This paper presents the experimental and numerical studies that had been conducted to investigate the stress concentration around a circular cutout in an isotropic material. Test specimens with circular holes were loaded in tension and bending. The tension test specimen was loaded in an Instron test machine. By mounting a set of strain gages orthogonal to the applied loading direction, the longitudinal strain measurements in the vicinity of the hole can be performed. The strains obtained by the series of strain gages placed at varying distances from the hole were extrapolated to the edge of the hole to determine the peak stress at the hole. These peak stresses were divided by the corresponding nominal far field stresses to obtain the stress concentration factors for specimen loaded in tension.
The bending case was investigated with a cantilever beam with a hole at its mid span. A hole was located so that the nominal stress at the fixed end was the same as the one at the location of the hole. Strain gages were placed at varying distances from the edge of the hole, one being directly adjacent to the edge. Known amounts of load were applied at the free end of the beam. The peak strains at the hole were extrapolated from the strain gage readings similar to what was done for the tension case. The stress concentration factor is the peak strain at the hole divided by the nominal strain at the same location.
The experimental results on stress concentrations were compared with finite element solutions performed on the specimen geometries and loadings similar to the ones used in the experiments.
Stress Concentration in a Tensile Specimen
1. An aluminum plate with a central circular hole will be subjected to a tensile load. 2. Strain distributions are measured using strain gages attached to different positions on the plate. 3. Local tangent strains are measured at five positions along the ligament. 4. Load the plate in
tension to a pressure of 500 psi.
5. Record data
Stress Concentration in a Cantilever Specimen
1. High-strength aluminum alloy beam, 1/4 x 1 x 12 ½ (3.2 x 25 x 318 mm), outfitted with preinstalled strain gages and a ¼” diameter stress concentration hole is used in the flexure frame. 2. Three very small strain gages are placed at varying distances from the hole, with one directly adjacent to the edge. 3. Plot strains on a graph sheet at the locations of the gage centerlines, and draw a smooth curve. 4. Extrapolate the data to the edge to obtain the peak strain. STRESS CONCENTRATION FACTORS
Any physical discontinuity in a structural member or a sudden change in the geometric form of a part leads to a region of stress concentration. The abrupt change in cross sections cause the stress “flow lines” to crowd causing high stress concentration. To mitigate this phenomenon, smoother changes such as fillet radii are introduced in structural members that make the “flow lines” less crowded causing lower stress concentrations. The theoretical stress concentration factor, Kt is defined in terms of maximum (or peak) stress, σmax and nominal (or average or far-field) stress, σnom as:
The theoretical stress concentration factor is a function of component geometry and loading. Stress concentration factor Kt for plate with hole under tension is shown as a function of the diameter to width ratio, d/b in Figure 1. For a plate with hole under bending the stress concentration factors as a function of the diameter to width ratio, d/b for various depth-to-thickness ratios d/h are shown in Figure 2. For the two specimen geometries used in the experiments, the experimental results as well as the results from the numerical solutions was compared with the corresponding analytical results displayed in Figures 1 and 2.
Figure 1: Stress Concentration Factor for a Plate with a Hole Loaded in Tension
Figure 2: Stress Concentration Factor for a Plate with a Hole Loaded in Bending
PRELIMINARY EXPERIMENTAL STUDIES
In order to appreciate the stress flow lines around a discontinuity, such as a circular hole, two preliminary demonstrations are introduced. The first one is a latex sheet containing a circular hole is used. Ruled parallel lines are drawn on the sheet orthogonal to the direction of pull. Because of the abrupt change in geometry at the hole, the lines curve around, leading to higher non-uniform stresses and strains. Figure 3 shows the latex sheet before (left) and after (right) the pull.
Figure 3 Latex Sheet with a Hole in Tension showing the stress flow lines
EXPERIMENTAL STUDIES ON A BAR WITH HOLE IN TENSION
A steel bar 0.125 in thick and 0.75 in wide is held between grips of an INSTRON machine with the length between the grips as 1.5 in. The bar contains a through thickness hole of 0.25 in diameter drilled through the center of the bar. The bar is pulled with a force of 30 lbs. The first step of analysis is to estimate the stress concentration factor using the analytical method. Far from the applied load area, stress is evenly distributed throughout the cross-section, which is called the nominal stress. From the definition of stress, this nominal stress can be calculated by ignoring the hole, as
In the experiment a set of strain gages is placed at varying distances from the edge of the hole, with one of the gages directly adjacent to the edge. Since the strain gages cannot be applied directly to the maximum stress location, remote stresses are used to determine the peak stress. A strain gage is placed very near to the grips, which provided information on nominal strain (and hence nominal stress) on the bar. The strains obtained by the series of strain gages placed at varying distances from the hole were extrapolated to the edge of the hole to determine the peak stress at the hole. These peak stresses were divided by the corresponding nominal far field stresses to obtain the stress concentration factors for specimen loaded in tension.
EXPERIMENTAL STUDIES ON A CANTILEVER BEAM WITH HOLE
Figure 5 presents the geometrical details of the specimen used in the study and Figure 6 shows the set-up used. In Figure 5, Location 1 corresponds to the region adjacent to the clamped end. Location 2 corresponds to that of the hole. At Location 1, the stress distribution is uniform across the beam width. The bending stress at the location 1 is given by
At this location, the bending moment M1 = W L where L is the length of the beam span, and W is the applied load. If the beam width is b and the beam thickness is t, then the moment of inertia at location 1, I1 = (1/12) (b t3). We also have c = t/2. Substituting these values in equation (1) we have,
At location 2, the bending moment M2 = W l where l is the distance of the hole center line from the free end where the load W is applied. With d as the hole diameter, the moment of inertia at location 2, I2 = (1/12) (b-d)( t3). At location 2, the nominal bending stress is therefore given by,
The students noted that the bending stress σ2 above is a nominal stress and contains no effect of stress concentration. The distance l was selected such that the nominal bending stresses at locations 1 and 2 are the same, that is σ1 = σ2, and thus from equations (4) and (5) it follows,
If the maximum bending stress at the edge of the hole is Σ2, which contains the stress concentration effect, then the elastic stress concentration factor Kt at the hole is given by (noting that σ1 = σ2),
If the stresses are below the proportional limit, the stresses and the strains will be proportional. Therefore the stress concentration factor is also the strain concentration factors at the two locations. Thus the stress concentration factor Kt at the hole is the ratio of the maximum strain at location 2, E2 to the nominal strain at location 2, ε2 and with our particular choice of l, becomes the ratio of maximum strain at location 2 to the nominal strain at location 1, ε1. Thus
Figure 5: Dimensions of the Cantilever Beam with Strain Gage Locations
Figure 6 illustrates the experimental setup used in the study. The beam is loaded such that a nominal strain ε1 reaches about 0.2%. This is important since the strain at the edge of the hole will be considerably higher and would produce yielding locally. The actual strains around the hole are measured by three small strain gages at varying distances from the edge of the hole in the transverse direction, with one gage placed directly on the edge of the hole.
The dimensions are, L = 254 mm, l = 178 mm, b = 25.4 mm, t = 6.35 mm, d = 6.35 mm
The strains at the three locations are plotted and extrapolated to the edge of the hole to obtain an estimate of the peak strain at the hole. Then by determining the nominal strain at the clamped end and using the peak strain at the hole, the stress concentration factor is calculated using equation (6).
Figure 6: Flexor Setup
NUMERICAL STUDIES ON A BAR WITH HOLE IN TENSION
Figure 7: Application of Loads and Boundary Conditions on the Quadrant Model
Figure 8: Stress Profile for the X-Stress
Figure 7 shows the load applied and the applicable boundary conditions. Tensile load is applied as a uniform pressure of 100 psi applied on the vertical edge to the right. The bottom edge of the quadrant is input as a zero UY displacement and the vertical edge (left) is input as a zero UX displacement. Figure 8 shows the typical stress profile for the X-stress. The nominal stress and the peak stress information are used to obtain the stress concentration factor.
NUMERICAL STUDIES ON A CANTILEVER BEAM WITH HOLE IN BENDING For cantilever beam with the same dimensions as the specimen, one end of the beam is fixed for all displacements and rotations simulating the clamped end. On the free end a load of 14.7 N (1.5 kgf) is applied. Figure 11 shows the von-Mises nodal stresses and Figure 12 the corresponding strain results. The maximum stress was found to be 33.7 MPa, at the edge of the hole, and the nominal stress at that section is around 12 MPa. This gives a stress concentration factor of around 2.8 at the hole. The same value is obtained when the strains are compared in Figure 12. The analytical value obtained for a plate with d/h =1.0 and d/t =0.25 is obtained from Figure 2 as 1.85. The analytical method as shown is Figure 2 is for a pure bending, whereas the finite element method that correctly represents cantilever bending and that explains the discrepancy in the results. The experimental results are closer to the finite element solution results.
Figure 9: Finite Element Stress Results
Figure 10: Finite Element Strain Results
1. The activities described in this paper are accomplished as two separate modules in the laboratory session for the course strength of materials offered at a sophomore level. One is on tensile loading and the other was on cantilever bending. Both modules involve experimental and numerical efforts. 2. The experimental methods use some preliminary studies to develop a feel for the stress flow, and are followed up with mounting strain gages at the appropriate locations in the specimen. 3. Students are able to visualize the effects of stress concentration around a circular hole for the case of both a bar in tension as well as a cantilever beam in bending.
4. The peak strain at the discontinuity (hole) is measured and compared with the nominal strain and the ratio, which measured the stress concentration factors were found to be close to the numerical solutions. 5. This activity provides the students an intuitive appeal into the detrimental effect of stress raisers caused by discontinuities. 6. The finite element studies give a perspective of how the loads and stresses get distributed around discontinuities. 7. The meshes are created automatically using ANSYS and the perspectives of the convergence of the solutions are also emphasized by mentioning that the mesh size refinement leads to progressively correct solutions.
1. Prof. Harijono’s Lecture notes, Experimental Stress Analysis-Ch.8, Strain and Its Measurement. 2. Stress concentration factor, http://en.wikipedia.org/wiki/Stress_concentration 3. http://www.scribd.com/doc/12965732/Petersons-Stress-Concentration-Factors-
EAS 3403 – EXPERIMENTAL STRESS ANALYSIS
Studies on Stress Concentration Using Experimental and Numerical Methods
MOHD NA’IM ABDULLAH155360
LECTURER :PROF. HARIJONO DJOJODIHARDJO