Line-spring Element Research Articles

Multiple-crack identification in a channel section steel beam using a combined response surface methodology and genetic algorithm

Summary The present study outlines a sequentially integrated finite-element method (FEM)–response surface method (RSM)–genetic algorithm (GA) framework and implements this to predict the crack parameters, namely, crack location and crack depth ratio. A central composite face centered response surface design of the RSM technique is used to establish the direct relationships between the input parameters (crack location and crack depth ratio) and responses (natural frequencies) to build the response surface function (RSF). Multiple edge cracks are considered, which exist on the top flange of a thin-walled channel section cantilever beam and modeled as line spring elements. In order to obtain RSFs of the first five natural frequencies in terms of process factors such as crack position and crack depth ratio, a number of numerical experiments based on FEM are conducted by using the design-of-experiment approach. An objective function obtained as the square of the difference between RSF and experimentally measured natural frequency has been minimized using GA to find out the optimum crack parameters. Twenty-four steel channel beam specimens have been tested in the laboratory to extract the crack parameters from measured natural frequencies using the proposed approach. The results of the study indicate that the proposed approach performs remarkably, yielding crack parameters with great precision. Copyright © 2015 John Wiley & Sons, Ltd.

Determination of Shakedown Boundary and Failure-Assessment-Diagrams of Cracked Pipe Bends

Determination of shakedown (SD) boundaries of 90-degree plain smooth pipe bends has recently received substantial attention by several researchers. However, scarce or almost no solid information is found within the literature regarding the determination of the shakedown boundary of cracked pipe bends. The current research presents two additions to the literature, namely, determination of shakedown boundary for a circumferentially cracked 90-degree pipe bend via a simplified technique utilizing the finite element (FE) method and introduction of failure-assessment diagrams (FADs) in compliance with the API 579 failure-for-service assessment of pressure vessel and piping components. The analyzed cracked pipe bend is subjected to the combined effect of steady internal pressure spectrum and cyclic in-plane closing (IPC) and opening (IPO) bending moments. Line spring elements (LSEs) are embedded in quadratic shell elements to model part-through cracks. FAD is obtained through linking the J-integral fracture mechanics parameter with the shakedown limit moments of the analyzed cracked 90-degree pipe bend. The LSE outcomes illustrated satisfactory results in comparison to the results of two verification studies: stress intensity factor (SIF) and limit load. Additionally, full elastic-plastic (ELPL) cyclic loading finite element analyses are conducted and the outcomes revealed very good correlation with the results obtained via the simplified technique. The maximum load carrying capacity (limit moment) and the elastic domain are also computed thereby generating a Bree diagram for the cracked pipe bend. Finally, Crack growth analysis is presented to complement the FAD.

Fullerenes as mass sensors: A numerical investigation

This paper investigates the novel development of a mass sensitive nanosensor based on the use of individual spherical fullerenes. The main advantage of the mass sensing ability of spherical fullerenes in comparison with other nanomaterials such as carbon nanotubes (CNTs) or graphene nanoribbons (GNRs) is the fact that they present almost perfect geometric symmetry and thus a unique vibrational behavior which is independent from the location of the externally added nanoparticle. The study is conducted by the use of a computationally effective numerical scheme based on the adoption of appropriate three dimensional line spring elements as well as point mass elements to simulate the atomistic structure of fullerenes and interatomic interactions appearing between carbon atoms. The free vibration of C20, C60, C80 and C180 molecules is analyzed without and with an external nanoparticle of specific mass attached on their structure to calculate the arisen change in their natural frequencies and corresponding shape modes. A parametric study concerning the magnitude and location of the added mass is performed in order to evaluate the mass sensing ability of the fullerenes under consideration.

CYCLIC PLASTICITY MODELLING FOR ANDES THIN SHELL AND LINE-SPRING FINITE ELEMENTS

This paper presents a proposed methodology to account for cyclic plastic response of the thin shell ANDES and line-spring finite elements. A through thickness integration scheme is employed for the shell element and stress resultant plasticity is used for the line-spring element. A simplified contact formulation to account for crack closure in the line-spring element is also presented. Numerical comparisons between the proposed models and detailed 3D analyses (pipes) are carried out and presented herein. A comparison between the present implementation and large scale experiment of a surface cracked pipe subjected to large cyclic plastic strains is also presented. The purpose of the presented implementation is to account for cyclic loading in pipeline technology where significant amount of plasticity in the loading cycles occurs.

Free vibration analysis of axially loaded cracked Timoshenko beam structures using the dynamic stiffness method

In this article, the purpose is to investigate the changes in the magnitude of natural frequencies and modal response introduced by the presence of a crack on an axially loaded uniform Timoshenko beam using a particular member theory. A new and convenient procedure based on the coupling of dynamic stiffness matrix and line-spring element is introduced to model the cracked beam. The application of the theory is demonstrated by two illustrative examples of bending–torsion coupled beams with different end conditions, for which the influence of axial force, shear deformation and rotatory inertia on the natural frequencies is studied. Moreover, a parametric study to investigate the effect of the crack on the modal characteristics of the beam is conducted.

Two-parameter fracture mechanics and circumferential crack growth in surface cracked pipelines using line-spring elements

A procedure for constraint correction of crack growth resistance curves for single edge notched specimens and for pipe geometries is presented. The procedure is based on FE models with the combination of shell- and line-spring finite elements. Crack tip opening displacement and T-stress are employed, and ductile crack growth is accounted for. Experimental crack growth resistance curves are obtained for both single edge notched tension- and bending-specimens for different crack depths to cover significantly different constraint levels. To account for different constraint levels, a method to scale the resistance curve using the T-stress is implemented. The analyses include ductile crack growth in both the circumferential and thickness directions. The effect of circumferential crack growth with biaxial loading is also presented. The results from the line-spring model are compared with detailed 3D-models for verification of the implementation of circumferential crack growth. The importance of including crack growth in circumferential direction is discussed based on numerical parametric studies. A measure to quantify the importance of circumferential crack growth is proposed.

Finite element study of partial-depth cracks in restrained PCC slabs

An existing finite element program—ISLAB2000—was modified to allow for the analysis of horizontally restrained and unrestrained portland cement concrete (PCC) pavement slabs with partial depth cracks. To simulate a partial-depth crack, special line spring elements were added to the ISLAB2000 finite element code. Previously published relationships between spring elements stiffness and crack depth were used for formulating the element stiffness matrix of the line spring elements and to calculate stress intensity factors after solving the system of finite element equations. Several example problems of partially cracked PCC slabs subject to typical highway loadings were solved to verify this newly developed finite element model termed ISLAB2005. The impact of crack depth and base friction parameters on slabs stresses and deflections was investigated. On the basis of this analysis, ISLAB2005 was found to be a reliable tool for analysis of partially cracked pavements that are either restrained or unrestrained from horizontal movements.

Closed form line spring yield surfaces for deep and shallow cracks: formulation and numerical performance

The line-spring finite element is a versatile numerical tool for performing engineering fracture mechanics analysis of surface cracked shells. The accuracy of the line-spring finite element solution for deep/medium-sized cracks has shown to be higher than the accuracy for the shallow-sized ones. Proper treatment of shallow cracks is important because they are the ones most frequently encountered in engineering practice. Accurate yield surfaces of plane-strain single edge-cracked specimens having shallow, as well as deep, cracks are developed here in order to improve the overall performance of the line-spring element. The yield surface is represented by equations that automatically satisfy the convexity requirement and that fit the result of limit load analyses. The present study addresses, furthermore, the performance of the backward Euler return algorithm for one of the accurate yield surface formulated here, by means of iso-error maps.

Fracture analysis of strength-mismatched welded wide plates by line spring elements

Higher utilization of structural materials leads to a need for accurate numerical tools for reliable prediction of structural response. Analysis with line spring finite elements is efficient in performing approximate fracture mechanics analysis of surface cracked shells. The present study addresses the performance of the line spring finite element utilized for describing a surface crack in a bimaterial elastic plastic wide plate structure. Some outlines for the application of the line spring in a yield strength mismatch situation are obtained from a parameter study, and utilized for the two-dimensional (2D) discretization of welded wide plates. Large scale test results are compared to the results obtained from both a 2D and 3D numerical analysis.

Damage Detection and Vibration Control in Smart Plates: Towards Multifunctional Smart Structures

This paper demonstrates the concept of sensitivity enhancing control (SEC) to aid in damage detection in smart structures through both experimental and simulation evaluation. Methods of implementing state estimate feedback using point measurements of strain along the structure are described, and an initial proof-of-concept laboratory experiment demonstrating enhancement of modal frequency shifts due to tip mass damage in a cantilevered beam is reported. Simulation results focus on applying state feedback control to finite-element models of a cantilevered structure with slot, through-surface crack, or surface crack damage. The simulation analysis ascertains the ability to enhance sensitivity of modal frequency shifts due to realistic damage cases that are difficult to evaluate experimentally. The simulation also ascertains the potential for using the same sensors and actuators for implementing both sensitivity enhancing control laws and vibration damping control laws that are insensitive to damage. In the control model with which SEC laws are designed, damage consists of simple reductions in thickness over a small area of the structure. Finite-element models to which control laws are applied are developed using commercial software (ABAQUS TM ) that more accurately models stiffness damage by releasing element connections or by using line spring elements to model fatigue cracks. Experimental results show that enhancement in sensitivity of modal frequencies to damage can be achieved using a single piezoceramic actuator and multiple piezoelectric strain sensors along the beam. Simulation results indicate that feedback control laws can be designed for either sensitivity enhancement or vibration suppression using identical hardware, providing for multifunctional smart structures. In addition, analysis demonstrates that commercial finite-element software is useful for model-based simulation of damaged controlled structures.

Finite element modelling of cracked inelastic shells with large deflections: two-dimensional and three-dimensional approaches

Higher utilization of structural materials leads to a need for accurate numerical tools for reliable predictions of structural response. In some instances, both material and geometrical non-linearities are allowed for, typically in assessments of structural collapse or residual strength in damaged conditions. The present study addresses the performance of surface-cracked inelastic shells with out-of-plane displacements not negligible compared to shell thickness. This situation leads to non-linear membrane force effects in the shell. Hence, a cracked part of the shell will be subjected to a non-proportional history of bending moment and membrane force. An important point in the discretization of the problem is whether a two-dimensional model describes the structural performance sufficiently, or a three-dimensional model is required. Herein, the two-dimensional modelling is performed by means of a Mindlin shell finite element. The cracked parts are accounted for by means of inelastic line spring elements. The three-dimensional models employ eight-noded solid elements. These models also account for ductile crack growth due to void coalescence by means of a modified Gurson–Tvergaard constitutive model, hence providing detailed solutions that the two-dimensional simulations can be tested against. Using this, the accuracy of the two-dimensional approach is checked thoroughly. The analyses show that the two-dimensional modelling is sufficient as long as the cracks do not grow. Hence, using fracture initiation as a capacity criterion, shell elements and line springs provide acceptable predictions. If significant ductile tearing occurs before final failure, the line spring ligaments have to be updated due to crack growth.

An engineering method for constraint based fracture assessment of welded structural components with surface cracks

In this study it has been shown that accurate descriptions of crack-tip stress-fields in surface cracked welded plates can be obtained without large 3D FEA models. When a fracture mechanics FE analysis is required in a large construction, existing shell models can be used in combination with a plane strain submodel. The 2D plane strain model is driven by displacements from the global shell model. This technique has been used to simulate crack-tip stress-fields in a surface cracked plate. The crack-tip stress fields are characterised with the J-integral and the constraint parameter, Q. The crack in the global shell model was simulated with line-spring elements. The global behaviour as well as the crack-tip stress-fields of the plane strain submodel have been compared to a 3D solid model. Initially, the crack-tip stress-fields in the plane strain model and the 3D model with surface crack were compared, using the same in-plane mesh and element type. It was found that when first order elements were used, the constraint was higher in 3D than in plane strain. For second order elements, however, the trend was the opposite. By using a correction factor for the load, the load vs. J behaviour and the crack-tip stress-fields of a surface cracked plate can be predicted from a shell analysis with line-spring elements and a plane strain model. Accurate predictions of J and Q were obtained using the shell + submodel technique for homogeneous material and for a weldment with fusion line crack. The shell + submodelling technique was used to assess brittle fracture in two steel weldments with a surface crack using the RKR failure criterion by Ritchie et al. [16]. For the investigated case, the toughness requirements could be relaxed significantly based on the two parameter analysis compared to conventional fracture mechanics analyses.

Numerical analysis of cracked inelastic shells with large displacements or mixed mode loading

The trend in higher utilization of structural materials leads to a need for accurate numerical tools for reliable predictions of structural response. In some instances both material and geometrical nonlinearities are allowed for, typically in assessments of structural collapse or residual strength in damaged conditions. Dynamically loaded structures are prone to fatigue cracking; this has to be accounted for when computing nonlinear structural response. The present study addresses the performance of cracked inelastic shells with out-of-plane displacement not negligible compared to shell thickness. This situation leads to membrane force effects in the shell. Hence, a cracked part of the shell will be subjected to a nonproportional history of bending moment and membrane force, and e.g. fracture mechanics parameters ( J-integral ) are affected. A Mindlin shell finite element based nonlinear program is developed and utilized herein. The cracked parts are accounted for by means of inelastic line spring elements. These elements also account for possible mode II deformations.

Numerical studies on crack growth in a steel tubular T-joint

The results of a numerical investigation on fatigue crack growth of an offshore tubular T-joint under the action of axial, in-plane and out-of-plane bending loads are presented in this paper. Extensive stress analysis has been carried out to determine the location of the hot spot stress along the brace—chord intersection for each load case. Semi-elliptical cracks with varying crack lengths and crack depths were introduced at the hot spot location by means of line spring elements for stress intensity factor evaluation. The line springs were properly constrained to prevent the problem of crack surface penetration. The stress intensity factors obtained are then used in a crack growth law for life estimation.

Finite-Element Analysis of Portland Cement Concrete Pavements with Cracks

It was verified that finite-element modeling could be successfully used to analyze concrete pavements with partial-depth cracks. An existing finite-element program, ILLI-SLAB, was modified (ILSL97) to allow for partial-depth crack analysis. To model a partial-depth crack, a special line spring element was added to the finite-element code. The line spring elements mimic the behavior of a crack by acting as a rotational hinge between two continuous slabs. By using available fracture mechanics techniques, a relationship was derived between the amount of moment load transfer across a crack and the crack depth. This analytical solution was then used to formulate the element stiffness matrix for the line spring element. The deflections predicted by the new finite-element program are correct, but the stresses in the vicinity of the crack tip needed to be corrected to match the stress singularity zone in front of cracks. Several example problems were used to verify the proposed finite-element model, and an example of a typical highway loading condition was analyzed.

A mixed mode I/II inelastic line spring

In many practical situations a cracked component may be subjected to mixed mode deformations. One important situation where this is relevant is surface cracked tubular joints applied in the offshore industry. Mixed mode affects both the deformation characteristics and fracture initiation capacity of the component, hence, an analysis model for numerical simulation should account for this. A finite element model consisting of solid elements is one approach to solve such problems. For complex components like a tubular joint, a very efficient model is obtained by means of shell finite elements. The cracked sections may be accounted for, utilizing line spring elements. Earlier work has concentrated on the Mode I deformation of the line spring. In the present study a mixed mode I/II elastic-plastic line spring is derived. The performance of the simple model is compared to continuum solutions obtained by means of detailed plane strain FE analyses. Both load vs deformation and J-integral performance of the line spring are considered in the comparisons.

Dynamic Response of Tubular T-Joints Under the Influence of Propagating Cracks

Abstract This paper presents an analytical study on the vibration response of tubular T-joints for detecting the existence of cracks along their intersections. The ABAQUS finite element program was utilized for carrying out the analysis. Frequency response functions were obtained for a joint with and without cracks. The joint was modeled with 8-node degenerate shell elements having 5 degrees of freedom per node. Line spring elements were used to model the crack. The exact crack configuration (semielliptical shape, Fig. 5(b)), as observed from numerous experimental fatigue crack investigations at the critical location, has been achieved through a mapping function, that allows a crack in a planar element to be mapped on to the tube surface. The natural frequency changes with respect to crack depth show little changes, being 4.82 percent for a 83-percent crack depth for the first mode. On the other hand, significant changes have been observed for bending moment and curvature as a function of crack depth. For an 83-percent chord thickness crack, a 97-percent change in bending moment at points around the crack vicinity, and 34.15 to 78 percent change in bending moments, for those locations far away from the crack location, have been observed. Natural frequency change should be combined with other modal parameters such as “bending moment (or bending strain)” and “curvature” changes for crack detection. The presence of the crack can be detected at locations far away from the crack location using such sensors as strain gages.

Crack Interaction Criteria in Pressure Vessels and Pipe

An attempt was made to define a new crack interaction criterion for pressurized cylinders with two co-planar surface cracks. Elastic-plastic finite element method with line spring concept (line spring element method) was used to verify the validity of the new interaction criterion and to establish the relative conservatism built into various codes/standards. The crack interaction criteria of two co-planar surface cracks as defined by ASME Section XI and BS PD6493 were studied and a new interaction criterion which accounts for crack shape and load factor was introduced. The basic idea behind the crack interaction criteria for co-planar surface cracks was the plastic zone and stress interaction near crack tips. To verify the new crack interaction criterion, comparisons of J-integral values were made for various crack sizes with different distances between cracks and loading conditions. Based upon these comparisons, the new crack interaction criteria, comparing a physical distance, s, to a characteristic distance d=(σ/σy)2(c1Q1 + c2Q2), proved to be a reasonable parameter for indication of the crack driving force interaction for co-planar cracks. The characteristic distance also represents a rigorous measure of an equivalent crack driving force for interacting cracks.

Enhanced elastic-plastic line-spring finite element

The elastic-plastic line-spring finite element of Parks and White (1982), as implemented in the ABAQUS finite element code (1988b), is modified in several ways for a better evaluation of crack-tip deformation parameters such as K1, J, crack-tip opening displacement and crack-tip stress triaxiality, the last of which is inferred or measured by the so-called T-stress and Q-stress parameters. An effective crack length formulation is introduced to capture the nonlinearity due to contained plane strain crack-tip plastic deformation, the size and orientation of which are known to be sensitive to the sign and magnitude (relative to yield strength) of the T-stress. More accurate yield surfaces, obtained from finite element limit analyses of plane strain single-edge cracked specimens, are employed to more sharply define the behavior of the line-spring model in the fully plastic region. More accurate relationships between incremental plastic crack-tip opening displacement and incremental load point displacement/rotation are also incorporated into the model for shallow crack configurations under fully plastic loading. The modified line-spring model is then applied to surface cracked plate problems having a wide range of crack-tip constraints, and the feasibility of implementing J-T(Q) based two-parameter characterizations of elastic-plastic crack front fields via simplified line-spring elements is examined.

A finite element study of stress fields and stress intensity factors in tubular joints

This paper reports a study on the determination of stress intensity factors in tubular joints in offshore jacket structures. Using finite elements, information on stress concentration factors and through-thickness stress distributions was first obtained from uncracked geometries. This was correlated with the stress intensity factors in joints containing semi-elliptical cracks which were modelled with line-spring elements. The validity of the numerical models was established, using a simple T-joint, by comparing the results with existing experimental data and results from three-dimensional finite element analyses. Various modelling assumptions used in previous numerical work were critically examined. The multi-planar effects in tubular joints were simulated by subjecting the out-of-plane braces to various loadings and restraints. It was found that a relationship exists between the stress concentration factor, the degree of bending and the stress intensity factor for the various loading and restraint cases considered, and that the stress intensity factors in multi-planar tubular joints can be estimated by suitably modifying an existing empirical equation for surface cracks in plain plates.