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Tubular joints are important connecting parts of a welded steel tube structure. The S-N curves based on the hot spot stress (HSS) method are often used to evaluate the fatigue life of tubular joints in practical engineering. The stress concentration factor (SCF) is a key parameter to calculate HSS. In this paper, stress concentration tests of hollow-section and concrete-filled double-skin tubular (CFDST) K-joints were carried out, respectively, and then finite element models of K-joints considering the weld were established. The developed models were validated with the experimental results. The influence of key geometrical parameters, such as the diameter ratio of brace to chord β, the diameter to thickness ratio of chord γ, the wall thickness ratio of brace to chord τ, brace angle θ, and hollow section ratio ζ on the distribution and key position of SCFs along the weld toe, was discussed. Parametric studies were conducted to obtain the calculating equations for the SCF values of CFDST K-joints. The results demonstrate that infill concrete can effectively reduce SCFs along the weld on the chord. When the hollow section ratio was reduced to 0.317, the SCF was reduced by 77.2%. Notably, the SCF reduction rate was sensitive to γ and θ, with a decrease observed as γ increased. The hollow section ratio ζ had a less pronounced effect on SCF distribution patterns, but as ζ decreased, the chord's stiffness improved, suggesting a potential approach to enhance joint performance. The distribution of SCFs is similar for joints of the same type but different geometric configurations. The innovatively integrated hollow section ratio in the CFDST design equation significantly simplifies and enhances the precision of SCF calculations for CFDST K-joints. [ABSTRACT FROM AUTHOR]

Tubular joints are important connecting parts of a welded steel tube structure. The S-N curves based on the hot spot stress (HSS) method are often used to evaluate the fatigue life of tubular joints in practical engineering. The stress concentration factor (SCF) is a key parameter to calculate HSS. In this paper, stress concentration tests of hollow-section and concrete-filled double-skin tubular (CFDST) K-joints were carried out, respectively, and then finite element models of K-joints considering the weld were established. The developed models were validated with the experimental results. The influence of key geometrical parameters, such as the diameter ratio of brace to chord β, the diameter to thickness ratio of chord γ, the wall thickness ratio of brace to chord τ, brace angle θ, and hollow section ratio ζ on the distribution and key position of SCFs along the weld toe, was discussed. Parametric studies were conducted to obtain the calculating equations for the SCF values of CFDST K-joints. The results demonstrate that infill concrete can effectively reduce SCFs along the weld on the chord. When the hollow section ratio was reduced to 0.317, the SCF was reduced by 77.2%. Notably, the SCF reduction rate was sensitive to γ and θ, with a decrease observed as γ increased. The hollow section ratio ζ had a less pronounced effect on SCF distribution patterns, but as ζ decreased, the chord’s stiffness improved, suggesting a potential approach to enhance joint performance. The distribution of SCFs is similar for joints of the same type but different geometric configurations. The innovatively integrated hollow section ratio in the CFDST design equation significantly simplifies and enhances the precision of SCF calculations for CFDST K-joints.

The determination of degree of bending (DoB) is critical to improve the accuracy of estimating fatigue life using stress life (S-N) method. Based on the truss structure of jack-up unit, the numerical model of tubular KK-joints including rack plate (thickness is 180mm) is established. In the present research, the data extracted from 378 FE analyses on 108 tubular KK-joints models were used to study the effect of geometric parameters on DoB values of KK-joints under six different types of axial loading, IPB and OPB loadings. The geometric parameters affecting the DoB of tubular KK-joints are analyzed by multiple nonlinear regression analysis, and 21 parameter design formulas are obtained to calculate the DoB values of tubular KK-joints under axial loadings, IPB and OPB loadings. The established geometric parameter equation can effectively evaluate the fatigue characteristics of truss legs with rack plate of jackup unit. [ABSTRACT FROM AUTHOR]

Helical surfaces are deployable and non-deployable. The first is the surface of a helical conoid widely used in technology, known as the screw. The second one is the surface of an unfolding helicoid or torso-helicoid. In both cases, the surface is formed from a blank - a sweep in the form of a flat ring. If in the first case the sweep can be calculated approximately, since the exact one does not exist due to the properties of non-expanded surfaces, then in the second case it is calculated exactly. However, this does not mean that it is just as possible to form a ready-made product from it - a twist of a helicoid torso. In contrast to non-expanded surfaces, during the manufacture of which the workpiece is subjected to complex stretching and compression deformations, an expanded surface can be obtained by bending with minimal plastic deformations, the magnitude of which depends on the thickness of the sheet. Bending occurs along rectilinear generators, which are theoretically located on the workpiece. In the process of bending, the location of these generators should not change. In the theory of differential geometry, such a process can be described analytically and is called continuous bending. With regard to the manufacture of the twist of the torso-helicoid, this means a gradual increase in the step to the desired value. The work contains parametric equations that describe this method of bending. At the same time, not only stretching of the workpiece along the axis of the coil occurs, but also its twisting around the axis. A comparison of these two movements was made and it was found that the relationship between them is not linear. The work gives a formula describing this dependence. According to it, when the workpiece is uniformly stretched along the axis, the angle of its twist around the axis increases according to a dependence close to quadratic. In the work, examples of the use of the torso-helicoid are given, the surface is visualized, and graphs are constructed. Confirmation of the reliability of the obtained results is the first found quadratic form of the torso-helicoid, which does not change when the surface is bent

The article describes how to derive the equation of curves that are obtained from string art. Conversely, if the equation of a curve is given, one can find the relation between the intercepts either on a rectangular axes, skewed axis, or a circle to trace out these curves. Some of these curves can be easily traced out using the string art if they are represented in their parametric form. Certain exercises are also provided for teachers and students to extend the concept discussed in the article. [ABSTRACT FROM AUTHOR]

Clipping has great importance in computer graphics. the clip window is generally of rectangular shape but it may be in any shape. for rectangular clip window we have many popular algorithms like Cohan-Sutherland algorithm. Cyrus Beck algorithm. Liang Barsky algorithms, but it is not necessary that clip window will be always in shape of rectangle it may be of any type of convex region. One of the most common geometrical entities is circle if clip window is in circular shape, then how clipping can be done? How we can identify the position of line segment with respect to circular clip window? Thi paper presents a simple algorithm using parametric equation of the line to determine the position of the line with respect to the circular clip window and discuss the method to clip the clipping candidate line. The proposed algorithm is very simple and less computationally expensive. [ABSTRACT FROM AUTHOR]

The prediction of thermal behavior is essential for an efficient initial design of thermal management systems which equip energy sources based on electrochemical cells. In this study, the surface temperature of various cylindrical types of Li-ion batteries is monitored at multiple points during discharge. Three different battery chemistries and two sizes (18650 and 21700) are considered in this study, allowing the comparison of the influence these parameters have on the temperature rise considering different discharge rates (1C, 2C and 3C). Based on repeated experimental measurements, a simple equation that describes the thermal behavior of batteries is proposed and further used to create 3D thermal maps for each analyzed battery (generally error is below 1 °C but never exceeds 3 °C). The practical utility of such an equation is that it can drastically reduce the time spent with experimental measurements required to characterize the thermal behavior of cylindrical Li-ion batteries, necessary for the initial design process of energy sources' thermal management system. [ABSTRACT FROM AUTHOR]

Helical surfaces are deployable and non-deployable. The first is the surface of a helical conoid widely used in technology, known as the screw. The second one is the surface of an unfolding helicoid or torso-helicoid. In both cases, the surface is formed from a blank - a sweep in the form of a flat ring. If in the first case the sweep can be calculated approximately, since the exact one does not exist due to the properties of non-expanded surfaces, then in the second case it is calculated exactly. However, this does not mean that it is just as possible to form a ready-made product from it - a twist of a helicoid torso. In contrast to non-expanded surfaces, during the manufacture of which the workpiece is subjected to complex stretching and compression deformations, an expanded surface can be obtained by bending with minimal plastic deformations, the magnitude of which depends on the thickness of the sheet. Bending occurs along rectilinear generators, which are theoretically located on the workpiece. In the process of bending, the location of these generators should not change. In the theory of differential geometry, such a process can be described analytically and is called continuous bending. With regard to the manufacture of the twist of the torso-helicoid, this means a gradual increase in the step to the desired value. The work contains parametric equations that describe this method of bending. At the same time, not only stretching of the workpiece along the axis of the coil occurs, but also its twisting around the axis. A comparison of these two movements was made and it was found that the relationship between them is not linear. The work gives a formula describing this dependence. According to it, when the workpiece is uniformly stretched along the axis, the angle of its twist around the axis increases according to a dependence close to quadratic. In the work, examples of the use of the torso-helicoid are given, the surface is visualized, and graphs are constructed. Confirmation of the reliability of the obtained results is the first found quadratic form of the torsohelicoid, which does not change when the surface is bent. [ABSTRACT FROM AUTHOR]

A total of 270 nonlinear steady-state finite element (FE) analyses were performed on 54 FE models of two-planar CHS KK-joints subjected to axial loading at five different temperatures (20°C, 200°C, 400°C, 550°C, and 700°C). The main objective was to study the influence of temperature and dimensionless geometrical parameters on the ultimate load, failure modes, and initial stiffness of the KK-joints. Results indicated that on an average basis, the ultimate load of a two-planar tubular KK-joint at 200°C, 400°C, 550°C, and 700°C is 90%, 75%, 45%, and 16% of the joint's ultimate load at ambient temperature, respectively. Outcomes of the parametric study showed that replacing the yield stress at ambient temperature with the corresponding value at elevated temperature, to apply the EN 1993-1-8 equations for the calculation of the joint's ultimate load at elevated temperatures, may lead to highly unconservative results that might endanger the safety of the structure. Results of the parametric study were then used to develop a set of design formulas, through nonlinear regression analyses, to calculate the ultimate load of two-planar tubular KK-joints subjected to axial loading at elevated temperatures. [ABSTRACT FROM AUTHOR]

In the present paper, the fire resistance of two-planar tubular KK-joints reinforced with collar plates is studied and formulated. After establishing a finite element (FE) model and its verification against the results of experimental tests, a total of 1440 numerical simulations were performed on 360 FE models of collar-plate-stiffened two-planar tubular KK-joints at four different temperatures (20, 400, 550, and 700°C). The studied joints were analysed under the axial loading and the reinforcing effect of the collar plate was investigated. Results showed that the resistance enhancement coefficient (Rs) increases by increasing the γ, θ, and τ; while it decreases by increasing the β. Results of the parametric investigation were utilised to establish new design equations to determine the bearing capacity of collar-plate-stiffened two-planar tubular KK-joints under the brace axial loads at fire condition. [ABSTRACT FROM AUTHOR]

The constantly growing demands of consumers, the creation of new materials, require designers to react quickly and offer the most modern items. When creating modern products, it is necessary to take into account aspects of different cultures in design. The solution to this problem is the use of curved shapes. This type of shapes have been used since ancient times to the present day, in the creation of items of clothing. Adapted, researched and used in the work is a software tool for complex, express, automated production of geometric elements with an oval shape, in a combination of current color trends, as well as for their application in modern textile design. The results of this study can be applied in the training of future specialists in the study of shapes and design of clothing.

The prediction of thermal behavior is essential for an efficient initial design of thermal management systems which equip energy sources based on electrochemical cells. In this study, the surface temperature of various cylindrical types of Li-ion batteries is monitored at multiple points during discharge. Three different battery chemistries and two sizes (18650 and 21700) are considered in this study, allowing the comparison of the influence these parameters have on the temperature rise considering different discharge rates (1C, 2C and 3C). Based on repeated experimental measurements, a simple equation that describes the thermal behavior of batteries is proposed and further used to create 3D thermal maps for each analyzed battery (generally error is below 1 °C but never exceeds 3 °C). The practical utility of such an equation is that it can drastically reduce the time spent with experimental measurements required to characterize the thermal behavior of cylindrical Li-ion batteries, necessary for the initial design process of energy sources’ thermal management system.

The effective notch stress approach for evaluating the fatigue strength of rib—deck welds requires notch stress concentration factors obtained from complex finite element analysis. To improve the efficiency of the approach, the notch stress concentration factors for three typical fatigue-cracking modes (i.e., root—toe, root—deck, and toe—deck cracking modes) were thoroughly investigated in this study. First, we developed a model for investigating the effective notch stress in rib—deck welds. Then, we performed a parametric analysis to investigate the effects of multiple geometric parameters of a rib—deck weld on the notch stress concentration factors. On this basis, the multiple linear stepwise regression analysis was performed to obtain the optimal regression functions for predicting the notch stress concentration factors. Finally, we employed the proposed formulas in a case study. The notch stress concentration factors estimated from the developed formulas show agree well with the finite element analysis results. The results of the case study demonstrate the feasibility and reliability of the proposed formulas. It also shows that the fatigue design curve of FAT225 seems to be conservative for evaluating the fatigue strength of rib—deck welds. [ABSTRACT FROM AUTHOR]

Energy-Dissipative Evolutionary Deep Operator Neural Network is an operator learning neural network. It is designed to seek numerical solutions for a class of partial differential equations instead of a single partial differential equation, such as partial differential equations with different parameters or different initial conditions. The network consists of two sub-networks, the Branch net and the Trunk net. For an objective operator G , the Branch net encodes different input functions f at a fixed number of sensors x i , i = 1 , 2 , ⋯ , m , and the Trunk net evaluates the output function at any location. By minimizing the error between the evaluated output q and the expected output G (f) (y) at the test point y , DeepONet generates a good approximation of the operator G. A key distinction in our methodology is the utilization of DeepONet for the training of the initial state, which operates as a multi-parametric operator. Further, the evolution of parameters in our model is facilitated by the Scalar Auxiliary Variable (SAV) method, leading to a formulated iterative process for the parameter's progression over time. The SAV approach is adopted to preserve essential physical properties of PDEs, particularly the Energy Dissipation Law. It introduces a kind of modified energy and establishes unconditional energy dissipation law in the discrete level. By treating the parameters of the well-trained DeepONet as a representation of the initial operator and evolving them by a dynamic system, our network can predict the accurate solution at any further time, while the training data is only the initial state. In order to validate the accuracy and efficiency of our neural networks, we provide numerical simulations of several partial differential equations, including heat equations, parametric heat equations, Allen-Cahn equations and a reaction–diffusion equation in three dimensions. • A novel evolutionary operator neural network is designed to solve PDEs. • EDE-DeepONet is capable of solving PDEs with initial and boundary conditions. • EDE-DeepONet solves the parametric PDEs. • The solution predicted by EDE-DeepONet keeps the unconditionally energy dissipation law. • Adaptive time step is employed in EDE-DeepONet to speed up the training. [ABSTRACT FROM AUTHOR]

The constantly growing demands of consumers, the creation of new materials, require designers to react quickly and offer the most modern items. When creating modern products, it is necessary to take into account aspects of different cultures in design. The solution to this problem is the use of curved shapes. This type of shapes have been used since ancient times to the present day, in the creation of items of clothing. Adapted, researched and used in the work is a software tool for complex, express, automated production of geometric elements with an oval shape, in a combination of current color trends, as well as for their application in modern textile design. The results of this study can be applied in the training of future specialists in the study of shapes and design of clothing. [ABSTRACT FROM AUTHOR]

Aiming at the problem that Gauss projection in the line engineering project needs to be frequent⁃ ly strapped and has low precision when designing the plane construction drawing of the project, this paper proposes a large elliptical Gaussian projection of an elliptical line projecting from the ellipsoid elliptical line of the route for the new central meridian, and analyzes the parametric theoretical model of the elliptic ellip⁃ soid. Firstly, it derives the meridian arc length formula by using the naturalized latitude instead of the geo⁃ detic latitude as a parameter. Secondly, according to the quadratic curve invariant theory, linear algebra, calculus and other knowledge, it deduces the model of solving the basic geometric parameters of the large ellipse ellipsoid with the plane equation coefficient as the parameter. Furthermore, a direct conversion model of the geodetic coordinates between the base ellipsoid and the large ellipse ellipsoid is deduced. Finally, based on some actual engineering data, the correctness and superiority of the theoretical model deduced in this paper are verified to be popularized in long⁃term projects. [ABSTRACT FROM AUTHOR]

Abstract The fatigue life of tubular joints commonly found in offshore structures is not only dependent on the value of hot-spot stress (HSS), but is also significantly influenced by the through-the-thickness stress distribution characterized by the degree of bending (DoB). In the present research, data extracted from the finite element (FE) analyses carried out on 81 two-planar tubular DT-joint models was used to study the effects of geometrical parameters on the DoB values in DT-joints subjected to six different types of loading including two types of axial loading, two types of in-plane bending (IPB) moment loading, and two types of out-of-plane bending (OPB) moment loading. Generated FE models were validated using available parametric equations. The determination of DoB values in a tubular joint is essential for improving the accuracy of fatigue life estimation using the stress-life (S N) method and particularly for predicting the fatigue crack growth based on the fracture mechanics (FM) approach. Geometrically parametric investigation was followed by a set of nonlinear regression analyses to develop three DoB parametric formulas for the fatigue analysis and design of two-planar DT-joints. Highlights • Geometrical effects on the DoB values in two-planar tubular DT-joints were studied. • Altogether, 81 FE models were generated and analyzed to extract the DoB values. • Six types of axial, IPB moment, and OPB moment loadings were considered for the parametric investigation. • A set of fatigue design equations was developed for the calculation of the DoB in two-planar DT-joints. [ABSTRACT FROM AUTHOR]

Abstract Due to being increasingly used in infrastructure, studying stress concentration in concrete-filled steel tubular joints is necessary. In this study, the variation of the maximum Stress Concentration Factor (SCF max) with non-dimensional geometric parameters in steel circular hollow section brace welded to concrete-filled circular hollow section chord (CHS-to-CFCHS) T-joints under in-plane and out-of-plane bending moment has been investigated. A database of the maximum SCFs in CHS-to-CFCHS T-joints under in-plane and out-of-plane bending moment is developed based on three-dimensional (3D) finite element (FE) models. The 3D FE models developed using ABAQUS software, have been verified using experimental results. Graphs showing variation of the maximum SCF, in CHS-to-CFCHS T-joints, with non-dimensional geometric parameters have been produced and compared with those for non-filled empty T-joints. Parametric equations for predicting the maximum SCFs in CHS-to-CFCHS T-joints under in-plane and out-of-plane bending moment have been developed in a multiple nonlinear regression analysis. There is a good agreement between the maximum SCFs predicted by the parametric equations and those determined from the experiments. Highlights • The effect of the concrete infill in reducing the SCF max is moderate under in-plane bending and its effect is significant under out-of-plane bending. • Parameter β has very little effect on the SCF max under in-plane bending and has significant effect under out-of-plane bending. • The trend in variation of the SCF max under in-plane and out-of-plane bending with parameter τ in CHS-to-CFCHS T-joints is similar to that in empty T-joints yet its effect is less under out-of-plane bending in the CHS-to-CFCHS T-joints due to significant effect of the concrete infill. • The trend in variation of the SCF max under in-plane and out-of-plane bending with parameter γ in CHS-to-CFCHS T-joints is similar to that in empty T-joints yet its effect is less under out-of-plane bending in the CHS-to-CFCHS T-joints due to significant effect of the concrete infill. • Parameter α has no effect on the SCF max under out-of-plane bending. [ABSTRACT FROM AUTHOR]

The human tricuspid valve, one of the key cardiac structures, plays an important role in the circulatory system. However, there are few mathematical models to accurately simulate it.In this paper, firstly, we consider the tricuspid valve as an elastic shell with a specific shape and establish its novel geometric model. Concretely, the anterior, the posterior and the septal leaflets of the valve are supposed to be portions of the union of two interfacing semi-elliptic cylindrical shells when they are fully open.Next, we use Koiter's linear shell model to describe the tricuspid valve leaflets in the static case, and provide a numerical scheme for this elastostatics model. Specifically, we discretize the space variable, i.e., the two tangent components of the displacement are discretized by using conforming finite elements (linear triangles) and the normal component of the displacement is discretized by using conforming Hsieh-Clough-Tocher triangles (HCT triangles).Finally, we make numerical experiments for the tricuspid valve and analyze the outcome. The numerical results show that the proposed mathematical model describes well the human tricuspid valve subjected to applied forces. [ABSTRACT FROM AUTHOR]