1. A CONTINUUM MECHANICS APPROACH OF DERIVING STRESS TENSOR COMPONENTS OF DOUBLE SHEAR-PLANE REVOLUTE JOINTS IN THE ELASTIC DOMAIN CHRISTOPHER STUBBS A THESIS SUBMITTED TO THE GRADUATE FACULTY OF RENSSELAER POLYTECHNIC INSTITUTE IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MECHANICAL ENGINEERING

2. INTRODUCTION The current engineering community uses finite element analyses in the development and design of double shear-plane clevis connections. Although accurate, finite element analyses of this nature are: Non-linear in nature Time consuming Computationally intensive. This thesis presents an approach for sizing frictionless double shear- plane clevis connections to be under their material yield strengths for their given application. Simple empirical formulae No specialized personnel required to develop finite element models, Shorter analysis time Shorter iteration time to understand what dimensions and variables are most critical for a given clevis system.

3. BACKGROUND A clevis connection is defined herein as a revolute joint, comprising of two of more lugs and a single pin. A double shear-plane clevis connection is defined as a clevis connection with three lugs, and therefore two shear planes. Widespread applications in mechanical systems, ranging from submarine applications to house-hold door hinges. Although both single- and multi-shear-plane clevis joints exist, double shear-plane joints are by far the most common in practical applications (balance of robustness/complexity). –ability and symmetry –simplicity of fabrication, installation, and maintenance.

4. HISTORY Cozzone & Melcon (1950): first to be able to correlate empirical data, destructive testing, and close-form formulae for predicting failure in the lug Maddux (1953): evaluated both single shear and multi shear lug designs, for use in designing components to be under their ultimate strength Rao (1978): evaluated the non-linearities of an initial clearance, and estimated the stresses in the pin, plate, and contact interface Wearing (1985): used finite element analysis to transition from a pin within an infinite plate, to a pin within a clevis lug joint Stenman (2008): re-evaluated Maddux’s conclusions for contact pressure To (2008): evaluated conforming bushing designs Antoni (2010): evaluated non-conforming bushing desings Strozzi (2011): further investigated initial clearance Kwon (2013): further investigated Maddux’s and Stenman’s findings on contact pressure Leveraging off FEA

5. PROCEDURE Step 1 Perform a convergence study on a finite element model to ensure accuracy and validity of analysis Step 2 Perform a carefully designed suite of analyses varying the parameters to be used in the closed-form solution – one parameter at a time Step 3 Perform regression analysis on the computed results and derive equations using best fit curves to the computed data Step 4 Verify solutions by comparing closed-form solutions with an additional analysis

6. STEP 1: CONVERGENCE STUDY A plot is produced of mesh density vs. stress, and then is normalized to the most dense mesh result. von Mises stresses of the pin are within 10% accuracy with a mesh-radius ratio of 1, and within 5% accuracy with a mesh-radius ratio of 2 The von Mises stresses of the lug are within 10% accuracy with a mesh-radius ratio of 6, and within 5% accuracy with a mesh-radius ratio of 8 Therefore, a mesh-radius ratio of 8 is maintained throughout the study.

7. PROCEDURE Step 1 Perform a convergence study on a finite element model to ensure accuracy and validity of analysis Step 2 Perform a carefully designed suite of analyses varying the parameters to be used in the closed-form solution – one parameter at a time Step 3 Perform regression analysis on the computed results and derive equations using best fit curves to the computed data Step 4 Verify solutions by comparing closed-form solutions with an additional analysis

8. STEP 2: MODEL DEVELOPMENT The finite element model consists of: One-quarter pin One-half outer lug One-quarter middle lug All elements are 3- dimensional 20-noded hexahedral reduced integration continuum elements, denoted in ABAQUS as C3D20R.

9. BOUNDARY CONDITIONS Symmetry boundary conditions are applied at mid-length of the inner lug and the mid-length of the pin 0-180 azimuth plane of the pin and both lugs Fixed at bottom of outer lug

10. LOAD AND INTERACTIONS Load is as a anti-pressure at the top of the inner lug One contact interaction constraint is imposed in the model, between the pin and the lugs 3D surface smoothing Instances are places in initial full- closure Normal contact: classical Lagrange multiplier method standard pressure-overclosure relationship. Tangential contact: zero-penalty (frictionless / full-slip condition)

11. PARAMETRIC ANALYSES ParametricUnits Minimu m Value Maximu m Value Increment Value Analyses Performed Loadlbf400400040010 Lug Widthin0.520.257 Lug Gapin00.90.110 Young's Modulus of Pin psi 1.00E+0 7 6.00E+071.00E+076 Poisson's Ratio of Pin -0.050.550.059

12. EXAMPLE OUTPUT

13. PROCEDURE Step 1 Perform a convergence study on a finite element model to ensure accuracy and validity of analysis Step 2 Perform a carefully designed suite of analyses varying the parameters to be used in the closed-form solution – one parameter at a time Step 3 Perform regression analysis on the computed results and derive equations using best fit curves to the computed data Step 4 Verify solutions by comparing closed-form solutions with an additional analysis

14. REGRESSION ANALYSES Through the use of regression analyses, multiplication factors are determined for each variable, and a final set of closed form expressions representing the finite element results are developed. Equations are presented for von Mises stress in the lug, shear tear-out in the lug, contact pressure in the lug, von Mises stress in the pin, shear stress in the pin, and contact pressure in the pin. For each stress mode, a general equation based upon the load applied is presented and then stress multiplication factors for each variable are presented. K 1 is the multiplication factor from the axial gap between lugs K 2­ is the multiplication factor from the Young’s Moduli K 3 is the multiplication factor from the Poisson’s ratios K 4 is the multiplication factor from the lug width. The equations were derived by creating a best fit curve for the data, evaluating them at the baseline condition (0.5” lug gap, 2” lug width, and Epin/Elug = Vpin/Vlug = 1), and dividing the equation by the result of the baseline equation, i.e. normalizing each equation such that at the baseline condition, the multiplication factor is equal to 1.

15. REGRESSION ANALYSES: EXAMPLE The various analyses are post-process, and best-fit curves are created The curves are created using a polynomial function of the power needed to create a good correlation (R 2 > 0.98) The curve for load applied was set about a y-intercept of 0, as no stress exists with zero load applied The equations taken from each curve, except for the load applied curve, are then evaluated for the baseline configuration The equations are then normalized by their value at the baseline configuration. This is done to set each equation to 1.0 at the baseline configuration, such that the equations can be used as multiplication factors, or factors that affect the stress as a function of the system’s deviation from the baseline configuration.

16. REGRESSION ANALYSES: EXAMPLE

19. These equations are then combined with the curve for the applied load, such that each multiplication factor is multiplied together, and then multiplied with the load applied curve. This yields a final equation for von Mises stress in lug of:

20. PROCEDURE Step 1 Perform a convergence study on a finite element model to ensure accuracy and validity of analysis Step 2 Perform a carefully designed suite of analyses varying the parameters to be used in the closed-form solution – one parameter at a time Step 3 Perform regression analysis on the computed results and derive equations using best fit curves to the computed data Step 4 Verify solutions by comparing closed-form solutions with an additional analysis

21. VERIFICATION ANALYSES Parameter Value Verification 1Verification 2Verification 3 Load (lbf)40080080 Gap (in)0.30.50.4 Epin (psi)6.00E+074.00E+073.00E+07 Elug (psi)3.00E+072.00E+073.00E+07 v_pin0.150.30.2 v_lug0.30.20.4 Lug Width (in) 11.41.2

22. VERIFICATION ANALYSES: EXAMPLE Verification 1 showed a maximum von Mises stress in the lug of 1612 psi For that configuration, the computed multiplication factors, and computed von Mises stress is 1590 psi (1.36% Error)

23. VERIFICATION ANALYSES It is found that the average error percentage among all stress components is less than 6.8%, within the acceptable limits of accuracy Stress Component Average Error (%) Lug von Mises4.2 Lug shear tear-out3.7 Lug contact pressure5.7 Pin von Mises6.8 Pin shear1.4 Pin contact pressure5.7

24. CONCLUSIONS This thesis presented an approach for sizing frictionless double shear-plane clevis connections to be under their material yield strengths for their given application Finite element analysis was utilized to simulate testing for purposes of developing empirical formulae based on the load through the connection, lug widths, lug gaps, Young’s moduli, and Poisson ratios Regression analysis was then used to derive closed-form empirical formulae, using multiplication factors based upon each parametric evaluated A verification analysis was performed to evaluate the error of the empirical formulae, and acceptable levels of accuracy were verified.

25. FUTURE WORK This thesis laid the groundwork and approach that can be used in developing a closed-form solution for the full range of clevis connections In this thesis, a number of stress components and a suite of variables were examined. This approach can be extended to study the effects of other variations on clevis connections, such as the existence of bearings, pin clearance, pin radius, and lug outer radius In addition, stress components such as pin bending and lug hoop stress can be evaluated further using this same technique.