A comparative study of usefulness for pad reinforcement in cylindrical vessels under external load on nozzle

The main purpose of this paper is to perform a comparative study of strength behavior for cylindrical shell intersections with and without pad reinforcement under out-of-plane moment loading on nozzle. Three pairs of full-scale test vessels with different d/D ratios were designed and fabricated for testing and analysis. A three-dimensional nonlinear finite element numerical analysis was also performed. The maximum elastic stress for each vessel under per unit moment on nozzle is provided. The plastic limit moment on nozzle is obtained by load–displacement and load–strain curves for each test vessel. The results indicate that the effect of pad reinforcement on decreasing maximum elastic stress and increasing plastic limit load is obviously effective. The study results will serve as the available data for understanding the usefulness of pad reinforcements and as the basis for developing an advanced design method by limit analysis for pad-reinforced cylindrical vessels under external loads on nozzle.     

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.     

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.     

Limit load and J estimates of a centre cracked plate with an asymmetric crack in a mismatched weld

The limit load and J estimates of a centre cracked plate with an asymmetric crack in the tensile properties mismatched weld were investigated. A limit load expression was derived on the basis of a simplified slip-line field. A good agreement between the predictions of the expression and finite element (FE) results was found for ratios of half-weld width to the crack ligament, H/l, of less than 0.5. The equivalent stress–strain relationship method (ESSRM) was used to predict elastic–plastic J values. Results from FE analyses show that the ESSRM is accurate for the crack with asymmetry in the mismatched weld provided an accurate theoretical or numerical value of the limit load of the same specimen is available. Defect assessment methods are discussed, and it is found that the failure assessment diagram (FAD) of an asymmetrically cracked mismatched weld can be constructed from the equivalent stress–strain relationship for the same mismatched geometry with a symmetric crack. The effect of an asymmetric crack on the FAD may then be covered by the limit load solution for the asymmetrically cracked mismatched weld.     

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.     

Derivation of ‘γ’ parameter from limit load expression of cracked component to evaluate J–R curve

Experimental evaluation of the J-integral requires the ‘η_pl’ function, proposed by Rice et al. [Progress in flaw growth and fracture toughness testing (1973) 231], to multiply the area under the load vs. plastic load-line-displacement curve. However, the J-integral, thus evaluated, requires modification if crack growth occurs. A ‘γ’ term was proposed by Hutchinson and Paris [Elastic–plastic fracture (1979) 37] and later generalised by Ernst et al. [Fracture mechanics (1979) 581] and Ernst and Paris [Techniques of analysis of load–displacement records by J-integral methods (1980)] to correct the J-integral to account for crack growth. The η_pl and γ functions are available for very few geometries under specific loading conditions. A limit load-based general expression of η_pl was given by Roos et al. [Int J Pres Ves Piping 23 (1986) 81], but no such expression is available for γ functions. The advantage of having limit load-based general expressions for η_pl and γ functions is that the limit load for a particular geometry subjected to a specific loading condition is easily available in the open literature. In the present paper, a limit load-based general expression for the γ function is derived. The general expression is then validated by deriving the known γ functions of various geometries subjected to various loading conditions, which are available in the open literature. The general expressions are then used to derive new η_pl and γ functions for same pipe and elbow geometries with various crack configurations under different loading conditions, for which no solutions are available in the open literature. Finally, experiments have been carried out on 200 mm nominal bore (NB) elbows with throughwall circumferential cracks under in-plane bending moment. The proposed new expressions of η_pl and γ functions for this geometry are used to obtain the J–R curve from the experimental load vs. load-line-displacement and load vs. crack growth data.     

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.     

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.     

Limit and burst pressures for a cylindrical shell intersection with intermediate diameter ratio

This paper provides results of inelastic stress analysis for a vessel–nozzle intersection with intermediate diameter ratio (d/D=0.526) under increasing internal pressure loading from experimental and non-linear finite element methods. The determination of the limit load due to internal pressure is performed by an experimental study for comparison with finite element analysis. The spread of plastic area for the analytical model vessel is provided. Also a burst test of the model vessel is carried out to provide some data to justify the existing design method and the basis for developing an advanced design guideline for shell intersections under internal pressure loading.     

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.     

Numerical analysis of limit load and reference stress of defective pipelines under multi-loading systems

The concepts of limit load and reference stress have been widely used in structural engineering design and component integrity assessment, especially in Nuclear Electric's (formerly CEGB) R5 and R6 procedures. The reference stress method has been proven to be successful in problems pertaining to creep growth, rupture damage, creep buckling, and more recently, elastic–plastic fracture toughness. An approximate method of reference stress determination relies on prior knowledge of limit loads for various configurations and loadings. However, determination of the limit loads for the problems with complicated geometric forms and loading conditions is not a simple task. In the present paper, a numerical solution method for radial loading is presented, the mathematical programming formulation is derived for the kinematic limit analysis of 3D structures under multi-loading systems, and moreover, a direct iterative algorithm used to determine the reference stress is proposed which depends on the evaluation of limit load. The numerical procedure is applied to determine the limit load and reference stress of defective pipelines under multi-loading systems. The effects of four kinds of typical part-through slots on the collapse loads of pipelines are investigated and evaluated in detail. Some typical failure modes corresponding to different configurations of slots and loading forms are studied.     

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.     

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.     

The effect of mechanical heterogeneity and limit load of a weld joint with longitudinal weld crack on the J-integral and failure assessment curve

According to the CEGB R6 (Rev.3) approach, the influence of strength mis-matching and the limit load of a weld joint with a longitudinal weld crack on the J-integral and failure assessment curve can be studied by using an elastic–plastic finite element method for Center-Cracked Panel (CCP) specimens. The results indicate that the values of the J-integral and the shapes and positions of the failure assessment curves are greatly affected by the strength mis-matching factor M, a ratio of weld metal yield strength to that of base metal. If the limit load of the base metal is adopted to normalize the applied load, then the greater the value of M, the larger the safe area is in the failure assessment curve (FAC). However, if the limit load of the weld metal is adopted to normalize the applied load, then the greater the value of M, the smaller the safe area is. Therefore, for the undermatched and evenmatched joints, it is safer to choose the limit load of the base metal as the normalized load, and for the overmatched joints, it is safer to choose the limit load of the weld metal as the normalized load. Moreover, when M is less than 0.8 for the former situation, the option 1 curve of the R6 is not a conservative assessment curve. Considering that there is no simple theoretical formula which is suitable for calculating the limit load of a mechanical heterogeneous weld joint under plane stress and a variety of crack geometries, it is recommended that no matter what the strength of the overmatched or undermatched weld joint is, it is safer to use the limit load of that metal which has the higher strength grade of base metal and weld metal as the normalized load.     

Some observations on the calculation of stresses due to local loads on cylindrical shells using BS 5500, Appendix G

The application of the BS 5500 Appendix G method of calculating stresses due to radial load and moments on rectangular areas on cylindrical shells is quite complex. A parametric survey attempting to simplify the calculations showed that:

1. (i) The limitations on the geometrical parameters required for the validity of the Appendix G method put considerable restrictions on the range of applicability of the method.

2. (ii) The method of calculation of stresses due to a circumferential moment leads to anomalies associated with the neglect of stresses caused by loads on the second loaded area.

3. (iii) For longitudinal moment on a rectangular area offset from the mid-length of the vessel, Appendix G requires a calculation of different effective lengths for the two loaded areas but this refinement does not appear to be necessary.

4. (iv) Some experimentally measured stresses are compared with the predictions of the Appendix G method which is shown to be conservative except for one series of tests.

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.     

Post-buckling behavior during earthquakes and seismic margin of FBR main vessels

Shaking-table tests of cylindrical shells were performed in order to examine the buckling and post-buckling characteristics of FBR main vessels under seismic shear loads. Static buckling tests were also performed, and it is confirmed that there is no significant difference between the static and dynamic load-displacement relations. Based on the test results, hysteresis rules of restoring force were formulated for both elastic and plastic shear-buckling. Non-linear dynamic-response analyses of the single-degree-of-freedom (SDOF) system were then carried out by using the hysteresis rules. The analyses were able to simulate the dynamic test results, especially energy-absorption capacity due to hysteresis behavior. Finally, the non-linear SDOF analysis was applied to the FBR main-vessel cylinder. It is pointed out that the hysteresis behavior could absorb a considerable amount of energy input from seismic motion, which would contribute to the seismic margin of FBR main vessels.     

Evaluating the optimum load range for box-type solar cookers

This paper introduces a new concept of Optimum Load Range (OLR) for solar cookers. OLR gives the load values for which cooker preferably shows good thermal as well as good cooking performance; it may be considered a crucial parameter for solar cookers. This OLR concept is based on the dependence of rate of rise of load temperature on different heat transfer processes between load and cooker interior. This concept illustrates solar cooking in two simple steps. The total time required to complete these steps puts an essential constraint for cooking of any load amount. The maximum value of load (upper limit of OLR) till which cooker shows satisfactory cooking may be determined from this constraint. This constraint requires determination of two OLR parameters which are t_{step I} and t_{step II}. The load for which cooker remain almost 30% efficient, may be referred as lower limit (minimum value) of OLR. For the verification of OLR, experimental studies have been conducted with a solar cooker named SFSC. The OLR parameters along with different thermal performance parameters (TPPs) (second figure of merit (F₂), utilization efficiency (η_u) etc.) suggested by different researches for solar cookers in water load condition have been computed from the measured thermal profiles of different loads (0.8–3.0 kg). From the curve analysis of different TPPs with load, the existence of upper limit of OLR is observed. The values of rate of rise of load temperature at water temperatures 80, 85 and 90 °C for different loads also confirm the same. The OLR of SFSC is found to be 1.2–1.6 kg.     

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.     

Validation of BS7910:2005 failure assessment diagrams for cracked square hollow section T-, Y- and K-joints

This paper describes the usage of finite element (FE) analyses results to validate the standard BS7910 assessment procedure for the safe design of cracked square hollow section (SHS) T-, Y- and K-joints. In the study, the actual 3D surface cracks obtained from previous fatigue tests have been included in the FE models. An automatic mesh generation program is then developed and used to produce the failure assessment diagram (FAD) through the J-integral method. The ultimate strength of uncracked SHS joints with reduced load bearing areas have been referenced to derive the plastic collapse loads of cracked SHS joints for the development of FAD. These loads have been validated against the previous experimental results. In comparison with the existing standard BS7910 Level 2A/3A FAD curve and the proposed assessment procedure for circular hollow section joints, it is found that a plastic collapse load with a penalty factor of 1.05 will be sufficient for the safe assessment of cracked SHS T, Y, and K-joints under brace end axial loading.