Beam Of Rectangular Cross-section Research Articles

Finite Element Solutions of Deep Beams

The finite element solutions of deep beams are presented. Two types of deep beam elements are introduced. The first element simulates the exact analytical solution of the deep beam of rectangular cross section. The element stiffness matrix is developed by solving the plane stress elasticity equations with the selection of the proper boundary conditions. The second element simulates the approximate solution of the deep beam of arbitrary cross section as it assumes a cubic normal strain distribution along the beam cross section. The element stiffness matrix is derived by applying the principle of virtual displacements of the system. The two elements are found to yield similar results for beam aspect ratios which are greater than 1.0. The deep beam element solutions were compared to the conventional engineering theory of beam solutions for simple (cantilever) structures and complex (turbine generator pedestal) structures; the results of the two methods of solution were found to differ significantly.

Failure Time of Loaded Wooden Beams During Fire

A model is presented for predicting the failure time of loaded wooden beams of rectangular cross-section exposed to elevated temperatures or to fire. Failure times calculated by the model were compared to failure times measured in this study using 19.05 mm x 19.05 mm simply supported southern pine beams, and to failure times measured by the National Bureau of Standards during the fire of a full scale room. Reasonable agreements were found between the calculated failure times and the data.

A Monte Carlo model for absorbed dose calculations in computed tomography.

A Monte Carlo program has been devised that estimates the total absorbed energy and mean dose in defined regions within phantoms exposed to rotating x-ray sources. The calculations deal with fanlike beams of rectangular cross section projected onto elliptic cylindrical phantoms, which may be inhomogeneous. The effects of coherent scattering, which may be significant in a narrow-beam geometry, can be studied. The code can score the exit flux, allowing study of the relationship between dose and image quality. The model is described, the code is verified, and typical results are presented.

Stability of columns with a single crack subjected to follower and vertical loads

A general flexibility matrix is developed which expresses the local flexibility of a beam of rectangular cross section with a single edge crack. The dominant term in this matrix is used to study the stability of the cracked column to follower and vertical loads. The kinetic method is used and eigencurves are developed to study the system stability.For follower type loads, the Beck column, stability charts are given for several crack locations and sizes. Flutter type of instability is always encountered. For vertical loads the same type of analysis reveals the divergent type of instability, reported already in the literature. It was found that cracks can make this system have flutter type of instability for vertical loads.

Scatter dose decrement values for rectangular fields.

The scatter dose decrement value (SDV) has been studied by calculational methods for 60Co gamma and 4- and 8-MV x-ray beams of rectangular cross section. In a principal plane and inside the field, the SDV is nearly independent of field size, energy, and depth when the position of the point considered is expressed as the fractional distance to the field edge. From this, it follows that the scatter dose profiles are quite similar in planes parallel to a principal plane. A second corollary is that the SDV for an arbitrary point approximately equals the product of the two SDV values for the projected points in the principal planes. Outside the field, the SDV loses this independence of the various beam conditions, and as a consequence thereof, the "product rule" leads to errors. Mathematical expressions have been derived for the SDV inside and outside small fields. The errors resulting when these small-field formulas are used for large fields have been studied. Expressed as fractions of the total dose on the centerline, these errors are of the order of +/- 1% inside the field and +/- 2% outside.

Stability of a pretwisted tapered cantilever beam subjected to dissipative and follower forces

Stability of a pretwisted tapered cantilever beam of rectangular cross-section subjected to a follower force at its free end is investigated. The effects of internal and external damping are included in the study. The non-self-adjoint boundary value problem is formulated with the Euler-Bernoulli theory and an associated adjoint boundary value problem is introduced. Approximate values of the critical load are calculated on the basis of a suitable adjoint variational principle for several values of the geometric and material parameters of the beam. The results are shown in graphs.

Plane Curved Beam Equations

It is demonstrated that plane curved beam (rectangular cross-section) equilibrium equations can be obtained by reducing the established equations for thin shells. The necessary equations are presented for the special case of curved beams of rectangular cross section having the reference surface located eccentrically to the centroidal surface.

A new rectangular beam theory

A new theory for beams of rectangular cross-section which includes warping of the cross-sections is presented in the present work. By satisfying the shear-free conditions on the lateral surfaces of the beam a pair of coupled equations of motion are obtained such that no arbitrary shear coefficient is required. It is shown that the uncoupled equation for the transverse displacement is the same as the corresponding equation in Timoshenko beam theory provided that for the Timoshenko equation the shear coefficient is taken to be 5 6 ; this value lies within the range of values, 0·822–0·870, appearing in the literature for the beam of rectangular cross-section. Results for two typical static examples are given for both the new theory and Timoshenko beam theory. These results are compared with the solutions of the comparable problems in the linear theory of elasticity. For the end loaded cantilever beam the new theory predicts the same result for the neutral surface deflection as does the linear theory of elasticity while Timoshenko beam theory underestimates the shear correction term by 20%. For the uniformly loaded and simply supported case both beam theories provide the same overestimate of the central deflection when compared with the theory of elasticity solution.

Effect of angle of attack on the stability of a rotating non-uniform cantilever with a tip mass subjected to dissipative and non-conservative forces

The influence of angle of attack on the stability of a rotating viscoelastic tapered cantilever beam of rectangular cross-section carrying a tip mass and subjected to a circulatory force at its free end is investigated. The effect of external damping is included in the study. The non-self-adjoint boundary value problem is formulated with the Euler-Bernoulli hypothesis and an appropriate adjoint boundary value problem is introduced. Approximate values of the critical flutter load are calculated on the basis of an adjoint variational principle for several values of geometric and material parameters of the system. The results are presented through a series of graphs.

Note on a Nontrivial Simple Example of Higher-Order One-Dimensional Beam Theory

Abstract : One-dimensional beam equations are derived for a rectangular-cross section beam consisting of shear webs and corner stringers with one cross-sectional axis of symmetry. It is shown that a systematic statement of this problem involves one more equation than elementary theory, with this additional equation involving the concept of cross-sectional warping and bi-moment. As an application of the general results of the paper, new conclusions are deduced concerning the location of the center of shear of the beam. (Author)

振動するはりから放射する音圧の解析

This investigation is the fundamental study to clear the mechanism of sound radiation from a machine structure and an architecture. In this paper, the sound radiation of an unbaffled beam of rectangular cross section is cleared. The relation between the amplitude of velocity and the sound pressure of an unbaffled beam is theoretically analyzed in consideration of the following cases. 1) The plane wave is radiated from a vibrating beam. 2) The mechanism of sound radiation from a vibrating beam is equivalent to that from two spherical sound sources. The analytic results are verified by experiments. It is concluded that in these experimental results, the sound pressure of an unbaffled beam is influenced by the amplitude of velocity, excitation frequency, the dimension of rectangular cross section of beam, and the distance from origin to observation point. In the latter, an agreement between the experimental results and the numerical results is obtained, but it is not so in the former.

Pure bending of an I-section beam having initial curvature

From the two ‘approximate’ theories of bending—Elementary Theory and Winkler available to designers for the determination of bending stresses in beams having initial curvature, the Winkler approximate solution is the one closer to the exact solution development by Golovin.Timoshenko and Goodier show a comparison of stresses resulting from the three theories for a beam having rectangular cross-section. It is shown here that the values of bending stresses given by Winkler theory are very close to Golovin's exact results. The difference between the two theories lies in the range of 0·1–3·1 % increasing with the ratio, c/a, of the outer to inner radius of curvature of the beam. The above close relationship between the two theories for a beam of rectangular cross-section is also reported by Ford.

End Effects in Long Curved Beam

Eigenfunction solutions of the biharmonic equation have been used to investigate the Saint-Venant boundary region in a long curved beam of rectangular cross section. The participation constants involved in the eigenfuction expansions are determined by expanding each of the nonorthogonal eigenfunctions in terms of an orthogonal set of real functions. The numerical convergence of the associated eigenfunction expansions and the decay properties have been studied for some prescribed self-equilibriated end loading.

Large amplitude free vibrations of tapered beams

The Galerkin method is used to investigate the large-amplitude free vibrations of simply supported and clamped tapered beams of rectangular cross-section which are frequently encountered in practical structures. Two types of linear tapers are considered: breadth and depth tapers. Two solutions are obtained for each type of taper, using trigonometric and polynomial displacement distributions. Frequency-amplitude relationships are obtained for all these cases for the fundamental flexural mode. The results indicate good agreement between trigonometric and polynomial solutions for all cases. The nonlinearity is noted to be always of the hardening type and, as expected, it is severe for beams with depth taper as compared with beams with breadth taper.

A method for calculating the reduction in stress intensity factor due to an elastic foundation

Stress intensity factors for an edge-cracked beam of rectangular cross section on a Winkler foundation are calculated for free, pinned and clamped end conditions. The solution is an approximation, valid for slender beams. The method which is used requires that the intensity factors be known for the beam without a foundation. The same technique may be used for various crack geometries, beam cross sections and types of foundation.

Transverse vibrations of pre-twisted sandwich beams

A set of equations of motion governing the bending and extensional displacements of a pre-twisted sandwich beam of rectangular cross-section are derived by using Hamilton's principle. The middle viscoelastic core is assumed to deform mainly through the classical shearing mechanism. The eigenvalues and loss factors of simply supported pre-twisted sandwich beams are computed by using the variational method. Analysis of the results revealed that pre-twisting the beam increases the real part of the eigenvalue by as much as 20% while reducing the loss factor by as much as 30 %. The loss factor of very soft, thickcored beams is especially sensitive to even small angles of pre-twist: e.g., a 22· 5° pre-twist may reduce the loss factor by as much as 80%. The effect of pre-twist is, however, shown not to be appreciable for soft, thin-cored beams. In any case, pre-twisting of the beam has a detrimental effect on the maximum loss factor that one can obtain for a specific size of the beam when only the shear parameter of the beam is changed.

Mass optimization of non-conservative cantilever beams with internal and external damping

An adjoint variational principle has been developed for a non-conservatively loaded cantilever beam with Kelvin-Voigt internal and linear external damping and is applied to a beam with a linearly distributed tangential load acting along the centerline of the beam. Relative mass optimization for beams of both rectangular and circular crosssections is considered from a graphical standpoint and from the viewpoint of a computer optimization routine with data given and discussed in both instances. In going to a Rosenbrock optimization routine for beams of rectangular cross-section with a minimum tip thickness constraint imposed it was quite clear that mass ratio reductions in the range 14·9 % to 38 % are possible and that the values of internal and external damping appear influential in determining just how much of a mass reduction is possible. Similarly, for beams of circular cross-section a Rosenbrock optimization routine with a minimum tip diameter constraint imposed showed that mass ratio reductions of the order of 27 % are possible.

Second-order treatment of combined effects of space-charge and external fields on plane-symmetric zero-emittance beams of charged particles

An earlier first-order treatment of the space-charge spread of zero-emittance charged particle beams under the influence of plane-symmetric external fields is revised and extended to include second-order effects. Elliptic and rectangular cross-section beams are investiged and conditions for parallel beam transport are carried out.

Theoretical limitations of electron beams of rectangular cross section

Two separate aspects of beams of rectangular cross section have been considered, namely the maximum space-charge-limited current which can be passed through rectangular slot apertures, and the maximum current density which can be obtained in collimated and focused beams. Both these limiting effects must be considered when attempting to maximize the current in beams of rectangular cross section. In both cases it has been found that the maximum current density is approximately the same for doubly focused rectangular beams, for singly focused ribbon beams, and for axially symmetric beams.

Plane-Stress yielding of cantilevers in bending due to combined shear and axial load

A plane-stress slip-line field analysis has been made of the possible plastic deformation modes of a built-in cantilever beam of rectangular cross-section subjected to a shear force at the free end with or without an axial load. Plastic-collapse loads for the several types of deformation patterns suggested are computed and the results presented in detail in the form of curves. Some simple upper- and lower-bound estimates to the collapse load have also been determined and are compared with both the plane-stress slip-line results and those presented by Green (1), plane-strain conditions being assumed. The agreement between the plane-stress slip-line fields and the assumed lower-bound results is found to be excellent.