Basic Load Cases Research Articles

General Exact Analytical Expressions for Rotation and Displacement of a Timoshenko Beam Under Variable Loads with Validation

The researchers involved in the field of applied mathematics mostly project the efficient and viable solutions of those problems in which have practical applications in science and engineering. The Timoshenko beam model (TBM) is described by a system of ordinary differential equations where the expressions of rotation and displacement of the beam are ultimately required. Most of the work on analytical solutions of beam problems in literature focuses the elastic and Euler-Bernoulli beams, whereas for the TBM the numerical solutions are usually preferred. The analytical expressions for the TBM exist for only very basic load cases and are also not general for any load function. In this paper, we attempt to suggest a novel protocol based on expressing a load function by a polynomial or its power series development, and then use it to develop general analytical expressions for the rotation and displacement of a fixed TBM which is not load specific. The proposed general equations can provide ease of access and handling for the practitioners working in applied mathematics, structural engineering and mechanical vibrations as the developed equations can be used with only constant inputs to tailor particular expressions of rotation and displacement profiles of a fixed TBM under any variable load. For performance evaluations of the proposed general equations, we have also obtained particular expressions for some important variable loads, like: linearly varying loads ((LVLs): triangular and trapezoidal and quadratically varying loads (QVLs): parabolic/square and circular loads. Finally, the proposed protocol for generalization has been validated for fixed elastic beams under uniformly distributed loads (UDLs), and the results match exactly with those expressions available in literature. The contributions of this study, on one hand provide ready, direct and exact general expressions for the rotation and displacement profiles of a fixed TBM, while on the other hand the solution of such problem is achieved with quite negligible computational overhead, execution time and software implementations.

Strength analysis of welded joints in support structures of heavy machinery subjected to changeable loading conditions

The durability of bearing units of large machines depends mainly on the condition of their welded joints. With this in mind, we developed numerical models of the analyzed bearing units, for which we performed FEM simulations of the stresses in welded joints in several basic load cases. In each of the respective variants the technical condition of the bearing nodes was different and it corresponded to the severity of the degradation processes. Different positions of the superstructure in relation to the undercarriage were also taken into account. The simulations used the hot-spot method dedicated to FEM analyses of complex welded structures. We discovered that the loads have a significant influence on the values and distribution of von Mises principal stresses and their axial components. Based on the carried out analyses, we identified the most unfavorable load cases that generate the highest stresses in the welded joints of the assessed nodes. We also demonstrated that the applied method effectively assesses the stresses of welded joints subjected to variable working loads.

An Artificial Neural Network Model for the Stress Concentration Factors in KT-Joints Subjected to Axial Compressive Load

Stress concentration factor (SCF) is usually used to estimate the fatigue life of an offshore joint. Historically, parametric equations were used to estimate SCF based on a statistical analysis of experimental and finite element analysis (FEA) results, to reduce cost and time. These equations give the SCF at the saddle/crown position for simple joints and basic load cases. However, for modified or defective joints, the location of the maximum SCF can change. In such circumstances, the single-point SCF equation cannot be used to estimate the maximum value of SCF, as its location may have changed from saddle/crown. To our knowledge, there are no general expressions to estimate SCF around the brace axis accurately. As artificial neural networks (ANN) can approximate the trend of complex phenomena better than conventional data fitting, a mathematical model based on ANN is proposed to estimate SCF based on the weights and biases of trained ANN. Nine hundred thirty-seven finite element simulations were performed to generate SCF data for training the ANN. This ANN was used to model an empirical equation for SCF. The proposed empirical model can estimate SCF around the brace axis with less than 5% error. The current study provides a roadmap to using FEA and ANN for empirical modeling of SCF in tubular joints, and this approach can be applied to any joint type, with or without design modification or damage. Once a database of similar equations is available, it can be utilized for quickly estimating SCF instead of costly experimentation and FEA. Optimization of the ANN can further improve the accuracy of the developed mathematical model.

SIMPLIFIED MODELING OF V-SHAPED BRIDGE PIERS

DDesigning a steel, V-shaped bridge pillar requires a complex and detailed finite element calculation model. The calculation can be simplified by using a simple one-dimensional beam model, but then the stress concentrations are not determined in detail. The use of a stress concentration factor, SCF, to make this possible. Before using this method, it is first verified whether both models behave correspondingly, by calculating the internal forces certain sections. Five basic load cases are used to perform this verification. Afterwards the stress concentration factors are determined for those same load cases. This method induces inconsistencies, some predictable but others unpredictable. The conclusion can however be that this simplified meth gives unreliable results. Therefore, a second method is used, where the high stresses in the curvatures of the pillar are related to the stresses in a nearby neutral section. Such a neutral section is defined as a section where the stress peaks due to the geometric effects caused by the actual geometry of the nodes of the V-shaped pilar are not present anymore. This method provides better results for the SCF; however, some issues remain, which are discuss in more detail in the paper. Finally, the impact of a varying radius of curvature on the stresses is also studied. Some clear trends are observed in this paper Overall, this paper discusses a possible simplified design method for V-shaped bridge pillars, at the same time listing the relevant design difficulties and possible pitfalls of a too simplified model.

Experimental Study of Hot Spot Stress for Spatial CHS KK-Joints

In offshore structures such as offshore wind turbine jacket foundations, prolonged exposure to wind and wave loads can lead to fatigue failure, especially at the joint connections. Currently, international codes primarily evaluate the fatigue performance of tubular joints using the hot spot stress (HSS) method, with stress concentration factor (SCF) being the most crucial parameter. Numerous studies have investigated the SCF of tubular joints and proposed corresponding calculation formulas. However, most of these studies have focused on simple basic loading cases. To comprehensively understand the distribution pattern of the HSS method at spatial tubular joints, this study selects the spatial CHS KK-joints as the research subject and performs the HSS testing on spatial KK-joints. The research indicates that, in CHS joints, the distribution of the HSS on the chord side remains consistent under tensile or compressive loading conditions along the brace axis. Additionally, within spatial joints, the loading conditions of neighboring braces near the reference brace significantly affect the distribution of HSS on the chord side and exhibit varying spatial effects on the SCF. Furthermore, this study applies recommended calculation formulas from codes to analyze the experimental results. The findings reveal significant discrepancies between the SCF calculation formulas recommended by CIDECT and the test results, rendering it inadequate for accurately calculating the SCF of spatial KK-joints. Therefore, further research is required to develop suitable calculation formulas for SCFs in spatial KK-joints.

Imperfection Sensitivity of Cylindrical Shells Under Shear Loading

AbstractOne of the basic load cases relevant for the design of shell structures is torsion, which can occur exemplary in silos and tanks under seismic actions or even under wind loading. To address the stability problem of cylindrical shells under this load case, the relevant design standard EN 1993‐1‐6 and the European Recommendation ECCS EDR5 provide simple approaches for the buckling design of cylindrical shells. In order to improve the current hand calculation rules given in the above documents, this numerical study attempts to include the imperfection sensitivity of cylindrical shells subjected to uniform torsion.In the present paper, theoretical, numerical, and experimental results from the literature are reviewed, and new systematic computational analyses and evaluations are introduced regarding the imperfection sensitivity of cylindrical shells under shear loading. For this purpose, a comprehensive computational study was performed using linear and nonlinear buckling analyses on perfect and imperfect shells. Since the influence of imperfections on the buckling strength of cylindrical shells is highly dependent on the imperfection pattern used in the numerical models, nonlinear buckling analyses were performed on imperfect cylindrical shells with three imperfection types: (i) eigenmode‐affine shape from torsion, (ii) longitudinal eigenmode‐affine pattern, and (iii) eigenmode‐affine shape from external pressure.

Experimental study of hot spot stress for three-planar tubular Y-joint: I. Basic loads

Various tubular joints of offshore wind turbines (OWTs) have seen rapid development in the past decades. However, experimental studies on multiplanar tubular joints have mainly been underexplored. This study designed and constructed a multiplanar loading experimental system for multiplanar tubular joints. A model test was performed using this system to investigate the hot spot stress (HSS) of a three-planar tubular Y-joint, which is critical for the fatigue design of a tripod substructure. The HSS distribution along the weld seams under the 12 basic load cases was obtained from the test analysis. The geometric stresses at the key positions of the chord and brace sides were obtained and analysed. The peak values of stress concentration factor (SCF) were obtained and compared with those calculated using the method recommended by the codes. An indirect method that explicitly considers the interaction between multiple braces was proposed and verified to calculate the SCF distribution of multiplanar tubular joints.

Riveted joints in composites, a practical tool to estimate stresses around the rivet hole

Riveted joints have different failure mechanisms, some associated to the failure of the rivet itself, and others associated to the materials to be joined. The failures associated to the material are mainly due to the stress concentrations at the rivet holes. In joints with metallic (ductile) materials, these stress concentrations might not play a relevant role in the failure initiation, but with unidirectional composite (brittle) materials, they do control the failure. Both the orthotropic behavior of joined materials and the non-linear contact conditions between the rivet and the hole, make that the problem does not have a general solution. In the present work, several finite element analyses have been carried out for basic load cases, different materials, and orientations of the fiber. Furthermore, a two-steps least squares adjustment has been performed to obtain a polynomial expression of the stress distributions for all cases. As a result, a tool is provided, allowing the users to easily estimate the stress distribution (σrr, σθθ) along the hole contour, and σxy along the tangential lines to the hole circumference. Such stress ditribution can be used for failure predictions. All input and output data are available to readers, by means of a spreadsheet, to estimate mentioned stress distributions.

Reliability and applications of a new design method for calculating hot-spot stress in CHS double K-joints under arbitrary combined loading

This paper presents further development of a simple and accurate new method for calculating hot-spot stress in circular hollow section (CHS) double K (DKK) joints to enable safe application for practical design. Such joints are widely used in the construction of offshore wind turbine (OWT) jackets and use of this design approach offers the opportunity to reduce the weight of these structures. The new hot-spot stress calculation method decomposes any arbitrary brace load combination (axial load, in-plane bending, out-of-plane bending) into six basic load cases and employs only 11 key stress concentration factors (SCFs) to define stress distributions around the weld toes of all four brace-chord connections in the DKK joint. This paper applies the UK Department of Energy (UK DEn) criteria for acceptance to ensure that the new method is safe. This design method is then applied to design two representative jacket structures in the North Sea (Seagreen and East Anglia wind farms) and the structures are compared to those designed using the current industry practice of calculating hot-spot stress. This comparative study indicates that steel weight savings of 10.43% and 7.07% may be achieved using the new hot-spot stress calculation method whilst achieving the same predicted service life.

A numerical framework for determination of stress concentration factor distributions in tubular joints

Fatigue design of welded tubular joints implies the calculation of stress concentration factors at distinct weld toe locations using parametric formulas for basic loading cases. A Python based framework for automated calculation of hot spot stress and stress concentration factor distributions along the entire weld is presented. Hot spot stresses are calculated based on extrapolation of surface stresses extracted from a finite element analysis output database. For each node of the weld toes, the extrapolation direction and location dependent distances are first determined. Next the complete stress tensor in each read-out point is extrapolated towards the weld toe. The hot spot stress is determined as the maximum value of three critical stress values following the guidelines of DNV.A validation study is carried out in which the framework results are compared to both experimental data and parametric equations from literature for three different load cases: axial loading, in-plane bending and out-of-plane bending. The results obtained from the developed framework show good agreement with the experimental results and capture the shape of the stress concentration factor distribution much better than reference parametric equations. Finally, the framework is used to demonstrate the importance of including the weld geometry in the finite element model of the tubular joint. The results show that using a model without a weld can result in overly conservative estimations of stress concentration factors.

Micromechanical evaluation of failure models for unidirectional fiber-reinforced composites

In this paper, micromechanical simulations are employed to evaluate the performance of the Tsai–Wu and Hashin failure criteria for fiber-reinforced composites, especially in stress states whose experimental reproduction is complicated. Micromechanical responses are generated using a finite element model of a representative volume element, in which only the matrix material experiences damage and the fibers are assumed to be elastic. Micromechanical simulations of basic load cases are used to calibrate macrolevel criteria. Finally, the response of the micromodel and macromodels is compared for various load combinations. Despite a good agreement between Tsai–Wu criterion predictions and micromodel results in a wide range of stress states, some stress combinations are highlighted for which the strength is not predicted accurately. Additionally, accuracy of the Hashin criterion suffers from ignoring the influence of stress in fiber direction on matrix failure.

Efficient frequency-domain based fatigue life estimation of spot welds in vehicle components

The estimation of the damage and fatigue life of spot welds in vehicle components with hundreds or thousands thereof within reasonable computation time is a challenge even for recent computers in automotive industry. Hence, an advanced procedure in the frequency-domain is proposed. Whereas the general frequency-domain approach is well-established, it nevertheless becomes efficient for a large number of spot welds only with a package of measures. Among others, a two-scale FE-method using a pre-processed data base for various spot weld/sheet geometries and specified basic load cases, application of the cutting-plane method, and a priori stress estimations to reduce the number of spot welds to be checked completely are suggested. By the example of a tank of a cooling medium in a vehicle it is shown that the proposed algorithm may reduce the computation time as compared to time-domain calculations substantially.

A novel framework for deriving the unified SCF in multi-planar overlapped tubular joints

For the past few decades, a significant number of research efforts have been devoted to the studies of SCF values and parametric design formulae at certain positions in various uniplanar and multi-planar tubular joints. However, very few investigations on the multi-planar overlapped tubular joints have been reported. The main reason may lie in the complexity of combining effects between multi-planarity and overlapping. In this paper, a novel framework for deriving the unified SCFs is proposed by reducing joint modeling from multi-planar out-of-plane overlapping to equivalent uniplanar non-overlapping. By integrating the equivalent beam model with solid finite element modeling for the overlapped tubular KK-joint in offshore jack-ups, the procedures for computing the unified SCF values in the equivalent beam stick model are successfully developed. Verification of the proposed framework is obtained by comparing the unified SCFs based on basic load cases with those based on actual wave loading typically experienced by the jack-up.

Generalized T-stress solutions for penny-shaped cracks in transversely isotropic piezoelectric media

A novel set of analytical and semi-analytical solutions is obtained for the generalized T-stress for both electrically impermeable and permeable penny-shaped cracks embedded in a transversely isotropic, linear piezoelectric, infinite medium under uniform far-field loading and general mechanical and electrical crack-face loading. The generalized stress field of the crack for two basic loading cases, viz. uniformly distributed self-equilibrated crack-face loading and a pair of crack-face self-equilibrated unit normal point forces and unit point electrical charges, are employed together with the asymptotic near-front expansion to establish fundamental solutions for the generalized T-stress. Such fundamental solutions are explicit and involve only elementary functions, and are then utilized to form an explicit integral formula for computing the generalized T-stress of a penny-shaped crack under general crack-face loading. The exact integration of the integral formula is also established for specific loading conditions, whereas its evaluation via an efficient quadrature for general loading conditions is also addressed. A selected set of numerical results is presented to verify the proposed quadrature and significant and useful findings are also reported.

A piecewise linear transverse shear transfer model for bolted side-plated beams

The performance of bolted side-plated (BSP) beams is affected by the degree of transverse partial interaction, which is a result of the interfacial slip caused by transverse shear transfer between the bolted steel plates and the reinforced concrete beams. However, explicit formulae for the transverse shear transfer profile have yet to be derived. In this paper, a simplified piecewise linear shear transfer model was proposed based on force superposition principle and simplification of shear transfer profiles derived from a previous numerical study. The magnitude of shear transfer was determined by force equilibrium and displacement compatibility condition. A set of design formulae for BSP beams under several basic load cases was also derived. Then the model was verified by test results. A worked example was also provided to illustrate the application of the proposed design formulae. This paper sheds some light on the shear force transfer mechanism of anchor bolts in BSP beams, and offers a practical method to evaluate the influence of transverse partial interaction in strengthening design.

Vertical Axis Wind Turbine Design Load Cases Investigation and Comparison with Horizontal Axis Wind Turbine

The paper studies the applicability of the IEC 61400-1 ed.3, 2005 International Standard of wind turbine minimum design requirements in the case of an onshore Darrieus VAWT and compares the results of basic Design Load Cases (DLCs) with those of a 3-bladed HAWT. The study is based on aeroelastic computations using the HAWC2 aero-servo-elastic code A 2-bladed 5 MW VAWT rotor is used based on a modified version of the DeepWind rotor For the HAWT simulations the NREL 3-bladed 5 MW reference wind turbine model is utilized Various DLCs are examined including normal power production, emergency shut down and parked situations, from cut-in to cut-out and extreme wind conditions. The ultimate and 1 Hz equivalent fatigue loads of the blade root and turbine base bottom are extracted and compared in order to give an insight of the load levels between the two concepts. According to the analysis the IEC 61400-1 ed.3 can be used to a large extent with proper interpretation of the DLCs and choice of parameters such as the hub-height. In addition, the design drivers for the VAWT appear to differ from the ones of the HAWT. Normal operation results in the highest tower bottom and blade root loads for the VAWT, where parked under storm situation (DLC 6.2) and extreme operating gust (DLC 2.3) are more severe for the HAWT. Turbine base bottom and blade root edgewise fatigue loads are much higher for the VAWT compared to the HAWT. The interpretation and simulation of DLC 6.2 for the VAWT lead to blade instabilities, while extreme wind shear and extreme wind direction change are not critical in terms of loading of the VAWT structure. Finally, the extreme operating gust wind condition simulations revealed that the emerging loads depend on the combination of the rotor orientation and the time stamp that the frontal passage of gust goes through the rotor plane.

Micromechanical validation of a mesomodel for plasticity in composites

In this paper, the performance of a recent homogenized orthotropic plasticity model for fiber reinforced composites (Vogler et al. (2013)) is investigated by comparing the model response against a micromechanical model. It is assumed that the micromechanical model which contains a recent plasticity model for polymers (Melro et al. (2013a,b)) offers a realistic representation of the plastic deformation of composite materials. Under that assumption, the performance of the homogenized model can be assessed based on the question how well it reproduces micromechanical results. Improved consistent tangent formulations for both plasticity models are presented and a study into the representative volume element size for elasto-plastic micromechanical analysis is performed. Micromechanical simulations of basic load cases are used to generate input for the homogenized model. Next, the response of the two models is compared for a range of different load cases. The loss of accuracy due to necessary simplifications in the homogenized model is quantified in the comparison. These simplifications in the mesomodel include ignoring the stress in fiber direction, disregarding the influence stress orientation under combined longitudinal shear and transverse loading, and the use of a single constant plastic Poisson ratio.

Computational geometrical and mechanical modeling of woven ceramic composites at the mesoscale

Woven composite materials are receiving particular attention in a wide range of specialized aeronautical applications. Reliable numerical prediction tools based on computational modeling are required to quantitatively characterize the role of the microstructure and damage mechanisms at the mesoscale. In this paper, such a computational strategy is illustrated on a generic SiC/SiC plain weave composite with chemical vapor infiltrated matrix. Matrix and tows damage mechanisms are respectively introduced through the use of an anisotropic damage model, and an homogenized model based on a micromechanical model on the fiber scale. The latter is presented in this paper for the first time. Particular attention is paid to the generation of accurate hexahedral meshes, compatible at the tow–tow and tow–matrix interfaces. The mesh quality is analyzed using an error estimator variable based on the strain energy density. Damage predictions obtained using tetrahedral and hexahedral meshes are compared for basic loading cases, illustrating the need for using high quality meshes in the growing community of woven composites computational modeling.

Behaviour and design of members with monosymmetric cross-section

Columns, beams and beam–columns with monosymmetric cross-sections are very frequently employed in steel structures. In this paper, the realistic, elasto-plastic buckling strength of such members is studied by means of non-linear numerical analyses and compared with the design rules found in Eurocode 3, the British Standard BS5950 and a new design proposal implemented in the Austrian national annex to Eurocode 3. The conservatism of the code regulations is highlighted, along with the codes' inability to capture the characteristic buckling behaviour of monosymmetric sections. For the basic load cases of pure axial compression, N, and pure bending moment, My, the source of the inaccuracy is shown to lie in the use of the buckling curves for flexural buckling as the basis of design. For beam–columns (load case N + My), it is shown that the sign of the bending moment (which can cause compression in either the smaller or the larger flange), in combination with an axial compression force, can cause remarkable differences when compared to the behaviour of double-symmetric sections.

Subsampling and homogenization to investigate variability of composite material mechanical properties

This paper presents an innovative homogenization sampling technique applied to multiscale modelling of composite materials. The goal is to build efficiently statistical variability of mechanical properties at mesoscopic scale from the heterogeneous media analysis at microscopic scale. It is applied to the transverse elastic properties of a unidirectional Long Fibres Reinforced Composite (LFRC). A large representative part of the ply – the cell – is modelled from a micrography and studied at microscopic scale with the Finite Element Analysis (FEA) under 2D plane strain hypothesis. The study consists in estimating the effective elastic properties of subcells, subparts of the previous cell, thanks to a specifically developed numerical procedure.A unique calculation is computed on the entire ply reduced to three basic loading cases is applied to the cell. Subsamples taken into the simulation cell are homogenized at post-processing level of strain and stress fields. A standard mechanics approach was considered. Various subsampling schemes are performed with various size and spatial distribution to generate variability functions of effective elastic properties at mesoscopic scale. A statistical inference is highlighted: the variability parameters vary with the way of sampling. Dispersion functions are finally obtained and discussed.