Showing papers on "Bending moment published in 1970"
01 Jan 1970
TL;DR: In this paper, a new strip theory is presented for predicting heave, pitch, sway, roll, and yaw motions as well as wave-induced vertical and horizontal shear forces, bending moments, and torsional moments for a ship advancing at constant speed with arbitrary heading in regular waves.
Abstract: A new strip theory is presented for predicting heave, pitch, sway, roll, and yaw motions as well as wave-induced vertical and horizontal shear forces, bending moments, and torsional moments for a ship advancing at constant speed with arbitrary heading in regular waves. A computer program based on this theory and with accurate close-fit section representation has been developed. Comparisons between computed values and experimental data show satisfactory agreement in general. In particular, very good agreement is shown for the heave and pitch motions and the vertical loads. Accurate results are also obtained for the coupled sway-roll motions in beam waves. Although comparisons are not yet available for the sway-roll-yaw motions in oblique waves, the satisfactory agreement shown for the horizontal loads in oblique waves suggests that the theory may also predict the horizontal motions quite well.
726 citations
82 citations
TL;DR: In this paper, the bending moments of semi-fixed partial dentures were found to be higher than those estimated for fixed part dentures attached at both ends, and the relationship of design to restorative materials was discussed in several papers.
Abstract: lh e principles of engineering design should be used in the structural design of fixed partial dentures as well as the biologic, esthetic, and mechanical restrictions of the oral cavity.’ The conventional structural design is primarily concerned with the analysis of given structures, using the conventional equations of strength of materials.2z 3 Bending moments in models representing fixed partial dentures were studied,’ and it was found that bending moments of semi-fixed partial dentures were higher than those estimated for fixed partial dentures attached at both ends. Load carrying capacities of dental beams were theoretically determined by Brumfield”g 6 who asserted that the most important factor was depth. The relationship of design to restorative materials was generally discussed in several papers.‘-g The first report concerning the measurement of stresses in fixed partial dentures using a brittle coating technique was published in 1965. lo Strain gages were also used to study the stress distribution on gold and chromium alloy bridges.ll
58 citations
TL;DR: Impulsively loaded rigid plastic continua deformation lower bounds lower bound calculation, considering beams and plates as discussed by the authors, considering both rigid and non-rigid materials. But this lower bound is not applicable to rigid materials.
Abstract: Impulsively loaded rigid plastic continua deformation lower bounds calculation, considering beams and plates
28 citations
TL;DR: In this article, a uniform Bernoulli-Euler beam of finite length is supported by a Pasternak-type foundation and subjected to a harmonic force F = F 0 e jwt, concentrated at the midpoint.
27 citations
TL;DR: In this article, the behavior of the lumped flexibility (considered as a black box) is governed by a prescribed plastic potential function, and three numerical examples are given in which interactions of the axial force, two bending moments, and torsional moment are considered.
Abstract: Associated with each joint is a rigid body (having six degrees of freedom) formed by a lumping procedure applied to the mass of the incident members. As long as elasticity is not exceeded, the members are considered to possess continuous flexibility. When yielding occurs at a member end, a small portion of the member is replaced by an elasto-plastic segment with lumped flexibility. If the segment length is set equal to zero, it corresponds to the concept of a plastic hinge. The behavior of the lumped flexibility (considered as a black box) is governed by a prescribed plastic potential function. Three numerical examples are given in which interactions of the axial force, two bending moments, and torsional moment are considered. One example deals with the response of a 2-story, 2-bay steel frame to earthquake shocks.
26 citations
01 Nov 1970
TL;DR: In this paper, the authors present a study of the behavior of 2-inch-diagrams with varying densities in a 2-inch-dimentioned sand.
Abstract: THE RESULTS ARE PRESENTED OF A STUDY OF THE BEHAVIOR OF PILES SUBJECTED TO AXIAL AND LATERAL FORCES. PILE- SOIL INTERACTION CAN BE REPRESENTED BY FAMILIES OF INTERACTION CURVES; CRITERIA FOR DESCRIBING SUCH FAMILIES OF INTERACTION CURVES FROM SOIL PROPERTIES ARE INVESTIGATED. SPECIFICALLY, ATTENTION IS DIRECTED TOWARD FORMULATION OF CRITERIA FOR PILES IN SAND. TO STUDY THE INTERACTION, 2-INCH-DIAMETER PILES WERE BURIED IN SUBMERGED SAND WITH CONTROLLED DENSITY. INSTRUMENTATION WAS PROVIDED FOR THE MEASUREMENT OF AXIAL LOAD AND BENDING MOMENT DISTRIBUTIONS. FROM THESE DISTRIBUTIONS, AXIAL AND LATERAL INTERACTION CURVES WERE GENERATED AND CORRELATED WITH MEASURED SOIL PROPERTIES. BASED ON THE CORRELATIONS, CRITERIA FOR DESCRIBING THE INTERACTION CURVES WERE FORMULATED. AXIAL AND LATERAL INTERACTION CURVES, GENERATED USNG THE PROPOSED CRITERIA, ARE USED TO PREDICT ANALYTICALLY THE RESPONSE OF THE TEST PILES. THE PROPOSED CRITERIA ARE ALSO USED TO COMPUTE THE RESPONSE OF A NUMBER OF PILES WHICH HAVE BEEN TESTED AND THE RESULTS REPORTED IN THE LITERATURE. THE COMPUTED RESPONSE OF THE PILES IS COMPARED WITH THE MEASURED RESPONSE OF THE PILES TO CHECK THE VALIDITY OF THE PROPOSED CRITERIA. /AUTHOR/
19 citations
Patent•
18 May 1970
TL;DR: In this article, a road vehicle suspension system has a beam pivotally connected with an axle at each end and one or more rubber blocks secured to a triangularly shaped block pivotally mounted on the beam, the rubber springs at their other ends being nonpivotally mounted on a frame of the vehicle, the pivot point between the rubber blocks and the beam lying on or about the horizontal center plane through the metal plate secured to the pivotally secured block.
Abstract: A road vehicle suspension system has a beam pivotally connected with an axle at each end and one or more rubber springs secured to a triangularly shaped block pivotally mounted on the beam, the rubber springs at their other ends being nonpivotally mounted on the frame of the vehicle, the pivot point between the rubber springs and the beam lying on or about the horizontal center plane through the metal plate secured to the pivotally secured block. This invention relates to suspension systems for road vehicles, and is particularly, but not necessarily exclusively, related to suspension systems for vehicles designed for heavy loading. It has been proposed to replace conventional suspension systems employing leaf springs which systems are heavy and relatively complex and costly, with a suspension system employing blocks of rubber or the like material having metal plates bonded to the ends of the blocks and serving to connect the blocks to the frame of the vehicle and to the axle (or axles in the case of a bogey or similar assembly). It is the case that in such suspension systems, the rubber or the like blocks are mounted so that they are subjected to combined shear and compression under load, giving greater flexibility and therefore better traction, and it is important to the correct functioning of the system that there should be no restriction imposed on the ability of the blocks to be compressed by the vertical and longitudinal components of the load, for otherwise the system would not have the ability to resiliently support the loads for which the system has been designed. The object of the present invention is to provide such, so called, rubber suspension systems with improved ability to withstand bending moments that can be set up in the rubber blocks under certain extreme conditions, for example when an unloaded vehicle is running empty and is braked violently. According to the present invention a road vehicle suspension system has a beam pivotally connected with an axle at each end and one or more rubber springs are pivotally mounted on the beam and nonpivotally mounted on the frame of the vehicle, the rubber springs each comprising metal plates, separated by and bonded to blocks of rubber or the like material mounted between the frame and the beam, such that the rubber portions are subjected to combined shear and compression under load, the pivot point between the rubber springs and the beam lying on or about the horizontal center plane through the metal plate pivotally secured to the beam. As had hitherto been proposed the pivotal connection between the rubber spring and the beam, with the vehicle unloaded lay below the horizontal center plane through the plate at that end of the rubber spring so that when running empty, severe braking or rapid acceleration could produce horizontal forces acting on the bottom edge of the lower plate with the result that severe bending moments could be set up in the rubber blocks thus increasing the possibility of tearing of the rubber blocks. This condition in the suspension system does not arise when the vehicle is loaded because when loaded, the effective line of action of any braking or acceleration forces has been effectively raised by the lowering of the frame and the load so that the braking and acceleration forces are taken centrally by the plate secured to the beam and, accordingly, no or exceedingly small, bending moments are applied to the rubber blocks. By arranging the pivot point of the rubber block to the beam to lie on or about the center plane thRough the plate on the rubber block secured to the beam, even when running empty, the braking forces and acceleration forces are taken centrally by that plate, and accordingly, the rubber blocks are not subjected to any bending moment at all, or are only subjected to such small bending moments that are well able to be resisted by the rubber blocks. When the vehicle is loaded, the effective line of action is raised above the center plane of the plate secured to the beam and any accelerating or braking forces in that plane do not subject the rubber blocks to bending moments.
18 citations
01 Jul 1970
TL;DR: In this paper, a simplified method for the Torsional analysis of SINGLE-SPAN or CONTINUOUS CURVED BOX Girders is presented.
Abstract: A SIMPLIFIED METHOD IS PRESENTED FOR THE TORSIONAL ANALYSIS OF SINGLE-SPAN OR CONTINUOUS CURVED BOX GIRDERS, WHICH, BY VIRTUE OF THEIR EXCELLENT STRENGTH IN RESISTING TORSION, ARE GENERALLY RECOGNIZED AS IDEAL SUPPORTING ELEMENTS FOR HORIZONTALLY CURVED STRUCTURES. UNDER MOST CONDITIONS ENCOUNTERED IN PRACTICE, THE BENDING AND TORSIONAL ANALYSIS OF CURVED BOX GIRDERS MAY BE UNCOUPLED AND INVESTIGATED INDEPENDENTLY. BY STRAIGHTENING THE CURVED GIRDER TO ITS FULL DEVELOPED LENGTH, THE BENDING MOMENTS AND VERTICAL SHEAR FORCES CAN BE READILY DETERMINED AS CUSTOMARILY DONE. THE PROPOSED METHOD SUGGESTS THAT THE TORSIONAL ANALYSIS CAN BE CARRIED OUT IN A SIMILAR MANNER, EXCEPT THAT (1) A STRAIGHT CONJUGATE BEAM SUBJECTED TO A DISTRIBUTED LOAD IN THE TORSIONAL MOMENT ANALYSIS, AND (2) A STRAIGHT CONJUGATE BEAM SUBJECTED TO A DISTRIBUTED LOAD FOR THE DETERMINATION OF THE ANGLES OF TWIST ARE TO BE CONSIDERED, WHERE M IS THE BENDING MOMENT OBTAINED IN THE APPROXIMATE FLEXURAL ANALYSIS, R IS THE RADIUS OF CURVATURE. THE APPLIED DISTRIBUTED TORQUE IN THE SPANWISE DIRECTION IS ALSO CONSIDERED. SINCE M/R IS A PARAMETER IN THE FORCING FUNCTION IN THE ANALYSIS, THE APPROXIMATE METHOD IS CALLED THE M/R-METHOD TO DIFFERENTIATE IT FROM THE CLASSICAL CONJUGATE BEAM METHOD FOR THE DETERMINATION OF BEAM DEFLECTIONS. THE ACCURACY AND LIMITATIONS OF THE APPROXIMATE METHOD ARE EXAMINED AS WELL AS SEVERAL OTHER ASPECTS RELATED TO THE PROBLEM IN GENERAL. IT IS FOUND THAT WITHIN THE LIMITS SPECIFIED, THE METHOD WILL GIVE RESULTS WITH SUFFICIENT ACCURACY FOR PRACTICAL PURPOSES.
18 citations
TL;DR: In this paper, the complementary variational principle has been used to derive the differential equations and the associated boundary conditions of the vibrating plate in terms of bending moments, and it is shown that the plate possesses an infinite number of zero frequency modes in which the plate remains in a state of constant strain under a set of self-equilibrating bending moments.
Abstract: The complementary variational principle has been used to derive the differential equations and the associated boundary conditions of the vibrating plate in terms of bending moments. It is shown that in this formulation, the plate possesses an infinite number of zero frequency modes in which the plate remains in a state of constant strain under a set of self-equilibrating bending moments. In applying the Rayleigh Ritz procedure for the non-zero frequency modes of the plate, it is shown that it the assumed functions are orthogonal to only a finite number of zero frequency modes, then one may obtain frequencies which are lower than the true frequencies of the plate. An iliustrative example is given in the paper.
TL;DR: In this paper, the results of tests on 18 rectangular reinforced concrete beams subjected to combined bending torsion and shear are reported, where the principal variables were the ratio of twisting to bending moment, the transverse shear force and the reinforcement configuration.
Abstract: The results of tests on 18 rectangular reinforced concrete beams subjected to combined bending torsion and shear are reported. The principal variables were the ratio of twisting to bending moment, the transverse shear force, and the reinforcement configuration. The beams failed by one of three different modes which were characterized by the formation of a hinge adjacent to one face of the beam and yielding of the reinforcement adjacent to the face opposite to the hinge. In beams subjected to a constant twisting moment and a varying bending moment, failures with the hinge located adjacent to the bottom face occurred in the region subjected to the lowest bending moment and failures with the hinge located adjacent to the top face occurred in the region of highest bending moment. Idealized failure surfaces are defined and expressions for the strength of these beams are derived using an equilibrium approach. The results are presented both in tabular form and in interaction diagrams. Good correlation is obtained between the predicted and observed mode of failure and strength.
TL;DR: A n-interaction diagram between bending moment and shear force in a beam is not a proper yield surface, and the convexity property of plasticity theory does not necessarily hold as discussed by the authors.
Abstract: A n “interaction diagram” between bending moment and shear force in a beam is not a proper yield surface, and the convexity property of plasticity theory does not necessarily hold. An empirical curve is, however, satisfactory for design in the practical range, although improvement to the theory remains possible for very short beams.
TL;DR: In this paper, the authors present a solution to the multiply connected and mixed boundary value problem, which is obtained through a recently developed modification to the Rayleigh-Ritz method which has very general application and renders the solution mathematically valid up to the internal corner points where the bending moments are singular.
Abstract: Considerable attention has been devoted in the literature on numerical methods towards securing energy convergence of solutions for, say, linearly elastic plate bending problems. Although energy convergence is necessary it by no means follows that the derived bending moments and shearing forces converge uniformly at a given point and it is this kind of feature which the engineer is really seeking.
This question is examined in the context of a problem which is of particular interest to the civil engineering field and concerns the bending of a square plate under uniformly distributed load; the plate has simply supported edges and contains a central square hole with free edges. The solution to this multiply connected and mixed boundary value problem is obtained through a recently developed modification to the Rayleigh–Ritz method which has very general application and renders the solution mathematically valid up to the internal corner points where the bending moments are singular. Use is made of triangular equilibrium finite elements in conjunction with continuous eigenfunctions. Although it is already known that the order (i.e. the eigenvalue) of the singularity at the internal corners is available by inspection, it is an interesting feature of the present solution that a good approximation to the amplitude is also obtained by an inspection of the finite element results.
TL;DR: In this article, a variational method is used to reduce the partial differential equations to a set of ordinary differential equations, which are then solved by numerical integration, and the results from the method compare favorably with existing solutions.
Abstract: An approximate solution procedure is presented for rectangular orthotropic sandwich plates under general boundary conditions. A variational method is used to reduce the partial differential equations to a set of ordinary differential equations, which are then solved by numerical integration. The method is used to obtain solutions for a sandwich plate with clamped boundaries. The results from the method compare favorably with existing solutions. A series of graphs suitable for design purposes is presented depicting the variation of the maximum values of the bending moments, shear forces and deflections in square sandwich plates with orthotropic cores for different shear stiffnesses.
TL;DR: In this paper, the response of rigid-plastic structures to a certain class of dynamic loadings is investigated taking into account changes in geometry and interactions between bending moments and membrane forces.
Abstract: Response of rigid-plastic structures to a certain class of dynamic loadings is investigated taking into account changes in geometry and interactions between bending moments and membrane forces. Starting from the usual assumption of the theory of moderately large deflection of shells and introducing a suitably modified principle of virtual velocities, bounds on permanent deflections of structures are established. In the limiting case of small deflections the present solution reduces to the earlier result due to Martin. The general procedure is explained on the example of simply supported circular plate loaded impulsively. The obtained bound on central deflection is compared with exact solution of the same problem and with Florence experimental data on aluminum plates. It is shown that considerable improvement is achieved relative to the simple bending solution. The present bounding curve closely follows the trend of experimental points over the whole range of applied impulse.
TL;DR: In this article, it is shown that bending moments can be evaluated at any point on the interferogram by moire interference between two transparencies, rotated 180° about the point in question, which not only yields the bending moments but also indicates the principal axes of bending and the relative sign between the two principal components.
TL;DR: In this paper, the three-dimensional elasto-plastic stress distributions along the plane of symmetry adjacent to the notch for two levels of applied bending moment were determined along with the associated plastic-zone sizes.
Abstract: Knowledge of the three-dimensional stress distribution near a notch for a strain-hardening material in an elasto-plastic state is limited, to say the least. This experimental investigation is concerned with obtaining some insight concerning the three0dimensional elastoplastic stress distributions and the associated plastic-zone sizes. A notched-bar configuration (scaled-up Charpy specimen) subjected to flexure was selected for this purpose. The three-dimensional elasto-plastic stress distributions were determined along the plane of symmetry adjacent to the notch for two levels of applied bending moment.
TL;DR: In this article, a novel integral equation technique is employed for the analysis of dynamic stability problems, where the governing equation of the linearized parametric resonance problem is transformed into an integral equation.
Abstract: A novel integral equation technique is employed for the analysis of dynamic stability problems. The governing equation of the linearized parametric resonance problem is transformed into an integral equation. The kernel of the integral equation is computed as the influence function for the deflection and/or bending moment of a corresponding beam.
The highest derivative of the governing function (in our case fourth derivative of the displacement function) is chosen as the basic unknown. Using the formal analogy with the differential equation of the beam flexure this highest derivative is comprehended as some unknown transverse ‘load’. The distribution of this ‘load’ is a priori assumed to be polygonal. Using elementary methods of structural analysis, the displacements due to the assumed ‘load’ are determined. These displacements, arrayed into a square matrix, approximate the kernel of the governing integral equation.
The subsequent procedure via Hill's determinant is a conventional one. The results prove to be accurate enough even for a very modest number of points of integration. This reflects the fact that the method is based on numerical integration rather than on numerical differentiation.
01 Jan 1970
TL;DR: In this paper, a probabilistic study of the ULTIMATE BENDING STRENGTH and DUCTILITY RATIO for RECTANGULAR CONCRETE SECTIONS is presented.
Abstract: A PROBABILISTIC STUDY IS MADE OF THE ULTIMATE BENDING STRENGTH AND DUCTILITY RATIO FOR RECTANGULAR CONCRETE SECTIONS REINFORCED IN TENSION ONLY. IT IS DIVIDED INTO THREE MAIN PARTS. THE FIRST PART DEVELOPS PREDICTION EQUATIONS FOR ULTIMATE MOMENT AND DUCTILITY RATIO. THE SECOND PART IS CONCERNED WITH THE VALUES EXPECTED IN PRACTICE (I.E. THE PROBABILITY DISTRIBUTIONS) OF THE PARAMETERS ENTERING INTO A PREDICTION OF ULTIMATE MOMENT AND DUCTILITY RATIO. THE THIRD PART GIVES PROBABILITY DISTRIBUTIONS OF THE ULTIMATE MOMENT AND DUCTILITY RATIO WHICH ARE OBTAINED BY MEANS OF THE PREDICTION EQUATIONS FROM THE PROBABILITY DISTRIBUTIONS OF THE PARAMETERS. /RRL/
TL;DR: In this article, a thin shell design for minimum bending stresses and transverse shear for composite materials with low interlaminar shear strength was proposed for composite material with low cross-correlations.
Abstract: Thin shell design for minimum bending stresses and transverse shear for composite materials with low interlaminar shear strength
TL;DR: In this article, the problem of total rupture of a beam of rectangular cross section under a bending moment and an axial tensile force has been considered based upon two fundamental physical laws: Odqvist's creep law and Kachanov's brittle rupture law.
TL;DR: In this article, an analytical expression is derived which describes the steady-state deflection of a long, thin, tubular structure with a locked overlapped cross section subjected to solar heating in a 0 g environment.
Abstract: An analytical expression is derived which describes the steady-state deflection of a long, thin, tubular structure with a locked overlapped cross section subjected to solar heating in a 0 g environment. By transforming the angular coordinate to correspond to the solar direction, it is possible to obtain a single expression which describes the circumferential temperature distribution. Thermal bending is found by closed form integration of the temperature-induced loading about the principal axes as dictated by overlap geometry. Maximum thermal bending is evaluated for various overlap angles as a function of dimensionless groups which include all thermal and geometric parameters. It is concluded that an optimum design is one which incorporates an overlap angle at 155°. The results are presented in graphical form suitable for engineering design, and a specific example is discussed. Nomenclature a = cross-sectional area of tube A = temperature parameter B — dimensionless parameter e = thermal coefficient of expansion E = Young's modulus of elasticity f(08) = function defined by Eq. (16) F = heat input function / = area moment of inertia k — thermal conductivity M = thermal bending moment r = element radius
TL;DR: In this paper, the results of tests on single-panel square reinforced concrete flat plate structures supported on four corner columns are presented, and the results indicate that existing methods for predicting the strength of column-slab connections of interior columns, when extended to corner columns, will produce extremely conservative results.
Abstract: The results of tests on single-panel square reinforced concrete flat plate structures supported on four corner columns are presented. The study aims at an investigation of the behavior and strength of the column-slab connection in the presence of axial load and biaxial bending. The distribution of the bending moments across the width of the slabs as well as the load-deflection behavior of the slabs up to collapse were observed and analyzed. The results indicate that existing methods for predicting the strength of column-slab connections of interior columns, when extended to corner columns, will produce extremely conservative results.
TL;DR: In this article, a method for analyzing light gage cold-formed steel beams with buckled flanges is described, where the points of zero bending moment are not known a priori and their determination is part of the solution.
Abstract: A method for analyzing light gage cold-formed steel beams with buckled flanges in situations when the bending moment changes sign is described. If the bending moment changes sign, the standard procedure using the same effective width for each buckled flange over the whole length of the beam breaks down. Instead, separate effective widths are established for the compression flanges of each part of the beam with a bending moment of the same sign. The points of zero bending moment are not known a priori and their determination is part of the solution. The governing equations are highly nonlinear and have to be solved by a numerical process of successive approximations. The approximations were found to converge very rapidly.
Patent•
19 May 1970
TL;DR: In this paper, a hanging roof structure in which the roof covering is supported by a plurality of lines and tensile forces acting in the lines are taken up by an annular beam of lattice construction is described.
Abstract: A hanging roof structure in which the roof covering is supported by a plurality of lines and tensile forces acting in the lines are taken up by an annular beam of lattice construction, thereby relieving supporting pillars from bending moments
TL;DR: The concept of equivalent smooth bend is examined in this article, where measurements of strain, transverse deflection, and change of radius have been taken on single unreinforced pipe bends subjected to in-plane bending moments.
Abstract: Measurements of strain, transverse deflection, and change of radius have been taken on single unreinforced pipe bends subjected to in-plane bending moments. Specimens having three different mitre angles for each of two sizes of pipe have been used. Observations were made with different combinations of leg lengths for each specimen.Stresses and flexibilities are compared with those calculated by the formulae given in the A.S.A. Piping Code B.31.1 and the concept of equivalent smooth bend is examined. Alternative methods of assessment are considered and improved methods are suggested.