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- 1. 8/8/2015 I-BEAM A Study in Stress Vs. Fastener Location jitesh.bhogal@gmail.com
- 3. Table of Contents Table OF Figures _______________________________________________ 1 INTRODUCTION _______________________________________________ 1 METHODS ___________________________________________________ 2 Tetrahedron Element___________________________________________ 2 Case 1: 2 Fasteners ____________________________________________ 3 Case 2: 3 Fasteners ____________________________________________ 3 Results ______________________________________________________ 4 Case 1 _____________________________________________________ 4 Case 1 Revision _______________________________________________ 4 Case 2:_____________________________________________________ 5 ANALYSIS____________________________________________________ 6 Along the Depth of the Beam______________________________________ 6 Along the Length of the Beam _____________________________________ 6 CONCLUSION _________________________________________________ 7 APPENDIX __________________________________________________ A0
- 4. jitesh.bhogal@gmail.com Table OF Figures Figure 1 : Loading Conditions ___________________________________________________1 Figure 2 : I-Beam Cross Section _________________________________________________ 2 Figure 3 : Tetrahedron Element _________________________________________________ 2 Figure 4 : Coarse Mesh Sample Stress Plot _________________________________________ 2 Figure 5 : Coarse Mesh Stress Peak ______________________________________________ 2 Figure 6 : Mesh Refinement At Critical Features _____________________________________ 3 Figure 7 : Mesh Refinement Max Stress Plot ________________________________________ 3 Figure 8 : Case 1 Start Location _________________________________________________ 3 Figure 9 : Case 1 End Location__________________________________________________ 3 Figure 10 : Case 2 Fasteners ___________________________________________________ 4 Figure 11 : Case 1 Results Plot __________________________________________________ 4 Figure 12 : Case 1 Adjusted Path_________________________________________________ 4 Figure 13: Case 1 Rev Car Load Plot _______________________________________________5 Figure 14 : Case 2 Rev Sample Stress Plot ___________________________________________5 Figure 15 : Case 2 Rev Stress Plot _________________________________________________5 Figure 16 : Stress Reduction Benefits _____________________________________________ 6 Table 1 : I-Beam Dimensions ___________________________________________________ 2 Table 2 : Case 1 Rev. Coarse Data________________________________________________ 4 Table 3 : Case 1 All Load Cases_________________________________________________ A0 Table 4 : Case 1 Rev Car Load Case ______________________________________________A1 Table 5 : Case 1 Rev Car Load Case Benefit Data At Y E.D 2.5 ___________________________ A2 Table 6 : Case 2 Rev All Load Cases Stress Data _____________________________________ A3 Table 7 : Stress Reduction Benefit Case 2 Rev at Y ED 2________________________________ A3
- 5. jitesh.bhogal@gmail.com Figure 1 : Loading Conditions I-BEAM A Study in Stress Vs. Fastener Location Consider the I-Beam (Figure 1). The Beam has a length L = 10 ft and other relevant dimension shown in Figure 2 and Table 1. For this analysis we are using a standard industrial Grade-8 1 inch bolt as a reference for the holes sizes. The Beam will be transversely loaded with limits corresponding to loads experienced in a bridge (Figure 1). The stress will be measured at the fastener locations. For the purpose of this report only the MAX stress experienced will be relevant. This report will outline the relationship between beam stress concentrations and bolt placement, specifically edge distance. Loading will be distributed to represent vehicles standing idle over the beam. We will be using Aluminum instead of steel for simplicity and to eliminate the material composition of steel as a defining factor. The purpose of this report is solely to analyze the effects of moving fasteners in the Maximum stress area under distributed loading conditions. INTRODUCTION Bridges are an important part of interconnecting roads across the world. I-Beams are widely used in the construction of bridges for their ability to resist shear forces. We will be analyzing the effect of fastener location on stress around the fastener holes on a clamped beam clamped at the fastener holes. There will be 3 Load cases: Car weight (approx. 5000 Lbs), Empty Truck (18000 Lbs), and Max Loaded Truck (30000 Lbs). A single beam will never experience the entire load from vehicles passing over the bridge, as such for simplicity we will load the beam with 50% (Gross over estimation of the loading but will reduces computational complexity) of the vehicle weight and constraint it on the edges at fastener holes. The Isotropic properties of aluminum are as follows: E=1.015e+007psi Poisson Ratio = 0.346 Density=0.098lb_in3 Yield Strength =13778psi
- 6. jitesh.bhogal@gmail.com 2 The beam cross-Section is shown in Figure 2 and corresponding values can be found in Table 1. This is a typical I-Beam as per ATSM A6 standard METHODS Using finite element analysis we capture stress data for various loading cases and are able to capture the behavior of our model over a wide array of varying Edge Distances. Tetrahedron Element Below is the element which will be used to model the I-Beam. The I-Beam will be broken down by the Meshing tool into several Tet-Elements (Figure 3). This is done to approximate results within a certain confidence interval of the actual solution. Reducing the element volume close to ZERO will produce more accurate results however; doing so will also significantly increase compute time and restrict large or very fine parameter sweeps. Figure 3 : Tetrahedron Element Figure 4 shows the Von Misses stress for a coarse mesh at Initial Conditions under Max loading. Here the Mesh is restricted to an element length of 200 mm globally. As it can be seen the coarse mesh yields in high spikes in stress concentration around the Clamped Fastener holes as shown in Figure 5. To yield better results we have made local refinements to the mesh scheme around the fastener holes and the fillet Radius. Here we only compare the effects of meshing on results from one case which will determine the meshing scheme that will be used for the remainder of the Report. Figure 6 and 7 show the Stress using the refined Mesh. Figure 4 : Coarse Mesh Sample Stress Plot Figure 5 : Coarse Mesh Stress Peak D (in) Tw (in) Bf (in) Tf (in) R (in) 12 0.35 5 .35 .25 Table 1 : I-Beam Dimensions Figure 2 : I-Beam Cross Section
- 7. jitesh.bhogal@gmail.com Figure 6 : Mesh Refinement at Critical Features Figure 7 : Mesh Refinement Max Stress Plot The mesh refinement constitutes subjecting all fastener holes and the fillet radiuses to a max element length of 5mm. Note that Max Von Misses stress contour is now much smoother and contains no highly concentrated stress area spikes; particularly around the fastener holes. Further refinement did not yield more accurate results and increased the computing time of the solutions. Edge Distance (ED) is defined as the distance from the edge of a feature (Flat or Plane surface) to the center of the faster hole. An ED of 1 corresponds to the center of the fastener hole being 1 Diameter length away from the Edge of the feature. In this Report we will analyze the effects of varying ED on stress concentration around the Clamping holes on the I-Beam. The resulting max stress will be plotted as a function of ED to determine if a relationship exists between ED and Max stress experienced. Further we will also analyze the change in Max stress with each step to determine if an optimal range exists beyond which increasing ED yields negligible gains. Case 1: 2 Fasteners Using a single column of two fasteners we will step the ED by constant increments. I hypothesize that the max stress should reduce as ED increases; however the beam should fail under Truck loading conditions. Figures 8 & 9 show the start and stop locations of Case 1. Figure 8 : Case 1 Start Location Figure 9 : Case 1 End Location Case 2: 3 Fasteners Using a single row of three fasteners we will step the ED by constant increments. The max value is derived from the physical limitations of installing fasteners; the bolt head size and the tool size must be accounted for during installation. As such fasteners closer than 1.5D cannot physically be installed. I hypothesize that the beam will be able to take a greater load than in Case 1. Start location for case 2 is similar to that shown in Figure 8 with the addition of a third fastener hole in the center of the two existing holes. Figure 10 shows the final position of case 2.
- 8. jitesh.bhogal@gmail.com 4 Figure 10 : Case 2 Fasteners Results Case 1 As expected the Beam experiences failure under the truck load cases. However contrary to expected results, the max stress increased with increasing ED. Figure 11 shows the Max stress for all the Load cases. Figure 11 : Case 1 Results Plot Case 1 Revision To investigate this behavior a secondary set of measurements were made using the following changes. The initial fastener locations are the same as Figure 8 however with each step along the length of the beam we now measure the stress and deflection stepwise along the depth of the beam. In other words the fastener location will NOT change diagonally each step but will be varied in one direction at a time as shows in Figure 12. Figure 12 : Case 1 Adjusted Path Table 2 shows the coarse results from this new step method. NOTE all following measurements will now be taken according to the revised methodology. The Columns represent the ED in the Y (along the Depth) direction while the Rows represent the ED in the X (along the Length) direction. The Stress values are in psi. ED Y direction X 2 2.5 3 3.5 2.5 5.74E+03 6.34E+03 7.50E+03 8.31E+03 3 5.62E+03 6.09E+03 6.84E+03 8.84E+03 3.5 5.21E+03 5.92E+03 6.52E+03 7.83E+03 4.2 4.94E+03 6.45E+03 6.80E+03 7.30E+03 Table 2 : Case 1 Rev. Coarse Data Here it can be seen that the stress behaves as predicted. Keeping the ED along the beam depth constant and increasing the ED along the beam length reduces the overall strain experienced by the fastener locations. More importantly the stress seems to Dip around the highlighted area in the table before rising again. To investigate this nonlinear behavior a more in-depth analysis is performed and detailed in Figure 13. From the trend of the gathered data it can be seen that there seem to be points of nonlinear behavior. These data points are taken using the Car loading case. The cause of these sudden peaks is not clear and requires an in depth analysis of the system architecture to determine whether the peaks are true stress concentration points or an artifact inherent within the software. Such an analysis is out of the scope of this report.
- 9. jitesh.bhogal@gmail.com Overall it can be seen from figure 3 that moving fastener holes towards the center of the web increases the stress concentration at the fastener location. Additionally the data seems to indicate that stresses behave linearly in relation of Edge Distance. This behavior is parallel to reducing the effective length of the beam to increase its ability to take on more loading. In industrial design the beam dimensions would be altered more significantly instead of increasing the Edge Distance. Case 2: As can be seen from Figure 15 the third hole alleviates some of the stress and allows for a stronger fixture. Table 6 in the appendix shows the data plotted in Figure 15. From the graph it can be seen that an increase of ED from the Flanges of the I-Beam continue to result in a net increase of MAX stress experience around the fastener locations. Similarly an increase in ED along the length of the beam reduces the max stress experience. Note that the beam appears to behave non linearly under Load Case 2 and Load Case 3. If we were to continue accumulating data points the limit of the minimum stress should correspond to an equivalent of clamping the beam at the midpoint length. Thus this provides support to the assumption that beyond a certain ED along the beam gains are negligible. Figure 14 : Case 2 Rev Sample Stress Plot Figure 15 : Case 2 Rev Stress Plot Figure 13: Case 1 Rev Car Load Plot
- 10. jitesh.bhogal@gmail.com 6 ANALYSIS The primary purpose of this report is to determine the effects of fastener location on an I-Beam. Doing so allows us to quantify the benefits of increasing ED. In Case 2 we saw an exaggerated non-linear behavior of the beam when experiencing failure under loading conditions of Case 2 and Case 3. Along the Depth of the Beam As we learned, increasing ED in the vertical direction, along the depth of the beam, results in a significantly positive and seemingly linear increase in stress concentrations. Therefore the optimal position along the depth of the beam for fastener locations is at the minimum required edge distance from features which will not restrict installation or assembly of the beam onto the final structure. In the case of multiple fasteners such as in Case 2 the designer should maintain symmetry along the fastener axis for optimum results. Along the Length of the Beam Increasing ED in along the length of the I-Beam results in a somewhat linear reduction of the stress. Although there is some interesting non- linear behavior (stress spikes) which occurs at varying locations this behavior is out of scope for the purpose of the report. The overall linear behavior is parallel to reducing the effective length of the beam. From a design point of view choosing an ED beyond 3d should trigger a re- design of the beam and result in a shortening of the length or other dimensional alterations to accommodate for the stress concentrations. Figure 16 and Figure 17 show graphs which show the benefit (a reduction of MAX stress concentrations on the fasteners) of increasing the ED along the beam with two fasteners (Figure 16) and three fasteners (Figure 17). Tabulated data for these graphs can be found in the Appendix. The important takeaway from these graphs is that the largest reduction occurs between 1ED and 2 ED after which the data seems to hover around zero. The increases near 3.5D are most likely due to inherent factors of the software which can be classified as outliers temporarily. A more detailed parameter sweep was not done due to time constraints. Figure 16 : Stress Reduction Benefits Case 1 As expected the stress at 0.75D are significantly higher than at higher ED’s. However with each step the reduction in Max stress experienced is reduced and seems to approach the zero. As such when designing the connections for an I-Beam or any similar beam which experiences transverse loading the designer should choose an ED between 1.5D-2.5D. Figure 17 : Stress Reduction Benefit Case 2
- 11. jitesh.bhogal@gmail.com CONCLUSION It can be concluded from the benefits graph in figures 16 and 17 that the stress reduction of the beam with respect to ED along the length of the beam is in-fact non-linear. This result is not entirely surprising as at the mid-point of the beam an increase in ED would effectively result in a decrease. The reduction of max stress is either linear with a minimum value at the mid- point or parabolic with a minimum at the mid- point. Our data supports the latter which suggests that gains from increasing ED approach as we approach the mid-point of the beam. As such fasteners should be placed within approximately 3D of the edge. To this there are additional constraints such as the connecting element. This element is usually a flat plate or an L-Flange. Although outside the scope of the report these connecting elements also have an optimal range. For example: the L- Flange. Using an ED greater than 3D would require a second column of fasteners to be placed at a smaller ED to prevent creating an enormous moment arm on the L-Flange. In conclusion this report analyzed the effects of Edge Distance on stress concentrations at fastener locations on an I-Beam. Through the analysis presented it can be concluded that there exists an optimal range of Edge Distances at which fasteners should be placed. If the Max stress experienced remains beyond the design threshold the designer should re-evaluate the beam design and redesign the Beam to accommodate the stress concentrations.
- 12. jitesh.bhogal@gmail.com 8 This Page is Intentionally Left Blank
- 14. APPENDIX Load 15000 9000 2500 Ed(X=Y) Stress (psi) Deflection (mm) Stress (psi) Deflection (mm) Stress (psi) Deflection (mm) 1 3.64E+04 1.74 2.18E+04 1.04 6.06E+03 0.29 1.1 1.2 3.68E+04 1.76 2.21E+04 1.06 6.14E+03 0.294 1.3 1.4 3.71E+04 1.81 2.23E+04 1.09 6.19E+03 0.302 1.5 1.6 3.71E+04 1.84 2.22E+04 1.1 6.18E+03 0.306 1.7 1.8 3.96E+04 1.88 2.40E+04 1.13 6.67E+03 0.313 1.9 2 3.75E+04 1.92 2.25E+04 1.15 6.25E+03 0.32 2.1 2.2 3.83E+04 1.96 2.30E+04 1.18 6.38E+03 0.327 2.3 2.4 4.25E+04 2.01 2.55E+04 1.21 7.09E+03 0.335 2.5 2.6 3.86E+04 2.05 2.32E+04 1.23 6.43E+03 0.342 2.7 2.8 4.14E+04 2.12 2.49E+04 1.27 6.91E+03 3.53 2.9 3 4.10E+04 2.18 2.46E+04 1.31 6.84E+03 0.363 3.1 3.2 4.23E+04 2.23 2.54E+04 1.34 7.04E+03 0.372 3.3 3.4 4.40E+04 2.31 2.64E+04 1.39 7.33E+03 0.385 3.5 3.6 4.28E+04 2.39 2.57E+04 1.43 7.13E+03 0.398 3.7 3.8 4.61E+04 2.48 2.77E+04 1.49 7.68E+03 0.414 3.9 4 5.04E+04 2.59 3.02E+04 1.55 8.40E+03 0.431 4.1 4.2 4.70E+04 2.66 2.82E+04 1.6 7.84E+03 0.444 4.3 4.89E+04 2.74 2.93E+04 1.64 8.15E+03 0.456 Table 3 : Case 1 All Load Cases
- 15. jitesh.bhogal@gmail.com Y direction CAR LC 2 2.25 2.5 2.75 3 3.25 3.5 3.75 x Dir Stress (psi) Stress (psi) Stress (psi) Stress (psi) Stress (psi) Stress (psi) Stress (psi) Stress (psi) 0.75 8.43E+03 1 8.42E+03 1.25 8.24E+03 1.5 7.70E+03 1.75 7.06E+03 2 6.81E+03 2.25 6.53E+03 2.5 5.74E+03 6.34E+03 7.50E+03 8.31E+03 2.75 5.65E+03 6.14E+03 6.16E+03 6.70E+03 7.47E+03 7.57E+03 8.14E+03 8.62E+03 3 5.62E+03 6.09E+03 6.84E+03 8.84E+03 3.25 6.07E+03 6.70E+03 7.96E+03 3.5 5.21E+03 5.92E+03 6.52E+03 7.83E+03 3.65 6.01E+03 6.50E+03 3.75 5.96E+03 6.51E+03 3.85 5.88E+03 6.66E+03 4 5.70E+03 6.31E+03 7.50E+03 4.1 5.87E+03 6.70E+03 4.15 5.94E+03 6.22E+03 4.2 4.94E+03 6.45E+03 6.80E+03 7.30E+03 4.5 5.15E+03 5.42E+03 6.10E+03 5 5.11E+03 5.36E+03 5.94E+03 6.66E+03 5.5 5.39E+03 6 5.04E+03 5.28E+03 5.88E+03 6.32E+03 7 4.61E+03 5.19E+03 5.71E+03 6.26E+03 8 4.48E+03 4.99E+03 5.42E+03 5.91E+03 Table 4 : Case 1 Rev Car Load Case
- 16. jitesh.bhogal@gmail.com 2 X start Delta Stress (psi) % Reduction 0.75 1.20E+01 0.19% 1 1.75E+02 2.76% 1.25 5.39E+02 8.50% 1.5 6.38E+02 10.06% 1.75 2.53E+02 3.99% 2 2.77E+02 4.37% 2.25 1.91E+02 3.01% 2.5 1.81E+02 2.85% 2.75 7.30E+01 1.15% 3 1.40E+01 0.22% 3.25 1.50E+02 2.37% 3.5 -8.70E+01 -1.37% 3.65 5.30E+01 0.84% 3.75 8.00E+01 1.26% 3.85 1.83E+02 2.89% 4 -1.77E+02 -2.79% 4.1 -7.00E+01 -1.10% Table 5 : Case 1 Rev Car Load Case Benefit Data At Y E.D 2.5
- 17. jitesh.bhogal@gmail.com 3 holes Loaded Truck Truck Car EdY EdY EdY Stress (psi) Edx 2 2.5 3 2 2.5 3 2 2.5 3 1 4.29E+04 4.86E+04 5.20E+04 2.57E+04 2.92E+04 3.12E+04 7.15E+03 8.10E+03 8.66E+03 1.5 2 3.72E+04 3.99E+04 4.74E+04 2.23E+04 2.39E+04 2.84E+04 6.20E+03 6.65E+03 7.90E+03 2.5 3 3.41E+04 3.70E+04 4.07E+04 2.05E+04 2.22E+04 2.44E+04 5.69E+03 6.17E+03 6.78E+03 3.5 4 3.10E+04 3.43E+04 3.78E+04 1.86E+04 2.06E+04 2.27E+04 5.17E+03 5.71E+03 6.30E+03 Table 6 : Case 2 Rev All Load Cases Stress Data Edx car Truck loaded Tuck 1 20.41% 13.18% 13.20% 1.5 6.69% 7.19% 7.18% 2 6.41% 7.27% 7.28% Table 7 : Stress Reduction Benefit Case 2 Rev at Y ED 2