Out-of-Plane Deformation Analysis of the Thin-Walled Closed Curved Box Girder under the Temperature Gradient

For calculating the thin-walled closed curved box girder caused by the temperature gradient of the internal force and displacement, based on the fundamental differential equation of the curve beam and the principle of minimum energy, set a reverse statically indeterminate simply supported curve beam as the basic structure, consider the warping effect of the closed curve box girder, and put forward a kind of plane curve beam temperature deformation simple analytical calculation method. Compared with the finite element calculation results, the relative error of the analytical calculation results is less than 5%. It is concluded that the analytical method has sufficient accuracy in calculating the out-of-plane deformation of the thin-walled closed curved box girder under the temperature gradient.

Two- and three-dimensional elastic networks with rigid junctions: modeling within the theory of micropolar shells and solids

For two- and three-dimensional elastic structures made of families of flexible elastic fibers undergoing finite deformations, we propose homogenized models within the micropolar elasticity. Here we restrict ourselves to networks with rigid connections between fibers. In other words, we assume that the fibers keep their orthogonality during deformation. Starting from a fiber as the basic structured element modeled by the Cosserat curve beam model, we get 2D and 3D semi-discrete models. These models consist of systems of ordinary differential equations describing the statics of a collection of fibers with certain geometrical constraints. Using a specific homogenization technique, we introduce two- and three-dimensional equivalent continuum models which correspond to the six-parameter shell model and the micropolar continuum, respectively. We call two models equivalent if their approximations coincide with each other up to certain accuracy. The two- and three-dimensional constitutive equations of the networks are derived and discussed within the micropolar continua theory.

Simultaneous Generation of Complex Structured Curve Beam

At present, people are using holographic technologies to shape complex optical beams for both fundamental research and practical applications. However, most of the reported works are focusing on the generation of a single beam pattern based on the computer-generated hologram (CGH). In this paper, we present a method for simultaneously shaping the multiple beam lattice where the intensity and phase of each individual beam can be prescribed along an arbitrary geometric curve. The CGH that is responsible for each individual beam is calculated by using the holographic beam shaping technique, afterwards all the CGHs are multiplexed and encoded into one phase-only hologram by adding respective linear phase grating such that different curves are appeared in different positions of the focal regions. We experimentally prove that the simultaneous generation of multiple beams can be readily achieved. The generated beams are especially useful for applications such as multitasking micro-machining and optical trapping.

COMPARATIVE ANALYSIS OF THE GEOMETRICAL CONTENT IN THE NEW BULGARIAN TEXTBOOKS OF MATHEMATICS FOR THIRD GRADE

Starting with 2018/2019 school year in Bulgaria, the math education in the third grade is implemented through new training kits. They were developed on the basis of the new third-grade mathematics curriculum, approved by Order No. РД 09-1093 / 25.01.2017 of the Minister of Education and Science, Annex No. 8, supplemented by Order No. РД 09-2555 / 15.06.2018 of the Minister of Education and Science. Training kits are approved by the Ministry of Education and Science and are 7 in total. Geometric learning content in new math textbooks is the second most important element after arithmetic content. It is combined with the arithmetic learning content, and by this the foundation of the successful study of geometry in the next school grades is laid. The new geometry knowledge that is included in the third grade curriculum is the following: straight line, curve, beam, angle, right angle, obtuse angle, acute angle, right triangle, acute triangle, obtuse triangle; naming geometric figures with Latin alphabet letters [11]78. It is important in a modern mathematics textbook to have a rich and varied geometric content. It is important that the new types of geometry tasks are introduced with rich visualization using a specific-inductive approach. The relative number of tasks of a given type is an important prerequisite for the successful formation and improvement of skills for solving geometric problems in pupils at the age of 9-10. This article will present a comparative analysis of the geometric content in the seven approved Bulgarian third-grade mathematics textbooks, which are used in the mass practice of this school year. For the purpose of the study, a classification of all types of tasks and exercises with geometric content has been developed. Then the tasks in the seven textbooks are systematized by the so chosen classification. The data are statistically processed taking into account the relative share of tasks of each type within a textbook, as well as a comparison between the relative shares of the geometric tasks in the different textbooks. The established differences in the number and relative share of different types of geometric tasks make it possible for the analyzed textbooks to be ranked. Such a study has not been published so far. It has a relation to the assessment of the quality of the textbooks offered. The conclusions formulated in this article can help primary teachers in their choice of textbooks to teach to their third grade students.

Determination of Material Property for Non-Pneumatic Tire Spokes by Inverse Method

The radial spokes of non-pneumatic tire have been developed to absorb impacts. In order to obtain its property for the further developments, it had to cut into the curve beam specimens. The 3-point bending was selected to test referring to ASTM D790. Subsequently, the finite element method was employed to simulate the 3-point bending test of specimens. The inverse method was used to determine the modulus of elasticity for specimen material. The gradient based on optimization scheme was used to optimize the modulus of elasticity by the input and output condition which was the vertical deflection and force, respectively. The optimized process was terminated at the desirable force tolerance of 0.00071 N. The elastic modulus of spoke was implemented in the finite element model of the 3-point bending test. There was found that the simulation result of vertical displacement obtained an average error of 4.87% by comparing with physical experiment.

Differential Equations of Motion for Naturally Curved and Twisted Composite Space Beams

The differential equations of motion for naturally curved and twisted elastic space beams made of anisotropic materials with noncircular cross sections, being a coupled system consisting of 14 second-order partial differential equations with variable coefficients, are derived theoretically. The warping deformation of beam’s cross section, as a new design factor, is incorporated into the differential equations in addition to the anisotropy of material, the curvatures of the rod axis, the initial twist of the cross section, the rotary inertia, and the shear and axial deformations. Numerical examples show that the effect of warping deformation on the natural frequencies of the beam is significant under certain geometric and boundary conditions. This study focuses on improving and consummating the traditional theories to build a general curve beam theory, thereby providing new scientific research reference and design principle for curve beam designers.

The Simplified Model and the Critical Angle of the Curved Bridges with Multi-Dimensional Earthquake Excitation

The existing methods of curve beam bridge is the finite element method, the modeling and analysis of complex, heavy workload, not suitable for the proposed design method of regularity. According to the character of curved bridge, the bridge superstructure and piers is simplified as packing quality model that have two levels of degrees of freedom and a surrounding mass center shaft torsion degree of freedom, then the analysis model of double mass simplified 6 degrees of freedom is established, studying the dynamic characteristics and the structure vibration behavior through the dynamics equation of curved bridges. Example show that,the results of double mass model with the traditional of finite element model is consistent, double mass model has simple calculation method and can consider the influence of sensitive factors, with the help of programming is more suitable for the proposed design method of regularity. At last, Application MATLAB programming discussed the critical angle of the curved bridges under multi-dimensional earthquake based on double mass model, after change the curve bridge related parameters only one operation can achieve the most disadvantaged angle.

The Simulation of the Interaction between SCR and Seabed

The simulation of the interaction between steel caternary risers (SCRs) and seabed was studied based on beam on elastic foundation theory and the node-spring system foundation mehod for comparison, with the SCR modeled by extensible flexibility cable curve beam. Not only the bottom support and the friction forces were taken into account, but also the damping force of seabed was included in the model of the interaction between SCR and seabed. The results show that compared with using the node-spring system foundation, the effect of the element length of the model on analytical result is insignificant and a longer element could be used for the model, when the beam on elastic foundation model is used. So, the analysis is not time-consuming and the memory requirement is not so large by using the beam on elastic foundation model.

Elastic Modulus Reduction Method Based Limit Analysis of Steel Lined Reinforced Concrete Penstocks Containing Flaws

In the context of ultimate bearing capacity of steel lined reinforced concrete penstock (SLRCP) containing flaws, the use of the elastic modulus reduction method (EMRM) in conjunction with curve beam based internal force calculation is presented. The corrosion models for steel bar and lining are introduced from the literature. The SLRCP is considered bearing load together with lining and reinforced concrete, and its internal force is determined based on a simplified curve beam model. The element bearing ratio of SLRCP is given through the internal force and sectional strength, and is employed by the EMRM. The change of ultimate bearing capacity of some typical SLRCP within their operating life is then evaluated.