Analysis of Deformation and Stress Characteristics of

Research ArticleAnalysis of Deformation and Stress Characteristics ofAnchored-Frame Structures for Slope Stabilization

Wei-na Ye Yong Zhou and Shuai-hua Ye

School of Civil Engineering Lanzhou University of Technology Lanzhou China

Correspondence should be addressed to Wei-na Ye yeweina163163com

Received 29 June 2020 Revised 18 November 2020 Accepted 28 November 2020 Published 15 December 2020

Academic Editor Qiang Tang

Copyright copy 2020 Wei-na Ye et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In recent years anchored-frame structures are widely being used in road slopes for stabilization and improvemente technologyof frame structure with anchors is becoming more and more mature but the pertinent theory lags behind the application Whilemore attention is being paid to the control of deformation there is still no uniform solution to the calculation of deformation inthe anchored-frame structures According to the classical laterla earth pressure theory and static equilibrium this paper improvesthe calculation method of lateral earth pressure and derives the calculation formula of slope-induced lateral earth pressure At thesame time based on the elastic foundation beammodel the columns and beams are treated as a whole system and the appropriateelastic frame beammodel is established e formula of the deformation and bending moments for the columns and beams in theanchored-frame structures are derived Additionally the calculated results based on the abovementioned newly derived formulasare compared with those of finite element simulations for a simulated case study e results of simulation and analyticalcalculation are basically consistent which prove the feasibility of the new analytical method

1 Introduction

With the acceleration of road construction road slopes areincreasing drastically In the last few decades the framestructures with anchors have been widely used in slopeengineering particularly in road slopes where the theo-retical research lags behind the application [1ndash7] In fact themajority of attention has been paid to the strength anddeformation behavior in slope stability analyses [8ndash12]ough the strength-related stability of slopes has receivedwidespread attentions in many engineered slopes there isstill no unified analysis method for the deformation of theframe structures with anchors [13ndash16] Only limited workhas been performed on the calculation method of the de-formation of the frame structures with anchors In thoseprevious studies as discussed by Zhou et al [17] the de-formation of prestressed anchors was in contrast with thedeformation of slope However the deformation of the soilbehind the retaining wall which is part of the framestructure was neglected so it may cause certain errors Somework (eg work of Liang et al [18] Tang et al [19] and Xiao

et al [20]) do not treat the frame structure with prestressedanchors as a unified system while others (eg work of Hanet al [21] Fang et al [22] Zhang et al [23] and Bringkgreveand Vermeer [24]) simulated an actual engineering caseusing finite element modeling engineering [25 26] Forexample He et al [27] analyzed the interactions between theprestressed anchored foundation beam and foundation bymeans of the elastic foundation beam model [28] which wasproposed by E Winkler in 1876 Zenkour [29] investigatedthe state of stressed and displacement of elastic plates usingsimple and mixed shear deformation theories Liew et al[30] explored the differential quadrature method forMindlin plates on Winkler foundations

is paper describes an improved analytical method forearth pressure determination and derives formula of earthpressure suitable for the slope using the classical elasticitytheory [31 32] e improved method takes into account thesurface load on the slope top and the angle of friction be-tween the retaining wall and the soil [33ndash36] Consideringthe compressive deformation of the soil the beam and thecolumn are treated as a whole system en the elastic

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 8870802 12 pageshttpsdoiorg10115520208870802

foundation beam model of the frame structure with anchorsis established by using the elastic foundation beam modele deformation and bending moment formula of thecolumn and beam are derived at the least In addition thecalculation is compared with the simulation results whichproves the correctness and reliability of the calculationmethod and provides a valuable reference for similar en-gineering design in the future

2 Deformation and Bending Moment inFrame Structures

21 Deformation Calculation of the Frame Structures eframe structure is composed a series of parallel beams andcolumns that are orthogonally intersected and the intersectionpoints are anchored into the slope e deformation andbendingmoments of the frame structure with anchors is causedby the lateral earth pressure and the prestresses of the anchorse deformation of the column and beam are as follows

Δx Δxp + δx

Δy Δyp + δy

⎧⎨

⎩ (1)

where Δx and Δy are the deformation of column and beamrespectivelyΔxp and Δyp are the deformation of column andbeam under the lateral earth pressure respectively and δx

and δy are the deformation of column under the prestress ofanchors respectively

22 Bending Moment Calculation of the Frame Structurese bending moments of the column and beam are asfollows

Mx Mxp + mx

My Myp + my

⎧⎨

⎩ (2)

where Mx and My are the bending moment of the columnand beam respectively Mxp and Myp are the bendingmoment of column and beam under earth pressure re-spectively and mx and my are the bending moment ofcolumn under the prestress of anchors respectively

3 Deformation and Bending MomentCalculation under Lateral Earth Pressure

31 Improved Lateral Earth Pressure Calculation Methode magnitude and distribution on the wall of the ad-ditional lateral earth pressure are related to the

magnitude and distribution of the additional surface orsurcharge load on the top of the slope e load should beconsidered in design and stability analysis In this paperthe static equilibrium method is adopted Because thereare few studies of determination of the critical slopesurface about the slope supported by anchored-framestructures the potential slip surface is assumed as astraight line for getting the maximum of lateral earthpressure considering the friction between the soil and thewall and the surface load on the slope top as shown inFigure 1

If the force Pa acts on the slope then the force Pa and soilweight surface load on the slope top and frictional resis-tance satisfy the static equilibrium

clAC + N tanφ minus Pa sin(θ + δ minus α) FW sin θ (3)

in which

FW qlBC + G q +cH

21113874 1113875

H sin(α minus θ)

sin α sin θ

lAC H

sin θ

lBC H sin(α minus θ)

sin α sin θ

G c times12

timesH

sin θtimes

H

sin αtimes sin(α minus θ)

cH2 sin(α minus θ)

2 sin α sin θ

N FW cos θ + Pa sin(θ + δ minus α)

(4)

where c is the cohesion of the soil φ is the angle of internalfriction of the soil H is the height of the slope θ is the anglebetween the critical sliding surface and the horizontal planeα is the angle between the anchor and the horizontal plane cis the unit weight of the soil q is the surface load on the topof the slope and δ is the angle of friction between the soil andretaining wall

e determination of the slip surface or the positionof point C depends on θ When pa reaches maximum θalso reaches its maximum so the position of point Cdepends onθ In order to reduce the difficulty of calcu-lation an indirect variable x is introduced in this paperthat is IAD x We can take the derivative of x get the xwhen the maximum of pa and then get the maximum ofθ

θ arctan1

minus ab +

a2b2 +(a + bη) +(a tanφ + b)cot αb(tanφ + η)

1113969 (5)

2 Advances in Civil Engineering

where

a minus sin(δ minus α)tanφ minus cos(δ minus α)

b minus sin(δ minus α) + cos(δ minus α)tanφ

η 2c(2q + cH)

⎧⎪⎪⎨

⎪⎪⎩(6)

e final calculation result of the lateral earth pressure is

Pa qH +cH

2

21113888 1113889

(cot θ minus cot α)sin(θ minus φ)

cosφ(a sin θ + b cos θ)minus

ηsin θ(a sin θ + b cos θ)

1113890 1113891 (7)

e earth pressure calculation of a homogeneous soil canbe directly calculated according to formula (7) is papermainly uses homogeneous soil slope as the research Due tothe complexity and variability of stratified soils the slipsurface is difficult to determine In order to simplify thederivation we also used a straight line to represent thecritical slip surfacee soil weight internal angle of frictionand cohesion of the stratified soils can be calculated by theweighted average method en we use equation (5) tocalculate the angle θ between the critical slip surface and thehorizontal plane

en we calculate angle θ between the critical slipsurface and the horizontal plane and the weight G en wesubstitute θ and G into equation (7) to obtain earth pressure

32 Displacement Calculation under Earth Pressure In thecalculation of Earth pressure the surface load on the slopetop is taken as a part of the soil weight However there is no

uniform form of Earth pressure distribution in the studiesabout the frame structure with anchors with considering theslope load In this paper according to the distribution formin the Technical Code for Building Slope Engineering(GB50330-2013) [37] the earth pressure distribution behindthe frame structure with anchors is shown in Figure 2(a)eEarth pressure of the slope is as follows

eah Eah

0875lx (8)

When the earth pressure acts on the column the columnis a statically indeterminate structure subjected to a thedistributed load During the period of designing the framestructure with anchors the force method is used to calculatethe unknown force

e column is considered as a cantilever beam ebending moment subjects to the uniform load can be de-scribed by follows

Mxqx

minusehk

3lxsx x + s0( 1113857

3 0le xle 025lx minus s0

minusehk

16sxlx x + s0 minus

lx

61113888 1113889 minus

ehk

4sx x + s0 minus

lx

41113888 1113889

2

025lx minus s0 lexle lx minus s0

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(9)

q

cl

B C

NGα

θ

A Dx

δ Pa

Figure 1 Earth pressure calculation

Advances in Civil Engineering 3

where ehk is the standard value of the lateral earth pressureresultant force acting on the slope-supporting structure sx isthe vertical spacing among each anchor s0 is the distancebetween the first row anchors and the slope top and lx is theslope height

e bending moment of the column under a unit force isshown as follows

Mxi

0 0le xle ( i minus 1 )sx

x minus ( i minus 1 )sx ( i minus 1 )sx le xle lx minus s0

⎧⎪⎨

⎪⎩

Δip 1113946MxpMxidx

ExIx

(10)

where Mxi is the bending moment of the column subject tothe unit force and Δip is the displacement of the columnunder the uniform load

δii 1113944 1113946MxiMxidx

ExIx

δix 1113944 1113946MxiMxqx

dx

ExIx

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(11)

In the meanwhile the displacement is zero at the point ofanchor

Δip + δFxi 0 (12)

where δii is the coefficient at the point ith when the unit forceacts on the ith point δix is the coefficient at some point whenthe unit force acts on the ith point δ is the matrix of co-efficient Fxi is the concentrated force at the ith point Ex isthe elastic modulus of the column and Ix is the inertiamoment of the column

en Fxi is known and the column is a statically de-terminate structure now e displacement of the staticallydeterminate structure subject to the uniform load is asfollows

δyqy Δip (13)

e displacement of the statically determinate structuresubject to the Fxi is as follows

δxiFxi

Fxiy2

6ExIx

3axi minus x( 1113857 0le xle axi

Fxia2xi

6ExIx

3x minus axi( 1113857 axi le xle lx

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(14)

where qx 07syehk lx is the length of column and axi is thedistance between the fixed end and Fxi

Finally the displacement of the column under earthpressure is as follows

Δxp δxqxminus 1113944

n

i1δxiFxi

(15)

At the same time the beam becomes a statically de-terminate structure e displacement of the statically de-terminate structure subject to the uniform load is as follows

δyqy

qyy

24EyIy

l3y + y

3+ 2lyy

21113872 1113873 (16)

e displacement of the statically determinate structuresubject to the Fyj is as follows

δyjFyj

Fyj ly minus ayj1113872 1113873

6lyEyIy

l2y minus r

2y minus y

21113872 1113873 0leyle ayj

Fyjayj ly minus y1113872 1113873

6lyEyIy

2lyy minus y2

minus a2yj1113872 1113873 ayj leyle ly

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

where qy 07sxehk ly is the length of beam and ayj is thedistance between the fixed end and Fyj R lyminus ayj

e displacement of the beam under earth pressure is asfollows

Δyp δyqyminus 1113944

m

j1δyjFyj

(18)

33 Bending Moment Calculation under Lateral EarthPressure When the force Fxi acts alone on the column thebending moment MFxi

is as follows

ehk

S0 S1Sx

025lx

Sx

(a)

ehk

SySy Sy

(b)

Figure 2 Calculation model (a) Column (b) Beam


MFxi

0 0lexle ( i minus 1 )sx

Fxi lx minus s0 minus x( 1113857 ( i minus 1 )sx lexle lx minus s01113896 (19)

e bending moment of the column underlateral earthpressure is as follows

Mxp Mxqxminus MFxi

(20)

e bending moment of the beam underlateral earthpressure can be described as follows

Myp αMqys2y (21)

where αM is the coefficient of bending moment

4 Deformation and Bending MomentCalculation under the Prestress of Anchors

With the deformation of the column and beam there is acertain compression deformation on the soil behind thesupporting structuree columns and beams are tied togetherby steel bars so the columns and beams of the frame structureare working as a whole system to bear the load e previousstudies not only neglected the deformation of the soil but alsocalculated the beam and column separately which will affectthe analysis results of the structure In this paper the columnsand beams are regarded as the bidirectional elastic foundationbeam As shown in Figure 3 the columns and beams are takenas the foundation beam the soil behind the structure is treatedas the foundation and the anchors are treated as forces Sincethe errors caused by the torque are small the influence of thetorque is ignored in the calculation [38]

41 Displacement Calculation under the Prestress of AnchorsAs illustrated in Figure 4 during the designing of the framestructure with anchors the bottom of the column is gen-erally regarded as a fixed end erefore the column istreated as a semi-infinite long beam and the beam is taken asthe free infinite beam [39]

Subjected to the concentrated force Pxij at the intersectionof the column and the beam the displacement at some point ofthe column and the beam can be expressed as follows

zij(x) Pxijλx

2k0bx

AλXx+ 2Dλax

Dλx + CλaxCλx1113872 1113873

zij(y) Pyijλy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

Xx x minus ax

11138681113868111386811138681113868111386811138681113868

Xy y minus ay

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎨

⎪⎪⎩

(22)

where Pij is the prestress of anchor at the intersection ofthe column and the beam Pxij and Pyij are the distributedforce on the column and beam λx and λy are the rigidflexible eigenvalue of the column and beam bx and by arethe section width of the column and beam Xx is the

distance between Pxij and some point on the column ax isthe distance between Pxij and the end of the column Xy isthe distance between Pyij and some point on the columnand beam and ay is the distance between Pyij and the endof the beam and k0 is the coefficient of subgrade reaction

e value of the coefficient of subgrade reaction dependson many factors such as elastic modulus Poissonrsquos ratioand the beamrsquos area e coefficient of subgrade reaction ofsoil can be obtained as follows [40]

k0 E0

088(1 minus μ)A

radic (23)

where E0 is the elastic modulus of soil μ is Poissonrsquos ratio ofsoil and A is the area of beam

For the anchored-frame structures the column andbeam are subjected to multiple concentrated forces at thesame time which should be superimposed to calculate thedisplacement of the entire column and beam

δx 1113944n

i1Pxij

λx

2k0bx

AλXx+ 2Dλax

Dλx minus CλaxCλx1113872 1113873

δy 1113944m

j1Pyij

λy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)

Furthermore to facilitate the expression the coefficientof displacement is represented as follows

ηxij λx

2k0bx

AλXx+ 2Dλax

Dλx + CλaxCλx1113872 1113873

ηyji λy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(25)

e displacement of the column and beam are as follows

δx 1113944n

i1Pxijηxij

δy 1113944

m

j1Pyijηyij

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)

z

Anchor

Column

Beam

Slope top

Slope top

x

y

Figure 3 e plan of anchored-frame structures


where δx is the displacement of the column subject to nconcentrated forces δy is the displacement of the beamsubject to n concentrated forces ηxij is the displacementcoefficient of the column under m concentrated forces andηyij is the displacement coefficient of the beam under mconcentrated forces

42 Bending Moment Calculation under the Prestress ofAnchors e calculation of bending moment is similar tothat of deformation e bending moment of the columnand beam under the prestress of anchors are as follows

mx 1113944n

i1ηmxi

Pxji

my 1113944m

j1ηmyj

Pyji

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(27)

in which

ηmxi

14λx

CλXxminus Cλax

Aλx minus 2DλaxBλx1113872 1113873

ηmyj

14λy

CλXy

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

where mx is the bending moment of the column subject to nconcentrated forces ηmxi

is the bending moment coefficientof the column subject to n concentrated forces my is thebending moment of the beam subject to m concentratedforces and ηmyj

is the bending moment coefficient of thebeam subject to m concentrated forces

43 Distribution of the Prestresses from the AnchorsActually the prestresses Pij of the anchor distribute Pxij tothe column and Pyij to the beam e key problem is how todistribute the force to the directions of the column andbeam e column and beam are orthogonal and the de-formation of column is equal to the deformation of beam atthe anchored points e coordination conditions are asfollows

Pij Pxij + Pyij

δxij δyij

⎧⎨

⎩ (29)

where

x lx minus s0 minus (i minus 1)sx

y (j minus 1)sy

⎧⎨

⎩ (30)

Because the prestress of anchor does not act on thecolumn or beam alone it is necessary to use the deformationand the coordination condition in formula (29) and calculateequations (26) and (29) with MATLAB software to get Pxij

or Pyij

5 Total Deformation and Bending Moment ofthe Frame Structures

51 Total Deformation Calculation of the Frame StructuresSubstituting equations (15) and (26) into equation (1)can yield the deformation calculation formula of thecolumn

Δx δxqxminus 1113944

n

i1δxiFxi

minus 1113944

n

i1ηxjiPxji (31)

Substituting equations (18) and (26) into equation (1)can yield the deformation calculation formula of thebeam

Δx δyqyminus 1113944

m

j1δyjFyj

minus 1113944m

j1ηyijPyij (32)

52 Total Bending Moment Calculation of the FrameStructures Substituting equations (20) and (27) intoequation (2) the bending moment of column is as follows

Mx Mxqxminus MFxi

minus 1113944n

i1ηmxi

Pxji (33)

Substituting equations (21) and (27) into equation (2)the bending moment of beam is as follows

My αMqys2y minus 1113944

m

j1ηmyj

Pyji (34)

ax

Ox

Pxij

Xx

(a)

ay

O y

Pyij

Xy

(b)

Figure 4 Elastic foundation beam model (a) Column (b) Beam


6 A Case Study

61ProjectOverview A real road slope in Lanzhou China isused here as a case study which has a height of 121m and asloping angle of 76deg ere are no existing buildings aroundthe slope According to the site investigation and geologicalsurvey the soil of the slope is a fill whose properties aresummarized in Table 1

In accordance with the design and calculation thissection of the slope reinforcement scheme determines theframe structure with anchors e beam and column cross-sectional area is 300mmtimes 300mm the concrete strengthgrade is C25 and the elastic modulus of the beam andcolumn both is 28times106 kNm3 e design parameters ofanchors are shown in Table 2

62 lte Finite Element Model is paper combines thenumerical simulation and the engineering examples to verifyand analyze the rationality of the deformation and bendingmoment calculation formula As shown in Figures 5 and 6 itis the model established by using PLAXIS 3D software

e FEM model uses the MohrndashCoulomb failure cri-terion for the considered soil and the model of the anchorand beam is an elastic-plastic model e soil physical pa-rameters are shown in Table 1 e elastic modulus of soil is11times 104 kPa and Poissonrsquos ratio of soil is 033 [41] eslope is at an angle of 76deg and the support type is framestructure with anchors ere is the pile foundation with adiameter of 800mm at the slope toe which is using theldquoEmbedded Beamrdquo e material of the columns and beamsis ldquoBeamrdquo e anchors are located at the junction of thecolumns and beams e material of the free section of theanchor is a ldquoPoint-To-Point Anchorrdquo and the anchor sectionis ldquoEmbedded Beamrdquoe prestress parameters are shown inTable 2 e contact stiffness between the anchors and thesoil and the contact stiffness of the column and beam are setto ldquomanualrdquo and the input value Rinter of the reductionfactor corresponding to the interface angle of friction andcohesion is 23

63 Comparison of Calculation Results and SimulationResults In order to verify the rationality of the deformationresults of the frame structure with anchor calculated by theelastic foundation beam method the numerical simulationwas carried out by PLAXIS 3D software and the analysis wascarried out by MATLAB software e calculation resultsand simulation results are shown in Figure 7 e parameterAij refers to the number of the jth anchor on the ith row beamwhile the parameter Bji refers to the number on the ithanchor of the jth row column

e distributed force on the beam and column is shownin Figure 7 Figure 7(a) is the distributed force in the columnand beam direction on the first row beam It is found that

there is some distribution rule for the distributed force onthe column and beam e distributed force on the beam issmaller than the counterpart on the column on the first rowbeam e force on the beam decreases from the middle toboth ends On the other hand the force on the columnincreases from the middle to both ends However inFigure 7(b) it is opposite in the fifth row beam e dis-tributed force of the beam is greater than the counterpart ofthe column on the fifth row beam Moreover the forceincreases from the middle to both ends in the beam di-rection Figure 7(c) shows that the distributed force is closeto each other not only between the beam and column butalso in the same direction Meanwhile the distributed forceis symmetrical in the beam direction Figure 7(d) shows thedistributed force in the column direction In the columndirection the distributed force on the column decreases withthe slope depth increase while the distributed force of thebeam is increased with the slope depth increase In additionthe distributed force is not symmetrical in the column di-rection which is different from the distributed force in thebeam direction As a result the elastic foundation beammodel should be different from the counterpart of thecolumn model

Further comparison between the calculation resultsand the simulation results for the third row beam and thesecond row column deformation is shown in Figure 8Figure 8(a) is the deformation of the second row columnbetween the calculation results and simulation results etrend of the curves is basically the same With the increasein slope height the deformation of the column becomesmore and more great e deformation of the slope is thelargest near the second row anchor while the deformationis the most small at the slope toe e deformation of theslope is larger at anchor points Figure 8(b) presents thedeformation of the third row beam between the calcu-lation results and the simulation results e deformationat the points of anchors is larger than the others and onthe other hand the deformation on the other points issmaller which leads to the deformation fluctuation Asshown in Figures 8(a) and 8(b) the simulation results aregreater than the analytical results because the lateral earthpressure determination is based on the limit state and theearth pressure may result in the reverse displacementwhile the lateral earth pressure in the FEM simulation isunder normal service conditions that may not involvefailure or the limit state of the soil In other words thelateral earth pressure in the analytical method is greaterand hence causes certain error in the result e char-acteristics of the shape indicates that the prestressinganchors can effectively control the development of largeslope deformation and the analyses mentioned aboveshow the validity and feasibility of the analytical method

e comparison of bending moments on the third rowbeam and the second row column is shown in Figure 9Obviously the trend of the curves is basically the same


Table 1 Soil properties of the fill in the studied slope

Soil ickness (m) Unit weight c (kNm3) Cohesion c (kPa) Angle of internal friction φ (deg)Fill gt20 16 150 230

Surface load

Beam

Soil221

m

10m

37m

Column

Plate

16m

12m

56m

Figure 5 Slope support model

Anchor

Plate

Beam

Column

Pile

Figure 6 Structure model

Table 2 Anchor parameters

Anchor number Anchor position Hm Free section length Lfm Anchorage section Lam Anchor diameter Dmm Prestress F0kN1 106 4 11 150 1002 81 4 13 150 1003 56 5 13 150 1004 31 6 10 150 1005 06 7 7 150 100


However owing to the difference in the lateral earth pressureon the frame structure the analytical result is generallyslightly greater than that from the FEM simulation eframe structure in the analytical result is subjected to the fulllateral earth pressure on the slope while the frame structuredoes not support the total lateral earth pressure in the FEM

simulation More interestingly the bending moment on thebeam is more uniformly distributed e reason is that theforces on the third row beam are more uniformly distributedand the analytical model is symmetrical In general theresult proves that the analytical method is viable andaccurate

0

20

40

60

80

100Fo

rce (

kN)

A12 A13 A14 A15A11

Anchor number

PxPy

(a)

0

20

40

60

80

100

Forc

e (kN

)

A52 A53 A54 A55A51

Anchor number

PxPy

(b)

0

20

40

60

80

100

Forc

e (kN

)

A32 A33 A34 A35A31

Anchor number

PxPy

(c)

0

20

40

60

80

100Fo

rce (

kN)

B22 B23 B24 B25B21

Anchor number

PxPy

(d)

Figure 7 Distributed force (a) e first row beam (b) e fifth row beam (c) e third row beam (d) e second row column


7 Conclusions

In this paper the analytical method for the lateral earthpressure calculation under the anchored-frame structure isimproved e soil behind the frame structure with anchorsis treated as a semi-infinite elastic body e elastic foun-dation beam model is used to derive the deformation andbending moment calculation formula of the beam andcolumn At the same time the finite element softwarePLAXIS 3D is used for the road slope engineering e caseand calculation results were compared and analyzed econclusions are as follows

(1) e analytical method for the calculation of thedistributed forces deformation and bending

moments on the beam and column in the anchored-frame structure is implemented by Matlab and re-sults are compared with the FEM simulation byPLAXIS 3D It is found that the calculation result isgreater than the simulation result but they are veryclose to each other which indicates the correctnessand feasibility of the analytical method

(2) e distributed force deformation and bendingmoment are symmetrical along the direction ofbeam e reason is that the infinitely long elasticfoundation beam model used to the beam issymmetrical

(3) Because the earth pressure is greater than simulationthe calculation result of deformation and bending

0

2

4

6

8

10

12

14Co

lum

n (m

)

ndash14 ndash12 ndash10 ndash8 ndash6 ndash4 ndash2 0 2 4ndash16Deformation (mm)

SimulateCalculate

(a)

ndash18

ndash15

ndash12

ndash9

ndash6

ndash3

0

Def

orm

atio

n (m

m)

2 4 6 8 100Beam (m)

SimulateCalculate

(b)

Figure 8 Deformation curve (a) Column (b) Beam

0

2

4

6

8

10

12

14

Colu

mn

(m)

ndash10 ndash5 0 5 10 15ndash15Bending moment (kNmiddotm)

SimulateCalculate

(a)

2 4 6 8 100Beam (m)

ndash20

ndash15

ndash10

ndash5

0

5

10

Bend

ing

mom

ent (

kNmiddotm

)

SimulateCalculate

(b)

Figure 9 Bending moment curve (a) Column (b) Beam


moment is larger Furthermore the column dis-placement increases when the slope high increases

(4) e analysis in this paper does not consider theeffects of other events such as earthquakes andcomplex geological conditions which will be furtherstudied and discussed in subsequent investigations

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of Interest

Acknowledgments

e work described in this paper was fully supported by twogrants from the National Natural Science Foundation ofChina (Award nos 51568042 and 51768040)

References

[1] K Shi X Wu Z Liu and S Dai ldquoCoupled calculation modelfor anchoring force loss in a slope reinforced by a frame beamand anchor cablesrdquo Engineering Geology vol 260 p 1052452019 in English

[2] G Yang Z Zhong Y Zhang and X Fu ldquoOptimal design ofanchor cables for slope reinforcement based on stress anddisplacement fieldsrdquo Journal of Rock Mechanics and Geo-technical Engineering vol 7 no 4 pp 411ndash420 2015 inEnglish

[3] C S Desai A Muqtadir and F Scheele ldquoInteraction analysisof anchormdashsoil systemsrdquo Journal of Geotechnical Engineeringvol 112 no 5 pp 537ndash553 1986 in English

[4] R D Hryciw ldquoAnchor design for slope stabilization bysurface loadingrdquo Journal of Geotechnical Engineering vol 117no 8 pp 1260ndash1274 1991 in English

[5] Y Tian M J Cassidy and W Liu ldquoOptimising the di-mensions of suction embedded anchors by investigating theopening mechanismrdquo Ocean Engineering vol 183 pp 350ndash358 2019 in English

[6] M F Randolph C Gaudin S M Gourvenec D J WhiteN Boylan and M J Cassidy ldquoRecent advances in offshoregeotechnics for deep water oil and gas developmentsrdquo OceanEngineering vol 38 no 7 pp 818ndash834 2011 in English

[7] C Gaudin M J Cassidy C D OrsquoLoughlin Y Tian D Wangand S Chow ldquoRecent advances in anchor design for floatingstructuresrdquo International Journal of Offshore and Polar En-gineering vol 27 no 1 pp 44ndash53 2017 in English

[8] S Ye G Fang and Y Zhu ldquoModel establishment and re-sponse analysis of slope reinforced by frame with prestressedanchors under seismic considering the prestressrdquo Soil Dy-namics and Earthquake Engineering vol 122 pp 228ndash2342019 in English

[9] S H Ye G W Fang and X R Ma ldquoReliability analysis ofgrillage flexible slope supporting structure with anchorsconsidering fuzzy transitional interval and fuzzy randomnessof soil parametersrdquo Arabian Journal for Science and Engi-neering vol 44 no 6 pp 8849ndash8857 2019 in English

[10] S H Ye and Z F Zhao ldquoSeismic response of pre-stressedanchors with frame structurerdquo Mathematical Problems in

Engineering vol 202015 pages 2020 in English Article ID9029045

[11] S-H Ye and A-P Huang ldquoSensitivity analysis of factorsaffecting stability of cut and fill multistage slope based onimproved grey incidence modelrdquo Soil Mechanics and Foun-dation Engineering vol 57 no 1 pp 8ndash17 2020 in English

[12] S H Ye and Z F Zhao ldquoAllowable displacement of slopesupported by frame structure with anchors under earth-quakerdquo International Journal of Geomechanics vol 20 no 10in English Article ID 04020188 2020

[13] J F Zou L Li and B Ruan ldquoAnalysis of displacement anddeformation of anchor bolt under elastic staterdquo China Rail-way Science vol 25 no 5 pp 94ndash96 2004 in Chinese

[14] Z H Tang and W Y Tang ldquoDiscussion on calculation of soilanchor deformationrdquo Geotechnical engineering vol 10 no 9pp 27-28 2007 in Chinese

[15] M H Zhao J L Liu and Z Long ldquoDeformation analysis ofbolt and calculation of critical anchorage lengthrdquo JournalArchitecture Civil Engineering vol 25 no 3 pp 17ndash21 2008

[16] Y Zhou ldquoAnalysis and design of frame prestressed anchorsupport structurerdquo M Sc esis Lanzhou University ofTechnology Lanzhou China 2004 in Chinese

[17] Y Zhou Y P Zhu and Y Z Ren ldquoAnchor deformation offlexible supporting system with prestressed anchorsrdquo ChinaRailway Science vol 36 no 3 pp 58ndash65 2015 in Chinese

[18] Y Liang D P Zou and G Zhao ldquoDesign of support of framebeam and prestressed anchorrdquo Chinese Journal of Rock Me-chanics and Engineering vol 25 no 2 pp 318ndash322 2006 inChinese

[19] H M Tang Y Z Xu and X S Cheng ldquoResearch on designtheory of lattice frame anchor structure in landslide controlengineeringrdquo Rock and Soil Mechanics vol 25 no 11pp 1683ndash1687 2004 in Chinese

[20] S G Xiao and D P Zhou ldquoInternal force calculating methodof prestressed-cable beam-on-foundation for high rocksloperdquo Chinese Journal of Rock Mechanics and Engineeringvol 22 no 2 pp 250ndash253 2003 in Chinese

[21] A M Han J G Li J H Xiao and H Z Xu ldquoMechanicalbehaviors of frame beam supporting structure with pre-stressed anchorsrdquo Rock and Soil Mechanics vol 31 no 9pp 2894ndash2900 2010 in Chinese

[22] W Fang and X H Liu ldquoNumerical simulation of rein-forcement effect of high red clay cutting sloperdquo Journal ofHighway and Transportation Research and Developmentvol 29 no 3 pp 22ndash28 2012 in Chinese

[23] H Zhang Y Lu and Q Cheng ldquoNumerical simulation ofreinforcement for rock slope with rockbolt (anchor cable)frame beamrdquo Journal of Highway and Transportation Researchand Development (English Edition) vol 25 no 1 pp 21ndash262008 in Chinese

[24] R Bringkgreve and P Vermeer PLAXIS-finite Element Codefor Soil and Rock Analysis Plaxis BV Delft Netherlands inEnglish 1998

[25] A P S Selvadurai and G M L Gladwell ldquoElastic analysis ofsoil-foundation interactionrdquo Journal of Applied Mechanicsvol 47 no 1 in English 1979

[26] G M L Gladwell Contact Problems in the Classical lteory ofElasticity Springer Berlin Germany in English 1980

[27] S M He X L Yang and Y J Zhou ldquoAnalysis of interactionof prestressed anchor rope foundation beam and foundationrdquoRock and Soil Mechanics vol 27 no 1 pp 83ndash88 2006 inChinese


[28] M A Biot ldquoBengding of an infinite beam on an elasticfoundationrdquo Journal of Applied Mechanics vol 59 ASME inEnglish pp A1ndashA7 1937

[29] A M Zenkour ldquoA state of stress and displacement of elasticplates using simple and mixed shear deformation theoriesrdquoJournal of Engineering Mathematics vol 44 no 1 pp 1ndash202002 in English

[30] K M Liew J-B Han Z M Xiao and H Du ldquoDifferentialquadrature method for Mindlin plates on Winkler founda-tionsrdquo International Journal of Mechanical Sciences vol 38no 4 pp 405ndash421 1996 in English

[31] K Terzaghi and R B Peck Soil Mechanics in EngineeringPractice John Wiley amp Sons New York NY USA in English2nd edition 1967

[32] AW Bishop ldquoe use of the slip circle in the stability analysisof slopesrdquo Geotechnique vol 5 no 1 pp 7ndash17 1955 inEnglish

[33] Z Y Chen M Chi P Sun and Y J Wang ldquoSimplifiedmethod of calculating active earth pressure for flexibleretaining wallsrdquo Chinese Journal of Geotechnical Engineeringvol 32 no 1 pp 22ndash27 2010 in Chinese

[34] X S Cao S G Xiao H He and S X Liu ldquoModel test oncharacteristics of slope pressure on multi-frame beams withanchor bolts used to stabilize high sloperdquo Chinese Journal ofUnderground Space and Engineering vol 11 no 5pp 1159ndash1163 2015 in Chinese

[35] S Y Zhao Y R Zheng P He and W Zhang ldquoDiscussion oncalculation methods of slope lateral loadrdquo Chinese Journal ofUnderground Space and Engineering vol 13 no 2 pp 434ndash441 2017 in Chinese

[36] L X Li and X Z Wang ldquoldquoResearch on earth pressureconsidering deformation pattern of foundation pit supportingstructurerdquo Chinese Journal of Underground Space and Engi-neering vol 14 no 4 pp 1024ndash1033 2018 in Chinese

[37] Peoplersquos Republic of China Industry Standards CompilationGroup Technical Code for Building Slope Engineering GB50330-2013 China Building Industry Press Beijing China2014 in Chinese

[38] Y Zhou and Y P Zhu ldquoCalculation of internal forces offramed flexible supporting structure with prestressed anchorsbased on torsional effects among beams and columnsrdquoJournal of Lanzhou University of Technology vol 35 no 2pp 116ndash121 2009 in Chinese

[39] G Zheng and X L Gu Advanced Basic Engineering Me-chanical Industry Press Beijing China 2007 in Chinese

[40] X F Chen Settlement Calculation lteory and EngineeringExamples Science Press Beijing China 2005 in Chinese

[41] L J He Y C Shi H L Yang M Q Hu and H L XinldquoAnalysis stability of loess slope during different groundmotionrdquo Northwest Seismology Journal vol 31 no 2pp 142ndash147 2009 in Chinese


foundation beam model of the frame structure with anchorsis established by using the elastic foundation beam modele deformation and bending moment formula of thecolumn and beam are derived at the least In addition thecalculation is compared with the simulation results whichproves the correctness and reliability of the calculationmethod and provides a valuable reference for similar en-gineering design in the future

2 Deformation and Bending Moment inFrame Structures

21 Deformation Calculation of the Frame Structures eframe structure is composed a series of parallel beams andcolumns that are orthogonally intersected and the intersectionpoints are anchored into the slope e deformation andbendingmoments of the frame structure with anchors is causedby the lateral earth pressure and the prestresses of the anchorse deformation of the column and beam are as follows

Δx Δxp + δx

Δy Δyp + δy

⎧⎨

⎩ (1)

where Δx and Δy are the deformation of column and beamrespectivelyΔxp and Δyp are the deformation of column andbeam under the lateral earth pressure respectively and δx

and δy are the deformation of column under the prestress ofanchors respectively

22 Bending Moment Calculation of the Frame Structurese bending moments of the column and beam are asfollows

Mx Mxp + mx

My Myp + my

⎧⎨

⎩ (2)

where Mx and My are the bending moment of the columnand beam respectively Mxp and Myp are the bendingmoment of column and beam under earth pressure re-spectively and mx and my are the bending moment ofcolumn under the prestress of anchors respectively

3 Deformation and Bending MomentCalculation under Lateral Earth Pressure

31 Improved Lateral Earth Pressure Calculation Methode magnitude and distribution on the wall of the ad-ditional lateral earth pressure are related to the

magnitude and distribution of the additional surface orsurcharge load on the top of the slope e load should beconsidered in design and stability analysis In this paperthe static equilibrium method is adopted Because thereare few studies of determination of the critical slopesurface about the slope supported by anchored-framestructures the potential slip surface is assumed as astraight line for getting the maximum of lateral earthpressure considering the friction between the soil and thewall and the surface load on the slope top as shown inFigure 1

If the force Pa acts on the slope then the force Pa and soilweight surface load on the slope top and frictional resis-tance satisfy the static equilibrium

clAC + N tanφ minus Pa sin(θ + δ minus α) FW sin θ (3)

in which

FW qlBC + G q +cH

21113874 1113875

H sin(α minus θ)

sin α sin θ

lAC H

sin θ

lBC H sin(α minus θ)

sin α sin θ

G c times12

timesH

sin θtimes

H

sin αtimes sin(α minus θ)

cH2 sin(α minus θ)

2 sin α sin θ

N FW cos θ + Pa sin(θ + δ minus α)

(4)

where c is the cohesion of the soil φ is the angle of internalfriction of the soil H is the height of the slope θ is the anglebetween the critical sliding surface and the horizontal planeα is the angle between the anchor and the horizontal plane cis the unit weight of the soil q is the surface load on the topof the slope and δ is the angle of friction between the soil andretaining wall

e determination of the slip surface or the positionof point C depends on θ When pa reaches maximum θalso reaches its maximum so the position of point Cdepends onθ In order to reduce the difficulty of calcu-lation an indirect variable x is introduced in this paperthat is IAD x We can take the derivative of x get the xwhen the maximum of pa and then get the maximum ofθ

θ arctan1

minus ab +

a2b2 +(a + bη) +(a tanφ + b)cot αb(tanφ + η)

1113969 (5)


where



η 2c(2q + cH)

⎧⎪⎪⎨

⎪⎪⎩(6)


Pa qH +cH

2

21113888 1113889




1113890 1113891 (7)





eah Eah

0875lx (8)



Mxqx

minusehk

3lxsx x + s0( 1113857


minusehk

16sxlx x + s0 minus

lx

61113888 1113889 minus

ehk

4sx x + s0 minus

lx

41113888 1113889

2


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(9)

q

cl

B C

NGα

θ

A Dx

δ Pa





Mxi



⎧⎪⎨

⎪⎩


ExIx

(10)



ExIx

δix 1113944 1113946MxiMxqx

dx

ExIx

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(11)


Δip + δFxi 0 (12)



δyqy Δip (13)


δxiFxi

Fxiy2

6ExIx


Fxia2xi

6ExIx


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(14)




n

i1δxiFxi

(15)


δyqy

qyy

24EyIy

l3y + y

3+ 2lyy

21113872 1113873 (16)


δyjFyj


6lyEyIy

l2y minus r

2y minus y

21113872 1113873 0leyle ayj


6lyEyIy

2lyy minus y2


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)




m

j1δyjFyj

(18)


is as follows

ehk

S0 S1Sx

025lx

Sx

(a)

ehk

SySy Sy

(b)



MFxi




Mxp Mxqxminus MFxi

(20)


Myp αMqys2y (21)






zij(x) Pxijλx

2k0bx

AλXx+ 2Dλax

Dλx + CλaxCλx1113872 1113873

zij(y) Pyijλy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

Xx x minus ax

11138681113868111386811138681113868111386811138681113868

Xy y minus ay

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎨

⎪⎪⎩

(22)




k0 E0

088(1 minus μ)A

radic (23)



δx 1113944n

i1Pxij

λx

2k0bx

AλXx+ 2Dλax


δy 1113944m

j1Pyij

λy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


ηxij λx

2k0bx

AλXx+ 2Dλax

Dλx + CλaxCλx1113872 1113873

ηyji λy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(25)


δx 1113944n

i1Pxijηxij

δy 1113944

m

j1Pyijηyij

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)

z

Anchor

Column

Beam

Slope top

Slope top

x

y





mx 1113944n

i1ηmxi

Pxji

my 1113944m

j1ηmyj

Pyji

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(27)

in which

ηmxi

14λx

CλXxminus Cλax


ηmyj

14λy

CλXy

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)





Pij Pxij + Pyij

δxij δyij

⎧⎨

⎩ (29)

where


y (j minus 1)sy

⎧⎨

⎩ (30)


or Pyij




n

i1δxiFxi

minus 1113944

n

i1ηxjiPxji (31)



m

j1δyjFyj

minus 1113944m

j1ηyijPyij (32)


Mx Mxqxminus MFxi

minus 1113944n

i1ηmxi

Pxji (33)



m

j1ηmyj

Pyji (34)

ax

Ox

Pxij

Xx

(a)

ay

O y

Pyij

Xy

(b)



6 A Case Study













Surface load

Beam

Soil221

m

10m

37m

Column

Plate

16m

12m

56m


Anchor

Plate

Beam

Column

Pile







0

20

40

60

80

100Fo

rce (

kN)

A12 A13 A14 A15A11

Anchor number

PxPy

(a)

0

20

40

60

80

100

Forc

e (kN

)

A52 A53 A54 A55A51

Anchor number

PxPy

(b)

0

20

40

60

80

100

Forc

e (kN

)

A32 A33 A34 A35A31

Anchor number

PxPy

(c)

0

20

40

60

80

100Fo

rce (

kN)

B22 B23 B24 B25B21

Anchor number

PxPy

(d)



7 Conclusions






0

2

4

6

8

10

12

14Co

lum

n (m

)


SimulateCalculate

(a)

ndash18

ndash15

ndash12

ndash9

ndash6

ndash3

0

Def

orm

atio

n (m

m)

2 4 6 8 100Beam (m)

SimulateCalculate

(b)


0

2

4

6

8

10

12

14

Colu

mn

(m)


SimulateCalculate

(a)

2 4 6 8 100Beam (m)

ndash20

ndash15

ndash10

ndash5

0

5

10

Bend

ing

mom

ent (

kNmiddotm

)

SimulateCalculate

(b)





Data Availability




Acknowledgments


References













































where



η 2c(2q + cH)

⎧⎪⎪⎨

⎪⎪⎩(6)


Pa qH +cH

2

21113888 1113889




1113890 1113891 (7)





eah Eah

0875lx (8)



Mxqx

minusehk

3lxsx x + s0( 1113857


minusehk

16sxlx x + s0 minus

lx

61113888 1113889 minus

ehk

4sx x + s0 minus

lx

41113888 1113889

2


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(9)

q

cl

B C

NGα

θ

A Dx

δ Pa





Mxi



⎧⎪⎨

⎪⎩


ExIx

(10)



ExIx

δix 1113944 1113946MxiMxqx

dx

ExIx

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(11)


Δip + δFxi 0 (12)



δyqy Δip (13)


δxiFxi

Fxiy2

6ExIx


Fxia2xi

6ExIx


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(14)




n

i1δxiFxi

(15)


δyqy

qyy

24EyIy

l3y + y

3+ 2lyy

21113872 1113873 (16)


δyjFyj


6lyEyIy

l2y minus r

2y minus y

21113872 1113873 0leyle ayj


6lyEyIy

2lyy minus y2


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)




m

j1δyjFyj

(18)


is as follows

ehk

S0 S1Sx

025lx

Sx

(a)

ehk

SySy Sy

(b)



MFxi




Mxp Mxqxminus MFxi

(20)


Myp αMqys2y (21)






zij(x) Pxijλx

2k0bx

AλXx+ 2Dλax

Dλx + CλaxCλx1113872 1113873

zij(y) Pyijλy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

Xx x minus ax

11138681113868111386811138681113868111386811138681113868

Xy y minus ay

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎨

⎪⎪⎩

(22)




k0 E0

088(1 minus μ)A

radic (23)



δx 1113944n

i1Pxij

λx

2k0bx

AλXx+ 2Dλax


δy 1113944m

j1Pyij

λy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


ηxij λx

2k0bx

AλXx+ 2Dλax

Dλx + CλaxCλx1113872 1113873

ηyji λy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(25)


δx 1113944n

i1Pxijηxij

δy 1113944

m

j1Pyijηyij

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)

z

Anchor

Column

Beam

Slope top

Slope top

x

y





mx 1113944n

i1ηmxi

Pxji

my 1113944m

j1ηmyj

Pyji

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(27)

in which

ηmxi

14λx

CλXxminus Cλax


ηmyj

14λy

CλXy

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)





Pij Pxij + Pyij

δxij δyij

⎧⎨

⎩ (29)

where


y (j minus 1)sy

⎧⎨

⎩ (30)


or Pyij




n

i1δxiFxi

minus 1113944

n

i1ηxjiPxji (31)



m

j1δyjFyj

minus 1113944m

j1ηyijPyij (32)


Mx Mxqxminus MFxi

minus 1113944n

i1ηmxi

Pxji (33)



m

j1ηmyj

Pyji (34)

ax

Ox

Pxij

Xx

(a)

ay

O y

Pyij

Xy

(b)



6 A Case Study













Surface load

Beam

Soil221

m

10m

37m

Column

Plate

16m

12m

56m


Anchor

Plate

Beam

Column

Pile







0

20

40

60

80

100Fo

rce (

kN)

A12 A13 A14 A15A11

Anchor number

PxPy

(a)

0

20

40

60

80

100

Forc

e (kN

)

A52 A53 A54 A55A51

Anchor number

PxPy

(b)

0

20

40

60

80

100

Forc

e (kN

)

A32 A33 A34 A35A31

Anchor number

PxPy

(c)

0

20

40

60

80

100Fo

rce (

kN)

B22 B23 B24 B25B21

Anchor number

PxPy

(d)



7 Conclusions






0

2

4

6

8

10

12

14Co

lum

n (m

)


SimulateCalculate

(a)

ndash18

ndash15

ndash12

ndash9

ndash6

ndash3

0

Def

orm

atio

n (m

m)

2 4 6 8 100Beam (m)

SimulateCalculate

(b)


0

2

4

6

8

10

12

14

Colu

mn

(m)


SimulateCalculate

(a)

2 4 6 8 100Beam (m)

ndash20

ndash15

ndash10

ndash5

0

5

10

Bend

ing

mom

ent (

kNmiddotm

)

SimulateCalculate

(b)





Data Availability




Acknowledgments


References















































Mxi



⎧⎪⎨

⎪⎩


ExIx

(10)



ExIx

δix 1113944 1113946MxiMxqx

dx

ExIx

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(11)


Δip + δFxi 0 (12)



δyqy Δip (13)


δxiFxi

Fxiy2

6ExIx


Fxia2xi

6ExIx


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(14)




n

i1δxiFxi

(15)


δyqy

qyy

24EyIy

l3y + y

3+ 2lyy

21113872 1113873 (16)


δyjFyj


6lyEyIy

l2y minus r

2y minus y

21113872 1113873 0leyle ayj


6lyEyIy

2lyy minus y2


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)




m

j1δyjFyj

(18)


is as follows

ehk

S0 S1Sx

025lx

Sx

(a)

ehk

SySy Sy

(b)



MFxi




Mxp Mxqxminus MFxi

(20)


Myp αMqys2y (21)






zij(x) Pxijλx

2k0bx

AλXx+ 2Dλax

Dλx + CλaxCλx1113872 1113873

zij(y) Pyijλy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

Xx x minus ax

11138681113868111386811138681113868111386811138681113868

Xy y minus ay

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎨

⎪⎪⎩

(22)




k0 E0

088(1 minus μ)A

radic (23)



δx 1113944n

i1Pxij

λx

2k0bx

AλXx+ 2Dλax


δy 1113944m

j1Pyij

λy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


ηxij λx

2k0bx

AλXx+ 2Dλax

Dλx + CλaxCλx1113872 1113873

ηyji λy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(25)


δx 1113944n

i1Pxijηxij

δy 1113944

m

j1Pyijηyij

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)

z

Anchor

Column

Beam

Slope top

Slope top

x

y





mx 1113944n

i1ηmxi

Pxji

my 1113944m

j1ηmyj

Pyji

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(27)

in which

ηmxi

14λx

CλXxminus Cλax


ηmyj

14λy

CλXy

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)





Pij Pxij + Pyij

δxij δyij

⎧⎨

⎩ (29)

where


y (j minus 1)sy

⎧⎨

⎩ (30)


or Pyij




n

i1δxiFxi

minus 1113944

n

i1ηxjiPxji (31)



m

j1δyjFyj

minus 1113944m

j1ηyijPyij (32)


Mx Mxqxminus MFxi

minus 1113944n

i1ηmxi

Pxji (33)



m

j1ηmyj

Pyji (34)

ax

Ox

Pxij

Xx

(a)

ay

O y

Pyij

Xy

(b)



6 A Case Study













Surface load

Beam

Soil221

m

10m

37m

Column

Plate

16m

12m

56m


Anchor

Plate

Beam

Column

Pile







0

20

40

60

80

100Fo

rce (

kN)

A12 A13 A14 A15A11

Anchor number

PxPy

(a)

0

20

40

60

80

100

Forc

e (kN

)

A52 A53 A54 A55A51

Anchor number

PxPy

(b)

0

20

40

60

80

100

Forc

e (kN

)

A32 A33 A34 A35A31

Anchor number

PxPy

(c)

0

20

40

60

80

100Fo

rce (

kN)

B22 B23 B24 B25B21

Anchor number

PxPy

(d)



7 Conclusions






0

2

4

6

8

10

12

14Co

lum

n (m

)


SimulateCalculate

(a)

ndash18

ndash15

ndash12

ndash9

ndash6

ndash3

0

Def

orm

atio

n (m

m)

2 4 6 8 100Beam (m)

SimulateCalculate

(b)


0

2

4

6

8

10

12

14

Colu

mn

(m)


SimulateCalculate

(a)

2 4 6 8 100Beam (m)

ndash20

ndash15

ndash10

ndash5

0

5

10

Bend

ing

mom

ent (

kNmiddotm

)

SimulateCalculate

(b)





Data Availability




Acknowledgments


References













































MFxi




Mxp Mxqxminus MFxi

(20)


Myp αMqys2y (21)






zij(x) Pxijλx

2k0bx

AλXx+ 2Dλax

Dλx + CλaxCλx1113872 1113873

zij(y) Pyijλy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

Xx x minus ax

11138681113868111386811138681113868111386811138681113868

Xy y minus ay

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎨

⎪⎪⎩

(22)




k0 E0

088(1 minus μ)A

radic (23)



δx 1113944n

i1Pxij

λx

2k0bx

AλXx+ 2Dλax


δy 1113944m

j1Pyij

λy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


ηxij λx

2k0bx

AλXx+ 2Dλax

Dλx + CλaxCλx1113872 1113873

ηyji λy

2k0by

AλXy

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(25)


δx 1113944n

i1Pxijηxij

δy 1113944

m

j1Pyijηyij

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(26)

z

Anchor

Column

Beam

Slope top

Slope top

x

y





mx 1113944n

i1ηmxi

Pxji

my 1113944m

j1ηmyj

Pyji

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(27)

in which

ηmxi

14λx

CλXxminus Cλax


ηmyj

14λy

CλXy

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)





Pij Pxij + Pyij

δxij δyij

⎧⎨

⎩ (29)

where


y (j minus 1)sy

⎧⎨

⎩ (30)


or Pyij




n

i1δxiFxi

minus 1113944

n

i1ηxjiPxji (31)



m

j1δyjFyj

minus 1113944m

j1ηyijPyij (32)


Mx Mxqxminus MFxi

minus 1113944n

i1ηmxi

Pxji (33)



m

j1ηmyj

Pyji (34)

ax

Ox

Pxij

Xx

(a)

ay

O y

Pyij

Xy

(b)



6 A Case Study













Surface load

Beam

Soil221

m

10m

37m

Column

Plate

16m

12m

56m


Anchor

Plate

Beam

Column

Pile







0

20

40

60

80

100Fo

rce (

kN)

A12 A13 A14 A15A11

Anchor number

PxPy

(a)

0

20

40

60

80

100

Forc

e (kN

)

A52 A53 A54 A55A51

Anchor number

PxPy

(b)

0

20

40

60

80

100

Forc

e (kN

)

A32 A33 A34 A35A31

Anchor number

PxPy

(c)

0

20

40

60

80

100Fo

rce (

kN)

B22 B23 B24 B25B21

Anchor number

PxPy

(d)



7 Conclusions






0

2

4

6

8

10

12

14Co

lum

n (m

)


SimulateCalculate

(a)

ndash18

ndash15

ndash12

ndash9

ndash6

ndash3

0

Def

orm

atio

n (m

m)

2 4 6 8 100Beam (m)

SimulateCalculate

(b)


0

2

4

6

8

10

12

14

Colu

mn

(m)


SimulateCalculate

(a)

2 4 6 8 100Beam (m)

ndash20

ndash15

ndash10

ndash5

0

5

10

Bend

ing

mom

ent (

kNmiddotm

)

SimulateCalculate

(b)





Data Availability




Acknowledgments


References















































mx 1113944n

i1ηmxi

Pxji

my 1113944m

j1ηmyj

Pyji

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(27)

in which

ηmxi

14λx

CλXxminus Cλax


ηmyj

14λy

CλXy

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)





Pij Pxij + Pyij

δxij δyij

⎧⎨

⎩ (29)

where


y (j minus 1)sy

⎧⎨

⎩ (30)


or Pyij




n

i1δxiFxi

minus 1113944

n

i1ηxjiPxji (31)



m

j1δyjFyj

minus 1113944m

j1ηyijPyij (32)


Mx Mxqxminus MFxi

minus 1113944n

i1ηmxi

Pxji (33)



m

j1ηmyj

Pyji (34)

ax

Ox

Pxij

Xx

(a)

ay

O y

Pyij

Xy

(b)



6 A Case Study













Surface load

Beam

Soil221

m

10m

37m

Column

Plate

16m

12m

56m


Anchor

Plate

Beam

Column

Pile







0

20

40

60

80

100Fo

rce (

kN)

A12 A13 A14 A15A11

Anchor number

PxPy

(a)

0

20

40

60

80

100

Forc

e (kN

)

A52 A53 A54 A55A51

Anchor number

PxPy

(b)

0

20

40

60

80

100

Forc

e (kN

)

A32 A33 A34 A35A31

Anchor number

PxPy

(c)

0

20

40

60

80

100Fo

rce (

kN)

B22 B23 B24 B25B21

Anchor number

PxPy

(d)



7 Conclusions






0

2

4

6

8

10

12

14Co

lum

n (m

)


SimulateCalculate

(a)

ndash18

ndash15

ndash12

ndash9

ndash6

ndash3

0

Def

orm

atio

n (m

m)

2 4 6 8 100Beam (m)

SimulateCalculate

(b)


0

2

4

6

8

10

12

14

Colu

mn

(m)


SimulateCalculate

(a)

2 4 6 8 100Beam (m)

ndash20

ndash15

ndash10

ndash5

0

5

10

Bend

ing

mom

ent (

kNmiddotm

)

SimulateCalculate

(b)





Data Availability




Acknowledgments


References













































6 A Case Study













Surface load

Beam

Soil221

m

10m

37m

Column

Plate

16m

12m

56m


Anchor

Plate

Beam

Column

Pile







0

20

40

60

80

100Fo

rce (

kN)

A12 A13 A14 A15A11

Anchor number

PxPy

(a)

0

20

40

60

80

100

Forc

e (kN

)

A52 A53 A54 A55A51

Anchor number

PxPy

(b)

0

20

40

60

80

100

Forc

e (kN

)

A32 A33 A34 A35A31

Anchor number

PxPy

(c)

0

20

40

60

80

100Fo

rce (

kN)

B22 B23 B24 B25B21

Anchor number

PxPy

(d)



7 Conclusions






0

2

4

6

8

10

12

14Co

lum

n (m

)


SimulateCalculate

(a)

ndash18

ndash15

ndash12

ndash9

ndash6

ndash3

0

Def

orm

atio

n (m

m)

2 4 6 8 100Beam (m)

SimulateCalculate

(b)


0

2

4

6

8

10

12

14

Colu

mn

(m)


SimulateCalculate

(a)

2 4 6 8 100Beam (m)

ndash20

ndash15

ndash10

ndash5

0

5

10

Bend

ing

mom

ent (

kNmiddotm

)

SimulateCalculate

(b)





Data Availability




Acknowledgments


References















































Surface load

Beam

Soil221

m

10m

37m

Column

Plate

16m

12m

56m


Anchor

Plate

Beam

Column

Pile







0

20

40

60

80

100Fo

rce (

kN)

A12 A13 A14 A15A11

Anchor number

PxPy

(a)

0

20

40

60

80

100

Forc

e (kN

)

A52 A53 A54 A55A51

Anchor number

PxPy

(b)

0

20

40

60

80

100

Forc

e (kN

)

A32 A33 A34 A35A31

Anchor number

PxPy

(c)

0

20

40

60

80

100Fo

rce (

kN)

B22 B23 B24 B25B21

Anchor number

PxPy

(d)



7 Conclusions






0

2

4

6

8

10

12

14Co

lum

n (m

)


SimulateCalculate

(a)

ndash18

ndash15

ndash12

ndash9

ndash6

ndash3

0

Def

orm

atio

n (m

m)

2 4 6 8 100Beam (m)

SimulateCalculate

(b)


0

2

4

6

8

10

12

14

Colu

mn

(m)


SimulateCalculate

(a)

2 4 6 8 100Beam (m)

ndash20

ndash15

ndash10

ndash5

0

5

10

Bend

ing

mom

ent (

kNmiddotm

)

SimulateCalculate

(b)





Data Availability




Acknowledgments


References















































0

20

40

60

80

100Fo

rce (

kN)

A12 A13 A14 A15A11

Anchor number

PxPy

(a)

0

20

40

60

80

100

Forc

e (kN

)

A52 A53 A54 A55A51

Anchor number

PxPy

(b)

0

20

40

60

80

100

Forc

e (kN

)

A32 A33 A34 A35A31

Anchor number

PxPy

(c)

0

20

40

60

80

100Fo

rce (

kN)

B22 B23 B24 B25B21

Anchor number

PxPy

(d)



7 Conclusions






0

2

4

6

8

10

12

14Co

lum

n (m

)


SimulateCalculate

(a)

ndash18

ndash15

ndash12

ndash9

ndash6

ndash3

0

Def

orm

atio

n (m

m)

2 4 6 8 100Beam (m)

SimulateCalculate

(b)


0

2

4

6

8

10

12

14

Colu

mn

(m)


SimulateCalculate

(a)

2 4 6 8 100Beam (m)

ndash20

ndash15

ndash10

ndash5

0

5

10

Bend

ing

mom

ent (

kNmiddotm

)

SimulateCalculate

(b)





Data Availability




Acknowledgments


References













































7 Conclusions






0

2

4

6

8

10

12

14Co

lum

n (m

)


SimulateCalculate

(a)

ndash18

ndash15

ndash12

ndash9

ndash6

ndash3

0

Def

orm

atio

n (m

m)

2 4 6 8 100Beam (m)

SimulateCalculate

(b)


0

2

4

6

8

10

12

14

Colu

mn

(m)


SimulateCalculate

(a)

2 4 6 8 100Beam (m)

ndash20

ndash15

ndash10

ndash5

0

5

10

Bend

ing

mom

ent (

kNmiddotm

)

SimulateCalculate

(b)





Data Availability




Acknowledgments


References















































Data Availability




Acknowledgments


References




























































Documents

Analysis of Deformation and Stress Characteristics of