Stress concentration

1. Introduction The FEM (Finite Element Method) is a way of obtaining a of finding a solution to a physical problem. It relies on discretizing a continuum domain into finite elements. The accuracy of the solution greatly depends on the number of elements used to represent the physical domain. As we progressively refine the mesh, the solution improves and given enough iterations it converges towards a specific result. If there is an analytical solution for the given problem, the mesh refinement procedure will converge towards the exact solution. However, there are situations where the solution does not converge with mesh refinement. Stress singularities are one of these situations. This article puts its focus on stress singularities and stress concentrations. What are they? When do they pose for concern? How should we, as FE analysts, deal with them? 2. Stress singularities In structural analysis, we are mainly concerned about displacements and their derivatives – the stresses. A stress singularity is a point of the mesh where the stress does not convergence towards a specific value. As we keep refinement the mesh, the stress at this point keeps increasing, and increasing, and increasing… Theoretically, the stress at the singularity is infinite. Typical situations where stress singularities occur are the appliance of a point load, sharp re-entrant corners, corners of bodies in contact and point restraints. As you can see, stress singularities are a common situation in FEA. ...

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The value of the stress concentration factor is calculated by: where the coefficients in the equation above are calculated from 0.1 ≤ h/r ≤ 2.0 2.0 ≤ h/r ≤ 50.0 C 1 0.955 + 2.169 √ h/r− 0.081 (h/r) 1.037 + 1.991 √ h/r+ 0.002 (h/r) C 2 −1.557 − 4.046 √ h/r+ 1.032 (h/r) −1.886 − 2.181 √ h/r− 0.048 (h/r) C 3 4.013 + 0.424 √ h/r− 0.748 (h/r) 0.649 + 1.086 √ h/r+ 0.142 (h/r) C 4 −2.461 + 1.538 √ h/r− 0.236 (h/r) 1.218 − 0.922 √ h/r− 0.086 (h/r) Sources: • • • The value of the stress concentration factor is calculated by: where the coefficients in the equation above are calculated from 0.1 ≤ h/r ≤ 2.0 2.0 ≤ h/r ≤ 50.0 C 1 1.024 + 2.092 √ h/r− 0.051 (h/r) 1.113 + 1.957 √ h/r C 2 −0.630 − 7.194 √ h/r+ 1.288 (h/r) −2.579 − 4.017 √ h/r− 0.013 (h/r) C 3 2.117 + 8.574 √ h/r− 2.160 (h/r) 4.100 + 3.922 √ h/r+ 0.083 (h/r) C 4 −1.420 − 3.494 √ h/r+ 0.932 (h/r) −1.528 − 1.893 √ h/r− 0.066 (h/r) Sources: • • • The value of the stress concentration factor is calculated by: where the coefficients in the equation above are calculated from 0.1 ≤ h/r ≤ 2.0 2.0 ≤ h/r ≤ 20.0 C 1 1.007 + 1.000 √ h/r− 0.031 (h/r) 1.042 + 0.982 √ h/r− 0.036 (h/r) C 2 −0.114 − 0.585 √ h/r+ 0.314 (h/r) −0.074 − 0.156 √ h/r− 0.010 (h/r) C 3 0.241 − 0.992 √ h/r− 0.271 (h/r) −3.418 + 1.220 √ h/r− 0.005 (h/r) C 4 −0.134 + 0.577 √ h/r− 0.012 (h/r) 3.450 − 2.046 √ h/r+ 0.051 (h/r) Sources: • • • The value of the stress concentration factor is calculated by: where the coefficients in the equation above are calculated from 0.1...

The fatigue strength of a component is known to highly depend on its surface quality, and it is thus necessary to develop a reliable and appropriate mathematical model for fatigue strength assessment that consider the effect of surface roughness. In this paper, different underlying physical mechanisms of the roughness effect at different regions of specimens were studied by fatigue testing of 7N01 aluminum alloy. For a quantitative analysis of the surface roughness effect, a revised stress field intensity approach for a fatigue strength assessment of microsized notches was proposed as a theoretical support. In the new model, a new form of weight function was built to adapt the characteristics of microsized notches. In addition, the effect of the field radius was fundamentally weakened on solution of the stress field intensity and the difficulty of fatigue failure region definition in the traditional method was overcome correspondingly in the proposed model, which made the calculated field strength accurate and objective. Finally, to demonstrate the validity of the revised approach quantitatively, specimens with conventionally sized notches were subjected to stress field intensity calculations. The results showed that the revised approach has satisfactory accuracy compared with the other two traditional approaches from the perspective of quantitative analysis. It is known that the fatigue strength of component depends largely on its surface quality, which is the key content...

From doing some research I've learned that internal stresses inside an object can "concentrate" on sharp edges, as shown in the above picture. For that reason we often make fillets so that the change in diameter is smoother and we get a smaller stress concentration. But why does the stress concentrate on sharp edges? From Googling I've found numerous inadequate answers, such as imagining "stress flow lines" getting packed at the sharp edge. Here I can see an analogy to electromagnetism, where we talk about flow lines of an electric field, but what is "flowing" here? What do the lines represent? Intuitively, I can imagine this: If we press the object on the left in the above picture, there is a higher stress after the edge, since there is now a smaller area enduring the same force. Hence the stress is higher. But this does not explain why the stress concentrates at the edge. I mean, even with the fillet, the radius decreases anyway, so why is there not a similarly high stress on the right of the fillet? Similar effect also occurs when we have a continuous piece of material with a hole in it. The stress is the highest around the hole. Once again, force lines, what do they represent? Again I can see the similarity to electromagnetism. But what kind of a force field are we talking about here? One source I found simply explained that we can use methods like finite element analysis to confirm this concentrating of stress does happen. But this is not an explanation either. Use of...

Stress Concentration ME 354, MECHANICS OF MATERIALS LABORATORY STRESS CONCENTRATIONS MGJ 27 OCT 96 PURPOSE The purpose of this exercise is to the study the effects of geometric discontinuities on the stress states in structures and to use photo elasticity to determine the stress concentration factor in a simple structure. EQUIPMENT Un-notched beam of birefringent material (an epoxy). Notched bend beam of the same birefringent material as the un-notched beam. Four-point flexure loading fixture with load pan and suitable masses for loading Circular polariscope with monochromatic light source PROCEDURE Part 1. Beam under Pure Bending to Determine the Stress-Optical Coefficient of the Material i) Install the un-notched beam (see Fig. 1) in the four-point flexure loading fixture ii) Attach the load pan (Note: The combined pan/fixture mass/weight is ~1.204 kg=2.66 lbf) iii) Apply 2 weights (~9.049 kg = 20 lbf each) one at a time to the load pan. iv) With the polarizer and analyzer crossed (dark field), focus the camera, and record the image using the thermal printer v) Determine the maximum fringe orders at the top and bottom of the beam including estimates of fractional fringes orders by counting the fringes. vi) The stress-optical coefficient can be calculated using the following relation: (1) where f is the stress-optical coefficient, is the fringe order, t is the model thickness, and are the plane-stress principal stresses. Part 2. Notched Beam under Pure Bending to Determin...

Normal and shear stress, as we have defined them, are measures of the average stressover a cross section. Simply put, the magnitude of stress at any place along the cross section is the same. This means the load is distributedover the entire cross section. Alternatively, if the external force is focused over a small region, it is referred to as a point load. A point load, unlike a distributed load, causes stress near the point of loading to be much higher than the average stress. This leads to very complicated deformations from very complicated states of stress. Describing this deformation is beyond the scope of this course. But, if you look at the illustrations of a distributed load vs. a point load below, what you will notice is that the deformations (and hence, the stress distributions) start to look similar once you get far away from the point load. A natural question is: how far away from the point load does the stress become evenly distributed (i.e. when are we safe to use our average stressdefinition)? The Saint-Venant Principle states that the average stress approximation is valid within the material for all points that are as far away from the load as the structure is wide. This statement may be easier to understand visually: Let’s switch gears for a moment, and return to the connection between stress and strain. Until now, our approach has been: 1. determine the external forces from a statics analysis, 2. calculate the internal stress, and 3. use Hooke’s law to d...