Maximum stress in a tensile strip with a large hole

Measurements were made to decide between two solutions of the title problem. In question was the limiting value of the stress-concentration factor at the edge of a large hole in a tensile strip as the hole diameter approached the strip width. The results indicate that the stress-concentration factor is near two and support one of the solutions with a minor qualification.     

Effect of two drille holes on the elastic and elastic-plastic strain distribution around a Charpy V-notch

Two small holes (0.0292 in.), appropriately drilled near the root of a Charpy V-notch, have been shown to reduce markedly the Charpy V-notch transition temperature of various steels. In the present study, three experimental techniques were used to define the effect of two holes on the mechanics of deformation and fracture of notched bars loaded in three- and four-point bending: (1) two-dimensional photoelastic stress analyses were performed on models of both the standard Charpy and drilled geometries; (2) a sensitive dislocation etch-pitting technique was used to observe directly the plastic-strain fields developed in V-notch samples of Fe?3% Si alloy loaded in slow bending; and (3) the Charpy striker was instrumented to record load-time curves during impact-bending and thereby determine the dynamic fracture strength of notched and drilled mildsteel samples. It was determined that two holes donot significantly reduce the elastic stress-concentration factor although they cause considerable redistribution of the local shear stresses around the notch. Consequently, the elastic-plastic state develops quite differently in the presence of two holes, and hole drilling can increase the load-carrying capacity of notched mild-steel bars by more than 100 percent even when bars fail by brittle cleavage prior to general yielding. The implications of these results with respect to other forms of “stress-relieving notches” are discussed.     

Hole-shape optimization in a finite plate in the presence of auxiliary holes

Minimizing the stress concentration around holes in uniaxially loaded finite plates is an important consideration in engineering design. One method for reducing the stress concentration around a central circular hole in a uniaxially loaded plate is to introduce smaller auxiliary holes on either side of the original hole to help smooth the flow of the tensile principal-stress trajectories past the original hole. This method has been demonstrated by Heywood and systematically studied by Erickson and Riley. Erickson and Riley show that for a central-hole diameter-to-plate width ratio of 0.222, the maximum stress reduction is up to 16 percent. In recent work, Durelliet al. show that the stress concentrations around holes in uniaxially loaded plates can be minimized by changing the hole shape itself till an optimum hole profile with constant stress values respectively on the tensile and compressive segments of the hole boundary is reached. By this technique the maximum stress reduction obtained for the above case is up to 20 percent. In the present work, starting with the optimum sizes and locations of central and auxiliary circular holes for a finite plate given by Erickson and Riley, a systematic study of the hole-shape optimization is undertaken. A two-dimensional photoelastic method is used. For a central-hole diameter-to-plate width ratio of 0.222, the reduction in stress-concentration factor obtained after hole-shape optimization is about 30 percent. It is also shown that it is possible to introduce the ‘equivalent ellipse’ concept for optimized holes.     

Stress concentrations in tensile strips with large circular holes

The purpose of this investigation was to determine more accurately the stress concentrations in tensile strips having large circular holes where the ratio of the hole diameter to strip width is greater than 0.5 Results of numerical elasticity analyses are presented for ratios of hole diameter to bar width ranging from 0.10 to 0.99, and photoelastic data are presented for the range from 0.40 to 0.94. Both numerical results and photoelastic data indicate that the stress-concentration factor based on net area approaches a value of one as the ratio of hole diameter to bar width approaches a value of one.     

Stress-concentration factors for elliptical holes near an edge

Experimental and theoretical stress-concentration factor and stress-intensity factor solutions exist for a large number of hole shapes and configurations. However, little work has been done on the interaction between holes and free edges. This paper reports the results of an experimental study which investigated elliptical holes close to each edge of a tension plate. The holes were symmetric with respect to the longitudinal axis of the models and had their major axes normal to the edge of the plate. The ellipse ratio and the distance between the center and the edge were varied. A statistical model is developed which shows that the stress concentrationK _g at both ends of the ellipses is linearly dependent on a geometric function ? such thatK _g = αΦ + γ. The values for the function ? are given.     

Investigation of stress-concentration factors in tensile strips

A photoelastic and a numerical investigation has been carried out to determine the stress-concentration factors at the edge of a central circular hole in a tensile strip for different ratios of hole diameter to width of the strip. The photoelastic data and the numerical results indicate that the stress-concentration factor at the minimum cross-sectional area tends to a value of two if the ratio of the hole diameter to the strip-width approaches a value of one.     

Effects of material and stacking sequence on behavior of composite plates with holes

Strain distributions to failure, tensile and compressive strain-concentration factors, and strength-reduction factors were determined for glass-, boron-, and graphite-epoxy plates with holes loaded in tension. Strain gages, photoelastic coatings and moiré techniques were used. Ten variations of layup and stacking sequence were studied. The boron-epoxy composite was found to be the stiffest and strongest of the three. The graphite laminate with the highest stress concentration and the most linear strain response exhibited the highest strength-reduction factor. In all cases, the maximum strain at failure on the hole boundary was higher than the ultimate tensile-coupon strain. In general, it was found that, the higher the stress-concentration factor, the higher the strength-reduction factor. Thus, the [0/90/0/90]_s layup with a stress-concentration factor of 4.82 had a strength-reduction factor of 3.18. At the other extreme, the most flexible layup [±45/±45]_s with the lowest stress-concentration factor of 2.06 had the lowest strength-reduction factor of 1.10. Stacking sequences associated with the tensile interlaminar normal stress or high interlaminar shear stress near the boundary, resulted in laminates 10 to 20 percent weaker than corresponding alternate stacking sequences. Furthermore, it was found that stacking-sequence variations can alter the mode of failure from catastrophic to noncatastrophic.     

Strain and stress concentrations in composite laminates containing a hole

The strain concentrations of orthotropic composite laminates containing a circular hole and subject to tensile loading were measured experimentally using strain gages. Then the stress concentrations were calculated using the strain distributions in the initial region of the stress-strain curve before any microdamages were developed. The graphite/epoxy AS4/3502 [O₂/±45]_2s and [45]_4s were chosen to represent fiber-dominated and matrix-dominated laminates, respectively. Several combinations of hole-diameter/plate-width ratio were designed to show the width effect. The conditions of the laminates, after the holes were drilled, were examined using X-ray techniques. Good correlation was obtained between theory and experimental result using specimens in good condition (without machining damages). A procedure for accurately determining the strain and stress concentrations is given. Paper was presented at the 1989 SEM Spring Conference on Experimental Mechanics, held in Cambridge, MA on May 29–June 1.     

The effect of low-frequency biaxial loads on stress-concentration factors in plates

It is generally recognized that stress-concentration factors under stress-wave loading are lower than those under static loads. In this work, the effect of low-range frequency of biaxial sinusoidally varying alternating stresses on the stress-concentration factors for circular and elliptical holes in Plexiglas plates is investigated. The experiments have been performed on a specially designed and built “biaxial cyclic-stress machine” and the results are presented in the form of curves. In the case of biaxial alternating stresses, the stress-concentration factor is defined as the ratio of amplitude of the maximum alternating stress around the geometrical discontinuity to the larger of the amplitudes of the two principal alternating stresses which would occur at the same point, if the geometrical discontinuity was not present. Both values are considered over a stress cycle. The results indicate a slight decrease in the values of stress-concentration factors with increase in frequency.     

Stress concentrations associated with circular holes in cylinders and bone in torsion

Experimental studies were undertaken to determine the torsional stress concentration factors (Kt) associated with circular holes in bone. Reflective photoelasticity was used to determine the stress field around a circular hole through one wall of the bone. A single adult sheep femur was used as the torsional model, in which six circular holes were concentrically machined through the posterior cortex. These holes ranged from 10.4 percent to 66.4 percent of the mediolateral bone diameter. From the photoelastic data, a stress concentration curve was developed for bone. The maximum stress location on the boundary of the hole was found to shift from the previously expected 45-deg location. Studies on tubes made of steel and plastic, both coated with photoelastic coating, were also performed. Three different pieces of steel tubing with similar inner to outer diameters were coated with different thicknesses of photoelastic coating. The variation in coating thickness did not appear to influence the stress-concentration factors in steel. TheKt in steel for 10 percent and 20 percent defects agreed with theKt associated with similar defects in bone. A single piece of plastic tubing was used in which six holes from 10 percent to 60 percent of the tube's outer diameter were concentrically machined through one wall. The location of the maximum stress around the boundary of the hole was found to shift, and this agreed with the maximum stress shift found in the bone. Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 8–11.     

Biaxial testing of [O2/±45] s graphite/epoxy plates with holes

An experimental investigation was conducted to study the behavior under biaxial-tensile loading of [O₂/±45] _s graphite/epoxy plates with circular holes and to determine the influence of hole diameter on failure. The specimens were 40-cm×40-cm (16-in.×16-in.) graphite/epoxy plates of [O₂/±45] _s layup. Four hole diameters, 2.54 cm (1.00 in.), 1.91 cm (0.75 in.), 1.27 cm (0.50 in.) and 0.64 cm (0.25 in.), were investigated. Deformations and strains were measured using strain gages and birefringent coatings. Biaxial tension in a 2∶1 ratio was applied by means of four whiffle-tree grip linkages and controlled with a servohydraulic system. Stress and strain redistributions occur around the hole at a stress level corresponding to localized failure around the 67.5-deg location and nonlinear strain response at the 0-deg location. Maximum measured strains at failure on the hole boundary are higher (approximately 0.016) than the highest ultimate strain of the unnotched laminate (0.010). Two basic patterns of failure were observed: (a) horizontal cracking initiating at points off the horizontal axis and accompanied by extensive delamination of the subsurface ±45 deg plies, and (b) vertical cracking along vertical tangents to the hole and accompanied by delamination of the outer 0-deg plies. The strength reduction ratios are lower than corresponding values for uniaxial loading by approximately 16 percent, although the stress-concentration factor under biaxial loading is lower.     

Stress-concentration factors for multiple semielliptical notches in beams under pure bending

Uniformly spaced semielliptical notches in rectangular beams under pure bending are examined photoelastically. Minimum stress-concentration factors, produced by multiple elliptical notching of beams, are obtained for wide ranges of notch width, semiminor elliptical axis, notch depth, notch pitch and depth of beam. In particular, the geometries of the optimum elliptical notch producing the least stress concentration are obtained for a practical range of parameters. Stress-concentration factors for beams with multiple semi-elliptical notches are compared to those for beams with single semielliptical notches and to those for beams with semicircular notches. The maximum reduction of stress-concentration factor for beams with multiple semielliptical notches is to approximately 37 percent of the stress-concentration factor for beams with single semielliptical notches. Within the range of parameters investigated, the stress-concentration factor for beams with multiple semielliptical notches was 15 to 37 percent less than that for multiple semicircular notches.     

Optimization of hole shapes in circular cylindrical shells under axial tension

Hole shapes are optimized in circular cylindrical shells subjected to axial load considering only the predominantly large membrane stresses present around the holes. Two-dimensional photoelastic isochromatics obtained with a special-purpose polariscope are utilized for the optimization process. The process leads to a significant decrease in the membrane stress-concentration factor and a modest decrease in weight, thus yielding a considerable increase in strength-to-weight ratio. This paper presents results for certain typical ratios of hole diameter to shell diameter. Previous theoretical and experimental studies for the circular hole have also been verified     

Photoelastic analysis of a thick plate with an elliptical hole subjected to simple out-of-plane bending

A three-dimensional photoelastic analysis using the stress freezing and slicing techniques was employed to study the stress distribution and the stress-concentration factors around an elliptical hole in a plate of finite thickness. The plate was subjected to simple out-of-plane bending. A special bending device was designed to produce uniform bending moment at the two opposite free edges of the plate. Six plates with various elliptical holes were studied. The stress variation across the plate thickness at the periphery of the elliptical hole was also investigated. The experimental results were correlated with the existing theoretical solutions.     

Photoelastic analysis of stresses around oblique holes

A three-dimensional photoelastic analysis was conducted to determine the magnitude and distribution of stresses around oblique holes in a uniaxially loaded plate. The holes were circular and inclined at angles of 45 deg and 60 deg with the faces of the plate. The plate-thickness-to-hole-diameter ratiot/D was 2.40. One end of each hole was blended to the face of the plate through a break radius equal to the radius of the hole. The plate dimensions were sufficiently large to simulate conditions of an infinite plate. The plates were loaded perpendicular to the plane of skewness. Stress distributions were obtained on sections perpendicular to the direction of loading. Results point to two critical areas of stress concentration: one at the acute-angle intersection of the hole and the surface of the plate and the other in the break-radius area. The stress concentrations in the latter area reach values of 4.6 and 6.7 compared to 3.6 and 4.5 at the acute-angle intersection, for the inclination angles of 45 deg and 60 deg, respectively. A simplified analysis used for the break-radius area gave results in agreement with the experiment. Thus, it was shown that break radii in oblique penetrations may have deleterious rather than beneficial effects. Comparison of results for the acute-angle intersection with existing theoretical and experimental values shows a definite and pronounced dependence of the stress-concentration factor on thickness-to-diameter ratio.     

Alleviation of the stress concentration with analogue reinforcement

In order to reduce the stress concentration around a hole in a plate, new, “analogue” reinforcements instead of reinforcing rings were used in this investigation. In two of these specimens, reinforcements with different volume fractions were arranged to coincide with the stress trajectories for an infinite plate with a hole under uniaxial tension. Two other specimens containing straight rectangular-grid-type reinforcements were made by using a photofabrication method. Specimens were then prepared by sandwiching these reinforcements between two epoxy-resin plates. Plane specimens, i.e., without reinforcement, were also made of the same epoxy resin for comparison. The stress concentrations at the edge of the hole under uniaxial tension were determined by photoelastic techniques. The measured stress-concentration factors were compared with well-known values for an infinite, isotropic, homogeneous plate containing a hole. Results were also compared with published data on [90/0/90/0]_s 7-ply laminated composite plates, and on plates strengthened with reinforcing rings. A definite reduction in stress concentration was observed on specimens containing analogue reinforcement.     

Elastic-plastic strain concentrations produced by various skew holes in a flat plate under uniaxial tension

The elastic and elastic-plastic surface strain fields around circular holes drilled at various skew angles to a flat plate have been experimentally evaluated in uniaxial tension. A photoelastic coating and moiré technique were used in the low- and high-strain regions, respectively. The maximum strain-concentration factor is shown to increase markedly with horizontal skew angle and decrease slightly with increasing vertical skew angle. Plastic deformation accentuates the differences between normal and skew holes, so that the angular dependence of the strain-concentration factor increases with nominal strain.     

Viscoelastic stress-concentration factors

Through the use of a new photoviscoelastic material, stereospecific polybutadiene, an exploratory photomechanical investigation was made of viscoelastic stress-concentration factors in bars loaded in dead-weight tension. Despite large creep deformations of the originally circular holes and circular arc fillets in the bars, the stress-concentration factors were observed to change by only about 25 percent of the initial values, which were in fair agreement with existing data on elastic concentrations. This brief report is a condensed summary of the technique and results of the investigation.     

Minimizing stress concentrations around circular holes in uniaxially loaded plates

The problem of determining stress distributions and reducing stress concentrations around holes in plates occurs in numerous design situations. One method for reducing the stress concentration around a central circular hole in a uniaxially loaded plate was demonstrated by Heywood. With this approach, smaller holes are introduced on either side of the original hole to help smooth the flow of the tensile principal-stress trajectories past the original hole. For the one case reported by Heywood (which did not produce the greatest reduction possible), the maximum stress was reduced to 84 percent of that due to a single hole. In the present program, a systematic study was undertaken using two-dimensional photoelasticity method to determine the optimum sizes and locations for the auxiliary holes for a number of plates with different central-hole diameter-to-plate width ratios. Maximum stress reductions from 13 to 21 percent were demonstrated for plates with hole diameter-to-plate width ratios between 0.1 and 0.6. With such reductions in maximum stress level, the improvement in fatigue life of a part can be very significant.