Preparation for reduction of pipes with tension. Investigation of local stability of thin-walled trapezoidal profiles with longitudinal-transverse bending Kholkin Evgeny Gennadievich Essence of the planned measure

where, p is the number of the current iteration; vt is the total speed of metal sliding over the tool surface; vn is the normal speed of metal movement; wn is the normal speed of the tool; st - friction stress;
- Yield stress as a function of the parameters of the deformable metal, at a given point; - Medium voltage; - Intensity of strain rate; x0 - strain rate of all-round compression; Kt - penalty factor for the speed of metal sliding over the tool (specified by the iteration method) Kn - penalty factor for metal penetration into the tool; m - conditional viscosity of the metal, refined by the method of hydrodynamic approximations; - tension tension or backwater during rolling; Fn - area cross section the end of the pipe to which tension or support is applied.
The calculation of the deformation-speed mode includes the distribution of the state of deformations along the stands along the diameter, the required value of the coefficient of plastic tension according to the state Ztot, the calculation of the drawing ratios, roll diameters of the rolls and the rotational speed of the main drive motors, taking into account the features of its design.
For the first stands of the mill, including the first stand that rolls, and for the last, placed after the last stand, rolls, the coefficients of plastic tension in them Zav.i are less than the required Ztot. Due to such a distribution of the plastic tension coefficients over all stands of the mill, the calculated wall thickness at the exit from it is greater than necessary along the reduction route. In order to compensate for the insufficient pulling capacity of the rolls of the stands located in the first and after the last stands that are rolled, it is necessary, using an iterative calculation, to find such a value Ztotal that the calculated and specified wall thicknesses at the exit from the state are the same. The greater the value of the required total coefficient of plastic tension according to the state Ztotal, the greater the error in its determination without iterative calculation.
After the iterative calculations have calculated the coefficients of the front and rear plastic tension, the thickness of the pipe wall at the inlet and outlet of the deformation cells along the stands of the reduction mill, we finally determine the position of the first and last stands that are rolled.
Of course, rolling the diameter is determined through the central angle qk.p. between the vertical axis of symmetry of the roll groove and the line drawn from the center of the pass, coincides with the rolling axis to a point on the surface of the pass groove, where the neutral line of the deformation zone is located on its surface, is conventionally located parallel to the rolling axis. The value of the angle qk.p., first of all, depends on the value of the coefficient of the rear Zset. and front Zper. tension, as well as the coefficient
hoods.
Determination of rolling diameter by the value of the angle qk.p. usually performed for a caliber, has the shape of a circle with a center in the rolling axis and a diameter equal to the average diameter of the caliber Dav.
The largest errors in determining the value of the rolling diameter without taking into account the actual geometric dimensions of the pass will be for the case when the rolling conditions determine its position either at the bottom or at the groove flange. The more the real shape of the caliber differs from the circle accepted in the calculations, the more significant this error will be.
The maximum possible range of change of the actual value of the diameter rolls the caliber is a roll cut. The greater the number of rolls forms a caliber, the greater the relative error in determining the rolling diameter without taking into account the actual geometric dimensions of the caliber.
With an increase in the partial reduction of the pipe diameter in the caliber, the difference between its shape and the round one grows. So, with an increase in the reduction of the pipe diameter from 1 to 10%, the relative error in determining the value of the rolling diameter without taking into account the actual geometric dimensions of the caliber increases from 0.7 to 6.3% for a two-roller, 7.1% for a three-roller and 7.4% - for a chotirio-roll "rolling" stand when, according to the kinematic conditions of rolling, rolling the diameter located along the bottom of the caliber.
Simultaneous increase in the same

INTRODUCTION

1 STATUS OF THE ISSUE ON THE THEORY AND TECHNOLOGY OF PROFILING MULTIFACETED PIPES BY DRAWING WITHOUT DRAWING (LITERARY REVIEW).

1.1 Range profile pipes with flat edges and their use in technology.

1.2 The main methods for the production of profile pipes with flat edges.

1.4 Drawing shaped tool.

1.5 Drawing multifaceted helical-twisted pipes.

1.6 Conclusions. Purpose and objectives of research.

2 DEVELOPMENT OF A MATHEMATICAL MODEL OF PIPE PROFILING BY DRAWING.

2.1 Basic provisions and assumptions.

2.2 Description of the deformation zone geometry.

2.3 Description of the power parameters of the profiling process.

2.4 Evaluation of the fillability of the corners of the drawing die and the tightening of the profile faces.

2.5 Description of the algorithm for calculating profiling parameters.

2.6 Computer analysis of force conditions for profiling square tubes without mandrel drawing.

2.7 Conclusions.

3 CALCULATION OF THE TOOL FOR STRENGTH FOR DRAWING PROFILE PIPES.

3.1 Statement of the problem.

3.2 Determination of the stress state of the die.

3.3 Construction of mapping functions.

3.3.1 Square hole.

3.3.2 Rectangular hole.

3.3.3 Plano-oval hole.

3.4 An example of calculating the stress state of a drawing die with a square hole.

3.5 An example of the calculation of the stress state of a drawing die with a round hole.

3.6 Analysis of the obtained results.

3.7 Conclusions.

4 EXPERIMENTAL STUDIES ON PROFILING OF SQUARE AND RECTANGULAR PIPES BY DRAWING.

4.1 Methodology of the experiment.

4.2 Profiling a square pipe by drawing in one transition into one die.

4.3 Profiling a square tube by drawing in one pass with counter tension.

4.4 Three-factor linear mathematical model of profiling square pipes.

4.5 Determination of the fillability of the corners of the drawing die and the tightening of the faces.

4.6 Improving the calibration of die channels for rectangular pipes.

4.7 Conclusions.

5 DRAWING OF PROFILE HELICALLY TWISTED PIPES.

5.1 Choice of technological parameters of drawing with torsion.

5.2 Determination of torque.

5.3 Determination of pulling force.

5.4 Experimental studies.

5.5 Conclusions.

Recommended list of dissertations

• Drawing thin-walled pipes with a rotating tool 2009, candidate of technical sciences Pastushenko, Tatyana Sergeevna

• Improving the technology of mandrelless drawing of thin-walled pipes into a block of drawing dies with a guaranteed wall thickness 2005, candidate of technical sciences Kargin, Boris Vladimirovich

• Improvement of processes and machines for the manufacture of cold profiled pipes based on the simulation of the deformation zone 2009, Doctor of Technical Sciences Parshin, Sergey Vladimirovich

• Modeling the process of profiling multifaceted pipes in order to improve it and select the parameters of the mill 2005, candidate of technical sciences Semenova, Natalya Vladimirovna

• Drawing pipes from anisotropic hardening material 1998, Ph.D. Chernyaev, Alexey Vladimirovich

Introduction to the thesis (part of the abstract) on the topic "Improving the process of profiling polyhedral pipes by mandrelless drawing"

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480 rub. | 150 UAH | $7.5 ", MOUSEOFF, FGCOLOR, "#FFFFCC",BGCOLOR, "#393939");" onMouseOut="return nd();"> Thesis - 480 rubles, shipping 10 minutes 24 hours a day, seven days a week and holidays

Kholkin Evgeny Gennadievich. Study of the local stability of thin-walled trapezoidal profiles with longitudinal-transverse bending: dissertation ... candidate of technical sciences: 01.02.06 / Kholkin Evgeny Gennadievich; [Place of protection: Ohm. state tech. un-t].- Omsk, 2010.- 118 p.: ill. RSL OD, 61 10-5/3206

Introduction

1. Overview of Stability Studies of Compressed Plate Structural Members 11

1.1. Basic definitions and methods for studying the stability of mechanical systems 12

1.1.1, Algorithm for studying the stability of mechanical systems by the static method 16

1.1.2. static approach. Methods: Euler, nonideality, energetic 17

1.2. Mathematical model and the main results of analytical studies of Euler stability. Stability factor 20

1.3. Methods for studying the stability of plate elements and structures made of them 27

1.4. Engineering methods for calculating plates and composite plate elements. The concept of the reduction method 31

1.5. Numerical studies of Euler stability by the finite element method: opportunities, advantages and disadvantages 37

1.6. Overview of experimental studies of the stability of plates and composite plate elements 40

1.7. Conclusions and tasks of theoretical studies of the stability of thin-walled trapezoidal profiles 44

2. Development of mathematical models and algorithms for calculating the stability of thin-walled plate elements of trapezoidal profiles: 47

2.1. Longitudinal-transverse bending of thin-walled plate elements of trapezoidal profiles 47

2.1.1. Problem statement, main assumptions 48

2.1.2. Mathematical model in ordinary differential equations. Boundary conditions, imperfection method 50

2.1.3. Algorithm for numerical integration, determination of critical

yarn and its implementation in MS Excel 52

2.1.4. Calculation results and their comparison with known solutions 57

2.2. Calculation of critical stresses for an individual plate element

in profile ^..59

2.2.1. A model that takes into account the elastic conjugation of the lamellar profile elements. Basic assumptions and tasks of numerical research 61

2.2.2. Numerical study of the stiffness of conjugations and approximation of the results 63

2.2.3. Numerical study of the buckling half-wavelength at the first critical load and approximation of the results 64

2.2.4. Calculation of the coefficient k(/3x,/32). Approximation of calculation results (A,/?2) 66

2.3. Assessment of the adequacy of calculations by comparison with numerical solutions by the finite element method and known analytical solutions 70

2.4. Conclusions and tasks of the pilot study 80

3. Experimental studies on local stability of thin-walled trapezoidal profiles 82

3.1. Description of prototypes and experimental setup 82

3.2. Sample testing 85

3.2.1. Methodology and content of tests G..85

3.2.2. Compressive test results 92

3.3. Findings 96

4. Accounting for local stability in calculations load-bearing structures from thin-walled trapezoidal profiles with a flat longitudinal - transverse bending 97

4.1. Calculation of critical stresses of local buckling of plate elements and the limiting thickness of a thin-walled trapezoidal profile 98

4.2. Permissible load area without taking into account local buckling 99

4.3. Reduction factor 101

4.4. Accounting for local buckling and reduction 101

Findings 105

Bibliographic list

Introduction to work

The relevance of the work.

Creating light, strong and reliable structures is an urgent task. One of the main requirements in mechanical engineering and construction is the reduction of metal consumption. This leads to the fact that structural elements must be calculated according to more accurate constitutive relations, taking into account the danger of both general and local buckling.

One of the ways to solve the problem of minimizing the weight is the use of high-tech thin-walled trapezoidal rolled profiles (TTP). Profiles are made by rolling thin sheet steel with a thickness of 0.4 ... 1.5 mm in stationary conditions or directly on the assembly site as flat or arched elements. Structures with the use of load-bearing arched coatings made of thin-walled trapezoidal profiles are distinguished by their lightness, aesthetic appearance, ease of installation and a number of other advantages compared to traditional types of coatings.

The main type of profile loading is longitudinal-transverse bending. Tone-

jfflF dMF" plate elements

profiles experiencing
compression in the middle plane
bones may lose space
new stability. local
buckling

Rice. 1. Example of local buckling

Yam,

^J

Rice. 2. Scheme reduced section profile

(MPU) is observed in limited areas along the length of the profile (Fig. 1) at significantly lower loads than the total buckling and stresses commensurate with the allowable ones. With MPU, a separate compressed plate element of the profile completely or partially ceases to perceive the load, which is redistributed between the other plate elements of the profile section. At the same time, in the section where the LPA occurred, the stresses do not necessarily exceed the allowable ones. This phenomenon is called reduction. reduction

is to reduce, in comparison with the real one, the cross-sectional area of ​​​​the profile when reduced to an idealized design scheme (Fig. 2). In this regard, the development and implementation of engineering methods for taking into account the local buckling of plate elements of a thin-walled trapezoidal profile is an urgent task.

Prominent scientists dealt with issues of plate stability: B.M. Broude, F. Bleich, J. Brudka, I.G. Bubnov, V.Z. Vlasov, A.S. Volmir, A.A. Ilyushin, Miles, Melan, Ya.G. Panovko, SP. Timoshenko, Southwell, E. Stowell, Winderberg, Khwalla and others. Engineering approaches to the analysis of critical stresses with local buckling were developed in the works of E.L. Ayrumyan, Burggraf, A.L. Vasilyeva, B.Ya. Volodarsky, M.K. Glouman, Caldwell, V.I. Klimanov, V.G. Krokhaleva, D.V. Martsinkevich, E.A. Pavlinova, A.K. Pertseva, F.F. Tamplona, ​​S.A. Timashev.

In the indicated engineering calculation methods for profiles with a cross section of a complex shape, the danger of MPU is practically not taken into account. At the stage of preliminary design of structures from thin-walled profiles, it is important to have a simple apparatus for assessing the bearing capacity of a particular size. In this regard, there is a need to develop engineering calculation methods that allow, in the process of designing structures from thin-walled profiles, to quickly assess their bearing capacity. The verification calculation of the bearing capacity of a thin-walled profile structure can be performed using refined methods using existing software products and, if necessary, adjusted. Such a two-stage system for calculating the bearing capacity of structures made of thin-walled profiles is the most rational. Therefore, the development and implementation of engineering methods for calculating the bearing capacity of structures made of thin-walled profiles, taking into account the local buckling of plate elements, is an urgent task.

The purpose of the dissertation work: study of local buckling in plate elements of thin-walled trapezoidal profiles during their longitudinal-transverse bending and development of an engineering method for calculating the bearing capacity, taking into account local stability.

To achieve the goal, the following research objectives.

Extension of analytical solutions for the stability of compressed rectangular plates to a system of conjugated plates as part of a profile.

Numerical study of the mathematical model of the local stability of the profile and obtaining adequate analytical expressions for the minimum critical stress of the MPC of the plate element.

Experimental evaluation of the degree of reduction in the section of a thin-walled profile with local buckling.

Development of an engineering technique for the verification and design calculation of a thin-walled profile, taking into account local buckling.

Scientific novelty work is to develop an adequate mathematical model of local buckling for a separate lamellar

element in the composition of the profile and obtaining analytical dependencies for calculating critical stresses.

Validity and reliability the obtained results are provided by basing on fundamental analytical solutions of the problem of stability of rectangular plates, correct application of the mathematical apparatus, sufficient for practical calculations, coincidence with the results of FEM calculations and experimental studies.

Practical significance is to develop an engineering methodology for calculating the bearing capacity of profiles, taking into account local buckling. The results of the work are implemented in LLC "Montazhproekt" in the form of a system of tables and graphical representations of the areas of permissible loads for the entire range of profiles produced, taking into account local buckling, and are used for preliminary selection of the type and thickness of the profile material for specific design solutions and types of loading.

Basic provisions for defense.

Mathematical model of flat bending and compression of a thin-walled profile as a system of conjugated plate elements and a method for determining the critical stresses of the MPU in the sense of Euler on its basis.

Analytical dependencies for calculating the critical stresses of local buckling for each lamellar profile element in a flat longitudinal-transverse bending.

Engineering method for verification and design calculation of a thin-walled trapezoidal profile, taking into account local buckling. Approbation of work and publication.

The main provisions of the dissertation were reported and discussed at scientific and technical conferences of various levels: International Congress "Machines, technologies and processes in construction" dedicated to the 45th anniversary of the faculty "Transport and technological machines" (Omsk, SibADI, December 6-7, 2007); All-Russian scientific and technical conference, "RUSSIA YOUNG: advanced technologies - in industry" (Omsk, Om-GTU, November 12-13, 2008).

Structure and scope of work. The dissertation is presented on 118 pages of text, consists of an introduction, 4 chapters and one appendix, contains 48 figures, 5 tables. The list of references includes 124 titles.

Mathematical model and main results of analytical studies of Euler stability. Stability factor

Any engineering project is based on the solution of differential equations of the mathematical model of motion and equilibrium mechanical system. The drafting of a structure, mechanism, machine is accompanied by some tolerances for manufacturing, in the future - imperfections. Imperfections can also occur during operation in the form of dents, gaps due to wear and other factors. All variants of external influences cannot be foreseen. The design is forced to work under the influence of random perturbing forces, which are not taken into account in the differential equations.

Factors not taken into account in the mathematical model - imperfections, random forces or perturbations can make serious adjustments to the results obtained.

Distinguish between the unperturbed state of the system - the calculated state at zero disturbances, and the perturbed - formed as a result of disturbances.

In one case, due to the perturbation, there is no significant change in the equilibrium position of the structure, or its motion differs little from the calculated one. This state of the mechanical system is called stable. In other cases, the equilibrium position or the nature of the movement differs significantly from the calculated one, such a state is called unstable.

The theory of the stability of motion and equilibrium of mechanical systems is concerned with the establishment of signs that make it possible to judge whether the considered motion or equilibrium will be stable or unstable.

A typical sign of the transition of a system from a stable state to an unstable one is the achievement by some parameter of a value called critical - critical force, critical speed, etc.

The appearance of imperfections or the impact of unaccounted for forces inevitably lead to the motion of the system. Therefore, in the general case, one should investigate the stability of the motion of a mechanical system under perturbations. This approach to the study of stability is called dynamic, and the corresponding research methods are called dynamic.

In practice, it is often enough to confine ourselves to a static approach, i.e. static methods for studying stability. In this case, the end result of the perturbation is investigated - a new established equilibrium position of the mechanical system and the degree of its deviation from the calculated, unperturbed equilibrium position.

The static statement of the problem assumes not to consider the forces of inertia and the time parameter. This formulation of the problem often makes it possible to translate the model from the equations of mathematical physics into ordinary differential equations. This significantly simplifies the mathematical model and facilitates the analytical study of stability.

A positive result of the analysis of equilibrium stability by the static method does not always guarantee dynamic stability. However, for conservative systems, the static approach in determining critical loads and new equilibrium states leads to exactly the same results as the dynamic one.

In a conservative system, the work of the internal and external forces of the system, performed during the transition from one state to another, is determined only by these states and does not depend on the trajectory of motion.

The concept of "system" combines a deformable structure and loads, the behavior of which must be specified. This implies two necessary and sufficient conditions for the conservatism of the system: 1) the elasticity of the deformable structure, i.e. reversibility of deformations; 2) conservatism of the load, i.e. independence of the work done by it from the trajectory. In some cases, the static method gives satisfactory results for non-conservative systems as well.

To illustrate the above, let's consider several examples from theoretical mechanics and strength of materials.

1. A ball of weight Q is in a recess in the support surface (Fig. 1.3). Under the action of the perturbing force 5P Q sina, the equilibrium position of the ball does not change, i.e. it is stable.

With a short-term action of the force 5P Q sina, without taking into account rolling friction, a transition to a new equilibrium position or oscillations around the initial equilibrium position is possible. When friction is taken into account, the oscillatory motion will be damped, that is, stable. The static approach allows to determine only the critical value of the perturbing force, which is equal to: Рcr = Q sina. The nature of the movement when the critical value of the perturbing action is exceeded and the critical duration of the action can be analyzed only by dynamic methods.

2. The rod is long / compressed by the force P (Fig. 1.4). From the strength of materials based on the static method, it is known that under loading within the limits of elasticity, there is a critical value of the compressive force.

The solution of the same problem with a follower force, the direction of which coincides with the direction of the tangent at the point of application, by the static method leads to the conclusion about the absolute stability of the rectilinear form of equilibrium.

Mathematical model in ordinary differential equations. Boundary conditions, imperfection method

Engineering analysis is divided into two categories: classical and numerical methods. Using classical methods, they try to solve the problems of distribution of stress and strain fields directly, forming systems of differential equations based on fundamental principles. An exact solution, if it is possible to obtain equations in a closed form, is possible only for the simplest cases of geometry, loads and boundary conditions. A fairly wide range of classical problems can be solved using approximate solutions to systems of differential equations. These solutions take the form of series in which the lower terms are discarded after convergence has been examined. Like exact solutions, approximate ones require a regular geometric shape, simple boundary conditions, and convenient application of loads. Accordingly, these solutions cannot be applied to most practical problems. The principal advantage of classical methods is that they provide a deep understanding of the problem under study. With the help of numerical methods, a wider range of problems can be investigated. Numerical methods include: 1) energy method; 2) method of boundary elements; 3) finite difference method; 4) finite element method.

Energy methods make it possible to find the minimum expression for the total potential energy of a structure over the entire given area. This approach only works well for certain tasks.

The boundary element method approximates the functions that satisfy the system of differential equations being solved, but not the boundary conditions. The dimension of the problem is reduced because the elements represent only the boundaries of the modeled area. However, the application of this method requires knowledge of the fundamental solution of the system of equations, which can be difficult to obtain.

The finite difference method transforms the system of differential equations and boundary conditions into the corresponding system of algebraic -equations. This method allows solving problems of analysis of structures with complex geometry, boundary conditions and combined loads. However, the finite difference method often turns out to be too slow due to the fact that the requirement for a regular grid over the entire study area leads to systems of equations of very high orders.

The finite element method can be extended to an almost unlimited class of problems due to the fact that it allows using elements of simple and various forms to get splits. The sizes of the finite elements that can be combined to obtain an approximation to any irregular boundaries in the partition sometimes differ by dozens of times. It is allowed to apply an arbitrary type of load to the elements of the model, as well as to impose any type of fastening on them. The main problem is the increase in costs to obtain results. One has to pay for the generality of the solution with the loss of intuition, since a finite element solution is, in fact, a set of numbers that are applicable only to a specific problem posed using a finite element model. Changing any significant aspect of the model usually requires a complete re-solving of the problem. However, this is not a significant cost, since the finite element method is often the only possible way her decisions. The method is applicable to all classes of field distribution problems, which include structural analysis, heat transfer, fluid flow, and electromagnetism. The disadvantages of numerical methods include: 1) the high cost of finite element analysis programs; 2) long training to work with the program and the possibility of full-fledged work only for highly qualified personnel; 3) quite often it is impossible to check the correctness of the result of the solution obtained by the finite element method by means of a physical experiment, including in nonlinear problems. t Review of experimental studies of the stability of plates and composite plate elements

The profiles currently used for building structures are made from metal sheets with a thickness of 0.5 to 5 mm and are therefore considered thin-walled. Their faces can be either flat or curved.

The main feature of thin-walled profiles is that faces with a high width-to-thickness ratio experience large buckling deformations under loading. A particularly intensive growth of deflections is observed when the magnitude of the stresses acting in the face approaches critical value. There is a loss of local stability, deflections become comparable with the thickness of the face. As a result, the cross section of the profile is strongly distorted.

In the literature on the stability of plates, a special place is occupied by the work of the Russian scientist SP. Timoshenko. He is credited with developing an energy method for solving problems of elastic stability. Using this method, SP. Timoshenko gave a theoretical solution to the problems of stability of plates loaded in the middle plane under different boundary conditions. The theoretical solutions were verified by a series of tests on freely supported plates under uniform compression. Tests confirmed the theory.

Assessment of the adequacy of calculations by comparison with numerical solutions by the finite element method and known analytical solutions

To check the reliability of the obtained results, numerical studies were carried out by the finite element method (FEM). Recently, numerical studies of the FEM have been increasingly used due to objective reasons, such as the lack of test problems, the impossibility of observing all conditions when testing on samples. Numerical methods make it possible to conduct research under "ideal" conditions, have a minimum error, which is practically unrealizable in real tests. Numerical studies were carried out using the ANSYS program.

Numerical studies were carried out with samples: a rectangular plate; U-shaped and trapezoidal profile element, having a longitudinal ridge and without a ridge; profile sheet (Fig. 2.11). We considered samples with a thickness of 0.7; 0.8; 0.9 and 1mm.

To the samples (Fig. 2.11), a uniform compressive load sgsh was applied along the ends, followed by an increase by a step Det. The load corresponding to the local buckling of the flat shape corresponded to the value of the critical compressive stress ccr. Then, according to the formula (2.24), the stability coefficient & (/? i, /? g) was calculated and compared with the value from table 2.

Consider a rectangular plate with a length a = 100 mm and a width 6 = 50 mm, compressed at the ends by a uniform compressive load. In the first case, the plate has a hinged fastening along the contour, in the second - a rigid seal along the side faces and a hinged fastening along the ends (Fig. 2.12).

In the ANSYS program, a uniform compressive load was applied to the end faces, and the critical load, stress, and stability coefficient &(/?],/?2) of the plate were determined. When hinged along the contour, the plate lost stability in the second form (two bulges were observed) (Fig. 2.13). Then the resistance coefficients k,/32) of the plates, found numerically and analytically, were compared. The calculation results are presented in Table 3.

Table 3 shows that the difference between the results of the analytical and numerical solutions was less than 1%. Hence, it was concluded that the proposed stability study algorithm can be used in calculating critical loads for more complex structures.

To extend the proposed method for calculating the local stability of thin-walled profiles to the general case of loading, numerical studies were carried out in the ANSYS program to find out how the nature of the compressive load affects the coefficient k(y). The research results are presented in a graph (Fig. 2.14).

The next step in checking the proposed calculation methodology was the study of a separate element of the profile (Fig. 2.11, b, c). It has a hinged fastening along the contour and is compressed at the ends by a uniform compressive load USZH (Fig. 2.15). The sample was studied for stability in the ANSYS program and according to the proposed method. After that, the results obtained were compared.

When creating a model in the ANSYS program, in order to uniformly distribute the compressive load along the end, a thin-walled profile was placed between two thick plates and a compressive load was applied to them.

The result of the study in the ANSYS program of the U-shaped profile element is shown in Figure 2.16, which shows that, first of all, the loss of local stability occurs at the widest plate.

Permissible load area without taking into account local buckling

For load-bearing structures made of high-tech thin-walled trapezoidal profiles, the calculation is carried out according to the methods of allowable stresses. An engineering method is proposed for taking into account local buckling in the calculation of the bearing capacity of structures made of thin-walled trapezoidal profiles. The technique is implemented in MS Excel, is available for wide application and can serve as the basis for appropriate additions to regulations regarding the calculation of thin-walled profiles. It is built on the basis of research and the obtained analytical dependences for calculating the critical stresses of local buckling of plate elements of a thin-walled trapezoidal profile. The task is divided into three components: 1) determining the minimum thickness of the profile (limiting t \ at which there is no need to take into account local buckling in this type of calculation; 2) determining the area of ​​​​allowable loads of a thin-walled trapezoidal profile, inside which the bearing capacity is provided without local buckling; 3) determination of the range of permissible values ​​NuM, within which the bearing capacity is provided in case of local buckling of one or more plate elements of a thin-walled trapezoidal profile (taking into account the reduction of the profile section).

At the same time, it is considered that the dependence of the bending moment on the longitudinal force M = f (N) for the calculated structure was obtained using the methods of resistance of materials or structural mechanics (Fig. 2.1). The allowable stresses [t] and the yield strength of the material cgt are known, as well as the residual stresses cst in plate elements. In calculations after local loss of stability, the "reduction" method was applied. In case of buckling, 96% of the width of the corresponding plate element is excluded.

Calculation of critical stresses of local buckling of plate elements and limiting thickness of a thin-walled trapezoidal profile A thin-walled trapezoidal profile is divided into a set of plate elements as shown in Fig.4.1. At the same time, the angle of mutual arrangement of neighboring elements does not affect the value of the critical stress of the local

Profile H60-845 CURVED buckling. It is allowed to replace curvilinear corrugations with rectilinear elements. Critical compressive stresses of local buckling in the sense of Euler for an individual /-th plate element of a thin-walled trapezoidal profile with width bt at thickness t, modulus of elasticity of the material E and Poisson's ratio ju in the elastic stage of loading are determined by the formula

The coefficients k(px, P2) and k(v) take into account, respectively, the influence of the rigidity of the adjacent plate elements and the nature of the distribution of compressive stresses over the width of the plate element. The value of the coefficients: k(px, P2) is determined according to Table 2, or calculated by the formula

Normal stresses in a plate element are determined in the central axes by the well-known formula for the resistance of materials. The area of ​​permissible loads without taking into account local buckling (Fig. 4.2) is determined by the expression and is a quadrangle, where J is the moment of inertia of the section of the profile period during bending, F is the sectional area of ​​the profile period, ymax and Umіp are the coordinates of the extreme points of the profile section (Fig. 4.1).

Here, the sectional area of ​​the profile F and the moment of inertia of the section J are calculated for a periodic element of length L, and the longitudinal force iV and the bending moment Mb of the profile refer to L.

The bearing capacity is provided when the curve of actual loads M=f(N) falls within the range of allowable loads minus the area of ​​local buckling (Fig. 4.3). Fig 4.2. Permissible load area without taking into account local buckling

The loss of local stability of one of the shelves leads to its partial exclusion from the perception of workloads - reduction. The degree of reduction is taken into account by the reduction factor

The bearing capacity is provided when the actual load curve falls within the range of permissible loads minus the load area of ​​local buckling. At smaller thicknesses, the line of local buckling reduces the area of ​​permissible loads. Local buckling is not possible if the actual load curve is placed in a reduced area. When the curve of actual loads goes beyond the line of the minimum value of the critical stress of local buckling, it is necessary to rebuild the area of ​​permissible loads, taking into account the reduction of the profile, which is determined by the expression