ALGORITHM FOR SHEAR FLOWS IN ARBITRARY CROSS-SECTIONS OF THIN-WALLED BARS Текст научной статьи по специальности «Медицинские технологии»

Magazine of Civil Engineering. 2019. 92(8). Pp. 3-26 Инженерно-строительный журнал. 2019. № 8(92). С. 3-26

Magazine of Civil Engineering issn

2071-0305

journal homepage: http://engstroy.spbstu.ru/

DOI: 10.18720/MCE.92.1

Algorithm for shear flows in arbitrary cross-sections of thin-walled bars

V. Yurchenko*

Kyiv National University of Construction and Architecture, Kyiv, Ukraine * E-mail: [email protected]

Keywords: thin-walled bar, arbitrary cross-section, shear forces flow, closed contour, graph theory, numerical algorithm, numerical examples, software implementation

Abstract. Development of a general computer program for the design and verification of thin-walled bar structural members remains an actual task. Despite the prevailing influence of normal stresses on the stress-strain state of thin-walled bars design and verification of thin-walled structural members should be performed taking into account not only normal stresses, but also shear stresses. Therefore, in the paper a thin-walled bar of an arbitrary cross-section which is undergone to the general load case is considered as investigated object. The main research question is development of mathematical support and knoware for numerical solution for the shear stresses problem with orientation on software implementation in a computer-aided design system for thin-walled bar structures. The problem of shear stresses outside longitudinal edges of an arbitrary cross-section (including open-closed multi-contour cross-sections) of a thin-walled bar subjected to the general load case has been considered in the paper. The formulated problem has been reduced to the searching problem for unknown shear forces flows that have the least value of the Castigliano's functional. Besides, constraints-equalities of shear forces flows equilibrium formulated for cross-section branch points, as well as equilibrium equation formulated for the whole cross-section relating to longitudinal axes of the thin-walled bar have been taken into account. A detailed numerical algorithm intended to solve the formulated problem has been proposed by the paper. The algorithm is oriented on software implementation in systems of computer-aided design of thin-walled bar structures. Developed algorithm has been implemented in SCAD Office environment by the program TONUS. Numerical examples for calculation of thin-walled bars with open and open-closed multi-contour cross-sections have been considered in order to validate developed algorithm and verify calculation accuracy for sectorial cross-section geometrical properties and shear stresses caused by warping torque and shear forces. Validity of the calculation results obtained using developed software has been proven by considered examples.

1. Introduction

To provide desired stiffness and strength in torsion, bridge superstructures are often constructed with a cross-section consisting of multiple cells (Figure 1) which have thin walls relative to their overall dimensions. When the cross-section contains multiple cells, they all provide resistance to applied torsion and for elastic continuity each cell must twist the same amount. With these considerations, equilibrium and compatibility conditions allow simultaneous equations to be formed and solved to determine the shear flow for each cell [1].

The behavior of single-box multi-cell box-girders with corrugated steel webs under pure torsion has been considered by Kongjian Shen et al. [2]. Experimental and numerical studies for considered structures have been also performed [3].

Dowell and Johnson proposed a relaxation method that distributes incremental shear flows back and forth between cells, reducing errors with each distribution cycle, until the final shear flows for all cells approximate the correct values. A closed-form approach has been introduced to determine, exactly, both the torsional constant and all shear flows for multi-cell cross-sections under torsion in the paper [1].

The problem of shear stresses determination for thin-walled bars has been also studied by Slivker in [4, 5] for the general loading case. His semi-sheared theory has been applied by Lalin et al. [6, 7] and Dyakov [8]

Yurchenko, V. Algorithm for shear flows in arbitrary cross-sections of thin-walled bars. Magazine of Civil Engineering. 2019. 92(8). Pp. 3-26. DOI: 10.18720/MCE.92.1

Юрченко В. Алгоритм определения потоков касательных усилий для произвольных сечений тонкостенных стержней // Инженерно-строительный журнал. 2019. № 8(92). С. 3-26. DOI: 10.18720/MCE.92.1

This work is licensed under a CC BY-NC 4.0

for the stability problems of thin-walled bars.

Further investigations in this area require the development of a detailed algorithm intended to software implementation in a computer-aided design system for thin-walled structures [9]. Such algorithm can be validated against benchmark examples as well as finite element results [10, 11]. It is reasonable to construct this algorithm using the mathematical apparatus of the graph theory as it is convenient to describe the topological properties of multi-cellular cross-section [12].

The graph algorithm used in this paper is given first by Tarjan [13]. Its application in analysis of thin-walled multi-cellular section is described by Alfano et al. [14], but the distribution of torsion stresses due to a change in normal stresses has not been considered. The graph theory has been also applied in [15, 16] to calculate the geometrical cross-sectional properties of thin-walled bars with hybrid (open-closed) types of cross-sections.

A simple computer program has been developed by Chai H. Yoo et al. [17] to evaluate the bending shear flow of any multiply-connected cellular sections. Prokic has developed a computer program for the determination of the torsional and flexural properties of thin-walled beams with arbitrary open-closed cross-section. In his paper [18] graph theory has been also applied to establish the topological properties of multicellular cross-section. Gurujee and Shah [19] presented a general purpose computer program capable of analyzing any planar frame made up of thin-walled structural members. Choudhary and Doshi proposed an algorithm for shear stress evaluation in ship hull girders [20].

Although many papers are published on the behavior of thin-walled bars, the development of a general computer program for the design and verification of thin-walled structural members remains an actual task. Despite the prevailing influence of normal stresses on the stress-strain state of thin-walled bars, the design and verification of thin-walled structural members should be performed taking into account not only normal stresses, but also shear stresses. Therefore, in this paper, a thin-walled bar of an arbitrary cross-section under the general load case is considered as investigated object. The main research question is the development of mathematical support and software for numerical solution for the shear stresses problem with orientation on software implementation in a computer-aided design system for the thin-walled structures.

2. Methods

2.1. Problem formulation

Let us consider the problem of shear stresses on longitudinal edges of an arbitrary section of a thin-walled bar that consists of several closed (connected and/or disconnected) contours and/or also open parts. Let us introduce in the plane of thin-walled cross-section a Cartesian coordinate system ycOzc with the origin in the center of mass C of the section, the direction of the coordinate system axes ycOzc coincides with the direction of principle axes of inertia. Let us also introduce in the plane of thin-walled cross-section a Cartesian coordinate system ysOzs with the origin in the shear center S of the section, the direction of the coordinate system axes ysOzs coincides with the direction of principle axes of inertia.

Let us introduce in further consideration the system of angular position coordinate with the origin in a certain (generally randomly selected) sectional point. Each considered sectional point can be associated with the angular position g. The value gshould be calculated as the geometrical length of the curve constructed from the origin to the considered sectional point taken along the sectional contour. We also assume that the increment of the angular position g corresponds to the positive direction of section path tracing.

We assume that the integral geometrical properties of the section are known: A is the cross-sectional area, Iy and Iz are the second moments of area relative to the main axes of inertia which coincide with axes of global Cartesian coordinate system ycOzc, Ia is the sectorial moment of inertia; It is the second moment of area for pure torsion. We also assume that Young's modulus E and shear modulus G are constant for the whole cross-section of the thin-walled bar.

Generally, the thin-walled bar is subjected to the action of eight force factors. Axial force N, bending moments My and Mz relative to the principle axes of inertia and warping bimoment B are applied at the center of mass C (see Figure 2) of the section and cause normal stresses in the cross-section oi(x, g):

ч N (x) Mv (x) . . M7 (x) , . B (x) , .

^ (x, g) = -r^-r^, (gh^ty, (gh^-1™, (g), (1.1)

A Ly Lz

where yi (g), zi (g), wi (g) are the coordinates and sectorial coordinate of the considered point in cross-section of a thin-walled bar.

Figure 1. Box-girder bridge multi-cell cross-section [23].

Figure 2. Cross-section of a thin-walled bar with representation of different angular positions as examples.

Shear forces Qy and Qz, total torque Mx and warping torque Ma are applied at the shear center S (see Figure 2) of the cross-section and cause shear stresses in the cross-section, which can be written in terms of shear forces flows Tj(x, g) as presented below:

( -, ç) =

T ( x,ç) (ç) '

(1.2)

where Sj (g) is the thickness of jth section element.

An arbitrary section of the thin-walled bar can be described by the set of sectional points P = {pp = {yp, zp } | p = 1, np } (yp and zp are the coordinates of pth sectional point in the global Cartesian

coordinate system yOz) and by the set of sectional segments S = {p = {pf, pe"d } | s = 1, ns}, which

connect some two adjacent sectional points (Figure 3), where np and ns are the numbers of the sectional points and segments, respectively.

TPs

Figure 3. Arbitrary cross-section of a thin-walled bar determined on the set of sectional points P and set of sectional segments S.

The specified segment thickness 5 = {Sx | s = 1, ns} corresponds to each sectional segment. The set

of sectorial coordinates «= {®P I p = 1,np} and the set of normalized sectorial coordinates

m = {mp | p = 1, np j of the section correspond to the set of the sectional points P, assuming that the values of the sectorial coordinates and normalized sectorial coordinates in each cross-sectional point are known.

The set of angular positions q = {pK = {gsK'art, geK"d } | k = 1, ng -1} is actually intended to implement a

numerical integration taken along the thin-walled section contour (e.g., when calculating geometrical properties of the cross-section, values of shear forces flows, etc.), where k is the number of a segment, ng - 1 is the number of the sectional segments. It should be noted that the angular positions are attributes of the ends of the sectional segments.

The initial data about the thin-walled section should be mapped onto the set of the angular positions ç, by means of corresponding sets of sectional segments

Sç = (^ = {çfrt, çf }: çs'art, çf ç ç), set of sectorial coordinates

ç - {àÇ ={a>sKtart, cae"d} :asJart, K"d Ç for the ends of sectional segments as well as the set of

K = 1, Ylç- 1

Ю' =

thicknesses ôç = {ôÇK ç ô} for the segments, к = 1, nç-1.

2.2. Distribution of shear forces flows along closed contours of an arbitrary cross-section of thin-walled bar

2.2.1. Construction of connected graph Gassociated with a section of a thin-walled bar

An arbitrary cross-section of a thin-walled bar can be associated with a planar connected non-oriented graph tf determined on the sets of tf = {V, R}, where V is the finite set of the graph vertices, R is the set of the graph edges or the set of unordered pairs on V (Figure 4) [21, 22]. Herewith, for each graph edge r = {u, v} e R we assume that u ^ v.

iH-J1

Figure 4. Graph tf associated with cross-section of thin-walled bar (the branch points are highlighted in red, while the end points are highlighted in blue).

The vertices of the graph tf are associated with characteristic sectional points only, which can be either:

1) branch points, i.e. sectional points connected with more than two sectional segments, vp = {pv | v = 1, nv }, here nv is the number of these points;

2) end points, i.e. sectional points connected with only one sectional segment vvend = {pg | g = 1, ng },

here ng is the number of these points.

The edges of the graph tf are associated with sectional parts located between characteristic sectional points (with unbranched sectional parts). An edge of the graph tf, as a rule, may contain several sectional segments, so the full information about edge R^ of the graph can be described by the set of sectional

segments sp, r = 1, ngrj, from the array Ss = {p = {gsK'ar', K } I K = 1, ng-1}, p e Ss, belonging to

considered graph edge, Ssp e Rj: Rp = {pp : sp e Ss a sp e Ry | r = 1, npjj }, here ngrj is the number of

segments for/h graph edge. The set of all the graph edges defined on the set of segments Sp can be expressed as Rs ={Rp | j = }.

We also assume that the arbitrary section of the thin-walled bar may contain some quantity of closed contours. Each closed contour is associated with a cycle of the graph tf or with a vertices sequence

Vo, v\, v'k,..., vkn, such that vk h^ vk+lVi o 3vf+1, where nk is the number of closed contours in the section

(the number of the graph tf cycles).

Magazine of Civil Engineering, 92(8), 2019

Some closed contour of a section r k (a basic cycle of the graph tf) can be definitely determined by the set of the graph edges Rj e Rs belonging to the considered contour r^ = {rj | j = 1, nagFk}, where nrrk is the number of the graph edges belonging to kh closed contour. Besides, it is convenient to have the mapping of the closed contour r^ onto the set of sectional segments s gm, Sgm e Ss, belonging to the considered closed contour, Vm = 1,n^ : r \ = {sS : s^ e Ss, 3R i c Ra : sgm c Ra a R^ c r^ }, here ngFk is the number of the sectional segments belonging to kh closed contour.

The closed contours (basic cycles of the graph tf) defined on the set of graph edges Ra and on the set of section segments Sa can be described as Om = {rrka | k = 1, nk} and O a = {r ?k | k = 1, nk}, respectively.

It should be noted that the identification of closed contours in the section and ^ m can be easily implemented using depth-first search algorithms on the graph.

Let us compose an incidence matrix i for the graph tf with dimensions nyx nr, i = {gij I i = 1, nv, j = 1, nr}. The components of the matrix take the following values: gj = 1, if ith graph vertex

is a start vertex for jth edge; gj = -1, if ith graph vertex is an end vertex for jth edge; gj = 0, otherwise. Let us

also introduce a matrix |i| = {|gj | i = 1, nv, j = 1, nr} composed of the modulus of elements gj of the matrix i.

Next, we can compose a matrix of basic graph cycles f with dimensions nk x nk, f = {fkJ}, k = 1, nk, j = 1, nr. The components of the matrix take the following values: fkj = 1, if jth graph edge belongs to kth basic graph cycle (Rj c r k) and the edge direction coincides with the positive direction of path tracing; fkj = -1,

if jth graph edge belongs to kth basic graph cycle (Rj c r k ) and the edge direction does not coincide with the positive direction of path tracing; fkj = 0,, if jth graph edge does not belong to kh basic graph cycle (Rj n r k =0).

2.2.2. Resolving equations relating to distribution of shear forces flows taken along closed contours for an arbitrary section of a thin-walled bar

Each jth edge Rj, j = 1, nr of the graph tf corresponds to a constant - edge weight,

VK:SÇK g Rj л sçK G S? :

r d r nrrj . d r nrrj 1 К nrr> lr

Pj = J = ^± !dr

(2.1)

s( a ) r=1, eRa s( a ) r=1 sK a r=1 si'

c a fc Kj K

Let us also compose the weighting matrix of unbranched sectional parts (edges of graph tf) - a square

matrix W with dimensions n x n and diagonal elementspj, j = 1, nr :

W =

Pi 0 ... 0 0 p2 ... 0

(2.2)

0 0 0 Pn

Besides, each jth graph edge Rj corresponds to the increment of the sectorial coordinate

АшГ = {С | j = 1Я}Г, Vk : £ g Rj л £ g Sç :

n r rj n r rj Гк+1 n r rj

Acorrtj = J pdr =J dc = ^ J dc=^J dc = ^Acr. (2.3)

r=1/r gR j r=1 Гк r=1

Each closed contour of the section r^, k = 1, n,, corresponds to the following constant - contour

weight, fj g F, Vj : R ç r? :

dç

dç

çrk

Pk r!Çi(ç) r^^O j=i

= Ê fkj- Pj •

(2.4)

'k ^j-^

Let us also introduce the weighting matrix of sectional contours - a square matrix K with dimensions

nk x nk:

K =

Pl1 -Pl2 -p21 p22

-pk1 -pk 2

"Pnk 1

^k 2

- Plk

- P2k

Pkk

- Pnkk

~Pmk

~P2nk

~Pk,

nk

nknk

(2.5)

where the diagonal elements of the matrix are the weights of kh closed contour, pkk = pk, k = 1, nk; other elements of the matrix paß take zero value paß = pßa = 0 when corresponded closed contours have no common edges: rga fl rß = 0, and the sum of the weights for all common edges:

paß = pßa=Tpr, Vr : Rr craAR^ crß.

Let us consider the problem of torsion for an arbitrary thin-walled section subjected to total torque Mx only. When the cross-section consists of a certain number of closed (connected and/or disconnected) contours, as well as open parts, the torsion problem for the cross-section of the thin-walled bar is statically indeterminate. Therefore, not only static equations but also strain compatibility conditions must be introduced to consideration.

Let us formulate the strain compatibility conditions considering Castigliano's functional. The latter can be identified with an expression for strain energy formulated in terms of stresses for an isotropic material [5]:

r r

C=-L 2G

Ê

J=1

zlr+l%) dç+/(r(ç))2 £(ç) dç

w

(2.6)

Besides, normal stresses can be omitted, as total torque acts only:

i. \

C =

J_ 2G

£{(r(ç))2 S(ç) dç

J =1 I ;

(2.7)

Let us rewrite Castigliano's functional C Equation (2.7) substituting shear stresses z(p) by their representation in terms of contour flows T = {Tk } , k = 1, nk

(ç) =

TM

¿k (ç) •

(2.8)

In this case we obtain the following expression for Castigliano's functional:

C

=IL& dç I IL^ dç I + ÎL& dç ZlZL f dç T T

J

i

T Tk , G J

dç

2G l,s(i)'2G £ ¿(ç).....2G £ ¿(ç) G rJ ¿(ç) G rJ %)

dç T T Jdç T dç T Tk'dç --

J

S(i) G l ¿(ç) G r,4 ¿(ç) "' G r k ¿(ç)

T2 Tk J

Tk-1Tk r dç

(2.9)

G

1,k ¿(ç)

Negative summands

T T ' i- i ' i

k-r k

dç

in Equation (2.9) take into account the mutual work of the

G tL

counter flows of shear stresses on the common parts of the thin-walled bar cross-section.

It is evident that the resulting torsional moment in the section caused by all contour flows of shear stresses T = {T} , k = 1, nk equals to the sum of the torsional moments caused by each of these flows [5]:

nk

M = 2 Tknk,

k=1

where £2k is the double area embraced by kh closed contour rgk of the section.

(2.10)

Let us present the formulated problem in the form of a mathematical programming task, namely as a problem for unknown contour shear forces flows T = {fk} , k = 1, nk that ensure the least value of the optimum criterion, i.e. Castigliano's functional C Equation (2.9) subject to equilibrium condition Equation (2.10). Let us present the solution of the formulated problem as follow:

M,

Tk = ak

a

(2.11)

nk

where ih is the double area for all closed contours of the section £20 =2&k; ak is the factor for the

k=i

distribution of shear forces flows along closed contour. Then Castigliano's functional Equation (2.9) can be rewritten as presented below:

C =

m 2

f

2GQl

a <j)

dç ~2

—a2

§

dç

+... + at

2aia2 Oh' 2aia2 i

dç

1 S(ç) S(ç) - kl ô{ç) 12rJ2 S(ç)

dç r dç _ r dç

dç

W)

2a1a3 i

. . ... - 2a,a, i / ^ - 2a2a3 i , . ... t(ç) ri t(ç) ri t(ç)

(2.11)

•••-2a2a4 j -^h-2a2ak i -TT^-•••-2ak-iak i x( ^ r24 %) rJ2k S(kk T k-i,k 5(k),

and the equilibrium equation Equation (2.10) can be presented by the following:

nk M M nk

Mx = !LaM nk = Mr T.aknk

rik dç

dç

k=i

a

a

0 k=1

or

"k

a=E akak.

(2.12)

k=i

So, the formulated problem can be presented as a searching problem for unknown distribution factors a = {ak }T , k = 1, nk of shear forces flows taken along closed contours of section that ensure the least value of Castigliano's functional C Equation (2.11) subject to equilibrium condition Equation (2.12).

The method of Lagrange multipliers can be used to reduce the problem Equations (2.11)-(2.12) to the searching for a stationary point of the following modified functional A(a, la), where Xa is the Lagrange

multiplier. Besides, the stationary conditions for the modified functional A(a, Xa) can be transformed to a system of linear algebraic equations with an order of nk + 1 presented below in the vector-matrix form:

(2.13)

I" K Q a " 0k !

(¿2)T 0 X A _ _a J

where Q = {A } , k = 1, nk is the column vector of double areas embraced by the closed contours of the thin-walled bar. The resolving system of equations Equation (2.13) to calculate distribution factors

a,

= |ak } , k = 1, nk of shear forces flows along the closed contours of the section is presented below:

Pii - P21

- P12

p22

Pk1 pk 2

~Pnk 1

A

nk 2

A

- Plk

- P2k

Pkk

~ P»kk A

- Pmk A' ax ' 0 "

- P2nk A a2 0

- Pknk A X ak = 0

P nknk A nk < 0

A nk 0 A0 _

(2.14)

where the diagonal elements of the matrix are the weights of kth closed contour,

_ nk

Pkk = Pk, k = 1 nk; A is double area embraced by kth closed contour rk, a = Z A ;

k=1

Ä is the Lagrange multiplier. Other elements of the matrix Partake zero value Paß = Pßa = 0 when corresponded closed contours have no common edges: Tga fl rß = 0, and the sum of weights for all common edges [5] is Paß = Pßa = ZPr, Vr : Rr ^ ra AR ^ rß.

r

The solution of the system of algebraic equations Equation (2.14) returns the column vector of factors ak = {ak | k = 1, nk} for the distribution of shear forces flows along the closed contours of the section. Based

on ak, we can generate the column vector of factors for the distribution of shear forces flows along the graph Ö" edges: Ar = {aj | j = 1, nr}, where each element should be determined as:

aj =

nk _

Z fkjak, fj e ¥ V =1

= 1, nr,

(2.15)

k=1

Since every graph edge R, j = 1, nr, is described by the set of sectional segments sp e S? as:

Rj = {sp : sp e S? a sp e Rj | r = 1, ngrj}, then it is possible to determine for each sectional segment sp e S? the value of piecewise constant distribution function for shear flows taken along section ag (p) as the set of a? = {ap | k = 1, ng-1} as follows: ap = aj, Vjc : sp R ^ 0, and ap = 0, otherwise.

2.3. Resolving equations for an arbitrary cross-section of a thin-walled bar

The search problem of shear forces flows for an arbitrary cross-section of a thin-walled bar (including open-closed multi-contour cross-sections) can be transformed into a minimization problem of Castigliano's functional C subject to constraints-equalities of shear forces flows equilibrium formulated for cross-section branch points, as well as subject to equilibrium equation for the whole cross-section relating to longitudinal axes of the thin-walled bar [5].

Let us present the formulated problem as a mathematical programming task, namely as searching for unknown values of shear forces flows at the start points of unbranched parts of a section:

T =

J- Q

j7;, j }T , j = ^ nr ,

which ensure the least value of the optimum criterion - Castigliano's functional C:

C* = C(f/) = min C(f5)

Ts enT v '

(3.1)

(3.2)

on a hyperplane of feasible decisions 3T described by the following system of constraints-equalities:

f T ) = { fv T ) = 0 | v = 1, "v -1} ;

fx T ) = 0,

(3.3)

where TS is the vector of design variables (searched shear flows); nr is the number of unknown shear flows;

rri *

TS is the optimum decision of the problem;

C* is the minimum value of Castigliano's functional;

fv is the function of the vector argument TS;

nv is the general number of constraints-equalities fv (Ts ) and f (TS ) which define the hyperplane of feasible decisions 3T in the sought space.

For Castigliano's functional C we will consider only those Euler's equations that define the strain

= 1, n„. Let us

compatibility conditions and are expressed depending on shear forces flows TS = {TS j } , j = 1

rewrite Castigliano's functional C Equation (2.6) replacing normal stresses o(ç) by Equation (1.1), and shear stresses z(ç) - by the dependence on shear forces flows Equation (1.2) as presented below:

T] (ç)=W)

TSJ - Q-Svj (ç)- fsozj (ç)-M^ Somj (ç)

(3.4)

(( (

C =

2G

z

j=\ - i VV v j

J oh -

f

2 (1 + v )

N M

— + —

A I

y

M,

B

Y

z, +—y, + — m,

j I j I j

y z m J

ôdç +

+ il ts2j -2ts,j -2Ts,j -2TS S

LS, j

om, j

dç

(3.5)

dç

+ JI &S + ^S + MmS

- J l T Soy,j - j Soz,j + j S om,j

where the functional dependence on the angular position gis omitted to simplify the presented formulas.

Let us leave in Equation (3.5) those summands that depend on shear forces flows values

TS = {TS j} , j = 1, nr, and also denote by the symbol ... all other summands that do not depend on the

vector TS• In this way we can obtain Castigliano's functional C in terms of shear forces flows TS = {TS j} [5] as presented below:

f

C = z

j=1

J\TL - T ,. iLS„. - T , - n Mm S.

2G

ls , j

\ j

GI " oy,j s ,j GI "oz,j S,j GI " om,j

dç

+ ...

C = z

j=1

fTS,j rd£-T dç-T T

2G J ô S,j GI J ^ ô Sj GI J oz,j ô S,j GT rom,j ô

Qz

dç ^ Q

dç

M.

dç

+...

(3.6)

(3.7)

where the integral J — can be calculated according to Equation (2.1), and the integrals J Soyj-

dç

J S — and J S0 — - using following Equations (3.8)-(3.10), respectively, Vk : sÇ e Rç. a sÇ e S?;

J oz, J <P J , J C K J K

dç

Sç (ç) dç nçrJ f Ç

o _ f oz,J ' ? _ ^K / o ç,start . A oç,mid . vç,end \

Shz ,J - J fi (ç) ^ ,K + Soz,K j

K=1 v

Sç (ç) dç nçrj f ç O _ f gy,J V^ / ^ _ \ ' ^K / oç,start . ,1 oç,mid . oç,end \ Shy, j -J fiç Soy ,K + 4 Soy,K + Soy,K j

Sç (ç)dç nçrJ f ç

S — f oo,J ' ^ — k / sç,start . 4 aç,mid . aç,end \

hm,J~ J S (^ç) ~ 6fiç ^ o®,K o®,^ om,K J

k—i v

(3.8)

(3.9)

(3.10)

Let us define the following column vectors consisting of n elements, Vj = 1, nr (according to the number of edges of the graph tf):

Shz —

Shz,1 Shy,1 Shw,1

Shz ,2 ; Shy — Shy,2 ; Shw — Sh®,2

Shz,nr S hy,nr S hw,nr

(3.11)

Using the weighting matrix of unbranched sectional parts W, Equation (2.2), as well as column vectors Shz, Shy and Shm, Equation (3.11), we can rewrite Castigliano's functional, Equation (3.7), as the following vector-matrix equation:

c ——TTwt; - tT-Q-S,^ - TT Qz -

2G

GJ

Q - M

z S,„ - TT Sh

GJ

GJ

(3.12)

Next, for each section branch point we can develop an equation of shear forces flows equilibrium in terms of projections on the longitudinal axis of the thin-walled bar. In order to obtain the general view for these equations (the system of equations by the number of branch points in the section), we can use the incidence matrices i and |i| introduced above, which reflect the topological structure of the considered cross-section of the thin-walled bar. In this case we obtain the following system of equations presented below in the matrix-vector form:

(T TE,! V2 I«

TE,5 V5 I;,9

^ TE,8^

TE,9 V7TS,10

TE 2 V3 Ts, TE,3 » TS,7

'Ts,

1E,2 v3 xSj E,3 ^

•V6

Sh.

Figure 5. Relating to formulate equilibrium equations for shear stresses flows in branch points of a thin-walled bar.

(I i+i )-(I i - i ) T'e — 0,

(3.13)

where TS — {ts j } , J — 1, nr is the vector of shear forces flows at the start points of unbranched sectional parts;

TE = {TE j } , j = 1, nr is the vector of shear forces flows at the end points of unbranched sectional parts:

TE - Ts -AT,

(3.14)

where AT = {ATj} , j = 1, nr is the vector of shear forces flows increments for each unbranched sectional part:

At = . + ^S,. . + ^S„ ..; (3.15)

I

Z, j

I

У, j

I

where the vectors Sz j, Sy j, Sm j are presented below:

S =

' Sz,1 " " Sy,1 _ r s„, 1

S z,2 II Sy,2 S II S„ 2 2

_Sz,nr _ _ Sy n _ S nr

(3.16)

and the components of vectors can be calculated as follow, Vk : sK g Rk a sK g :

J J? J J K J K

Sz, j = j yg (g) %) dg - 2 f ( yrrt + 2 avK

Sy,j - j zg (g)*(g) dg - § [ôgKlgK [zg/tart + 2 AzgK

f-rj = y

S„,j - j .g (g)ô(g)dg - 2fe (V™ + 2 A^g

(3.17)

(3.18)

(3.19)

Let us rewrite the system of equations Equation (3.13) substituting TE according to Equation (3.14). We obtain the following system of equations:

(Ii| + i) Ts -(i| - i)x(T, - AT) = 0; (Ii| - i) Ts -(|i\ - i) Ts +(|i| - i) AT = 0;

2 ifs +(| i - i ) Af = 0;

and taking into account Equaton (3.15):

m ^ i Qy

Qz

2iTS +(|il -i)x| ^sz, +^sv, +^sn/

s Ml z| I z,j I V'J i m'j

0.

(3.20)

(3.21)

(3.22)

(3.23)

The system of equations in Equation (3.23) in the matrix-vector form has nv equilibrium equations. The last equation is linear-dependent or a linear combination from the previous nv -1 equations. Let us rewrite Equation (3.23) excluding the last equilibrium equation:

2i'T +(| i'|- i')x

f

^ s.

A

j

I y'J I

y w J

0;

(3.24)

where I' is the incidence matrix of the graph tf truncated by the last row with dimensions (nv-l)xnr,

1 ' = { gj1 j =1, nv-1j =1, nr};

I

|I'| is the matrix composed using the modulus of elements gj of the truncated matrix i' as

I11 = {|gj 11 = 1 nv -1 j = 1 nr}.

It is possible to derive the last equilibrium equation relating to the longitudinal axis x - x of the thin-walled bar as a condition of the static equivalence of the torsion moment caused by the shear forces flows to the total torque Mx acting in the cross-section of the thin-walled bar:

Mx_E i T te) da = о

j=i t,

(3.25)

where Tj ( ç) is the shear forces flow at some point of the cross-section, which can be expressed depending on shear forces flow TS j ( ç) at the start point of the corresponded unbranched part of the section as follow:

T = T _ Qy S _ Qz S _ M m S

1 j ~ 1S, j j °oz, j I °oy, j I °o

(3.26)

where we omitted the functional dependence from the angular position (to simplify presented formulas). Then:

M _E fl T -Q^S _Q^S _S

Mx E i I TS,j J Soz,j J Soy,j J Si,j

\

I,

L

pdç = 0;

f

mx _E

j=i

TS, j iPdte_ ^ i Soz, _ Q i Soy, jPdte _ ^ i So^, Pdï

\ J

Finally, we obtain [5]:

= 0.

r . Q nr Q nr M ' f

E Ts, j i pdç _ -J- E i SOZJpdç_ Q E i S0yjpdç_ Ei Som,jpdç_ Mx = 0; (3.27)

J=1 t,. Jz J=11,. Jy J=1 t,. Ji j=1 t,.

nr nr nr

where integrals Ei Soz,pdç , Ei Soy,pdç and Ei S(Ш ,pdç can be calculated using

J=11, J=11, J=11,

Equations. (3.28)-(3.30), respectively, У к : sçK g RJ л sçK g Sç :

лтк

s,» = E i SÇ„J M) pdç = E E^r( sïT++si:;d )

J=1 с,- j=1 V к=1 6

vй

I n

Sp = E i Sçy, (m)pdç = E E^r (SçyT + 4SteyK + SlK)

J \

J=11 r

J=1V к=1 6

Sp,=E i si,j (m)pdç=e EMsçjt+4s^:Kd+si::dK )

J=1 С,- J=1 V к=1 6

(3.28)

(3.29)

(3.30)

Let us rewrite the constraints-equality Equation (3.27) using vector representation taking into account Equations (3.28)-(3.30) as presented below:

mTT„ _p _p _MiSnn_M = 0.

S j pz j py t pi x

z y rn

(3.31)

Thus, the formulated problem is presented as a mathematical programming task of searching for the unknown values of shear forces flows at the start points of the unbranched parts of the section:

Ts ={TS, J } , J = 1

= 1, n,

(3.32)

which ensure the least value of the following Castigliano's functional C Equation (3.12):

rop^eHKO B.

C = — Tf W f - ff Q 2G GI

Shz - TT

GL,

■S, - tT

mw

_w

Gl

S

min,

(3.33)

subject to the following equilibrium conditions Equations (3.24) and (3.31):

I °z,J ' I "y,J ' I ~w,J

2i'T +(|i'|- i')l QlS, +,.+s

A

0;

Qyc Q:

(3.34)

M„

ST T -— S„„ --wSw -M = 0.

I

pz

I

py

I

pw

The method of Lagrange multipliers can be used to reduce the mathematical programming task Equations (3.32)-(3.34) to the searching for the stationary point of the following modified functional

A (Ts , XT, \ ) :

a (ts , x, x ) = 21

= — tT wt - TT S,. - T

2G S S S GL

T Qz S - fT Mw o +

hz ^ s^t hy 1S s^t hw """

GI

y

GIw

+Xf

+ X

2i' TS +(| i' |- i')

f

Y

+ , Mw S

I S z, J + I S y, J + I S W,J

V z y w J

3fT - Qy_ - Q - Mw S -M

I

pz

I

py

I

pw

^ min,

(3.35)

where X = {Xf }, f = 1, nv -1 is the vector of Lagrange multipliers consisting of nv -1 elements; xn is an additional Lagrange multiplier.

The stationary conditions of the modified functional A(T, XT, Xn ), Equation (3.35), can be transformed into a system of n + nv linear algebraic equations and presented in vector-matrix form as follow [5]:

where

1W

G

()f

2i'T Awrs T 0nr

® nv-1, nv-1 0„v-1 x X = Mx x 0„v-1

f-1 0 1 J

Qy

S

hz

G

(i'-|i'l) Sz

S

pz

+ & x

Shy

G

(i'-M) Sy

S

py

Mw

+ —wx

S

hw

G

(i'-|i'l) Sw

S

pw

1/ W

/G W

2i'T

M = 21' 0 nvnv-1 0nv -

(A®? )T -1 0 _

M is a square matrix with dimensions (nr + nv )x(nr + nv), where n and nv are the numbers of edges

Yurchenko, V.

(3.36)

z

Инженерно-строительный журнал, № 8(92), 2019

and vertices of the graph tf, respectively; Aw^ is the column vector of sectorial coordinates increments

Аю^ = {Дсор. j | j = 1,nr} consisting of n. components calculated according to Equation (2.3); Sy, Sz, Sm are the column vectors Equation (3.16) with n. components calculated according to Equations (3.17)—(3.19) respectively; Shy, Shz, Sm are the column vectors Equation (3.11) with n. components calculated according

to Equations (3.8)-(3.10), respectively; Spy, Spz, Spm are the integral section properties calculated according to Equations (3.28)-(3.30), respectively.

The solution of the system of equations in Equation (3.36) determines the column vector of shear forces flows TS = {TS j} , j = 1, nr, at the start points of unbranched cross-section parts. The vector TS can be also presented as follow:

^ + Q-b + M

L z I„ y I„

Ts = Mxbx

-b„.

(3.37)

In this case, the system of algebraic equations, Equation (3.36), disintegrates and transforms into four systems of nr + nv algebraic equations relating to the column vectors bx, by, b2 and bw consisting of n elements [5] as presented below:

M x

Shy

Г ьх ~ 0nr " by " G

i = 0»v-1 ; M x i = (i'-li' )x Sy

1 \y Spy

M x

bz

i i

Shz

G

(H i'|)x Sz

S

pz

M x

к

i i

S

hm

G

(i'- |i'l)x Sm

Sr

pm

(3.38)

where

i = {i} ,i = {\,f} ,i = {i,f} ,k={K,f} ,f =1

= 1, n -1 are the unknown column

vectors of Lagrange multipliers consisting of nv - 1 elements;

Kvx, , \2, are the additional Lagrange multipliers.

The projection of the vector bx = {bx j | j = 1, nr} defined of the set of n unbranched sectional parts

into the set of sectional segments b1 = {b1 | k = 1, n -1} can be written as: bX K = br,Vk : sK z Rj; and

x ^ x,k 1 i j x, x,j j

bg= 0 Vk : SK H Rj = 0. Similarly, the column vectors b = {b | i = 1, n }, b = {b | i = 1, n } and

X,K K 11 j y I y, j I «•' ' r I ' 2 I 2, j \ J ' r I

bm ={bmJ | j = 1, nr} can be also projected into the set of sectional segments obtaining corresponded

column vectors bp = {by,K | K = 1, ng - 1}, bgz = \bgZK | K = 1, ng - 1} and bp = {bi K | K = ^ ng - 1}.

The following transformations for the first moments of inertia and for the sectorial moment of inertia

should be performed, У к = 1, ng -1:

Soz,K ^ {Sg>z,K - bl,K } • S

Sg ^ { s g - ь g }• v g

cm,к | cm,к m ,к J' cm,

ay,к

{sg ,к- ьу,к}

\sm ag /Ir

(3.39)

(3.40)

Magazine of Civil Engineering, 92(8), 2019

Let us define the sets of shear forces flows values for the start, middle and end points at the middle line of the sectional segments Tgst = {k}, Tg,mid = {r/'mid}, Tg,end = {r/'end}, k _ 1, ng-1, consisting of

n -1 elements (by the number of sectional segments) as presented below [22]:

T-rg,start _

k ~

rpgmid _

k ~

T^end _

Q Qy rig start Soz, K Qz r» g,start 'Soy,K " Mw

Iz Iy Iw

pH< Q Qy r» gmid 'Soz, K Qz r» g, mid 'Soy,K ' Mw

K Iy Iw

Q K Qy r» g, end 'Soz, K Qz r» g, end 'Soy,K " Mw

K Iy Iw

r»g,start. om,K '

cigmid . ow,K '

cigend ow,K '

(3.41)

(3.42)

(3.43)

where the first moments of inertia Sgz K, Sg, K and the sectorial moment of inertia Sgw K are calculated using

oz Oy ow, K

transformations in Equations (3.39) and (3.40), respectively.

The shear stresses for each Kh sectional segment Tg _ {rg _ \ K'""t, be calculated as presented below:

(_g,start _g, mid _g,end ) | i i _

,TK ,tK }}, K_ 1 , ng-1 , can

tl_<

c, start rpg, start K 1 -—) H5gK

iK SgK Ik

c,mid _ ygmid K 1 -—) H5gK

lK sgK Ik

_g,end _ ygend K I-—) H5gK

lK sgK Ik J

(3.44)

where the torsion moment of inertia Ix and the parameter — are calculated as:

Ix _ Ik + Ir_ 3 Z igM)) + Ir;

K_1

—_1-f.

The components

rrrg,start K T-rg,mid K and T-rg,end K

sg sg

(3.45)

(3.46)

in Equation (3.44) define shear stresses values for the

start, middle and end points at the middle line of Kh sectional segment, accordingly. Besides, transition from the shear stresses related to the middle line of th segment to the shear stresses at the outside longitudinal edges

of this segment can be performed by addition or subtraction of the member (—— H5gK.

Ik

3. Results and Discussion

3.1. Software implementation

The numerical algorithm developed and presented above has been implemented to the TONUS software (hereinafter - TONUS), which is a satellite of the SCAD Office environment [24], as shown in Figure 6. TONUS is intended to create cross-sections of thin-walled bars, to calculate their geometrical properties as well as to calculate normal, shear and equivalent stresses in these cross-sections [9]. TONUS allows to consider arbitrary (including open-closed) cross-sections of thin-walled bars. The cross-section of a thin-walled bar is constructed from the set of segments (stripes) by specifying node coordinates that define the position of segment ends as well as by specifying thicknesses for all segments.

Figure 6. TONUS main window.

In addition to the calculation of geometrical properties for the cross-sections of thin-walled bars, TONUS also presents a sectorial coordinates diagram as well as static moment diagrams Su, Sv and a first sectorial moment Sw diagram.

To present normal, shear and equivalent stresses diagrams in the section of a thin-walled bar, the user should specify internal forces acting in the section. Initial data to construct normal stresses diagram include bending moments Mu and Mv relating to the main axis of inertia of the thin-walled bar cross-section, axial force N applied at the center of mass of the section, as well as warping bimoment B. Initial data to construct shear stresses diagram are shear forces Qu and Qv applied at the center of mass of the cross-section as well as total torque Mx and warping torque Mm. In order to represent equivalent stresses diagram user should also specify a strength theory.

3.2. Example 1: open thin-walled cross-section

Let us consider an example of calculation of a thin-walled bar with open profile in order to validate the developed algorithm and verify the accuracy of the calculated sectorial cross-section properties and shear stresses caused by warping torsion.

Initial data for calculation are presented in Figure 7. The results of calculation, namely sectorial coordinates diagram m [cm2], and shear stresses diagram related to the value of warping torque TmM— x107 [cm-3], have been obtained in [18] and presented in Figure 8.

The results of calculation, namely sectorial coordinates w, sectorial moment of inertia Sw and shear stresses Tw caused by the warping torque Mm = 107 kN cm, have been also obtained using TONUS and presented in Figures 10-12.

a b

Figure 7. Dimensions [cm] Figure 8. Results of calculation according to [18]: a - sectorial °f the open thin-walled section. coordinate m [cm2]; b -shear stresses related to the warping

torque x107 [cm-3].

Figure 9. Considered cross-section with segments and points numbers.

Figure 10. Results obtained using TONUS: sectorial coordinate m [cm2].

Figure 11. Results obtained using TONUS: sectorial moment of inertia Sm [cm4].

Figure 12. Results obtained using TONUS: modulus of shear stresses Tm [kN/cm2] caused by warping torsion for the value of warping torque Mm = 107 kN cm.

Table 1. Comparison of the first sectorial moment and shear stresses caused by the warping torque for the considered cross-section.

Section segment number

Section point number

First sectorial moment Sm [cm4]

Shear stresses Tm [kN/cm2] (when Mm= 107 kN cm)

(Figure 9) (Figure 9) [18] TONUS Deviation,% [18] TONUS Deviation,%

1 1 32126 32140 0.04 1735 1736 0.06

1 2 0 0 0 0 0 0

2 1 32126 32140 0.04 3470 3472 0.06

2 8 30580 30585 0.02 3303 3304 0.06

3 8 30580 30585 0.02 2202 2202 0

3 4 7999 7985 0.18 576 575 0.17

4 4 6013 6019 0.1 433 432 0.23

4 5 0 0 0 0 0 0

5 4 14008 14004 0.03 1513 1513 0

5 3 15498 15498 0 1674 1674 0

6 6 0 0 0 0 0 0

6 3 25423 25443 0.08 1373 1374 0.07

7 3 9943 9945 0.02 537 537 0

7 7 0 0 0 0 0 0

Table 2. Comparison of sectorial coordinates for the considered cross-section.

Section point number Sectorial coordinate C [cm2]

(Figure 9) [18] TONUS Deviation, %

1 707 707 0

2 1436 1436 0

3 -258 -258 0

4 308 308 0

5 494 494 0

6 -1438 -1438 0

7 921 921 0

8 -810 -810 0

Sectorial first moment of inertia and shear stresses caused by warping torsion, as well as sectorial coordinates for considered thin-walled bar cross-section are presented in Tables 1 and 2. The comparisons have been made with some results presented in [18], which represent exact results for the considered example. As it can be seen, the deviations do not exceed 0.25 % in all cases. It proves the validity of the results obtained using the developed software.

3.3. Example 2: open-closed multi-contour thin-walled cross-section

Let us consider an example of calculation of a thin-walled bar with open-closed multi-contour profile in order to validate developed algorithm and verify calculation accuracy for geometrical cross-section properties and shear stresses caused by warping torsion, as well as shear force. The initial data for calculation are presented in Figure 13.

The calculation results, namely sectorial coordinates diagram w [cm2], diagram of shear stresses caused by warping torsion related to the value of warping torque TwMwW x107 [cm-3], as well as diagram of

shear stresses caused by acting of shear force related to the value of shear force TuQ— x105 [cm-2] have been obtained by Prokic [18] and presented in Figure 14.

100

100

so

100

0.5

100

100

100

0.5

50

Figure 13. Dimensions [cm] of the open-closed multi-contour section of the thin-walled bar.

Figure 14. Results of calculations according to [18]: a - sectorial coordinates diagram w [cm2]; b - shear stresses diagram caused by warping torsion related to the value of the warping torque

x107 [cm 3]; c - shear stresses diagram caused by shear force related to the value

of shear force TuM'U x105 [cm-2].

c

a 7 i i a ^ m M

g C i 3 ii D m

Figure 15. Cross-section with segments and points numbers.

The calculation results, namely sectorial coordinates m, static moment Sv relating to the main axes of

inertia v - v, first sectorial moment Sm, shear stresses t caused by shear force Qu = 105 kN, as well as

shear stresses Tm caused by warping torque Mm = 107 kN cm for the considered cross-section section have been obtained using TONUS and presented in Figure 16.

a

7e-014 V*3"1 1 - 7 ' 4e-C w i /

- 7

749 ■'349 00

W » № Mes

\ + /

» "-38S86 1

% >10

Y J

10 436 10 436 ï

Figure 16. Results obtained using TONUS: a - distribution diagram of normalized sectorial coordinates w [cm2]; b - distribution diagram of first sectorial moment Sw [cm4]; c - distribution

diagram of modulus of shear stresses Tw [kN/cm2], constructed depending on the value

of the warping torque Mw = 107 kN cm; d- distribution diagram for the first moment Sv [cm3]

relating to the principle axis v — v; e - distribution diagram of modulus of shear stresses Tu [kN/cm2], constructed depending on the value of shear force Qu = 105 kN.

b

c

d

e

First moment Sv and first sectorial moment Sm, shear stresses т and Tm caused by shear force Qu

and warping torque Mm, respectively, as well as sectorial coordinates m for the considered cross-section

are presented in Tables 3-5. The comparisons have been made with some results presented in [18], which represent exact results for the considered example. The deviations are no more than 0.3 % in all design cases. It proves the validity of the results obtained using the developed software.

Table 3. Comparison of first moments for considered cross-section.

Section segment

Section point

First sectorial moment [cm4]

First moment Sv [cm3]

mber number ure 15) (Figure 15) [18] TONUS Deviation,% [18] TONUS Deviation,%

1 1 0 0 0 0 0 0

1 2 87776 87892 0.13 3643 3634 0.25

2 2 65181 65296 0.18 740 741 0.14

2 3 63932 64036 0.16 2903 2899 0.14

3 3 67055 67159 0.16 1812 1817 0.28

6 7 26114 26164 0.19 3595 3606 0.3

6 8 26489 26517 0.11 - 10 -

7 8 44606 44666 0.13 3816 3819 0.08

9 2 22595 22595 0 4373 4369 0.09

9 7 26135 26164 0.11 3606 3606 0

10 3 3176 3177 0.03 4715 4716 0.02

10 8 18117 18149 0.15 4031 4033 0.05

Table 4. Comparison of shear stresses caused by the warping torque, as well as by the shear force for the considered cross-section.

Section Section Shear stresses Tm [kN/cm2] Shear stresses Tu [kN/cm2]

segment number point number (when Mm = 107 kN cm) (when Qu = 105 kN)

(Figure 15) (Figure 15) [18] TONUS Deviation, % [18] TONUS Deviation, %

1 1 0 0 0 0 0 0

1 2 843 844 0.12 197 197 0

2 2 626 627 0.16 40 40 0

2 3 614 615 0.16 157 157 0

3 3 644 645 0.16 98 98 0

6 7 209 209 0 162 163 0.6

6 8 212 212 0 - 10 0

7 8 357 357 0 172 172 0

9 2 434 434 0 473 473 0

9 7 502 503 0.20 390 390 0

10 3 61 61 0 510 510 0

10 8 348 349 0.29 436 436 0

Table 5. Comparison of normalized sectorial coordinate for the considered cross-section.

Section point number Sectorial coordinate Ш [cm2]

(Figure 15) [18] TONUS Deviation, %

1 +3241 +3241 0

2 -1483 -1483 0

3 -1102 -1102 0

7 -261 -261 0

8 +249 +249 0

4. Conclusions

The results of the presented study can be formulated as follow:

1. The searching problem of shear stresses outside longitudinal edges of an arbitrary cross-section (including open-closed multi-contour cross-sections) of a thin-walled bar subjected to the general load case has been considered in the paper.

2. The formulated problem has been transformed into a minimization problem of Castigliano's functional subject to constraints-equalities of shear forces flows equilibrium formulated for cross-section branch points as well as subject to an equilibrium equation for the whole cross-section relating to longitudinal axes of the thin-walled bar.

3. A detailed numerical algorithm intended to solve the searching problem of shear forces flows for an arbitrary cross-section of a thin-walled bar subjected to the general loading case using the mathematical apparatus of the graph theory has been developed. The algorithm is oriented on software implementation in systems of computer-aided design of the thin-walled structures.

4. The developed algorithm has been implemented to the TONUS software, which is a satellite of the SCAD Office environment.

5. Numerical examples for calculation of the thin-walled bars with open and open-closed multi-contour cross-sections have been considered in order to validate developed algorithm and verify calculation accuracy for sectorial cross-section geometrical properties and shear stresses caused by warping torque and shear forces.

6. Validity of the calculation results obtained using the developed software has been proven by considered examples.

References

1. Dowell, R.K., Johnson, T.P. Closed-form shear flow solution for box-girder bridges under torsion [Online]. Engineering Structures. 2012. No. 34. Pp. 383-390. URL: https://doi.org/10.10167j.engstruct.2011.09.023

2. Shen, K., Wan, S., Mo, Y.L., Jiang, Z., Li, X. Behavior of single-box multi-cell box-girders with corrugated steel webs under pure torsion. Part II: Theoretical model and analysis [Online]. Thin-Walled Structures. 2018. No. 129. Pp. 558-572. URL: https://doi.org/10.10167j.tws.2017.12.023

3. Shen, K., Wan, S., Mo, Y.L., Jiang, Z., Li, X.. Behavior of single-box multi-cell box-girders with corrugated steel webs under pure torsion. Part I: Experimental and numerical studies [Online]. Thin-Walled Structures. 2018. No. 129. Pp. 542-557. URL: https://doi.org/10.1016/j.tws.2017.10.038

4. Perelmuter, A.V., Slivker, V.I. Numerical structural analysis: models, methods and pitfalls. Springer-Verlag Berlin Heidelberg, 2003. 600 p.

5. Slivker, V.I. Mechanics of structural elements. Theory and applications. Springer-Verlag Berlin Heidelberg. 2007. 786 p.

6. Lalin, V., Rybakov, V., Sergey, A. The finite elements for design of frame of thin-walled beams. Applied Mechanics and materials. 2014. Vol. 578-579. Pp. 858-863. DOI: 10.4028/www.scientific.net/AMM.578-579.858

7. Lalin, V.V., Rybakov, V.A., Diakov, S.F., Kudinov, V.V., Orlova, E.S. The semi-shear theory of V.I. Slivker for the stability problems of thin-walled bars. Magazine of Civil Engineering. 2019. Vol. 87. Is. 3. Pp. 66-79.

8. Dyakov, S.F. Comparing the results of the thin-walled bar torsion problem according to Vlasov and Slivker theories [Online]. Structural mechanics of engineering constructions and buildings. 2013. No. 1. Pp. 24-31. URL: http://journals.rudn.ru/structural-mechanics/article/view/11128 (rus)

9. Yurchenko, V.V. Proyektirovaniye karkasov zdaniy iz tonkostennykh kholodnognutykh profiley v srede SCAD Office [Designing of steel frameworks from thin-walled cold-formed profiles in SCAD Office] [Online]. Magazine of Civil Engineering. 2010. No. 8. Pp. 3846. URL: http://engstroy.spbstu.ru/index_2010_08/yurchenko_LSTK.pdf (rus)

10. Adany, S., Schafer, B.W. Generalized constrained finite strip method for thin-walled members with arbitrary cross-section: Primary modes [Online]. Thin-Walled Structures. 2014. No. 84. Pp. 150-169. URL: https://doi.org/10.1016/j.tws.2014.06.001

11. Jonsson, J. Determiantion of shear stresses, warping functions and section properties of thin-walled beams using finite elements. Computer and Structures. 1998. No. 68. Pp. 393-410.

12. Sharafi, P., Teh, L.H., Hadi, M.N.S. Shape optimization of thin-walled steel sections using graph theory and ACO algorithm. Journal of Constructional Steel Research. 2014. No. 101. Pp. 331-341.

13. Tarjan, R. Depth-first search and linear graph algorithms. SIAM Journal Computing. 1972. No. 1. Pp. 146-60.

14. Alfano, G., Marotti de Sciarra, F., Rosati, L. Automatic analysis of multicell thin-walled sections. Computer and Structures. 1996. No. 59. Pp. 641-655.

15. Waldron, P. Sectorial properties of straight thin-walled beams. Computers and Structures. 1986. Vol. 24. Is. 1. Pp. 147-156.

16. Yoo, C.H. Cross-sectional properties of thin-walled multi-cellular section. Computer and Structures. 1986. No. 22. Pp. 53-61.

17. Yoo, C.H., Kang, J., Kim, K., Lee, K.C. Shear flow in thin-walled cellular sections [Online]. Thin-Walled Structures. 2011. 49(11). Pp. 1341-1347. URL: https://doi.org/10.1016/j.tws.2011.02.014

18. Prokic, A. Computer program for determination of geometrical properties of thin-walled beams with open-closed section. Computers and Structures. 2000. No. 74. Pp. 705-715.

19. Gurujee, C.S., Shah, K.R. A computer program for thin-walled frame analysis. Advances in Engineering Software. 1989. No. 11. Pp. 58-70.

20. Choudhary, G.K., Doshi, K.M. An algorithm for shear stress evaluation in ship hull girders [Online]. Ocean Engineering. 2015. 108(1). Pp. 678-691. URL: https://doi.org/10.1016/j.oceaneng.2015.08.048

21. Perelmuter, A., Yurchenko, V. Shear stresses in hybrid thin-walled section: development of detail numerical algorithm based on the graph theory. Proceedings of 3rd Polish Congress of Mechanics and 21st International Conference on Computer Methods in Mechanics. Short Papers. 2015. Vol. 2. Pp. 943-944.

22. Yurchenko, V. Searching shear forces flows for an arbitrary cross-section of a thin-walled bar: development of numerical algorithm based on the graph theory. International journal for computational civil and structural engineering. 2019. No. 15(1). Pp. 153-170.

23. San Francisco-Oakland Bay Bridge New East Span Skyway - San Francisco and Oakland, California. American Segmental Bridge Institute [Online]. URL: http://www.asbi-assoc.org/projects

24. URL: www.scadsoft.com

Contacts:

Vitalina Yurchenko, +38(063)8926491; [email protected]

© Yurchenko, V., 2019

Инженерно-строительный журнал

сайт журнала: http://engstroy.spbstu.ru/

ISSN

2071-0305

DOI: 10.18720/MCE.92.1

Алгоритм определения потоков касательных усилий для произвольных сечений тонкостенных стержней

В. Юрченко*

Киевский национальный университет строительства и архитектуры, г. Киев, Украина * E-mail: [email protected]

Ключевые слова: тонкостенный стержень, произвольное сечение, потоки касательных усилий, замкнутый контур, теория графов, численный алгоритм, численные примеры, программная реализация

Аннотация. Разработка универсального программного комплекса для расчета и проектирования тонкостенных стержневых элементов конструкций остается актуальной задачей. Несмотря на превалирующее влияние нормальных напряжений на напряженно-деформированное состояние тонкостенных стержней, проверка несущей способности таких элементов должна выполняться, принимая во внимание также и значения касательных напряжений. В связи с этим рассмотрена задача поиска значений потоков касательных усилий для произвольного сечения (открыто-замкнутого многоконтурного сечения) тонкостенного стержня для общего случая нагружения. Сформулированная задача приведена к задаче математического программирования, а именно к задаче поиска значений неизвестных потоков касательных напряжений, обеспечивающих наименьшее значение функционала Кастильяно при удовлетворении ограничений равновесия потоков в точках ветвления сечения, а также при удовлетворении уравнения равновесия всего сечения тонкостенного стержня относительно продольной оси. Разработан детальный алгоритм численного решения сформулированной задачи с использованием математического аппарата теории графов, ориентированный на программную реализацию в системах автоматизированного проектирования тонкостенных стержневых систем. Выполнена программная реализация разработанного алгоритма в среде вычислительного комплекса SCAD Office в программе ТОНУС. С целью верификации разработанного алгоритма и проверки точности вычислений геометрических характеристик и касательных напряжений рассмотрены примеры расчета тонкостенных стержневых элементов открытого и открыто-замкнутого многоконтурного сечений. На рассмотренных примерах доказана достоверность результатов, получаемых при использовании разработанного программного обеспечения.

1. Dowell R.K., Johnson T.P. Closed-form shear flow solution for box-girder bridges under torsion [Электронный ресурс] // Engineering Structures. 2012. No. 34. Pp. 383-390. URL: https://doi.org/10.1016/j.engstruct.2011.09.023

2. Shen K., Wan S., Mo Y.L., Jiang Z., Li X. Behavior of single-box multi-cell box-girders with corrugated steel webs under pure torsion. Part II: Theoretical model and analysis [Электронный ресурс] // Thin-Walled Structures. 2018. No. 129. Pp. 558-572. URL: https://doi.org/10.1016/j.tws.2017.12.023

3. Shen K., Wan S., Mo Y.L., Jiang Z., Li X. Behavior of single-box multi-cell box-girders with corrugated steel webs under pure torsion. Part I: Experimental and numerical studies [Электронный ресурс] // Thin-Walled Structures. 2018. No. 129. Pp. 542-557. URL: https://doi.org/10.1016/j.tws.2017.10.038

4. Perelmuter A.V., Slivker V.I. Numerical structural analysis: models, methods and pitfalls. Springer-Verlag Berlin Heidelberg, 2003. 600 p.

5. Slivker V.I. Mechanics of structural elements. Theory and applications. Springer-Verlag Berlin Heidelberg, 2007. 786 p.

6. Lalin V., Rybakov V., Sergey A. The finite elements for design of frame of thin-walled beams [Электронный ресурс] // Applied Mechanics and materials. 2014. Vol. 578-579. Pp. 858-863. URL: 10.4028/www.scientific.net/AMM.578-579.858

7. Лалин В.В., Рыбаков В.А., Дьяков С.Ф., Кудинов В.В., Орлова Е.С. Полусдвиговая теория В.И. Сливкера в задачах устойчивости тонкостенных стержней // Инженерно-строительный журнал. 2019. № 3(87). С. 66-79. DOI: 10.18720/MCE.87.6

8. Дьяков С.Ф. Сравнительный анализ задачи кручения тонкостенного стержня по моделям Власова и Сливкера [Электронный ресурс] // Строительная механика инженерных конструкций и сооружений. Обзорно-аналитический и научно-технический журнал. 2013. № 1. С. 24-31. URL: http://journals.rudn.ru/structural-mechanics/article/view/11128

9. Юрченко В.В. Проектирование каркасов зданий из тонкостенных холодногнутых профилей в среде SCAD Office // Инженерно-строительный журнал. 2010. № 8(18). С. 38-4. DOI: 10.18720/MCE.18.7.

10. Adany S., Schafer B.W. Generalized constrained finite strip method for thin-walled members with arbitrary cross-section: Primary modes [Электронный ресурс] // Thin-Walled Structures. 2014. No. 84. Pp. 150-169. URL: https://doi.org/10.1016/j.tws.2014.06.001

Литература

11. Jonsson J. Determiantion of shear stresses, warping functions and section properties of thin-walled beams using finite elements // Computer and Structures. 1998. No. 68. Pp. 393-410.

12. Sharafi P., Teh L.H., Hadi M.N.S. Shape optimization of thin-walled steel sections using graph theory and ACO algorithm // Journal of Constructional Steel Research. 2014. No. 101. Pp. 331-341.

13. Tarjan R. Depth-first search and linear graph algorithms // SIAM Journal Computing. 1972. Np. 1. Pp. 146-60.

14. Alfano G., Marotti de Sciarra F., Rosati L. Automatic analysis of multicell thin-walled sections // Computer and Structures. 1996. No. 59. Pp. 641-55.

15. Waldron P. Sectorial properties of straight thin-walled beams // Computers and Structures. 1986. Vol. 24. Is. 1. Pp. 147-156.

16. Yoo C.H. Cross-sectional properties of thin-walled multi-cellular section // Computer and Structures. 1986. No. 22. Pp. 53-61.

17. Yoo C.H., Kang J., Kim K., Lee K.C. Shear flow in thin-walled cellular sections [Электронный ресурс] // Thin-Walled Structures. 2011. No. 49(11). Pp. 1341-1347. URL: https://doi.org/10.1016/j.tws.2011.02.014

18. Prokic A. Computer program for determination of geometrical properties of thin-walled beams with open-closed section // Computers and Structures. 2000. No. 74. Pp. 705-715.

19. Gurujee C.S., Shah K.R. A computer program for thin-walled frame analysis // Advances in Engineering Software. 1989. No. 11. Pp. 58-70.

20. Choudhary G.K., Doshi K.M. An algorithm for shear stress evaluation in ship hull girders [Электронный ресурс] // Ocean Engineering. 2015. No. 108(1). Pp. 678-691. URL: https://doi.org/10.1016/j.oceaneng.2015.08.048

21. Perelmuter A., Yurchenko V. Shear stresses in hybrid thin-walled section: development of detail numerical algorithm based on the graph theory // Proceedings of 3rd Polish Congress of Mechanics and 21st International Conference on Computer Methods in Mechanics. Short Papers. 2015. Vol. 2. Pp. 943-944.

22. Yurchenko V. Searching shear forces flows for an arbitrary cross-section of a thin-walled bar: development of numerical algorithm based on the graph theory // International journal for computational civil and structural engineering. 2019. No. 15(1). Pp. 153-170.

23. San Francisco-Oakland Bay Bridge New East Span Skyway - San Francisco and Oakland, California [Электронный ресурс]. American Segmental Bridge Institute. URL: http://www.asbi-assoc.org/projects.

24. URL: www.scadsoft.com

Контактные данные:

Виталина Юрченко, +38(063)8926491; Эл. почта: [email protected]

© Юрченко В., 2019