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The photo-elastic method has been employed to determine stress concentration factor (SCF) for square plates containing holes and inclined slots when the plate edges are subjected to in-plane tension combined with compression. Analyses given of the isochromatic fringe pattern surrounding the hole provides the SCF conveniently. The model material is calibrated from the known solution to the stress raiser arising from a small circular hole in a plate placed under biaxial tension-compression. These results also compare well with a plane stress FE analysis. Consequently, photo-elasticity has enabled SCF’s to be determined experimentally for a biaxial stress ratio, nominally equal to –4, in plates containing a long, thin slot arranged to be in alignment with each stress axis. The two, principal stresses lying along axes of symmetry in the region surrounding the notch are separated within each isochromatic fringe by the Kuske method [1]. FE provides a comparable full-field view in which contours of maximum shear stress may be identified with the isochromatic fringe pattern directly. The principal stress distributions referred to the plate axes show their maximum concentrations at the notch boundary. Here up to a fourfold magnification occurs in the greater of the two nominal stresses under loads applied to the plate edges. Thus, it is of importance to establish the manner in which the tangential stress is distributed around the slot boundary. Conveniently, it is shown how this distribution is also revealed from an isochro-matic fringe pattern, within which lie the points of maximum tension and maximum compression.

It is well known that small holes and slots raise the stress in loaded plates locally by factors of 3 or 4. Clearly this becomes important to an assessment of fatigue life when, in localised regions of high stress, cyclic loading accelerates the crack initiation process. The various design rules [2,3] require the SCF to be known but often estimates are made for unusual geometries. The problem of crack initiation from holes and slots is important to understand for ensuring the safety of many structures bearing load. The prediction of cyclic life is possible when the stress raiser is quantified with a stress intensity factor. The finite element technique has been used [4-6] to estimate stress concentration factors in various engineering components where fatigue cracks occur. Holes and slots are less severe than pre-existing cracks but are always present in designs involving fittings, sharp radii, connections and attachments [7-12]. Here, in common with much of the early work on quantifying stress raisers [13-15], we shall examine the influence of slots experimentally by the photo-elastic method. Firstly, the technique adopted is verified by two alternative methods: 1) from the analytical solution to the stress concentration around a hole in a bi-axially stressed plate and 2) from a numerical FE simulation. Both methods 1) and 2) can provide the contours of maximum shear stress in the surrounding material which photoelasticity reveals within its isochromatic fringe pattern. The shear stresses are separated into major and minor principal stresses along axes of symmetry and around the notch boundary in providing agreement between the three methods. Thereafter, a slot is arranged to lie parallel to each of the perpendicular stress axes in turn to establish the severity of its concentration experimentally from photoelasticity. The degree of stress concentration (SC) is revealed from locating points of maximum tension and compression around the notch boundary.

Overall, the scope of this study is to provide SCF’s for a slot with aspect ratio of 5 (nominally) in fixed orientations to various in-plane biaxial stress states. Here the present investigation quantifies SCF’s for slots with vertical and horizontal dispositions in a stress field for which tension is combined with compression. The information given provides for the apparent omission of this geometry in the library of published SCF available for a multitude of alternative geometries [

Holes and slots 1.0 mm wide and with a maximum length dimension of 2a = 6.2 mm were milled into the centre of Makralon and araldite CT 200 photoelastic sheets. Two methods of bi-axial loading square testpieces were employed [

The shear linkage frame (reported here) was loaded in tension along the square’s vertical diagonal, thereby inducing compression along its horizontal diagonal. _{x}/σ_{y} = –3.8 in which the co-ordinates x and y are aligned with the square’s horizontal and vertical diagonals. Q is the negative, central principal stress ratio that applies in the absence of a notch. The compressive stress induced across the horizontal diagonal was increased by the stated ratio (i.e. σ_{x} = Qσ_{y} = –3.8 σ_{x} from the contacts made along the four sloping sides. In the frame four equal length links were allowed to rotate upon their 12 mm diameter end connecting pins. A 2 mm groove machined along the inside of each link provided the register for a 75 mm square, Makralon photoelastic model containing the notch (either a hole or a slot) at its centre. Corners were chamfered for ease of assembly (see

Isochromatic fringe patterns under a series of increasing loads were recorded with a Practica 35 mm SLR camera with bellows attachment for close-ups. Typical, 2 s exposures, at an aperture setting of f5.6 with Kodak 100 ASA film, are reproduced here as line diagrams. The shear frame’s stress ratio Q was found from loading an un-notched aluminium plate with a 0˚/90˚ strain gauge bonded to its centre. It will be seen that a more precise value of Q follows from the classical solution to the stress concentration arising from a hole in a plate under biaxial loading.

There are two coefficients used in photoelasticity to convert the fringe count (N, fringes) into a stress magnitude. The supplier provides a Material Fringe Stress Coefficient f, independent of the sheet thickness. The user adopts a Model Fringe Stress Coefficient F in which thickness is accounted for. Either coefficient is found from a calibration upon strips cut from the sheet in which the fringes are counted within a known stress field, usually from a simple tension test or a beam in four-point loading. In the former method a unit increase in the fringe number corresponds to an alternating light and dark fringe pattern when the tensile stress (= load/section area) is uniform under a purely axial load. A bending calibration is less prone to experimental error but the stress field varies linearly between maximum tension and compression corresponding to (say) a hogging beam’s top and bottom edges. With equal, maximum stress magnitudes the fringes are counted at one edge where the stress is calculated from theory. Normally, the gradient of a plot between calculated stress (in MPa) and the fringes count N (= 1, 2, 3 etc.) provides F directly, from which f = Ft. Here a tension test conducted upon a parallel strip of test material between the polariser and analyser revealed a Material Fringe Stress Coefficient f = 13.9 N/mm/fringe [

For a model of thickness t, photo-elasticity theory [13-15] gives the magnitude of the difference between the principal stresses p and q

where it is seen that the model stress coefficient F = f/t (MPa/fringe) provides a direct conversion from a numerical fringe order value N to the magnitude of the principal stress difference. The special principal stress symbols p and q that appear universally in photoelastic analyses identify here with the hoop and radial stress around a slot boundary and along axes of symmetry.

Counting the fringe order N around a notch boundary allows the major principal (tangential) stress p and the maximum stress concentration S to be found directly from F. This is because the minor principal (radial) stress q is zero normal to the boundary. Equation (1) simplifies to give the boundary’s circumferential (or hoop) stress p = NF where N is counted around the notch boundary starting from the zero-order fringe location. In the Figures 1, 3 and 5 that follow the N = 0 location is shown to mark those boundary points at which there is an interchange between hoop tension and compression.

To find the stress distribution along the x, y axes of symmetry in the body of the plate, beyond the notch (a hole or slot), equi-spaced points n are taken along each axis. The fringe number N is then counted at radii r_{n} from the centre of the notch, separated by Δr. These symmetry axes coincide with principal stress directions where the shear stress component is absent. For the calculation of p and q it is convenient to convert to polar co-ordinates (r, θ) at each point n upon the xand y-axes so that the principal stress difference becomes:

Using Kuske’s method [_{r} increase by an amount δσ_{r} across a radial increment δr and apply the radial equilibrium equation [_{r} as

The incremental change (Δσ_{r})_{n} in σ_{r} between adjacent pairs of points n – 1 and n is estimated from the average of the their two gradients times their radial separation:

By placing n – 1 = 0 at a notch boundary, where σ_{r} = 0, the radial stress at “body” positions n = 1, 2, follows from successive summations:

where (Δσ_{r})_{n} is given in Equation (3). The hoop stress follows from Equations (2a) and (4a) as

The FE analysis of slotted plates was conducted using the Abaqus code [

Recently, a number of solutions have appeared to the SCF’s in plates [19,20] thick cylinders [

The shear frame’s linkages apply a vertical tensile force in combination with horizontal compressive force to the diagonals of a square plate. Here it needs to be recognised that these remote forces produce compressive and tensile stresses at the respective vertical and horizontal positions upon the boundary of a hole located at the centre of the plate. This means that a hole elongates in the direction of tension putting its North and South points under compression and its East and West under tension. _{90}/σ_{0} = –2. This hoop stress ratio applies to each angular position within the hole’s boundary. An average stress ratio of σ_{90}/σ_{0} = –1.83 was found from fringe patterns under numerous loads when fractional order fringes were accounted for. Elasticity theory [_{x}/σ_{y} imposed by the frame:

when σ_{90}/σ_{0} = –1.83, Equation (5) shows Q = –3.84. To check Q, an un-notched 2 mm thick aluminium alloy plate replaced the testpiece having 0˚/90˚ strain gauges bonded at its centre in alignment with the plate’s x, y-diagonals. When a state of plane stress is assumed the stress ratio Q at the plate centre becomes [

Under an increasing vertical load, the 0˚/90˚ strains responded linearly in a constant ratio ε_{y}/ε_{x} = –0.561, in which ε_{y} is positive and ε_{x} is negative. Substituting into Equation (6) with v = 1/3 gives Q = –3.53, which is in acceptable agreement with Q found from Equation (5).

The hoop stress variation measured around the hole is shown in _{θ} = NF to the fringe count around the hole. The variation found is seen to agree fairly well with the theoretical stress-function prediction [

where S = σ_{y} is the “applied” tensile stress. The latter is found from dividing the applied load by the area of the plate section through its horizontal diagonal. The “applied”, horizontal compressive stress follows as QS = σ_{x}.

Separating p from q by the method outlined above in para (3.1), provides the principal stresses along the x, y axes of symmetry (Figures 3(a) and (b) apply with a common legend). The experimental distributions shown here agree well with FE predictions and the stress function solution to this problem [

where a is the hole radius. The stress ordinates in Figures 3(a) and (b) are normalised with the remote stress S to show that two maximum stress concentration factors (SCF) arise at x and y on the boundary under an applied stress ratio Q = –3.84. The greater of these occurs in com-

pression with SCF = 14/3.84 = 3.65.

Given its validation from within (a), (b) and (c) above, we may now use photoelasticity to determine stress distributions where a theoretical solution is unavailable. In particular, slots 6.2 mm long × 1 mm wide, in alignment with the yand x-directions, will replace the hole at the testpiece centre. Correspondingly, the following vertical and horizontal slot analyses apply.

_{θ} = NF, as derived from the fringe count N, is shown in _{90}/σ_{0} = –5 between the maximum compressive and tensile stresses. Separating the principal stresses beyond the slot, along the more highly stressed y-axis, leads to the distribution shown in