Stocky Beams Research Articles

An innovative method for surmounting the constrained torsional problems of stocky beams with arbitrary noncircular cross-sectional shapes and with arbitrary elastic material properties

The constrained torsion problems of stocky beams, being still an unsurmountable challenge in solid mechanics, will be solved by the innovative method proposed, and the computational results of numerical examples signifying that the beam model proposed is reasonable and the numerical analysis methodology of nodal line method is effective. The creative cruxes are as followings:•a type of innovative beam model—stocky beam in solid mechanics analysis is defined, which uses the stress state of the beam to distinguish whether a structural member is a beam or not, and the axis of the stocky beam is the locus of the minimum torsional stiffness centre of all its cross sections;•the warpages of all cross sections of a stocky beam are depicted as a mathematical expression that signifies a family of curved surfaces evolved from Prandtl's membrane shapes;•the constrained torsion problem of a stocky beam can be transformed into the boundary value problem of a set of ordinary differential equations by using the nodal line method developed.

The torsional centre position of stocky beams with arbitrary noncircular cross-sectional shapes and with arbitrary elastic material properties

The aim of this study focuses on torsional centre position of the cross section of a complex section stocky beam (CSSB), a new type of model for structural members with arbitrary noncircular cross-sectional shapes and with arbitrary elastic material properties and with length-to-height or length-to-width ratio no more than 3 encountered in modern civil engineering structures. The nodal-line method is utilized to describe the shapes of the out-of-plane warpages of all cross sections of a CSSB under restrained torsional conditions by employing n unknown displacement functions of nodal lines, together with n quadratic interpolation functions inspired and evolved by the shape of Prandtl’ membrane. The governing ordinary differential equations are established by employing the principle of energy in elasticity. Based on the fact that the torsional stiffness of cross section of the same CSSB stays unchangeable with the change of the torque modes exerted to it, and on the computational results of the change rate of torsional angle due to the individual action of each torque corresponding to three torque modes, a formula for calculating two coordinates of the torsional centre position of the cross section is formulated. The computational results of numerical examples are demonstrated to verify the rationality and correctness of the formula.

Vibrations of rotationally restrained Timoshenko beam at hinged supports during an earthquake

Abstract The present paper describes an analytic solution procedure for flexural vibration of a rotationally restrained hinged-hinged Timoshenko beam at the supports during an earthquake. Focusing on maximal magnitudes of internal loads such as bending moment and shearing force under wide variations of two parameters, k L / E I and κ G A L 2 / E I , various beams under synchronous and asynchronous support motions are simulated. The simulations under asynchronous support motions show the following facts. The variations of the maximal magnitudes of internal loads of stocky beams due to the variation of k L / E I from zero to infinity show much wider variations than those of slender beams as κ G A L 2 / E I decrease s . The maximal magnitudes of internal loads of a beam tend to be governed by their static components as k L / E I increases and κ G A L 2 / E I decreases. When the internal loads are governed by their static components, maximal magnitudes of internal loads of the stocky tend to increase monotonically as the value of k L / E I increases. However, the simulations under synchronous support motions show the static components of the internal loads vanish and the internal loads are governed by dynamic components irrespective of the two parameters.

First-order, buckling and post-buckling behaviour of GFRP pultruded beams. Part 1: Experimental study

Although fibre reinforced polymers exhibit several advantages over traditional materials, their widespread acceptance is being delayed by the lack of appropriate design codes. In fact, additional and comprehensive experimental data are needed to assess the accuracy of recently developed analytical and numerical design tools. This work reports an experimental study on the first-order, buckling and post-buckling behaviours of I-section beams made of GFRP pultruded profiles. Tests were first carried out on small-scale (coupon) specimens, in order to determine the most relevant material mechanical properties. Full-scale tests were then conducted on (i) simply supported beams with spans varying from 1.0 m to 4.0 m under 3-point bending and (ii) cantilevers with spans ranging from 2.0 m to 4.0 m subjected to a tip point load applied at the end cross-section centroid or top/bottom flange mid-point. While the first series is aimed at investigating the flexural behaviour under service and failure conditions (including the local buckling of the top flange), the objective of the second series is to study the collapse behaviour stemming from lateral-torsional buckling. The results obtained confirm that, due to the GFRP low Young’s modulus and high strength, the beam structural integrity is often governed by excessive deformation and/or local and global buckling phenomena, rather than by material strength limitations. Moreover, the low shear-to-Young’s modulus ratio implies that the role played by the shear deformation is quite relevant, particularly in stocky beams. The experimental data presented here is used to validate and assess the accuracy of numerical simulations reported in a companion paper (Part 2).

Structural Responses of Axially Restrained Steel Beams with Semirigid Moment Connection in Fire

This paper investigates the structural responses of generally restrained steel beams under fire conditions by the finite-element method. The axial restraint is represented by a linear elastic spring, whereas the rotational restraint is simulated as a semirigid rotational spring. The developments of beam internal forces, cross-sectional stresses and strains, together with displacements are examined. Moreover, the effects of some important factors such as load utilization factor, beam slenderness ratio, and axial and flexural restraint ratios, are also studied. Numerical analyses show that the critical temperature is reduced by axial restraint but increased by semirigid rotational restraint. Some stocky beams subjected to low utilization factors buckle at elevated temperature. The later part of the paper focuses on investigating the effect of cross-sectional thermal gradient; the associated thermal bowing is detrimental for a slender beam. This paper also tabulates the finite-element (FE) predictions of the lower and upper bounds of beam critical temperature for design purposes. Besides FE analyses, the simplified methods from EC3 Pt.1.2 and BS5950 Pt.8 for critical temperature are also examined.

Lateral post-buckling analysis of beams

The nonlinear lateral buckling response of perfect stocky beams in the vicinity of the critical bifurcational state is discussed. Attention is focused on the initial post-buckling response. This depends on the nature of the critical branching point which is explored by using a nonlinear (lateral) bending-curvature relationship. It is found that the plane of loading (associated with the major moment of inertia) looses its stability through a stable symmetric bifurcation point. Hence, the above beams are not sensitive to imperfections, exhibiting post-buckling strength. However, the post-buckling equilibrium path is quite shallow, so that the load-carrying capacity of such beams above the critical state is rather limited. The analysis is supplemented by illustrative examples forI-beams for which the effect of various parameters on the initial postbuckling path is also discussed.

Vibration of shear‐deformable beams using a spline‐function approach

AbstractB‐spline functions are used as trial functions in a Rayleigh‐Ritz analysis of the free vibration of shear‐deformable Timoshenko beams. In a first approach it is demonstrated in numerical applications that when the lateral deflection and the cross‐sectional rotation are represented by functions of equal order the calculated natural frequencies are of good accuracy for stocky beams but can overestimate the true frequencies very considerably for slender beams. This is identified as a shear‐locking difficulty and consideration of its causes points clearly to the adoption of a new displacement field in which the deflection is represented by a B‐spline function which is one order higher than that used to represent the rotation. Numerical results using this new displacement field demonstrate good accuracy for both stocky and slender beams: the shear‐locking difficulty is completely eliminated. This has clear significance for the analysis of shear‐deformable plates and shells when using B‐spline functions.

EFFECTS OF MOMENT DISTRIBUTION ON LATERAL-TORSIONAL BUCKLING STRENGTH OF ROLLED STEEL I-BEAMS

Inelastic lateral-torsional buckling strength of steel beams depends markedly on the location and extent of high moment regions causing the reductions in sectional regidities.Rolled I-sections with two different sizes are tested carefully under a concentrated load at the central point and two equal concentrated loads at the one third points of simply supported beams, respectively. The test results of load-deformation curves and strength variations are compared with both loading conditions. Appropriate design strength formulas for the various load patterns of beams are also discussed by a review of the available experimental data on the ultimate strength of stocky beams.

The stability and strength of stocky beams

A brief analysis has been made of the ultimate strengths of short span I-beams bent in their planes of maximum stiffness by pure terminal couples. In the cases treated, primary failure would occur by lateral buckling in all but the most stocky members. The critical applied moments were less in magnitude than the fully plastic moment, the practical collapse load coinciding with that causing full plasticity of the flanges. The principle discussed by Shanley (1947) has been assumed applicable to problems of beam stability and the theory has been extended to beams the materials of which exhibit both discontinuous and continuous stress-strain relationships. Estimates of the buckling moments of typical stocky beams have been compared with the results of a series of tests, the theory predicting a lower limit to the range of possible collapse loads.