A theoretical analysis of the local buckling in thin-walled bars with open cross-section subjected to warping torsion

Thin-Walled Structures 76 (2014) 42–55

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A theoretical analysis of the local buckling in thin-walled bars with open cross-section subjected to warping torsion Andrzej Szychowski n Faculty of Civil Engineering and Architecture, Kielce University of Technology, Al. Tysiąclecia Państwa Polskiego 7, 25-314 Kielce, Poland

art ic l e i nf o

a b s t r a c t

Article history: Received 3 July 2013 Received in revised form 7 November 2013 Accepted 7 November 2013 Available online 30 November 2013

Results of a theoretical analysis of the local buckling in thin-walled bars with open cross-section subjected to warping torsion are presented. The local critical bimoment, which generates local buckling of a thin-walled bar and constitutes the limit of the applicability of the classical Vlasov theory, is defined. A method of determining local critical bimoment on the basis of critical warping stress is developed. It is shown that there are two different local critical bimoments with regard to absolute value for bars with an unsymmetrical cross-section depending on the sense of torsion load (sign of bimoment). However, for bars with bisymmetrical and monosymmetrical sections, the determined absolute values of local critical bimoments are equal to each other, irrespective of the sense of torsional load. Critical warping stresses, local critical bimoments and local buckling modes for selected cases of thin-walled bars with open crosssection are determined. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Thin-walled bars Open cross-section Warping torsion Local buckling Local critical bimoment Theoretical analysis

1. Introduction Cold-formed thin-walled bars with open cross-section belong to groups of members in which limit load-carrying capacity is pretedermined by local or distorsional buckling. Bending and nonuniform torsion occur in thin-walled steel beams in which transverse load acts off the shear center of the cross-section. Torsional moments and bimoments appear in cross-sections in the process of lateraltorsional buckling or flexural-torsional buckling of thin-walled members with geometrical (general and local) imperfections. In this case, torsion moments and bimoments are generated by an amplification of displacements and angles of rotation “along the directions” of geometrical imperfection. Contemporary cold-formed steel members, or beams welded from thin sheet metals with open section, are characterized by small thickness of walls and relatively small torsional rigidity. Bimoment caused by warping torsion can be an essential component of section load for this class of thin-walled members. For the purpose of precise description of phenomena occurring in thin-walled bars with open cross-section subjected to warping torsion, the following definitions will be applied in the further part of the study:


A thin-walled bar “with rigid cross-section contour” – a thinwalled bar whose cross-sections during load increment are

n

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subject to warping displacement, but maintain the original shape of section contour; A thin-walled bar “with flexible cross-section contour” – a bar built from flat walls (thin plates) in which, after local or distortional critical stresses are reached, local deflections of component plates or displacements of stiffened edges of walls occur. As a result, the geometry of the cross-section contour of a thin-walled bar is changed; “A thin-walled bar segment” – the section of a bar between transverse stiffenings (diaphragms, ribs, etc.), which assure a rigid cross-section contour in place of their location. “The constructional system of a thin-walled bar” – the mutual geometrical arrangement of component plates (walls), transverse stiffenings (diaphragms, ribs), and local and overall boundary conditions of a thin-walled bar.

The Vlasov theory [1] refers to thin-walled bars with a rigid cross-section contour. This fact limits the possibility of its application to the estimation of the limit load-carrying capacity of currently used thin-walled bars with a flexible cross-section contour in which phenomena caused by local or distorsional buckling of walls occur. It is not possible to analyze postbuckling load-carrying capacity reserves from the Vlasov theory because local or distorsional buckling generates change in the geometry of the cross-section contour. Invariability of the crosssection contour is the fundamental assumption of the Vlasov theory [1].

A. Szychowski / Thin-Walled Structures 76 (2014) 42–55

Nomenclature coefficients of power polynomials Ainp bs ; t s width, thickness of a plate (wall s) BðxÞ bimoment function Bcr;L ; Bcr;R local critical bimoment (“left” – positive, “right” – negative) By first yield bimoment Ds plate flexural rigidity (wall s) E Young's modulus of elasticity f ins ; f jqs dimensionless, free parameters of deflection function of a plate (wall s) G shear modulus of elasticity i, j, n, q, p natural number subscript io the number of half-waves of the sine function in the direction of the plate (or the segment) length Iω warping section constant It St-Venant torsion constant kω coefficient of critical warping stress ls length of a thin-walled bar segment, length of a plate (wall s) Ls work done by external forces Linjq component elements of the work done by external forces function m coefficient which characterizes the longitudinal stress variation according to (19) M t;L ; M t;R load of a concentrated torsional moment (“left” – positive, “right” – negative) M t;cr critical torsional moment from the condition of local buckling M 1 ðxs Þ; M 2 ðxs Þ moments of elastically restrained longitudinal edges (No. 1, 2) of the component plate (wall s) in a thin-walled bar segment no degree of the polynomial, number of polynomials U sum of the total potential energy Vs strain energy of the bending of a plate (wall s) V injq component elements of the bending strain energy function

In practice, we can distinguish three types of sections of thinwalled bars built from flat walls whose behavior under load generating normal stresses is shown schematically in Fig. 1. Bars with a rigid cross-section contour, in which dependences: sc o sLcr and sc o sD cr (Fig. 1b) occur, i.e. bars with a flexible cross-section contour from the condition of local buckling: sLcr osc o sD cr or sLcr o sD cr o sc (Fig. 1c), and bars with a flexible cross-section

43

ws xs ; ys ; zs x; y; z Xs Y in

deflection function of a plate (wall s) Cartesian coordinates of a plate (wall s) Cartesian coordinates of a thin-walled bar segment longitudinal body forces power polynomials (9) with previously determined coefficients Ainp αs coefficient of stress distribution in the direction of the width of a plate (wall s) βðxs Þ; ðβðxÞÞ function of normal stress (or bimoment) distribution in the direction of the plate (or the segment) length pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κ ¼ GI s =EI ω flexural – torsional coefficient of a cross-section parameter describing load of a plate (wall s) on the χs edge containing the center of local coordinate system ðys ¼ 0Þ φ1 ðxs Þ; φ2 ðxs Þ angles of a component plate's rotation (wall s) on longidudinal edges (No. 1, 2) at the connection of adjacent plates ϕ angle of twist rotation v Poisson's ratio ρs ; δs geometrical parameters of a cross-section according to (4) λps ¼ bs =t s slenderness of a plate (wall s) sc ; st normal stresses (compression, tension) sω ; τω warping stresses (normal, shear) sLcr ; sD local buckling stress, distortional buckling stress cr sLω;cr critical warping stress from the condition of local buckling (positive) ωi sectorial coordinates ωc a sectorial coordinate corresponding to critical stress sLω;cr sE;s Euler's stress for a plate (wall s) s0 comparative edge stress in the cross-section of a thinwalled bar ∇2 Laplace's operator Λ Lagrange's function for a thin-walled bar segment ψq Lagrange's multipliers μi multipliers of edge stresses

contour from the condition of distortional buckling for which L D L sD cr osc o scr or scr o scr o sc (Fig. 1d). The necessity of the distinction of thin-walled bars with a rigid cross-section contour (Fig. 1b) from thin-walled bars with a flexible cross-section contour (Fig. 1c,d) under a load exceeding local or distortional critical stresses has an essential significance for the correct interpretation of phenomena

Fig. 1. Types of sections of thin-walled bars with flat walls (a) distribution of warping normal stresses, (b) rigid cross-section contour, (c) the flexible cross-section contour from the local buckling condition, and (d) the flexible cross-section contour from the distortional buckling condition.

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A. Szychowski / Thin-Walled Structures 76 (2014) 42–55

accompanying obtainment of a load-carrying capacity. A theoretical determination of critical stresses ðsLcr ; sD cr Þ in simple and composite states of load with regard to warping torsion (generated e.g. by eccentric transverse load in relation to the shear center of cross-section) is necessary for a more exact design of thin-walled steel members based e.g. on the effective width method. The investigations of local stability of thin-walled bars with open cross-sections, described in the literature, refer primarily to axially or eccentrically compressed, bended and sheared sections without accounting for warping torsion. Only a few studies, e.g. [2–5], describe investigations on open thin-walled bars sensitive to local or distorsional buckling under the load of transverse bending and torsion. Murray and Lau [2] examined a cantilever thin-walled bar with C-section sensitive to local stability loss at transverse loading acting away from the shear center of the section. Kavanagh and Ellifritt [3] analyzed bended thin-walled beams with channel section with a single edge fold stiffeners. The investigated members were supported by transverse brace. The plane of transverse load did not coincide with the shear center of the sections and generated additional torsion of beams. Put, Pi and Trahair [4] presented an experimental analysis of bended and torsioned cold-formed channel beams. The investigation concerned simply supported beams transversely bended by concentrated force at the mid-span. An analysis of the failure load of a bended and torsioned bar depending on the “sign” and eccentricity magnitude, understood as the distance of the line of the activity of transverse load from the shear center of the section, was made. The influence of the local buckling of sections on the failure load and the failure mode of test models was observed. Gotluru et al. [5] compared experimental results [6–8] of bended and torsioned thin-walled steel beams with lipped channel sections with the results of numeric calculations. Computer programs based on the finite elements method (CU-BEAM [8], ABAQUS [9]) and the finite strip method (CU-FSM [7]) were used in the numeric analysis of the thin-walled bars. The influence of the stiffness of a warping spring of support sections of the investigated models on angles of twist rotation and torsion stiffness was also analyzed. In the above cited investigations [2–5], warping normal stresses, generated by the warping torsion in a thin-walled bar, did not exceed the remaining components of the section load generated, for example, by transverse bending. In the present design standards for cold-formed steel members, e.g. [10,11], warping normal stresses are determined on the basis of geometrical characteristics of gross cross-section. In the above design regulations, the possibility of occurrence of local buckling in the thin-walled member with open cross-section, subjected to warping torsion, is disregarded. Szychowski [12] proved experimentally and confirmed theoretically the occurrence of the local buckling of nonuniform torsioned thin-walled bars with Z-section without the participation of other components of load. A “local critical bimoment” occurred in experimental investigations, i.e. bimoment which generated the local buckling of a thin-walled bar. In [12], local critical bimoment is defined as bimoment which generates local buckling in the thin-walled bar with open cross-section, subjected to warping torsion, without the participation of other components of the section load. This study presents a method of the theoretical determination of critical warping stresses ðsLω;cr Þ and local critical bimoments ðBcr Þ from the condition of the local buckling of thin-walled bars with any open cross-section built from flat walls (thin plates). Note: in the further part of the study the upper subscript “L” is omitted in the denotation sLω;cr .

2. Assumptions Theoretical analysis was applied to a thin-walled bar (Fig. 2) with a finite number of diaphragms (or transverse ribs), which assures a “rigid cross-section contour” only at the site of their location. It was assumed that diaphragms are infinitely rigid in their planes, and simultaneously, they do not hinder warping of sections. Diaphragms in the number p divide the thin-walled bar into p 1 segments. Length ls of particular segments is determined by the spacing of neighboring diaphragms. The minimal number of diaphragms is p ¼2. In this case, a thin-walled bar is built from one segment. The geometry of a thin-walled bar segment with open cross-section and the adopted coordinate system of the segment ðx; y; zÞ and component plates ðxs ; ys ; zs Þs ¼ 1;2;3 are shown in Fig. 3. The centers of all the coordinate systems are located in this crosssection (in the axis of the diaphragm), in which the extreme value of bimoment occurs. A segment consists of so component plates (walls) connected by longitudinal edges. On the ith edge, r number of plates (where: r rso ) are connected. The minimum number of connected plates on one edge is r¼ 2. Segments built from two types of plates are considered: the internal plate (I type) and the cantilever plate (II type). The I-type plate (Fig. 4a) is supported on all edges. The II-type plate (Fig. 4b) is supported on three edges with one longitudinal free (not supported) edge. It was assumed that the boundary conditions of component plates on extreme diaphragms of segment (x ¼ 0, x ¼ ls ) are equal to a simple support. Technical examples of the important sections of open thin-walled

Fig. 2. The “constructional system” of a thin-walled bar with open cross-section.

Fig. 3. The separate segment of thin-walled bar.

A. Szychowski / Thin-Walled Structures 76 (2014) 42–55

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Fig. 4. The separated component plates of a thin-walled bar segment (a) internal plate (I-type) and (b) cantilever plate (II-type).

diaphragms (or transverse ribs), which is in agreement with the construction of thin-walled steel members, the assumption is closer to physical reality: εy ffi  νεx and sy ffi 0; where it results: sx ¼ Eεx [13]. 3. The local critical bimoment Experimental investigations [12] unambiguously indicate that warping normal stresses caused by warping torsion (generating bimoment) can be the cause of local buckling of a thin-walled bar with open cross-section. The bimoment which generates local buckling is called “local critical bimoment”. Local critical bimoment can be determined on the basis of critical warping stress ðsω;cr Þ from the formula Bcr ¼

sω;cr I ω ωc

ð1Þ

The problem of the theoretical determination of local critical bimoment is reduced to the analysis of the local stability of a thinwalled bar segment loaded with warping normal stresses, whose intensity in the direction of the bar length depends on the function of bimoment B(x).

4. Critical warping stress Fig. 5. Diagrams of warping stresses in exemplary thin-walled bars composed of I- and II-type plates (a) unsymmetric section (in relation to axis), (b) monosymmetric, and (c) bisymmetric.

bars, consisting of the above-mentioned two types of plates, are shown in Fig. 5. It was assumed that in pre-buckling range of load, the following hypotheses are important: (1) a thin-walled bar maintains a “rigid contour” in the plane of every cross-section; (2) negligibly small shear deformations in the plane of component plates (bar walls) are not taken into consideration; and (3) the distribution of warping normal stresses is in conformity with the hypothesis of the warping function, according to [1]. Under the above assumptions, the bimoment fuction can be determined on the basis of the integration of the well-known differential equation of the nonuniform torsion according to the classical Vlasov theory [1]. Vlasov's theoretical assumptions [1] concerning the “continuous” distribution of diaphragms, which were to eliminate the local buckling of a thin-walled bar, impose the following conditions: ~ x , where: E~ ¼ εy ¼ 0 and sy ¼  νsx , from which results sx ¼ Eε E=ð1  ν2 Þ. Under the assumption of the finite number of

Critical warping stress (within elastic range), obtained from the condition of the local buckling of the thin-walled bar with open cross-section subjected to warping torsion, is represented by the formula: sω;cr ¼ kω sE;s

ð2Þ

where sE;s ¼

 2 π2 E ts 2 12ð1  ν Þ bs

ð3Þ

In order to theoretically determine sω;cr , the local stability of segment (cf. Fig. 3) built from the component plates (walls s) connected by longitudinal edges was analyzed. The length of the segment ls is univocally determined by the spacing of diaphragms (or transverse ribs) aside from the spontaneously formed node lines ðws ¼ 0Þ of local buckling. Geometrical characteristics of the cross-section and the function of bimoment were determined in the system of Cartesian coordinates ðx; y; zÞ. Deflections of the subsequent component plates of number s are described in local, right-hand coordinate sets ðxs ; ys ; zs Þ separately for each plate (cf. Fig. 3). The geometry of a cross-section is described by

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dimensionless values which refer to a chosen (comparative) plate marked by number 1 ρs ¼

bs ; b1

δs ¼

ts ; t1

ð4Þ

The method of the separation of a thin-walled bar segment into component plates, known from studies concerning the local stability of members with closed sections [14–19], was used for the analysis of the local buckling of a thin-walled bar with open cross-section. Solution of the problem consists in a detailed analysis and “composition” of the separated plates (walls) according to the structure of a thin-walled section with regard to the static and kinematic boundary conditions on the edges of connection of adjacent plates. It was assumed that after the local buckling of a thin-walled bar: (1) longitudinal edges of the connection of adjacent plates remain straight; (2) angles between the adjoining plates remain unchanged (but a corner can rotate); and (3) continuity of displacements (rotations) and the equilibrium of edge bending moments on longitudinal edges of the connection of adjacent plates are maintained.

being a component part of the thin-walled bars with open crosssection subjected to warping torsion. Polynomials Yin were used in the following form: Y i1 ¼ η  2η3 þ η4 ; Y i2 ¼ η  10η3 þ 15η4  6η5 ; Y i3 ¼  η þ 26η3 73η4 þ72η5 24η6 ; Y i4 ¼ 2η  3η2 þ η3 ; Y i5 ¼ η  η3 ; where η ¼ ys =bs . For a II-type plate the deflection function, derived and tested in [21] (for the parameter no ¼5), was used in the following form:  n  n    io o y iπxs sin : ð10Þ ws ðxs ; ys Þ ¼ t s ∑ ∑ f ins s bs ls i¼1 n¼1 The expression ðys =bs Þn in the first parenthesis of the series (10) can be treated in practice as one-dimensional polynomial of the form (9) with coefficient Ainp ¼ 1. In the case of elastic restraint of a II-type plate in a thin-walled bar segment, all fins are nonzero parameters of the deflection function, and their initial values result from static and kinematic boundary conditions on the conection of adjacent plates [21]. The strain energy of the bending of a I- or II-type component plate (s) was determined from the formula "  2 2 #) Z Z ( Ds ls bs ∂2 ws ∂ 2 ws ∂ ws 2 2 Vs ¼ ð∇ ws Þ 2ð1  νÞ  dxs dys 2 0 0 ∂xs ∂ys ∂xs 2 ∂ys 2 ð11Þ

4.1. Boundary conditions of component plates The following boundary conditions occur on longitudinal edges of component plates of a thin-walled bar segment: (a) I-type plate (Fig. 4a): ws ¼ 0;

∂2 ws ∂2 ws þν ¼ M 1 ðxs Þ; ∂ys 2 ∂xs 2

M 2 ðxs Þ for ys ¼ 0; ys ¼ bs ;

ð5Þ

(b) II-type plate (Fig. 4b): ws ¼ 0;

∂2 ws ∂2 ws þν ¼ M 1 ðxs Þ for ys ¼ 0 ∂ys 2 ∂xs 2

∂2 ws ∂ 2 ws ∂3 ws ∂ 3 ws þν ¼ 0; þ ð2  νÞ 2 ¼ 0 for ys ¼ bs 2 2 3 ∂ys ∂xs ∂ys ∂xs ∂ys

ð6Þ

ð7Þ

Boundary conditions ws ¼ 0 on longitudinal edges ðys ¼ 0; ys ¼ bs Þ of a I-type plate and on the supported edge ðys ¼ 0Þ of a II-type plate fulfill the conditions of deflections caused by the local buckling of a thin-walled bar [20]. 4.2. Functions of deflections and the strain energy Polynomial-sinusoidal series are used for the description of the deflections of component plates (I- and II-type) caused by the local buckling of a thin-walled bar. The proposal of Jakubowski [17–19], who used function in the form  n    io o iπxs ws ðxs ; ys Þ ¼ t s ∑ ; ð8Þ ∑ f ins Y in sin ls i¼1 n¼1 for the description of deflections of component plates (walls) with closed section, was applied to a I-type plate, where  p 7 y Y in ¼ ∑ Ainp s : ð9Þ bs p¼0 In [17,18], over 20 different polynomials were tested, and their formulas together with physical interpretation, as well as reference numbers were provided. On the basis of tests performed in the present study, polynomials of numbers 2, 10, 13, 14 and 16, according to [17,18], were found useful to the description of deflection of the I-type plate,

Proceeding analogously as in [17–19,21], and putting the deflection function (8) for a I-type plate and the function (10) for a II-type plate to dependence (11), we receive separately for each plate (wall s) Vs ¼

Ds t 2s io io no no f f ∑ ∑ ∑ ∑ V 4bs ls i ¼ 1 j ¼ 1 n ¼ 1 q ¼ 1 injq in jq

ð12Þ

Elements V injq of the square form (12) depend on polynomial coefficients, Poisson's ratio (v) and the plate aspect ratio ðls =bs Þ. Due to the fact that the adopted deflection functions ((8) for a Itype plate and (10) for a II-type plate) are polynomial-sinusoidal series with coefficients Ainp determined in advance, elements V injq can be calculated from the sequence of formulas derived for plates supported on all edges in [17]. 4.3. Membrane stress state in component plates Thin-walled bars with a flexible cross-section contour, considered in the present study, are characterized by considerable slenderness ðλps Þ of component plates (cf. Fig. 1c). For this class of members, the warping of cross-sections is primarily generated by the geometrical variation in an open thin-walled bar with a negligibly small influence of shape strains. The secondary influence of warping shear stresses on the warping of thin-walled bars with an flexible cross-section contour makes it possible to accept a hypothesis, put forward by Vlasov [1], about the unlimited shape stiffness of walls ðγ xy ¼ 0Þ. Examples of the thin-walled sections and graphs of warping (normal and shear) stresses caused by warping torsion are shown in Fig. 5. An analysis of the statics and the local stability of thin-walled bars with a flexible cross-section contour, subjected to warping torsion, confirmed by the results of the experimental investigations [12], makes it possible to state that warping normal stresses ðsω Þ have a decisive effect on the local buckling of this class of members. The influence of warping shear stresses ðτω Þ in typical, technically important sections is secondary. The example of the basic analysis of stresses in Z-section is given in the further part of the present study (Example 3). The variation of normal stresses in the direction of the length of a bar segment depends on the bimoment function BðxÞ. Without accounting for the influence of warping shear stresses on the local buckling of a thin-walled bar, the variation of normal stresses in

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47

Fig. 6. The membrane stress state in a component plate (wall s) at eccentric compression and a longitudinal stress variation caused by (a) warping shear stresses τω and (b) longitudinal body forces Xs.

the direction of the segment length can be obtained by the introduction of longitudinal body forces Xs (which are active in the middle planes of component plates – Fig. 6), according to the concept presented in [14,21]. Such an approach allows for the reduction of the deflection function by decreasing the number of the expressions of series (8 and 10) necessary for the approximation of the buckling mode of a thin-walled bar segment. Component plates (I- or II-type, Fig. 4) are loaded with warping normal stresses (according to the hypothesis on the warping function [1]), and their distribution (for plate s) can be presented in the form   y sω;s ðxs ; ys Þ ¼ χ s s0 1  αs s βðxs Þ: ð13Þ bs The distribution of body forces generating the longitudinal variation of warping normal stresses, according to Fig. 6b, assumes the form   ∂sω;s y X s ðxs ; ys Þ ¼  ¼  χ s s0 1  αs s β′ðxs Þ: ð14Þ ∂xs bs In the further part of the study, the influence of warping shear stresses ðτω Þ on the local buckling of a thin-walled bar segment with open cross-section is disregarded. The variation of warping normal stresses, according to (13), in the direction of the length of a plate, was obtained by the introduction of longitudinal body forces, according to the formula (14).

x2 β2 ðxs Þ ¼ 1  m 2s ; ls where m ¼ 1  s1 =s0 :

The work done by external forces for a component plate (wall s) was determined from the formula Ls ¼ 

ts 2

Z

ls 0

Z

bs

0

    ∂ws 2 sω;s xs ; ys dxs dys ∂xs

ð15Þ

Substituting into formula (15) stresses according to dependence (13), and the deflection function (8) for a I-type plate, or (10) for a II-type plate, we receive separately for each plate (wall s): Ls ¼ 

χ s s0 bs t 3s io io no no ∑ ∑ ∑ ∑ Linjq f in f jq ls i¼1j¼1n¼1q¼1

ð16Þ

Elements Linjq depend on the coefficient αs and the longitudinal distribution of warping normal stresses along the length of a plate depending on the bimoment function. Formulas for the elements Linjq were derived in [21] for: (1) the linear distribution of stress, and (2) the nonlinear distribution of stress (in accordance with 2nd degree parabola) in the direction of plate length, for which function βðxs Þ can be presented, respectively, in the form xs β1 ðxs Þ ¼ 1  m ; ls

ð17Þ

ð19Þ

4.5. Solution of the problem The problem of the determination of critical warping stress from the condition of the local buckling of a thin-walled bar segment was solved with the application of the energy metod. Strain energy of the bending Vs and work done by external forces Ls in individual plates as the function of parameters fins are determined from formulas ((12) and (16)). The sum of the potential energy of component plates is calculated from the formula so

U ¼ ∑ ðV s  Ls Þ s¼1

ð20Þ

In order to determine the total potential energy of a thinwalled bar segment, static and kinematic boundary conditions on all the junction edges of adjacent plates are taken into account. The condition of the equilibrium of bending moments on the ith edge of connection of r plates leads to one equation r

∑ M s ðxs Þ ¼ 0

s¼1

4.4. Work done by external forces

ð18Þ

ð21Þ

The continuity condition of displacements (angles of rotation) on the ith edge of connection of r plates leads to (r 1) equations in the form φs ðxs Þ  φs þ 1 ðxs þ 1 Þ ¼ 0

ð22Þ

Edge moments M s ðxs Þ of the elastic restraint and angles of rotation φs ðxs Þ on longitudinal contact edges can be expressed on the basis of formulas derived in [17]. Boundary conditions of component plates can be written in the uniform form: Rq ðf ins Þ ¼ 0 for q ¼ 1; 2; ::::; qo ;

ð23Þ

where qo – the total number of Eq. (23) which result from conditions (21) and (22) and numbers (io) of half waves of the sine function applied for the approximation of the deflection function in the direction of the segment length. The search for the minimum of the total potential energy in relation to Eq. (20) and boundary condtions (23) is carried out with use of the Lagrange multipliers method. Lagrange's function is created in the form qo

Λ ¼ U þ ∑ ψ q Rq ðf ins Þ q¼1

ð24Þ

where Rq ðf ins Þ – left sides of Eq. (23) from the boundary conditions of component plates.

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A. Szychowski / Thin-Walled Structures 76 (2014) 42–55

Critical warping stress ðsω;cr Þ of a thin-walled bar segment with open cross-section is determined from the solution of the equation set ( ∂Λ ¼0 ∂f ins ð25Þ Rq ðf ins Þ ¼ 0

The method of determination of critical warping stresses, local critical bimoments and the local buckling modes of a thin-walled bar segment with open cross-section is presented by selected examples in the further part of the study.

The equation set (25), which is linear and homogeneous in relation to f ins , represents a generalized problem of eigenvalues. The critical warping stress (eigenvalue) and the local buckling mode (eigenvector) are determined from the zeroing condition of the characteristic determinant of this set.

5. Coefficients (kω) of critical warping stresses In the present work coefficients (kω) of critical warping stresses according to Eq. (2) are determined for segments of thin-walled bars with Z-section (Fig. 7), C-section (Fig. 8) and I-section (Fig. 9).

Fig. 7. Distribution of warping normal stresses in Z-section generated by (a) bimoment BL – “left” and (b) bimoment BR – “right”.

Fig. 8. Distribution of warping normal stresses in C-section generated by (a) bimoment BL – “left” and (b) bimoment BR – “right”.

Fig. 9. Distribution of warping normal stresses in I-section generated by (a) bimoment BL – “left” and (b) bimoment BR – “right”.

A. Szychowski / Thin-Walled Structures 76 (2014) 42–55

49

The following (simplified) convention of the denotation of bimoments was applied for considered sections. Bimoment, which generates warping compressing stresses (positive) on free edges of a comparative plate b1 (flange) was marked as BL – “left” (with the sign “ þ”). Bimoment, which generates, on this same edge, warping tension stresses (negative) was marked as BR – “right” (with the sign “-”). The following cases of bimoment ðBi ¼ L;R Þ distribution in the direction of the segment length were considered: (1) constant, (2) linear, and (3) nonlinear in accordance with 2nd degree parabola. The function of bimoments is written, respectively, in the form: Bi;0 ðxÞ ¼ Bcr;i;0 ;

ð26Þ

Bi;1 ðxÞ ¼ Bcr;i;1 β1 ðxÞ;

ð27Þ

Bi;2 ðxÞ ¼ Bcr;i;2 β2 ðxÞ;

ð28Þ

where β1 ðxÞ according to (17) and β2 ðxÞ according to (18) for xs  x. For a univocal distinction of bimoments BL and BR, the coefficients ðkω;L ; kω;R Þ were referred to selected edge compressive stresses ðsω;cr;L ; sω;cr;R Þ in the most loaded cross-section of the thin-walled bar segment (cf. Figs. 7, 8 and 9). On the basis of dependences derived above, the computational programs: “Z_LBcr.nb”, “C_LBcr.nb” and “I_LBcr.nb” were developed in the environment of the package Mathematica [22]. The programs make it possible to determine: critical warping stresses (eigenvalues), coefficients kω, and local buckling modes (eigenvectors) for a thin-walled bars segment with: Z-section (Fig. 7), C-section (Fig. 8), or I-section (Fig. 9) at arbitrary geometrical parameters, according to Eq. (4). The segments can be loaded with bimoment BL or BR of longitudinal distribution, according to formulas (26), (27) or (28). Selected results of calculations (in the form of graphs of coefficients kω) executed for the value of parameters: E ¼205 GPa and ν ¼ 0:3 are presented below. On the basis of critical warping stresses, determined according to Eq. (2), from formula (1), it is possible to determine the eigenvalues of local critical bimoments (Bcr;L or Bcr;R ). Respective examples are shown in Section 6.

Fig. 10. Comparison of graphs of coefficients (kω) of critical warping stresses in a thin-walled bar segment with Z-section (ρ2 ¼ 2; ρ3 ¼ 1, δ2 ¼ δ3 ¼ 1), with buckling coefficients (k) for separate plates of limit boundary conditions on supported edges: (a) kω;L and k for plate b1 and (b) kω;R and k for plate b2.

5.1. The graphs of coefficients kω 5.1.1. The Z-section In Fig. 10a, the graph of the coefficient kω,L of the critical warping stress in a thin-walled bar segment with Z-section (for ρ2 ¼ 2, ρ3 ¼ 1, δ2 ¼ δ3 ¼ 1) loaded with a constant bimoment BL along length is shown by a continuous line. In order to exemplify the degree of elastic restraint of component plate b1 in a bar segment, dashed lines show the graphs of buckling coefficients (k) for a separate plate b1, which is built-in on one-side (upper line) or simply supported on one-side (lower line). Coefficient kω,L for a segment were determined by the program “Z_LBcr.nb”, whereas buckling coefficients (k) for an eccentrically compressed plate b1 (dashed lines) were calculated by the program “Ncr_plate_free.nb”, according to [21] for parameter α1 ¼ 4. In Fig. 10b, the graph of the coefficient kω,R of the critical warping stress is represented by a continuous line for the same segment loaded with a constant bimoment BR on its length. Dashed lines represent graphs of the buckling coefficients (k), known from the literature, e.g. [15] for a separate plate b2, which is longitudinally built-in on both sides (upper line) and simply supported (lower line) and loaded with axial compression. Graphs represented in Fig. 10 (for m¼ 0) have a character of “garland” curves, which makes it possible to estimate the number of half-waves of the local buckling depending on relation ls =b1 . For example for Bcr;R by ls =b1 ¼ 5, three uniform half-wavelengths

Fig. 11. Diagrams of coefficient kω;L for a thin-walled bar segment with Z-section (ρ2 ¼ 2; ρ3 ¼ 1, δ2 ¼ δ3 ¼ 1) under the load of bimoment BL;1 ðxÞ, according to (26) or (27) – continuous line and BL;2 ðxÞ, according to (28) – dashed line.

will be formed between the diaphragms of the segment (in the direction of axis x), whereas for Bcr;L , only 2 half-wavelengths will be formed. The above observation leads to the conclusion, that the optimum distribution (from the point of view of the local stability of a thin-walled bar) of the diaphragms (or transverse ribs) depends on the sense of torsional load which generates local critical bimoments Bcr;R or Bcr;L . In the considered case, for the obtainment of the maximum critical warping stress, diaphragms should be spaced at approx. ls ¼ 3:3b1 for bimoment Bcr;L (cf. Fig. 10a), or approx. ls ¼ 2:3b1 for bimoment Bcr;R (cf. Fig. 10b).

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In the case of variation of bimoment along the length of a thinwalled bar segment, the accuracy of the determined coefficients depends on parameter io, which determines the number of halfwaves of the sine function in the direction of axis xs (formulas (8),

(10)) and relation ls =b1 . On the basis of the analysis of the convergence of results, in practical applications io ¼ 8 was assumed for further calculations of coefficients kω;L and kω;R , where ls =b1 r 6, which gives a sufficient accuracy from the technical of standpoint with a simultaneous reduction of calculations. Figs. 11 and 12 show graphs of coefficients kω;L and kω;R of critical warping stresses at longitudinal variation of bimoment as the function of parameters m and ls =b1 . Graphs refer to a thinwalled bar segment with Z-section (for: ρ2 ¼ 2; ρ3 ¼ 1, δ2 ¼ δ3 ¼ 1), under a load which generates the distribution of bimoment BL (Fig. 11) or BR (Fig. 12) according to formula (26) or (27) – continuous line, and formula (28) – dashed line. The analysis of graphs presented in Figs. 11 and 12 indicates that for unsymmetrical Z-section (in relation to the axis), coefficients (kω;L , kω;R ) and critical warping stresses (sω;cr;L and sω;cr;R ) differ considerably depending on the sense of the torsional load (sign of bimoment). This results from the specificity of stress distribution in Z-section caused by bimoments BL and BR (cf. Fig. 7), and the fact that free edges of plates b1 and b3 (II-type) are far less resistant to compressive stresses in comparison with supported edges of plate b2 (I-type).

Fig. 12. Diagrams of coefficient kω;R for a thin-walled bar segment with Z-section (ρ2 ¼ 2; ρ3 ¼ 1, δ2 ¼ δ3 ¼ 1) under the load of bimoment BR;1 ðxÞ, according to (26) or (27) – continuous line and BR;2 ðxÞ, according to (28) – dashed line.

Fig. 14. Diagrams of coefficients kω;L and kω;R for a thin-walled bar segment with C-section (ρ2 ¼ 2; ρ3 ¼ 1, δ2 ¼ δ3 ¼ 1) under the load generating the distribution of bimoment BL or BR according to (26) or (27) – continuous line, and according to (28) – dashed line.

Fig. 13. A comparison of coefficients of critical warping stresses at linear distribution of bimoment BL or BR according to formula (27) when m¼ 0.5 for different geometrical proportions of the Z-section according to formulas (4): (a) coefficient kω;L , (b) coefficient kω;R . (Note: curve 1: ρ2 ¼ 2; ρ3 ¼ 1, δ2 ¼ 2, δ3 ¼ 1; curve 2: ρ2 ¼ 2; ρ3 ¼ 5=4, δ2 ¼ δ3 ¼ 1; curve 3:ρ2 ¼ 2;ρ3 ¼ 1, δ2 ¼ δ3 ¼ 1 (dashed line); curve 4:ρ2 ¼ 2; ρ3 ¼ 3=4, δ2 ¼ δ3 ¼ 1; and curve 5:ρ2 ¼ 2; ρ3 ¼ 1, δ2 ¼ 1=2, δ3 ¼ 1.).

Fig. 15. A comparison of coefficients kω;L and kω;R at nonlinear distribution of bimoment BL or BR according to formula (28) when m¼ 0.25 for different geometrical proportions of the C-section according to formulas (4). (Note: curve 1: ρ2 ¼ 2; ρ3 ¼ 1, δ2 ¼ 2, δ3 ¼ 1; curves 2L and 2R: ρ2 ¼ 2; ρ3 ¼ 4=3, δ2 ¼ δ3 ¼ 1; curve 3: ρ2 ¼ 2;ρ3 ¼ 1, δ2 ¼ δ3 ¼ 1 (dashed line); curves 4L and 4R: ρ2 ¼ 2; ρ3 ¼ 2=3, δ2 ¼ δ3 ¼ 1; and curve 5: ρ2 ¼ 2; ρ3 ¼ 1, δ2 ¼ 1=2, δ3 ¼ 1).

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Fig. 13 compares the coefficients kω;L (Fig. 13a) and kω;R (Fig. 13b) of critical warping stresses at linear distribution of bimoment BL or BR according to the formula (27) when m ¼ 0:5 for different geometrical proportions of the Z-section according to formulas (4). 5.1.2. The C-section Fig. 14 shows graphs of coefficients kω;L and kω;R of critical warping stresses in a thin-walled bar segment with C-section (for: ρ2 ¼ 2; ρ3 ¼ 1, δ2 ¼ δ3 ¼ 1) under a load generating the distribution of bimoment BL or BR according to formula (26) or (27) – continuous line, and formula (28) – dashed line. The analysis of graphs presented in Fig. 14 shows that for a monosymmetrical C-section (ρ3 ¼ 1, δ3 ¼ 1), the graphs of coefficients (kω;L and kω;R ) are superimposed, and critical warping stresses (sω;cr;L and sω;cr;R ) are equal to one another, irrespective of the sense of torsional load (sign of bimoment). This results from the antisymmetry of signs and values of sectorial coordinates in relation to the axis of cross-section symmetry. Fig. 15 shows the comparison of coefficients kω;L and kω;R at a non-linear distribution of bimoment BL or BR according to formula (28) when m ¼ 0:25 for different geometric proportions of C-section according to formulas (4). For unsymmetrical sections, graphs of coefficients kω;L and kω;R were differentiated by denoting the number of curve with a respective index (L or R). In the case of unsymmetrical C-section (cf. Fig. 15 – curves 2L and 2R as well as 4L and 4R), similarly as in Z-section, coefficients (kω;L and kω;R ) and critical warping stresses (sω;cr;L and sω;cr;R ) are differentiated depending on the sense of the torsional load (sign of bimoment). However, these differences are considerably smaller than those for Z-section. 5.1.3. The I-section Fig. 16 shows graphs of coefficients kω;L and kω;R of critical warping stresses in a thin-walled bar segment with I-section (for: ρ2;4;5 ¼ 1, ρ3 ¼ 2, δ2;3;4;5 ¼ 1 ), under a load generating the distribution of bimoment BL or BR according to formula (26) or (27) – continuous line, and formula (28) – dashed line. The analysis of graphs presented in Fig. 16 indicates that for the bisymmetrical I-section, graphs of coefficients (kω;L and kω;R ) are superimposed, and critical warping stresses (sω;cr;L and sω;cr;R ) are equal to one another, irrespective of the sense of torsional load (sign of bimoment). This results, similarly as for the monosymmetrical C-section, from the antisymmetry of signs and values

Fig. 16. Diagrams of coefficients kω;L and kω;R for a thin-walled bar segment with I-section (ρ2;4;5 ¼ 1, ρ3 ¼ 2, δ2;3;4;5 ¼ 1) under the load generating the distribution of bimoment BL or BR according to (26) or (27) – continuous line, and according to (28) – dashed line.

Fig. 17. A comparison of coefficients kω;L and kω;R at linear distribution of bimoment BL or BR according to formula (27) when m¼ 0.75 for different geometrical proportions of the I-section according to formulas (4). (Note: curve 1: ρ2;4;5 ¼ 1, ρ3 ¼ 2, δ2;4;5 ¼ 1, δ3 ¼ 2; curves 2L and 2R:ρ2 ¼ 5=6, ρ3 ¼ 2, ρ4 ¼ 2=3, ρ5 ¼ 1=2, δ2;3;4;5 ¼ 1; curve 3: ρ2;4;5 ¼ 1, ρ3 ¼ 2, δ2;3;4;5 ¼ 1 (dashed line); curve 4: ρ2 ¼ 1, ρ3 ¼ 2, ρ4;5 ¼ 2=3, δ2;3;4;5 ¼ 1; curve 5: ρ2;4;5 ¼ 1, ρ3 ¼ 2, δ2;4;5 ¼ 1, δ3 ¼ 1=2; and curve 6: ρ2 ¼ 1, ρ3 ¼ 2, ρ4;5 ¼ 1=2, δ2;4;5 ¼ 1, δ3 ¼ 1=4).

Fig. 18. The segment of thin-walled bar with: (a) Z-section, (b) C-section, and (c) I-section under the load generating the linear distribution of bimoment BL or BR according to (27) when m ¼0.5.

of sectorial coordinates in relation to the axis of cross-section symmetry. Fig. 17 presents the comparison of coefficients kω;L and kω;R at the linear distribution of bimoment BL or BR according to formula (27) when m ¼ 0:75 for different geometric proportions of Isection according to formulas (4). For the unsymmetrical section, graphs of coefficients kω;L and kω;R were differentiated by denoting the number of curve with a respective index (L or R). Also in the case of the unsymmetrical I-section (cf. Fig. 17 – curves 2L and 2R), coefficients kω;L and kω;R are differentiated depending on the sense of torsional load (sign of bimoment). However in this instance, similarly as for the C-section, these differences are considerably smaller than those for Z-section. The analysis of graphs presented in Figs. 11, 12, 14 and 16 demonstrates that coefficients of critical warping stresses increase and the “garland” character of curves, characteristic of m¼ 0, disappears with an increasing value of the parameter m (from 0 to 1). For the same value of parameters m (eg. m ¼ 1) and ls =b1 load of a thin-walled bar segment with a bimoment BL;2 (nonlinear distribution) gives a smaller critical warping stress in relation to load with a bimoment BL;1 (linear distribution). A similar regularity is also found for a bimoment BR (cf. Fig. 12).

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6. Example of determining local critical bimoments

Example 1. Problem. Determine local critical bimoments for a segment of a thin-walled bar with (a) Z-, (b) C-, and (c) I-sections subjected to warping torsion which generates a linear distribution of bimoment BL or BR over the segment length according to Eq. (27) for m ¼ 0:5 (Fig. 18). Compare values received with the first yield bimoment By (neglecting local buckling) for the abovementioned exchanged sections fabricated from S355 steel according to [10] (fyb ¼ 355 MPa). The length of the bar segment is ls ¼360 mm. (Note: for comparison, sections are characterized by comparable geometric dimensions). (a) The Z-section (cf. Fig. 7): 1) dimensions of the midline of cross-section: b1,3 ¼ 60 mm; b2 ¼120 mm; t1,2,3 ¼ 1.2 mm; 2) sectorial coordinates and warping constant: ω1 ¼ 27 cm2 ; ω2 ¼ 9 cm2 ; I ω ¼ 389 cm6 ; 3) Euler's stresses for a plate b1 and b2 according to formula (3): sE;1 ¼ 74:1 MPa; sE;2 ¼ 18:5 MPa, 4) for ls =b1 ¼ 6 according to the program “Z_LBcr.nb”: kω;L ¼ 1:528; kω;R ¼ 6:342; (cf. Figs. 11 and 12); 5) critical warping stresses for the compressed edges of Z-section: sω;cr;L ¼ μ1 s0;cr ¼ kω;L sE;1 ¼ 113:2 MPa; sω;cr;R ¼ s0;cr ¼ kω;R sE;2 ¼ 117:3 MPa; 6) local critical bimoments according to formula (1):

with unsymmetrical section (with respect to the axis) depends on the sense of the external torsional load (sign of bimoment), or otherwise, it depends on the settings of the section in relation to the sense of torsion load. For the considered Z-section, the absolute value of local critical bimoment Bcr;R (tension on the free edges of plates b1 and b3) is approx. 3.11 times greater than that of bimoment Bcr;L which causes compression of free edges of plates b1 and b3. However, the absolute value of the first yield bimoment (By), determined on the basis of the Vlasov theory [1] (neglecting local buckling), does not depend on the sense of torsional load (sign of bimoment). For comparison, in the considered Z-section the absolute value of the first yield moment (By) is approx. 3.13 times greater than Bcr;L and approximately equal to Bcr;R . In the case of a thin-walled bar segment with monosymmetrical C-section (Example 1b) or bisymmetrical I-section (Example 1c), the absolute values of local critical bimoments (Bcr;L , Bcr;R ) are equal to one another irrespective of the sense of external torsional load (sign of bimoment). However, the absolute value of the first yield bimoment (By) is approxim. 2.8 times greater than Bcr;L;R for C-section and approx. 3.12 times greater than Bcr;L;R for I-section. It should be noted that in spite of the fact that the considered I-section (Example 1c) is characterized by the greatest relation I ω =ωmax ¼ 34:6 cm4 , the greatest absolute value of local critical bimoment was obtained for Z-section (Example 1a: I ω =ωmax ¼ 14:4 cm4 ) at warping torsion that generates Bcr;R . This results from the specificity of stress distribution in Z-section for which bimoment Bcr;R generates the tension of the free edges of plates b1 and b3 (IItype) and the compression of plate b2 stiffened on both sides (I-type).

2

Bcr;L ¼ sω;cr;L I ω =ω1 ¼ 163 kNcm ; 2

Bcr;R ¼ sω;cr;R I ω =ω2 ¼  507 kNcm ; 7) first yield bimoment (neglecting local buckling): By ¼ 2 f yb I ω =ωmax ¼ 7511 kNcm ; 8) comparison of bimoments: jBcr;R =Bcr;L j ¼ 3:11; By =Bcr;L ¼ 3:13; By =Bcr;R ¼ 1:01; 9) interval of applicability of the Vlasov theory:  507 o B o163 [kNcm2]. (b) The C-section (cf. Fig. 8): 1) b1;3 ¼ 60 mm; b2 ¼ 120 mm; t 1;2;3 ¼ 1:2 mm; 2) ω1 ¼ 22:5 cm2 ; ω4 ¼  22:5 cm2 ; I ω ¼ 272 cm6 ; 3) sE;1;3 ¼ 74:1 MPa; 4) for ls =b1 ¼ 6 according to the program “C_LBcr.nb”: kω;L ¼ kω;R ¼ 1:706; (cf. Fig. 14); 5) sω;cr;L ¼ μ1 s0;cr ¼ kω;L sE;1 ¼ 126:4 MPa; sω;cr;R ¼ μ3 s0;cr ¼ kω;R sE;3 ¼ 126:4 MPa 2 6) Bcr;L ¼ sω;cr;L I ω =ω1 ¼ 153 kNcm ; 2 Bcr;R ¼ sω;cr;R I ω =ω4 ¼  153 kNcm ; 2 7) By ¼ f yb I ω =ωmax ¼ 7 429 kNcm ; 8) jBcr;R =Bcr;L j ¼ 1; By =Bcr;L ¼ 2:8; By =Bcr;R ¼ 2:8; 2 9)  153 o B o 153 ½kNcm . (c) The I-section (cf. Fig. 9): 1) b1;2;4;5 ¼ 60 mm; b3 ¼ 120 mm; t 1C5 ¼ 1:2 mm; 2) ω1 ¼ 36 cm2 ; ω2 ¼  36 cm2 ; I ω ¼ 1244 cm6 ; 3) sE;1;2 ¼ 74:1 MPa; 4) for ls =b1 ¼ 6 according to the program, “I_LBcr.nb”: kω;L ¼ kω;R ¼ 1:536; (cf. Fig. 16); 5) sω;cr;L ¼ μ1 s0;cr ¼ kω;L sE;1 ¼ 113:8 MPa; sω;cr;R ¼ μ2 s0;cr ¼ kω;R sE;2 ¼ 113:8 MPa; 2 6) Bcr;L ¼ sω;cr;L I ω =ω1 ¼ 393 kNcm ; 2 Bcr;R ¼ sω;cr;R I ω =ω2 ¼  393 kNcm ; 2 7) By ¼ f yb I ω =ωmax ¼ 7 1227 kNcm ; 8) jBcr;R =Bcr;L j ¼ 1; By =Bcr;L ¼ 3:12; By =Bcr;R ¼ 3:12; 2 9)  393 o B o 393 ½kNcm . The results of the Example 1a confirm the thesis that the absolute value of the local critical bimoment in a thin-walled bar

7. The example of determining a local buckling mode In the case of the constant distribution of bimoment (for parameter m ¼ 0) over the length of a thin-walled bar segment, the half-waves of buckling which are formed in critical state have an equal length and a constant amplitude, and their number depends on relation ls =b1 . Besides, the symmetry or antisymmetry of local deflections in relation to the transverse axis of the segment is maintained. In the analyzed cases of the variable distribution of bimoment (for m 4 0), the longitudinal variation of warping normal stresses influences the local buckling mode of a thin-walled bar segment. “Half-waves” of buckling, which are generated between diaphragms (along axis xs), have different lengths and decreasing amplitudes. Besides, the conditions of the symmetry or antisymmetry of local deflections in relation to the transverse axis of the segment are not maintained. Example 2. Problem. Determine local buckling mode of component plates for segments of thin-walled bars considered in Example 1a and b. Computational modules of programs: “Z_LBcr.nb” (Z-section) and “C_LBcr.nb” (C-section) were used for the theoretical determination of the local buckling mode, assuming the distribution of bimoment according to Eq. (27) for m ¼0.5. Fig. 19a shows the buckling mode of component plates b1, b2 and b3 of a segment with Z-section (according to Example 1a) under warping torsion, which generates a “left” local critical bimoment ðBcr;L Þ. Deflections were shown in local coordinate sets ðxs ; ys ; zs Þs ¼ 1;2;3 , whose center was assumed to be in this crosssection where the maximum value of bimoment occurs. The eigenvector of the buckling mode was normalized in such a way that maximmum deflections of plates b1 and b3 were 1. Fig. 19b shows, in corresponding sets of coordinates, the buckling mode of component plates of the same segment with Z-section under warping torsion, which generates a “right” local critical bimoment

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Fig. 19. Local buckling mode of component plates b1, b2 and b3 of thin-walled bar segment with: (a) Z-section under the load Bcr;L , (b) Z-section under the load Bcr;R , and (c) C-section under the load Bcr;L , for the data according to Example 1a,b.

Fig. 20. The static system of the thin-walled bar with Z-section subjected to warping torsion (Example 3).

ðBcr;R Þ. In this case, the eigenvector of the buckling mode was normalized in such a way that maximum deflections of plate b2 were 1. In turn, Fig. 19c shows the buckling mode of component plates b1, b2 and b3 of a segment with monosymmetrical C-section (according to Example 1b) under warping torsion which generates the “left” local critical bimoment ðBcr;L Þ. In this case, the eigenvector of the buckling mode was normalized so that the maximum deflection of plate b1 was 1. However, the buckling mode of the same segment under warping torsion, which generates the “right” local critical bimoment ðBcr;R Þ, is antisymmetric in relation to the axis of the cross-section symmetry. The comparison of the theoretically determined local buckling modes of the considered sections shows that local stability loss is

determined, to a large extent, by slenderness of plates and their boundary conditions on the most compressed longitudinal edges. The local buckling mode of a segment with Z-section for Bcr;L (Fig. 19a), demonstrates that its stability loss is determined by eccentrically compressed plates b1 and b3 (II-type), which extort the symmetrical form of deflection of the tensioned plate b2. In the case of the same segment with the Z-sections for Bcr;R (Fig. 19b), its local buckling is generated by a compressed plate b2 (I-type). Characteristic deflections of plates b1 and b3 result in this case from the effect of the buckled plate b2, which is limited when they approach the free edge, due to the “straightening” effect of warping tensile stresses. In turn, the local buckling mode of a segment with Csection for Bcr;L (Fig. 19c) indicates that its stability loss is determined by the eccentrically compressed plate b1 (II-type), which extorts the unsymmetrical form of deflection of plate b2 (web).

8. A theoretical estimation of critical torsional load In any arbitrary case of multisegment thin-walled bar (containing p Z3 diaphragms or transverse ribs), the conservative estimation of the critical torsional load (from the condition of local buckling) can be obtained from an analysis of the stability of separate segments. The accuracy of results obtained in this way

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will be greater, the greater is the length of component segments. In this case, the effect of mutual elastic restraint of component plates (walls) of adjacent segments decreases. However, in the case of symmetrically supported and loaded, and symmetrically transversely stiffened thin-walled bars composed of an even number of segments, deflections of component plates generated by a local buckling are antisymmetric in relation to the diaphragm which stiffens the cross-section at the mid-span of a member. The critical torsional load can be determined on the basis of local critical bimoment for the most loaded segment. Local boundary conditions of component plates on central segment joints are, in this case, conformable to simple support. A theoretical determination of critical torsional load of a thin-walled bar with use of the symmetry of the construction and load is shown in Example 3. Example 3. Problem. For a warping torsion in thin-walled bar (Fig. 20) with Z-section according to Example 1a, determine the critical value of torsional load from the condition of local buckling depending on the sense of external torsional moment (M t;L or M t;R ). Data: L ¼ 1440 mm, l1 ¼ 360 mm (transverse rib spacing), b1;3 ¼ 60 mm; b2 ¼ 120 mm; t 1;2;3 ¼ 1:2 mm; ω1 ¼ 27 cm2 ; ω2 ¼ 9 cm2 ; I ω ¼ 389 cm6 ; I t ¼ 0:01365 cm4 ; E ¼ 205 GPa; G ¼ 80 GPa; ν ¼ 0:3. The constructional system of a thin-walled bar consists, in this case, of four segments at ls =b1 ¼ 6. Deflections generated by the local buckling of matching component plates (walls) of central segments are antisymmetric in relation to the central diaphragm, in the plane of which the external torsion moment is active. The differential equation of the nonuniform torsion [1], when the thin-walled bar is supported and loaded according to Fig. 20, can be represented in the form EI ω ϕIV  GI t ϕII þ M t δx ¼ L=2 ¼ 0

ð29Þ

The bimoment function determined from Eq. (29) by boundary conditions according to Fig. 20 has the form [1] BðxÞ ¼

M t sinhðκxÞ 2κ coshðκL=2Þ

ð30Þ

In the case of thin-walled members, which are characterized by considerable slenderness ðλps Þ of component plates, torsional stiffness depends mostly on the stiffness of warping torsion ðEI ω Þ. At a small thickness of walls, the stiffness of St. Venant's uniform torsion ðGI t Þ is negligibly small, and the flexural-torsional coefficient (κ) of the cross-section shows very small values p (near zero). ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi For the considered cross-section, it is only: κ ¼ GI t =EI ω ¼ 3:7  10  3 cm  1 . On the basis of the above statements, the function of bimoment for a thin-walled bar, supported and loaded according to the diagram shown in Fig. 20 at κ-0, can be determined from the formula   M t sinhðκxÞ Mt x ¼ ð31Þ BðxÞ ¼ lim κ-0 2κ coshðκL=2Þ 2 In this case, the absolute value of bimoment at the mid-span is: BðL=2Þ ¼

Mt L 4

are: M t;cr;L ¼ 4 U Bcr;L =L ¼ 4:5 kNcm and M t;cr;R ¼ 4 U Bcr;R =L ¼ 14:1 kNcm, respectively. Determined on the basis of the Vlasow theory [1] (within the pre-buckling range), the maximum warping shear stresses are: τω;L ¼ 3:5 MPa under load M t;cr;L and τω;R ¼  11 MPa under load M t;cr;R . Small values of warping shear stresses confirm the thesis concerning their secondary effect on the local buckling of open thin-walled bars with a flexible cross-section contour. Results of Example 3 also confirm the thesis that the absolute value of the critical torsional load of a thin-walled bar with an unsymmetrical section depends on its settings in relation to the sense of torsional load. Similarly, as for local critical bimoments (Bcr;L , Bcr;R , cf. Example 1a), the absolute value of the critical torsional moment M t;cr;R is approx. 3.11 times greater than the absolute value M t;cr;L . In the case of a thin-walled bar with an corresponding static system, as shown in Fig. 20, but containing only the diaphragm at mid-span (in the plane of external torsional load) and transverse stiffeners on supports, its constructional system consists of two symmetrical segments of ls =b1 ¼ 12. In this case, coefficients of critical warping stresses, determined by means of “Z_LBcr.nb” program (for bimoment distribution according to formula (27) at m ¼ 1) amounted to: kω;L ¼ 1:528; kω;R ¼ 6:342, and are practically equal to coefficients determined in Example 1a for the segment with ls =b1 ¼ 6 for m ¼0.5. (Note: in calculations for the segment with ls =b1 ¼ 12, the value of the parameter i0 ¼ 16 was adopted). The above result shows that local buckling of the thin-walled bar with a flexible cross-section contour subjected to warping torsion is determined by its zones maximally loaded with bimoment. Consequently, the principle of the correct construction of this class of thin-walled bars is to locate stiffeners (diaphragms or ribs) in these cross-sections in which an extreme value of bimoment occurs.

ð32Þ

For the linear distribution of bimoment in central segments (according to formula (27) for m ¼ 0.5), local critical bimoments 2 according to Example 1a are, respectively: Bcr;L ¼ 163 kNcm for 2 load M t;L , and Bcr;R ¼  507 kNcm for load M t;R . Critical torsional moments from the condition of the local buckling of a thin-walled bar depending on the sense of torsion load and calculated after the transformation of formula (32)

9. Conclusions from the theoretical analysis The load-carrying capacity of thin-walled bars with a flexible cross-section contour (according to Fig. 1c), subjected to warping torsion, is preceded and conditioned by the phenomena of local buckling. An objective measure of local buckling resistance of this class of members with open cross-section is local critical bimoment. Solution of the problem of the local stability loss in a thinwalled bar segment makes it possible to determine the critical warping stresses, which can be used to determine local critical bimoments. The local buckling modes are determined by the slendernerss of component plates and their boundary conditions on the most compressed longitudinal edges. Two different local critical bimoments with regard to absolute value (Bcr;L , Bcr;R ) are found in open thin-walled bars with unsymmetrical cross-section (in relation to axis) depending on the sense of torsional load (sign of bimoment). The way of the arrangement of the section in relation to the sense of torsional load determines the occurrence of a “greater” or “smaller” local buckling resistance of a thin-walled bar in this case. In thin-walled bars with at least one axis of the symmetry of cross-section, in which the antisymmetry of the signs and the value of warping normal stresses in relation to this axis occurs, the absolute values of local critical bimoments ðBcr;L ; Bcr;R Þ are equal to one another, irrespective of the sense of torsional load (sign of bimoment). The variation of bimoment distribution in the direction of a thin-walled bar length generates the unsymmetrical form of local buckling. “Half-waves” with variable lengths and variable (decreasing) amplitudes are formed between diaphragms (or

A. Szychowski / Thin-Walled Structures 76 (2014) 42–55

transverse ribs). The greatest deflections of walls are found on the side of the maximum load of a segment under the action of bimoment. The calculation of the warping torsion in thin-walled bars with a flexible cross-section contour (Fig. 1c) on the basis of the Vlasov theory in the whole elastic range ( By o B o By ) may lead to considerable errors caused by local buckling. The Vlasov theory is useful in the analysis of stress distribution in the cross-section of a thin-walled bar under warping torsion causing bimoment contained in the range: Bcr;R o B o Bcr;L , (cf. Example 1). After exceeding the critical load, a thin-walled bar surrenders to local buckling and the basic assumption of the Vlasov theory [1] concerning the rigid cross-section contour loses its validity. Finer spacing of transverse stiffeners in a thin-walled bar with a flexible cross-section contour leads to an increase of the range of the applicability of the Vlasov theory. However, finer spacing of diaphragms (or ribs) in thin-walled steel members currently used so that the Vlasov theory could be applied in the whole elastic range is not rational. For example, for the C-section (according to Example 1b), in order to apply the Vlasov theory to the whole elastic range, diaphragms should be spaced at approx. 37 mm intervals. However, the rule of the correct construction of this class of thin-walled members is to locate stiffeners (diaphragms or ribs) in these cross-sections in which an extreme bimoment value occurs. References [1] Vlasow VZ. Thin-walled elastic beams. Jerusalem: Israel Program for Scientific Translations; 1961. [2] Murray NW, Lau YC. The behavior of a channel cantilever under combined bending and torsional loads. Thin-Walled Struct 1983;1:55–74. [3] Kavanagh KT, Ellifritt DS. Design strengths of cold-formed channels in bending and torsion. J Struct Eng ASCE 1994;120:5. [4] Put BM, Pi YL, Trahair NS. Bending and torsion of cold-formed channel beams. J Struct Eng ASCE 1999;125(5):540–6.

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