A unified procedure for the vibration analysis of elastically restrained Timoshenko beams with variable cross sections

2020 ◽  
Vol 68 (1) ◽  
pp. 38-47
Author(s):  
Gang Wang ◽  
Wen L. Li ◽  
Wanyou Li ◽  
Zhihua Feng ◽  
Junfang Ni
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Vibration Analysis ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Timoshenko Beams ◽  
Unified Approach ◽  
Cross Section Area ◽  
Section Area ◽  
Elastically Restrained

A generalized analytical method is developed for the vibration analysis of Timoshenko beams with elastically restrained ends. For a beam with any variable cross section along the lengthwise direction, the finite element method is the only unified approach to handle those kinds of problems, since the analytical solutions could not be obtained by the governing equations when the cross section area and the second moment of area changing variably lengthwise. In this article, a unified approach is proposed to study the Timoshenko beam with any variable cross sections. The cross section area and second moment of area of the beam are both expanded into cosine series, which are mathematically capable of representing any variable cross section. The translational displacement and rotation of cross section are expressed in the Fourier series by adding some admissible functions which are used to handle the elastic boundary conditions with more accuracy and high convergence rate. By using Hamilton's principle, the eigenvalues and the coefficients of the Fourier series are both obtained. Some examples are presented to illustrate the excellent accuracy of this method. Analytical solutions of the vibration of the beam are achieved for different combinations of boundary conditions including classical and elastically restrained ones. The derived results can be used as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of Timoshenko beams with any variable cross section.

Simulation and Analysis on Operating Characteristics of Central Exhaust System in High-Rise Residential

2012 ◽  
Vol 614-615 ◽  
pp. 656-664 ◽  
Author(s):  
Gang Li ◽  
Yue Ren Wang ◽  
Guo Hui Feng ◽  
Ming Zhi Jiang ◽  
Qian Liu
Keyword(s):  
Cross Section ◽  
Exhaust System ◽  
Variable Cross Section ◽  
Maximum Pressure ◽  
Variable Cross ◽  
Main Duct ◽  
Operating Characteristics ◽  
Cross Section Area ◽  
Section Area ◽  
Operation Rate

Operating characteristics of variable cross section area of main duct in the main-branch central exhaust system was simulated based on various stories, operation rate and cross section area of main duct by FLUENT. The results show that there are certain obvious induced flow in the lower part of the system when the users in the upper floor work in the central exhaust system,and the internal pressure of main duct appears low- high-low trend in the system from the bottom to the top floor. The maximum pressure position is not in the bottom, but in the middle and lower floor of the system, where is the most disadvantageous floors of the exhaust system. It indicates that 400 mm x 500 mm is the necessary cross section area of main exhaust duct for 10 layers buildings,and 400 mm x 750 mm is the necessary one for 20 layers buildings.

scholarly journals Exact Longitudinal Vibration Characteristics of Rods with Variable Cross-Sections

2011 ◽  
Vol 18 (4) ◽  
pp. 555-562 ◽  
Author(s):  
Bulent Yardimoglu ◽  
Levent Aydin
Keyword(s):  
Differential Equation ◽  
Cross Section ◽  
Cross Sections ◽  
Longitudinal Vibration ◽  
Transformation Method ◽  
Variable Cross ◽  
Cross Sectional ◽  
Cross Section Area ◽  
Section Area ◽  
Sinusoidal Functions

Longitudinal natural vibration frequencies of rods (or bars) with variable cross-sections are obtained from the exact solutions of differential equation of motion based on transformation method. For the rods having cross-section variations as power of the sinusoidal functions ofax+b, the differential equation is reduced to associated Legendre equation by using the appropriate transformations. Frequency equations of rods with certain cross-section area variations are found from the general solution of this equation for different boundary conditions. The present solutions are benchmarked by the solutions available in the literature for the special case of present cross-sectional variations. Moreover, the effects of cross-sectional area variations of rods on natural characteristics are studied with numerical examples.

Heat Transfer Characteristic Inside Two-Pass Variable Cross-Section Channels With Ribs and Bleed Holes

2004 ◽  
Author(s):  
Xiangyun Liu ◽  
Shuiting Ding ◽  
Zhi Tao ◽  
Guoqiang Xu
Keyword(s):  
Heat Transfer ◽  
Cross Section ◽  
Transfer Characteristic ◽  
Reynolds Numbers ◽  
Variable Cross Section ◽  
Heat Transfer Process ◽  
Variable Cross ◽  
Relative Roughness ◽  
Cross Section Area ◽  
Section Area

Heat transfer characteristics were experimentally studied for two-pass channels with variable cross-section with and without ribs. Rectangular ribs were placed symmetrically on two opposite walls with attack angles of 90° and 60° to the main flow. The bleed holes were only distributed on the rib roughened surface along the second pass. The cross-section area of the channel varied in the way that it increased along the first pass and decreased along the second. The relative roughness heights were 0.092 and 0.123; the relative roughness pitches were 10 and 20. The experiments were carried out at Reynolds numbers from 7100 to 60000. Analysis of the experimental results may lead to the following conclusions: (1) The heat transfer was enhanced by the presence of variable cross section. (2) The 60° ribs was not necessarily the optimistic option as published in the literature, test showed that 90° ribs produced a hither heat transfer for the relative roughness height of 0.092. (3) For up to 30% of bleeding were outlet, the existence of bleed holes enhanced the heat transfer process.

scholarly journals The Impact of Damage in Inconel 718 on Hardness Measured by the Vickers Method

2015 ◽  
Vol 2015 (7) ◽  
pp. 69-79
Author(s):  
Maciej Malicki ◽  
Bartosz Madejski
Keyword(s):  
Cross Section ◽  
Inconel 718 ◽  
Variable Cross Section ◽  
Damage Parameter ◽  
Variable Cross ◽  
Cross Section Area ◽  
Hardness Tests ◽  
Section Area ◽  
Individual Specimen ◽  
The Impact

Abstract To prevent failure of machine components it is necessary to measure material damage generated in a component throughout its entire lifetime. Damage can be quantified by means of damage parameters. This paper considers the usefulness of hardness measurements to evaluate damage parameter in Inconel 718. Vickers hardness tests were performed on a specimen with a variable cross section area after tensile testing. The specimen’s geometry enabled the evaluation of damage parameter in respect of hardness measurements made on one individual specimen.

scholarly journals Development of a Micro Swimming Robot Using Optimised Giant Magnetostrictive Thin Films

2006 ◽  
Vol 3 (3) ◽  
pp. 161-170 ◽  
Author(s):  
Y. Zhang ◽  
G. Liu ◽  
H. Li
Keyword(s):  
Thin Films ◽  
Cross Section ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Propulsive Force ◽  
Fish Robot ◽  
Caudal Fin ◽  
Cross Section Area ◽  
Section Area ◽  
Micro Robot

A fish-like swimming micro robot is developed using an optimised fin actuator made of giant magnetostrictive films (GMFs). The force oscillation dynamic model of a GMF cantilever with variable cross-section area is derived, and the propulsive model of the fish robot in liquid is established. A discrete variate method for optimising caudal fin configuration is proposed to optimise its propulsive force and drive efficiency under the constraints of fixed surface area and sufficient fin end strength. Both theoretical analysis and experimental results have confirmed that the optimised caudal fin configuration can generate more powerful propulsion and improved efficiency.

scholarly journals INFLUENCE OF RUNNER CROSS-SECTION ON AIR ENTRAPMENT IN PRESSURE DIE CASTS VOLUME

2021 ◽  
Vol 2021 (2) ◽  
pp. 4491-4495
Author(s):  
STEFAN GASPAR ◽  
◽  
JAN MAJERNIK ◽  
TOMAS CORANIC ◽  
JAN PASKO ◽  
...  
Keyword(s):  
Mechanical Properties ◽  
Cross Section ◽  
Variable Cross Section ◽  
System Structure ◽  
Variable Cross ◽  
Gating System ◽  
Air Entrapment ◽  
Porosity Reduction ◽  
Cross Section Area ◽  
Section Area

Technology of metal die casting is characterized by production of casts complicated as to shape yet with positive mechanical properties and with high repeatability of production. However, casts are porous to a certain extent which eventually reduces their mechanical properties. One of the significant methods of porosity reduction of casts rests in correct design of a gating system. The submitted paper studies the influence of cross-section area of a runner on air entrapped in the cast volume. Seven alternatives of runners with the identical structural organization and variable cross-section area were compared. In case of a gating system design there was an assumption made that the runner with the largest cross section would deliver the lowest possible velocity to the melt before reaching the runner which would result in the lowest possible values of air entrapment. The air entrapment in the cast volume is evaluated behind the cores which were evaluated as critical points with regards to further processing. The results reached during examination of the melt flowing through runners proved the aforementioned assumption, yet the values of air entrapment in die casts volume did not show remarkable differences. In its final part, the paper clarifies the reached results and recommendations which should be taken into consideration when designing the gating system structure.

scholarly journals Failure Analysis of Intersections of Large-Scale Variable Cross-Section Roadways in Deep Soft Rock and Study of Integrated Control Technology of Bolting and Grouting

2021 ◽  
Author(s):  
Shengrong Xie ◽  
Yiyi Wu ◽  
Dongdong Chen ◽  
Ruipeng Liu ◽  
Xintao Han ◽  
...  
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Large Scale ◽  
Integrated Control ◽  
Soft Rock ◽  
Variable Cross Section ◽  
Surrounding Rock ◽  
Variable Cross ◽  
Geological Environment ◽  
Cable Bolt

Abstract In deep underground mining, achieving stable support for roadways along with long service life is critical and the complex geological environment at such depths frequently presents a major challenge. Owing to the coupling action of multiple factors such as deep high stress, adjacent faults, cross-layer design, weak lithology, broken surrounding rock, variable cross-sections, wide sections up to 9.9 m, and clusters of nearby chambers, there was severe deformation and breakdown in the No. 10 intersection of the roadway of large-scale variable cross-section at the − 760 m level in the Nanfeng working area of the Wuyang Coal Mine. As there are insufficient examples in engineering methods pertaining to the geological environment described above, the numerical calculation model was oversimplified and support theory underdeveloped; therefore, it is imperative to develop an effective support system for the stability and sustenance of deep roadways. In this study, a quantitative analysis of the geological environment of the roadway through field observations, borehole peeking, and ground stress testing is carried out to establish the FLAC 3D variable cross-section crossing roadway model. This model is combined with the strain softening constitutive (surrounding rock) and Mohr-Coulomb constitutive (other deep rock formations) models to construct a compression arch mechanical model for deep soft rock, based on the quadratic parabolic Mohr criterion. An integrated control technology of bolting and grouting that is mainly composed of a high-strength hollow grouting cable bolt equipped with modified cement grouting materials and a high-elongation cable bolt is developed by analyzing the strengthening properties of the surrounding rock before and after bolting, based on the Heok-Brown criterion. As a result of on-site practice, the following conclusions are drawn: (1) The plastic zone of the roof of the cross roadway is approximately 6 m deep in this environment, the tectonic stress is nearly 30 MPa, and the surrounding rock is severely fractured. (2) The deformation of the roadway progressively increases from small to large cross-sections, almost doubling at the largest cross-section. The plastic zone is concentrated at the top plate and shoulder and decreases progressively from the two sides to the bottom corner. The range of stress concentration at the sides of the intersection roadway close to the passageway is wider and higher. (3) The 7 m-thick reinforced compression arch constructed under the strengthening support scheme has a bearing capacity enhanced by 1.8 to 2.3 times and increase in thickness of the bearing structure by 1.76 times as compared to the original scheme. (4) The increase in the mechanical parameters c and φ of the surrounding rock after anchoring causes a significant increase in σc and σt; the pulling force of the cable bolt beneath the new grouting material is more than twice that of ordinary cement grout, and according to the test, the supporting stress field shows that the 7.24 m surrounding rock is compacted and strengthened in addition to providing a strong foundation for the bolt (cable). On-site monitoring shows that the 60-day convergence is less than 30 mm, indicating that the stability control of the roadway is successful.

scholarly journals Load bearing capacity of thin-walled rectangular and I-shaped steel sections of short both empty and concrete-filled columns

2021 ◽  
Vol 15 (58) ◽  
pp. 77-85
Author(s):  
Amor Bouaricha ◽  
Naoual Handel ◽  
Aziza Boutouta ◽  
Sarah Djouimaa
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Load Capacity ◽  
Cross Sectional ◽  
Short Columns ◽  
Cross Section Area ◽  
Section Area ◽  
Specimen Height ◽  
Steel Sections ◽  
Thin Walled

In this experimental work, strength results obtained on short columns subjected to concentric loads are presented. The specimens used in the tests have made of cold-rolled, thin-walled steel. Twenty short columns of the same cross-section area and wall thickness have been tested as follows: 8 empty and 12 filled with ordinary concrete. In the aim to determine the column section geometry with the highest resistance, three different types of cross-sections have been compared: rectangular, I-shaped unreinforced and, reinforced with 100 mm spaced transversal links. The parameters studied are the specimen height and the cross-sectional steel geometry. The registered experimental results have been compared to the ultimate loads intended by Eurocode 3 for empty columns and by Eurocode 4 for compound columns. These results showed that a concrete-filled composite column had improved strength compared to the empty case. Among the three cross-section types, it has been found that I-section reinforced is the most resistant than the other two sections. Moreover, the load capacity and mode of failure have been influenced by the height of the column. Also, it had noted that the experimental strengths of the tested columns don’t agree well with the EC3 and EC4 results.

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