Limit loads for thin-walled piping branch junctions under internal pressure and in-plane bending

The present work presents plastic limit load solutions for thin-walled branch junctions under internal pressure and in-plane bending, based on detailed three-dimensional (3-D) finite element (FE) limit analyses using elastic–perfectly plastic materials. To assure reliability of the FE limit loads, modelling issues are addressed first, such as the effect of kinematic boundary conditions and branch junction geometries on the FE limit loads. Then the FE limit loads for branch junctions under internal pressure and in-plane bending are compared with existing limit load solutions, and new limit load solutions, improving the accuracy, are proposed based on the FE results. The proposed solutions are valid for ratios of the branch-to-run pipe radius and thickness from 0.4 to 1.0, and the mean radius-to-thickness ratio of the run pipe from 10.0 to 20.0.     

Effects of attached straight pipes on finite element limit analysis for pipe bends

The effect of the length of an attached straight pipe on the plastic limit load of a 90° pipe bend under combined pressure and bending is quantified, based on finite element (FE) limit analyses using elastic–perfectly plastic materials with the small geometry change option. Systematic FE limit analyses of pipe bends with various lengths of the attached pipe are performed. It is shown that the effect of the length of the attached straight pipe on plastic limit loads can be significant, and the limit loads tend to decrease with decrease of the length of the attached straight pipe. In the limiting case of no attachment, the limit loads are found to be close to existing analytical solutions.     

Limit loads for piping branch junctions under internal pressure and in-plane bending—Extended solutions

The authors have previously proposed plastic limit load solutions for thin-walled branch junctions under internal pressure and in-plane bending, based on finite element (FE) limit loads resulting from three-dimensional (3-D) FE limit analyses using elastic–perfectly plastic materials [Kim YJ, Lee KH, Park CY. Limit loads for thin-walled piping branch junctions under internal pressure and in-plane bending. Int J Press Vessels Piping 2006;83:645–53]. The solutions are valid for ratios of the branch-to-run pipe radius and thickness from 0.4 to 1.0, and for the mean radius-to-thickness ratio of the run pipe from 10.0 to 20.0. Moreover, the solutions considered the case of in-plane bending only on the branch pipe. This paper extends the previous solutions in two aspects. Firstly, plastic limit load solutions are given also for in-plane bending on the run pipe. Secondly, the validity of the proposed solutions is extended to ratios of the branch-to-run pipe radius and thickness from 0.0 to 1.0, and the mean radius-to-thickness ratio of the run pipe from 5.0 to 20.0. Comparisons with FE results show good agreement.     

Limit and plastic collapse loads for un-reinforced mitred bends under pressure and bending

Approximate limit and plastic collapse load solutions for un-reinforced mitred bends under internal pressure and under bending are proposed in this paper, based on three-dimensional finite element analysis and approximate solutions for smooth bends. Solutions are given for single- and multi-mitred bends (mainly for single and double segmented bends) with the pipe mean radius-to-thickness ratio (r/t) ranging from r/t = 5 to r/t = 50, and the bend radius-to-mean radius ratio (R/r) from R/r = 2 to R/r = 4. Internal pressure, in-plane bending and out-of-plane bending loads are considered, but not their combination. It is found that the essential features of limit and plastic collapse loads for mitred bends are similar to those for smooth bends, and thus existing solutions for smooth elbows can be used to construct limit loads and plastic collapse for mitred bends.     

Net-section limit moments and approximate J estimates for circumferential cracks at the interface between elbows and pipes

This paper firstly presents net-section limit moments for circumferential through-wall and part-through surface cracks at the interface between elbows and attached straight pipes under in-plane bending. Closed-form solutions are proposed based on fitting results from small strain FE limit analyses using elastic–perfectly plastic materials. Net-section limit moments for circumferential cracks at the interface between elbows and attached straight pipes are found to be close to those for cracks in the centre of elbows, implying that the location of the circumferential crack within an elbow has a minimal effect on the net-section limit moment. Accordingly it is also found that the assumption that the crack locates in a straight pipe could significantly overestimate the net-section limit load (and thus maximum load-carrying capacity) of the cracked component. Based on the proposed net-section limit moment, a method to estimate elastic–plastic J based on the reference stress approach is proposed for circumferential cracks at the interface between elbows and attached straight pipes under in-plane bending.     

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.     

Plastic collapse moment of 90° long radius elbows with internal circumferential surface crack at intrados under in-plane bending

Piping elbows under in-plane bending moment are vulnerable to cracking. The crack initiates at the surface and eventually reaches through the thickness and may lead to failure. The structural integrity assessment requires knowledge of the limit load. Limit load solutions for elbows with through-wall crack configurations are available in the open literature. But solutions for surface crack are not available. This paper presents a closed form expression for the plastic collapse moment (PCM) of 90°, long radius elbows with circumferential surface cracks at the intrados, under in-plane bending moment. The expression is derived, based on the results of non-linear (geometric and material) FE analyses covering a wide range of geometries and crack sizes. These plastic collapse moments evaluated herein will help in structural integrity assessment.     

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.     

Shakedown and limit analysis of 90° pipe bends under internal pressure,cyclic in-plane bending and cyclic thermal loading

The Linear Matching Method is used to create the shakedown limit and limit load interaction curves of 90° pipe bends for a range of bend factors. Two load cases are considered i) internal pressure and in-plane bending (which includes opening, closing and reversed bending) and ii) internal pressure and a cyclic through wall temperature difference giving rise to thermal stresses. The effects of the ratios of bend radius to pipe mean radius (R/r) and mean radius to wall thickness (r/t) on the limit load and shakedown behaviour are presented.     

Elastic–plastic J and COD estimates for axial through-wall cracked pipes

This paper proposes engineering estimation equations of elastic–plastic J and crack opening displacement (COD) for axial through-wall cracked pipes under internal pressure. On the basis of detailed 3D finite element (FE) results using deformation plasticity, the plastic influence functions for fully plastic J and COD solutions are tabulated as a function of the mean radius-to-thickness ratio, the normalised crack length, and the strain hardening. On the basis of these results, the GE/EPRI-type J and COD estimation equations are proposed and validated against 3D FE results based on deformation plasticity. For more general application to general stress–strain laws or to complex loading, the developed GE/EPRI-type solutions are re-formulated based on the reference stress (RS) concept. Such a re-formulation provides simpler equations for J and COD, which are then further extended to combined internal pressure and bending. The proposed RS based J and COD estimation equations are compared with elastic–plastic 3D FE results using actual stress–strain data for Type 316 stainless steels. The FE results for both internal pressure cases and combined internal pressure and bending cases compare very well with the proposed J and COD estimates.     

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.     

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.     

Limit moment of local wall thinning in pipe under bending

In engineering practice, pipe containing local wall thinning may be subjected to bending load. The existence of local wall thinning on pipe surface impairs the load-carrying capacity of pipe. In order to maintain the integrity of the pipe containing local wall thinning, it is very important to develop a method to evaluate such a pipe with local wall thinning under bending. In this paper, the limit moment of local wall thinning pipe under pure bending is computed employing 3D elastic–plastic finite element analysis. The results show that the limit moment of pipe is affected not only by the width of defect but also by the longitudinal length of defect. When the longitudinal length of defect overpasses some critical value, the results from net-section collapse criterion (NSC) are in very reasonable agreement with the results from finite element analysis. Therefore, the NSC formula can conservatively be used to assess the limit load-carrying capability of local wall thinning pipe under bending.     

Effect of reinforcement on plastic limit loads of branch junctions

This paper provides effects of reinforcement shape and area on plastic limit loads of branch junctions under internal pressure and in-plane/out-of-plane bending, via detailed three-dimensional finite element limit analysis assuming elastic-perfectly plastic material behaviour. It is found that reinforcement is most effective when (in-plane/out-of-plane) bending is applied to the branch pipe. When bending is applied to the run pipe, reinforcement is less effective when bending is applied to the branch pipe. The reinforcement effect is the least effective for internal pressure.     

A global limit load solution for plates with surface cracks under combined end force and cross-thickness bending

A global limit load solution for rectangular surface cracks in plates under combined end force and cross-thickness bending is derived, which allows any combination of positive/negative end force and positive/negative cross-thickness moment. The solution is based on the net-section plastic collapse concept and, therefore, gives limit load values based on the Tresca yielding criterion. Solutions for both cases with and without crack face contact are derived when whole or part of the crack is located in the compressive stress zone. From the solution, particular global limit load solutions for plates with extended surface cracks and through-thickness cracks under the same loading conditions are obtained. The solution is consistent with the limit load solution for surface cracks in plates under combined tension and positive bending due to Goodall & Webster and Lei when both the applied end force and bending moment are positive. The solution reduces to the limit load solution for plain plates under combined end force and cross-thickness bending when the crack vanishes.     

Plastic limit pressures for cracked pipes using finite element limit analyses

Based on detailed finite element (FE) limit analyses, the present paper provides approximations for plastic limit pressure solutions for plane strain pipes with extended inner axial cracks; axi-symmetric (inner) circumferential cracks; axial through-wall cracks; axial (inner) surface cracks; circumferential through-wall cracks; and circumferential (inner) surface cracks. In particular, for surface crack problems, the effect of the crack shape, semi-elliptical or rectangular, on the limit pressure is quantified. Comparisons with existing analytical and empirical solutions show a large discrepancy for short circumferential through-wall cracks and for surface cracks (both axial and circumferential). Being based on detailed 3D FE limit analysis, the present solutions are believed to be accurate, and thus to be valuable information not only for plastic collapse analysis of pressurised piping but also for estimating non-linear fracture mechanics parameters based on the reference stress approach.     

Net-section limit load approach for failure strength estimates of pipes with local wall thinning

The net-section limit load approach is typically used in assessment of pipes with local wall thinning, based on which a maximum load carrying capacity is easily estimated from one equation that includes two terms associated with the effect of the defect geometry and the material's resistance (strength). To better understand the applicability of the net-section limit load approach to pipes with local wall thinning, four different limit load expressions for pipes with local wall thinning under pure bending are considered, together with two different definitions of the material's resistance. Estimated failure moments are then compared with full-scale pipe test data. It is found that the use of an appropriate limit load solution reduces not only the degree of conservatism but also the dependence of the assessment results on the wall thinning geometry, and thus gives the best results. Therefore, finding such solutions for pipes with local wall thinning is an important issue.     

Parametric peak stress functions of 90° pipe bends with ovality under steady-state creep conditions

Stress-based life prediction techniques are commonly used to estimate the failure life of pressurised pipe-related components, such as welds and bends, under creep conditions. Previous research has shown that reasonable life predictions can be obtained, based on the steady-state peak stresses, compared with the life predictions obtained from creep damage modelling. In this work, a series of parametric steady-state peak rupture stress functions of right-angled pipe bends with ovality are presented, which are based on the results obtained from finite element (FE) analyses, covering a number of material property and geometry parameters in practical ranges. Methods used to determine the stress functions are described. The FE analyses have been performed using axisymmetric models, subjected to internal pressure only, with a Norton creep law. Typical examples of parametric peak stress curve fitting are shown. In particular, the accuracy of the interpolation and extrapolation abilities of the stress functions is assessed. The results show that in most cases the interpolated and extrapolated peak stresses are accurate to within ±3% and ±5%, respectively.     

Limit loads and fracture mechanics parameters for thick-walled pipes

In this paper, information on plastic limit loads and both elastic and elastic-plastic fracture mechanics parameters is given for cracked thick-walled pipes with mean radius-to-thickness ratios ranging from two to five. It is found that existing limit load expressions for thin-walled pipes can be applied to thick-walled pipes, provided that they are normalized with respect to the corresponding un-cracked thick-walled pipe values. For elastic fracture mechanics parameters, FE values of the influence functions for the stress intensity factor and the crack opening displacement are tabulated. For elastic-plastic J, it is shown that existing reference stress based J estimates can be applied, provided that a proper limit load for thick-walled pipes is used.     

Limit analysis based on elastic compensation method of branch pipe tee connection under internal pressure and out-of-plane moment loading

A new method of calculating the limit load of a structure via a sequence of incompressible elastic finite element calculations with variable Young's moduli converging to the rigid perfectly plastic problem is used to study the limit load of branch pipe tee connections. Several models of branch pipe tee connection are meshed with shell elements and submitted to internal pressure with end axial load effect or out-of-plane moment. Results are compared with lower and upper bound analytical solutions and experimental results reported in the literature. Computations with 20 noded cubic elements are also proposed to validate shell studies. The J integral is also calculated by a simplified method with the limit load, using an example of a defective branch pipe tee connection.