Numerical Analysis and Measurement of Non-Uniform Torsion

Transcription

1 Paper Civil-Comp Press, 0 Proceedings of the Eleventh International Conference on Computational Structures Technology, B.H.V. Topping, (Editor), Civil-Comp Press, Stirlingshire, Scotland Numerical Analysis and easurement of Non-Uniform Torsion J. urin, T. Sedlar, V. Kralovic, V. Goga, A. Kalas and. Aminaghai Department of Applied echanics and echatronics UEA Faculty of Electrical Engineering and Information Technology Slova University of Technology in Bratislava, Slovaia Institute for echanics of aterials and Structures Vienna University of Technology, Austria Astract The suject of the present paper is an innovative approach to the investigation of the non-uniform torsional ehaviour of prismatic eams taing into account the secondary torsional moment deformation effects. This investigation contains computational and experimental parts for oth open and closed thin-walled crosssections. Results of original experimental measurements of some warping-torsion features will e presented and compared with the ones otained analytically and numerically. The necessity of considering the warping effects for closed thin-walled eams are studied and evaluated. The accuracy and the correctness of the results otained are evaluated and discussed. The main novelty in this study is the measurement of some of the most important warping parameters, especially for closed cross-sections. These measurements prove the importance of warping consideration in non-uniform torsion of open and closed thin-walled profiles, and also the necessity of considering the secondary torsion moment deformation effect, especially in closed cross-sections. Keywords: non-uniform torsion, warping, open and closed cross-sections, measurement and numerical calculation, secondary torsional moment deformation effect. Introduction The effect of warping must e assumed in stress and deformation analyses of structures aove all with thin-walled cross-sections loaded y torsion. Warping effects occur mainly at the points of action of the concentrated torsion moments (except for free eam ends) and at sections with free-warping restrictions. Special theories of torsion with warping non-uniform torsion have een used to solve such prolems analytically (e.g. []). The analogy etween the nd order eam theory (with tensional axial force) and torsion with warping is also very often used

2 (e.g. [], []). One has to point out that in the literature and practice, as well as in the EC- [4] and EC-9 [5] guidelines, strong warping is assumed to occur in open crosssections only. Warping-ased stresses and deformations in closed sections, however, are assumed to e insignificant and have een therefore neglected. According to the aove mentioned theory of torsion with warping and the analogy, special D-eam finite elements have een designed and implemented into the finite element codes (e.g. [6], [7]). The warping effect is included through an additional degree of freedom at each nodal point - the first derivative of the angle of twisting of the eam cross-section. Important progress in the solution of torsion with warping has een reached in papers [8] and [9] where a comination of oundary and finite element method was used allowing a warping analysis for composite eams with longitudinally varying cross-section. However, latest theoretical results have shown that the effect of warping must e considered in the case of non-uniform torsion of closed-section eams [0]. For prismatic eams, the analogy etween the torsion with warping (including the secondary torsion moment deformation effect) and the nd order eam theory (including the shear force deformation effect) has to e used. This approach was implemented into the computer code IQ-00 []. This analogy does not hold for non-prismatic eams []. According to the last research results in this area ([0], [], [], [4]), the local stiffness relation of a new two-node finite element for torsion with warping of straight eam structures is presented in contriution [5]; again ased on the aove-mentioned analogy. The warping part of the first derivative of the twist angle has een considered as the additional degree of freedom in each node at the element ends which can e regarded as part of the twist angle curvature caused y the warping moment. This new finite element can e used in non-uniform torsion analyses of constant open and closed cross-section eams. Finally in [6], the oundary element method has een applied in the non-uniform torsion analysis of simply or multiply connected ars of douly symmetrical aritrary constant cross-section, taing into account secondary torsional moment deformation effects. Necessity of the non-uniform torsion effect in the analysis of close shaped cross-section has een confirmed. It has een concluded in various papers and studies (e.g. [], [5], [6]) that results given y the classical thin tue theory and numerical analyses can differ significantly in the stress calculation especially in the analysis of the closed crosssections if the effect of secondary torsion moment deformation effect is neglected. Additionally, the inaccuracy of the thin tue theory in calculating the secondary torsion constant even for closed thin-walled cross-sections has een oserved in [6], [7]. Hence, it is straightforward to carry out measurements of the most significant features of non-uniform torsion: the total torsion angle, warping displacement and the normal stress caused y imoment. The measurement results can then e used in evaluation of the calculated results otained y several theories and commercial software outputs. It is worth here noting that there is a lac of relevant experimental results availale in literature. This was a strong motivation for the authors to deal with design of a new measurement device which would e ale to measure some deformation effects and normal stress caused y non-uniform torsion. The aim of

3 this paper is also the presentation of this new measurement device [8] which was used in verification and evaluation of the results otained y the aove mentioned theories. Predominantly, the measurement of some major warping effects will ring proof for the necessity of normal (imoment) stress consideration in the design and analysis of closed thin-walled eams. Design of the measurement device The measurement device (Figure ) consists of a welded ase frame to which the measured eam is attached via a set of supporting clamps enaling fixed (clamped) inding. At this particular position, all six DOFs are restrained. Four closed-section races are welded onto the ase holding a circular earing jewel in which a radial all-earing is set. A hollow two-arm-hu is rotating in the earing. Dimensions of the measured eam are limited y the hole of the hu. An exchangeale steel hudis is designed to enale torsion moment application via a twisting force couple onto individual eams. The force application is realized y two teflon screws with spherical endings. The hu-disc is fastened to the hu via four safety olts. The centre-line of the hu s arms runs through the centre of the measured sample. Both arms have specially shaped cut-offs with splineways and holes. The tension wires are guided through splineways, set of rollers to a tension gear. This mechanism enales the torque application and restrictions at one end of the eam result in nonuniform torsion. The measurement device provides opportunities to measure some parameters of non-uniform torsion. A laser-device is fixed on one of the rotating hug-arms while its projection (e.g. onto a wall) enales the total torsion angle calculation. The total warping displacements is measured via a micro-displacementmeter. Longitudinal normal (imoment) stress distriution is calculated from measured values of strain via a set of attached resistant strain gauges and a logger. Results are then compared with theoretical predictions. Figure : Prototype of the measurement device the mechanical parts.

4 echatronic set-up (Figure ) used for the measurement of non-uniform torsion deformation and stress is as follows: linear strain gauges (HB 6/LY type), universal amplifier QuantumX X440 A and a laptop with software for acquisition of measurement data Catman Easy. Figure : The measurement parts and equipments. The measurement device was designed and faricated at Department of Applied echanics and echatronics of IEAE of FEI STU in Bratislava. Theoretical ase for torsion with warping including the secondary torsion moment deformation effect. Semi-analytical approach As previously mentioned, the effect of warping must e considered also in the case of torsion of constant closed-shaped cross-sections [], [], [4]. The analogy etween the torsion with warping (including the secondary torsion moment deformation effect) and the nd order eam theory with tensional axial force (including the shear force deformation effect) can e used in this case. The analogical variales and constants, differential expressions and equations, oundary conditions and expressions for stress and deformations calculations are descried in [4] in full length. 4

5 Torsion with warping i m T e T,, Tp Ts T x ϑ e P Bending of a eam (with tensional force) i m e Q,w N, u q, ϕ Fig.. Analogical state variales Similar to the derivation of transfer matrices for the nd order eam theory for constant cross-section under shear force deformation effect, and after implementation of the analogical variales according to Figure, the transfer matrix for torsion with warping has een constructed []: ϑ ϑ = 0 0 Ts 0 a 0 0 κ 0 K 0 ϑi ϑi i Tsi () with GIT K = κ and f = K eing the transfer functions j derived as: EI sinh( fx) 0 = cosh( fx), =, for f and with a 0 =, for j x j : a j =. j! j : j = j a K j These transfer functions can e calculated also y the special exponential functions descried in [0]. Here: ϑ = ϑ( x) - is the twist angle transfer function and ϑ i is its value at node i; ϑ = ϑ ( x) - is the warping part of the first derivative of the twist angle transfer function and ϑ i is its value at node i; = ( x) - is the imoment transfer function and i is its value at node i; Ts = Ts ( x) is the 5

6 secondary torsion moment transfer function and Tsi is its value at node i. The stiffness constants are: = ; EI = EI ; =. EI GI Ts The aritrary cross-sectional characteristics are descried y: A cross-sectional area [m ]; I T torsion constant [m 4 ]; I Ts secondary torsion constant [m 4 ]; I - warping constant [m 6 ]. aterial properties: E elasticity modulus; G shear modulus. The secondary torsion moment deformation effect is encountered through constant I = + T κ and the transfer constants j, j 0,. Parameter κ = if this effect is I Ts neglected. This is usually made in the case of open form cross-sections where the effect of the secondary torsion moment has een assumed insignificant. Deformation effect of the secondary torsion moment must e considered first of all in the case of the closed form cross-sections as demonstrated in []. The expressions for calculation of the secondary torsion constant (denoted as I Ts ) depend on the cross-section type. These can e found in [8] and [], for example. This approach was implemented into the computer code IQ-00 [], which can e used in analysis of torsion with warping of the closed and opened thin walled crosssections.. Finite and oundary element method approach The aove-descried transfer functions with x = L (L is the length of the rod) have een originally used to estalish the local stiffness relation of the finite element for torsion of a straight eam structure with warping [5]. Despite authors efforts, no other finite element designed for warping-torsion including the secondary torsion deformation effect has een found in literature to the present date. Significant contriution, from point of view of secondary torsion moment consideration and verification of necessity of the warping inclusion into the torsion of the solid and closed cross section analysis, was presented in papers [6] and [7]. In these papers, the Boundary Element ethod (BE) has een used for the solution of such prolems inclusive calculation of the secondary torsion and warping constants. Figure 4 shows a prismatic eam element of length L with two nodes i and, and with appropriated geometric, static, inematics and material quantities as descried aove y the semi-analytical approach. In comparison with Figure, the classical finite element method definition has een used for positive orientation of the nodal variales for this approach. In order to include the warping, a new degree of freedom is added to the classical nodal variales in the stiffness matrix in each element nodal point. The warping part of the first derivative of the twist angle ( ϑ ) has een originally considered as the additional degree of freedom in each node at the element ends. This can e regarded as part of the twist angle curvature which is caused y the warping moment. 6

7 7 Figure 4: Prismatic finite element for torsion. This choice rings advantages when applying the oundary conditions. If the secondary torsion moment deformation effect is not considered ( ) ( ) x x ϑ ϑ =. The nodal displacement vector in the local coordinate system, as shown in Fig. 4, is { } { } i i T e u ϑ ϑ ϑ ϑ = () and the respective nodal load vector is { } { } T i Ti T e F = () where Ti and T are the torsion moments, i and are the warping moments at the nodal points. The geometrical meaning of all these inematics and static variales is evident from Figure 4. Using the aforementioned transfer matrix relations () (after some mathematical manipulations) the element local stiffness relation for torsion with warping (including the secondary torsion moment deformation effect) has een derived. The resultant relation can e written as [5]: = i i T i Ti ϑ ϑ ϑ ϑ κ κ κ κ κ κ κ 0 0 (4)

8 For the straight eam structures the local relation (4) coincides with the gloal one. The implementation of the expression (4) into the local equation of the D-eam finite element is straightforward, and it will e done in our future wor. When the inematic and static variales in (4) are nown at the nodal points, the primary and secondary torsion moment, normal and shear stress can e calculated in a usual way [9]. The nodal points secondary torsion moments are: Tsi = κ ( GI ϑ ); κ ( GI ϑ ) Ti T i Ts = (5) T T The primary torsion moments at the nodal points are: Tpi = Ti Tsi; Tp = T Ts (6) Expressions for the shear and normal stress calculation depend on the crosssectional area type. This prolem has een descried in detail in [0]. These expressions will e now used for stress calculations in our numerical examples. This new finite element can e used for analysis of torsion with warping of constant oth open and closed-shaped cross-sections. 4 Numerical experiment and measurement of nonuniform torsion of cantilever eams The aim of this chapter is the experimental verification of theoretical findings that the analogy etween the torsion with warping (including the secondary torsion moment deformation effect) and the nd order eam theory with tensional axial force (including the shear force deformation effect) holds. The most important verification is the experimental proof of the necessity of warping consideration in closed thin-walled cross-sections loaded y torsion. Torsion of clamped prismatic eams of oth open and closed cross-sections will e studied. Otained results will e evaluated and discussed. 4. U-eam loaded y torsion with warping The aim of the following investigation is the measurement and calculation of the total twisting angle, total warping displacement at the unrestrained end of the eam. The normal stress flow along the we of the U-eam caused y the imoment will e determined. A prismatic cantilever aluminium eam of length L = 0.8 m and constant U- shaped cross-section (E = 79 GPa Young modulus, G = E/((+ν )) =. GPa shear modulus, ν = Poisson ratio), loaded y torsion moment = 5.9 Nm at its free end is depicted in Fig. 5. 8

9 Figure 5: Geometrical parameters, normal and shear stress distriution. According Figure 5, the following cross-section parameters have een chosen and calculated [], [9]: h = 0.07 m, = m, t = s = 0.00 m, A = A s + A G = hs+ +t = m the cross-sectional area containing the area of the we A s and the area of flange plates A G, e = = the auxiliary constant, 6 + A S / A G e = = m - the eccentricity of the shear centre, = eh / = AS + A G = m, = ( e) h / = m - warping ordinate at S corners and, I = ( + ) A + A G = m 6 - the IT = AG t + AS s = m 4 the torsion constant, e 8 S = A G = m 4, e S0 = S = m 4, e α = e = and β = e = auxiliary constants, 9 warping constant; ( ) ( βh) 7 I Ts = = the secondary torsion α α h β t 5s constant. Parameters S and S 0 are used in shear stress calculation. The eam was analyzed with program ANSYS [6], IQ-00 [], and with new effective eam element [5]. Prior to the clamping and setting of the eam within the measurement device, a set of linear gauges needs to e glued onto the eam surface. At this point, one has to point out that the normal stress (and strain) is a function of all coordinates, i.e. imoment changes with the longitudinal direction and warping ordinate changes m 4 9

10 along the camer line. Therefore a special care has to e given to this process. In our experiment, six linear strain gauges (T T 6 ) are glued to the tension side of the aluminium eam along its length (Figure 6). Three of them have (T, T and T 5 ) have a pair gauge (T 7, T 8 and T 9 ) placed exactly on the opposite (compression) side of the eam as a mirror image. The correct set-up is affirmed when each pair of gauges shows identical asolute values of stress. y T,i T T 4 T 5 T 6 T T x T,i x y T 9 L T 8 T 7 z Figure 6: Positioning of the strain gauges. The twisting force couple (torque) was applied via insertion of weights to the tension mechanism of the measurement device. Four values of twisting angle were generated and each measurement was repeated ten times. Average values were then used for the comparison with theoretical results. In theoretical analysis, the outputs eing the imoment, primary and secondary torsional moment. These values were determined using the software IQ-00. Normal and shear stresses (Figure 5) at any point of the eam can e calculated according to [], [], [] and [8], respectively. For this particular U-eam the calculated parameters are as follows: maximal normal stress σ =. Pa, maximal primary shear stress τ p, f =. 04 Pa in the flange, maximal secondary shear stress o τ s, w =.8 Pa in the middle of the we. The total torsion angle is ϕ = σ ( x) aximum warping displacement calculated is x = dx = 0 µm. ( l ) E The numerical calculations were otained using the Beam88 element specially designed within the ANSYS software [6]. A fine mesh was used in order to acquire the imoment flow as well as normal and shear stress distriution. Geometrical and warping parameters are identical with the ones calculated analytically (see aove) hence the normal stress σ, AN flow will e derived from ANSYS imoment output. aximum normal stress is σ, AN = Pa. The inputs for shear stress calculation, e.g. the primary and secondary torsion moments are not availale, i.e. only a summation of primary and secondary shear stresses is generated for each 0

11 element. Therefore, at this stage, the experimental analysis will also e restrained to normal stresses only. The measured sample (Figure 6) has een set within the measurement device according to the process descried in the measurement set-up section aove. The e applied twisting force couple generating a torque of T = 5. 9 Nm and the restrictions at the clamped end result in its non-uniform torsion (and imoment distriution). The measurement of normal stress flow along the eam's length has een carried out at six positions (Figure 6). Due to the nature and size of the linear gauges it is impossile to position them at the very end of the camer line. Their positions can e estimated with the precision of aout 0.5 mm and need to e taen into account when used in comparisons with other methods. Gauges' positions (x T,i - longitudinal, y T,i - distance from camer end), measured normal stressσ, ex, analytical calculations - σ, an (in accordance with [], [], [], [5], respectively) and FE calculations with ANSYS - σ, FE, as well as the difference evaluations δ ex/an are present in Ta.. gauge analyzed position normal stress difference x T,i [mm] y T,i [mm] σ,ex [Pa] σ,an [Pa] σ,fe [Pa] δ ex/an [%] T / T T / T T T T 5 / T T Tale : Comparison of calculated and measured normal stresses in eam's longitudinal direction. The difference value is calculated as: δ ex / an = 00 ( σ, an σ, ex) / σ, an. The first four measurements are in good agreement with the theory and numerical results. For the last two, the total difference is of aout.5 Pa. This can refer to the well-nown fact that for small strains linear gauges do not give accurate results. The total torsion o angle measured is ϕ = 6. and the maximum warping displacement x = 97 µm. Both agree well with analytical and numerical predictions. A more visual interpretation of the normal stress flow is in Figure 7. Theoretical and numerical results do not show any significant difference among themselves.

12 Figure 7: Normal stress flow along the eam length: measured values in comparison with numerical (ANSYS Beam88) and analytical results at gauges' positions. 4. Closed-section eam loaded y non-uniform torsion As discussed earlier, in literature and papers there is a lac of availale experimental evidence of non-uniform torsion ehaviour for closed-section eams. In this suchapter, the authors present a complex analytical, numerical and experimental study of a closed-section prismatic eam loaded y non-uniform torsion ringing an experimental proof of the necessity of warping consideration for such structures. In addition, a large disproportion in calculations of torsion and warping constants using various analytical and numerical approaches will e shown and discussed. In the analysis, measurements and calculations for the total twisting angle, the total warping displacement at the unrestrained end and normal stress flow will e discussed. The measured material properties of the prismatic cantilever aluminium eam (Figure 8) are as follows: Young modulus E = 79 GPa, shear modulus G = E/((+ν )) =. GPa and Poisson ration ν = Figure 8: Geometrical parameters, normal and shear stress distriution.

13 According Figures 8 and 9, the following geometrical parameters have een chosen and calculated [], [9]: L = 0.8 m, L = 0.5 m, h = m, = 0.08 m, t = s = 0.00 m, A = A s + A G = (hs+ t) = m - the cross-sectional area containing the area of the wes A s and flanges A G, h ht s 6 R = = m A - warping ordinate at the corners, I = R = 4 ht + s ( h) 8 - =.97 0 m 6 - the warping constant; I T = = m 4 - the h / s + / t 0( h / s + / t) I A torsion constant, ITs = = ( Ah) ( h + ) + AS AG torsion constant. The eam was analyzed using the ANSYS [6] and IQ-00 [] software, respectively and with new effective eam element [5]. The clamping process has een followed exactly as in section 4.. In our analysis, five linear strain gauges (T T 5 ) are glued to the tension side of the aluminium eam along its length (Figure 9). Gauge T has a pair gauge T 6 placed exactly on the opposite (compression) side of the eam as a mirror image. The correct set-up is affirmed when the pair shows identical asolute values m 4 - the secondary e The twisting force couple (torque) generating a torque of = 00 Nm was applied T via the tension-wire mechanism of the measurement device at position L. In comination with restrictions at the clamped end and position of the torque, this setup resulted in non-uniform torsion and imoment distriution (Figure 0). Four values of twisting angle were generated and each measurement was repeated ten times. Average values were then used for the comparison with theoretical results. Figure 9: Positioning of the strain gauges. In theoretical analysis, the outputs eing the imoment, primary and secondary torsional moment. These values were determined using the classical TTT [] and the software IQ-00. Normal and shear stresses (Fig. 8) at any point of the eam can e calculated according to [], [], [] and [8], respectively. For this particular

14 e eam and the torsion moment = 00 Nm, the calculated parameters are as T follows: maximal imoment = 0.4 Nm, normal stress σ = R, an = I = 9.77 Pa, maximal primary shear stress τ = =. 95 Pa, maximal p, max ht Ts R secondary shear stress τ s, max = = 6. 7 Pa. The total I A h + t G + 4 6( h / s + / t) o torsion angle is ϕ =.56. aximum warping displacement calculated is σ ( x) x = dx = 0.85 µm; this value is strongly influenced y the torque position ( l ) E and imoment distriution. The numerical calculations were also otained using the ANSYS Beam88 element [6]. A fine mesh was used in order to acquire the imoment flow as well as normal and shear stress distriution. Warping parameters, in particular the warping constant - I, FE =.5 0 m 6, differ from analytical calculations (as pointed out in e.g. [6], [7]). oreover, the element tale output data for imoment at the clamped end give,fe = 0.8 Nm and the corresponding maximum normal σ, FE = 6. 4Pa - twice the value determined analytically [] and []. The inputs for shear stress calculation are not availale (see section 4.). Comparison of analytical and numerical calculations of the cross-sectional and warping parameters, maximum imoment and normal stress values are presented in Tale. The imoment flow is depicted in Figure 0. Tp ANSYS TTT TTT parameter Beam88 Ruin classical I T [mm 4 ] I TS [mm 4 ] I [mm 6 ] R [mm ] [Nm ] σ max [Pa] Tale : Cross-sectional and warping parameters, maximum imoment and normal stress comparisons: numerical and analytical calculations. 4

15 Figure 0: Bimoment flow: FE calculation otained via ANSYS Beam88, classical TTT [] and semi-analytically [],[]. The measurement of normal stress flow along the eam's length has een carried out at 5 positions; gauges positions x T,i and y T,i (Figure 9), corresponding measured normal stressσ, ex, analytical calculations σ, an [] and FE calculations with ANSYS Beam88 σ, FE as well as the difference evaluations δ ex/an are present in Tale. The difference value is calculated as δ ex / an = 00 ( σ, an σ, ex) / σ, an. gauge analyzed position normal stress difference x T,i [mm] y T,i [mm] σ,ex [Pa] σ,an [Pa] σ,fe [Pa] δ ex/an [%] T / T T T T T Tale : Comparison of calculated and measured normal stresses in eam's longitudinal direction. A more visual interpretation of the normal stress flow is depicted in Figure showing significant differences etween analytical and numerical (ANSYS) results. Experimental investigation has confirmed the proposed analytical and semianalytical predictions well. 5

16 Figure : Normal stress distriution along the eam s longitudinal direction: measured values in comparison with numerical (ANSYS Beam88) and theoretical results re-calculated to gauges' positions. The first four measurements are in very good agreement with the theory. However, there is an almost Pa difference in the last measured value. Also, the total o torsion angle measured is ϕ =.5 which gives a 5. % difference in comparison with theoretical result. Both discrepancies can e caused y imperfection in the external torque application which will e improved with device-modifications currently in process. The small values of warping displacement (see aove) were immeasurale with the micro-displacement-meter. 6

17 5 Conclusions This paper represents a valuale contriution in the analysis of thin-walled eams suject to non-uniform torsions. Authors novel approach includes semi-analytical and numerical analyses and measurement of some ey warping-torsion features such as normal (imoment) stress flow, torsion angle and warping displacement for oth open and closed cross-sections. For the measurement a specially designed measurement device has een uilt. The majority of the otained results are in a good agreement with theoretical predictions. However, numerical analysis of the closed cross-sections, as carried out using commercial software [6], differs significantly from measurement. As expected, the importance of the inclusion of warping in the analysis of oth open and closed cross-sections has een confirmed experimentally. This is in contradiction with the widely used eurocodes [4], [5]. The measurements have also verified the necessity of the secondary torsion moment deformation effect consideration especially in closed cross-sections analyses. In their future wor, the authors will focus on a complex analysis of thin-walled structures that will include the Thin Tue Theory, Boundary Element ethod and Finite Element ethod using eam, shell and solid elements. In addition, the disproportion in calculations of the warping constant and secondary torsion constant will e analyzed and evaluated. Acnowledgement: This paper has een supported y Grant Agency VEGA (grant No. /054/). References [] V.Z. Vlasov, Tenostěnné pružné prúty, Praha, SNTL, 96. [] K. Roi, G. Sedlace, Theorie der Wölrafttorsion unter Berücsichtigung der seundären Schuverformungen - Analogieetrachtung zur Berechnung des querelasteten Zugstaes, Stahlau, 5, 4, 966. [] H. Ruin, Wölrafttorsion von Durchlaufträgern mit onstantem Querschnitt unter Berücsichtigung seundärer Schuverformung, Stahlau, 74, Heft, 86, 005. [4] EC- Eurocode. Design of steel structures. [5] EC-9 Eurocode 9. Design of aluminium structures. [6] ANSYS Swanson Analysis System, Inc., 0 Johnson Road, Houston, PA 54/00, USA. [7] RSTAB, Ingenieur - Software Dlual GmH, Tiefenach, 006. [8] E.J Sapountzais, V.G. oos, -D eam element of composite cross-section including warping and shear deformation effect, Computers and Structures, 85, 0,

18 [9] V.G. oos, E.J. Sapountzais, -D eam element of variale composite cross-section including warping effect, Acta echanica, 7 (-4), 70, 004. [0] H. Ruin, Torsions-Querschnittswerte für rechtecige Hohlprofile nach EN 00-: 006, Stahlau, 76, Heft, 007. [] H. Ruin, Zur Wölrafttorsion geschlossener Querschnitte und ihrer Irrtümer - Grundlagen, 9th Stahlauseminar, Band 40, ISSN: , 007. [] H. Ruin,. Aminaghai, Wölrafttorsion ei veränderlichen, offenem Querschnitt - hat die Biegezugstaanalogie noch Gültigeit?, Stahlau, 76, 747, 007. [] H. Ruin,. Aminaghai, H. Weier, IQ-00. The civil engineering structures program, TU Vienna, Bautaellen für Ingenieure, Werner Verlag, 7. Auflage, 006. [4] H. Ruin, Baustati, anusript der Vorlesung. Institute of Structural Analysis, Vienna University of Technology, 007. [5] J. urin, V. Kutis, An Effective Finite Element for Torsion of Constant Cross-Sections Including Warping with Secondary Torsion oment Deformation Effect, Engineering Structures, ISSN , Vol. 0, Iss. 0, 76-7, 008. [6] V.G. oos, E.J. Sapountzais, Secondary torsional moment deformation effect y BE, International Journal of echanical Science. IJS-0R. Accepted for pulication, 0. [7] V.G. oos, E.J. Sapountzais, A Three-Dimensional Beam including Torsional Warping and Shear Deformation Effects arising from Shear Forces and Secondary Torsional oments, in B.H.V. Topping, Y. Tsompanais, (Editors), Proceedings of the Thirteenth International Conference on Civil, Structural and Environmental Engineering Computing, Civil-Comp Press, Stirlingshire, UK, Paper 7, 0. doi:0.40/ccp.96.7 [8] J. urin, T. Sedlar, A. Kalas, V. Kralovic, A device for measurement of nonuniform torsion of the thin-walled cross-sections, Patent 607 Slova Repulic, 0 (In Slova). [9] H. Ruin, Vorlesung: BAUSTATIK. Grundlegende Formeln der Theorie II. Ordnung, Institut für Baustati TU Wien,