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УДК 621.88-21-026.564-034.14
ANALYSIS OF STABILITY OF STEEL MEMBERS WITH ACCOUNT OF THE FIXING CONDITIONS AND SUPPORTS STIFFNESS
V. Kondel, PhD., Assoc. Prof., D. Loboda, Student, Poltava V.G. Korolenko National Pedagogical University
Abstract. The authors have studied the stability of compressed steel structures and parts of machines with account of support conditions. The analysis of experimental and theoretical data of the behaviour of compressed structures and their members is provided. The new approaches to the critical-load design with account of supports stiffness are described.
Rey words: compressed steel members, buckling analysis, fixing conditions, supports stiffness.
АНАЛ1З СТ1ЙКОСТ1 СТАЛЕВИХ ЕЛЕМЕНТ1В З УРАХУВАННЯМ УМОВ IX КР1ПЛЕННЯ ТА ЖОРСТКОСТ1 ОПОР
В.М. Кондель, доц., к.т.н., Д.О. Лобода, студ., Полтавський нацюнальний педагопчний ушверситет ¡меш В.Г. Короленка
Анотаця. Робота присвячена до^дженню стткостг сталевих стиснутих елементiв кон-струкцт та деталей машин з урахуванням умов гх кртлення. Запропоновано новi npo^i фор-мули для розрахунюв стиснутих елементiв за критичними навантаженнями з урахуванням жорсткостi опор.
Ключов1 слова: стиснутi сталевi елементи, розрахунки на стiйкiсть, умови кртлення, жор-стюсть опор.
АНАЛИЗ УСТОЙЧИВОСТИ СТАЛЬНЫХ ЭЛЕМЕНТОВ С УЧЕТОМ УСЛОВИЙ ИХ КРЕПЛЕНИЯ И ЖЕСТКОСТИ ОПОР
В.Н. Кондель, доц., к.т.н., Д.А. Лобода, студ., Полтавский национальный педагогический университет имени В.Г. Короленко
Аннотация. Работа посвящена исследованию устойчивости стальных сжатых элементов конструкций и деталей машин с учетом условий их крепления. Предложены новые простые формулы для расчетов сжатых элементов по критическим нагрузкам с учетом жесткости опор.
Ключевые слова: сжатые стальные элементы, расчеты на устойчивость, условия крепления, жесткость опор.
Introduction
Throughout the service life of the components, the members of the structures and parts of machines undergo various damage, the nature of which depends on their stress-strain state and working conditions (tension or compression, bending, torsion, torsional bending, variable load, etc.). To ensure the strength, reliability and durability of these members, numerous investi-
gations are conducted on the processes of accumulation of damage, in particular, the origin and development of cracks in materials from which the structures and parts of machines are made. The main directions of preventing these negative phenomena or reducing their impact on the work of structures and parts of machines in general is to conduct timely and qualitative diagnostics, that is, to determine the technical condition of the object, research and design of rational cross-
sectional forms, analysis of the fixing conditions and supports stiffness.
Among all the members of structures and parts of machines, significant parts are those that work on compression: columns, racks, upper belt trusses, screws in screw-nut transfers, screw springs, and others. They are designated not only for strength, but also for stability, because buckling is a very dangerous phenomenon, resulting in uncontrolled growth of deformation, and the compressed rod is destroyed almost instantly, which excludes any possibility of taking effective measures to prevent accidents and disasters. That is why, for a long time the stability of compressed members has been studied to develop measures to prevent such emergencies in the future. Since the main tasks of technical diagnostics are to control and predict the technical condition of the object, as well as to identify the causes of the violation of the normal operation of the structures compressed members and machines parts, additional studies should be carried out on the effect of the conditions of fastening the ends of the members and the supports stiffness on the values of critical or dangerous loading.
Analysis of Publications
Prominent scientists L. Euler, F. Engesser, F. Yasinsky, L. Tetmayer, T. Karman, F. Shenli and others [1, 2] contributed greatly to solving the problem of the stability of compressed members. They proposed formulas for determining the buckling load and stress that are unsafe for a compressed rod, depending on its flexibility, material, ways of fixing the ends and type of load.
Experiments have shown that taking into account the possible danger of instantaneous destruction of compressed members of structures and parts of machines due to uncontrolled growth of deformations at the marginal loading, these members should be calculated not only on the strength of the condition
but also for stability
F г i A L 1
F
F
n
(1)
(2)
where the coefficient of stock nst is taken depending on the material, the shape of the section, the responsibility of the structure and the member (for steel from 1.8 to 3.5, for iron from 5.0 to 6.0, for wood from 2.8 to 3.2) [1, 2].
When calculating the allowable load on condition (2) it is necessary to find a buckling load Fcr , that is, the least axial force that can withstand a barely curved compressed rod in equilibrium.
For the first time, this problem was solved by L. Euler [1, 2], who proposed a formula for determining the buckling load for a rod with a hinged fix at both ends (Fig. 1)
Fr =
It1 EI„
(3)
lF
ц = 0,5 ц = 0,7
Ц = 1
yffffffff
ц = 1
Fig. 1. Dependence of coefficients ^ from the method of fixing the ends of the rod
Further studies have shown that the value of buckling load Fcr for rods significantly depends on the ways of fixing their ends, so F. Yasinsky proposed the notion of a buckling length ^i [1, 2], therefore formula (3) has the form
F =
ГС1 EIш
(Ц )2
(4)
st
Coefficient of buckling length of the rod ^ ranges from 0.5 to 2 (Fig. 1), which means that, by equation (4), the rod, pinched from both ends (^ = 0,5), can withstand 16 times the load than the same rod, pinched from one end and free from the other (^ = 2).
It should be noted that the above cases of attachment of the ends of the rod (Fig. 1) in
1
l
practice, are rarely found in their pure form. The most widespread cases are the fixation, when one end of the rod is stiffly fixed, and the other is elastically stuck (^ = 0,7) or when both ends are elastically secured (^ = 1) [1, 2].
In order to carry out qualitative technical diagnostics of compressed members of structures and parts of machines, one should consider the impact of the stiffness supports on the value of the coefficient of buckling length ^ and the
magnitude of the buckling load Fcr .
Purpose and Tasks
The purpose of this article is to study the stability of compressed steel members of structures and parts of machines taking into account the conditions for fixing their ends and supports stiffness.
Investigation of Stability of Compressed Steel Members with Account of Fixing Conditions and Supports Stiffness
Let us consider the case where one end of the rod is rigidly fixed, and the other one is elas-tically obturated (Fig. 2). After the buckling, the elastically obturated end of the rack moves vertically to a magnitude f, with the result of an
elastic reaction of the support Rb, which, in
turn, is proportional to the deviation f
Rb = cfb,
2
Ц + k2y = k2fb(1 -4-)-k2 . (7) -2
dxz
У
F„
F„
УГ-0 ^ В FCr : ^
X
I vv> Rb
X
Fig. 2. Compressed rod, pinched from one end and elastically stuck on the other
The solution of this differential equation is dependence
cl cf
y = C sin kx + D cos kx + fb (1--) +—x .(8)
F F
cr cr
The constants integration C and D and the buckling load Fcr will be determined from the following three boundary conditions: when x = 0 displacement and the angle of rotation of the section are zero:
y (0) = Ул = 0;
(9)
where c is coefficient of the stiffness of the support B.
-f = ф(0) = фл = 0;
dx
(10)
We form the differential equation of the elastic line of the compressed rod after the buckling d 2
EImn = Fcr (fb - y) - cfb (l - x). (5) dx
Let us denote
2 F
k2 =- cr
FIm
(6)
Then equation (5) will look
d-Xy=k2(fb -У) - f (l - x)
dx
FI
J-'± min
or
when x = l displacement is equal to deviation
fb (Fig. 2)
y (l ) = Jb = fb . (11)
By the condition (9) we find from equation (8)
D = -fb (1 - ^l ).
Fcr
To use the boundary condition (10) we define the derivative of the displacement y
— = kC cos kx - kD sin kx H--fb
dx Fcr
where when x = 0
c c
kC +—fb = 0 or C =--fb.
F kF b
Fcr =
ГС2 EImil
(0,7/)2
We substitute the expressions for constants integration C and D in the formula (8) and obtain the final equation of the curved axis of the compressed rod
cf c/ y ( x) =--— sin kx + fb (1--)cos kx +
kFcr Fcr
+fb(1 - FT) + #x.
Fcr Fcr
(12)
To determine the buckling load Fcr, we use the boundary condition (11). Thus, for x = l with equation (12) we have
y (l ) = - f sin kl + fb (1 - )cos kl +
kFcr Fcr
or
,(1 - )+fl = fb
c cl --sinkl - (1--)coskl = 0 .
kFcr Fcr
After simple mathematical transformations, we obtain the equation
tgkl = kl (1 - Fcrr- ), cl
(13)
solving which, that is, determining the least root k , we can find the buckling load, since
Fr = k2 EI „
(14)
Let us consider two limiting cases. When c = 0, we get tgki = <x or ki = rc / 2, we have a case where the left end of the rod is stiffly fixed, and the right one is free, then ^ = 2 (Fig. 1) and the buckling load
Fcr =
ГС2 EIm (2l )2
If c = m (very stiff support), we get tgki = ki. Having solved this equation, we have ki = rc / a , where a = 0.699155659643...« 0.7, which gives the formula for the rod, one end of which is pinched, and the other is hinged (Fig. 1), then ^ = 0,7 and the buckling load
Thus, if the coefficient of stiffness of the support c varies from zero to infinity, then this can be taken into account by the coefficient of buckling length ^, which, accordingly, varies from 2 to 0.7. This means that as the stiffness of the support increases, the coefficient ^ decreases together with the stability of the compressed rod increases and the buckling strength value it can withstand [1-5]. We deduce the dependence of the coefficient of buckling length ^ on the stiffness of the support c.
To study, we choose a steel compressed rack with a square cross-section of 5^5 cm. Let us define its length, taking into account that the Euler's formula (4) is applied only to flexible rods in which flexibility X > 100 [1, 2]. Since
X =
ц/
(15)
where /min is the minimum radius of inertia of the square cross-section of the rod
12a2 2л/3 '
hi hd then the length of the rack l = - min =
ц 273Ц
l 100 • 5 2062 l = —¡=-= 206,2 cm.
2^3 • 0.7
For calculations we accept a stable length i = 2,1 m. Then the smallest and most flexible rod respectively makes up
X „!„ = °,699 • ^^ = 101,7 and
X _ = = 291,0.
Stiffness of the rod with buckling
0 054
EImin = 2-105 -= 0,10417 МН м2,
min 12
and buckling load, which will withstand the stability with minimal stiffness, that is, at X min = 101,7 a rate of
4
a
a
F1 =
3,141592 • 0,10417
(0,699 • 2,1)2 = 0,4771МН = 477,1 кН.
Parameter
k =
Fr
'EI m
V
л2 EI m
(pl)2 EImin pl '
3,14159 2140
k =-= 2,140m ,
0,699 • 2,1
and the stiffness of the support c can be determined from the equation (13)
c =
kFr
c =-
kl - tgkl ' 2,140 • 477,1
2,140 • 2,1 - tg (2,140 • 2,1) = 177497,3 KN/m.
Similarly, we find the value of the buckling load Fcr, the parameter k and the stiffness of the support c at various values of the coefficient of buckling length from 0.7 to 2.0 at intervals of 0.1. The results of calculations are presented in the Table, in which pexp and i pth - the experimental and theoretical values of the coefficients of buckling length.
Table Results of calculations of the rod for stability
ber (reducing the coefficient of buckling length p ), it is necessary to increase the stiffness of the support c .
c, KN/m
250 -r-
Fig. 3. Dependence of the stiffness of the support c from the coefficient of buckling length p
This graph has the form of hyperbola, therefore, taking into account the condition that, when c = 0 p = 2 and as well c = oo p = 0,699, as the tabular data of the studies of the stability of the compressed members, a function p = f (c) with sufficient for practical calculations can be accurately presented in the form
p = 0,699 +
1,301
0,025c +1
(16)
where c is the stiffness of the support, which is measured in kN/m.
pexp X F , cr ' KN k, -1 м c, KN/m p* l5l , %
0,699 101,7 477,1 2,140 177497,3 0,699 0,00
0,700 101,8 475,8 2,137 9528,8 0,704 0,57
0,800 116,4 364,3 1,870 232,7 0,890 11,25
0,900 130,9 287,8 1,662 153,0 0,969 7,67
1,000 145,5 233,1 1,496 111,0 1,044 4,40
1,100 160,0 192,7 1,360 83,2 1,121 1,91
1,200 174,6 161,9 1,247 63,2 1,203 0,25
1,300 189,1 137,9 1,151 48,1 1,290 0,77
1,400 203,7 118,9 1,069 36,3 1,381 1,36
1,500 218,2 103,6 0,997 27,0 1,476 1,60
1,600 232,8 91,1 0,935 19,4 1,575 1,56
1,700 247,3 80,7 0,880 13,2 1,677 1,35
1,800 261,9 72,0 0,831 8,06 1,782 1,00
1,900 276,4 64,6 0,787 3,71 1,890 0,53
2,000 291,0 58,3 0,748 0,000161 2,000 0,00
Using equation (16), we construct a graph of the dependence of the coefficient of buckling length of the rod p from the stiffness of the support (Fig. 4).
И
c, кН/м
According to the table data, we construct a plot of the dependence of the stiffness support c on the coefficient p (Fig. 3), which confirms that to increase the stability of the compressed mem-
Fig. 4. Dependence the coefficient of buckling length p from the stiffness of the support c
The curve ^ = f (c), which as a function c = f , takes a form of hyperbola, as evidence of a reduction coefficient ^ (increased stability of the compressed rod) with increasing stiffness of the support. The value of theoretical
coefficients of buckling length ^th, obtained by the formula (16), is not much different from experimental factors ^,exp in Table (average arithmetic deviation 5m = 2,28%, average square deviation - cm = 3,82 %), and if you do not consider the data in the range ^ = 0,8...1,0 , these deviations generally are within one percent (5m = 0,91% and cm = 1,10%). This means that calculating buckling load from the known formula (4), the coefficient of buckling length of the rod ^ can be determined by equation (16), taking into account the stiffness of the support c . It is also important to note that the coefficient ^ significantly influences the values of buckling load, since in the Euler formula (4) it is in the square. This means that in the case when c = oo (^ = 0,7), the stable will withstand load
22/0,72 = 8,2 times more than with c = 0 (^ = 2). The graph ^ = f (c) shows that the increase in the stiffness of the support c from zero to 100 kN/m (Fig. 4) affects more significantly the value of the coefficient ^ increases the
buckling load (22/1,0712 = 3,5times), than 100 to 300 kN/m (1,0712/0,8522 = 1,6times), which should be taken into account when conducting technical diagnostics of compressed rods.
The graph shows that the increase in the stiffness of the support from zero to 100 kN/m (Fig. 4) affects more significantly the value of the coefficient and increase of the buckling load (times), than the increase from 100 to 300 kN/m (times), which should be taken into account when conducting technical diagnostics of compressed rods.
Conclusion
When investigating the compressed members of structures and parts of machines, one should pay attention not only to the conditions of fastening the ends of the rods (Fig. 1), but also the stiffness of the support, which can be in the range from zero to infinity, which is taken into account by the coefficient of buckling length ^ (from 2 to 0.7).
Calculating the buckling load by formula (4), as well as the flexibility of the rod X according to formula (15), the coefficient of buckling length of the rod ц, sufficient for practical calculations can be determined by equation (16), taking into account the stiffness of the support.
To increase the stability of the compressed member, it is necessary to increase the stiffness of the support, taking into account the features of the graph ц = f (c) for reducing the impact on the coefficient ц of significant values of stiffness (Fig. 4).
Data on the conditions for fixing the ends of the rod and stiffness of the supports can be used in further studies of the stability of the members, in particular, in determining the stress reduction factor [6-10], designing compressed members of constant and variable cross-sections [4], rational forms of solid and composite cross-sections for compressed structures and their members [5], etc. These scientific developments will improve the reliability and durability of structural members and parts of machines and will avoid many accidents and disasters in the future.
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Рецензент: В.П. Кожушко, профессор, д.т.н.,
ХНАДУ.
