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17Magazine of Civil Engineering. 2021. 107(7). Article No. 10706
Magazine of Civil Engineering
journal homepage: http://engstroy.spbstu.ru/
ISSN 2712-8172
DOI: 10.34910/MCE. 107.6
Algorithm of correcting bimoments in calculations of thin-walled bar systems
I.N. Serpik* , R.O. Shkolyarenko
Bryansk State University of Engineering and Technology, Bryansk, Russia *E-mail: inserpik@gmail.com
Keywords: frames, thin-walled bars, restrained torsion, bimoments, bar connection nodes, finite element method, iterative process
Abstract. An algorithm developed for enhancing the accuracy of the calculation of frames formed by thin-walled open-section bars is presented. The existing bar models for analysis of frame systems consisting of open-section bars subjected to restrained torsion require improvement. Some authors have shown that the traditional premise of a balance of bimoments at the junction of such bars may be violated in many cases. The methodology described in this article is formulated on the condition that the disbalance on bimoments in connecting nodes of rods reinforced with transversal ribs can be taken into account on the basis of the eccentric moments transfer on the bar junctions. An approach based on the Lagrange variational principle to the construction of equations of finite element analysis while taking into account such disbalances is proposed. Herewith, some additional nodal bimoments are introduced. They allow us to correct the solution of the problem and do not affect the global stiffness matrix of the finite element system. A presented rapidly converging iterative process makes it possible to estimate the values of such bimoments. The performance of the suggested methodology has been illustrated via an example of the calculation of frames made of I-beams and U-beams. The comparison of the results of bimoments definition using the developed bar calculation schemes and shell models have shown that the suggested algorithm allows describing the disbalance of bimoments in bar connection nodes to a fairly high degree of precision for practical goals. This result may have significant importance for improving computer modelling of deformations of the thin-walled open-section bar structures.
Calculations of thin-walled bars and bar systems while taking into account torsion can be performed efficiently using shell models [1-3] or three-dimensional analysis [4]. However, the implementation of such approaches for real constructions, especially for carrying out multivariant calculations is often associated with fairly lengthy computational process working hours. Using bar design models is more promising for engineering practice.
It should be noted that the modern regulatory requirements for steel structures (Russian State Standard SP 16.13330.2017 "Steel Structures. Updated revision of SNiP II-23-81*") stipulates consideration of bimoments when determining normal stresses in bar cross sections. First of all, this factor can be significant for thin-walled open-section bars if there is restrained warping of cross sections during torsion. Theories of calculating thin-walled bars while taking into account restrained torsion within a one-dimensional approach are described in sufficient detail in scientific literature. The best known theory is the shear theory by A.A. Umansky, the shearless theory by V.Z. Vlasov, and the semi-shear theory by V.I. Slivker [5-7]. Much attention was also paid to the development of bar finite elements for thin-walled bars based on models of various types [8-26].
Several approaches to taking into account restrained torsion using the finite-element method based on bar models are presented in [8]. For the shearless theory, a double-node open-section finite element has been considered with approximation of rotation angle using Hermite cubic polynomials. For the semi-shear theory,
Serpik, I.N., Shkolyarenko, R.O. Algorithm of correcting bimoments in calculations of thin-walled bar systems. Magazine of Civil Engineering. 2021. 107(7). Article No. 10706. DOI: 10.34910/MCE.107.6
© Serpik, I.N., Shkolyarenko, R.O., 2021. Published by Peter the Great St. Petersburg Polytechnic University. This work is licensed under a CC BY-NC 4.0
1. Introduction
some stiffness matrices of open- and closed-section thin-walled bar finite elements have been constructed for three models of description of rotation angles and measures of warping of cross sections: linear approximations of rotation angle and measure of warping in a double-node finite element, quadratic approximation of rotation angle and linear approximation of measure of warping in a three-node finite element, quadratic approximations of rotation angle, and measure of warping for a three-node finite element. In [9] precision of finite-element analysis has been studied using the shearless and semi-shear theories for thin-walled open-section bars. A double-node finite element constructed within the shearless theory with cubic approximation of rotation angle and offset of the line along which the approximation of longitudinal displacements was performed is considered during analysis of deformations of plate-and-bar systems in [10]. In [11] an issue of calculation has been elaborated using a finite-element method for thin-walled open-section curved bars.
Different variants of thin-walled open-section finite elements have been constructed using analytical solutions of differential equations [13, 14, 19, 20]. In [21] an approach has been considered for implementation of the shearless theory in finite-element analysis using a double-bar finite element. By introduction of the main and dummy bar, a possibility is ensured to set the seventh degree of freedom in the node, which allows taking into account the restrained torsion within the existing software systems which traditionally use six degrees of freedom in a node.
However, the frame calculation methodology taking into account the strain warping needs to be further developed. The most widely used supposition is that presented in [6], concerning the balance of bimoments and the equality of measures of warping at bar junctions. The analysis on this basis of flat-space frames made of open-section bars [27, 28] was considered. However, [29, 30] note that such conditions are often violated, and bar interaction behavior depends significantly on their connection design. This problem can be fundamentally solved using a combined approach, where bar finite elements are introduced outside the junction node zone, and the junction strains are described using shell finite elements [31]. At the same time, the structural designs are complicated significantly in this case.
In [32-34] the regularity of the transfer of internal force factors in bar connection nodes equipped with transversal ribs has been studied in respect to disbalance of bimoments. It has been noted that such disbalance can be taken into account based on the consideration of eccentric moments transfer in the bar junctions. In [34] a step-by-step scheme for accounting of this phenomenon has been introduced within the finite element method by changing position of auxiliary linking elements between the bars. Fairly high precision of this methodology has been illustrated by an example of calculation of thin-walled structures consisting of two and three channel bars. At the same time, this approach supposes re-forming the matrix of a system of resulting equations during each iterative process step.
The aim of this work is the development of a rapidly converging, iterative scheme for accounting of physical prerequisites of [32-34] by introducing some additional nodal bimoments which do not influence the global stiffness matrix of the finite element method and do not require any changes of positions of auxiliary connection links.
Let us consider a linearly elastic frame made of thin-walled open-section bars equipped with transversal ribs. We will assume that the frame bar has a longitudinally uniform cross-section and can be generally subjected to tension-compression, cross bending in two principal planes, and restrained torsion. Let us deem Vlasov's restrained torsion theory to be true for bars. We will discretise the object using thin-walled bar finite elements placing nodal points in the bending centers of their extreme cross sections. Herewith, we initially form bar system S , where thin-walled bars T are located between nodes U of the finite-element model, and the external load is reduced to such nodes. In system S bars T can be directly pairwise connected on nodes U or using stiff inserts D (Fig. 1). Let us assume that measures of warping cross sections are transferred through such inserts without any changes. The potential energy for bar T (Fig. 2) can be written as
2. Methods
2.1. Definition of the bimoment relationship condition
£ (Rxiui + Ryivi + Rziwi + mxAi + myi$yi + mzAi + BmA )
i = 1
where l is the bar length,
n is a longitudinal force,
Serpik, I.N., Shkolyarenko, R.O.
du „
sx = — is the strain per unit of length along axis Ox ,
dx
u is the cross section center-of-gravity displacement vector projection on axis Ox ,
My, Mz are the bending moments in relation to the main central axes Oy , Oz of cross section,
d2 w d 2v ^ „
Xy = —, xz = —2 are the bar bending strains in relation to axes Oy and Oz ,
dx dx
w, v are the cross section center of bending displacement vector projections on axes Oz and Oy ,
Mq is a pure torsion moment,
Q = d<x/dx is a measure of warping,
<x is the cross section rotation angle relative to axis Ox ,
Bm is a bimoment,
Rxi is the force acting on the bar along axis Ox in cross section from node i (i = 1, 2), ui, vi, wi are the values u, v, w in node i,
Ryi, Rzi are the forces applied to the bar from node i in the center of bending Pi of cross section hi, mxi, myi, mzi, Bmi are the axial moments and a bimoment acting on the bar from node i, <xi,<yi,<zi are the cross section Ht rotation angles relative to axes Ox, Oy, Oz , Qi is the measure of warping for node i.
Figure 1. Example of introduction of a stiff insert D between nodes of thin-walled bars I and II: Q , Cii , Pi , Pii are respectively centers of gravity and centers of bending of cross sections
of bars to be joined.
Figure 2. Bar Tof system S by the example of a channel section: Hi, Hi are the nodal cross sections with centers of gravity Ci, Ci and bending centers Pi, Pi; f is a vector connecting the cross section center of gravity and center of bending.
Let us present the bimoment of node i as
B^t = Bv( myi) + Bv( mzi ) + Bc
mi • BRmiaRi,
(2)
where Bw (my), Bm (m^ ) are bimoments created in such a node by moments myt, mzi respectively with due allowance for the actual conditions of their application to the bar,
Bmi is a bimoment which we will treat as one conditioned via transfer of bimoments from neighboring
bars,
BRati is an external bimoment applied in cross section Ht,
aRi is the relative share of the bimoment BRmi taken up by the finite element.
For example, as shown by calculations [33], for I-section (Fig. 3a), connected with some bar L through flange n, moment myi from such a bar will be actually transferred in the plane spaced from plane Cixz
approximately at distance d = 0.6ha , where ha is half the distance between the middle planes of the flanges. Let us introduce a self-balanced system of force couples acting in plane Cixz with moments mA, mB
mB =
. Herewith, moment mA bends the bar, and the self-balanced system of
provided |mA| = |
moments myi, mB will be treated as a bimoment Bm (myi), the modulus of which
Bc( myi)
m
yi
d.
(3)
Figure 3. Transfer of a moment to I-section (a) and channel section (b): R are the transversal ribs.
Figure 4. Principal sectorial coordinates for a channel section.
For channel section (Fig. 3b) during transfer of moment mzi through the channel web n it can be approximately assumed that it acts in the middle plane of this web [34]. Let us consider moments mA, mB , which are equal in absolute value to moment |mzi| and which are in the plane parallel to plane Cixy and spaced from the middle plane of the channel web n by distance t. This distance corresponds to the position of points with zero values of principal sectorial coordinates [6] shown in Fig. 4, where coa , cob are coordinates depending on the dimensions of the cross section.
Then we get
' ' 1 = Kit. (4)
K( mzi )|
According to the researches of [33, 34], we assume that in system S at the junction of two or more bars T connected directly on nodes Uor using inserts D the following relationship on bimoments is fulfilled:
kO , ~ x*
£ (Bci(k) + BRci(k) ) = 0 (5)
k =1
where kO is the number of bars to be connected,
Bci(k), BRCi(k) are magnitudes Bci, BRCi for bar k in the connecting node,
*
() is a designation indicating that the signs of the bimoments in brackets are adjusted according to
Fig. 5.
a. b. c.
Figure 5. Sign rules for bimoments in terms of their action on connecting nodes U when applied relative to axes X u Y(a), Y u Z (b), and X u Z (c).
2.2. Forming the finite-element model and iterative problem solution process
Let us discretise the bar system S using the concept of the finite element method within the displacement method based on approximations used in [8, 10]. We will consider the next scheme of the description of displacements in the bar finite element of bar T (see Fig. 2). Let us represent the vector of generalised strains of the finite element as follows:
{Se } = "|sx Xy Xz Q Q . (6)
Vector of generalised stresses corresponding to vector {s}
k} = {n My Mz Mq Bc}T.
(7)
Let us set down the vector of generalised displacements of finite element node i as
T
{Si } = {u v w <xi <yi < Qi} (i =1,2). (8)
Magazine of Civil Engineering, 107(7), 2021
Taking into account Equations (6) and (7), let us represent the finite element elasticity matrix [ De ] determined by relationship {ae} = [ De ]{ee) [35] as follows:
[ De ] = diag {EA EIy EIZ GIt EIa}, (9)
where E, G are material elasticity modulus and shear modulus, A is the bar cross section area,
I y and Iz are the cross-sectional moments of inertia relative to axes Cy and Cz , It is the geometrical stiffness factor for pure torsion, Im is the principal sectorial moment of inertia.
Let us approximate displacement u along axis Ox using linear law, and displacements v, w and rotation angle <x using third-degree polynomials. We represent the vector of nodal displacement of the finite
element as
[s. } =
-M
№}J'
(10)
Then taking into account Equations (6), (8), and (10), let us report the finite element strain matrix [ Be ] determined by expression {£e } = [ Be ]{t?e} [35] in terms of
[Be ] = [[Be1 ] [Be2 ]], (11)
where
[ Be1 ] =
_ 1
_ l
0 0
0 0
0 0
6 12 x
l2 l3
0 0
6 12 x
l
2 l3
0 0
0 0
6 x 6 x
l2 +13 6 12 x
l2 l3
4 6 x
_ 4+7
0
0 0
0 0
4 6x l l2
0 0
0 0
---^ 0
, 4 x 3x
1--+
l l2
4 6x
_ 4+T"
[Be2 ] =
0 0
6 12 x l2 l3
0 0
_ 12x
_ 7+7"
0 0
0 0
6 x 6 x
I2 _ ~T
6 12x
l2 I3
2 6x
_ 2+7 0
0 0
0 0
2 6 x
1 ~ 7
0 0
0 0
---^ 0
2 x 3x
--+ -
l l2 2 6x
_ 2+7
Let us calculate the finite element stiffness matrix using numerical integration based on Gaussian quadrature on three points. In such a case, if an auxiliary variable Z = 2(x-1/2)/1 is used, it can be set down as
/ 3 r -.J r
[Ke ] = - I Wj Be (Zj )] [De ]Be (Zj )
2 j = 1
(12)
where Wi = W3 = 5/9 , W2 = 8/9 , Zi = -Z3 = >/0.6 , Z2 = 0 are coefficients and coordinates of Gaussian integration points.
When a finite element system is formed, one should transfer to nodal points in bending centers Pi. Let us note the correctness of the equality
ui =UPi +<ify -<yifz'
(13)
where upi is the projection of the displacement vector of cross section Ht bending center on axis Ox ; fy , fz are the projections of vector f on axes Oy and Oz (see Fig. 2).
Taking into account Equations (8) and (13), let us express vector {Si} through vector {¿Pi} of generalized displacements for node in the point Pi:
{Si } = №Pf}.
(14)
where
[A] =
1 0 0 0 -fz fy 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
T
{5Pi } = {uPi vi wi <xi <yi <zi Oi} (i = 1,2).
Then the finite element stiffness matrix for nodal points in bending centers will be determined by relationship
[ Kpb ] = [Q]-1 [ Ke ][Q]
(15)
where matrix
[Q]:
[A] 0 0 [A]
Taking into account Equations (1), (2), (10), (14), and (15), the system of equations for the finite element formed on the basis of the variational principle of Lagrange can be represented a follows:
([Kpe ]{5Pe }) =([Q]-1 {Qe } + Rm } + {R }) ,
where {¿Pe } is the finite element displacement vector for nodes in points P:
S } = {{SP2l},
{Qe } = {Rx1 Ryi Rzi mx1 my1 mzi Bc1 Ry2 Ry2 Ry2 mx2
{Rbm } = {0 0 0 0 0 0 Bmc1 0 0 0 0 0 0
BMci = Bc (myi) + Bc (mzi) (i = 1 2),
{Re } is the vector of parts of external nodal forces applied in cross sections Hx, H2.
Taking into account relationships (5) and (16) let us set down the system of linear algebraic equations for the finite-element model of bar system as
(K ]{sT=({rbm }+{r})*, (17)
where [ K ] is the global stiffness matrix,
{S} is a vector of nodal displacement of the finite-element model,
{Rbm } is a vector formed on the basis of bimoments Bc (myi) and Bc (mzi),
{R} is a vector of external generalized nodal forces.
Let us introduce the following iterative process for solution the system of Equations (17):
[K]{S}(s) = {R} + {Rbm}(s-1) (s = 1,2..), (18)
where s is an iteration number,
{Rbm }(S 1) is the vector {Rbm } obtained from the results of iteration s -1 with {Rbm }(0) = 0.
As shown by calculations, iterative process (18) usually practically converges on highest values of internal force factors in 3 to 6 iterations. In the first iteration one can perform LU decomposition [36] the matrix of system of equations. Then contribution of subsequent iterations into the overall labor intensity of the solution of the problem will be insignificant.
3. Results and Discussion
Let us provide the results of the calculations using the suggested methodology for two examples. In example 1 a steel deformable system formed by bars I and II was considered (Fig. 6). Bar I is made of I-section No. 20B1 according to Russian State Standard GOST R 57837-2017, bar 2 is made of channel section No. 10P according to Russian State Standard GOST 8240-97. The bars have transversal ribs R. The system has rigid fixing H and loading with force couple with moment M. This object was calculated using a shell finite element model (Fig. 7) in the finite element analysis program Autodesk NEi Nastran (license of Federal State Budget Educational Institution of Higher Education "Bryansk State Engineering Technological University", No. PR-05918596) and using a bar model (Fig. 8). 6450 quadrangular shell-type finite elements and 10 thin-walled bar finite elements were considered, respectively. Further refinement of both meshes did not lead in any significant changes of the calculation results. Fig. 9 illustrates a diagram of bimoments in bars obtained using a shell model. Fig. 10 and 11 illustrate bimoments calculated using the considered bar finite element without adjustment on bimoments and with implementation of the iterative process (18). Here the
signs of bimoments correspond to local coordinate systems xiyizi (i = I, II) for the beams (see Fig. 6). For
the shell model, during calculation of bimoments according to Fig. 4 and 12 an approximate relationship was used
n
Bc= £ OjtjljCj, (19)
j=1
mx 2 mz 2
BMc2 }
Bc2} ,
Magazine of Civil Engineering, 107(7), 2021
where n is the number of finite elements separated by the cross section in question into line segments A, of its middle surfaces,
oj is the membrane stress for this cross section in finite element j averaged on line segment A j, tj is the thickness of finite element j , lj is the length of line segment A j,
C0j is the principal sectorial coordinate of the center of this line segment.
Figure 6. Double-bar system.
Figure 7. System of shell finite elements for example 1.
Figure 8. Splitting of example 1 object into bar Figure 9. Diagram of bimoments according to finite elements: U are the finite element nodes. calculation results in Autodesk NEi Nastran
(N-m2).
Figure 10. Bimoment calculation results for bar model without considering the disturbance on bimoments (N-m2).
Figure 11. Diagram of bimoments obtained upon introduction of correcting nodal bimoments (N-m2).
Figure 12. Principal sectorial coordinates for I-section: o)d is a value depending on the dimensions of the cross section.
Fig. 9-11 show that on the maximum absolute value of bimoment in I-section the result obtained in the bar model without implementation of iterative process (18) is different from the shell model by 26 %, and on bimoment in channel section by 28 %. When the correcting bimoments were used, the respective discrepancies amounted only 6.5 % and 6.3 %. Fig. 13 illustrates the graphs of the behavior of these bimoments during the iterative process. It shows that the convergence on them is actually achieved in 3 iterations.
b.
3
Iteration.
Figure 13. Bimoments in example 1 for the joined node in I-section (a) and channel section (b) depending on the iteration number.
In example 2 the steel frame (Fig. 14) is calculated. Its columns are channel sections, and the cross bar is I-section. The dimensions of cross sections are assumed to be the same as for the dimensionally identical sections in example 1. The bars are reinforced with transversal ribs R. The columns are fixed rigidly at the bottom in supports H. The system is loaded with moments M1, M2 , M3 , concentrated forces F, F2 , F3, acting in the median planes of the channel webs, and distributed load q , acting in the main vertical plane of the crossbar.
The calculation results were compared using shell and bar models. For the shell scheme (Fig. 15) 31,850 finite elements were used. When the bar model was formed, 92 thin-walled bar finite elements were introduced. By analogy with example 1, the results of the calculation of bimoments using Autodesk NEi Nastran are provided in Fig. 16. The bimoments obtained for the bar model based on iterative adjustment are shown in Fig. 17. The signs of bimoments in these diagrams were assumed in accordance with the position of the local coordinate axes shown in Fig. 14. Comparing Fig. 16 and 17 one can conclude that in terms of the bimoment value maximum for joined nodes, the result obtained using the suggested methodology is different from the shell model by 6.6 %, and in terms of the maximum value of bimoment in the columns, by 5.2 %. The bimoment values calculated in the iterative process for joined node A are shown in Fig. 18, where it is clear that in terms of these magnitudes, the convergence was actually achieved in 4 to 6 iterations.
Figure 14. Double-span frame: I, II, III are the structurally identical columns; IV is the cross bar; A, B, C are the bar connection nodes.
Figure 15. System of shell finite elements for example 2.
Figure 16. The results of bimoments calculation based on the shell model (N-m2).
Figure 17. The results of bimoments calculation using a correcting iterative scheme in the bar model (N-m2).
a. b-
Figure 18. The change of bimoments of node A for column (а) and cross bar (b) in the iterative process.
It should be noted that in the structures where the bar connection nodes have significant reinforcement, including that using inclined ribs, some additional disturbances in terms of bimoments may appear. At the same time, such disturbance types may appear also in some straight bars if there are any design factors determining the local restrained warpings [5]. In such cases, in addition to the suggested approach, both for frames and individual bars, it is necessary to introduce into the design model some stiffening elements resistant to the bimoment transfer. According to [33, 34], when using only transversal ribs with thickness in accordance with the requirements of Russian State Standard SP 16.13330.2017, such additional disturbances are insignificant. At the same time, as follows from results of the presented work, when the bar models are used, taking into account the disbalance on bimoments in bar connection nodes caused by moment transfer behavior can increase the calculation precision significantly.
4. Conclusions
1. An algorithm of correcting bimoments has been developed. It allows taking into account the disbalance on bimoments in joined nodes of frames which are generated via open-section bars equipped with transversal ribs based on a rapidly converging iterative process. In this calculation scheme, the same matrix
of system of equations is used during each iteration. It determines the main labor intensity of the calculation process for execution of the first iteration.
2. Strains of each bar are simulated within V.Z. Vlasov's shearless theory. Based on the approach of V.V. Lalin and V.A. Rybakov, stiffness matrices have been constructed for double-node bar finite elements in which the cross section rotation angles are described using cubic law.
3. Using the variational principle of Lagrange and taking into account the conditions of the moment transfer at bar junctions, the finite element method resulting system of equations is formed. It includes the global stiffness matrix constructed in the supposition of the bimoment balance in the bar connection nodes, and on the right hand side of the equations, the correcting bimoments are included. They are determined during iterations.
4. The results are provided for finite element method calculation on bimoment values for two frames using shell models and the suggested algorithm. When corrective bimoments were used, the iterative process practically converged in 3 to 6 iterations. Herewith, the deviation of the calculation results in the presented bar models as compared with the shell schemes amounted to no more than 7 % of the maximum absolute bimoment values.
5. Acknowledgement
The reported study was funded by RFBR according to the research project No. 18-08-00567.
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Contacts:
Igor Serpik, inserpik@gmail.com
Roman Shkolyarenko, shkroman130@mail.ru
