4. Methodology
4.3 Steps of the simulation
4.3.1 Geometry Definition
4.2 Tools Employed in the Study
This section discusses the software used to perform the FEM analysis as well as the formulas used to validate the results. These software tools were already specified within the scope of the study since one of the goals of the study is to established if Franc2D is able to provide realistic estimations of the stress intensity factors (SIF).
4.2.1 Casca
Casca is a pre-processing platform to generate the geometry. It also provides the options to divide the geometry into different segments. The output file from Casca is the input file which can be imported in Franc2D to further apply the boundary conditions and run the simulations [19].
4.2.2 Franc2D
Fracture Analysis Code (Franc2D) is a freeware software developed by Cornell Fracture Group [19]. The program uses Linear Elastic Fracture Mechanics (LEFM) concepts and modified crack closure techniques to perform the numerical analysis for stress intensity factors. This approach corresponds to the recommended approach defined within the scope of this study, i.e., crack propagation theory.
4.3 Steps of the simulation
FEM analysis was required to obtain the SIF values for the crack propagation path. The analysis was performed on Franc2D. Following sections explain the step-by-step process used to obtain the SIF values.
4.3.1 Geometry Definition
The geometry construction is done using pre-processor called Casca (see section 4.2.1). A single quarter model is used for all the non-load carrying joints since the throat thicknesses for all the specimens are the same (Table 1).
However, different models are constructed for load-carrying joints because of varying throat thicknesses (see Table 2). Quarter models are then divided into three different segments to incorporate different mesh densities throughout the specimen. A refined mesh is introduced in the vicinity of weld throat for non-load carrying specimens and around weld root for
load-Figure 12: Meshed non-load carrying specimen.
Refined Mesh
Methodology
17 carrying joints (Figure 12). The mesh elements used in the study are simple quadrilateral Q4 elements which are enough for the analysis of cruciform joints.
The model generated by Casca was imported in Franc2D. The loads, material, and boundary conditions were defined as discussed in section 4.4. After the definition of the boundary conditions, analysis was performed for the un-cracked non-load carrying specimen. The maximum tensile stresses (in Y direction) were obtained at the weld toe, as indicated in Figure 13. Hence, the weld toe, in line with the literature studies, has been used as the stress intensive location and also the spot for crack initiation in the study.
In accordance with the maximum stresses, a crack of 0.1 mm was induced at the location of weld toe. This crack was then grown for 49 steps with each step of 0.01mm of crack increment. For crack growth, automatic crack propagation option was used in Franc2D which grows the crack in the direction of maximum hoop stresses around the crack tip (Section 4.4.1). Once, the cack is induced, the program automatically refines the mesh to better predict the SIF values [26], see Figure 14a. The automatic crack propagation is also followed by automatic analysis after each step of crack growth, resulting in an SIF value
for each iteration of crack growth. The crack growth path has been shown in Figure 14b.
A sample SIF curve generated by Frnac2D has been shown in Figure 14c.
The SIF values given by Franc2D were then exported out of the program in the form of .txt file to do the post-processing in excel. The integrated form of Paris Law, as discussed
Figure 13: Tensile stresses in a loaded non-load carrying specimen.
Figure 14: Non Load Carrying Joint. a) Crack initiation. b) Crack propagation path. c) SIF history.
14a) 14b) 14c)
Methodology
18
in section 3.1.1, was applied to obtain the fatigue life results. These results were then compared to the laboratory test results, as discussed in next sections in detail.
For load-carrying joint, an initial crack of 5mm was introduced at the left edge of the specimen in order for the crack tip to lie at the weld root. The crack tip served as non-penetrated weld gap creating a location for high stresses at the weld root, which is also the case for load carrying cruciform joints. The high stress values are shown in Figure 15.
From the crack tip of the initial crack, as indicated in Figure 16a, the crack was grown over 60 small steps with each steps leading to a crack growth of 0.01 mm. Like non-load carrying specimen, automatic crack growth option was used to propagate the crack. As seen in Figure 16b, the crack path curved at around 45° as the crack grows further towards fracture.
The SIF values generated from Franc2D, as shown in Figure 16c, were used as inputs to the Paris integration equation (see section 3.1.1) along with C and m values from BS7910 [5] to calculate the fatigue life for load carrying joints. These results are then compared to the laboratory test results to obtain a correlation, if possible. The detailed results are discussed in later parts of the study.
Figure 15: High stress location at weld root for load-carrying specimen.
Figure 16: Load carrying cruciform joint. a) Crack initiation. b) crack growth path. c) SIF history.
16b)
16a) 16c)
Methodology
19 4.4 Specifications of the Model
The scope of this study involves investigation of two types of joints. Non-load carrying joints, as discussed previously in section 2.5.1, fail at weld toe where the stress concentrations are high. On the other hand, load carrying joints, see section 2.5.2, fail from the weld root. The material and geometry parameters of non-load carrying joints are given in 4.4.1. The model specifications of load carrying joints are described in 4.4.2.
4.4.1 Non-Load Carrying Joints
The material used in the study is a steel alloy with Modulus of Elasticity of “E” 206 GPa and poison ration “v” of 0.3. The material specifications remain consistent throughout the study and do not change. Non-load carrying joints are modeled in such a way that a distributed tensile load is applied at the top edge. Crack is initiated at the weld toe. The direction of the load and the crack propagation path are indicated in Figure 17. Due to the symmetry of the joint, only a quarter model is used. The node on the bottom left is contrained in both x and y directions, whereas, the bottom and left edges are constrained in y and x directions, respectively, as shown in Figure 18.
Figure 17: Non Load Carrying Specimen Figure 18: Quarter model with x and y constraints.
Four models with varying nominal stresses and throat length have been employed for the study. The specimens are shown in Table 1.
Table 1: Specifications Non-Load Carrying Joints
Parameter Unit S235-T-1-1 S235-T-1-2 S235-T-1-3 S235-T-1-4 Nominal Stress MPa 241.14 159.94 260.30 134.14
Throat Thickness mm 5.0 5.0 5.0 5.0
Test
Temperature
°C -20 RT -20 RT
Crack propagation path
∆𝜎 = Nominal Stress ∆𝜎 = Nominal Stress
Methodology
20 4.4.2 Load Carrying Joints
The material properties for load carrying joints are same as mentioned in 4.4.1. Also, the quarter model and the constraints mentioned in section 4.4.1 have been used.
Like load carrying joints, non-load carrying joints are also modelled with tensile loads applied at the top edge. As discussed in 2.5.2, load carrying joints have high stress intensities at the weld throat due to lack of penetration. To incorporate the lack of penetration in the model, an initial crack is induced at the left edge in such a way that the crack tip of the induced initial crack is located at the weld throat. Crack is then propagated in small steps from initial crack yielding a crack path at 45° of angle, as shown in Figure 19.
Four specimens with different nominal stress values and throat thickness are analysed for the study, as shown in Table 2.
Table 2: Specifications of Load Carrying Joints.
Parameter Unit S235-C-1-1 S235-C-1-14 S235-C-3-14 S235--1-2 Nominal Stress MPa 295.73 111.13 126.30 175.59 Throat
Thickness
mm 5.70 5.96 5.48 5.77
Test
Temperature
°C -20 -20 -50 RT
Crack Propagation Path
∆𝜎 = Nominal Stress
Initial Crack
Figure 19: Load Carrying Cruciform Joint.
Methodology
21
4.5 Measurement of Throat Thickness for Weld Geometries
Welded structures are prone to stress concentrations at the weld locations which directly influence the fatigue life of the welded components. In order to arrive at accurate estimations of stress concentration factors, a precise measurement of geometry is vital.
One of the most relevant geometry factors for such fatigue life investigations of fillet welded joints is the throat thickness. Hence, it is felt important to mention the exact method used to measure the exact throat thicknesses for the welded joints under consideration in this study. The throat thicknesses for the joints referred in this study have already been presented in Table 1and Table 2.
The fillet welded joints under consideration have been measured using a manual method at institute M10 of the Technical University of Hamburg (TUHH). This method is based on collecting the dense point cloud using a 3D scanning device which is imported into a scan-to-3D software called rapidform. The point cloud is further converted into a closed space with 3D triangulation approach leading to polygon meshes. Flank angle and toe radius make the basis to characterize the specimen geometry in the concerned method.
Toe radius is measured by making equally spaced cuts (Figure 20) on the surface of the created model. Flank angle is determined by fitting a circle at the weld toe, whereas, the flank angle is determined by two tangent lines which are linearly fitted to the base plate and the weld flank, see Figure 21 [27].
Figure 20: Equally spaced cuts made after triangulation. [27]
Figure 21: Weld toe radius and flank angle determination. 21a) Circle fitting for toe radius. 21b) Tangent lines fitting for flank angle. [27]
21a) 21b)
Results and Discussion
22
5. Results and Discussion
In this section, the results of the study have been discussed in detail. The stepwise method followed for FEM analysis to obtain SIF values is given in section 4.3. The results for SIF values as well as fatigue life have been presented 5.1 followed by validation of SIF values through the of formulas in 5.2.
5.1 Fatigue Life Results
The SIF values obtained from numerical simulations in section 4.3 have been used in the crack propagation formulae (presented in section 3.1.1) to obtain the fatigue life values (results identified as Franc2D in Figure 20). The results obtained from crack propagation formula have been compared against the fatigue life values from destructive tests, data referred to as Real Time Tests.
As seen in Figure 20, there are high errors in the fatigue life values obtained from crack propagation approach and those of the test results. For specimen S235-T-1-1, the crack propagation approach underestimates the fatigue life by 70%. Similarly, for specimen S235-T-1-2 and S235-T-1-3, the fatigue life predicted by the formula is lower by 86% and 52%, respectively. Unlike former three specimens, for S235-T-1-4 the crack propagation formula overestimates the fatigue life by 82%. The major difference between the specifications of S235-T-1-4 with the rest of the specimens is the nominal stress value.
The cyclic stress applied on S235-T-1-4 is significantly lower than the rest of the specimens as presented in Table 1. This lower stress value results in higher number of
Figure 22: Fatigue life results of crack propagation approach (Franc2D) vs. destructive laboratory tests for non-load carrying joints.
Results and Discussion
23
cycles towards lower crack lengths of the analysis, hence, resulting in overestimation of the number of fatigue cycles that this particular specimen can endure before fracture.
Fatigue life comparison of formula and test results for load carrying joints are presented in Figure 21. The trends shown in this comparison are similar to the non-load carrying specimens. Paris Law Eq. 3.2 underestimates the fatigue life for all the load-carrying specimens under consideration. The results calculated from the formula exhibit significantly lower fatigue life values with errors ranging 56-72 % as compare to the test results.
Hence, it can be said that the crack propagation approach can either underestimate the fatigue life or overestimate the values, depending upon the magnitude of cyclic stress applied on the model. The errors ranging from 56% to 86% for non-load carrying joints and 56 to 72 % for load-carrying joints suggest a further investigation into the SIF values obtained from Franc2D. The model specifications for FE analysis as well as the SIF values obtained from Franc2D are hence validated in the next section.
5.2 Validation of the FE Model and SIF values
As discussed in previous section, the fatigue life results obtained from Paris equation are significantly different from the lab test results, which leads to the question of validity of the SIF values obtained from Franc2D. These values are validated by calculating SIF values from the analytical formulas proposed in section 3.1 (Eq. 3.3 to Eq. 3.7) and then these values are compared against Franc2D values.
Figure 23: Fatigue life results of crack propagation approach (Franc2D) vs. destructive laboratory tests for load-carrying joints.
Results and Discussion
24
As seen in Figure 24, the trend is consistent for all the non-load carrying models, i.e., the values estimated by Franc2D remain slightly below those of obtained from analytical formulas and become higher towards the higher crack lengths. Hence, it can be said that the SIF values estimated by Franc2D are very close to the ones obtained from the formulas, at least for the crack lengths below 3.7 mm.
The SIF values obtained from Franc2D for load carrying joints are compared against the results obtained from the equations 3.8 to 3.10, as mentioned in section 3.1.3. As evident from Figure 25, SIF values are constantly underestimated by the formulas for the first two specimens Figure 25a and Figure 25b. However, the error between the formula and Franc2D increases as the crack growth values approach final fracture. For specimens given in Figure 25c and Figure 25d, a sudden overestimation of the SIF values can be seen by the formulas. This upward trend gives an indication to the limitations of the formula used here.
24 d) 24 c)
24 b) 24 a)
Figure 24: Crack growth trends from analytical formulae and Franc2D for non-load carrying models, 24a) Specimen S235-T-1-1, 24b) Specimen S235-T-1-2. 24c) Specimen S235-T-1-3. 24d) Specimen S235-T-1- .
Results and Discussion
25
Based on the results presented in Figure 24, it can be said that the SIF values for non-load carrying joints stand validated since the results from the formulas ´closely correlate with those of Franc2D. Although for load-carrying joints, Figure 25 shows larger errors towards bigger crack lengths, the errors are below 40% for almost 70 % of the crack growth path. Based on these results, the SIF values predicted by Franc2D for load carrying joints can also be considered valid and accurate.
Although the validity of SIF values obtained from Franc2D is already established, the question of large errors for fatigue strength, raised in section 5.1, still remains open. The underestimation of the fatigue life by crack propagation approach can be linked to three factors related to the 2D simulation, temperature effects and the chosen values for C and m constants in Paris equation.
Regarding the effect of temperature on fatigue life values for both load-carrying and non-load carrying joints, some studies have been performed at sub-zero temperatures. In the studies by Braun et al [24] and Bridges et al [28], the fatigue strength of the welded steel
23 c) 23 c)
23 d) 23 d)
25 d) 25 c)
25 a) 25 b)
Figure 25: Crack growth trends from analytical formulae and Franc2D for load carrying joints, 23a) Specimen S235-C-1-1, 23b) Specimen S235-C-1-14. 23c) Specimen S235-C-3-14. 23d) Specimen S235-C-1-2.
Results and Discussion
26
joints have been found to increase with a decrease in temperature. This temperature factor can explain a small portion of the errors between the calculated and tested fatigue strengths indicated by the fatigue life results, as some of the specimens have been tested on sub-zero temperatures as indicated in Table 1 and Table 2. However, this increase in fatigue life has been reported to be around 8% for -20°C and 15% for -50°C [24], which means that the testing temperature is not the single factor for the large errors under discussion.
The other factor to which can be associated with the errors are the values of constants in Paris Law. As seen in Eq. 3.1, fatigue life is dependent upon range of SIF values, C and m factors. As SIF values have already been verified, the large errors can be associated with the values chosen for constants C and m.
Another factor that can be associated with the underestimation of fatigue life by crack propagation approach is the 2D simulation factor. A similar study of fillet welded joints conducted by Fischer et.al [1] proposed similar results, where crack propagation approach has been identified as predicting very small fatigue life as compared to the fatigue life tests. These conservative assessments have been linked to the simplified 2D simulations and crack closures in the study. The study argues that the crack shapes can be semi-elliptical which is neglected in 2D simulations leading to conservative fatigue life assessments.
Conclusion and Suggestions
27
6. Conclusion and Suggestions
In an effort to evaluate the crack propagation approach when applied in a 2D configuration to welded fillet joints, this study utilized a 2D numerical simulator called Franc2D to grow the cracks from an initial crack length 𝑎𝑖 of 0.1 mm to a final fracture. The SIF values obtained from Franc2D along with C and m values from IIW recommendations were used in the integrated form of Paris Law to predict the fatigue life values. Based on the comparison of results from crack propagation approach against the test results, following conclusions can be withdrawn.
Franc2D can predict accurate crack growth curves for both load carrying and non-load carrying fillet welds.
SIF values obtained from Franc2D can be considered as valid after a very close correlation shown between the values obtained from Franc2D and that of analytical formulas proposed in the literature.
It can be deduced from the comparison of fatigue values from crack propagation approach against test results that crack propagation approach underestimates the fatigue life and gives a very conservative number.
Crack propagation approach underestimates or overestimates (in some cases) the fatigue life for non-load carrying joints with errors ranging from 52% to as high as 86%.
For load-carrying joints, crack propagation approach has been found to be constantly underestimating the fatigue life from 56% to 72 %.
Since the SIF values stand validated, one of the factors which could be linked to the underestimation of fatigue life by crack propagation approach is the values of C and m constants in Paris equation.
Another reason for underestimation of fatigue life is considered to be an over simplification of the crack growth path in 2D simulation. The crack growth path is thought to be semi-elliptical in reality, which is ignored in 2D simulation.
Since, fatigue life values obtained from crack propagation approach are not reasonable, further investigation into the temperature dependence of fatigue life was not plausible. However, because of the fact that some of the actual tests were conducted at sub-zero temperatures, this could be an additional factor in the increased fatigue life in actual tests.
As to which factor contributes largest to the underestimation of fatigue life, further investigations are necessary. A 3D simulation of crack growth path could be a next step into validation of the elliptical crack path argument. Further investigations are needed regarding C and m values to be used for welded fillet joints. Finally, if the results show acceptable errors after the implementation of aforementioned steps, a further investigation into sub-zero temperature effects could be feasible.
References
28
References
[1] C. Fischer, O. Feltz, W. Fricke, and P. Lazzarin, “Application of the Notch Stress
[1] C. Fischer, O. Feltz, W. Fricke, and P. Lazzarin, “Application of the Notch Stress