Stability of the cross-sectional profile during pipe reduction. Development of methods for calculating the deformation-speed modes of hot reduction with tension of pipes of increased accuracy n. Assessment of the adequacy of calculations by comparison with numerical solutions
UDC 621.774.3
STUDY OF THE DYNAMICS OF CHANGES IN THE PIPE WALL THICKNESS DURING REDUCTION
K.Yu. Yakovleva, B.V. Barichko, V.N. Kuznetsov
The results of an experimental study of the dynamics of changes in the wall thickness of pipes during rolling, drawing in monolithic and roller dies are presented. It is shown that with an increase in the degree of deformation, a more intense increase in the thickness of the pipe wall is observed in the processes of rolling and drawing in roller dies, which makes their use promising.
Key words: cold-formed pipes, thick-walled pipes, pipe drawing, pipe wall thickness, pipe inner surface quality.
The existing technology for the manufacture of cold-formed thick-walled pipes of small diameter from corrosion-resistant steels provides for the use of cold rolling processes on cold rolling mills and subsequent mandrelless drawing in monolithic dies. It is known that the production of pipes of small diameter by cold rolling is associated with a number of difficulties due to a decrease in the rigidity of the "rod-mandrel" system. Therefore, to obtain such pipes, a drawing process is used, mainly without a mandrel. The nature of the change in the pipe wall thickness during mandrelless drawing is determined by the ratio of wall thickness S and outer diameter D, and the absolute value of the change does not exceed 0.05-0.08 mm. In this case, wall thickening is observed at the ratio S/D< 0,165-0,20 в зависимости от наружного диаметра заготовки . Для данных соотношений размеров S/D коэффициент вытяжки д при волочении труб из коррозионно-стойкой стали не превышает значения 1,30 , что предопределяет многоцикличность известной технологии и требует привлечения новых способов деформации.
The aim of the work is a comparative experimental study of the dynamics of changes in the wall thickness of pipes in the processes of reduction by rolling, drawing in a monolithic and roller die.
Cold-formed pipes were used as blanks: 12.0x2.0 mm (S/D = 0.176), 10.0x2.10 mm (S/D = 0.216) from steel 08Kh14MF; dimensions 8.0x1.0 mm (S / D = 0.127) from steel 08X18H10T. All pipes were annealed.
Drawing in monolithic dies was carried out on a chain drawing bench with a force of 30 kN. For roller drawing, we used a die with offset pairs of rollers BP-2/2.180. Drawing in a roller die was carried out using an oval-circle gauge system. Pipe reduction by rolling was carried out according to the “oval-oval” calibration scheme in a two-roll stand with rolls 110 mm in diameter.
At each stage of deformation, samples were taken (5 pcs. for each study option) to measure the outer diameter, wall thickness, and roughness of the inner surface. Measurement of the geometric dimensions and surface roughness of the pipes was performed using an electronic caliper TTTC-TT. electronic point micrometer, profilometer Surftest SJ-201. All tools and devices have passed the necessary metrological verification.
The parameters of cold deformation of pipes are given in the table.
On fig. 1 shows graphs of the dependence of the relative increase in wall thickness on the degree of deformation e.
Analysis of the graphs in fig. 1 shows that during rolling and drawing in a roller die, in comparison with the process of drawing in a monolithic die, a more intense change in the thickness of the pipe wall is observed. This, according to the authors, is due to the difference in the scheme of the stress state of the metal: during rolling and roller drawing, the tensile stresses in the deformation zone are smaller. The location of the wall thickness change curve during roller drawing is lower than the wall thickness change curve during rolling due to slightly higher tensile stresses during roller drawing due to the axial application of the deformation force.
The extremum of the function of the change in wall thickness as a function of the degree of deformation or relative reduction along the outer diameter observed during rolling corresponds to the value S/D = 0.30. By analogy with hot reduction by rolling, where a decrease in wall thickness is observed at S/D > 0.35, it can be assumed that cold reduction by rolling is characterized by a decrease in wall thickness at a ratio of S/D > 0.30.
Since one of the factors determining the nature of the change in wall thickness is the ratio of tensile and radial stresses, which in turn depends on the parameters
Pass No. Pipe dimensions, mm S,/D, Si/Sc Di/Do є
Reduction by rolling (pipes made of steel grade 08X14MF)
О 9.98 2.157 О.216 1.О 1.О 1.О О
1 9.52 2.23O 0.234 1.034 0.954 1 .30 80.04
2 8.1O 2.35O O.29O 1.O89 O.812 1.249 O.2O
Z 7.01 2.324 O.332 1.077 O.7O2 1.549 O.35
Reduction by rolling (pipes made of steel grade 08X18H10T)
О 8,О6 1,О2О О,127 1,О 1,О 1,О О
1 7.OZ 1.13O O.161 1.1O8 O.872 1.O77 O.O7
2 6.17 1.225 0.199 1.201 0.766 1.185 0.16
C 5.21 1.310 0.251 1.284 0.646 1.406 0.29
Reducing by drawing in a roller die (pipes made of steel grade 08X14MF)
О 12.ОО 2.11 О.176 1.О 1.О 1.О О
1 10.98 2.20 0.200 1.043 0.915 1.080 0.07
2 1O.O8 2.27 O.225 1.O76 O.84O 1.178 O.15
Z 9.O1 2.3O O.2O1 1.O9O O.751 1.352 O.26
Reducing by drawing in a monolithic die (pipes made of steel grade 08X14MF)
О 12.ОО 2.11О О.176 1.О 1.О 1.О О
1 1O.97 2.135 0.195 1.O12 O.914 1.1O6 O.1O
2 9.98 2.157 O.216 1.O22 O.832 1.118 O.19
C 8.97 2.160 0.241 1.024 0.748 1.147 0.30
Di, Si - respectively, the outer diameter and wall thickness of the pipe in aisle.
Rice. 1. Dependence of the relative increase in pipe wall thickness on the degree of deformation
ra S/D, it is important to study the influence of the S/D ratio on the position of the extremum of the function of changing the pipe wall thickness in the process of reduction. According to the data of the work, at smaller S/D ratios, the maximum value of the pipe wall thickness is observed at large deformations. This fact was studied on the example of the process of reduction by rolling pipes with dimensions of 8.0x1.0 mm (S/D = 0.127) of steel 08Kh18N10T in comparison with the data on rolling pipes with dimensions of 10.0x2.10 mm (S/D = 0.216) of steel 08Kh14MF. The measurement results are shown in fig. 2.
The critical degree of deformation at which the maximum value of the wall thickness was observed during pipe rolling with the ratio
S/D = 0.216 was 0.23. When rolling pipes made of steel 08Kh18N10T, the extremum of the increase in wall thickness was not reached, since the ratio of pipe dimensions S/D, even at the maximum degree of deformation, did not exceed 0.3. An important circumstance is that the dynamics of the increase in wall thickness during the reduction of pipes by rolling is inversely related to the ratio of the dimensions S/D of the original pipe, which is demonstrated by the graphs shown in Fig. 2, a.
Analysis of curves in fig. 2b also shows that the change in the S/D ratio during the rolling of pipes made of steel grade 08Kh18N10T and pipes made of steel grade 08Kh14MF has a similar qualitative character.
S0/A)=0.127 (08X18H10T)
S0/00=0.216 (08X14MF)
Degree of deformation, b
VA=0;216 (08X14MF)
(So/Da=0A21 08X18H10T) _
Degree of deformation, є
Rice. Fig. 2. Changes in wall thickness (a) and S/D ratio (b) depending on the degree of deformation during rolling of pipes with different initial S/D ratios
Rice. Fig. 3. Dependence of the relative value of the roughness of the inner surface of pipes on the degree of deformation
In the process of reduction different ways the roughness of the inner surface of the pipes was also evaluated by the arithmetic mean deviation of the microroughness height Ra. On fig. Figure 3 shows the graphs of the dependence of the relative value of the parameter Ra on the degree of deformation when pipes are reduced by rolling and drawing in monolithic dies
woolness of the inner surface of the pipes in the i-th passage and on the original pipe).
Analysis of curves in fig. 3 shows that in both cases (rolling, drawing) an increase in the degree of deformation during reduction leads to an increase in the parameter Ra, that is, it worsens the quality of the inner surface of the pipes. The dynamics of change (increase) in the roughness parameter with an increase in the degree of deformation in the case of
ducting of pipes by rolling in two-roll calibers significantly (about two times) exceeds the same indicator in the process of drawing in monolithic dies.
It should also be noted that the dynamics of changes in the roughness parameter of the inner surface is consistent with the above description of the dynamics of changes in wall thickness for the considered reduction methods.
Based on the research results, the following conclusions can be drawn:
1. The dynamics of changes in pipe wall thickness for the considered cold reduction methods is the same - intense thickening with an increase in the degree of deformation, subsequent slowdown in the increase in wall thickness with the achievement of a certain maximum value at a certain ratio of pipe dimensions S / D and a subsequent decrease in the increase in wall thickness.
2. The dynamics of changes in the wall thickness of pipes is inversely related to the ratio of the dimensions of the original pipe S/D.
3. The greatest dynamics of the increase in wall thickness is observed in the processes of rolling and drawing in roller dies.
4. An increase in the degree of deformation during reduction by rolling and drawing in monolithic dies leads to a deterioration in the state of the inner surface of the pipes, while the increase in the roughness parameter Ra during rolling occurs more intensively than during drawing. Taking into account the conclusions drawn and the nature of the change in wall thickness during deformation, it can be argued that for drawing pipes in roller dies,
The change in the Ra parameter will be less intense than for rolling, and more intense in comparison with monolithic drawing.
The information obtained about the regularities of the cold reduction process will be useful in designing routes for the manufacture of cold-formed pipes from corrosion-resistant steels. At the same time, the use of the drawing process in roller dies is promising for increasing the thickness of the pipe wall and reducing the number of passes.
Literature
1. Bisk, M.B. cold deformation steel pipes. In 2 hours, Part 1: Preparation for deformation and drawing / M.B. Bisk, I.A. Grekhov, V.B. Slavin. -Sverdlovsk: Mid-Ural. book. publishing house, 1976. - 232 p.
2. Savin, G.A. Pipe drawing / G.A. Savin. -M: Metallurgy, 1993. - 336 p.
3. Shveikin, V.V. Technology of cold rolling and reduction of pipes: textbook. allowance / V.V. Shveikin. - Sverdlovsk: Publishing House of UPI im. CM. Kirova, 1983. - 100 p.
4. Technology and equipment for pipe production /V.Ya. Osadchiy, A.S. Vavilin, V.G. Zimovets and others; ed. V.Ya. Osadchy. - M.: Intermet Engineering, 2007. - 560 p.
5. Barichko, B.V. Basics technological processes OMD: lecture notes / B.V. Barichko, F.S. Dubinsky, V.I. Krainov. - Chelyabinsk: Publishing House of SUSU, 2008. - 131 p.
6. Potapov, I.N. Theory of pipe production: textbook. for universities / I.N. Potapov, A.P. Kolikov, V.M. Druyan. - M.: Metallurgy, 1991. - 424 p.
Yakovleva Ksenia Yuryevna, junior researcher, Russian Research Institute of the Pipe Industry (Chelyabinsk); [email protected]
Barichko Boris Vladimirovich, Deputy Head of the Seamless Pipe Department, Russian Research Institute of the Pipe Industry (Chelyabinsk); [email protected]
Kuznetsov Vladimir Nikolaevich, head of the cold deformation laboratory of the central plant laboratory, Sinarsky Pipe Plant OJSC (Kamensk-Uralsky); [email protected]
Bulletin of the South Ural State University
Series "Metallurgy" ___________2014, vol. 14, no. 1, pp. 101-105
STUDY OF DYNAMIC CHANGES OF THE PIPE WALL THICKNESS IN THE REDUCTION PROCESS
K.Yu. Yakovleva, The Russian Research Institute of the Tube and Pipe Industries (RosNITI), Chelyabinsk, Russian Federation, [email protected],
B.V. Barichko, The Russian Research Institute of the Tube and Pipe Industries (RosNITI), Chelyabinsk, Russian Federation, [email protected],
V.N. Kuznetsov, JSC "Sinarsky Pipe Plant", Kamensk-Uralsky, Russian Federation, [email protected]
The results of the experimental study of dynamic changes for the pipe wall thickness during rolling, drawing both in single-piece and roller dies are described. The results show that with the deformation increasing the faster growth of the pipe wall thiknness is observed in rolling and drawing with the roller dies. The conclusion can be drawn that the usage of roller dies is the most promising one.
Keywords: cold-formed pipes, thick-wall pipes, pipe drawing, pipe wall thickness, quality of the inner surface of pipe.
1. Bisk M.B., Grekhov I.A., Slavin V.B. Kholodnaya deformatsiya stal "nykh trub. Podgotovka k deformatsii i volochenie. Sverdlovsk, Middle Ural Book Publ., 1976, vol. 1. 232 p.
2 Savin G.A. Volochenie tube. Moscow, Metallurgiya Publ., 1993. 336 p.
3. Shveykin V.V. Tekhnologiya kholodnoy prokatki i redutsirovaniya trub. Sverdlovsk, Ural Polytechn. Inst. Publ., 1983. 100 p.
4. Osadchiy V.Ya., Vavilin A.S., Zimovets V.G. et al. Tekhnologiya i obrudovanie trubnogo proizvodstva. Osadchiy V.Ya. (Ed.). Moscow, Intermet Engineering Publ., 2007. 560 p.
5. Barichko B.V., Dubinskiy F.S., Kraynov V.I. Osnovy tekhnologicheskikh protsessov OMD. Chelyabinsk Univ. Publ., 2008. 131 p.
6. Potapov I.N., Kolikov A.P., Druyan V.M. Teoriya trubnogo proizvodstva. Moscow, Metallurgiya Publ., 1991. 424 p.
3.2 Calculation of the rolling table
The basic principle of constructing the technological process in modern installations is to obtain pipes of the same constant diameter on a continuous mill, which allows the use of a billet and a sleeve of also a constant diameter. Obtaining pipes of the required diameter is ensured by reduction. Such a system of work greatly facilitates and simplifies the setting of the mills, reduces the stock of tools and, most importantly, allows you to maintain high productivity of the entire unit even when rolling pipes of a minimum (after reduction) diameter.
We calculate the rolling table against the rolling progress according to the method described in. The outer diameter of the pipe after reduction is determined by the dimensions of the last pair of rolls.
D p 3 \u003d (1.010..1.015) * D o \u003d 1.01 * 33.7 \u003d 34 mm
where D p is the diameter of the finished pipe after the reduction mill.
The wall thickness after continuous and reduction mills must be equal to the wall thickness of the finished pipe, i.e. S n \u003d Sp \u003d S o \u003d 3.2 mm.
Since a pipe of the same diameter comes out after a continuous mill, we take D n \u003d 94 mm. In continuous mills, the calibration of the rolls ensures that in the last pair of rolls the inner diameter of the pipe is 1-2 mm larger than the diameter of the mandrel, so that the diameter of the mandrel will be equal to:
H \u003d d n - (1..2) \u003d D n -2S n -2 \u003d 94-2 * 3.2-2 \u003d 85.6 mm.
We take the diameter of the mandrels equal to 85 mm.
The inner diameter of the sleeve must ensure the free insertion of the mandrel and is taken 5-10 mm larger than the diameter of the mandrel
d g \u003d n + (5..10) \u003d 85 + 10 \u003d 95 mm.
We accept the wall of the sleeve:
S g \u003d S n + (11..14) \u003d 3.2 + 11.8 \u003d 15 mm.
The outer diameter of the sleeves is determined based on the value of the inner diameter and wall thickness:
D g \u003d d g + 2S g \u003d 95 + 2 * 15 \u003d 125 mm.
The diameter of the used workpiece D h =120 mm.
The diameter of the mandrel of the piercing mill is selected taking into account the amount of rolling, i.e. rise in the inner diameter of the sleeve, which is from 3% to 7% of the inner diameter:
P \u003d (0.92 ... 0.97) d g \u003d 0.93 * 95 \u003d 88 mm.
The drawing coefficients for piercing, continuous and reduction mills are determined by the formulas:
,
The overall draw ratio is:
The rolling table for pipes 48.3×4.0 mm and 60.3×5.0 mm in size was calculated in a similar way.
The rolling table is presented in Table. 3.1.
Table 3.1 - TPA-80 rolling table
|
Size of finished pipes, mm |
Workpiece diameter, mm |
Piercing mill |
Continuous mill |
reduction mill |
Overall elongation ratio |
||||||||||
|
Outside diameter |
Wall thickness |
Sleeve size, mm |
Mandrel diameter, mm |
Draw ratio |
Pipe dimensions, mm |
Mandrel diameter, mm |
Draw ratio |
Pipe size, mm |
Number of stands |
Draw ratio |
|||||
|
Wall thickness |
Wall thickness |
Wall thickness |
|||||||||||||
3.3 Calculation of the calibration of the reduction mill rolls
Roll calibration is important integral part calculation of the operating mode of the mill. It largely determines the quality of the pipes, tool life, load distribution in the working stands and the drive.
Roll calibration calculation includes:
distribution of partial deformations in the stands of the mill and calculation of average diameters of calibers;
determination of the dimensions of the rolls.
3.3.1 Partial strain distribution
According to the nature of the change in partial deformations, the stands of the reduction mill can be divided into three groups: the head one at the beginning of the mill, in which the reductions increase intensively during rolling; calibrating (at the end of the mill), in which the deformations are reduced to a minimum value, and a group of stands between them (middle), in which partial deformations are maximum or close to them.
When rolling pipes with tension, the values of partial deformations are taken on the basis of the stability condition of the pipe profile at a plastic tension value that ensures the production of a pipe of a given size.
The coefficient of total plastic tension can be determined by the formula:
,
where 
- axial and tangential strains taken in logarithmic form; T is the value determined in the case of a three-roll caliber by the formula
where (S/D) cp is the average ratio of wall thickness to diameter over the period of pipe deformation in the mill; k-factor taking into account the change in the degree of thickness of the pipe.
,
,
where m is the value of the total deformation of the pipe along the diameter.
.
The value of the critical partial reduction at such a coefficient of plastic tension, according to , can reach 6% in the second stand, 7.5% in the third stand and 10% in the fourth stand. In the first cage, it is recommended to take in the range of 2.5-3%. However, to ensure a stable grip, the amount of compression is generally reduced.
In the pre-finishing and finishing stands of the mill, the reduction is also reduced, but to reduce the load on the rolls and improve the accuracy of the finished pipes. In the last stand of the sizing group, the reduction is taken equal to zero, the penultimate one - up to 0.2 from the reduction in the last stand of the middle group.
AT middle group stands practice uniform and non-uniform distribution of partial deformations. With a uniform distribution of compression in all stands of this group, they are assumed to be constant. The uneven distribution of particular deformations can have several variants and be characterized by the following patterns:
compression in the middle group is proportionally reduced from the first stands to the last - falling mode;
in the first few stands of the middle group, partial deformations are reduced, while the rest are left constant;
compression in the middle group is first increased and then reduced;
in the first few stands of the middle group, partial deformations are left constant, and in the rest they are reduced.
With decreasing deformation modes in the middle group of stands, the differences in the magnitude of the rolling power and the load on the drive decrease, caused by an increase in the resistance to deformation of the metal during rolling, due to a decrease in its temperature and an increase in the strain rate. It is believed that reducing the reduction towards the end of the mill also improves the quality of the outer surface of the pipes and reduces the transverse wall variation.
When calculating the calibration of the rolls, we assume a uniform distribution of reductions.
The values of partial deformations in the stands of the mill are shown in fig. 3.1.
Crimp Distribution
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Ilyashenko A.V. – Associate Professor of the Department of Structural Mechanics
Moscow State Construction University,
candidate of technical sciences
The study of the bearing capacity of compressed thin-walled elastic rods that have an initial deflection and have undergone local buckling is associated with the determination of the reduced cross section rod. The main provisions adopted for the study of the stress-strain state in the supercritical stage of compressed non-ideal thin-walled rods are given in the works. This article discusses the supercritical behavior of rods, which are presented as a set of jointly working elements - plates with an initial perish, simulating the operation of corner, tee and cruciform profiles. These are the so-called shelves-plates with one elastically pinched edge and the other free (see figure). In the works, such a plate is referred to as type II.
It was found that the breaking load, which characterizes the bearing capacity of the rod, significantly exceeds the load P cr (m), at which there is a local buckling of the imperfect profile. From the graphs presented in , it can be seen that the deformations of the longitudinal fibers along the perimeter of the cross section in the supercritical stage become extremely unequal. In fibers far from the ribs, compression strains decrease with increasing load, and at loads close to the limit, due to a sharp curvature of these fibers due to initial bends and ever-increasing arrows of longitudinal half-waves formed after local loss stability, tensile strains appear and grow rapidly.
Sections of the cross section with curved longitudinal fibers release stresses, as if they are switched off from the operation of the rod, weakening the effective section and reducing its rigidity. So, the bearing capacity of a thin-walled profile is not limited to local buckling. The full load, perceived by more rigid (less curved) sections of the cross section, can significantly exceed the value of P cr (m) .
We will obtain an effective, reduced section, excluding non-working sections of the profile. To do this, we use the expression for the stress function Ф k (x, y), which describes the stress state of the kth plate of type II (see).
Let's move on to supercritical stresses σ kx (in the direction of the external compressive force), determined in the most unfavorable section of the rod (x=0). Let's write them down in general view:
σ kx =∂ 2 Ф k (A km ,y, f kj , f koj , β c,d , β c,d,j ,ℓ, s) ∕ ∂ y 2 , (1)
where the integration constants А km (m=1,2,…,6) and the arrows of the acquired deflection components f kj (j=1,2) are determined from the solution of the system of resolving equations . This system of equations includes nonlinear variational equations and boundary conditions that describe the joint operation of non-ideal profile plates. Arrows f koj (j=1,2,…,5) components of the initial deflection k-th plate are determined experimentally for each profile type;
ℓ is the length of the half-wave formed during local buckling;
s is the plate width;
β c,d = cs 2 + dℓ 2 ;
β c,d,j = cs 4 + dl 2 s 2 + gl 4 ;
c, d, j are positive integers.
The reduced or effective width of the reduced section of the plate-shelf (type II) is denoted by s p. To determine it, we write out the conditions for the transition from the actual cross section of the rod to the reduced one:
1. The stresses in the longitudinal fibers at the initial face of the plate (at y=0) adjacent to the rib (see figure) remain the same as those obtained by the nonlinear theory (1):
where F 2 kr =f 2 kr +2f k0r f kr .
To determine the stress σ k2 =σ k max it is necessary to substitute in (1) the ordinate of the most loaded longitudinal fiber, which is found from the condition: ∂σ kx /∂y=0.
2. The sum of internal forces in the plate during the transition to the reduced section in the direction of the compressive force does not change:
3. The moment of internal forces relative to the axis passing through the initial face (y=0) perpendicular to the plane of the plate remains the same:
From the figure, it is obvious that
σ ′ k2 = σ k1 + y p (σ k2 -σ k1) / (y p + s p). (5)
We write down the system of equations for determining the reduced width of the plate s p. To do this, we substitute (1) and (5) into (3) and (4):
where α=πs/ℓ ; F kr,ξ =f kr f koξ +f kr f kξ +f kor f kξ ;
r, ξ are positive integers.
The resulting system of equations (6) and (7) makes it possible to determine the reduced width s p of each of the plates-shelves that make up a compressed thin-walled rod that has undergone local buckling. Thus, the actual cross section of the profile was replaced by a reduced one.
The proposed technique seems to be useful both in theoretical and practical terms when calculating the bearing capacity of compressed pre-curved thin-walled rods, in which local wave formation is permissible according to operational requirements.
Bibliographic list
- Ilyashenko A.V., Efimov I.B. Stress-strain state after local buckling of compressed thin-walled rods, taking into account the initial deflection. Stroitel'nye konstruktsii i materialy. Corrosion protection. - Ufa: Works of in-ta NIIpromstroy, 1981. - P.110-119.
- Ilyashenko A.V. To the calculation of thin-walled tee, angle and cruciform profiles with initial camber // Pile foundations. - Ufa: Sat. scientific tr. Niipromstroy, 1983. - S. 110-122.
- Ilyashenko A.V., Efimov I.B. Experimental study of thin-walled rods with curved lamellar elements // Organization and production construction works. - M .: Tsentr.Buro n.-t. Information of Minpromstroy, 1983.
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 bend: dissertation... candidate of technical sciences: 01.02.06 / Kholkin Evgeniy 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 the calculations of load-bearing structures made of 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 of the reduced section of the 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 an insignificant cost, since the finite element method is often the only possible way to solve it. 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 a separate /-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 quadrilateral, 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