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Eastern-European Journal of Enterprise Technologies ISSN 1729-3774
APPLIED MEGHAMICS
Викладений алгоритм виршення задач стiйкостi плоског форми згину елементiв спещальних вантажотдйомних механiзмiв у виглядi кругових арок з перерiзами, що мають двi вс симетри. Проттегрована система диференщальних рiвнянь стш-костi елементiв у виглядi кругових арок. Побудован варiанти систем фундамен-тальних функщй диференщальних рiвнянь стiйкостi арок з постшними коефщента-ми. Задачi стiйкостi запропоновано вирi-шувати МГЕ
Ключовi слова: сттк1сть, система диференщальних рiвнянь зi змтними коефщен-тами, фундаментальн функци, МГЕ
Изложен алгоритм решения краевых задач устойчивости плоской формы изгиба элементов специальных грузоподъемных механизмов в виде круговых арок с сечениями, имеющими две оси симметрии. Проинтегрирована система дифференциальных уравнений устойчивости элементов в виде круговых арок. Построены варианты систем фундаментальных функций дифференциальных уравнений устойчивости арок с постоянными коэффициентами. Задачи устойчивости предложено решать МГЭ
Ключевые слова: устойчивость, система дифференциальных уравнений с переменными коэффициентами, фундаментальные функции, МГЭ
UDC 539.3
|DOI: 10.15587/1729-4061.2018.125490|
STABILITY OF
structural ELEMENTS of special LIFTING
mechanisms in the
FORM OF CIRCuLAR
ARCHES
V. Orobey
Doctor of Technical Sciences, Professor* E-mail: v.f.orobey@opu.ua O. Daschenko Doctor of Technical Sciences, Professor* E-mail: daschenko.o.f@opu.ua L. Kolomiets Doctor of Technical Sciences, Professor Department of Standardization, Conformity Assessment and Quality Odessa State Academy of Technical Regulation and Quality Kovalska str., 15, Odessa, Ukraine, 65020 E-mail: leonkolom61@gmail.com O. Lymarenko PhD, Associatet Professor* E-mail: a.m.limarenko@opu.ua *Department of dynamics, durability of machines and resistance of materials Odessa National Polytechnic University Shevchenka ave., 1, Odessa, Ukraine, 65044
1. Introduction
Booms of special lifting machines have the form of circular arches. The use of circular arches is due to the advantages over rectilinear rods in strength and rigidity. In this regard, arch elements of crane structures very often have a large ratio of the axial moments of inertia of cross-sections. In this case, the design meets the requirements of strength and rigidity, but at the same time, there is a risk of lateral-torsional buckling. After buckling, the rod experiences two bends and torsion. Significant cross-section displacements often lead to various accidents.
The phenomenon of buckling can be prevented by calculation. However, this requires appropriate, sufficiently accurate and reliable mathematical models of buckling processes. At present, theoretical developments of stability of the simple bending of the circular arch are rudimentary and do not allow solving important practical problems in the
needed amount. Thus, the problem of creating computational models of stability problems of circular arches is relevant and necessary for practice [1-7].
2. Literature review and problem statement
The problem of stability of the simple bending of rectilinear beams with sections in the form of a narrow strip has been posed as early as the 19th century. Much later, the theory of spatial stability of plane and spatial rods and rod systems has been generalized [1].
The constructed theory could not be used for a long time because the corresponding differential equations had variable coefficients and integration encountered serious mathematical difficulties [2]. There are known solutions to various problems of calculating the curves of rods in the form of circular arches taking into account only bending deformation [3].
© V Orobey, O. Daschenko, L. Kolomiets, O. Lymarenko, 2018
This problem has found the effective resolution only with the advent of a numerical-analytic version of the boundary element method (BEM). This method allows mathematically rigorous and exact solution of boundary value problems for the linear homogeneous and inhomogeneous differential equations with variable coefficients [4, 5].
Various solutions of differential stability equations are accumulated for rectilinear rods, while for circular arches there are no fundamental solution functions for Cauchy problems of stability of the simple bending. The problems of stability of the simple bending of circular arches can be solved by means of professional packages of the finite element method (FEM) such as Ansys, Solid Works, Abaqus, etc. At this time, the FEM is the most common numerical method, has a rather simple algorithm logic and a large number of arithmetic operations [6]. However, the lack of an exact stiffness matrix of the problems of stability of the simple bending of structural elements in the form of circular arches does not allow obtaining accurate and reliable results with an arbitrarily large sampling of the structure. The application of the BEM algorithm compares favorably [7]. It uses an exact system of differential equations of the problem, a mathematically rigorous procedure for constructing its solution, and a very logically simple process of forming a resolving system of linear algebraic equations of the boundary value stability problem [8]. In addition, as shown in [9], the BEM allows obtaining exact values of the problem parameters (forces, displacements, stresses, natural vibration frequencies [10, 11], buckling forces) both at the boundary and within the region. Moreover, the BEM has the simplest algorithm logic among other numerical methods, good convergence of the solution, high stability of arithmetic operations, and a very small accumulation of rounding errors in numerical operations [12]. At the same time, the method is characterized by the simplicity of the algorithm logic [10-12], good convergence of the minimum error of the solution results and high stability.
In this regard, the literature review logically leads to the following formulation of the aim and objectives of the study.
3. The aim and objectives of the study
The aim of this paper is to construct a system of fundamental orthonormal functions for problems of stability of the simple bending of circular arches with sections with two or more axes of symmetry.
To achieve the aim, the following objectives were set:
- to simplify the general differential equations of stability of circular arches with allowance for the symmetry of their sections;
- to obtain the resolving ordinary differential equation of the problems under consideration;
- to construct the systems of fundamental orthonormal functions of the differential equation for the two most important cases of the roots of the characteristic equation;
- to present practical recommendations on the application of the resulting calculated ratios of boundary value problems of stability of arches.
4. Development of software
The system of equations of stability of the simple bending of a circular rod, after taking into account the symmetry of the section, is reduced to the following form [1]:
EIwIV (a)+^ qIV (a) + R
EI raeIV (a)-GId e11 (a)+
MZ (a). R
/ \ EIy
mz (a)-R
7 (a) = 0; (a) = 0,
where EIy - rigidity of the section in the horizontal xOz plane; w(a) - flexural motion of the rod axis along the Oz axis (Fig. 1); EIw - sectorial rigidity of the section under the constrained torsion; R - radius of the axis of the circular rod; 0(a) - angle of torsion of the section around the Ox axis; MZ(a) - bending moment in the section caused by a given transverse load; GId - rigidity of the section under torsion; a - angular coordinate of the current section.
Fig. 1. Design scheme of the problem of stability of a circular rod
It can be seen that the system (1) has variable coefficients in the form of the bending moment MZ(a). Considering that this is generally a set of simple functions, the difficulties that will be encountered in the integration of this system become obvious.
The problem can be substantially simplified if we use the numerical-analytical version of the BEM [2-5].
In this method, it is necessary to have a solution of the Cauchy problem for equations (1), but with constant coefficients. We now describe the procedure for integrating the simplified system of equations. The initial parameters of the constrained torsion and bending in the horizontal plane are as follows:
GIde(0) - torsion angle, kNm2;
GIdeI (0) - derivative of the torsion angle, kNm;
m
(0) - bimoment, kNm2;
Bra(o)=- Gf e"
k =
tion, —; m
flexural-torsional characteristic of the
Mœ(0) = -
GI
(0) - flexural-torsional moment, kNm;
motion of the section towards the Oz axis,
EIyW (0) kNm3;
EIyw' (0)= EI^(0) kNm2;
EIyw''(0)=-My (0) plane, kNm;
EIyw"'(0)—Qz (0) plane, kN.
These initial parameters and the system of equations (1) form the Cauchy problem of stability of the plane of the bending shape of the circular rod. To form fundamental solutions of the Cauchy problem, we perform a number of transformations.
angle of rotation of the section, bending moment in the horizontal transverse force in the horizontal
sec-
From the second equation of the system (1), it follows that (MZ = const):
The general solution of the equation (5) can be written in the form:
w"(a) =
1
my-
R
[-EImQlV (a) + GIdQn (a)]. (2)
0(a) = C1 + C2 a + C3ch a a + + C4sh a a + C5 ■ cos b a + C6sin b a,
(9)
By double integration of this expression, we obtain a connection between the flexural motion w(a) and the torsion angle 8(a):
w (a) =
1
M7 - Ely 7 R
[-EIX (a) + GId 0(a)] +
+ (A a + B)
1
M - ^
R
where the integration constants are equal to EI
B =
A =
M7 —^ 7 R
EI
»
-EIm011 (0)-GId 0(0);
M7 -
R
-(0) + EI m0 m (0)-GId 0'(O).
- z., q^s+z0 0IV+Zo = 0,
1 (a) 2 (a) 3 (a) '
where
J-z2 +<J z; + 4 Z1Z3 lz2 + ^ z2 + 4 Z1Z3
-2z1
2z1
(10)
(3)
(4)
(5)
By five-time differentiation of the expression (9), taking into account the ratios between the initial parameters and expression (3), we can form a system of linear algebraic equations for the integration constants C1-C2:
If we substitute w"(a) from (2) into the first equation of the system (1), we obtain the resolving differential equation of stability of the simple bending the circular rod:
f 0(0)
0'(o)
f 1 1 1 \ f C1 Ï Bm(0)k2
1 a b C2 GId
a2 -b2 C3 Mm(0)k2
a3 -b3 C4 GId
A53 A55 C5 M y (0)
v A64 A66 y J GIy Q. (0) EI,,
, (11)
where
z = EIy EIw
Z1 = f EI
M7--y
x 7 R
z3 = M7 - ÇLl .
3 7 R
= EIy GId
Z2 ~f EI
M7--y
, 7 R
EL. ;
R
(6)
The equation (5) is classified as the sixth-order linear homogeneous differential equation with constant coefficients. Its solution can be obtained according to the standard scheme. The characteristic equation for (5) has the form:
(-z1 )t6 + z2t4 + z1t2 = 0.
where elements of the coefficient matrix of the equation (11) have the form:
A53 =
a2(-EIma2 + GId) A _-b2(mb2 + GId)
; A55 =
M7 - ^
7 R
MZ - ^ 7 R
A64 =
_a3 (-EIma2 + GId) . =-b3 (EI mb2 + GId )
EI ' A66 = EI ■
M--y- M--y
(12)
R
R
(7)
The integration constants after solving the system of the equations (11) are written in the form:
Its roots are of various kinds. Consider the two most important combinations of the roots. First case.
t1,2 = 0 - valid multiples;
C1 = 0(0)
C2 = 0/M
a2 + b2 My(0)
x1b2 x a2 EIy
a2 + b2 Qz(0)
x1b2 - X2a2 EIy
x, + x2 B
x1b2 x a2 _ GId _
x nx2 Mw(0)k2 "
x1b2 -x2a2 GId
- -sjz^ + 4z12
-2Z1
two valid roots;
= ±ij—
^ + 4 Z1Z
2z1
two imaginary roots.
(8)
C=
b2
C4 =
b2
M
(0)
EI
B
GI
a (x1b2 - x2a2)
Qz(0)
EI,,
a (x1b2 - x2a2
M
<0)k2
GL
C5 = 2 2
M
EI
B
GI
(13)
a=
a /
+
x
w
¿3,4 =±
x
w
where the following are denoted:
_ a2 (-EIma2 + GId)
xi _ EI '
M7--y
Z R
b (EI b + GId)
X2_ ET ■
Mz -7 R
(14)
1a - A13 - A14 - A17 - A18 ^
1 - A23 A24 - A27 - A28
A33 A34 A37 A38
A43 A44 A47 A48
- A53 - A54 1a - A57 - A58
A63 - A64 1 - A67 - A68
A73 A74 A77 A78
A83 A84 A87 A88 )
(15)
-(a + b )c + b —1 chaa + a —2cosba A _ v ' a_a2_; C _ GI d
_ — b2 — —a2 ' ~
-ab(a2 + b2 )ca + b3 —2 shaa + a3 —2sin ba
^ ' /72 /72
Mr - ^ z R
A4 _"
ab (x1b2 -j
The constants C1-C6 are substituted into the expression for the torsion angle 0(a) (9) and then four bending parameters (using the expression (3)) and four parameters of the constrained torsion relative to the corresponding initial parameters can be formed. After rationing of the fundamental functions, it is convenient to present these expressions in the matrix form as follows:
EIyW (a)
EIy j(0) My (a) Q? (a) GId 0(a)
GId 0'(a) B» M,„ (a)
A17 _"
k2 (x1 + x2 )c - k2x2 —1 chaa + k2x1 —2 cos ba + (b2 - —2a2 )c ei
GI
^18 _"
k2ab(x1 + x2)ca-k2■ x2b—2shaa-k2ax1 —2sinba+ab(x1b2-x2a2)ca ei
ab (x1b2 - —a"
GId
a __a
A23 _
x1b —1 shaa - x2a sin ba
ab (x1b2 - —a
; A24 _ A3; A34 _ A23;
-k2xxbshaa + k2——a sin ba EIy A —_1 2_1 2_ _4 — 4 •
— . 1 —)j- ? ^44 _ 33>
ab(x1b2 - —a"
GI
. _ x1b2chaa - x2a2cos ba; A _ [x1x2b2 (chaa- cos ba)]k2 EIy;
A33 _ ¡"2 2 ; A37 _ ¡"2 2 ;
x1b - x2a x1b - x2a GId
_ xxab2shaa + x2a2sin ba; A _-x1x2b2 (ashaa + bsin ba)k2 EIy ;
A43 _ ¡"2 2 ; A47 _ ¡"2 2 ;
x1b - x2a x1b - x2a GId
A28 _ A7; A38 _ A27; A48 _ A37; A53 _
-b2 (1 - chaa)- a2 (1 - cos ba) GId
EIy
-b3 (aa- shaa)- a3 (ba- sin ba) GId
A54_ .. EI";
aa(x1b2 - x2a2)
Elyw (0)
EIy j(0)
My (0)
Qz (0) GId 0(0) GId 0'(0) Bm(0)
Mw(0) ,
From this expression, it follows that when solving the problems of stability of circular arches by the BEM, it is necessary to solve only eight equations, with an error of less than 1 % [11]. According to the FEM, as the experiment shows [12], it will be required to derive a thousand equations, with an error of 5 % or more.
The fundamental orthonormal functions of the equation (15) take the form:
[x2 (1 - chaa) + x1 (1 - cos ba)k2
A57 _ x b2 - X (2 '
_ [bx2 (aa-shaa) + ax1 (1 - cos ba)] k2 _ ab2shaa-a2b sin ba GId ;
A™ _ ~ ~ A ; A63 _ T2 2 ;
"21 x1b - x2a EI
A _ A • A _
^64 _ 53' 67 _
ib (x1b2 -: (- x2ashaa + x1b sin ba)k2
A — A • A —
68 57 73
a2b2 (chaa- cos ba)k2 GId
x1b2 - x2a2 k2EIy '
A63 _ _ -x2a2chaa + x1b2 cos ba _
"F" "
A A -
A74 r,2 ; A77 ^ ,2 2
A _ A67 a _ a3b2shaa + a2b3 sin ba GId A78 _ -pr"' A«3 _ x1b2 -x2a2 k2^;
- x2a3.shaa- x1b3sin ba
A84 _ A73; A87 _ T2 2 ; A88 _ A77.
X
The expression (15) is the resolving equation of the BEM for solving boundary value problems of stability of the simple bending of structures in the form of individual arches, rings, ring systems, and combined arch systems.
Second case.
The roots are valid multiple and imaginary:
r4 + s4 > 0; s4 < 0; r4 < 0.
b1 = V-r2 -Vr4 + s4; b2 = \/-r2 Wr4 + s4;
„2 _ z2 . .4 _ Z3 .
r —; s — — ;
2z1 Z1
= ElyEIw =
Z1 = EI ' Z2 =
Mr--y
z R
za— |Mz-
z R
EIyGId EL
Mr - EIy z R
R
(17)
A3 = "
b22 - bf ) c - b22 —2 cos b1a - bf cos b2a
bi_b_;
—1b22 - —2b2 '
A4 =
b1b2 (b22 - b2 ) ca - b23 —3 sin b1a + bf —fsin b2a _5_b_;
—3b23 - —4b3 ;
A7
-(—2 - —1 )c+—2 T^cos b1a -
b2
- —1 cos b2a
b2
k +(—1b22 - —2b2 )c
— b - — b
12 ^2^1
Ely
GL
^18
-(b1 —4 - b2—3 )ca + + —4 —Min b1a- —3 —rsin b2a
4 b3 b3
k2 +(—3b23 - —4b13 )c
—1b2 — 2b1
Ely
G/,,
The general solution of the equation (5) takes the form:
0 (a) = C1 + C2a + C3 ■ cos b1a + + C4 sin b1a + C5 cos b2a + C6 sin b2a.
(18)
The integration constants, expressed through the initial parameters of the equation (11) for this case, have the form:
C1 = 0(o) +
b2 - b2 " My(0)l —2 - —1 Bw(0)k2
—b2 -—2b2 EIy —b - — 2b2 GId
A3 =
A4 =
A27 =
b22 — sin b1a - b2 — sin b2a b1 ^ b2 ;
b1b2 ( b22 - b2 ) c - b23 —3 cos b1a + b13 —4 cos b2a _b_b_;
—3b23 - —4b3 ;
-—2 — sin b1a + —1 —2 sin b2a b1 b2
—1b22 - —2b2
EE. ;
GI/
C2 = 0(O) + "
C =__b2_
°3 —1b22 - —2b2
bb (t ^ - b2 ) Gz(0) b1 —1 b2 — 3
—3b2 - —4b3 L EIy \ —3b3 - —4b3
M (0)k2 w(0)
GI,
M
y(o)
EI,,
—1b22 - —2b2
B tJ2 w(0)
GI,
A0Q —
A33 =
- (b1—4 - b2—3 ) c+—4 —3 cos b1a - —3 —4 cos b2a b1 b2
—3b23 - —4b1 b22 —1 cos b1a - b2—2 cos b2a _
EE
GI,
b3
C =__b2_
°4 ~ .. U3 „ Î.3
b12
Q
'z(O)
EI,,
—3b23 - —4b3
C5 b2 b2
C b3
C —3b23 - —4b3
M
y(o)
EI
Oz(o) EI,,
Mm(0)k2 " GI,
Brn(0)k2
—3b23 - —4b3
GI
M (0)k2 w(0)
GI
b2 (EI mb2 + GId) ^ = b22 ( + GIt
Mr - EE-
z R
Mz - EE z R
(19)
The fundamental orthonormal functions of the equation (15) after all transformations are written in the form:
b23 —3 sin b1a - b3 —4 sin b2a A _b2 ;
A34 = —3b23 - —b '
A37 =
A38 =
A43 —
A44 —
A47 =
[-x^cosb1a + —1—2cosb2a]k2 EIy
—1b22 - —b
-—4 —3 sin fea + — 3 —4 sin b2a b1 b2
GI/
EI
GI
—3b23 - —4b3 -b1b22 —1 sin b1a + b12b2—2 sin b2a
b23 —3 cos b1a - b13 —4 cos b2a —3b23 - —14b13
[1x2b1 sinb1a - ——2b2 sin b2a]k2 EIy —1b22 - —2b12 GId
+
X
X
4
X
3
A,o =
- x3 x4 ^cos ha + x3 x4 ^cos b2a b h
xh - x4h3
=
b22 (1 - cos b1a)- bf (1 - cos b2a) GId
£I„'
A, =
b23 (b1a - sin b1a) - b3 (b2a - sin b2a) GId
x3b23 - x4b13
EI..'
A, =
[-x2 (1 - cos b1a) + x1 (1 - cos b2a)] k2 xtf - x2b12 ;
A58 = ^63 : ^64 = ^67 =
^68 = A73 =
A74 = A77 =
A78 = A83 =
A* =
[-x4 (b1a - sin b1a) + x3 (b2a - sin b2a)] k2
x3b23 - x4b3 ''
b1b22 sin b1a - b12b2 sin b2a GId
b1b23 (1 - cos b1a) - b13b2 (1 - cos b2a) GId
x3% - xh EIy'
[-x2b1 sin b1a + x1b2 sin b2a]k2
x^2 - x2h2 ;
[-x4b1 (1 - cos b1a) + x3b2 (1 - cos b2a)] k2
xh - x b! ;
b12b22cos b1a- b12b22cos b2a GId (x1b22 - x2b12)k2 EIy'
bh sin b1a - b13b22 sin b2a GId
k (23 x4b!
EI
A=
A=
c=-
- x2b3 cos b1a + x1b22 cos b2a
x1b22 - x2b12 ''
- x4b12 sin bxa + x3b22 sin b2a
x3b23 - x4b13
-b13b23sin b1a+ b12b23sin b2a GId k2 (x1b22 - x2b12) EIy'
b13b23 cos b1a - b13b23 cos b2a GId _
k2 ( -x4b3 ) EIy''
x2b13 sin b1a - x1b23 sin b2a
(x3b2 - x2b12 )
_ -x4b13 cos ^a + x3b23 sin b2a _
x3b3 - x4b13
GI
M, - ^ z £
(20)
These fundamental functions, as well as the expressions (16), serve as the initial mathematical model of stability problems of circular arches.
5. Discussion of the proposed approach to solving stability problems
5. 1. The case when MZ=const
This case for circular arches is very rare and is possible only with the hinge support and loading by concentrated equal bending moments. In this case, equation (15) can be used directly for the entire structure using the BEM algorithm [2-6].
5. 2. The case when MZ is some function of the angular coordinate a
This is the most common case for arch structures. Here it is necessary to have an analytical expression for the MZ(a) function. This function can be constructed most simply by the BEM algorithm [5, 6], where the procedure for calculating the MZ(a) function from the existing loads is described exhaustively. Then the arch is broken into n parts [7-9]. In each part, the values of the bending moment MZ are calculated from the known expression so that the area of the step figure MZ is equal to the area of the valid plot MZ. If this condition is met, then for n > 30 almost exact results of critical loads Mcr, Fcr, qcr are obtained [10-12].
It should be noted that the conducted studies have removed the problems of mathematical modeling of very complex problems of stability of structural elements of lifting machines.
6. Conclusions
1. When solving the problems of stability of the simple bending of the arch by the FEM, it is necessary to solve about 1,000 linear algebraic equations. The error of the solution will be about 5 %. To solve the problems of stability of arches by the BEM, it will be required to solve only eight equations and the error of the results will be less than 1 %.
2. The simplified system of differential equations of problems of stability of the simple bending of rods in the form of circular arches with variable coefficients is presented. Horizontal motions and angles of torsion of the axis of circular arches serve as unknowns.
3. The sixth-order ordinary differential equation with constant coefficients for the considered stability problems and use of the BEM technology is derived. The resulting equation allows constructing an exact analytical solution of the problems of stability of circular arches according to the known theory.
4. The matrix equation of boundary value problems of stability of the simple bending of circular arches by the BEM is formed. This equation makes it possible to substantially simplify the logic of solving stability problems and obtain exact values of critical loads.
The analysis of the presented material shows that in the framework of the algorithm of the numerical-analytical version of the BEM it is possible to construct the resolving equation of stability problems of the simple bending of circular rods. This equation can be applied to the solution of very complex problems of stability of various structures containing rods, outlined along the circle arch.
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