# Failure assessment of cracked uni-planar square hollow section T-, Y- and K-joints using the new BS 7910:2013+A1:2015

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## Abstract

This paper covers the validation of standard safety assessment procedure in the new BS 7910:2013+A1:2015 for cracked uni-planar square hollow section (SHS) T-, Y- and K-joints using the finite element analyses. The procedure is based on the failure assessment diagram (FAD) method. A completely new and robust finite element mesh generator is developed, and it is validated using the full-scale experimental test results. FAD curves are constructed using the elastic *J*-integral (\(J_{\mathrm{e}}\)), the elastic-plastic *J*-integral (\(J_{\mathrm{ep}}\)) and the plastic collapse load (\(P_{\mathrm{c}}\)) values calculated using the 3D cracked models of the joints. The results reveal that the standard Option 1 FAD curve of the new BS code is not always safe in assessing the safety and integrity of cracked uni-planar SHS joints. Therefore, a penalty factor of 1.2 for plastic collapse load is recommended to move all the constructed Option 3 FAD curves above the standard Option 1 curve. The new Option 3 FAD curves are found to generate optimal solutions for cracked uni-planar SHS T-, Y- and K-joints.

## Keywords

Failure assessment diagram Finite element analysis Surface crack Square hollow section joints## List of symbols

- \(J_{\mathrm{e}}\)
Elastic

*J*-integral- \(J_{\mathrm{ep}}\)
Elastic-plastic

*J*-integral- \(K_{\mathrm{I}}\)
Stress intensity factor

- \(K_{\mathrm{IC}}\)
Fracture toughness

- \(K_{\mathrm{r}}\)
Fracture ratio using stress intensity factor

- \(L_{\mathrm{r}}\)
Ratio of applied load to plastic collapse load

- \(P_{\mathrm{a}}\)
Applied load

- \(P_{\mathrm{c}}\)
Plastic collapse load

- \(\sigma _{\mathrm{ref}}\)
Reference stress

- \(\sigma _{\mathrm{y}}\)
Yield stress

## 1 Introduction

Several assessment methods based on fracture mechanics have been developed during the last 40 years. The widely used assessment technique that is followed in recent times is the failure assessment diagram (FAD) approach. The basis of the FAD approach was originally developed by Dowling and Townley [1]. According to this approach, a structure can fail by either of two mechanisms, namely brittle fracture and plastic collapse. A design curve labelled as failure assessment curve is used to interpolate between the two failure criterions as shown in Fig. 1. The integrity of the structure is determined by the relative position of the assessment point on the FAD. The relative position is derived from two separate set of calculations covering both brittle fracture and plastic collapse. The final assessment procedure is quite straightforward, i.e. if the relative position of the point is inside the FAD curve, the structure is considered to be safe.

However, if the relative position of the point is outside the FAD curve, the structure is considered to be unsafe. The reliability of the result produced by FAD approach depends on the accuracy by which the failure assessment curve is plotted which in turn depends upon the structural geometry, nature of loading and the crack size. Complex three-dimensional geometry makes it challenging to create the FE mesh model of a cracked SHS joint by using commercial FE softwares. The current study develops a new FE mesh generator in order to construct reliable FAD curves. The details of the new FE mesh generator and its validation using the full-scale experimental test results are explained in the subsequent sections.

## 2 The new BS 7910:2013+A1:2015 approach

The Option 1 and Option 2 FAD curves of BS 7910:2013+A1:2015 [2] are intended for all types of structures and may not essentially provide optimal solutions for different types of structures such as cracked uni-planar SHS T-, Y- and K-joints. Option 3 FAD curve gives the most accurate representation of the structural behaviour of a component [2]. There are many existing structures such as bridges, buildings and offshore structures which are fabricated using square hollow section (SHS) joints. In this study, Option 3 FAD curves are constructed for cracked uni-planar SHS T-, Y- and K-joints to validate the standard Option 1 curve of BS 7910:2013+A1:2015 [2] and to recommend optimal solutions that can safely be used to assess cracked uni-planar SHS T-, Y- and K-joints.

## 3 New finite element mesh generator and validation

*r*strain singularity, which relates to the actual crack tip strain field for a totally plastic material.

Broadly, two approaches are used to model crack tip elements for elastic-plastic analyses. Figure 2b illustrates the first method where the crack tip nodes are coincident and untied. The drawback for using this approach is the blunting at the crack tip caused due to the formation of large strains during the elastic-plastic analyses. This can be ignored if the deformation at the crack is comparatively small [6]. However, for large-scale deformations, the collapse of the crack tip elements takes place leading to inaccurate elastic-plastic *J*-integral (\(J_{\mathrm{ep}}\)) outcomes. Moreover, there are convergence issues associated with the large-scale plastic deformation analyses. Anderson [7] introduced the concept of using a keyhole, depicted in Fig. 2c, for which there is a finite radius for the plastic zone located at the crack tip. Modelling using keyhole ensures the convergence of solutions for large-scale plastic deformation as well as it prevents the collapse of elements adjacent to crack tip region. The selection of finite radius for the keyhole requires special attention. Very small values of finite radius lead to collapse of the adjacent elements around crack tip while the large values lead to inaccurate results since the value must be within the plastic zone. For the current research works, the value of keyhole radius is chosen according to the provisions specified in API RP579 [8], whereby it is at least five times smaller than the tip radius in the distorted state. The value is set as a variable so that the user is able to vary it appropriately for each case. A value of 0.1mm is found to be sufficient to avoid the collapse of elements at the crack tip for the large-scale nonlinear deformation. The authors recommend a finite radius of keyhole from 0.05mm to 0.15mm with 0.1mm as the most appropriate value for load values in the range of 1000 kN.

*J*-integral results from the new FE mesh generator are compared with the results produced from the mesh generated using \({\hbox {FEACrack}}^{\mathrm{TM}}\) software [9] (Fig. 5). The maximum percentage difference among both cases is 0.09% at the deepest point of the crack and 7.89% at the crack ends. The variation trend observed for

*J*-integral values with respect to different crack front angle values compliments each other. In addition, convergence of solutions reconfirms the reliability of using the new FE mesh generator. The experimental test rigs for cracked uni-planar SHS T-, Y- and K-joints are shown in Fig. 6. The load–displacement curves point out the analogous trends between experimental and numerical models (Fig. 7). The tendency of the load–displacement curve obtained from numerical analyses being slightly above the experimental curve is most visible in the case of SHS K-joint. This is due to the fact that the stress–strain curve used for numerical analyses may not be exactly same as the actual condition in the full-scale experimental tests. For instance, the stress–strain curve at the corner of the SHS joints is generally higher due to the strain hardening in the corner. It is also observed that the post-peak unloading curves observed in the experimental results deviate from numerical curves. This deviation does not affect the accuracy of the results produced using the new FE mesh generator as there is good agreement between the experimental and numerical results for the load–displacement curves in the plastic collapse region. Detailed discussion on the experimental test setup and the loading protocol is outside the main purview of the current manuscript. Further in-depth discussion about the validation of the new FE mesh generator is also available in other previously published works of the authors [10, 11, 12, 13, 14].

## 4 FADs of cracked uni-planar SHS joints

*J*-integral (\(J_{\mathrm{ep}}\)) values are obtained, respectively. The

*J*-integral is calculated using the virtual crack extension (VCE) technique which is built in ABAQUS software [15]. It interprets the

*J*-integral as a function of location along the crack front which provides the reduction in total potential energy of the loaded structure due to an increase in the crack opening area at that same position on the crack front.

*s*is the coordinate which identifies location along the crack front,

*J*(

*s*) is the

*J*-integral value, and \(\delta l(s)\) is the advance of the crack front normal to itself and in the local plane of the crack at this point. The value of

*J*(

*s*) and the VCE are then interpolated from the value at crack front nodal position through the identical order of interpolation as used in the elements adjoining the crack front. Thus, Eq. (5) becomes, in the FE model, a set of equations which relate the set of total potential energy variations produced by crack advances at each crack front nodal position to the

*J*-integral values of the same crack front nodal positions. After these values are determined, the equations can be solved for these nodal positions values of

*J*-integral. The method is specifically suitable because it is simple to use, cost beneficial and ensures good accuracy [16].

There are two methods for the application of VCE techniques in the 3D settings as illustrated in Fig. 9. In Fig. 9a, all the crack front nodes are perturbed, while in Fig. 9b only one node is perturbed. In the first approach, elliptical crack is reduced to two degrees of freedom by transforming elliptical cracks to be distorted into other elliptical cracks, changing only the size parameters (semi major and minor axes). In the second approach, the local node is disturbed alone leading to a good estimate of the local value of *J*-integral or stress intensity factors *K*. This results in the increase in the number of degrees of freedom. For the current study, the second approach (local perturbation) is used to calculate the *J*-integral along the crack front.

*J*-integral (\(J_{\mathrm{ep}}\)) values and the plastic collapse load (\(P_{\mathrm{c}}\)). The \(P_{\mathrm{c}}\) load is determined using the twice elastic compliance (TEC) criterion. Several previous studies used the TEC criterion to determine the \(P_{\mathrm{c}}\) load of cracked tubular joints [17, 18]. This plastic criterion is described in ASME VIII Division 2 [19] as the twice elastic slope (TES) criterion. The TEC criterion is established on the load–deformation response of a structure in the plastic analysis as shown in Fig. 10. The \(P_{\mathrm{c}}\) load is the load corresponding to the intersection of the load–deformation curve and the twice elastic compliance line. The twice elastic compliance line starts from the origin of the load–deformation curve and has twice the slope of the initial elastic response, which is calculated using the equation

*J*-integral (\(J_{\mathrm{ep}}\)) values and the plastic collapse load (\(P_{\mathrm{c}}\)) values, Option 3 FAD curves are constructed for cracked uni-planar SHS T-, Y- and K-joints. It is observed that some of the constructed curves tend to fall below the standard Option 1 curve of BS 7910:2013+A1:2015 [2] (Fig. 11). This is especially prominent towards the right side of the FAD curves which is near to the plastic collapse region in the standard Option 1 curve of BS 7910:2013+A1:2015 [2]. This trend is not desirable as it is clear from Fig. 1 that the integrity of the structure is estimated by the relative position of the assessment point on the standard FAD curve. Therefore, if an assessment point is located in the intermediate region (the region below the standard curve and above the constructed curves in Fig. 11), it may be assumed to be safe, but it may not be truly safe [6]. A straightforward approach to solve this is by introducing a penalty factor for plastic collapse load which is described in following paragraph. The results also reveal that the constructed FAD curves for cracked uni-planar SHS T-, Y- and K-joints with low brace to chord width ratio (\(\upbeta \)) appear to fall below the Option 1 curve of BS 7910:2013+A1:2015 [2] when compared to the models with higher \(\upbeta \) ratio. This is attributed to increasing plastic collapse load of the uni-planar SHS T-, Y- and K-joints with increasing value of \(\upbeta \). It is to be noted that the curve shown in Fig. 11 from Lie et al. [20] is based on the previous version of codes as currently there are no published works depicting the FAD curves for cracked uni-planar SHS T-, Y- and K-joints using the latest BS 7910:2013+A1:2015 [2].

In order to obtain a conservative and safe prediction, a penalty factor of 1.20 is recommended for cracked uni-planar SHS T-, Y- and K-joints. The newly proposed penalty factor is introduced to Eq. (3) whereby the plastic collapse load of the flawed structure is divided by a factor of 1.20 so that a conservative value of the plastic collapse load is used to construct the FAD curves. This procedure ensures that there are no more assessment points which are located at the intermediate region (region corresponding to below the standard curve and above the constructed curves in Fig. 11). Figure 12 depicts the constructed FAD curves for cracked uni-planar SHS T-, Y- and K-joints with the newly introduced penalty factor of 1.20. It can clearly be seen that all the newly constructed FAD curves are located above the standard Option 1 curve of BS 7910:2013+A1:2015 [2]. To summarize, in order to assess the safety and integrity of cracked uni-planar SHS T-, Y- and K-joints in practice using the standard Option 1 curve of BS 7910:2013+A1:2015 [2], a penalty factor of 1.20 is recommended to use while calculating the plastic collapse load values, thereby producing optimal solutions.

## 5 Conclusions

Complex three-dimensional geometry makes it challenging and time-consuming to generate the finite element (FE) mesh models of cracked SHS joints by using commercial FE softwares. In order to assist in failure assessment of damaged uni-planar SHS T-, Y- and K-joints, the present study develops an entirely new and robust FE mesh generator. Mesh refinement and zoning techniques are extensively used in the new FE mesh generator so that it is robust enough to model different types of cracked uni-planar SHS welded joints within a short period of time. The new FE mesh generator is validated using the full-scale experimental test results as well as with the commercial software results. Subsequently, the highest level Option 3 FAD curves of cracked SHS joints are constructed using the elastic (\(J_{\mathrm{e}}\)) and elastic-plastic *J*-integral (\(J_{\mathrm{ep}}\)) values and the plastic collapse load (\(P_{\mathrm{c}}\)) values. The constructed Option 3 FAD curves are further used to validate the standard Option 1 FAD curve of the BS 7910:2013+A1:2015 [2]. It is found that Option 1 FAD curve of the BS 7910:2013+A1:2015 [2] is not always safe in assessing the safety and integrity of cracked uni-planar SHS T-, Y- and K-joints. For obtaining a conservative safe prediction, a penalty factor of 1.20 is recommended for cracked uni-planar SHS T-, Y- and K-joints. The newly recommended penalty factor is used to modify the plastic collapse load of the cracked T-, Y- and K-joints through dividing by a factor of 1.20 so that a conservative value of the plastic collapse load is used to construct the FAD curves. It can be seen that all the newly constructed FAD curves using the penalty factor are located just above the standard Option 1 FAD curve of the BS 7910:2013+A1:2015 [2]. These new Option 3 FAD curves are able to provide optimal solutions for cracked uni-planar SHS T-, Y- and K-joints.

## Notes

### Acknowledgements

The authors would like to thank the Maritime Port Authority of Singapore for funding this research project under the Grant No. MPA 23/04.15.03 – RDP 005/06/031.

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