The regular prismatic bar under torsion: A new approach based on caustics

Summary A new approach to the general torsion problem of the regular prismatic bar was introduced, based on extensive and meticulous experimental evidence, indicating that the twisted prismatic bars, except the cylindrical ones, present, a longitudinal warping, following a sinusoidal type of curves for the warping displacements with their extrema displaced towards the corners.As a guide for the establishment of the method the solutions of the elliptical and the equiangular triangular bars were used. It was found that the regions inside the inscribed circles tangent to the sides of their undeformed cross-sections constituted regions whose warped surfaces constitute undulated forms following sinusoidal surfaces. This property was proved exact for the exact solutions for the elliptic and triangular cross-sections and it is assumed by induction valid for any regular cross-section. Comparing the warping displacements along the inscribed circle to a square or regular polygon cross section with the classical Timoshenko solution for the square bar it was found an exact coincidence in these displacements. It was further shown that these circles and all concentric full circles constitute the initial curves of the caustics formed from reflections of the warped cross-sections of the twisted bars.The differences between the stresses and displacements between the existing approximate solutions and the new approach appear in the partly curved triangles lying outside the inscribed circles. For the square cross-section it was shown that thew-displacements between Timoshenko's solution and the present one were less than 5 percent. Experimental evidence with the Moiré method and comparison with the exact solution for the equiangular triangular cross section proved that the present approach is closer to reality and therefore presents higher accuracy than any approximate method. Furthermore, it presents the advantage that the full circles inscribed inside the cross-section constitute the initial curves forming caustics in space when a parallel light beam is reflected on the warped cross-section of the twisted bars. interesting properties of caustics, proved in this paper, help down-rightly for a thorough study of the form and properties of the twisted bars.With 16 Figures     

Postbuckling of a twisted prismatic rod under terminal thrust

The three-dimensional post-buckling behaviour of a geometrically nonlinear but linearly elastic rod under the simultaneous actions of twist and thrus tis studied. It is assumed that the ellipse of inertia of the cross-section of the rod is a circle. If the ratio d of twist to thrust is small the postbuckling behaviour is supercritical and if d is large itis subcritical. The aim of this paper is to study the intermediate degenerate case when d = d_c. The analysis is considerably simplified by making use of the symmetry propertiesof the rod equations     

Effects of fibers and conventional reinforcement contents on fibers scaterring in prismatic bars

This paper presents results from a second stage of an experimental study of the dependence of steel fibers distribution along RFC prismatic specimens on the conventional reinforcement ratio and on the total amount of fibers in the concrete mix. The experimental program included two types of prismatic specimens with 30- and 60- kg/m³ of hooked-end steel fibers. Each specimen was sawed into equal segments and the numbers of fibers appearing at the cross-sections were counted and used for a further statistical analysis. This analysis comprised calculations of the average value and standard deviation of a non-dimensional variable, which represents the distribution of the total steel along the specimen. They were used to calibrate a theoretical model, which had been previously proposed by the authors. The test results showed reasonable to good agreement with the theoretical model. A comparison between the results of the 30- and 60-kg/m³ fibers shows that as the conventional reinforcement ratio increases, the standard deviations for the different mixtures approach each other.     

The method of caustics in environmental cracking

A simple and powerful crack growth criterion under combined environmental conditions and applied loads is formulated for mixed-mode situations based on the method of caustics. It is established that the axis of symmetry of the caustic represents the crack growth direction, while the crack growth condition is expressed in terms of the diameter of the caustic. Caustics can easily be obtained by illuminating the specimen in the vicinity of the crack tip by a laser light beam and recording the optical pattern on a viewing screen at some distance from the specimen. An elegant procedure for the experimental study of environmental cracking is thus developed.     

The non-linear solution of the mixed-mode caustics in birefringent materials

The purpose of this note is to study the transmitted and rear face reflected caustics in birefringent plates for a mixed-mode stress field at the crack tip.     

The influence of geometrical imperfections in the method of caustics

The influence of locally varying specimen thickness on the shape of the shadow spot and, hence, on the stress intensity factor is of first order but for all practical cases comparable with the experimental data scatter.     

The effect of biaxiality of loading in the orientation of caustics

The influence of biaxiality of loading on the orientation of caustics formed from the tips of a slant crack in an infinite plate was studied. Some intricate results on the subject were, also, emphasized. The correlation between characteristic values of stress-intensity factors and several combinations of the biaxiality ratio and the angle of inclination was established.     

Torsional rod wave scattering from tapering between regions of different cross-sections

Previously (Engan 1998) we have studied the scattering of torsional waves in circular rods with a free surface where the rod diameter is subject to abrupt diameter changes. Here, we discuss the scattering of torsional waves from tapered regions and, in particular, transmission of the fundamental torsional mode. The calculations are based on a mode expansion technique in which propagating as well as cut-off modes are taken into account. The tapered region is approximated by numerous steps separated by uniform rod parts arranged so that they closely render the taper shape. Transfer matrices for all steps and uniform parts are combined to yield the desired transmission coefficient through the taper region. Numerical results of the frequency variation of fundamental mode transmission are presented, depending on geometrical parameters, such as taper length and the diameters of the two uniform adjacent rod parts. The dependence of the transmission on taper shape is demonstrated by calculating the response of various shapes.     

The method of caustics for mixed-mode crack loading with higher order term effects

The method of caustics or shadow-spot method for mixed-mode crack loading is investigated within the framework of Westergaard stress function analysis. The effect of higher order terms in the stress functions onto the shape and size of the caustic is studied for optically isotropic materials. The classical evaluation procedure is generalized and it is shown that the relationship between stress intensity factor and caustic diameter is of the general form K∞ D^(5/2)+n.