Research ArticleResponse Analysis of Frame Supporting Structure ofSlope under Harmonic Vibration

Jian Duan12 Zhi-xin Yan12 Rui-jian Guo12 and Zhi-hua Ren123

1 School of Civil Engineering and Mechanics Lanzhou University Lanzhou 730000 China2 Key Laboratory of Mechanics on Disaster and Environment in Western China (Lanzhou University) Ministry of EducationLanzhou 730000 China

3 Institute of Science and Technology of Yunnan Province Highway Kunming 650051 China

Correspondence should be addressed to Zhi-xin Yan yzx10163com

Received 16 June 2013 Revised 9 October 2013 Accepted 10 October 2013 Published 9 January 2014

Academic Editor Evangelos J Sapountzakis

Copyright copy 2014 Jian Duan et al This 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

Based on certain assumptions the dynamic mechanical model for frame supporting structure of slope is established the dynamicequilibrium governing equation for vertical beam under forced vibration is derived and hence its analytical solutions to harmonicforced vibration are obtainedWhat is more the finite difference format and corresponding calculation procedure for vertical beamunder forced vibration are given and programmed by usingMATLAB language In the case studies comparative analyses have beenperformed to the response of vertical beam under horizontal harmonic forced vibration by using different calculating methods andwith anchoring systemdamping effect neglected or considered As a result the feasibility correctness and characteristics of differentmethods can be revealed and the horizontal forced vibration law of vertical beam can be unveiled as well

1 Introduction

Frame supporting techniques of slope have been widelyapplied due to their outstanding advantages [1] The force-transferring mechanism is that the frame supporting struc-ture transmits the earth pressure or additional forces (suchas seismic loading) it bears to the bolt then transmitsthem to deep stable ground and guarantees the safety ofslope It is obvious that frame supporting structure is avery important component in the whole system Framesupporting structure always consists of horizontal beam andvertical beam which are casted by concrete and shelvedor mounted on slope surface It functions by fixing theintersection of horizontal beam and vertical beam throughanchor Even though previous earthquake damage surveyshave revealed that frame supporting techniques of slopeshow good seismic performance [2] the additional seismicstress triggered by strong earthquake is still high enoughto cause the failure of frame supporting structure such asinclined section shear failure or normal section bendingfailure and cause secondary geological hazards In terms ofthe mechanical characteristics of frame supporting structure

of slope quite a lot of researches have been conducted Forexample comparing with field tests Yang et al [3] have putup forward the calculating model and equation for framesupporting structure with anchor based on the principleof elastic foundation beam Tian et al [4] have given thefinite difference format for the internal force calculation offrame supporting structurewith bolt based onWinkler elasticfoundation model and node deformation compatibility Lin[5] has analyzed the influence of slope rate and anchoringforce and so forth to the internal force through laboratorymodel test of frame supporting structure Zhu et al [6]have studied the distribution law of frame internal force andnodal force between horizontal beam and vertical beam byin situ test Based onWinkler elastic foundation beam theoryand certain assumptions Dong et al [7] have established adynamic computing model for frame supporting structurewith prestressed anchor and given the analytical solution toharmonic forced vibration by using modal analysis methodHowever the mechanical characteristics of frame supportingstructure of slope not only involve the mutual interactionamong beams ground and anchors but also should considerthe deformation compatibility within the structure What is

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 603863 13 pageshttpdxdoiorg1011552014603863

2 Mathematical Problems in Engineering

more current studies only focus on statistic problems whilefew researches have paid attention to dynamic problems andno related code can be referred to as well Therefore thedynamic behaviors of frame supporting structure of slopeunder harmonic forced vibration are explored in this paperand this research can provide some theoretical support tothe calculation of dynamic force and reinforcement of thestructure

2 Dynamic Mechanical Model of FrameSupporting Structure of Slope

21 Basic Assumptions Under the horizontal seismic load-ing the dynamic mechanical model of frame supportingstructure of slope is established based on following assump-tions

(1) Beams and ground of slope are considered as con-tinuous isotropic and elastic and only the elasticdynamic problems are analyzed

(2) Space torsional effect of beams is neglected horizon-tal beam and vertical beam are viewed as indepen-dent continuous beam respectively Meanwhile themutual interaction among horizontal beam verticalbeam and anchor is considered through node defor-mation compatibility

(3) Under the horizontal seismic loading 1198861015840119892(119905) from slope

bottom the beams and the ground of slope keepcontact all the time and the interaction between themis simulated as horizontal linear spring model themechanical effect of which is similar to Winkler elas-tic foundation beam [8ndash13] Therefore the stiffnesscoefficient 119896

0can be determined by referring to the

theory of elastic foundation beam The bottom ofvertical beam is considered as sliding support whichonly subjected to vertical constraint for its horizontalconstraint is so slight that can be neglected

(4) Influenced by slope height the particle load accelera-tion of frame is 119886

119892(119905) = 120595(119910 + 119867

0)1198861015840

119892(119905) in which the

value for the amplification coefficient of acceleration120595(119910 + 119867

0) refers to literature [14] where 119910 is the

vertical height from the particle to lower platform and1198670is the vertical length from lower platform to slope

bottom

So the dynamic mechanical model of horizontal beamand vertical beam is almost the same and the only differencebetween them lies in the characteristic of particle loadacceleration function More specifically the correspondingparameter of horizontal beam 120595(ℎ

119894+ 1198670) is constant In

other words the horizontal seismic acceleration for the wholehorizontal beam is the same at the same time However forthe vertical beam the corresponding parameter of slope-height effect would change with the change of particleheight as a result the particle load acceleration along thevertical beam would vary at the same moment Thereforethe dynamic response of horizontal beam can be viewedas a special case of vertical beam namely that the particle

acceleration of beam is the same so the analyses for verticalbeam can be extended to horizontal beamDynamic responseanalysis for vertical beam is conducted as follows

22 Dynamic Governing Equation According to structuraldynamic theory and above basic assumptions the dynamicmechanical model of vertical beam is shown in Figure 1

Based on the layout scheme of slope bolts the verticalbeam of the frame supporting structure can be divided into119899 sections To 119894th section the segment of infinitesimal length119889119910 sin120573 is chosen to perform the dynamic response analysisjust as the revelation of Figure 1 According to DrsquoAlembertprinciple the dynamic equilibrium equation for this segmentof infinitesimal length can be expresses as

119876119894+

120597119876119894

120597119910

119889119910 +

1205972119908119894

1205971199052119889119898 +

1198960119887119908119894119889119910

sin120573

= 119886119892(119905) 119889119898 + 119876

119894 (1)

where 119876119894and 119908

119894represent the shear force and horizontal

displacement of vertical beam respectively 120573 is slope angleand the mass of the segment of infinitesimal length is 119889119898 =

(120588119887ℎ sin120573)119889119910 where 119887 ℎ and 120588 are the width height anddensity of vertical beam respectively Equation (1) can besimplified as

120597119876119894

120597119910

+

120588119887ℎ

sin120573

1205972119908119894

1205971199052

+

1198960119887

sin120573

119908119894=

120588119887ℎ

sin120573

119886119892(119905) (2)

Since 119872119894= 119864119868(120597

21199081198941205971199102) 119876119894= 120597119872

119894120597119910 = 119864119868(120597

31199081198941205971199103)

and 119868 = 119887ℎ312sin3120573 the dynamic equilibrium can be further

simplified as

1205974119908119894

1205971199104

+

120588119887ℎ

119864119868 sin120573

1205972119908119894

1205971199052

+

1198960119887

119864119868 sin120573

119908119894=

120588119887ℎ

119864119868 sin120573

119886119892(119905) (3)

Given that 1205721

= 120588119887ℎ119864119868 sin120573 and 1205722

= 1198960119887119864119868 sin120573 the

dynamic governing equation can be transformed into

1205974119908119894

1205971199104

+ 1205721

1205972119908119894

1205971199052

+ 1205722119908119894= 1205721119886119892(119905) (4)

23 Boundary Condition According to the above basic assu-mptions the boundary condition for the dynamic responseof vertical beam could be obtained

1198721

1003816100381610038161003816119910=0

= 119864119868

12059721199081

1205971199102

100381610038161003816100381610038161003816100381610038161003816119910=0

= 0

1198761

1003816100381610038161003816119910=0

= 119864119868

12059731199081

1205971199103

100381610038161003816100381610038161003816100381610038161003816119910=0

= 0

119872119899

1003816100381610038161003816119910=119867

= 119864119868

1205972119908119899

1205971199102

100381610038161003816100381610038161003816100381610038161003816119910=119867

= 0

119876119899

1003816100381610038161003816119910=119867

= 119864119868

1205973119908119899

1205971199103

100381610038161003816100381610038161003816100381610038161003816119910=119867

= 0

(5)

Meanwhile according to the principle of deformationcompatibility in the outside endpoint of anchor namely that

Mathematical Problems in Engineering 3

Upper platform (slope crest)

n minus 1th row anchor

ith row anchor

k0

k0

k0Free segment

Anchoring segment

1th row anchor

120573

ag(t) Lower platform (slope toe)

n

i + 1

ui2(Li2

Li2

t)

i

1

o

y

hi

H

Mi +120597Mi

120597ydy

Qi +120597Qi

120597ydy1205972wi

120597t2dm

k0bwidy

sin120573 Qi

Mi

H0

Md

ag(t)dm

dy

Figure 1 Dynamic mechanical model of vertical beam

the demarcation point of vertical beam the displacement 119908119894

rotation angle 120579119894 and bending moment 119872

119894should meet the

following conditions

119908119894

1003816100381610038161003816119910=ℎ119894

= 119908119894+1

1003816100381610038161003816119910=ℎ119894

= 1199061198942(1198711198942 119905) cos120593

119894(119894 = 1 sim 119899 minus 1)

120579119894

1003816100381610038161003816119910=ℎ119894

= 120579119894+1

1003816100381610038161003816119910=ℎ119894

=

120597119908119894

120597119910

10038161003816100381610038161003816100381610038161003816119910=ℎ119894

=

120597119908119894+1

120597119910

10038161003816100381610038161003816100381610038161003816119910=ℎ119894

(119894 = 1 sim 119899 minus 1)

119872119894

1003816100381610038161003816119910=ℎ119894

= 119872119894+1

1003816100381610038161003816119910=ℎ119894

= 119864119868

1205972119908119894

1205971199102

100381610038161003816100381610038161003816100381610038161003816119910=ℎ119894

= 119864119868

1205972119908119894+1

1205971199102

100381610038161003816100381610038161003816100381610038161003816119910=ℎ119894

(119894 = 1 sim 119899 minus 1)

(6)

where 120593119894is the inclined angle of the 119894th row anchor and

1199061198942(1198711198942 119905) is the displacement of 119894th row anchor of which

under horizontal seismic loading 1198861015840

119892(119905) from slope bottom

the specific formula can be found in the literature [15] whichis based on some assumptions of using the Kelvin-Voigtmodel to simulate the interaction between anchoring sectionand the surrounding rock-soil mass the inertial dynamicaction of slope structure which exerts on the free section canbe simplified as the equivalent lumped mass 119872

119889that acts on

the end of free section just as illustrated in the 2th row anchorof Figure 1

3 Analytical Solutions to Vertical Beam underHarmonic Forced Vibration

31 Theoretical Analysis Coordination is considered andsteady-state complex method is used to the derivation ofharmonic forced vibration response of vertical beam It isassumed that the solution to the governing equation (4) is1199081015840

119894(119910 119905) = 119882

119894(119910)119890119894120596119892119905 Then

1198821015840101584010158401015840

119894(119910) + (120572

2minus 12057211205962

119892)119882119894(119910) = 120572

11198861015840

119892120595 (119910 + 119867

0) (7)

Given that 1205741= 120573(1 minus 119894) 120574

2= 120573(1 + 119894) 120574

3= 120573(minus1 + 119894) and

1205744= 120573(minus1 minus 119894) the complex solution to (7) is

119882119894(119910) = 119862

11989411198901205741119910

+ 11986211989421198901205742119910

+ 11986211989431198901205743119910

+ 11986211989441198901205744119910

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

(8)

where119862119894111986211989421198621198943 and119862

1198944are constant complex coefficients

which can be decided by specific boundary conditionLikewise the horizontal displacement bending moment

and shear force of the 119894th section in the vertical beam can beexpressed as

1199081015840

119894(119910 119905)

= 119882119894(119910) 119890119894120596119892119905

= (11986211989411198901205741119910

+ 11986211989421198901205742119910

+ 11986211989431198901205743119910

+ 11986211989441198901205744119910

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

)

times 119890119894120596119892119905

1198721015840

119894(119910 119905)

= 119864119868 (1205742

111986211989411198901205741119910

+ 1205742

211986211989421198901205742119910

+ 1205742

311986211989431198901205743119910

+ 1205742

411986211989441198901205744119910

) 119890119894120596119892119905

1198761015840

119894(119910 119905)

= 119864119868 (1205743

111986211989411198901205741119910

+ 1205743

211986211989421198901205742119910

+ 1205743

311986211989431198901205743119910

+ 1205743

411986211989441198901205744119910

) 119890119894120596119892119905

(9)

Meanwhile according to literature [15] the displacementresponse of the outside endpoint of anchor under horizontalharmonic loading can be simplified as 119906

1015840

1198942(1198711198942 119905) = 120581

119894119890119894120596119892119905

4 Mathematical Problems in Engineering

It is assumed that 119880119894(ℎ119894) = 120581

119894cos120593119894 Substitute them into

boundary condition equations (5) and (6) Then

1205742

111986211

+ 1205742

211986212

+ 1205742

311986213

+ 1205742

411986214

= 0

1205743

111986211

+ 1205743

211986212

+ 1205743

311986213

+ 1205743

411986214

= 0

11986211989411198901205741ℎ119894

+ 11986211989421198901205742ℎ119894

+ 11986211989431198901205743ℎ119894

+ 11986211989441198901205744ℎ119894

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

= 119880119894(ℎ119894)

119862(119894+1)1

1198901205741ℎ119894

+ 119862(119894+1)2

1198901205742ℎ119894

+ 119862(119894+1)3

1198901205743ℎ119894

+ 119862(119894+1)4

1198901205744ℎ119894

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

= 119880119894(ℎ119894)

120574111986211989411198901205741ℎ119894

+ 120574211986211989421198901205742ℎ119894

+ 120574311986211989431198901205743ℎ119894

+ 120574411986211989441198901205744ℎ119894

= 1205741119862(119894+1)1

1198901205741ℎ119894

+ 1205742119862(119894+1)2

1198901205742ℎ119894

+ 1205743119862(119894+1)3

1198901205743ℎ119894

+ 1205744119862(119894+1)4

1198901205744ℎ119894

1205742

111986211989411198901205741ℎ119894

+ 1205742

211986211989421198901205742ℎ119894

+ 1205742

311986211989431198901205743ℎ119894

+ 1205742

411986211989441198901205744ℎ119894

= 1205742

1119862(119894+1)1

1198901205741ℎ119894

+ 1205742

2119862(119894+1)2

1198901205742ℎ119894

+ 1205742

3119862(119894+1)3

1198901205743ℎ119894

+ 1205742

4119862(119894+1)4

1198901205744ℎ119894

1205742

111986211989911198901205741119867

+ 1205742

211986211989921198901205742119867

+ 1205742

311986211989931198901205743119867

+ 1205742

411986211989941198901205744119867

= 0

1205743

111986211989911198901205741119867

+ 1205743

211986211989921198901205742119867

+ 1205743

311986211989931198901205743119867

+ 1205743

411986211989941198901205744119867

= 0

(10)

The above linear equations can be solved by matrixmethod It is assumed that 119860

119894= 1198901205741ℎ119894 119861

119894= 1198901205742ℎ119894 119862

119894= 1198901205743ℎ119894

119863119894= 1198901205744ℎ119894 and 119864

119894= 119880119894(ℎ119894) minus (120572

11198861015840

119892120595(119910 + 119867

0)(1205722minus 12057211205962

119892))

Then (10) can be expressed as matrix format

1198624119899times1

= 119861minus1

4119899times41198991198604119899times1

(11)

where

119860 = [0 0 1198641

1198641

0 0 sdot sdot sdot 119864119894

119864119894

0 0 sdot sdot sdot 119864119899minus1

119864119899minus1

0 0 0 0]

119879

119862 = [11986211

11986212

11986213

11986214

sdot sdot sdot 1198621198941

1198621198942

1198621198943

1198621198944

sdot sdot sdot 1198621198991

1198621198992

1198621198993

1198621198994]

119879

119861 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1205742

11205742

21205742

31205742

40 0 0 0

1205743

11205743

21205743

31205743

40 0 0 0

1198601

1198611

1198621

1198631

0 0 0 0

0 0 0 0 1198601

1198611

1198621

1198631

12057411198601

12057421198611

12057431198621

12057441198631

minus12057411198601

minus12057421198611

minus12057431198621

minus12057441198631

1205742

11198601

1205742

21198611

1205742

31198621

1205742

41198631

minus1205742

11198601

minus1205742

21198611

minus1205742

31198621

minus1205742

41198631

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

d

sdot sdot sdot 119860119894

119861119894

119862119894

119863119894

sdot sdot sdot 0 0 0 0

sdot sdot sdot 1205741119860119894

1205742119861119894

1205743119862119894

1205744119863119894

sdot sdot sdot 1205742

1119860119894

1205742

2119861119894

1205742

3119862119894

1205742

4119863119894

d

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

Mathematical Problems in Engineering 5

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

119860119894

119861119894

119862119894

119863119894

minus1205741119860119894

minus1205742119861119894

minus1205743119862119894

minus1205744119863119894

minus1205742

1119860119894

minus1205742

2119861119894

minus1205742

3119862119894

minus1205742

4119863119894

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

d

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

d

sdot sdot sdot 119860

119899minus1119861119899minus1

119862119899minus1

119863119899minus1

0 0 0 0

sdot sdot sdot 0 0 0 0 119860119899minus1

119861119899minus1

119862119899minus1

119863119899minus1

sdot sdot sdot 1205741119860119899minus1

1205742119861119899minus1

1205743119862119899minus1

1205744119863119899minus1

minus1205741119860119899minus1

minus1205742119861119899minus1

minus1205743119862119899minus1

minus1205744119863119899minus1

sdot sdot sdot 1205742

1119860119899minus1

1205742

2119861119899minus1

1205742

3119862119899minus1

1205742

4119863119899minus1

minus1205742

1119860119899minus1

minus1205742

2119861119899minus1

minus1205742

3119862119899minus1

minus1205742

4119863119899minus1

sdot sdot sdot 0 0 0 0 1205742

1119860119899

1205742

2119861119899

1205742

3119862119899

1205742

4119863119899

sdot sdot sdot 0 0 0 0 1205743

1119860119899

1205743

2119861119899

1205743

3119862119899

1205743

4119863119899

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(12)

After getting constant complex coefficient 1198624119899times1

it isassumed that119862

1198941= 1198881198941+11988811989421198941198621198942

= 1198881198943+11988811989441198941198621198943

= 1198881198945+1198881198946119894 and

1198621198944

= 1198881198947

+ 1198881198948119894 Substituting them into (9) the displacement

can be simplified as

1199081015840

119894(119910 119905)

= (1198881198941119890119910120573

+ 1198881198942119890119910120573

119894)

times [cos (120596119892119905 minus 119910120573) + 119894 sin (120596

119892119905 minus 119910120573)]

+ (1198881198943119890119910120573

+ 1198881198944119890119910120573

119894) [cos (120596119892119905 + 119910120573) + 119894 sin (120596

119892119905 + 119910120573)]

+ (1198881198945119890minus119910120573

+ 1198881198946119890minus119910120573

119894) [cos (120596119892119905 + 119910120573) + 119894 sin (120596

119892119905 + 119910120573)]

+ (1198881198947119890minus119910120573

+ 1198881198948119890minus119910120573

119894) [cos (120596119892119905 minus 119910120573) + 119894 sin (120596

119892119905 minus 119910120573)]

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

(cos120596119892119905 + 119894 sin120596

119892119905)

(13)

According to the calculating principle of complexmethod the corresponding imaginary part is the steady-statesolution to the mechanical response of vertical beam underharmonic seismic loading Then

119908119894(119910 119905)

= 119890120573119910

[1198881198941sin (120596

119892119905 minus 120573119910) + 119888

1198942cos (120596

119892119905 minus 120573119910)

+ 1198881198943sin (120596

119892119905 + 120573119910) + 119888

1198944cos (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[1198881198945sin (120596

119892119905 + 120573119910) + 119888

1198946cos (120596

119892119905 + 120573119910)

+ 1198881198947sin (120596

119892119905 minus 120573119910) + 119888

1198948cos (120596

119892119905 minus 120573119910)]

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

sin120596119892119905

119872119894(119910 119905)

= 21205732119864119868 119890120573119910

[minus1198881198941cos (120596

119892119905 minus 120573119910) + 119888

1198942sin (120596

119892119905 minus 120573119910)

+ 1198881198943cos (120596

119892119905 + 120573119910) minus 119888

1198944sin (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[minus1198881198945cos (120596

119892119905 + 120573119910) + 119888

1198946sin (120596

119892119905 + 120573119910)

+ 1198881198947cos (120596

119892119905 minus 120573119910) minus 119888

1198948sin (120596

119892119905 minus 120573119910)]

119876119894(119910 119905)

= 21205733119864119868 119890120573119910

[(1198881198942minus 1198881198941) sin (120596

119892119905 minus 120573119910)

minus (1198881198941+ 1198881198942) cos (120596

119892119905 minus 120573119910)

minus (1198881198943+ 1198881198944) sin (120596

119892119905 + 120573119910)

+ (1198881198943minus 1198881198944) cos (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[(1198881198945minus 1198881198946) sin (120596

119892119905 + 120573119910)

+ (1198881198945+ 1198881198946) cos (120596

119892119905 + 120573119910)

+ (1198881198947+ 1198881198948) sin (120596

119892119905 minus 120573119910)

+ (1198881198948minus 1198881198947) cos (120596

119892119905 minus 120573119910)]

(14)

6 Mathematical Problems in Engineering

32 Case Studies Take a slope as an example Its height is12m the slope angle is 65∘ and its physical and mechanicalparameters are 120588

119904= 1600 kgsdotmminus3 119864

119904= 012MPa 120583

119904=

03 119888119904

= 20 kPa and 120601119904

= 25∘ The seismic precautionary

intensity in the local region is 8 degrees and the peakground acceleration is 02 gThe dynamic safety factor is 1015calculated by Fellenius method hence its support is neededA preliminary supporting scheme is using fixed end anchorsand frame supporting structure The dynamic safety factoris considered as 12 calculated by the pseudostatic methodand the design scheme is shown in Figure 2 The anchor usescategory II steel bar diameter size is 28mm and its spaceis 33m the dip120593 = 10

∘ four rows of anchors have beenarranged both lengths of anchoring section and free sectionare 5m The mortar grade M30 is used as bonding agent thediameter of anchor hole is 119863 = 130mm The section size ofthe frame beam is 119887timesℎ = 045 times 05m concrete is gradeC30120588 = 2500 kgm3 119864 = 255GPa Earthquake accelerationis 1198861015840

119892(119905) = 119886

1015840

119892sin120596119892119905 = 2 sin 4120587119905 the duration is 25 s and

1198960= 2 times 10

3 kNm3From Figures 3 4 and 5 the displacement bending

moment and shear force response of vertical beam have beengiven when damping effect of anchorage system is neglectedwhile from Figures 6 7 8 9 10 11 12 and 13 correspondingresponse has been given at special sections (intersection andmidspan section) and typical moment (according to momentof acceleration peak) It is found that the displacement bend-ingmoment and shear force response of vertical beamvibratesynchronously with earthquakeThe law of bending momentand shear force response shows agreement with the har-monic forced vibration characteristic of two-end cantileveredcontinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuous beamMorespecifically the bending moment response at intersectionopposite to the one at midspan section and the shear forceresponse show a jumping from the left hand to the righthand at the intersection of vertical beam just as illustratedin Figures 12 and 13 What is more the cantilevered lengthat the two ends of vertical beam has noticeable influenceon the mechanical response of frame supporting structureThe larger the cantilevered length is the more the bendingmoment of intersection is and the greater the difference ofbending moment between intersection and midspan sectionis Examining the frame supporting structure independentlythe bending moment can keep good balance for the wholevertical beamwhen its cantilevered length is 02 or 025 timesthe anchor adjacent spacing referring to the design expe-rience of two-end cantilevered continuous beam Howeverthe cantilevered length of vertical beam is still related to thearranging scheme of anchors actually the length even decidesthe arrangement of anchors and directly influences the slopesupporting effect of anchors Hence in specific engineeringdesign the determination of cantilevered length should alsoconsider the factors such as mechanical characteristic ofvertical beam and slope supporting effect of anchors

Just as shown from Figures 14 15 and 16 the dis-placement bending moment and shear force response ofvertical beam with damping effect considered have been

given as well The overall trend is similar to the previous onewith damping effect neglected When considering anchoragesystem damping effect the solutions can be obtained byusing steady-state complex method and the peak valuesof the displacement bending moment and shear force aresmaller than the ones with damping effect neglected Eventhough the response shows some attenuation characteristicof amplitude in terms of magnitude it does not demonstratethe attenuation characteristic of amplitude in terms of timeWhat is more the mechanical responses show certain initialphases which can be attributed to the dynamic behaviors ofvertical beam under steady-state vibration with damping

4 Finite Difference Method for Vertical Beamunder Forced Vibration

41 Finite Difference Format for Dynamic Response AnalysisThe numerical solutions to dynamic governing equation (4)could be also achieved by using finite difference methodAccording to the basic principles of finite difference method[16] the research area119860(0 le 119905 le 119879 0 le 119910 le 119867) can be dividedinto 119899 subregions119860

119894Thewhole region of119860

119894can be redivided

into many grids 119860119894Δ119905Δ1198971

(119872119873119894minus119873119894minus1

+ 1) which is shown inFigure 17 Meanwhile in order to guarantee the convergenceand stability of calculation the ratio of the grids should meetthe following condition 119903

119894= Δ119905Δ

2119897119894le radic120572

1(119894 = 1 sim 119899)

The governing equation (4) can be expanded accordingto central difference formula and the difference format ofdynamic equation for grid node 119908

119894(119895 119896) can be obtained

(6119908119894(119895 119896) minus 4 [119908

119894(119895 119896 + 1) + 119908

119894(119895 119896 minus 1)]

+119908119894(119895 119896 + 2) + 119908

119894(119895 119896 minus 2)) (Δ119897

4

119894)

minus1

+ 1205721

119908119894(119895 + 1 119896) + 119908

119894(119895 minus 1 119896) minus 2119908

119894(119895 119896)

Δ1199052

+ 1205722119908119894(119895 119896) = 120572

1120595119894(119896) 1198861015840

119892[119895Δ119905]

(15)

where 120595119894(119896) = 120595[(119896 minus119873

119894minus1)Δ119897119894+sum119911=119894minus1

119911=1Δ119897119911(119873119911minus119873119911minus1

) +1198670]

Likewise the boundary equations (5) and (6) can beexpanded by endpoint derivative differential formula and thecorresponding differential equation can be attained

119864119868

21199081(119895 0) minus 5119908

1(119895 1) + 4119908

1(119895 2) minus 119908

1(119895 3)

Δ1198972

1

= 0

119864119868

minus1199081(119895 0) + 3119908

1(119895 1) minus 3119908

1(119895 2) + 119908

1(119895 3)

Δ1198973

1

= 0

119864119868 (2119908119899(119895119873119899) minus 5119908

119899(119895119873119899minus 1)

+4119908119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)) (Δ119897

2

119899)

minus1

= 0

119864119868 (119908119899(119895119873119899) minus 3119908

119899(119895119873119899minus 1)

+ 3119908119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)) (Δ119897

3

119899)

minus1

= 0

119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

Mathematical Problems in Engineering 7

5050

50

5050

50

Free section

Anchoring section

Anchor

5050

15

15

120

045 045

165 16533 33

Grass

Anchor

Horizontal beam Vertical beam

33

30

30

30

65∘

Figure 2 Design scheme for slope (unit m)

02

46

810

12

005115225minus003minus002minus001

0

002001

003

w (y

t)

(m)

t (s) y(m)2

46

810

11525y(m)

Figure 3 Displacement response of vertical beam

times104

3

minus3

2

minus2

1

252

151

050 0

24

6y(m)

810

12

minus10

M(yt)(

Nmiddotm

)

t(s)

Figure 4 Bending moment response of vertical beam

times104

4

minus4

2

252

151

050 0

24

68

1012minus2

0

Q(yt)(

N)

y(m)t(s)

Figure 5 Shear force response of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Mi(

Nmiddotm

)

y = 15my = 45m

y = 75my = 105m

t(s)

Figure 6 Bending moment response at intersection of verticalbeam

2119908119894(119895119873119894) minus 3119908

119894(119895119873119894minus 1) + 119908

119894(119895119873119894minus 2)

Δ119897119894

=

minus2119908119894+1

(119895119873119894) + 3119908

119894+1(119895119873119894+ 1) minus 119908

119894+1(119895119873119894+ 2)

Δ119897119894+1

8 Mathematical Problems in Engineering

minus12000minus10000minus8000minus6000minus4000minus2000

02000400060008000

100001200014000

000 025 050 075 100 125 150 175 200 225 250

y = 3my = 6my = 9m

Mi(

Nmiddotm

)

t(s)

Figure 7 Bending moment response at midspan section of verticalbeam

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 8 Shear force response at left hand of intersection of verticalbeam

(2119908119894(119895119873119894) minus 5119908

119894(119895119873119894minus 1)

+ 4119908119894(119895119873119894minus 2) minus 119908

119894(119895119873119894minus 3)) (Δ119897

2

119894)

minus1

= (2119908119894+1

(119895119873119894) minus 5119908

119894+1(119895119873119894+ 1)

+ 4119908119894+1

(119895119873119894+ 2) minus 119908

119894+1(119895119873119894+ 3)) (Δ119897

2

119894+1)

minus1

(16)

The displacement response of 119894th column grid nodes shouldalso satisfy the following initial conditions

119908119894[0 119896] = 0

120597119908119894

120597119905

10038161003816100381610038161003816100381610038161003816119905=0

=

119908119894[1 119896] minus 119908

119894[0 119896]

Δ119905

= minus2V0

(17)

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 9 Shear force response at right hand of intersection ofvertical beam

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

000 025 050 075 100 125 150 175 200 225 250minus4000

Qi(

N)

y = 3my = 6my = 9m

t(s)

Figure 10 Shear force response atmidspan section of vertical beam

In (17) V0is the initial velocity of vertical beam If the

harmonic vibration is needed then the initial velocity can bedefined as V

0= minus119886119892120596119892

Equations (15)ndash(17) can be simplified and the finite dif-ference format for the dynamic response of vertical beam isshown as

119908119894[0 119896] = 0

119908119894[1 119896] = minus2Δ119905V

0

(18)

119908119894(119895 + 1 119896)

= (2 minus

6Δ1199052

1205721Δ1198974

119894

minus

1205722Δ1199052

1205721

)119908119894(119895 119896)

+

Δ1199052

1205721Δ1198974

119894

[4119908119894(119895 119896 + 1) + 4119908

119894(119895 119896 minus 1)

Mathematical Problems in Engineering 9

00 15 30 45 60 75 90 105 120minus003

minus002

minus001

000

001

002

003wi(

m)

t = 1125 st = 1375 s

y(m)

Figure 11 Displacement response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120minus30000

minus20000

minus10000

0

10000

20000

30000

t = 1125 st = 1375 s

Mi(

Nmiddotm

)

y(m)

Figure 12 Bending moment response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120

minus30000

minus40000

minus20000

minus10000

0

10000

20000

30000

40000

t = 1125 st = 1375 s

Qi(

N)

y(m)

Figure 13 Shear force response along vertical beam at typicalmoment

times3

minus3

2

minus2

1

25 2 15 1 05 0 02

46

810

12minus10

w(yt)(

m)

y(m)

t(s)

10minus3

Figure 14 Displace response of vertical beam

minus 119908119894(119895 119896 + 2) minus 119908

119894(119895 119896 minus 2)]

minus 119908119894(119895 minus 1 119896) + Δ119905

2120595119894(119896) 1198861015840

119892(119895Δ119905)

(19)1199081(119895 0) = 3119908

1(119895 2) minus 2119908

1(119895 3)

1199081(119895 1) = 2119908

1(119895 2) minus 119908

1(119895 3)

119908119899(119895119873119899minus 1) = 2119908

119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)

119908119899(119895119873119899) = 3119908

119899(119895119873119899minus 2) minus 2119908

119899(119895119873119899minus 3)

(20)119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

119908119894+1

(119895119873119894+ 1)

= ([

6Δ1198972

119894

Δ1198972

119894+1

+

10 (Δ119897119894+ Δ119897119894+1

)

Δ119897119894+1

minus 6]119908119894(119895119873119894)

minus 7119908119894(119895119873119894minus 2) + 3119908

119894(119895119873119894minus 3)

+ [

12Δ1198972

119894

Δ1198972

119894+1

+

5Δ119897119894

Δ119897119894+1

]119908119894+1

(119895119873119894+ 2)

minus

3Δ1198972

119894

Δ1198972

119894+1

119908119894+1

(119895119873119894+ 3))

times (15 [

Δ119897119894

Δ119897119894+1

+

Δ1198972

119894

Δ1198972

119894+1

])

minus1

119908119894(119895119873119894minus 1)

=

2 (Δ119897119894+ Δ119897119894+1

)

3Δ119897119894+1

119908119894(119895119873119894) +

1

3

119908119894(119895119873119894minus 2)

minus

Δ119897119894

Δ119897119894+1

119908119894+1

(119895119873119894+ 1) +

Δ119897119894

3Δ119897119894+1

119908119894+1

(119895119873119894+ 2)

(21)

According to the finite difference method mentionedabove for a specific engineering problem the displacementof grid points in the initial and the first row can be calculatedby (18) For the second row the displacement of all thegrid nodes in interior columns can be decided by (19) thegrid nodes in exterior boundary columns and their adjacentcolumns (columns 0 1119873

119899minus1 and119873

119899) can be decided by (20)

and based on (21) the node displacement of all the interiorboundary adjacent columns can be achieved Likewise the

10 Mathematical Problems in Engineering

30002000

minus2000minus3000

1000

25 215 1

050 0

24

68

1012minus1000

0

y(m)

M(yt)(

Nmiddotm

)t(s)

Figure 15 Bending moment response of vertical beam

4000

minus4000

2000

252

151

050 0

24

68

1012

minus20000

Q(yt)(

N)

y(m)

t(s)

Figure 16 Shear force response of vertical beam

displacement for all the grid nodes from row 3 to row119872 canbe decided

After getting the displacement of all grid nodes accordingto 119872119894= 119864119868(120597

21199081198941205971199102) and 119876

119894= 119864119868(120597

31199081198941205971199103) the numerical

solutions to the bending moment and shear force responseof vertical beam can be obtained by conducting the finitedifference to the displacement of all grid nodes again

42 Comparative Analysis and Discussion Similarly basedon previous case study the slope is considered under hori-zontal harmonic seismic loading and with anchoring systemdamping neglected or considered respectively The numer-ical solutions to the dynamic response of vertical beam canbe achieved by using finite difference method and MATLABprogramming and 119903

119894= Δ119905Δ

2119897119894= 000001(025 times 025) =

000016 le radic1205721= 0002 (119894 = 1 sim 5)

In order to verify the correctness of the two methodsabove the solutions of intersection 119910 = 15m by differentmethods andwith damping effect neglected or considered areshown in Figures 18 and 19

When damping effect is neglected according to Figure 18it is found that the calculation results by the three methodsshare similar vibration trend and only show slight differencein amplitude generally speaking the finite element solutionmentioned previously is the smallest due to the size of finitedifference grids and it achieved by method of Literature [7] issecond because of the calculated amount of high order vibra-tionmode and the theoretical solutionmentioned previouslyis the largest and the most reasonable but the accuracyof two previous methods can be respectively improved bydecreasing grid ratio and increasing the calculated amount

so that the results of the three methods show more excellentagreement which indicates the reasonability and feasibilityof two proposed methods under the condition of neglectingdamping effect

When damping effect is considered as shown inFigure 19 the proposed theoretical solution and literature [7]solution are also well consistent and the former is slightlylarger than the latter Through the comparison and analysisof Figures 14 15 16 and 19 it is found that when usingthe finite difference method the attenuation characteristicof vertical beam response can be clearly shown in the firstsecond and then with the passing of time the vibrationmode is gradually degenerated into constant state whileno attenuation characteristic is shown and the calculatingresult keeps steady state all the time if using the theoreticalmethods So in the first second the calculating results usedthe finite difference method and theoretical methods haveconsiderable difference which is attributed to the theoreticalsolution based on steady-state complex method Specificallythe calculating result using finite difference method is largerthan the result using theoretical methods in the first secondAfter the first second the vibration pattern and the ampli-tude are consistent by using these three methods Besidesthe finite difference method not only can demonstrate theattenuation characteristic of amplitude in terms ofmagnitudein the damping anchorage system but also can illustrate theattenuation characteristic of amplitude in terms of time andits calculating result will be more reliable and reasonable

Generally speaking the finite difference method cannot only be applied to the dynamic response analysis offrame supporting structure under harmonic earthquake but

Mathematical Problems in Engineering 11

t

M

j

1Δt

0(N0) 1 Δl1 N1 Niminus1 Δli k Ni Nnminus1 Δln Nn y

H

A1 Ai An

Low

er p

latfo

rm (s

lope

toe)

w1

1th

row

anch

or

wi(j k) wn

Upp

er p

latfo

rm (s

lope

cres

t)

ith section1th section nth section

ith ro

w an

chor

n minus

1th

row

anch

or

i minus 1

th ro

w an

chor

Figure 17 Finite difference grids of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical methodProposed finite difference method

minus40000

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 18 Solutions of vertical beam with damping effect neglected by different methods

000 025 050 075 100 125 150 175 200 225 250minus15000minus12500minus10000minus7500

minus2500minus5000

0250050007500

10000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

Proposed finite difference method

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical method

minus15000

minus20000

minus5000

minus10000

0

5000

10000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 19 Solutions of vertical beam with damping effect considered by different methods

12 Mathematical Problems in Engineering

also be used to the analysis of random seismic loadingBesides it can demonstrate the attenuation characteristics ofamplitude in terms of both magnitude and time in otherwords it can indicate the damping effect better Thereforethe calculating result obtained by finite difference methodwould be more approximate to actual conditions and hencebe more reliable and reasonable However the analyticalmethod is only confined to the analysis of frame supportingstructure under harmonic earthquake Besides it can onlyreflect the attenuation characteristics of amplitude in termsof magnitude and the peak value of mechanical response gotby analytical method would be underestimated as a resultpotential hazards may be brought aboutWhat is more basedon steady-state complex method its mechanical responseshave initial phase values which are clearly contradictory tothe reality

5 Conclusions

Through the dynamic response analysis of frame supportingstructure of slope under harmonic forced vibration theconclusion follows

(1) Based on certain assumptions and force analysis ofsegment of infinitesimal length the dynamic govern-ing equation of vertical beam under forced vibrationis derived and the analytical solutions to the dynamicresponse of vertical beam under harmonic forcedvibration are obtainedMeanwhile according to finitedifference method the finite difference format andcorresponding calculating procedure for dynamicresponse of vertical beam under forced vibrationare established and finally programmed by usingMATLAB language

(2) In the case studies three methods have been usedin the dynamic response analysis and with dampingeffect neglected or considered respectivelyThe feasi-bility and correctness of the two proposed methodshave been examined and the characteristic of eachmethod has been summarized

(3) Under horizontal harmonic seismic loading themechanical response of vertical beam would syn-chronize correspondingly Its response law showsagreement with the one of two-end cantilevered con-tinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuousbeam In specific engineering design the factors suchas mechanical characteristic of vertical beam andslope supporting effect of anchors should also beconsidered

(4) Compared with analytical solutions the finite differ-ence method can simulate the dynamic response offrame supporting structure under various earthquakeconditions and it also can indicate the damping effectof anchorage system very well hence the calculatingresult obtained ismore reliable and reasonableMore-over the finite differencemethod is achieved based onprogramming calculation its calculation speed and

accuracy can be guaranteed and hence it has broadapplication prospect

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by Project (51308273 41372307) sup-ported by National Natural Science Foundation of China andProject (2010(A)06-b) supported by Science and Technol-ogy Project supported by Yunan Provincial CommunicationDepartment

References

[1] L K Cheng ldquoPresent status and development of groundanchoragesrdquo China Civil Engineering Journal vol 34 no 3 pp7ndash16 2001 (Chinese)

[2] D P Zhou J J Zhang and Y Tang ldquoSeismic damage analysis ofroad slopes in Wenchuan earthquakerdquo Chinese Journal of RockMechanics and Engineering vol 29 no 3 pp 565ndash576 2010(Chinese)

[3] M Yang H T Hu C J Lu et al ldquoCalculation of internal forcesfor prestressed anchor cable frame used in reinforced roadcutsoil sloperdquo Chinese Journal of Rock Mechanics and Engineeringvol 21 no 9 pp 1383ndash1386 2002 (Chinese)

[4] Y H Tian J K Liu and Y F Zhang ldquoInternal force calculationof prestressed anchor cable frame by finite difference methodrdquoJournal of Beijing Jiaotong University vol 31 no 4 pp 22ndash252007 (Chinese)

[5] G H Lin ldquoModel test of anchor grid beam for cut sloperdquoTechnology of Highway and Transport vol 3 pp 19ndash22 2004(Chinese)

[6] B L Zhu M Yang H T Hu et al ldquoTesting study on internalforces for prestressed anchor cable frame in reinforced soilsloperdquo Chinese Journal of Rock Mechanics and Engineering vol24 no 4 pp 697ndash702 2005 (Chinese)

[7] J H Dong Y P Zhu Y Zhou et al ldquoDynamic calculationmodel and seismic response for frame supporting structurewith prestressed anchorsrdquo Science China Technological Sciencesvol 53 no 7 pp 1957ndash1966 2010

[8] S Catal ldquoSolution of free vibration equations of beam onelastic soil by using differential transform methodrdquo AppliedMathematical Modelling vol 32 no 9 pp 1744ndash1757 2008

[9] M Balkaya M O Kaya and A Saglamer ldquoAnalysis of thevibration of an elastic beam supported on elastic soil using thedifferential transform methodrdquo Archive of Applied Mechanicsvol 79 no 2 pp 135ndash146 2009

[10] M Ansari E Esmailzadeh and D Younesian ldquoFrequencyanalysis of finite beams on nonlinear KelvinVoight foundationunder moving loadsrdquo Journal of Sound and Vibration vol 330no 7 pp 1455ndash1471 2011

[11] O Demirdag and Y Yesilce ldquoSolution of free vibration equationof elastically supported Timoshenko columns with a tip massby differential transform methodrdquo Advances in EngineeringSoftware vol 42 no 10 pp 860ndash867 2011

Mathematical Problems in Engineering 13

[12] MH Yas andN Samadi ldquoFree vibrations and buckling analysisof carbon nanotube-reinforced composite Timoshenko beamson elastic foundationrdquo International Journal of Pressure Vesselsand Piping vol 98 pp 119ndash128 2012

[13] R A Jafari-Talookolaei M Abedi M H Kargarnovin et alldquoAn analytical approach for the free vibration analysis ofgenerally laminated composite beams with shear effect androtary inertiardquo International Journal of Mechanical Sciences vol65 no 1 pp 97ndash104 2012

[14] S W Qi F Q Wu F Z Yan et al Dynamic Response Analysesof Rock Slope Science Press Beijing China 2007 Chinese

[15] J H Dong and Y P Zhu ldquoDynamic calculation model andseismic response for the system of soil nailing and surroundingsoilrdquo Chinese Journal of Theoretical and Applied Mechanics vol41 no 2 pp 236ndash242 2009

[16] Z L Xu Elasticity Higher Education Press Beijing China2009 Chinese

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2 Mathematical Problems in Engineering

more current studies only focus on statistic problems whilefew researches have paid attention to dynamic problems andno related code can be referred to as well Therefore thedynamic behaviors of frame supporting structure of slopeunder harmonic forced vibration are explored in this paperand this research can provide some theoretical support tothe calculation of dynamic force and reinforcement of thestructure

2 Dynamic Mechanical Model of FrameSupporting Structure of Slope

21 Basic Assumptions Under the horizontal seismic load-ing the dynamic mechanical model of frame supportingstructure of slope is established based on following assump-tions

(1) Beams and ground of slope are considered as con-tinuous isotropic and elastic and only the elasticdynamic problems are analyzed

(2) Space torsional effect of beams is neglected horizon-tal beam and vertical beam are viewed as indepen-dent continuous beam respectively Meanwhile themutual interaction among horizontal beam verticalbeam and anchor is considered through node defor-mation compatibility

(3) Under the horizontal seismic loading 1198861015840119892(119905) from slope

bottom the beams and the ground of slope keepcontact all the time and the interaction between themis simulated as horizontal linear spring model themechanical effect of which is similar to Winkler elas-tic foundation beam [8ndash13] Therefore the stiffnesscoefficient 119896

0can be determined by referring to the

theory of elastic foundation beam The bottom ofvertical beam is considered as sliding support whichonly subjected to vertical constraint for its horizontalconstraint is so slight that can be neglected

(4) Influenced by slope height the particle load accelera-tion of frame is 119886

119892(119905) = 120595(119910 + 119867

0)1198861015840

119892(119905) in which the

value for the amplification coefficient of acceleration120595(119910 + 119867

0) refers to literature [14] where 119910 is the

vertical height from the particle to lower platform and1198670is the vertical length from lower platform to slope

bottom

So the dynamic mechanical model of horizontal beamand vertical beam is almost the same and the only differencebetween them lies in the characteristic of particle loadacceleration function More specifically the correspondingparameter of horizontal beam 120595(ℎ

119894+ 1198670) is constant In

other words the horizontal seismic acceleration for the wholehorizontal beam is the same at the same time However forthe vertical beam the corresponding parameter of slope-height effect would change with the change of particleheight as a result the particle load acceleration along thevertical beam would vary at the same moment Thereforethe dynamic response of horizontal beam can be viewedas a special case of vertical beam namely that the particle

acceleration of beam is the same so the analyses for verticalbeam can be extended to horizontal beamDynamic responseanalysis for vertical beam is conducted as follows

22 Dynamic Governing Equation According to structuraldynamic theory and above basic assumptions the dynamicmechanical model of vertical beam is shown in Figure 1

Based on the layout scheme of slope bolts the verticalbeam of the frame supporting structure can be divided into119899 sections To 119894th section the segment of infinitesimal length119889119910 sin120573 is chosen to perform the dynamic response analysisjust as the revelation of Figure 1 According to DrsquoAlembertprinciple the dynamic equilibrium equation for this segmentof infinitesimal length can be expresses as

119876119894+

120597119876119894

120597119910

119889119910 +

1205972119908119894

1205971199052119889119898 +

1198960119887119908119894119889119910

sin120573

= 119886119892(119905) 119889119898 + 119876

119894 (1)

where 119876119894and 119908

119894represent the shear force and horizontal

displacement of vertical beam respectively 120573 is slope angleand the mass of the segment of infinitesimal length is 119889119898 =

(120588119887ℎ sin120573)119889119910 where 119887 ℎ and 120588 are the width height anddensity of vertical beam respectively Equation (1) can besimplified as

120597119876119894

120597119910

+

120588119887ℎ

sin120573

1205972119908119894

1205971199052

+

1198960119887

sin120573

119908119894=

120588119887ℎ

sin120573

119886119892(119905) (2)

Since 119872119894= 119864119868(120597

21199081198941205971199102) 119876119894= 120597119872

119894120597119910 = 119864119868(120597

31199081198941205971199103)

and 119868 = 119887ℎ312sin3120573 the dynamic equilibrium can be further

simplified as

1205974119908119894

1205971199104

+

120588119887ℎ

119864119868 sin120573

1205972119908119894

1205971199052

+

1198960119887

119864119868 sin120573

119908119894=

120588119887ℎ

119864119868 sin120573

119886119892(119905) (3)

Given that 1205721

= 120588119887ℎ119864119868 sin120573 and 1205722

= 1198960119887119864119868 sin120573 the

dynamic governing equation can be transformed into

1205974119908119894

1205971199104

+ 1205721

1205972119908119894

1205971199052

+ 1205722119908119894= 1205721119886119892(119905) (4)

23 Boundary Condition According to the above basic assu-mptions the boundary condition for the dynamic responseof vertical beam could be obtained

1198721

1003816100381610038161003816119910=0

= 119864119868

12059721199081

1205971199102

100381610038161003816100381610038161003816100381610038161003816119910=0

= 0

1198761

1003816100381610038161003816119910=0

= 119864119868

12059731199081

1205971199103

100381610038161003816100381610038161003816100381610038161003816119910=0

= 0

119872119899

1003816100381610038161003816119910=119867

= 119864119868

1205972119908119899

1205971199102

100381610038161003816100381610038161003816100381610038161003816119910=119867

= 0

119876119899

1003816100381610038161003816119910=119867

= 119864119868

1205973119908119899

1205971199103

100381610038161003816100381610038161003816100381610038161003816119910=119867

= 0

(5)

Meanwhile according to the principle of deformationcompatibility in the outside endpoint of anchor namely that

Mathematical Problems in Engineering 3

Upper platform (slope crest)

n minus 1th row anchor

ith row anchor

k0

k0

k0Free segment

Anchoring segment

1th row anchor

120573

ag(t) Lower platform (slope toe)

n

i + 1

ui2(Li2

Li2

t)

i

1

o

y

hi

H

Mi +120597Mi

120597ydy

Qi +120597Qi

120597ydy1205972wi

120597t2dm

k0bwidy

sin120573 Qi

Mi

H0

Md

ag(t)dm

dy

Figure 1 Dynamic mechanical model of vertical beam

the demarcation point of vertical beam the displacement 119908119894

rotation angle 120579119894 and bending moment 119872

119894should meet the

following conditions

119908119894

1003816100381610038161003816119910=ℎ119894

= 119908119894+1

1003816100381610038161003816119910=ℎ119894

= 1199061198942(1198711198942 119905) cos120593

119894(119894 = 1 sim 119899 minus 1)

120579119894

1003816100381610038161003816119910=ℎ119894

= 120579119894+1

1003816100381610038161003816119910=ℎ119894

=

120597119908119894

120597119910

10038161003816100381610038161003816100381610038161003816119910=ℎ119894

=

120597119908119894+1

120597119910

10038161003816100381610038161003816100381610038161003816119910=ℎ119894

(119894 = 1 sim 119899 minus 1)

119872119894

1003816100381610038161003816119910=ℎ119894

= 119872119894+1

1003816100381610038161003816119910=ℎ119894

= 119864119868

1205972119908119894

1205971199102

100381610038161003816100381610038161003816100381610038161003816119910=ℎ119894

= 119864119868

1205972119908119894+1

1205971199102

100381610038161003816100381610038161003816100381610038161003816119910=ℎ119894

(119894 = 1 sim 119899 minus 1)

(6)

where 120593119894is the inclined angle of the 119894th row anchor and

1199061198942(1198711198942 119905) is the displacement of 119894th row anchor of which

under horizontal seismic loading 1198861015840

119892(119905) from slope bottom

the specific formula can be found in the literature [15] whichis based on some assumptions of using the Kelvin-Voigtmodel to simulate the interaction between anchoring sectionand the surrounding rock-soil mass the inertial dynamicaction of slope structure which exerts on the free section canbe simplified as the equivalent lumped mass 119872

119889that acts on

the end of free section just as illustrated in the 2th row anchorof Figure 1

3 Analytical Solutions to Vertical Beam underHarmonic Forced Vibration

31 Theoretical Analysis Coordination is considered andsteady-state complex method is used to the derivation ofharmonic forced vibration response of vertical beam It isassumed that the solution to the governing equation (4) is1199081015840

119894(119910 119905) = 119882

119894(119910)119890119894120596119892119905 Then

1198821015840101584010158401015840

119894(119910) + (120572

2minus 12057211205962

119892)119882119894(119910) = 120572

11198861015840

119892120595 (119910 + 119867

0) (7)

Given that 1205741= 120573(1 minus 119894) 120574

2= 120573(1 + 119894) 120574

3= 120573(minus1 + 119894) and

1205744= 120573(minus1 minus 119894) the complex solution to (7) is

119882119894(119910) = 119862

11989411198901205741119910

+ 11986211989421198901205742119910

+ 11986211989431198901205743119910

+ 11986211989441198901205744119910

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

(8)

where119862119894111986211989421198621198943 and119862

1198944are constant complex coefficients

which can be decided by specific boundary conditionLikewise the horizontal displacement bending moment

and shear force of the 119894th section in the vertical beam can beexpressed as

1199081015840

119894(119910 119905)

= 119882119894(119910) 119890119894120596119892119905

= (11986211989411198901205741119910

+ 11986211989421198901205742119910

+ 11986211989431198901205743119910

+ 11986211989441198901205744119910

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

)

times 119890119894120596119892119905

1198721015840

119894(119910 119905)

= 119864119868 (1205742

111986211989411198901205741119910

+ 1205742

211986211989421198901205742119910

+ 1205742

311986211989431198901205743119910

+ 1205742

411986211989441198901205744119910

) 119890119894120596119892119905

1198761015840

119894(119910 119905)

= 119864119868 (1205743

111986211989411198901205741119910

+ 1205743

211986211989421198901205742119910

+ 1205743

311986211989431198901205743119910

+ 1205743

411986211989441198901205744119910

) 119890119894120596119892119905

(9)

Meanwhile according to literature [15] the displacementresponse of the outside endpoint of anchor under horizontalharmonic loading can be simplified as 119906

1015840

1198942(1198711198942 119905) = 120581

119894119890119894120596119892119905

4 Mathematical Problems in Engineering

It is assumed that 119880119894(ℎ119894) = 120581

119894cos120593119894 Substitute them into

boundary condition equations (5) and (6) Then

1205742

111986211

+ 1205742

211986212

+ 1205742

311986213

+ 1205742

411986214

= 0

1205743

111986211

+ 1205743

211986212

+ 1205743

311986213

+ 1205743

411986214

= 0

11986211989411198901205741ℎ119894

+ 11986211989421198901205742ℎ119894

+ 11986211989431198901205743ℎ119894

+ 11986211989441198901205744ℎ119894

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

= 119880119894(ℎ119894)

119862(119894+1)1

1198901205741ℎ119894

+ 119862(119894+1)2

1198901205742ℎ119894

+ 119862(119894+1)3

1198901205743ℎ119894

+ 119862(119894+1)4

1198901205744ℎ119894

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

= 119880119894(ℎ119894)

120574111986211989411198901205741ℎ119894

+ 120574211986211989421198901205742ℎ119894

+ 120574311986211989431198901205743ℎ119894

+ 120574411986211989441198901205744ℎ119894

= 1205741119862(119894+1)1

1198901205741ℎ119894

+ 1205742119862(119894+1)2

1198901205742ℎ119894

+ 1205743119862(119894+1)3

1198901205743ℎ119894

+ 1205744119862(119894+1)4

1198901205744ℎ119894

1205742

111986211989411198901205741ℎ119894

+ 1205742

211986211989421198901205742ℎ119894

+ 1205742

311986211989431198901205743ℎ119894

+ 1205742

411986211989441198901205744ℎ119894

= 1205742

1119862(119894+1)1

1198901205741ℎ119894

+ 1205742

2119862(119894+1)2

1198901205742ℎ119894

+ 1205742

3119862(119894+1)3

1198901205743ℎ119894

+ 1205742

4119862(119894+1)4

1198901205744ℎ119894

1205742

111986211989911198901205741119867

+ 1205742

211986211989921198901205742119867

+ 1205742

311986211989931198901205743119867

+ 1205742

411986211989941198901205744119867

= 0

1205743

111986211989911198901205741119867

+ 1205743

211986211989921198901205742119867

+ 1205743

311986211989931198901205743119867

+ 1205743

411986211989941198901205744119867

= 0

(10)

The above linear equations can be solved by matrixmethod It is assumed that 119860

119894= 1198901205741ℎ119894 119861

119894= 1198901205742ℎ119894 119862

119894= 1198901205743ℎ119894

119863119894= 1198901205744ℎ119894 and 119864

119894= 119880119894(ℎ119894) minus (120572

11198861015840

119892120595(119910 + 119867

0)(1205722minus 12057211205962

119892))

Then (10) can be expressed as matrix format

1198624119899times1

= 119861minus1

4119899times41198991198604119899times1

(11)

where

119860 = [0 0 1198641

1198641

0 0 sdot sdot sdot 119864119894

119864119894

0 0 sdot sdot sdot 119864119899minus1

119864119899minus1

0 0 0 0]

119879

119862 = [11986211

11986212

11986213

11986214

sdot sdot sdot 1198621198941

1198621198942

1198621198943

1198621198944

sdot sdot sdot 1198621198991

1198621198992

1198621198993

1198621198994]

119879

119861 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1205742

11205742

21205742

31205742

40 0 0 0

1205743

11205743

21205743

31205743

40 0 0 0

1198601

1198611

1198621

1198631

0 0 0 0

0 0 0 0 1198601

1198611

1198621

1198631

12057411198601

12057421198611

12057431198621

12057441198631

minus12057411198601

minus12057421198611

minus12057431198621

minus12057441198631

1205742

11198601

1205742

21198611

1205742

31198621

1205742

41198631

minus1205742

11198601

minus1205742

21198611

minus1205742

31198621

minus1205742

41198631

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

d

sdot sdot sdot 119860119894

119861119894

119862119894

119863119894

sdot sdot sdot 0 0 0 0

sdot sdot sdot 1205741119860119894

1205742119861119894

1205743119862119894

1205744119863119894

sdot sdot sdot 1205742

1119860119894

1205742

2119861119894

1205742

3119862119894

1205742

4119863119894

d

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

Mathematical Problems in Engineering 5

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

119860119894

119861119894

119862119894

119863119894

minus1205741119860119894

minus1205742119861119894

minus1205743119862119894

minus1205744119863119894

minus1205742

1119860119894

minus1205742

2119861119894

minus1205742

3119862119894

minus1205742

4119863119894

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

d

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

d

sdot sdot sdot 119860

119899minus1119861119899minus1

119862119899minus1

119863119899minus1

0 0 0 0

sdot sdot sdot 0 0 0 0 119860119899minus1

119861119899minus1

119862119899minus1

119863119899minus1

sdot sdot sdot 1205741119860119899minus1

1205742119861119899minus1

1205743119862119899minus1

1205744119863119899minus1

minus1205741119860119899minus1

minus1205742119861119899minus1

minus1205743119862119899minus1

minus1205744119863119899minus1

sdot sdot sdot 1205742

1119860119899minus1

1205742

2119861119899minus1

1205742

3119862119899minus1

1205742

4119863119899minus1

minus1205742

1119860119899minus1

minus1205742

2119861119899minus1

minus1205742

3119862119899minus1

minus1205742

4119863119899minus1

sdot sdot sdot 0 0 0 0 1205742

1119860119899

1205742

2119861119899

1205742

3119862119899

1205742

4119863119899

sdot sdot sdot 0 0 0 0 1205743

1119860119899

1205743

2119861119899

1205743

3119862119899

1205743

4119863119899

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(12)

After getting constant complex coefficient 1198624119899times1

it isassumed that119862

1198941= 1198881198941+11988811989421198941198621198942

= 1198881198943+11988811989441198941198621198943

= 1198881198945+1198881198946119894 and

1198621198944

= 1198881198947

+ 1198881198948119894 Substituting them into (9) the displacement

can be simplified as

1199081015840

119894(119910 119905)

= (1198881198941119890119910120573

+ 1198881198942119890119910120573

119894)

times [cos (120596119892119905 minus 119910120573) + 119894 sin (120596

119892119905 minus 119910120573)]

+ (1198881198943119890119910120573

+ 1198881198944119890119910120573

119894) [cos (120596119892119905 + 119910120573) + 119894 sin (120596

119892119905 + 119910120573)]

+ (1198881198945119890minus119910120573

+ 1198881198946119890minus119910120573

119894) [cos (120596119892119905 + 119910120573) + 119894 sin (120596

119892119905 + 119910120573)]

+ (1198881198947119890minus119910120573

+ 1198881198948119890minus119910120573

119894) [cos (120596119892119905 minus 119910120573) + 119894 sin (120596

119892119905 minus 119910120573)]

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

(cos120596119892119905 + 119894 sin120596

119892119905)

(13)

According to the calculating principle of complexmethod the corresponding imaginary part is the steady-statesolution to the mechanical response of vertical beam underharmonic seismic loading Then

119908119894(119910 119905)

= 119890120573119910

[1198881198941sin (120596

119892119905 minus 120573119910) + 119888

1198942cos (120596

119892119905 minus 120573119910)

+ 1198881198943sin (120596

119892119905 + 120573119910) + 119888

1198944cos (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[1198881198945sin (120596

119892119905 + 120573119910) + 119888

1198946cos (120596

119892119905 + 120573119910)

+ 1198881198947sin (120596

119892119905 minus 120573119910) + 119888

1198948cos (120596

119892119905 minus 120573119910)]

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

sin120596119892119905

119872119894(119910 119905)

= 21205732119864119868 119890120573119910

[minus1198881198941cos (120596

119892119905 minus 120573119910) + 119888

1198942sin (120596

119892119905 minus 120573119910)

+ 1198881198943cos (120596

119892119905 + 120573119910) minus 119888

1198944sin (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[minus1198881198945cos (120596

119892119905 + 120573119910) + 119888

1198946sin (120596

119892119905 + 120573119910)

+ 1198881198947cos (120596

119892119905 minus 120573119910) minus 119888

1198948sin (120596

119892119905 minus 120573119910)]

119876119894(119910 119905)

= 21205733119864119868 119890120573119910

[(1198881198942minus 1198881198941) sin (120596

119892119905 minus 120573119910)

minus (1198881198941+ 1198881198942) cos (120596

119892119905 minus 120573119910)

minus (1198881198943+ 1198881198944) sin (120596

119892119905 + 120573119910)

+ (1198881198943minus 1198881198944) cos (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[(1198881198945minus 1198881198946) sin (120596

119892119905 + 120573119910)

+ (1198881198945+ 1198881198946) cos (120596

119892119905 + 120573119910)

+ (1198881198947+ 1198881198948) sin (120596

119892119905 minus 120573119910)

+ (1198881198948minus 1198881198947) cos (120596

119892119905 minus 120573119910)]

(14)

6 Mathematical Problems in Engineering

32 Case Studies Take a slope as an example Its height is12m the slope angle is 65∘ and its physical and mechanicalparameters are 120588

119904= 1600 kgsdotmminus3 119864

119904= 012MPa 120583

119904=

03 119888119904

= 20 kPa and 120601119904

= 25∘ The seismic precautionary

intensity in the local region is 8 degrees and the peakground acceleration is 02 gThe dynamic safety factor is 1015calculated by Fellenius method hence its support is neededA preliminary supporting scheme is using fixed end anchorsand frame supporting structure The dynamic safety factoris considered as 12 calculated by the pseudostatic methodand the design scheme is shown in Figure 2 The anchor usescategory II steel bar diameter size is 28mm and its spaceis 33m the dip120593 = 10

∘ four rows of anchors have beenarranged both lengths of anchoring section and free sectionare 5m The mortar grade M30 is used as bonding agent thediameter of anchor hole is 119863 = 130mm The section size ofthe frame beam is 119887timesℎ = 045 times 05m concrete is gradeC30120588 = 2500 kgm3 119864 = 255GPa Earthquake accelerationis 1198861015840

119892(119905) = 119886

1015840

119892sin120596119892119905 = 2 sin 4120587119905 the duration is 25 s and

1198960= 2 times 10

3 kNm3From Figures 3 4 and 5 the displacement bending

moment and shear force response of vertical beam have beengiven when damping effect of anchorage system is neglectedwhile from Figures 6 7 8 9 10 11 12 and 13 correspondingresponse has been given at special sections (intersection andmidspan section) and typical moment (according to momentof acceleration peak) It is found that the displacement bend-ingmoment and shear force response of vertical beamvibratesynchronously with earthquakeThe law of bending momentand shear force response shows agreement with the har-monic forced vibration characteristic of two-end cantileveredcontinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuous beamMorespecifically the bending moment response at intersectionopposite to the one at midspan section and the shear forceresponse show a jumping from the left hand to the righthand at the intersection of vertical beam just as illustratedin Figures 12 and 13 What is more the cantilevered lengthat the two ends of vertical beam has noticeable influenceon the mechanical response of frame supporting structureThe larger the cantilevered length is the more the bendingmoment of intersection is and the greater the difference ofbending moment between intersection and midspan sectionis Examining the frame supporting structure independentlythe bending moment can keep good balance for the wholevertical beamwhen its cantilevered length is 02 or 025 timesthe anchor adjacent spacing referring to the design expe-rience of two-end cantilevered continuous beam Howeverthe cantilevered length of vertical beam is still related to thearranging scheme of anchors actually the length even decidesthe arrangement of anchors and directly influences the slopesupporting effect of anchors Hence in specific engineeringdesign the determination of cantilevered length should alsoconsider the factors such as mechanical characteristic ofvertical beam and slope supporting effect of anchors

Just as shown from Figures 14 15 and 16 the dis-placement bending moment and shear force response ofvertical beam with damping effect considered have been

given as well The overall trend is similar to the previous onewith damping effect neglected When considering anchoragesystem damping effect the solutions can be obtained byusing steady-state complex method and the peak valuesof the displacement bending moment and shear force aresmaller than the ones with damping effect neglected Eventhough the response shows some attenuation characteristicof amplitude in terms of magnitude it does not demonstratethe attenuation characteristic of amplitude in terms of timeWhat is more the mechanical responses show certain initialphases which can be attributed to the dynamic behaviors ofvertical beam under steady-state vibration with damping

4 Finite Difference Method for Vertical Beamunder Forced Vibration

41 Finite Difference Format for Dynamic Response AnalysisThe numerical solutions to dynamic governing equation (4)could be also achieved by using finite difference methodAccording to the basic principles of finite difference method[16] the research area119860(0 le 119905 le 119879 0 le 119910 le 119867) can be dividedinto 119899 subregions119860

119894Thewhole region of119860

119894can be redivided

into many grids 119860119894Δ119905Δ1198971

(119872119873119894minus119873119894minus1

+ 1) which is shown inFigure 17 Meanwhile in order to guarantee the convergenceand stability of calculation the ratio of the grids should meetthe following condition 119903

119894= Δ119905Δ

2119897119894le radic120572

1(119894 = 1 sim 119899)

The governing equation (4) can be expanded accordingto central difference formula and the difference format ofdynamic equation for grid node 119908

119894(119895 119896) can be obtained

(6119908119894(119895 119896) minus 4 [119908

119894(119895 119896 + 1) + 119908

119894(119895 119896 minus 1)]

+119908119894(119895 119896 + 2) + 119908

119894(119895 119896 minus 2)) (Δ119897

4

119894)

minus1

+ 1205721

119908119894(119895 + 1 119896) + 119908

119894(119895 minus 1 119896) minus 2119908

119894(119895 119896)

Δ1199052

+ 1205722119908119894(119895 119896) = 120572

1120595119894(119896) 1198861015840

119892[119895Δ119905]

(15)

where 120595119894(119896) = 120595[(119896 minus119873

119894minus1)Δ119897119894+sum119911=119894minus1

119911=1Δ119897119911(119873119911minus119873119911minus1

) +1198670]

Likewise the boundary equations (5) and (6) can beexpanded by endpoint derivative differential formula and thecorresponding differential equation can be attained

119864119868

21199081(119895 0) minus 5119908

1(119895 1) + 4119908

1(119895 2) minus 119908

1(119895 3)

Δ1198972

1

= 0

119864119868

minus1199081(119895 0) + 3119908

1(119895 1) minus 3119908

1(119895 2) + 119908

1(119895 3)

Δ1198973

1

= 0

119864119868 (2119908119899(119895119873119899) minus 5119908

119899(119895119873119899minus 1)

+4119908119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)) (Δ119897

2

119899)

minus1

= 0

119864119868 (119908119899(119895119873119899) minus 3119908

119899(119895119873119899minus 1)

+ 3119908119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)) (Δ119897

3

119899)

minus1

= 0

119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

Mathematical Problems in Engineering 7

5050

50

5050

50

Free section

Anchoring section

Anchor

5050

15

15

120

045 045

165 16533 33

Grass

Anchor

Horizontal beam Vertical beam

33

30

30

30

65∘

Figure 2 Design scheme for slope (unit m)

02

46

810

12

005115225minus003minus002minus001

0

002001

003

w (y

t)

(m)

t (s) y(m)2

46

810

11525y(m)

Figure 3 Displacement response of vertical beam

times104

3

minus3

2

minus2

1

252

151

050 0

24

6y(m)

810

12

minus10

M(yt)(

Nmiddotm

)

t(s)

Figure 4 Bending moment response of vertical beam

times104

4

minus4

2

252

151

050 0

24

68

1012minus2

0

Q(yt)(

N)

y(m)t(s)

Figure 5 Shear force response of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Mi(

Nmiddotm

)

y = 15my = 45m

y = 75my = 105m

t(s)

Figure 6 Bending moment response at intersection of verticalbeam

2119908119894(119895119873119894) minus 3119908

119894(119895119873119894minus 1) + 119908

119894(119895119873119894minus 2)

Δ119897119894

=

minus2119908119894+1

(119895119873119894) + 3119908

119894+1(119895119873119894+ 1) minus 119908

119894+1(119895119873119894+ 2)

Δ119897119894+1

8 Mathematical Problems in Engineering

minus12000minus10000minus8000minus6000minus4000minus2000

02000400060008000

100001200014000

000 025 050 075 100 125 150 175 200 225 250

y = 3my = 6my = 9m

Mi(

Nmiddotm

)

t(s)

Figure 7 Bending moment response at midspan section of verticalbeam

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 8 Shear force response at left hand of intersection of verticalbeam

(2119908119894(119895119873119894) minus 5119908

119894(119895119873119894minus 1)

+ 4119908119894(119895119873119894minus 2) minus 119908

119894(119895119873119894minus 3)) (Δ119897

2

119894)

minus1

= (2119908119894+1

(119895119873119894) minus 5119908

119894+1(119895119873119894+ 1)

+ 4119908119894+1

(119895119873119894+ 2) minus 119908

119894+1(119895119873119894+ 3)) (Δ119897

2

119894+1)

minus1

(16)

The displacement response of 119894th column grid nodes shouldalso satisfy the following initial conditions

119908119894[0 119896] = 0

120597119908119894

120597119905

10038161003816100381610038161003816100381610038161003816119905=0

=

119908119894[1 119896] minus 119908

119894[0 119896]

Δ119905

= minus2V0

(17)

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 9 Shear force response at right hand of intersection ofvertical beam

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

000 025 050 075 100 125 150 175 200 225 250minus4000

Qi(

N)

y = 3my = 6my = 9m

t(s)

Figure 10 Shear force response atmidspan section of vertical beam

In (17) V0is the initial velocity of vertical beam If the

harmonic vibration is needed then the initial velocity can bedefined as V

0= minus119886119892120596119892

Equations (15)ndash(17) can be simplified and the finite dif-ference format for the dynamic response of vertical beam isshown as

119908119894[0 119896] = 0

119908119894[1 119896] = minus2Δ119905V

0

(18)

119908119894(119895 + 1 119896)

= (2 minus

6Δ1199052

1205721Δ1198974

119894

minus

1205722Δ1199052

1205721

)119908119894(119895 119896)

+

Δ1199052

1205721Δ1198974

119894

[4119908119894(119895 119896 + 1) + 4119908

119894(119895 119896 minus 1)

Mathematical Problems in Engineering 9

00 15 30 45 60 75 90 105 120minus003

minus002

minus001

000

001

002

003wi(

m)

t = 1125 st = 1375 s

y(m)

Figure 11 Displacement response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120minus30000

minus20000

minus10000

0

10000

20000

30000

t = 1125 st = 1375 s

Mi(

Nmiddotm

)

y(m)

Figure 12 Bending moment response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120

minus30000

minus40000

minus20000

minus10000

0

10000

20000

30000

40000

t = 1125 st = 1375 s

Qi(

N)

y(m)

Figure 13 Shear force response along vertical beam at typicalmoment

times3

minus3

2

minus2

1

25 2 15 1 05 0 02

46

810

12minus10

w(yt)(

m)

y(m)

t(s)

10minus3

Figure 14 Displace response of vertical beam

minus 119908119894(119895 119896 + 2) minus 119908

119894(119895 119896 minus 2)]

minus 119908119894(119895 minus 1 119896) + Δ119905

2120595119894(119896) 1198861015840

119892(119895Δ119905)

(19)1199081(119895 0) = 3119908

1(119895 2) minus 2119908

1(119895 3)

1199081(119895 1) = 2119908

1(119895 2) minus 119908

1(119895 3)

119908119899(119895119873119899minus 1) = 2119908

119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)

119908119899(119895119873119899) = 3119908

119899(119895119873119899minus 2) minus 2119908

119899(119895119873119899minus 3)

(20)119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

119908119894+1

(119895119873119894+ 1)

= ([

6Δ1198972

119894

Δ1198972

119894+1

+

10 (Δ119897119894+ Δ119897119894+1

)

Δ119897119894+1

minus 6]119908119894(119895119873119894)

minus 7119908119894(119895119873119894minus 2) + 3119908

119894(119895119873119894minus 3)

+ [

12Δ1198972

119894

Δ1198972

119894+1

+

5Δ119897119894

Δ119897119894+1

]119908119894+1

(119895119873119894+ 2)

minus

3Δ1198972

119894

Δ1198972

119894+1

119908119894+1

(119895119873119894+ 3))

times (15 [

Δ119897119894

Δ119897119894+1

+

Δ1198972

119894

Δ1198972

119894+1

])

minus1

119908119894(119895119873119894minus 1)

=

2 (Δ119897119894+ Δ119897119894+1

)

3Δ119897119894+1

119908119894(119895119873119894) +

1

3

119908119894(119895119873119894minus 2)

minus

Δ119897119894

Δ119897119894+1

119908119894+1

(119895119873119894+ 1) +

Δ119897119894

3Δ119897119894+1

119908119894+1

(119895119873119894+ 2)

(21)

According to the finite difference method mentionedabove for a specific engineering problem the displacementof grid points in the initial and the first row can be calculatedby (18) For the second row the displacement of all thegrid nodes in interior columns can be decided by (19) thegrid nodes in exterior boundary columns and their adjacentcolumns (columns 0 1119873

119899minus1 and119873

119899) can be decided by (20)

and based on (21) the node displacement of all the interiorboundary adjacent columns can be achieved Likewise the

10 Mathematical Problems in Engineering

30002000

minus2000minus3000

1000

25 215 1

050 0

24

68

1012minus1000

0

y(m)

M(yt)(

Nmiddotm

)t(s)

Figure 15 Bending moment response of vertical beam

4000

minus4000

2000

252

151

050 0

24

68

1012

minus20000

Q(yt)(

N)

y(m)

t(s)

Figure 16 Shear force response of vertical beam

displacement for all the grid nodes from row 3 to row119872 canbe decided

After getting the displacement of all grid nodes accordingto 119872119894= 119864119868(120597

21199081198941205971199102) and 119876

119894= 119864119868(120597

31199081198941205971199103) the numerical

solutions to the bending moment and shear force responseof vertical beam can be obtained by conducting the finitedifference to the displacement of all grid nodes again

42 Comparative Analysis and Discussion Similarly basedon previous case study the slope is considered under hori-zontal harmonic seismic loading and with anchoring systemdamping neglected or considered respectively The numer-ical solutions to the dynamic response of vertical beam canbe achieved by using finite difference method and MATLABprogramming and 119903

119894= Δ119905Δ

2119897119894= 000001(025 times 025) =

000016 le radic1205721= 0002 (119894 = 1 sim 5)

In order to verify the correctness of the two methodsabove the solutions of intersection 119910 = 15m by differentmethods andwith damping effect neglected or considered areshown in Figures 18 and 19

When damping effect is neglected according to Figure 18it is found that the calculation results by the three methodsshare similar vibration trend and only show slight differencein amplitude generally speaking the finite element solutionmentioned previously is the smallest due to the size of finitedifference grids and it achieved by method of Literature [7] issecond because of the calculated amount of high order vibra-tionmode and the theoretical solutionmentioned previouslyis the largest and the most reasonable but the accuracyof two previous methods can be respectively improved bydecreasing grid ratio and increasing the calculated amount

so that the results of the three methods show more excellentagreement which indicates the reasonability and feasibilityof two proposed methods under the condition of neglectingdamping effect

When damping effect is considered as shown inFigure 19 the proposed theoretical solution and literature [7]solution are also well consistent and the former is slightlylarger than the latter Through the comparison and analysisof Figures 14 15 16 and 19 it is found that when usingthe finite difference method the attenuation characteristicof vertical beam response can be clearly shown in the firstsecond and then with the passing of time the vibrationmode is gradually degenerated into constant state whileno attenuation characteristic is shown and the calculatingresult keeps steady state all the time if using the theoreticalmethods So in the first second the calculating results usedthe finite difference method and theoretical methods haveconsiderable difference which is attributed to the theoreticalsolution based on steady-state complex method Specificallythe calculating result using finite difference method is largerthan the result using theoretical methods in the first secondAfter the first second the vibration pattern and the ampli-tude are consistent by using these three methods Besidesthe finite difference method not only can demonstrate theattenuation characteristic of amplitude in terms ofmagnitudein the damping anchorage system but also can illustrate theattenuation characteristic of amplitude in terms of time andits calculating result will be more reliable and reasonable

Generally speaking the finite difference method cannot only be applied to the dynamic response analysis offrame supporting structure under harmonic earthquake but

Mathematical Problems in Engineering 11

t

M

j

1Δt

0(N0) 1 Δl1 N1 Niminus1 Δli k Ni Nnminus1 Δln Nn y

H

A1 Ai An

Low

er p

latfo

rm (s

lope

toe)

w1

1th

row

anch

or

wi(j k) wn

Upp

er p

latfo

rm (s

lope

cres

t)

ith section1th section nth section

ith ro

w an

chor

n minus

1th

row

anch

or

i minus 1

th ro

w an

chor

Figure 17 Finite difference grids of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical methodProposed finite difference method

minus40000

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 18 Solutions of vertical beam with damping effect neglected by different methods

000 025 050 075 100 125 150 175 200 225 250minus15000minus12500minus10000minus7500

minus2500minus5000

0250050007500

10000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

Proposed finite difference method

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical method

minus15000

minus20000

minus5000

minus10000

0

5000

10000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 19 Solutions of vertical beam with damping effect considered by different methods

12 Mathematical Problems in Engineering

also be used to the analysis of random seismic loadingBesides it can demonstrate the attenuation characteristics ofamplitude in terms of both magnitude and time in otherwords it can indicate the damping effect better Thereforethe calculating result obtained by finite difference methodwould be more approximate to actual conditions and hencebe more reliable and reasonable However the analyticalmethod is only confined to the analysis of frame supportingstructure under harmonic earthquake Besides it can onlyreflect the attenuation characteristics of amplitude in termsof magnitude and the peak value of mechanical response gotby analytical method would be underestimated as a resultpotential hazards may be brought aboutWhat is more basedon steady-state complex method its mechanical responseshave initial phase values which are clearly contradictory tothe reality

5 Conclusions

Through the dynamic response analysis of frame supportingstructure of slope under harmonic forced vibration theconclusion follows

(1) Based on certain assumptions and force analysis ofsegment of infinitesimal length the dynamic govern-ing equation of vertical beam under forced vibrationis derived and the analytical solutions to the dynamicresponse of vertical beam under harmonic forcedvibration are obtainedMeanwhile according to finitedifference method the finite difference format andcorresponding calculating procedure for dynamicresponse of vertical beam under forced vibrationare established and finally programmed by usingMATLAB language

(2) In the case studies three methods have been usedin the dynamic response analysis and with dampingeffect neglected or considered respectivelyThe feasi-bility and correctness of the two proposed methodshave been examined and the characteristic of eachmethod has been summarized

(3) Under horizontal harmonic seismic loading themechanical response of vertical beam would syn-chronize correspondingly Its response law showsagreement with the one of two-end cantilevered con-tinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuousbeam In specific engineering design the factors suchas mechanical characteristic of vertical beam andslope supporting effect of anchors should also beconsidered

(4) Compared with analytical solutions the finite differ-ence method can simulate the dynamic response offrame supporting structure under various earthquakeconditions and it also can indicate the damping effectof anchorage system very well hence the calculatingresult obtained ismore reliable and reasonableMore-over the finite differencemethod is achieved based onprogramming calculation its calculation speed and

accuracy can be guaranteed and hence it has broadapplication prospect

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by Project (51308273 41372307) sup-ported by National Natural Science Foundation of China andProject (2010(A)06-b) supported by Science and Technol-ogy Project supported by Yunan Provincial CommunicationDepartment

References

[1] L K Cheng ldquoPresent status and development of groundanchoragesrdquo China Civil Engineering Journal vol 34 no 3 pp7ndash16 2001 (Chinese)

[2] D P Zhou J J Zhang and Y Tang ldquoSeismic damage analysis ofroad slopes in Wenchuan earthquakerdquo Chinese Journal of RockMechanics and Engineering vol 29 no 3 pp 565ndash576 2010(Chinese)

[3] M Yang H T Hu C J Lu et al ldquoCalculation of internal forcesfor prestressed anchor cable frame used in reinforced roadcutsoil sloperdquo Chinese Journal of Rock Mechanics and Engineeringvol 21 no 9 pp 1383ndash1386 2002 (Chinese)

[4] Y H Tian J K Liu and Y F Zhang ldquoInternal force calculationof prestressed anchor cable frame by finite difference methodrdquoJournal of Beijing Jiaotong University vol 31 no 4 pp 22ndash252007 (Chinese)

[5] G H Lin ldquoModel test of anchor grid beam for cut sloperdquoTechnology of Highway and Transport vol 3 pp 19ndash22 2004(Chinese)

[6] B L Zhu M Yang H T Hu et al ldquoTesting study on internalforces for prestressed anchor cable frame in reinforced soilsloperdquo Chinese Journal of Rock Mechanics and Engineering vol24 no 4 pp 697ndash702 2005 (Chinese)

[7] J H Dong Y P Zhu Y Zhou et al ldquoDynamic calculationmodel and seismic response for frame supporting structurewith prestressed anchorsrdquo Science China Technological Sciencesvol 53 no 7 pp 1957ndash1966 2010

[8] S Catal ldquoSolution of free vibration equations of beam onelastic soil by using differential transform methodrdquo AppliedMathematical Modelling vol 32 no 9 pp 1744ndash1757 2008

[9] M Balkaya M O Kaya and A Saglamer ldquoAnalysis of thevibration of an elastic beam supported on elastic soil using thedifferential transform methodrdquo Archive of Applied Mechanicsvol 79 no 2 pp 135ndash146 2009

[10] M Ansari E Esmailzadeh and D Younesian ldquoFrequencyanalysis of finite beams on nonlinear KelvinVoight foundationunder moving loadsrdquo Journal of Sound and Vibration vol 330no 7 pp 1455ndash1471 2011

[11] O Demirdag and Y Yesilce ldquoSolution of free vibration equationof elastically supported Timoshenko columns with a tip massby differential transform methodrdquo Advances in EngineeringSoftware vol 42 no 10 pp 860ndash867 2011

Mathematical Problems in Engineering 13

[12] MH Yas andN Samadi ldquoFree vibrations and buckling analysisof carbon nanotube-reinforced composite Timoshenko beamson elastic foundationrdquo International Journal of Pressure Vesselsand Piping vol 98 pp 119ndash128 2012

[13] R A Jafari-Talookolaei M Abedi M H Kargarnovin et alldquoAn analytical approach for the free vibration analysis ofgenerally laminated composite beams with shear effect androtary inertiardquo International Journal of Mechanical Sciences vol65 no 1 pp 97ndash104 2012

[14] S W Qi F Q Wu F Z Yan et al Dynamic Response Analysesof Rock Slope Science Press Beijing China 2007 Chinese

[15] J H Dong and Y P Zhu ldquoDynamic calculation model andseismic response for the system of soil nailing and surroundingsoilrdquo Chinese Journal of Theoretical and Applied Mechanics vol41 no 2 pp 236ndash242 2009

[16] Z L Xu Elasticity Higher Education Press Beijing China2009 Chinese

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Upper platform (slope crest)

n minus 1th row anchor

ith row anchor

k0

k0

k0Free segment

Anchoring segment

1th row anchor

120573

ag(t) Lower platform (slope toe)

n

i + 1

ui2(Li2

Li2

t)

i

1

o

y

hi

H

Mi +120597Mi

120597ydy

Qi +120597Qi

120597ydy1205972wi

120597t2dm

k0bwidy

sin120573 Qi

Mi

H0

Md

ag(t)dm

dy

Figure 1 Dynamic mechanical model of vertical beam

the demarcation point of vertical beam the displacement 119908119894

rotation angle 120579119894 and bending moment 119872

119894should meet the

following conditions

119908119894

1003816100381610038161003816119910=ℎ119894

= 119908119894+1

1003816100381610038161003816119910=ℎ119894

= 1199061198942(1198711198942 119905) cos120593

119894(119894 = 1 sim 119899 minus 1)

120579119894

1003816100381610038161003816119910=ℎ119894

= 120579119894+1

1003816100381610038161003816119910=ℎ119894

=

120597119908119894

120597119910

10038161003816100381610038161003816100381610038161003816119910=ℎ119894

=

120597119908119894+1

120597119910

10038161003816100381610038161003816100381610038161003816119910=ℎ119894

(119894 = 1 sim 119899 minus 1)

119872119894

1003816100381610038161003816119910=ℎ119894

= 119872119894+1

1003816100381610038161003816119910=ℎ119894

= 119864119868

1205972119908119894

1205971199102

100381610038161003816100381610038161003816100381610038161003816119910=ℎ119894

= 119864119868

1205972119908119894+1

1205971199102

100381610038161003816100381610038161003816100381610038161003816119910=ℎ119894

(119894 = 1 sim 119899 minus 1)

(6)

where 120593119894is the inclined angle of the 119894th row anchor and

1199061198942(1198711198942 119905) is the displacement of 119894th row anchor of which

under horizontal seismic loading 1198861015840

119892(119905) from slope bottom

the specific formula can be found in the literature [15] whichis based on some assumptions of using the Kelvin-Voigtmodel to simulate the interaction between anchoring sectionand the surrounding rock-soil mass the inertial dynamicaction of slope structure which exerts on the free section canbe simplified as the equivalent lumped mass 119872

119889that acts on

the end of free section just as illustrated in the 2th row anchorof Figure 1

3 Analytical Solutions to Vertical Beam underHarmonic Forced Vibration

31 Theoretical Analysis Coordination is considered andsteady-state complex method is used to the derivation ofharmonic forced vibration response of vertical beam It isassumed that the solution to the governing equation (4) is1199081015840

119894(119910 119905) = 119882

119894(119910)119890119894120596119892119905 Then

1198821015840101584010158401015840

119894(119910) + (120572

2minus 12057211205962

119892)119882119894(119910) = 120572

11198861015840

119892120595 (119910 + 119867

0) (7)

Given that 1205741= 120573(1 minus 119894) 120574

2= 120573(1 + 119894) 120574

3= 120573(minus1 + 119894) and

1205744= 120573(minus1 minus 119894) the complex solution to (7) is

119882119894(119910) = 119862

11989411198901205741119910

+ 11986211989421198901205742119910

+ 11986211989431198901205743119910

+ 11986211989441198901205744119910

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

(8)

where119862119894111986211989421198621198943 and119862

1198944are constant complex coefficients

which can be decided by specific boundary conditionLikewise the horizontal displacement bending moment

and shear force of the 119894th section in the vertical beam can beexpressed as

1199081015840

119894(119910 119905)

= 119882119894(119910) 119890119894120596119892119905

= (11986211989411198901205741119910

+ 11986211989421198901205742119910

+ 11986211989431198901205743119910

+ 11986211989441198901205744119910

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

)

times 119890119894120596119892119905

1198721015840

119894(119910 119905)

= 119864119868 (1205742

111986211989411198901205741119910

+ 1205742

211986211989421198901205742119910

+ 1205742

311986211989431198901205743119910

+ 1205742

411986211989441198901205744119910

) 119890119894120596119892119905

1198761015840

119894(119910 119905)

= 119864119868 (1205743

111986211989411198901205741119910

+ 1205743

211986211989421198901205742119910

+ 1205743

311986211989431198901205743119910

+ 1205743

411986211989441198901205744119910

) 119890119894120596119892119905

(9)

Meanwhile according to literature [15] the displacementresponse of the outside endpoint of anchor under horizontalharmonic loading can be simplified as 119906

1015840

1198942(1198711198942 119905) = 120581

119894119890119894120596119892119905

4 Mathematical Problems in Engineering

It is assumed that 119880119894(ℎ119894) = 120581

119894cos120593119894 Substitute them into

boundary condition equations (5) and (6) Then

1205742

111986211

+ 1205742

211986212

+ 1205742

311986213

+ 1205742

411986214

= 0

1205743

111986211

+ 1205743

211986212

+ 1205743

311986213

+ 1205743

411986214

= 0

11986211989411198901205741ℎ119894

+ 11986211989421198901205742ℎ119894

+ 11986211989431198901205743ℎ119894

+ 11986211989441198901205744ℎ119894

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

= 119880119894(ℎ119894)

119862(119894+1)1

1198901205741ℎ119894

+ 119862(119894+1)2

1198901205742ℎ119894

+ 119862(119894+1)3

1198901205743ℎ119894

+ 119862(119894+1)4

1198901205744ℎ119894

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

= 119880119894(ℎ119894)

120574111986211989411198901205741ℎ119894

+ 120574211986211989421198901205742ℎ119894

+ 120574311986211989431198901205743ℎ119894

+ 120574411986211989441198901205744ℎ119894

= 1205741119862(119894+1)1

1198901205741ℎ119894

+ 1205742119862(119894+1)2

1198901205742ℎ119894

+ 1205743119862(119894+1)3

1198901205743ℎ119894

+ 1205744119862(119894+1)4

1198901205744ℎ119894

1205742

111986211989411198901205741ℎ119894

+ 1205742

211986211989421198901205742ℎ119894

+ 1205742

311986211989431198901205743ℎ119894

+ 1205742

411986211989441198901205744ℎ119894

= 1205742

1119862(119894+1)1

1198901205741ℎ119894

+ 1205742

2119862(119894+1)2

1198901205742ℎ119894

+ 1205742

3119862(119894+1)3

1198901205743ℎ119894

+ 1205742

4119862(119894+1)4

1198901205744ℎ119894

1205742

111986211989911198901205741119867

+ 1205742

211986211989921198901205742119867

+ 1205742

311986211989931198901205743119867

+ 1205742

411986211989941198901205744119867

= 0

1205743

111986211989911198901205741119867

+ 1205743

211986211989921198901205742119867

+ 1205743

311986211989931198901205743119867

+ 1205743

411986211989941198901205744119867

= 0

(10)

The above linear equations can be solved by matrixmethod It is assumed that 119860

119894= 1198901205741ℎ119894 119861

119894= 1198901205742ℎ119894 119862

119894= 1198901205743ℎ119894

119863119894= 1198901205744ℎ119894 and 119864

119894= 119880119894(ℎ119894) minus (120572

11198861015840

119892120595(119910 + 119867

0)(1205722minus 12057211205962

119892))

Then (10) can be expressed as matrix format

1198624119899times1

= 119861minus1

4119899times41198991198604119899times1

(11)

where

119860 = [0 0 1198641

1198641

0 0 sdot sdot sdot 119864119894

119864119894

0 0 sdot sdot sdot 119864119899minus1

119864119899minus1

0 0 0 0]

119879

119862 = [11986211

11986212

11986213

11986214

sdot sdot sdot 1198621198941

1198621198942

1198621198943

1198621198944

sdot sdot sdot 1198621198991

1198621198992

1198621198993

1198621198994]

119879

119861 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1205742

11205742

21205742

31205742

40 0 0 0

1205743

11205743

21205743

31205743

40 0 0 0

1198601

1198611

1198621

1198631

0 0 0 0

0 0 0 0 1198601

1198611

1198621

1198631

12057411198601

12057421198611

12057431198621

12057441198631

minus12057411198601

minus12057421198611

minus12057431198621

minus12057441198631

1205742

11198601

1205742

21198611

1205742

31198621

1205742

41198631

minus1205742

11198601

minus1205742

21198611

minus1205742

31198621

minus1205742

41198631

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

d

sdot sdot sdot 119860119894

119861119894

119862119894

119863119894

sdot sdot sdot 0 0 0 0

sdot sdot sdot 1205741119860119894

1205742119861119894

1205743119862119894

1205744119863119894

sdot sdot sdot 1205742

1119860119894

1205742

2119861119894

1205742

3119862119894

1205742

4119863119894

d

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

Mathematical Problems in Engineering 5

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

119860119894

119861119894

119862119894

119863119894

minus1205741119860119894

minus1205742119861119894

minus1205743119862119894

minus1205744119863119894

minus1205742

1119860119894

minus1205742

2119861119894

minus1205742

3119862119894

minus1205742

4119863119894

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

d

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

d

sdot sdot sdot 119860

119899minus1119861119899minus1

119862119899minus1

119863119899minus1

0 0 0 0

sdot sdot sdot 0 0 0 0 119860119899minus1

119861119899minus1

119862119899minus1

119863119899minus1

sdot sdot sdot 1205741119860119899minus1

1205742119861119899minus1

1205743119862119899minus1

1205744119863119899minus1

minus1205741119860119899minus1

minus1205742119861119899minus1

minus1205743119862119899minus1

minus1205744119863119899minus1

sdot sdot sdot 1205742

1119860119899minus1

1205742

2119861119899minus1

1205742

3119862119899minus1

1205742

4119863119899minus1

minus1205742

1119860119899minus1

minus1205742

2119861119899minus1

minus1205742

3119862119899minus1

minus1205742

4119863119899minus1

sdot sdot sdot 0 0 0 0 1205742

1119860119899

1205742

2119861119899

1205742

3119862119899

1205742

4119863119899

sdot sdot sdot 0 0 0 0 1205743

1119860119899

1205743

2119861119899

1205743

3119862119899

1205743

4119863119899

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(12)

After getting constant complex coefficient 1198624119899times1

it isassumed that119862

1198941= 1198881198941+11988811989421198941198621198942

= 1198881198943+11988811989441198941198621198943

= 1198881198945+1198881198946119894 and

1198621198944

= 1198881198947

+ 1198881198948119894 Substituting them into (9) the displacement

can be simplified as

1199081015840

119894(119910 119905)

= (1198881198941119890119910120573

+ 1198881198942119890119910120573

119894)

times [cos (120596119892119905 minus 119910120573) + 119894 sin (120596

119892119905 minus 119910120573)]

+ (1198881198943119890119910120573

+ 1198881198944119890119910120573

119894) [cos (120596119892119905 + 119910120573) + 119894 sin (120596

119892119905 + 119910120573)]

+ (1198881198945119890minus119910120573

+ 1198881198946119890minus119910120573

119894) [cos (120596119892119905 + 119910120573) + 119894 sin (120596

119892119905 + 119910120573)]

+ (1198881198947119890minus119910120573

+ 1198881198948119890minus119910120573

119894) [cos (120596119892119905 minus 119910120573) + 119894 sin (120596

119892119905 minus 119910120573)]

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

(cos120596119892119905 + 119894 sin120596

119892119905)

(13)

According to the calculating principle of complexmethod the corresponding imaginary part is the steady-statesolution to the mechanical response of vertical beam underharmonic seismic loading Then

119908119894(119910 119905)

= 119890120573119910

[1198881198941sin (120596

119892119905 minus 120573119910) + 119888

1198942cos (120596

119892119905 minus 120573119910)

+ 1198881198943sin (120596

119892119905 + 120573119910) + 119888

1198944cos (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[1198881198945sin (120596

119892119905 + 120573119910) + 119888

1198946cos (120596

119892119905 + 120573119910)

+ 1198881198947sin (120596

119892119905 minus 120573119910) + 119888

1198948cos (120596

119892119905 minus 120573119910)]

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

sin120596119892119905

119872119894(119910 119905)

= 21205732119864119868 119890120573119910

[minus1198881198941cos (120596

119892119905 minus 120573119910) + 119888

1198942sin (120596

119892119905 minus 120573119910)

+ 1198881198943cos (120596

119892119905 + 120573119910) minus 119888

1198944sin (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[minus1198881198945cos (120596

119892119905 + 120573119910) + 119888

1198946sin (120596

119892119905 + 120573119910)

+ 1198881198947cos (120596

119892119905 minus 120573119910) minus 119888

1198948sin (120596

119892119905 minus 120573119910)]

119876119894(119910 119905)

= 21205733119864119868 119890120573119910

[(1198881198942minus 1198881198941) sin (120596

119892119905 minus 120573119910)

minus (1198881198941+ 1198881198942) cos (120596

119892119905 minus 120573119910)

minus (1198881198943+ 1198881198944) sin (120596

119892119905 + 120573119910)

+ (1198881198943minus 1198881198944) cos (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[(1198881198945minus 1198881198946) sin (120596

119892119905 + 120573119910)

+ (1198881198945+ 1198881198946) cos (120596

119892119905 + 120573119910)

+ (1198881198947+ 1198881198948) sin (120596

119892119905 minus 120573119910)

+ (1198881198948minus 1198881198947) cos (120596

119892119905 minus 120573119910)]

(14)

6 Mathematical Problems in Engineering

32 Case Studies Take a slope as an example Its height is12m the slope angle is 65∘ and its physical and mechanicalparameters are 120588

119904= 1600 kgsdotmminus3 119864

119904= 012MPa 120583

119904=

03 119888119904

= 20 kPa and 120601119904

= 25∘ The seismic precautionary

intensity in the local region is 8 degrees and the peakground acceleration is 02 gThe dynamic safety factor is 1015calculated by Fellenius method hence its support is neededA preliminary supporting scheme is using fixed end anchorsand frame supporting structure The dynamic safety factoris considered as 12 calculated by the pseudostatic methodand the design scheme is shown in Figure 2 The anchor usescategory II steel bar diameter size is 28mm and its spaceis 33m the dip120593 = 10

∘ four rows of anchors have beenarranged both lengths of anchoring section and free sectionare 5m The mortar grade M30 is used as bonding agent thediameter of anchor hole is 119863 = 130mm The section size ofthe frame beam is 119887timesℎ = 045 times 05m concrete is gradeC30120588 = 2500 kgm3 119864 = 255GPa Earthquake accelerationis 1198861015840

119892(119905) = 119886

1015840

119892sin120596119892119905 = 2 sin 4120587119905 the duration is 25 s and

1198960= 2 times 10

3 kNm3From Figures 3 4 and 5 the displacement bending

moment and shear force response of vertical beam have beengiven when damping effect of anchorage system is neglectedwhile from Figures 6 7 8 9 10 11 12 and 13 correspondingresponse has been given at special sections (intersection andmidspan section) and typical moment (according to momentof acceleration peak) It is found that the displacement bend-ingmoment and shear force response of vertical beamvibratesynchronously with earthquakeThe law of bending momentand shear force response shows agreement with the har-monic forced vibration characteristic of two-end cantileveredcontinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuous beamMorespecifically the bending moment response at intersectionopposite to the one at midspan section and the shear forceresponse show a jumping from the left hand to the righthand at the intersection of vertical beam just as illustratedin Figures 12 and 13 What is more the cantilevered lengthat the two ends of vertical beam has noticeable influenceon the mechanical response of frame supporting structureThe larger the cantilevered length is the more the bendingmoment of intersection is and the greater the difference ofbending moment between intersection and midspan sectionis Examining the frame supporting structure independentlythe bending moment can keep good balance for the wholevertical beamwhen its cantilevered length is 02 or 025 timesthe anchor adjacent spacing referring to the design expe-rience of two-end cantilevered continuous beam Howeverthe cantilevered length of vertical beam is still related to thearranging scheme of anchors actually the length even decidesthe arrangement of anchors and directly influences the slopesupporting effect of anchors Hence in specific engineeringdesign the determination of cantilevered length should alsoconsider the factors such as mechanical characteristic ofvertical beam and slope supporting effect of anchors

Just as shown from Figures 14 15 and 16 the dis-placement bending moment and shear force response ofvertical beam with damping effect considered have been

given as well The overall trend is similar to the previous onewith damping effect neglected When considering anchoragesystem damping effect the solutions can be obtained byusing steady-state complex method and the peak valuesof the displacement bending moment and shear force aresmaller than the ones with damping effect neglected Eventhough the response shows some attenuation characteristicof amplitude in terms of magnitude it does not demonstratethe attenuation characteristic of amplitude in terms of timeWhat is more the mechanical responses show certain initialphases which can be attributed to the dynamic behaviors ofvertical beam under steady-state vibration with damping

4 Finite Difference Method for Vertical Beamunder Forced Vibration

41 Finite Difference Format for Dynamic Response AnalysisThe numerical solutions to dynamic governing equation (4)could be also achieved by using finite difference methodAccording to the basic principles of finite difference method[16] the research area119860(0 le 119905 le 119879 0 le 119910 le 119867) can be dividedinto 119899 subregions119860

119894Thewhole region of119860

119894can be redivided

into many grids 119860119894Δ119905Δ1198971

(119872119873119894minus119873119894minus1

+ 1) which is shown inFigure 17 Meanwhile in order to guarantee the convergenceand stability of calculation the ratio of the grids should meetthe following condition 119903

119894= Δ119905Δ

2119897119894le radic120572

1(119894 = 1 sim 119899)

The governing equation (4) can be expanded accordingto central difference formula and the difference format ofdynamic equation for grid node 119908

119894(119895 119896) can be obtained

(6119908119894(119895 119896) minus 4 [119908

119894(119895 119896 + 1) + 119908

119894(119895 119896 minus 1)]

+119908119894(119895 119896 + 2) + 119908

119894(119895 119896 minus 2)) (Δ119897

4

119894)

minus1

+ 1205721

119908119894(119895 + 1 119896) + 119908

119894(119895 minus 1 119896) minus 2119908

119894(119895 119896)

Δ1199052

+ 1205722119908119894(119895 119896) = 120572

1120595119894(119896) 1198861015840

119892[119895Δ119905]

(15)

where 120595119894(119896) = 120595[(119896 minus119873

119894minus1)Δ119897119894+sum119911=119894minus1

119911=1Δ119897119911(119873119911minus119873119911minus1

) +1198670]

Likewise the boundary equations (5) and (6) can beexpanded by endpoint derivative differential formula and thecorresponding differential equation can be attained

119864119868

21199081(119895 0) minus 5119908

1(119895 1) + 4119908

1(119895 2) minus 119908

1(119895 3)

Δ1198972

1

= 0

119864119868

minus1199081(119895 0) + 3119908

1(119895 1) minus 3119908

1(119895 2) + 119908

1(119895 3)

Δ1198973

1

= 0

119864119868 (2119908119899(119895119873119899) minus 5119908

119899(119895119873119899minus 1)

+4119908119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)) (Δ119897

2

119899)

minus1

= 0

119864119868 (119908119899(119895119873119899) minus 3119908

119899(119895119873119899minus 1)

+ 3119908119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)) (Δ119897

3

119899)

minus1

= 0

119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

Mathematical Problems in Engineering 7

5050

50

5050

50

Free section

Anchoring section

Anchor

5050

15

15

120

045 045

165 16533 33

Grass

Anchor

Horizontal beam Vertical beam

33

30

30

30

65∘

Figure 2 Design scheme for slope (unit m)

02

46

810

12

005115225minus003minus002minus001

0

002001

003

w (y

t)

(m)

t (s) y(m)2

46

810

11525y(m)

Figure 3 Displacement response of vertical beam

times104

3

minus3

2

minus2

1

252

151

050 0

24

6y(m)

810

12

minus10

M(yt)(

Nmiddotm

)

t(s)

Figure 4 Bending moment response of vertical beam

times104

4

minus4

2

252

151

050 0

24

68

1012minus2

0

Q(yt)(

N)

y(m)t(s)

Figure 5 Shear force response of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Mi(

Nmiddotm

)

y = 15my = 45m

y = 75my = 105m

t(s)

Figure 6 Bending moment response at intersection of verticalbeam

2119908119894(119895119873119894) minus 3119908

119894(119895119873119894minus 1) + 119908

119894(119895119873119894minus 2)

Δ119897119894

=

minus2119908119894+1

(119895119873119894) + 3119908

119894+1(119895119873119894+ 1) minus 119908

119894+1(119895119873119894+ 2)

Δ119897119894+1

8 Mathematical Problems in Engineering

minus12000minus10000minus8000minus6000minus4000minus2000

02000400060008000

100001200014000

000 025 050 075 100 125 150 175 200 225 250

y = 3my = 6my = 9m

Mi(

Nmiddotm

)

t(s)

Figure 7 Bending moment response at midspan section of verticalbeam

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 8 Shear force response at left hand of intersection of verticalbeam

(2119908119894(119895119873119894) minus 5119908

119894(119895119873119894minus 1)

+ 4119908119894(119895119873119894minus 2) minus 119908

119894(119895119873119894minus 3)) (Δ119897

2

119894)

minus1

= (2119908119894+1

(119895119873119894) minus 5119908

119894+1(119895119873119894+ 1)

+ 4119908119894+1

(119895119873119894+ 2) minus 119908

119894+1(119895119873119894+ 3)) (Δ119897

2

119894+1)

minus1

(16)

The displacement response of 119894th column grid nodes shouldalso satisfy the following initial conditions

119908119894[0 119896] = 0

120597119908119894

120597119905

10038161003816100381610038161003816100381610038161003816119905=0

=

119908119894[1 119896] minus 119908

119894[0 119896]

Δ119905

= minus2V0

(17)

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 9 Shear force response at right hand of intersection ofvertical beam

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

000 025 050 075 100 125 150 175 200 225 250minus4000

Qi(

N)

y = 3my = 6my = 9m

t(s)

Figure 10 Shear force response atmidspan section of vertical beam

In (17) V0is the initial velocity of vertical beam If the

harmonic vibration is needed then the initial velocity can bedefined as V

0= minus119886119892120596119892

Equations (15)ndash(17) can be simplified and the finite dif-ference format for the dynamic response of vertical beam isshown as

119908119894[0 119896] = 0

119908119894[1 119896] = minus2Δ119905V

0

(18)

119908119894(119895 + 1 119896)

= (2 minus

6Δ1199052

1205721Δ1198974

119894

minus

1205722Δ1199052

1205721

)119908119894(119895 119896)

+

Δ1199052

1205721Δ1198974

119894

[4119908119894(119895 119896 + 1) + 4119908

119894(119895 119896 minus 1)

Mathematical Problems in Engineering 9

00 15 30 45 60 75 90 105 120minus003

minus002

minus001

000

001

002

003wi(

m)

t = 1125 st = 1375 s

y(m)

Figure 11 Displacement response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120minus30000

minus20000

minus10000

0

10000

20000

30000

t = 1125 st = 1375 s

Mi(

Nmiddotm

)

y(m)

Figure 12 Bending moment response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120

minus30000

minus40000

minus20000

minus10000

0

10000

20000

30000

40000

t = 1125 st = 1375 s

Qi(

N)

y(m)

Figure 13 Shear force response along vertical beam at typicalmoment

times3

minus3

2

minus2

1

25 2 15 1 05 0 02

46

810

12minus10

w(yt)(

m)

y(m)

t(s)

10minus3

Figure 14 Displace response of vertical beam

minus 119908119894(119895 119896 + 2) minus 119908

119894(119895 119896 minus 2)]

minus 119908119894(119895 minus 1 119896) + Δ119905

2120595119894(119896) 1198861015840

119892(119895Δ119905)

(19)1199081(119895 0) = 3119908

1(119895 2) minus 2119908

1(119895 3)

1199081(119895 1) = 2119908

1(119895 2) minus 119908

1(119895 3)

119908119899(119895119873119899minus 1) = 2119908

119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)

119908119899(119895119873119899) = 3119908

119899(119895119873119899minus 2) minus 2119908

119899(119895119873119899minus 3)

(20)119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

119908119894+1

(119895119873119894+ 1)

= ([

6Δ1198972

119894

Δ1198972

119894+1

+

10 (Δ119897119894+ Δ119897119894+1

)

Δ119897119894+1

minus 6]119908119894(119895119873119894)

minus 7119908119894(119895119873119894minus 2) + 3119908

119894(119895119873119894minus 3)

+ [

12Δ1198972

119894

Δ1198972

119894+1

+

5Δ119897119894

Δ119897119894+1

]119908119894+1

(119895119873119894+ 2)

minus

3Δ1198972

119894

Δ1198972

119894+1

119908119894+1

(119895119873119894+ 3))

times (15 [

Δ119897119894

Δ119897119894+1

+

Δ1198972

119894

Δ1198972

119894+1

])

minus1

119908119894(119895119873119894minus 1)

=

2 (Δ119897119894+ Δ119897119894+1

)

3Δ119897119894+1

119908119894(119895119873119894) +

1

3

119908119894(119895119873119894minus 2)

minus

Δ119897119894

Δ119897119894+1

119908119894+1

(119895119873119894+ 1) +

Δ119897119894

3Δ119897119894+1

119908119894+1

(119895119873119894+ 2)

(21)

According to the finite difference method mentionedabove for a specific engineering problem the displacementof grid points in the initial and the first row can be calculatedby (18) For the second row the displacement of all thegrid nodes in interior columns can be decided by (19) thegrid nodes in exterior boundary columns and their adjacentcolumns (columns 0 1119873

119899minus1 and119873

119899) can be decided by (20)

and based on (21) the node displacement of all the interiorboundary adjacent columns can be achieved Likewise the

10 Mathematical Problems in Engineering

30002000

minus2000minus3000

1000

25 215 1

050 0

24

68

1012minus1000

0

y(m)

M(yt)(

Nmiddotm

)t(s)

Figure 15 Bending moment response of vertical beam

4000

minus4000

2000

252

151

050 0

24

68

1012

minus20000

Q(yt)(

N)

y(m)

t(s)

Figure 16 Shear force response of vertical beam

displacement for all the grid nodes from row 3 to row119872 canbe decided

After getting the displacement of all grid nodes accordingto 119872119894= 119864119868(120597

21199081198941205971199102) and 119876

119894= 119864119868(120597

31199081198941205971199103) the numerical

solutions to the bending moment and shear force responseof vertical beam can be obtained by conducting the finitedifference to the displacement of all grid nodes again

42 Comparative Analysis and Discussion Similarly basedon previous case study the slope is considered under hori-zontal harmonic seismic loading and with anchoring systemdamping neglected or considered respectively The numer-ical solutions to the dynamic response of vertical beam canbe achieved by using finite difference method and MATLABprogramming and 119903

119894= Δ119905Δ

2119897119894= 000001(025 times 025) =

000016 le radic1205721= 0002 (119894 = 1 sim 5)

In order to verify the correctness of the two methodsabove the solutions of intersection 119910 = 15m by differentmethods andwith damping effect neglected or considered areshown in Figures 18 and 19

When damping effect is neglected according to Figure 18it is found that the calculation results by the three methodsshare similar vibration trend and only show slight differencein amplitude generally speaking the finite element solutionmentioned previously is the smallest due to the size of finitedifference grids and it achieved by method of Literature [7] issecond because of the calculated amount of high order vibra-tionmode and the theoretical solutionmentioned previouslyis the largest and the most reasonable but the accuracyof two previous methods can be respectively improved bydecreasing grid ratio and increasing the calculated amount

so that the results of the three methods show more excellentagreement which indicates the reasonability and feasibilityof two proposed methods under the condition of neglectingdamping effect

When damping effect is considered as shown inFigure 19 the proposed theoretical solution and literature [7]solution are also well consistent and the former is slightlylarger than the latter Through the comparison and analysisof Figures 14 15 16 and 19 it is found that when usingthe finite difference method the attenuation characteristicof vertical beam response can be clearly shown in the firstsecond and then with the passing of time the vibrationmode is gradually degenerated into constant state whileno attenuation characteristic is shown and the calculatingresult keeps steady state all the time if using the theoreticalmethods So in the first second the calculating results usedthe finite difference method and theoretical methods haveconsiderable difference which is attributed to the theoreticalsolution based on steady-state complex method Specificallythe calculating result using finite difference method is largerthan the result using theoretical methods in the first secondAfter the first second the vibration pattern and the ampli-tude are consistent by using these three methods Besidesthe finite difference method not only can demonstrate theattenuation characteristic of amplitude in terms ofmagnitudein the damping anchorage system but also can illustrate theattenuation characteristic of amplitude in terms of time andits calculating result will be more reliable and reasonable

Generally speaking the finite difference method cannot only be applied to the dynamic response analysis offrame supporting structure under harmonic earthquake but

Mathematical Problems in Engineering 11

t

M

j

1Δt

0(N0) 1 Δl1 N1 Niminus1 Δli k Ni Nnminus1 Δln Nn y

H

A1 Ai An

Low

er p

latfo

rm (s

lope

toe)

w1

1th

row

anch

or

wi(j k) wn

Upp

er p

latfo

rm (s

lope

cres

t)

ith section1th section nth section

ith ro

w an

chor

n minus

1th

row

anch

or

i minus 1

th ro

w an

chor

Figure 17 Finite difference grids of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical methodProposed finite difference method

minus40000

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 18 Solutions of vertical beam with damping effect neglected by different methods

000 025 050 075 100 125 150 175 200 225 250minus15000minus12500minus10000minus7500

minus2500minus5000

0250050007500

10000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

Proposed finite difference method

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical method

minus15000

minus20000

minus5000

minus10000

0

5000

10000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 19 Solutions of vertical beam with damping effect considered by different methods

12 Mathematical Problems in Engineering

also be used to the analysis of random seismic loadingBesides it can demonstrate the attenuation characteristics ofamplitude in terms of both magnitude and time in otherwords it can indicate the damping effect better Thereforethe calculating result obtained by finite difference methodwould be more approximate to actual conditions and hencebe more reliable and reasonable However the analyticalmethod is only confined to the analysis of frame supportingstructure under harmonic earthquake Besides it can onlyreflect the attenuation characteristics of amplitude in termsof magnitude and the peak value of mechanical response gotby analytical method would be underestimated as a resultpotential hazards may be brought aboutWhat is more basedon steady-state complex method its mechanical responseshave initial phase values which are clearly contradictory tothe reality

5 Conclusions

Through the dynamic response analysis of frame supportingstructure of slope under harmonic forced vibration theconclusion follows

(1) Based on certain assumptions and force analysis ofsegment of infinitesimal length the dynamic govern-ing equation of vertical beam under forced vibrationis derived and the analytical solutions to the dynamicresponse of vertical beam under harmonic forcedvibration are obtainedMeanwhile according to finitedifference method the finite difference format andcorresponding calculating procedure for dynamicresponse of vertical beam under forced vibrationare established and finally programmed by usingMATLAB language

(2) In the case studies three methods have been usedin the dynamic response analysis and with dampingeffect neglected or considered respectivelyThe feasi-bility and correctness of the two proposed methodshave been examined and the characteristic of eachmethod has been summarized

(3) Under horizontal harmonic seismic loading themechanical response of vertical beam would syn-chronize correspondingly Its response law showsagreement with the one of two-end cantilevered con-tinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuousbeam In specific engineering design the factors suchas mechanical characteristic of vertical beam andslope supporting effect of anchors should also beconsidered

(4) Compared with analytical solutions the finite differ-ence method can simulate the dynamic response offrame supporting structure under various earthquakeconditions and it also can indicate the damping effectof anchorage system very well hence the calculatingresult obtained ismore reliable and reasonableMore-over the finite differencemethod is achieved based onprogramming calculation its calculation speed and

accuracy can be guaranteed and hence it has broadapplication prospect

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by Project (51308273 41372307) sup-ported by National Natural Science Foundation of China andProject (2010(A)06-b) supported by Science and Technol-ogy Project supported by Yunan Provincial CommunicationDepartment

References

[1] L K Cheng ldquoPresent status and development of groundanchoragesrdquo China Civil Engineering Journal vol 34 no 3 pp7ndash16 2001 (Chinese)

[2] D P Zhou J J Zhang and Y Tang ldquoSeismic damage analysis ofroad slopes in Wenchuan earthquakerdquo Chinese Journal of RockMechanics and Engineering vol 29 no 3 pp 565ndash576 2010(Chinese)

[3] M Yang H T Hu C J Lu et al ldquoCalculation of internal forcesfor prestressed anchor cable frame used in reinforced roadcutsoil sloperdquo Chinese Journal of Rock Mechanics and Engineeringvol 21 no 9 pp 1383ndash1386 2002 (Chinese)

[4] Y H Tian J K Liu and Y F Zhang ldquoInternal force calculationof prestressed anchor cable frame by finite difference methodrdquoJournal of Beijing Jiaotong University vol 31 no 4 pp 22ndash252007 (Chinese)

[5] G H Lin ldquoModel test of anchor grid beam for cut sloperdquoTechnology of Highway and Transport vol 3 pp 19ndash22 2004(Chinese)

[6] B L Zhu M Yang H T Hu et al ldquoTesting study on internalforces for prestressed anchor cable frame in reinforced soilsloperdquo Chinese Journal of Rock Mechanics and Engineering vol24 no 4 pp 697ndash702 2005 (Chinese)

[7] J H Dong Y P Zhu Y Zhou et al ldquoDynamic calculationmodel and seismic response for frame supporting structurewith prestressed anchorsrdquo Science China Technological Sciencesvol 53 no 7 pp 1957ndash1966 2010

[8] S Catal ldquoSolution of free vibration equations of beam onelastic soil by using differential transform methodrdquo AppliedMathematical Modelling vol 32 no 9 pp 1744ndash1757 2008

[9] M Balkaya M O Kaya and A Saglamer ldquoAnalysis of thevibration of an elastic beam supported on elastic soil using thedifferential transform methodrdquo Archive of Applied Mechanicsvol 79 no 2 pp 135ndash146 2009

[10] M Ansari E Esmailzadeh and D Younesian ldquoFrequencyanalysis of finite beams on nonlinear KelvinVoight foundationunder moving loadsrdquo Journal of Sound and Vibration vol 330no 7 pp 1455ndash1471 2011

[11] O Demirdag and Y Yesilce ldquoSolution of free vibration equationof elastically supported Timoshenko columns with a tip massby differential transform methodrdquo Advances in EngineeringSoftware vol 42 no 10 pp 860ndash867 2011

Mathematical Problems in Engineering 13

[12] MH Yas andN Samadi ldquoFree vibrations and buckling analysisof carbon nanotube-reinforced composite Timoshenko beamson elastic foundationrdquo International Journal of Pressure Vesselsand Piping vol 98 pp 119ndash128 2012

[13] R A Jafari-Talookolaei M Abedi M H Kargarnovin et alldquoAn analytical approach for the free vibration analysis ofgenerally laminated composite beams with shear effect androtary inertiardquo International Journal of Mechanical Sciences vol65 no 1 pp 97ndash104 2012

[14] S W Qi F Q Wu F Z Yan et al Dynamic Response Analysesof Rock Slope Science Press Beijing China 2007 Chinese

[15] J H Dong and Y P Zhu ldquoDynamic calculation model andseismic response for the system of soil nailing and surroundingsoilrdquo Chinese Journal of Theoretical and Applied Mechanics vol41 no 2 pp 236ndash242 2009

[16] Z L Xu Elasticity Higher Education Press Beijing China2009 Chinese

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

It is assumed that 119880119894(ℎ119894) = 120581

119894cos120593119894 Substitute them into

boundary condition equations (5) and (6) Then

1205742

111986211

+ 1205742

211986212

+ 1205742

311986213

+ 1205742

411986214

= 0

1205743

111986211

+ 1205743

211986212

+ 1205743

311986213

+ 1205743

411986214

= 0

11986211989411198901205741ℎ119894

+ 11986211989421198901205742ℎ119894

+ 11986211989431198901205743ℎ119894

+ 11986211989441198901205744ℎ119894

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

= 119880119894(ℎ119894)

119862(119894+1)1

1198901205741ℎ119894

+ 119862(119894+1)2

1198901205742ℎ119894

+ 119862(119894+1)3

1198901205743ℎ119894

+ 119862(119894+1)4

1198901205744ℎ119894

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

= 119880119894(ℎ119894)

120574111986211989411198901205741ℎ119894

+ 120574211986211989421198901205742ℎ119894

+ 120574311986211989431198901205743ℎ119894

+ 120574411986211989441198901205744ℎ119894

= 1205741119862(119894+1)1

1198901205741ℎ119894

+ 1205742119862(119894+1)2

1198901205742ℎ119894

+ 1205743119862(119894+1)3

1198901205743ℎ119894

+ 1205744119862(119894+1)4

1198901205744ℎ119894

1205742

111986211989411198901205741ℎ119894

+ 1205742

211986211989421198901205742ℎ119894

+ 1205742

311986211989431198901205743ℎ119894

+ 1205742

411986211989441198901205744ℎ119894

= 1205742

1119862(119894+1)1

1198901205741ℎ119894

+ 1205742

2119862(119894+1)2

1198901205742ℎ119894

+ 1205742

3119862(119894+1)3

1198901205743ℎ119894

+ 1205742

4119862(119894+1)4

1198901205744ℎ119894

1205742

111986211989911198901205741119867

+ 1205742

211986211989921198901205742119867

+ 1205742

311986211989931198901205743119867

+ 1205742

411986211989941198901205744119867

= 0

1205743

111986211989911198901205741119867

+ 1205743

211986211989921198901205742119867

+ 1205743

311986211989931198901205743119867

+ 1205743

411986211989941198901205744119867

= 0

(10)

The above linear equations can be solved by matrixmethod It is assumed that 119860

119894= 1198901205741ℎ119894 119861

119894= 1198901205742ℎ119894 119862

119894= 1198901205743ℎ119894

119863119894= 1198901205744ℎ119894 and 119864

119894= 119880119894(ℎ119894) minus (120572

11198861015840

119892120595(119910 + 119867

0)(1205722minus 12057211205962

119892))

Then (10) can be expressed as matrix format

1198624119899times1

= 119861minus1

4119899times41198991198604119899times1

(11)

where

119860 = [0 0 1198641

1198641

0 0 sdot sdot sdot 119864119894

119864119894

0 0 sdot sdot sdot 119864119899minus1

119864119899minus1

0 0 0 0]

119879

119862 = [11986211

11986212

11986213

11986214

sdot sdot sdot 1198621198941

1198621198942

1198621198943

1198621198944

sdot sdot sdot 1198621198991

1198621198992

1198621198993

1198621198994]

119879

119861 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1205742

11205742

21205742

31205742

40 0 0 0

1205743

11205743

21205743

31205743

40 0 0 0

1198601

1198611

1198621

1198631

0 0 0 0

0 0 0 0 1198601

1198611

1198621

1198631

12057411198601

12057421198611

12057431198621

12057441198631

minus12057411198601

minus12057421198611

minus12057431198621

minus12057441198631

1205742

11198601

1205742

21198611

1205742

31198621

1205742

41198631

minus1205742

11198601

minus1205742

21198611

minus1205742

31198621

minus1205742

41198631

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

d

sdot sdot sdot 119860119894

119861119894

119862119894

119863119894

sdot sdot sdot 0 0 0 0

sdot sdot sdot 1205741119860119894

1205742119861119894

1205743119862119894

1205744119863119894

sdot sdot sdot 1205742

1119860119894

1205742

2119861119894

1205742

3119862119894

1205742

4119863119894

d

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

sdot sdot sdot 0 0 0 0

Mathematical Problems in Engineering 5

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

119860119894

119861119894

119862119894

119863119894

minus1205741119860119894

minus1205742119861119894

minus1205743119862119894

minus1205744119863119894

minus1205742

1119860119894

minus1205742

2119861119894

minus1205742

3119862119894

minus1205742

4119863119894

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

d

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

d

sdot sdot sdot 119860

119899minus1119861119899minus1

119862119899minus1

119863119899minus1

0 0 0 0

sdot sdot sdot 0 0 0 0 119860119899minus1

119861119899minus1

119862119899minus1

119863119899minus1

sdot sdot sdot 1205741119860119899minus1

1205742119861119899minus1

1205743119862119899minus1

1205744119863119899minus1

minus1205741119860119899minus1

minus1205742119861119899minus1

minus1205743119862119899minus1

minus1205744119863119899minus1

sdot sdot sdot 1205742

1119860119899minus1

1205742

2119861119899minus1

1205742

3119862119899minus1

1205742

4119863119899minus1

minus1205742

1119860119899minus1

minus1205742

2119861119899minus1

minus1205742

3119862119899minus1

minus1205742

4119863119899minus1

sdot sdot sdot 0 0 0 0 1205742

1119860119899

1205742

2119861119899

1205742

3119862119899

1205742

4119863119899

sdot sdot sdot 0 0 0 0 1205743

1119860119899

1205743

2119861119899

1205743

3119862119899

1205743

4119863119899

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(12)

After getting constant complex coefficient 1198624119899times1

it isassumed that119862

1198941= 1198881198941+11988811989421198941198621198942

= 1198881198943+11988811989441198941198621198943

= 1198881198945+1198881198946119894 and

1198621198944

= 1198881198947

+ 1198881198948119894 Substituting them into (9) the displacement

can be simplified as

1199081015840

119894(119910 119905)

= (1198881198941119890119910120573

+ 1198881198942119890119910120573

119894)

times [cos (120596119892119905 minus 119910120573) + 119894 sin (120596

119892119905 minus 119910120573)]

+ (1198881198943119890119910120573

+ 1198881198944119890119910120573

119894) [cos (120596119892119905 + 119910120573) + 119894 sin (120596

119892119905 + 119910120573)]

+ (1198881198945119890minus119910120573

+ 1198881198946119890minus119910120573

119894) [cos (120596119892119905 + 119910120573) + 119894 sin (120596

119892119905 + 119910120573)]

+ (1198881198947119890minus119910120573

+ 1198881198948119890minus119910120573

119894) [cos (120596119892119905 minus 119910120573) + 119894 sin (120596

119892119905 minus 119910120573)]

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

(cos120596119892119905 + 119894 sin120596

119892119905)

(13)

According to the calculating principle of complexmethod the corresponding imaginary part is the steady-statesolution to the mechanical response of vertical beam underharmonic seismic loading Then

119908119894(119910 119905)

= 119890120573119910

[1198881198941sin (120596

119892119905 minus 120573119910) + 119888

1198942cos (120596

119892119905 minus 120573119910)

+ 1198881198943sin (120596

119892119905 + 120573119910) + 119888

1198944cos (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[1198881198945sin (120596

119892119905 + 120573119910) + 119888

1198946cos (120596

119892119905 + 120573119910)

+ 1198881198947sin (120596

119892119905 minus 120573119910) + 119888

1198948cos (120596

119892119905 minus 120573119910)]

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

sin120596119892119905

119872119894(119910 119905)

= 21205732119864119868 119890120573119910

[minus1198881198941cos (120596

119892119905 minus 120573119910) + 119888

1198942sin (120596

119892119905 minus 120573119910)

+ 1198881198943cos (120596

119892119905 + 120573119910) minus 119888

1198944sin (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[minus1198881198945cos (120596

119892119905 + 120573119910) + 119888

1198946sin (120596

119892119905 + 120573119910)

+ 1198881198947cos (120596

119892119905 minus 120573119910) minus 119888

1198948sin (120596

119892119905 minus 120573119910)]

119876119894(119910 119905)

= 21205733119864119868 119890120573119910

[(1198881198942minus 1198881198941) sin (120596

119892119905 minus 120573119910)

minus (1198881198941+ 1198881198942) cos (120596

119892119905 minus 120573119910)

minus (1198881198943+ 1198881198944) sin (120596

119892119905 + 120573119910)

+ (1198881198943minus 1198881198944) cos (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[(1198881198945minus 1198881198946) sin (120596

119892119905 + 120573119910)

+ (1198881198945+ 1198881198946) cos (120596

119892119905 + 120573119910)

+ (1198881198947+ 1198881198948) sin (120596

119892119905 minus 120573119910)

+ (1198881198948minus 1198881198947) cos (120596

119892119905 minus 120573119910)]

(14)

6 Mathematical Problems in Engineering

32 Case Studies Take a slope as an example Its height is12m the slope angle is 65∘ and its physical and mechanicalparameters are 120588

119904= 1600 kgsdotmminus3 119864

119904= 012MPa 120583

119904=

03 119888119904

= 20 kPa and 120601119904

= 25∘ The seismic precautionary

intensity in the local region is 8 degrees and the peakground acceleration is 02 gThe dynamic safety factor is 1015calculated by Fellenius method hence its support is neededA preliminary supporting scheme is using fixed end anchorsand frame supporting structure The dynamic safety factoris considered as 12 calculated by the pseudostatic methodand the design scheme is shown in Figure 2 The anchor usescategory II steel bar diameter size is 28mm and its spaceis 33m the dip120593 = 10

∘ four rows of anchors have beenarranged both lengths of anchoring section and free sectionare 5m The mortar grade M30 is used as bonding agent thediameter of anchor hole is 119863 = 130mm The section size ofthe frame beam is 119887timesℎ = 045 times 05m concrete is gradeC30120588 = 2500 kgm3 119864 = 255GPa Earthquake accelerationis 1198861015840

119892(119905) = 119886

1015840

119892sin120596119892119905 = 2 sin 4120587119905 the duration is 25 s and

1198960= 2 times 10

3 kNm3From Figures 3 4 and 5 the displacement bending

moment and shear force response of vertical beam have beengiven when damping effect of anchorage system is neglectedwhile from Figures 6 7 8 9 10 11 12 and 13 correspondingresponse has been given at special sections (intersection andmidspan section) and typical moment (according to momentof acceleration peak) It is found that the displacement bend-ingmoment and shear force response of vertical beamvibratesynchronously with earthquakeThe law of bending momentand shear force response shows agreement with the har-monic forced vibration characteristic of two-end cantileveredcontinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuous beamMorespecifically the bending moment response at intersectionopposite to the one at midspan section and the shear forceresponse show a jumping from the left hand to the righthand at the intersection of vertical beam just as illustratedin Figures 12 and 13 What is more the cantilevered lengthat the two ends of vertical beam has noticeable influenceon the mechanical response of frame supporting structureThe larger the cantilevered length is the more the bendingmoment of intersection is and the greater the difference ofbending moment between intersection and midspan sectionis Examining the frame supporting structure independentlythe bending moment can keep good balance for the wholevertical beamwhen its cantilevered length is 02 or 025 timesthe anchor adjacent spacing referring to the design expe-rience of two-end cantilevered continuous beam Howeverthe cantilevered length of vertical beam is still related to thearranging scheme of anchors actually the length even decidesthe arrangement of anchors and directly influences the slopesupporting effect of anchors Hence in specific engineeringdesign the determination of cantilevered length should alsoconsider the factors such as mechanical characteristic ofvertical beam and slope supporting effect of anchors

Just as shown from Figures 14 15 and 16 the dis-placement bending moment and shear force response ofvertical beam with damping effect considered have been

given as well The overall trend is similar to the previous onewith damping effect neglected When considering anchoragesystem damping effect the solutions can be obtained byusing steady-state complex method and the peak valuesof the displacement bending moment and shear force aresmaller than the ones with damping effect neglected Eventhough the response shows some attenuation characteristicof amplitude in terms of magnitude it does not demonstratethe attenuation characteristic of amplitude in terms of timeWhat is more the mechanical responses show certain initialphases which can be attributed to the dynamic behaviors ofvertical beam under steady-state vibration with damping

4 Finite Difference Method for Vertical Beamunder Forced Vibration

41 Finite Difference Format for Dynamic Response AnalysisThe numerical solutions to dynamic governing equation (4)could be also achieved by using finite difference methodAccording to the basic principles of finite difference method[16] the research area119860(0 le 119905 le 119879 0 le 119910 le 119867) can be dividedinto 119899 subregions119860

119894Thewhole region of119860

119894can be redivided

into many grids 119860119894Δ119905Δ1198971

(119872119873119894minus119873119894minus1

+ 1) which is shown inFigure 17 Meanwhile in order to guarantee the convergenceand stability of calculation the ratio of the grids should meetthe following condition 119903

119894= Δ119905Δ

2119897119894le radic120572

1(119894 = 1 sim 119899)

The governing equation (4) can be expanded accordingto central difference formula and the difference format ofdynamic equation for grid node 119908

119894(119895 119896) can be obtained

(6119908119894(119895 119896) minus 4 [119908

119894(119895 119896 + 1) + 119908

119894(119895 119896 minus 1)]

+119908119894(119895 119896 + 2) + 119908

119894(119895 119896 minus 2)) (Δ119897

4

119894)

minus1

+ 1205721

119908119894(119895 + 1 119896) + 119908

119894(119895 minus 1 119896) minus 2119908

119894(119895 119896)

Δ1199052

+ 1205722119908119894(119895 119896) = 120572

1120595119894(119896) 1198861015840

119892[119895Δ119905]

(15)

where 120595119894(119896) = 120595[(119896 minus119873

119894minus1)Δ119897119894+sum119911=119894minus1

119911=1Δ119897119911(119873119911minus119873119911minus1

) +1198670]

Likewise the boundary equations (5) and (6) can beexpanded by endpoint derivative differential formula and thecorresponding differential equation can be attained

119864119868

21199081(119895 0) minus 5119908

1(119895 1) + 4119908

1(119895 2) minus 119908

1(119895 3)

Δ1198972

1

= 0

119864119868

minus1199081(119895 0) + 3119908

1(119895 1) minus 3119908

1(119895 2) + 119908

1(119895 3)

Δ1198973

1

= 0

119864119868 (2119908119899(119895119873119899) minus 5119908

119899(119895119873119899minus 1)

+4119908119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)) (Δ119897

2

119899)

minus1

= 0

119864119868 (119908119899(119895119873119899) minus 3119908

119899(119895119873119899minus 1)

+ 3119908119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)) (Δ119897

3

119899)

minus1

= 0

119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

Mathematical Problems in Engineering 7

5050

50

5050

50

Free section

Anchoring section

Anchor

5050

15

15

120

045 045

165 16533 33

Grass

Anchor

Horizontal beam Vertical beam

33

30

30

30

65∘

Figure 2 Design scheme for slope (unit m)

02

46

810

12

005115225minus003minus002minus001

0

002001

003

w (y

t)

(m)

t (s) y(m)2

46

810

11525y(m)

Figure 3 Displacement response of vertical beam

times104

3

minus3

2

minus2

1

252

151

050 0

24

6y(m)

810

12

minus10

M(yt)(

Nmiddotm

)

t(s)

Figure 4 Bending moment response of vertical beam

times104

4

minus4

2

252

151

050 0

24

68

1012minus2

0

Q(yt)(

N)

y(m)t(s)

Figure 5 Shear force response of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Mi(

Nmiddotm

)

y = 15my = 45m

y = 75my = 105m

t(s)

Figure 6 Bending moment response at intersection of verticalbeam

2119908119894(119895119873119894) minus 3119908

119894(119895119873119894minus 1) + 119908

119894(119895119873119894minus 2)

Δ119897119894

=

minus2119908119894+1

(119895119873119894) + 3119908

119894+1(119895119873119894+ 1) minus 119908

119894+1(119895119873119894+ 2)

Δ119897119894+1

8 Mathematical Problems in Engineering

minus12000minus10000minus8000minus6000minus4000minus2000

02000400060008000

100001200014000

000 025 050 075 100 125 150 175 200 225 250

y = 3my = 6my = 9m

Mi(

Nmiddotm

)

t(s)

Figure 7 Bending moment response at midspan section of verticalbeam

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 8 Shear force response at left hand of intersection of verticalbeam

(2119908119894(119895119873119894) minus 5119908

119894(119895119873119894minus 1)

+ 4119908119894(119895119873119894minus 2) minus 119908

119894(119895119873119894minus 3)) (Δ119897

2

119894)

minus1

= (2119908119894+1

(119895119873119894) minus 5119908

119894+1(119895119873119894+ 1)

+ 4119908119894+1

(119895119873119894+ 2) minus 119908

119894+1(119895119873119894+ 3)) (Δ119897

2

119894+1)

minus1

(16)

The displacement response of 119894th column grid nodes shouldalso satisfy the following initial conditions

119908119894[0 119896] = 0

120597119908119894

120597119905

10038161003816100381610038161003816100381610038161003816119905=0

=

119908119894[1 119896] minus 119908

119894[0 119896]

Δ119905

= minus2V0

(17)

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 9 Shear force response at right hand of intersection ofvertical beam

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

000 025 050 075 100 125 150 175 200 225 250minus4000

Qi(

N)

y = 3my = 6my = 9m

t(s)

Figure 10 Shear force response atmidspan section of vertical beam

In (17) V0is the initial velocity of vertical beam If the

harmonic vibration is needed then the initial velocity can bedefined as V

0= minus119886119892120596119892

Equations (15)ndash(17) can be simplified and the finite dif-ference format for the dynamic response of vertical beam isshown as

119908119894[0 119896] = 0

119908119894[1 119896] = minus2Δ119905V

0

(18)

119908119894(119895 + 1 119896)

= (2 minus

6Δ1199052

1205721Δ1198974

119894

minus

1205722Δ1199052

1205721

)119908119894(119895 119896)

+

Δ1199052

1205721Δ1198974

119894

[4119908119894(119895 119896 + 1) + 4119908

119894(119895 119896 minus 1)

Mathematical Problems in Engineering 9

00 15 30 45 60 75 90 105 120minus003

minus002

minus001

000

001

002

003wi(

m)

t = 1125 st = 1375 s

y(m)

Figure 11 Displacement response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120minus30000

minus20000

minus10000

0

10000

20000

30000

t = 1125 st = 1375 s

Mi(

Nmiddotm

)

y(m)

Figure 12 Bending moment response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120

minus30000

minus40000

minus20000

minus10000

0

10000

20000

30000

40000

t = 1125 st = 1375 s

Qi(

N)

y(m)

Figure 13 Shear force response along vertical beam at typicalmoment

times3

minus3

2

minus2

1

25 2 15 1 05 0 02

46

810

12minus10

w(yt)(

m)

y(m)

t(s)

10minus3

Figure 14 Displace response of vertical beam

minus 119908119894(119895 119896 + 2) minus 119908

119894(119895 119896 minus 2)]

minus 119908119894(119895 minus 1 119896) + Δ119905

2120595119894(119896) 1198861015840

119892(119895Δ119905)

(19)1199081(119895 0) = 3119908

1(119895 2) minus 2119908

1(119895 3)

1199081(119895 1) = 2119908

1(119895 2) minus 119908

1(119895 3)

119908119899(119895119873119899minus 1) = 2119908

119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)

119908119899(119895119873119899) = 3119908

119899(119895119873119899minus 2) minus 2119908

119899(119895119873119899minus 3)

(20)119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

119908119894+1

(119895119873119894+ 1)

= ([

6Δ1198972

119894

Δ1198972

119894+1

+

10 (Δ119897119894+ Δ119897119894+1

)

Δ119897119894+1

minus 6]119908119894(119895119873119894)

minus 7119908119894(119895119873119894minus 2) + 3119908

119894(119895119873119894minus 3)

+ [

12Δ1198972

119894

Δ1198972

119894+1

+

5Δ119897119894

Δ119897119894+1

]119908119894+1

(119895119873119894+ 2)

minus

3Δ1198972

119894

Δ1198972

119894+1

119908119894+1

(119895119873119894+ 3))

times (15 [

Δ119897119894

Δ119897119894+1

+

Δ1198972

119894

Δ1198972

119894+1

])

minus1

119908119894(119895119873119894minus 1)

=

2 (Δ119897119894+ Δ119897119894+1

)

3Δ119897119894+1

119908119894(119895119873119894) +

1

3

119908119894(119895119873119894minus 2)

minus

Δ119897119894

Δ119897119894+1

119908119894+1

(119895119873119894+ 1) +

Δ119897119894

3Δ119897119894+1

119908119894+1

(119895119873119894+ 2)

(21)

According to the finite difference method mentionedabove for a specific engineering problem the displacementof grid points in the initial and the first row can be calculatedby (18) For the second row the displacement of all thegrid nodes in interior columns can be decided by (19) thegrid nodes in exterior boundary columns and their adjacentcolumns (columns 0 1119873

119899minus1 and119873

119899) can be decided by (20)

and based on (21) the node displacement of all the interiorboundary adjacent columns can be achieved Likewise the

10 Mathematical Problems in Engineering

30002000

minus2000minus3000

1000

25 215 1

050 0

24

68

1012minus1000

0

y(m)

M(yt)(

Nmiddotm

)t(s)

Figure 15 Bending moment response of vertical beam

4000

minus4000

2000

252

151

050 0

24

68

1012

minus20000

Q(yt)(

N)

y(m)

t(s)

Figure 16 Shear force response of vertical beam

displacement for all the grid nodes from row 3 to row119872 canbe decided

After getting the displacement of all grid nodes accordingto 119872119894= 119864119868(120597

21199081198941205971199102) and 119876

119894= 119864119868(120597

31199081198941205971199103) the numerical

solutions to the bending moment and shear force responseof vertical beam can be obtained by conducting the finitedifference to the displacement of all grid nodes again

42 Comparative Analysis and Discussion Similarly basedon previous case study the slope is considered under hori-zontal harmonic seismic loading and with anchoring systemdamping neglected or considered respectively The numer-ical solutions to the dynamic response of vertical beam canbe achieved by using finite difference method and MATLABprogramming and 119903

119894= Δ119905Δ

2119897119894= 000001(025 times 025) =

000016 le radic1205721= 0002 (119894 = 1 sim 5)

In order to verify the correctness of the two methodsabove the solutions of intersection 119910 = 15m by differentmethods andwith damping effect neglected or considered areshown in Figures 18 and 19

When damping effect is neglected according to Figure 18it is found that the calculation results by the three methodsshare similar vibration trend and only show slight differencein amplitude generally speaking the finite element solutionmentioned previously is the smallest due to the size of finitedifference grids and it achieved by method of Literature [7] issecond because of the calculated amount of high order vibra-tionmode and the theoretical solutionmentioned previouslyis the largest and the most reasonable but the accuracyof two previous methods can be respectively improved bydecreasing grid ratio and increasing the calculated amount

so that the results of the three methods show more excellentagreement which indicates the reasonability and feasibilityof two proposed methods under the condition of neglectingdamping effect

When damping effect is considered as shown inFigure 19 the proposed theoretical solution and literature [7]solution are also well consistent and the former is slightlylarger than the latter Through the comparison and analysisof Figures 14 15 16 and 19 it is found that when usingthe finite difference method the attenuation characteristicof vertical beam response can be clearly shown in the firstsecond and then with the passing of time the vibrationmode is gradually degenerated into constant state whileno attenuation characteristic is shown and the calculatingresult keeps steady state all the time if using the theoreticalmethods So in the first second the calculating results usedthe finite difference method and theoretical methods haveconsiderable difference which is attributed to the theoreticalsolution based on steady-state complex method Specificallythe calculating result using finite difference method is largerthan the result using theoretical methods in the first secondAfter the first second the vibration pattern and the ampli-tude are consistent by using these three methods Besidesthe finite difference method not only can demonstrate theattenuation characteristic of amplitude in terms ofmagnitudein the damping anchorage system but also can illustrate theattenuation characteristic of amplitude in terms of time andits calculating result will be more reliable and reasonable

Generally speaking the finite difference method cannot only be applied to the dynamic response analysis offrame supporting structure under harmonic earthquake but

Mathematical Problems in Engineering 11

t

M

j

1Δt

0(N0) 1 Δl1 N1 Niminus1 Δli k Ni Nnminus1 Δln Nn y

H

A1 Ai An

Low

er p

latfo

rm (s

lope

toe)

w1

1th

row

anch

or

wi(j k) wn

Upp

er p

latfo

rm (s

lope

cres

t)

ith section1th section nth section

ith ro

w an

chor

n minus

1th

row

anch

or

i minus 1

th ro

w an

chor

Figure 17 Finite difference grids of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical methodProposed finite difference method

minus40000

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 18 Solutions of vertical beam with damping effect neglected by different methods

000 025 050 075 100 125 150 175 200 225 250minus15000minus12500minus10000minus7500

minus2500minus5000

0250050007500

10000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

Proposed finite difference method

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical method

minus15000

minus20000

minus5000

minus10000

0

5000

10000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 19 Solutions of vertical beam with damping effect considered by different methods

12 Mathematical Problems in Engineering

also be used to the analysis of random seismic loadingBesides it can demonstrate the attenuation characteristics ofamplitude in terms of both magnitude and time in otherwords it can indicate the damping effect better Thereforethe calculating result obtained by finite difference methodwould be more approximate to actual conditions and hencebe more reliable and reasonable However the analyticalmethod is only confined to the analysis of frame supportingstructure under harmonic earthquake Besides it can onlyreflect the attenuation characteristics of amplitude in termsof magnitude and the peak value of mechanical response gotby analytical method would be underestimated as a resultpotential hazards may be brought aboutWhat is more basedon steady-state complex method its mechanical responseshave initial phase values which are clearly contradictory tothe reality

5 Conclusions

Through the dynamic response analysis of frame supportingstructure of slope under harmonic forced vibration theconclusion follows

(1) Based on certain assumptions and force analysis ofsegment of infinitesimal length the dynamic govern-ing equation of vertical beam under forced vibrationis derived and the analytical solutions to the dynamicresponse of vertical beam under harmonic forcedvibration are obtainedMeanwhile according to finitedifference method the finite difference format andcorresponding calculating procedure for dynamicresponse of vertical beam under forced vibrationare established and finally programmed by usingMATLAB language

(2) In the case studies three methods have been usedin the dynamic response analysis and with dampingeffect neglected or considered respectivelyThe feasi-bility and correctness of the two proposed methodshave been examined and the characteristic of eachmethod has been summarized

(3) Under horizontal harmonic seismic loading themechanical response of vertical beam would syn-chronize correspondingly Its response law showsagreement with the one of two-end cantilevered con-tinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuousbeam In specific engineering design the factors suchas mechanical characteristic of vertical beam andslope supporting effect of anchors should also beconsidered

(4) Compared with analytical solutions the finite differ-ence method can simulate the dynamic response offrame supporting structure under various earthquakeconditions and it also can indicate the damping effectof anchorage system very well hence the calculatingresult obtained ismore reliable and reasonableMore-over the finite differencemethod is achieved based onprogramming calculation its calculation speed and

accuracy can be guaranteed and hence it has broadapplication prospect

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by Project (51308273 41372307) sup-ported by National Natural Science Foundation of China andProject (2010(A)06-b) supported by Science and Technol-ogy Project supported by Yunan Provincial CommunicationDepartment

References

[1] L K Cheng ldquoPresent status and development of groundanchoragesrdquo China Civil Engineering Journal vol 34 no 3 pp7ndash16 2001 (Chinese)

[2] D P Zhou J J Zhang and Y Tang ldquoSeismic damage analysis ofroad slopes in Wenchuan earthquakerdquo Chinese Journal of RockMechanics and Engineering vol 29 no 3 pp 565ndash576 2010(Chinese)

[3] M Yang H T Hu C J Lu et al ldquoCalculation of internal forcesfor prestressed anchor cable frame used in reinforced roadcutsoil sloperdquo Chinese Journal of Rock Mechanics and Engineeringvol 21 no 9 pp 1383ndash1386 2002 (Chinese)

[4] Y H Tian J K Liu and Y F Zhang ldquoInternal force calculationof prestressed anchor cable frame by finite difference methodrdquoJournal of Beijing Jiaotong University vol 31 no 4 pp 22ndash252007 (Chinese)

[5] G H Lin ldquoModel test of anchor grid beam for cut sloperdquoTechnology of Highway and Transport vol 3 pp 19ndash22 2004(Chinese)

[6] B L Zhu M Yang H T Hu et al ldquoTesting study on internalforces for prestressed anchor cable frame in reinforced soilsloperdquo Chinese Journal of Rock Mechanics and Engineering vol24 no 4 pp 697ndash702 2005 (Chinese)

[7] J H Dong Y P Zhu Y Zhou et al ldquoDynamic calculationmodel and seismic response for frame supporting structurewith prestressed anchorsrdquo Science China Technological Sciencesvol 53 no 7 pp 1957ndash1966 2010

[8] S Catal ldquoSolution of free vibration equations of beam onelastic soil by using differential transform methodrdquo AppliedMathematical Modelling vol 32 no 9 pp 1744ndash1757 2008

[9] M Balkaya M O Kaya and A Saglamer ldquoAnalysis of thevibration of an elastic beam supported on elastic soil using thedifferential transform methodrdquo Archive of Applied Mechanicsvol 79 no 2 pp 135ndash146 2009

[10] M Ansari E Esmailzadeh and D Younesian ldquoFrequencyanalysis of finite beams on nonlinear KelvinVoight foundationunder moving loadsrdquo Journal of Sound and Vibration vol 330no 7 pp 1455ndash1471 2011

[11] O Demirdag and Y Yesilce ldquoSolution of free vibration equationof elastically supported Timoshenko columns with a tip massby differential transform methodrdquo Advances in EngineeringSoftware vol 42 no 10 pp 860ndash867 2011

Mathematical Problems in Engineering 13

[12] MH Yas andN Samadi ldquoFree vibrations and buckling analysisof carbon nanotube-reinforced composite Timoshenko beamson elastic foundationrdquo International Journal of Pressure Vesselsand Piping vol 98 pp 119ndash128 2012

[13] R A Jafari-Talookolaei M Abedi M H Kargarnovin et alldquoAn analytical approach for the free vibration analysis ofgenerally laminated composite beams with shear effect androtary inertiardquo International Journal of Mechanical Sciences vol65 no 1 pp 97ndash104 2012

[14] S W Qi F Q Wu F Z Yan et al Dynamic Response Analysesof Rock Slope Science Press Beijing China 2007 Chinese

[15] J H Dong and Y P Zhu ldquoDynamic calculation model andseismic response for the system of soil nailing and surroundingsoilrdquo Chinese Journal of Theoretical and Applied Mechanics vol41 no 2 pp 236ndash242 2009

[16] Z L Xu Elasticity Higher Education Press Beijing China2009 Chinese

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

119860119894

119861119894

119862119894

119863119894

minus1205741119860119894

minus1205742119861119894

minus1205743119862119894

minus1205744119863119894

minus1205742

1119860119894

minus1205742

2119861119894

minus1205742

3119862119894

minus1205742

4119863119894

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

d

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

sdot sdot sdot 0 0 0 0 0 0 0 0

d

sdot sdot sdot 119860

119899minus1119861119899minus1

119862119899minus1

119863119899minus1

0 0 0 0

sdot sdot sdot 0 0 0 0 119860119899minus1

119861119899minus1

119862119899minus1

119863119899minus1

sdot sdot sdot 1205741119860119899minus1

1205742119861119899minus1

1205743119862119899minus1

1205744119863119899minus1

minus1205741119860119899minus1

minus1205742119861119899minus1

minus1205743119862119899minus1

minus1205744119863119899minus1

sdot sdot sdot 1205742

1119860119899minus1

1205742

2119861119899minus1

1205742

3119862119899minus1

1205742

4119863119899minus1

minus1205742

1119860119899minus1

minus1205742

2119861119899minus1

minus1205742

3119862119899minus1

minus1205742

4119863119899minus1

sdot sdot sdot 0 0 0 0 1205742

1119860119899

1205742

2119861119899

1205742

3119862119899

1205742

4119863119899

sdot sdot sdot 0 0 0 0 1205743

1119860119899

1205743

2119861119899

1205743

3119862119899

1205743

4119863119899

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(12)

After getting constant complex coefficient 1198624119899times1

it isassumed that119862

1198941= 1198881198941+11988811989421198941198621198942

= 1198881198943+11988811989441198941198621198943

= 1198881198945+1198881198946119894 and

1198621198944

= 1198881198947

+ 1198881198948119894 Substituting them into (9) the displacement

can be simplified as

1199081015840

119894(119910 119905)

= (1198881198941119890119910120573

+ 1198881198942119890119910120573

119894)

times [cos (120596119892119905 minus 119910120573) + 119894 sin (120596

119892119905 minus 119910120573)]

+ (1198881198943119890119910120573

+ 1198881198944119890119910120573

119894) [cos (120596119892119905 + 119910120573) + 119894 sin (120596

119892119905 + 119910120573)]

+ (1198881198945119890minus119910120573

+ 1198881198946119890minus119910120573

119894) [cos (120596119892119905 + 119910120573) + 119894 sin (120596

119892119905 + 119910120573)]

+ (1198881198947119890minus119910120573

+ 1198881198948119890minus119910120573

119894) [cos (120596119892119905 minus 119910120573) + 119894 sin (120596

119892119905 minus 119910120573)]

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

(cos120596119892119905 + 119894 sin120596

119892119905)

(13)

According to the calculating principle of complexmethod the corresponding imaginary part is the steady-statesolution to the mechanical response of vertical beam underharmonic seismic loading Then

119908119894(119910 119905)

= 119890120573119910

[1198881198941sin (120596

119892119905 minus 120573119910) + 119888

1198942cos (120596

119892119905 minus 120573119910)

+ 1198881198943sin (120596

119892119905 + 120573119910) + 119888

1198944cos (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[1198881198945sin (120596

119892119905 + 120573119910) + 119888

1198946cos (120596

119892119905 + 120573119910)

+ 1198881198947sin (120596

119892119905 minus 120573119910) + 119888

1198948cos (120596

119892119905 minus 120573119910)]

+

12057211198861015840

119892120595 (119910 + 119867

0)

1205722minus 12057211205962

119892

sin120596119892119905

119872119894(119910 119905)

= 21205732119864119868 119890120573119910

[minus1198881198941cos (120596

119892119905 minus 120573119910) + 119888

1198942sin (120596

119892119905 minus 120573119910)

+ 1198881198943cos (120596

119892119905 + 120573119910) minus 119888

1198944sin (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[minus1198881198945cos (120596

119892119905 + 120573119910) + 119888

1198946sin (120596

119892119905 + 120573119910)

+ 1198881198947cos (120596

119892119905 minus 120573119910) minus 119888

1198948sin (120596

119892119905 minus 120573119910)]

119876119894(119910 119905)

= 21205733119864119868 119890120573119910

[(1198881198942minus 1198881198941) sin (120596

119892119905 minus 120573119910)

minus (1198881198941+ 1198881198942) cos (120596

119892119905 minus 120573119910)

minus (1198881198943+ 1198881198944) sin (120596

119892119905 + 120573119910)

+ (1198881198943minus 1198881198944) cos (120596

119892119905 + 120573119910)]

+ 119890minus120573119910

[(1198881198945minus 1198881198946) sin (120596

119892119905 + 120573119910)

+ (1198881198945+ 1198881198946) cos (120596

119892119905 + 120573119910)

+ (1198881198947+ 1198881198948) sin (120596

119892119905 minus 120573119910)

+ (1198881198948minus 1198881198947) cos (120596

119892119905 minus 120573119910)]

(14)

6 Mathematical Problems in Engineering

32 Case Studies Take a slope as an example Its height is12m the slope angle is 65∘ and its physical and mechanicalparameters are 120588

119904= 1600 kgsdotmminus3 119864

119904= 012MPa 120583

119904=

03 119888119904

= 20 kPa and 120601119904

= 25∘ The seismic precautionary

intensity in the local region is 8 degrees and the peakground acceleration is 02 gThe dynamic safety factor is 1015calculated by Fellenius method hence its support is neededA preliminary supporting scheme is using fixed end anchorsand frame supporting structure The dynamic safety factoris considered as 12 calculated by the pseudostatic methodand the design scheme is shown in Figure 2 The anchor usescategory II steel bar diameter size is 28mm and its spaceis 33m the dip120593 = 10

∘ four rows of anchors have beenarranged both lengths of anchoring section and free sectionare 5m The mortar grade M30 is used as bonding agent thediameter of anchor hole is 119863 = 130mm The section size ofthe frame beam is 119887timesℎ = 045 times 05m concrete is gradeC30120588 = 2500 kgm3 119864 = 255GPa Earthquake accelerationis 1198861015840

119892(119905) = 119886

1015840

119892sin120596119892119905 = 2 sin 4120587119905 the duration is 25 s and

1198960= 2 times 10

3 kNm3From Figures 3 4 and 5 the displacement bending

moment and shear force response of vertical beam have beengiven when damping effect of anchorage system is neglectedwhile from Figures 6 7 8 9 10 11 12 and 13 correspondingresponse has been given at special sections (intersection andmidspan section) and typical moment (according to momentof acceleration peak) It is found that the displacement bend-ingmoment and shear force response of vertical beamvibratesynchronously with earthquakeThe law of bending momentand shear force response shows agreement with the har-monic forced vibration characteristic of two-end cantileveredcontinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuous beamMorespecifically the bending moment response at intersectionopposite to the one at midspan section and the shear forceresponse show a jumping from the left hand to the righthand at the intersection of vertical beam just as illustratedin Figures 12 and 13 What is more the cantilevered lengthat the two ends of vertical beam has noticeable influenceon the mechanical response of frame supporting structureThe larger the cantilevered length is the more the bendingmoment of intersection is and the greater the difference ofbending moment between intersection and midspan sectionis Examining the frame supporting structure independentlythe bending moment can keep good balance for the wholevertical beamwhen its cantilevered length is 02 or 025 timesthe anchor adjacent spacing referring to the design expe-rience of two-end cantilevered continuous beam Howeverthe cantilevered length of vertical beam is still related to thearranging scheme of anchors actually the length even decidesthe arrangement of anchors and directly influences the slopesupporting effect of anchors Hence in specific engineeringdesign the determination of cantilevered length should alsoconsider the factors such as mechanical characteristic ofvertical beam and slope supporting effect of anchors

Just as shown from Figures 14 15 and 16 the dis-placement bending moment and shear force response ofvertical beam with damping effect considered have been

given as well The overall trend is similar to the previous onewith damping effect neglected When considering anchoragesystem damping effect the solutions can be obtained byusing steady-state complex method and the peak valuesof the displacement bending moment and shear force aresmaller than the ones with damping effect neglected Eventhough the response shows some attenuation characteristicof amplitude in terms of magnitude it does not demonstratethe attenuation characteristic of amplitude in terms of timeWhat is more the mechanical responses show certain initialphases which can be attributed to the dynamic behaviors ofvertical beam under steady-state vibration with damping

4 Finite Difference Method for Vertical Beamunder Forced Vibration

41 Finite Difference Format for Dynamic Response AnalysisThe numerical solutions to dynamic governing equation (4)could be also achieved by using finite difference methodAccording to the basic principles of finite difference method[16] the research area119860(0 le 119905 le 119879 0 le 119910 le 119867) can be dividedinto 119899 subregions119860

119894Thewhole region of119860

119894can be redivided

into many grids 119860119894Δ119905Δ1198971

(119872119873119894minus119873119894minus1

+ 1) which is shown inFigure 17 Meanwhile in order to guarantee the convergenceand stability of calculation the ratio of the grids should meetthe following condition 119903

119894= Δ119905Δ

2119897119894le radic120572

1(119894 = 1 sim 119899)

The governing equation (4) can be expanded accordingto central difference formula and the difference format ofdynamic equation for grid node 119908

119894(119895 119896) can be obtained

(6119908119894(119895 119896) minus 4 [119908

119894(119895 119896 + 1) + 119908

119894(119895 119896 minus 1)]

+119908119894(119895 119896 + 2) + 119908

119894(119895 119896 minus 2)) (Δ119897

4

119894)

minus1

+ 1205721

119908119894(119895 + 1 119896) + 119908

119894(119895 minus 1 119896) minus 2119908

119894(119895 119896)

Δ1199052

+ 1205722119908119894(119895 119896) = 120572

1120595119894(119896) 1198861015840

119892[119895Δ119905]

(15)

where 120595119894(119896) = 120595[(119896 minus119873

119894minus1)Δ119897119894+sum119911=119894minus1

119911=1Δ119897119911(119873119911minus119873119911minus1

) +1198670]

Likewise the boundary equations (5) and (6) can beexpanded by endpoint derivative differential formula and thecorresponding differential equation can be attained

119864119868

21199081(119895 0) minus 5119908

1(119895 1) + 4119908

1(119895 2) minus 119908

1(119895 3)

Δ1198972

1

= 0

119864119868

minus1199081(119895 0) + 3119908

1(119895 1) minus 3119908

1(119895 2) + 119908

1(119895 3)

Δ1198973

1

= 0

119864119868 (2119908119899(119895119873119899) minus 5119908

119899(119895119873119899minus 1)

+4119908119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)) (Δ119897

2

119899)

minus1

= 0

119864119868 (119908119899(119895119873119899) minus 3119908

119899(119895119873119899minus 1)

+ 3119908119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)) (Δ119897

3

119899)

minus1

= 0

119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

Mathematical Problems in Engineering 7

5050

50

5050

50

Free section

Anchoring section

Anchor

5050

15

15

120

045 045

165 16533 33

Grass

Anchor

Horizontal beam Vertical beam

33

30

30

30

65∘

Figure 2 Design scheme for slope (unit m)

02

46

810

12

005115225minus003minus002minus001

0

002001

003

w (y

t)

(m)

t (s) y(m)2

46

810

11525y(m)

Figure 3 Displacement response of vertical beam

times104

3

minus3

2

minus2

1

252

151

050 0

24

6y(m)

810

12

minus10

M(yt)(

Nmiddotm

)

t(s)

Figure 4 Bending moment response of vertical beam

times104

4

minus4

2

252

151

050 0

24

68

1012minus2

0

Q(yt)(

N)

y(m)t(s)

Figure 5 Shear force response of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Mi(

Nmiddotm

)

y = 15my = 45m

y = 75my = 105m

t(s)

Figure 6 Bending moment response at intersection of verticalbeam

2119908119894(119895119873119894) minus 3119908

119894(119895119873119894minus 1) + 119908

119894(119895119873119894minus 2)

Δ119897119894

=

minus2119908119894+1

(119895119873119894) + 3119908

119894+1(119895119873119894+ 1) minus 119908

119894+1(119895119873119894+ 2)

Δ119897119894+1

8 Mathematical Problems in Engineering

minus12000minus10000minus8000minus6000minus4000minus2000

02000400060008000

100001200014000

000 025 050 075 100 125 150 175 200 225 250

y = 3my = 6my = 9m

Mi(

Nmiddotm

)

t(s)

Figure 7 Bending moment response at midspan section of verticalbeam

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 8 Shear force response at left hand of intersection of verticalbeam

(2119908119894(119895119873119894) minus 5119908

119894(119895119873119894minus 1)

+ 4119908119894(119895119873119894minus 2) minus 119908

119894(119895119873119894minus 3)) (Δ119897

2

119894)

minus1

= (2119908119894+1

(119895119873119894) minus 5119908

119894+1(119895119873119894+ 1)

+ 4119908119894+1

(119895119873119894+ 2) minus 119908

119894+1(119895119873119894+ 3)) (Δ119897

2

119894+1)

minus1

(16)

The displacement response of 119894th column grid nodes shouldalso satisfy the following initial conditions

119908119894[0 119896] = 0

120597119908119894

120597119905

10038161003816100381610038161003816100381610038161003816119905=0

=

119908119894[1 119896] minus 119908

119894[0 119896]

Δ119905

= minus2V0

(17)

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 9 Shear force response at right hand of intersection ofvertical beam

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

000 025 050 075 100 125 150 175 200 225 250minus4000

Qi(

N)

y = 3my = 6my = 9m

t(s)

Figure 10 Shear force response atmidspan section of vertical beam

In (17) V0is the initial velocity of vertical beam If the

harmonic vibration is needed then the initial velocity can bedefined as V

0= minus119886119892120596119892

Equations (15)ndash(17) can be simplified and the finite dif-ference format for the dynamic response of vertical beam isshown as

119908119894[0 119896] = 0

119908119894[1 119896] = minus2Δ119905V

0

(18)

119908119894(119895 + 1 119896)

= (2 minus

6Δ1199052

1205721Δ1198974

119894

minus

1205722Δ1199052

1205721

)119908119894(119895 119896)

+

Δ1199052

1205721Δ1198974

119894

[4119908119894(119895 119896 + 1) + 4119908

119894(119895 119896 minus 1)

Mathematical Problems in Engineering 9

00 15 30 45 60 75 90 105 120minus003

minus002

minus001

000

001

002

003wi(

m)

t = 1125 st = 1375 s

y(m)

Figure 11 Displacement response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120minus30000

minus20000

minus10000

0

10000

20000

30000

t = 1125 st = 1375 s

Mi(

Nmiddotm

)

y(m)

Figure 12 Bending moment response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120

minus30000

minus40000

minus20000

minus10000

0

10000

20000

30000

40000

t = 1125 st = 1375 s

Qi(

N)

y(m)

Figure 13 Shear force response along vertical beam at typicalmoment

times3

minus3

2

minus2

1

25 2 15 1 05 0 02

46

810

12minus10

w(yt)(

m)

y(m)

t(s)

10minus3

Figure 14 Displace response of vertical beam

minus 119908119894(119895 119896 + 2) minus 119908

119894(119895 119896 minus 2)]

minus 119908119894(119895 minus 1 119896) + Δ119905

2120595119894(119896) 1198861015840

119892(119895Δ119905)

(19)1199081(119895 0) = 3119908

1(119895 2) minus 2119908

1(119895 3)

1199081(119895 1) = 2119908

1(119895 2) minus 119908

1(119895 3)

119908119899(119895119873119899minus 1) = 2119908

119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)

119908119899(119895119873119899) = 3119908

119899(119895119873119899minus 2) minus 2119908

119899(119895119873119899minus 3)

(20)119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

119908119894+1

(119895119873119894+ 1)

= ([

6Δ1198972

119894

Δ1198972

119894+1

+

10 (Δ119897119894+ Δ119897119894+1

)

Δ119897119894+1

minus 6]119908119894(119895119873119894)

minus 7119908119894(119895119873119894minus 2) + 3119908

119894(119895119873119894minus 3)

+ [

12Δ1198972

119894

Δ1198972

119894+1

+

5Δ119897119894

Δ119897119894+1

]119908119894+1

(119895119873119894+ 2)

minus

3Δ1198972

119894

Δ1198972

119894+1

119908119894+1

(119895119873119894+ 3))

times (15 [

Δ119897119894

Δ119897119894+1

+

Δ1198972

119894

Δ1198972

119894+1

])

minus1

119908119894(119895119873119894minus 1)

=

2 (Δ119897119894+ Δ119897119894+1

)

3Δ119897119894+1

119908119894(119895119873119894) +

1

3

119908119894(119895119873119894minus 2)

minus

Δ119897119894

Δ119897119894+1

119908119894+1

(119895119873119894+ 1) +

Δ119897119894

3Δ119897119894+1

119908119894+1

(119895119873119894+ 2)

(21)

According to the finite difference method mentionedabove for a specific engineering problem the displacementof grid points in the initial and the first row can be calculatedby (18) For the second row the displacement of all thegrid nodes in interior columns can be decided by (19) thegrid nodes in exterior boundary columns and their adjacentcolumns (columns 0 1119873

119899minus1 and119873

119899) can be decided by (20)

and based on (21) the node displacement of all the interiorboundary adjacent columns can be achieved Likewise the

10 Mathematical Problems in Engineering

30002000

minus2000minus3000

1000

25 215 1

050 0

24

68

1012minus1000

0

y(m)

M(yt)(

Nmiddotm

)t(s)

Figure 15 Bending moment response of vertical beam

4000

minus4000

2000

252

151

050 0

24

68

1012

minus20000

Q(yt)(

N)

y(m)

t(s)

Figure 16 Shear force response of vertical beam

displacement for all the grid nodes from row 3 to row119872 canbe decided

After getting the displacement of all grid nodes accordingto 119872119894= 119864119868(120597

21199081198941205971199102) and 119876

119894= 119864119868(120597

31199081198941205971199103) the numerical

solutions to the bending moment and shear force responseof vertical beam can be obtained by conducting the finitedifference to the displacement of all grid nodes again

42 Comparative Analysis and Discussion Similarly basedon previous case study the slope is considered under hori-zontal harmonic seismic loading and with anchoring systemdamping neglected or considered respectively The numer-ical solutions to the dynamic response of vertical beam canbe achieved by using finite difference method and MATLABprogramming and 119903

119894= Δ119905Δ

2119897119894= 000001(025 times 025) =

000016 le radic1205721= 0002 (119894 = 1 sim 5)

In order to verify the correctness of the two methodsabove the solutions of intersection 119910 = 15m by differentmethods andwith damping effect neglected or considered areshown in Figures 18 and 19

When damping effect is neglected according to Figure 18it is found that the calculation results by the three methodsshare similar vibration trend and only show slight differencein amplitude generally speaking the finite element solutionmentioned previously is the smallest due to the size of finitedifference grids and it achieved by method of Literature [7] issecond because of the calculated amount of high order vibra-tionmode and the theoretical solutionmentioned previouslyis the largest and the most reasonable but the accuracyof two previous methods can be respectively improved bydecreasing grid ratio and increasing the calculated amount

so that the results of the three methods show more excellentagreement which indicates the reasonability and feasibilityof two proposed methods under the condition of neglectingdamping effect

When damping effect is considered as shown inFigure 19 the proposed theoretical solution and literature [7]solution are also well consistent and the former is slightlylarger than the latter Through the comparison and analysisof Figures 14 15 16 and 19 it is found that when usingthe finite difference method the attenuation characteristicof vertical beam response can be clearly shown in the firstsecond and then with the passing of time the vibrationmode is gradually degenerated into constant state whileno attenuation characteristic is shown and the calculatingresult keeps steady state all the time if using the theoreticalmethods So in the first second the calculating results usedthe finite difference method and theoretical methods haveconsiderable difference which is attributed to the theoreticalsolution based on steady-state complex method Specificallythe calculating result using finite difference method is largerthan the result using theoretical methods in the first secondAfter the first second the vibration pattern and the ampli-tude are consistent by using these three methods Besidesthe finite difference method not only can demonstrate theattenuation characteristic of amplitude in terms ofmagnitudein the damping anchorage system but also can illustrate theattenuation characteristic of amplitude in terms of time andits calculating result will be more reliable and reasonable

Generally speaking the finite difference method cannot only be applied to the dynamic response analysis offrame supporting structure under harmonic earthquake but

Mathematical Problems in Engineering 11

t

M

j

1Δt

0(N0) 1 Δl1 N1 Niminus1 Δli k Ni Nnminus1 Δln Nn y

H

A1 Ai An

Low

er p

latfo

rm (s

lope

toe)

w1

1th

row

anch

or

wi(j k) wn

Upp

er p

latfo

rm (s

lope

cres

t)

ith section1th section nth section

ith ro

w an

chor

n minus

1th

row

anch

or

i minus 1

th ro

w an

chor

Figure 17 Finite difference grids of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical methodProposed finite difference method

minus40000

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 18 Solutions of vertical beam with damping effect neglected by different methods

000 025 050 075 100 125 150 175 200 225 250minus15000minus12500minus10000minus7500

minus2500minus5000

0250050007500

10000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

Proposed finite difference method

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical method

minus15000

minus20000

minus5000

minus10000

0

5000

10000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 19 Solutions of vertical beam with damping effect considered by different methods

12 Mathematical Problems in Engineering

also be used to the analysis of random seismic loadingBesides it can demonstrate the attenuation characteristics ofamplitude in terms of both magnitude and time in otherwords it can indicate the damping effect better Thereforethe calculating result obtained by finite difference methodwould be more approximate to actual conditions and hencebe more reliable and reasonable However the analyticalmethod is only confined to the analysis of frame supportingstructure under harmonic earthquake Besides it can onlyreflect the attenuation characteristics of amplitude in termsof magnitude and the peak value of mechanical response gotby analytical method would be underestimated as a resultpotential hazards may be brought aboutWhat is more basedon steady-state complex method its mechanical responseshave initial phase values which are clearly contradictory tothe reality

5 Conclusions

Through the dynamic response analysis of frame supportingstructure of slope under harmonic forced vibration theconclusion follows

(1) Based on certain assumptions and force analysis ofsegment of infinitesimal length the dynamic govern-ing equation of vertical beam under forced vibrationis derived and the analytical solutions to the dynamicresponse of vertical beam under harmonic forcedvibration are obtainedMeanwhile according to finitedifference method the finite difference format andcorresponding calculating procedure for dynamicresponse of vertical beam under forced vibrationare established and finally programmed by usingMATLAB language

(2) In the case studies three methods have been usedin the dynamic response analysis and with dampingeffect neglected or considered respectivelyThe feasi-bility and correctness of the two proposed methodshave been examined and the characteristic of eachmethod has been summarized

(3) Under horizontal harmonic seismic loading themechanical response of vertical beam would syn-chronize correspondingly Its response law showsagreement with the one of two-end cantilevered con-tinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuousbeam In specific engineering design the factors suchas mechanical characteristic of vertical beam andslope supporting effect of anchors should also beconsidered

(4) Compared with analytical solutions the finite differ-ence method can simulate the dynamic response offrame supporting structure under various earthquakeconditions and it also can indicate the damping effectof anchorage system very well hence the calculatingresult obtained ismore reliable and reasonableMore-over the finite differencemethod is achieved based onprogramming calculation its calculation speed and

accuracy can be guaranteed and hence it has broadapplication prospect

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by Project (51308273 41372307) sup-ported by National Natural Science Foundation of China andProject (2010(A)06-b) supported by Science and Technol-ogy Project supported by Yunan Provincial CommunicationDepartment

References

[1] L K Cheng ldquoPresent status and development of groundanchoragesrdquo China Civil Engineering Journal vol 34 no 3 pp7ndash16 2001 (Chinese)

[2] D P Zhou J J Zhang and Y Tang ldquoSeismic damage analysis ofroad slopes in Wenchuan earthquakerdquo Chinese Journal of RockMechanics and Engineering vol 29 no 3 pp 565ndash576 2010(Chinese)

[3] M Yang H T Hu C J Lu et al ldquoCalculation of internal forcesfor prestressed anchor cable frame used in reinforced roadcutsoil sloperdquo Chinese Journal of Rock Mechanics and Engineeringvol 21 no 9 pp 1383ndash1386 2002 (Chinese)

[4] Y H Tian J K Liu and Y F Zhang ldquoInternal force calculationof prestressed anchor cable frame by finite difference methodrdquoJournal of Beijing Jiaotong University vol 31 no 4 pp 22ndash252007 (Chinese)

[5] G H Lin ldquoModel test of anchor grid beam for cut sloperdquoTechnology of Highway and Transport vol 3 pp 19ndash22 2004(Chinese)

[6] B L Zhu M Yang H T Hu et al ldquoTesting study on internalforces for prestressed anchor cable frame in reinforced soilsloperdquo Chinese Journal of Rock Mechanics and Engineering vol24 no 4 pp 697ndash702 2005 (Chinese)

[7] J H Dong Y P Zhu Y Zhou et al ldquoDynamic calculationmodel and seismic response for frame supporting structurewith prestressed anchorsrdquo Science China Technological Sciencesvol 53 no 7 pp 1957ndash1966 2010

[8] S Catal ldquoSolution of free vibration equations of beam onelastic soil by using differential transform methodrdquo AppliedMathematical Modelling vol 32 no 9 pp 1744ndash1757 2008

[9] M Balkaya M O Kaya and A Saglamer ldquoAnalysis of thevibration of an elastic beam supported on elastic soil using thedifferential transform methodrdquo Archive of Applied Mechanicsvol 79 no 2 pp 135ndash146 2009

[10] M Ansari E Esmailzadeh and D Younesian ldquoFrequencyanalysis of finite beams on nonlinear KelvinVoight foundationunder moving loadsrdquo Journal of Sound and Vibration vol 330no 7 pp 1455ndash1471 2011

[11] O Demirdag and Y Yesilce ldquoSolution of free vibration equationof elastically supported Timoshenko columns with a tip massby differential transform methodrdquo Advances in EngineeringSoftware vol 42 no 10 pp 860ndash867 2011

Mathematical Problems in Engineering 13

[12] MH Yas andN Samadi ldquoFree vibrations and buckling analysisof carbon nanotube-reinforced composite Timoshenko beamson elastic foundationrdquo International Journal of Pressure Vesselsand Piping vol 98 pp 119ndash128 2012

[13] R A Jafari-Talookolaei M Abedi M H Kargarnovin et alldquoAn analytical approach for the free vibration analysis ofgenerally laminated composite beams with shear effect androtary inertiardquo International Journal of Mechanical Sciences vol65 no 1 pp 97ndash104 2012

[14] S W Qi F Q Wu F Z Yan et al Dynamic Response Analysesof Rock Slope Science Press Beijing China 2007 Chinese

[15] J H Dong and Y P Zhu ldquoDynamic calculation model andseismic response for the system of soil nailing and surroundingsoilrdquo Chinese Journal of Theoretical and Applied Mechanics vol41 no 2 pp 236ndash242 2009

[16] Z L Xu Elasticity Higher Education Press Beijing China2009 Chinese

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

32 Case Studies Take a slope as an example Its height is12m the slope angle is 65∘ and its physical and mechanicalparameters are 120588

119904= 1600 kgsdotmminus3 119864

119904= 012MPa 120583

119904=

03 119888119904

= 20 kPa and 120601119904

= 25∘ The seismic precautionary

intensity in the local region is 8 degrees and the peakground acceleration is 02 gThe dynamic safety factor is 1015calculated by Fellenius method hence its support is neededA preliminary supporting scheme is using fixed end anchorsand frame supporting structure The dynamic safety factoris considered as 12 calculated by the pseudostatic methodand the design scheme is shown in Figure 2 The anchor usescategory II steel bar diameter size is 28mm and its spaceis 33m the dip120593 = 10

∘ four rows of anchors have beenarranged both lengths of anchoring section and free sectionare 5m The mortar grade M30 is used as bonding agent thediameter of anchor hole is 119863 = 130mm The section size ofthe frame beam is 119887timesℎ = 045 times 05m concrete is gradeC30120588 = 2500 kgm3 119864 = 255GPa Earthquake accelerationis 1198861015840

119892(119905) = 119886

1015840

119892sin120596119892119905 = 2 sin 4120587119905 the duration is 25 s and

1198960= 2 times 10

3 kNm3From Figures 3 4 and 5 the displacement bending

moment and shear force response of vertical beam have beengiven when damping effect of anchorage system is neglectedwhile from Figures 6 7 8 9 10 11 12 and 13 correspondingresponse has been given at special sections (intersection andmidspan section) and typical moment (according to momentof acceleration peak) It is found that the displacement bend-ingmoment and shear force response of vertical beamvibratesynchronously with earthquakeThe law of bending momentand shear force response shows agreement with the har-monic forced vibration characteristic of two-end cantileveredcontinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuous beamMorespecifically the bending moment response at intersectionopposite to the one at midspan section and the shear forceresponse show a jumping from the left hand to the righthand at the intersection of vertical beam just as illustratedin Figures 12 and 13 What is more the cantilevered lengthat the two ends of vertical beam has noticeable influenceon the mechanical response of frame supporting structureThe larger the cantilevered length is the more the bendingmoment of intersection is and the greater the difference ofbending moment between intersection and midspan sectionis Examining the frame supporting structure independentlythe bending moment can keep good balance for the wholevertical beamwhen its cantilevered length is 02 or 025 timesthe anchor adjacent spacing referring to the design expe-rience of two-end cantilevered continuous beam Howeverthe cantilevered length of vertical beam is still related to thearranging scheme of anchors actually the length even decidesthe arrangement of anchors and directly influences the slopesupporting effect of anchors Hence in specific engineeringdesign the determination of cantilevered length should alsoconsider the factors such as mechanical characteristic ofvertical beam and slope supporting effect of anchors

Just as shown from Figures 14 15 and 16 the dis-placement bending moment and shear force response ofvertical beam with damping effect considered have been

given as well The overall trend is similar to the previous onewith damping effect neglected When considering anchoragesystem damping effect the solutions can be obtained byusing steady-state complex method and the peak valuesof the displacement bending moment and shear force aresmaller than the ones with damping effect neglected Eventhough the response shows some attenuation characteristicof amplitude in terms of magnitude it does not demonstratethe attenuation characteristic of amplitude in terms of timeWhat is more the mechanical responses show certain initialphases which can be attributed to the dynamic behaviors ofvertical beam under steady-state vibration with damping

4 Finite Difference Method for Vertical Beamunder Forced Vibration

41 Finite Difference Format for Dynamic Response AnalysisThe numerical solutions to dynamic governing equation (4)could be also achieved by using finite difference methodAccording to the basic principles of finite difference method[16] the research area119860(0 le 119905 le 119879 0 le 119910 le 119867) can be dividedinto 119899 subregions119860

119894Thewhole region of119860

119894can be redivided

into many grids 119860119894Δ119905Δ1198971

(119872119873119894minus119873119894minus1

+ 1) which is shown inFigure 17 Meanwhile in order to guarantee the convergenceand stability of calculation the ratio of the grids should meetthe following condition 119903

119894= Δ119905Δ

2119897119894le radic120572

1(119894 = 1 sim 119899)

The governing equation (4) can be expanded accordingto central difference formula and the difference format ofdynamic equation for grid node 119908

119894(119895 119896) can be obtained

(6119908119894(119895 119896) minus 4 [119908

119894(119895 119896 + 1) + 119908

119894(119895 119896 minus 1)]

+119908119894(119895 119896 + 2) + 119908

119894(119895 119896 minus 2)) (Δ119897

4

119894)

minus1

+ 1205721

119908119894(119895 + 1 119896) + 119908

119894(119895 minus 1 119896) minus 2119908

119894(119895 119896)

Δ1199052

+ 1205722119908119894(119895 119896) = 120572

1120595119894(119896) 1198861015840

119892[119895Δ119905]

(15)

where 120595119894(119896) = 120595[(119896 minus119873

119894minus1)Δ119897119894+sum119911=119894minus1

119911=1Δ119897119911(119873119911minus119873119911minus1

) +1198670]

Likewise the boundary equations (5) and (6) can beexpanded by endpoint derivative differential formula and thecorresponding differential equation can be attained

119864119868

21199081(119895 0) minus 5119908

1(119895 1) + 4119908

1(119895 2) minus 119908

1(119895 3)

Δ1198972

1

= 0

119864119868

minus1199081(119895 0) + 3119908

1(119895 1) minus 3119908

1(119895 2) + 119908

1(119895 3)

Δ1198973

1

= 0

119864119868 (2119908119899(119895119873119899) minus 5119908

119899(119895119873119899minus 1)

+4119908119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)) (Δ119897

2

119899)

minus1

= 0

119864119868 (119908119899(119895119873119899) minus 3119908

119899(119895119873119899minus 1)

+ 3119908119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)) (Δ119897

3

119899)

minus1

= 0

119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

Mathematical Problems in Engineering 7

5050

50

5050

50

Free section

Anchoring section

Anchor

5050

15

15

120

045 045

165 16533 33

Grass

Anchor

Horizontal beam Vertical beam

33

30

30

30

65∘

Figure 2 Design scheme for slope (unit m)

02

46

810

12

005115225minus003minus002minus001

0

002001

003

w (y

t)

(m)

t (s) y(m)2

46

810

11525y(m)

Figure 3 Displacement response of vertical beam

times104

3

minus3

2

minus2

1

252

151

050 0

24

6y(m)

810

12

minus10

M(yt)(

Nmiddotm

)

t(s)

Figure 4 Bending moment response of vertical beam

times104

4

minus4

2

252

151

050 0

24

68

1012minus2

0

Q(yt)(

N)

y(m)t(s)

Figure 5 Shear force response of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Mi(

Nmiddotm

)

y = 15my = 45m

y = 75my = 105m

t(s)

Figure 6 Bending moment response at intersection of verticalbeam

2119908119894(119895119873119894) minus 3119908

119894(119895119873119894minus 1) + 119908

119894(119895119873119894minus 2)

Δ119897119894

=

minus2119908119894+1

(119895119873119894) + 3119908

119894+1(119895119873119894+ 1) minus 119908

119894+1(119895119873119894+ 2)

Δ119897119894+1

8 Mathematical Problems in Engineering

minus12000minus10000minus8000minus6000minus4000minus2000

02000400060008000

100001200014000

000 025 050 075 100 125 150 175 200 225 250

y = 3my = 6my = 9m

Mi(

Nmiddotm

)

t(s)

Figure 7 Bending moment response at midspan section of verticalbeam

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 8 Shear force response at left hand of intersection of verticalbeam

(2119908119894(119895119873119894) minus 5119908

119894(119895119873119894minus 1)

+ 4119908119894(119895119873119894minus 2) minus 119908

119894(119895119873119894minus 3)) (Δ119897

2

119894)

minus1

= (2119908119894+1

(119895119873119894) minus 5119908

119894+1(119895119873119894+ 1)

+ 4119908119894+1

(119895119873119894+ 2) minus 119908

119894+1(119895119873119894+ 3)) (Δ119897

2

119894+1)

minus1

(16)

The displacement response of 119894th column grid nodes shouldalso satisfy the following initial conditions

119908119894[0 119896] = 0

120597119908119894

120597119905

10038161003816100381610038161003816100381610038161003816119905=0

=

119908119894[1 119896] minus 119908

119894[0 119896]

Δ119905

= minus2V0

(17)

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 9 Shear force response at right hand of intersection ofvertical beam

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

000 025 050 075 100 125 150 175 200 225 250minus4000

Qi(

N)

y = 3my = 6my = 9m

t(s)

Figure 10 Shear force response atmidspan section of vertical beam

In (17) V0is the initial velocity of vertical beam If the

harmonic vibration is needed then the initial velocity can bedefined as V

0= minus119886119892120596119892

Equations (15)ndash(17) can be simplified and the finite dif-ference format for the dynamic response of vertical beam isshown as

119908119894[0 119896] = 0

119908119894[1 119896] = minus2Δ119905V

0

(18)

119908119894(119895 + 1 119896)

= (2 minus

6Δ1199052

1205721Δ1198974

119894

minus

1205722Δ1199052

1205721

)119908119894(119895 119896)

+

Δ1199052

1205721Δ1198974

119894

[4119908119894(119895 119896 + 1) + 4119908

119894(119895 119896 minus 1)

Mathematical Problems in Engineering 9

00 15 30 45 60 75 90 105 120minus003

minus002

minus001

000

001

002

003wi(

m)

t = 1125 st = 1375 s

y(m)

Figure 11 Displacement response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120minus30000

minus20000

minus10000

0

10000

20000

30000

t = 1125 st = 1375 s

Mi(

Nmiddotm

)

y(m)

Figure 12 Bending moment response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120

minus30000

minus40000

minus20000

minus10000

0

10000

20000

30000

40000

t = 1125 st = 1375 s

Qi(

N)

y(m)

Figure 13 Shear force response along vertical beam at typicalmoment

times3

minus3

2

minus2

1

25 2 15 1 05 0 02

46

810

12minus10

w(yt)(

m)

y(m)

t(s)

10minus3

Figure 14 Displace response of vertical beam

minus 119908119894(119895 119896 + 2) minus 119908

119894(119895 119896 minus 2)]

minus 119908119894(119895 minus 1 119896) + Δ119905

2120595119894(119896) 1198861015840

119892(119895Δ119905)

(19)1199081(119895 0) = 3119908

1(119895 2) minus 2119908

1(119895 3)

1199081(119895 1) = 2119908

1(119895 2) minus 119908

1(119895 3)

119908119899(119895119873119899minus 1) = 2119908

119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)

119908119899(119895119873119899) = 3119908

119899(119895119873119899minus 2) minus 2119908

119899(119895119873119899minus 3)

(20)119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

119908119894+1

(119895119873119894+ 1)

= ([

6Δ1198972

119894

Δ1198972

119894+1

+

10 (Δ119897119894+ Δ119897119894+1

)

Δ119897119894+1

minus 6]119908119894(119895119873119894)

minus 7119908119894(119895119873119894minus 2) + 3119908

119894(119895119873119894minus 3)

+ [

12Δ1198972

119894

Δ1198972

119894+1

+

5Δ119897119894

Δ119897119894+1

]119908119894+1

(119895119873119894+ 2)

minus

3Δ1198972

119894

Δ1198972

119894+1

119908119894+1

(119895119873119894+ 3))

times (15 [

Δ119897119894

Δ119897119894+1

+

Δ1198972

119894

Δ1198972

119894+1

])

minus1

119908119894(119895119873119894minus 1)

=

2 (Δ119897119894+ Δ119897119894+1

)

3Δ119897119894+1

119908119894(119895119873119894) +

1

3

119908119894(119895119873119894minus 2)

minus

Δ119897119894

Δ119897119894+1

119908119894+1

(119895119873119894+ 1) +

Δ119897119894

3Δ119897119894+1

119908119894+1

(119895119873119894+ 2)

(21)

According to the finite difference method mentionedabove for a specific engineering problem the displacementof grid points in the initial and the first row can be calculatedby (18) For the second row the displacement of all thegrid nodes in interior columns can be decided by (19) thegrid nodes in exterior boundary columns and their adjacentcolumns (columns 0 1119873

119899minus1 and119873

119899) can be decided by (20)

and based on (21) the node displacement of all the interiorboundary adjacent columns can be achieved Likewise the

10 Mathematical Problems in Engineering

30002000

minus2000minus3000

1000

25 215 1

050 0

24

68

1012minus1000

0

y(m)

M(yt)(

Nmiddotm

)t(s)

Figure 15 Bending moment response of vertical beam

4000

minus4000

2000

252

151

050 0

24

68

1012

minus20000

Q(yt)(

N)

y(m)

t(s)

Figure 16 Shear force response of vertical beam

displacement for all the grid nodes from row 3 to row119872 canbe decided

After getting the displacement of all grid nodes accordingto 119872119894= 119864119868(120597

21199081198941205971199102) and 119876

119894= 119864119868(120597

31199081198941205971199103) the numerical

solutions to the bending moment and shear force responseof vertical beam can be obtained by conducting the finitedifference to the displacement of all grid nodes again

42 Comparative Analysis and Discussion Similarly basedon previous case study the slope is considered under hori-zontal harmonic seismic loading and with anchoring systemdamping neglected or considered respectively The numer-ical solutions to the dynamic response of vertical beam canbe achieved by using finite difference method and MATLABprogramming and 119903

119894= Δ119905Δ

2119897119894= 000001(025 times 025) =

000016 le radic1205721= 0002 (119894 = 1 sim 5)

In order to verify the correctness of the two methodsabove the solutions of intersection 119910 = 15m by differentmethods andwith damping effect neglected or considered areshown in Figures 18 and 19

When damping effect is neglected according to Figure 18it is found that the calculation results by the three methodsshare similar vibration trend and only show slight differencein amplitude generally speaking the finite element solutionmentioned previously is the smallest due to the size of finitedifference grids and it achieved by method of Literature [7] issecond because of the calculated amount of high order vibra-tionmode and the theoretical solutionmentioned previouslyis the largest and the most reasonable but the accuracyof two previous methods can be respectively improved bydecreasing grid ratio and increasing the calculated amount

so that the results of the three methods show more excellentagreement which indicates the reasonability and feasibilityof two proposed methods under the condition of neglectingdamping effect

When damping effect is considered as shown inFigure 19 the proposed theoretical solution and literature [7]solution are also well consistent and the former is slightlylarger than the latter Through the comparison and analysisof Figures 14 15 16 and 19 it is found that when usingthe finite difference method the attenuation characteristicof vertical beam response can be clearly shown in the firstsecond and then with the passing of time the vibrationmode is gradually degenerated into constant state whileno attenuation characteristic is shown and the calculatingresult keeps steady state all the time if using the theoreticalmethods So in the first second the calculating results usedthe finite difference method and theoretical methods haveconsiderable difference which is attributed to the theoreticalsolution based on steady-state complex method Specificallythe calculating result using finite difference method is largerthan the result using theoretical methods in the first secondAfter the first second the vibration pattern and the ampli-tude are consistent by using these three methods Besidesthe finite difference method not only can demonstrate theattenuation characteristic of amplitude in terms ofmagnitudein the damping anchorage system but also can illustrate theattenuation characteristic of amplitude in terms of time andits calculating result will be more reliable and reasonable

Generally speaking the finite difference method cannot only be applied to the dynamic response analysis offrame supporting structure under harmonic earthquake but

Mathematical Problems in Engineering 11

t

M

j

1Δt

0(N0) 1 Δl1 N1 Niminus1 Δli k Ni Nnminus1 Δln Nn y

H

A1 Ai An

Low

er p

latfo

rm (s

lope

toe)

w1

1th

row

anch

or

wi(j k) wn

Upp

er p

latfo

rm (s

lope

cres

t)

ith section1th section nth section

ith ro

w an

chor

n minus

1th

row

anch

or

i minus 1

th ro

w an

chor

Figure 17 Finite difference grids of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical methodProposed finite difference method

minus40000

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 18 Solutions of vertical beam with damping effect neglected by different methods

000 025 050 075 100 125 150 175 200 225 250minus15000minus12500minus10000minus7500

minus2500minus5000

0250050007500

10000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

Proposed finite difference method

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical method

minus15000

minus20000

minus5000

minus10000

0

5000

10000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 19 Solutions of vertical beam with damping effect considered by different methods

12 Mathematical Problems in Engineering

also be used to the analysis of random seismic loadingBesides it can demonstrate the attenuation characteristics ofamplitude in terms of both magnitude and time in otherwords it can indicate the damping effect better Thereforethe calculating result obtained by finite difference methodwould be more approximate to actual conditions and hencebe more reliable and reasonable However the analyticalmethod is only confined to the analysis of frame supportingstructure under harmonic earthquake Besides it can onlyreflect the attenuation characteristics of amplitude in termsof magnitude and the peak value of mechanical response gotby analytical method would be underestimated as a resultpotential hazards may be brought aboutWhat is more basedon steady-state complex method its mechanical responseshave initial phase values which are clearly contradictory tothe reality

5 Conclusions

Through the dynamic response analysis of frame supportingstructure of slope under harmonic forced vibration theconclusion follows

(1) Based on certain assumptions and force analysis ofsegment of infinitesimal length the dynamic govern-ing equation of vertical beam under forced vibrationis derived and the analytical solutions to the dynamicresponse of vertical beam under harmonic forcedvibration are obtainedMeanwhile according to finitedifference method the finite difference format andcorresponding calculating procedure for dynamicresponse of vertical beam under forced vibrationare established and finally programmed by usingMATLAB language

(2) In the case studies three methods have been usedin the dynamic response analysis and with dampingeffect neglected or considered respectivelyThe feasi-bility and correctness of the two proposed methodshave been examined and the characteristic of eachmethod has been summarized

(3) Under horizontal harmonic seismic loading themechanical response of vertical beam would syn-chronize correspondingly Its response law showsagreement with the one of two-end cantilevered con-tinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuousbeam In specific engineering design the factors suchas mechanical characteristic of vertical beam andslope supporting effect of anchors should also beconsidered

(4) Compared with analytical solutions the finite differ-ence method can simulate the dynamic response offrame supporting structure under various earthquakeconditions and it also can indicate the damping effectof anchorage system very well hence the calculatingresult obtained ismore reliable and reasonableMore-over the finite differencemethod is achieved based onprogramming calculation its calculation speed and

accuracy can be guaranteed and hence it has broadapplication prospect

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by Project (51308273 41372307) sup-ported by National Natural Science Foundation of China andProject (2010(A)06-b) supported by Science and Technol-ogy Project supported by Yunan Provincial CommunicationDepartment

References

[1] L K Cheng ldquoPresent status and development of groundanchoragesrdquo China Civil Engineering Journal vol 34 no 3 pp7ndash16 2001 (Chinese)

[2] D P Zhou J J Zhang and Y Tang ldquoSeismic damage analysis ofroad slopes in Wenchuan earthquakerdquo Chinese Journal of RockMechanics and Engineering vol 29 no 3 pp 565ndash576 2010(Chinese)

[3] M Yang H T Hu C J Lu et al ldquoCalculation of internal forcesfor prestressed anchor cable frame used in reinforced roadcutsoil sloperdquo Chinese Journal of Rock Mechanics and Engineeringvol 21 no 9 pp 1383ndash1386 2002 (Chinese)

[4] Y H Tian J K Liu and Y F Zhang ldquoInternal force calculationof prestressed anchor cable frame by finite difference methodrdquoJournal of Beijing Jiaotong University vol 31 no 4 pp 22ndash252007 (Chinese)

[5] G H Lin ldquoModel test of anchor grid beam for cut sloperdquoTechnology of Highway and Transport vol 3 pp 19ndash22 2004(Chinese)

[6] B L Zhu M Yang H T Hu et al ldquoTesting study on internalforces for prestressed anchor cable frame in reinforced soilsloperdquo Chinese Journal of Rock Mechanics and Engineering vol24 no 4 pp 697ndash702 2005 (Chinese)

[7] J H Dong Y P Zhu Y Zhou et al ldquoDynamic calculationmodel and seismic response for frame supporting structurewith prestressed anchorsrdquo Science China Technological Sciencesvol 53 no 7 pp 1957ndash1966 2010

[8] S Catal ldquoSolution of free vibration equations of beam onelastic soil by using differential transform methodrdquo AppliedMathematical Modelling vol 32 no 9 pp 1744ndash1757 2008

[9] M Balkaya M O Kaya and A Saglamer ldquoAnalysis of thevibration of an elastic beam supported on elastic soil using thedifferential transform methodrdquo Archive of Applied Mechanicsvol 79 no 2 pp 135ndash146 2009

[10] M Ansari E Esmailzadeh and D Younesian ldquoFrequencyanalysis of finite beams on nonlinear KelvinVoight foundationunder moving loadsrdquo Journal of Sound and Vibration vol 330no 7 pp 1455ndash1471 2011

[11] O Demirdag and Y Yesilce ldquoSolution of free vibration equationof elastically supported Timoshenko columns with a tip massby differential transform methodrdquo Advances in EngineeringSoftware vol 42 no 10 pp 860ndash867 2011

Mathematical Problems in Engineering 13

[12] MH Yas andN Samadi ldquoFree vibrations and buckling analysisof carbon nanotube-reinforced composite Timoshenko beamson elastic foundationrdquo International Journal of Pressure Vesselsand Piping vol 98 pp 119ndash128 2012

[13] R A Jafari-Talookolaei M Abedi M H Kargarnovin et alldquoAn analytical approach for the free vibration analysis ofgenerally laminated composite beams with shear effect androtary inertiardquo International Journal of Mechanical Sciences vol65 no 1 pp 97ndash104 2012

[14] S W Qi F Q Wu F Z Yan et al Dynamic Response Analysesof Rock Slope Science Press Beijing China 2007 Chinese

[15] J H Dong and Y P Zhu ldquoDynamic calculation model andseismic response for the system of soil nailing and surroundingsoilrdquo Chinese Journal of Theoretical and Applied Mechanics vol41 no 2 pp 236ndash242 2009

[16] Z L Xu Elasticity Higher Education Press Beijing China2009 Chinese

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

5050

50

5050

50

Free section

Anchoring section

Anchor

5050

15

15

120

045 045

165 16533 33

Grass

Anchor

Horizontal beam Vertical beam

33

30

30

30

65∘

Figure 2 Design scheme for slope (unit m)

02

46

810

12

005115225minus003minus002minus001

0

002001

003

w (y

t)

(m)

t (s) y(m)2

46

810

11525y(m)

Figure 3 Displacement response of vertical beam

times104

3

minus3

2

minus2

1

252

151

050 0

24

6y(m)

810

12

minus10

M(yt)(

Nmiddotm

)

t(s)

Figure 4 Bending moment response of vertical beam

times104

4

minus4

2

252

151

050 0

24

68

1012minus2

0

Q(yt)(

N)

y(m)t(s)

Figure 5 Shear force response of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Mi(

Nmiddotm

)

y = 15my = 45m

y = 75my = 105m

t(s)

Figure 6 Bending moment response at intersection of verticalbeam

2119908119894(119895119873119894) minus 3119908

119894(119895119873119894minus 1) + 119908

119894(119895119873119894minus 2)

Δ119897119894

=

minus2119908119894+1

(119895119873119894) + 3119908

119894+1(119895119873119894+ 1) minus 119908

119894+1(119895119873119894+ 2)

Δ119897119894+1

8 Mathematical Problems in Engineering

minus12000minus10000minus8000minus6000minus4000minus2000

02000400060008000

100001200014000

000 025 050 075 100 125 150 175 200 225 250

y = 3my = 6my = 9m

Mi(

Nmiddotm

)

t(s)

Figure 7 Bending moment response at midspan section of verticalbeam

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 8 Shear force response at left hand of intersection of verticalbeam

(2119908119894(119895119873119894) minus 5119908

119894(119895119873119894minus 1)

+ 4119908119894(119895119873119894minus 2) minus 119908

119894(119895119873119894minus 3)) (Δ119897

2

119894)

minus1

= (2119908119894+1

(119895119873119894) minus 5119908

119894+1(119895119873119894+ 1)

+ 4119908119894+1

(119895119873119894+ 2) minus 119908

119894+1(119895119873119894+ 3)) (Δ119897

2

119894+1)

minus1

(16)

The displacement response of 119894th column grid nodes shouldalso satisfy the following initial conditions

119908119894[0 119896] = 0

120597119908119894

120597119905

10038161003816100381610038161003816100381610038161003816119905=0

=

119908119894[1 119896] minus 119908

119894[0 119896]

Δ119905

= minus2V0

(17)

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 9 Shear force response at right hand of intersection ofvertical beam

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

000 025 050 075 100 125 150 175 200 225 250minus4000

Qi(

N)

y = 3my = 6my = 9m

t(s)

Figure 10 Shear force response atmidspan section of vertical beam

In (17) V0is the initial velocity of vertical beam If the

harmonic vibration is needed then the initial velocity can bedefined as V

0= minus119886119892120596119892

Equations (15)ndash(17) can be simplified and the finite dif-ference format for the dynamic response of vertical beam isshown as

119908119894[0 119896] = 0

119908119894[1 119896] = minus2Δ119905V

0

(18)

119908119894(119895 + 1 119896)

= (2 minus

6Δ1199052

1205721Δ1198974

119894

minus

1205722Δ1199052

1205721

)119908119894(119895 119896)

+

Δ1199052

1205721Δ1198974

119894

[4119908119894(119895 119896 + 1) + 4119908

119894(119895 119896 minus 1)

Mathematical Problems in Engineering 9

00 15 30 45 60 75 90 105 120minus003

minus002

minus001

000

001

002

003wi(

m)

t = 1125 st = 1375 s

y(m)

Figure 11 Displacement response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120minus30000

minus20000

minus10000

0

10000

20000

30000

t = 1125 st = 1375 s

Mi(

Nmiddotm

)

y(m)

Figure 12 Bending moment response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120

minus30000

minus40000

minus20000

minus10000

0

10000

20000

30000

40000

t = 1125 st = 1375 s

Qi(

N)

y(m)

Figure 13 Shear force response along vertical beam at typicalmoment

times3

minus3

2

minus2

1

25 2 15 1 05 0 02

46

810

12minus10

w(yt)(

m)

y(m)

t(s)

10minus3

Figure 14 Displace response of vertical beam

minus 119908119894(119895 119896 + 2) minus 119908

119894(119895 119896 minus 2)]

minus 119908119894(119895 minus 1 119896) + Δ119905

2120595119894(119896) 1198861015840

119892(119895Δ119905)

(19)1199081(119895 0) = 3119908

1(119895 2) minus 2119908

1(119895 3)

1199081(119895 1) = 2119908

1(119895 2) minus 119908

1(119895 3)

119908119899(119895119873119899minus 1) = 2119908

119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)

119908119899(119895119873119899) = 3119908

119899(119895119873119899minus 2) minus 2119908

119899(119895119873119899minus 3)

(20)119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

119908119894+1

(119895119873119894+ 1)

= ([

6Δ1198972

119894

Δ1198972

119894+1

+

10 (Δ119897119894+ Δ119897119894+1

)

Δ119897119894+1

minus 6]119908119894(119895119873119894)

minus 7119908119894(119895119873119894minus 2) + 3119908

119894(119895119873119894minus 3)

+ [

12Δ1198972

119894

Δ1198972

119894+1

+

5Δ119897119894

Δ119897119894+1

]119908119894+1

(119895119873119894+ 2)

minus

3Δ1198972

119894

Δ1198972

119894+1

119908119894+1

(119895119873119894+ 3))

times (15 [

Δ119897119894

Δ119897119894+1

+

Δ1198972

119894

Δ1198972

119894+1

])

minus1

119908119894(119895119873119894minus 1)

=

2 (Δ119897119894+ Δ119897119894+1

)

3Δ119897119894+1

119908119894(119895119873119894) +

1

3

119908119894(119895119873119894minus 2)

minus

Δ119897119894

Δ119897119894+1

119908119894+1

(119895119873119894+ 1) +

Δ119897119894

3Δ119897119894+1

119908119894+1

(119895119873119894+ 2)

(21)

According to the finite difference method mentionedabove for a specific engineering problem the displacementof grid points in the initial and the first row can be calculatedby (18) For the second row the displacement of all thegrid nodes in interior columns can be decided by (19) thegrid nodes in exterior boundary columns and their adjacentcolumns (columns 0 1119873

119899minus1 and119873

119899) can be decided by (20)

and based on (21) the node displacement of all the interiorboundary adjacent columns can be achieved Likewise the

10 Mathematical Problems in Engineering

30002000

minus2000minus3000

1000

25 215 1

050 0

24

68

1012minus1000

0

y(m)

M(yt)(

Nmiddotm

)t(s)

Figure 15 Bending moment response of vertical beam

4000

minus4000

2000

252

151

050 0

24

68

1012

minus20000

Q(yt)(

N)

y(m)

t(s)

Figure 16 Shear force response of vertical beam

displacement for all the grid nodes from row 3 to row119872 canbe decided

After getting the displacement of all grid nodes accordingto 119872119894= 119864119868(120597

21199081198941205971199102) and 119876

119894= 119864119868(120597

31199081198941205971199103) the numerical

solutions to the bending moment and shear force responseof vertical beam can be obtained by conducting the finitedifference to the displacement of all grid nodes again

42 Comparative Analysis and Discussion Similarly basedon previous case study the slope is considered under hori-zontal harmonic seismic loading and with anchoring systemdamping neglected or considered respectively The numer-ical solutions to the dynamic response of vertical beam canbe achieved by using finite difference method and MATLABprogramming and 119903

119894= Δ119905Δ

2119897119894= 000001(025 times 025) =

000016 le radic1205721= 0002 (119894 = 1 sim 5)

In order to verify the correctness of the two methodsabove the solutions of intersection 119910 = 15m by differentmethods andwith damping effect neglected or considered areshown in Figures 18 and 19

When damping effect is neglected according to Figure 18it is found that the calculation results by the three methodsshare similar vibration trend and only show slight differencein amplitude generally speaking the finite element solutionmentioned previously is the smallest due to the size of finitedifference grids and it achieved by method of Literature [7] issecond because of the calculated amount of high order vibra-tionmode and the theoretical solutionmentioned previouslyis the largest and the most reasonable but the accuracyof two previous methods can be respectively improved bydecreasing grid ratio and increasing the calculated amount

so that the results of the three methods show more excellentagreement which indicates the reasonability and feasibilityof two proposed methods under the condition of neglectingdamping effect

When damping effect is considered as shown inFigure 19 the proposed theoretical solution and literature [7]solution are also well consistent and the former is slightlylarger than the latter Through the comparison and analysisof Figures 14 15 16 and 19 it is found that when usingthe finite difference method the attenuation characteristicof vertical beam response can be clearly shown in the firstsecond and then with the passing of time the vibrationmode is gradually degenerated into constant state whileno attenuation characteristic is shown and the calculatingresult keeps steady state all the time if using the theoreticalmethods So in the first second the calculating results usedthe finite difference method and theoretical methods haveconsiderable difference which is attributed to the theoreticalsolution based on steady-state complex method Specificallythe calculating result using finite difference method is largerthan the result using theoretical methods in the first secondAfter the first second the vibration pattern and the ampli-tude are consistent by using these three methods Besidesthe finite difference method not only can demonstrate theattenuation characteristic of amplitude in terms ofmagnitudein the damping anchorage system but also can illustrate theattenuation characteristic of amplitude in terms of time andits calculating result will be more reliable and reasonable

Generally speaking the finite difference method cannot only be applied to the dynamic response analysis offrame supporting structure under harmonic earthquake but

Mathematical Problems in Engineering 11

t

M

j

1Δt

0(N0) 1 Δl1 N1 Niminus1 Δli k Ni Nnminus1 Δln Nn y

H

A1 Ai An

Low

er p

latfo

rm (s

lope

toe)

w1

1th

row

anch

or

wi(j k) wn

Upp

er p

latfo

rm (s

lope

cres

t)

ith section1th section nth section

ith ro

w an

chor

n minus

1th

row

anch

or

i minus 1

th ro

w an

chor

Figure 17 Finite difference grids of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical methodProposed finite difference method

minus40000

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 18 Solutions of vertical beam with damping effect neglected by different methods

000 025 050 075 100 125 150 175 200 225 250minus15000minus12500minus10000minus7500

minus2500minus5000

0250050007500

10000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

Proposed finite difference method

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical method

minus15000

minus20000

minus5000

minus10000

0

5000

10000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 19 Solutions of vertical beam with damping effect considered by different methods

12 Mathematical Problems in Engineering

also be used to the analysis of random seismic loadingBesides it can demonstrate the attenuation characteristics ofamplitude in terms of both magnitude and time in otherwords it can indicate the damping effect better Thereforethe calculating result obtained by finite difference methodwould be more approximate to actual conditions and hencebe more reliable and reasonable However the analyticalmethod is only confined to the analysis of frame supportingstructure under harmonic earthquake Besides it can onlyreflect the attenuation characteristics of amplitude in termsof magnitude and the peak value of mechanical response gotby analytical method would be underestimated as a resultpotential hazards may be brought aboutWhat is more basedon steady-state complex method its mechanical responseshave initial phase values which are clearly contradictory tothe reality

5 Conclusions

Through the dynamic response analysis of frame supportingstructure of slope under harmonic forced vibration theconclusion follows

(1) Based on certain assumptions and force analysis ofsegment of infinitesimal length the dynamic govern-ing equation of vertical beam under forced vibrationis derived and the analytical solutions to the dynamicresponse of vertical beam under harmonic forcedvibration are obtainedMeanwhile according to finitedifference method the finite difference format andcorresponding calculating procedure for dynamicresponse of vertical beam under forced vibrationare established and finally programmed by usingMATLAB language

(2) In the case studies three methods have been usedin the dynamic response analysis and with dampingeffect neglected or considered respectivelyThe feasi-bility and correctness of the two proposed methodshave been examined and the characteristic of eachmethod has been summarized

(3) Under horizontal harmonic seismic loading themechanical response of vertical beam would syn-chronize correspondingly Its response law showsagreement with the one of two-end cantilevered con-tinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuousbeam In specific engineering design the factors suchas mechanical characteristic of vertical beam andslope supporting effect of anchors should also beconsidered

(4) Compared with analytical solutions the finite differ-ence method can simulate the dynamic response offrame supporting structure under various earthquakeconditions and it also can indicate the damping effectof anchorage system very well hence the calculatingresult obtained ismore reliable and reasonableMore-over the finite differencemethod is achieved based onprogramming calculation its calculation speed and

accuracy can be guaranteed and hence it has broadapplication prospect

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by Project (51308273 41372307) sup-ported by National Natural Science Foundation of China andProject (2010(A)06-b) supported by Science and Technol-ogy Project supported by Yunan Provincial CommunicationDepartment

References

[1] L K Cheng ldquoPresent status and development of groundanchoragesrdquo China Civil Engineering Journal vol 34 no 3 pp7ndash16 2001 (Chinese)

[2] D P Zhou J J Zhang and Y Tang ldquoSeismic damage analysis ofroad slopes in Wenchuan earthquakerdquo Chinese Journal of RockMechanics and Engineering vol 29 no 3 pp 565ndash576 2010(Chinese)

[3] M Yang H T Hu C J Lu et al ldquoCalculation of internal forcesfor prestressed anchor cable frame used in reinforced roadcutsoil sloperdquo Chinese Journal of Rock Mechanics and Engineeringvol 21 no 9 pp 1383ndash1386 2002 (Chinese)

[4] Y H Tian J K Liu and Y F Zhang ldquoInternal force calculationof prestressed anchor cable frame by finite difference methodrdquoJournal of Beijing Jiaotong University vol 31 no 4 pp 22ndash252007 (Chinese)

[5] G H Lin ldquoModel test of anchor grid beam for cut sloperdquoTechnology of Highway and Transport vol 3 pp 19ndash22 2004(Chinese)

[6] B L Zhu M Yang H T Hu et al ldquoTesting study on internalforces for prestressed anchor cable frame in reinforced soilsloperdquo Chinese Journal of Rock Mechanics and Engineering vol24 no 4 pp 697ndash702 2005 (Chinese)

[7] J H Dong Y P Zhu Y Zhou et al ldquoDynamic calculationmodel and seismic response for frame supporting structurewith prestressed anchorsrdquo Science China Technological Sciencesvol 53 no 7 pp 1957ndash1966 2010

[8] S Catal ldquoSolution of free vibration equations of beam onelastic soil by using differential transform methodrdquo AppliedMathematical Modelling vol 32 no 9 pp 1744ndash1757 2008

[9] M Balkaya M O Kaya and A Saglamer ldquoAnalysis of thevibration of an elastic beam supported on elastic soil using thedifferential transform methodrdquo Archive of Applied Mechanicsvol 79 no 2 pp 135ndash146 2009

[10] M Ansari E Esmailzadeh and D Younesian ldquoFrequencyanalysis of finite beams on nonlinear KelvinVoight foundationunder moving loadsrdquo Journal of Sound and Vibration vol 330no 7 pp 1455ndash1471 2011

[11] O Demirdag and Y Yesilce ldquoSolution of free vibration equationof elastically supported Timoshenko columns with a tip massby differential transform methodrdquo Advances in EngineeringSoftware vol 42 no 10 pp 860ndash867 2011

Mathematical Problems in Engineering 13

[12] MH Yas andN Samadi ldquoFree vibrations and buckling analysisof carbon nanotube-reinforced composite Timoshenko beamson elastic foundationrdquo International Journal of Pressure Vesselsand Piping vol 98 pp 119ndash128 2012

[13] R A Jafari-Talookolaei M Abedi M H Kargarnovin et alldquoAn analytical approach for the free vibration analysis ofgenerally laminated composite beams with shear effect androtary inertiardquo International Journal of Mechanical Sciences vol65 no 1 pp 97ndash104 2012

[14] S W Qi F Q Wu F Z Yan et al Dynamic Response Analysesof Rock Slope Science Press Beijing China 2007 Chinese

[15] J H Dong and Y P Zhu ldquoDynamic calculation model andseismic response for the system of soil nailing and surroundingsoilrdquo Chinese Journal of Theoretical and Applied Mechanics vol41 no 2 pp 236ndash242 2009

[16] Z L Xu Elasticity Higher Education Press Beijing China2009 Chinese

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

minus12000minus10000minus8000minus6000minus4000minus2000

02000400060008000

100001200014000

000 025 050 075 100 125 150 175 200 225 250

y = 3my = 6my = 9m

Mi(

Nmiddotm

)

t(s)

Figure 7 Bending moment response at midspan section of verticalbeam

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 8 Shear force response at left hand of intersection of verticalbeam

(2119908119894(119895119873119894) minus 5119908

119894(119895119873119894minus 1)

+ 4119908119894(119895119873119894minus 2) minus 119908

119894(119895119873119894minus 3)) (Δ119897

2

119894)

minus1

= (2119908119894+1

(119895119873119894) minus 5119908

119894+1(119895119873119894+ 1)

+ 4119908119894+1

(119895119873119894+ 2) minus 119908

119894+1(119895119873119894+ 3)) (Δ119897

2

119894+1)

minus1

(16)

The displacement response of 119894th column grid nodes shouldalso satisfy the following initial conditions

119908119894[0 119896] = 0

120597119908119894

120597119905

10038161003816100381610038161003816100381610038161003816119905=0

=

119908119894[1 119896] minus 119908

119894[0 119896]

Δ119905

= minus2V0

(17)

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

000 025 050 075 100 125 150 175 200 225 250minus40000

y = 15my = 45m

y = 75my = 105m

Qi(

N)

t(s)

Figure 9 Shear force response at right hand of intersection ofvertical beam

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

000 025 050 075 100 125 150 175 200 225 250minus4000

Qi(

N)

y = 3my = 6my = 9m

t(s)

Figure 10 Shear force response atmidspan section of vertical beam

In (17) V0is the initial velocity of vertical beam If the

harmonic vibration is needed then the initial velocity can bedefined as V

0= minus119886119892120596119892

Equations (15)ndash(17) can be simplified and the finite dif-ference format for the dynamic response of vertical beam isshown as

119908119894[0 119896] = 0

119908119894[1 119896] = minus2Δ119905V

0

(18)

119908119894(119895 + 1 119896)

= (2 minus

6Δ1199052

1205721Δ1198974

119894

minus

1205722Δ1199052

1205721

)119908119894(119895 119896)

+

Δ1199052

1205721Δ1198974

119894

[4119908119894(119895 119896 + 1) + 4119908

119894(119895 119896 minus 1)

Mathematical Problems in Engineering 9

00 15 30 45 60 75 90 105 120minus003

minus002

minus001

000

001

002

003wi(

m)

t = 1125 st = 1375 s

y(m)

Figure 11 Displacement response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120minus30000

minus20000

minus10000

0

10000

20000

30000

t = 1125 st = 1375 s

Mi(

Nmiddotm

)

y(m)

Figure 12 Bending moment response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120

minus30000

minus40000

minus20000

minus10000

0

10000

20000

30000

40000

t = 1125 st = 1375 s

Qi(

N)

y(m)

Figure 13 Shear force response along vertical beam at typicalmoment

times3

minus3

2

minus2

1

25 2 15 1 05 0 02

46

810

12minus10

w(yt)(

m)

y(m)

t(s)

10minus3

Figure 14 Displace response of vertical beam

minus 119908119894(119895 119896 + 2) minus 119908

119894(119895 119896 minus 2)]

minus 119908119894(119895 minus 1 119896) + Δ119905

2120595119894(119896) 1198861015840

119892(119895Δ119905)

(19)1199081(119895 0) = 3119908

1(119895 2) minus 2119908

1(119895 3)

1199081(119895 1) = 2119908

1(119895 2) minus 119908

1(119895 3)

119908119899(119895119873119899minus 1) = 2119908

119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)

119908119899(119895119873119899) = 3119908

119899(119895119873119899minus 2) minus 2119908

119899(119895119873119899minus 3)

(20)119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

119908119894+1

(119895119873119894+ 1)

= ([

6Δ1198972

119894

Δ1198972

119894+1

+

10 (Δ119897119894+ Δ119897119894+1

)

Δ119897119894+1

minus 6]119908119894(119895119873119894)

minus 7119908119894(119895119873119894minus 2) + 3119908

119894(119895119873119894minus 3)

+ [

12Δ1198972

119894

Δ1198972

119894+1

+

5Δ119897119894

Δ119897119894+1

]119908119894+1

(119895119873119894+ 2)

minus

3Δ1198972

119894

Δ1198972

119894+1

119908119894+1

(119895119873119894+ 3))

times (15 [

Δ119897119894

Δ119897119894+1

+

Δ1198972

119894

Δ1198972

119894+1

])

minus1

119908119894(119895119873119894minus 1)

=

2 (Δ119897119894+ Δ119897119894+1

)

3Δ119897119894+1

119908119894(119895119873119894) +

1

3

119908119894(119895119873119894minus 2)

minus

Δ119897119894

Δ119897119894+1

119908119894+1

(119895119873119894+ 1) +

Δ119897119894

3Δ119897119894+1

119908119894+1

(119895119873119894+ 2)

(21)

According to the finite difference method mentionedabove for a specific engineering problem the displacementof grid points in the initial and the first row can be calculatedby (18) For the second row the displacement of all thegrid nodes in interior columns can be decided by (19) thegrid nodes in exterior boundary columns and their adjacentcolumns (columns 0 1119873

119899minus1 and119873

119899) can be decided by (20)

and based on (21) the node displacement of all the interiorboundary adjacent columns can be achieved Likewise the

10 Mathematical Problems in Engineering

30002000

minus2000minus3000

1000

25 215 1

050 0

24

68

1012minus1000

0

y(m)

M(yt)(

Nmiddotm

)t(s)

Figure 15 Bending moment response of vertical beam

4000

minus4000

2000

252

151

050 0

24

68

1012

minus20000

Q(yt)(

N)

y(m)

t(s)

Figure 16 Shear force response of vertical beam

displacement for all the grid nodes from row 3 to row119872 canbe decided

After getting the displacement of all grid nodes accordingto 119872119894= 119864119868(120597

21199081198941205971199102) and 119876

119894= 119864119868(120597

31199081198941205971199103) the numerical

solutions to the bending moment and shear force responseof vertical beam can be obtained by conducting the finitedifference to the displacement of all grid nodes again

42 Comparative Analysis and Discussion Similarly basedon previous case study the slope is considered under hori-zontal harmonic seismic loading and with anchoring systemdamping neglected or considered respectively The numer-ical solutions to the dynamic response of vertical beam canbe achieved by using finite difference method and MATLABprogramming and 119903

119894= Δ119905Δ

2119897119894= 000001(025 times 025) =

000016 le radic1205721= 0002 (119894 = 1 sim 5)

In order to verify the correctness of the two methodsabove the solutions of intersection 119910 = 15m by differentmethods andwith damping effect neglected or considered areshown in Figures 18 and 19

When damping effect is neglected according to Figure 18it is found that the calculation results by the three methodsshare similar vibration trend and only show slight differencein amplitude generally speaking the finite element solutionmentioned previously is the smallest due to the size of finitedifference grids and it achieved by method of Literature [7] issecond because of the calculated amount of high order vibra-tionmode and the theoretical solutionmentioned previouslyis the largest and the most reasonable but the accuracyof two previous methods can be respectively improved bydecreasing grid ratio and increasing the calculated amount

so that the results of the three methods show more excellentagreement which indicates the reasonability and feasibilityof two proposed methods under the condition of neglectingdamping effect

When damping effect is considered as shown inFigure 19 the proposed theoretical solution and literature [7]solution are also well consistent and the former is slightlylarger than the latter Through the comparison and analysisof Figures 14 15 16 and 19 it is found that when usingthe finite difference method the attenuation characteristicof vertical beam response can be clearly shown in the firstsecond and then with the passing of time the vibrationmode is gradually degenerated into constant state whileno attenuation characteristic is shown and the calculatingresult keeps steady state all the time if using the theoreticalmethods So in the first second the calculating results usedthe finite difference method and theoretical methods haveconsiderable difference which is attributed to the theoreticalsolution based on steady-state complex method Specificallythe calculating result using finite difference method is largerthan the result using theoretical methods in the first secondAfter the first second the vibration pattern and the ampli-tude are consistent by using these three methods Besidesthe finite difference method not only can demonstrate theattenuation characteristic of amplitude in terms ofmagnitudein the damping anchorage system but also can illustrate theattenuation characteristic of amplitude in terms of time andits calculating result will be more reliable and reasonable

Generally speaking the finite difference method cannot only be applied to the dynamic response analysis offrame supporting structure under harmonic earthquake but

Mathematical Problems in Engineering 11

t

M

j

1Δt

0(N0) 1 Δl1 N1 Niminus1 Δli k Ni Nnminus1 Δln Nn y

H

A1 Ai An

Low

er p

latfo

rm (s

lope

toe)

w1

1th

row

anch

or

wi(j k) wn

Upp

er p

latfo

rm (s

lope

cres

t)

ith section1th section nth section

ith ro

w an

chor

n minus

1th

row

anch

or

i minus 1

th ro

w an

chor

Figure 17 Finite difference grids of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical methodProposed finite difference method

minus40000

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 18 Solutions of vertical beam with damping effect neglected by different methods

000 025 050 075 100 125 150 175 200 225 250minus15000minus12500minus10000minus7500

minus2500minus5000

0250050007500

10000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

Proposed finite difference method

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical method

minus15000

minus20000

minus5000

minus10000

0

5000

10000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 19 Solutions of vertical beam with damping effect considered by different methods

12 Mathematical Problems in Engineering

also be used to the analysis of random seismic loadingBesides it can demonstrate the attenuation characteristics ofamplitude in terms of both magnitude and time in otherwords it can indicate the damping effect better Thereforethe calculating result obtained by finite difference methodwould be more approximate to actual conditions and hencebe more reliable and reasonable However the analyticalmethod is only confined to the analysis of frame supportingstructure under harmonic earthquake Besides it can onlyreflect the attenuation characteristics of amplitude in termsof magnitude and the peak value of mechanical response gotby analytical method would be underestimated as a resultpotential hazards may be brought aboutWhat is more basedon steady-state complex method its mechanical responseshave initial phase values which are clearly contradictory tothe reality

5 Conclusions

Through the dynamic response analysis of frame supportingstructure of slope under harmonic forced vibration theconclusion follows

(1) Based on certain assumptions and force analysis ofsegment of infinitesimal length the dynamic govern-ing equation of vertical beam under forced vibrationis derived and the analytical solutions to the dynamicresponse of vertical beam under harmonic forcedvibration are obtainedMeanwhile according to finitedifference method the finite difference format andcorresponding calculating procedure for dynamicresponse of vertical beam under forced vibrationare established and finally programmed by usingMATLAB language

(2) In the case studies three methods have been usedin the dynamic response analysis and with dampingeffect neglected or considered respectivelyThe feasi-bility and correctness of the two proposed methodshave been examined and the characteristic of eachmethod has been summarized

(3) Under horizontal harmonic seismic loading themechanical response of vertical beam would syn-chronize correspondingly Its response law showsagreement with the one of two-end cantilevered con-tinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuousbeam In specific engineering design the factors suchas mechanical characteristic of vertical beam andslope supporting effect of anchors should also beconsidered

(4) Compared with analytical solutions the finite differ-ence method can simulate the dynamic response offrame supporting structure under various earthquakeconditions and it also can indicate the damping effectof anchorage system very well hence the calculatingresult obtained ismore reliable and reasonableMore-over the finite differencemethod is achieved based onprogramming calculation its calculation speed and

accuracy can be guaranteed and hence it has broadapplication prospect

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by Project (51308273 41372307) sup-ported by National Natural Science Foundation of China andProject (2010(A)06-b) supported by Science and Technol-ogy Project supported by Yunan Provincial CommunicationDepartment

References

[1] L K Cheng ldquoPresent status and development of groundanchoragesrdquo China Civil Engineering Journal vol 34 no 3 pp7ndash16 2001 (Chinese)

[2] D P Zhou J J Zhang and Y Tang ldquoSeismic damage analysis ofroad slopes in Wenchuan earthquakerdquo Chinese Journal of RockMechanics and Engineering vol 29 no 3 pp 565ndash576 2010(Chinese)

[3] M Yang H T Hu C J Lu et al ldquoCalculation of internal forcesfor prestressed anchor cable frame used in reinforced roadcutsoil sloperdquo Chinese Journal of Rock Mechanics and Engineeringvol 21 no 9 pp 1383ndash1386 2002 (Chinese)

[4] Y H Tian J K Liu and Y F Zhang ldquoInternal force calculationof prestressed anchor cable frame by finite difference methodrdquoJournal of Beijing Jiaotong University vol 31 no 4 pp 22ndash252007 (Chinese)

[5] G H Lin ldquoModel test of anchor grid beam for cut sloperdquoTechnology of Highway and Transport vol 3 pp 19ndash22 2004(Chinese)

[6] B L Zhu M Yang H T Hu et al ldquoTesting study on internalforces for prestressed anchor cable frame in reinforced soilsloperdquo Chinese Journal of Rock Mechanics and Engineering vol24 no 4 pp 697ndash702 2005 (Chinese)

[7] J H Dong Y P Zhu Y Zhou et al ldquoDynamic calculationmodel and seismic response for frame supporting structurewith prestressed anchorsrdquo Science China Technological Sciencesvol 53 no 7 pp 1957ndash1966 2010

[8] S Catal ldquoSolution of free vibration equations of beam onelastic soil by using differential transform methodrdquo AppliedMathematical Modelling vol 32 no 9 pp 1744ndash1757 2008

[9] M Balkaya M O Kaya and A Saglamer ldquoAnalysis of thevibration of an elastic beam supported on elastic soil using thedifferential transform methodrdquo Archive of Applied Mechanicsvol 79 no 2 pp 135ndash146 2009

[10] M Ansari E Esmailzadeh and D Younesian ldquoFrequencyanalysis of finite beams on nonlinear KelvinVoight foundationunder moving loadsrdquo Journal of Sound and Vibration vol 330no 7 pp 1455ndash1471 2011

[11] O Demirdag and Y Yesilce ldquoSolution of free vibration equationof elastically supported Timoshenko columns with a tip massby differential transform methodrdquo Advances in EngineeringSoftware vol 42 no 10 pp 860ndash867 2011

Mathematical Problems in Engineering 13

[12] MH Yas andN Samadi ldquoFree vibrations and buckling analysisof carbon nanotube-reinforced composite Timoshenko beamson elastic foundationrdquo International Journal of Pressure Vesselsand Piping vol 98 pp 119ndash128 2012

[13] R A Jafari-Talookolaei M Abedi M H Kargarnovin et alldquoAn analytical approach for the free vibration analysis ofgenerally laminated composite beams with shear effect androtary inertiardquo International Journal of Mechanical Sciences vol65 no 1 pp 97ndash104 2012

[14] S W Qi F Q Wu F Z Yan et al Dynamic Response Analysesof Rock Slope Science Press Beijing China 2007 Chinese

[15] J H Dong and Y P Zhu ldquoDynamic calculation model andseismic response for the system of soil nailing and surroundingsoilrdquo Chinese Journal of Theoretical and Applied Mechanics vol41 no 2 pp 236ndash242 2009

[16] Z L Xu Elasticity Higher Education Press Beijing China2009 Chinese

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

00 15 30 45 60 75 90 105 120minus003

minus002

minus001

000

001

002

003wi(

m)

t = 1125 st = 1375 s

y(m)

Figure 11 Displacement response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120minus30000

minus20000

minus10000

0

10000

20000

30000

t = 1125 st = 1375 s

Mi(

Nmiddotm

)

y(m)

Figure 12 Bending moment response along vertical beam at typicalmoment

00 15 30 45 60 75 90 105 120

minus30000

minus40000

minus20000

minus10000

0

10000

20000

30000

40000

t = 1125 st = 1375 s

Qi(

N)

y(m)

Figure 13 Shear force response along vertical beam at typicalmoment

times3

minus3

2

minus2

1

25 2 15 1 05 0 02

46

810

12minus10

w(yt)(

m)

y(m)

t(s)

10minus3

Figure 14 Displace response of vertical beam

minus 119908119894(119895 119896 + 2) minus 119908

119894(119895 119896 minus 2)]

minus 119908119894(119895 minus 1 119896) + Δ119905

2120595119894(119896) 1198861015840

119892(119895Δ119905)

(19)1199081(119895 0) = 3119908

1(119895 2) minus 2119908

1(119895 3)

1199081(119895 1) = 2119908

1(119895 2) minus 119908

1(119895 3)

119908119899(119895119873119899minus 1) = 2119908

119899(119895119873119899minus 2) minus 119908

119899(119895119873119899minus 3)

119908119899(119895119873119899) = 3119908

119899(119895119873119899minus 2) minus 2119908

119899(119895119873119899minus 3)

(20)119908119894(119895119873119894) = 119908119894+1

(119895119873119894) = 1199061198942(1198711198942 119895Δ119905) cos120593

119894

119908119894+1

(119895119873119894+ 1)

= ([

6Δ1198972

119894

Δ1198972

119894+1

+

10 (Δ119897119894+ Δ119897119894+1

)

Δ119897119894+1

minus 6]119908119894(119895119873119894)

minus 7119908119894(119895119873119894minus 2) + 3119908

119894(119895119873119894minus 3)

+ [

12Δ1198972

119894

Δ1198972

119894+1

+

5Δ119897119894

Δ119897119894+1

]119908119894+1

(119895119873119894+ 2)

minus

3Δ1198972

119894

Δ1198972

119894+1

119908119894+1

(119895119873119894+ 3))

times (15 [

Δ119897119894

Δ119897119894+1

+

Δ1198972

119894

Δ1198972

119894+1

])

minus1

119908119894(119895119873119894minus 1)

=

2 (Δ119897119894+ Δ119897119894+1

)

3Δ119897119894+1

119908119894(119895119873119894) +

1

3

119908119894(119895119873119894minus 2)

minus

Δ119897119894

Δ119897119894+1

119908119894+1

(119895119873119894+ 1) +

Δ119897119894

3Δ119897119894+1

119908119894+1

(119895119873119894+ 2)

(21)

According to the finite difference method mentionedabove for a specific engineering problem the displacementof grid points in the initial and the first row can be calculatedby (18) For the second row the displacement of all thegrid nodes in interior columns can be decided by (19) thegrid nodes in exterior boundary columns and their adjacentcolumns (columns 0 1119873

119899minus1 and119873

119899) can be decided by (20)

and based on (21) the node displacement of all the interiorboundary adjacent columns can be achieved Likewise the

10 Mathematical Problems in Engineering

30002000

minus2000minus3000

1000

25 215 1

050 0

24

68

1012minus1000

0

y(m)

M(yt)(

Nmiddotm

)t(s)

Figure 15 Bending moment response of vertical beam

4000

minus4000

2000

252

151

050 0

24

68

1012

minus20000

Q(yt)(

N)

y(m)

t(s)

Figure 16 Shear force response of vertical beam

displacement for all the grid nodes from row 3 to row119872 canbe decided

After getting the displacement of all grid nodes accordingto 119872119894= 119864119868(120597

21199081198941205971199102) and 119876

119894= 119864119868(120597

31199081198941205971199103) the numerical

solutions to the bending moment and shear force responseof vertical beam can be obtained by conducting the finitedifference to the displacement of all grid nodes again

42 Comparative Analysis and Discussion Similarly basedon previous case study the slope is considered under hori-zontal harmonic seismic loading and with anchoring systemdamping neglected or considered respectively The numer-ical solutions to the dynamic response of vertical beam canbe achieved by using finite difference method and MATLABprogramming and 119903

119894= Δ119905Δ

2119897119894= 000001(025 times 025) =

000016 le radic1205721= 0002 (119894 = 1 sim 5)

In order to verify the correctness of the two methodsabove the solutions of intersection 119910 = 15m by differentmethods andwith damping effect neglected or considered areshown in Figures 18 and 19

When damping effect is neglected according to Figure 18it is found that the calculation results by the three methodsshare similar vibration trend and only show slight differencein amplitude generally speaking the finite element solutionmentioned previously is the smallest due to the size of finitedifference grids and it achieved by method of Literature [7] issecond because of the calculated amount of high order vibra-tionmode and the theoretical solutionmentioned previouslyis the largest and the most reasonable but the accuracyof two previous methods can be respectively improved bydecreasing grid ratio and increasing the calculated amount

so that the results of the three methods show more excellentagreement which indicates the reasonability and feasibilityof two proposed methods under the condition of neglectingdamping effect

When damping effect is considered as shown inFigure 19 the proposed theoretical solution and literature [7]solution are also well consistent and the former is slightlylarger than the latter Through the comparison and analysisof Figures 14 15 16 and 19 it is found that when usingthe finite difference method the attenuation characteristicof vertical beam response can be clearly shown in the firstsecond and then with the passing of time the vibrationmode is gradually degenerated into constant state whileno attenuation characteristic is shown and the calculatingresult keeps steady state all the time if using the theoreticalmethods So in the first second the calculating results usedthe finite difference method and theoretical methods haveconsiderable difference which is attributed to the theoreticalsolution based on steady-state complex method Specificallythe calculating result using finite difference method is largerthan the result using theoretical methods in the first secondAfter the first second the vibration pattern and the ampli-tude are consistent by using these three methods Besidesthe finite difference method not only can demonstrate theattenuation characteristic of amplitude in terms ofmagnitudein the damping anchorage system but also can illustrate theattenuation characteristic of amplitude in terms of time andits calculating result will be more reliable and reasonable

Generally speaking the finite difference method cannot only be applied to the dynamic response analysis offrame supporting structure under harmonic earthquake but

Mathematical Problems in Engineering 11

t

M

j

1Δt

0(N0) 1 Δl1 N1 Niminus1 Δli k Ni Nnminus1 Δln Nn y

H

A1 Ai An

Low

er p

latfo

rm (s

lope

toe)

w1

1th

row

anch

or

wi(j k) wn

Upp

er p

latfo

rm (s

lope

cres

t)

ith section1th section nth section

ith ro

w an

chor

n minus

1th

row

anch

or

i minus 1

th ro

w an

chor

Figure 17 Finite difference grids of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical methodProposed finite difference method

minus40000

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 18 Solutions of vertical beam with damping effect neglected by different methods

000 025 050 075 100 125 150 175 200 225 250minus15000minus12500minus10000minus7500

minus2500minus5000

0250050007500

10000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

Proposed finite difference method

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical method

minus15000

minus20000

minus5000

minus10000

0

5000

10000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 19 Solutions of vertical beam with damping effect considered by different methods

12 Mathematical Problems in Engineering

also be used to the analysis of random seismic loadingBesides it can demonstrate the attenuation characteristics ofamplitude in terms of both magnitude and time in otherwords it can indicate the damping effect better Thereforethe calculating result obtained by finite difference methodwould be more approximate to actual conditions and hencebe more reliable and reasonable However the analyticalmethod is only confined to the analysis of frame supportingstructure under harmonic earthquake Besides it can onlyreflect the attenuation characteristics of amplitude in termsof magnitude and the peak value of mechanical response gotby analytical method would be underestimated as a resultpotential hazards may be brought aboutWhat is more basedon steady-state complex method its mechanical responseshave initial phase values which are clearly contradictory tothe reality

5 Conclusions

Through the dynamic response analysis of frame supportingstructure of slope under harmonic forced vibration theconclusion follows

(1) Based on certain assumptions and force analysis ofsegment of infinitesimal length the dynamic govern-ing equation of vertical beam under forced vibrationis derived and the analytical solutions to the dynamicresponse of vertical beam under harmonic forcedvibration are obtainedMeanwhile according to finitedifference method the finite difference format andcorresponding calculating procedure for dynamicresponse of vertical beam under forced vibrationare established and finally programmed by usingMATLAB language

(2) In the case studies three methods have been usedin the dynamic response analysis and with dampingeffect neglected or considered respectivelyThe feasi-bility and correctness of the two proposed methodshave been examined and the characteristic of eachmethod has been summarized

(3) Under horizontal harmonic seismic loading themechanical response of vertical beam would syn-chronize correspondingly Its response law showsagreement with the one of two-end cantilevered con-tinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuousbeam In specific engineering design the factors suchas mechanical characteristic of vertical beam andslope supporting effect of anchors should also beconsidered

(4) Compared with analytical solutions the finite differ-ence method can simulate the dynamic response offrame supporting structure under various earthquakeconditions and it also can indicate the damping effectof anchorage system very well hence the calculatingresult obtained ismore reliable and reasonableMore-over the finite differencemethod is achieved based onprogramming calculation its calculation speed and

accuracy can be guaranteed and hence it has broadapplication prospect

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by Project (51308273 41372307) sup-ported by National Natural Science Foundation of China andProject (2010(A)06-b) supported by Science and Technol-ogy Project supported by Yunan Provincial CommunicationDepartment

References

[1] L K Cheng ldquoPresent status and development of groundanchoragesrdquo China Civil Engineering Journal vol 34 no 3 pp7ndash16 2001 (Chinese)

[2] D P Zhou J J Zhang and Y Tang ldquoSeismic damage analysis ofroad slopes in Wenchuan earthquakerdquo Chinese Journal of RockMechanics and Engineering vol 29 no 3 pp 565ndash576 2010(Chinese)

[3] M Yang H T Hu C J Lu et al ldquoCalculation of internal forcesfor prestressed anchor cable frame used in reinforced roadcutsoil sloperdquo Chinese Journal of Rock Mechanics and Engineeringvol 21 no 9 pp 1383ndash1386 2002 (Chinese)

[4] Y H Tian J K Liu and Y F Zhang ldquoInternal force calculationof prestressed anchor cable frame by finite difference methodrdquoJournal of Beijing Jiaotong University vol 31 no 4 pp 22ndash252007 (Chinese)

[5] G H Lin ldquoModel test of anchor grid beam for cut sloperdquoTechnology of Highway and Transport vol 3 pp 19ndash22 2004(Chinese)

[6] B L Zhu M Yang H T Hu et al ldquoTesting study on internalforces for prestressed anchor cable frame in reinforced soilsloperdquo Chinese Journal of Rock Mechanics and Engineering vol24 no 4 pp 697ndash702 2005 (Chinese)

[7] J H Dong Y P Zhu Y Zhou et al ldquoDynamic calculationmodel and seismic response for frame supporting structurewith prestressed anchorsrdquo Science China Technological Sciencesvol 53 no 7 pp 1957ndash1966 2010

[8] S Catal ldquoSolution of free vibration equations of beam onelastic soil by using differential transform methodrdquo AppliedMathematical Modelling vol 32 no 9 pp 1744ndash1757 2008

[9] M Balkaya M O Kaya and A Saglamer ldquoAnalysis of thevibration of an elastic beam supported on elastic soil using thedifferential transform methodrdquo Archive of Applied Mechanicsvol 79 no 2 pp 135ndash146 2009

[10] M Ansari E Esmailzadeh and D Younesian ldquoFrequencyanalysis of finite beams on nonlinear KelvinVoight foundationunder moving loadsrdquo Journal of Sound and Vibration vol 330no 7 pp 1455ndash1471 2011

[11] O Demirdag and Y Yesilce ldquoSolution of free vibration equationof elastically supported Timoshenko columns with a tip massby differential transform methodrdquo Advances in EngineeringSoftware vol 42 no 10 pp 860ndash867 2011

Mathematical Problems in Engineering 13

[12] MH Yas andN Samadi ldquoFree vibrations and buckling analysisof carbon nanotube-reinforced composite Timoshenko beamson elastic foundationrdquo International Journal of Pressure Vesselsand Piping vol 98 pp 119ndash128 2012

[13] R A Jafari-Talookolaei M Abedi M H Kargarnovin et alldquoAn analytical approach for the free vibration analysis ofgenerally laminated composite beams with shear effect androtary inertiardquo International Journal of Mechanical Sciences vol65 no 1 pp 97ndash104 2012

[14] S W Qi F Q Wu F Z Yan et al Dynamic Response Analysesof Rock Slope Science Press Beijing China 2007 Chinese

[15] J H Dong and Y P Zhu ldquoDynamic calculation model andseismic response for the system of soil nailing and surroundingsoilrdquo Chinese Journal of Theoretical and Applied Mechanics vol41 no 2 pp 236ndash242 2009

[16] Z L Xu Elasticity Higher Education Press Beijing China2009 Chinese

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

30002000

minus2000minus3000

1000

25 215 1

050 0

24

68

1012minus1000

0

y(m)

M(yt)(

Nmiddotm

)t(s)

Figure 15 Bending moment response of vertical beam

4000

minus4000

2000

252

151

050 0

24

68

1012

minus20000

Q(yt)(

N)

y(m)

t(s)

Figure 16 Shear force response of vertical beam

displacement for all the grid nodes from row 3 to row119872 canbe decided

After getting the displacement of all grid nodes accordingto 119872119894= 119864119868(120597

21199081198941205971199102) and 119876

119894= 119864119868(120597

31199081198941205971199103) the numerical

solutions to the bending moment and shear force responseof vertical beam can be obtained by conducting the finitedifference to the displacement of all grid nodes again

42 Comparative Analysis and Discussion Similarly basedon previous case study the slope is considered under hori-zontal harmonic seismic loading and with anchoring systemdamping neglected or considered respectively The numer-ical solutions to the dynamic response of vertical beam canbe achieved by using finite difference method and MATLABprogramming and 119903

119894= Δ119905Δ

2119897119894= 000001(025 times 025) =

000016 le radic1205721= 0002 (119894 = 1 sim 5)

In order to verify the correctness of the two methodsabove the solutions of intersection 119910 = 15m by differentmethods andwith damping effect neglected or considered areshown in Figures 18 and 19

When damping effect is neglected according to Figure 18it is found that the calculation results by the three methodsshare similar vibration trend and only show slight differencein amplitude generally speaking the finite element solutionmentioned previously is the smallest due to the size of finitedifference grids and it achieved by method of Literature [7] issecond because of the calculated amount of high order vibra-tionmode and the theoretical solutionmentioned previouslyis the largest and the most reasonable but the accuracyof two previous methods can be respectively improved bydecreasing grid ratio and increasing the calculated amount

so that the results of the three methods show more excellentagreement which indicates the reasonability and feasibilityof two proposed methods under the condition of neglectingdamping effect

When damping effect is considered as shown inFigure 19 the proposed theoretical solution and literature [7]solution are also well consistent and the former is slightlylarger than the latter Through the comparison and analysisof Figures 14 15 16 and 19 it is found that when usingthe finite difference method the attenuation characteristicof vertical beam response can be clearly shown in the firstsecond and then with the passing of time the vibrationmode is gradually degenerated into constant state whileno attenuation characteristic is shown and the calculatingresult keeps steady state all the time if using the theoreticalmethods So in the first second the calculating results usedthe finite difference method and theoretical methods haveconsiderable difference which is attributed to the theoreticalsolution based on steady-state complex method Specificallythe calculating result using finite difference method is largerthan the result using theoretical methods in the first secondAfter the first second the vibration pattern and the ampli-tude are consistent by using these three methods Besidesthe finite difference method not only can demonstrate theattenuation characteristic of amplitude in terms ofmagnitudein the damping anchorage system but also can illustrate theattenuation characteristic of amplitude in terms of time andits calculating result will be more reliable and reasonable

Generally speaking the finite difference method cannot only be applied to the dynamic response analysis offrame supporting structure under harmonic earthquake but

Mathematical Problems in Engineering 11

t

M

j

1Δt

0(N0) 1 Δl1 N1 Niminus1 Δli k Ni Nnminus1 Δln Nn y

H

A1 Ai An

Low

er p

latfo

rm (s

lope

toe)

w1

1th

row

anch

or

wi(j k) wn

Upp

er p

latfo

rm (s

lope

cres

t)

ith section1th section nth section

ith ro

w an

chor

n minus

1th

row

anch

or

i minus 1

th ro

w an

chor

Figure 17 Finite difference grids of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical methodProposed finite difference method

minus40000

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 18 Solutions of vertical beam with damping effect neglected by different methods

000 025 050 075 100 125 150 175 200 225 250minus15000minus12500minus10000minus7500

minus2500minus5000

0250050007500

10000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

Proposed finite difference method

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical method

minus15000

minus20000

minus5000

minus10000

0

5000

10000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 19 Solutions of vertical beam with damping effect considered by different methods

12 Mathematical Problems in Engineering

also be used to the analysis of random seismic loadingBesides it can demonstrate the attenuation characteristics ofamplitude in terms of both magnitude and time in otherwords it can indicate the damping effect better Thereforethe calculating result obtained by finite difference methodwould be more approximate to actual conditions and hencebe more reliable and reasonable However the analyticalmethod is only confined to the analysis of frame supportingstructure under harmonic earthquake Besides it can onlyreflect the attenuation characteristics of amplitude in termsof magnitude and the peak value of mechanical response gotby analytical method would be underestimated as a resultpotential hazards may be brought aboutWhat is more basedon steady-state complex method its mechanical responseshave initial phase values which are clearly contradictory tothe reality

5 Conclusions

Through the dynamic response analysis of frame supportingstructure of slope under harmonic forced vibration theconclusion follows

(1) Based on certain assumptions and force analysis ofsegment of infinitesimal length the dynamic govern-ing equation of vertical beam under forced vibrationis derived and the analytical solutions to the dynamicresponse of vertical beam under harmonic forcedvibration are obtainedMeanwhile according to finitedifference method the finite difference format andcorresponding calculating procedure for dynamicresponse of vertical beam under forced vibrationare established and finally programmed by usingMATLAB language

(2) In the case studies three methods have been usedin the dynamic response analysis and with dampingeffect neglected or considered respectivelyThe feasi-bility and correctness of the two proposed methodshave been examined and the characteristic of eachmethod has been summarized

(3) Under horizontal harmonic seismic loading themechanical response of vertical beam would syn-chronize correspondingly Its response law showsagreement with the one of two-end cantilevered con-tinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuousbeam In specific engineering design the factors suchas mechanical characteristic of vertical beam andslope supporting effect of anchors should also beconsidered

(4) Compared with analytical solutions the finite differ-ence method can simulate the dynamic response offrame supporting structure under various earthquakeconditions and it also can indicate the damping effectof anchorage system very well hence the calculatingresult obtained ismore reliable and reasonableMore-over the finite differencemethod is achieved based onprogramming calculation its calculation speed and

accuracy can be guaranteed and hence it has broadapplication prospect

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by Project (51308273 41372307) sup-ported by National Natural Science Foundation of China andProject (2010(A)06-b) supported by Science and Technol-ogy Project supported by Yunan Provincial CommunicationDepartment

References

[1] L K Cheng ldquoPresent status and development of groundanchoragesrdquo China Civil Engineering Journal vol 34 no 3 pp7ndash16 2001 (Chinese)

[2] D P Zhou J J Zhang and Y Tang ldquoSeismic damage analysis ofroad slopes in Wenchuan earthquakerdquo Chinese Journal of RockMechanics and Engineering vol 29 no 3 pp 565ndash576 2010(Chinese)

[3] M Yang H T Hu C J Lu et al ldquoCalculation of internal forcesfor prestressed anchor cable frame used in reinforced roadcutsoil sloperdquo Chinese Journal of Rock Mechanics and Engineeringvol 21 no 9 pp 1383ndash1386 2002 (Chinese)

[4] Y H Tian J K Liu and Y F Zhang ldquoInternal force calculationof prestressed anchor cable frame by finite difference methodrdquoJournal of Beijing Jiaotong University vol 31 no 4 pp 22ndash252007 (Chinese)

[5] G H Lin ldquoModel test of anchor grid beam for cut sloperdquoTechnology of Highway and Transport vol 3 pp 19ndash22 2004(Chinese)

[6] B L Zhu M Yang H T Hu et al ldquoTesting study on internalforces for prestressed anchor cable frame in reinforced soilsloperdquo Chinese Journal of Rock Mechanics and Engineering vol24 no 4 pp 697ndash702 2005 (Chinese)

[7] J H Dong Y P Zhu Y Zhou et al ldquoDynamic calculationmodel and seismic response for frame supporting structurewith prestressed anchorsrdquo Science China Technological Sciencesvol 53 no 7 pp 1957ndash1966 2010

[8] S Catal ldquoSolution of free vibration equations of beam onelastic soil by using differential transform methodrdquo AppliedMathematical Modelling vol 32 no 9 pp 1744ndash1757 2008

[9] M Balkaya M O Kaya and A Saglamer ldquoAnalysis of thevibration of an elastic beam supported on elastic soil using thedifferential transform methodrdquo Archive of Applied Mechanicsvol 79 no 2 pp 135ndash146 2009

[10] M Ansari E Esmailzadeh and D Younesian ldquoFrequencyanalysis of finite beams on nonlinear KelvinVoight foundationunder moving loadsrdquo Journal of Sound and Vibration vol 330no 7 pp 1455ndash1471 2011

[11] O Demirdag and Y Yesilce ldquoSolution of free vibration equationof elastically supported Timoshenko columns with a tip massby differential transform methodrdquo Advances in EngineeringSoftware vol 42 no 10 pp 860ndash867 2011

Mathematical Problems in Engineering 13

[12] MH Yas andN Samadi ldquoFree vibrations and buckling analysisof carbon nanotube-reinforced composite Timoshenko beamson elastic foundationrdquo International Journal of Pressure Vesselsand Piping vol 98 pp 119ndash128 2012

[13] R A Jafari-Talookolaei M Abedi M H Kargarnovin et alldquoAn analytical approach for the free vibration analysis ofgenerally laminated composite beams with shear effect androtary inertiardquo International Journal of Mechanical Sciences vol65 no 1 pp 97ndash104 2012

[14] S W Qi F Q Wu F Z Yan et al Dynamic Response Analysesof Rock Slope Science Press Beijing China 2007 Chinese

[15] J H Dong and Y P Zhu ldquoDynamic calculation model andseismic response for the system of soil nailing and surroundingsoilrdquo Chinese Journal of Theoretical and Applied Mechanics vol41 no 2 pp 236ndash242 2009

[16] Z L Xu Elasticity Higher Education Press Beijing China2009 Chinese

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

t

M

j

1Δt

0(N0) 1 Δl1 N1 Niminus1 Δli k Ni Nnminus1 Δln Nn y

H

A1 Ai An

Low

er p

latfo

rm (s

lope

toe)

w1

1th

row

anch

or

wi(j k) wn

Upp

er p

latfo

rm (s

lope

cres

t)

ith section1th section nth section

ith ro

w an

chor

n minus

1th

row

anch

or

i minus 1

th ro

w an

chor

Figure 17 Finite difference grids of vertical beam

000 025 050 075 100 125 150 175 200 225 250minus30000minus25000minus20000minus15000minus10000minus5000

05000

100001500020000250003000035000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical methodProposed finite difference method

minus40000

minus30000

minus20000

minus10000

0

10000

20000

30000

40000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 18 Solutions of vertical beam with damping effect neglected by different methods

000 025 050 075 100 125 150 175 200 225 250minus15000minus12500minus10000minus7500

minus2500minus5000

0250050007500

10000

Proposed theoretical methodProposed finite difference methodLiterature [7] method

My=15

m(N

middotm)

t(s)

(a) Bending moment response at intersection 119910 = 15m

Proposed finite difference method

000 025 050 075 100 125 150 175 200 225 250

Proposed theoretical method

minus15000

minus20000

minus5000

minus10000

0

5000

10000

Literature [7] method

Qy=15

m(N

)

t(s)

(b) Shear force response at left hand of intersection 119910 = 15m

Figure 19 Solutions of vertical beam with damping effect considered by different methods

12 Mathematical Problems in Engineering

also be used to the analysis of random seismic loadingBesides it can demonstrate the attenuation characteristics ofamplitude in terms of both magnitude and time in otherwords it can indicate the damping effect better Thereforethe calculating result obtained by finite difference methodwould be more approximate to actual conditions and hencebe more reliable and reasonable However the analyticalmethod is only confined to the analysis of frame supportingstructure under harmonic earthquake Besides it can onlyreflect the attenuation characteristics of amplitude in termsof magnitude and the peak value of mechanical response gotby analytical method would be underestimated as a resultpotential hazards may be brought aboutWhat is more basedon steady-state complex method its mechanical responseshave initial phase values which are clearly contradictory tothe reality

5 Conclusions

Through the dynamic response analysis of frame supportingstructure of slope under harmonic forced vibration theconclusion follows

(1) Based on certain assumptions and force analysis ofsegment of infinitesimal length the dynamic govern-ing equation of vertical beam under forced vibrationis derived and the analytical solutions to the dynamicresponse of vertical beam under harmonic forcedvibration are obtainedMeanwhile according to finitedifference method the finite difference format andcorresponding calculating procedure for dynamicresponse of vertical beam under forced vibrationare established and finally programmed by usingMATLAB language

(2) In the case studies three methods have been usedin the dynamic response analysis and with dampingeffect neglected or considered respectivelyThe feasi-bility and correctness of the two proposed methodshave been examined and the characteristic of eachmethod has been summarized

(3) Under horizontal harmonic seismic loading themechanical response of vertical beam would syn-chronize correspondingly Its response law showsagreement with the one of two-end cantilevered con-tinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuousbeam In specific engineering design the factors suchas mechanical characteristic of vertical beam andslope supporting effect of anchors should also beconsidered

(4) Compared with analytical solutions the finite differ-ence method can simulate the dynamic response offrame supporting structure under various earthquakeconditions and it also can indicate the damping effectof anchorage system very well hence the calculatingresult obtained ismore reliable and reasonableMore-over the finite differencemethod is achieved based onprogramming calculation its calculation speed and

accuracy can be guaranteed and hence it has broadapplication prospect

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by Project (51308273 41372307) sup-ported by National Natural Science Foundation of China andProject (2010(A)06-b) supported by Science and Technol-ogy Project supported by Yunan Provincial CommunicationDepartment

References

[1] L K Cheng ldquoPresent status and development of groundanchoragesrdquo China Civil Engineering Journal vol 34 no 3 pp7ndash16 2001 (Chinese)

[2] D P Zhou J J Zhang and Y Tang ldquoSeismic damage analysis ofroad slopes in Wenchuan earthquakerdquo Chinese Journal of RockMechanics and Engineering vol 29 no 3 pp 565ndash576 2010(Chinese)

[3] M Yang H T Hu C J Lu et al ldquoCalculation of internal forcesfor prestressed anchor cable frame used in reinforced roadcutsoil sloperdquo Chinese Journal of Rock Mechanics and Engineeringvol 21 no 9 pp 1383ndash1386 2002 (Chinese)

[4] Y H Tian J K Liu and Y F Zhang ldquoInternal force calculationof prestressed anchor cable frame by finite difference methodrdquoJournal of Beijing Jiaotong University vol 31 no 4 pp 22ndash252007 (Chinese)

[5] G H Lin ldquoModel test of anchor grid beam for cut sloperdquoTechnology of Highway and Transport vol 3 pp 19ndash22 2004(Chinese)

[6] B L Zhu M Yang H T Hu et al ldquoTesting study on internalforces for prestressed anchor cable frame in reinforced soilsloperdquo Chinese Journal of Rock Mechanics and Engineering vol24 no 4 pp 697ndash702 2005 (Chinese)

[7] J H Dong Y P Zhu Y Zhou et al ldquoDynamic calculationmodel and seismic response for frame supporting structurewith prestressed anchorsrdquo Science China Technological Sciencesvol 53 no 7 pp 1957ndash1966 2010

[8] S Catal ldquoSolution of free vibration equations of beam onelastic soil by using differential transform methodrdquo AppliedMathematical Modelling vol 32 no 9 pp 1744ndash1757 2008

[9] M Balkaya M O Kaya and A Saglamer ldquoAnalysis of thevibration of an elastic beam supported on elastic soil using thedifferential transform methodrdquo Archive of Applied Mechanicsvol 79 no 2 pp 135ndash146 2009

[10] M Ansari E Esmailzadeh and D Younesian ldquoFrequencyanalysis of finite beams on nonlinear KelvinVoight foundationunder moving loadsrdquo Journal of Sound and Vibration vol 330no 7 pp 1455ndash1471 2011

[11] O Demirdag and Y Yesilce ldquoSolution of free vibration equationof elastically supported Timoshenko columns with a tip massby differential transform methodrdquo Advances in EngineeringSoftware vol 42 no 10 pp 860ndash867 2011

Mathematical Problems in Engineering 13

[12] MH Yas andN Samadi ldquoFree vibrations and buckling analysisof carbon nanotube-reinforced composite Timoshenko beamson elastic foundationrdquo International Journal of Pressure Vesselsand Piping vol 98 pp 119ndash128 2012

[13] R A Jafari-Talookolaei M Abedi M H Kargarnovin et alldquoAn analytical approach for the free vibration analysis ofgenerally laminated composite beams with shear effect androtary inertiardquo International Journal of Mechanical Sciences vol65 no 1 pp 97ndash104 2012

[14] S W Qi F Q Wu F Z Yan et al Dynamic Response Analysesof Rock Slope Science Press Beijing China 2007 Chinese

[15] J H Dong and Y P Zhu ldquoDynamic calculation model andseismic response for the system of soil nailing and surroundingsoilrdquo Chinese Journal of Theoretical and Applied Mechanics vol41 no 2 pp 236ndash242 2009

[16] Z L Xu Elasticity Higher Education Press Beijing China2009 Chinese

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

also be used to the analysis of random seismic loadingBesides it can demonstrate the attenuation characteristics ofamplitude in terms of both magnitude and time in otherwords it can indicate the damping effect better Thereforethe calculating result obtained by finite difference methodwould be more approximate to actual conditions and hencebe more reliable and reasonable However the analyticalmethod is only confined to the analysis of frame supportingstructure under harmonic earthquake Besides it can onlyreflect the attenuation characteristics of amplitude in termsof magnitude and the peak value of mechanical response gotby analytical method would be underestimated as a resultpotential hazards may be brought aboutWhat is more basedon steady-state complex method its mechanical responseshave initial phase values which are clearly contradictory tothe reality

5 Conclusions

Through the dynamic response analysis of frame supportingstructure of slope under harmonic forced vibration theconclusion follows

(1) Based on certain assumptions and force analysis ofsegment of infinitesimal length the dynamic govern-ing equation of vertical beam under forced vibrationis derived and the analytical solutions to the dynamicresponse of vertical beam under harmonic forcedvibration are obtainedMeanwhile according to finitedifference method the finite difference format andcorresponding calculating procedure for dynamicresponse of vertical beam under forced vibrationare established and finally programmed by usingMATLAB language

(2) In the case studies three methods have been usedin the dynamic response analysis and with dampingeffect neglected or considered respectivelyThe feasi-bility and correctness of the two proposed methodshave been examined and the characteristic of eachmethod has been summarized

(3) Under horizontal harmonic seismic loading themechanical response of vertical beam would syn-chronize correspondingly Its response law showsagreement with the one of two-end cantilevered con-tinuous beam and the anchor to the frame supportingstructure is just like the bearing to the continuousbeam In specific engineering design the factors suchas mechanical characteristic of vertical beam andslope supporting effect of anchors should also beconsidered

(4) Compared with analytical solutions the finite differ-ence method can simulate the dynamic response offrame supporting structure under various earthquakeconditions and it also can indicate the damping effectof anchorage system very well hence the calculatingresult obtained ismore reliable and reasonableMore-over the finite differencemethod is achieved based onprogramming calculation its calculation speed and

accuracy can be guaranteed and hence it has broadapplication prospect

Conflict of Interests

The authors declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by Project (51308273 41372307) sup-ported by National Natural Science Foundation of China andProject (2010(A)06-b) supported by Science and Technol-ogy Project supported by Yunan Provincial CommunicationDepartment

References

[1] L K Cheng ldquoPresent status and development of groundanchoragesrdquo China Civil Engineering Journal vol 34 no 3 pp7ndash16 2001 (Chinese)

[2] D P Zhou J J Zhang and Y Tang ldquoSeismic damage analysis ofroad slopes in Wenchuan earthquakerdquo Chinese Journal of RockMechanics and Engineering vol 29 no 3 pp 565ndash576 2010(Chinese)

[3] M Yang H T Hu C J Lu et al ldquoCalculation of internal forcesfor prestressed anchor cable frame used in reinforced roadcutsoil sloperdquo Chinese Journal of Rock Mechanics and Engineeringvol 21 no 9 pp 1383ndash1386 2002 (Chinese)

[4] Y H Tian J K Liu and Y F Zhang ldquoInternal force calculationof prestressed anchor cable frame by finite difference methodrdquoJournal of Beijing Jiaotong University vol 31 no 4 pp 22ndash252007 (Chinese)

[5] G H Lin ldquoModel test of anchor grid beam for cut sloperdquoTechnology of Highway and Transport vol 3 pp 19ndash22 2004(Chinese)

[6] B L Zhu M Yang H T Hu et al ldquoTesting study on internalforces for prestressed anchor cable frame in reinforced soilsloperdquo Chinese Journal of Rock Mechanics and Engineering vol24 no 4 pp 697ndash702 2005 (Chinese)

[7] J H Dong Y P Zhu Y Zhou et al ldquoDynamic calculationmodel and seismic response for frame supporting structurewith prestressed anchorsrdquo Science China Technological Sciencesvol 53 no 7 pp 1957ndash1966 2010

[8] S Catal ldquoSolution of free vibration equations of beam onelastic soil by using differential transform methodrdquo AppliedMathematical Modelling vol 32 no 9 pp 1744ndash1757 2008

[9] M Balkaya M O Kaya and A Saglamer ldquoAnalysis of thevibration of an elastic beam supported on elastic soil using thedifferential transform methodrdquo Archive of Applied Mechanicsvol 79 no 2 pp 135ndash146 2009

[10] M Ansari E Esmailzadeh and D Younesian ldquoFrequencyanalysis of finite beams on nonlinear KelvinVoight foundationunder moving loadsrdquo Journal of Sound and Vibration vol 330no 7 pp 1455ndash1471 2011

[11] O Demirdag and Y Yesilce ldquoSolution of free vibration equationof elastically supported Timoshenko columns with a tip massby differential transform methodrdquo Advances in EngineeringSoftware vol 42 no 10 pp 860ndash867 2011

Mathematical Problems in Engineering 13

[12] MH Yas andN Samadi ldquoFree vibrations and buckling analysisof carbon nanotube-reinforced composite Timoshenko beamson elastic foundationrdquo International Journal of Pressure Vesselsand Piping vol 98 pp 119ndash128 2012

[13] R A Jafari-Talookolaei M Abedi M H Kargarnovin et alldquoAn analytical approach for the free vibration analysis ofgenerally laminated composite beams with shear effect androtary inertiardquo International Journal of Mechanical Sciences vol65 no 1 pp 97ndash104 2012

[14] S W Qi F Q Wu F Z Yan et al Dynamic Response Analysesof Rock Slope Science Press Beijing China 2007 Chinese

[15] J H Dong and Y P Zhu ldquoDynamic calculation model andseismic response for the system of soil nailing and surroundingsoilrdquo Chinese Journal of Theoretical and Applied Mechanics vol41 no 2 pp 236ndash242 2009

[16] Z L Xu Elasticity Higher Education Press Beijing China2009 Chinese

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

[12] MH Yas andN Samadi ldquoFree vibrations and buckling analysisof carbon nanotube-reinforced composite Timoshenko beamson elastic foundationrdquo International Journal of Pressure Vesselsand Piping vol 98 pp 119ndash128 2012

[13] R A Jafari-Talookolaei M Abedi M H Kargarnovin et alldquoAn analytical approach for the free vibration analysis ofgenerally laminated composite beams with shear effect androtary inertiardquo International Journal of Mechanical Sciences vol65 no 1 pp 97ndash104 2012

[14] S W Qi F Q Wu F Z Yan et al Dynamic Response Analysesof Rock Slope Science Press Beijing China 2007 Chinese

[15] J H Dong and Y P Zhu ldquoDynamic calculation model andseismic response for the system of soil nailing and surroundingsoilrdquo Chinese Journal of Theoretical and Applied Mechanics vol41 no 2 pp 236ndash242 2009

[16] Z L Xu Elasticity Higher Education Press Beijing China2009 Chinese

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of