Use of local and global limit load solutions for plates with surface cracks under tension

Some available experimental results for the ductile failure of plates with surface cracks under tension are reviewed. The response of crack driving force, J, and the ligament strain near the local and global limit loads are investigated by performing elastic-perfectly plastic finite element (FE) analysis of a plate with a semi-elliptical crack under tension. The results show that a ligament may survive until the global collapse load is reached when the average ligament strain at the global collapse load, which depends on the uniaxial strain corresponding to the flow stress of the material and the crack geometry, is less than the true fracture strain of the material obtained from uniaxial tension tests. The FE analysis shows that ligament yielding corresponding to the local limit load has little effect on J and the average ligament strain, whereas approach to global collapse corresponds to a sharp increase in both J and the average ligament strain. The prediction of the FE value of J using the reference stress method shows that the global limit load is more relevant to J-estimation than the local one.     

Analysis of interacting semi-elliptical surface cracks in finite thickness plates under remote bending load

Interaction effects of two coplanar self-same shallow and deep semi-elliptical surface cracks in finite thickness plates subjected to remote tension have been previously investigated by Sethuraman et al. Using the finite element based force method. In the present study, the effect of remote bending load on interacting semi-elliptical surface cracks in a finite thickness plate is analyzed. Stress intensity factors are evaluated along the entire crack front using a modified force method based on the three-dimensional finite element solution. The line spring model has also been used to evaluate stress intensity factors at the deepest point of a crack using shell finite element analysis. Parametric studies involving a wide ranges of geometric dimensions and crack configurations viz. crack shape aspect ratio (0.2≤a/c≤1.0), crack depth ratio (0.2≤a/t≤0.9), relative crack location (0.2≤2c/d≤0.9) and normalized location on the crack front (0≤2ϕ/π≤2) are carried out for numerical estimation of crack interaction factors. Due to the crack interaction, the stress intensity factor distribution is observed to be asymmetric along the crack front. The interaction is also observed to cease when the distance between two cracks is more than five times the crack width (i.e. 2c/d less than 0.2) irrespective of crack shape aspect ratio. Finally, an empirical relation is proposed for the evaluation of crack interaction crack interaction factors for the range of parameters considered. For the ranges considered, the proposed empirical relation predicts the crack interaction factors at the inner and outer surface points of the crack within ±4% of the three-dimensional finite element solutions.     

Compound cracks in pipes under combined bending and tension

Limit load and J-resistance curve solutions are developed for a compound crack in a pipe under combined tension and bending. The solution is based on a thick-walled cylinder assumption and the J solution can be applied with load-displacement data from one pipe test. Material resistance curves are developed for compound cracks in Type-304 stainless steel, Inconel 600 and A106 GrB base materials and a 304 stainless steel TIG weld. Tearing instability analyses are performed to assess the load-carrying capacity of pipes containing compound cracks and to evaluate nonconservatism associated with the use of C(T) specimen J-R data.     

Linear and non-linear analyses for semi-elliptical surface cracks in pipes under bending

In this paper, the J integral was calculated for semi-elliptical surface cracks in pipes under bending using three-dimensional finite element analysis. The computations were performed for elastic and elastic-plastic behaviours. For the elastic case, the numerical results allowed the extrapolation of shape functions for analytical determination of the J integral. The results are in a good agreement with those in the literature if the ratio between the radius and the thickness of the pipe (R/t) is from 1 to 10. The analysis was extended to values of the ratio R/t higher than 10. For the elastic-plastic, the numerical results are in good agreement with the analytical solution found in the literature for thick pipes (R/t ≥ 10). The effect of the ratio R/t becomes sensible when the ratio of the applied moment to the moment of reference (M/Mor) exceeds 0.9.     

Rules for the assessment of interacting surface cracks under mode I load

A review of the current widely used methods for the assessment of adjacent and coplanar surface cracks under LEFM conditions shows them to be neither realistic nor always safe, having little scientific basis. Since such cracks can interact they cannot necessarily be assumed to be independent, so improved methods of accounting for the interaction are required and are provided by this work. In a companion paper a continuous method of estimating the interaction factors for two coplanar surface cracks was developed and proven. This enables realistic predictions of crack interaction and growth and is used as a basis for comparison of simpler assessment methods, the crack interaction estimation being somewhat cumbersome for general use. The assessment methods developed are thus shown to be safe. Furthermore they are realistic and straightforward to use. Cracks under tension, bending, and combined tension and bending load are considered.     

A review of limit load solutions for cylinders with axial cracks and development of new solutions

Limit load solutions for axially cracked cylinders are reviewed and compared with available finite element (FE) results. New limit solutions for thick-walled cylinders with axial cracks under internal pressure are developed to overcome problems in the existing solutions. The newly developed limit load solutions are a global solution for through-wall cracks, global solutions for internal/external surface cracks and local solutions for internal/external surface cracks. The newly developed limit pressure solutions are compared with available FE data and the results show that the predictions agree well with the FE results and are generally conservative.     

The interaction of two parallel non-coplanar identical surface cracks under tension and bending

In this paper, the interaction between two identical, non-coplanar, semi-elliptical surface cracks is investigated. The interacting cracks are assumed to be in an infinite plate subjected to remote tension or to pure bending loads. The stress intensity factors (SIFs) for these cracks are calculated using three-dimensional linear finite element analysis. A parametric study involving the relative horizontal and vertical separation distance between the two surface cracks is carried out for a specific crack shape and crack depth to plate thickness ratios of 0.3 and 0.2, respectively. An empirical formula is derived that relates the effects of the relative positions of these cracks to their SIFs.     

Limit loads for pipe bends under combined pressure and in-plane bending based on finite element limit analysis

Approximate plastic limit load solutions for pipe bends under combined internal pressure and bending are obtained from detailed three-dimensional (3-D) FE limit analyses based on elastic-perfectly plastic materials with the small geometry change option. The FE results show that existing limit load solutions for pipe bends are lower bounds but can be very different from the present FE results in some cases, particularly for bending. Accordingly closed-form approximations are proposed for pipe bends under combined pressure and in-plane bending based on the FE results.     

Plastic limit load of cylindrical shells with cutouts subject to pure bending moment

In this paper, the results of limit analyses of thin-walled cylindrical shells with a circular hole under the action of a pure bending moment are presented in dimensionless form for a wide range of geometric parameters. Analytical estimation of lower bound limit load is carried out using the feasible sequential quadratic programming (FSQP) technique. The finite element calculations of limit load consist of elastic–plastic and lower and upper bound predictions by elastic compensation methods. A testing device was made to perform experiments to obtain limit bending moment of cylinders with circular openings. The analytical and finite element calculations are compared with experimental results and their correlation is discussed. The finite element calculation results were found to be in good agreement with lower bound estimations by the nonlinear mathematical programming (FSQP) method and the formula proposed by Shu.     

Shakedown limit loads for elbows under internal pressure and cyclic in-plane bending

This paper presents elastic, shakedown and plastic limit loads for 90° elbows under constant internal pressure and cyclic in-plane bending, via finite element (FE) analysis. Effects of the elbow geometry (the bend radius to mean radius ratio and the mean radius-to-thickness ratio) and of the large geometry change are systematically investigated. By normalizing the in-plane bending moment by the plastic limit load solution of Calladine, the shakedown diagram is found to be close to unity up to a certain value of normalized pressure (normalized with respect to the limit pressure) and then to decrease almost linearly with increasing normalized pressure. The value up to which shakedown limit loads remain constant depends on the elbow geometry and the large geometry change effect. Effects of the elbow geometry and the large geometry change on shakedown diagrams are discussed.     

Estimating the elastic interaction factors of two coplanar surface cracks under mode I load

Two cracks sufficiently close together will interact by causing the stress intensity factor at the tips of each to increase. The amount of this increase can be quantified through the use of interaction factors. This paper studies the form of these for the important and practical special case where the cracks have a semi-elliptical shape. A continuous method of approximate solution is proposed in this paper, and it is shown to be effective in the closeness of its predictions to those in the literature, and to our own finite element analyses. The work is a precursor to an assessment when cracks are close enough for interaction to need evaluation. This forms the subject of a companion paper.     

The limit analysis of an axisymmetric circular cylinder of arbitrary length with one end free, under applied shear force and moment and zero axial force

Limit interaction surfaces have been obtained for a cylinder made of rigid-perfectly plastic material, subjected to shear and moment at one end only. It is shown that beyond a certain critical length, depending on the applied moment and shear, there is no further strengthening effect. This result has applications in the design of protrusion lengths for nozzles in pressure vessels.     

A comparison of plastic collapse and limit loads for single mitred pipe bends under in-plane bending

This paper presents a comparison of the plastic collapse loads from experimental in-plane bending tests on three 90° single un-reinforced mitred pipe bends, with the results from various 3D solid finite element models. The bending load applied reduced the bend angle and in turn, the resulting cross-sectional ovalisation led to a recognised weakening mechanism. In addition, at maximum load there was a reversal in stiffness, characteristic of buckling. This reversal in stiffness was accompanied by significant ovalisation and plasticity at the mitre intersection. Both the weakening mechanism and the post-buckling behaviour are only observable by testing or by including large displacement effects in the plastic finite element solution. A small displacement limit solution with an elastic-perfectly plastic material model overestimated the collapse load by more than 40% and could not reproduce the buckling behaviour.     

Assessment of cracks under secondary and combined loading

Following a recent presentation of methods for the assessment of defects under primary loading, proposals are made here for the extension of these methods to combined primary and secondary loading, without necessarily involving the excessive pessimism of treating the secondary loading as primary, but allowing for strain enhancement at reduced ligaments. Finite element calculations have shown that the proposals provide a pessimistic bound on the value of the contour integral J in the cases considered for a number of work-hardening relationships and geometries, including cases of elastic follow-up.     

Analytical and experimental fracture assessment of carbon steel piping with circumferential through-wall cracks under bending

In this paper the true stress-strain curves of carbon steel pipes are described by a three-segment fitting method rather than the Ramberg-Osgood relation for their yield plateau characteristics. Failure assessment curves for carbon steel piping containing circumferential through-wall cracks were obtained by different options in the CEGB/R6 approach. The J-integral was calculated using a new expression proposed by the authors. Initiation and maximum moments of piping under bending were predicted by several methods and measured by experiments. Calculated moments were quite close to test results.     

Stress intensity factor equations for mixed-mode surface and corner cracks in finite-thickness plates subjected to tension loads

Normalized mixed-mode stress intensity factor equations are presented for deflected and inclined circular surface and corner cracks in finite-thickness plates under uniform remote tensile loading. The equations are obtained by performing non-linear regression analyses on the data from previous numerical solutions based on three-dimensional enriched finite elements. In the equations, the effects of deflection/inclination angles and plate thickness on mixed-mode stress intensity factors are included. The comparisons of normalized stress intensity factors from the equations with those of the finite element analyses show good agreement. Thus, it is concluded that, as a reasonable approximation, the presented equations can be used to assess stress intensity factors and fracture conditions of mixed-mode circular surface and corner cracks in finite-thickness plates.     

Ligament yielding of a plate with semi-elliptical surface cracks under uniform tension

The ligament yielding of a plate with a semi-elliptical surface crack under tension loading is calculated using the Dugdale model. The stress intensity factor for the strip load is obtained using the weight-function method. The necessary crack opening displacement is calculated with an approach proposed by Petroski and Achenbach.     

Plastic collapse of pipe bends under combined internal pressure and in-plane bending

Plastic collapse of pipe bends with attached straight pipes under combined internal pressure and in-plane closing moment is investigated by elastic–plastic finite element analysis. Three load histories are investigated, proportional loading, sequential pressure–moment loading and sequential moment–pressure loading. Three categories of ductile failure load are defined: limit load, plastic load (with associated criteria of collapse) and instability loads. The results show that theoretical limit analysis is not conservative for all the load combinations considered. The calculated plastic load is dependent on the plastic collapse criteria used. The plastic instability load gives an objective measure of failure and accounts for the effects of large deformations. The proportional and pressure–moment load cases exhibit significant geometric strengthening, whereas the moment–pressure load case exhibits significant geometric weakening.