- Email: [email protected]
Finite Element for Calculation of Structures Made of Thin-Walled Open Profile Rods
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 150 (2016) 1673 – 1679
International Conference on Industrial Engineering, ICIE 2016
Finite Element for Calculation of Structures Made of Thin-Walled Open Profile Rods A. Tusnin* Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russia
Abstract
Structures of the open thin-walled profile are usually designed so to avoid twisting the individual elements. However, for some systems it is impossible to completely prevent twisting due to the torsion of a thin-walled open section, sectional warping due to tightness, additional sectoral stresses. The torsional stiffness of the thin-walled open section of the rod is much higher than their pure torsion rigidity. The structural analysis of the open thin-walled profile can be performed using a finite element shell. This requires careful selection and design of the grid partition and greatly increases the number of nodes and elements in the calculation scheme. Thin-walled rod finite elements can be used to calculate the complex spatial structures of a thin-walled open section with the account of clean and restrained torsion. © 2016 2016The TheAuthors. Authors. Published Elsevier © Published by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the organizing committee of ICIE 2016. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016 Keywords: restrained
torsion; warping; bimoment; thin-walled rod
1. Background Development of finite element structural analysis of thin-walled open section considering restrained torsion number of papers [1-16] . Finite elements have from 8 to 14 degrees of freedom, and allow for calculation of the rod systems. Thin-walled finite elements are used in some computer systems [17, 18] With the use of thin-walled finite element calculation problem solved in the elastic stage and with the development of plastic deformation, the stability of thin-walled open section. These studies showed the possibility of the successful application of thin-walled finite
* Corresponding author. Tel.: +7-916-115-1421; fax:. E-mail address: [email protected]
1877-7058 © 2016 The Authors. 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.2016.07.149
1674
A. Tusnin / Procedia Engineering 150 (2016) 1673 – 1679
element calculation of thin-walled open section. Insufficiently addressed issues of constructing stiffness matrices of thin-walled rod finite element exposed and the transition from the stiffness matrix of the rod in the local coordinate system to the stiffness matrix in the global coordinate system based on a constructive solution for interface rods. 2. Methods Also taken into account when calculating the normal rod systems, degrees of freedom at each node: three linear and three angular, for construction of thin-walled open section introduces the seventh degree of freedom of assembly - sectional warping. Thus, the thin-walled rod end element with an open profile nodes in the beginning and the end 14 has degrees of freedom. Figure 1 shows a finite element movable nodes. Each rod is connected local coordinate system ( axis X1, Y1, Z1) randomly oriented relative to the total system ( axis X, Y, Z). Coordinate system right. Following notations displacements in the local coordinate system: u1- linear movement along the axis X1; v1- linear movement along the axis Y1; w1- linear movement along the axis Z1; D1 G angle of rotation about the axis X1; E1 - angle of rotation about the axis Y1; J1 - angle relative to the axis Z1; 1 sectional warping . Corresponding displacements in the global coordinate system: u, v, w, D, E, J, į. Having index 1 indicates that the movement considered in the local coordinate system , and the subscripts "H" and "K" is used to designate the parameters of displacement and respectively the beginning and end of the rod. In the study of stress-strain state of thin-walled open section, the following possible boundary conditions at the ends of thin-walled bar: a rigid support, the hinge support, consolidation of torsion rod with free warping, twisting and consolidation of warping [19]
Fig.1. Possible move nodes thin-walled finite element
Design calculation reduces to determining the unknown displacements of nodes, which are then determined efforts in the elements. To determine the movements necessary to solve a system of linear algebraic equations:
Ro U
Ɋ,
(1)
where Ro - stiffness matrix in the global coordinate system using the boundary conditions; U - the displacement vector components design; Ɋ - load vector with the boundary conditions. The system (1) to each of the nodes corresponds to design seven equilibrium equations: six equations corresponding to concentrated forces and moments and equilibrium equation bimoment node. Stiffness matrix Ro is formed from the stiffness matrices of individual rods. The system ( 1) can be represented as:
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A. Tusnin / Procedia Engineering 150 (2016) 1673 – 1679
R1,1 R2,1 # Ri ,1 # Rn ,1
R1,2 R2,2 # Ri ,2 # Rn ,2
where Ri , j
! R1, j ! R2, j # ! Ri , j # ! Rn , j
ri , j ri 1, j ri 2, j ri 3, j ri 4, j ri 5, j ri 6, j
! R1, n U1 ! R2, n U 2 # # u ! Ri , n U i # # ! Rn , j U n
ri , j 1 ri 1, j 1 ri 2, j 1 ri 3, j 1 ri 4, j 1 ri 5, j 1 ri 6, j 1
ri , j 2 ri 1, j 2 ri 2, j 2 ri 3, j 2 ri 4, j 2 ri 5, j 2 ri 6, j 2
P1 P2 #
(2)
Pi # Pn
ri , j 3 ri 1, j 3 ri 2, j 3 ri 3, j 3 ri 4, j 3 ri 5, j 3 ri 6, j 3
ri , j 4 ri 1, j 4 ri 2, j 4 ri 3, j 4 ri 4, j 4 ri 5, j 4 ri 6, j 4
ri , j 5 ri 1, j 5 ri 2, j 5 ri 3, j 5 ri 4, j 5 ri 5, j 5 ri 6, j 5
ri , j 6 ri 1, j 6 ri 2, j 6 ri 3, j 6 ri 4, j 6 ri 5, j 6 ri 6, j 6
(3)
elements of the stiffness matrix representing the reaction due to the movement in the global coordinate system of node number i of the node number of individual movements node number j;
ui vi wi Ui
D i ; Ɋi Ei Ji Gi
Fxi Fyi Fzi M xi M yi
(4, 5)
M zi Bi
U i - displacement vector in the global coordinate system of node number i; Ɋi - vector of external loads acting on the node number in the global coordinate system . In expression (5) Fxi , Fyi , Fzi , - the concentrated forces in the direction of axes X, Y, Z, respectively; M xi , M yi , M zi , - focused point relative to the axes X, Y, Z, respectively; ȼi - bimoment representing two equal magnitude and oppositely directed torque acting in parallel planes at some distance from each other. When considering the balance bimoment node used the following simplifying assumptions: x the stress at node are accepted in accordance with the theory of thin elastic rods Vlasov; x local stress at the node is not taken into account; x warping ( the first derivative of the twist angle ) cross sections in the site for all terminals within the same node ( the condition is satisfied when the appropriate design for interface). Longitudinal displacement of the points are proportional to the cross section at the warping sectorial area. For the most common cross-section in the form of symmetric and asymmetric I-beams, channels, etc. sections of straight line segments, diagrams sectorial areas linear. In each section of the straight sections sectorial area always seems to be linear law [20]
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A. Tusnin / Procedia Engineering 150 (2016) 1673 – 1679
Consider the balance bimoment node. Let node in equilibrium, received additional infinitesimal warping. Then the work of the external bimoment applied to the node, this move will be:
W
B pG
(6)
Longitudinal displacement of the points of the rod room while i will be:
ui
G Zi
(7)
where - Zi the sectorial area ( coordinates ) of the points of the rod number i. Job normal stresses on the displacements caused by an additional warping node (increase in the internal energy), all rods connecting node is:
U
¦ ³ V G Z dA i
i
where Bi
i
A
G ¦ ³ V i Zi dA G ¦ Bi , i
A
(8)
i
³ V Z dA - in the rod bimoment i. i
i
A
At equilibrium, the equality node:
W
U
or B p
(9)
¦B
i
,
(10)
i
bimoment means that the equilibrium in the node In determining external bimoment should consider eccentricities application of concentrated forces and bending moments. Constraints imposed on the design, you can consider deleting from the system (2) equations corresponding to the movements, which are superimposed on the communication of these movements and equating to zero in the remaining equations. After determining the displacements of nodes are moving construction start and end of each rod in the local coordinate system, and on local movements, forces in the bars. For successful implementation of the numerical calculation is necessary to develop the stiffness matrix for thinwalled open section with all the features of web design. Most just get the stiffness matrix for thin-walled bar open profile with two axes of symmetry (welded and rolled I-beams), widely used in spatial beam structures. For problems in the linear formulation can be considered the motions for bending and compression rod separately from movements causing twisting and warping. Thus, the task of developing the thin-wall stiffness matrix of a finite element is reduced to a certain combination of the stiffness matrix, taking into account the linear movement and rotation angles about the axes, with the stiffness matrix, taking into account the angle of rotation about the axis and the sectional warping. For the geometric characteristics of thin-walled open section of the rod, the following notation: A - cross-sectional area of the rod; I y
2 1
³ z dA - moment of inertia about the axis Y
1
of the rod; I z
A
the axis Z1 of the rod; I t
3 i i
¦b t 3
2 1
³ y dA - moment of inertia about A
- the moment of inertia of the rod to clean torsion, where bi and ti - according
to the width and thickness of the sheets, forming section; IZ
2 1
³ Z dA - sectorial moment of inertia of the rod, A
where Zi - sectorial area. The elastic modulus E, the shear modulus G. Length of the rod – l.
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A. Tusnin / Procedia Engineering 150 (2016) 1673 – 1679
Development of the stiffness matrix of thin-walled finite element based on the Vlasov theory [20], which is good experimental and theoretical confirmation. Differential equation describing the stress- strain state of a rod under torsion has the following form:
EI ZD1IV GI tD1s m ( x1 ) bc( x1 )
0,
(11)
where m( x1 ) - the intensity of the external torque distribution, b(x1 ) - the intensity of the external distributed bimoment. Equation (11) can be written as:
D1IV k 2D1s
where k
m( x1 ) bc( x1 ) EIZ
(12)
GIt - flexural- torsional response of the rod. EIZ
(13)
Solutions of equation (12) using the method of initial parameters obtained Bychkov D.V. [19] The values of the reaction due to the angle of torsion and warping for rod ends secured by twisting and warping under the given unit and warping torsion angles at the ends of the rod. For components of the stiffness matrix of thin-walled open section of the rod caused by twisting and warping , consider rod ends secured by twisting and warping. The stiffness matrix comprises the reaction of connections possible when the unit moves, as which, in this case, considered the rotation angle relative to the longitudinal axis of the rod and warping. Rotation angle connection is considered positive if when looking from the end of the rotation axis X1 is counterclockwise. Warping is considered positive, in which the closest to the observer shelf rotates clockwise. Positive torque is directed in the same connection, as a positive angle. Positive bimoment acts of links so that when viewed along the shoulder closest to the observer bimoment moment acting clockwise. Given the previously established patterns [19] matrix torsional stiffness and warping thin-walled element with two axes of symmetry in the local coordinate system has the form:
D 1ɧ
G 1ɧ
D 1ɧ
O EI Z
G 1ɧ
aEI Z l2
l3
D 1ɤ aEI Z l2
O EI Z l3
aEI Z l2
P EI Z
aEI Z l2
gEI Z l
l
D 1ɤ O EI Z aEI Z l3
l2
G 1ɤ aEI Z
gEI Z l
l2
where, ɚ
k 2l 2 (ch(kl ) 1) ; kl sh(kl ) 2ch(kl ) 2
G 1ɤ
O EI Z aEI Z l3 aEI Z l2
l2
P EI Z l
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A. Tusnin / Procedia Engineering 150 (2016) 1673 – 1679
g
kl ( sh(kl ) kl ) ; kl sh(kl ) 2ch(kl ) 2
O
k 3l 3 sh ( kl ) ; kl sh ( kl ) 2ch ( kl ) 2
P
kl (kl ch( kl ) sh(kl )) kl sh(kl ) 2ch(kl ) 2
Combining the stiffness matrix of the torsion and warping of known stiffness matrix of linear displacement and rotation angles relative to the axis, can be obtained stiffness matrix of thin-walled finite element, which has a dimension of 14x14. Components of the stiffness matrix are the reactions of links arising from the movements of individual bonds. Considered positive reaction, the direction of which coincides with the positive direction of the corresponding displacement. Table 1 shows the structure of a stiffness matrix of a finite element of thin walled open section. In the blank cells in Table 1 are arranged zero. The stiffness matrix is symmetrical about the main diagonal, so the Table 1 shows only the elements located to the right and above the stiffness matrix. Table 1. Stiffness matrix of thin-walled finite element with two axes of symmetry u1ɧ u1
ɧ
v1
ɧ
w1ɧ D1 ɧ E1
v1ɧ
w1ɧ
D1 ɧ
E1ɧ
r2,2
r2,6 r3,3
v1ɤ
w1ɤ
D1 ɤ
E1ɤ
r2,9
r3,5
J1 ɤ
G1ɤ
r2,13 r3,10
r4,4
r4,7 r5,10 r6,6
w1ɤ D1 ɤ E1ɤ J1 ɤ G1ɤ
Nonzero elements of the stiffness matrix are:
r4,14 r5,12
r6,9
r6,13 r7,11
r7,7 Symmetrically with respect to main diagonal
r3,12 r4,11
r5,5
G1ɧ u1
u1ɤ r1,8
J1 ɧ
v1ɤ
G1ɧ
r1,1
ɧ
ɤ
J1 ɧ
r7,14
r8,8 r9,9
r9,13 r10,10
r10,12 r11,11
r11,14 r12,12 r13,13 r14,14
A. Tusnin / Procedia Engineering 150 (2016) 1673 – 1679
r1,1
r8,8
EA / l ,
r2,6
r2,13
6 EI z / l 2 ,
r3,3
r10,10
12 EI y / l 3 , r3,5
r10,12
6 EI y / l 2 ,
r4,11
O EIZ / l 3 ,
r5,10
6EI y / l 2 ,
r6,9 6EI z / l , 2
ɚ EIZ / l ,
EA / l ,
r11,14
r9,9 12EI z / l 3 ,
r2,2
12 EI z / l 3 ,
r2,9 r4,4
2
r7,11
r1,8
r11,11
6 EI z / l 2 ,
r9,13
6 EI y / l 2 ,
r3,12
O EIZ / l 3 ,
ɚ EIZ / l 2 ,
12 EI y / l 3 ,
r3,10
r4,7 r5,5
r4,14 r12,12
ɚ EIZ / l 2 , 4 EI y / l ,
r5,12
2 EI y / l ,
r6,6
r13,13
4 EI z / l ,
r6,13
2 EI z / l ,
r7,7
r14,14
P EIZ / l ,
r7,14
1679
gEIZ / l
3. Conclusions Analysis of previously executed research and design practices showed that for the calculation of thin-walled structures considering restrained torsion can be successfully applied finite element method. The author shows the stiffness matrix method of constructing a thin-walled rod of open profile with two axes of symmetry. Matrix presented successfully used in computing complex STK, which is used for the calculation and design of spatial beam structures considering restrained torsion. References [1] V.A. Postnov, I.Y. Harhurim, The finite element method in the calculation of ship structures, 1974. [2] Program: Biegetorsionstheorie II, order BT II, Stuttgart, 1991. [3] S. Rajasekaran, Finite element analysis of thin-walled for open cross sections, Structural Engineering Report. 34 (1971) 144160. [4] S. Rajasekaran, D.W. Murray, Finite element solution of inelastic beam equations, Journal of Structural Divigion. (1999) 10251041. [5] S. Rajasekaran, Instability of tapered thin-walled beams of generic section, Journal of Engineering Mechanics. 8 (1994) 16301640. [6] G. Gluck, J. Kalev, Computer method for analysis of multistory structures, Computer and Structures 56 (1972) 2532. [7] M.A. Grisfield, Finite element methods for the analysis of multicellular structures, Proc. Civil Engineering. 48 (1971) 151162. [8] S.L. Chan, S. Kitipornchai, Geometric nonlinear analysis of asymmetrical thin-walled beams-columns, Engineering Structure. 9 (1987) 243 254. [9] Y.B. Yang, Linear and nonlinear analysis of space frames with nonuniform torsion using interactive computer graphics, Ph.D. diss., Cjrnell U., Ithaca Y, 1984. [10] G. Turkalj, S. Kravanja, E.B. Merdanovic, Numerical Simulation of Large-Displacement Behaviour of Thin-Walled Frames Incorporating Joint Action, in: Proceeding of Design, Fabrication and Economy of Metal Structures, International Conference Prosedings 2013, Miskolc, Hungary. (2013) 127132. [11] V.A. Rybakov, O.S. Gamayunova, The stress-strain state of frame constructions’ elements from thin-walled cores, Internet Journal Construction of Unique Buildings and Structures. 12 (2013) 79123. [12] N.I. Vatin, J. Havula, L. Martikainen, A. Sinelnikov, A. Orlova, S. Salamakhin, Thin-walled cross-sections and their joints: tests and FEMmodelling, Advanced Materials Research. 945949 (2014) 12111215. [13] V.V. Lalin, V.A. Rybakov, S.A. Morozov, The Finite Elements Research for Calculation of Thin-Walled Bar Systems, Magazine of Civil Engineering. 1 (2012) 5373. [14] N.I. Vatin, J. Havula, L. Martikainen, A. Sinelnikov, A. Orlova, S. Salamakhin, Thin-walled cross-sections and their joints: tests and FEMmodelling, Advanced Materials Research. 945949 (2014) 12111215. [15] B.W. Schafer, T. Peköz, Computational modeling of cold-formed steel: characterizing geometric imperfections and residual stresses, Journal of Constructional Steel Research. 47 (1998) 193210. [16] N.I. Vatin, J. Havula, L. Martikainen, A. Sinelnikov, A. Orlova, S. Salamakhin, Thin-walled cross-sections and their joints: tests and FEMmodelling, Advanced Materials Research. 945949 (2014) 12111215. [17] MSC Nastran 2012 Linear Static Analysis User’s Guide, MSC Software Corporation, 2012. [18] Programmer’s Manual for Mechanical APDL, 2009. [19] D.V. Bychkov, Calculation of beam and frame beam systems made of thin-walled elements, Moscow, 1948. [20] V.Z. Vlasov, Thin elastic rods, Fizmatgiz, Moscow, 1959.
ScienceDirect Procedia Engineering 150 (2016) 1673 – 1679
International Conference on Industrial Engineering, ICIE 2016
Finite Element for Calculation of Structures Made of Thin-Walled Open Profile Rods A. Tusnin* Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russia
Abstract
Structures of the open thin-walled profile are usually designed so to avoid twisting the individual elements. However, for some systems it is impossible to completely prevent twisting due to the torsion of a thin-walled open section, sectional warping due to tightness, additional sectoral stresses. The torsional stiffness of the thin-walled open section of the rod is much higher than their pure torsion rigidity. The structural analysis of the open thin-walled profile can be performed using a finite element shell. This requires careful selection and design of the grid partition and greatly increases the number of nodes and elements in the calculation scheme. Thin-walled rod finite elements can be used to calculate the complex spatial structures of a thin-walled open section with the account of clean and restrained torsion. © 2016 2016The TheAuthors. Authors. Published Elsevier © Published by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the organizing committee of ICIE 2016. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016 Keywords: restrained
torsion; warping; bimoment; thin-walled rod
1. Background Development of finite element structural analysis of thin-walled open section considering restrained torsion number of papers [1-16] . Finite elements have from 8 to 14 degrees of freedom, and allow for calculation of the rod systems. Thin-walled finite elements are used in some computer systems [17, 18] With the use of thin-walled finite element calculation problem solved in the elastic stage and with the development of plastic deformation, the stability of thin-walled open section. These studies showed the possibility of the successful application of thin-walled finite
* Corresponding author. Tel.: +7-916-115-1421; fax:. E-mail address: [email protected]
1877-7058 © 2016 The Authors. 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.2016.07.149
1674
A. Tusnin / Procedia Engineering 150 (2016) 1673 – 1679
element calculation of thin-walled open section. Insufficiently addressed issues of constructing stiffness matrices of thin-walled rod finite element exposed and the transition from the stiffness matrix of the rod in the local coordinate system to the stiffness matrix in the global coordinate system based on a constructive solution for interface rods. 2. Methods Also taken into account when calculating the normal rod systems, degrees of freedom at each node: three linear and three angular, for construction of thin-walled open section introduces the seventh degree of freedom of assembly - sectional warping. Thus, the thin-walled rod end element with an open profile nodes in the beginning and the end 14 has degrees of freedom. Figure 1 shows a finite element movable nodes. Each rod is connected local coordinate system ( axis X1, Y1, Z1) randomly oriented relative to the total system ( axis X, Y, Z). Coordinate system right. Following notations displacements in the local coordinate system: u1- linear movement along the axis X1; v1- linear movement along the axis Y1; w1- linear movement along the axis Z1; D1 G angle of rotation about the axis X1; E1 - angle of rotation about the axis Y1; J1 - angle relative to the axis Z1; 1 sectional warping . Corresponding displacements in the global coordinate system: u, v, w, D, E, J, į. Having index 1 indicates that the movement considered in the local coordinate system , and the subscripts "H" and "K" is used to designate the parameters of displacement and respectively the beginning and end of the rod. In the study of stress-strain state of thin-walled open section, the following possible boundary conditions at the ends of thin-walled bar: a rigid support, the hinge support, consolidation of torsion rod with free warping, twisting and consolidation of warping [19]
Fig.1. Possible move nodes thin-walled finite element
Design calculation reduces to determining the unknown displacements of nodes, which are then determined efforts in the elements. To determine the movements necessary to solve a system of linear algebraic equations:
Ro U
Ɋ,
(1)
where Ro - stiffness matrix in the global coordinate system using the boundary conditions; U - the displacement vector components design; Ɋ - load vector with the boundary conditions. The system (1) to each of the nodes corresponds to design seven equilibrium equations: six equations corresponding to concentrated forces and moments and equilibrium equation bimoment node. Stiffness matrix Ro is formed from the stiffness matrices of individual rods. The system ( 1) can be represented as:
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A. Tusnin / Procedia Engineering 150 (2016) 1673 – 1679
R1,1 R2,1 # Ri ,1 # Rn ,1
R1,2 R2,2 # Ri ,2 # Rn ,2
where Ri , j
! R1, j ! R2, j # ! Ri , j # ! Rn , j
ri , j ri 1, j ri 2, j ri 3, j ri 4, j ri 5, j ri 6, j
! R1, n U1 ! R2, n U 2 # # u ! Ri , n U i # # ! Rn , j U n
ri , j 1 ri 1, j 1 ri 2, j 1 ri 3, j 1 ri 4, j 1 ri 5, j 1 ri 6, j 1
ri , j 2 ri 1, j 2 ri 2, j 2 ri 3, j 2 ri 4, j 2 ri 5, j 2 ri 6, j 2
P1 P2 #
(2)
Pi # Pn
ri , j 3 ri 1, j 3 ri 2, j 3 ri 3, j 3 ri 4, j 3 ri 5, j 3 ri 6, j 3
ri , j 4 ri 1, j 4 ri 2, j 4 ri 3, j 4 ri 4, j 4 ri 5, j 4 ri 6, j 4
ri , j 5 ri 1, j 5 ri 2, j 5 ri 3, j 5 ri 4, j 5 ri 5, j 5 ri 6, j 5
ri , j 6 ri 1, j 6 ri 2, j 6 ri 3, j 6 ri 4, j 6 ri 5, j 6 ri 6, j 6
(3)
elements of the stiffness matrix representing the reaction due to the movement in the global coordinate system of node number i of the node number of individual movements node number j;
ui vi wi Ui
D i ; Ɋi Ei Ji Gi
Fxi Fyi Fzi M xi M yi
(4, 5)
M zi Bi
U i - displacement vector in the global coordinate system of node number i; Ɋi - vector of external loads acting on the node number in the global coordinate system . In expression (5) Fxi , Fyi , Fzi , - the concentrated forces in the direction of axes X, Y, Z, respectively; M xi , M yi , M zi , - focused point relative to the axes X, Y, Z, respectively; ȼi - bimoment representing two equal magnitude and oppositely directed torque acting in parallel planes at some distance from each other. When considering the balance bimoment node used the following simplifying assumptions: x the stress at node are accepted in accordance with the theory of thin elastic rods Vlasov; x local stress at the node is not taken into account; x warping ( the first derivative of the twist angle ) cross sections in the site for all terminals within the same node ( the condition is satisfied when the appropriate design for interface). Longitudinal displacement of the points are proportional to the cross section at the warping sectorial area. For the most common cross-section in the form of symmetric and asymmetric I-beams, channels, etc. sections of straight line segments, diagrams sectorial areas linear. In each section of the straight sections sectorial area always seems to be linear law [20]
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A. Tusnin / Procedia Engineering 150 (2016) 1673 – 1679
Consider the balance bimoment node. Let node in equilibrium, received additional infinitesimal warping. Then the work of the external bimoment applied to the node, this move will be:
W
B pG
(6)
Longitudinal displacement of the points of the rod room while i will be:
ui
G Zi
(7)
where - Zi the sectorial area ( coordinates ) of the points of the rod number i. Job normal stresses on the displacements caused by an additional warping node (increase in the internal energy), all rods connecting node is:
U
¦ ³ V G Z dA i
i
where Bi
i
A
G ¦ ³ V i Zi dA G ¦ Bi , i
A
(8)
i
³ V Z dA - in the rod bimoment i. i
i
A
At equilibrium, the equality node:
W
U
or B p
(9)
¦B
i
,
(10)
i
bimoment means that the equilibrium in the node In determining external bimoment should consider eccentricities application of concentrated forces and bending moments. Constraints imposed on the design, you can consider deleting from the system (2) equations corresponding to the movements, which are superimposed on the communication of these movements and equating to zero in the remaining equations. After determining the displacements of nodes are moving construction start and end of each rod in the local coordinate system, and on local movements, forces in the bars. For successful implementation of the numerical calculation is necessary to develop the stiffness matrix for thinwalled open section with all the features of web design. Most just get the stiffness matrix for thin-walled bar open profile with two axes of symmetry (welded and rolled I-beams), widely used in spatial beam structures. For problems in the linear formulation can be considered the motions for bending and compression rod separately from movements causing twisting and warping. Thus, the task of developing the thin-wall stiffness matrix of a finite element is reduced to a certain combination of the stiffness matrix, taking into account the linear movement and rotation angles about the axes, with the stiffness matrix, taking into account the angle of rotation about the axis and the sectional warping. For the geometric characteristics of thin-walled open section of the rod, the following notation: A - cross-sectional area of the rod; I y
2 1
³ z dA - moment of inertia about the axis Y
1
of the rod; I z
A
the axis Z1 of the rod; I t
3 i i
¦b t 3
2 1
³ y dA - moment of inertia about A
- the moment of inertia of the rod to clean torsion, where bi and ti - according
to the width and thickness of the sheets, forming section; IZ
2 1
³ Z dA - sectorial moment of inertia of the rod, A
where Zi - sectorial area. The elastic modulus E, the shear modulus G. Length of the rod – l.
1677
A. Tusnin / Procedia Engineering 150 (2016) 1673 – 1679
Development of the stiffness matrix of thin-walled finite element based on the Vlasov theory [20], which is good experimental and theoretical confirmation. Differential equation describing the stress- strain state of a rod under torsion has the following form:
EI ZD1IV GI tD1s m ( x1 ) bc( x1 )
0,
(11)
where m( x1 ) - the intensity of the external torque distribution, b(x1 ) - the intensity of the external distributed bimoment. Equation (11) can be written as:
D1IV k 2D1s
where k
m( x1 ) bc( x1 ) EIZ
(12)
GIt - flexural- torsional response of the rod. EIZ
(13)
Solutions of equation (12) using the method of initial parameters obtained Bychkov D.V. [19] The values of the reaction due to the angle of torsion and warping for rod ends secured by twisting and warping under the given unit and warping torsion angles at the ends of the rod. For components of the stiffness matrix of thin-walled open section of the rod caused by twisting and warping , consider rod ends secured by twisting and warping. The stiffness matrix comprises the reaction of connections possible when the unit moves, as which, in this case, considered the rotation angle relative to the longitudinal axis of the rod and warping. Rotation angle connection is considered positive if when looking from the end of the rotation axis X1 is counterclockwise. Warping is considered positive, in which the closest to the observer shelf rotates clockwise. Positive torque is directed in the same connection, as a positive angle. Positive bimoment acts of links so that when viewed along the shoulder closest to the observer bimoment moment acting clockwise. Given the previously established patterns [19] matrix torsional stiffness and warping thin-walled element with two axes of symmetry in the local coordinate system has the form:
D 1ɧ
G 1ɧ
D 1ɧ
O EI Z
G 1ɧ
aEI Z l2
l3
D 1ɤ aEI Z l2
O EI Z l3
aEI Z l2
P EI Z
aEI Z l2
gEI Z l
l
D 1ɤ O EI Z aEI Z l3
l2
G 1ɤ aEI Z
gEI Z l
l2
where, ɚ
k 2l 2 (ch(kl ) 1) ; kl sh(kl ) 2ch(kl ) 2
G 1ɤ
O EI Z aEI Z l3 aEI Z l2
l2
P EI Z l
1678
A. Tusnin / Procedia Engineering 150 (2016) 1673 – 1679
g
kl ( sh(kl ) kl ) ; kl sh(kl ) 2ch(kl ) 2
O
k 3l 3 sh ( kl ) ; kl sh ( kl ) 2ch ( kl ) 2
P
kl (kl ch( kl ) sh(kl )) kl sh(kl ) 2ch(kl ) 2
Combining the stiffness matrix of the torsion and warping of known stiffness matrix of linear displacement and rotation angles relative to the axis, can be obtained stiffness matrix of thin-walled finite element, which has a dimension of 14x14. Components of the stiffness matrix are the reactions of links arising from the movements of individual bonds. Considered positive reaction, the direction of which coincides with the positive direction of the corresponding displacement. Table 1 shows the structure of a stiffness matrix of a finite element of thin walled open section. In the blank cells in Table 1 are arranged zero. The stiffness matrix is symmetrical about the main diagonal, so the Table 1 shows only the elements located to the right and above the stiffness matrix. Table 1. Stiffness matrix of thin-walled finite element with two axes of symmetry u1ɧ u1
ɧ
v1
ɧ
w1ɧ D1 ɧ E1
v1ɧ
w1ɧ
D1 ɧ
E1ɧ
r2,2
r2,6 r3,3
v1ɤ
w1ɤ
D1 ɤ
E1ɤ
r2,9
r3,5
J1 ɤ
G1ɤ
r2,13 r3,10
r4,4
r4,7 r5,10 r6,6
w1ɤ D1 ɤ E1ɤ J1 ɤ G1ɤ
Nonzero elements of the stiffness matrix are:
r4,14 r5,12
r6,9
r6,13 r7,11
r7,7 Symmetrically with respect to main diagonal
r3,12 r4,11
r5,5
G1ɧ u1
u1ɤ r1,8
J1 ɧ
v1ɤ
G1ɧ
r1,1
ɧ
ɤ
J1 ɧ
r7,14
r8,8 r9,9
r9,13 r10,10
r10,12 r11,11
r11,14 r12,12 r13,13 r14,14
A. Tusnin / Procedia Engineering 150 (2016) 1673 – 1679
r1,1
r8,8
EA / l ,
r2,6
r2,13
6 EI z / l 2 ,
r3,3
r10,10
12 EI y / l 3 , r3,5
r10,12
6 EI y / l 2 ,
r4,11
O EIZ / l 3 ,
r5,10
6EI y / l 2 ,
r6,9 6EI z / l , 2
ɚ EIZ / l ,
EA / l ,
r11,14
r9,9 12EI z / l 3 ,
r2,2
12 EI z / l 3 ,
r2,9 r4,4
2
r7,11
r1,8
r11,11
6 EI z / l 2 ,
r9,13
6 EI y / l 2 ,
r3,12
O EIZ / l 3 ,
ɚ EIZ / l 2 ,
12 EI y / l 3 ,
r3,10
r4,7 r5,5
r4,14 r12,12
ɚ EIZ / l 2 , 4 EI y / l ,
r5,12
2 EI y / l ,
r6,6
r13,13
4 EI z / l ,
r6,13
2 EI z / l ,
r7,7
r14,14
P EIZ / l ,
r7,14
1679
gEIZ / l
3. Conclusions Analysis of previously executed research and design practices showed that for the calculation of thin-walled structures considering restrained torsion can be successfully applied finite element method. The author shows the stiffness matrix method of constructing a thin-walled rod of open profile with two axes of symmetry. Matrix presented successfully used in computing complex STK, which is used for the calculation and design of spatial beam structures considering restrained torsion. References [1] V.A. Postnov, I.Y. Harhurim, The finite element method in the calculation of ship structures, 1974. [2] Program: Biegetorsionstheorie II, order BT II, Stuttgart, 1991. [3] S. Rajasekaran, Finite element analysis of thin-walled for open cross sections, Structural Engineering Report. 34 (1971) 144160. [4] S. Rajasekaran, D.W. Murray, Finite element solution of inelastic beam equations, Journal of Structural Divigion. (1999) 10251041. [5] S. Rajasekaran, Instability of tapered thin-walled beams of generic section, Journal of Engineering Mechanics. 8 (1994) 16301640. [6] G. Gluck, J. Kalev, Computer method for analysis of multistory structures, Computer and Structures 56 (1972) 2532. [7] M.A. Grisfield, Finite element methods for the analysis of multicellular structures, Proc. Civil Engineering. 48 (1971) 151162. [8] S.L. Chan, S. Kitipornchai, Geometric nonlinear analysis of asymmetrical thin-walled beams-columns, Engineering Structure. 9 (1987) 243 254. [9] Y.B. Yang, Linear and nonlinear analysis of space frames with nonuniform torsion using interactive computer graphics, Ph.D. diss., Cjrnell U., Ithaca Y, 1984. [10] G. Turkalj, S. Kravanja, E.B. Merdanovic, Numerical Simulation of Large-Displacement Behaviour of Thin-Walled Frames Incorporating Joint Action, in: Proceeding of Design, Fabrication and Economy of Metal Structures, International Conference Prosedings 2013, Miskolc, Hungary. (2013) 127132. [11] V.A. Rybakov, O.S. Gamayunova, The stress-strain state of frame constructions’ elements from thin-walled cores, Internet Journal Construction of Unique Buildings and Structures. 12 (2013) 79123. [12] N.I. Vatin, J. Havula, L. Martikainen, A. Sinelnikov, A. Orlova, S. Salamakhin, Thin-walled cross-sections and their joints: tests and FEMmodelling, Advanced Materials Research. 945949 (2014) 12111215. [13] V.V. Lalin, V.A. Rybakov, S.A. Morozov, The Finite Elements Research for Calculation of Thin-Walled Bar Systems, Magazine of Civil Engineering. 1 (2012) 5373. [14] N.I. Vatin, J. Havula, L. Martikainen, A. Sinelnikov, A. Orlova, S. Salamakhin, Thin-walled cross-sections and their joints: tests and FEMmodelling, Advanced Materials Research. 945949 (2014) 12111215. [15] B.W. Schafer, T. Peköz, Computational modeling of cold-formed steel: characterizing geometric imperfections and residual stresses, Journal of Constructional Steel Research. 47 (1998) 193210. [16] N.I. Vatin, J. Havula, L. Martikainen, A. Sinelnikov, A. Orlova, S. Salamakhin, Thin-walled cross-sections and their joints: tests and FEMmodelling, Advanced Materials Research. 945949 (2014) 12111215. [17] MSC Nastran 2012 Linear Static Analysis User’s Guide, MSC Software Corporation, 2012. [18] Programmer’s Manual for Mechanical APDL, 2009. [19] D.V. Bychkov, Calculation of beam and frame beam systems made of thin-walled elements, Moscow, 1948. [20] V.Z. Vlasov, Thin elastic rods, Fizmatgiz, Moscow, 1959.