1

A Finite Element Study of the Effect of Geometric Dimensioning and Tolerancing on Bolt Design Stresses in Torque Carrying Bolted FlangesbyCarney AndersonAn Engineering Project Submitted to the GraduateFaculty of Rensselaer Polytechnic Institutein Partial Fulfillment of theRequirements for the degree ofMASTER OF ENGINEERING IN MECHANICAL ENGINEERING

Approved:

_________________________________________Ernesto Gutierrez-Miravete, Project Adviser

Rensselaer Polytechnic InstituteHartford, CTApril, 2012(For Graduation May, 2012)

CONTENTSLIST OF TABLESiiiLIST OF FIGURESivABSTRACTv1.- INTRODUCTION12.- METHODOLOGY32.1- GEOMETRY32.2- FEA MODELING62.3- MATERIALS103. - RESULTS AND DISCUSSION113.1 - ASSEMBLY BEHAVIOR113.2 - BOLT STRESS/LOADS123.3 - VALIDATION184.- CONCLUSIONS20REFERENCES22

LIST OF TABLESTableDescriptionPage1Typical Mechanical Properties for Inco 718102Stress/Load Results Summary123Comparison of Fine vs. Coarse Mesh Results19LIST OF FIGURESFigureDescriptionPage1Torque Carrying Bolted Shaft Assembly223D CAD Assembly33Tolerance Conditions4-54Assembly Cross Section/Boundary Conditions75FEA Component Mesh76Gap and Penetration in FEA ContactModeling97Load Step Geometry Progression108Loaded Flange Deflection Results11-129Bolt Shank Stress Plots14-1510Tolerance/Stress Sensitivity Plots16-1711Fine/Coarse Mesh Comparison1812Coarse Model Loaded Flange Deflection Results19

ABSTRACTThis paper reports on a Finite Element (FE) modeling study investigating the contributions of tolerances towards the limiting bolt stresses in torque carrying bolted flanges. For a given flange layout, with set boundary conditions, the peak stresses experienced by the bolts are directly related to the dimensional variations permitted during the manufacturing of the flange components. By comparing stress results from Finite Element Analysis (FEA) models of flange assemblies both in the nominal condition, and with dimensional error imposed, the Geometric and Dimensional Tolerancing (GD&T) controls which stresses are most dependent on may be identified. Furthermore, by running models with varying magnitudes of dimensional error, a trend curve may be built to better understand this sensitivity. This allows a designer to better predict the actual worst case stresses a given design may experience, and better balance tight tolerance requirements with nominal configuration changes. A parts price is directly tied to the tolerances applied to it, therefore this could reduce a parts manufacturing cost significantly, while better utilizing the design materials. A list of key tolerances will be identified, and a stress vs. tolerance curve for each (based on a given flange configuration) will be provided. Critical design stresses will be predicted via 3D FEA models of the bolted flange assemblies.

ii

1.- INTRODUCTIONBolted joints are utilized to transmit torque in a number of high torque applications like automotive drive trains and jet engine rotor stacks. In these applications it is important to be able to predict bolt stresses. Although there are a number of studies using the Finite Element Method (FEM) on the various methods available for modeling bolted flanges (see e.g. [1]), few consider the effects of tolerance on the bolt stresses. In order to reduce part cost, a designer may decide to increase the Geometric Dimensioning and Tolerancing (GD&T) allowances for their flange holes. However, without understanding the GD&T contribution to stress, the designer may inadvertently inflict a durability shortfall onto the part. This study outlines a method by which to quantify the increases in bolt stresses from variations in GD&T allowances. Improving a designers understanding of GD&T on part stresses should allow to better balance design constraints like cost, producibility, and durability.

The objective of the study is to predict the level of sensitivity of bolt design stresses to the principal tolerances for torque carrying bolted flanges. This will be accomplished by producing trend curves of tolerance vs. stress. The intent being that with this information, flanges may be more accurately designed to the required strength capability, and without unnecessarily adding cost. This trend info will be summarized by stress vs. tolerance curves for a typical flange configuration. Bolt hole diameter and bolt hole true position are the tolerances that will be assessed (bolt diameter was considered, but it was dropped as it essentially has the same effect as hole diameter, and fastener vendors typically hold tight size tolerances relative to the hole tolerances held by shaft or disk manufacturers), since they are the major cost drivers when producing bolt holes. Figure 1 shows a 3D Computer Aided Drafting (CAD) representation of a torque carrying bolted flange. There are 2 discreet torque carrying shafts connected in the middle by a piloted flange, and a plurality of bolts. The bolts both hold the shafts together axially, and are the mechanism by which torque is transmitted between the 2 shafts.

22

Figure 1: Torque carrying bolted shaft assembly (part configuration and loading scenario to be analyzed via 3D FEA in this study)2.- METHODOLOGYIn order to run this study, a number of 3D FEA models were built for a series of slightly varying bolted flange/shaft assemblies. The 3D geometry was built in the Computer Aided Drafting (CAD) program Unigraphics, and the FEA modeling and post processing was done in Ansys. The models were all run at room temperature, and all the parts were modeled as nickel alloy. The following subchapters provide the specifics of, and the reasoning behind, the geometry of the different cases, the mesh and boundary conditions applied, and the material properties that were used.

2.1- GEOMETRY

The geometry was modeled using the FEM, with the interfaces modeled as contacts (see figure 4 for an axisymmetric representation). All the models consist of two short shafts with their flanges snapped together, and bound axially by pre-loaded bolt members.

Figure 2: 3D Cad model of the bolted flange assembly.

In order to populate the trend curves, 9 separate FE models were constructed, a baseline case, 4 cases with varying hole size tolerance (with geometric deviations ranging from 0.001 to 0.004), and 4 cases with varying hole true position tolerance (with geometric deviations ranging from 0.001 to 0.004). For all conditions, the bolts were modeled with nominal geometry. For the baseline condition, both flanges were modeled in a nominal condition (the hole diameters were consistent at 0.251, and were equally spaced circumferentially). For the tolerance cases, shaft 1 was left at nominal, and the dimensional deviations from nominal geometry were applied to the holes of shaft 2. The application of tolerances was done in a worst case fashion; for bolt hole diameter (figure 3a), a single hole was modeled at the minimum diameter, while the remainder of the holes in that flange had maximum diameter holes; for true position (figure 3b), one hole in the shaft 2 flange was offset the full tolerance circumferentially (see figure 3 for a picture of the tolerance conditions).

Figure 3a: Hole size tolerance condition

Figure 3b: True position tolerance condition

Both of these tolerance conditions cause one bolts shank to contact its bolt hole prior to the others, therefore forcing that bolt to bear the torque load until it deflects enough that the remaining bolts shanks come into contact with their associated flange holes. As a result, the toleranced holes bolt carries more torque than the remaining bolts, which is what drives the relatively high stress into said bolt (these trends can be seen in the load/stress results in table 2). The sliding contacts allow for the hole edges to contact the bolt shank sections, and accurately represent the compressive contact that occurs in real parts. Maximum principal stress, and maximum bearing stress (via minimum principal stress on the surface of the bolt shank) were checked, as they are the typical stresses that limit part life. Von-Mises stress was checked, in order to understand the likely extent of plasticity in the bolt. The nominal flange model was built to baseline the stresses, and to verify that the model functions properly and produces reasonable results. Next, subsequent models with discrete geometry variations were run in order to evaluate tolerance sensitivity. A range of models (see table 2) were run in order to gather data to populate the trend curves. Since the stress results from the models are being compared, it is important to make sure that the mesh is approximately the same for all. Specifically, the mesh density at the contact locations and high stress locations needs to match. Variations in mesh density will drive variations in stress, which can be seen in the results shown in table 3.

2.2- FEA MODELINGThe FEA analysis was performed in Ansys. All parts are meshed entirely with 8 noded bricks (solid 45 in Ansys). The membrane section of one flange is fixed from moving circumferentially, axially and radially, while the second flanges membrane section has tangentially oriented loads applied to it in order to apply the torque. A full 360 degree model was built instead of opting for a simpler partial model with symmetry constraints applied for two main reasons: a) The symmetry constraints would artificially hold the 2 flange parts on center, instead of allowing the 2 parts to re-center themselves to shifted bolt circle as it would in reality b) It would force multiple holes to be out of tolerance based on number of sectors specified (i.e. if a sector was used, the symmetric geometry assumption would be that 4 bolts were at the tolerance condition), and this is an attempt to look at a worst case condition. The geometry was modeled in Unigraphics (see figure 2), and then imported into Ansys. A cross section of the assembly, and the applied boundary conditions and interfaces can be seen in figure 4. Figure 5 shows the mesh of the various components; in the assembly view, you can see that the mesh density in the bolt is much finer than that in the most of the flange body, since the study is primarily concerned with the bolt results. The picture of the hole mesh shows the relatively fine and uniform mesh around the bolt holes. This was created by dividing out the volume around the hole, and sweeping the mesh from front to back. It is important to have good mesh density in this portion of the flange in order to ensure good results for the contact solutions. The bolt mesh view shows the uniform bolt mesh that was created by sweeping axially down the center of the bolt, and circumferentially around the outside of the bolt.

Figure 4: Diagram showing where boundary conditions and contacts are applied on a cross sectional view of the bolted flange assembly.

Figure 5: FEA MeshThese FEA models are complex and difficult to solve for a number of reasons. The first being the large quantity of moving parts included in the model. After the assembly load case is solved, in which the applied interference between the bolt head (see figure 7a) and one of the flange faces is resolved (stretching the bolt, and compressing the flange material under the bolt head, see figure 7b) a bolt pre-load is achieved; then a new time point is initiated where the torque is applied. As stated earlier, the free end of shaft 1 is locked in position, and a circumferential load is applied to the other end. In order to resolve the torque step, shaft 2 (the loaded shaft) must spin about the diametral snap fit until the bolt hole barrel faces contact the individual bolt shanks (see figures 7c, and 7d). Next, the loaded flange forces the bolts to bend within the fixed flange until the shank contacts the barrels of those holes. Finally, once these contacts are all established, and somewhat settled, the system can start resolving the load balance in a semi-linear manner. The contacts themselves significantly increase the difficulty of solving the model. Contact models are inherently non-linear since they essentially change boundary conditions depending on the amount of deflection within the system. Once contact is established, there is all of a sudden another constraint on the model which didnt exist in the previous solve iteration step. In this analysis, this complexity is exacerbated by the sheer number of contacts that need to be resolved (each bolt shank with 2 separate flange holes, each bolt and nut washer face with the flange faces, and finally the flanges to each other). The original intent was to use surface to surface contact at all of the interfaces, but the model was too unstable, and would not solve. For the sake of solving, the interface between shaft 1 (the fixed shaft) and the bolt heads on that side were modeled as bonded contacts. Bonded contact is created first by creating node pairs (one from each part associated with the interface) where said nodes are aligned within a specified tolerance, then by coupling the deflections of the two nodes in each set, in all degrees of freedom (DOFs). Its somewhat unrealistic, since it does not allow any sliding between the two surfaces. In order to keep this constraint from affecting the results, the stress values and load summations were all pulled from the free half (shaft #2 side) of the bolts.

Modeling contacts starts by identifying likely contact locations between parts (see figure 6). During the solving algorithm, at each step, the gap between the established locations is quantified. If at any point this gap comes back as negative, contact has been established, and the resultant forces between the two contacting bodies are added to the equilibrium equations. The local contact forces are driven by the deformation of the mesh required to resolve the identified penetration. The new applied load (essentially an added boundary condition) complicates equilibrium resolution, especially because the level of penetration and contacting area change with each solution iteration. Contact may be modeled node to node, node to surface, or surface to surface. In this case we are using surface to surface due to the relatively large displacements of the bolts. In the case of surface to surface contacts, 2D surface elements are modeled on each side of the areas identified as possible contact locations. The Gauss points (located along the element edges) are used to calculate the gap between the two surfaces.

Figure 6: Gap and Penetration measurements of meshed bodies in near-contact and contact, respectively, for surface to surface contact (references 1 and 2).

Figure 7: a) Shows the initial step, where the bolt head interferes with the flange b) shows the geometry after said initial fit has been resolved; c) shows a bolt shank relative to a corresponding hole in flange 2 prior to torque application, while d) shows where the bolt shank ends up within the bolt hole after the torque load has been resolved

2.3- MATERIALS

Inconel 718 (see table 1 for properties) was used for all parts, since it is an extremely common material in moderate temperature, torque loaded applications.

Table 1: Typical Mechanical Properties for Precip. Hardened Inco 718 (ref 4)

3. - RESULTS AND DISCUSSIONThe following section provides an overview of the loaded flange behavior, detailed bolt stress results and kts, and the method/results for the validation of the FEA practices used.

3.1 - ASSEMBLY BEHAVIORThe deflection results are as expected, with uniform hoop deflection (Uy of RSYS 5, in Ansys), about the circumference, of the bolts and flanges in the baseline case (displacement increases with radius since the angular displacement of the flange is constant), and skewed displacement of the bolt and flange section near the tolerance hole in the tolerance cases (see figure 8). The head of the bolt in the toleranced hole is deflecting significantly more than that of the rest of the bolts, indicating that it is undergoing greater shear load (as was intended). Alternatively, the flange section adjacent to the tolerance hole is deflecting less, since its small hole contacts the bolt shank prior to that of the other holes.

Figure 8a: Hoop deflection plot, in inches, of the free flange and the free side bolt heads for the baseline case

Figure 8b: Hoop deflection plot, in inches, of the free flange and the free side bolt heads for the hole +.002 case3.2 - BOLT STRESS/LOADSThe resultant total torque loading (checked by querying the reaction forces on the nodes of shaft 1 that were locked into position) matches within 0.1% of the intended applied loading. The baseline case also shows reasonable stress results (see table 2), some stresses are a bit high, but this is likely due to the relatively coarse mesh in the bolt head to shank fillet, and the fact that it is not unlikely to yield bolts slightly from pre-load alone.

Table 2: Test Case array, and corresponding stress/load results.

This highlights one of the difficulties in modeling edge of contact bearing stress with FEA; since the effective contact area is extremely small when 2 round surfaces contact, elastic stresses spike significantly (see bearing stress plot in figure 9a). In the real world, a small area of each part would yield until the contact area was large enough to reduce the stresses to elastic levels. A better result may be obtained by using elastic-plastic material models, and extremely fine mesh at the high stress locations. However, this would significantly increase the complexity of the models (due to added non-linearity of the material models, and the increase in model size associated with mesh refinement). Figure 9a shows that the max bearing stress occurs at the center of the bolt shank, where the two flanges contact. The nodal reaction load vectors, shown in figure 9c, align with the max bearing stress results, showing the nodal reaction loads peaking at the axial center of the bolt, and falling off towards the ends. The highest tensile loads (illustrated in figures 9b and 9d) are seen to occur in the bolt head to shank fillet, which is not surprising since there is a significant stress concentration associated with a small fillet, and the bending that the bolt head induces. For the purpose of studying the stress trends, it is sufficient to look at the relative stresses between cases in spite of the fact that they are over yield, as long as the modeling method remains consistent. Additionally, it should be understood that an elastically predicted stress greater than the tensile strength for a given material, does not necessarily indicate a likely failure. Elastically predicted bending stresses appear artificially high when they reach the plastic regime, because they continue to grow linearly, while in reality the material would relax, and the actual stress would increase much slower. The shear load summation on the highly loaded bolt also provides a clear picture of how much additional load the bolt is taking. This load alone could be used to assess how much additional stress that bolt is experiencing via hand calculations. It is important to note that the intent of this study is not to create stress bounds for design, but instead to elaborate on the relative effects of component stress and tolerance allowance. These relationships are illustrated best in the trend curves shown in figures 10a and 10b, which show the estimated stress increases relative to varying GD&T error. The horizontal axes represent deviation from nominal shape in inches, while the vertical axes show the corresponding percent increase in bolt stress or load over that of the nominal configuration. The origins of the plots represent the nominal case, and 4 data points of increasing dimensional deviation were used to build the trend lines. The trend lines were created using 5th order polynomials in excel.

Figure 9a: Bearing Stress (Min Principal Stress, in psi) plot of the bolt shank section 2 surface elements for the baseline stress case.

Figure 9b: Max Running Tensile Stress (Max Principal Stress, in psi) plot of the bolt shank section 2 surface elements for the baseline stress case.

Figure 9c: Bearing stress (psi) plot with nodal reaction loads on the bolt shank sections 1 and 2 surface elements for the baseline stress case.

Figure 9d: Max Von-Mises stress (psi) plot of the bolt shank section 2 surface elements for the baseline stress case.

Figure 10a: Trend curves for critical stresses/loads vs. hole tolerance

Figure 10b: Trend curves for critical stresses vs. true position tolerance

3.3 - VALIDATIONIn order to investigate the sensitivity of the results to the mesh density, a side study was performed. The bolt mesh was coarsened significantly (0.05 bolt shank elements in the coarse case vs. 0.025 elements in the standard case, see figure 11) in the baseline case and the 0.004 true position case, and the models were run and post-processed as before.

Figure 11: Mesh comparison of the finely meshed bolt (used for the primary models) and the coarsely meshed bolt (used for the mesh sensitivity study models).

The deflection results of the coarse models were slightly greater than that of the fine models. This was to be expected because contact penetration sensitivity goes down with contact surface mesh density, since theres significantly fewer Gauss points. However, the deflection contours are very similar (see figure 12). The stress results of the coarse models show a relatively small change in the results relative to the large change in mesh density (see table 3); only -10% to -14% delta stress changes when element size doubled.

Figure 12: Hoop deflection plot, in inches, of the free flange and the free side bolt heads for the baseline-coarse case.

Table 3: Results for the Mesh Density Sensitivity Testing.

4.- CONCLUSIONSThe trend curves (see figures 10a and 10b) show that both hole size and true position tolerance have significant influences on the bolt limiting stresses. The results shown are worst case, and it should be understood that the large majority of parts machined to a given tolerance will be well within that tolerance, except in the case that the applied tolerance is far tighter than the manufacturing process capability. It is even more unlikely that the distribution of tolerances within a feature set will end up as were modeled in this study (worst case as described in the method section). However, this does yield insight into the effect of tolerances on a given parts stresses, and it is reasonable to assume that in a mass production situation, a certain fraction of parts produced will be similar to those modeled. These curves can be used to estimate the likely worst case stresses that may be seen in a design with a given nominal stress and bolt hole tolerance allowances.

Bolt hole tolerances are shown to have a slightly larger effect on stress than true position (see relative kts between trend curves in figure 10a vs 10b). Fortunately, hole size is likely more easily controlled than hole position; as hole size is largely a function of the cutting/drilling tool size, and position is a function of many smaller components. Based on these two factors it could be concluded that a relatively cheap and robust design may be achieved by holding hole size to a relatively tight tolerance, and true position to something less so.

Some factors that affect the results of a study like this are material properties, flange geometry, and flange loading. A material with a low Youngs modulus will be softer, and will likely result in better load sharing between bolts with all else being equal. Nominal bolt hole size relative to the bolt will likely have a large affect on both bearing stress (based on Hertzian contact theory) and tensile stresses in the bolt (since a larger bolt hole will put more bending into the head of the bolt for a given bolt size and pre-load). If the flange torque loading is not significant enough to deflect the bolts until they share load, then the stress spikes will be much higher (the entire torque load may be taken by one bolt alone). It is therefore important to understand the differences between an actual design, and a study model like the one used here.REFERENCES1. Finite element analysis and modeling of structure with bolted joints by Jeong Kim, Joo-Cheol Yoon, and Beom-Soo Kang; Applied Mathematical Modelling 31 (2007) 8959112. The Finite Element Method for Solid and Structural Mechanics: Sixth edition, O.C. Zienkiewicz, R.L. Taylor; First published in 1967 by McGraw-Hill, Fifth edition published by Butterworth-Heinemann 2000, Reprinted 2002, Sixth edition 20053. Ansys Users Manual: Contact Technology Guide, Ansys, Inc; Release 12.0 April 2009: http://www1.ansys.com/customer/content/documentation/120/ans_ctec.pdf4. Inconel Alloy 718 General Information provided by Special Metals http://www.specialmetals.com/documents/Inconel%20alloy%20718.pdf

Hole MeshBolt Mesh

Assembly Cut-Away View

Fine Bolt Mesh

Coarse Bolt Mesh