Monosymmetric Cross-sections Research Articles

Series solutions for spatially coupled buckling anlaysis of thin-walled Timoshenko curved beam on elastic foundation

The spatially coupled buckling, in-plane, and lateral bucking analyses of thin-walled Timoshenko curved beam with non-symmetric, double-, and mono-symmetric cross-sections resting on elastic foundation are performed based on series solutions. The stiffness matrices are derived rigorously using the homogeneous form of the simultaneous ordinary differential equations. The present beam formulation includes the mechanical characteristics such as the non-symmetric cross-section, the thickness-curvature effect, the shear effects due to bending and restrained warping, the second-order terms of semitangential rotation, the Wagner effect, and the foundation effects. The equilibrium equations and force-deformation relationships are derived from the energy principle and expressions for displacement parameters are derived based on power series expansions of displacement components. Finally the element stiffness matrix is determined using force-deformation relationships. In order to verify the accuracy and validity of this study, the numerical solutions by the proposed method are presented and compared with the finite element solutions using the classical isoparametric curved beam elements and other researchers` analytical solutions.

DYNAMIC STIFFNESS MATRIX FOR FLEXURAL-TORSIONAL, LATERAL BUCKLING AND FREE VIBRATION ANALYSES OF MONO-SYMMETRIC THIN-WALLED COMPOSITE BEAMS

This paper presents the elastic strain energy, the potential energy with the second order terms of finite rotations, and the kinetic energy with rotary inertia effect for thin-walled composite beams of mono-symmetric cross-section. The equations of motion and force-displacement relationships are derived from the energy principle and explicit expressions for displacement parameters are given based on power series expansions of displacement components. The exact dynamic stiffness matrix is determined using the force-displacement relationships. In addition, the finite element model based on Hermitian interpolation polynomial is developed. In order to verify the accuracy and validity of the formulation, numerical examples are solved and the solutions are compared with results from ABAQUS's shell elements, analytical solutions from previous researchers and the finite element solutions using the Hermitian beam elements. The influence of constant and linearly variable axial forces, fiber orientation, and boundary conditions on the vibration behavior of composite beam are also investigated.

Vibration analysis with second-order effects of pultruded FRP frames using locking-free elements

Vibrations of pultruded (FRP) thin-walled beams and frames are analysed. A second-order approximation of the displacement field is adopted, accounting for shear strain effects due to both nonuniform bending and torsion. Geometric nonlinearities arising from concentrated or distributed external loads are considered. A simple two-node finite element model using polynomial interpolation is adopted. In particular, the interpolating polynomials follow from the solutions to the homogeneous Timoshenko-beam problem and to the warping-torsion problem for a beam with doubly-symmetrical cross-section. Such shape functions contain parameters which tend to zero when the influence of shear deformations becomes negligible, resulting in a locking- free formulation. The convergence properties of the adopted element in the presence of flexural–torsional vibrations are highlighted with reference to mono-symmetrical cross-section carbon-FRP profiles. Vibration frequencies and mode shapes of pultruded glass-FRP portal frames are extensively evaluated as a function of the applied loads and sudden exchanges of fundamental modes are pointed out. The influence of frame geometry, shear deformations, participating masses and geometrical effects due to distributed external loads is investigated.

Elastic flexural-torsional buckling of thin-walled cantilevers

Previous studies by the authors revealed that the two representative theories with slight differences between, widely used in investigating the flexural-torsional buckling of thin-walled beams, have led to two different solutions in well-known literature for assessing critical loads of simply supported beams of monosymmetric cross section. With these two solutions, significant differences in critical loads may be found for these monosymmetric beams. Based on the classical variational principle for buckling analyses, a new theory on the flexural-torsional buckling of thin-walled members was proposed by the authors. In this paper, this new theory as well as the other two typical theories is employed to investigate the flexural-torsional buckling of cantilevers. This paper first gives a brief review and a careful comparative study on the flexural-torsional buckling of thin-walled cantilevers employing three different buckling theories. Differences between these theories are demonstrated with investigations on buckling of cantilevers under pure bending and two typical transverse loads. Explicit solutions, capable of considering variations of beam length and loading position along the vertical axis of cross section, are presented for predicting the critical loads of doubly symmetric cantilevers under two typical transverse loads. Advantages of presented solutions, such as good accuracy and ease of use, are exploited through the comparisons of critical results with those from existing solutions and finite element analyses.

Shear-flexible thin-walled element for composite I-beams

A shear-flexible finite element based on an orthogonal Cartesian coordinate system is developed for the flexural and buckling analyses of thin-walled composite I-beams with both doubly and mono-symmetrical cross-sections. Using the first-order shear deformable beam theory, the derived element includes both the transverse shear and the restrained warping induced shear deformations. Governing equations are derived from the principle of minimum total potential energy. Three different types of finite elements, namely, linear, quadratic and cubic elements are developed to solve the governing equations. The geometric stiffness for the buckling analysis of axially loaded, thin-walled composite beams is developed. The resulting linearized buckling problem is solved using a shifted inverse iteration algorithm. A parametric study of the effects of the aspect ratio and the fibre orientation on the tip displacement is presented. The convergence of the elements is also investigated. The elastic buckling loads for mono- and doubly-symmetric I-beam cross-sections are compared with other results available in the literature and with solutions using shell elements in a commercially available finite element program.

Lateral buckling analysis of thin-walled laminated composite beams with monosymmetric sections

Lateral buckling of thin-walled composite beams with monosymmetric sections is studied. A general geometrically nonlinear model for thin-walled laminated composites with arbitrary open cross-section and general laminate stacking sequences is given by using systematic variational formulation based on the classical lamination theory. All the stress resultants concerning bar and shell forces are defined, and nonlinear strain tensor is derived. General nonlinear governing equations are given, and the lateral buckling equations are derived by linearizing the nonlinear governing equations. Based on the analytical model, a displacement-based one-dimensional finite element model is developed to formulate the problem. Numerical examples are obtained for thin-walled composite beams with monosymmetric cross-sections and angle-ply laminates. The effects of fiber orientation, location of applied load, modulus ratio, and height-to-span ratio on the lateral buckling load are investigated. The torsion parameter and a newly-defined composite monosymmetry parameter are also investigated for various cases.

Flexural–torsional vibration of simply supported open cross-section steel beams under moving loads

The present work deals with linearized modal analysis of the combined flexural-torsional vibration of simply supported steel beams with open monosymmetric cross-sections, acted upon by a load of constant magnitude, traversing its span eccentrically with constant velocity. After thoroughly investigating the free vibrations of the structure, which simulates a commonly used highway bridge, its forced motions under the aforementioned loading type are investigated. Utilizing the capabilities of symbolic computations within modern mathematical software, the effect of the most significant geometrical and cross-sectional beam properties on the free vibration characteristics of the beam are established and presented in tabular and graphical form. Moreover, adopting realistic values of the simplified vehicle model adopted, the effects of eccentricity, load magnitude and corresponding velocity are assessed and interesting conclusions for structural design purposes are drawn. The proposed methodology may serve as a starting point for further in-depth study of the whole scientific subject, in which sophisticated vehicle models, energy dissipation and more complicated bridge models may be used.

Dynamic response of axially loaded monosymmetrical thin-walled Bernoulli–Euler beams

The dynamic bending–torsion coupled vibrations of elastic axially loaded slender thin-walled beams with monosymmetrical cross-sections are investigated by using normal mode method. The Bernoulli–Euler beam theory is employed and the effects of warping stiffness and axial force are included in the present formulations. The theoretical expressions for the displacement response of axially loaded slender thin-walled beams subjected to concentrated or distributed loads are presented. The method is illustrated by its application to two test examples to describe the effects of warping stiffness and axial force on the dynamic behavior of thin-walled beams. The numerical results for the dynamic bending displacements and torsional displacements are given. The proposed theory is fairly general and can be used for thin-walled beam assemblage of arbitrary boundary conditions subjected to various kinds of loads.

Flexural–torsional buckling of thin-walled beam members based on shell buckling theory

This paper summarized currently available techniques of setting up flexural–torsional buckling theory of thin-walled members. It is found that all the existing methods introduced a nonlinear load potential in their total potentials, while based on the classical variational principle for stability of a solid structure, no such load potential should be included. This situation has led to an inconsistency between some widely referenced monographs in buckling theories of beams with mono-symmetrical cross-sections. This paper provides a new theory for flexural–torsional buckling of thin-walled members based on the classical variational principle and the theory for thin-walled shells. No nonlinear load potential is included, but a new term: nonlinear strain energy from transverse stresses, which has been neglected in previous theories of thin-walled members, is introduced in. It is found that the nonlinear load potential is not equivalent to the contribution of transverse stresses for beams with mono-symmetrical cross-sections, which causes the inconsistency mentioned above. The comparison shows that the proposed theory and the traditional theory are the same for most cases encountered in practice.

Coupled bending and torsional vibration of axially loaded Bernoulli–Euler beams including warping effects

The dynamic transfer matrix method for determining natural frequencies and mode shapes of the bending-torsion coupled vibration of axially loaded thin-walled beams with monosymmetrical cross sections is developed by using a general solution of the governing differential equations of motion based on Bernoulli–Euler beam theory. This method takes into account the effect of warping stiffness and gives allowance to the presence of axial force. The dynamic transfer matrix is derived in detail. Two illustrative examples on the application of the present theory are given for bending-torsion coupled beams with thin-walled open cross sections. The effects of axial load and warping stiffness on coupled bending-torsional frequencies are discussed. Compared with those available in the literature, numerical results demonstrate the accuracy and effectiveness of the proposed method.

An Analysis of Current Stability Theories for Thin-Walled Members

There are two different formulae available for calculating the critical moments of thin-walled beams under transverse loads. They give quite different results for beams with monosymmetrical cross-sections. This paper reviews the stability theories for thin-walled beams and identifies some points which might be present and have led to the current situation of the buckling moment formulae for the beams. Finally a new theory proposed by the authors is briefly introduced and a rational way for establishing a general buckling theory for thin-walled members is proposed.

Out-of-plane buckling and bracing requirement in double-angle trusses

Truss members built-up with double angles back-to-back have monosymmetric cross-section and twisting always accompanies flexion upon the onset of buckling about the axis of symmetry. Approximate formulae for calculating the buckling capacity are presented in this paper for routine design purpose. For a member susceptible only to flexural buckling, its optimal cross-section should consist of slender plate elements so as to get larger radius of gyration. But, occurrence of twisting changes the situation owing to the weakness of thin plates in resisting torsion. Criteria for limiting the leg slenderness are discussed herein. Truss web members in compression are usually considered as hinged at both ends for out-of-plane buckling. In case one (or both) end of member is not supported laterally by bracing member, its adjoining members have to provide an elastic support of adequate stiffness in order not to underdesign the member. The stiffness provided by either compression or tension chords in different cases is analyzed, and the effect of initial crookedness of compression chord is taken into account. Formulae are presented to compute the required stiffness of chord member and to determine the effective length factor for inadequately constrained compressive diagonals.

Elastic limit state flexural–torsional postbuckling analysis of bars with open thin-walled cross-sections under axial thrust

This work deals with the elastic limit state flexural–torsional postbuckling analysis of simply supported bars with open thin-walled asymmetric cross-sections under axial thrust. As it is well known stocky bars with the above type of cross-sections always fail by flexural–torsional buckling in the case of asymmetric cross-sections (whose centroid does not coincide with the shear centre), while in the case of monosymmetric cross-sections the failure may occur either through flexural (Euler) buckling or flexural–torsional buckling depending on the geometric characteristics of the bars. In all the above three cases the critical state is associated with postbuckling strength. In this paper attention focuses on the first yielding occurring at the initial part of the post-critical path of flexural–torsional buckling. This is, in case of bars made from ideal elastic–ideal plastic material, associated with the maximum combined normal stress, due to axial compression, bending and warping, which, along with the nonlinear equilibrium equation, yield the maximum (ultimate) elastic load-carrying capacity. The elastic limit state postbuckling analysis given here is demonstrated with the aid of equal-leg angles commonly used in trusses.

Instantaneous and time-dependent analysis of composite wood-concrete cross-sections using Dischinger’s equations of state: Part 2-time-dependent analysis

In the second and final part of this paper, the time-dependent analysis of a monosymmetric composite wood-concrete cross-section is presented. The relaxation procedure is used to assemble the stress and strain vectors due to creep and shrinkage effects. These are solved using the finite difference method and finally an illustrative example is used to demonstrate the method.

Analysis of a mono-symmetric core wall structure coupled with connecting beams

An open mono-symmetric core wall coupled with connecting beams is analysed under lateral loading, using Vlasov's thin-walled beam theory. Shearing deformations in the plane of the wall are taken into account, considering geometric compatibility. Coupling beams are treated as a medium with an equivalent stiffness property. Effect of connecting beam stiffness and of shearing deformations on the rotation of the whole system are presented. Inclusion of shearing deformations increases rotation of the structure; this effect being more outstanding in core walls with mono-symmetrical cross-section and those with very stiff connecting beams.

General non-dimensional equation for lateral buckling

A new approximate non-dimensional equation for the elastic lateral buckling load of straight prismatic beams of monosymmetric cross-section with general boundary conditions and general loading, which includes the effects of initial bending curvature prior to buckling, is derived. For sections where the flexural rigidity about the axis of initial bending, EI zz , is smaller than the flexural rigidity about the other principal centroidal axis, EI yy , it is shown that lateral buckling is still a possibility

Buckling of monosymmetric arches under point loads

A finite element procedure is described for analysing the flexural-torsional buckling of arches of monosymmetric cross-section. First an inplane analysis is performed to obtain the distributions of axial force and bending moment in the arch. These are then substituted into the buckling equation for monosymmetric arches, and the resulting eigenvalue problem is solved numerically. Higher order quintic shape functions are used to describe the element displacement fields. The effects of the distances from the arch shear centre axis of point and uniformly distributed loads are also included in the analysis. Flexural-torsional buckling tests on circular aluminium arches of monosymmetric I-section are also described. The test results for arches subjected to central concentrated point loads are compared with the finite element theory.

Flexural‐Torsional Buckling of Monosymmetric Arches

A flexural‐torsional buckling theory is developed for arches of monosymmetric cross section. Nonlinear strain‐displacement relations are derived for the axial and shear strains, and these are substituted into the second variation of the total potential to obtain the buckling equation. Closed form solutions are obtained for arches subjected to equal and opposite end moments and for arches and rings subjected to a uniformly distributed radial load. The solutions are compared with previous theoretical results.

Columns: Static and Dynamic Interactive Buckling

The interaction of local and overall buckling in thin‐walled columns is considered. A finite strip method in conjunction with the theory of mode interaction is developed to study the problem. The method is applicable to doubly symmetric cross sections carrying uniform stress or monosymmetric cross sections subjected to uniform end compression. Imperfection‐sensitivity surfaces are presented for stiffened panels and I‐section columns. A brief parametric study on I‐section columns is presented to illustrate the effect of flange slenderness and the ratio of local to Euler critical load on the maximum load carried. The effect of dynamic loads in the form of suddenly applied end compression is also investigated. It is found that in the presence of overall imperfections likely to occur in practice, the suddenness of load application can cause a reduction of about 10% of the maximum static load carrying capacity. A test program on I‐section columns carried out in Cornell University is cited for an evaluation of ...

Dynamic Stability of Monosymmetrical Thin-Walled Structures

A study is made on the dynamic stability of thin-walled beams of monosymmetrical cross section. Under the action of axial periodic loads, the parametric excitation of the coupled flexural-torsional type of vibrations is investigated. The nonlinear differential equations and the appropriate boundary conditions used in the analysis of the problem were obtained using the energy formulation. The effect of large angles of rotation and the effect of longitudinal deformations were taken into account. The parametric stability of a thin-walled beam of monosymmetrical split ring section is studied. The study included the establishment of the regions of parametric instability. The steady-state response and the transient growth of the amplitude of oscillations are investigated when the system is parametrically excited. Attention is focused on the response when the coupled flexural-torsional type of vibrations of predominant torsional characteristics is excited. The effect of viscous damping on the steady-state and the transient responses is also included in the analysis.