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Method of Calculation of Strengthening of the Loaded Rod Structures Taking into Account Plastic Deformations
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 150 (2016) 1741 – 1747
International Conference on Industrial Engineering, ICIE 2016
Method of Calculation of Strengthening of the Loaded Rod Structures Taking into Account Plastic Deformations M.N. Serazutdinova, M.N. Ubaydulloyeva,* a
Kazan National Research Technological University, K. Marks st., 68, Kazan, 420015, Russia
Abstract Article presents the variation method of calculation of rod systems, strengthened by the increase of the size of the cross-sections of elements, change of the structural scheme and the stress state of the constructions. The methods of accounting the influence on the stress-strain state of repair stresses, the introduction of the new pre-stressed structural elements, changing the calculation scheme of strengthened structures are described. Relations used to calculate the strengthened rod systems with elastic and elasticplastic deformations are shared. © 2016 2016Published The Authors. Published by Elsevier Ltd. © by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016. Peer-review under responsibility of the organizing committee of ICIE 2016 Keywords: rod structures; strengthening; variation method
1. Introduction Many of the existing buildings and facilities need to be repaired and strengthened. The main reasons that cause the need to strengthen the exploited structures are the deterioration and damage of structural elements, the reconstruction of buildings and increasing load on existing construction of buildings. Increase of loads often requires strengthening of structural elements and increase of their load capacity in advance. Various methods are used: the increase of size of the cross sections of elements; introduction of new prestressed structural elements or removal of some existing structural elements.
* Corresponding author. Tel.: +7-927-249-1250. E-mail address: [email protected]
1877-7058 © 2016 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016
doi:10.1016/j.proeng.2016.07.164
1742
M.N. Serazutdinov and M.N. Ubaydulloyev / Procedia Engineering 150 (2016) 1741 – 1747
A representative feature of strengthening and repairs of most buildings is that they are carried out without the complete withdrawal of structures from the stress state [1 - 3]. This factor determines the specifics of calculations and design of structural elements of the strengthened structures. Nowadays, the work of many Russian and foreign scientists are devoted to the questions of research on strengthened load designs [3-11]. Study [1] and methodological recommendations [4, 5] are reviewing the questions of calculation of statically determinate structures, strengthened without withdrawing them from stress state. Please note that these studies were performed taking into account only elastic deformation and can be applied to the structural elements mostly of the simple rectangular cross-sectional shape. Studies [6, 7] are devoted to the development of design methodology of strengthening reinforced concrete structures by changing their structural scheme. Theoretical and experimental studies of reinforced concrete structures strengthened by the combined system are presented in publications [6, 8, 9]. Methods for determination of stress-strain state strengthened under load of metal structures presented in works [2, 10 - 13]. Please note that the works mentioned above do not pay enough attention to studying the effect of some factors that influence the strengthening efficiency designs. These factors include, for example, the presence of initial stresses in structural elements, the occurrence of mounting strengths and stresses min the period of strengthening, redistribution of efforts in strengthening the statically indeterminate structures, change in the balance of strength and elastic characteristics of the main and strengthening materials, the occurrence of elastic-plastic deformations. Therefore, nowadays the development of a universal method for determining the stress-strain state of strengthened structures under stress in order to assess the effectiveness of strengthening taking into account various factors is important. 2. Variational method of calculating rod structures strengthened in the stress state The report describes the variational method for calculating rod systems strengthened using the following methods: the increase of cross-section of elements, changes of the structural scheme and the stress state of structures [14-16]. The results of studies of the effect on the stress-strain state of the strengthened rod systems with elastic and elastic-plastic deformations of repair stress, introduction of new prestressed structural elements which change the calculation scheme of strengthened structures, execution sequence of strengthening stages. The presented mathematical model of the rod and the variational method have great flexibility and allow to calculate rod system, elements of which have, in general, the spatial curved axis with different forms of crosssectional shapes. Some of the cross-sections types are presented on Fig. 1. The main provisions of the method used for calculating the stress-strain state are described in works [14-17]. To calculate the stress and deformation rods theory, based on the model of Timoshenko is used [18].
Fig. 1. The rod and its cross-sections
The expressions for the strains are determined by the formulas
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M.N. Serazutdinov and M.N. Ubaydulloyev / Procedia Engineering 150 (2016) 1741 – 1747
Hx
dI du 1 dI 2 y 3 z ; J xy dx dx dx
du 2 dI I 3 z 1 , J xz dx dx
du 3 dx
I2 y
dI 1 , dx
(1)
where u1 , u2 , u3 , I1 , I2 , I3 - movements and rotation angles of the cross-sections of the rod relative to the coordinate axes Ox , Oy , Oz . In some cases warping of cross-sections occurs during the torsion of rods. In such cases the use of relations (1) leads to incorrect results. To account for the cross-sections warping arising during the torsion of the rod, refined relations for the strain are used. The cases where the cross section of the rod (Fig. 1) has the shape of a rectangle or the rod is thin-walled have been considered. For the cross section in the form of a rectangle
Hx
dI du1 dI d E ( x) y 3 z 2 Id ( y , z ) dx dx dx dx
J xy
du2 dI wI I3 z 1 d E ( x) ; J xz dx dx wy
(2) du3 d I wI I2 y 1 d E ( x) ; J yz wz dx dx
0.
(3)
Where Id - function, which is determined by the solution of the problem of torsion of a rectangular cross-section of the rod using the elasticity relations; E ( ɯ) - warping measure function.
Fig. 2. Thin-walled rod scheme
To determine the VAT of thin rods the key assumptions of Tymoshenko theories and hypotheses adopted in the theory of thin-walled rods taking into account shifts are used. In cross sections of the bar is introduced local coordinate system Ɉxyz, with an axis perpendicular to the cross section plane (Fig. 2). It is believed that the torsion center coincides with the start of Ɉxyz coordinate system, and the unit vectors of the coordinate system. In cross-sections of the rod the local coordinate system Ɉxyz is introduced, with axis Ɉx, perpendicular to the cross section plane (Fig. 2). We believed that the torsion center coincides with the start of Ɉxyz coordinate system, and i , j , k - the unit vectors of the coordinate system. Strain and stress are calculated at a fixed point in the local coordinate system, Myt zt (fig. 2) ) using formulas:
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Hx
J
J xyk t
dI ( x) d E ( ɯ) du1 ( x) dI 2 ( x) Z ( s) , z y 3 dx dx dx dx
(4)
du ( x) du2 ( x ) dI ( x) ty 3 tz 1 U ( y ɦ , z ɦ ) I 2 ( x ) t z I3 ( x ) t y E ( ɯ ) U ( y ɦ , z ɦ ) dɯ dɯ dɯ dI1 ( x) zt . dɯ
(5)
(6)
k Where J - constant wall thickness averaged shift; J xyt - shift during pure torsion, distributed linearly on wall
thickness; U ( y ɦ , z ɦ )
y ɦ t z z ɦ t y – the perpendicular length from the origin Ɉxyz to a line tangent to O1 M ;
s
Z ( y, z ) Z ( s) ³ U ds ; Z ( y, z ) – sectorial profile area; t y
dy / ds, t z
dz / ds - the directing cosines of the
o
axes Ɇ yt , Ɇ zt ; y
y ɦ yt t y zt t z , z
z ɦ yt t z - zt t y .
Calculation of stress-strain state of the strengthened loaded rod structure consists of the following steps: 1) calculation of the stress-strain state of the original rod structure under repair loads; 2) determination of the stress-strain state of the structure after strengthening, under the additional operating loads, taking into account the occurrence of elastic-plastic deformations, changes in cross-section sizes of the rods, the accession of new prestressed elements. During the first stage of calculations the stress-strain state of an existing structure from the effects of the repair load is defined. We assume that the stresses and strains are related by Hooke's law:
Vx
EH x , W xy
GJ xy , W xz
GJ xz .
(7)
Movements of rod structural elements during strengthening period are determined from the variational equation
G U GW
0,
(8)
where the variation of strain energy of the rod system G U and the variation of the work of external forces G W are determined by the following relations:
GU
³ ³³ V xGH x W yGJ xy W zGJ xz dA dl ,
(9)
lC Aɩ
GW
ɪ ɪ ɪ ɪ ɪ ɪ ³ (q1 G u1 q2 G u2 q3 G u3 ) dl ¦ ( F1n G u1 ( xn ) F2 n G u2 ( xi ) F3n G u3 ( xn ))
lq
n
¦ ( M 1pk I1k ( xk ) M 2pk I2 k ( xk ) M 3pk I3k ( xk ))
(10)
k
Where lC , Ⱥɩ – length and cross-sectional areas of rods of a damaged construction during repairs; q1ɪ , q2ɪ , q3ɪ , F1ɪn , F2ɪn , F3ɪn , Ɇ 1ɪk , Ɇ 2ɪk , Ɇ 3ɪk – distributed loads, concentrated forces and moments active in the period of repair (strengthening).
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M.N. Serazutdinov and M.N. Ubaydulloyev / Procedia Engineering 150 (2016) 1741 – 1747
Fig. 3. Prandti’s diagram
At the second stage the stress-strain state of the structure, strengthened by the increase of the size of crosssections of the elements, change of the structural scheme and the stress state of the structure is defined. Calculation of loaded strengthened structure is carried out taking into account the occurrence of elastic-plastic deformations. We believed that in the event of plastic deformation behavior of the material is characterized by Prandtl diagram (Fig. 3). For an ideal elastic-plastic material, relation between the intensity of the stress V i and the intensity of deformation H i while H i d H ɬ determined by Hooke’s law V i EH i , and while H i t H ɬ defined as Vi V ɬ . The intensity of the stresses and strains are determined by expressions
Vi
V x2 3 W xy2 W xz2 , H i
1 4H x2 3 J xy2 J xz2 . 3
(11)
In determining the stress-strain state of strengthened rod structure variational equation (2) can be written as:
G U con G U pl G W
0.
(12)
Where G U con - variation of strain energy rod system in the elastic deformation zone; G U pl - variation of strain energy in the zone of plastic deformation. The expressions for G U con and G U pl are recorded taking into the account the existence of repair loads
V xɪ , W xyɪ , W xzɪ in then structural elements of strengthened construction, changes in the geometric sizes of the cross sections of rods and the accession of new rod elements
G U con
ɪ ɪ ɪ ³ > ³³ V x GH x W xyGJ xy W xzGJ xz dA ³³ V x GH x W xyGJ xy W xzGJ xz dA @ dl
l con
Au
Ap
³ > ³³ V xGH x W xyGJ xy W xzGJ xz dA
l usd
Ausd
(13)
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M.N. Serazutdinov and M.N. Ubaydulloyev / Procedia Engineering 150 (2016) 1741 – 1747 ɪ ɪ ɪ ³ > ³³ V x GH x W xyGJ xy W xzGJ xz dA ³³ V x GH x W xyGJ xy W xzGJ xz dA
G U pl
u Aupr
l pl
p Aupr
³³ V xpl GH x W xypl GJ xy W xzpl GJ xz dA u Aupr
@ dl ³ > ³³ V xGH x W xyGJ xy W xzGJ xz dA usd usd l
(14)
Aupr
pl
³³ V xpl GH x W xypl GJ xy W xzpl GJ xz dA Ausd pl
@ dl
p Where Au – cross-sections area of the rods after strengthening; Aupr – area of the zone of elastic deformation the
u main construction material domain; Acon , Aplu - respectively, the areas of the elastic and plastic deformations of usd cross sections of the main rods after strengthening; l usd , Ausd - length and area of additionally introduced rods; lcon , usd usd l usd - length and area of elastic and elastic-plastic deformation zones of the additional strengthened pl , Aupr , Apl
rods; V xpl , W xypl , W xzpl - normal and shear stresses arising in the plastic deformation of rods domain. The stresses active in the plastic deformation domain are determined by expressions
V xpl
V x / K,
W xypl
W xzpl
W xy / K ,
W xz / K ,
(15)
where K V i / V ɬ . The variation of the work of external forces
GW
ɭ ɭ ɭ ɭ ɭ ɭ ³ (q1 G u1 q2 G u2 q3 G u3 ) dl ¦ ( F1i G u1 ( xi ) F2i G u2 ( xi ) F3i G u3 ( xi )) i
lq
(16)
¦ ( Ɇ 1ɭk I1k ( xk ) Ɇ 2ɭk I2 k ( xk ) Ɇ 3ɭk I3k ( xk )). k
Where q1ɭ , q2ɭ , q3ɭ , F1iɭ , F2ɭi , F3ɭi , Ɇ 1ɭk , Ɇ 2ɭk , Ɇ 3ɭk - loads active on the structure after strengthening. The components of the displacement of the rod points are defined in the global coordinate system 0x y z and expressed as u
^ u1 ( x),
ɬ
u2 ( x), u3 ( x)` ; I
ɬ
ɬ
^ I ( x), I ( x), I ( x)` , E ^ E ( x), 0, 0` . 1
2
3
1
Solving the problem rod system is broken down into N sections, in each of which the components of the displacement vector u , rotation angles I and warping measures E are presented in the form of rows: uk
uki
M
i f m t , Ik Iki ¦ Ckm
m 1
M
i f m t , Ek ¦ Dkm
m 1
Eki
M
i fm t . ¦ Bkm
(17)
m 1
Where function of the form f m t with the number from the row Ɇ = 1, 2, 3, 4,…, m determined by expression f1 t 1 3 t 2 2t 3 , f 2 t m
3t 2 2t 3 , f 3 t
t 1 t 2 l , f4 t
i i i , Dkm , Bkm – unknown constants; t 5, M ; Ckm
x / li
t
3
t 2 l , f m t
0 d t d 1 ;
1 t
2
t ( m 3) , where
x – the length of the longitudinal axis of the
rod, measured from the beginning of the section to the considered point; li – the length of section of the rod; k
1, 2, 3 ; i 1, N .
M.N. Serazutdinov and M.N. Ubaydulloyev / Procedia Engineering 150 (2016) 1741 – 1747
1747
3. Conclusions x Mathematical models of deformation of the loaded rod systems, strengthened by the increase of the size of the cross-sections of elements, change of the structural scheme and the stress state of the structure have been developed. x On the basis of this method a computer program which can be used to determine the stress-strain state rod structures in stress state assuming occurrence of plastic deformations has been created. References [1] A.Yu. Budin, M.V. Chekreneva, Strengthening of port constructions, Transport, 1983. [2] I.S. Rebrov, Strengthening of rod metal structures, Stroyizdat, Leningrad, 1988. [3] S.V. Bondarenko, R.S. Sanzharovsky, Strengthening of ferroconcrete structures at reconstruction of buildings, Stroyizdat, Moscow, 1990. [4] Methodical instructions on increase of the bearing capacity and determination of efficiency of repair and strengthening of port hydraulic engineering constructions, Transport, Leningrad, 1982. [5] Methodical instructions on operation and strengthening of the berthing engineering constructions having local damages, Transport, Leningrad, 1977. [6] I.S. Guchkin, A.V. Pankov, The ferroconcrete beams strengthened by the combined system from steel elements, Regional architecture and building, The Penza state university of arkhitecture and engineering. 2 (2010) 94100. [7] A.I. Tatarenkov, Strength and deformability of the bent ferroconcrete structures strengthened under loading, Ph.D. diss., Orel, 2005. [8] A.V. Pankov, Strength and deformability of the ferroconcrete beams strengthened by the combined system with various design data, Ph.D. diss., Penza, 2011. [9] V.V. Cheryachukin, I.S. Guchkin, A.V. Pankov, Design of strengthening of ferroconcrete rafter beams by the combined system, Vestnik VolGASU, Series: Construction and architecture. 36 (2010) 913. [10] I.K. Rodionov, About some questions of strengthening by method of increase in section of the squeezed cores of steel angular farms, in: Proceeding of Collection of scientific works Science, Equipment and Education of Tolyatti and Volga Region, part 2. (2000) 165169. [11] Y. Cheng, B.W. Schafer, Simulation of cold-formed steel beams in local and distortional bucking with applications to the direct strength method, Journal of Constructional Steel Research. 63 (2007) 581590. [12] J. Heander, Strengthening of a steel railway bridge and its impact on the dynamic response to passing trains, Engineering Structures. 33 (2011) 635–646. [13] L. Gannon, Finite element study of steel beams reinforced while under load, Engineering Structures. 31 (2009) 2630–2647. [14] M.N. Serazutdinov, M.N. Ubaydulloyev, H.A. Abragim, Calculation of the strengthened loaded structures by a variation method, News of higher educational institutions, Construction. 7 (2010) 118124. [15] M.N. Serazutdinov, F.S. Khayrullin, Metod of calculation of curvilinear rods, News of higher educational institutions, Construction and architecture. 5 (1991) 104108. [16] M.N. Ubaydulloyev, M.N. Serazutdinov, Modeling of the intense deformed condition of the strengthened rod systems, Scientifically technical journal Structural Mechanics of Engineering Constructions and Buildings. (2012) 4351. [17] M.N. Serazutdinov, M.F.Garifullin, Shells of complex shape, Soviet Applied Mechanics. 11 (1991) 10771082. [18] S.P. Tymoshenko, Mekhanik of materials, Mir, 1976.
ScienceDirect Procedia Engineering 150 (2016) 1741 – 1747
International Conference on Industrial Engineering, ICIE 2016
Method of Calculation of Strengthening of the Loaded Rod Structures Taking into Account Plastic Deformations M.N. Serazutdinova, M.N. Ubaydulloyeva,* a
Kazan National Research Technological University, K. Marks st., 68, Kazan, 420015, Russia
Abstract Article presents the variation method of calculation of rod systems, strengthened by the increase of the size of the cross-sections of elements, change of the structural scheme and the stress state of the constructions. The methods of accounting the influence on the stress-strain state of repair stresses, the introduction of the new pre-stressed structural elements, changing the calculation scheme of strengthened structures are described. Relations used to calculate the strengthened rod systems with elastic and elasticplastic deformations are shared. © 2016 2016Published The Authors. Published by Elsevier Ltd. © by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016. Peer-review under responsibility of the organizing committee of ICIE 2016 Keywords: rod structures; strengthening; variation method
1. Introduction Many of the existing buildings and facilities need to be repaired and strengthened. The main reasons that cause the need to strengthen the exploited structures are the deterioration and damage of structural elements, the reconstruction of buildings and increasing load on existing construction of buildings. Increase of loads often requires strengthening of structural elements and increase of their load capacity in advance. Various methods are used: the increase of size of the cross sections of elements; introduction of new prestressed structural elements or removal of some existing structural elements.
* Corresponding author. Tel.: +7-927-249-1250. E-mail address: [email protected]
1877-7058 © 2016 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016
doi:10.1016/j.proeng.2016.07.164
1742
M.N. Serazutdinov and M.N. Ubaydulloyev / Procedia Engineering 150 (2016) 1741 – 1747
A representative feature of strengthening and repairs of most buildings is that they are carried out without the complete withdrawal of structures from the stress state [1 - 3]. This factor determines the specifics of calculations and design of structural elements of the strengthened structures. Nowadays, the work of many Russian and foreign scientists are devoted to the questions of research on strengthened load designs [3-11]. Study [1] and methodological recommendations [4, 5] are reviewing the questions of calculation of statically determinate structures, strengthened without withdrawing them from stress state. Please note that these studies were performed taking into account only elastic deformation and can be applied to the structural elements mostly of the simple rectangular cross-sectional shape. Studies [6, 7] are devoted to the development of design methodology of strengthening reinforced concrete structures by changing their structural scheme. Theoretical and experimental studies of reinforced concrete structures strengthened by the combined system are presented in publications [6, 8, 9]. Methods for determination of stress-strain state strengthened under load of metal structures presented in works [2, 10 - 13]. Please note that the works mentioned above do not pay enough attention to studying the effect of some factors that influence the strengthening efficiency designs. These factors include, for example, the presence of initial stresses in structural elements, the occurrence of mounting strengths and stresses min the period of strengthening, redistribution of efforts in strengthening the statically indeterminate structures, change in the balance of strength and elastic characteristics of the main and strengthening materials, the occurrence of elastic-plastic deformations. Therefore, nowadays the development of a universal method for determining the stress-strain state of strengthened structures under stress in order to assess the effectiveness of strengthening taking into account various factors is important. 2. Variational method of calculating rod structures strengthened in the stress state The report describes the variational method for calculating rod systems strengthened using the following methods: the increase of cross-section of elements, changes of the structural scheme and the stress state of structures [14-16]. The results of studies of the effect on the stress-strain state of the strengthened rod systems with elastic and elastic-plastic deformations of repair stress, introduction of new prestressed structural elements which change the calculation scheme of strengthened structures, execution sequence of strengthening stages. The presented mathematical model of the rod and the variational method have great flexibility and allow to calculate rod system, elements of which have, in general, the spatial curved axis with different forms of crosssectional shapes. Some of the cross-sections types are presented on Fig. 1. The main provisions of the method used for calculating the stress-strain state are described in works [14-17]. To calculate the stress and deformation rods theory, based on the model of Timoshenko is used [18].
Fig. 1. The rod and its cross-sections
The expressions for the strains are determined by the formulas
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M.N. Serazutdinov and M.N. Ubaydulloyev / Procedia Engineering 150 (2016) 1741 – 1747
Hx
dI du 1 dI 2 y 3 z ; J xy dx dx dx
du 2 dI I 3 z 1 , J xz dx dx
du 3 dx
I2 y
dI 1 , dx
(1)
where u1 , u2 , u3 , I1 , I2 , I3 - movements and rotation angles of the cross-sections of the rod relative to the coordinate axes Ox , Oy , Oz . In some cases warping of cross-sections occurs during the torsion of rods. In such cases the use of relations (1) leads to incorrect results. To account for the cross-sections warping arising during the torsion of the rod, refined relations for the strain are used. The cases where the cross section of the rod (Fig. 1) has the shape of a rectangle or the rod is thin-walled have been considered. For the cross section in the form of a rectangle
Hx
dI du1 dI d E ( x) y 3 z 2 Id ( y , z ) dx dx dx dx
J xy
du2 dI wI I3 z 1 d E ( x) ; J xz dx dx wy
(2) du3 d I wI I2 y 1 d E ( x) ; J yz wz dx dx
0.
(3)
Where Id - function, which is determined by the solution of the problem of torsion of a rectangular cross-section of the rod using the elasticity relations; E ( ɯ) - warping measure function.
Fig. 2. Thin-walled rod scheme
To determine the VAT of thin rods the key assumptions of Tymoshenko theories and hypotheses adopted in the theory of thin-walled rods taking into account shifts are used. In cross sections of the bar is introduced local coordinate system Ɉxyz, with an axis perpendicular to the cross section plane (Fig. 2). It is believed that the torsion center coincides with the start of Ɉxyz coordinate system, and the unit vectors of the coordinate system. In cross-sections of the rod the local coordinate system Ɉxyz is introduced, with axis Ɉx, perpendicular to the cross section plane (Fig. 2). We believed that the torsion center coincides with the start of Ɉxyz coordinate system, and i , j , k - the unit vectors of the coordinate system. Strain and stress are calculated at a fixed point in the local coordinate system, Myt zt (fig. 2) ) using formulas:
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M.N. Serazutdinov and M.N. Ubaydulloyev / Procedia Engineering 150 (2016) 1741 – 1747
Hx
J
J xyk t
dI ( x) d E ( ɯ) du1 ( x) dI 2 ( x) Z ( s) , z y 3 dx dx dx dx
(4)
du ( x) du2 ( x ) dI ( x) ty 3 tz 1 U ( y ɦ , z ɦ ) I 2 ( x ) t z I3 ( x ) t y E ( ɯ ) U ( y ɦ , z ɦ ) dɯ dɯ dɯ dI1 ( x) zt . dɯ
(5)
(6)
k Where J - constant wall thickness averaged shift; J xyt - shift during pure torsion, distributed linearly on wall
thickness; U ( y ɦ , z ɦ )
y ɦ t z z ɦ t y – the perpendicular length from the origin Ɉxyz to a line tangent to O1 M ;
s
Z ( y, z ) Z ( s) ³ U ds ; Z ( y, z ) – sectorial profile area; t y
dy / ds, t z
dz / ds - the directing cosines of the
o
axes Ɇ yt , Ɇ zt ; y
y ɦ yt t y zt t z , z
z ɦ yt t z - zt t y .
Calculation of stress-strain state of the strengthened loaded rod structure consists of the following steps: 1) calculation of the stress-strain state of the original rod structure under repair loads; 2) determination of the stress-strain state of the structure after strengthening, under the additional operating loads, taking into account the occurrence of elastic-plastic deformations, changes in cross-section sizes of the rods, the accession of new prestressed elements. During the first stage of calculations the stress-strain state of an existing structure from the effects of the repair load is defined. We assume that the stresses and strains are related by Hooke's law:
Vx
EH x , W xy
GJ xy , W xz
GJ xz .
(7)
Movements of rod structural elements during strengthening period are determined from the variational equation
G U GW
0,
(8)
where the variation of strain energy of the rod system G U and the variation of the work of external forces G W are determined by the following relations:
GU
³ ³³ V xGH x W yGJ xy W zGJ xz dA dl ,
(9)
lC Aɩ
GW
ɪ ɪ ɪ ɪ ɪ ɪ ³ (q1 G u1 q2 G u2 q3 G u3 ) dl ¦ ( F1n G u1 ( xn ) F2 n G u2 ( xi ) F3n G u3 ( xn ))
lq
n
¦ ( M 1pk I1k ( xk ) M 2pk I2 k ( xk ) M 3pk I3k ( xk ))
(10)
k
Where lC , Ⱥɩ – length and cross-sectional areas of rods of a damaged construction during repairs; q1ɪ , q2ɪ , q3ɪ , F1ɪn , F2ɪn , F3ɪn , Ɇ 1ɪk , Ɇ 2ɪk , Ɇ 3ɪk – distributed loads, concentrated forces and moments active in the period of repair (strengthening).
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M.N. Serazutdinov and M.N. Ubaydulloyev / Procedia Engineering 150 (2016) 1741 – 1747
Fig. 3. Prandti’s diagram
At the second stage the stress-strain state of the structure, strengthened by the increase of the size of crosssections of the elements, change of the structural scheme and the stress state of the structure is defined. Calculation of loaded strengthened structure is carried out taking into account the occurrence of elastic-plastic deformations. We believed that in the event of plastic deformation behavior of the material is characterized by Prandtl diagram (Fig. 3). For an ideal elastic-plastic material, relation between the intensity of the stress V i and the intensity of deformation H i while H i d H ɬ determined by Hooke’s law V i EH i , and while H i t H ɬ defined as Vi V ɬ . The intensity of the stresses and strains are determined by expressions
Vi
V x2 3 W xy2 W xz2 , H i
1 4H x2 3 J xy2 J xz2 . 3
(11)
In determining the stress-strain state of strengthened rod structure variational equation (2) can be written as:
G U con G U pl G W
0.
(12)
Where G U con - variation of strain energy rod system in the elastic deformation zone; G U pl - variation of strain energy in the zone of plastic deformation. The expressions for G U con and G U pl are recorded taking into the account the existence of repair loads
V xɪ , W xyɪ , W xzɪ in then structural elements of strengthened construction, changes in the geometric sizes of the cross sections of rods and the accession of new rod elements
G U con
ɪ ɪ ɪ ³ > ³³ V x GH x W xyGJ xy W xzGJ xz dA ³³ V x GH x W xyGJ xy W xzGJ xz dA @ dl
l con
Au
Ap
³ > ³³ V xGH x W xyGJ xy W xzGJ xz dA
l usd
Ausd
(13)
1746
M.N. Serazutdinov and M.N. Ubaydulloyev / Procedia Engineering 150 (2016) 1741 – 1747 ɪ ɪ ɪ ³ > ³³ V x GH x W xyGJ xy W xzGJ xz dA ³³ V x GH x W xyGJ xy W xzGJ xz dA
G U pl
u Aupr
l pl
p Aupr
³³ V xpl GH x W xypl GJ xy W xzpl GJ xz dA u Aupr
@ dl ³ > ³³ V xGH x W xyGJ xy W xzGJ xz dA usd usd l
(14)
Aupr
pl
³³ V xpl GH x W xypl GJ xy W xzpl GJ xz dA Ausd pl
@ dl
p Where Au – cross-sections area of the rods after strengthening; Aupr – area of the zone of elastic deformation the
u main construction material domain; Acon , Aplu - respectively, the areas of the elastic and plastic deformations of usd cross sections of the main rods after strengthening; l usd , Ausd - length and area of additionally introduced rods; lcon , usd usd l usd - length and area of elastic and elastic-plastic deformation zones of the additional strengthened pl , Aupr , Apl
rods; V xpl , W xypl , W xzpl - normal and shear stresses arising in the plastic deformation of rods domain. The stresses active in the plastic deformation domain are determined by expressions
V xpl
V x / K,
W xypl
W xzpl
W xy / K ,
W xz / K ,
(15)
where K V i / V ɬ . The variation of the work of external forces
GW
ɭ ɭ ɭ ɭ ɭ ɭ ³ (q1 G u1 q2 G u2 q3 G u3 ) dl ¦ ( F1i G u1 ( xi ) F2i G u2 ( xi ) F3i G u3 ( xi )) i
lq
(16)
¦ ( Ɇ 1ɭk I1k ( xk ) Ɇ 2ɭk I2 k ( xk ) Ɇ 3ɭk I3k ( xk )). k
Where q1ɭ , q2ɭ , q3ɭ , F1iɭ , F2ɭi , F3ɭi , Ɇ 1ɭk , Ɇ 2ɭk , Ɇ 3ɭk - loads active on the structure after strengthening. The components of the displacement of the rod points are defined in the global coordinate system 0x y z and expressed as u
^ u1 ( x),
ɬ
u2 ( x), u3 ( x)` ; I
ɬ
ɬ
^ I ( x), I ( x), I ( x)` , E ^ E ( x), 0, 0` . 1
2
3
1
Solving the problem rod system is broken down into N sections, in each of which the components of the displacement vector u , rotation angles I and warping measures E are presented in the form of rows: uk
uki
M
i f m t , Ik Iki ¦ Ckm
m 1
M
i f m t , Ek ¦ Dkm
m 1
Eki
M
i fm t . ¦ Bkm
(17)
m 1
Where function of the form f m t with the number from the row Ɇ = 1, 2, 3, 4,…, m determined by expression f1 t 1 3 t 2 2t 3 , f 2 t m
3t 2 2t 3 , f 3 t
t 1 t 2 l , f4 t
i i i , Dkm , Bkm – unknown constants; t 5, M ; Ckm
x / li
t
3
t 2 l , f m t
0 d t d 1 ;
1 t
2
t ( m 3) , where
x – the length of the longitudinal axis of the
rod, measured from the beginning of the section to the considered point; li – the length of section of the rod; k
1, 2, 3 ; i 1, N .
M.N. Serazutdinov and M.N. Ubaydulloyev / Procedia Engineering 150 (2016) 1741 – 1747
1747
3. Conclusions x Mathematical models of deformation of the loaded rod systems, strengthened by the increase of the size of the cross-sections of elements, change of the structural scheme and the stress state of the structure have been developed. x On the basis of this method a computer program which can be used to determine the stress-strain state rod structures in stress state assuming occurrence of plastic deformations has been created. References [1] A.Yu. Budin, M.V. Chekreneva, Strengthening of port constructions, Transport, 1983. [2] I.S. Rebrov, Strengthening of rod metal structures, Stroyizdat, Leningrad, 1988. [3] S.V. Bondarenko, R.S. Sanzharovsky, Strengthening of ferroconcrete structures at reconstruction of buildings, Stroyizdat, Moscow, 1990. [4] Methodical instructions on increase of the bearing capacity and determination of efficiency of repair and strengthening of port hydraulic engineering constructions, Transport, Leningrad, 1982. [5] Methodical instructions on operation and strengthening of the berthing engineering constructions having local damages, Transport, Leningrad, 1977. [6] I.S. Guchkin, A.V. Pankov, The ferroconcrete beams strengthened by the combined system from steel elements, Regional architecture and building, The Penza state university of arkhitecture and engineering. 2 (2010) 94100. [7] A.I. Tatarenkov, Strength and deformability of the bent ferroconcrete structures strengthened under loading, Ph.D. diss., Orel, 2005. [8] A.V. Pankov, Strength and deformability of the ferroconcrete beams strengthened by the combined system with various design data, Ph.D. diss., Penza, 2011. [9] V.V. Cheryachukin, I.S. Guchkin, A.V. Pankov, Design of strengthening of ferroconcrete rafter beams by the combined system, Vestnik VolGASU, Series: Construction and architecture. 36 (2010) 913. [10] I.K. Rodionov, About some questions of strengthening by method of increase in section of the squeezed cores of steel angular farms, in: Proceeding of Collection of scientific works Science, Equipment and Education of Tolyatti and Volga Region, part 2. (2000) 165169. [11] Y. Cheng, B.W. Schafer, Simulation of cold-formed steel beams in local and distortional bucking with applications to the direct strength method, Journal of Constructional Steel Research. 63 (2007) 581590. [12] J. Heander, Strengthening of a steel railway bridge and its impact on the dynamic response to passing trains, Engineering Structures. 33 (2011) 635–646. [13] L. Gannon, Finite element study of steel beams reinforced while under load, Engineering Structures. 31 (2009) 2630–2647. [14] M.N. Serazutdinov, M.N. Ubaydulloyev, H.A. Abragim, Calculation of the strengthened loaded structures by a variation method, News of higher educational institutions, Construction. 7 (2010) 118124. [15] M.N. Serazutdinov, F.S. Khayrullin, Metod of calculation of curvilinear rods, News of higher educational institutions, Construction and architecture. 5 (1991) 104108. [16] M.N. Ubaydulloyev, M.N. Serazutdinov, Modeling of the intense deformed condition of the strengthened rod systems, Scientifically technical journal Structural Mechanics of Engineering Constructions and Buildings. (2012) 4351. [17] M.N. Serazutdinov, M.F.Garifullin, Shells of complex shape, Soviet Applied Mechanics. 11 (1991) 10771082. [18] S.P. Tymoshenko, Mekhanik of materials, Mir, 1976.