Shape Memory Elements in Bending: Influence of the Shape of their Cross-Section

2000 ◽  
Vol 11 (12) ◽  
pp. 977-984 ◽  
Author(s):  
Vratislav Kafka ◽  
David Vokoun
Keyword(s):  
Shape Memory ◽  
Cross Section ◽  
Elastic Limit ◽  
Rectangular Cross Section ◽  
Circular Cross Section ◽  
Bending Moment ◽  
Internal Variables ◽  
Memory Recovery ◽  
The Cross ◽  
The Mathematical Model

The effect of the shape of the cross-section of a bent prismatic bar on its shape memory recovery moment is investigated. The analysis is based on the mathematical model of the first author (Kafka, 1994a, 1994b, 2001). The area of the cross-section of the bar is assumed to be constant, the shape of the cross-section is varied. The investigated shapes are rectangles with various relations of their sides, and a circular cross-section. It is assumed that the rod is bent above elastic limit and unloaded at room temperature, which results in macroscopic residual stresses giving zero bending moment, and in residual internal variables descriptive of the change of the state of the material. Then, the resulting form is held fixed and temperature of the rod is raised. Due to the increase of temperature there arise recovery stresses resulting in recovery moments. These moments—depending on the shape of the cross-section—are calculated, and in this way the effectiveness of the shape of the cross-section is evaluated. In the case of a rectangular cross-section the effect of the relation of the sides is strongly non-linear, the effect of the circular cross-section is lower by 20% than that of a square cross-section.

scholarly journals Cross-Sectional Analysis of the Resistance of Rc Members Subjected to Bending With/without Axial Force

2021 ◽  
Author(s):  
Marek Lechman
Keyword(s):  
Cross Section ◽  
Axial Force ◽  
Rectangular Cross Section ◽  
Bending Moment ◽  
Equilibrium Equations ◽  
Cross Sectional ◽  
Cross Sectional Analysis ◽  
Nonlinear Stress ◽  
The Cross ◽  
Sectional Analysis

The paper presents section models for analysis of the resistance of RC members subjected to bending moment with or without axial force. To determine the section resistance the nonlinear stress-strain relationship for concrete in compression is assumed, taking into account the concrete softening. It adequately describes the behavior of RC members up to failure. For the reinforcing steel linear elastic-ideal plastic model is applied. For the ring cross-section subjected to bending with axial force the normalized resistances are derived in the analytical form by integrating the cross-sectional equilibrium equations. They are presented in the form of interaction diagrams and compared with the results obtained by testing conducted on RC columns under eccentric compression. Furthermore, the ultimate normalized bending moment has been derived for the rectangular cross-section subjected to bending without axial force. It was applied in the cross-sectional analysis of steel and concrete composite beams, named BH beams, consisting of the RC rectangular core placed inside a reversed TT welded profile. The comparisons made indicated good agreements between the proposed section models and experimental results.

Combined Bending and Compression of a Pressurized Circular Cylindrical Membrane Column

1965 ◽  
Vol 87 (3) ◽  
pp. 372-378
Author(s):  
W. E. Jahsman
Keyword(s):  
Compressive Strength ◽  
Cross Section ◽  
Compressive Load ◽  
Circular Cross Section ◽  
Bending Moment ◽  
Maximum Load ◽  
Bending Moments ◽  
Lateral Deflection ◽  
The Cross

Load-lateral deflection curves are developed for a pressurized tube of circular cross section under combined bending and compression. The tube walls are assumed to have negligible compressive strength so that wrinkling develops if the stress tends to become negative. It is found that for a given bending moment, the load increases monotonically with deflection until a maximum is reached beyond which the load decreases with increasing deflection. An interaction curve of the maximum load versus bending moment shows that the presence of only a small amount of bending significantly decreases the maximum compressive load below the classical Euler load. Conversely, for bending moments which produce almost complete wrinkling of the cross section, only very small amounts of compressive load can be supported.

scholarly journals Cross-Section Deformation and Bending Moment of a Steel Square Tubular Section

Materials ◽  
2020 ◽  
Vol 13 (22) ◽  
pp. 5170
Author(s):  
Stanisław Kut ◽  
Feliks Stachowicz
Keyword(s):  
Numerical Modeling ◽  
Cross Section ◽  
Circular Cross Section ◽  
Bending Moment ◽  
Experimental Tests ◽  
Cross Sectional ◽  
Cold Bending ◽  
Bending Process ◽  
The Cross ◽  
Sectional Shape

When bending thin-walled profiles, significant distortion of the cross-section occurs, which has a significant impact on the course of the bending moment characteristics and on the value of allowable bending curvatures. This paper presents the results of experimental and numerical modeling of the box profile bending process, which was carried out in order to determine the dependence of the cross-sectional shape and bending moment of bending curvature. Extensive numerical calculations were used to model the process of shaping a square pipe from a circular tube and to model the bending process, especially when taking into account the effects of such a deformation path. The pure bending moment characteristics and the deformation of the cross-section were performed for a 25 × 25 × 2 mm square tube made of S235JR structural steel. The innovative approach for determining the parameters of cold bending square tubes pertained to considering the stress state in the preserved material in individual areas of their cross-section. The results of numerical modeling—after considering the history of deformation (i.e., the process of forming a square pipe from a pipe with a circular cross-section)—gave a satisfactory agreement with the results of experimental tests, both in terms of the degree of pipe wall deflection and the characteristics of the bending moment.

scholarly journals Measurements of the absolute indices of refraction in strained glass

1910 ◽  
Vol 83 (566) ◽  
pp. 572-579 ◽  
Keyword(s):  
Plane Wave ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Vertical Plane ◽  
Bending Moment ◽  
The Other ◽  
Double Refraction ◽  
Light Passing ◽  
The Absolute ◽  
The Cross

1. In June, 1907, the author described a method by which the double-refraction in strained glass could be measured by observing the deviation of a ray of light passing through a slab of glass under flexure. If a slab or beam of glass of rectangular cross-section be bent in a vertical plane under a bending moment M, and if a plane wave be transmitted through the glass in a direction perpendicular to the plane of flexure, the light is broken up into two components, one polarised horizontally ( i. e . perpendicular to the cross-section and along the line of stress) and the other vertically.

scholarly journals Analytical modeling of the shape memory effect in SMA beams with rectangular cross section under reversed pure bending

2021 ◽  
pp. 1045389X2098878
Author(s):  
Enrico Radi
Keyword(s):  
Martensitic Transformation ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Axial Stress ◽  
Bending Moment ◽  
Integral Form ◽  
Loading Path ◽  
Martensite Variant ◽  
Young’S Moduli ◽  
The Cross

An analytical model is developed for a prismatic SMA beam with rectangular cross section subjected to alternating bending at temperature below the austenitic transformations. The loading path consists in a loading-unloading cycle under bending and then under reversed bending. Two opposite martensitic variants take place, whose volume fractions evolve linearly with the axial stress. Different Young’s moduli are taken for the austenitic and martensitic phases. As the bending moment is increased, the martensitic transformation starts from the top and bottom and then it extends inwards. If the maximum applied bending moment is large enough, then the complete Martensitic transformation takes place at the upper and lower parts of the cross section. During unloading and the following reversed bending, reorientation of the Martensite variant into the opposite one takes place starting from the boundary between the fully martensitic region and the intermediate transforming region. Special attention is devoted to calculate analytically the axial stress and Martensite variant distributions within the cross section at each stage of the process. A closed form moment-curvature relation is provided for loading and elastic unloading and in integral form for the rest of the process. The approach is then validated by comparison with analytical results available in the literature.

scholarly journals Designing for People’s Safety on Flooded Streets: Uncertainties and the Influence of the Cross-Section Shape, Roughness and Slopes on Hazard Criteria

Water ◽  
2021 ◽  
Vol 13 (15) ◽  
pp. 2119
Author(s):  
Luís Mesquita David ◽  
Rita Fernandes de Carvalho
Keyword(s):  
Cross Section ◽  
Overland Flow ◽  
Rectangular Cross Section ◽  
Safety Hazard ◽  
Cross Section Shape ◽  
Section Shape ◽  
Flood Flows ◽  
The Cross ◽  
Flooding Risk

Designing for exceedance events consists in designing a continuous route for overland flow to deal with flows exceeding the sewer system’s capacity and to mitigate flooding risk. A review is carried out here on flood safety/hazard criteria, which generally establish thresholds for the water depth and flood velocity, or a relationship between them. The effects of the cross-section shape, roughness and slope of streets in meeting the criteria are evaluated based on equations, graphical results and one case study. An expedited method for the verification of safety criteria based solely on flow is presented, saving efforts in detailing models and increasing confidence in the results from simplified models. The method is valid for 0.1 m2/s 0.5 m2/s. The results showed that a street with a 1.8% slope, 75 m1/3s−1 and a rectangular cross-section complies with the threshold 0.3 m2/s for twice the flow of a street with the same width but with a conventional cross-section shape. The flow will be four times greater for a 15% street slope. The results also highlighted that the flood flows can vary significantly along the streets depending on the sewers’ roughness and the flow transfers between the major and minor systems, such that the effort detailing a street’s cross-section must be balanced with all of the other sources of uncertainty.

scholarly journals Asymptotic Behavior of Stable Structures Made of Beams

2021 ◽  
Author(s):  
Georges Griso ◽  
Larysa Khilkova ◽  
Julia Orlik ◽  
Olena Sivak
Keyword(s):  
Asymptotic Behavior ◽  
Cross Section ◽  
Cross Sections ◽  
Linear Elasticity ◽  
Circular Cross Section ◽  
Linear Elasticity Theory ◽  
Linear Elastic ◽  
The Cross ◽  
Small Rotation ◽  
Stable Structures

AbstractIn this paper, we study the asymptotic behavior of an $\varepsilon $ ε -periodic 3D stable structure made of beams of circular cross-section of radius $r$ r when the periodicity parameter $\varepsilon $ ε and the ratio ${r/\varepsilon }$ r / ε simultaneously tend to 0. The analysis is performed within the frame of linear elasticity theory and it is based on the known decomposition of the beam displacements into a beam centerline displacement, a small rotation of the cross-sections and a warping (the deformation of the cross-sections). This decomposition allows to obtain Korn type inequalities. We introduce two unfolding operators, one for the homogenization of the set of beam centerlines and another for the dimension reduction of the beams. The limit homogenized problem is still a linear elastic, second order PDE.

scholarly journals Validation of Welding Simulations Using Thermal Strains Measured with DIC

2011 ◽  
Vol 70 ◽  
pp. 129-134 ◽  
Author(s):  
Maarten De Strycker ◽  
Pascal Lava ◽  
Wim Van Paepegem ◽  
Luc Schueremans ◽  
Dimitri Debruyne
Keyword(s):  
Residual Stresses ◽  
Cross Section ◽  
Circular Cross Section ◽  
Welding Process ◽  
Weld Bead ◽  
Image Correlation ◽  
Steel Tubes ◽  
Element Analysis ◽  
Strain Evolution ◽  
The Cross

Residual stresses can affect the performance of steel tubes in many ways and as a result their magnitude and distribution is of particular interest to many applications. Residual stresses in cold-rolled steel tubes mainly originate from the rolling of a flat plate into a circular cross section (involving plastic deformations) and the weld bead that closes the cross section (involving non-uniform heating and cooling). Focus in this contribution is on the longitudinal weld bead that closes the cross section. To reveal the residual stresses in the tubes under consideration, a finite element analysis (FEA) of the welding step in the production process is made. The FEA of the welding process is validated with the temperature evolution of the thermal simulation and the strain evolution for the mechanical part of the analysis. Several methods for measuring the strain evolution are available and in this contribution it is investigated if the Digital Image Correlation (DIC) technique can record the strain evolution during welding. It is shown that the strain evolution obtained with DIC is in agreement with that found by electrical resistance strain gauges. The results of these experimental measuring methods are compared with numerical results from a FEA of the welding process.

Analysis of Twisting Stiffness for a Multifiber Composite Layer

1974 ◽  
Vol 41 (3) ◽  
pp. 658-662 ◽  
Author(s):  
C. W. Bert ◽  
S. Chang
Keyword(s):  
Cross Section ◽  
Least Squares Method ◽  
Rectangular Cross Section ◽  
Torsional Rigidity ◽  
Composite Layer ◽  
Circular Cross Section ◽  
Relaxation Method ◽  
Fiber Composites ◽  
Numerical Results ◽  
Torsion Problem

The twisting stiffness of a rectangular cross section consisting of a single row of solid circular cross-section fibers embedded in a matrix is analyzed. The problem is formulated as a Dirichlet torsion problem of a multielement region and solved by the boundary-point least-squares method. Numerical results for a single-fiber square cross section compare favorably with previous relaxation-method results. New numerical results for three and five-fiber composites suggest that the torsional rigidity of a multifiber composite can be approximated from the torsional rigidities of single and three-fiber models.

Bending Vibrations of Variable Section Beams

1956 ◽  
Vol 23 (1) ◽  
pp. 103-108
Author(s):  
E. T. Cranch ◽  
Alfred A. Adler
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Circular Cross Section ◽  
Beam Theory ◽  
Cantilever Beams ◽  
Constant Depth ◽  
Bending Vibrations ◽  
Variable Section

Abstract Using simple beam theory, solutions are given for the vibration of beams having rectangular cross section with (a) linear depth and any power width variation, (b) quadratic depth and any power width variation, (c) cubic depth and any power width variation, and (d) constant depth and exponential width variation. Beams of elliptical and circular cross section are also investigated. Several cases of cantilever beams are given in detail. The vibration of compound beams is investigated. Several cases of free double wedges with various width variations are discussed.

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