Rectangular Cross Section Beams Calculation on the Stability of a Flat Bending Shape Taking into Account the Initial Imperfections

2019 ◽  
Vol 974 ◽  
pp. 551-555 ◽  
Author(s):  
I.M. Zotov ◽  
Anastasia P. Lapina ◽  
Anton S. Chepurnenko ◽  
B.M. Yazyev
Keyword(s):  
Cross Section ◽  
Basic Equation ◽  
Applied Load ◽  
Rectangular Cross Section ◽  
Twist Angle ◽  
Stability Loss ◽  
Lateral Buckling ◽  
Initial Imperfections ◽  
Beam Stability ◽  
The Stability

The article presents the derivation of the resolving equation for the calculation of lateral buckling of rectangular beams. When deriving the basic equation, the initial imperfections of the beam are taken into account, which are specified in the form of the eccentricity of the applied load, the initial deflection in the plane of least stiffness and the initial twist angle. The influence of initial imperfections on the process of beam stability loss is investigated.

scholarly journals Flat bending shape stability of the beams with variable section width

2020 ◽  
Vol 164 ◽  
pp. 02016
Author(s):  
Anastasia Lapina ◽  
Serdar Yazyev ◽  
Anton Chepurnenko ◽  
Irina Dubovitskaya
Keyword(s):  
Trigonometric Series ◽  
Cross Section ◽  
Secular Equation ◽  
Rectangular Cross Section ◽  
Twist Angle ◽  
Shape Stability ◽  
Lateral Buckling ◽  
Concentrated Force ◽  
Angle Function ◽  
Variable Section

The paper proposes a methodology for calculating lateral buckling of beams of variable rectangular cross section based on the energy approach. The technique is considered on the example of a cantilever beam of variable width with two sections under the action of a concentrated force. The twist angle function was set in the form of a trigonometric series. As a result, the problem is reduced to a generalized secular equation.

The lateral instability of yielded mild steel beams of rectangular cross-section

1950 ◽  
Vol 242 (846) ◽  
pp. 197-242 ◽  
Keyword(s):  
Mild Steel ◽  
Critical Load ◽  
Cross Section ◽  
Applied Load ◽  
Principal Axis ◽  
Rectangular Cross Section ◽  
Torsional Rigidity ◽  
Steel Beams ◽  
Lateral Instability ◽  
Lateral Buckling

The critical load causing secondary failure of a deep beam by lateral buckling may be calculated by standard methods for those cases in which the beam behaves elastically under the applied load. When, however, the load is sufficiently great to cause partial yield of the beam, these methods give an estimate for the critical load which is too high. In the present paper the phenomenon of lateral buckling in deep mild steel beams of rectangular cross-section is studied from both a theoretical and an experimental standpoint. The paper is divided into three parts. In part I the critical lateral buckling load is shown to depend on the flexural rigidity of the beam about its weaker principal axis while the applied load, causing flexure about its stronger principal axis, is held constant. The dependence of this rigidity on the extent to which the beam has yielded is calculated, and the results are confirmed by tests on beams of rectangular and circular cross-section. It is also shown that the critical load depends on the initial torsional rigidity of the beam, defined as the initial slope of the torque against angle of twist per unit length relation for torsion about the longitudinal axis of the beam while the applied bending load is held constant. In part II it is first shown that in a beam which has partially yielded the shear force due to the variation of the applied bending moment along the length of the beam is carried entirely in the central elastic core of the beam. Using the theory of combined elastic and plastic deformation, it is then shown that the initial torsional rigidity remains constant at its value for elastic torsion, and experimental evidence in favour of this conclusion is presented. Using the results of parts I and II, the conditions causing lateral instability in deep mild steel beams of rectangular cross-section are determined in part III. For a beam bent by pure terminal couples these conditions may be deduced directly, but for the cases of beams subjected to central concentrated loads and of cantilevers a step by step solution of the governing differential equation is necessary. Experimental confirmation is given for the case of pure bending.

scholarly journals WOODEN BEAM FLAT BENDING SHAPE STABILITY TAKING THE CREEP INTO ACCOUNT

2021 ◽  
Vol 9 (2) ◽  
pp. 6-10
Author(s):  
Anastasiya Lapina
Keyword(s):  
Numerical Solution ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Critical Time ◽  
Shape Stability ◽  
Lateral Buckling ◽  
Initial Imperfections ◽  
New Criterion

The article deals with the problem of lateral buckling of a wooden beam of rectangular cross-section, taking into account the initial imperfections under creep conditions. An algorithm for the numerical solution is presented. The linear Maxwell-Thompson equation is used as the creep law. The character of the growth of the deflection of the beam at various load levels is investigated and a new criterion is introduced to determine the critical time.

Effect of Slenderness Ratio on the Natural Frequencies of Pre-Twisted Cantilever Beams of Uniform Rectangular Cross-Section

1971 ◽  
Vol 13 (1) ◽  
pp. 51-59 ◽  
Author(s):  
B. Dawson ◽  
N. G. Ghosh ◽  
W. Carnegie
Keyword(s):  
Differential Equations ◽  
Shear Deformation ◽  
Cross Section ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Slenderness Ratio ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Rotary Inertia ◽  
Cantilever Beams

This paper is concerned with the vibrational characteristics of pre-twisted cantilever beams of uniform rectangular cross-section allowing for shear deformation and rotary inertia. A method of solution of the differential equations of motion allowing for shear deformation and rotary inertia is presented which is an extension of the method introduced by Dawson (1)§ for the solution of the differential equations of motion of pre-twisted beams neglecting shear and rotary inertia effects. The natural frequencies for the first five modes of vibration are obtained for beams of various breadth to depth ratios and lengths ranging from 3 to 20 in and pre-twist angle in the range 0–90°. The results are compared with those obtained by an alternative method (2), where available, and also to experimental results.

scholarly journals Calculation of wooden beams on the stability of a flat bending shape enhancement

2018 ◽  
Vol 196 ◽  
pp. 01003 ◽  
Author(s):  
Anton Chepurnenko ◽  
Vera Ulianskaya ◽  
Serdar Yazyev ◽  
Ivan Zotov
Keyword(s):  
Differential Equation ◽  
Cross Section ◽  
Stability Problem ◽  
Secular Equation ◽  
Rectangular Cross Section ◽  
Center Of Gravity ◽  
Critical Force ◽  
Distributed Load ◽  
Matlab Package ◽  
The Stability

Flat bending stability problem of constant rectangular cross section wooden beam, loaded by a distributed load is considered. Differential equation is provided for the cases when load is located not in the center of gravity. The solution of the equation is performed numerically by the method of finite differences. For the case of applying a load at the center of gravity, the problem reduces to a generalized secular equation. In other cases, the iterative algorithm developed by the authors is implemented, in the Matlab package. A relationship between the value of the critical force and the position of the load application point is obtained. A linear approximating function is selected for this dependence.

scholarly journals CALCULATION OF THE FLAT BENDING SHAPE STABILITY OF RECTANGULAR CROSS SECTION BEAMS WITH REGARD TO CREEP

2019 ◽  
Vol 46 (1) ◽  
pp. 169-176
Author(s):  
I. M. Zotov ◽  
A. S. Chepurnenko ◽  
S. B. Yazyev
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Rectangular Cross Section ◽  
Twist Angle ◽  
Euler Method ◽  
Shape Stability ◽  
Concentrated Force ◽  
Flat Shape ◽  
The Stability

Objectives. The article presents the conclusion of the resolving equation for calculating the stability of the flat form of deformation of prismatic beams, taking into account the rheological properties of the material.Method. The problem is reduced to a second-order differential equation for the twist angle, which is solved numerically by the finite difference method in combination with the Euler method.Result. The obtained differential equation allows one to take into account the presence of initial imperfections in the form of the initial deflection of the beam, the initial angle of twist, and also the eccentricity of the applied load. The solution of the test problem for a cantilever polymer beam under the action of a concentrated force is presented. The non-linear Maxwell-Gurevich equation is used as the creep law. The value of the long-term critical load is introduced and it is shown that with a load less than the long-term critical creep is limited. It has been established that, as with the squeezed rods, with a load less than the long-term critical, the growth rate of the displacements with time decays. When F = F_dl, the displacements grow at a constant speed, and when F> F_dl, the growth rate of displacements increases with time. The results obtained confirm the validity of the chosen method.Conclusion. A universal resolving equation is obtained for calculating the stability of a flat shape of bending of rectangular beams, suitable for arbitrary creep laws.

Recommendations for Designing Wooden Arches on Metal-toothed Plates

2021 ◽  
Author(s):  
Bohdan Demchyna ◽  
◽  
Yaroslav Shydlovskyi
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Laboratory Testing ◽  
Horizontal Plane ◽  
Thrust Force ◽  
Rectangular Cross Section ◽  
Rectangular Section ◽  
Pilot Studies ◽  
Different Types ◽  
The Stability

This paper presents the findings of the pilot studies and recommendations for designing of two-hinged wooden arches. The prototype models of wooden arches with the span of 6mand the rise of 1m were designed. The models had a rectangular cross-section of 180x40mm and a T-section of 180x40mm with a plywood plate with the thickness of 6 mm and the width of 500mm. The main objective of the T-section was to ensure the stability of the arch. Each arch was composed of six segments –boards joined by clamping plates. The bowstring truss including two inclined tie bars enables carrying asymmetric loads and provides in-plane stability of the arch. A methodology for laboratory testing of the prototype models of wooden arches subjected to different types of loads was developed. Two prototypes of wooden arches were tested with rectangular cross-sections and two T-section ones subjected to the loading across the span, and two prototypes subjected to the half-span loading. In total, eight arches were tested. Deflections of arches, cross-section deformations and arch thrust force were recorded. The arches were tested until failure. The results of testing revealed insufficient stability of the arches with rectangular cross-section in the horizontal plane. For the arches with T-section the whole arch rib was damaged, the in-plane stability was ensured by the T-section. The collapsing force of the T-section arch was about 1.3 times greater than the collapsing force of the rectangular section arches.

Modal Curves of Pre-Twisted Beams of Rectangular Cross-Section

1969 ◽  
Vol 11 (1) ◽  
pp. 1-13 ◽  
Author(s):  
B. Dawson ◽  
W. Carnegie
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Twist Angle ◽  
Ritz Method ◽  
Transformation Method ◽  
Mode Shapes ◽  
Modes Of Vibration ◽  
Vibrational Characteristics ◽  
Theoretical Results ◽  
Good Agreement

An important aspect of the theoretical study of the vibrational characteristics of turbine and compressor blading is the prediction of the modal curves from which the stresses along the length of the blading can be determined. The accurate prediction of the modal curves allowing for such factors as pre-twist, camber, size of cross-section, centrifugal tensile effects, aerodynamic effects, etc., is still not possible. However, a better understanding of the effects of some of these parameters can be obtained by a study of the modal curves of relatively simple idealized models. In this work the theoretical mode shapes of vibration of pre-twisted rectangular cross-section beams for various width to depth ratios and pre-twist angle in the range 0-90° are examined. The theoretical results are obtained by the transformation method given by Carnegie, Dawson and Thomas (1)† and the accuracy of these results is verified by comparison with results obtained by Dawson (2) using the Ritz method. The theoretical results are compared to modal curves determined experimentally and good agreement is shown between them. A physical explanation of the effects of the pre-twist angle upon the modal curves is given for the first three modes of vibration.

Equilibrium, Stability, and Dynamics of Rectangular Liquid-Filled Vessels

2011 ◽  
Vol 6 (4) ◽  
Author(s):  
Russell Trahan ◽  
Tamás Kalmár-Nagy
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Mathieu Equation ◽  
Liquid Volume ◽  
Position Vector ◽  
Equilibrium Stability ◽  
Bifurcation Diagrams ◽  
Equilibrium Configurations ◽  
Theoretical Predictions ◽  
The Stability

Here we focus on the stability and dynamic characteristics of a rectangular, liquid-filled vessel. The position vector of the center of gravity of the liquid volume is derived and used to express the equilibrium angles of the vessel. Analysis of the potential function determines the stability of these equilibria, and bifurcation diagrams are constructed to demonstrate the co-existence of several equilibrium configurations of the vessel. To validate the results, a vessel of rectangular cross section was built. The results of the experiments agree well with the theoretical predictions of stability. The dynamics of the unforced and forced systems with a threshold constraint is discussed in the context of the nonlinear Mathieu equation.

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