Non-linear buckling and postbuckling behavior of thin-walled beams considering shear deformation

The static stability of thin-walled composite beams, considering shear deformation and geometrical non-linear coupling, subjected to transverse external force has been investigated in this paper. The theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field (accounting for bending and warping shear) considering moderate bending rotations and large twist. This non-linear formulation is used for analyzing the prebuckling and postbuckling behavior of simply supported, cantilever and fixed-end beams subjected to different load condition. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results show that the beam loses its stability through a stable symmetric bifurcation point and the postbuckling strength is in relation with the buckling load value. Classical predictions of lateral buckling are conservative when the prebuckling displacements are not negligible and the non-linear buckling analysis is required for reliable solutions. The analysis is supplemented by investigating the effects of the variation of load height parameter. In addition, the critical load values and postbuckling response obtained with the present beam model are compared with the results obtained with a shell finite element model (Abaqus).     

Approximate vibration analysis of laminated curved panel using higher-order shear deformation theory

An approximate analysis for free vibration of a laminated curved panel (shell) with four edges simply supported (SS2), is presented in this paper. The transverse shear deformation is considered by using a higher-order shear deformation theory. For solving the highly coupled partial differential governing equations and associated boundary conditions, a set of solution functions in the form of double trigonometric Fourier series, which are required to satisfy the geometry part of the considered boundary conditions, is assumed in advance. By applying the Galerkin procedure both to the governing equations and to the natural boundary conditions not satisfied by the assumed solution functions, an approximate solution, capable of providing a reliable prediction for the global response of the panel, is obtained. Numerical results of antisymmetric angle-ply as well as symmetric cross-ply and angle-ply laminated curved panels are presented and discussed.     

Nonlinear deformation of thin-walled shells with local loads

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 10, pp. 50–55, October, 1994.     

A new non-linear higher-order shear deformation theory for large-amplitude vibrations of laminated doubly curved shells

A consistent higher-order shear deformation non-linear theory is developed for shells of generic shape, taking geometric imperfections into account. The geometrically non-linear strain-displacement relationships are derived retaining full non-linear terms in the in-plane displacements; they are presented in curvilinear coordinates in a formulation ready to be implemented. Then, large-amplitude forced vibrations of a simply supported, laminated circular cylindrical shell are studied (i) by using the developed theory, and (ii) keeping only non-linear terms of the von Kármán type. Results show that inaccurate results are obtained by keeping only non-linear terms of the von Kármán type for vibration amplitudes of about two times the shell thickness for the studied case.     

An asymptotic non-linear model for thin-walled rods with strongly curved open cross-section

In this paper, we present a non-linear one-dimensional model for thin-walled rods with open strongly curved cross-section, obtained by asymptotic methods. A dimensional analysis of the non-linear three-dimensional equilibrium equations lets appear dimensionless numbers which reflect the geometry of the structure and the level of applied forces. For a given force level, the order of magnitude of the displacements and the corresponding one-dimensional model are deduced by asymptotic expansions.     

Variational formulation of stability problems for thin-walled members

Summary The governing equations for thin-walled members subjected to axial force, biaxial bending and torsion are formulated by variational method. The resulting equations differ slightly with currently accepted governing equations. Numerical examples are solved for simple problems to illustrate some of the features of the proposed equations.

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Übersicht Die beschreibenden Differentialgleichungen für dünnwandige Bauglieder, die durch achsiale Kräfte, biachsiale Biegung und Torsion beansprucht sind, werden nach einem Variationsverfahren aufgestellt. Die so erhaltenen Gleichungen unterscheiden sich etwas von den zur Zeit üblichen Gleichungen. Um die Eigenschaften der jetzt vorgeschlagenen Gleichungen zu zeigen, werden numerische Ergebnisse für einige einfache Aufgaben angegeben.

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On leave from University of Tokyo, Tokyo Japan.     

Geometrical nonlinear analysis of thin-walled composite beams using finite element method based on first order shear deformation theory

Based on a seven-degree-of-freedom shear deformable beam model, a geometrical nonlinear analysis of thin-walled composite beams with arbitrary lay-ups under various types of loads is presented. This model accounts for all the structural coupling coming from both material anisotropy and geometric nonlinearity. The general nonlinear governing equations are derived and solved by means of an incremental Newton–Raphson method. A displacement-based one-dimensional finite element model that accounts for the geometric nonlinearity in the von Kármán sense is developed to solve the problem. Numerical results are obtained for thin-walled composite beam under vertical load to investigate the effects of fiber orientation, geometric nonlinearity, and shear deformation on the axial–flexural–torsional response.     

Interfacial stresses in curved members bonded with a thin plate

The use of steel plates or externally bonded fibre-reinforced polymer laminates for flexural strengthening of concrete, masonry, timber or metallic structures is a technique that has become very popular. The effectiveness of this technique hinges heavily on the performance of the bond between the strengthening plate and the substrate, which has been the subject of many existing studies. In particular, the interfacial stresses between a beam and a soffit plate within the linear elastic range have been addressed by numerous analytical investigations. Surprisingly, none of these investigations has examined interfacial stresses in members with a curved soffit, despite that such members are often found in practice. This paper presents an analytical model for the interfacial stresses between a curved member of uniform section size and a thin plate bonded to its soffit. The governing differential equations for the interfacial shear and normal stresses are formulated and then solved with appropriate simplifying assumptions. Two numerical examples are presented to illustrate the effect of the curvature of the member on the interfacial stress distributions in a simply supported curved beam for the two cases of a point load and a uniformly distributed load. The analytical solution is verified by comparing its predictions with those from a finite element model.     

Kinematic analysis of a rolling tensegrity structure with spatially curved members

In this work, a tensegrity structure with spatially curved members is applied as rolling locomotion system. The actuation of the structure allows a variation of the originally cylindrical shape to a conical shape. Moreover, the structure is equipped with internal movable masses to control the position of the center of mass of the structure. To control the locomotion system a reliable actuation strategy is required. Therefore, the kinematics of the system considering the nonholonomic constraints are derived in this paper. Based on the resulting insight in the locomotion behavior a feasible actuation strategy is designed to control the trajectory of the system. To verify this approach kinematic analyses are evaluated numerically. The simulation data confirm the path following due to an appropriate shape change of the tensegrity structure. Thus, this system enables a two-dimensional rolling locomotion.

     

Measurement techniques in the testing of thin-walled structural members

The paper describes methods for measuring displacements and end moments in the testing of thin-walled columns. The complete spatial deformation of a column can be captured by using local and overall deformation measurement frames. The local frame involves spring-loaded levers and roller bearings to mount the frame on to the specimen. The overall frame slides along high-precision shafts using linear ball bearings. Fixed-ended bearings are used to measure minor and major axis bending moments as well as the applied load. The bearings allow the line of action of the applied force to be determined.     

Linear viscoelastic analysis of straight and curved thin-walled laminated composite beams

This paper is devoted to study the behavior, in the range of linear viscoelasticity, of shear flexible thin-walled beam members constructed with composite laminated fiber-reinforced plastics. This work appeals to the correspondence principle in order to incorporate in unified model the motion equations of a curved or straight shear-flexible thin-walled beam member developed by the authors, together with the micromechanics and macromechanics of the reinforced plastic panels. Then, the analysis is performed in the Laplace or Carson domains. That is, the expressions describing the micromechanics and macromechanics of a plastic laminated composites and motion equations of the structural member are transformed into the Laplace or Carson domains where the relaxation components of the beam structure (straight or curved) are obtained. The resulting equations are numerically solved by means of finite element approaches defined in the Laplace or Carson domains. The finite element results are adjusted with a polynomial fitting. Then the creep behavior is obtained by means of a numerical technique for the inverse Laplace transform. Predictions of the present methodology are compared with experimental data and other approaches. New studies are performed focusing attention in the flexural–torsional behavior of shear flexible thin-walled straight composite beams as well as for thin-walled curved beams and frames.     

薄壁构件空间变形计算理论浅议

通过传统方法和“翘曲理论”的新方法在几组薄壁构件的算例中的对比,对薄壁构件空间变形计算理论进行一些初步的讨论．“翘曲理论”与现行的两大理论相比,当两的覆盖范围相同时,计算结果是完全相同的,效率大大提高,而且更精确．     

Physically and geometrically nonlinear deformation of thin-walled conical shells with a curvilinear hole

The elastoplastic state of thin conical shells with a curvilinear (circular) hole is analyzed assuming finite deflections. The distribution of stresses, strains, and displacements along the hole boundary and in the zone of their concentration are studied. The stress-strain state around a circular hole in shells subject to internal pressure of prescribed intensity is analyzed taking into account two nonlinear factors __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 73–79, April 2007.     

Free vibration of horizontally curved thin-walled beams with rectangular hollow sections considering two compatible displacement fields

A finite element is presented for vibration analyses of horizontally curved thin-walled rectangular hollow beams. Eight cross-section deformation modes are employed to describe the mid-surface contour displacement field with the modal superposition method. Focused on the in-plane moment equilibrium condition and the displacement continuity condition, two compatible displacement fields are constructed to calculate the strain energy and the kinetic energy of the beam, respectively. With the application of Hamilton’s principle the dynamic governing equations are formulated, and then approximated for the finite element implementation. Finally, numerical examples are illustrated to verify the validity of the present theory.     

An asymptotically correct model for initially curved and twisted thin-walled composite beams

The variational–asymptotic method has been applied to develop an asymptotically correct model for initially curved and twisted, thin-walled, composite beams of arbitrary cross-sectional shapes and arbitrary anisotropic materials. In a two-step asymptotic reduction procedure, the three-dimensional strain energy is asymptotically reduced first to a two-dimensional shell strain energy and then to a one-dimensional beam strain energy. This is a new attempt where initially curved and twisted, thin-walled, composite beams, with open or closed sections, have been modeled in an asymptotically correct unified framework.     

Constitutive model for plastic deformation of nanocrystalline materials with shear band

A phase mixture model was used to study the plastic deformation behaviors in hardening stage of nanocrystalline materials. The material was considered as a composite of grain interior phase and grain boundary (GB) phase. The constitutive equations of the two phases were determined in term of their main deformation mechanisms. In softening stage, a shear band deformation mechanism was presented and the corresponding constitutive relation was established. Numerical simulations have shown that the predications fit well with experimental data. The investigation using the finite-element method (FEM) provided a direct insight into quantifying shear localization effect in nanocrystalline materials.