shrinkage cracking of thin irregular shotcrete … · shrinkage cracking of thin irregular...

9th International Conference on Fracture Mechanics of Concrete and Concrete StructuresFraMCoS-9

V. Saouma, J. Bolander, and E. Landis (Eds)

SHRINKAGE CRACKING OF THIN IRREGULAR SHOTCRETE SHELLSUSING MULTIPHYSICS MODELS

ANDREAS SJOLANDER∗, TOBIAS GASCH†, ANDERS ANSELL †† AND RICHARDMALM †††

KTH Royal Institute of Technology, Department of Civil and Architectural EngineeringStockholm, Sweden

∗ email: [email protected]

† email: [email protected]

†† email: [email protected]

††† email: [email protected]

Key words: Cracking, Fibre Reinforced Shotcrete, Multi-physics, Shrinkage

Abstract. Shotcrete (sprayed concrete) is commonly used to support tunnels in good quality hardrock. Including a drainage system often results in end-restrained sections of shotcrete, which havecreated problems with shrinkage induced cracking. In this paper a multi-physical material model withcoupled behaviour between thermal actions, moisture transportation and mechanical strain has beenused to model and describe the complex behaviour and effects of shrinkage of such a structure. Themodel was first calibrated against a free shrinkage test and then used to simulate an experimental set-up for testing of end-restrained shrinkage. The first results lead to a need of tuning of the parameterscontrolling the drying of the shotcrete to accurately describe the experimental results. This tuningcould be an indication that the shrinkage behaviour differs between a restrained and an un-restrainedsample. However, further research about possible changes in the pore structure as well as moredetailed measurements of the early shrinkage behaviour is needed before any such conclusions canbe drawn.

1 INTRODUCTION

Tunnels in good quality hard rock arenormally supported with plain, unreinforcedshotcrete (sprayed concrete), or a combinationof fibre reinforced shotcrete, FRS, and rockbolts. The commonly used excavation methodfor hard rock, drill and blast, will naturally re-sult in a shotcrete shell with highly varyingthickness due to the irregular shape of the rocksurface. For shotcrete that are not continu-ously bonded to the rock, problems related toshotcrete shrinkage cracking have been reported

[1, 2]. Shrinkage and strength development ofshotcrete are complex processes, depending onfactors such as the composition, of shotcretevariations in relative humidity and temperatureand is also highly influenced by the creep be-haviour. In this paper an attempt to improve theaccuracy and reliability in modeling the effectsof shrinkage is made by using a multi-physicalmaterial model as presented by Gasch et al. [3].The model has coupled behaviour between ther-mal actions as well as moisture transportation,i.e., drying and wetting of the shotcrete, and

1

DOI 10.21012/FC9.140

Andreas Sjolander, Tobias Gasch, Anders Ansell and Richard Malm

time-dependent deformations. An experimen-tal setup for end-restrained shotcrete slabs sub-jected to shrinkage, as tested by Bryne et al. [4],will be used for comparison with the numericalresults.

1.1 Shotcrete sprayed on soft drains

For tunnels, leaking water may cause prob-lems with deterioration of shotcrete and corro-sion of steel fibres. For tunnels built in cold cli-mates, the formation of ice can cause spalling ofthe already cracked shotcrete. To increase thetechnical life span, as well as limit the down-time of the tunnel due to maintenance work,a drainage system must therefore be installed.One example of such a solution is to cover waterleading cracks with a system of synthetic drainmats that will transport the water to a drainagesystem. The drain mats are fixed to the rock sur-face with rock bolts and covered with shotcrete,see Figure 1. The shotcrete is only bonded tothe rock surface at the end of each drain mat andthis, in combination with the low stiffness of themat, creates an end-restrained slab. Shotcretesprayed on drains are therefore always rein-forced with fibres, which commonly are of steelor glass. In-situ investigations [2, 5] as well asexperimental [6, 7] and numerical results [8, 9]have shown that end-restrained shotcrete slabsare prone to shrinkage cracking. The strain-softening behaviour of FRS leads to the forma-tion of few cracks whose width usually exceedsthe limits according to standards with respect todurability [10].

Irregular Rock Surface

Fibre Reinforced Shotcrete

Plain Shotcrete

Drain Mat

Rock Bolt

Figure 1: Build up of section with shotcrete sprayed onsoft drains

1.2 Shrinkage of shotcreteAutogenous and drying shrinkage of

shotcrete are caused by chemical processesin the cement and evaporation of water. Forshotcrete the fast development of bond strengthand of internal ettringite crystals leads toshrinkage within a stiff structure. Movementsin this structure will increase the porosity andtherefore increase the rate of the drying shrink-age [11]. The use of shotcrete for rock supportalso implies that it will be loaded at an earlystage due to the movement in the rock massafter excavation which further complicates theproblem. This is, however, out of the scoop forthis paper.

1.3 Irregular thickness of shotcreteIn-situ studies of the actual thickness of the

applied shotcrete show that the standard devia-tion could be up to 50 % of the intended thick-ness [2, 12]. Differences in applied thicknesswill mainly be due to problems with rebound,the quality of the substrate and skill of the op-erator [13]. If an end-restrained beam with ir-regular thickness is subjected to an evenly dis-tributed tensile stress, the change in center ofgravity along the beam will introduce effectsof bending moment. An irregular geometricshape will also introduce local stress concentra-tions. Furthermore, the one sided drying condi-tion of the shotcrete will introduce a strain gra-dient through the thickness. Since the rate ofdrying depends on the thickness a strain gradi-ent will also exist in the plane of the thickness.

2 BACKGROUNDThe material model used within this paper is

based on the work by Gasch et al. [3] where amodel with coupled behaviour between mois-ture, external heat and time-dependent defor-mation was presented. The model is basedon the Microprestress-Solidification, MPS, the-ory as originally presented by Bazant et al.[14, 15, 16, 17] and later refined and improvedby Jirsek and Havlsek [18]. The model has beenimplemented in the FE code Comsol Multi-physics [19] and is restricted to simulation of

2


moisture transport under isothermal conditions,i.e., there is no coupling between moisture andthermal fields. The main concepts of the modelare briefly described below and for a more thor-ough description the reader is refereed to givenreferences.

2.1 Moisture transport in concreteThe physical mechanism of moisture ex-

change within a specimen and to its surround-ing environment can be expressed with Fick’sfirst law, which in its general form for two ormore dimension can be written as in Eq. 1

J = −Dh ·∆φ (1)

where J is the flux, i.e. the change in moisturecontent over the area,Dh is the diffusivity of theconcrete and the difference in relative humidity,∆φ, is the driving force of the diffusion. Thediffusivity describes the rate of change in rel-ative humidity and is here described as a non-linear function of the relative humidity. Thisequation was first presented in [20] and is nowalso recognized in the FiB Model Code [21].For a relative humidity of 0 ≤ H ≤ 1 this equa-tion can be simplified to Eq. 2:

Dh = −D1[α0 +1− α0

1 + ( 1−H1−Hc

n)] (2)

The diffusivity is described by D1 for H = 1.0and with D1 · α0 when H = 0. At a relative hu-midity equal to Hc the rate of diffusion rapidlychange with a magnitude described by the pa-rameter n. The diffusivity is plotted in Figure2.

Figure 2: Relationship between diffusivity and relativehumidity

Fick’s first law describes the change in rel-ative humidity during steady state. A transientevent, such as drying of a concrete specimen,can be described with Fick’s second law, as:

∂H

∂t= Dh∇φ (3)

The change in relative humidity over time, ∂H/ ∂t, is in Eq. 3 described by the second deriva-tive of the relative humidity,∇φmultiplied withthe diffusivity. The moisture exchange betweenthe specimen and the ambient air is describedwith a mixture of natural and essential bound-ary conditions described by Eq. 4

−J · n = βh · (H −Henv) (4)

Here n is the normal to the surface, where βhdescribe the rate of exchange in moisture be-tween the surface and the ambient air whichhave a relative humidity equal to Henv. Finally,the relation between relative humidity withinthe shotcrete and its shrinkage is described bya single parameter, ksh, which is determinedby fitting experimental results to the numericalmodel.

2.2 Damage modelThe effective stresses are described by an

isotropic continuum damage model, based onthe work by Oliver et. al [22]. Here the effec-tive stress tensor, σ, is calculated based on thesingle damage scalar, ω.

σ = (1− ω)Di : εi (5)

In Eq. 5, Di and εi are the elastic stiffness andstrain tensors, respectively. The evolution ofdamage is here described by an exponential lawin which the softening depends on the tensilestrength and fracture energy.

2.3 Experimentally workIn a recent project carried out at the KTH

Royal Institute of Technology focus was puton development of material parameters ofshotcrete as well as the bond between youngshotcrete and hard rock [23]. In addition to

3


this, an experimental setup for testing of end-restrained shrinkage, see Figure 3, was devel-oped and presented by Bryne et al. [4, 7].The aim with this experiment was to simulatethe shrinkage behaviour of shotcrete sprayedon soft drains. A granite slab with dimensions1100x400x100 mm3 was used as a substrateand two layers of plastic film was placed tocreate a de-bonded area of 700x400 mm2 forthe shotcrete, simulating the effect of the drainmats. Shotcrete was then sprayed over the gran-ite slab with a target thickness of 20 mm. Atthe centre of the granite slab two strain gaugeswere placed 10 mm from the top and bottom,respectively. The restrained movement of theshotcrete slab could thus be monitored throughthe strains in the granite slab. The slabs werethen placed in a climate chamber with T = 20◦C and RH = 50 %. To reduce early dryingshrinkage the slab was kept under a wet jutecloth covered by a thin plastic sheet for the firstthree days. At day four, the plastic sheet andjute cloth were removed and the measurementsstarted. Thereby, early movement due to auto-genous and drying shrinkage as well as thermalactions due to cement hydration was, unfortu-nately, not monitored. A hypothesis for the ex-periment was that the addition of glass fibres inthe shotcrete could delay or prevent cracking.The experiment resulted in five cracked slabsand one failure due to de-bonding. Four of theslabs cracked after 6-7 days and one after 16days. The time of failure was measured fromthe time of spraying. In Table 1 the composi-tion of the shotcrete is presented.

Table 1: Composition of shotcrete [4]

Material Content [kg/m3]Cement 495Densified silica 19.8Water 220Superplasticiser 3.5Glass fibre 0 / 5Aggregate 0-2 mm 394Aggregate 0-8 mm 1183

3 NUMERICAL SIMULATIONSFor the numerical simulations, results from

two of the six slabs were used. Measurementsof the actual thickness of the applied shotcretewere used to create accurate geometric models.The measured min/max/mean thickness of slab1 and 2 was 12/38/25 mm and 25/45/37 mm, re-spectively. For slab 1 plain shotcrete was usedwhile slab 2 was reinforced with 5 kg/m3 ofglass fibres. The two slabs were chosen due totheir differences in time at failure; slab 1 failedafter 6 days and slab 2 after 16 days. The dif-ference is believed too partly be related to thegeometry and it is therefore interesting to inves-tigate if the numerical simulations can capturethis behaviour.

700 200200

400

100

h

Plastic sheet Strain gauges

Bonding zones

GraniteShotcrete

Pinned supportRoller support

x

y

Figure 3: Experimental set-up for end-restrained shrink-age test from Bryne et. al [4]

Shrinkage was applied in two different waysin the models to investigate the effects, and pos-sible importance, of non-linear shrinkage. First,the measured free shrinkage after 50 days, see3.3, was recalculated to an equivalent tempera-ture load according to Eq. 6:

Teq =εshα

(6)

Here α is the coefficient of thermal expansionfor shotcrete, see Table 2. With a free shrink-age, εsh of -360 µε the equivalent temperature

4


Table 2: Mechanical properties of granite and shotcrete

Parameter Value Unit CommentEg 70 GPa Young’s modulus of graniteνg 0.2 - Poisson’s ratio of graniteρg 2600 kg/m3 Density of graniteEc0 38(38) GPa Young’s modulus of plain shotcrete at 28 daysν 0.2 - Poisson’s ratio of shotcreteρg 2300 kg/m3 Density of shotcretefck0 61(59) MPa Compressive strength plain shotcrete at 28 daysfctm0 4.4(4.3) MPa Tensile strength of plain shotcrete at 28 daysGf 153(152) Nm Fracture energy of plain shotcrete at 28 dayssf 0.575 mm/d Moisture surface factorksh 0.001 - Shrinkage related to relative humidityD1 1.9*10−10 m2/s Diffusivity of shotcrete at H=1.0H c 0.80 - Relative humidityα0 0.05 - Relation between diffusivity at H=1.0 and H=0n 15 - Describing rate of change in diffusivity at H c

become −36◦C, the change in temperature wasassumed linear over the time period of 50 days.Shrinkage was then simulated with the multi-physical model where the relative humidity ofthe shotcrete and ambient air was set to 100 %and 50 %, respectively.

3.1 Numerical model

A 3D finite element model was created usinga free tetrahedral mesh. To increase the accu-racy in describing the irregular geometry and al-low for a minimum of four elements through thethickness of the shotcrete slab the element sizewas set to 3-8 mm. The geometry of the gran-ite slab was considered to be perfectly flat anda linear elastic material model and an elementsize of 20-30 mm was used. The interaction be-tween the shotcrete and granite slab at the bond-ing zone, see Fig 3, was modelled using a tiecondition and hence de-bonding was not con-sidered. Between the bonding zones, a contactsurface was used between the two slabs. Thecontact in the xy-plane, see Figure 3, was fric-tion less and separation was allowed due to ten-sile forces acting perpendicular to the xy-plane.Drying of the shotcrete slab was allowed alongall free sides and the boundary conditions forthe slab were modelled with a roller and pinned

support, see Fig. 3. Rotation at the bound-ary conditions was allowed around the x-axis.The gravity load from shotcrete and granite wasconsidered and each model resulted in a total ofapproximately 800.000 DOF. An attempt to in-clude the effects of creep resulted in around 25million DOF and due to limitations in time andcomputational power, creep was not considered.

3.2 Mechanical propertiesThe evolution of compressive strength for

plain, fck0, and glass fire reinforced (GFR),fck5, shotcrete presented in Eq. 7-8 are based onthe work presented in [7]. Tensile strength andfracture energy were not tested and estimationsof Young’s modulus based on flexural tests werebelieved to underestimate the stiffness. Themechanical properties of these parameters are,therefore, based on their relation to the com-pressive strength as given in Eurocode 2 and theFiB Model Code [21, 24].

fck0(t) = 76.68e−1.70/(t)0.60

(7)

fck05(t) = 81.41e−1.86/(t)0.53

(8)

The added glass fibres had a length of 6 mm andbased on regular four point bending test per-formed by [23] and presented by [9] it could

5


be concluded that the increase of fracture en-ergy was small and it was therefore omitted inthese simulations. All mechanical properties ofthe granite and shotcrete are presented in Table2 together with parameters used to describe themoisture transport in the shotcrete. Numberswithin brackets are for 5 kg/m3GRF shotcrete.

3.3 Verification of free shrinkageFree shrinkage tests of plain and glass fi-

bre reinforced shotcrete as well as a referencetest with plain, un-reinforced, concrete is pre-sented in [4]. The shotcrete was first sprayedinto a box-shaped mould and left to cure un-der water for three days. After 24 hours themould was removed and after 72 hours speci-mens with a dimension of 100x100x400 mm3

was sawn out. The specimens were then placedinto a climate chamber with T = 20 ◦C andRH = 50 % where drying was allowed alongall sides. The free shrinkage of plain shotcreteis described by Eq. 9 which, due to uncertain-ties in the measurements of the young shotcrete,is valid for 7 ≤ t ≤ 112 days [4]. The devi-ation in the measured free shrinkage for plainand GFR shotcrete is negligible for the studiedtime of 20 days and the free shrinkage for plainshotcrete was therefore used.

εsh = 0.085e−4.22/(t−7)0.42

(9)

In the plot of the free shrinkage in Figure 4 thenumerical model, red solid line, shows goodagreement with the test results, blue solid line.In an attempt to match the timeline for therestrained shrinkage tests, which started afterthree days of curing, measured values from day4-7 were added. To match the experimental re-sults, a ramp function was added to the bound-ary condition of the humidity, i.e., the externalhumidity was assumed to vary linearly fromRH = 100 % to RH = 50 % during a periodof 10 days. The slow initial shrinkage couldbe due to that the drying of the surface of thespecimen took some time after the water cur-ing process or the mentioned inaccuracy in themeasurements. If the free shrinkage instead wasassumed to start at day 7, from which reliable

measurements exist, the ramp function could beomitted and the resulting simulated free shrink-age is presented by the red dashed line in Figure4. This model will be used for simulation of theend-restrained shrinkage.

Time [days]0 10 20 30 40 50

Fre

e S

hrin

kage

[%]

0

0.01

0.02

0.03

0.04

0.05

Figure 4: Free shrinkage of shotcrete from experimentsand numerical simulations. Blue line is result from mea-surement and red lines are from numerical simulations.Red solid and dashed lines are with and without the rampfunction starting at day four and seven, respectively.

4 RESULTSIn Figure 5 results from the numerical sim-

ulations of slab-1 and 2, respectively, are pre-sented. The strains are presented at the locationof the strain gauges, as previously shown inFigure 3, and plotted at the top of the figure.Dashed lines, in Figure 5, are the mean valuesfrom the experiment where strains were mea-sured on both sides of the slab. Blue solid linesare results from the multi-physical model andred lines are results when shrinkage was appliedas an equivalent temperature load. Due to thelack of early measurements, the initial strain inthe granite slab, at t = 3 days, were estimatedby assuming that the strains in the granite slabafter cracking would be equal to zero. The bot-tom of the figure shows the location of cracksin the shotcrete, plotted on a contour plot thatrepresents the thickness of the shotcrete. Thethin black solid lines show the end of the bond-ing zones. As can be seen in top of Figure 5, thelevels of strains show good correspondence be-tween the experiments and the simulations, ex-cept for the results from the temperature model

6


Figure 5: Results from experiment, dashed line, temperature model, red line, and multi-physical model, blue line, forslab-1 (left) and slab-2 (right). Top figure shows strain in the granite and bottom figure shows crack patterns plotted on acontour plot representing the thickness of the shotcrete slab.

for slab-1. This indicates that the mechani-cal properties of shotcrete and granite are quiteaccurate. The localization of the crack, as pre-sented at the bottom of Figure 5, showed goodagreement for slab-1 meanwhile it localizedvery close to the bonding zone for slab-2. Thetime of failure for the temperature model willdepend on how the change in temperature wasmodelled and a linearly decreasing temperatureover 50 days clearly underestimates the effectsof early shrinkage. The fact that the failure inthe multi-physical model occurs much later in-dicates that the shrinkage of the restrained slabscould not be accurately described by the resultsfrom the free shrinkage test.

5 DISCUSSIONThe large difference in results could be due

to several reasons and first the possible inac-curacy and errors in the measurements of thefree shrinkage must be considered. The startof measurements at day four together with un-reliable results until day seven makes it diffi-cult to accurately predict the failure of slab-1, which occurred after six days. The slabs

were water cured but not stored under water, asthe specimens for the free shrinkage test. Thiswould likely have resulted in faster initial dry-ing shrinkage of the slabs since these were notfully saturated at the start of the drying process.This was, as described previously, accountedfor by excluding the ramp function used for theboundary conditions. This is indicated by thered dashed curve in Figure 4. Even thoughthis slow initiation of the drying process wasomitted in the numerical analysis, the time offailure was around three times higher than theexperiments. The positive stress-relieving ef-fects of creep, which had to be omitted in themodel, should have further increased the differ-ence in time. However, to better match the ex-perimental results two approaches were under-taken. The first one was based on adjusting theparameters controlling the moisture transporta-tion and for the second one, an initial state ofstress in the shotcrete was assumed at the startof the drying process.

7


5.1 Tuning of the moisture transportationmodel

To tune the moisture transportation modelthe diffusivity, D1, was first increased whichspeed up the moisture transportation within theshotcrete. When shrinkage occurs in the stiffshotcrete structure the porosity will change [11]which would increase the diffusivity. Sinceboth the slabs and the test specimens for freeshrinkage were shotcreted, this effect should beaccounted for. The difference between the testswas thus that the slabs were end-restrained andit could be possible that this increased stiffnessof the structure could further increase the poros-ity. The normal bond strength, during similarconditions as to the experimental setup, can ac-cording to Bryne et al. reach 1 MPa within24 hours [25]. However, further research mustbe conducted before any conclusion could bedrawn. It should be added that the model doesnot account for an increased moisture trans-portation through the cracks which could havean effect on the results. Secondly, the surfacefactor, sf , was increased which control the rateof exchange in relative humidity between thesurface of the shotcrete and the ambient air.This increase the rate of change in relative hu-midity, which is the driving force of the diffu-sion, between the surface and the inside of theshotcrete slab. Both the free shrinkage test andend-restrained test were performed within a cli-mate chamber. The environmental conditionswere therefore controlled and the increased sur-face factor could be explained by the shotcreteslab not being fully saturated at the start of dry-ing. To fit the experimental results both Dh andsf had to be increased with a factor of four. Thestrains at the location for the strain gauge, seeFigure 3, are plotted in Figure 6 where solidand dashed lines represent numerical and exper-imental results, respectively. Even though anincrease of both the parameters can be justifiedphysically, a factor of four feels rather large.However, the tuned moisture model captures therate of strain and the time of failure of the ex-periment very well.

5.2 Initial state of stressDue to autogenous shrinkage and thermal ef-

fects of the hydrated cement it is reasonable toassume an initial state of stress in the shotcreteat the start of monitoring, i.e., at t = 3 days.Estimations of the initial strains in the graniteslab were, as previously described, performedby [23]. Strains for the six slabs were estimatedto be between 0 − 10 µε and 0 − (−20) µε forthe lower and upper strain gauge, respectively.These assumed initial strains corresponds ratherwell to measured strains presented by [7]. How-ever, in that test cast shotcrete slabs were used,i.e., concrete with properties of sprayed con-crete, and the environmental conditions werenot controlled. A uniform distributed temper-ature load of −7 ◦C and −3.5 ◦C was appliedto shotcrete slab-1 and 2 to generate an initialstress in the shotcrete, as well as an initial strainthe granite. In Figure 7 the strains from the ex-periment and the numerical analysis are plot-ted. Dashed and solid lines are from experi-ments and numerical analysis, respectively. Theapplied initial strains for slab-1, blue lines inFigure 7, corresponds well to estimated strainsbut was overestimated for slab-2, red lines inFigure 7. The time of failure and the rate ofstrains were underestimated for both slabs. Thisindicates that the rate of strain described bythe model must be changed to accurately de-scribe the experiment. A combination of an ini-tial stress and a tuned moisture transportationmodel could most likely be used to more accu-rately simulate the experiment.

0 5 10 15 20−80

−60

−40

−20

0

20

40

Time (days)

Str

ain

(10−

6 )

Lower Side

Upper Side

Figure 6: Strains in granite slab from experiments,dashed lines, and simulations, solid lines. Blue line isfor slab 1 and red line for slab 2. Parameters controllingthe moisture transportation have been tuned.

8


0 3 6 9 12 15 18 21 24 27 30 33−80

−60

−40

−20

0

20

40

Time (days)

Str

ain

(10−

6 )

Lower Side

Upper Side

Figure 7: Strains in granite slab from experiments,dashed lines, and simulations, solid lines. Blue line isfor slab 1 and red line for slab 2. An initial state of stressat t = 3 days have been assumed.

5.3 Temperature modelThe temperature model can as well be fine-

tuned to accurately describe the time of failureof the shotcrete slab, by simply changing thetime over which the temperature decreases. Byusing an exponential decreasing function theeffects of early shrinkage can also be accountedfor. However, the temperature model is limitedto simulate shrinkage with the same constantrelative humidity as the test it has been verifiedagainst. Since shrinkage in the multi-physicalmodel is rather described by the physical pro-cess of drying, it can also be used for simula-tions with different environmental conditions.

5.4 Damage modelThe results have shown some non-unique so-

lutions with respect to crack patterns, which fur-ther influence the levels of strain and time offailure. This can be seen by comparing the sim-ulated levels of strain at failure for slab-2, pre-sented in Figure 6and 7. The damaged areas inthe slabs are similar for different analysis, butthe localization of the final crack varies. The ir-regular geometry introduces effects of bendingmoment in the slabs and damage will thereforedevelop in thinner sections first. Damage willthen either propagate in this section, or start toform in other thin sections. This enables the for-mation of damage in several small areas of theslab before the final crack is localized. This isbelieved to be the reasons for the variation ofcrack localization.

6 CONCLUSIONSIn this paper a coupled multi-physical mate-

rial model has been used in an attempt to simu-late experimental results for an end-restrainedshotcrete slab subjected to shrinkage. A freeshrinkage test was used to tune the moisturetransportation model which was then used tosimulate the experiment. First results lead to aneed of tuning of the model which, possibly, isa combined effect of an initial state of stress inthe shotcrete and an increased rate of strain.The presented results indicate that for fu-ture studies, more detailed measurements areneeded. To enable the moisture transportationmodel to be more accurately tuned, tempera-ture and relative humidity within the shotcreteshould be measured. Measurements should startas soon as possible after spraying to capture theeffects of autogenous shrinkage and thermal ef-fects due to the hydration of cement. The pres-ence of cracks will locally increase the mois-ture transportation and their occurrence shouldtherefore be more thoroughly monitored duringthe experiment. Future studies should also fo-cus on whether or not a restrained sample willshrink faster compared to an unrestrained sam-ple. Such a study should also include the struc-ture of the material, i.e., the porosity.It can be concluded that both types of models,the multi-physical and the uncoupled, can betuned to accurately simulate the experimentalresults. However, besides being a more phys-ical correct solution to the shrinkage problemthe multi-physical model will also be able tosimulate the behaviour with various boundaryconditions, i.e., different relative humidity. Themulti-physical model is therefore preferable butsome further improvements are needed. Be-cause shrinkage cracks in the experiment oc-curred within 16 days, the multi-physical modelmust, clearly, be improved to also account fordevelopment of thermal stresses due to the hy-dration of cement as well as autogenous shrink-age. The effects of cracks with respect to mois-ture transportation should also be added. Fi-nally, it would be desirable to increase the ro-bustness of the damage model in terms of local-

9


ization of the cracks.

ACKNOWLEDGEMENTSThe work presented in this paper is part of

a larger research project focusing on numeri-cal simulations of irregular shotcrete and theinteraction between shotcrete and rock. Theproject is sponsored by BeFo, the Rock Engi-neering Research Foundation and their supportis hereby greatly acknowledged.

References[1] Holmgren J B. Shotcrete research and

practice in Sweden a development over35 years. In: Shotcrete: Elements of asystem. TSE Pty. Ltd. Sydney Australia:CRC Press/Balkema; 2010. p. 135–142.

[2] Ansell A. Investigation of shrinkagecracking in shotcrete on tunnel drains.Tunnelling and Underground Space Tech-nology. 2010 sep;25(5):607–613.

[3] Gasch T, Malm R, Ansell A. Modellingtime-dependent deformations of concretesubjected to variable environmental con-ditions using a coupled hygro-thermo-mechanical model. To be published.2015;.

[4] Bryne E L, Holmgren J, Ansell A. Shrink-age testing of end-restrained shotcrete ongranite slabs. Magazine of Concrete Re-search. 2014;p. 1–11.

[5] Malmgren L, Nordlund E, Rolund S. Ad-hesion strength and shrinkage of shotcrete.Tunnelling and Underground Space Tech-nology. 2005;20(1):33–48.

[6] Carlsward J. Shrinkage cracking ofsteel fibre reinforced self compacting con-crete overlays Test methods and theoreti-cal modelling; 2006.

[7] Bryne E L, Ansell A, Holmgren J. Inves-tigation of restrained shrinkage crackingin partially fixed shotcrete linings. Tun-nelling and Underground Space Technol-ogy. 2014 may;42:136–143.

[8] Groth P. Fibre Reinforced Concrete. LuleaUniversity of Technology; 2010.

[9] Sjolander A, Ansell A, Bryne E L. Nu-merical simulations of restrained shrink-age cracking of shotcrete. To be pub-lished. 2015;.

[10] Holmgren J, Ansell A. Test on restrainedshrinkage of shotcrete with steel fibres andglas fibres. In: Federation NC, editor. Pro-ceedings of the XX Symposium on NordicConcrete Research; 2008. .

[11] Lagerblad B, Fjallberg L, Vogt C.Shrinkage and durability of shotcrete.In: Shotcrete: Elements of a system.TSE Pty. Ltd. Sydney Australia: CRCPress/Balkema; 2010. p. 173–180.

[12] Malmgren L, Nordlund E. Interaction ofshotcrete with rock and rock boltsA nu-merical study. International Journal ofRock Mechanics and Mining Sciences.2008 jun;45(4):538–553.

[13] Brekke L T, Einstein H H, Mason E R.State-of-the-art review on shotcrete [Finalreport]; 1976.

[14] Bazant P Z, Prasannan S. Solidifica-tion theory for concrete creep. I: Formu-lation. Journal of Engineering Mechanics.1989;115(8):1691–1703.

[15] Bazant Z, Prasannan S. Solidification the-ory for concrete creep. II: Verification andApplication. Journal of Engineering Me-chanics. 1989;115(8):1704–1725.

[16] Bazant Z, Hauggaard A, Baweja S, UlmF. Microprestress-solidification theory forconcrete creep 1: Aging and drying ef-fects. Journal Of Engineering Mechanics.1997;123(11):1188–1194.

[17] Bazant P Z, Hauggaard B A, BawejaS. Microprestress-solidification theory forconcrete creep 2: Algorithm and verifica-tion. Journal Of Engineering Mechanics.1997;123(11):1195–1201.

10


[18] Jirasek M, Havlasek P. Microprestressso-lidification theory of concrete creep: Re-formulation and improvement. Cementand Concrete Research. 2014;60:51–62.

[19] COMSOL. COMSOL, Multiphysics ver5.1 Documentation; 2015.

[20] Bazant P Z, Najjar J L. Nonlinear waterdiffusion in nonsaturated concrete. Matri-aux et Construction. 1972;(1):3–20.

[21] fib. fib Model Code for ConcreteStrucutres. Wiley-VCH Verlag GmbH Co;2010.

[22] Oliver J, Cervera M, Oller S, LublinerJ. Oliver-1990-Isotropic damage models

and smeare.pdf. Computer Adided Anal-ysis and Design of Concrete Strucutres.1990;p. 945–957.

[23] Bryne E L. Time Dependent MaterialProperties of Shotcrete for Hard RockTunneling; 2014.

[24] CEN. Eurocode 2: Design of ConcreteStrucutre -Part 1-1: General rules andrules for buildings; 2004.

[25] Bryne E L, Ansell A, Holmgren J. Labo-ratory testing of early age bond strengthof shotcrete on hard rock. Tunnellingand Underground Space Technology. 2014mar;41:113–119.

11

shrinkage cracking of thin irregular shotcrete … · shrinkage cracking of thin irregular...

Documents