

Principal tensile stresses region

A = ndash0315E + 07 B = ndash0233E + 07 C = ndash0150E + 07 D = ndash678928 E = 145595 F = 970118 G = 0179E + 07 H = 0262E + 07 I = 0344E + 07

(d)

Figure 7 Isosurface of principal stresses distribution under vertical load or combined load (a) Principal compressive stresses under verticalload (b) Principal tensile stresses under vertical load (c) Principal compressive stresses under combined load (d) Principal tensile stressesunder combined load


railway cap is recommended with considering the strengthsafety factors of the material [24]

41 Reaction ForceCalculation at Pile Top Reaction forces atpile top under vertical load are not equal ie the cap is notabsolutely rigid Checking calculation of bearing capacityusing formulas for rigid cap produces certain blindnesswhich may cause potential safety hazards to practical en-gineering applications e pile reaction force is related tocap rigid angle load transfer path geological conditionsload forms on cap top piles arrangement and the shearingeffects of soil between piles

FEM can be used to calculate accurate reaction force atpile top of railway caps by establishing pile cap model basedon the linear elastic assumption e spring stiffness coef-ficient calculated by ldquomrdquo method is applied to the nodes ofpiles model to simulate the pile foundation constraint e

tie bar force in the spatial truss model is calculated accordingto the static equilibrium by the maximum reaction force atpile top in the same row piles e strut force is the largervalue of the result according to the static equilibrium bymaximum reaction force at pile top in the same row piles andthe reaction force at middle pile top

42 Bearing Capacity of Tie Bar We found that the effectivestresses of longitudinal steel bars mainly concentrated withinpile diameter by analyzing the reinforcement stresses distri-bution under different reinforcement ratios Considering thestrength safety factor the allowable stress value of steel bar wasdetermined [24] and the bearing capacity of tie bar should be

TleAs σs1113858 1113859 (1)

where T is the design value of the tie bar force under loadwhich is calculated according to the static equilibrium of

012 reinforcement ratio020 reinforcement ratio

Tran

sver

se st

eel b

ars s

tres

s (M

Pa)

40

60

80

100

120

140

160

180

200

ndash5 ndash4 ndash3 ndash2 ndash1 0 1 2 3 4 5 6ndash6Longitudinal position (m)

(a)


Long

itudi

nal s

teel

bar

s str

ess (

MPa

)

ndash8 ndash6 ndash4 ndash2 0 2 4 6 8 10ndash10Transverse position (m)

60

80

100

120

140

160

180

200

220

240

260

280

(b)

Figure 8 Stress values and distribution of steel bars (a) Transverse steel bars along the longitudinal position (b) Longitudinal steel barsalong the transverse position

Tie bars Bottom nodes

Strut

Top junction surface

q

(a)


Strut


q

P

M

(b)

Figure 9 Construction of the spatial truss model under different loads (a) Vertical load (b) Combined load


spatial truss model As is the total area of longitudinal steelbars within pile diameter at cap bottom and [σs] is theallowable stress value of steel bar

43 Bearing Capacity of Strut e bearing capacity ofconcrete strut was checked according to the followingformula

FleAc σc1113858 1113859 (2)

where F is the design value of strut force under load which iscalculated according to the static equilibrium of spatial trussmodelAc is the effective area of concrete strut and [σc] is theallowable stress value of concrete

According to principal compressive stress trajectoriesthe cross section of strut is elliptic and the strut is a cylinderStrut section height (presented in Figure 10) was calculatedaccording to (3) and (4) referring to the provisions inAASHTO LRFD

t D sinθs + ha cos θs (3)

ha s + 6db (4)

where t is the calculated height of strut D is the pile di-ameter ha is the length of the back face of a node θs is theinclination degree of the strut s is the distance between thesteel bar center at the top layer and the cap bottom and db isthe steel bar diameter

When deducing formula of calculating splitting bearingcapacity it is considered that the compressive stress diffu-sion angle is diffused according to the formulaθ arctan(12) e centroid of the bottom compressionarea is at 12 of the effective height so the dimension of strutperpendicular to tie bar is D + 05ha e calculation of strutheight along tie bar cannot consider diffusion range due tothe anchorage of steel bar strut height can be calculatedaccording to (3) erefore the cross-sectional area of strutin the cap is calculated as

DReaction force C

at pile top

Strut force F

h a

6db

s

Tie bars totalforce T

t

Strut

NodeTie bars

ha cos θs

Dsin θs

θs

Figure 10 Schematic diagram of calculation size of strut

Table 4 Comparison of ultimate load of caps between FEM and the present method

Calculation model Ultimate load(2)(1)a

(1) e result of FEM (106N) (2) e result of present method (106N)

Concrete strength grade

C25 2495 2388 0957C30 3064 2809 0917C35 3400 3301 0971C40 3745 3792 1013C45 4177 4214 1009

Pile spacing (m)

42 2961 3031 102454 2843 2570 090460 2608 2347 090066 2222 2152 0968

Cap thickness (m)

20 2464 2236 090725 2728 2572 094335 3122 2987 095740 3334 3114 0934

e average value of the ratio is 0954 with a variance of 0042Notee calculation model of cap thickness 30m and pile spacing 48m is the same as that of concrete strength grade C30 therefore not all of them arelisted in Table 4 aRatio of the result of present method (2) and FEM (1)


Ac π D sin θs + ha cos θs( 1113857 D + 05ha( 1113857

4 (5)

e concrete allowable stress of central compressivemember is calculated according to the ultimate compressivestrength fc of concrete considering the strength safety factore concrete strut is a cylinder-like central compressionmember It is necessary to convert ultimate compressivestrength of concrete to that of a cylinder to obtain theeffective compressive strength of strut ie the allowablestress of concrete strut

e relationship between fc and fcuk is approximated as

fc 067fcuk (6)

e relationship between fcyk and fcuk is approximated as

fcyk 08fcuk (7)

According to (6) and (7) the relationship between fcykand fc is as follows

fcyk 12fc (8)

With a 25 safety coefficient of central compressive memberthe allowable stress of concrete strut is calculated as follows

σc1113858 1113859 β12fc

25 048βfc (9)

where fc is the ultimate compressive strength of concretefcuk is the compressive strength standard value of 150mmconcrete cube fcyk is the compressive strength standardvalue of φ150mmtimes 300mm concrete cylinder and β is theincrease coefficient of allowable stress under different loadcombinations

44 Validation of Calculation Method e Results of thecalculation method presented in this study were comparedwith FEM of different concrete strength grade longitudinalpile spacing and cap thickness values as shown in Table 4e average ratio of our values to that of FEMwas 0954 witha variance of 0042 showing good agreement between themStrength safety coefficient derived from our method isconservative which is beneficial to the structure safetyerefore the calculation method this paper proposed canbe used to calculate the ultimate load of cap and further todesign and check the cap

5 Conclusions

FEM was used to analyze the mechanical behavior for fif-teen-pile thick cap of mixed passenger and freight railwaybridge Concrete strength grade cap thickness pile spacingreinforcement ratio and reinforcement distributed formdirectly affected bearing capacity of cap e cap showedpunching failure as αlt 45deg Reaction forces at pile top undervertical load were not equal the force was larger at themiddle pile top than that at the edge pile top and showed itsminimum at the corner pile top suggesting that the cap isnot absolutely rigid e principal compressive stress

trajectories in the concrete were mainly transmitted alongthe line between pile and pier outer edge while the effectivestresses of reinforcement were mainly distributed in the pilediameter range between pile and pile connection

According to the failure mechanism principal stresstrajectories in the concrete and stresses distribution in thereinforcement this paper proposed a spatial truss modelwhich was applicable to the design of railway caps espatial truss model was composed of concrete struts and steeltie bars interconnected at nodes concrete between pier outeredge and piles inside the cap served as struts and the steelbars at cap bottom within the pile diameter range were tiebars With reference to the relevant specifications of capdesign the formulas for calculating the bearing capacity ofstrut-and-tie bar were presented A comparison between ourresults and that of FEM indicated that our method could beused to calculate the ultimate load of cap and further todesign and check cap of railway bridge

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon reasonablerequest

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was partly supported by the Fundamental Re-search Funds for the Central Universities (31920200059)

References

[1] H L Guo and B Jiang ldquoStudy of pile cap failure type dis-criminationrdquo Engineering Mechanics vol 30 no 6 pp 142ndash147 2013

[2] R Souza D Kuchma J W Park and T BittencourtldquoAdaptable strut-and-tie model for design and verification offour-pile capsrdquo ACI Structural Journal vol 106 no 2pp 142ndash150 2009

[3] X B Huang C Y Liu S Hou et al ldquoAn analysis of the impactexerted on bearing capacity of pier and pile after increasingpile cap heightrdquo Shock and Vibration vol 2018 p 9 2018

[4] A G Bloodworth J Cao andM Xu ldquoNumerical modeling ofshear behavior of reinforced concrete pile capsrdquo Journal ofStructural Engineering vol 138 no 6 pp 708ndash717 2012

[5] M Victoria O M Querin and P Martı ldquoGeneration of strut-and-tie models by topology design using different materialproperties in tension and compressionrdquo Structural andMultidisciplinary Optimization vol 44 no 2 pp 247ndash2582011

[6] P Chetchotisak and J Teerawong ldquoReliability-based assess-ment of RC pile cap design methods and proposals for theirstrength resistance factorsrdquo KSCE Journal of Civil Engineer-ing vol 23 no 8 pp 3372ndash3382 2019

[7] ACI Building Code Requirements for Structural Concreteand Commentary American Concrete InstituteFarmingtonHills MI USA 2014


[8] British Standards Institution Structural Use of Concrete Part1 Code of Practice for Design and Construction BritishStandards Institution London UK 1997

[9] K S Abdul-Razzaq and M A Farhood ldquoDesign-orientedtesting and modeling of reinforced concrete pile capsrdquo KSCEJournal of Civil Engineering vol 23 no 8 pp 3509ndash35242019

[10] J Park D Kuchma and R Souza ldquoStrength predictions ofpile caps by a strut-and-tie model approachrdquo CanadianJournal of Civil Engineering vol 35 no 12 pp 1399ndash14132008

[11] H L Guo ldquoSpacial strut and tie model for punching loadtransfer mechanism analysis of pile-caprdquo Journal of BuildingStructures vol 30 no 1 pp 147ndash156 2009 in Chinese

[12] C F Sun Q Gu and S M Peng ldquoExperimental research onsteel fiber reinforced concrete two-pile thick capsrdquo Journal ofBuilding Structures vol 31 no 2 pp 117ndash124 2010 inChinese

[13] Y M Yun and J A Ramirez ldquoStrength of concrete struts inthree-dimensional strut-tie modelsrdquo Journal of StructuralEngineering vol 142 no 11 2016

[14] Y Zhou and G-L Dai ldquoParameter study of a new strut-and-tie model for a thick cap with six pilesrdquo International Journalof Geomechanics vol 16 no 1 2016

[15] C E Broms ldquoStrut-and-tie model for punching failure ofcolumn footings and pile capsrdquo ACI Structural Journalvol 115 no 3 pp 689ndash698 2018

[16] Y M Yun B Kim and J A Ramirez ldquoree-dimensionalgrid strut-and-tie model approach in structural concretedesignrdquoAci Structural Journal vol 115 no 1 pp 15ndash26 2018

[17] CSA Group Canadian Highway Bridge Design Code Cana-dian Standards Association Mississauga Canada 2016

[18] China Communications Press Specifications for Design ofHighway Reinforced Concrete and Prestressed Concrete Bridgesand Culverts China Communications Press Co Ltd BeijingChina 2018

[19] American Association of State Highway and TransportationOfficials Bridge Design Specifications American Associationof State Highway and Transportation Officials WashingtonDC USA 8 edition 2017

[20] China Railway Publishing House Code for Design on Subsoiland Foundation of Railway Bridge and Culvert China RailwayPublishing House Co Ltd Beijing China 2017

[21] China Architecture and Building Press Code For Design ofConcrete Structures (Ce 2015 Version) China Architectureand Building Press Beijing China 2015

[22] H B Wang Z S Xu and W X Ren ldquoAnalysis of naturalfrequency about 3-dimensional pile group-pile cap-pierrdquoJournal of Changsha Railway University vol 17 no 2pp 74ndash79 1999

[23] H N He G L Dai and Y L Diao ldquoSpatial truss model testand numerical simulation on thick caps of large-scale grouppiled foundationrdquo China Civil Engineering Journal vol 48no 8 pp 102ndash109 2015

[24] China Railway Publishing House Code for Design of ConcreteStructures of Railway Bridge and Culvert China RailwayPublishing House Co Ltd Beijing China 2017


318 Existing research on the application of STM mainlyfocused on the few pile caps although STM was modifiedand improved by model tests and FEM and a higher pre-cision STM was proposed [2 10ndash16] In the literature thecompatibility and tensile contributions of concrete materials[2] modifying the node and tie parameters [14] the sizeeffect (decreasing strength while increasing member size)[15] or the constitutive laws and strain compatibility ofcracked reinforced concrete [10] were considered

Most of the aforementioned research is based on few pilecaps in the building structure CANCSA S6-14 [17] JTG3362-2018 [18] and AASHTO LRFD-2017 [19] all putforward STM applicable to the calculation of pile caps andmade specific provisions on the calculation parameters ofstrut and tie However at present there are few studiesdealing with railway caps In order to meet the requirementsof bearing capacity and stiffness thick cap and group pilesfoundation are mostly adopted in railway bridge Me-chanical characteristics of railway bridge caps are differentfrom those of few pile caps in building structure ethickness reinforcement and concrete strength grade ofrailway caps in China are generally determined according tothe force the pile cap is usually designed by controlling therigid angle and setting steel bars mesh at the bottom [20]Although this method has proved to be reliable in long-termengineering construction practices it is unclear how theforce was calculated To date there is no better calculationmethod for analyzing the caps In this study finite elementmethod (FEM) was used to study the influencing factors ofbearing capacity and mechanical behavior of caps based onthe fifteen-pile thick cap of mixed passenger and freightrailway For more accurate mechanical characteristics of thecap ANSYS software was used to establish the analyticalmodel of the prototype cap Moreover by taking into ac-count factors such as concrete strength grade pile spacingcap thickness reinforcement ratio reinforcement distrib-uted form load form and their variations this paper an-alyzed the influence of each factor on bearing capacityfailure mechanism inside the cap stress trajectories in theconcrete stresses distribution in the reinforcement and lawof the reaction force at pile top is study was seeking todetermine whether STM method is suitable for the calcu-lation of railway cap and proposed the calculation method ofcap bearing capacity

2 Establishment of Finite ElementAnalysis Model

A fifteen-pile cap of mixed passenger and freight railwaycontains two-floor caps e plane dimension of the first-floor and second-floor cap is 186mtimes 122m and134mtimes 70m respectively the thickness of each floor cap is3m and the concrete strength grade is C30 e cap isdesigned according to the reinforcement on six sides withsteel bars being evenly distributed orthogonally at its bot-tom e steel bar grade is HRB335 with a diameter of25mm and a spacing of 10 cm and the reinforcement ratio is017 e cap is connected with the bored pile with adiameter of 150 cm e concrete strength grade of pile and

pier is C30 and C35 respectively e main dimensions ofthe cap and number of piles are shown in Figure 1

According to the structural characteristics and calcula-tion requirements of railway caps the stress-strain rela-tionship of concrete under uniaxial compression[Figure 2(a)] was referring to the curve specified in GB50010-2010 (2015 edition) [21] Concrete was simulatedusing SOLID65 3D solid element in ANSYS software andthe failure criterion was selected based on the five-parametermodel of WillamndashWarnke criterion

SOLID65 can be used to calculate the cracking andcrushing of concrete with the special cracking and crushingcapabilities Combined with the material characteristics ofreinforced concrete the shear transfer coefficients are in-troduced and the uniaxial tensile strength ft and uniaxialcompressive strength fc of concrete are added to simulate thecracking and crushing of concrete In this paper sheartransfer coefficient for an open crack βt 05 shear transfercoefficient for a closed crack βc 095 uniaxial tensilestrength ft and uniaxial compressive strength fc were de-termined according to the concrete strength grade

Constitutive relation of steel bar was simplified asuniaxial stress and the ideal elastic-plastic model wasadopted to ignore the stress strengthening stage[Figure 2(b)] When constructing the finite element modelthe steel bars were diffused in the concrete elementsaccording to the volume ratio of reinforcement and thecoordinate system and the concrete element was regarded asa continuous uniform material without taking into con-sideration the bond and slip between steel bar and concrete

e pile-soil interaction model adopted the linear elastichypothesis and used spring to simulate soil stiffness [22]at is to say it took into consideration the lateral resistanceof soil on the pile side and its distribution the vertical re-sistance of soil and the compression of pile itself econstraint of soil on pile foundation was equivalent to aseries of discontinuous springs which were simulated byspring element COMBIN14 in ANSYS software Springstiffness coefficient was calculated by the ldquomrdquo method basedon the linear elastic hypothesis Figure 3 shows the finiteelement model

3 Analysis of Influencing Factors on Cap

In order to analyze the effects of some factors such asconcrete strength grade pile spacing cap thickness bottomreinforcement ratio reinforcement distributed form andload form on the mechanical behavior of the cap rigid angleα was introduced which is defined as the angle of the ex-tension of the outer edge of the pier root to the outer edge ofthe pile top e change of pile spacing or cap thickness wasconsidered as the change of α e parameters of eachinfluencing factor were changed according to Table 1 andother unchanged parameters can be seen in Figure 1 InTable 1 the concentrated reinforcement was within the pilediameter range e combined load containing horizontalforce vertical force and bendingmoment was considered byapplying the seismic force at the pier bottom to the cap topthrough the seismic calculation of the bridge structure






1860

130 4 times 400 130

130

2 times

480

130

1220

700

1340No 1

No 2 No 3 No 4

No 5


No 11No 12 No 13 No 14

No 15

Unit cm

(a)

Unit cm

300

300

Pier

hei

ght h

Pile

leng

th l

21deg

α = 34deg

700

1220

55 330 330 55

Pier

Second-floor cap

First-floor cap

Piles

150 150 150

(b)





σ

ε

fc

05 fc

0 εc εcu


(a)

σ

ε

fs

εs0


(b)



First-floor cap

Second-floor cap

Pier











Ulti

mat

e loa

d (1

06 N)

200

220

240

260

280

300

320

340

360

380

400


(a)

Ulti

mat

e loa

d (1

06 N)

180

200

220

240

260

280

300


(b)

Ulti

mat

e loa

d (1

06 N)

220

240

260

280

300

320


(c)

Ulti

mat

e loa

d (1

06 N)

240

250

260

270

280

290

300


(d)








Cracks

(a)

Cracks

(b)

Cracks

(c)

Cracks

(d)








Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

100 150 200 250 30050Load value (106 N)

(a)Re

actio

n fo

rce p

ropo

rtio

n at

pile

top

()



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(b)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(c)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

100 150 200 250 300 35050Load value (106 N)

(d)






Cap thickness


Pile spacing




stresses region


(a)

Principaltensile

stresses region


(b)

Figure 7 Continued






stresses region


(c)



(d)








TleAs σs1113858 1113859 (1)



Tran

sver

se st

eel b

ars s

tres

s (M

Pa)

40

60

80

100

120

140

160

180

200


(a)


Long

itudi

nal s

teel

bar

s str

ess (

MPa

)


60

80

100

120

140

160

180

200

220

240

260

280

(b)



Strut


q

(a)


Strut


q

P

M

(b)





FleAc σc1113858 1113859 (2)




ha s + 6db (4)



DReaction force C

at pile top

Strut force F

h a

6db

s


t

Strut

NodeTie bars

ha cos θs

Dsin θs

θs






C25 2495 2388 0957C30 3064 2809 0917C35 3400 3301 0971C40 3745 3792 1013C45 4177 4214 1009

Pile spacing (m)

42 2961 3031 102454 2843 2570 090460 2608 2347 090066 2222 2152 0968

Cap thickness (m)

20 2464 2236 090725 2728 2572 094335 3122 2987 095740 3334 3114 0934




4 (5)



fc 067fcuk (6)


fcyk 08fcuk (7)


fcyk 12fc (8)


σc1113858 1113859 β12fc

25 048βfc (9)



5 Conclusions




Data Availability




Acknowledgments


References































1860

130 4 times 400 130

130

2 times

480

130

1220

700

1340No 1

No 2 No 3 No 4

No 5


No 11No 12 No 13 No 14

No 15

Unit cm

(a)

Unit cm

300

300

Pier

hei

ght h

Pile

leng

th l

21deg

α = 34deg

700

1220

55 330 330 55

Pier

Second-floor cap

First-floor cap

Piles

150 150 150

(b)





σ

ε

fc

05 fc

0 εc εcu


(a)

σ

ε

fs

εs0


(b)



First-floor cap

Second-floor cap

Pier











Ulti

mat

e loa

d (1

06 N)

200

220

240

260

280

300

320

340

360

380

400


(a)

Ulti

mat

e loa

d (1

06 N)

180

200

220

240

260

280

300


(b)

Ulti

mat

e loa

d (1

06 N)

220

240

260

280

300

320


(c)

Ulti

mat

e loa

d (1

06 N)

240

250

260

270

280

290

300


(d)








Cracks

(a)

Cracks

(b)

Cracks

(c)

Cracks

(d)








Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

100 150 200 250 30050Load value (106 N)

(a)Re

actio

n fo

rce p

ropo

rtio

n at

pile

top

()



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(b)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(c)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

100 150 200 250 300 35050Load value (106 N)

(d)






Cap thickness


Pile spacing




stresses region


(a)

Principaltensile

stresses region


(b)

Figure 7 Continued






stresses region


(c)



(d)








TleAs σs1113858 1113859 (1)



Tran

sver

se st

eel b

ars s

tres

s (M

Pa)

40

60

80

100

120

140

160

180

200


(a)


Long

itudi

nal s

teel

bar

s str

ess (

MPa

)


60

80

100

120

140

160

180

200

220

240

260

280

(b)



Strut


q

(a)


Strut


q

P

M

(b)





FleAc σc1113858 1113859 (2)




ha s + 6db (4)



DReaction force C

at pile top

Strut force F

h a

6db

s


t

Strut

NodeTie bars

ha cos θs

Dsin θs

θs






C25 2495 2388 0957C30 3064 2809 0917C35 3400 3301 0971C40 3745 3792 1013C45 4177 4214 1009

Pile spacing (m)

42 2961 3031 102454 2843 2570 090460 2608 2347 090066 2222 2152 0968

Cap thickness (m)

20 2464 2236 090725 2728 2572 094335 3122 2987 095740 3334 3114 0934




4 (5)



fc 067fcuk (6)


fcyk 08fcuk (7)


fcyk 12fc (8)


σc1113858 1113859 β12fc

25 048βfc (9)



5 Conclusions




Data Availability




Acknowledgments


References





























σ

ε

fc

05 fc

0 εc εcu


(a)

σ

ε

fs

εs0


(b)



First-floor cap

Second-floor cap

Pier











Ulti

mat

e loa

d (1

06 N)

200

220

240

260

280

300

320

340

360

380

400


(a)

Ulti

mat

e loa

d (1

06 N)

180

200

220

240

260

280

300


(b)

Ulti

mat

e loa

d (1

06 N)

220

240

260

280

300

320


(c)

Ulti

mat

e loa

d (1

06 N)

240

250

260

270

280

290

300


(d)








Cracks

(a)

Cracks

(b)

Cracks

(c)

Cracks

(d)








Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

100 150 200 250 30050Load value (106 N)

(a)Re

actio

n fo

rce p

ropo

rtio

n at

pile

top

()



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(b)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(c)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

100 150 200 250 300 35050Load value (106 N)

(d)






Cap thickness


Pile spacing




stresses region


(a)

Principaltensile

stresses region


(b)

Figure 7 Continued






stresses region


(c)



(d)








TleAs σs1113858 1113859 (1)



Tran

sver

se st

eel b

ars s

tres

s (M

Pa)

40

60

80

100

120

140

160

180

200


(a)


Long

itudi

nal s

teel

bar

s str

ess (

MPa

)


60

80

100

120

140

160

180

200

220

240

260

280

(b)



Strut


q

(a)


Strut


q

P

M

(b)





FleAc σc1113858 1113859 (2)




ha s + 6db (4)



DReaction force C

at pile top

Strut force F

h a

6db

s


t

Strut

NodeTie bars

ha cos θs

Dsin θs

θs






C25 2495 2388 0957C30 3064 2809 0917C35 3400 3301 0971C40 3745 3792 1013C45 4177 4214 1009

Pile spacing (m)

42 2961 3031 102454 2843 2570 090460 2608 2347 090066 2222 2152 0968

Cap thickness (m)

20 2464 2236 090725 2728 2572 094335 3122 2987 095740 3334 3114 0934




4 (5)



fc 067fcuk (6)


fcyk 08fcuk (7)


fcyk 12fc (8)


σc1113858 1113859 β12fc

25 048βfc (9)



5 Conclusions




Data Availability




Acknowledgments


References































Ulti

mat

e loa

d (1

06 N)

200

220

240

260

280

300

320

340

360

380

400


(a)

Ulti

mat

e loa

d (1

06 N)

180

200

220

240

260

280

300


(b)

Ulti

mat

e loa

d (1

06 N)

220

240

260

280

300

320


(c)

Ulti

mat

e loa

d (1

06 N)

240

250

260

270

280

290

300


(d)








Cracks

(a)

Cracks

(b)

Cracks

(c)

Cracks

(d)








Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

100 150 200 250 30050Load value (106 N)

(a)Re

actio

n fo

rce p

ropo

rtio

n at

pile

top

()



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(b)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(c)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

100 150 200 250 300 35050Load value (106 N)

(d)






Cap thickness


Pile spacing




stresses region


(a)

Principaltensile

stresses region


(b)

Figure 7 Continued






stresses region


(c)



(d)








TleAs σs1113858 1113859 (1)



Tran

sver

se st

eel b

ars s

tres

s (M

Pa)

40

60

80

100

120

140

160

180

200


(a)


Long

itudi

nal s

teel

bar

s str

ess (

MPa

)


60

80

100

120

140

160

180

200

220

240

260

280

(b)



Strut


q

(a)


Strut


q

P

M

(b)





FleAc σc1113858 1113859 (2)




ha s + 6db (4)



DReaction force C

at pile top

Strut force F

h a

6db

s


t

Strut

NodeTie bars

ha cos θs

Dsin θs

θs






C25 2495 2388 0957C30 3064 2809 0917C35 3400 3301 0971C40 3745 3792 1013C45 4177 4214 1009

Pile spacing (m)

42 2961 3031 102454 2843 2570 090460 2608 2347 090066 2222 2152 0968

Cap thickness (m)

20 2464 2236 090725 2728 2572 094335 3122 2987 095740 3334 3114 0934




4 (5)



fc 067fcuk (6)


fcyk 08fcuk (7)


fcyk 12fc (8)


σc1113858 1113859 β12fc

25 048βfc (9)



5 Conclusions




Data Availability




Acknowledgments


References





























Cracks

(a)

Cracks

(b)

Cracks

(c)

Cracks

(d)








Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

100 150 200 250 30050Load value (106 N)

(a)Re

actio

n fo

rce p

ropo

rtio

n at

pile

top

()



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(b)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(c)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

100 150 200 250 300 35050Load value (106 N)

(d)






Cap thickness


Pile spacing




stresses region


(a)

Principaltensile

stresses region


(b)

Figure 7 Continued






stresses region


(c)



(d)








TleAs σs1113858 1113859 (1)



Tran

sver

se st

eel b

ars s

tres

s (M

Pa)

40

60

80

100

120

140

160

180

200


(a)


Long

itudi

nal s

teel

bar

s str

ess (

MPa

)


60

80

100

120

140

160

180

200

220

240

260

280

(b)



Strut


q

(a)


Strut


q

P

M

(b)





FleAc σc1113858 1113859 (2)




ha s + 6db (4)



DReaction force C

at pile top

Strut force F

h a

6db

s


t

Strut

NodeTie bars

ha cos θs

Dsin θs

θs






C25 2495 2388 0957C30 3064 2809 0917C35 3400 3301 0971C40 3745 3792 1013C45 4177 4214 1009

Pile spacing (m)

42 2961 3031 102454 2843 2570 090460 2608 2347 090066 2222 2152 0968

Cap thickness (m)

20 2464 2236 090725 2728 2572 094335 3122 2987 095740 3334 3114 0934




4 (5)



fc 067fcuk (6)


fcyk 08fcuk (7)


fcyk 12fc (8)


σc1113858 1113859 β12fc

25 048βfc (9)



5 Conclusions




Data Availability




Acknowledgments


References
































Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

100 150 200 250 30050Load value (106 N)

(a)Re

actio

n fo

rce p

ropo

rtio

n at

pile

top

()



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(b)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

10

100 150 200 25050Load value (106 N)

(c)

Reac

tion

forc

e pro

port

ion

at p

ile to

p (

)



4

5

6

7

8

9

100 150 200 250 300 35050Load value (106 N)

(d)






Cap thickness


Pile spacing




stresses region


(a)

Principaltensile

stresses region


(b)

Figure 7 Continued






stresses region


(c)



(d)








TleAs σs1113858 1113859 (1)



Tran

sver

se st

eel b

ars s

tres

s (M

Pa)

40

60

80

100

120

140

160

180

200


(a)


Long

itudi

nal s

teel

bar

s str

ess (

MPa

)


60

80

100

120

140

160

180

200

220

240

260

280

(b)



Strut


q

(a)


Strut


q

P

M

(b)





FleAc σc1113858 1113859 (2)




ha s + 6db (4)



DReaction force C

at pile top

Strut force F

h a

6db

s


t

Strut

NodeTie bars

ha cos θs

Dsin θs

θs






C25 2495 2388 0957C30 3064 2809 0917C35 3400 3301 0971C40 3745 3792 1013C45 4177 4214 1009

Pile spacing (m)

42 2961 3031 102454 2843 2570 090460 2608 2347 090066 2222 2152 0968

Cap thickness (m)

20 2464 2236 090725 2728 2572 094335 3122 2987 095740 3334 3114 0934




4 (5)



fc 067fcuk (6)


fcyk 08fcuk (7)


fcyk 12fc (8)


σc1113858 1113859 β12fc

25 048βfc (9)



5 Conclusions




Data Availability




Acknowledgments


References






























Cap thickness


Pile spacing




stresses region


(a)

Principaltensile

stresses region


(b)

Figure 7 Continued






stresses region


(c)



(d)








TleAs σs1113858 1113859 (1)



Tran

sver

se st

eel b

ars s

tres

s (M

Pa)

40

60

80

100

120

140

160

180

200


(a)


Long

itudi

nal s

teel

bar

s str

ess (

MPa

)


60

80

100

120

140

160

180

200

220

240

260

280

(b)



Strut


q

(a)


Strut


q

P

M

(b)





FleAc σc1113858 1113859 (2)




ha s + 6db (4)



DReaction force C

at pile top

Strut force F

h a

6db

s


t

Strut

NodeTie bars

ha cos θs

Dsin θs

θs






C25 2495 2388 0957C30 3064 2809 0917C35 3400 3301 0971C40 3745 3792 1013C45 4177 4214 1009

Pile spacing (m)

42 2961 3031 102454 2843 2570 090460 2608 2347 090066 2222 2152 0968

Cap thickness (m)

20 2464 2236 090725 2728 2572 094335 3122 2987 095740 3334 3114 0934




4 (5)



fc 067fcuk (6)


fcyk 08fcuk (7)


fcyk 12fc (8)


σc1113858 1113859 β12fc

25 048βfc (9)



5 Conclusions




Data Availability




Acknowledgments


References































stresses region


(c)



(d)








TleAs σs1113858 1113859 (1)



Tran

sver

se st

eel b

ars s

tres

s (M

Pa)

40

60

80

100

120

140

160

180

200


(a)


Long

itudi

nal s

teel

bar

s str

ess (

MPa

)


60

80

100

120

140

160

180

200

220

240

260

280

(b)



Strut


q

(a)


Strut


q

P

M

(b)





FleAc σc1113858 1113859 (2)




ha s + 6db (4)



DReaction force C

at pile top

Strut force F

h a

6db

s


t

Strut

NodeTie bars

ha cos θs

Dsin θs

θs






C25 2495 2388 0957C30 3064 2809 0917C35 3400 3301 0971C40 3745 3792 1013C45 4177 4214 1009

Pile spacing (m)

42 2961 3031 102454 2843 2570 090460 2608 2347 090066 2222 2152 0968

Cap thickness (m)

20 2464 2236 090725 2728 2572 094335 3122 2987 095740 3334 3114 0934




4 (5)



fc 067fcuk (6)


fcyk 08fcuk (7)


fcyk 12fc (8)


σc1113858 1113859 β12fc

25 048βfc (9)



5 Conclusions




Data Availability




Acknowledgments


References
































TleAs σs1113858 1113859 (1)



Tran

sver

se st

eel b

ars s

tres

s (M

Pa)

40

60

80

100

120

140

160

180

200


(a)


Long

itudi

nal s

teel

bar

s str

ess (

MPa

)


60

80

100

120

140

160

180

200

220

240

260

280

(b)



Strut


q

(a)


Strut


q

P

M

(b)





FleAc σc1113858 1113859 (2)




ha s + 6db (4)



DReaction force C

at pile top

Strut force F

h a

6db

s


t

Strut

NodeTie bars

ha cos θs

Dsin θs

θs






C25 2495 2388 0957C30 3064 2809 0917C35 3400 3301 0971C40 3745 3792 1013C45 4177 4214 1009

Pile spacing (m)

42 2961 3031 102454 2843 2570 090460 2608 2347 090066 2222 2152 0968

Cap thickness (m)

20 2464 2236 090725 2728 2572 094335 3122 2987 095740 3334 3114 0934




4 (5)



fc 067fcuk (6)


fcyk 08fcuk (7)


fcyk 12fc (8)


σc1113858 1113859 β12fc

25 048βfc (9)



5 Conclusions




Data Availability




Acknowledgments


References





























FleAc σc1113858 1113859 (2)




ha s + 6db (4)



DReaction force C

at pile top

Strut force F

h a

6db

s


t

Strut

NodeTie bars

ha cos θs

Dsin θs

θs






C25 2495 2388 0957C30 3064 2809 0917C35 3400 3301 0971C40 3745 3792 1013C45 4177 4214 1009

Pile spacing (m)

42 2961 3031 102454 2843 2570 090460 2608 2347 090066 2222 2152 0968

Cap thickness (m)

20 2464 2236 090725 2728 2572 094335 3122 2987 095740 3334 3114 0934




4 (5)



fc 067fcuk (6)


fcyk 08fcuk (7)


fcyk 12fc (8)


σc1113858 1113859 β12fc

25 048βfc (9)



5 Conclusions




Data Availability




Acknowledgments


References




























4 (5)



fc 067fcuk (6)


fcyk 08fcuk (7)


fcyk 12fc (8)


σc1113858 1113859 β12fc

25 048βfc (9)



5 Conclusions




Data Availability




Acknowledgments


References













































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MechanicalBehaviorandCalculationMethodforRCFifteen-Pile ... · 2020. 9. 18. · A ﬁfteen-pile cap of mixed passenger and freight railway contains two-ﬂoor caps. e plane dimension