FINITE GEOMETRY CORRECTION FACTORS FOR THE STRESS FIELD AND STRESS INTENSITIES AT TRASNVERSE FILLET WELDS

Transcription

1 Riggenbach i FINITE GEOMETRY CORRECTION FACTORS FOR THE STRESS FIELD AND STRESS INTENSITIES AT TRASNVERSE FILLET WELDS by KANE RYAN RIGGENBACH Submitted in partial fulfillment of the requirements For the Degree of Master of Science Thesis Adviser: Dr. Brian Metrovich Department of Civil Engineering CASE WESTERN RESERVE UNIVERSITY August, 2012

2 Riggenbach ii CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis of Kane Ryan Riggenbach candidate for the Master of Science degree. Dr. Brian Metrovich Dr. Michael Pollino Dr. Dario Gasparini Date of Defense June 6 th, 2012 *We also certify that written approval had been obtained for any proprietary material contained therein.

3 Riggenbach iii Table of Contents List of Tables:... iv List of Figures:... v Introduction:... 1 Background:... 2 Wedge Stress Analysis Basis Procedure: Modeling: Models: Material Model: Modeling Method: Modeling Choices: Comparison 1: Comparison 2: Modeling Side Notes: Data and Analysis: Accuracy of Individual Correction Factors Surface Fitting Approach Conclusions Future Work Works Cited... 74

4 Riggenbach iv List of Tables: Table 1: Typical Bulkhead and Stiffener Dimensions Table 2: Comparison of Partition Modeling Schemes Table 3: Comparison of Modeling Partition Schemes Table 4: Finite Length of Attachment vs. K Table 5: Finite Length Correction Factor Values Table 6: Finite Height Stress Intensity Factors and Correction Factor Values [2D] Table 7: Residuals from Finite Height Correction Factor [2D] Table 8: Finite Thickness Stress Intensity Factors and Correction Factors [2D] Table 9: Residuals from Finite Thickness Correction Factor [2D] Table 10: Stress Intensity Factors for varying Length and Thickness Table 11: Comparison of Predicted Stress Intensity Factors from 2D fits vs. Measured Stress Intensity Factors Table 12: Normalized Thickness Length Stress Intensity Factors F(t,L) Table 13: Goodness of Fit Comparison for F(t,L) Surface Table 14: Finite Thickness Correction Factor Data Table 15: Length Correction Factor Values F(L) Table 16: Predicted Stress Intensity Factors by F(t) and F(L) curve fits Table 17: Interaction Thickness Length Correction Values Table 18: Residuals and Residual Percentages for Thickness Length Interaction Surface Fit Table 19: Stress Intensity Factors for all Height and Lengths of Attachments Table 20: Varying Height and Length Stress Intensity Factors [3D] Table 21: Normalized Height and Length Stress Intensity Factors Table 22: Residuals and Residual Percentages from Height Length Surface Fit Table 23: Comparison of Fitting Method for thickness, length, and height data from Table

5 Riggenbach v List of Figures: Figure 1: Weld Profile showing Wedge Geometry and Coordinate System Figure 2: Strength of the singularity as a function of wedge angle (Barber, 2010) Figure 3: Representation of Bueckner's Principle Figure 4: Bulkhead Attachment (Left) and Transverse Stiffener (Right) Models Figure 5: Realistic Fillet Weld Profile Figure 6: Boundary Conditions and Loading Pattern Modeled Figure 7: Base Reactions for Typical Bulkhead Configurations Figure 8: Circular partition with collapsed quarter point quad elements (Left) and rectangular partition with quarter point quad elements (Right) Figure 9: Circular Partition with collapsed quarter point quad elements (Left) and rectangular partition with quart point quad elements (Right) Figure 10: Circular Partition with no analysis warnings (Left) and with 6 analysis warnings (Right) Figure 11: Lack of Fusion Plane where welding would not penetrate Figure 12: Stress Intensity Factor vs. Height in developing the Finite Length Figure 13: Finite Length Correction Factor Plot [2D] Figure 14: Residual % Plot for Finite Length Correction 2D Figure 15: Finite Height Correction Factor Plot [2D] Figure 16: Finite Thickness Correction Factor Plot [2D] Figure 17: Varying Length of Attachment versus Stress Intensity Factor for Varying Thickness Curves Figure 18: Varying Thickness vs. Stress Intensity Factor for Varying Length Curves Figure 19: Surface Plot of Stress Intensity Factors as a function of Thickness and Length Figure 20: Plot of Predicted Stress Intensity Factors from 2D Correction Factors Figure 21: Comparison of Predicted to Calculated Stress Intensity Factors [2D] Figure 22: Stress Intensity Factor as a function of Thickness and Length (Surface Fit) Figure 23: Finite Thickness Correction Factor (2D) Figure 24: Residual % Plot of Thickness Correction Factor Figure 25: Finite Length Correction Factor Figure 26: Residual % of Length Correction Factor F(L) Figure 27: Interaction Thickness Length Correction Factor F(t,L)*... 62

6 Riggenbach vi Figure 28: Height of Attachment vs. Stress Intensity Factor Figure 29: Height of the Attachment vs. Stress Intensity Factor (Length Curves) Figure 30: Height and Length Correction Factor Surface Fit [3D]... 67

7 Riggenbach vii Acknowledgments I would first like to thank my advisor, Dr. Metrovich, for his patience, guidance, and wisdom throughout this learning process. Without his leadership, this project would have been impossible to complete or even undertake. I am sincerely grateful for the opportunity to work with and learn from him. I would also like to thank the members of my defense committee, Dr. Dario Gasparini and Dr. Michael Pollino, for their time, comments, questions, and suggestions. Their insight and input is greatly appreciated. I would also like to thank my family for all of their support throughout not only this project, but my entire college experience.

8 Riggenbach viii Finite Geometry Correction Factors for the Stress Field and Stress Intensities at Transverse Fillet Welds by KANE RYAN RIGGENBACH Abstract This study presents the numerical development of corrections factors for finite geometry effects on the stress field and stress intensity factors at transverse fillet weld toes. The correction factors have direct application to fatigue life estimated and fracture mechanics strength calculations of transverse stiffeners, short cover plates, bulkhead attachments, cruciform section, and other filler welded plate configurations. The stress field around the weld toe is based on Williams Eigenvalue Expansion for a wedge, although it does not currently contain corrections for finite boundaries. The correction factors were developed using finite element analyses modeling small cracks for multiple series of dimensions to assess the behavior of stress intensity factors. The stress intensity factors were measured with J-Contour Integrals, normalized to a baseline stress intensity factor, and fit using numerical methods. The final result was two multi-variable correction factors that accurately capture the behavior of the stress intensity factor and stress field.

9 Riggenbach 1 Introduction: The numerical development of finite dimension correction factors applied to closed form expressions of the stress field and stress intensity factor for cracks emanating from a weld toe (wedge profile) is presented. These correction factors are limited to two dimensional configurations and specifically developed for transverse fillet welded details such as bulkhead attachments, transverse stiffeners, and short cover plates. The purpose of these correction factors is to more accurately assess and predict finite base plate thickness effects and finite attachment height and length effects on the stress field and fatigue life in the weld toe region of transverse welded details. The high stress concentration at the weld toe region is where the majority of the fatigue life is consumed and thus more precise predictions of local stresses will permit more accurate estimations of fatigue life. Closed form expressions of the stress field and stress intensity factor of a crack propagating from transverse weld toes have previously been developed (Metrovich, 2007). However, these closed form expressions currently lack corrections for finite thicknesses, heights of attachments, and length of attachments. The solutions are based on Williams Eigenvalue expansion (Williams, 1952), which is an infinite series Airy Stress Function expansion describing the stress field around the vertex of an un-cracked elastic wedge. To modify this solution for a cracked geometry, Bueckner s principle of superposition was employed (Bueckner, 1958). This principle applies a Green s function to the stress field of the un-cracked body to correct for the solution of a cracked body, thus allowing the evaluation of the stress intensity factor for a crack. In this case, the

10 Riggenbach 2 Green s function was developed by Stallybrass for an edge crack in a semi-infinite plane subjected to arbitrary loading on the crack surface (Stallybrass, 1970). The basic methodology for the development of the correction factor was based on modeling various finite and infinite dimensions of the base plate and typical transverse welded details using a two dimensional finite element model to determine characteristic behavior in stress intensity factors. The resulting trends could then be analyzed using numerical methods to determine an appropriate correction factor form for a given range of dimensions. Background: As a result of high stress concentration and singularity at the weld toe, this location presents an ideal initiation site for fatigue cracks. Studies in the past have shown that the large majority of cracks in transverse welded structures emanate from the weld toe region (Metrovich & et al., 2003) (Shen & Clayton, 1996). It is well known that cracks spend the majority of their cyclic lifetime while short and very few cycles at longer lengths. As a result, understanding the stress state for the stress concentration in the un-cracked configuration and understanding the stress intensities in the short cracked configuration is imperative to better comprehending and estimating the fatigue life of these details. The field of Fracture Mechanics was developed to describe and predict the driving force behind crack nucleation and propagation, and is useful for predicting fatigue life estimations at these stress hot spots. There are two alternate approaches in the application of fracture mechanics. The first is the energy criterion

11 Riggenbach 3 which was initially proposed by Griffith during the 1920s and later developed by Irwin and Orowan in the 1950s (Griffith, 1920) (Irwin, 1948) (Orowan, 1948). The energy approach is based on the idea that fracture requires the local breaking of bonds and a global energy balance. The balance compares the energy available to grow a crack (function of crack size and stress level) against the material resistance to crack growth (function of surface energy, plastic work, or any other means of energy dissipation). Irwin defined the energy release rate (G) as a measure of the energy available for an increment of crack extension. In other words, (G) is a measure of the energy required to form new crack surfaces. Energy approaches describe the global behavior of the structure (Anderson, 2005). The second approach relies on the concept of the stress intensity factor to fully describe the stress field near the crack tip in linear elastic materials. The stress intensity factors should not be confused with stress concentration factors, which describe the finite stress elevation near features, such as notches or holes, to the applied nominal stress in elastic bodies. In the case of a crack, as the distance from the tip approaches zero, a stress concentration factor becomes unbounded. In contrast, stress intensity factors completely define the amplitude of the strains, displacements, and stresses at the crack tip, where the stress concentration is undefined. While the energy and stress intensity factors approaches are equivalent as noted in Equation (1) shown below, in practice stress intensity factors have become the standard for calculating the stress state because they directly define the stress amplitudes locally at the crack tip.

12 Riggenbach KI (1 v ) KI G I (Plane Stress) G I = (Plane Strain) Equation (1) E E As a result of their ease in application, stress intensity factors have been compiled for numerous geometries, loading schemes, and boundary conditions over the years (Murakami, 1987), (Tada, Paris, & Irwin, 1985), (Sih, 1973), (Wu & Carlsson, 1991), (Fett & Munz, 1997), and (Rooke & Cartwright, 1976). Nevertheless, even though these compendiums and handbooks have been amassed, the number of possible geometrical and loading configurations is much larger than the number of available solutions. As a result a large number of correction factors have yet to be defined. For the correction factors that have been defined, few closed form solutions have actually been derived. The majority of stress intensity factor solutions have been numerically fit, calibrated, or generated through other empirical schemes. The first step in the process of determining correction factors is determination of the stress intensity factors themselves. In a journal article by D.P. Rooke, author of a stress intensity handbook above, the author lists and compares the time required to generate stress intensity formulas and approximations for several methods (Rooke, Barrata, & Cartwright, 1981). The simplest methods of generating stress intensity factors are by super-position (such as Bueckner s Theorem), manipulation schemes of stress concentration formulas, Green s functions, calibration, and finally weight functions. Some of the more advanced methods of generating stress intensity factors are by collocation (mapping), integral transforms, boundary integrals, and of course finite element analyses. Several of these methods are briefly discussed to provide background

13 Riggenbach 5 for understanding where stress intensity factor formulas and correction factors originate, and also to provide background for future derivations. The first technique for generating stress intensity factors is through calibration to accommodate various geometries. Churchmen presented a procedure for using influence functions from a edge dislocation (simulation of a crack on the grain level) along projection lines extending from the wedge vertex combined with Williams semiinfinite wedge solution to accommodate changes in the geometry (Churchmen & Hills, 2005). This solution requires that the stress distribution and displacement functions for an edge dislocation be known. The edge displacements can then be computed for similar finite geometries with edge dislocations present and a collocation method can be used to calibrate the known stress field to the second geometry enabling the stress intensity factor to be calculated. On the one hand, calibration produces the approximately correct stress intensities, however it requires that the displacements and stress distribution from similar edge dislocation be known. In the end, it is just a numeric technique to view how varying finite dimensions will affect the displacements and stress distribution. To utilize this method the size of finite dimensions that can be calibrated is limited. As the cracks grow larger, calibrating the stresses and displacements from infinite geometries would be inaccurate because finite dimension effects become more pronounced for smaller dimensions and larger cracks. The second method of generating stress intensity factors is through manipulation of stress concentration factors. As previously noted, stress concentration formulas can be found in a multitude of handbooks. One method to correlate stress

14 Riggenbach 6 concentrations to stress intensity factors is to express the stress concentration formula in an equivalent series form. When the stress concentration is expanded into a series form, in some cases by taking the limit of the stress concentration as a notch tip radius for example approaches zero, there will be several unknown coefficients. These coefficients can be found by fitting the series to data generated for small notches using the original stress concentration formula or to data extrapolated from the original stress concentrations (Hasebe & Kutanda, 1978). In effect, this method is simply breaking up the stress concentration into series form so that a finite number of terms can be considered at once rather than the entire series thus limiting the magnitude of the stresses predicted. Then, the finite number of terms can be fit to data as the notch tip radius approaches zero which is a crack. The process limits the stress concentrations from rapidly increasing toward infinity at the crack tip singularity. The advantage of this method is that a stress concentration may be computed for a certain geometry that has no current stress intensity factor solution but that has a similar stress concentration factor. The accuracy of the stress intensity solution is limited by the accuracy of the stress concentration factor. Most importantly, the stress concentration formula must be accurate as the notch tip radius approaches zero, which is often difficult to find. A third method of generating stress intensity factors is through weight functions. The weight function method was first proposed by Bueckner and later revised by Rice (Bueckner 1970, Rice 1972). The equations for this process are not shown but rather the concept is presented. If the stress intensity factor K (1) and the crack face displacements u(a,x) (1) are known for a particular loading pattern (1) on an infinite body, then the

15 Riggenbach 7 stress intensity factor K (2) can be found for any other load pattern by integrating a weighting function of displacements from the (1) load system and the stress distribution from the (2) load system. Both load patterns must be symmetric with respect to the crack line. Originally, weight functions were only valid for prescribed surface tractions though these limitations have since been relaxed. Now combinations of prescribed surface tractions and displacement boundary conditions can be utilized. (Wu & Carlsson, 1983) An important point to note is that the boundary conditions for the first load system (geometry, prescribed displacement locations, and prescribed traction locations) must be the same as the second load system in order for the method to be applicable. The magnitudes of these quantities do not have to be the same, but their location must be. One of the drawbacks of the method is that it relies on a valid solution of the crack face displacements for a load system. For finite geometries this limitation becomes restrictive because finite geometry is often very complicated and highly sensitive to alterations. In addition, complete solutions to finite dimension problems are extremely limited. Today, the most common method of generating stress intensity factors is through the use of finite elements. Increasing accuracy of measurement, precision of meshing algorithms, and added functionality such as direct measurement of crack tip opening displacement and the advancement of J-Contour Integral have allowed finite elements to become the standard. To generate the stress intensity factors in this paper, the J Contour Integral method was utilized. The J Contour Integral, which was originally developed

16 Riggenbach 8 independently by Cherepanov and Rice, will be presented in detail in the modeling section (Cherepanov, 1967) (Rice, 1968). All methods presented above can provide extremely accurate formulations for calculating and determining stress intensity factors. However, many of these methods are difficult to implement, so finite element modeling became the method of choice. When using some of the derived methods, the correction factors go hand in hand with the derivation. However, for the finite element method, accurately determining the stress intensity factors is only part of the solution. In addition, one must understand the geometrical component to correction factors in the predicted stress intensities. In practice, correction factors for geometrical changes and loading conditions are ultimately applied by users to the base stress intensity factor. Reference handbooks and compendiums on stress intensity factors typically compile this type of information along with the total stress intensity factor. Just as with stress intensity factors, there are a couple ways of eliciting these correction factors from the stress intensity factors. One of the methods of generating these geometrical corrections is by utilizing theoretical stress functions and manipulating them to derive the geometrical effects. Prime examples of this method were developed for the geometrical constraints on center cracks plates (Isida 1971, Isida 1973). Typically, these geometrical correction factors can be approximated by normalizing the numerically obtained data to a theoretical baseline stress intensity factor. One well known geometrical correction factor is the secant correction factor (Tada, Paris, & Irwin, 1985).

17 Riggenbach 9 a a F sec t 2t Equation (2) The dimensionless factor is a finite width correction dependent on the ratio of crack length (a) to thickness of the plate (t). The factor was developed to correct for large values of the ratio of (a/t). As the ratio (a/t) goes to zero, the correction factor goes to 1. As the ratio (a/t) increases toward 0.9, the correction approaches 3. Beyond a ratio of 0.9 the correction factor rapidly increases because the net section stresses become exceedingly large. For small crack sizes and dimensions, similar to cracks in this study, the secant correction factor was insignificant because the crack sizes modeled were so small that the ratio of (a/t) never attained a value greater than 1. The relevant aspect of the secant correction factor is that it is an example of a simple numerically developed correction factor. Originally, there was a tangent correction factor used to describe finite width corrections. The secant correction factor was an improvement on the analytically developed tangent correction factor, which error under larger a/t ratios. The goal of the present study is to numerically determine the influence of finite dimensions on stress intensity factors for cracks growing out of a wedge vertex. In the present case, the wedge vertex is the weld toe of a transverse fillet weld attachment. Ideally, the correction factors should be accurate, simple to apply, and sufficiently versatile to cover a range of geometries. Finite element analyses using the J Contour Integral were used to calculate stress intensities for a number of finite dimensions. The stress intensity factors were normalized and the behavior was analyzed using curve fits and surface fits. The resulting fits are empirical corrections for finite dimensional effects

18 Riggenbach 10 on the stress field and stress intensity factor. Furthermore, the resulting correction factors will benefit future work by saving time and effort in determining many finite geometry stress intensity factors and the behavior of the finite dimensional effects. Thus, more effort could be focused on solving the analytical relationship between the finite dimensions and the stress intensity factor. Lastly, this work provides a guide to the process of studying further finite dimensions and features affecting the crack stress intensity factors. Wedge Stress Analysis Basis A weld toe, as seen in Figure 1 below, can be modeled as a generalized twodimensional wedge. The stress field in generalized plane elastic wedge problems has been studied in the past by (Wieghardt, 1907) (Williams, 1952) (Tada, Paris, & Irwin, 1985) (Szabo, 1988) (Vasilopoulos, 1988) As a result of the sharp change in geometry at the wedge vertex, a stress concentration exists. In the limit, as the radius (r) from the wedge vertex (weld toe) tends toward zero, the stresses conversely tend toward infinity under linear elastic conditions. This phenomenon is known as a stress singularity.

19 Riggenbach 11 Figure 1: Weld Profile showing Wedge Geometry and Coordinate System One possible way of describing the stress field around the wedge vertex is with an Airy Stress Function. Specifically, Williams developed a method of exploring the nature of the stress field near the singularity by defining a set of polar coordinates centered on the corner and expanding the stress as an asymptotic series in powers of r (Barber, 2010). For the purposes of this paper, a brief overview will be presented of Williams Stress Field though the full derivation can be found in (Metrovich, 2007) or (Barber, 2010). Equation 3 below shows the general series solution form of the stresses: n i 1 cir f ( i,, ) Equation (3) i0 The function f(λ,α,θ) is dependent on the stresses being considered. For example, the function will be different for the stresses in the θθ than in the rr or θr directions.

20 Riggenbach 12 Williams solution was derived on the boundary condition of straight traction free wedge straight surfaces. Consequently, the solution is highly accurate within a finite region near the wedge vertex, but at larger distances from the vertex the accuracy is dubious because the initial geometry (described above) the solution was derived on is usually altered. The solution to the wedge problem is comprised of two orthogonal systems of equations; the symmetric stress components (Mode 1) and the antisymmetric stress (Mode 2) about the wedge bisector (ϴ=0 or along the x axis) (Shown in Figure 1). The magnitude of the stress singularity is controlled by the Eigenvalue (λ) in Equation (2). When λ<1, a singular stress field is present because the radius term will move to the denominator. At most, only the first two terms of the expansion, corresponding to the symmetric Mode 1 stresses and to the anti-symmetric Mode 2 stresses can produce singular stresses. The higher order stress terms are for the higher non-singular Eigenvalues, larger than 1. Therefore, magnitude of the Eigenvalues, and thus the stress singularity, are dependent on the wedge opening angle (α). Figure 2 below graphically illustrates the first two Eigenvalues for various wedge opening angles (Barber, 2010).

21 Riggenbach 13 Figure 2: Strength of the singularity as a function of wedge angle (Metrovich, 2007) Figure 2 shows the strength of the symmetric singularity (Mode 1) is always greater than the anti-symmetric singularity (Mode 2). By expanding Equation (1), the first expansion corresponding to the Mode 1 stress distribution can be shown as (Williams, 1952): Mode I c ( 1) r cos( 1) Q cos( 1) c r ( 1) cos( 1) Q cos( 1) rr c r ( 1) cos( 1) Q ( 1) cos( 1) c r ( 1) sin( 1) Q ( 1) sin( 1) r where : Q 1 cos( 1) sin( 1) 2 2 ( 1) and ( 1) cos( 1) sin( 1) 2 2

22 Riggenbach 14 And the second expansion corresponding to the Mode 2 anti-symmetric stress distribution can be shown below as: Mode II c ( 1) r sin( 1) Q sin( 1) c r ( 1) sin( 1) Q sin( 1) rr c r ( 1) sin( 1) Q ( 1) sin( 1) c r ( 1) sin( 1) Q ( 1) sin( 1) r where : Q 2 cos( 1) sin( 1) ( 1) 2 2 and ( 1) cos( 1) sin( 1) 2 2 The most important thing to note from the above distributions is the stress field can be entirely described at the vertex by the undetermined coefficient (c i ). The value of this coefficient may be found analytically for simple geometries. For more complex geometries, a brute force extraction of either stresses or displacements from a finite element analysis is necessary to numerically determine the coefficient. The value of the undetermined coefficient is dependent on the wedge angle (α) and the remote boundary conditions far from the wedge vertex. In the limit, as the wedge angle (α) approaches 2π, the wedge geometry collapses to the sharp crack geometry and the undetermined coefficients are equivalent to the conventional stress intensity factors K I and K II. Now that the first two terms of the Eigenvalue Series Equation (3) have been expanded, the normal stress acting at the wedge vertex is: n (1) (2) i 1 cir f ( i,, ) i3 Equation (4)

23 Riggenbach 15 Recall from above that as the wedge opening angle (α) approaches 2π, the Eigenvalues approach their lowest values of 0.5 and thus the Mode 1 and Mode 2 stress fields become singular. As a result, near the wedge vertex (i.e. crack tip), the Mode 1 and Mode 2 stresses completely describe stress field and additional non-singular series terms are not needed. Conversely, as the distance (r) from the wedge vertex increases the amount of terms (n) necessary to fully describe the stress field will increase and more series terms must be considered. The stress field can now be applied to evaluate the stress intensity factor for cracks emanating from the wedge vertex. Stallybrass has previously derived the solution for the stress intensity factor of an edge crack in a semi-infinite plane subjected to arbitrary pressure loading on the crack surface (Stallybrass, 1970). The required pressure distribution over the crack surface must be defined as: yy ( 1) ( ) r r p p a a Equation (5) In the pressure distribution above the power term (γ) can be any positive real number, there is no requirement that (γ) be an integer. P is the magnitude of the pressure distribution at the crack tip. The resulting stress intensity factor may be expressed as: KI p a FG FE Equation (6) Where F G is a correction factor for the gradient of the pressure distribution over the crack surface and F E is the free surface correction factor. These correction factors are given below (Tada, Paris, & Irwin, 1985):

24 Riggenbach 16 F E ( 1) Equation (7) t x1 e t dt for x>0 2 0 FG where ( x) 1 x1 nn! lim for any x 2 2 n x( x 1)( x 2)... x n 1! Equation (8) In order to combine the exact stress distribution for the wedge geometry given by Williams Eigenvalue Expansion with Stallybrass s formula, Bueckner s principle was utilized (Bueckner, 1958). Bueckner s principle can be visually seen in Figure 3. Figure 3: Representation of Bueckner's Principle Bueckner s principle states that once the stress distribution for the un-cracked geometry subjected to far field stress is known (Figure 3 Left), the stress distribution at the location of the crack can be applied as a pressure loading on the crack surface (Figure 3 Middle). The stress intensity factor can then be calculated for each geometry (K=0 for the un-cracked geometry). Finally, the addition of these two stress intensity factors gives the stress intensity factor for the cracked geometry subjected to far field

25 Riggenbach 17 loading. In terms of the current problem, the un-cracked stress distribution is given by Williams Eigenvalue solution. The pressure loading on the cracked wedge geometry is provided by setting Williams Eigenvalue Stress Distribution equal to the Stallybrass s stress distribution and solving for the pressure (p) in Equation (4). This pressure serves as input into the solution for the stress intensity factor given in Equation (5). The only issue in this process is that Equation (5) was derived for a crack growing out of a straight edge, and does not account for the additional constraint provided by the wedge geometry. However, this issue can be accounted for with the inclusion of a wedge correction factor (F H ). No known solutions could be found in the literature for this correction factor, but previous numerical work based on similar correction factors approximates the correction as F H =0.8 (Metrovich, 2004). The number was specifically based on correcting the half plane geometry to a ¾ plane symmetry (α=270 ). The final correction for this solution corrects for non-symmetrical crack growth out of the wedge vertex, at an angle β away from the wedge bisector. Again, no solution was available in the literature for this correction factor, so it was numerically developed using finite element models to compare stress intensity factors for angles β=0 and 45. The angle β=45 was specifically chosen because the cracks in typical transverse fillet welded attachments have been found to propagate along a path vertically through the flange. The value of the correction factor for this angle was estimated to be F β = The stress intensity factor for a crack emanating from a wedge vertex is thus given: K F F F F p a I G E H i where p c f (, ) a i i 1 Equation (9)

26 Riggenbach 18 Recall that there are both Mode I and Mode II stress components in Williams series solution; a pressure and stress intensity factor component can be evaluated for each stress distribution. It is important to clarify that when referring to Mode I and Mode II this is relevant to the stress distribution along the crack, symmetric versus antisymmetric, and both of these terms contribute to the same stress intensity factor. This nomenclature should not be confused with Mode I and Mode II stress intensity factors from classical fracture mechanics. In fracture mechanics, there are three different types of stress intensity factors which refer to the method of crack opening with Mode I is tensile opening, Mode II is in plane shearing, and Mode III is out of plane shearing. Mode I stress intensity factors are known to dominate fatigue crack behavior and thus are the main focus of the current study. The total stress intensity factor for a crack emanating from the wedge vertex is given by the superposition of stress intensity factor from the series stress field: K K K K Mode 1 Mode II Higher Order Terms I I I I where: (, ) Mode I 1 1 K F F F F c f a a I G E H 1 1 Mode II 2 1 K F F F F c f (, ) a a I G E H 3 2 Equation (10) As can be seen above, the correction factors applied to the stress intensity factor solution are also directly correcting the undetermined coefficients (c i ). The work of the present study is to better understand finite dimensional effects on these undetermined coefficients which control the magnitude of the stress intensity factor. Previous studies have shown that the undetermined coefficient is also a function of the far field boundaries. For example, the thickness of the flange (modeled as the base plate), the

27 Riggenbach 19 length of attachment, and the height of the attachment all affect the magnitude of the undetermined coefficient. These are not the only factors that have an effect on the coefficient, but these factors were chosen because the finite dimensions have been shown to have the substantial effects on the coefficients. Procedure: In order to determine how each finite dimension affects the stress intensity factor, it was necessary to isolate each variable. This meant that for each dimension, a set of models was created varying only the chosen dimension. The primary dimensions varied in this study were the length of the attachment (L), height of the attachment (H), thickness of the base plate or flange (t), and the crack size (a). To limit the number of models and data generated, it was necessary to place dimensional limits on each variable that would focus on realistic configurations. The general procedure was to sketch the general 2D geometry, choose one or two specific dimensions to isolate, create a finite element mesh, analyze the geometry to determine the stress intensity factor, then return to the original geometry and apply either a larger of smaller value for the isolated dimension, and repeat the analysis to eventually generate a set of stress intensity factors for the varying dimension(s), normalize the data set by a baseline, and finally use numerical methods to curve and possibly surface fit the correction factors as a function of the primary dimensions. The exact modeling process for generating and analyzing the models will be covered in the following section.

28 Riggenbach 20 Modeling: Models: The two dimensional geometries of typical stiffener and bulkhead attachments can be seen below in Figure 4. Two dimensional configurations for the attachment are also shown in Table 1 following. These geometries correspond to typical welded details studied in National Cooperative Highway Research Programs (NCHRP) test programs (NCHRP Report 147, 1974) (NCHRP Report 102, 1970). Figure 4: Bulkhead Attachment (Left) and Transverse Stiffener (Right) Models L [in] H [in] t [in] Bulkhead Varies Stiffener Varies Table 1: Typical Bulkhead and Stiffener Dimensions Cracks in these configurations typically grow as semi-elliptical cracks out of the weld toe through the thickness of the tension flange. In order to simplify the modeling process, only a 2-D cross section of these models was considered. As a result, none of the

29 Riggenbach 21 possible 3-D effects such as flange bending effects or finite out of plane dimensions were considered. It is believed that as the height and length of the attachment increases the stress intensity factor increases accordingly. These trends stem from the fact that for larger heights and longer lengths more stress can flow higher into the attachment and thus more stress must funnel out in the weld toe region (a shear flow effect). For short length attachments the height configuration matters less than for longer attachments because the stress flow has less horizontal distance (length) to distribute vertically (height) into the attachment. Moreover, as the length of the attachment increases, the amount of horizontal distance that the stress has to distribute vertically increases so in these case the height configuration is much more important. Figure 4 above shows a typical fillet weld illustration with a 45 weld profile angle. Classically, 45 weld angles are used in strength calculations because the throat thickness is minimized thus giving the lowest strength estimate. However in real welds, the initial weld angle (I.E. wedge reentry angle) at the toe approaches 90. Figure 5 below shows a more realistic weld profile. Figure 5: Realistic Fillet Weld Profile

30 Riggenbach 22 Previous studies by Metrovich on the fatigue behavior of stainless steel welded attachments found that the lower bound fatigue data occurred for geometries with steeper weld angles approaching 90 degrees (Metrovich, 2004). Furthermore, in order to utilize Williams Eigenvalue Series, it requires straight traction free boundaries for the wedge. Future work may be to develop a correction from Williams straight boundary requirement to allow the curved weld profile. Nevertheless, it was decided to model the fillet welds as squares to give the lowest bound on fatigue life and allow the use of Williams Eigenvalue series directly. The lower bound on fatigue life should also correlates to the upper bound for stress intensity factors of cracks at the weld toe. The loading was an applied pure tensile traction of 10 ksi for all analyses run. The magnitude was chosen because the results can easily be scaled to other loadings and it provided a more significant load to accurately access the stress intensity factors rather than applying a unit load. To ensure pure tensile loading of the model and to restrain the model from rigid body translations, horizontal roller supports were applied across the entire bottom length of the base plate and the vertical and horizontal displacements were restrained in the bottom left corner. Figure 6 following depicts the applied boundary conditions and loading pattern for a general bulkhead attachment model.

31 Riggenbach 23 Weld Detail Roller Supports Figure 6: Boundary Conditions and Loading Pattern Modeled If the roller supports had not been applied to the bottom of the structure there would have been additional bending stresses induced due to the eccentricity of the geometry. Even though there is only a tensile traction applied to the base plate because of stress flow into and out of the attachment, some bending forces are created. As the attachment size increases, the bending forces (in this case the base reactions across the bottom represent the bending forces) will also grow larger because the increasing stiffness of the attachment makes the base plate s tendency to wrap around the attachment greater. These bending forces can affect the stress intensity factor at the weld toe by contributing additional stresses. For short dimensioned attachments these bending forces should be relatively small, because the stiffness of the attachment is small. Also, in realistic configurations the web of a beam would be present to handle these bending forces.

32 Riggenbach 24 The goal of the present study was to analyze finite base thicknesses and as a result the base thickness dimensions were varied to very small dimensions while the attachment dimensions remained large. These configurations could result in large bending forces. Because the goal of the present study was to focus on the finite dimensional effects, the bending force effects would simply complicate the problem making it hard to isolate the different effects. Thus the bottom roller supports were added. Unlike the height and length dimensions, the finite thickness parameter can have numerous effects because of these bending forces. In some previous studies it was found that smaller thickness had shorter fatigue lives. For pure tensile loading, thinner plates should have had lower stress intensity factors if the same stress was applied. However, it is possible these additional bending and warping forces present in the thinner sections could have affected the stress intensity factors. In the future, the bottom surface could be modeled as free of restraints to evaluate these additional bending effects and their effect on finite thicknesses. Simply to illustrate the bending forces, the base reactions in the vertical direction were plotted from an analysis. Figure 7 below exhibits the base reactions for the typical bulkhead configuration (L=8.2 ), on the thickest base plate (t=40 ), for a tall attachment (H=8 ).

33 Riggenbach 25 Figure 7: Base Reactions for Typical Bulkhead Configurations Material Model: The material modeled in this study was an idealized isotropic elastic steel for both the base metal, attachment, and the weld material. The elastic modulus (E) used was 30,000 ksi and Poisson s Ratio (ν) was 0.3. No other material properties were utilized because a linear elastic analysis was performed. Modeling Method: In order to determine the finite dimension effects on the stress intensity factor, the general 2-D model was created based on the geometry shown on Figure 4. After the base geometry was modeled, a crack was inserted as a seam. In ABAQUS, a seam is defined as an edge or face with overlapping nodes that has the capability of opening or separating during an analysis (ABAQUS 6.9 Online Documentation, 2009). When a seam is defined there are no attractive forces between overlapping nodes so that the seam is effectively modeled as being a closed crack free to open under normal force or stress. For these models a pure tensile loading (Mode 1 loading) was considered. As a result, crack closure contact properties were not required, although ABAQUS defines the seam edges as friction free hard contact.

34 Riggenbach 26 In order to run a contour integral analysis around the seam crack, the tip of the seam was defined as the crack front. A contour integral analysis defines contours or rings of elements around the crack tip that ABAQUS automatically selects to determine the energy release rate from the crack tip. While ABAQUS says that it will carry out a contour integral analysis, the divergence theorem allows ABAQUS to actually use an area integral. Numerically, area integrals are just as accurate but are faster to evaluate in finite elements than contour integrals. The type of contour integral method applied is called the J-Integral. The J- Integral provides a measure of the strain energy released at crack tip during a crack extension. J is the value of an energetic contour integral around the crack tip. The path this contour takes around the crack tip is independent, resulting in any contour around the crack tip providing the same value of J. For linear elastic analyses, ABAQUS uses Equation 11 to compute the stress intensity factor. Equation 11 is only applicable for homogenous isotropic materials. Equation also shows how Abaqus would combine stress intensity factors for Mode I, Mode II, and Mode III, although is these analyses Mode I stress intensity factors are the only ones present. 1 1 J ( K K ) K E 2G I II III Equation (11) (For plane stress configurations E E and for plane strain configurations E E 2 / (1 ) ) If the model were under mixed mode loading, ABAQUS would use an interaction integral method to solve the individual stress intensity factors.

35 Riggenbach 27 Once the contour integral history output has been requested, the last task was to mesh the model. Contour integrals require quadrilateral finite elements be used for the meshing. When the crack front is defined as the seam tip, the quadratic quadrilateral elements around the crack tip can be collapsed into triangular quadratic elements. The mid-side node on the quadratic triangular element may be located at the quarter point of the side nearest the crack tip, by doing so, the 1/ r singularity associated with linear elastic fracture mechanics may be captured. In order to mesh the model, a great deal of seed transitioning was necessary. The mesh sizing usually ranged over a scale of six orders of magnitude. To accomplish this large transition, concentric circular partitions were added around the crack tip which allowed the mesh size to gradually increase in orders of magnitude from the smallest size of 1e-6 inches to the largest size of 1 inch. The elements used throughout the modeling process were CPS8R ABAQUS standard elements. These elements are 8-node bi-quadratic plane stress quadrilateral elements with reduced integration. Modeling Choices: Comparison 1: One of the greatest challenges encountered when modeling was meshing at the tip of very small cracks. In order to accurately capture the stress intensity factors using J- Integrals the mesh near the crack tip had to be extremely fine. For cracks sizes on the order of 1e-4 to 1e-5 inches, using quarter point triangular elements swept through a circular partition often became difficult. The difficulty arose because the circular

36 Riggenbach 28 element node locations were at the numeric limit of the CAD system used by ABAQUS. An alternative to the circular partition was simply to partition the model in rectangular fashion. Rectangular partitions and meshes were easier to apply at the limit of the CAD system because the nodes of these elements align more easily with the rectangular grid embedded in ABAQUS s CAD. In order to ensure that this meshing scheme was as accurate as the circular partitions, several comparisons were conducted. The first comparison evaluated the stress intensity factor for an edge crack of 5e-4 inches in a half plane subjected to uniform tension loading of 1 ksi. The two meshing schemes are shown below. Figure 8: Circular partition with collapsed quarter point quad elements (Left) and rectangular partition with quarter point quad elements (Right) The size of the elements in the circular partition is 7e-5 inches. The size of the elements in the rectangular partition above is 2e-5 inches. Table 2 below shows the value of the stress intensity factor from each J Contour Integral around the crack tip, the average of the 2 nd through 10 th contours, and the associated standard deviation. The first contour is usually inaccurate and was discarded from the data.

37 Riggenbach 29 Circular Partiton Rectangular Patition Contour Number K K Average Contours Standard Deviation E Table 2: Comparison of Partition Modeling Schemes Table 2 above shows that the results given by the two methods differ by % which illustrates that both modeling methods produce nearly identical results. The difference in their stress intensity factors is purely due to numerical error. One may also note that the precision, given by the standard deviation, was higher for the circular partitions. Equation 12 shows how the theoretical result was calculated for this comparison. K F a E Equation (12) In Equation 11, F E is the free edge correction factor (F E = 1.12), σ is the nominal stress applied on the model (σ = 1.0 ksi), and (a) is the crack length (a = in.). Substituting these values into Equation 1 gives K equals ksi in. This theoretical result is % different from the circular partition and % different

38 Riggenbach 30 from the rectangular partition. Thus, this shows the finite element models produce identical results to those produced by classical fracture mechanics. To further justify the use of rectangular partitioning, another comparison of the same type was run for the stiffener model. Stress intensity factors were determined for the stiffener geometry using both circular and rectangular partitions. Figure 2a and 2b below show the two meshes. The crack size studied in the model was 0.01 [in]. This geometry addresses the effect of high geometrical stress gradients from a wedge tip on the SIF of a crack. Figure 9: Circular Partition with collapsed quarter point quad elements (Left) and rectangular partition with quart point quad elements (Right) The size of the elements in the circular partition Figure 9 (Left) are 5e-5 inches and the sizes of the elements in the rectangular partitions (Figure 9 Right) are 1e-5 inches. Once again, the same conclusion was reached. Table 3 below shows that the two meshing techniques give nearly identical results differing by only ksi in or 0.52%.

39 Riggenbach 31 Average SIF from contours 2-10 Standard Deviation Circular Partiton (Fig. 9 (Left)) E-05 Rectangular Patition (Fig. 9 (Right)) Table 3: Comparison of Modeling Partition Schemes Table 3 illustrates again illustrates that the precision of the integrals, shown by the standard deviation between stress intensity factors, was higher for the circular partition. The comparisons show it is acceptable to use rectangular partitions when the crack sizes become too small or difficult to mesh with circular partitions. This result was expected because theoretically the J-integral is independent of contour path. From a numerical viewpoint, it is important to understand some complex contour paths can introduce more numerical error. For the majority of analyses run in the paper, crack sizes larger than 1e-4 [in] were run with circular partitions while rectangular partitions were applied to smaller cracks. Comparison 2: Another difficulty arose when decreasing the size of the elements size in the mesh. Once the meshing scheme had been decided, rectangular or circular, the size of the elements had to be optimized through a trial and error process until the smallest element size could be attained. If the elements size were made too small, some elements would be skewed to conform to the CAD system and to the overall mesh. Skewed elements decrease the accuracy and precision of the J contour integrals, although avoiding analysis warnings like skewed elements altogether is a difficult task. Rather than try to eliminate the analysis warnings completely, an acceptable amount of warnings was defined. To clarify, Abaqus defines analysis warnings as finite elements

40 Riggenbach 32 with either shape and/or size related warnings such as: corner angles less than 10 degrees, angles larger than 160 degrees, aspect ratios greater than 10. In order to determine the acceptable amount of warnings that would not significantly decrease the accuracy or precision of the stress intensity factors, multiple analyses were run. Figure 10 is a comparison of a mesh with no analysis warnings and a mesh with analysis warnings around the crack tip. The largest source for analysis warnings was the smallest elements around the crack tip as shown on the right because this area required a very fine mesh to capture the stress field and give accurate J-integral values. Figure 10: Circular Partition with no analysis warnings (Left) and with 6 analysis warnings (Right) The size of element around the crack tip in Figure 10 (Left) are 4.5e-5 inches compared with 4e-5 inches in Figure 10 (Right). The crack size in this model is inches. Table 3 below provides the stress intensity factor results from contours 2-10 for these two meshes. Average SIF Standard Deviation Figure 10 (Left) E-06 Figure 10 (Right) E-05 Table 3: Comparison of no analysis warnings vs. 6 analysis warnings

41 Riggenbach 33 Table 3 illustrates that the presence of analysis warnings in the model doesn t significantly impact the stress intensity factor. The difference between the stress intensity factors is %. The existence of analysis warnings did decrease the precision of the contour integrals by 8.738e-6. The loss in precision is something to be aware of, but as long as the number of analysis warnings is kept to a minimum it should not affect the results. After comparing SIF results between these two cases, it appears that only several warnings have a negligible effect on the SIF. Similar comparisons were run for the bulkhead model with the same results as well as for analysis warnings in the transition region of the model. The transition region is the radius around the crack tip where the smallest elements around the crack tip increase in size to the largest elements in the model. Again, as long as the analysis warnings combined from the transition region and crack tip region were kept to a minimum the results were negligibly impacted. After running multiple comparisons, it was decided that an acceptable amount for analysis warnings should be 10 element warnings. Modeling Side Notes: When constructing the model including the crack and partitions, the precision of the geometry and mesh can be drastically affected by the placement of the origin. Section of the ABAQUS user manual discusses preserving nodal precision. As previously mentioned ABAQUS had difficulty in generating circular partitions that actually were circular and honoring seeding sizes around the smallest crack sizes (1e-5). The precision in generation of these circular partitions was dependent on the distance

42 Riggenbach 34 from the origin of the base sketch. If the partition was located at a great distance from the origin there would be diminished precision and the circular partitions generally became highly skewed. Similarly, fine meshes far from the origin would often generate element warnings because precision was lost in nodal placement. By default, ABAQUS Standard (Implicit) only stores the nodal coordinates in single precision. A simple solution to these problems was to always make sure that the origin of the base feature in the model had the crack front or tip placed exactly at the origin. By doing this, the precision was greatly enhanced. Another possible solution that was not utilized would have to develop two models. A large geometry model such as the one analyzed where the local stresses around the weld toe from each analysis could then be recorded and applied to a sub-model. The resulting sub-model would have different scaling than the original model, size of cracks analyzed could have been reduced even further (ABAQUS 6.9 Online Documentation, 2009) Data and Analysis: The idea postulated in the introduction was that correction factors could be extracted for finite dimension effects: specifically, finite length of attachment (L), finite thickness of base plate/flange (t), and finite height of attachment (H). The governing relationship for the stress intensity factor would then be as obtained as: KFinite t,h,l KInfinite Dimension F( H) F( L) F( t) Equation (13) The infinite dimension stress intensity factor (K) in the above equation should theoretically be from when the height of attachment, length of attachment, and

43 Riggenbach 35 thickness of base are all infinite. However, modeling three simultaneous infinite dimension isn t reasonable because it does not define a unique loading case. In the following data it will be shown that one of these three dimensions will always limit the stress intensity factor from reaching its infinite dimension solution. As a result, bounds were placed on the dimensions to conform to realistic modeling conditions. The bounds had to be high enough to capture a large number of realistic configurations for attachments. The bounds were based on previous modeling data as will be seen in the following sections. The first finite correction factor that was developed was for the length of attachment. The upper bound on the length variable was set at 80 inches. In order to capture the finite length effects, a set of stress intensity factors were attained where the length of the attachment was the limiting dimension and the height and thickness were in their infinite configurations relative to the length, to minimize their effects on the length correction factor. The crack size was held constant at 1e-4 inches, the smallest crack size that could be efficiently modeled with collapsed quadrilateral crack tip elements. More importantly, for cracks of this size and smaller, the first term (Mode I symmetric stresses) from Williams Eigenvalue Series completely dominates the stress intensity factor. As a result, no additional expansions (i.e. terms) are needed to accurately describe the stress field or stress intensity factor. Thus, correction factors attained from this crack length are also direct correction factors on the first coefficient (c 1 ). To study the possible additional corrections on the second term (c 3 ), larger crack sizes could be studied in the future and compared with the results of this study. From a

44 Riggenbach 36 practicality standpoint, cracks of this size are extremely difficult if not impossible to identify outside of laboratory settings. However, the goal of this research is to capture the correction on the first term of the Eigenvalue expansion. By correcting the first term, it will affect every crack size, so modeling a crack any larger or smaller than this dimension was not necessary. All of the models also had a constant lack of fusion size of 0.25 inches modeled behind the fracture fillet weld at each end. This dimension was based on the lack of fusion of plane present directly under the flanges or stiffener cross-section where welding did not penetrate. An illustration of the lack of fusion plane is shown in Figure 11. Figure 11: Lack of Fusion Plane where welding would not penetrate In the future, the lack of fusion size could also be varied to evaluate its effects on the stress field at the weld toe because it would also control the amount of stress flow into and out of the attachment. The first step in the process was to determine the minimum thickness and height dimensions that make these dimensions behave as if they were infinite when compared

45 K [ksi (in)] Riggenbach 37 to the length. In other words the length of the attachment was the limiting dimension and the other two dimensions would not be contributing any correction factors. A set of models were created with L = 80 while the height of the attachment and thickness were individually increased until the stress intensity factor no longer increased, which indicated the length was finally the limiting variable. Below is a plot illustrating the process for varying height of attachment against the measured stress intensity factor Height of Attachment vs K 6 L=80" Height of Attachment [in] Figure 12: Stress Intensity Factor vs. Height in developing the Finite Length Figure 12 shows once the height of attachment reaches 24 inches, the stress intensity factor becomes asymptotic and thus an increase in height no longer affects the correction factor. The same process was repeated for the thickness of attachment, and resulted in an infinite thickness of 40 inches. Because both infinite configurations could not be developed simultaneously, after the stress intensity factor leveled off for the

46 Riggenbach 38 thickness, the height of attachment was then increased again to ensure it was still in its infinite configuration and that the length of attachment was the limiting dimension. After this recursive process had determined the infinite height (H=24 inches) and thickness (t = 40 inches), the length of the attachment was varied to smaller dimensions and the stress intensity factors were measured. The data from this process can be seen in Table 4. Length [in] K [ksi in ] Table 4: Finite Length of Attachment vs. K The next step in developing the correction factor was to normalize the data by the stress intensity factor for the infinite condition. As previously noted, the infinite condition can not be reached for all three dimensions concurrently. To capture the variation of stress intensity factor with length, the infinite condition became the case when L was at its upper bound and the thickness and height were in the infinite condition. In this case, L=80 inches, t=40 inches, and H=24 inches. By normalizing the data in this way it gives the value of the correction factor for shorter lengths of attachments and returns a value of 1 when the length of attachment is at its upper bound. The results of this normalization can be seen in below.

47 Riggenbach 39 Length [in] F(L) Table 5: Finite Length Correction Factor Values The final step in developing the finite length correction factor was simply to fit the data from Table 5. The data was fit using least square techniques in Matlab and Tablecurve 2D. TableCurve 2D was utilized because the software has a large internal database of functions that it fits to the data and then presents a table of all the fits with filtering options for the user to choose the best function for their application. In this case, the criterion for the correction factor was first to be as accurate as possible and second to do so with the fewest number of coefficients so the correction factor would be easy to apply. The equation and plot of the fit can be seen below in Figure 13. Figure 13: Finite Length Correction Factor Plot [2D]

48 Riggenbach 40 Figure 13 chosen fit was a combination of a square root term and natural logarithms. Only three coefficients were required to accurately fit the data. The coefficient of determination (r 2 ), a measure of the fitted outcomes to those predicted by the data, for this fit was which is extremely close to the ideal fit of 1. The highest percent residual (error) from the fit was only 0.742% at the smallest length (L=1 inch). To illustrate the goodness of fit, the residual (error) percent at each data point was plotted in Figure 14. Figure 14: Residual % Plot for Finite Length Correction 2D At this point the correction factor for the length has been defined with height and thickness at infinite configurations under uniform tension. The next step was to repeat this same procedure for the height correction factor. For the height of 24 inches, the relative infinite thickness and length configurations had to be determined so that in this case the height would be the limiting finite dimension affecting the stress intensity factor. After these infinite configurations had been captured, the height of the attachment was varied to smaller dimensions to capture the behavior of the finite

49 Riggenbach 41 height correction factor. The stress intensity factors were normalized by the infinite thickness, infinite length, and upper bound finite height stress intensity factor. The resulting stress intensity factors and height correction factor values can be seen in Table 6. Again, the correction factor values were determined by dividing the stress intensity factors by the largest height so that the correction factor goes to 1 at the height upper limit. Height [in] K [ksi in ] F(H) Values Table 6: Finite Height Stress Intensity Factors and Correction Factor Values [2D] It should be noted that while the stress intensity factors in the Table 6 are greater than the length stress intensity factors (Table 4), the important takeaway are the correction factor values because ultimately the behavior of the stress intensity with height is what will be utilized in predicting future stress intensity values. To develop the height correction factor then, the values above were plotted and fit using Matlab and Tablecurve 2D. The same criterion for the best equation was utilized in the selection of the height correction factor. The resulting best fit plot and equation can be seen in Figure 15 below.

50 Riggenbach 42 Figure 15: Finite Height Correction Factor Plot [2D] The coefficient of determination for this fit was which is a very good fit. To illustrate the goodness of fit the residuals and percent residuals were again computed and are shown in Table 7 below. Height [in] F(H) Value Fit F(H) Residual Residual% Table 7: Residuals from Finite Height Correction Factor [2D] This concludes the finite height correction factor. The last step is to determine the finite thickness correction factor by using the now defined process. The upper bound on the thicknesses was set at 40 inches. Once the infinite height and length were determined for the upper thickness bound, the thickness dimension was varied. The stress intensity and correction factors for varying finite thickness can be seen below.

51 Riggenbach 43 Thickness [in] K [ksi in ] F(t) Table 8: Finite Thickness Stress Intensity Factors and Correction Factors [2D] The best fit for the finite thickness correction can be seen plotted below in Figure 16. In order to accurately fit the thickness correction factor, four coefficients were necessary. The best fit equation was again a combination of polynomial terms and a natural logarithm. Figure 16: Finite Thickness Correction Factor Plot [2D]

52 Riggenbach 44 The coefficient of determination for this fit was To further illustrate the goodness of fit the residuals and residual percentages were calculated and can be seen below in Table 9. Thickness [in] F(t) Value Fit F(t) Residual Residual% Table 9: Residuals from Finite Thickness Correction Factor [2D] Table 9 shows that largest percent residual is %. This concludes the development of the thickness correction. Accuracy of Individual Correction Factors The next step was to test the accuracy of these correction factors at predicting a set of data. A set of models was created for varying thickness and length while the height was held constant. The height was held constant so that a three dimensional plot could be made of the data to visually see how the data and correction factors behavior compares. The thickness ranged from a lower limit of inches to an upper limit of 40 inches. Likewise, the lengths ranged from a lower limit of 1 inch to an upper limit of 80 inches. The constant height of the attachment was arbitrarily set at 4 inches. Table

53 Riggenbach below contains the stress intensity factors against the thickness (horizontal) and length (vertical). Length of Attachment [in] K [ksi (in)] Thickness of Base [in] Table 10: Stress Intensity Factors for varying Length and Thickness The best way to illustrate the trends in the data set above are by plotting the two 2-D curves: one plot of stress intensity factors as a function of the length for multiple thickness curves (Figure 17) and one plot of stress intensity factors as a function of thickness for multiple length curves (Figure 18).

54 K [ksi(in)^0.5] Riggenbach Length of Attachment vs. K Length of Attachment [in] t=40" t=30" t=20" t=15" t=10" t=5" t=2" t=1" t=0.75" t=0.5" t=0.25" t=0.125" Figure 17: Varying Length of Attachment versus Stress Intensity Factor for Varying Thickness Curves Figure 17 shows all of the stress intensity factors have leveled off by the upper limits of 80 inches which indicates that the infinite length configuration has been reached. For the smallest thickness of inches, any finite length effects were below the lowest length evaluated. The thickness effects can also be seen in Figure 17 by looking at one specific length and then looking vertically upward at all of the thickness data points. For example, at L=80 inches moving up the curves shows that as the thickness approaches its upper limit of 40 inches the data points converge indicating the thickness is also reaching its infinite configuration. The next figure better illustrates the stress intensity factors behavior with respect to thickness. Figure 18 reverses the places of length and thickness from Figure 17.

55 K [ksi(in)^0.5] Riggenbach 47 Thickness of Attachment vs. K L=80" L=60" L=30" L=20" L=8.2" L=1" Thickness of Attachment [in] Figure 18: Varying Thickness vs. Stress Intensity Factor for Varying Length Curves The only real trend that can be elicited from Figure 18 at this point is the infinite thickness trend. Similar to the length configuration, the various length curves are all asymptotic. The last step to visualize this data set is to combine Figure 17 and Figure 18 into a surface plot of thickness versus length versus stress intensity factor. Figure 19 below illustrates this surface plot. Figure 19: Surface Plot of Stress Intensity Factors as a function of Thickness and Length

56 Riggenbach 48 To assess the accuracy of the finite correction factors, the predicted stress intensity factors were computed and the error between the predicted and measured stress intensity factors was calculated. The results of this process can be seen below. K Predicted by L [in] t [in] H [in] F(H) F(t) F(L) K(inf)*F(L)*F(H)*F(t) Actual K Error % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

57 Riggenbach 49 K Predicted by L [in] t [in] H [in] F(H) F(t) F(L) K(inf)*F(L)*F(H)*F(t) Actual K Error % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Table 11: Comparison of Predicted Stress Intensity Factors from 2D fits vs. Measured Stress Intensity Factors

58 Riggenbach 50 The data in Table 11 illustrates that isolating the finite dimension correction factors and multiplying them together does not accurately predict the stress intensity factors. The largest percent error from the data set was % and the smallest percent error was 1.938%. The reason for these large discrepancies is simply due to the shift from finite to infinite behavior. Each individual correction was derived on the basis that the other two dimensions were in their infinite configuration and it was assumed that interaction between two finite dimensions would be simply modeled by multiplying their correction factors. However, as one dimension gets smaller, it lowers the infinite configuration for the other two dimensions. For example at L=80 inches, the infinite height configuration wasn t reached until H=24 inches. As the length becomes smaller, the height required to reach the infinite configuration also shrinks. Thus for a length of 10 inches, the height required to reach the infinite height configuration is also lowered to possibly 10 inches. Yet, the current height correction factor would be applying a correction factor still for all heights greater than 10 inches until 24 inches because it assumes that H is behaving in a finite way. All the correction factors were derived with an absolute reference to their upper limit so that any height less than 24 inches will have a correction factor, and length less than 80 inches will have a correction factor, and so on. The scaling complication occurs for all the dimensional relationship. Returning to the data set, all but a few of the predicted stress intensity factors lie below the actual data. The predicted stress intensity factors that are most accurate are the ones closest to the original upper limits because these correction factors were all

59 Riggenbach 51 near 1 so their multiplication doesn t introduce much error. The predicted stress intensity factor surface for the given set of data can be seen below in Figure 20. Figure 20: Plot of Predicted Stress Intensity Factors from 2D Correction Factors The predicted stress intensity surface (Figure 20) can be compared with the actual measured stress intensity factor surface (Figure 19). The most noticeable comparison is the shape of the individual curves. Taking 2D slices of the each surface and plotting them on the same graph illustrates the difficulty in capturing the true interaction of these dimensions. Below curves of the predicted and measured data are plotted for two length curves (20 and 1 ).

60 Riggenbach 52 Figure 21: Comparison of Predicted to Calculated Stress Intensity Factors [2D] As the thickness increases in Figure 21, the predicted stress intensities gradually rise while the measured stress intensity factors rise at a very steep slope and then levels off. The L=1 curve in the measured data illustrates that when the thickness is extremely small (t <0.75 inches), it is in a finite configuration, where there is a thickness configuration, and as the thickness gets larger (t >0.75 inches) it quickly reaches an infinite configuration, where there is no thickness correction factor. On the other hand, the predicted stress intensity factors were derived on the fact that the thickness dimension would be in a finite configuration so the thickness correction factor is always present over the whole range of thickness less than the upper bound of 40 inches. For the L=20 curve the measured data doesn t reach an infinite configuration until approximately 8 inches. Also, it should be noted that for the longer length in this figure the transition to the infinite thickness is much more

61 Riggenbach 53 gradual whereas for the shorter length attachment this transition from finite to infinite is sharp point. This illustrates that not only does the point at which the infinite configuration is reached depend on the other two dimensions but also the form of the behavior is dependent. One of the other options that was available when creating these curves was to instead create curve fits each dimensions going from finite to infinite configuration so that rather than having as the correction factor reaches its upper limit and beyond the correction factor never goes above one. Again though, the same problem would be present because the point at which the infinite configuration is reached is always going to be dependent on the other dimensions. To summarize, the difficulty in accurately predicting the stress intensity factor for multiple variable dimensions is complicated by the fact that whether each dimension is in its infinite or finite configuration is dependent on the other two dimensions. To accurately predict the stress intensity factor, the relationships between the dimensions to predict whether a dimension is in its infinite or finite configuration would first have to be determined. The situation is then more complicated because the form of the finite behavior is not constant. The form or function governing the finite behavior is also dependent on the other two dimensions. Surface Fitting Approach Until the relationship between the dimensions can be determined, solution is to vary only two dimensions at a time. By varying only two dimensions, it allows a surface to be fit to the data over a range that captures when each dimension is in the infinite and finite configuration. As a result, the empirical function to fit the data will be

62 Riggenbach 54 captured in a surface and the varying transition point from finite to infinite behavior will be captured. The data in Table 10 is an example of this type of data. In this case the height was held constant while the thickness and length go through both infinite and finite configurations. By fitting the data, any thickness and length combination less than their respective upper bounds can be predicted for a height of 4 inches. Additionally, this thickness and length combination can then be combined with a second surface fit that incorporates varying height. If the thickness was held constant, the height and length dimensions could be varied to provide a second surface fit. This second surface would allow one to predict the stress intensity factor for any height and length combination by capturing the full finite and infinite relationship between H and L. Logically, a surface of varying thickness and height could also be created to capture the variation although only one of these surfaces is necessary. The process will be further described in the following data. The fitting of surfaces can be accomplished in two ways: first by fitting the entire surface with one plane equation, or secondly by fitting the surface with two curves and one plane equation. Both methods will be shown in the following data. Initially, it may seem that 1 surface equation would be easier to apply than 2 curves and surface equation, which is true in terms of coefficients and overall equation length. However, curves usually provide higher accuracy in fitting than surface fitting because less data is required. First, fitting the entire surface with one equation will be shown. All of the stress intensity factors from Table 10 (pg 38) were normalized by the largest stress intensity

63 Riggenbach 55 factor, which for this height of attachment the largest stress intensity factor was for an infinite thickness and infinite length. The result of the normalization process can be seen below in Table 12. Length of Attachment [in] F(t,L) Thickness of Base [in] Table 12: Normalized Thickness Length Stress Intensity Factors F(t,L) Surface fitting is generally more difficult than curve fitting because the equations needed to fit the data increase in coefficients, thus requiring larger data sets to accurately fit the coefficients and data. Frequently, an interpolation process of the data is necessary to generate additional intermittent data or to convert sparse data sets into uniform grids. Sparse data sets allow the fitting surface greater freedom to behave in any manner between data points. In this case though the resulting surface fits showed no improvement in accuracy with interpolation methods. The chosen best fit of the data was a series of natural logarithmic terms. The log fit had a total of 10 coefficients. Figure 22 shows the resulting logarithmic fit of the data followed by the fit equation. To clarify, some of the points on the extreme left side of the surface do not look to be on the surface. This was simply a graphical problem because the surface is broken up into many evenly distributed segments (I.e. the lines running perpendicular to the thickness axis and the length axis). At the far left, the stress intensity factors is changing

64 Riggenbach 56 very rapidly, so the only way to get the surface to extend down into this region would have been to drastically increase the density of the lines. The increased density would have made the rest of the graph illegible. One can simply plug points into the fit equation to check that indeed, the surface does accurately predict these points. Figure 22: Stress Intensity Factor as a function of Thickness and Length (Surface Fit) The size of the equation necessary to capture the data was complex, which may pose as a limitation for the application of this correction factor, though it was not wholly unreasonable. Table 13 below showcases the goodness of fit for the logarithmic fit of the surface from Figure 22.

65 Riggenbach 57 Ln() Series r^2 = Residual Residual % Max Min Average E Table 13: Goodness of Fit Comparison for F(t,L) Surface The residuals shown in Table 13 are obviously very small but the correction factor values only vary between 0 and 1 so the percent errors are much higher with a maximum error of 7%. Larger residuals occur at the lowest length and thickness values where the gradients were the highest. This concludes development of the surface length and thickness correction factor. The second method of fitting the data, using two curve-fits (one for the thickness correction and one for the length correction) and one interaction surface will now be illustrated. For the thickness correction factor, the data at L= 80 inches was normalized by the largest thickness which was the infinite thickness for the length (t=40 inches). This data provides the correction factor values for the thickness fit. The result of this normalization process can be seen below in Table 14. The data from Table 14 was then curve fit using Matlab and TableCurve 2D. The resulting correction factor can be seen below in Figure 23.

66 Riggenbach 58 K [ksi in] F(t) F(t) Thickness of Base [in] Table 14: Finite Thickness Correction Factor Data The title bar of Figure 23 also shows the polynomial function form with five coefficients. Figure 23: Finite Thickness Correction Factor (2D) Figure 23 shows that the polynomial equation fits the data extremely well. However, to more accurately measure the goodness of fit the residual percent (or error percent of the original correction factor values) at each data point were plotted in Figure 24. The title bar of Figure 24 shows the coefficient of determination (r 2 ), a measure of the fitted outcomes to those predicted by the data, for this correction, as which is extremely close to the ideal perfect fit of 1. The largest residual percent is only 0.293% which indicates a very good fit for the thickness correction factor.

67 Riggenbach 59 Figure 24: Residual % Plot of Thickness Correction Factor This process was repeated for the length correction factor. The data can be seen as the final column in Table 10. Because there were fewer lengths than thicknesses initially analyzed in Table 10, a number of additional lengths were analyzed for the infinite thickness of 40 inches to provide a better curve fit of the data. The length correction factor values were found by normalizing the data by the infinite length of 80 inches. The resulting normalized values data and stress intensity factors can be seen below in Table 15. K [ksi in] F(L) F(L) Length of Attachment [in] Table 15: Length Correction Factor Values F(L) The data in Table 15 was again fit using MatLab and Tablecurve 2D. The resulting fit of the data and equation can be seen below in Figure 25. The fit chosen was of polynomial form, although to accurately fit this data eight coefficients were required.

68 Riggenbach 60 Figure 25: Finite Length Correction Factor Figure 25 above also looks to accurately fit the data but to check the goodness of fit the residual percents were plotted in Figure 26. The coefficient of determination for this fit was The largest percent residual in this case was slightly larger than the thickness fit at %. Figure 26: Residual % of Length Correction Factor F(L) Lastly, to determine the interaction surface for the given set of data, or the variation in infinite length point, the predicted stress intensity factors by the curve fits alone was found. The equation used to calculate these stress intensity factors can be seen below.

69 Riggenbach 61 KFinite Thickness and Length KInfinite Thickness and Length F( L) F( t) Equation (14) The infinite thickness and infinite length stress intensity factor is obtained from the data set of varying thickness and length for constant height. The value of the infinite stress intensity factor was ksi in. The predicted finite stress intensity factors were computed and are shown in Table 16. K*F(t)*F(L) Thickness of Base [in] F(L) F(t) Length of Attachm ent [in] Table 16: Predicted Stress Intensity Factors by F(t) and F(L) curve fits As expected, the predicted stress intensity factors are much below the measured finite element ones at the lower limits because as the correction factors approach their lower limits they are predicting finite behavior for both dimensions when in actuality infinite behavior for each is more easily reached. To determine the interaction surface correction factor values, the measured stress intensity factors were divided by those predicted stress intensity factors in Table 16. The data for the interaction correction factors can be seen below.

70 Riggenbach 62 Length of Attachm ent [in] Interaction Thickness Length Correction Factor Values Thickness of Base [in] Table 17: Interaction Thickness Length Correction Values The values in Table 17 were then fit using Tablecurve 3D. The resulting surface fit and equation are shown in Figure 27. Figure 27: Interaction Thickness Length Correction Factor F(t,L)*