On Torsion and Transverse Flexure of Orthotropic Elastic Plates

1980 ◽  
Vol 47 (4) ◽  
pp. 855-860 ◽  
Author(s):  
E. Reissner

The equations of transverse bending of shear-deformable plates are used for the derivation of a system of one-dimensional equations for beams with unsymmetrical cross section, with account for warping stiffness, in addition to bending, shearing, and twisting stiffness. Significant results of the analysis include the observation that the rate of change of differential bending moment is given by the difference between torque contribution due to plate twisting moments and torque contribution due to plate shear stress resultants; a formula for shear center location which generalizes a result by Griffith and Taylor so as to account for transverse shear deformability and end-section warping restraint; a second-order compatibility equation for the differential bending moment; a contracted boundary condition of support for unsymmetrical cross-section beam theory in place of an explicit consideration of the warping deformation boundary layer; and construction of a problem where the effect of the conditions of support of the beam is such as to give noncoincident shear center and twist center locations.

Author(s):  
Lawrence N Virgin

Locating the shear, or flexural, center of non-symmetric cross-sectional beams is a key element in the teaching of structural mechanics. That is, establishing the point on the plane of the cross-section where an applied load, generating a bending moment about a principal axis, results in uni-directional deflection, and no twisting. For example, in aerospace structures it is particularly important to assess the propensity of an airfoil section profile to resist bending and torsion under the action of aerodynamic forces. Cross-sections made of thin-walls, whether of open or closed form are of special practical importance and form the basis of the material in this paper. The advent of 3D-printing allows the development of tactile demonstration models based on non-trivial geometry and direct observation.


2013 ◽  
Vol 4 (4) ◽  
pp. 133-144 ◽  
Author(s):  
Šarūnas Kelpša ◽  
Mindaugas Augonis

When the various reinforced concrete structures are designed according to EC2 and STR, the difference of calculation results, is quite significant. In this article the calculations of shear strength of bending reinforced concrete elements are investigated according to these standards. The comparison of such calculations is also significant in the sense that the shear strength calculations are carried out according to different principles. The STR regulations are based on work of the shear reinforcement crossing the oblique section and the compressed concrete at the end of the section. In this case, at the supporting zone, the external bending moment and shear force should be in equilibrium with the internal forces in reinforcement and compressed concrete, i.e., the cross section must be checked not only from the external shear force, but also from bending moment. In EC2 standard, the shear strengths are calculated according to simplified truss model, which consists of the tension shear reinforcement bars and compressed concrete struts. The bending moment is not estimated. After calculation analysis of these two methods the relationships between shear strength and various element parameters are presented. The elements reinforced with stirrups and bends are investigated additionally because in EC2 this case is not presented. According to EC2 the simplified truss model solution depends on the compression strut angle value θ, which is limited in certain interval. Since the component of tension reinforcement bar directly depends on the angle θ and the component of compression strut depends on it conversely, then exists some value θ when the both components are equal. So the angle θ can be found when such two components will be equated. However, such calculation of angle θ became complicated if the load is uniform, because then the components of tension bar are estimated not in support cross section but in cross section that are displaced by distance d. So, the cube equation should be solved. For simplification of such solution the graphical method to find out the angle θ and the shear strength are presented. In these graphics the intersection point of two components (shear reinforcement and concrete) curves describes the shear strength of element. Santrauka Straipsnyje apžvelgtos ir palygintos STR ir EC2 įstrižojo pjūvio stiprumo skaičiavimo metodikos stačiakampio skerspjūvio elementams. Normatyve neapibrėžtas EC2 metodikos santvaros modelio spyrių posvyrio kampo skaičiavimas, lemiantis galutinį įstrižojo pjūvio stiprumą. Straipsnyje pateikiamos kampo θ apskaičiavimo lygtys, atsižvelgiant į apkrovimo pobūdį. Norint supaprastinti pateiktų lygčių sprendimą siūlomas grafoanalitinis sprendimo būdas, pritaikant papildomus koeficientus. EC2 neapibrėžia skaičiavimo išraiškų, kai skersinis armavimas yra apkabos ir atlankos. Minėtos išraiškos suformuluotos ir pateiktos straipsnyje. Nustačius EC2 metodikos dėsningumus siūlomas alternatyvus apytikslis skaičiavimo būdas atlankomis ir apkabomis armuotiems elementams. Straipsnyje apžvelgtos abi – STR ir EC2 – metodikos, išskiriant pagrindinius skirtumus ir dėsningumus.


2019 ◽  
Vol 275 ◽  
pp. 01026
Author(s):  
Chenjie Zhao ◽  
Xiaohong Xiong ◽  
Zhenhua Xiong ◽  
Kangwen Wu ◽  
Zhen Cao ◽  
...  

Six specimens were made and tested to study the mechanical properties of LBL beams. The mean ultimate loading value is 68.39 MPa with a standard deviation of 6.37 MPa, giving a characteristic strength (expected to be exceeded by 95% of specimens) of 57.91 MPa, and the mean ultimate deflection is 53.3 mm with a standard deviation of 5.5 mm, giving the characteristic elastic modulus of 44.3 mm. The mean ultimate bending moment is 20.18 kN.m with a standard deviation of 1.88 kN.m, giving the characteristic elastic modulus of 17.08 kN.m. The mean elastic modulus is 9688 MPa with a standard deviation of 1765 MPa, giving the characteristic elastic modulus of 6785 MPa, and the mean modulus of rupture is 93.3 MPa with a standard deviation of 8.6 MPa, giving the characteristic elastic modulus of 79.2 MPa. The strain across the cross-section for all LBL beams is basically linear throughout the loading process, following standard beam theory.


1981 ◽  
Vol 48 (3) ◽  
pp. 601-605 ◽  
Author(s):  
E. Reissner

Equations for small finite displacements of shear-deformable plates are used to derive a one-dimensional theory of finite deformations of straight slender beams with one cross-sectional axis of symmetry. The equations of this beam theory are compared with the corresponding case of Kirchhoff’s equations, and with a generalization of Kirchhoff’s equations which accounts for the deformational effects of cross-sectional forces. Results of principal interest are: 1. The equilibrium equations are seven rather than six, in such a way as to account for cross-sectional warping. 2. In addition to the usual six force and moment components of beam theory, there are two further stress measures, (i) a differential plate bending moment, as in the corresponding linear theory, and (ii) a differential sheet bending moment which does not occur in linear theory. The general results are illustrated by the two specific problems of finite torsion of orthotropic beams, and of the buckling of an axially loaded cantilever, as a problem of bending-twisting instability caused by material anisotropy.


2005 ◽  
Vol 10 (2) ◽  
pp. 151-160 ◽  
Author(s):  
J. Kala ◽  
Z. Kala

Authors of article analysed influence of variability of yield strength over cross-section of hot rolled steel member to its load-carrying capacity. In calculation models, the yield strength is usually taken as constant. But yield strength of a steel hot-rolled beam is generally a random quantity. Not only the whole beam but also its parts have slightly different material characteristics. According to the results of more accurate measurements, the statistical characteristics of the material taken from various cross-section points (e.g. from a web and a flange) are, however, more or less different. This variation is described by one dimensional random field. The load-carrying capacity of the beam IPE300 under bending moment at its ends with the lateral buckling influence included is analysed, nondimensional slenderness according to EC3 is λ¯ = 0.6. For this relatively low slender beam the influence of the yield strength on the load-carrying capacity is large. Also the influence of all the other imperfections as accurately as possible, the load-carrying capacity was determined by geometrically and materially nonlinear solution of very accurate FEM model by the ANSYS programme.


2018 ◽  
Author(s):  
Miguel Abambres

Original Generalized Beam Theory (GBT) formulations for elastoplastic first and second order (postbuckling) analyses of thin-walled members are proposed, based on the J2 theory with associated flow rule, and valid for (i) arbitrary residual stress and geometric imperfection distributions, (ii) non-linear isotropic materials (e.g., carbon/stainless steel), and (iii) arbitrary deformation patterns (e.g., global, local, distortional, shear). The cross-section analysis is based on the formulation by Silva (2013), but adopts five types of nodal degrees of freedom (d.o.f.) – one of them (warping rotation) is an innovation of present work and allows the use of cubic polynomials (instead of linear functions) to approximate the warping profiles in each sub-plate. The formulations are validated by presenting various illustrative examples involving beams and columns characterized by several cross-section types (open, closed, (un) branched), materials (bi-linear or non-linear – e.g., stainless steel) and boundary conditions. The GBT results (equilibrium paths, stress/displacement distributions and collapse mechanisms) are validated by comparison with those obtained from shell finite element analyses. It is observed that the results are globally very similar with only 9% and 21% (1st and 2nd order) of the d.o.f. numbers required by the shell finite element models. Moreover, the GBT unique modal nature is highlighted by means of modal participation diagrams and amplitude functions, as well as analyses based on different deformation mode sets, providing an in-depth insight on the member behavioural mechanics in both elastic and inelastic regimes.


Materials ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3438
Author(s):  
Michał Szczecina ◽  
Andrzej Winnicki

This paper discusses a choice of the most rational reinforcement details for frame corners subjected to opening bending moment. Frame corners formed from elements of both the same and different cross section heights are considered. The case of corners formed of elements of different cross section is not considered in Eurocode 2 and is very rarely described in handbooks. Several reinforcement details with both the same and different cross section heights are presented. The authors introduce a new reinforcement detail for the different cross section heights. The considered details are comprised of the primary reinforcement in the form of straight bars and loops and the additional reinforcement in the form of diagonal bars or stirrups or a combination of both diagonal stirrups and bars. Two methods of static analysis, strut-and-tie method (S&T) and finite element method (FEM), are used in the research. FEM calculations are performed with Abaqus software using the Concrete Damaged Plasticity model (CDP) for concrete and the classical metal plasticity model for reinforcing steel. The crucial CDP parameters, relaxation time and dilatation angle, were calibrated in numerical tests in Abaqus. The analysis of results from the S&T and FE methods allowed for the determination of the most rational reinforcement details.


2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Giselle Nevares ◽  
Felipe Xavier ◽  
Luciana Gominho ◽  
Flávia Cavalcanti ◽  
Marcely Cassimiro ◽  
...  

This study aimed to analyse and compare apical extrusion of debris in canals instrumented with systems used in reciprocating and continuous motion. Sixty mandibular premolars were randomly divided into 3 groups (n=20): the Reciproc (REC), WaveOne (WO), and HyFlex CM (HYF) groups. One Eppendorf tube per tooth was weighed in advance on an analytical balance. The root canals were instrumented according to the manufacturer’s instructions, and standardised irrigation with 2.5% sodium hypochlorite was performed to a total volume of 9 mL. After instrumentation, the teeth were removed from the Eppendorf tubes and incubated at 37°C for 15 days to evaporate the liquid. The tubes were weighed again, and the difference between the initial and final weight was calculated to determine the weight of the debris. The data were statistically analysed using the Shapiro-Wilk, Wilcoxon, and Mann-Whitney tests (α=5%). All systems resulted in the apical extrusion of debris. Reciproc produced significantly more debris than WaveOne (p<0.05), and both systems produced a greater apical extrusion of debris than HyFlex CM (p<0.001). Cross section and motion influenced the results, despite tip standardization.


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