End Slope Research Articles

An Analysis of Large Deflection in a Symmetrical Three-point Bending of Beam

The large deflection problem of a thin elastic simply supported beam is analysed for a symmetrical three-point bending .The derived nonlinear differential equation governing beam deflections is solved by applying the numerical method (R-K-G method) and the analytical method based on Legendre-Jacobi form's elliptic integrals of the first and the second kinds. Moreover, a reduction technique is proposed to estimate representative flexural quantities such as a maximum deflection, an end slope, and a maximum bending stress in large deflection states from the conventional linear bending theory in place of the exact large deflection theory. An experiment is also performed to confirm the applicability of the proposed large deflection theory. The experimental results agree well with those obtained from the exact large deflection theory.

An Analysis of Large Deflections in a Four-point Bending : In the Case of the Presence of Friction at Loading Supports

Concerning a symmetrical four-point bending, the nonlinear large deflection problem of a thin elastic beam in the presence of friction between beam and the loading supports is investigated analytically by applying Legendre-Jacobi form's elliptic integrals of the first and the second kinds. A reduction technique is proposed to estimate easily the maximum deflection, the end slope, and the maximum bending stress in large flexural states from the conventional linear bending theory. An experiment is also performed to confirm the applicability of the presented large deflection theory. The experimental results agree well with the solutions based on the large deflection theory.

大たわみ変形の簡易推定法についての一提案 四点曲げ

It is very important industrially to estimate the degree of large deflection appearing in such members as a plate spring, plastics and so on. In this report, the problem of large deflection of a simply supported beam in a symmetrical four-point bending test was treated. It was shown that the desired specific quantities such as the maximum deflection, maximum curvature, maximum bending stress and end slope, which are especially matters of great concern in the material testing, were able to be predicted easily from a conventional linear bending theory and three reduction factors derived here newly.For the convenience of applications, the nomographs of the reduction factors were given graphically in terms of the vertical load, the large deflection at midpoint and the end slope.The experiment of large deflection in which a PVC thin plate was utilized was carried out to confirm the availability of the newly proposed reduction method. The experimental results obtained were in good agreement with the predicted ones by the reduction method. Therefore, the present simple reduction method can be put into practical use.

大たわみ変形の簡易推定法についての一提案 三点曲げ試験

It is very important industrially to estimate the degree of large deflection appearing in such members as a plate spring, plastics and so on. In this report, the problem of large deflection of a simply supported beam in a symmetrical three-point bending test was treated. It was shown that the desired specific quantities such as maximum deflection, maximum curvature, maximum bending stress and end slope, which are especially matters of great concern in the material testing, were able to be predicted easily from a conventional linear bending theory and two reduction factors derived here newly.For the convenience of applications, the nomographs of the reduction factors were described graphically in terms of the vertical load, the large deflection at midpoint and the end slope.An experimental investigation of large deflection of a PVC thin plate was performed to confirm the availability of the newly proposed reduction method. The experimental results obtained were in good agreement with the predicted ones by the reduction method. Therefore, the simple reduction method can be put into practical use.

An analysis of large deflections in a four-point bending with friction at all supports.

In a symmetrical four-point bending with friction at both loading supports and reaction supports, the problem of nonlinear large deflections of a thin elastic simply supported beam is analysed by the numerical method (R-K-G method) and the analytical method using the Legendre-Jacobi form's elliptic integrals of the first and second kinds. Moreover, the conversion factors are proposed to estimate the maximum deflection, end slope, maximum bending stress in large flexural states with friction at loading and reaction supports from the conventional linear bending theory in place of the exact large deflection theory. The experimental results conform well to ones obtained by the presented large deflection theory.

An analysis of large deflections in a four-point bending (In the case of the presence of friction at loading supports)

In symmetrical four-point bending, the problem of nonlinear large deflections of a thin elastic beam in the presence of friction at loading supports is investigate analytically by applying the Legendre-Jacobi's elliptic integrals of the first and second kinds. Moreover, the reduction factors are presented to estimate easily the maximum deflection, end slope and maximum bending stress in large flexural states. The experiment is made in order to confirm the applicability of the proposed large deflection theory and the experimental results agree well with the theoretical ones.

Circadian control of singing in crickets: Two different pacemakers for early-evening and before-dawn activity

(1) Under LD 12:12 and constant temperature, calling song patterns of the Australian Field Cricket Teleogryllus commodus were frequently divided into two more or less distinct activity components. The first began 1–3 h before the L D -transition and ended with the sudden lights-off, whereas the second was restricted mainly to the second half of the night. In subsequent continuous light (LL) the circadian activity pattern was usually a unimodal band. Its onset slope was always a continuation of the first onsets in the preceding LD-cycle. In those cases where shortly after the LD LL transition the rhythm split into two components, both corresponded to those in LD. (2) In “atypical” LD-aatterns only the second component appeared during the late night. In such cases, the onset slope of the subsequent free-running rhythm did not start at the preceding LD-entrained onsets, but was advanced by several hours. (3) In free-running rhythms the mean value of the onset slopes ( τ o = 25.1 ± 0.07 h ) did not differ significantly from that of the end slopes ( τ e = 25.0 ± 0.07 h ). However, in individual activity patterns, simultaneous differences in onset and end slopes of up to 0.7 h were found. τ o- and τ e-values recorded directly after the LD LL -transition did not correlate with the preceding phase angle difference between the entrained activity and the zeitgeber. (4) Under constant conditions after-effects such as spontaneous period changes and phase shifts in the slopes of activity onset and/or end occurred mainly between five and 20 days after the LD LL -transition . (5) Period changes and phase delays in the onset and end slopes were also found when crickets were exposed to low temperature pulses (2 ± 2°C, 2h duration). Following the cold treatment both period changes and phase shifts in the onset slope frequently differed from those in the end slope. (6) The results are consistent with a concept of two activity components controlled by weakly coupled circadian pacemakers. This is clearly evident in split rhythms but is also true for unimodal patterns. Normally the two components partly overlap and their existence is concealed in the undivided activity band. They are only revealed in unimodal patterns if individual oscillatory properties such as different period lengths or phase shifts appear in the onset or the end slope.

Lateral thinking at gloucester leads to new ground rules : Parkinson, J New Civ Engr, N506, Sept 1982, P14–15

It was anticipated that severe settlement of the roughly 8M thick layer of alluvium in the River Severn flood plain would also lead to lateral forces acting on piles supporting a new crossing of the navigable section of the river carrying the A40 trunk road away from the city centre. The embankment was formed two years early under an advanced contract. Extensive soil tests checked the decrease in shear strength, and the lateral movement, but it was found to be well within the estimated 20mm allowed in the design. Hollow U shaped bridge abutments, formed half way up the end slope of the embankment were used. Very good agreement has been found between the reassessed rate of vertical settlement and values predicted, using data on load application, by a TRRL finite element computer program. The viaduct structure has an in-situ prestressed cellular deck with pre-cast drop beams over the river. Because major plant and heavy falsework foundation loads could not be allowed on the flood plain, temporary work supports had to be separately piled and Bailey bridge sections have been used to support a complete span plus a 12M overlay. (Author/TRRL)

A bifurcation problem involving elastica

A flexible inextensible elastica has been considered, with fixed end points and fixed equal end slopes, and varying length. The existence of at least one elastic curve has been proved when the elastica length is greater than the distance between the end points. A critical value of the length has been found, as a function of the given end slope only, corresponding to some elastic curves. From each of them two new elastic curves branch when the length is greater than the critical value.

A beam bending problem with a free boundary

The stress analysis of a pipeline lifted for jointing or repair is discussed. The behaviour of the raised part of the line is governed by ordinary beam theory, in the absence of axial loads. However, the analysis is non-trivial because the total length of line lifted from the foundation is unknown, introducing a free boundary into the problem. Two practical situations are considered. In the first, the line is lifted by a single sling to obtain a specified height at the end of the pipe. In the second, the line is supported at two places, achieving a specified height and zero end slope. Computed results, non-dimensionalised in terms of a particular case for which a closed form solution exists, are presented for the two situations.

Post-buckling analysis of spring-hinged cantilever columns

The post-buckling behaviour of spring-hinged cantilever columns has been investigated in this paper. The governing nonlinear differential equation with variable coefficients has been solved using Galerkin finite element method. Three loading conditions, namely, a concentrated tip load, a uniformly distributed load and a linearly varying load, acting on the column are considered in the present analysis. Very accurate estimates of the axial load parameters have been presented for different end slopes and for different spring constants.

Post-buckling analysis of tapered cantilever columns

Abstract The post-buckling behaviour of tapered cantilever columns is investigated in this paper. The governing nonlinear differential equation with variable coefficients is solved using the Galerkin finite element method. Three loading conditions on the column are considered in the analysis - a concentrated tip load, a uniformly distributed load and a linearly varying load. Very accurate estimates of the axial load parameters are presented for various end slopes and for different taper parameters.

Results of monitoring and observation of opencast mine ‘Kosovo’ - belacevac end slope landslide (in Serbo-Croat) : Obradovic, R; Ilic, D; Stamatovic, A Rud Glas, N4, 1977, P15–21

Results of monitoring and observation of opencast mine ‘Kosovo’ - belacevac end slope landslide (in Serbo-Croat) : Obradovic, R; Ilic, D; Stamatovic, A Rud Glas, N4, 1977, P15–21

Landslide of opencast mine ‘Kosovo’ belacevac west end slope (in Serbo-Croat) : Obradovic, R; Ilic, D; Dutina, A Rud Glas, N3, 1977, P5–15

Landslide of opencast mine ‘Kosovo’ belacevac west end slope (in Serbo-Croat) : Obradovic, R; Ilic, D; Dutina, A Rud Glas, N3, 1977, P5–15

A simple specification of the parametric cubic segment

A parametric cubic segment may be specified in many ways but most specifications require quantities peculiar to the parametric representation rather than the geometric properties of the curve segment. This paper shows that a three-dimensional cubic segment is specified completely by its end points, end slopes and one intermediate point.

Convex Cubic Splines

Control curves used in industry are usually planar cubic splines, continuous and single-valued for a specific independent variable. If large slopes are specified at certain points, the spline coefficients, which are computed in order to preserve second derivative continuity of the spline, invariably lead to wildly oscillatory curves. Normally the large slopes occur at the ends of the curve, and this paper deals with such cases, developing methods which have been used successfully to combat the problems encountered. The end point where the difficulty occurs is ignored for the moment and a curve (spline or not) is designed for the remaining points. Then, with the point and slope adjacent to the difficult point known, the design is completed with one or more spline segments so that the resulting curve, which connects the difficult point to the next point, is convex (or concave), and has at least first derivative continuity. For the large finite slope case, two methods are described. The first method constructs the desired curve as a sequence of spline segments with finite slope everywhere, whereas the second expands the interval so that a single spline segment with infinite end slope passes through the difficult point with the required slope. The case of infinite end slope is also treated. In this and the preceding cases, second derivative continuity can be preserved at the juncture under certain conditions.

Post-buckling of uniform cantilever columns—Galerkin finite element solution

Post-buckling analysis of uniform cantilever columns subjected to three types of loading, namely a concentrated tip load, a uniformly distributed load and a linearly varying load has been presented using the Galerkin Finite Element Method. The load parameters are presented for various end slopes and the results are compared with published results. It is noted that the results compare very well.

Approximations for large deflection of a cantilever beam

Rational approximations for the end slope and deflection of a cantilever beam are derived from the exact solution in terms of elliptic integrals [1] by asymptotic methods.

End slope and fundamental frequency of vibrating fuel rods

An analytical tool is presented for calculating both the fundamental frequency and the end slope per unit amplitude of a rod supported by two adjacent grids for arbitrary support fixities.

On the Diffusion of Load From a Transverse Tension Bar Into a Semi-Infinite Elastic Sheet

This paper deals with the load diffusion from a tension bar of finite length and uniform cross section into a semi-infinite sheet, the axis of the bar being perpendicular to the edge of the sheet. The bar is regarded as a one-dimensional elastic continuum, whereas the elastic sheet is treated within the two-dimensional theory of generalized plane stress. Three alternative models for the stringer-attachment are considered: (a) line-contact, (b) area-contact based on matching the axial stringer-strain and the corresponding average sheet-strain across the width of the strip of adhesion, and (c) area-contact based on matching the stringer strain and the corresponding sheet-strain along the center line of the strip of adhesion. It is shown that the line-contact model, in contrast to both area-contact models, does not admit the transmission of portions of the applied load through forces concentrated at the ends of the adhering bar segment. Further, asymptotic estimates are deduced for the end slopes of the load-diffusion curves appropriate to the three models under consideration. The integro-differential equation for the stringer-force in Case (b) and Case (c) is reduced to a standard Fredholm integral equation, which is solved numerically. The results thus obtained are compared with available experimental findings.