Stresses at and in the neighborhood of a near-edge hole in a plate subjected to an offset load from measured temperatures

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Stresses at and in the neighborhood of a near-edge hole in a plate subjected

to an offset load from measured temperatures

S.J. Lin

a

, D.R. Matthys

b

, S. Quinn

c

, J.P. Davidson

d

, B.R. Boyce

e

, A.A. Khaja

d

, R.E. Rowlands

d,* a_{National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan}

b_{Marquette University, Milwaukee, WI, USA} c_{University of Southampton, Southampton, UK}

d_{Mechanical Engineering, University of Wisconsin, 1415 Engineering Drive, Madison, WI 53706, USA} e_{Stress Photonics Inc., Madison, WI, USA}

a r t i c l e i n f o

Article history:

Received 23 December 2011 Accepted 3 September 2012 Available online 3 October 2012 Keywords: Thermoelasticity Stress functions Holes Stress analysis

a b s t r a c t

An Airy stress function is used to process the associated temperature data and thereby determine the individual stresses in a plate containing a near-edge circular hole and which is subjected to a concen-trated edge load away from the hole. Formulating the stress function so its origin is at the center of the hole enables the traction-free conditions to be imposed analytically on the edge of the hole. This significantly reduces the number of coefficients that must be retained in the stress function. Results agree with those from measured strains and thefinite element method. The capability developed is applicable beyond the present situation, including with other measured quantities such as strains or displacements. Ó 2012 Elsevier Masson SAS. All rights reserved.

1. Introduction

This paper combines measured temperature information and an applicable series representation of the Airy stress function to determine the individual stresses in a plate which contains a near-edge circular hole and is subjected to a concentrated load away from the hole,Fig. 1. The general approach is called ther-moelastic stress analysis (TSA, or therther-moelasticity). The plate is supported along the bottom edge, C’DC, and subjected to an offset concentrated edge load, P*. Interest in the problem is motivated by applications to mechanical components and buried structures. The semi-infinite plate (half-plane) is approximated by afinite plate. Classical thermoelasticity only gives informa-tion related to the sum of the principal stresses. However, the coefficients of the stress function, and hence the individual stress components, are obtainable from thermoelastically-measured isopachic data and local boundary conditions. Situa-tions involving fatigue, for example, require knowledge of the

individual components of stress. Imposing the traction-free conditions analytically on the edge of the hole reduces the number of coefﬁcients in the stress function series and enables them all to be determined from the measured temperatures. This circumvents having to supplement the recorded temperature information with some other measured data. While theoretical stress analyses have employed stress functions extensively for many years, their combination with measured information is much more recent.

Thermoelasticity provides stress information of an actual structure in its operating environment with a resolution compa-rable to that of foil strain gauges (Dulieu-Barton and Stanley, 1998; Greene et al., 2008; Patterson and Rowlands, 2008). In a cyclically loaded member experiencing adiabatic and reversible conditions, the change in the local temperature is proportional to the change in stress. The temperature ﬂuctuations, which are related to the associated varying stresses by thermodynamic principles, are recorded by a sensitive infrared detector. Features of the method include the ability to determine the individual stresses at and near a hole in a member without knowing the loading, material constitutive properties or the boundary condi-tions beyond those at a hole. The fact that the technique does not involve differentiating the measured temperature data is advantageous. As knowledge of the distant geometry or external boundary conditions, including the concentrated load, P (per unit

* Corresponding author. Tel.: þ1 608 262 3205; fax: þ1 608 265 8213. E-mail addresses: kathysjlin@cc.kuas.edu.tw (S.J. Lin), don.matthys@ marquette.edu (D.R. Matthys), s.quinn@soton.ac.uk (S. Quinn), jpdavidson@ wisc.edu (J.P. Davidson), bboyce@stressphotonics.com (B.R. Boyce), khaja@ wisc.edu(A.A. Khaja),rowlands@engr.wisc.edu(R.E. Rowlands).

Contents lists available atSciVerse ScienceDirect

European Journal of Mechanics A/Solids

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m/ l o ca t e / e j m s o l

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thickness), is not required, the method is capable of solving inverse problems.

The present region of interest is at and in the vicinity of the hole,

Fig. 1. The general approach is applicable irrespective of the form or magnitude of the loading and stresses are provided without employing measured data on, or immediately near, the boundary of the hole. This is important since it is often difﬁcult to obtain accurate measured information at these locations. The TSA results agree with those from strain gauges and FEM. This is the ﬁrst known case where recorded stress, strain or displacement data have been processed by a stress function in the absence of at least one axis of symmetry. While used here to solve a particular engi-neering problem, the currently developed concepts are applicable to more complicated situations and other techniques of experi-mental mechanics.

Wang et al. (2005) analyzed a somewhat similar problem photoelastically. However the fact that their load was immediately above the hole significantly simplified the analysis. Also, an important feature of utilizing TSA-measured isopachics over photoelastic isochromatics is that TSA necessitates only linear least squares; the isochromatic method requires non-linear least squares. Unlike the method of Wang et al., 2005, the present technique is applicable tofinite plates having an arbitrarily shaped hole and unknown external geometry.Lin et al., 2009,2011are the only previously published hybridization of measured temperatures and a stress function in real variables whereby the traction-free conditions at a cutout are imposed analytically. Unlike the present approach, that ofLin et al., 2009is restricted mathemat-ically to a central circular hole in a uniaxially-loaded rectangular plate and that analysis also benefitted from a previously published stress function. The analysis of Lin et al., 2011 enjoyed the advantage of symmetry about the vertical axis. The authors are unaware of any published TSA study in which the three inde-pendent stresses are evaluated using real variables without having symmetry about either the x- or y-axis. In addition to the present lack of such symmetry necessitating many more Airy coefficients than the case ofLin et al., 2011, the offset load ofFig. 1required the formulation of a more complicated stress function in terms of polar coordinates and whose origin is at the center of the hole (Lin, 2007). Having a stress function in terms of polar coordinates centered at the middle of the hole enables imposing the traction-free conditions analytically at the hole and reduces the number of Airy coefficients required. This is in contrast to the approach described inRyall et al. (1992). The latter formulated a TSA concept

for stress analyzing perforated plates in which the origin of the stress function was away from the hole. Such an arrangement greatly hampers the prospects of satisfying boundary conditions at the hole. Extremely few previously published analyses have correlated TSA results with other independently measured data such as recorded strains.

The present analysis of the plate ofFig. 1wherein the hole and load are offset both horizontally and vertically from each other signiﬁcantly extends thermoelastic applicability beyond prior capabilities. The current study is the most complicated and comprehensive, and yet most general, stress analysis formulation to date where temperature measurements are synergized with an Airy stress function.

s

qq (3)

in which S* is the thermoelastically-detected signal, K is the thermo-mechanical coefﬁcient and

D

S represents the associated

Fig. 1. Schematic geometry of a plate containing a near-edge hole subjected to an offset concentrated load.

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s

rr ¼

s

rq ¼ 0 are imposed analytically (i.e., for

all values of

q

) along a radius r¼ R (thereby creating a traction-free hole of radius R located a distance D below the top of the plate and a horizontal distance E from the load,Fig. 1), several of the original Airy coefﬁcients become functions of other Airy coefﬁcients. The expressions for the stresses then become (Lin, 2007)

s

rr ¼ 2$R2 r2 þ 2 $c0þ 2$R4 r3 þ 2$r $sin

q$d

1þ 2$R4 r3 þ 2$r $cos

q$d

0 1 þh6$R4_6$r4_$c0 2þ 4$R2_4$r2_$d0 2 i $cosð2$qÞ þ F6$ 4$R3_$sinð6$qÞ ½F4$sinð3$qÞ þ F5$cosð3$qÞþ F₁$6$r 4$r3_$cosð3$qÞ $b0 3 þ ( F6$ 12$R5_$sinð6$qÞ ½F4$sinð3$qÞ þ F5$cosð3$qÞþ F₂$6$r 12$r5_$cosð3$qÞ ) $c0 3 þ ( F6$ 8$R3_$sinð6$qÞ ½F4$sinð3$qÞ þ F5$cosð3$qÞþ F3$6$r 10$r3 $cosð3,qÞ ) $d0 3 þ ( F6$ ð1Þ$R2_$cosð3$qÞ ½F4$sinð3$qÞ þ F5$cosð3$qÞþ R2$F4$6$r$cosð3,qÞ 6$R$½F4$sinð3$qÞ þ F5$cosð3$qÞ ) $A0 XN n¼ 4;5;6. nh ð1Þ$n$n2 1$R2n$rðn2Þþ n$ðn þ 1Þ$ðn 2Þ$Rð2nþ2Þ$rn_{þ n$ðn þ 1Þ$r}ðnþ2Þi_$c0 n þhð1Þ$n2_{$ðn 1Þ$R}ð22nÞ_$rðn2Þ_þ_n2₁_{$ðn 2Þ$R}2n_$rn_{þ ðn 1Þ$ðn þ 2Þ$r}ni_$d0 n o $cosðn$qÞ XN n¼ 2;3;4. nh n2 1$rðn2Þ_$ð1Þ$R2_{þ ðn þ 1Þ$ðn 2Þ$r}n_{þ ðn þ 1Þ$r}ðnþ2Þ_$Rð2$nþ2Þi_$b n þhðn 1Þ$rðn2Þ_$ð1Þ$Rð2$nþ2Þ_n2₁_$rðnþ2Þ_$R2_{þ ðn 1Þ$ðn þ 2Þ$r}ni_$d n o $sinðn$qÞ (4)

s

qq ¼ 2$R2 r2 þ 2 $c0þ 2$R4 r3 þ 6$r $sin

q$d

1þ 2$R4 r3 þ 6$r $cos

q$d

0 1 þh 6$R4_{þ 24$R}6_$r2_{þ 6$r}4_$c0 2þ 4$R2_{þ 12$R}4_$r2_$d0 2 i $cosð2$qÞ þ F7$ 4$R3_$sinð6$qÞ ½F4$sinð3$qÞ þ F5$cosð3$qÞþ h ð1Þ$F1$6$r þ 20$r3 i $cosð3$qÞ $b0 3 þ ( F7$ 12$R5_$sinð6$qÞ ½F4$sinð3$qÞ þ F5$cosð3$qÞþ h ð1Þ$F2$6$r þ 12$r5 i $cosð3$qÞ ) $c0 3 þ ( F7$ 8$R3_$sinð6$qÞ ½F4$sinð3$qÞ þ F5$cosð3$qÞþ h ð1Þ$F3$6$r þ 2$r3 i $cosð3$qÞ ) $d0 3 þ ( F7$ ð1Þ$R2_$cosð3$qÞ ½F4$sinð3$qÞ þ F5$cosð3$qÞþ R2$ð1Þ$F4$6$r$cosð3$qÞ 6$R$½F4$sinð3$qÞ þ F5$cosð3$qÞ ) $A0 þ XN n¼ 4;5;6. nh ð1Þ$n$n2₁_$R2n_$rðn2Þ_{þ n$ðn þ 1Þ$ðn þ 2Þ$R}ð2nþ2Þ_$rn_{þ n$ðn þ 1Þ$r}ðnþ2Þi_$c0 n þhð1Þ$n2_{$ðn 1Þ$R}ð22nÞ_$rðn2Þ_þ_n2₁_{$ðn þ 2Þ$R}2n_$rn_{þ ðn 1Þ$ðn 2Þ$r}ni_$d0 n o $cosðn$qÞ þ XN n¼ 2;3;4. nh n2 1$rðn2Þ$ð1Þ$R2_{þ ðn þ 1Þ$ðn þ 2Þ$r}n_{þ ðn þ 1Þ$r}ðnþ2Þ_$Rð2$nþ2Þi_$b n þhðn 1Þ$rðn2Þ_$ð1Þ$Rð2$nþ2Þ_n2₁_$rðnþ2Þ_$R2_{þ ðn 1Þ$ðn 2Þ$r}ni_$d n o $sinðn$qÞ (5)

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Moreover, S of Equation(3)becomes

Coefﬁcients F1through F9of Equations(4)e(7)are functions of

the polar coordinates r and

q

, and the geometric quantities R, D and E ofFig. 1. Details are given in theAppendix A.

With due respect to the expressions for F1through F9, all Airy

coefﬁcients appearing in the individual components of stress of Equations(4)e(6)now also appear in the expression for S in Equation

(7). The individual components of stress can therefore be evaluated from the measured temperatures without involving far-ﬁeld condi-tions or necessitating any other experimental technique or data.

The stresses of Equations (4)e(7)depend only on Airy coef ﬁ-cients c0, d1, b0₃, Ao, cn0 for n 2, d0nfor n 1 and bnand dnfor n 2

(which now depend indirectly on P) but exclude P, an, a0n, b0and cn

of Equation (1). The absence of P in the stress components of Equations (4)e(6) indicates this approach could handle inverse problems. For example, if the present analysis were extended to cover the entire width, w, of the plate ofFig. 1, the magnitude of P could be obtained by integrating the vertical stress,

s

xx, over the

entire horizontal cross-section of the plate at some value of x.

3. Thermoelastic stress analysis

3.1. Experimental details

The aluminum (6061 T6511, ultimate strength of 275e311 MPa (40e45 ksi) and yield point of 241e275 MPa (35e40 ksi)) plate of

Fig. 1was sprayed with Krylonﬂat black paint prior to testing to provide an enhanced and uniform surface emissivity. Fig. 2 is a photograph of the experimental setup. Compatible with the assumptions behind Equation (3), the plate was compressed sinusoidally between 222 N (50 lb) and 1112 N (250 lb) at 20 Hz. The top load was applied through a short piece of drill rod (equivalent to silver steel in the UK).Fig. 2also shows the liquid-nitrogen-cooled Stress Photonics DeltaTherm DT1410 infrared camera (sensor array of 256 256 pixels) used to record the load-induced temperature variations. The thermoelastic image ofFig. 3

qq

¼ 4$c0þ 8$r$sin

q$d

1þ 8$r$cos

q$d

10 þ 24$r2$R6$cosð2$qÞ$c02þ

12$r2_$R4_4$r2_{$cosð2$qÞ$d}0 2 þ F9$ð4Þ$R3$sinð6$qÞ ½F4$sinð3$qÞ þ F5$cosð3$qÞþ 16$r 3_$cosð3$qÞ $b0 3þ F9$12$R5$sinð6$qÞ ½F4$sinð3$qÞ þ F5$cosð3$qÞ$c 0 3 þ ( F₉$8$R3_$sinð6$qÞ ½F4$sinð3$qÞ þ F5$cosð3$qÞ 8$r 3_$cosð3$qÞ ) $d0 3þ ( F₉$ð1Þ$R2_$cosð3$qÞ ½F4$sinð3$qÞ þ F5$cosð3$qÞ ) $A0 þ XN n¼ 4;5;6;. n 4$n$ðn þ 1Þ$rn_$Rð2nþ2Þ_{$cosðn$qÞ$c}0 nþ h 4$n2 1 $rn_$R2n_{4$ðn 1Þ$r}ni_{$cosðn$qÞ$d}0 n o þ XN n¼ 2;3;4;. h 4$ðn þ 1Þ$rn_$b n 4$ðn 1Þ$rn$dn i $sinðn$qÞ ð7Þ

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the entire plate contains approximately 15,000 discrete tempera-ture values.

3.2. Evaluating the Airy coefﬁcients

By using discretely measured TSA stress information [

D

S¼ S*/ K¼

D

(

s

rrþ

s

qq)] in the neighborhood of the hole and analytically

imposing the traction-free boundary conditions,

s

rr¼

s

rq¼ 0, on

the boundary of the hole to give Equations(4)e(7), one obtains the following set of linear isopachic equations containing the unknown Airy coefﬁcients,

where A is the m by k Airy matrix composed of a m set of isopachic expressions given in Equation(7) having k Airy coefficients, c is a vector consisting of the k unknown Airy coefficients, and vector d contains the thermoelastically-measured values of S at m input locations corresponding to the set of equations in isopachic matrix A. Knowing A and d, Equation(8)can be solved to evaluate the Airy coefficients, c, using least-squares. Once these coefficients are ob-tained, the individual stresses of Equations(4)e(6)are available. 3.3. Measured input data and the number of Airy coefficients

Fig. 4shows the source locations of the 849 (of the approxi-mately 15,000 available) input temperature values ofFig. 3used for stress analyzing around the hole of the plate. Since recorded values

of S* within three or four pixels of an edge are usually unreliable, the data of Fig. 4 originate at least 4 pixels (approximately 3 mm¼ 0.1 inches) away from the boundary of the hole and the top edge ofFig. 1. The magnitudes of the Airy coefﬁcients are available from Equation(8)and the 849 measured isopachics,

D

S (¼S*/K), at the locations indicated in Fig. 4. The analysis by the described method, i.e., based on Equations(3)e(8), and evaluating the Airy coefﬁcients from the aforementioned measured thermoelastic input data,

D

S¼ S*/K. is denoted as TSA.

The source locations of the 849 input values indicated inFig. 4

are largely arbitrary, but selected based on experience and the high stresses between the hole and the offset load, P (Greene et al., 2008;Patterson and Rowlands, 2008;Joglekar, 2009). Although the present model DT1410 staring-array camera records a full-ﬁeld image virtually instantaneously, there is aﬁnite pixel resolution. The unreliable temperatures at and near an edge mentioned in the

preceding paragraph occurs because at a boundary the detector “sees” a pixel which is partly on the stressed specimen and partly on the stress-free background. The quality of an edge signal can be further reduced by the cyclic motion of the structure such that the detector receives different data from the different spatial positions. A realistic number of Airy coefficients, k, to use in Equations(4)e (7)must be chosen. Too few coefficients (i.e., too small a value of N) can produce inaccurate results, while too many coefficients can cause the Airy matrix, A, of Equation(8)to become unstable or even singular due to computer round-off errors. The amount of input data needed can depend on k, so more coefficients could necessi-tate more measured values of S*. An appropriate number, k, of Airy coefficients to use was selected based on the condition number, C, of the Airy matrix, the RMS of the discrepancy between the

Amxkckx1 ¼ dmx1 2 6 6 6 6 6 6 6 4 S_r₁_;_q₁c₀; d1; d01; c02; d02; ; b03c30; d03; A0; c04; d04; ...c0N; d0N; b2; d2...bN; dN S_r₂_;_q₂c0; d1; d01; c02; d02; ; b03c30; d03; A0; c04; d04; ...c0N; d0N; b2; d2...bN; dN « « « S_r_m_;_q_mc0; d1; d01; c02; d02; ; b03c30; d03; A0; c04; d04; ...c0N; d0N; b2; d2...bN; dN 3 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 c0 d1 d0₁ « bN dN 3 7 7 7 7 7 7 5 ¼ 2 6 6 6 6 6 6 4 S1 S2 S3 « « Sm 3 7 7 7 7 7 7 5 (8)

Fig. 2. Experimental setup.

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calculated data d0and the input values of d of Equation(8), and contour plots of the reconstructed S*¼ K

D

S (determined using the now known Airy coefficients and Equation (1)). These various assessments indicate k¼ 25 as a realistic number of coefficients to use (Lin, 2007). This corresponds to N¼ 6 in Equations(4)e(7). Having determined the coefficients existing in the stress expres-sions, i.e., the 25 TSA-determined Airy coef_{ficients, the magnitudes} of the individual stresses are available from Equations(4) through (6), including on the edge of the hole, even without employing measured stress information there.

Fig. 5compares the recorded signal, S*, with that reconstructed by the described TSA approach (experimentally evaluated Airy coefﬁcients and Equations(4)e(8)) along horizontal lines ab and a0b0and vertical lines cd and c0d0ofFig. 6. As well as further vali-dating the selection of k ¼ 25, the results shown in Fig. 5

demonstrate the unreliability of measured TSA data within three or four recorded locations of an edge.

4. FEM analysis

In addition to using FEM (ANSYS)-predicted stresses with which to compare the TSA (TSA-determined Airy coefﬁcients) determined stresses (i.e., called TSA), FEM-predicted values of S (simulated TSA isopachics) were employed. Results based on ANSYS-generated isopachic values of

s

rrþ

s

qq to simulate measured TSA inputs are denoted as TSA (ANS) and they serve as a check on numerical robustness of the approach. The ANSYS model of the loaded plate of

Fig. 1was meshed by 3896 isoparametric elements with 13,332 nodes (Lin, 2007). Small elements were utilized in the regions of high stress. Elements between the top edge of the plate and the circular hole are as small as 0.64 mm (0.02500) and the edge of the hole is meshed by 99 elements (element size is approximately 0.61 mmw 3.6_{per element). The lack of symmetry required a full}

FE model ofFig. 1. Unlike the described TSA approach, FEM neces-sitates complete boundary information. Recognizing that the aluminum plate was supported along its bottom edge physically by a stationary steel platen, a roller constraint (no vertical motion) was applied numerically along the lower edge CDC0ofFig. 1.

The TSA(ANS) calculations used FEM (ANSYS)-simulated input values of S at 288 source locations along three concentric circles about the origin having rw 12.45 mm (0.4900_{), 14.22 mm (0.56}00_),

and 16 mm (0.6300), respectively. Using these FE generated isopachic magnitudes of S¼

s

rrþ

s

_qq, it was possible to solve the Airy matrix

equation Ac ¼ d of Equation(8), where A is an m (¼288) by k (¼25) matrix containing a set of 288 linear isopachic equations with 25 independent coefﬁcients at each of the 288 FE source locations and

R

x

y

2R

a

b

a

b

c

d

c

d

′ ′ ′ ′

Fig. 6. Source locations of S* for TSA measured and the described approach TSA predictions ofFig. 5

Fig. 7. Normalized hoop stress around the boundary of the hole.

y

x

r P*

Fig. 4. Source locations of the 849 measured input values.

Fig. 5. S* measured by TSA (TSA measured) and predicted by the described approach, (TSA) with k¼ 25 (A) along lines ab and a0_b0_{and (B) along lines cd and c}0_d0_of_{Fig. 6}

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vector d was composed of the 288 ANSYS-generated isopachic values corresponding to the isopachic equations in matrix A. After determining the vector c containing all of the Airy coefﬁcients, individual components of stress based on TSA(ANS) were available from Equations(4) through (6).

5. Experimental results

Fig. 7shows the normalized hoop stresses along the edge of the hole.Figs. 9e11contain individual stresses located further away from the hole boundary, where rw 1.24R, seeFig. 8. The utilized locations for r/Rs 1 ofFigs. 8e11are not particularly signiﬁcant, but were selected to illustrate the continuing reliability of the TSA upon moving away from the edge of the hole. All actual stresses are normalized here by a uniform stress

s

0¼ 1.05 MPa (152.4 psi), i.e.,

the concentrated edge load, P*¼ 890 N (200 lbs), divided by the gross cross-sectional area of 88.9 mm (3.500) wide 9.53 mm (3/800₎

thick. Based on the measured strains from strain gauges (Vishay Micromeasurements type CEA-13-032UW-120) bonded on the internal curved surface of the hole at

q

¼ 0_{(top of the hole) and}

90(right side of the hole), values of the normalized hoop stress at P*¼ 890 N (200 pounds) at these locations are 1.5 and 11.2, respectively, seeFig. 7. The strain gauge and FEM results of Figs.7

and9e11show the reliability of the TSA technique.

Although the details are omitted here, a subsequent TSA analysis was conducted in which

s

xy¼

s

x¼ 0 were also imposed

at 80 locations along the top edge of the plate (Lin, 2007). These

s

xyand

s

xstresses were transformed at each of the 80 positions

into the corresponding equations in polar coordinates (so now include Airy coefficients) and the additional expressions added to the now modified Equation (8). The resulting stresses at, and in the vicinity of, the hole (based on 2 80 þ 849 ¼ 1009 input/side conditions and k ¼ 25) are insignificantly different from the original TSA results ofFig. 7. This provides additional confidence in the analysis.

6. Summary, discussion and conclusions

A major contribution of this paper is the developed ability to combine measured temperatures and an Airy stress function to determine the stresses in a plate containing an arbitrarily located hole and an arbitrarily located edge load in the absence of symmetry. Formulating an Airy stress function whose origin is at the center of the hole facilitates imposing the traction-free condi-tions analytically at the edge of the hole. In addition to evaluating the Airy coefficients from the temperature data, this reduces the number of independent coef_{ficients in the stress function and} hence in the expressions for the individual stresses. Arranging the origin of the coordinate system to be within the hole overcomes a major shortcoming of the technique described by Ryall et al. (1992). Attention is paid to how many coefficients to retain in the present stress function series. The source locations of the currently employed temperature data were influenced by previous experi-ence, but Joglekar, 2009considered (for different situations) the effects of varying the source locations and amount of measured thermoelastic input data. Although demonstrated here for the specific case of a near-edge hole in a plate which is subjected to an offset edge load, the general technique is applicable well beyond the present situation.

The authors are unaware of any previously published technique which processes measured information with an Airy stress function to evaluate the individual stresses at and in the neighborhood of a hole in the absence of at least some symmetry, let alone for a near-edge hole which is offset from a concentrated load. Some experi-mental stress analyses have employed stress functions in terms of complex variables (Huang and Rowlands, 1991;Lin and Rowlands, 1995). In addition to being mathematically simpler, using a stress function in real variables enables the individual stresses to be determined at, and around, the entire boundary of a hole or notch

Fig. 9. Normalizedsrrfor locations shown inFig. 8

Fig. 10. Normalizedsqqfor locations shown inFig. 8

Fig. 11. Normalizedsrqfor locations shown inFig. 8

y

x

r = 1.2R ~

1.27R

R

R = 9.53 mm

(3/8 )

″

(8)

in one operation. The methods ofHuang and Rowlands (1991)and

Lin and Rowlands (1995)have to be applied incrementally around the edge of a cutout. Unlike the cited complex-variable stress function techniques, the use of a stress function in terms of real variables enables application to loaded as well as traction-free cutouts (Foust and Rowlands, 2003). The present paper empha-sizes afinite geometry, but the method can be extended (restricted) to infinitely large members by appropriate modifications to the stress function.

Advantages of the described thermoelastic approach over other methods of stress/strain analysis for such a problem include (i) ability to obtain reliable stresses at the edge of geometric discontinuities despite unreliable measured data at such locations, (ii) virtually no component preparation is required, (iii) it is unnecessary to differentiated the measured information, (iv) only linear (rather than non-linear) least-squares is involved and (v) full-field capability (but digital information also available). The success of the present stress analysis approach without requiring knowledge of the constitutive properties, or far-field shape or boundary conditions is significant. Purely boundary-collocation, theoretical or numerical (FEM, finite difference) methods neces-sitate knowing the applied loads, but such information is often insufficiently well known in practice. Having said that, traction-free conditions were additionally applied along a portion of the top of the plate of Fig. 1, but with insignificant consequences. While not done so, one could also impose the traction-free conditions along the left and right vertical edges of the plate of

Fig. 1. Recognizing that the applied load P* does not appear explicitly in Equations (4)e(7), the present situation could be treated as an inverse problem and P* evaluated by integrating

s

xx

across a horizontal area of the plate.

Although the agreement between the ANSYS and experimental thermoelastic and strain gauge results in Figs.7and9e11is good, the discrepancies between FEM and measured results might well represent a case where the assumed, versus actual, boundary conditions cause the ANSYS-predicted stresses not to represent exactly physical reality. Fig. 3 illustrates that the stress is not uniform along the bottom edge of the plate. The elastic modulus of the aluminum plate is one-third that of steel of the more massive supporting platen so there could be some nonuniformly distributed vertical displacement (and stresses) along the bottom edge of the plate. The ANSYS assumption of zero vertical motion of the bottom of the plate might therefore be incorrect. However, when a sheet of lead and a sheet of compliant paper were each separately inserted between the bottom surface of the vertically loaded plate and the supporting steel platen in attempts to assess any nonuniformity in the vertical motion (or loading) along the bottom edge CDC0of the plate ofFig. 1, none was detected.

The present physical plate isfinite in size and rectangular in shape, but the method can handle large, arbitrarily shaped members. Confidence in the thermoelastic results is provided by comparing them with those from strain gauges and FEM. While influenced by Wang et al. (2005), the present TSA method provides individual components of stress at and in the neigh-borhood of the hole. Wang et al., 2005 does not display any individual stress components. The analyses ofWang et al. (2005),

Lin et al. (2009)andLin et al. (2011)involve a traction-free, round hole which is co-linear with the applied load, whereas the current concentrated load is offset horizontally from the hole. Although demonstrated here to a single traction-free round hole, ongoing research at the University of Wisconsin indicates that thermo-elasticity can be applied to loaded or neighboring multiple holes as well as to non-circular cutouts. The approach could handle the situation if both the hole and load were offset from the horizontal center of the plate, or if the perforated plate were loaded virtually

anywhere along its external edge. Multiple edge loadings could be accommodated. This paper synergizes recorded temperatures with a stress function, but the latter can also be used advanta-geously with measured strains (Foust and Rowlands, 2003). Other than having to approximate the boundary condition along the bottom edge of the plate, the external loading of the physical plate was known so FEM modeling was straight forward. However engineering cases often involve unknown loads rendering it difﬁcult to reliably model the situation numerically, thereby necessitating such experimental capabilities such as presented here.

Acknowledgments

The authors wish to thank the US Air Force for providing funds (Grant #FOSR FA9550-05-1-0289) with which to purchase the TSA equipment, and John Dreger and Weston Skye, University of Wis-consin, for their technical assistance. Dr. Quinn’s participation was funded by the Worldwide Universities Network (WUN), which enabled him to visit the University of WisconsineMadison under the Global Exchange Programme. Assistant Professor Lin acknowl-edges the kind support of the Robert M. Bolz Wisconsin Distin-guished Graduate Fellowship.

Notation

an, bn, cn, dn, a0n, b0n, c0n, d0nand Ao Airy coefﬁcients

A Airy matrix

c vector of Airy coefﬁcients C condition number

d thermoelastic S input vector d0 vector of evaluated S

D location of hole below top edge of plate E horizontal distance between hole and load F1,...., F9 coefﬁcients

k number of Airy coefﬁcients K thermo-mechanical coefﬁcient m number of input values P load per thickness P* concentrated load r,

q

polar coordinates R radius of hole

S* recorded thermoelastic signal S normal or isopachic stress (¼

s

rrþ

s

qq)

t thickness of plate

D

change

f

Airy stress function

s

shear stress

Appendix A. (Lin and Rowlands, 1995)

F1 ¼ R3_$½12$F 4$sinð3$qÞ þ 4$F5$cosð3$qÞ 6$R$½F4$sinð3$qÞ þ F5$cosð3$qÞ (A1) F2 ¼ 12$R 5_{$½ F} 4$sinð3$qÞ þ F5$cosð3$qÞ

6$R$½F4$sinð3$qÞ þ F5$cosð3$qÞ (A2)

F3 ¼ R

3_$½6$ð1Þ$F

4$sinð3$qÞ þ 10$F5$cosð3$qÞ

6$R$½F4$sinð3$qÞ þ F5$cosð3$qÞ

(9)

References

Dulieu-Barton, J.M., Stanley, P., 1998. Development and applications of thermoelastic stress analysis. Journal of Strain Analysis for Engineering Design 33 (2), 93e104. Foust, B.E., Rowlands, R.E., 2003. Inverse stress analysis of pinned connections using strain gages. In: Tanaka, M. (Ed.), Proc. Inverse Problems in Engineering Mechanics (ISIP 2003), Nagano, Japan, pp. 323e332.

Greene, R.J., Patterson, E.A., Rowlands, R.E., 2008. Thermoelastic stress analysis. In: Sharpe, W.N. (Ed.), Handbook of Solid Experimental Mechanics. Elsevier (Chapter 26).

Huang, Y.M., Rowlands, R.E., 1991. Quantitative stress analysis based on the measured trace of the stress tensor. Journal of Strain Analysis for Engineering Design 26 (1), 55e63.

Joglekar, N., 2009. Separating Stresses Using Airy’s Stress Function and TSA: Effects of Varying the Amount and Source Locations of the Input Measured TSA Data and the Number of Airy Coefﬁcients, MSc thesis, University of Wisconsin, Madison, WI, USA. Lin, S.-T., Rowlands, R.E., 1995. Thermoelastic stress analysis of orthotropic

composites. Experimental Mechanics 35 (3), 257e265.

Lin, S.-J., 2007. Two- and Three-dimensional Hybrid PhotomechanicaleNumerical Stress Analysis, PhD thesis, University of Wisconsin, Madison, WI, USA. Lin, S.-J., Matthys, D.R., Rowlands, R.E., 2009. Separating stresses thermoelastically

in a central circularly perforated plate using an Airy stress function. Strain 45 (6), 516e526.

Lin, S.-J., Quinn, S., Matthys, D.R., New, A.M., Kincaid, I.M., Boyce, B.B., Khaja, A.A., Rowlands, R.E., 2011. Thermoelastic determination of individual stresses in vicinity of a near-edge hole beneath a concentrated load. Experimental Mechanics 51 (6), 1441e1452.

Patterson, E.A., Rowlands, R.E., 2008. Determining individual stresses individually. Journal of Strain Analysis for Engineering Design 43 (6), 519e527.

Ryall, T.G., Heller, M., Jones, R., 1992. Determination of stress components from thermoelastic data without boundary conditions. Journal of Applied Mechanics 59 (4), 841e847.

Soutas-Little, R.W., 1998. Elasticity. Dover Publications, Inc., Mineola, New York. Wang, W.-C., Chen, Y.-M., Lin, M.-S., Wu, C.-P., 2005. Investigation of the

stressﬁeld of a near-surface circular hole. Experimental Mechanics 45 (3), 244e249.

F4¼ 2$ðD R$cos

qÞ

(A4) F5 ¼ 2$ðD R$cos

qÞ

p$

R2_{þ D}2_{þ E}2_2$R$ðE$sin

_q

_{þ D$cos}

_qÞ

2$ 1 2$ðD R$cos

qÞ

2_$sinð2$qÞ1 2$ðR$sin

q

EÞ 2_$sinð2$qÞ

þ ðR$sin

q

_ðr$sin

_q

_{EÞ$ðD r$cos}

_{qÞ$sinð2$qÞ}

i

(A7) F8 ¼ 2$ðD r$cos

qÞ

p$

q

_{þ D$cos}

qÞ

2$ 1 2$ðD r$cos

qÞ

2_$sinð2$qÞ1 2$ðr$sin

q

EÞ 2_$sinð2$qÞ

þ ðr$sin

q

EÞ$ðD r$cos