Variation Of Bending Moment Research Articles

Applications of holography and speckle photography to the measurement of displacement and strain

Holographic interferometry was used to investigate the behaviour of a plastic pipe joint when subjected to internal pressure. By a combination of ‘live’ and double-exposure techniques the variation of bending moment along an axial surface generator of the joint can be evaluated. To obtain the surface strain, an interferometric technique using speckle photography was used. An important practical factor is the method of mounting the object. For the measurement of much larger surface movements, such as occur when a cantilever is loaded, simple speckle photography, with subsequent point-by-point analysis of the recorded image, is used.

Energetics of Hovering Flight in Hummingbirds and in Drosophila

ABSTRACT Expressions have been derived for an estimate of the average coefficient of lift, for the variation in bending moment or torque caused by wind forces and by inertia forces, and for the power output during hovering flight on one spot when the wings move according to a horizontal figure-of-eight. In both hummingbirds and Drosophila the flight is consistent with steady-state aerodynamics, the average lift coefficient being 1·8 in the hummingbird and o-8 in Drosophila. The aerodynamic or hydraulic efficiency is 0·5 in the hummingbird and 0·3 in Drosophila, and in both types the aerodynamic power output is 22–24 cal/g body weight/h. The total mechanical power output is 39 cal g−1 h−1 in the hummingbird because of the extra energy needed to accelerate the wing-mass. It is 24 cal g−1 h−1 in Drosophila in which the inertia term is negligible because the wing-stroke frequency is reduced to the lowest possible value for sustained flight. In both animals the mechanical efficiency of the flight muscles is 0·2. It is argued that the tilt of the stroke plane relative to the horizontal is an adaptation to the geometrically unfavourable induced wind and to the relatively large lift/drag ratio seen in many insects. The vertical movements at the extreme ends may serve to reduce the interaction between the shed ‘stopping’ vortex and the new bound vortex of opposite sense which has to be built up during the early part of the return stroke. Two additional non-steady flow situations may exist at either end of the stroke, delayed stall and delayed build-up of circulation (Wagner effect), but since they have opposite effects it is probable that the resultant force is of about the same magnitude as that estimated for a steady-state situation. Most insects have an effective elastic system to counteract the adverse effect of wing-inertia, but small fast-moving vertebrates have not. It is argued that the only material available for this purpose in this group is elastin and that it is unsuited at the rates of deformation required because recent measurements have shown that the damping is relatively high, probably due to molecular factors.

Concrete Beams in Bending, Torsion and Shear

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.

The dynamic lateral instability of beams

The vibrations of a deep slender beam, bent to uniform curvature by in variant moments acting in a vertical plane, which is also the plane of maximum stiffness, have been studied. It is shown that the moments couple up the lateral bending and torsional modes of the beam, those modes being replaced by two independent modes, each involving torsion and flexure. One of these m odes is associated with a frequency which decreases with increasing bending moment, the frequency becoming zero when the moment reaches the critical value for lateral instability. The other mode is associated with a frequency which increases with bending moment. Experiments were carried out on an I-section cantilever carrying an end mass. Owing to the varying bending moment, the theoretical analysis of this case is more complicated, and an iterative method, originated by Schwarz (1890), has been employed. Results are in reasonable agreement with experiment.

On the stability under shearing forces of a flat elastic strip

1. The investigation relates to flat elastic strip, of uniform breadth, thickness and material, upon which a uniform shear is imposed by tangential tractions applied at its edges and in its plane. The tractions appear in the expression for the change of potential energy which occurs when the strip is bent, and they must therefore affect both the modes and the frequencies of its free transverse vibrations. If sufficiently intense, they will bring about a condition of limiting elastic stability, since they can neutralize, in certain types of distortion, the restoring effects of the flexural stresses. The results have some bearing on the stability of the webs of deep plate girders, which take the greater part of the total shear transmitted. The correspondence must not, however, be pressed unduly, because in a girder uniform shear will be accompanied by a varying bending moment which imposes additional stresses upon the web. It is more accurate to describe the sheared strip (of which the length, in this paper, has been assumed to be infinite) as the limiting case either of a narrow annular disc, or of a short tube, subjected to torsion. The similarity of the three problems is illustrated by the specimens shown in fig. 1, which have buckled under conditions of limiting elastic stability.