2008 International ANSYS Conference

Transcription

1 2008 International ANSYS Conference Using FEA Results To Determine Stress Concentration Factors W.D. Growney Engineering Specialist Eaton Aerospace, Conveyance Systems Division 2008 ANSYS, Inc. All rights reserved. 1 ANSYS, Inc. Proprietary

2 Agenda Overview of Eaton Corporation Background Stress Concentration Factors from FEA results Example: Finite Plate with a Central Hole Example: Square Shoulder With Fillet (Pure Bending) Summary 2008 ANSYS, Inc. All rights reserved. 2 ANSYS, Inc. Proprietary

3 Vocabulary A F FEA K t P σ max σ nom Area Force Finite Element Analysis Stress Concentration Factor Pressure Maximum Stress Nominal Stress 2008 ANSYS, Inc. All rights reserved. 3 ANSYS, Inc. Proprietary

4 Eaton Worldwide Founded in 1911 by J.O. Eaton World Headquarters in Cleveland, Ohio USA Customers in more than 150 countries More than 70,000 employees worldwide Chairman & CEO Alexander M. Cutler 2008 ANSYS, Inc. All rights reserved. 4 ANSYS, Inc. Proprietary

5 Eaton Business Groups Electrical Fluid Power Truck Automotive A premier diversified industrial manufacturer A global leader in: Electrical systems and components for power quality, distribution and control Fluid power systems and services for industrial, mobile and aircraft equipment Intelligent truck drivetrain systems for safety and fuel economy Automotive engine air management systems, powertrain solutions and specialty controls for performance, fuel economy and safety 2008 ANSYS, Inc. All rights reserved. 5 ANSYS, Inc. Proprietary

6 Background Stress Concentration factors (K t s) for numerous simple geometries have been determined by researchers (analytical equations) Roark and Peterson have compiled these into easy to use tables Using simple K t s for complex geometries can induce error Determining stress concentration factors (K t ) for complex geometries can be difficult and expensive If strain gages cannot be applied to the maximum stress location, remote stresses must be used to determine the peak stress value (induces error) 2008 ANSYS, Inc. All rights reserved. 6 ANSYS, Inc. Proprietary

7 Using FEA Results To Determine K t s K t = σ max / σ nom FEA results can be used to easily determine the maximum stress (σ max ) Determining the nominal stress (σ nom ) can be more difficult Knowledge of the stress gradient provides a means of determining the nominal stress (σ nom ) 2008 ANSYS, Inc. All rights reserved. 7 ANSYS, Inc. Proprietary

8 Finite Plate With A Central Hole Quarter-section Finite Element (FE) model. Example (from Roark s Formulas for Stress and Strain, 7 TH edition): A 2 = in 2 Length = in A 1 = in 2 r = in Width = in FEA Geometry (see Figure, quarter model shown): Length = in (Length (Roark) = infinite) Width = in (D (Roark) = in) r = in (r (Roark) = in) t = in A 1 = in * in = in 2 A 2 = in * in = in 2 Inputs (Area for applied load (pressure): P 1 = 320 lbf/in 2 F = 320 lbf/in 2 * in 2 = 30 lbf 2008 ANSYS, Inc. All rights reserved. 8 ANSYS, Inc. Proprietary

9 Uni-Axial Stress Gradient Finite Plate w/hole Stress Gradient 1200 σ(lbf/in 2 ) Stress Gradient Uniform Stress Field The graph depicts the stress gradient for a finite plate with a central hole and a uniform stress gradient The uniform stress gradient is the nominal stress (σ nom ) That would exist if the stress concentration were not present σ nom for a finite plate with a central hole is simply: σ nom = P end * A 1 / A 2 or σ nom = F end / A 2 σ nom = 320 lbf/in 2 * in 2 / in 2 = 480 lbf/in Distance (in), from hole free surface 2008 ANSYS, Inc. All rights reserved. 9 ANSYS, Inc. Proprietary

10 Area Under Stress Gradient Curves 1200 Finite Plate w/hole Stress Gradient σ(lbf/in 2 ) Stress Gradient Uniform Stress Field Distance (in), from hole free surface The area under the two stress gradient curves (stress concentration and uniform stress field) must be equal (equivalent energy) The area under the uniform stress field curve is: A Kt = σ nom * Distance (from hole free surface) 2008 ANSYS, Inc. All rights reserved. 10 ANSYS, Inc. Proprietary

11 Area Under Stress Gradient Curve Table 1 ANSYS Integration Operation Results (A Kt ), Plate w/hole S SX A Kt S SX A Kt S SX A Kt ANSYS Post1 (post processor) provides integration calculations via path operations A Kt (ANSYS integration) = 240 lbf/in Solve for σ nom : σ nom = A Kt / Distance = 240 lbf/in / 0.5 in = 480 lbf/in ANSYS, Inc. All rights reserved. 11 ANSYS, Inc. Proprietary

12 Calculating The K t (Finite Plate w/hole) The Example from Roark s Formulas for Stress and Strain, 7 TH edition (pgs ), provided photo-elastic stress analysis results with σ max = 1130 lbf/in 2 The Roark (photo-elastic) K t for a finite plate with a central hole is: K t (photo-elastic) = σ max / σ nom = 1130 lbf/in 2 / 480 lbf/in 2 = 2.35 The Roark (formula) K t for a finite plate with a central hole is: K t (formula) = (2r/D) (2r/D) (2r/D) 3 = 2.31 The FEA K t for a finite plate with a central hole is: K t (FEA) = σ max /σ nom = 1115 lbf/in 2 / 480 lbf/in 2 = 2.32 Error (photo-elastic to FEA): % error = {(K t (photo-elastic) K t (FEA)) / K t (photo-elastic)} * 100 % error = {( ) / 2.35} * 100 = 1.3% Error (formula to FEA): % error = {(K t (formula) K t (FEA)) / K t (formula)} * 100 % error = {( ) / 2.31} * 100 = -0.43% 2008 ANSYS, Inc. All rights reserved. 12 ANSYS, Inc. Proprietary

13 Conclusion (uniform stress K t from FEA) FEA results (path operations and integration to determine the area under the stress gradient curve) provides an accurate method of determining the nominal stress (σ nom ) The K t from FEA results provide a value with less than 2% error for a uniform stress field Reference documents (K t analytical equations) are not required K t s are for actual geometry Further verification/validation of the methodology is required 2008 ANSYS, Inc. All rights reserved. 13 ANSYS, Inc. Proprietary

14 Square Shoulder With Fillet (Pure Bending) Roark s Analytical model Square shoulder with fillet in a member of rectangular cross-section Example (from Roark s Formulas for Stress and Strain, 7th edition) 2008 ANSYS, Inc. All rights reserved. 14 ANSYS, Inc. Proprietary

15 Square Shoulder With Fillet (Pure Bending) Half-section Finite Element (FE) model. Square shoulder with fillet in a member of rectangular cross-section Example (from Roark s Formulas for Stress and Strain, 7 TH edition): L/2 (half model) r h (3 places) Geometry (see Figure, quarter model shown): D = in (Roark D = in) r = in (Roark r = in) h = in (Roark h = in) L/2 = in (Roark L = in) D Inputs (applied load (force): F y = lbf 2008 ANSYS, Inc. All rights reserved. 15 ANSYS, Inc. Proprietary

16 Square Shoulder With Fillet (Pure Bending) Roark s Analytical model The h/r ratio = 2.0, therefore, the K t was calculated using the two forms of the analytical equation 0.1 < h/r < 2.0 and 2.0 < h/r < 20.0 Square shoulder example (from Roark s Formulas for Stress and Strain, 7 th edition) Both forms of the equation provide a K t of K t Theoretical Calculations (from Roark's "Formulas for Stress and Strain", 7 th edition) 0.1 h/r h/r 20.0 h 1.0 L/D 1.0 C C D 3.0 2h/D C C r 0.5 h/r 2.0 C C L 3.0 L/D C C K t K t ANSYS, Inc. All rights reserved. 16 ANSYS, Inc. Proprietary

17 Square Shoulder With Fillet (Pure Bending) FEA model The max stress (12,242 lbf/in 2 ) occurs adjacent to the beginning of the fillet For ease of calculation (known cross-section and moment arm) the K t is determined at the beginning of the fillet where the max stress is 10,900 lbf/in 2 For pure bending σ max = MY/I = σ nom (stress if K t did not exist) σ nom = MY/(bh/12) = in * (1000 lbf) * (0.500 in) / ((1.000 in)*(1.000 in) / 12)) σ nom = 9000 lbf/in 2 F y = lbf 2008 ANSYS, Inc. All rights reserved. 17 ANSYS, Inc. Proprietary

18 Square Shoulder With Fillet (Pure Bending) Stress Gradients Bending Stress (lbf/in2) The area under the pure bending stress gradient is: A Kt = ½ σ max * Distance to neutral axis σ max = 2 * A Kt / Distance Using path operations and integration, the area under the Rect_Bar_Gradient curve is: A Kt = lbf/in σ max = σ nom = 2 * A Kt / Distance = 2 * ( lbf/in) / (0.5 in) = lbf/in 2 Rectangular Bar Stress Gradient Rect_Bar_Gradient Rect_Bar (IF Kt=1) Distance to Neutral Axis (in) Table 1 ANSYS Integration Operation Results (A Kt ), Rectangular Bar S σ x A Kt S σ x A Kt S σ x A Kt ANSYS, Inc. All rights reserved. 18 ANSYS, Inc. Proprietary

19 Square Shoulder With Fillet (Pure Bending) FEA K t Using the FEA smax (10,900 lbf/in 2 ) at the beginning of the fillet and snom from the area under the curve, the FEA Kt for the square shoulder with a fillet geometry is: K t (FEA) = σ max FEA / σ nom = lbf/in 2 / lbf/in 2 = Analytical K t (Roark) The K t using the analytical formula from Roark is: K t = Error (formula to FEA): % error = {(K t (formula) K t (FEA)) / K t (formula)} * 100 % error = {( ) / 1.222} * 100 = -2.95% 2008 ANSYS, Inc. All rights reserved. 19 ANSYS, Inc. Proprietary

20 Conclusion (Pure Bending stress K t from FEA) The K t obtained from the equivalent energy method has an error of less than 3% to the analytical K t Analytical K t s are obtained from empirical data Reference documents (K t analytical equations) are not required K t s are for actual geometry Further verification/validation of the methodology is required 2008 ANSYS, Inc. All rights reserved. 20 ANSYS, Inc. Proprietary

21 Summary The preliminary work presented for determining K t s from FEA results is very promising Obtaining actual geometry K t s from FEA results eliminates the need to find, and the error associated with, using empirically based analytical K t s for simple geometry Minimal error (less than 3%) between empirically based analytical K t s and K t s obtained from FEA results is demonstrated for uni-axial and pure bending stress fields Additional verification/validation of the methodology is required Expansion of the methodology to include multi-axial states of stress will be investigated 2008 ANSYS, Inc. All rights reserved. 21 ANSYS, Inc. Proprietary