A STUDY OF THE APPROXIMATE THEORY OF AN ELASTIC THICK BEAM

1957 ◽  
Vol 35 (6) ◽  
pp. 757-765 ◽  
Author(s):  
Seiichi Higuchi ◽  
Hideo Saito ◽  
Chiaki Hashimoto

The flexural vibration of a thick beam is analyzed two-dimensionally by using the general equation of flexural motion. The characteristic equations and the frequencies of thick beams for the various end conditions are obtained and comparisons are made between the frequencies obtained from the present calculations and those from the elementary theory of a thin beam.

1965 ◽  
Vol 87 (3) ◽  
pp. 303-308 ◽  
Author(s):  
D. S. Griffin

A solution is obtained using the linear theory of elasticity for the stresses in a long, thick, uniformly curved plate due to pressures on the curved surfaces and forces or deflections specified on the straight edges. The maximum stresses are compared with those obtained by the elementary thin beam theory, showing the range of applicability of the simpler theory. In general, the accuracy of the elementary theory depends on the total plate angle (α) as well as the mean radius-to-thickness ratio (rm/t).


2011 ◽  
Vol 95 (533) ◽  
pp. 235-239
Author(s):  
Juan Pla

In this note we start by exploring a type of solution of the equation in positive integersfor a given p, which will enable us easily to derive a class of solutions in integers of the more general equation in positive integersfor any positive integers p and n.In another part of this note we explore some connections between the formula we find and a particular chapter in the elementary theory of numbers.


Studies of calcite twin lamellae in an experimental thick-beam fold, a multilithologic, layered thin-beam buckle, and in several multilithologic faulted and drape-folded specimens provide a test of the use of these lamellae to map the strain and stress fields developed under known, yet complex, boundary conditions. Even though the number of grains available for sampling per domain of homogeneous deformation is small (8 to 100), we find that in 24 of 28 domains studied the derived orientations and relative magnitudes of the principal strain axes agree to + 30 with the corresponding inferred principal stress axes, with those inferred from accompanying microfractures and faults, and with those expected from the bending and the boundary conditions. The absolute differences in strain between those inferred from the twin lamellae and those calculated from bedding-thickness changes or from the surface strains recovered from distorted grid-lines averages ±0.01. The poorest correlations are for domains where the sample size is < 20. In addition, the twin-lamellae strain analyses permit better definitions of superposed deformation and anticlastic bending than do the corresponding stress analyses. For strains up to about 0.15, calcite twin lamellae are remarkably good indicators of the strain and stress histories of deformations.


2016 ◽  
Vol 24 (5) ◽  
pp. 839-859 ◽  
Author(s):  
Lourdes Rubio ◽  
José Fernández-Sáez ◽  
Antonino Morassi

We consider the identification of a single open crack in a simply supported beam having nonuniform smooth profile and undergoing infinitesimal in-plane flexural vibration. The profile is assumed to be symmetric with respect to the mid-point of the beam axis. The crack is modeled by inserting a rotational linearly elastic spring at the damaged cross-section. We establish sufficient conditions for the unique identification of the crack by a suitable pair of natural frequency data, and we present a constructive algorithm for determining the damage parameters. The result is proved under a technical a priori assumption on the zeros of a suitable function determined in terms of the eigenfunctions of the problem. Extensions to beams under different sets of end conditions are also discussed. Theoretical results are confirmed by an extensive numerical investigation, both on simulated and experimental data.


The lateral buckling of an I-section beam bent by pure couples in the plane of its web is considered. The method of Reissner and Grober, who discussed beams of rectangular section, is combined with that of Timoshenko who gave approximate theory for an I-section. In this way, the critical moment of an I-section beam supported at its ends in the most general manner is obtained. The result shows that the critical moment depends upon the fixing of the individual flanges as well as of the whole beam at the ends, and has been verified by experiment. When the principal flexural rigidities of the section are of the same order, so that the curvature in the plane of bending is large, a modification of the end-conditions given by previous writers for both I-and rectangular-section beams is necessary. This modification is not important in practical cases.


This paper is concerned with qualitative properties of the natural frequencies and principal modes of some basic vibrating systems: the taut, non-uniform string; slender non-uniform rods in longitudinal and torsional vibration; the slender beam in flexural vibration; the circular membrane. These systems may be modelled by lumped-mass systems, finite elements, or by differential equations. The paper summarizes results on the distinctness of natural frequencies, interlacing of frequencies corresponding to different end conditions, nodal patterns for modes, etc. It also briefly describes research on the reconstruction of such systems from natural frequency or principal mode data.


2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Emem Ayankop Andi ◽  
Sunday Tunbosun Oni

This paper investigates the flexural vibration of a finite nonuniform Rayleigh beam resting on an elastic foundation and under travelling distributed loads. For the solution of this problem, in the first instance, the generalized Galerkin method was used. The resulting Galerkin’s equations were then simplified using the modified asymptotic method of Struble. The simplified second-order ordinary differential equation was then solved using the method of integral transformation. The closed form solution obtained was analyzed and results show that, an increase in the values of foundation moduli K and rotatory inertia correction factor R0 reduces the response amplitudes of both the clamped-clamped nonuniform Rayleigh beam and the clamped-free nonuniform Rayleigh beam. Also for the same natural frequency, the critical speed for the moving distributed mass problem is smaller than that for the moving distributed force problem. Hence resonance is reached earlier in the former. Furthermore, resonance conditions for the dynamical system are attained significantly by both R0 and K for the illustrative end conditions considered.


1957 ◽  
Vol 24 (4) ◽  
pp. 509-514
Author(s):  
A. A. Betser ◽  
M. M. Frocht

Abstract Simply supported short Castolite beams of uniform rectangular cross section were subjected to central transverse impact by a heavy mass. Photoelastic streak photographs were taken of the transverse section of symmetry for a wide range of spans, widths, and impact velocities at exposures of less than 1 microsec. The maximum tensile stresses were determined. Comparison with the elementary theory for long beams shows that while this theory is satisfactory for long beams, it does not agree with the results from short beams. An approximate theory for short beams under central impact is developed which gives satisfactory agreement. The duration of impact also was determined and the appearance of isotropic points is discussed.


TAPPI Journal ◽  
2012 ◽  
Vol 11 (10) ◽  
pp. 9-17
Author(s):  
ALESSANDRA GERLI ◽  
LEENDERT C. EIGENBROOD

A novel method was developed for the determination of linting propensity of paper based on printing with an IGT printability tester and image analysis of the printed strips. On average, the total fraction of the surface removed as lint during printing is 0.01%-0.1%. This value is lower than those reported in most laboratory printing tests, and more representative of commercial offset printing applications. Newsprint paper produced on a roll/blade former machine was evaluated for linting propensity using the novel method and also printed on a commercial coldset offset press. Laboratory and commercial printing results matched well, showing that linting was higher for the bottom side of paper than for the top side, and that linting could be reduced on both sides by application of a dry-strength additive. In a second case study, varying wet-end conditions were used on a hybrid former machine to produce four paper reels, with the goal of matching the low linting propensity of the paper produced on a machine with gap former configuration. We found that the retention program, by improving fiber fines retention, substantially reduced the linting propensity of the paper produced on the hybrid former machine. The papers were also printed on a commercial coldset offset press. An excellent correlation was found between the total lint area removed from the bottom side of the paper samples during laboratory printing and lint collected on halftone areas of the first upper printing unit after 45000 copies. Finally, the method was applied to determine the linting propensity of highly filled supercalendered paper produced on a hybrid former machine. In this case, the linting propensity of the bottom side of paper correlated with its ash content.


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