AnalyticalCalculationofCriticalAnchoringLengthofSteelBar ... · 2021. 1. 12. · Qingdao University of Technology, Qingdao 266033, China 3Qingdao Geo-Engineering Survering Institute,

Research ArticleAnalytical Calculation of Critical Anchoring Length of Steel Barand GFRP Antifloating Anchors in Rock Foundation

Nan Yan12 Xueying Liu1 Mingyi Zhang12 Xiaoyu Bai 12 Zheng Kuang1

Yongfeng Huang1 Desheng Jing1 Jun Yan34 Cuicui Li5 and Zhongsheng Wang34

1School of Civil Engineering Qingdao University of Technology Qingdao 266033 China2Cooperative Innovation Center of Engineering Construction and Safety Shandong Blue Economic ZoneQingdao University of Technology Qingdao 266033 China3Qingdao Geo-Engineering Survering Institute Qingdao 266071 China4)e Key Laboratory of Urban Geology and Underground Space ResourcesShandong Provincial Bureau of Geology and Mineral Resources Qingdao 266590 China5Qingdao Construction Group Qingdao 266071 China

Correspondence should be addressed to Xiaoyu Bai baixiaoyu538163com

Received 9 July 2020 Revised 23 December 2020 Accepted 31 December 2020 Published 12 January 2021

Academic Editor Jose J Muntildeoz

Copyright copy 2021NanYan et alis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Antifloating anchors are widely used during the construction of slab foundations to prevent uplift However existing methods forcalculating the critical length of these anchors have limited capabilities and therefore require further research As the mechanismswhich govern the displacement and stability of antifloating anchors are closely related to those of piles subject to uplift a simplifiedanchor model has been developed based on existing concentric thin-walled cylinder shear transfer models used for pile designAnalytical expressions for the critical length of the steel bar and GFRP (glass fiber reinforced polymer) antifloating anchors in rockare derived accordingly before demonstrating the validity of the method through engineering examplese research results showthat when the length of an antifloating anchor is less than a critical length shear slip failure occurs between the anchor andsurrounding material due to excessive shear stress When the length of an anchor approaches the critical length the shear stressgradually decreases to the undisturbed state If the anchor length is larger than the critical length the uplift loads are safelytransferred to the ground without causing failure e ratio of elastic modulus between the anchor and rock mass was found to bepositively correlated with the critical anchoring length Because the elastic modulus of GFRP bars is lower than that of steel barsthe critical anchoring length of GFRP bars is greater than that of steel bars under the same anchor-to-rock modulus ratio (EaEs)e results show that the proposed calculation method for the critical length of antifloating anchors appears valid and couldprovide a theoretical basis for the design of antifloating anchors after further refinement

1 Introduction

Antifloating anchors are widely used to prevent the uplift ofslab foundations as they cause only small stress concen-trations employ simple construction technology and arelow cost [1ndash3] Traditional steel bar antifloating anchors haveproven effective under a range of conditions but have shortdesign lives in harsh environments with corrosive groundwater or electric currents produced by rail infrastructureNonmetallic antifloating anchors have the potential to

overcome this limitation Within this class of anchor theglass fiber reinforced polymer (GFRP) type has becomeprominent due to the advantages of high tensile strengthcorrosion resistance electromagnetic interference resis-tance and low cost [4ndash7]

In recent years many scholars at home and abroad haveconducted studies on the performance of anchors Zhanget al [8] proposed a time-varyingmodel to describe the load-deformation characteristics of GFRP soil nails during thepullout process by observing the continuous interaction

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 7838042 10 pageshttpsdoiorg10115520217838042

between GFRP soil nails and sande tests were carried outto validate a proposed analytical model Additionally Trejoet al [9] studied GFRP bars embedded in concrete underunsubmerged conditions for 7 years In order to better assessthe capacity loss and ACI design reduction factors aprobabilistic model of the time-varying bearing capacity ofGFRP bars embedded in concrete is needed In order toimprove the anchoring performance of antifloating anchorthe anchoring length of anchor is usually increased How-ever Bai et al [10] found that there is a critical anchoringlength in the steel bar and GFRP antifloating anchor throughfield drawing test that is the anchor will no longer bestressed after reaching a certain depth is indicates thatincreasing the anchoring length without limit cannot con-tinuously improve the anchoring performance of the anti-floating anchor and will result in the consequence ofincreasing cost and material waste erefore it is of greatsignificance to determine the accurate calculation method ofcritical anchoring length of antifloating anchor with steeland GFRP bars for saving cost and improving constructionefficiency

At present there are few reports on the calculationmethod of critical anchoring length of steel and GFRPanchors in rock foundation In this paper by combining theideal concentric thin-walled cylinder shear model and thesimplified shear stress distribution model of the antifloatinganchor the critical anchoring length of the steel bar andGFRP antifloating anchor is derived Compared with anengineering example the feasibility of the above calculationmethod is verified and the influence of the ratio of elasticmodulus of anchor to rock mass on the critical anchoringlength is discussed

2 Basic Theory

21 Calculation ofGeotechnicalMovement Considering thatthe working mechanism of antifloating anchor is similar tothat of uplift pile the theory of uplift pile is used to analyzethe antifloating anchor Assuming that the rock and soilbody around the anchor is an ideal elastic body and theanchoring solid in contact with the anchor body has thesame properties as the surrounding rock and soil withoutconsidering the increase of vertical stress the soil defor-mation under the action of drawing load can be representedby the ideal concentric thin-walled cylindrical shear model(Figure 1) [11 12] e anchor interacts with the sur-rounding rock and soil to transfer the drawing load to theadjacent concentric cylinders and this process is transmittedto n cylinders Using the method of unit selection for axi-symmetric problems in space in the theory of elasticity twocylinders separated by dr two vertical planes forming dθand two horizontal planes bounded by a tiny hexahedronseparated by dz are selected as the units Because the con-centric cylinder model is an axisymmetric problem there areonly normal stress and axial shear stress on the two cylindersof the element body ere are only normal and radial shearstresses on the two planesere is only annular stress on thetwo vertical surfaces and the increment is 0e force on theelement body is shown in Figure 2

In Figures 1 and 2 P represents the drawing load on theanchor σr σθ and σz represent the normal stress along the rθ and z directions respectively τrz represents the shearstress acting on the cylinder along the z-axis r represents thedistance between the element body and the z-axis and θrepresents the rotation angle of the element body from the xaxis

As shown in Figure 1 the z-axis torque balance of themodel can be obtained by combining with the force exertedon the element in Figure 2

τ0r0 τrzr (1)

where τ0 is the shear stress at the interface between theanchor and the surrounding rock and soil and r0 is theanchor radius

In combination with Figure 2 ignoring the physical forceof the element the balance equation of spatial axisymmetricproblem of elastic mechanics can be obtained as follows

zσz

zz+

zτrz

zrminusτrz

r 0 (2)

z

r

P dr Unit

Antifloatinganchor

Figure 1 Concentric thin wall cylinder shear model

x

y

z

θ dθ r

dr

dz

σr

σz

σz + (partσzpartz)dz

σθ

σθ

τrz

τrz + (partτrzpartr)dr

σr + (partσrpartr)dr

Figure 2 Schematic diagram of the unit force

2 Mathematical Problems in Engineering

When the anchor is under tension the shear stressvariation of the element in the rock and soil is much greaterthan that of the vertical normal stress erefore zσzzz 0is considered which is ignored erefore equation (2)becomes

zτrz middot r

zr τrz (3)

Integrate both sides of equation (3) and combine withequation (1) to obtain

τrz τ0r0

r (4)

Assuming that the displacement of the unit in Figure 2along the z-axis direction is ω the displacement along ther-axis direction is μ and the hoop displacement is 0 thenits shear displacement can be expressed as

crz τrz

Gs

zμzz

+zωzr

(5)

where crz is the shear displacement of unit and Gs stands forshear modulus of rock and soil mass

e unit mainly has vertical displacement ereforeignoring its displacement along the direction of r-axisequation (5) becomes

τrz

Gs

zωzr

(6)

For uniform soil mass combined with equations (6) and(4) the following equation can be obtained

τ0r0rGs

zωzr

(7)

Integrating equation (7) the result is

ωs τ0r0Gs

1113946rm

r0

1rdr (8)

where ωs is the total displacement of rock and soil mass andrm is the effective influence radius of the drawing loadbeyond which the influence of the drawing load on thesurrounding rock and soil mass can be ignored

Cooke et al [13] found that the rock and soil mass hardlydeforms when the anchor radius is more than 20 times so

rm 20r0 (9)

where μs is Poissonrsquos ratio of rock and soil mass and Lc is thecritical anchoring length of anchor

By calculating equation (8) the total displacement of therock and soil mass can be expressed as

ωs τ0r0Gs

middot lnrm

r01113888 1113889 (10)

22 Displacement Calculation of Steel Antifloating AnchorA large number of test results showed that [14] under theaction of drawing load the shear stress between the steel bar

antifloating anchor and the anchoring solid reaches itsmaximum value at a shallow distance from the orifice thengradually develops downward and finally attenuates to 0 at acertain depth

To simplify the calculation the shear stress distributionof the steel bar antifloating anchor is regarded as an inverseright triangle distribution that is it reaches a peak value onthe upper surface of the rock and soil mass then decreaseslinearly downward and drops to 0 at the critical anchoringlength as shown in Figure 3 In Figure 3 P is the pullingload Lc is the critical anchoring length of anchor and τ1 isthe shear stress at the entrance namely the peak shear stress

In order to master the anchoring performance betweenthe anchor body and the anchoring solid as a whole Wonet al [15] proposed amethod to solve the average shear stressof the anchor e distribution mode is shown as dotted linein Figure 3 which is expressed by equation (11)

τ P

2πr0Lc

(11)

where τ is the average shear stress of the anchor and r0 is theanchor radius

In order to ensure that the total shear stress obtainedunder the inverted triangle model is equal to the averageshear stress model there are the following relationships

τ1 2τ (12)

us the shear stress distribution function of the anchorunder the inverted triangle model is

τ(x) τ1 middot 1 minusx

Lc

1113888 1113889 P

πr0Lc

middot 1 minusx

Lc

1113888 1113889 0le xleLc( 1113857

(13)

where τ (x) is the shear stress of the solid interface betweenanchor body and anchoring solid when the depth is x

e anchoring solid is regarded as an elastic body andthe distribution function of its axial force along the an-chorage depth is

P(x) 2πr0 middot 1113946Lc

x

P

πr0Lc

middot 1 minust

Lc

1113888 1113889 dt

P 1 minus2x

Lc

+x2

L2c

1113888 1113889

(14)

According to Hookersquos law the elastic displacement of thebar top can be expressed as

ωsa 1113946Lc

0

P(x)

πr20Esa

dx PLc

3πr20Esa

(15)

where Esa is the elastic modulus of the steel bar anchor

23 Displacement Calculation of GFRP Antifloating Anchore elastic modulus of GFRP bars is much lower than that ofsteel bars and the behavior of shear stress under pull-outload is different from that of steel barse results of relevanttests [16ndash19] show that compared with the steel antifloating

Mathematical Problems in Engineering 3

anchor the peak shear stress of the GFRP antifloating an-chor appears in a deeper place and the distribution law isquite different from that of the inverted triangle modelerefore it is necessary to improve the calculation methodof anchor displacement to make it more suitable for GFRPantifloating anchor

As shown in Figure 4 the peak shear stress of the GFRPantifloating anchor is located at position Lx away from theground and the peak value is still τ1 e shear stressdistribution is bounded by the peak point and graduallydecreases upward and decreases to 0 at the orifice edownward shear stress of the peak point also decreasesgradually and drops to 0 at the position of the critical an-choring length Lc

In order to ensure that the total shear stress is constantτ1 in the GFRP shear stress model still satisfies the rela-tionship of equations (11) and (12)

From Figure 4 the expression of GFRP antifloatinganchor shear stress distribution is

τ(x)

P

πr0LcLx

middot x 0le xlt Lx( 1113857

P

πr0Lc

middotLc minus x

Lc minus Lx

1113888 1113889 Lx lexle Lc( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(16)

Its axial force distribution is

P(x) 2πr0 1113946Lc

xτ(t) dt

P

Lc

Lc minusx2

Lx

1113888 1113889 0lexlt Lx( 1113857

P Lc minus x( 11138572

Lc Lc minus Lx( 1113857Lx le xleLc( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(17)

According to Hookersquos law the elastic displacement of therod top is

ωGa 1113946Lc

0

P(x)

πr20EGa

dx

P

3πr20EGa

Lc minus 5Lx +2L

2x

Lc

1113888 1113889

(18)

24 Calculation of Critical Anchoring Length of AntifloatingAnchor Assuming that the antifloating anchor deforms inharmony with the surrounding rock and soil the dis-placement of the anchor top should be equal to the dis-placement of the surrounding rock and soil e criticalanchoring lengths of the steel bar and GFRP antifloatinganchors are solved separately

(1) For the steel bar antifloating anchor with ωs ωsasubstituting equations (10) and (15) into equation(18) it can be obtained as follows

PLc

3πr20Esa

τ0r0Gs

middot lnrm

r01113888 1113889 (19)

For the elastic soil it can be obtained as follows

Gs Es

2 1 + μs( 1113857 (20)

where Es is the elastic modulus of rock and soil massSubstituting equations (11) (12) (20) into (19) andsimplifying

τ1L2c

3r20Esa

2τ0 1 + μs( 1113857

Es

middot lnrm

r01113888 1113889 (21)

It should be noted that τ1 on the left side of equation(21) represents the shear stress of the anchor at theorifice in the model shown in Figure 3 τ0 on the rightside of the equation represents the shear stress of aunit of the innermost concentric cylinder (that is the

xL c

P

Antifloatinganchor

τ ττ1

Figure 3 Simplified model for the shear stress of steel antifloating anchor


anchor) in the model shown in Figure 1 e shearstress of a certain unit for the unit near the orifice τ0has the same meaning as τ1 on the left side of theequation which is the shear stress of the antifloatinganchor at the orifice erefore equation (21) can besimplified to

L2c

3r20Esa

2 1 + μs( 1113857

Es

middot lnrm

r01113888 1113889 (22)

According to equation (9) the expression of criticalanchoring length of reinforced antifloating anchor isobtained after simplifying equation (22)

Lc r0

6 middot ln 20 middot 1 + μs( 1113857 middotEsa

Es

1113971

(23)

(2) For GFRP antifloating anchor ωs ωGa andsubstituting equations (10) and (18) into ωs ωGa itcan be obtained as follows

τ0r0Gs

middot lnrm

r01113888 1113889

P

3πr20EGa

Lc minus 5Lx +2L

2x

Lc

1113888 1113889 (24)

Using the same simplification method as the steel barantifloating anchor and substituting equations (11) (12) and(20) into (24) it can be obtained as follows

Lc minus52

Lx1113874 11138752

lnrm

r01113888 1113889 middot

6r20 1 + μs( 1113857EGa

Es

+174

L2x (25)

e critical anchoring length of GFRP antifloating an-chor is obtained as

Lc 52

Lx +

6 middot ln 20 middot 1 + μs( 1113857r20 middot

EGa

Es

+174

L2x

1113971

(26)

In addition since the simplified model of shear stressdistribution cannot determine the location of Lx this paperuses the following method to solve it

You [20] used the Mindlin displacement solution to findthe shear stress distribution of the antifloating anchor

τ(x) P

πr0middot

tx

21113874 1113875 middot e

minus tx22( ) (27)

where t 1(1 + μs)(3 minus 2μs)r20 middot EsEGa

Since the shear stress distribution of the GFRP anti-floating anchor has only one peak point the point where thederivative function of equation (27) is 0 can be used todetermine the location of Lx and the final result is

P

πr0middot

tx

21113874 1113875 middot e

minus tx22( )1113890 1113891prime

0

rArrx

1t

1113970

(28)

e critical anchoring length of GFRP antifloating an-chor rod is obtained by substituting equations (28) into (26)

Lc 52

1t

1113970

+


EGa

Es

+174t

1113971

(29)


3 Example Analysis

In this section the above theoretical method will be used tocalculate the critical anchoring length of the antifloatinganchor in different tests and compare it with the actual

x

P

Antifloatinganchor

τ

(a)

x

L c

L x

P

Antifloatinganchor

τ

τ1

(b)

Figure 4 Simplified model for the shear stress of GFRP antifloating anchor (a) eoretical distribution diagram (b) Simplified model ofshear stress distribution


anchoring length At the same time the rationality of thedesign of the anchoring length will be analyzed in con-junction with the distribution law of the shear stress of theanchor

One part of anchor test [21ndash23] was selected as theexample of steel bar antifloating anchor e other parts ofanchor tests [17 18 23 24] were selected as the example ofGFRP antifloating anchor e calculation parameters andanchoring length of each test anchor are shown in Tables 1and 2 and the distribution of shear stress along the an-choring depth is shown in Figures 5 and 6

In Table 1 it can be known from the pull-out test ofsteel bar anchors in [23] that the anchoring length of steelbar anchors is much lower than the theoretical criticalanchoring length Combined with the distribution law ofanchor shear stress shown in Figure 5(a) near the end ofantifloating anchor there is still a high shear stress betweenanchor body and anchoring solid which then rapidlydescends to 0 and there is no excess anchoring lengthreserve It is shown that the shorter anchoring length is notenough to provide enough contact area between anchorbody and anchoring solid and the bonding force betweenthe two cannot be played normally e drawing load canonly be transferred to the surrounding rock and soil in arelatively shallow range which leads to the shear stress ofthe interface between the anchor body and the anchoringsolid far higher than the normal level and eventually in-evitably leads to the interface reaching the ultimate shearstrength value prematurely and the anchor body and theanchoring solid have relative slip resulting in shear slipfailure e description of test anchor failure in [23] isconsistent with the above analysis results which can in-directly prove the rationality of the above critical anchoringlength calculation method

In Table 1 the pull-out test of the steel bar anchor in[21 22] shows that the actual anchor anchoring length islonger than the theoretical critical anchoring lengthCombined with the shear stress distribution of the anchorin Figures 5(b) and 5(c) it is shown that the antifloatinganchor in the two tests has partial anchoring length reservethe anchor body and anchoring solid can play their

bonding role normally and they have sufficient anchoringlength for the downward transfer of drawing load Inaddition the actual anchoring length of the tested anti-floating anchor in [22] is about 065m longer than thetheoretical critical anchoring length accounting for 81 ofthe actual anchoring length It shows that the test anchor istoo long to be anchored and a long part of the anchor doesnot play its role in the actual stress which results in thewaste of anchor material

For GFRP bars a similar conclusion can be obtained bycombining Table 2 and Figure 6 When the anchoringlength is lower than the theoretical critical anchoringlength there is no residual length reserve of the anchorunder the action of drawing load and the shear stressbetween the anchor and the anchoring solid is higher thanthe normal level which leads to the shear-slip failure be-tween the anchor and the anchoring solid Similarly an-chors whose anchoring length is higher than the criticalanchoring length have a partial length reserve and thebonding force between the anchor body and the anchoringsolid can play normally with sufficient length for thedownward transfer of the drawing load

In addition according to the comparison results of[17 23] in Table 2 for GFRP antifloating anchors whoseanchoring length is lower than the theoretical critical an-choring length as the actual anchoring length is close to thecritical anchoring length the shear stress gradually tends tobe normal and the shear stress value near the end of theanchor gradually decreases

e above results are summarized as follows when theanchoring length is lower than the critical anchoringlength the shear stress between the anchor and the an-choring solid under the action of drawing load is higherthan the normal level (the shear stress when the anchoringlength is long enough) and the antifloating anchor thussuffers shear slip failure As the actual anchoring lengthapproaches the critical anchoring length the shear stressbecomes closer to the normal level When the anchoringlength is higher than the critical anchoring length theanchor has some reserved length and enough length totransfer the load downward However the anchoring

Table 1 Calculating parameters and anchoring length of steel anchors

Anchor test source r0 (mm) μs Esa (GPa) Es (MPa) Test anchoring length (m) eoretical critical anchoring length (m)[21] 8 03 200 3times104 015 0100[22] 15 05 200 543times104 08 0149[23] 14 033 200 30 3 5589

Table 2 Calculating parameters and anchoring length of GFRP anchors

Anchor test source r0 (mm) μs Esa (GPa) Es (MPa) Test anchoring length (m) eoretical critical anchoring length (m)[17] 14 033 51 32 5 5871[18] 14 033 45 32 645 5514[23] 14 033 51 30 3 6063[24] 16 025 43 305times103 065 0624


Anc

horin

g le

ngth

(m)

P = 15kN

015

012

009

006

003

0000 2 4 6 8 10 12

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 29094kN

08

07

06

05

04

03

02

01

000 5 10 15 20 25 30 35

Shear stress (MPa)

(b)

Anc

horin

g le

ngth

(m)

P = 80kNP = 160kNP = 240kN

35

30

25

20

15

10

05

0000 04 08 12 16 20

Shear stress (MPa)

(c)

Figure 5 Distribution for the shear stress of steel anchors (a) [21] (b) [22] (c) [23]

Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

5

4

3

2

1

00 05 10 15 20 25 30

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 50kNP = 100kNP = 200kN

P = 250kNP = 400kNP = 450kN

6

5

4

3

2

1

000 05 10 15 20 25 30

Shear stress (MPa)

(b)

Figure 6 Continued


length has little effect on improving the anchoring per-formance instead most of the anchor will not play its rolecausing material waste

4 Influence of Elastic Modulus Ratio(EaEs) between Anchor and Rock Mass onCritical Anchorage Length

It can be seen from equations (23) and (29) that the ratio ofelastic modulus of anchor and rock mass has a great in-fluence on the calculation of critical anchoring length thatis the critical anchoring length of the same type of anti-floating anchor in different environments is also different

In practical application the diameter of antifloatinganchor is relatively large Assuming that the anchor is lo-cated in the fourth system with a radius of 14mm and itsPoissonrsquos ratio μs is usually 03 the ratio relationship be-tween the critical anchoring length and rock-soil mass elasticmodulus of the anchor is shown in Figure 7

As shown in Figure 7 EaEs of steel bar and GFRP bar arepositively correlated with critical anchoring length Underthe same conditions of EaEs the critical anchoring length ofGFRP bar is greater than that of steel bar With the increaseof EaEs the gap of critical anchoring length between themtends to increase

e reasons for the above phenomenon can be sum-marized as follows the elastic modulus of GFRP bar is much

Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

35

30

25

20

15

10

05

0000 06 12 18 24 30 36

Shear stress (MPa)

(c)

Anc

horin

g le

ngth

(m)

07

06

05

04

03

02

01

0000 05 10 15 20 25

Shear stress (MPa)

P = 40kN

(d)

Figure 6 Distribution for the shear stress of GFRP anchors (a) [17] (b) [18] (c) [23] (d) [24]

Criti

cal a

ncho

ring

leng

th (m

m)

Steel bar anti-floating anchorGFRP bar anti-floating anchor

0

50

100

150

200

250

300

350

400

450

500

550

1 2 3 4 5 6 7 8 9 100Ea Es

Figure 7 Influence of EaEs on critical anchoring length Ea is the elastic modulus of anchor


lower than that of steel bar (generally 4-5 times) and thedeformation of GFRP bar is greater under the same loadlevele test results show that the GFRP antifloating anchorbody and anchoring solid have better coordinated defor-mation ability the bond strength between the anchor bodyand the anchoring solid is lower and slip failure is morelikely to occur In order to prevent the GFRP antifloatinganchor from prematurely slipping and damaging the sta-bility of the antifloating structure it is necessary to lengthenthe anchoring length and increase the adhesion force be-tween the anchor and the anchoring solid (includingchemical adhesive force friction resistance and mechanicalbite force)

In addition with the increase of EaEs that is thestrength of rock and soil gradually decreases the ability ofcoordinated deformation between GFRP antifloating anchorand rock and soil further increases and the increase am-plitude is much greater than that of steel bar anchorresulting in the decrease amplitude of bonding force be-tween GFRP antifloating anchor and anchoring solid thanthat of steel bar anchor erefore the increase amplitude ofcritical anchoring length of GFRP antifloating anchor islarger than that of steel bar antifloating anchor

5 Conclusion

(1) Based on the ideal concentric thin-walled cylindershear model and the simplified shear stress distri-bution model of antifloating anchor the criticalanchoring length of antifloating anchor with steel barand GFRP bar is derived By comparing the theo-retical value of critical anchoring length with themeasured value of an engineering example andcombining with the experimental results and phe-nomenon in an engineering example the rationalityof the analytical calculation method and the basichypothesis are verified

(2) With the critical anchoring length as the boundarywhen the anchoring length is lower than the criticalanchoring length shear slip failure occurs betweenthe anchor and the anchoring solid due to excessiveshear stress under the action of drawing load Withthe approach of the actual anchoring length and thecritical anchoring length the shear stress graduallydecreases to the normal level When the anchoringlength is higher than the critical anchoring lengththe anchor has some length reserve and the load istransferred downward gradually because the anchorhas enough anchoring length To avoid waste theanchoring length should not be too long

(3) EaEs is positively correlated with the critical an-choring length Under the same condition of EaEsthe critical anchoring length of GFRP antifloatinganchor is greater than that of steel bar antifloatinganchor With the increase of EaEs the difference ofcritical anchoring length between them increasesgradually

Data Availability

e experimental data used to support the findings of thisstudy will be made available upon request

Conflicts of Interest

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

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (51708316 and 51778312) the ChinaPostdoctoral Science Foundation Funding (2018M632641)the Shandong Provincial Postdoctoral Innovation Programof China (201903043) Higher Educational Science andTechnology Program of Shandong Province (J16LG02)Qingdao Postdoctoral Applied Research Program (2018101)and Key Program of Natural Science Foundation of Shan-dong Province (ZR2020KE009)

References

[1] Z Achillides and K Pilakoutas ldquoBond behavior of fiberreinforced polymer bars under direct pullout conditionsrdquoJournal of Composites for Construction vol 8 no 2pp 173ndash181 2004

[2] X Bai X Liu M Zhang et al ldquoStress transfer properties anddisplacement difference of GFRP antifloating anchorrdquo Ad-vances in Civil Engineering vol 2020 p 18 Article ID8894720 2020

[3] H Ashrafi M Bazli A Vatani Oskouei et al ldquoEffect of se-quential exposure to UV radiation and water vapor con-densation and extreme temperatures on the mechanicalproperties of GFRP barsrdquo Journal of Composites for Con-struction vol 22 no 1 Article ID 04017047 2017

[4] P V Vijay and H V S Gangarao ldquoBending behavior anddeformability of glass fiber-reinforced polymer reinforcedconcrete membersrdquo ACI Structural Journal vol 98 no 6pp 834ndash842 2011

[5] V M Karbhari J W Chin D Hunston et al ldquoDurability gapanalysis for fiber-reinforced polymer composites in civil in-frastructurerdquo Journal of Composites for Construction vol 7no 3 pp 238ndash247 2003

[6] D-S Xu and J-H Yin ldquoAnalysis of excavation induced stressdistributions of GFRP anchors in a soil slope using distributedfiber optic sensorsrdquo Engineering Geology vol 213 pp 55ndash632016

[7] Z Kuang M-Y Zhang and X-Y Bai ldquoLoad-bearing char-acteristics of fibreglass uplift anchors in weathered rockrdquoProceedings of the Institution of Civil Engineers-GeotechnicalEngineering vol 173 no 1 pp 49ndash57 2020

[8] C-C Zhang H-H Zhu Q Xu B Shi and G-X Mei ldquoTime-dependent pullout behavior of glass fiber reinforced polymer(GFRP) soil nail in sandrdquo Canadian Geotechnical Journalvol 52 no 6 pp 671ndash681 2015

[9] D Trejo P Gardoni and J J Kim ldquoLong-Term performanceof glass fiber-reinforced polymer reinforcement embedded inconcreterdquo ACI Materials Journal vol 108 no 6 pp 605ndash6132011

[10] X Y Bai M Y Zhang L Zhu et al ldquoExperimental study onshear characteristics of interface of full-bonding glass fiber


reinforced polymer anti-floating anchorsrdquo Chinese Journal ofRockMechanics and Engineering vol 37 no 6 pp 1407ndash14182018

[11] R W Cooke and G Price Strains and Displacements AroundFriction Piles Building Research Station London UK 1978

[12] M F Randolph and C P Wroth ldquoAnalysis of deformation ofvertically loaded pilesrdquo Journal of Geotechnical and Geo-environmental Engineering vol 104 no 12 pp 465ndash4881978

[13] R W Cooke G Price and K Tarr ldquoJacked piles in Londonclay a study of load transfer and settlement under workingconditionsrdquo Geotechnique vol 29 no 2 pp 113ndash147 1979

[14] Y-S Kim H-J Sung H-W Kim and J-M Kim ldquoMoni-toring of tension force and load transfer of ground anchor byusing optical FBG sensors embedded tendonrdquo Smart Struc-tures and Systems vol 7 no 4 pp 303ndash317 2011

[15] J-P Won C-G Park H-H Kim S-W Lee and C-I JangldquoEffect of fibers on the bonds between FRP reinforcing barsand high-strength concreterdquo Composites Part B Engineeringvol 39 no 5 pp 747ndash755 2008

[16] N-K Kim ldquoPerformance of tension and compression an-chors in weathered soilrdquo Journal of Geotechnical and Geo-environmental Engineering vol 129 no 12 pp 1138ndash11502003

[17] X Y Bai M Y Zhang and H L Kou ldquoField experimentalstudy of load transfer mechanism of GFRP anti-floatinganchors based on embedded bare fiber bragg grating sensingtechnologyrdquo Engineering Mechanics vol 32 no 8 pp 172ndash181 2015

[18] H-L Kou W Guo and M-Y Zhang ldquoPullout performanceof GFRP anti-floating anchor in weathered soilrdquo Tunnellingand Underground Space Technology vol 49 pp 408ndash4162015

[19] G B Maranan A C Manalo W Karunasena andB Benmokrane ldquoPullout behaviour of GFRP bars with anchorhead in geopolymer concreterdquo Composite Structures vol 132pp 1113ndash1121 2015

[20] C A You ldquoMechanical analysis of fully-grouted anchorrdquoChinese Journal of Rock Mechanics and Engineering vol 19no 3 pp 339ndash341 2000

[21] S C Gu and X P Cui ldquoCharacteristic research on the an-chorage load transfer of anchor in concreterdquo Concrete vol 32no 10 pp 27ndash30 2010

[22] Y Z Zhang Z H Shi and J Zhang ldquoExperimental study ofload distribution of anchoring section for rock anchorsrdquo Rockamp Soil Mechanics vol 32 no 2 pp 184ndash188 2010

[23] X Y Bai M Y Zhang and N Yan ldquoField contrast test andmechanism analysis on anchorage performance of anti-floating anchors with two different materialsrdquo China CivilEngineering Journal vol 48 no 8 pp 38ndash46 2015

[24] Y C Kuang Y Xu and L W Ou ldquoResearch on the bondbehavior of a glass-fiber-reinforced plastic anchorrdquo Journal ofHarbin Engineering University vol 37 no 12 pp 1658ndash16642016


between GFRP soil nails and sande tests were carried outto validate a proposed analytical model Additionally Trejoet al [9] studied GFRP bars embedded in concrete underunsubmerged conditions for 7 years In order to better assessthe capacity loss and ACI design reduction factors aprobabilistic model of the time-varying bearing capacity ofGFRP bars embedded in concrete is needed In order toimprove the anchoring performance of antifloating anchorthe anchoring length of anchor is usually increased How-ever Bai et al [10] found that there is a critical anchoringlength in the steel bar and GFRP antifloating anchor throughfield drawing test that is the anchor will no longer bestressed after reaching a certain depth is indicates thatincreasing the anchoring length without limit cannot con-tinuously improve the anchoring performance of the anti-floating anchor and will result in the consequence ofincreasing cost and material waste erefore it is of greatsignificance to determine the accurate calculation method ofcritical anchoring length of antifloating anchor with steeland GFRP bars for saving cost and improving constructionefficiency

At present there are few reports on the calculationmethod of critical anchoring length of steel and GFRPanchors in rock foundation In this paper by combining theideal concentric thin-walled cylinder shear model and thesimplified shear stress distribution model of the antifloatinganchor the critical anchoring length of the steel bar andGFRP antifloating anchor is derived Compared with anengineering example the feasibility of the above calculationmethod is verified and the influence of the ratio of elasticmodulus of anchor to rock mass on the critical anchoringlength is discussed

2 Basic Theory

21 Calculation ofGeotechnicalMovement Considering thatthe working mechanism of antifloating anchor is similar tothat of uplift pile the theory of uplift pile is used to analyzethe antifloating anchor Assuming that the rock and soilbody around the anchor is an ideal elastic body and theanchoring solid in contact with the anchor body has thesame properties as the surrounding rock and soil withoutconsidering the increase of vertical stress the soil defor-mation under the action of drawing load can be representedby the ideal concentric thin-walled cylindrical shear model(Figure 1) [11 12] e anchor interacts with the sur-rounding rock and soil to transfer the drawing load to theadjacent concentric cylinders and this process is transmittedto n cylinders Using the method of unit selection for axi-symmetric problems in space in the theory of elasticity twocylinders separated by dr two vertical planes forming dθand two horizontal planes bounded by a tiny hexahedronseparated by dz are selected as the units Because the con-centric cylinder model is an axisymmetric problem there areonly normal stress and axial shear stress on the two cylindersof the element body ere are only normal and radial shearstresses on the two planesere is only annular stress on thetwo vertical surfaces and the increment is 0e force on theelement body is shown in Figure 2

In Figures 1 and 2 P represents the drawing load on theanchor σr σθ and σz represent the normal stress along the rθ and z directions respectively τrz represents the shearstress acting on the cylinder along the z-axis r represents thedistance between the element body and the z-axis and θrepresents the rotation angle of the element body from the xaxis

As shown in Figure 1 the z-axis torque balance of themodel can be obtained by combining with the force exertedon the element in Figure 2

τ0r0 τrzr (1)

where τ0 is the shear stress at the interface between theanchor and the surrounding rock and soil and r0 is theanchor radius

In combination with Figure 2 ignoring the physical forceof the element the balance equation of spatial axisymmetricproblem of elastic mechanics can be obtained as follows

zσz

zz+

zτrz

zrminusτrz

r 0 (2)

z

r

P dr Unit

Antifloatinganchor

Figure 1 Concentric thin wall cylinder shear model

x

y

z

θ dθ r

dr

dz

σr

σz

σz + (partσzpartz)dz

σθ

σθ

τrz

τrz + (partτrzpartr)dr

σr + (partσrpartr)dr

Figure 2 Schematic diagram of the unit force



zτrz middot r

zr τrz (3)


τrz τ0r0

r (4)


crz τrz

Gs

zμzz

+zωzr

(5)



τrz

Gs

zωzr

(6)


τ0r0rGs

zωzr

(7)


ωs τ0r0Gs

1113946rm

r0

1rdr (8)



rm 20r0 (9)



ωs τ0r0Gs

middot lnrm

r01113888 1113889 (10)





τ P

2πr0Lc

(11)



τ1 2τ (12)



Lc

1113888 1113889 P

πr0Lc

middot 1 minusx

Lc

1113888 1113889 0le xleLc( 1113857

(13)




x

P

πr0Lc

middot 1 minust

Lc

1113888 1113889 dt

P 1 minus2x

Lc

+x2

L2c

1113888 1113889

(14)


ωsa 1113946Lc

0

P(x)

πr20Esa

dx PLc

3πr20Esa

(15)








τ(x)

P

πr0LcLx


P

πr0Lc

middotLc minus x

Lc minus Lx

1113888 1113889 Lx lexle Lc( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(16)


P(x) 2πr0 1113946Lc

xτ(t) dt

P

Lc

Lc minusx2

Lx

1113888 1113889 0lexlt Lx( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(17)


ωGa 1113946Lc

0

P(x)

πr20EGa

dx

P

3πr20EGa

Lc minus 5Lx +2L

2x

Lc

1113888 1113889

(18)



PLc

3πr20Esa

τ0r0Gs

middot lnrm

r01113888 1113889 (19)


Gs Es

2 1 + μs( 1113857 (20)


τ1L2c

3r20Esa

2τ0 1 + μs( 1113857

Es

middot lnrm

r01113888 1113889 (21)


xL c

P

Antifloatinganchor

τ ττ1




L2c

3r20Esa

2 1 + μs( 1113857

Es

middot lnrm

r01113888 1113889 (22)


Lc r0


Es

1113971

(23)


τ0r0Gs

middot lnrm

r01113888 1113889

P

3πr20EGa

Lc minus 5Lx +2L

2x

Lc

1113888 1113889 (24)


Lc minus52

Lx1113874 11138752

lnrm

r01113888 1113889 middot

6r20 1 + μs( 1113857EGa

Es

+174

L2x (25)


Lc 52

Lx +


EGa

Es

+174

L2x

1113971

(26)



τ(x) P

πr0middot

tx

21113874 1113875 middot e

minus tx22( ) (27)



P

πr0middot

tx

21113874 1113875 middot e

minus tx22( )1113890 1113891prime

0

rArrx

1t

1113970

(28)


Lc 52

1t

1113970

+


EGa

Es

+174t

1113971

(29)


3 Example Analysis


x

P

Antifloatinganchor

τ

(a)

x

L c

L x

P

Antifloatinganchor

τ

τ1

(b)
















Anc

horin

g le

ngth

(m)

P = 15kN

015

012

009

006

003

0000 2 4 6 8 10 12

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 29094kN

08

07

06

05

04

03

02

01

000 5 10 15 20 25 30 35

Shear stress (MPa)

(b)

Anc

horin

g le

ngth

(m)

P = 80kNP = 160kNP = 240kN

35

30

25

20

15

10

05

0000 04 08 12 16 20

Shear stress (MPa)

(c)


Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

5

4

3

2

1

00 05 10 15 20 25 30

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 50kNP = 100kNP = 200kN

P = 250kNP = 400kNP = 450kN

6

5

4

3

2

1

000 05 10 15 20 25 30

Shear stress (MPa)

(b)

Figure 6 Continued








Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

35

30

25

20

15

10

05

0000 06 12 18 24 30 36

Shear stress (MPa)

(c)

Anc

horin

g le

ngth

(m)

07

06

05

04

03

02

01

0000 05 10 15 20 25

Shear stress (MPa)

P = 40kN

(d)


Criti

cal a

ncho

ring

leng

th (m

m)


0

50

100

150

200

250

300

350

400

450

500

550

1 2 3 4 5 6 7 8 9 100Ea Es





5 Conclusion




Data Availability




Acknowledgments


References





























zτrz middot r

zr τrz (3)


τrz τ0r0

r (4)


crz τrz

Gs

zμzz

+zωzr

(5)



τrz

Gs

zωzr

(6)


τ0r0rGs

zωzr

(7)


ωs τ0r0Gs

1113946rm

r0

1rdr (8)



rm 20r0 (9)



ωs τ0r0Gs

middot lnrm

r01113888 1113889 (10)





τ P

2πr0Lc

(11)



τ1 2τ (12)



Lc

1113888 1113889 P

πr0Lc

middot 1 minusx

Lc

1113888 1113889 0le xleLc( 1113857

(13)




x

P

πr0Lc

middot 1 minust

Lc

1113888 1113889 dt

P 1 minus2x

Lc

+x2

L2c

1113888 1113889

(14)


ωsa 1113946Lc

0

P(x)

πr20Esa

dx PLc

3πr20Esa

(15)








τ(x)

P

πr0LcLx


P

πr0Lc

middotLc minus x

Lc minus Lx

1113888 1113889 Lx lexle Lc( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(16)


P(x) 2πr0 1113946Lc

xτ(t) dt

P

Lc

Lc minusx2

Lx

1113888 1113889 0lexlt Lx( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(17)


ωGa 1113946Lc

0

P(x)

πr20EGa

dx

P

3πr20EGa

Lc minus 5Lx +2L

2x

Lc

1113888 1113889

(18)



PLc

3πr20Esa

τ0r0Gs

middot lnrm

r01113888 1113889 (19)


Gs Es

2 1 + μs( 1113857 (20)


τ1L2c

3r20Esa

2τ0 1 + μs( 1113857

Es

middot lnrm

r01113888 1113889 (21)


xL c

P

Antifloatinganchor

τ ττ1




L2c

3r20Esa

2 1 + μs( 1113857

Es

middot lnrm

r01113888 1113889 (22)


Lc r0


Es

1113971

(23)


τ0r0Gs

middot lnrm

r01113888 1113889

P

3πr20EGa

Lc minus 5Lx +2L

2x

Lc

1113888 1113889 (24)


Lc minus52

Lx1113874 11138752

lnrm

r01113888 1113889 middot

6r20 1 + μs( 1113857EGa

Es

+174

L2x (25)


Lc 52

Lx +


EGa

Es

+174

L2x

1113971

(26)



τ(x) P

πr0middot

tx

21113874 1113875 middot e

minus tx22( ) (27)



P

πr0middot

tx

21113874 1113875 middot e

minus tx22( )1113890 1113891prime

0

rArrx

1t

1113970

(28)


Lc 52

1t

1113970

+


EGa

Es

+174t

1113971

(29)


3 Example Analysis


x

P

Antifloatinganchor

τ

(a)

x

L c

L x

P

Antifloatinganchor

τ

τ1

(b)
















Anc

horin

g le

ngth

(m)

P = 15kN

015

012

009

006

003

0000 2 4 6 8 10 12

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 29094kN

08

07

06

05

04

03

02

01

000 5 10 15 20 25 30 35

Shear stress (MPa)

(b)

Anc

horin

g le

ngth

(m)

P = 80kNP = 160kNP = 240kN

35

30

25

20

15

10

05

0000 04 08 12 16 20

Shear stress (MPa)

(c)


Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

5

4

3

2

1

00 05 10 15 20 25 30

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 50kNP = 100kNP = 200kN

P = 250kNP = 400kNP = 450kN

6

5

4

3

2

1

000 05 10 15 20 25 30

Shear stress (MPa)

(b)

Figure 6 Continued








Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

35

30

25

20

15

10

05

0000 06 12 18 24 30 36

Shear stress (MPa)

(c)

Anc

horin

g le

ngth

(m)

07

06

05

04

03

02

01

0000 05 10 15 20 25

Shear stress (MPa)

P = 40kN

(d)


Criti

cal a

ncho

ring

leng

th (m

m)


0

50

100

150

200

250

300

350

400

450

500

550

1 2 3 4 5 6 7 8 9 100Ea Es





5 Conclusion




Data Availability




Acknowledgments


References
































τ(x)

P

πr0LcLx


P

πr0Lc

middotLc minus x

Lc minus Lx

1113888 1113889 Lx lexle Lc( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(16)


P(x) 2πr0 1113946Lc

xτ(t) dt

P

Lc

Lc minusx2

Lx

1113888 1113889 0lexlt Lx( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(17)


ωGa 1113946Lc

0

P(x)

πr20EGa

dx

P

3πr20EGa

Lc minus 5Lx +2L

2x

Lc

1113888 1113889

(18)



PLc

3πr20Esa

τ0r0Gs

middot lnrm

r01113888 1113889 (19)


Gs Es

2 1 + μs( 1113857 (20)


τ1L2c

3r20Esa

2τ0 1 + μs( 1113857

Es

middot lnrm

r01113888 1113889 (21)


xL c

P

Antifloatinganchor

τ ττ1




L2c

3r20Esa

2 1 + μs( 1113857

Es

middot lnrm

r01113888 1113889 (22)


Lc r0


Es

1113971

(23)


τ0r0Gs

middot lnrm

r01113888 1113889

P

3πr20EGa

Lc minus 5Lx +2L

2x

Lc

1113888 1113889 (24)


Lc minus52

Lx1113874 11138752

lnrm

r01113888 1113889 middot

6r20 1 + μs( 1113857EGa

Es

+174

L2x (25)


Lc 52

Lx +


EGa

Es

+174

L2x

1113971

(26)



τ(x) P

πr0middot

tx

21113874 1113875 middot e

minus tx22( ) (27)



P

πr0middot

tx

21113874 1113875 middot e

minus tx22( )1113890 1113891prime

0

rArrx

1t

1113970

(28)


Lc 52

1t

1113970

+


EGa

Es

+174t

1113971

(29)


3 Example Analysis


x

P

Antifloatinganchor

τ

(a)

x

L c

L x

P

Antifloatinganchor

τ

τ1

(b)
















Anc

horin

g le

ngth

(m)

P = 15kN

015

012

009

006

003

0000 2 4 6 8 10 12

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 29094kN

08

07

06

05

04

03

02

01

000 5 10 15 20 25 30 35

Shear stress (MPa)

(b)

Anc

horin

g le

ngth

(m)

P = 80kNP = 160kNP = 240kN

35

30

25

20

15

10

05

0000 04 08 12 16 20

Shear stress (MPa)

(c)


Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

5

4

3

2

1

00 05 10 15 20 25 30

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 50kNP = 100kNP = 200kN

P = 250kNP = 400kNP = 450kN

6

5

4

3

2

1

000 05 10 15 20 25 30

Shear stress (MPa)

(b)

Figure 6 Continued








Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

35

30

25

20

15

10

05

0000 06 12 18 24 30 36

Shear stress (MPa)

(c)

Anc

horin

g le

ngth

(m)

07

06

05

04

03

02

01

0000 05 10 15 20 25

Shear stress (MPa)

P = 40kN

(d)


Criti

cal a

ncho

ring

leng

th (m

m)


0

50

100

150

200

250

300

350

400

450

500

550

1 2 3 4 5 6 7 8 9 100Ea Es





5 Conclusion




Data Availability




Acknowledgments


References





























L2c

3r20Esa

2 1 + μs( 1113857

Es

middot lnrm

r01113888 1113889 (22)


Lc r0


Es

1113971

(23)


τ0r0Gs

middot lnrm

r01113888 1113889

P

3πr20EGa

Lc minus 5Lx +2L

2x

Lc

1113888 1113889 (24)


Lc minus52

Lx1113874 11138752

lnrm

r01113888 1113889 middot

6r20 1 + μs( 1113857EGa

Es

+174

L2x (25)


Lc 52

Lx +


EGa

Es

+174

L2x

1113971

(26)



τ(x) P

πr0middot

tx

21113874 1113875 middot e

minus tx22( ) (27)



P

πr0middot

tx

21113874 1113875 middot e

minus tx22( )1113890 1113891prime

0

rArrx

1t

1113970

(28)


Lc 52

1t

1113970

+


EGa

Es

+174t

1113971

(29)


3 Example Analysis


x

P

Antifloatinganchor

τ

(a)

x

L c

L x

P

Antifloatinganchor

τ

τ1

(b)
















Anc

horin

g le

ngth

(m)

P = 15kN

015

012

009

006

003

0000 2 4 6 8 10 12

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 29094kN

08

07

06

05

04

03

02

01

000 5 10 15 20 25 30 35

Shear stress (MPa)

(b)

Anc

horin

g le

ngth

(m)

P = 80kNP = 160kNP = 240kN

35

30

25

20

15

10

05

0000 04 08 12 16 20

Shear stress (MPa)

(c)


Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

5

4

3

2

1

00 05 10 15 20 25 30

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 50kNP = 100kNP = 200kN

P = 250kNP = 400kNP = 450kN

6

5

4

3

2

1

000 05 10 15 20 25 30

Shear stress (MPa)

(b)

Figure 6 Continued








Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

35

30

25

20

15

10

05

0000 06 12 18 24 30 36

Shear stress (MPa)

(c)

Anc

horin

g le

ngth

(m)

07

06

05

04

03

02

01

0000 05 10 15 20 25

Shear stress (MPa)

P = 40kN

(d)


Criti

cal a

ncho

ring

leng

th (m

m)


0

50

100

150

200

250

300

350

400

450

500

550

1 2 3 4 5 6 7 8 9 100Ea Es





5 Conclusion




Data Availability




Acknowledgments


References









































Anc

horin

g le

ngth

(m)

P = 15kN

015

012

009

006

003

0000 2 4 6 8 10 12

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 29094kN

08

07

06

05

04

03

02

01

000 5 10 15 20 25 30 35

Shear stress (MPa)

(b)

Anc

horin

g le

ngth

(m)

P = 80kNP = 160kNP = 240kN

35

30

25

20

15

10

05

0000 04 08 12 16 20

Shear stress (MPa)

(c)


Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

5

4

3

2

1

00 05 10 15 20 25 30

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 50kNP = 100kNP = 200kN

P = 250kNP = 400kNP = 450kN

6

5

4

3

2

1

000 05 10 15 20 25 30

Shear stress (MPa)

(b)

Figure 6 Continued








Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

35

30

25

20

15

10

05

0000 06 12 18 24 30 36

Shear stress (MPa)

(c)

Anc

horin

g le

ngth

(m)

07

06

05

04

03

02

01

0000 05 10 15 20 25

Shear stress (MPa)

P = 40kN

(d)


Criti

cal a

ncho

ring

leng

th (m

m)


0

50

100

150

200

250

300

350

400

450

500

550

1 2 3 4 5 6 7 8 9 100Ea Es





5 Conclusion




Data Availability




Acknowledgments


References




























Anc

horin

g le

ngth

(m)

P = 15kN

015

012

009

006

003

0000 2 4 6 8 10 12

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 29094kN

08

07

06

05

04

03

02

01

000 5 10 15 20 25 30 35

Shear stress (MPa)

(b)

Anc

horin

g le

ngth

(m)

P = 80kNP = 160kNP = 240kN

35

30

25

20

15

10

05

0000 04 08 12 16 20

Shear stress (MPa)

(c)


Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

5

4

3

2

1

00 05 10 15 20 25 30

Shear stress (MPa)

(a)

Anc

horin

g le

ngth

(m)

P = 50kNP = 100kNP = 200kN

P = 250kNP = 400kNP = 450kN

6

5

4

3

2

1

000 05 10 15 20 25 30

Shear stress (MPa)

(b)

Figure 6 Continued








Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

35

30

25

20

15

10

05

0000 06 12 18 24 30 36

Shear stress (MPa)

(c)

Anc

horin

g le

ngth

(m)

07

06

05

04

03

02

01

0000 05 10 15 20 25

Shear stress (MPa)

P = 40kN

(d)


Criti

cal a

ncho

ring

leng

th (m

m)


0

50

100

150

200

250

300

350

400

450

500

550

1 2 3 4 5 6 7 8 9 100Ea Es





5 Conclusion




Data Availability




Acknowledgments


References


































Anc

horin

g le

ngth

(m)

P = 100kNP = 200kN

P = 300kNP = 400kN

35

30

25

20

15

10

05

0000 06 12 18 24 30 36

Shear stress (MPa)

(c)

Anc

horin

g le

ngth

(m)

07

06

05

04

03

02

01

0000 05 10 15 20 25

Shear stress (MPa)

P = 40kN

(d)


Criti

cal a

ncho

ring

leng

th (m

m)


0

50

100

150

200

250

300

350

400

450

500

550

1 2 3 4 5 6 7 8 9 100Ea Es





5 Conclusion




Data Availability




Acknowledgments


References






























5 Conclusion




Data Availability




Acknowledgments


References












































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