Book Review: Elastic Beams and Frames, by J. D. Renton

The book represents a very interesting example of the possibility to combine in a single publication basic theory of structures and quite advanced topics on the same subject. The author fulfills this objective in a reasonable size book, less than 400 pages divided into 15 chapters averaging 20 pages each plus 9 short appendices. A diskette is also included in the book. This diskette contains training as well practical executable programs on different aspects of structural analysis, such as cross-sections properties,general-purpose computer programs for the static, dynamic and stability analysis of simple bar structures, etc. The book figures are didactic and have been carefully drawn.

A hybrid-mixed finite element formulation for the geometrically exact analysis of three-dimensional framed structures

This paper addresses the development of a hybrid-mixed finite element formulation for the quasi-static geometrically exact analysis of three-dimensional framed structures with linear elastic behavior. The formulation is based on a modified principle of stationary total complementary energy, involving, as independent variables, the generalized vectors of stress-resultants and displacements and, in addition, a set of Lagrange multipliers defined on the element boundaries. The finite element discretization scheme adopted within the framework of the proposed formulation leads to numerical solutions that strongly satisfy the equilibrium differential equations in the elements, as well as the equilibrium boundary conditions. This formulation consists, therefore, in a true equilibrium formulation for large displacements and rotations in space. Furthermore, this formulation is objective, as it ensures invariance of the strain measures under superposed rigid body rotations, and is not affected by the so-called shear-locking phenomenon. Also, the proposed formulation produces numerical solutions which are independent of the path of deformation. To validate and assess the accuracy of the proposed formulation, some benchmark problems are analyzed and their solutions compared with those obtained using the standard two-node displacement/ rotation-based formulation.

Transverse Vibrations in Beams Supported by a Piece-Wise Homogeneous Visco- Elastic Foundation

Transversal vibrations induced by a load moving uniformly along an infinite beam resting on a piece-wise homogeneous visco-elastic foundation are studied. Special attention is paid to the additional vibrations, conventionally referred to as transition radiations, which arise as the point load traverses the place of foundation discontinuity. The governing equations of the problem are solved by the normalmode analysis. The solution is expressed in a form of infinite sum of orthogonal natural modes multiplied by the generalized coordinate of displacement. The natural frequencies are obtained numerically exploiting the concept of the global dynamic stiffness matrix. This ensures that the frequencies obtained are exact. The methodology has restrictions neither on velocity nor on damping. The approach looks simple, though, the numerical expression of the results is not straightforward. A general procedure for numerical implementation is presented and verified. To illustrate the utility of the methodology parametric optimization is presented and influence of the load mass is studied. The results obtained have direct application in analysis of railway track vibrations induced by high-speed trains when passing regions with significantly different foundation stiffness.

An Analytical Solution for the Elastic Lateral-Distortional Buckling of I-section Beams

Lateral-distortional buckling may occur in I-section beams with slender webs and stocky flanges. A computationally efficient method is presented in this paper to study this phenomenon. Previous studies on distortional buckling have been on the use of 3(rd) and 5(th) order polynomials to model the displacements. The present study provides an alternative way, using Fourier Series, to model the behaviour. Beams of different cross-sectional dimensions, load cases and restraint conditions are examined and compared. The accuracy and versatility of the method are verified by calibrating against the results of other published studies. The present method is believed to be a simple and efficient way of determining the buckling load and mode shapes of I-section beams that are susceptible to lateral-distortional buckling modes.

Elastic scattering and total reaction cross section of (6)He+(120)Sn

The elastic scattering of (6)He on (120)Sn has been measured at four energies above the Coulomb barrier using the (6)He beam produced at the RIBRAS (Radioactive Ion Beams in Brasil) facility. The elastic angular distributions have been analyzed with the optical model and three- and four-body continuum-discretized coupled-channels calculations. The total reaction cross sections have been derived and compared with other systems of similar masses.

Elastic scattering and total reaction cross sections for the (8)Li+(12)C system

The elastic-scattering angular distribution for (8)Li on (12)C has been measured at E(LAB) = 23.9 MeV with (8)Li radioactive nuclear beam produced by the Radioactive Ion Beams in Brazil facility. This angular distribution was analyzed in terms of optical-model with Woods-Saxon and double-folding Sao Paulo potential. The roles of the breakup and inelastic channels were also investigated with cluster folding and deformed potentials, respectively, through coupled-channels calculations. The angular distribution for the proton-transfer (12)C((8)Li, (9)Be)(11)B reaction was also measured at the same energy. The spectroscopic factor for the <(9)Be|(8)Li + p > bound system was obtained and compared with shell-model calculations and with other experimental values. Total reaction cross sections for the present system were also extracted from the elastic-scattering analysis. A systematic of the reduced reaction cross sections obtained from the present and published data on (6,7,8)Li isotopes on (12)C was performed as a function of energy.

An extended BEM formulation for plates reinforced by rectangular beams

This article presents a BEM formulation developed particularly for analysis of plates reinforced by rectangular beams. This is an extended version of a Previous paper that only took into account bending effects. The problem is now re-formulated to consider bending and membrane force effects. The effects of the reinforcements are taken into account by using a simplified scheme that requires application of ail initial stress field to locally correct the bending and stretching stiffness of the reinforcement regions. The domain integrals due to the presence of the reinforcements are then transformed to the reinforcement/plate interface. To reduce the number of degrees of freedom related to the presence of the reinforcement, the proposed model was simplified to consider only bending and stretching rigidities in the direction of the beams. The complete model can be recovered by applying all six internal force correctors, corresponding to six degrees of freedom per node. Examples are presented to confirm the accuracy of the formulation and to illustrate the level of simplification introduced by this strong reduction in the number of degrees of freedom. (C) 2008 Elsevier Ltd. All rights reserved.

Elastically restrained Bernoulli-Euler beams applied to rotary machinery modelling

Facing the lateral vibration problem of a machine rotor as a beam on elastic supports in bending, the authors deal with the free vibration of elastically restrained Bernoulli-Euler beams carrying a finite number of concentrated elements along their length. Based on Rayleigh's quotient, an iterative strategy is developed to find the approximated torsional stiffness coefficients, which allows the reconciliation between the theoretical model results and the experimental ones, obtained through impact tests. The mentioned algorithm treats the vibration of continuous beams under a determined set of boundary and continuity conditions, including different torsional stiffness coefficients and the effect of attached concentrated masses and rotational inertias, not only in the energetic terms of the Rayleigh's quotient but also on the mode shapes, considering the shape functions defined in branches. Several loading cases are examined and examples are given to illustrate the validity of the model and accuracy of the obtained natural frequencies.

A hybrid procedure to identify the optimal stiffness coefficients of elastically restrained beams

The formulation of a bending vibration problem of an elastically restrained Bernoulli-Euler beam carrying a finite number of concentrated elements along its length is presented. In this study, the authors exploit the application of the differential evolution optimization technique to identify the torsional stiffness properties of the elastic supports of a Bernoulli-Euler beam. This hybrid strategy allows the determination of the natural frequencies and mode shapes of continuous beams, taking into account the effect of attached concentrated masses and rotational inertias, followed by a reconciliation step between the theoretical model results and the experimental ones. The proposed optimal identification of the elastic support parameters is computationally demanding if the exact eigenproblem solving is considered. Hence, the use of a Gaussian process regression as a meta-model is addressed. An experimental application is used in order to assess the accuracy of the estimated parameters throughout the comparison of the experimentally obtained natural frequency, from impact tests, and the correspondent computed eigenfrequency.

Mechanical characterization of old chestnut beams

The main objective of this work is to evaluate, by non-destructive techniques, seven old Chestnut beams. For that, after the geometric assessment and the detailed visual inspection that allowed to strength grade the beams, a series of non-destructive tests was setup. In a first step, non-destructive bending tests, under the elastic limit, were performed to quantify the modulus of elasticity in bending (MoE) of the seven beams. Then, Resistograph® and Pilodyn® tests were done to assess the superficial decay and to have aclearer idea of the voids dimensions. Then, two beams were tested in bending until failure to evaluate the bending strength. In a second step, end parts were cut from the beams, one per end of the beams, to perform Resistograph®, Pilodyn® and ultrasound tests, to quantify the density of the beams and to extract meso-specimens to be used in tension parallel to the grain tests

Effective stiffness behaviour of sandwich beams under uncoupled bending and torsion loadings

Sandwich geometries, mainly in the form of panels and beams, are commonly applied in various transportation industries, such as aerospace, aeronautic and automotive. Sandwich geometries represent important advantages in structural applications, namely high specific stiffness, low weight, and possibility of design optimization prior to manufacturing. The aim of this paper is to uncover the influence of the number of reinforcements (ribs), and of the thickness on the mechanical behavior of all-metal sandwich panels subjected to uncoupled bending and torsion loadings. In this study, four geometries are compared. The orientation of the reinforcements and the effect of transversal ribs are also considered in this study. It is shown that the all the relations are non-linear, despite the elastic nature of the analysis in the Finite Element software ANSYS MECHANICAL APDL.

Investigation of the Loss of Prestress in Prestressed Concrete Beams due to Steam Curing, HR-62, 1961

Mass production of prestressed concrete beams is facilitated by the accelerated curing of the concrete. The ·method most commonly used for this purpose is steam curing at atmospheric pressure. This requires concrete temperatures as high as 150°F. during the curing period. Prestressing facilities in Iowa are located out of doors. This means that during the winter season the forms are set and the steel cables are stressed at temperatures as low as 0°F. The thermal expansion of the prestressing cables should result in a reduction of the stress which was placed in them at the lower temperature. If the stress is reduced in the cables, then the amount of prestress ultimately transferred to the concrete may be less than the amount for which the beam was designed. Research project HR-62 was undertaken to measure and explain the difference between the initial stress placed in the cables and the actual stress which is eventually transferred to the concrete. The project was assigned to the Materials Department Laboratory under the general supervision of the Testing Engineer, Mr. James W. Johnson. A small stress bed complete with steam curing facilities was set up in the laboratory, and prestressed concrete beams were fabricated under closely controlled conditions. Measurements were made to determine the initial stress in the steel and the final stress in the concrete. The results of these tests indicate that there is a general loss of prestressing force in excess of that caused by elastic shortening of the concrete. The exact amount of the loss and the identification of the factors involved could not be determined from this limited investigation.

Scientific program of the Radioactive Ion Beams Facility in Brasil (RIBRAS)

The RIBRAS facility (Radioactive Ion Beams in Brasil) is installed in connection with the 8MV Pelletron tandem of the University of Sao Paulo Physics Institute. It consists of two superconducting solenoids which focalize light radioactive secondary beams of low energy, produced by transfer reactions. Recent experimental results include the elastic scattering and transfer reactions of (6)He halo nucleus on (9)Be, (27)Al, (51)V and (120)Sn targets. The elastic scattering and transfer of (8)Li and (7)Be on several targets is also being studied. The transfer reaction (8)Li(p,alpha)(5)He of astrophysical interest was also Studied in the E(cm)=0.2-2.5 MeV energy range.

Fiberglass-reinforced glulam beams: Mechanical properties and theoretical model

The glued-laminated lumber (glulam) technique is an efficient process for making rational use of wood. Fiber-Reinforced Polymers (FRPs) associated with glulam beams provide significant gains in terms of strength and stiffness, and also alter the mode of rupture of these structural elements. In this context, this paper presents a theoretical model for designing reinforced glulam beams. The model allows for the calculation of the bending moment, the hypothetical distribution of linear strains along the height of the beam, and considers the wood has a linear elastic fragile behavior in tension parallel to the fibers and bilinear in compression parallel to the fibers, initially elastic and subsequently inelastic, with a negative decline in the stress-strain diagram. The stiffness was calculated by the transformed section method. Twelve non-reinforced and fiberglass reinforced glulam beams were evaluated experimentally to validate the proposed theoretical model. The results obtained indicate good congruence between the experimental and theoretical values.