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Abstract

The study presents the results of theoretical investigations into lateral torsional buckling (LTB) of bi-symmetric I-beams, elastically restrained against warping at supports. Beam loading schemes commonly used in practice are taken into account. The whole range of stiffness of the support joints, from free warping to warping fully restrained, is considered. To determine the critical moment, the energy method is used. The function of the beam twist angle is described with power polynomials that have simple physical interpretation. Computer programs written in symbolic language for numerical analysis are developed. General approximation formulas are devised. Detailed calculations are performed for beams with end-plate joints. Critical moments determined with programs and approximation formulas are compared with the results obtained by other researchers and with those produced by FEM. Very good accuracy of results is obtained.

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Authors and Affiliations

R. Piotrowski
A. Szychowski
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Abstract

Closed form solutions for the flexural-torsional buckling of elastic beam-columns may only be obtained for simple end boundary conditions, and the case of uniform bending and compression. Moment gradient cases need approximate analytical or numerical methods to be used. Investigations presented in this paper deal with the analytical energy method applied for any asymmetric transverse loading case that produces a moment gradient. Part I of this paper is devoted entirely to the theoretical investigations into the energy based out-of-plane stability formulation and its general solution. For the convenience of calculations, the load and the resulting moment diagram are presented as a superposition of two components, namely the symmetric and antisymmetric ones. The basic form of a non-classical energy equation is developed. It appears to be a function dependent upon the products of the prebuckling displacements (knowfrom the prebuckling analysis) and the postbuckling deformation state components (unknowns enabling the formulation of the stability eigenproblem according to the linear buckling analysis). Firstly, the buckling state solution is sought by presenting the basic form of the non-classical energy equation in several variants being dependent upon the approximation of the major axis stress resultant M�� and the buckling minor axis stress resultant Mz. The following are considered: the classical energy equation leading to the linear eigenproblem analysis (LEA), its variant leading to the quadratic eigenproblem analysis (QEA) and the other non-classical energy equation forms leading to nonlinear eigenproblem analyses (NEA). The novel forms are those for which the stability equation becomes dependent only upon the twist rotation and its derivatives. Such a refinement is allowed for by using the second order out-of-plane bending differential equation through which the minor axis curvature shape is directly related to the twist rotation shape. Secondly, the effect of coupling of the in-plane and out-of-plane buckling forms is taken into consideration by introducing approximate second order bending relationships. The accuracy of the classical energy method of solving FTB problems is expected to be improved for both H- and I-section beam-columns. The outcomes of research presented in this part are utilized in Part II.
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Authors and Affiliations

Marian Giżejowski
1
ORCID: ORCID
Anna Barszcz
1
ORCID: ORCID
Paweł Wiedro
1
ORCID: ORCID

  1. Warsaw University of Technology, Faculty of Civil Engineering, Al. Armii Ludowej 16, 00-637 Warsaw, Poland
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Abstract

Elastic lateral-torsional buckling of double-tee section structural steelworks has been widely investigated with regard to the major axis bending of single structural elements as a result of certain loading conditions. No specific attention has been paid to the general formulation in which an arbitrary span load pattern was associated with unequal end moments as a result of the moment distribution between structural members of the load bearing system.Anumber of analytical solutionswere developed on the basis of the Vlasov theory of thin-walled members. Since the accurate closed-form solutions of lateral-torsional buckling (LTB) of beams may only be obtained for simple loading and boundary conditions, more complex situations are treated nowadays by using numerical finite element methods (FEM). Analytical and numerical methods are frequently combined for the purpose of: a) verification of approximate analytical formulae or b) presentation the results in the form of multiple curve nomograms to be used in design practice. Investigations presented in this paper deal with the energy method applied to LTB of any complex loading condition of elements of simple end boundary conditions, bent about the major axis. Firstly, a brief summary of the second-order based energy equation dealt with in this paper is presented and followed by its approximate solution using the so-called refined energy method that in the case of LTB coincides with the Timoshenko’s energy refinement. As a result, the LTB energy equation shape functions of twist rotation and minor axis displacement are chosen such that they cover both the symmetric and antisymmetric lateral-torsional buckling modes. The latter modes are chosen in relation to two lowest LTB eigenmodes of beams under uniform major axis bending. Finally, the explicit form of the general solution is presented as being dependent upon the dimensionless bending moment equations for symmetric and antisymmetric components, and the in-span loads. Solutions based on the present investigations are compared for selected loading conditions with those obtained in the previous studies and verified with use of the LTBeam software. Conclusions are drawn with regard to the application of obtained closed-form solutions in engineering practice.
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Authors and Affiliations

Anna Barszcz
1
ORCID: ORCID
Marian Giżejowski
1
ORCID: ORCID
Malwina Pękacka
2

  1. Warsaw University of Technology, Faculty of Civil Engineering, Al. Armii Ludowej 16, 00-637 Warsaw, Poland
  2. Warsaw University of Technology Graduate, Faculty of Civil Engineering, Al. Armii Ludowej 16, 00-637 Warsaw, Poland
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Abstract

The cable force of a cable-stayed bridge plays a vital role in its internal force state. Different cable forces on both sides of the main tower make the force characteristics of the polygonal-line tower quite different from those of the straight-line tower. Therefore, the determination of the cable force of the polygonal-line tower cable-stayed bridge is a crucial aspect of any evaluation of its mechanical characteristics. A single-cable plane prestressed concrete broken-line tower cable-stayed bridge is taken as a case study to conduct a model test and theoretical cable force determination. The reasonable cable force of the bridge is determined by the minimum bending energy method combined with false load and internal force balance methods. analysis includes a comparison between cable force calculation results, model test results, and the design value of the actual bridge. The distribution law of the dead load cable force of the completed bridge is determined accordingly.
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Authors and Affiliations

Yanfeng Li
1
ORCID: ORCID
Tianyu Guo
1
ORCID: ORCID
Longsheng Bao
1
ORCID: ORCID
Fuchun Wang
1

  1. School of Transportation Engineering, Shenyang Jianzhu University, Shenyang 110168, China

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