scholarly journals ОПТИМАЛЬНОЕ ПРОЕКТИРОВАНИЕ СТАТИЧЕСКИ ОПРЕДЕЛИМОЙ БАЛКИ ПРИ ОГРАНИЧЕНИИ НА МАКСИМАЛЬНЫЙ ПРОГИБ

2020 ◽  
pp. 28-34
Author(s):  
Сергей Сергеевич Куреннов
Keyword(s):  
Programming Problem ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Constant Width ◽  
Variable Cross Section ◽  
Solution Procedure ◽  
Linear Functions ◽  
Problem Solution ◽  
Variable Cross ◽  
Nodal Points

Here is solved the optimization problem for the longitudinal depth distribution in the beam with a limitation on the maximum value of deflection. A review of the references is done, and it is shown that the known solutions are either erroneous, because they are based on false hypotheses, or have a narrow field of application, limited only to symmetrical constructions for which the point of the maximum deflection is known a priori. The paper considers a beam of the rectangular cross-section of constant width. The beam is assumed to be statically determinate, and the load is arbitrary and asymmetric and multidirectional as well. The points (or point) of the beam maximum deflections are unknown in advance and would be determined in the problem-solution procedure. A linear problem is considered. The optimization criterion is the mass of the beam. To find the deflections of the beam, i.e. to solve the differential equation of a variable cross-section beam bending the finite difference method is used. The design problem is reduced to the required beam depths obtaining in the system of nodal points. In this case, the desired solution must satisfy the restriction system for the nodal points shift and the sign of variables as well. Since the restrictions of the shift of each node are considered separately and independently, so the proposed method allows flexible control of the beam shift restrictions. Using the change of variables proposed in the paper, the problem to be solved is reduced to a nonlinear programming problem where the criterion function is separable and restrictions are linear functions. Using linearization, this problem can be reduced to the linear programming problem relatively to new variables. The model problem is solved, and it is shown that the proposed algorithm efficiently allows us to solve the problems of the beam optimal design with the restrictions of the maximally allowed deflection. The proposed approach can be spread for the strength limitations, for beams of variable width, I-beam cross-section, etc.

scholarly journals First natural frequency of multi-segment floor joists with variable cross section

2019 ◽  
Vol 28 (4) ◽  
pp. 526-538
Author(s):  
Marek Chalecki ◽  
Jacek Jaworski ◽  
Olga Szlachetka
Keyword(s):  
Natural Frequency ◽  
Cross Section ◽  
Constant Width ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Uniform Load ◽  
Simply Supported ◽  
Made In

The Rayleigh’s method can be used to determine the first natural frequency of beams with variable cross-section. The authors analyse multi-segment simply supported beams, symmetrical with respect to their midpoint, having a constant width and variable height. The beams consist generally of five segments. It has been assumed that the neutral bar axis deflected during vibrations has a shape of a beam deflected by a static uniform load. The calculations were made in Mathematica environment and their results are very close to those obtained with FEM.

Effect of Bimodularity on the Stress State of a Variable Cross-Section Reinforced Beam

2019 ◽  
Vol 974 ◽  
pp. 646-652
Author(s):  
Aleksey N. Beskopylny ◽  
Elena Kadomtseva ◽  
Vadim Poltavskii ◽  
Mikhail Lukianenko
Keyword(s):  
Stress State ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Neutral Line ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Reinforcing Bars ◽  
Tensile Stresses ◽  
Simply Supported ◽  
Compressive Stresses

The article is dedicated to the effect of different modulus of the material on the stress state of a beam of the variable rectangular cross section. The height of the beam varies linearly along its length. Formulas for calculating the maximum compressive and tensile stresses and determining the neutral line are obtained. The maximum tensile and compressive stresses are determined for the clamped and simply supported beams. The dependence of the maximum normal stress on the number of reinforcing bars located in the stretched zones is numerically investigated. The stress state of the beam is compared with and without consideration of the bimodularity of the material for simply supported and cantilever beams. It is shown that taking into account the bimodularity of the material significantly affects the maximum tensile and compressive stresses. The magnitude of the tensile stresses is increased by 30%; the magnitude of the compressive stresses is reduced by 21%. As a bimodular material, fibro foam concrete is considered in work.

Cable stayed bridge over the Crati river in Cosenza

2021 ◽  
Author(s):  
Pierangelo Pistoletti ◽  
Marcello Vaccarezza ◽  
Simone Varni ◽  
Paolo Roggero ◽  
Marco Sciarra
Keyword(s):  
Reinforced Concrete ◽  
Cross Section ◽  
Concrete Structure ◽  
Constant Width ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Middle Plane ◽  
Reinforced Concrete Structure ◽  
Cable Stayed Bridge ◽  
The City

<p>The “Bridge over the Crati River” is a cable stayed bridge in a single span of 114.725m, designed by the architect Santiago Calatrava for the city of Cosenza. The main elements are the main tower, the deck, the cables, the reinforced concrete pier and the abutments; one of those is the restraint of the main rear cables. The steel tower has a variable cross section that tapers towards the top where the cables anchoring devices are located. It is inclined of 51° from the vertical and it is anchored by means of 2 anchor stays. The bridge is composed of two parts: a reinforced concrete structure with a span of 30 m that overhangs the railway and the composite cable-stayed bridge with a span of 114.725 m. The cross section of the bridge has a constant width of 24 m all over its length. The deck is sub-divided in four vehicle lanes, two in each direction, divided by a central pedestrian lane slightly raised in comparison with the vehicle lanes. There are 40 stays, placed symmetrically in respect to the middle plane of the structure and arranged according to a mixed geometry between an harp and a fan shape to facilitate the positioning of the anchoring devices at the top of the pylon.</p>

The Nonlinear Static Formulation of Variable Cross-Section Beam Element Based on Positional Description

2012 ◽  
Vol 557-559 ◽  
pp. 2367-2370
Author(s):  
Lv Zhou Ma ◽  
Jian Liu ◽  
Xun Lin Diao ◽  
Yu Qin Yan
Keyword(s):  
Cross Section ◽  
Large Deflection ◽  
Variable Cross Section ◽  
Beam Element ◽  
Solution Procedure ◽  
Variable Cross ◽  
Hyper Elastic ◽  
Nonlinear Static ◽  
Newton Raphson ◽  
Raphson Iteration

Based on positional FEM (finite element method), the nonlinear static formulation to treat large deflection of variable cross-section beam element is created by using the lowest potential energy theory. Adopting linear constitutive relation for hyper-elastic materials, the formulation and the solution procedure by Newton-Raphson iteration method are very simple.

Boundary element method in numerical study of three-dimensional Stokes flows in channels of arbitrary shape

2012 ◽  
Vol 9 (1) ◽  
pp. 94-97
Author(s):  
Yu.A. Itkulova
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Cross Section ◽  
Numerical Study ◽  
Three Dimensional ◽  
Cylindrical Tube ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Periodic Flows ◽  
Element Method

In the present work creeping three-dimensional flows of a viscous liquid in a cylindrical tube and a channel of variable cross-section are studied. A qualitative triangulation of the surface of a cylindrical tube, a smoothed and experimental channel of a variable cross section is constructed. The problem is solved numerically using boundary element method in several modifications for a periodic and non-periodic flows. The obtained numerical results are compared with the analytical solution for the Poiseuille flow.

Longitudinal oscillation of a rod with a variable cross section

2019 ◽  
Vol 14 (2) ◽  
pp. 138-141
Author(s):  
I.M. Utyashev
Keyword(s):  
Cross Section ◽  
Natural Frequencies ◽  
Liouville Problem ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Longitudinal Vibrations ◽  
Cross Sectional ◽  
Independent Functions ◽  
The Cross ◽  
The Right

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional area as an exponential function of a polynomial of degree n. Based on this idea, it was possible to formulate the Sturm-Liouville problem with boundary conditions of the third kind. The linearly independent functions of the general solution have the form of a power series in the variables x and λ, as a result of which the order of the characteristic equation depends on the choice of the number of terms in the series. The presented approach differs from the works of other authors both in the formulation and in the solution method. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given. From the analysis of the numerical results it follows that in a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod. It is shown that with an increase in the rigidity of fixation at the right end, the natural frequencies increase for all cross section profiles. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.

scholarly journals Theoretical and Experimental Studies on MEMS Variable Cross-Section Cantilever Beam Based Piezoelectric Vibration Energy Harvester

Micromachines ◽  
2021 ◽  
Vol 12 (7) ◽  
pp. 772
Author(s):  
Xianming He ◽  
Dongxiao Li ◽  
Hong Zhou ◽  
Xindan Hui ◽  
Xiaojing Mu
Keyword(s):  
Power Density ◽  
Cantilever Beam ◽  
Cross Section ◽  
Energy Harvester ◽  
Variable Cross Section ◽  
Vibration Energy ◽  
Variable Cross ◽  
Vibration Energy Harvester ◽  
Piezoelectric Vibration Energy Harvester ◽  
Piezoelectric Vibration

The piezoelectric vibration energy harvester (PVEH) based on the variable cross-section cantilever beam (VCSCB) structure has the advantages of uniform axial strain distribution and high output power density, so it has become a research hotspot of the PVEH. However, its electromechanical model needs to be further studied. In this paper, the bidirectional coupled distributed parameter electromechanical model of the MEMS VCSCB based PVEH is constructed, analytically solved, and verified, which laid an important theoretical foundation for structural design and optimization, performance improvement, and output prediction of the PVEH. Based on the constructed model, the output performances of five kinds of VCSCB based PVEHs with different cross-sectional shapes were compared and analyzed. The results show that the PVEH with the concave quadratic beam shape has the best output due to the uniform surface stress distribution. Additionally, the influence of the main structural parameters of the MEMS trapezoidal cantilever beam (TCB) based PVEH on the output performance of the device is theoretically analyzed. Finally, a prototype of the Aluminum Nitride (AlN) TCB based PVEH is designed and developed. The peak open-circuit voltage and normalized power density of the device can reach 5.64 V and 742 μW/cm3/g2, which is in good agreement with the theoretical model value. The prototype has wide application prospects in the power supply of the wireless sensor network node such as the structural health monitoring system and the Internet of Things.

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