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International Journal of Adhesion and Adhesives

journal homepage: www.elsevier.com/locate/ijadhadh

Spatially-degraded adhesive anchors under material uncertainty

R. Tipireddya, S. Kumarb,⁎

a Pacific Northwest National Laboratory, P.O. Box 999, MSIN K7-90, Richland, WA 99352, USAb Institute Center for Energy (iEnergy), Department of Mechanical and Materials Engineering, Masdar Institute of Science and Technology, PO Box 54224,Abu Dhabi, United Arab Emirates

A R T I C L E I N F O

Keywords:Spatially-degraded adhesivesPullout performanceInterfacial stressesCompositesStochastic materialMaterial uncertainty

A B S T R A C T

The classic problem of stress transfer in a cylinder through shear to a surrounding medium, is analyzed here inthe context of pullout of an anchor under material uncertainty. Assuming a log-normal distribution for therandom shear stiffness field of the adhesive, the stochastic differential equation (SDE) is formulated forspatially-tailored/degraded adhesive anchors. The stochastic shear stress distribution in the adhesive ispresented for various embedment lengths and adhesive thicknesses clearly demarcating the regime over whichfailure would initiate. The stochastic variation of maximum shear stress of the adhesive as a function ofembedment length and adhesive thickness is also presented. It is observed that the mean maximum shearstresses in the degraded adhesive for both fixed and free embedded-end cases converge and the influence ofboundary condition at the embedded-end on shear stress field disappears as the embedment length is increased.For the parameters considered here, about 45% longer embedment length is required compared to an intactbondline for shear-dominated load transfer, suggesting that the design of adhesive anchors should adequatelyaccount for likely in-service damage that causes material uncertainty to avoid premature failure.

1. Introduction

The generic mechanics problem of pullout of a fiber embedded in amatrix [1–6] is considered here in the context of adhesive anchorsunder uncertainty. Design of adhesive anchors requires that theadhesive is strong enough to resist the service loading and that thestiffness of adhesive effectively transfers stress along the bondline. Amajor concern in epoxy adhesive anchoring systems is the effect ofcreep deformations under sustained loading. As the adhesives have aviscous component, they exhibit nonlinear irreversible creep deforma-tions during service leading to degradation of the bondline especially inoperating conditions such as the presence of moisture, freeze-thawcycles or off-design cure. This reduces the stiffness and consequentlythe strength at every material point in the adhesive layer as a functionof time [7]. Usually adhesive anchors are designed to have embedmentlengths that are at least 10 times larger than the diameter of the anchorin order to ensure shear-dominated load transfer from the anchor tothe surrounding material. As long as the embedment length is largerthan the shear lag length, the free or fixed embedded-end condition ofthe anchor doesn't influence the pullout performance. Nevertheless,deterioration of adhesive properties during service would render theadhesive compliant, warranting longer embedment length than theoriginally designed length for complete shear stress transfer. In this

context, the shear lag length is more than the embedment length andtherefore the boundary condition at the embedded-end of the anchorbecomes important as it significantly influences the stress state in theadhesive layer and hence the pullout performance.

Free embedded-end condition may exist when the bottom of thehole is dry due to non-penetration of adhesive this far. The combina-tion of such deficient installation and irreversible creep deformations ofthe anchorage assembly led to the fatal collapse of a suspended ceilingsection in the Interstate 90 Connector Tunnel in Boston, Massachusettson July 10th 2006, provoking extensive research interest in predictingthe degradation of the adhesive bond and long-term performance ofbonded anchors under sustained loading [8]. The adhesive anchorsmay fail due to material damage (concrete cone failure or anchorbreakage) and surface failures such as shear failure along adhesive-concrete interface and tensile failure at the embedded-end of theanchor (anchor-adhesive interface or adhesive-concrete interface) [9].Among these, the pull-out failure, involving shearing of the adhesiveover the embedment length is the most important. Numerous theore-tical studies were proposed to predict the pullout performance of suchadhesively bonded anchors [10,9,11] with homogeneous bondline. Theidea of functionally graded adhesives has already been used in bondedrepairs [12,13]. The structural performance of stiffness-graded multi-material bonded systems was theoretically/numerically evaluated [14–

http://dx.doi.org/10.1016/j.ijadhadh.2017.02.010

⁎ Corresponding author.E-mail addresses: s.kumar@eng.oxon.org, skumaar@mit.edu (S. Kumar).

International Journal of Adhesion & Adhesives 76 (2017) 61–69

Available online 08 February 20170143-7496/ © 2017 Elsevier Ltd. All rights reserved.

MARK

23]. Such functionally graded interfaces (gradation of elastic propertiesover the bondline) were found helpful in minimizing the stresses andhence maximizing the load carrying capacity of adhesively bondedstructures. The objective of this study is to predict the stress field andthe pullout characteristics of spatially-degraded adhesive anchorsunder uncertainty.

Uncertainties in their structural performance of adhesively bondedsystems arise due to fabrication inaccuracies, degradation or naturalheterogeneity [24,25]. Material uncertainties include randomness inmaterial properties such as stiffness, yield strength etc., and geometricuncertainties involve randomness in parameters such as bondlength,adhesive thickness etc. Uncertainty in structural performance ofadhesively bonded joints have been studied for reliability analysis[26,24], but uncertainty in performance of adhesive anchors have notyet been discussed in literature. In the analysis of adhesive anchorssubjected to monotonic tensile loads, the axial stresses generated in theanchor is transferred to the concrete primarily through the shearing ofthe adhesive. Hence the shear modulus is an important parameter inthe stress analysis and design of adhesive anchors. Nevertheless, asdiscussed before, in service, irreversible creep deformation of thethermosetting adhesive coupled with the operating scenarios such aspresence of moisture, freeze-thaw cycles, or off-design cure can createspatial gradient properties of the bondline, leading to its variablestiffness along the bondline. Therefore, uncertainties in shear modulusand its gradient along the bondline are expected and the randomness inshear modulus should be taken into account for the design of suchadhesive anchors. Towards this end, the study here considers the shearstiffness of the adhesive as a random field and examines the shearstress distribution along the bondline employing a stochastic modelingframework.

The anchor is embedded in the concrete through an inhomogeneousadhesive layer. The concrete is assumed to be a semi-infinite half-spaceand is considered rigid since its Young's modulus is much higher thanthe epoxy adhesive [9]. The strain energy in the system is expressed asa function of axial deformation of the anchor and the energy functionalis minimized to obtain the SDE using a variational method. MonteCarlo methods are standard methods for uncertainty quantification,however, they converge slowly. Hence polynomial chaos [27] basedmethods are considered here. A log-normal distribution [28] isassumed for the random shear modulus. Log-normal random field isobtained by taking exponential of a Gaussian random field. Log-normaldistribution ensures non-negative values for the random shear mod-ulus. For computational purposes, random field must be discretized inboth spatial and stochastic domains. Hence the Gaussian part of theshear modulus is approximated in Karhunen-Loève (KL) expansion[29] and the shear modulus is approximated with Polynomial chaos(PC) expansion [27] in Hermite polynomials. This results in deforma-tions expanded in PC coefficients which are then solved numericallyusing stochastic Galerkin [27,30] or stochastic collocation [31,32]methods. Recent developments in this field include stochastic dimen-sion reduction methods such as Basis adaptation [33] and activesubspace methods [34] in order to reduce the computational cost insolving stochastic systems. In this work, the stochastic shear stress fieldin the system is obtained for both fixed and free bonded-in endconditions of the anchor and the results are discussed. The stochasticvariation of maximum shear stress of the adhesive as a function ofembedment length and adhesive thickness is also presented.

2. Problem formulation

Consider an anchor rod of diameter D and length l embedded in aconcrete half-space through an adhesive layer of thickness t, as shownin Fig. 1a. This assembly is referred to a Cylindrical coordinate system(r θ z, , ) placed at the center of the rod at the embedded end O as shownin Fig. 1a. Let E be the Young's modulus of the anchor. The shearmodulus of the adhesive G is assumed to vary as a function of the axial

coordinate z such that G G(0) = max and G l G( ) = min as shown in Fig. 1b,where G G≥max min. G G=max min represents a homogeneous bondlinewhile G G≠max min represents a tailored/degraded bondline. The anchoris subjected to an axial tensile load P at z=l. The load transfer from theanchor to the surrounding concrete primarily occurs through shearingof the adhesive. The concrete is considered rigid as it is much stifferthan the adhesive. Considering axial deformation of the anchor w z ω( , )and shear deformation of the adhesive due to pull-out load, the strainenergy in the anchor is expressed as follows [9]:

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟∫U EA dw z ω

dzπ D t

tG z ω w z ω dz=

2( , ) +

2( + ) ( , ) ( , )

l

0

22

(1)

where, A D= π4

2 is th area of anchor and A πt D t= ( + )a is the crosssectional area of the adhesive layer. The above energy functional isrendered stationary with respect to the unknown displacement w toobtain the following second order SDE:

π D tt

G z ω w z ω EA d w z ωdz

z l ω Ω( + ) ( , ) ( , ) − ( , ) = 0, ∈ [0, ], ∈ .2

2 (2)

Here, the shear modulus of the adhesive which varies as a function ofthe axial coordinate z (Fig. 1b) is considered as a random fieldG z ω ω Ω( , ), ∈ , where Ω( , , ) is a complete probability space withsample space Ω that contains all possible outcomes in a randomexperiment, σ-algebra that contain the set of events and probabilitymeasure that maps the events to the probabilities. Random shearmodulus, G z ω( , ) leads to random deformation w z ω( , ) and conse-quently to random shear stress τ z ω( , ). The shear strain γ z ω( , ) and theshear stress τ z ω( , ) in the adhesive are proportional to the deformationw z ω( , ) of the anchor and are given respectively by

γ w z ωt

τ w z ωt

G z ω= ( , ) , = ( , ) ( , )(3)

The axial strain ε and axial stress σ in the anchor, respectively are

ε dwdz

z ω σ dwdz

z ω E= ( , ), = ( , )(4)

Let G z ω H z ω( , ) = exp( ( , )), be the random shear modulus modeledwith log-normal distribution, where H z ω( , ) is the Gaussian randomfield with mean H z( )m and following covariance function,

⎛⎝⎜

⎞⎠⎟C z z σ z z

l( , ) = exp − ( − ) ,H H

z1 2

2 1 22

2(5)

where, σH is standard deviation of H and lz is its correlation length.H z ω( , ) is now approximated using a truncated Karhunen-Loève (KL)expansion to convert infinite dimensional random field into a finitedimensional one, as follows,

∑H z ω H z ξ H z H z λ ξ( , ) ≈ ( , ) = ( ) + ( )mi

d

i i i=1 (6)

where, ξ N∼ (0, 1)i are uncorrelated standard Gaussian random vari-ables and hence are also independent. Hm is the mean of the randomfield H z ω( , ), and H z{ ( )}i and λ{ }i are eigenfunctions and eigenvalues ofC z z( , )H 1 2 obtained by solving following integral eigenvalue problem,

∫ C z z H z dz λ H z( , ) ( ) = ( ).l

H i i i0

1 2 2 2 1 (7)

The eigenvalues λ{ }i are positive and non-increasing as shown in Fig. 3and hence the KL expansion in (6) can be truncated after d + 1 terms,and the eigenfunctions H z{ ( )}i are orthonormal, that is,

∫ H z H z dz δ( ) ( ) = ,l

i j ij0 (8)

where, δij is the Kronecker delta. Now, the shear modulus G z ω( , ) canbe approximated with a truncated polynomial chaos expansion asfollows,

R. Tipireddy, S. Kumar International Journal of Adhesion & Adhesives 76 (2017) 61–69

62

∑ξ ξG z ω G z G z G z ψ( , ) ≈ ( , ) = ( ) + ( ) ( ),mi

p

i i=1 (9)

where Gm is the mean, Gi are PC coefficients and ξψ ( )i are multi-variateHermite polynomials in ξ ξ ξ= [ , …, ]d

T1 , that are orthonormal with

respect to the probability density of ξ, that is,

∫ ξ ξ ξψ ψ d δ( ) ( ) = .Ω

i j ij (10)

Here, p + 1 = d Md M

( + ) !! !

is the number of terms in the expansion, where dis the dimension and M is the degree of polynomial chaos expansion.Deformation field w z ω( , ) can also be expanded using a truncatedpolynomial chaos as follows,

∑ξ ξw z ω w z w z w z ψ( , ) ≈ ( , ) = ( ) + ( ) ( ),mi

p

i i=1 (11)

where, wm is the mean and wi are the PC coefficients of the deformationfield w z ω( , ). Using polynomial chaos approach, (2) can be solved withintrusive (stochastic Galerkin) or non-intrusive (stochastic collocation)approaches.

2.1. Stochastic Galerkin method

When above expansion is substituted into SDE in (2), we get

∑ ∑ ∑ξ ξ ξπ D tt

G z ψ w z ψ EA d w zdz

ψ( + ) ( ) ( ) ( ) ( ) − ( ) ( ) = 0.i

p

i ii

p

i ii

pi

i=0 =0 =0

2

2(12)

Rewriting the above equation,

∑ ∑ ∑ξ ξ ξβ ψ ψ G z w z χ d w zdz

ψ( ) ( ) ( ) ( ) − ( ) ( ) = 0.j

p

i

p

i j i ji

pi

i=0 =0 =0

2

2(13)

where,

β π D tt

χ EA= ( + ) , and = .

Multiplying both sides by ξψ ( )k and taking expectation, we get followingcoupled system of ODEs

∑ ∑ ∑ξ ξ ξ ξ ξβ ψ ψ ψ G z w z χ d ω zdz

ψ ψ

k p

[ ( ) ( ) ( )] ( ) ( ) − ( ) [ ( ) ( )] = 0,

= 1, …,

j

p

i

p

i j j i ji

pi

i k=0 =0 =0

2

2

(14)

where [·] represents mathematical expectation. Since ξψ ( )i are ortho-normal, ξ ξψ ψ δ[ ( ) ( )] =i k ik, where δik is the Kronecker delta. Let

ξ ξ ξc ψ ψ ψ= [ ( ) ( ) ( )]ijk i j j , then above set of equations can be written as,

∑ ∑π D tt

c G z w z EA d w zdz

k p( + ) ( ) ( ) − ( ) = 0, = 1, …, .j

p

i

p

ijk i jk

=0 =0

2

2(15)

The deterministic coupled set of differential equations in (15) can besolved to get polynomial chaos coefficients wi(z) of ξw z( , ).

2.2. Stochastic collocation method

In non-intrusive approach, the governing Eq. (2) is solved at fewrealizations of the random variables called quadrature points corre-sponding to their probability measure. Let ξ q( ) be the quadrature pointsof the underlying random variables, ξ, then following deterministicequation is solved,

ξ ξ ξπ D tt

G z w z EA d w zdz

z l( + ) ( , ) ( , ) − ( , ) = 0, ∈ [0, ],q qq

( ) ( )2 ( )

2 (16)

for q n= 1, …, q, where nq is the total number of quadrature pointsrequired. PC coefficients w z( )i of the random field ξw z( , ) are obtainedby projecting ξw z( , ) on to the polynomial ξψ ( )i as

∫ ∑ξ ξ ξ ξ ξ ξ ξw z w z ψ w z ψ d w z ψ μ( ) = [ ( , ) ( )] = ( , ) ( ) ≈ ( , ) ( ) ,i iΩ

iq

nq

iq q

=1

( ) ( ) ( )q

(17)

where, ξ ξ ξ= [ , …, ]q qd

q T( )1( ) ( ) is the quadrature point and μ q( ) is the

corresponding weight.

3. Deterministic loading and boundary conditions

When the adhesive anchor is in an appropriate service condition,the embedded-end of the anchor is perfectly bonded to the concrete. In

Fig. 1. Adhesive anchors under axial tensile load P: a) Anchorage assembly with fully bonded/debonded embedded-end b) the variation of shear modulus of the adhesive G(z) along theembedment depth; Gmax is the maximum shear modulus of the degraded bondline, Gmin is the minimum shear modulus of the bondline and m is a positive integer.

R. Tipireddy, S. Kumar International Journal of Adhesion & Adhesives 76 (2017) 61–69

63

this case, the bonded-in end of the anchor is regarded to be fullyrestrained by the concrete. Therefore, the boundary conditions arew(0) = 0 and w l′( ) = P

EA . If debonding occurs at the embedded-end, therestraint of the concrete would be zero. On the contrast, the bottom ofthe hole would be dry if the adhesive had not penetrated this far due todeficient installation. So for the debonded/unbonded embedded-endcase, the boundary conditions are w′(0) = 0 and w l′( ) = P

EA . Thesolutions for the deformation of the anchor w z( )using these twodeterministic boundary conditions are obtained.

Numerical results for the SDE given by Eq. (2) is presented in thenext section. Both stochastic Galerkin and stochastic collocationapproaches result in same solution. Hence, for the sake of demonstra-tion, the stochastic collocation approach is employed here. Bothmethods have their own advantages and disadvantages and it is beyondthe scope of this work to compare the merits of these methods.However, it is worth mentioning that the stochastic collocationmethods can make use of legacy softwares readily without muchmodification in the code [35]. For numerical implementation, randomshear modulus G z ω H z ω( , ) = exp[ ( , )] is assumed to have a log-normal

distribution with mean⎛⎝⎜

⎞⎠⎟G z G G G z

l( ) = + ( − )m max min max

2

.

4. Numerical solution

A steel anchor (E=210 GPa, D=15 mm and l=100 mm) embeddedin the rigid half-space using a thermosetting adhesive of thicknesst=2.5 mm is subjected to an axial load P=1.786 kN. The adhesive isassumed to have a random shear modulus G z ω H z ω( , ) = exp[ ( , )] withG = 1400 MPamax and G = 140 MPamin . G z ω( , ) is assumed to have acoefficient of variation δ = 0.1. Fig. 2a shows the mean and variation ofshear modulus, and Fig. 2b shows mean and few realizations of G z ω( , ).Gaussian random field H x ω( , ) is assumed to have the followingcovariance function,

⎛⎝⎜

⎞⎠⎟C z z σ z z

l( , ) = exp − ( − ) ,H H

z1 2

2 1 22

2(18)

where σ δ= ln(1 + )H2 ,

⎛⎝⎜

⎞⎠⎟H z( ) = lnm

G z

δ

( )

1 +n

2, and l =z

l4 is the correlation

length of H z ω( , ). Random field H z ω( , ) is then expressed usingKarhunen-Loève expansion by solving Eq. (9) to obtain eigenfunctionsand eigenvalues. Eigenvalues are non-increasing as shown in Fig. 3 andthe KL expansion can be truncated to obtain finite dimensional system.Here, the KL expansion is truncated after d + 1 = 11 terms resulting ina d=10 dimensional system. It means that the number of input randomvariables ξ{ }i is 10. Then the shear modulus G z ω( , ), deformationw z ω( , ) and shear stress τ z ω( , ) are expanded using a polynomial chaos

(PC) expansion in d=10 dimensions and order M=3. To solve thesystem using non-intrusive approach, Smolyak sparsegrid method isemployed, in which for a dimension d=10 and sparsegrid level 5, totalnumber of deterministic solutions required nq=8761. After computingthe solution and quantities of interest (shear stress here), polynomialchaos coefficients, mean, standard deviation, probability density func-tion and cumulative density function are obtained.

5. Results and discussion

Fig. 4a and b show the mean and variation of normalized axialstress σA

Pin the anchor over the normalized embedment depth η for

bonded and debonded embedded-end conditions respectively. Here,blue solid line indicates the mean of the normalized axial stress as afunction of normalized embedment length. Stochastic variation isrepresented as lower and upper bounds, represented in dotted linesin magenta. Lower and upper bounds are computed as mean minusstandard deviation and mean plus standard deviation respectively.Note that for debonded embedded-end case, the deterministic tractionboundary conditions both at z=0 and z=l are strictly satisfied as shownin Fig. 4b. On the other hand, for bonded embedded-end case, the axialstress in the anchor at z=0 is non-zero as expected for l l< cr albeit thesolution satisfies the applied traction boundary condition at z=l.

Fig. 5a and b show the mean and variation of normalized shearstress in the adhesive along the bondline for the bonded and debondedembedded-end conditions respectively. The shear stress distribution

Fig. 2. (a) Mean and variation of the shear modulus G η( ), (b) samples of G η( ) for m=2.

Fig. 3. Eigenvalues of for covariance function of H η ω( , ).

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over the embedment depth in case of debonded embedded-end isdifferent to that of the bonded embedded-end case as the load transferin the former case to concrete occurs only through shearing of lateralsurface of the adhesive such that

∫S π D t τ η dη P= ( + ) ( ) =0

1

(19)

On the other hand, a portion of applied load is transferred to concretethrough tensile mode at the embedded-end for bonded embedded-endcase for l l< cr such that P S D σ= + π

c42 , where, σc is the tensile stress at

the bottom of the anchor. σc can be computed using Eq. (4) whichshould be less than the tensile strength of the concrete σtc. Therefore,permissible tensile stress at the embedded-end is given by

⎛⎝⎜

⎞⎠⎟σ P

πDSP

σ= 4 1 − ≤c ct

2 (20)

Note that → 1SP

for free embedded-end case irrespective of thechoice of l and hence σ → 0c . For the chosen adhesive withGmax=1400 MPa, the shear strength τcmax=6.2 MPa [9] and theposition dependent shear strength τ z( )c is assumed to be of thefollowing form:

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟⎛⎝⎜

⎞⎠⎟

⎤⎦⎥τ z G

Gzl

τ( ) = 1 + − 1c min

max

m

maxc

(21)

The variation in normalized shear strength of the adhesiveτ z( ) =c τ z A

P( )c

(based on Eq. (21)) is also shown in Fig. 5a and b toclearly demarcate the shear failure regime of the adhesive. FromFig. 5b, it can be seen that the shear stress in the adhesive at theembedded-end doesn't die down as the embedment length chosen isnot sufficient for complete stress transfer, i.e., l l< cr . Fig. 6a and bshow the probability density function (PDF) and cumulative densityfunction (CDF) respectively of the normalized shear stress (τ =∼ τA

P) at

z=0 for a debonded embedded-end case. It shows that the shear stressτ∼ at this point has minimum and maximum values of 0.15 and 0.31respectively and a mean value of 0.21 and standard deviation of 0.002.Figs. 7a and b show the PDF and CDF of τ∼ at the loaded end (at z=l) forthe fixed embedded-end case with minimum and maximum values of0.1 and 0.15 and mean and standard deviation of 0.12 and 0.0012respectively. The PDF and CDF at the location of maximum shearstress of the adhesive both in bonded and debonded cases are shown inFigs. 8 and 9 respectively. In Fig. 10 the variation in mean shear stressboth in fixed and free embedded-end cases for different embedmentlengths with l = {50, 100, 150, 200, 250} mm are shown along withposition dependent shear strength. It can be observed that themaximum shear stress in the adhesive decreases and its location

Fig. 4. Mean and variation of the axial stress σAP

in the anchor: a) bonded embedded-end b) debonded embedded-end. (For interpretation of the references to color in this figure, the

reader is referred to the web version of this article.)

Fig. 5. Mean and variation of the shear stress τAP

in the adhesive: a) bonded embedded-end b) debonded embedded-end. τ η( )maxc is the position dependent shear strength.

R. Tipireddy, S. Kumar International Journal of Adhesion & Adhesives 76 (2017) 61–69

65

changes as the embedment length is increased in both cases. Fig. 10also indicates the regime over which shear failure would occur uponshear stress τ z( ) in the adhesive reaching its position dependent shearstrength τ z( )c . Fig. 11 shows standard deviation of the shear stress

distribution for bonded and debonded embedded-end cases over theembedment length respectively for different choices of l. Figs. 12 and13 show variation in mean and standard deviation of the shear stressfor different thicknesses of the adhesive layer with

Fig. 6. The shear stress τAP

at z=0 for debonded embedded-end case: a) PDF and b) CDF.

Fig. 7. The shear stress τAP

at z=l for bonded embedded-end case: a) PDF and b) CDF.

Fig. 8. The shear stress τAP

at the location of maximum shear stress (z = 73.1 mmmax ) for bonded embedded-end case: a) PDF and b) CDF.

R. Tipireddy, S. Kumar International Journal of Adhesion & Adhesives 76 (2017) 61–69

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t = {2.5, 3.5, 4.5, 5.5, 6.5} mm. Fig. 14 shows the maximum shear stressin the adhesive as a function of the adhesive thickness for l=100 mmand we could still see the difference in shear stress for two differentcases as expected since the embedment length l=100 mm is notsufficient for complete stress transfer. Fig. 15 shows the stochasticvariation of maximum shear stress as a function of embedment length

for both bonded and debonded embedded-end cases for degraded/tailored bondline in comparison with that of intact bondline. It can beobserved that the mean maximum shear stresses for bonded anddebonded embedded-end cases converge as the embedment length isincreased and the influence of boundary condition at the embedded-end disappears for l=150 mm for degraded bondline. Furthermore,

Fig. 9. The shear stress τAP

at the location of maximum shear stress (z = 71 mmmax ) for debonded embedded-end case: a) PDF and b) CDF.

Fig. 10. Mean of the shear stress distribution in the adhesive as a function of embedment depth: a) bonded embedded-end b) debonded embedded-end.

Fig. 11. Standard deviation of the shear stress distribution in the adhesive as a function of embedment depth: a) bonded embedded-end b) debonded embedded-end.

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l D= 16.67cr (250 mm) [see Fig. 10] is the minimum embedment lengthrequired for the degraded bondline, while a minimum embedmentlength of l D= 11.33cr (170 mm) is required for an intact bondline(G G=max min=1400 MPa) to ensure complete shear stress transfer [19],for the parameters considered here. In service, with increase in level of

damage, l required for shear-dominated load transfer would increasecompared to that of intact bondline, leading to premature failure suchas the fatal collapse of a suspended ceiling section in the Interstate 90Connector Tunnel in Boston, Massachusetts on July 10th 2006.Foregoing discussion clearly indicates that the optimal embedment

Fig. 12. Mean of the shear stress distribution in the adhesive as a function adhesive thickness: a) bonded embedded-end b) debonded embedded-end.

Fig. 13. Standard deviation of the shear stress distribution in the adhesive as a function adhesive thickness: a) bonded embedded-end b) debonded embedded-end.

Fig. 14. Mean and variation of maximum shear stress in the adhesive as a function ofadhesive thickness for both bonded and debonded embedded-end cases for l = 100 mm.

Fig. 15. Mean and variation of maximum shear stress in the adhesive as a function ofembedment length for t = 2.5 mm .

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length l depends on the shear modulus profile, and the minimum andmaximum values of shear modulus of the adhesive chosen when allother parameters are fixed.

6. Conclusions

Pullout performance of spatially-degraded adhesive anchors underuncertain material properties is investigated through a stochasticmodeling framework. Uncertainties in the shear modulus of theadhesive is formulated as a random field with log-normal distribution.Truncated Karhunen-Loève expansion and truncated polynomial chaosexpansions are used to approximate the infinite dimensional randomfields in finite dimensional random space. Stochastic Galerkin andcollocation methods are presented for the SDE and finally thenumerical solution for the stochastic shear stress field in the adhesiveis obtained using stochastic collocation method. Uncertainties in theshear stress at different locations are presented through probabilitydensity function (PDF) and cumulative density function (CDF). Meanand standard deviation of the shear stress for different parameters suchas embedment length and adhesive thickness are presented clearlydemarcating the regime over which failure occur upon shear stressreaching its position dependent critical value. Finally, probabilisticvariation of maximum shear stress as a function of embedment lengthand adhesive thickness is presented for both bonded and debondedembedded-end cases. It is found that the shear-lag length requiredincreases with increase in in-service damage, warranting longerembedment length than that of an intact bondline. For the parametersconsidered here, about 45% longer embedment length of anchor isrequired compared to an intact bondline for shear-dominated loadtransfer indicating that the design of adhesive anchors should ade-quately account for likely in-service damage to avoid premature failuresuch as the fatal collapse of a suspended ceiling section in the Interstate90 Connector Tunnel in Boston, Massachusetts on July 10th 2006.Manufacturing of composite materials is a complex process and itresults in a large number of uncertainties. Current work focuses ondemonstrating propagation of one such an uncertainty, namely shearmodulus of the adhesive to the quantity of interest namely shear stressin the adhesive. Subsequent studies will focus on propagating multiplesources of uncertainty as in various composite materials and manu-facturing processes, including Additive Manufacturing [36].

Acknowledgment

SK gratefully acknowledges the financial support from the AbuDhabi National Oil Company (ADNOC) under Award No. EX2016-000010. RT would like to acknowledge the financial support of the U.S.Department of Energy, Office of Science, Office of Advanced ScientificComputing Research as part of the “Uncertainty Quantification ForComplex Systems Described by Stochastic Partial DifferentialEquations” project. Pacific Northwest National Laboratory is operatedby Battelle for the DOE under Contract DE-AC05-76RL01830. SK andRT would like to thank Professor Brian Wardle, MassachusettsInstitute of Technology for his valuable suggestions and commentson the manuscript.

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