A toughness based deformation limit for X- and K-joints under brace axial tension

Abstract

This study reports a deformation limit for the initiation of ductile fracture failure in fatigue-cracked circular hollow section (CHS) X- and K-joints subjected to brace axial tension. The proposed approach sets the deformation limit as the numerically computed crack driving force in a fatigue crack at the hot-spot location in the tubular joint reaches the material fracture toughness measured from standard fracture specimens. The calibration of the numerical procedure predicates on reported numerical computations on the crack driving force and previously published verification study against large-scale CHS X-joints with fatigue generated surface cracks. The development of the deformation limit includes a normalization procedure, which covers a wide range of the geometric parameters and material toughness levels. The lower-bound deformation limits thus developed follow a linear relationship with respect to the crack-depth ratio for both X- and K-joints. Comparison of the predicated deformation limit against experimental on cracked tubular X- and K-joints demonstrates the conservative nature of the proposed deformation limit. The proposed deformation limit, when extrapolated to a zero crack depth, provides an estimate on the deformation limits for intact X- and K-joints under brace axial loads.

Keywords

circular hollow section (CHS) / tubular joint / fracture failure / deformation limit / J-integral

Introduction

Unstable fracture failure has emerged as one of the critical failure mechanisms in both onshore and offshore steel tubular structures subjected to extreme overloading events [ 1– 3], e.g., earthquakes and hurricanes. The engineering design against unstable fracture failure in these steel structures predicates primarily on a material requirement, measured by the qualitative Charpy energy levels, as outlined in existing design guidelines [ 4, 5]. This significantly simplified material requirement aims to provide a uniform criterion for all structural components and all foreseeable loading conditions during the design life of the structure. However, such a simplified criterion does not provide a quantitative evaluation against the fracture failure, in line with the load and resistance factor design often deployed in developing the structural design codes [ 4, 5]. The Charpy energy requirement does not recognize the geometry-dependent fracture resistance in the critical structural component, and the fabrication-induced (e.g., welding) variations in the material fracture toughness. The engineering assessment of fracture failure in welded tubular connections require a detailed evaluation of the crack driving force in relation to the material fracture toughness measured from standard fracture specimens.

The evolution of the fracture mechanics [ 6] has provided a number of convenient parameters in measuring both the crack driving force and the material fracture toughness, for example, the stress-intensity factor (SIF) for the linear-elastic fracture mechanics, the crack-tip opening displacement (CTOD) and the energy release rate J-integral for elastic-plastic fracture mechanics. Previous researchers have used these crack-tip parameters to quantify the crack driving force in tubular connections under both mode I and mixed-mode conditions [ 7– 12], to assess the competing failure mechanisms between the plastic collapse and the brittle fracture in welded tubular connections [ 13– 16], and to estimate the fatigue life of tubular joints [ 17– 19]. Some researchers [ 20, 21] have integrated the crack driving force and the material fracture resistance in the load-deformation representation of the welded tubular joint to reflect the effect of fracture failure in the joint and subsequently on the global structure. These studies have leveraged on the fundamental fracture mechanics theory in assessing the fatigue and fracture failure for welded tubular connections and often require a detailed evaluation on the crack driving force (measured either by the stress intensity factor or the J-integral) through a carefully designed finite element mesh with a refined crack-front modeling.

Direct applications of the above crack-tip parameters in routine engineering design to assess unstable fracture failure remains infeasible due primarily to the huge amount of the preprocessing effort involved in generating a detailed crack-front mesh for the welded tubular connections with a complex topology. Deformation limits have evolved as a convenient engineering criterion for welded tubular joints [ 22, 23] corresponding to the ultimate limit state or the serviceability limit state. Recent research works have led to deformation limits corresponding to the fracture failure in welded tubular joints based on very limited experimental evidence [ 24– 26]. Ahmed and Qian [ 27, 28] have proposed two different, but consistent, deformation limits at the initiation of ductile tearing in fatigue-cracked circular hollow section (CHS) X-joints subjected to brace in-plane bending, based on both the material fracture toughness level and on the failure assessment diagram approach. These two deformation limits render similar non-dimensional deformation limits for different geometric dimensions of the tubular joints and provide an approximate deformation limit for intact X-joints under brace in-plane bending when extrapolated to a zero crack depth.

This study extends the fracture toughness based deformation limit to circular hollow section (CHS) X- and K-joints under brace axial loading. The deformation developed in this study corresponds to the deformation level at which the crack driving force computed from a large-deformation, elastic-plastic analysis of a fatigue-cracked CHS joint reaches the material fracture toughness (quantified by the critical energy release rate, J_{Ic} , measured from small-scale fracture specimens). This study develops, through a comprehensive numerical investigation, a normalized deformation limit, applicable to a wide range of joint geometries and material toughness levels.

This paper starts with a brief description of the basic assumptions and procedures in developing the deformation limit corresponding to the initiation of the fracture failure. The next section presents the details of the numerical procedure and the validation of the finite element model. The following section discusses the variation of the deformation limits with respect to the different material and geometric parameters. The subsequent section derives the normalized deformation parameter, considering the variation in material toughness levels and geometric parameters. The following section confirms the conservatism of the proposed deformation limit against experimental results. The final section summarizes the findings of the study.

Methods and scope

This study proposes a fracture-toughness based deformation limit for fatigue-cracked CHS X- and K-joints under the brace axial tension. This deformation limit corresponds to the onset of ductile tearing in the steel material of a CHS joint. A circular hollow section joint often comprises of a primary member (the chord) and one or more secondary members (the braces). A complete definition of the joint geometry often employs the following parameters: the outer diameter of the chord ( d_{\mathrm{0}} ), the brace diameter to chord diameter ratio ( \beta ), the chord outer radius over the wall thickness ratio ( \gamma ), the chord length over the chord radius ratio ( \alpha ), the brace wall thickness to chord wall thickness ratio ( \tau ). In addition to the geometric parameters, this study uses the Ramberg-Osgood relationship [ 29] to describe the uniaxial stress-strain relationship of steel,

(1) \epsilon ={\displaystyle \frac{\sigma }{E}}+\mathrm{0.002}{\left({\displaystyle \frac{\sigma }{{\sigma }_y}}\right)}^n,

where {\sigma }_y refers to the material yield strength, E denotes the Young’s modulus.

This study entails the following assumptions in developing the deformation limit for CHS X- and K-joints subjected to brace axial loading:

1) The deformation limit requires a material fracture toughness value, J_{Ic} , measured from the conventional small-scale fracture specimens following procedures outlined in ASTM E-1820 [ 30]. This J_{Ic} value represents a plane strain fracture toughness under high plasticity constraints, and therefore a lower bound toughness value for the fracture resistance in large-scale three dimensional joints. This remains a reasonable assumption due to the typical high constraint experienced by the fatigue cracks in tubular joints [ 18]. The fracture specimens used in measuring the fracture toughness should preferably utilize the material near the heat-affected zone along the brace-to-chord intersection, where fatigue cracks often nucleate and grow.

2) This study assumes a surface breaking fatigue crack at the hot-spot location along the brace-to-chord intersection to define the initial condition in determining the deformation limit. The crack driving force, used in the comparison against the material fracture toughness, corresponds to the energy release rate, J-value, at the deepest crack-front location.

3) The deformation value, at which the crack driving force at the deepest crack-front location reaches J_{Ic} , represents the deformation limit for the CHS X- and K-joints. Such a deformation limit is inherently both geometry and material dependent. Engineering practice necessitates a simple deformation criterion, applicable over a practical range of geometric and material parameters. This study proposes, therefore, a normalized deformation limit with respect to both the geometric and material toughness levels.

The investigation follows the steps below to derive the global deformation corresponding to the onset of ductile tearing:

1) Step 1: Determine the material fracture toughness J_{Ic} from standard fracture specimens [ 30] for materials near the heat-affected zone along the brace-to-chord intersection. The fracture specimens should comply with the small-scale yielding conditions prescribed in the testing standard.

2) Step 2: Determine the J-integral at the deepest location along the front of a fatigue surface crack at the hot spot location in CHS joints, using the domain-integral approach implemented in large deformation, elastic-plastic analyses for a wide range of geometric and material parameters. Coupling the computed J-value with the applied displacement leads to a relationship between the crack driving force versus the load-line displacement.

3) Step 3: Based on the J­{\Delta }_{LLD} relationship, calculate the critical deformation level at J=J_{Ic} . In this study the critical deformation, {\delta }_{cr} , refers to the critical load-line displacement (axial deformation of the joint) normalized by the chord diameter, d_{\mathrm{0}}.

4) Step 4: Normalize the deformation limit for each joint with respect to the geometric and material parameters, so that the normalized deformation limit, when plotted against the crack-depth ratio, minimizes the scatter among different geometric and material parameters.

5) Step 5: Develop a lower-bound deformation limit for the geometric and material parameters considered.

This study selects two types of CHS joints namely X-joints with \theta ={\mathrm{90}}^o and K-joints with \theta ={\mathrm{60}}^o as shown in Fig. 1, representing two typical configurations of the CHS joints in tubular structures. For X-joints, the semi-elliptical surface crack resides at the saddle point of the chord member near the end of a typical grinding profile of the weld toe. For K-joints, this study assumes a semi-elliptical surface crack located at the weld toe of the crown point of the chord connected to the tension brace. Figure 1(a) shows the configuration of a typical CHS X-joint with \theta ={\mathrm{90}}^o , fabricated by profiling and welding the branch members (the brace) to the main member (the chord). The crack aspect ratio considered in this study remains within the typical range of the crack aspect ratios measured in realistic large-scale tubular joints [ 31]. The weld geometry of the CHS X- and K-joints follows the AWS specification [ 32]. The surfaces of the fatigue cracks remain perpendicular to the chord outer surface. The current finite element model omits the material in the weld-grinding profile to avoid highly distorted elements around the brace-to-chord intersection [ 33]. Figure 1(b) shows the typical configuration of a CHS K-joint with \theta ={\mathrm{60}}^o , which includes a surface crack near the tension brace weld toe.

Using a validated finite element (FE) procedure, the current study covers a wide range of geometric and material parameters. Tables 1 and 2 list the geometric and material parameters studied in this investigation for X-and K-joints respectively. For CHS X-joints, the current study investigates the crack driving force corresponding to different geometric parameters with three \beta ratios (0.3, 0.6 and 0.9) and four \gamma ratios (10, 15, 20 and 25). For a selected \beta (0.6) and \gamma (15) ratio, the current study also investigates the effect of three \tau ratios (0.50, 0.75 and 1.00), three material yield strength {\sigma }_y (350 MPa, 460 MPa and 690 MPa) while fixing the strain hardening exponent n at 5 and three strain hardening exponents (5, 10 and 20) while maintaining a constant material yield strength {\sigma }_y of 690 MPa. For each of the joint dimensions considered, the numerical study covers three crack depth ratios a/t_{\mathrm{0}} (0.07, 0.30 and 0.50) with the crack aspect ratio maintained at a/c=\mathrm{0.2} . The chord outer diameter for all X-joints remains fixed at 406 mm, with the brace-to-chord intersection angle at \theta ={\mathrm{90}}^o . The chord length to the chord radius ratio \alpha equals 16 to avoid the constraining effect of the chord ends. Similarly, the brace has a length equal to four times the brace outer diameter. The material Young’s modulus and Poisson’s ratio equals 203 GPa and 0.3, respectively, for the chord, brace and the welds in this study. The elastic-plastic material property, following the uniaxial true stress-strain relationship of a typical steel S355, as plotted in Fig. 2, remains the same for the chord, brace and weld materials for all geometric parameter variation models.

For CHS K-joints with \theta ={\mathrm{60}}^o , the parametric study covers four \beta ratios (0.3, 0.6, 0.9 and 1.0), three \gamma ratios (15, 20 and 25) and three g\text{'} ratios (2, 6 and 10). The chord outer diameter for all models remains fixed at 406 mm, with the brace-to-chord intersection angle equal to \theta ={\mathrm{60}}^o . The ratio \tau remains as a constant, or 1.0 for K-joints. The chord length ratio \alpha equals 16 to avoid the constraining effect of the chord ends, while the brace has a length equal to four times the brace outer diameter. The parametric investigation also examines the effect of the crack depth ratios ( a/t_{\mathrm{0}}=\mathrm{0.2} , 0.5 and 0.7) for selected K-joints, with the crack aspect ratio fixed at a/c=\mathrm{0.25} . The material Young’s modulus equals 203 GPa and the material Poisson’s ratio equals 0.3 for the steel materials and the welds considered in this study. The elastic-plastic material property follows the uniaxial stress-strain relationship in Fig. 2 and remains the same for the chord, brace and weld materials.

The material fracture toughness (often measured by the J_{Ic} value) depends on the manufacturing process and the chemical compositions of the steel and may demonstrate significant variations even for the same steel grade delivered by different steel suppliers. This study thus strives to develop a dimensionless deformation limit, which includes the J_{Ic} value in the normalization procedure, so that the deformation criterion developed becomes independent of the J_{Ic} value over the range of values investigated. The parametric study hence considers five fracture toughness values (100, 150, 200, 250 and 300 {\mathrm{kJ/m}}^2 ) representing a practical and wide range of fracture toughness values for steels.

Numerical modeling and verification

3-D FE Modeling

Figure 3 illustrates a typical FE mesh used for CHS X-joints under brace axial tension. The fatigue crack locates at the weld toe near the saddle point along the brace-to-chord intersection. An automatic mesh generation procedure based on the Patran command language generates the global continuous mesh of the FE models [ 34]. The detailed crack-front mesh follows the procedure in a professional fracture mesh generator [ 35]. The presence of three planes of symmetry permits the use of a one-eighth model with the number of nodes varying between 7,000 and 19,000 for different X-joints. The FE models shown in Fig. 3 employ 20-node brick elements with a reduced integration. The large-deformation, elastic-plastic analysis utilizes the open-source research code WARP3D [ 36].

The local crack-front model contains a crack plane oriented along the brace-to-chord intersection curve, consistent with the real fatigue cracks observed during the experiments [ 37– 39]. Each crack-front node contains 10 rings of elements with 16 elements in each ring. The modeling of the crack tip includes an initial circular notch of 20 mm in radius to enhance the numerical convergence at large deformations. A mesh-tieing procedure available in the WARP3D research code [ 36] connects the local crack-front mesh to the global continuous mesh similar to the procedures adopted and verified in previous studies [ 40, 41].

Similarly, the FE mesh for a typical, half-symmetric K-joint with a weld-toe crack consists of two parts: the global mesh and the local crack-front model, as shown in Fig. 4. The use of 20-node solid elements with reduced integration facilitates an accurate representation of the weld profile which follows the specifications in AWS [ 32]. The numerical model utilizes a rigid plate at the end of the chord and brace members, with its thickness equal to the wall thickness of the corresponding member, and the Young’s modulus equal to 100 times that of the corresponding member. The boundary condition imposed on the center node of the rigid plate follows that shown in Fig. 4(d) hich provides a conservative representation of the framing effect on the K-joint by adjacent structures [ 42]. The balanced loading condition also develops a higher elastic-plastic driving force, compared to other boundary conditions investigated in [ 33].

The local crack-front model contains a block extracted from the global mesh at the crown-point near the tension brace, as shown in Fig. 4(b). The orientation of the crack plane follows naturally the brace-to-chord intersection curve, consistent with the real fatigue cracks observed in test specimens [ 43]. At each crack-front location, the current FE model employs 8 rings of elements and 16 elements along the circumferential direction in each ring with a small, initially circular root notch about 50µm in radius at the crack tip to enhance the numerical convergence at large deformations, as shown in Fig. 4(c). The size of the first ring is about 3% of the crack depth. A typical FE mesh shown in Fig. 4(a) includes approximately 160,000 nodes and 32,000 elements. The two models (global and crack-front block) are connected via the mesh-tieing procedure, as shown in Fig. 4(b).

J-domain integral

This investigation utilizes the domain-integral approach, developed by Shih et al. [ 44] to compute the path independent, energy release rate along the curved front of the surface crack based on the displacement-strain-stress fields obtained from the very detailed, 3-D finite element analyses. The local value of the energy release rate at a crack-front point under a static loading condition follows,

(2) J=\underset{\Gamma \to \mathrm{0}}{\lim }{\displaystyle {\int }_{\Gamma }\left(W{n}_{\mathrm{1}}-P_{ij}{\displaystyle \frac{\partial u_i}{\partial X_{\mathrm{1}}}}{n}_j\right)d\Gamma },

where Г denotes a vanishingly small contour in the plane normal to the crack front at the crack-front point, and n_j refers to the unit vector normal to the contour Г, W is the strain energy density, and P_{ij} is the component of 1st Piola-Kirchhoff stress tensor. X_{\mathrm{1}} represents the local coordinate in a plane perpendicular to the crack-front line.

Verification of FE analysis

The current study verifies the FE analysis procedure against a detailed numerical study by Zhang [ 45], who computes the crack driving force using ABAQUS [ 46] along the front of a surface crack located at the chord weld toe near the saddle point in CHS X-joints under brace tension. The verification considers an X-joint with \beta =\mathrm{0.6} , \gamma =\mathrm{20} and \tau =\mathrm{1.0} . The semi-elliptical surface crack located at the saddle point along the brace-to-chord intersection has a crack depth ratio of a/t_{\mathrm{0}}=\mathrm{0.2} and a crack aspect ratio of a/c=\mathrm{0.25} . Figure 5 shows the close agreement in the elastic-plastic crack driving forces computed from the current numerical procedure and those reported by Zhang [ 45]. In addition to the comparison in Fig. 5, our previous investigations [ 27, 28] have verified the same numerical procedure used in this study against experimental results on CHS X-joints subjected to brace in-plane bending.

Results and discussion

J-D_LLD relationship of CHS X-Joints

Figure 6 displays the evolution of the crack driving force with respect to the axial deformation ({\Delta }_{LLD}/d_{\mathrm{0}}) for CHS X-joints. The J-integral values plotted in Fig. 6 correspond to the energy release rate computed from the domain integral approach at the deepest crack-front location in the X-joint. Figure 6a and 6b examine the J-D_LLD evolutions for the CHS X-joints with three different crack-depth ratios ( a/t_{\mathrm{0}}=\mathrm{0.07} , 0.30, and 0.50) for two different yield strengths, {\sigma }_y=\mathrm{355}\mathrm{MPa} and 690 MPa, respectively.

The J-values at the deepest crack-front position exhibits a strong dependence on the crack-depth ratio as evidenced in Fig. 6(a). At a large deformation level, the increase in the crack driving force slows down due to the plastic deformation in adjacent materials along the brace-to-chord intersection. Such plastic deformations become manifested when the crack driving force exceeds 300 kJ/m² for the X-joints shown in Fig. 6. A very small crack depth ratio leads to a slightly lower energy release rate than does the larger crack depth ratio, as shown in Fig. 6(a). Figure 6(b) confirms the similar trend in the computed J-integrals for different crack-depth ratios in the same CHS X-joint with \theta ={\mathrm{90}}^o and an enhanced material strength ( {\sigma }_y=\mathrm{690}\mathrm{MPa} and n=\mathrm{20} ). Figure 6(c) presents the J-D_LLD relationship for X-joints with different material yield strengths but the same strain hardening exponent ( n=\mathrm{5} ) and the crack depth ratio ( a/t_{\mathrm{0}}=\mathrm{0.3} ). The crack driving force at the deepest crack-front position does not indicate noticeable differences among varying material yield strength at relatively small deformation levels ( ), and shows a larger value for higher material yield strength at large deformations. Figure 6(d) displays the J-D_LLD evolutions for varying strain hardening exponents for a yield strength of {\sigma }_y=\mathrm{690}\mathrm{MPa} and a crack depth ratio of a/t_{\mathrm{0}}=\mathrm{0.2} . The J-D_LLD curves show negligible dependence on the strain hardening exponent n.

Figure 7(a) explores the effect of \beta ratios ( \beta =\mathrm{0.9} , 0.6 and 0.3) for X-joints with \gamma =\mathrm{15} , \tau =\mathrm{1} and a/t_{\mathrm{0}}=\mathrm{0.3} , on the crack driving force at the deepest crack-front location. Figure 7(a) confirms that the J-value in an X-joint with a small \beta ratio demonstrates lower crack driving forces than those in an X-joint with a large \beta ratio, at the same brace extension. A large \beta ratio facilitates a direct load transfer between the two braces of an X-joint under brace axial tension, through the chord materials near the saddle point. This leads to a higher crack driving force in X-joints with a larger \beta ratio. Figure 7(b) examines the effect of \gamma ratios ( \gamma =\mathrm{10} , 15, 20, and 25) on the CHS X-joint with \beta =\mathrm{0.6} , \tau =\mathrm{1} and a/t_{\mathrm{0}}=\mathrm{0.3} . Figure 7(b) shows that the crack driving force at the deepest crack-front position remains much higher in an X-joint with a thick chord wall (a low \gamma ratio) than that in an X-joint with a thin chord wall (a high \gamma ratio), at the same level of joint deformation. The same amount of brace deformation causes much severer plastic deformations along the brace-to-chord intersection in a thick chord wall than the plastic deformations in a thin chord wall. This leads consequently to a larger crack driving force in an X-joint with a lower \gamma ratio. Figure 7(c) compares the effect of different t ratios ( \tau =\mathrm{1.0} , 0.75 and 0.50) for the X-joint with \beta =\mathrm{0.6} , \gamma =\mathrm{15} and a/t_{\mathrm{0}}=\mathrm{0.3} . The J-value shows a weak dependence on the t ratio.

J-D_LLD relationship of CHS K-joints

Figure 8 displays the evolution of the crack driving force with respect to the normalized axial deformation ({\Delta }_{LLD}/d_{\mathrm{0}}) for CHS K-joints under a balanced brace axial load (see Fig. 4(d)). Figure 8(a) examines the J-D_LLD evolution for CHS K-joints with three different crack-depth ratios ( a/t_{\mathrm{0}}=\mathrm{0.2} , 0.5, and 0.7) for \beta =\mathrm{0.3} , \gamma =\mathrm{15} , g\text{'}=\mathrm{6} and \tau =\mathrm{1.0} . The J-values at the deepest crack-front position exhibits some dependence on the crack-depth ratio as revealed in Fig. 8(a). As the crack depth ratio increases, the corresponding domain integral value at the same level of deformation decreases slightly for K-joints with a relatively large gap ratio. Zhang [ 45] has performed a detailed investigation on the decreasing crack driving force with an increasing crack-depth ratio. The unexpected decrease of the crack driving force with respect to an increasing crack-depth ratio occurs in K-joints with a relatively large gap ratio. A large gap ratio forces the redistribution of the stresses to the intact material near the free surface ( \phi =\mathrm{0} , see Fig. 1(c) for the definition of f), leading to a larger crack driving force near the free surface than that at the deepest crack-front location, \phi =\pi /\mathrm{2} [ 45]. Figure 8(b) indicates a clear distinction in the J-D_LLD relationships for K-joints with \gamma =\mathrm{20} and g\text{'}=\mathrm{6} corresponding to different \beta ratios. The bending interaction between two crown points in the chord wall increases significantly as the brace axial load applied on a K-joint increases. The decrease in the \beta ratio reduces the local dihedral angle near the saddle point and enhances the local bending and shearing action in the chord wall along a correspondingly smaller weld length. Therefore, the increased intensity of the plastic deformation near the crack surface for a small \beta ratio introduces a higher crack driving force than that for a larger \beta ratio. For joints with a very large \beta ratio ( \to \mathrm{1.0} ), the very large local dihedral angle at the saddle point ( \to {\mathrm{180}}^o ) forces a significantly elevated stress field in materials near the crown point between the two braces. This leads to an increase in the crack driving force, as evidenced by the reversal trend in the J-values as b increases from 0.9 to 1.0. Figure 8(c) demonstrates the effect of the \gamma ratio on the crack driving force. The change in \gamma ratios varies the chord wall thickness, thus the bending action in the chord material for joints with the same chord outer diameter d_{\mathrm{0}} and the same \tau ratio. The decrease in the \gamma ratio introduces a larger magnitude of the bending action over the gap region in the K-joint (under the same joint deformation level), and thus produces a higher value of the J-integral. On the other hand, a decrease in the \gamma ratio reduces the chord wall compliance against bending, and generates lower values of the elastic-plastic J-integrals. The competing action of these two effects reduces the variation of the elastic-plastic driving force with respect to the \gamma ratio, as shown in Fig. 8(c). Figure 8(d) plots the variation of the elastic-plastic J-values for different gap ratios ( g\text{'} ) ranging from 2 to 10 for a specific \beta ratio and \gamma ratio. The chord wall between the two crown points experiences bending, shearing and membrane actions at large deformation levels. A lower gap ratio reduces the eccentricity between the two braces of a K-joint, and leads to an enhanced axial load resistance, compared to a K-joint with a larger gap ratio. Therefore, the crack located at the weld toe in a K-joint with a smaller gap experiences a higher crack driving force.

Normalization of elastic-plastic crack driving force

J-Integral variations

Figure 9 plots the variations of J-values at three different axial deformations ( {\Delta }_{LLD}/d_{\mathrm{0}}=\mathrm{0.01} , 0.015 and 0.02) with respect to the variations in the crack-depth ratios and non-dimensional geometric parameters for X-joints. Figure 9(a) indicates the strong dependence of the crack driving force on the crack-depth ratio for X-joints with \beta =\mathrm{0.6} , \gamma =\mathrm{15} and \tau =\mathrm{1.0} . The J-value increases substantially as the crack-depth ratio ( a/t_{\mathrm{0}} ) increases to 0.3. However, the crack driving force maintains an approximately constant value as the crack-depth ratio increases further from 0.3 to 0.5, implying that plastic deformations in materials adjacent to the deepest crack-front location hinders the development the crack driving force. Figure 9(b) shows the variation of the J-values at constant joint deformation levels with respect to the \beta ratio for X-joints with \gamma =\mathrm{15} , \tau =\mathrm{1} and a/t_{\mathrm{0}}=\mathrm{0.3} . Figure 9(c) elucidates an approximately linear relationship between the J-values (at fixed joint deformation levels) and the \mathrm{1}/\gamma ratio for X-joints with \beta =\mathrm{0.6} , \tau =\mathrm{1} and a/t_{\mathrm{0}}=\mathrm{0.3} .

Figure 10 plots the variation of J-values at three constant joint deformations ( {\Delta }_{LLD}/d_{\mathrm{0}}=\mathrm{0.01} , 0.015 and 0.02) with respect to the variations in the crack-depth ratios and non-dimensional geometric parameters for K-joints. Figure 10(a) indicates the strong dependence of the crack driving force on the crack-depth ratio for K-joints with \beta =\mathrm{0.3} , \gamma =\mathrm{15} , g\text{'}=\mathrm{6} and \tau =\mathrm{1.0} . The crack driving force exhibits an inversely proportional relationship with the crack depth ratio, as discussed above for Fig. 8(a). Figure 10(b) and (c) show the variation of the J-values at three fixed joint deformation levels with respect to the \beta and \gamma ratios, respectively, for K-joints with g\text{'}=\mathrm{6} , \tau =\mathrm{1} and a/t_{\mathrm{0}}=\mathrm{0.2} . The crack driving forces do not exhibit a distinctive trend with respect to the \beta or \gamma ratios of K-joints. Figure 10(d) demonstrates an approximately linear relationship between the J-values and the g\text{'} ratio for K-joints with \beta =\mathrm{0.6} , \gamma =\mathrm{15} , \tau =\mathrm{1} and a/t_{\mathrm{0}}=\mathrm{0.2} .

Normalized J for CHS X-joints

The critical deformation in this study refers to the deformation level at which the crack-driving force at the deepest crack-front position equals the material fracture toughness. As demonstrated by the numerical results in Fig. 9, this critical deformation exhibits strong dependence both on the geometric variations and on the measured material fracture toughness. To cover a wide range of geometric parameters and a practical range of toughness values, this study develops a dimensionless deformation criterion normalized against the geometric parameters and J-values.

Since the critical deformation depends on the crack driving force, J, the normalization procedure separates naturally into two parts: 1) the normalization of the J-value into a dimensionless parameter; and 2) the normalization against other dimensionless geometric parameters, e.g., b, g, g' and t. A previous study [ 47] has confirmed the insignificance of the chord length ratio \alpha in the computed crack driving force. This study, therefore, excludes the a ratio in the normalization process.

The normalization of the J-values aims to cover a practical range of the fracture toughness J_{Ic} values, which are often measured using the standard fracture specimens following the protocols outlined in material testing standard [ 30]. As the J_{Ic} measures the fracture resistance at the onset of ductile tearing in the fracture specimen [ 30], the plastic zone near the crack tip remains relatively small compared to other dimensions of the specimen, e.g., the thickness of the specimen. The small-scale yielding condition implies that the crack-tip condition has not deviated significantly from the theoretical framework of the linear-elastic fracture mechanics. The current study therefore normalizes the J value against the material’s elastic modulus, E, instead of the material yield strength ( {\sigma }_y ) as,

(3) J_n={\displaystyle \frac{J_{Ic}}{E\times y}}f(\beta ,\gamma ,\tau ),

where y refers to a dimensional parameter and f(\beta ,\gamma ,\tau ) defines a function based on the dimensionless geometric parameters. The numerical results in Fig. 6 confirm the relative insignificance of {\sigma }_y and n on the computed J-integrals, for small to intermediate J-values ( \le \mathrm{300}{\mathrm{kJ/m}}^2 ).

The normalization against other non-dimensional parameters follows the approximate relationships between the J-value and the geometric parameters illustrated in Fig. 9. This leads to,

(4) J_n={\displaystyle \frac{J_{Ic}\gamma }{\beta Ey}}.

The effect of \gamma translates essentially to the effect of the chord wall thickness on the computed J-values. The current study therefore simplifies Eq. (4) by integrating the effect of \gamma in the parameter y,

(5) J_n={\displaystyle \frac{J_{Ic}}{E\beta t_{\mathrm{0}}}}.

Normalized J for CHS K-joints

The normalized critical deformation for CHS K-joints follows a similar procedure as CHS X-joints. Following the correlations in Fig. 10, the critical deformation anticipates strong dependence both on the geometric variations and on the measured material fracture toughness. To cover a wide range of geometric parameters and a practical range of toughness values, the dimensionless deformation criterion includes both the geometric parameters and J-values.

Following the same rationale as CHS X-joints, the normalization against geometric parameters of K-joints follows the approximate relationships between the J-value and the geometric parameters illustrated in Fig. 10. This leads to,

(6) J_n={\displaystyle \frac{J_{Ic}\beta g\text{'}\gamma }{Ey}}.

The effect of \gamma translates essentially to the effect of the chord wall thickness on the computed J-values. The current study therefore simplifies Eq. (6) by integrating the effect of \gamma in the parameter y,

(7) J_n={\displaystyle \frac{J_{Ic}\beta g\text{'}}{E{t}_{\mathrm{0}}}}.

A fracture based deformation criterion

Figure 11(a) presents the critical deformation ( {\delta }_{cr}={\Delta }_{LLD}/d_{\mathrm{0}} ) corresponding to five different fracture toughness values and different crack depth ratios ( a/t_{\mathrm{0}} ) for CHS X-joints. For the same material and geometric properties, the critical deformation escalates with a decreasing crack size, at an increasing rate for small crack depth ratios. Figure 11(b) and (c) illustrate the critical deformation for X-joints with different geometric parameters. The critical deformation exhibits a strong scatter with respect to the material fracture toughness values, as indicated in Fig. 11(a), and with respect to the joint geometric parameters \beta and \gamma , as demonstrated in Fig. 11(b) to (c).

Figure 12(a) presents the critical deformation values corresponding to five different fracture toughness values and different crack depth ratios for CHS K-joints with \theta ={\mathrm{60}}^o . For the same material and geometric properties, the critical deformation escalates slowly with an increasing crack depth. Figure 12(b), (c) and (d) illustrate the critical deformations for K-joints against varying geometric parameters. The critical deformation exhibits pronounced scatters among both the material fracture toughness values and the geometric parameters, as observed in Fig. 12.

To minimize the dependence of the critical deformation of X- and K-joints on the material toughness and joint geometric parameters, this study normalizes the critical deformation against the geometry dependent, non-dimensional material toughness discussed in Section 5. In linear-elastic fracture mechanics, the elastic energy release rate remains proportional to the square of the remotely applied displacements, J\propto {\Delta }^{\mathrm{2}} . This study therefore normalizes the critical deformation against the non-dimensional crack driving forces for X- and K-joints presented above. The normalized critical deformation at the onset of ductile tearing thus follows,

(8) {{\displaystyle \delta }}_n={\displaystyle \frac{{{\displaystyle ({{\displaystyle \Delta }}_{LLD}/{{\displaystyle d}}_{\mathrm{0}})}}_{cr}}{{{\displaystyle J}}_n^{\frac{\mathrm{1}}{\mathrm{2}}}}}.

Substituting Eqs. (5) or (7) into Eq. (8), the critical deformations for X- and K-joints become,

(9) {\bar{\delta }}_{cr}={\displaystyle \frac{{\delta }_{cr}}{{\left({\displaystyle \frac{J_{Ic}}{E\times \beta \times t_{\mathrm{0}}}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}}}\mathrm{for\; X-joints},

(10) {\bar{\delta }}_{cr}={\displaystyle \frac{{\delta }_{cr}}{{\left({\displaystyle \frac{J_{Ic}\beta g\text{'}}{E{t}_{\mathrm{0}}}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}}}\mathrm{for\; K-joints}.

Figures 13 and 14 present the normalized critical deformation ( {\bar{\delta }}_{cr} ) based on Eqs. (9) and (10) for X- and K-joints, respectively. The normalized critical deformations show significantly reduced scatters among the material fracture toughness levels and joint geometric parameters.

Despite the normalization procedure, the critical deformations in Figs. 13 and 14 still indicate a certain degree of scatters among different fracture toughness values for the same crack depth ratios. This study therefore proposes a lower-bound deformation limit to estimate the onset of ductile tearing in both X- and K-joints, based on the normalized critical deformation values. The proposed lower-bound fracture deformation limit for X-joints under brace axial tension corresponds to the mean minus twice the standard deviation values for 236 numerical data points computed in this study, and follows a linear relationship with the crack-depth ratio,

(11) {\bar{\delta }}_{cr}=\mathrm{1.114}-\mathrm{1.096}\left({\displaystyle \frac{a}{t_{\mathrm{0}}}}\right).

Following the same procedure, the lower-bound deformation limit for K-joints under balanced brace axial loads, corresponding to the mean minus twice the standard deviation value for 210 numerical data points, retains a linear relationship with respect to the crack-depth ratio,

(12) {\bar{\delta }}_{cr}=\mathrm{0.244}+\mathrm{0.039}\left({\displaystyle \frac{a}{t_{\mathrm{0}}}}\right).

Figure 15 compares Eq. (11) and (12) against the numerical data for all X- and K-joints considered in the numerical investigation. Equations (11) and (12) show a conservative estimate compared to the numerical results.

Comparison with reported test data

This section presents a brief comparison against the experimental data reported in previous works [ 48– 50] for cracked X- and K-joints. Table 3 lists the geometric parameters of the CHS X- and K-joints extracted from previous studies. The crack area indicated in Table 3 refers to the percentage of the crack area over the brace-to-chord intersection area. For the CHS X-joints, the reported experimental study utilizes two different types of steels [ 48], each with a different fracture toughness level, 178 {\mathrm{kJ/m}}^2 and 315 {\mathrm{kJ/m}}^2 respectively. Figure 16(a) compares the experimentally observed critical deformation limit prior to the significant unloading caused by ductile tearing. The critical deformation limit calculated based on the proposed formula yields very conservative estimates as compared to the critical deformations observed during the experiments [ 48].

For CHS K-joints, the experimental study [ 49, 50] does not provide any measurement on the fracture toughness of the material. This study therefore assumes two different values of the material fracture toughness, 150 {\mathrm{kJ/m}}^2 and 300 {\mathrm{kJ/m}}^2 . Figure 16(b) confirms the conservative estimation of the proposed deformation limit when compared to the experimental data.

Summary and conclusions

Th e tearing in circular hollow section X- and K-joints with a surface crack located at the weld toe of the hot-spot position in the chord member. The proposed deformation limit provides an explicit measure to quantify the ductile fracture failure in engineering designs for a wide range of geometric parameters and material toughness levels. The approach described in this study remains adaptable to other types of welded connections, which experience ductile facture failure under extreme loads. The above study supports the following conclusions:

1) The critical deformation limit corresponding to different fracture toughness values (from {\mathrm{100\; kJ/m}}^2 to {\mathrm{300\; kJ/m}}^2 ) exhibits strong dependence on both the geometric parameters ( a/t_{\mathrm{0}} , \beta , \gamma and g\text{'} ratios) and the material fracture toughness levels. The deformation limit, however, indicates marginal dependence on the material yield strength and the hardening exponent for small to intermediate fracture toughness levels. The dependence of the deformation limit on the geometric parameters reflects directly the dependence of the computed J-integral values on these parameters.

2) This study proposes a dimensionless crack driving force, based on a detailed examination of the J-integral variation with respect to different geometric parameters for the X-joint and K-joints. The dimensionless J-value, J_n , incorporates the dependence of the geometric parameters into the normalization procedure and shows significantly reduced variations in the J-values among different geometric parameters. The normalized deformation limit, equal to the ratio of the deformation limit divided by the square root of the dimensionless crack driving force, indicates significantly reduced scatter among the geometric parameters of the CHS X- and K-joint and the fracture toughness levels.

3) The proposed deformation limits represent lower-bound deformation limits corresponding to the onset of ductile tearing for CHS X- and K-joints with a surface crack at the hot-spot position of the chord weld toe. This deformation limit shows an approximately linear relationship with respect to the crack depth ratio and provides a conservative estimation of the fracture deformation limit for intact joints, when the crack-depth ratio approaches zero.

4) The proposed deformation limit provides conservative estimations on the critical deformation levels at the fracture failure for both CHS X- and K-joint tests reported in previous studies [ 48– 50].

Nomenclature

E= Young’s modulus

J= elastic-plastic crack driving force

J_{Ic}= fracture toughness measured by the critical energy release rate

J_n= normalized fracture toughness

P_{ij}= 1st Piola-Kirchhoff stress tensor

W= strain energy density

a= depth of the crack

d_{\mathrm{0}}= outer diameter of the chord

d_{\mathrm{1}}= outer diameter of the brace

g= gap between two braces for CHS K-joints

g\text{'}= non-dimensional gap ratio for CHS K-joints, (g/t_{\mathrm{0}})

l_{\mathrm{0}}= length of the chord

l_{\mathrm{1}}= length of the brace

n= material strain hardening exponent

n_j= unit vector ( j=\mathrm{1},\mathrm{2}\mathrm{or\; 3} )

t_{\mathrm{0}}= thickness of the chord

t_{\mathrm{1}}= thickness of the brace

\Gamma = a counter-clockwise path around a crack tip

\alpha = chord length to chord radius ratio ( \mathrm{2}{l}_{\mathrm{0}}/d_{\mathrm{0}} )

\beta = brace diameter to chord diameter ratio ( d_{\mathrm{1}}/d_{\mathrm{0}} )

\gamma = chord radius to chord wall thickness ratio ( d_{\mathrm{0}}/\mathrm{2}{t}_{\mathrm{0}} )

{\Delta }_{LLD}= load-line displacement

\epsilon = strain

\theta = angle between the longitudinal axes of the brace and the chord

{\delta }_{cr}= critical axial deformation limit

{\bar{\delta }}_{cr}= normalized critical axial deformation limit

\sigma = stress

{\sigma }_y= material yield strength

\tau = brace wall thickness over chord wall thickness ratio ( \tau =t_{\mathrm{1}}/t_{\mathrm{0}} )

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