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2010 45: 647The Journal of Strain Analysis for Engineering DesignC Trautner, M McGinnis and S Pessiki

determination of in-situ stresses in concrete structuresAnalytical and numerical development of the incremental core-drilling method of non-destructive

  

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Analytical and numerical development of theincremental core-drilling method of non-destructivedetermination of in-situ stresses in concrete structuresC Trautner1*, M McGinnis2, and S Pessiki3

1Simpson, Gumpertz, and Heger, Waltham, Massachusetts, USA2Civil Engineering Departrment, University of Texas, Tyler, Texas, USA3Civil and Environmental Engineering Department, Lehigh University, Bethlehem, Pennsylvania, USA

The manuscript was received on 21 September 2009 and was accepted after revision for publication on 12 May 2010.

DOI: 10.1243/03093247JSA600

Abstract: The incremental core-drilling method (ICDM) is a technique to assess in-situ stressesin concrete structures. These stresses may be either constant or vary through the thickness of themember under investigation. In this method, a core is drilled into a concrete structure incre-mentally. The displacements which occur locally around the perimeter of the core at each incre-ment are related to the in-situ stresses by an elastic calculation process known as the influencefunction method. This paper presents the analytical and numerical techniques necessary forpractical use of the ICDM. In particular, influence function coefficients for the specific geometryof a core hole are calculated using finite element techniques.

Keywords: in-situ stresses, concrete, non-destructive evaluation, core-drilling method,influence function

1 INTRODUCTION

Reliable information about the in-situ state of stress

in the concrete of an existing structure can be critical

to an evaluation of that structure. This evaluation

may be performed as part of a load rating determin-

ation, or it may be performed to determine whether

repair or replacement of the structure is necessary.

Often, as in the case of a structural member subjected

to bending or eccentric prestressing, the stresses in

the member vary through the thickness of the mem-

ber, i.e. they vary as a function of through-thickness

position. Currently available methods of investigat-

ing stresses in concrete structures, including the core-

drillingmethod and the direct and indirect core-drilling

methods, are limited to stresses that do not vary with

depth [1–3].

The influence function (IF) method has been de-

veloped over the past two decades as a method of

relating non-uniform residual stresses in steel struc-

tures to strains acquired at the surface [4–7]. In this

method, a matrix of IF coefficients is developed to

allow the change in strain as the hole is drilled in

successively deeper increments to be correlated to the

stress in the steel as a function of depth. The strains

are traditionally acquired from an ASTM-standard

rosette of radially oriented strain gauges [10]. To

determine residual stresses that vary as a function of

depth, the hole must be drilled in successive incre-

ments. The strains acquired at each depth increment

are then used to construct the stress distribution

through the thickness of the member.

The present paper seeks to combine elements of the

currently available method of concrete stress inves-

tigation known as the core-drilling method with the

IF method to create a general, non-destructive tech-

nique of investigating stresses in concrete structures.

To accomplish this goal, analytical formulations of the

IF method as adapted to the geometry and measure-

ment configuration used in the core-drilling method

are presented. Finite element models used to calibrate

these IFs are described. Modelling inputs, including

material, geometrical, and load properties are de-

scribed in detail. The solution of the IF matrices is

*Corresponding author: Engineering Mechanics and Infrastruc-

ture, Simpson, Gumpertz, and Heger, Waltham, MA 02453, USA.

email: catrautner@sgh.com

647

JSA600 J. Strain Analysis Vol. 45

then described. Sources of error in both themodelling

procedure and solution technique are described and

quantified.

2 ANALYTICAL FORMULATION OF THE IFMETHOD FOR THE INCREMENTAL CORE-DRILLING METHOD

The incremental core-drilling method (ICDM) is a

linear-elastic application of the IF method. The IF

method has been applied to a number of different situ-

ations that use the ASTM hole-drilling method [4–7].

Although the current application of the method in-

volves concrete (rather than steel) and a core hole

rather than a standard hole, it is subject to several of

the same assumptions. Specifically, it is assumed that

the following conditions are satisfied for the member

under investigation [9].

1. The material under investigation can be idealized

as a linear-elastic, homogeneous continuum.

2. Plane stress conditions exist.

3. The stresses under investigation vary only with re-

spect to the through-thickness position z (i.e. they

do not vary within the plane of the object).

In contrast to the strains used in the ASTM hole-

drilling method [10], the basis for the development

of the ICDM is displacement measurement. These

displacements are acquired along a ‘measurement

circle’ – an imaginary circle offset from the outer edge

of the core hole at the surface along which displace-

ments aremeasured. Themeasurement circle is shown

in Fig. 1.

A Cartesian reference frame is defined at the centre

of the core hole of outside radius a, with the x-axis

typically aligned with the longitudinal axis of the

member under investigation. Displacements at any

point along the measurement circle are measured in

terms of their radial (u) and tangential (v) compo-

nents. The measurement circle is defined by radius

rm, which is typically on the order of 25mm larger

than the outside core hole radius a [1, 2]. For the

current work, awas taken as 75mm and the core hole

width tb was taken as 5mm. These dimensions reflect

the radius and blade thickness of commonly avail-

able coring bits for concrete. The through-thickness

position z is measured positive into the member, with

zmax denoting the coordinate of the deepest core

depth considered. To make the IFs for the ICDM as

general as possible and to reduce the number of units

in calculations, it is convenient to measure through-

thickness position in terms of normalized depth H,

where H5 z/rm.

Any two-dimensional (2-D) stress distribution can

be decomposed into a mean stress component P, a

deviatoric stress component Q, and a shear stress

component t. In the ICDM, each of these compo-

nents is considered separately, and each of these

components is assumed to be a function of through-

thickness depth. The following notation is used

P Hð Þ~ sxx Hð Þzsyy Hð Þ2

ð1Þ

Fig. 1 Core hole configuration

648 C Trautner, M McGinnis, and S Pessiki

J. Strain Analysis Vol. 45 JSA600

Q Hð Þ~ sxx Hð Þ{syy Hð Þ2

ð2Þ

tXY Hð Þ~tXY Hð Þ ð3Þ

Once these functions are known, any general stress

distribution can be represented by superimposing

these components.

The IFs provide the solution for these stress func-

tions by correlating displacements found during the

coring procedure to in-situ stresses. Within an arbi-

trarily small increment at H, the IF GA provides the

relieved radial displacement that would occur under

a unit equibiaxial stress field if the core depth is h

duR h,Hð Þ~ 1

EGA h,Hð ÞP0 dh ð4Þ

The total relieved displacement at any given core

hole depth is due not only to the relieved displace-

ment in the last increment, but also in previous

increments. Therefore, the total displacement at a

particular hole depth is the summation or integral

over all the increments up to that point. Displace-

ment and stress are then related by the integral of

the IF. This can be written as

uR h,Hð Þ~ 1

E

ðh0

GA h,Hð ÞP Hð ÞdH ð5Þ

IFs for the deviatoric stress Q and shear stress txy can

be developed in a similar manner. However, because

deviatoric stress and shear stress in a core hole are

not axisymmetric, the displacement patterns they pro-

duce are not constant around themeasurement circle.

Application ofQ(H)51 produces sinusoidal radial and

tangential displacements around themeasurement cir-

cle. Therefore, the relieved displacements due to a unit

deviatoric stress can be written in terms of radial and

tangential components

uR h,H , hð Þ~ 1

E

ðh0

GB h,Hð ÞQ Hð Þ cos 2hð ÞdH ð6Þ

vR h,H , hð Þ~ 1

E

ðh0

GC h,Hð ÞQ Hð Þ sin 2hð ÞdH ð7Þ

where h is the angular position of the point being

measured. The IFs GB and GC relate a unit deviatoric

stress to relieved radial and tangential displacements,

respectively. The application of a pure shear stress

will also produce radial and tangential displacements

that vary as a function of angular position on themeas-

urement circle. The same IFs developed for deviatoric

stress can be used to relate these displacements

uR h,H , hð Þ~ 1

E

ðh0

GB h,Hð Þt Hð Þ sin 2hð ÞdH ð8Þ

vR h,H , hð Þ~ 1

E

ðh0

{GC h,Hð Þt Hð Þ cos 2hð ÞdH ð9Þ

Superposition can then be used to represent the

measurements that would be taken in the drilling

procedure

uR h,H , hð Þ~ 1

E

ðh0

GA h,Hð ÞP Hð ÞdH

z1

E

ðh0

GB h,Hð ÞQ Hð Þ cos 2hð ÞdH

z1

E

ðh0

GB h,Hð Þt Hð Þ sin 2hð ÞdH ð10Þ

vR h,H , hð Þ~ 1

E

ðh0

GC h,Hð ÞQ Hð Þ sin 2hð ÞdH

z1

E

ðh0

{GC h,Hð Þt Hð Þ cos 2hð ÞdH

ð11Þ

Since the stress functions and IFs are unknown insidethe integral, equations (10) and (11) are not solvable inclosed form. Instead, these equations are solved byevaluating the IFs numerically through a least-squarestechnique.

For the numerical solution of the IFs, the three IFs

can be represented by a double-power expansion

GA~Xnk~1

Xml~1

aklhi{1Hj{1 ð12Þ

GB~Xnk~1

Xml~1

bklhi{1Hj{1 ð13Þ

Analytical and numerical development of the incremental core-drilling method 649

JSA600 J. Strain Analysis Vol. 45

GC~Xnk~1

Xml~1

cklhi{1Hj{1 ð14Þ

where akl, bkl, and ckl are the individual IF coeffi-

cients. The double-summations of equations (12),

(13), and (14), once evaluated, reduce to a single

value for each case. This means that for a given hole

depth and loading depth, the IF G takes on a single

value. The IF matrices a, b, and c are dependent on

the core hole geometric property a and the specimen

material property nu.

3 DETERMINATION OF IFS BY FINITE ELEMENTANALYSIS

3.1 Mean stress influence coefficient matrix a

The determination of the IF coefficients is accom-

plished by finite element analyses in which the core-

drilling process is simulated by removing subsequent

layers of elements in a simulated structure. The core

hole is then loaded from the inside, resulting in dis-

placement at the measurement circle. The displace-

ments vary as the core hole depth and loading depth

change, and the influence function coefficients are

calibrated from these displacements. The influence

function coefficients calculated in this manner can

then be used to correlate relieved displacements to

in-situ stresses.

As stated in the previous section, the influence

coefficient akl correlates relieved radial displacement

to mean in-situ stress P. The portion of radial dis-

placement caused by mean in-situ stress is given by

the first portion of equation (10). Integrating this

expression from z5 0 to z5h (over the entire core

hole depth) produces

uRp h,Hð Þ~ 1

E

ðh0

Xnk~1

Xml~1

aklHl{1hk{1P Hð ÞdH ð15Þ

To calculate the IF coefficients, the stress variation

through depth is discretized into layers. The stress

within each of these layers is constant, so the inte-

gration in equation (15) reduces to

uRp h,Hð Þ~ 1

E

Xnk~1

Xml~1

aklk

hl{1HkP ð16Þ

Equation (16) gives the relieved displacement on the

measurement circle for a hole of depth h loaded

from zero to H. To determine the IF coefficients, the

stress within each layer is set to unity. Equation (16)

can then be rewritten to give the relieved displace-

ment at the measurement circle due to unit loading

uRpij~1

E

Xnk~1

Xml~1

aklk

hl{1i hk

j ð17Þ

where hi represents the non-dimensional core hole

depth and hj the non-dimensional loading depth.

This equation is solved for the influence coefficient

matrix, a, based on a matrix of relieved displace-

ments, uR. Equation (17) represents an overdeter-

mined system of simultaneous equations, and is

solved using a least-squares technique. Each entry in

the uR matrix is indexed by the core hole depth (row

i) and the loading depth (column j). Figure 2 shows

the core hole depth and loading depth for two

example relieved-displacement entries in the uR

matrix. Because the loading depth can never exceed

the core hole depth (that is, hj is always smaller than

hi), the matrix uR is lower triangular.

To produce the matrix of relieved radial displace-

ments uR, an axisymmetric finite element model

was employed. Twenty-five partial core depths were

simulated, with a maximum core depth of 150mm.

The ABAQUS 6.8.1 finite element code was used,

running on a 64-bit Intel Xeon-based computer.

Linear elastic behaviour of concrete with a com-

pressive strength of approximately 45 was modelled

using an elastic modulus of 32 000MPa and a

Poisson ratio of 0.20. As shown by the presence of

the elastic modulus E in equation (17), the IF

matrices developed using these material properties

can be used for concrete of any elastic modulus by

simple scaling. Poisson ratio exerts a subtle influ-

ence on the accuracy of the IF coefficients which

cannot be isolated in the solution procedure [7, 11].

This effect is outside the scope of the current paper,

and n5 0.20 was chosen as typical for common

normal weight concrete mixes [8].

Based on an extensive refinement study [11], the

area around the core was meshed with approxi-

mately 2000 eight-node biquadratic axisymmetric

elements (ABAQUS CAX8R), a relatively coarse mesh

of four-node bilinear axisymmetric elements (ABA-

QUS CAX4R) was used in the remaining area. The

geometric incompatibility between the two types of

elements was enforced with a kinematic tie con-

straint, which was determined to have an insignif-

icant impact on the accuracy of the relieved dis-

placements [11]. Element aspect ratios of up to 8:1

were permitted outside the immediate vicinity of the

core and were found to have a negligible effect on

650 C Trautner, M McGinnis, and S Pessiki

J. Strain Analysis Vol. 45 JSA600

the results [11]. A schematic diagram of the mesh in

the core region is shown in Fig. 3. Displacement in

the vertical (z) direction was constrained at the

bottom of the centre of the core. No other boundary

conditions were used.

The matrix of relieved displacements produced

using this model is too large to be presented numeri-

cally in this paper. However, a graphical presenta-

tion of this matrix is shown in Fig. 4. The magnitude

of the displacements initially increases with increas-

ing core depth, then remains relatively constant, then

starts to decrease for non-dimensional core depths

larger than unity (for this case, approximately 100mm).

This suggests that, depending on the stress distribution

under investigation, there may be a point in the drilling

procedure where displacements measured between

successive increments will be too small to be useful.

Using the matrix of relieved displacements, equa-

tion (17) can be solved for the individual IF coef-

ficients akl. The size of the matrix a is an important

parameter, as a smaller matrix gives more numerical

stability while a larger matrix gives a higher potential

accuracy. Appropriate sizes for this matrix have been

reported as large as 10610 and as small as 666 [7, 9].

Based on an extensive accuracy study, the size for

the current work was chosen as 969 [11]. The accu-

racy of the proposed solution can be expressed in

terms of the relative residual e, which is a measure of

how accurately the solution reproduces the calibra-

tion displacements. It is defined as

eij~uRpij{ 1=Eð ÞPn

k~1

Pml~1 akl=kð Þhl{1

i hkj

uRijð18Þ

Fig. 3 Axisymmetric finite element model

Fig. 2 Relieved displacement matrix entries

Analytical and numerical development of the incremental core-drilling method 651

JSA600 J. Strain Analysis Vol. 45

The matrix e therefore gives an error measurement

in the solution for every non-zero entry in the uR

matrix. The error matrix for the mean stress IF

matrix a is shown in Fig. 5. The relative residual

matrix shows that error in the solution is concen-

trated at small core hole depths, but the highest error

is still less than 0.5 per cent. The agreement between

the calibration displacements and the calculated

displacements verifies the solution procedure. The

matrix a is given in Appendix 2.

3.2 Deviatoric/shear stress IF coefficient matricesb and c

The IF matrices b and c relate the deviatoric and

shear stresses to radial and tangential displacement,

respectively. Equations (10) and (11) indicate that, at

a minimum, two IFs are necessary to solve for an un-

known stress distribution if only radial or only tan-

gential components of displacement are used. This

is similar to the ASTM hole-drilling technique [10],

where two IFs are used to solve for a stress distri-

bution based on radial strains. However, the use of a

third IF (i.e. the deviatoric-stress tangential-displac-

ement IF matrix c) allows the use of the entire dis-

placement field, which can decrease the sensitivity

of the technique to noise in the displacement field.

This can enhance the overall reliability and accuracy

of the technique.

The production of the calibration displacements

for the computation of the matrices b and c requires

the application of a unit deviatoric stress to the

inside of the core hole. The equations relating unit

deviatoric stress to radial and tangential displace-

ment are developed similarly to equations (15) and

(17). Setting the deviatoric stress within each incre-

ment to unity, the following equations can be solved

for the IF matrices b and c, respectively

Fig. 5 Plot of relative residual e

Fig. 4 Relieved mean-stress radial-displacement matrix uRp

652 C Trautner, M McGinnis, and S Pessiki

J. Strain Analysis Vol. 45 JSA600

uRqij~1

E

Xnk~1

Xml~1

bklk

hl{1i hk

j cos 2hð Þ ð19Þ

vRqij~1

E

Xnk~1

Xml~1

cklk

hl{1i hk

j sin 2hð Þ ð20Þ

Since deviatoric stress distorts the area around the

core hole in a manner which is not constant around

the circumference, displacements are necessarily a

function of the tangential position h. To solve these

equations, it is convenient to take radial and tan-

gential displacements for the solution of b and c

from the finite element simulations at h5 0u and

h5 45u, respectively, so that the trigonometric argu-

ment is equal to unity.

Production of the finite element displacements due

to deviatoric stress can be accomplished either by the

use of axisymmetric-harmonic elements [7] or by the

use of three-dimensional (3D) continuum elements. In

the current work, continuum elements were used so

that the calibration model could be reused to model

experimental specimens created during a later portion

of the work. The use of 3D elements readily allows

modelling of in-situ stresses in a Cartesian coordinate

system representative of beam bending or prestress-

ing. Such stresses were applied to the model and used

to check the solution accuracy as described later. A

quarter-symmetric model was employed, using a rela-

tively finemesh of ABAQUSC3D20R 20-node reduced-

integration bricks around the vicinity of the core.

The dimensions of themodel were 75067506150mm

(dimensions were determined by sensitivity analysis).

A kinematic tie was used to connect this mesh to a

relatively coarse mesh of C3D8R eight-node reduced-

integration bricks used to model infinite boundary

conditions. The model mesh can be seen in Fig. 6.

Appropriate quarter-symmetric boundary condi-

tions were imposed at the free faces observable in

Fig. 6, and displacement in the z-direction was con-

strained at a point at the bottom of the centre of the

core hole. Material properties were identical to those

used in the axisymmetric model. To load each core

hole increment with a unit deviatoric stress, a com-

bination of normal (pressure) loading and shear-

traction-type loading was used. Plots of the matrices

of relieved displacements are shown in Fig. 7 and

Fig. 8.

As with the displacements calculated for the mean

stress influence coefficient matrix a, the increase in

displacement between each successive increment

becomes smaller and smaller, especially after the

non-dimensional hole depth is greater than about

one. This further strengthens the notion that there is

a practical limit to how deep a core can be made be-

fore the displacements measured between successive

increments are too small to be useful. The matrices b

and c were calculated from these displacements and

are shown in Appendix 2.

4 ACCURACY OF THE PROPOSED IF SOLUTION

The relative residual measurement e provides a check

of the solution procedure used in the IF problem, but

provides no insight to the accuracy of the calculated

Fig. 6 A 3D finite element model

Analytical and numerical development of the incremental core-drilling method 653

JSA600 J. Strain Analysis Vol. 45

IF matrices as applied to a practical problem. A direct

comparison between a known stress distribution and

the stress distribution calculated using measured

displacements and the IFs is necessary. Experimental

verification is outside the scope of the current paper,

but the experimental technique can be simulated

using a 3D finite element model similar to that used

to calibrate the IF matrices b and c. To simulate the

procedure, a linearly varying stress sxx was applied at

one end of the model, as shown in Fig. 9.

Other potentially measurable stresses (syy and txy)

were set to zero. For this case, the outermesh of C3D8R

elements was changed to C3D20R elements. Since

the C3D20R element can represent a state of linear

strain, the linear gradient in the x-direction propa-

gates through the model producing both constant-

curvature bending and axial shortening in the plate.

At minimum, a set of three displacements must be

acquired at each coring depth, so that the three un-

known stress components (P,Q, and txy) can be com-

puted. For this example, a set of three radial dis-

placements at h5 0u, 45u, and 90uwere acquired along

a measurement circle of rm5 100mm at non-dimen-

sional core depths of 0.75 and 1.5. The acquired dis-

Fig. 8 Relieved tangential displacements due to deviatoric stress

Fig. 9 Linear stress gradient application

Fig. 7 Relieved radial displacements due to deviatoric stress

654 C Trautner, M McGinnis, and S Pessiki

J. Strain Analysis Vol. 45 JSA600

placements (normalized to the measurement circle

radius of 100mm) are

D1~{0:000 383 273

{0:045 890 900

" #

D2~{0:000 0845 788

{0:000 0736 378

" #

D3~0:000 076 8292

0:000 144 0682

" #

D1 corresponds to h5 0u, D2 to h5 45u, etc. Note that

the magnitude of the acquired displacements is small

(on the order of a few tens of micrometres), and is

typical for practical applications of the ICDM. Using

the IF matrices calculated in the previous sections, a

linear stress distribution was fit to the displacement

data. A plot of sxx is shown in Fig. 10. The relative

error cannot be plotted in the figure because the ap-

plied stress is zero at the bottom of the core. However,

the two curves are essentially indistinguishable and

the absolute difference between the solutions (plotted

as the raw error in the figure) is less than 0.1MPa, in-

dicating a highly accurate prediction of in-situ stresses.

Because the IF method is formulated in the space

of mean, deviatoric, and shear stress, the IF matrices

developed can be used to investigate any 2D stress

distribution. To illustrate the versatility and accuracy

of the proposed solution, the model described in the

previous example was subjected to a biaxial stress dis-

tribution as shown in Fig. 11. Stress in the x-direction

was varied linearly through the depth from 15MPa

at the top of the plate to zero at the bottom (at

H5 150mm). Stress in the y-direction was constant

versus depth at 7MPa. Displacements weremeasured

at core depths of H5 0.75 and H5 1.5 using the

measurement configuration described in the previous

example. The normalized displacement vectors were

D1~{0:000 337 011

{0:000 321 695

" #

D2~{0:000 237 669

{0:000 230 968

" #

D3~{0:000 138 778

{0:000 140 73

" #

It is interesting to note that the magnitude of D1 and

D2 actually decrease with depth, indicating there is no

clear pattern that should be expected in the displace-

ment data for a general loading case. The in-situ

stresses were again calculated as linear distributions,

and the results were plotted against the applied stress,

as shown in Fig. 12. As with the uniaxial example,

there is close agreement between the calculated and

applied stresses. Themaximumerror in each direction

is less than 3 per cent when normalized to the average

applied stress in that direction.

Fig. 10 Calculated and applied stresses for uniaxial loading

Analytical and numerical development of the incremental core-drilling method 655

JSA600 J. Strain Analysis Vol. 45

5 SUMMARY AND CONCLUSIONS

The analytical formulation for the IFmethod, as applied

to the ICDM, was presented and discussed. IF function

matrices for the practical implementation of the tech-

nique were calculated based on displacements from

axisymmetric and 3D finite element simulations. A

review of the calibration displacements indicated that

there is a limit to howdeep a coremay be drilled before

the difference in displacementmeasured between suc-

cessive increments becomes too small to be useful.

The solution procedure for the IFmatrices was verified

by the accurate reproduction of the calibration dis-

placements. The accuracy of the technique was veri-

fied outside the solution procedure by the accurate

calculation of in-situ stresses in a finite elementmodel.

The IF matrices calculated are given in Appendix 2.

F Authors 2010

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5 Schajer, G. S. Measurement of non-uniform resi-dual stresses using the hole-drilling method, partII-Practical Application of the Integral Method.Trans. ASME, 1988, 110(4), 344–349.

6 Beghini, M. and Bertini, L. Recent advances in thehole-drillingmethod for residual stressmeasurement.J. Mater. Engng Performance, 1998, 7(2), 163–172.

7 Beghini, M. Analytical expression of the influencefunctions for accuracy and versatility improvementin the hole-drilling method. J. Strain Analysis, 2000,35(2), 125–135.

8 Kosmatka, S., Kerkhoff, B., and Panarese, W. Designand control of concrete mixtures, 14th edition, 2003(Portland Cement Association, Skokie, Illinois).

9 Turker, H. Theoretical development of the core-drilling method for nondestructive evaluation ofstresses in concrete structures. PhD Thesis, LehighUniversity, 2003.

10 ASTM E837-08e1:2008 Standard test method fordetermining residual stresses by the hole-drillingstrain-gage method.

11 Trautner, C. Development of the incremental coredrilling method for non-destructive investigationof stresses in concrete structures. Master’s Thesis,Lehigh University, 2008.

Fig. 12 Computed and applied stresses (biaxial case)

Fig. 11 Plan view of biaxial stress distribution

656 C Trautner, M McGinnis, and S Pessiki

J. Strain Analysis Vol. 45 JSA600

APPENDIX 1

Notation

a outside radius of core hole

ckl coefficient for the dimensionless

tangential shear IF

duR infinitesimal radial displacement due

to unit equibiaxial stress P0

D1, D2, D3 relaxed displacement for a given

measurement configuration

e relative residual matrix or other error

measurement

Ec Young’s modulus

f9c compressive strength of concrete

G IF (mean radial, deviatoric/shear

radial, or deviatoric/shear tangential)

h dimensionless hole depth5 z/rmhi finite element simulated

dimensionless hole depth

hj finite element simulated dimension-

less extreme depth for interior loading

H dimensionless through-thickness

position5 z/rmi index for the finite element simulated

hole depth5 1, …, N

j index for the extreme depth position

of the finite element simulated

pressure loading5 1, …, i

k row index for the IF coefficient

matrix5 1, …, n

l column index for the IF coefficient

matrix5 1, …, m

m number of columns in the IF

coefficient matrix

n number of rows in the IF coefficient

matrix

N total number of finite element

simulated hole depths

P residual stress function of

through-thickness position

P0 unit equibiaxial traction (stress)

Q residual stress functions of

through-thickness position

R total number of increments hj used

in core-drilling procedure

rm measurement circle radius

tb width of core hole (i.e. core drill

blade thickness)

u measured radial displacement

uRp normalized finite element radial

displacement matrix due to unit

mean stress

uRq normalized finite element radial

displacement matrix due to unit

deviatoric stress

URij finite element radial displacement

for a hole of depth hi with loading

from the surface to depth

v measured tangential displacement

vRq normalized finite element tangential

displacement matrix due to unit

deviatoric stress

z through-thickness position

zmax maximum hole depth

akl coefficient for the dimensionless

radial equibiaxial IF

bkl coefficient for the dimensionless

radial deviatoric/shear IF

ckl coefficient for the dimensionless

tangential deviatoric/shear IF

h angle measured counterclockwise

from x-axis to point of interest

n Poisson ratio

sapp general term for the calculated

applied stress applied to a finite

element model or specimen

scalc stresses calculated from numerical

procedure

APPENDIX 2

Calculated IF matrices

Table 1 The IF matrix a

1.51061021 1.335610 23.087610 3.331610 29.172 21.595610 1.806610 27.568 1.19521.014610 1.0286102 26.1446102 1.8316103 23.0436103 2.9426103 21.6336103 4.7886102 25.66861025.844610 2.0766102 22.959 22.1586103 5.9046103 27.1956103 4.4866103 21.3586103 1.5106102

2.432610 2.7646102 3.010610 21.5456103 2.6486103 22.0076103 1.2216103 27.4046102 2.1036102

4.65761022 21.7316103 6.3316103 29.6086103 6.5796103 22.2426103 1.2666102 5.2346102 22.3756102

2.4826102 29.0726102 28.0186102 3.7886103 28.6426102 21.5106103 24.245610 4.7326102 25.4776103.06361021 2.6036103 24.5786103 3.4836102 1.5726103 25.3596102 1.1176103 29.8376102 2.0586102

25.6786102 24.821610 1.3126103 7.2416102 21.5836103 23.2746102 7.8616102 21.8856102 2.54125.545610 6.7016102 21.0586103 21.3396102 7.8296102 2.0506102 27.4946102 3.8086102 26.303610

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JSA600 J. Strain Analysis Vol. 45

Table 2 The IF matrix b

2.472 1.972 3.041 24.155610 1.1606102 21.5386102 1.0906102 23.995610 5.98321.4846102 7.9856102 21.9776103 3.5506103 25.1166103 5.2006103 23.2756103 1.1306103 21.6336102

1.9346103 29.6566103 1.7306104 21.5866104 1.0626104 28.1916103 5.7246103 22.2936103 3.7206102

21.0596104 5.3886104 29.4976104 7.6056104 22.6816104 2.1786103 4.5196102 1.4216102 25.6356102.2686104 21.1646105 1.8516105 21.0766105 23.660 2.0176104 24.0896103 26.6776102 1.2836102

21.7956104 1.0666105 21.4606105 2.9006104 5.5536104 22.0666104 21.0516104 5.5566103 24.1616102

24.7116103 21.5406104 1.1606104 3.4746104 21.8316104 22.6666104 1.8486104 5.9466102 21.6086103

1.0556104 22.0316104 2.3886104 21.1336104 21.9316104 1.7116104 1.0636104 21.4166104 3.6366103

22.1486103 2.0726103 3.1856103 21.0876104 1.2316104 28.329 21.0596104 7.4796103 21.5956103

Table 3 The IF matrix c

23.124 4.163 28.943 2.487610 24.592610 4.679610 22.627610 7.692 29.20961021

1.9216102 25.6186102 7.7176102 21.0346103 1.5256103 21.5766103 9.4916102 23.0426102 4.05561023.1056103 8.6816103 28.5516103 2.6756103 1.1376103 21.2476103 4.8286102 28.874610 1.4672.1006104 26.0226104 6.8226104 24.1496104 1.7206104 25.1826103 3.5726102 2.6506102 22.009610

26.6206104 1.7406105 21.6576105 6.4966104 23.6466103 27.6166103 4.1366103 21.6686102 23.5886102

1.0876105 22.4816105 1.6816105 23.7746102 24.0646104 1.6106104 22.2726102 23.9696103 1.5756103

29.0136104 1.5696105 22.7976104 26.1146104 22.3836102 3.2656104 21.4936104 5.0226103 21.4956103

3.1666104 22.1066104 25.1556104 2.2636104 4.9036104 22.2606104 21.8326104 1.2906104 21.9026103

23.1936102 21.8066104 3.1516104 3.5516103 22.7516104 7.053 1.9946104 21.1146104 1.8306103

658 C Trautner, M McGinnis, and S Pessiki

J. Strain Analysis Vol. 45 JSA600