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Розроблено методику оцтки напруже-но-деформованого стану позацентрово роз-тягнутих трубобетонних елементiв суцшь-ного поперечного перерву. Отримано тео-ретинш залежностi. Визначено значення напружень i деформацш в оболонц та бетонному ядрi залежно вiд геометричних характеристик i ф^ико-мехашнних властивостей матерiалiв. Ураховуеться об'емно-напруже-ний стан бетонного осердя
Ключовi слова: трубобетон, напруже-но-деформований стан, позацентровий роз-
тяг, теорiя пружностi, граничний стан □-□
Разработана методика оценки напряженно-деформированного состояния внецен-тренно растянутых трубобетонных элемен -тов сплошного поперечного сечения. Получены теоретические зависимости. Определены значения напряжений i деформаций в оболочке и бетонном ядре в зависимости от геометрических характеристик и физико-механических свойств материалов. Учитывается объемно-напряженное состояние бетонного ядра
Ключевые слова: трубобетон, напряженно-деформированное состояние, внецентрен-ное растяжение, теория упругости, предельное состояние
UDC 624.015.5
IDOI: 10.15587/1729-4061.2017.1068441
DEVELOPMENT OF A PROCEDURE FOR THE EVALUATION OF THE STRESSED-DEFORMED STATE OF PIPE-CONCRETE ELEMENTS THAT ARE STRETCHED OFF-CENTER
D. Yermolen ko
Doctor of Technical Sciences, Professor* E-mail: [email protected] V. Pe n ts PhD, Associate Professor* E-mail: [email protected] G. Golovko PhD, Associate Professor Department of Computer and information technology and systems** E-mail: [email protected] S . M u r z a PhD*
*Department of structures from metal, wood and plastic** **Poltava National Technical Yuri Kondratyuk University Pershotravneva ave., 24, Poltava, Ukraine, 36011
1. Introduction
There are several different stages of the stressed-strained state in the elements with indirect reinforcement, as well as in other structures, from the onset of loading up until destruction. These states are characterized by different magnitude and character of deformations and stresses.
Based on the studies already conducted [1-3], when characterizing the boundary state of the elements with indirect reinforcement, it is possible to argue that at low loads the shell deforms elastically while the concrete starts to exhibit plastic deformations. With an increase in the load, cracks form in concrete, the lateral pressure between the concrete and the shell increases. Upon further increase in loading, longitudinal stresses in the shell reach yield point, and there occur cracks in the concrete core in the planes, perpendicular to the plane of applied effort. Under this state, an element with indirect reinforcement is capable of receiving growing load. The boundary condition is determined by the magnitude of critical deformations. In the case of pipe-concrete, limit deformations imply yield point of the steel shell.
During operation of stretched elements made of pipe-concrete, there may appear conditions under which a structure
works in the elastic or plastic stage. At present, there are estimations of the stressed-strained state of pipe-concrete that works on compression, bending and central elongation; however, evaluation of the stressed-strained state of pipe-concrete on a non-center stretch is lacking. The implementation of effective pipe-concrete structures in building industry necessitates a more detailed study into the features of work under load. Results of the study could be used in designing the structures in which pipe-concrete elements work under stretching, in particular, the through columns, arches, trusses, etc. Thus, the task of the estimation of the stressed-strained state of pipe-concrete that is stretched off-center appears relevant.
2. Literature review and problem statement
It was established by experimental data [4] that in the pipe-concrete elements the concrete core works at stretching jointly with a pipe-shell. In this case, with an increase in the level of concrete load, complex stressed-strained state in the steel and concrete constantly changes [5]. The magnitude of stress is most significantly affected by the reinforcement
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coefficient and mechanical characteristics of materials [1] (deformation module and Poisson ratio). Characteristics E and v are not constant magnitudesn. With an increase in loading, elastic characteristics change magnitude.
Articles [6, 7] described theoretically substantiated procedures for studying the operation of pipe-concrete structures under load. Concrete is characterized by a nonlinear dependence between stresss and deformations at all levels of loading [8]. Curvilinear character of deformation dependence on stresses is explained by the development of fracture of concrete at the micro level [9]. The work of steel that the shell is made of is not confined to the work in the elastic stage as well [10].
Description of the stressed-strained state of structures whose material deforms nonlinearly in a general case runs into significant mathematical complexities [11]. For the elements with indirect reinforcement and, specifically, in pipe-concrete elements, these circumstances are complicated by t e concrete being exposed to the conditions of the triaxial stressed state [12]. This manifests itself regardless of the material of the outer shell [2, 13]. Most discussions addressed the issue of the joint work of the core and the shell at the boundary of contact [3]. In addition, still unresolved up to now is the issue of defining a single criterion for the strength of concrete core in the composition of pipe-concrete [14].
3. Research goal and objectives
The goal of present study is the development of a procedure for evaluating the stressed-deformed state of pipe-concrete elements that are stretched off-center from the onset of loading and up to their reaching the boundary state.
To accomplish the formulated goal, the following tasks have been set:
- based on the apparatus of the theory of mechanics of the deformed body, to develop an algorithm of determining the components of the stressed-deformed state of pipe-concrete elements that are stretched off-center;
- to apply the developed algorithm for determining the bearing capacity of pipe-concrete elements that are stretched off-center and to compare with experimental data.
4. Procedure for the estimation of the stressed-deformed state of pipe-concrete elements at stretching
In order to solve the set task, by employing the mathematical theory of elasticity we obtained a solution for the elastic stage of work of the stretched element, based on paper [12]. The stressed-deformed state of pipe-concrete elements that are stretched off-center is the sum of the stressed states of the element stretched centrally [15] and of the bent element. In this case, the latter works under conditions of pure bending.
The following prerequisites underlie the procedure:
- concrete is considered to be an isotropic material with elastic plasticity;
- a linear dependence is considered to exist between stresses and deformations;
- the hypothesis of flat cross-sections is considered to hold;
- geometrical and physical characteristics of the utilized materials lengthwise of the element are accepted to be constant;
- it is considered that the pipe and the concrete deform jointly;
- a two-digit stress epure in the first variant of the boundary condition on strength of the concrete and the pipe in the compacted and stretched zones has the shape of a triangle, while in the second variant - the shape of a rectangle;
- the static conditions are maintained;
- the shape of the bending of the element axis is accepted to be flat and corresponds to a sinusoid;
- concrete in the stretched zone is not taken into account in the work of the cross section; concrete of the compacted part of the cross-section is under conditions of volumetric compression.
We shall consider the stressed-deformed state of pipe-concrete elements working under pure bending for the two variants of boundary condition by strength.
In the first case, deformations in the most compressed fibers of cross-section reach values that match yield point of steel of the pipe. For this purpose, we shall split the section into three regions (Fig. 1).
dA,
Fig. 1. Estimated diagram of the cross section of a pipe-concrete element that works under bending
In line with the rule of superposition, equally acting internal efforts of the compressed and stretched zones of cross section is the sum of internal efforts of separate regions, and the moments are the sum of internal moments. Mathematical essence of internal effort for each region separately is the volume of some spatial figure. Moment is determined as the product of the corresponding volume Vz on the shoulder (the distance from the center of gravity of the body of volume Vz to the point of coordinate origin).
For section 1 (compressed concrete region), we obtain:
N =
2rV'
1+cos a
(n-a)cos a cos2 a sin a sin3 a
(1)
M ^
1 + cos a
cosasin a n-a sin4a
(2)
3 8 32
For section 2 (compressed pipe-shell region), we obtain: 2r 8,os
N 2
M =
1 + cos a 2r2S s a3'
[(n-a) cos a + sin a] ;
(3)
cos asma-
n + a sin 2a
1 + cos a ^ 2 4
For section 3 (stretched pipe-shell region), we obtain:
(4)
2rsSsasz , . ,
N3 =—^^^ (sm a-acos a); 3 1 - cosa
2r2S,a? (a sin 2a
M3 = "s
1 - cos a I 2
--cos a sin a .
(6)
We shall write the static conditions X Nz = 0, X Mz = 0 in the following form:
X Nz = N + N2 - N3 = 0; X Mz = M4 + M2 + M3 - M = 0.
(7)
(8)
By substituting equations (1), (2) and (3) in (4), we shall obtain:
X Nz = 2rs
1 + cos a
1 + cos a
[(rc - a) cos a + sin a] -
a
1 - cos a
X Mz = 2r;
(sin a-acos a)
= 0;
(9)
1+cos a
+S
a
1+cos a
cos a sin a + -
n + a sin 2a
aS ( a sin 2a
1 - cos a I 2
--cos a sin a
= M,
(10)
where &z is the stress in the stretched zone of the pipe; ab' ,
of are the stresses in the compressed zone of the concrete and the pipe-shell;
(n-a) cos a cos2 a sin a sin3 a
ra. = ----+-+-;
1 2 2 3
cos asin3 a n-a sin 4a
3
8 32
(11)
(12)
a. = n a. .
Fig. 1 shows that osz can be expressed through of in the following way:
(14)
a =
af (1 - cos a) 1 + cos a
(15)
Substitute equation (15) in equations (9) and (10), we shall obtain:
2 r a
2r as
X N =——— \nr ra,+Sncos a] = 0; (5) ^ z 1 + cosaL s 1 s J
X
», 2rs2af ( s n + 2a \ __
m =—I nrra, + S,- = M.
1 + cos a
(16)
(17)
We determine af from equation (17): Ne (1+cos a)
a: =-
2r I nrra, + S
s I s 2 s
rc + 2aV
(18)
We determine angle a from equation (16):
cos a = —
nS,
(19)
In the second case, the cross-section of a pipe-concrete element is divided into three regions, similar to that in the first variant of boundary condition.
For section 1 (compressed concrete region), we obtain:
N = 2r/of
1 + cos3 a^
M, = 2ra; I sin a-
sin4a
(20)
(21)
For section 2 (compressed pipe-shell region), we obtain:
N2 = 2rs Ss a (n-a); (22)
M2 = 2rs2Ssaf sin a. (23)
For section 3 (stretched pipe-shell region), we obtain:
N3 = 2rs S ^a ; (24)
M3 = 2rs2Ssaz sin a. (25)
We substitute equations (20)-(25) in the static conditions (7), (8), and we obtain:
Using the equation of compatibility of deformations, it is possible to write:
X Nz = 2r#
1+cos3 a^
(13) +2rs S s aZ' (n-a)- 2rs S s aZa = 0;
(26)
nr ra
i
E Mz = 2r?cbz
sin 4a
+2rs 8soS' sin a + 2rs 8soS sin a = M;
where
1 + cos3 a
3
ra = sin a--
sin4a
4
where
(27)
(28)
(29)
AT
(»4+4
o: =- N.
3 4
(42)
(43)
By using the equations of compatibility of deformations, as well as in line with Fig. 1, osz, osz', ob can be expressed through osz':
(30)
Thus, we obtained the value of stresses and deformations of the boundary fibers of cross-section at pure bending. The values of stresses and deformations of the appropriate fibers at central stretching is determined by procedure [15]. Complete stressed-deformed state by the transverse cross section of the pipe-concrete element that is stretched off-center is equal to the sum of the stressed-deformed states at axial compression from effort N and pure bend from the bending moment M.
Substitute expression (13) and (30) in equations (26) and (27), we shall obtain:
E N = 2r, o,' [nr, ra3 + 8, (n- 2a)] = 0; E Mz = 2rs2osz' (nrsrai + 28s sin a) = M;
o ' =-
Ne
2r (nrra, + 28 sin a)
" 4 5 4 5 )
(31)
(32)
(33)
In the case of a two-digit epure stresses at off-center stretching, a neutral line slightly changes its position, which is why we consider angle a to remain unchanged until the destruction, and the value of angle a is determined from (19).
Therefore, stresses and deformations of the pipe-concrete elements that are stretched off-center will equal to: a) in the most stretched fiber; - at stage I:
e z = v
- at stage II:
e Z =
b) in the most compressed fiber; - at stage I:
- at stage II:
S , o 7 e V = -§-
5. Comparison of theoretical and experimental data on the pipe-concrete elements that are stretched off-center
In order to compare with the experimental research, we employed pipe-concrete elements from article [1]. We accepted as a theoretical bearing capacity the magnitude ofstretching effort, which matched the occurrence in the most compressed, or stretched, fibers of the estimated cross section of deformation of yield strength of the steel of a pipe-shell. Comparison of the theoretical and experimental bearing capacity is given in Table 1.
(34)
(35)
(36)
(37)
(38)
(39)
(40)
Table 1
Comparison of the experimental and theoretical values of bearing capacity
Series of a sample Eccentricity, mm Bearing capacity, kN Difference, %
Experimental Theoretical
TR-1 50 204 214 -5.1
TR-2 50 144 154 -7.0
TR-3 25 221 241 -9.1
TR-3 50 149 159 -7.0
TR-4 50 112 122 -9.0
TB-11 50 224 244 -9.3
TB-21 50 158 178 -13.2
TB-31 25 243 263 -8.3
TB-31 50 164 174 -6.5
TB-32 50 164 173 -6.0
TB-33 50 164 174 -6.2
TB-41 50 124 134 -8.2
Average, % -7.91
Fig. 2 shows a characteristic diagram of the change in the stressed-deformed state of pipe-concrete elements that are stretched off-center.
s
s
s
s
N, kN
Fig. 2. Comparison of theoretical and experimental dependences of longitudinal and transverse deformations on the magnitude of loading by off-center stretching with an eccentricity of e=50 mm for a sample of the series TB-31
An analysis of the change in the stressed-deformed state of the pipe-concrete elements that are stretched off-center revealed a common character of deformation for all the examined samples. Thus, the character of dependence N-e at the initial stage displays an almost straight shape. Noticeable increase in the longitudinal and transverse deformations from the effort is observed at eaxial=(50-80)^10-5 and £lateral=(10-25>10-5. At this point, the curve N-e has a pronounced curvilinear character. At reaching eaxkl=(200-220)40-5, which corresponds to yield point of steel of the pipe, there is an increase in elateral. In the longitudinal and transverse directions the growth of deformations occurs in proportion, while the dependence is of a curvilinear character. Noteworthy is the fact that almost all the epures of deformation are curvilinear, that is, the hypothesis on the flat cross sections held.
6. Discussion of results of applying the proposed procedure for estimating bearing capacity of the pipe-concrete elements that are stretched off-center
The obtained functional dependences make it possible to establish the magnitudes of stresses and deformations in the pipe-shell and in the concrete of pipe-concrete elements that are stretched off-center depending on geometrical characteristics and physical-mechanical properties of the element materials. The procedure takes into account the elastic stage of work of the materials considering the volumetrically stressed state and an increase in the loading from zero to destruction. The disadvantage of this procedure is that it does not account for the plastic properties of the applied materials. The benefit of the developed procedure is the proposed simplified method
for solving the system of defining equations based on the hypothesis about a joint work of the components of pipe-concrete (steel shell and concrete core) at all stages of loading with an effort that is stretched off-center. This procedure is implemented through a successive solution of separate equations. The existing evaluation procedures of the stressed-deformed state of stretched pipe-concrete elements do not take into account the negative impact of bending moment, which is an additional force factor under conditions of the off-center stretching.
The proposed procedure of evaluation of the stressed-deformed state males it possible to determine with a sufficient accuracy the bearing capacity of pipe-concrete elements that are stretched off-center with different geometrical parameters and mechanical properties. Results of the calculation of bearing capacity satisfactorily match experimental data.
In order to increase the accuracy of theoretical results of the proposed procedure, it is necessary to improve it. Failing to consider plastic properties of steel and concrete has led to theoretical overestimation of the bearing capacity of pipe-concrete elements that are stretched off-center in comparison with experimental data. Further development of the proposed methodology should take into account the development of plastic properties of steel and concrete.
7. Conclusions
1. We developed a procedure for evaluating the stressed-strained state of elements that are stretched off-center, which make it possible to determine the value of bearing capacity, stresses and deformations in the shell and the concrete core depending on geometrical characteristics and physical-mechanical properties of the materials. Its special feature is in the fact that in order to to determine components of the stressed-strained state, the system of equations is solved not by selecting a particular function but rather by consistent solution of separate equations. It is shown that the procedure for determining components of the stressed-deformed state of pipe-concrete elements that are stretched off-center is significantly simplified due to this special feature.
2. We compared experimental values of the bearing capacity of pipe-concrete elements that are stretched off-center with theoretical values that were calculated by the developed procedure. Analysis of the results demonstrates that deviations were equal to 7.9 %. Such a convergence of comparison results confirms the hypothesis about joint work of the steel shell and the concrete core up until the moment when a pipe-concrete element that is stretched off-center reaches maximal bearing capacity.
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