Bounds for the Torsional Rigidity of Isotropic Beams

1962 ◽  
Vol 58 (2) ◽  
pp. 417-419 ◽  
Author(s):  
L. M. Milne-Thomson
Keyword(s):  
Cross Section ◽  
Isotropic Material ◽  
Torsional Rigidity ◽  
Connected Region ◽  
Closed Contour ◽  
Prismatic Beam ◽  
The Cross ◽  
Simply Connected Region ◽  
Simply Connected

Consider a cylindrical or prismatic beam of isotropic material. Let the cross-section of the beam be a simply-connected region S bounded by the closed contour C.

A Conformal Proof of a Jordan Curve Problem

1969 ◽  
Vol 12 (5) ◽  
pp. 673-674 ◽  
Author(s):  
G. Spoar ◽  
N.D. Lane
Keyword(s):  
Jordan Curve ◽  
Connected Region ◽  
Inversive Plane ◽  
Simply Connected Region ◽  
Simply Connected

The following theorem appears in [1].Let R be a closed simply connected region of the inversive plane bounded by a Jordan curve J, and let J be divided into three closed arcs A1, A2, A3. Then there exists a circle contained in R and having points in common with all three arcs.

scholarly journals The torsion and flexure of shafting with keyways or cracks

1932 ◽  
Vol 138 (836) ◽  
pp. 607-634 ◽  
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Torsional Rigidity ◽  
Circular Cross Section ◽  
Transverse Load ◽  
Torsion Problem ◽  
Circular Section ◽  
Physical Conditions ◽  
The Cross ◽  
Straight Lines

The object of the paper is to investigate the properties of shafts of circular cross-section into which keyways or slits have been cut, first when subjected to torsion, and second when bent by a transverse load at one end. The torsion problem for similar cases has been treated by several writers. Filon has worked out an approximation to the case of a circular section with one or two keyways ; in his method the boundary of the cross-section was a nearly circular ellipse and the boundaries of the keyways were confocal hyperbolas. In particular he considered the case when the hyperbola degenerated into straight lines starting from the foci. The solution for a circular section with one keyway in the form of an orthogonal circle has been obtained by Gronwall. In each case the solution has been obtained by the use of a conformal trans­formation and this method is again used in this paper, the transformations used being ρ = k sn 2 t . ρ = k 1/2 sn t , ρ = k 1/2 sn 1/2 t where ρ = x + iy , t = ξ + i η. No work appears to have been done on the flexure problem which is here worked out for several cases of shafts with slits. 2. Summary of the Problems Treated . We first consider the torsional properties of shafts with one and with two indentations. In particular cases numerical results have been obtained for the stresses at particular points and for the torsional rigidity. The results for one indentation and for two indentations of the same width and approximately the same depth have been compared. We next consider the solution of the torsion problem for one, two or four equal slits of any depth from the surface towards the axis. The values of the stresses have not been worked out in these cases since the stress is infinite at the bottom of the slits. This in stress occurs because the physical conditions are not satisfied at the bottom of the slits, but as had been pointed out by Filon this does not affect the validity of the values of the torsional rigidity. We compare the effect on the torsional rigidity of the shaft of one, two and four slits of the same depth in particular cases. We also compare the results for one slit with those obtained by Filon by another method, and find very good agreement which is illustrated by a graph. The reduction in torsional rigidity due to a semicircular keyway is compared with that due to a slit of approximately the same depth. Finally the distortion of the cross-sections at right angles to the planes is investigated, and in this, several interesting and perhaps unexpected features appear. The relative shift of the two sides of the slits is calculated in several cases.

Elastic Torsion in the Presence of Initial Axial Stress

1950 ◽  
Vol 17 (4) ◽  
pp. 383-387
Author(s):  
J. N. Goodier
Keyword(s):  
Cross Section ◽  
Axial Stress ◽  
Torsional Rigidity ◽  
Torsional Stiffness ◽  
Lateral Buckling ◽  
Initial Tension ◽  
Open Section ◽  
Thin Walled ◽  
The Cross

Abstract The torsional rigidity, for small elastic torsion, of bars of thin-walled open section, is, in general, altered by initial tension, compression, bending, or other axial stress. This appears in the increase of torsional stiffness of strips due to tension, in the decrease to zero in open sections which buckle torsionally as columns, and also has an influence on lateral buckling of beams. This paper contains an extension of the Saint Venant solution for ordinary torsion to the problem of torsion in the presence of initial axial stress with any distribution on the cross section. The results are confirmed by tests, and validate the intuitively derived formulas which are in use.

$\beta$-transformation, natural extension and invariant measure

2000 ◽  
Vol 20 (5) ◽  
pp. 1271-1285 ◽  
Author(s):  
GAVIN BROWN ◽  
QINGHE YIN
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Natural Extension ◽  
Connected Region ◽  
Two Dimensional ◽  
Constant Multiple ◽  
Unit Square ◽  
Simply Connected Region ◽  
Simply Connected ◽  
Dimensional Lebesgue Measure

For $\beta>1$, consider the $\beta$-transformation $T_\beta$. When $\beta$ is an integer, the natural extension of $T_\beta$ can be represented explicitly as a map on the unit square with an invariant measure: the corresponding two-dimensional Lebesgue measure. We show that, under certain conditions on $\beta$, the natural extension is defined on a simply connected region and an invariant measure is a constant multiple of the Lebesgue measure.We characterize those $\beta$ in terms of the $\beta$-expansion of one, and study the structure and size of the set of all such $\beta$.

scholarly journals Closed form solution in buckling optimization problem of twisted rod

2021 ◽  
Author(s):  
Vladimir Kobelev
Keyword(s):  
Closed Form ◽  
Cross Section ◽  
Cross Sections ◽  
Optimization Problem ◽  
Closed Form Solution ◽  
Form Solution ◽  
Optimal Shape ◽  
Actual Problem ◽  
The Cross ◽  
Simply Connected

Abstract The applications of this method for stability problems are illustrated in this manuscript. In the context of twisted rods, the counterpart for Euler’s buckling problem is Greenhill's problem, which studies the forming of a loop in an elastic bar under torsion (Greenhill, 1883). We search the optimal shape of the rod along its axis. A priori form of the cross-section remains unknown. For the solution of the actual problem the stability equations take into account all possible convex, simply connected shapes of the cross-section. Thus, we drop the assumption about the equality of principle moments of inertia for the cross-section. The cross-sections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the rod and vary along its axis. The distribution of material along the length of a twisted rod is optimized so that the rod is of the constant volume T and will support the maximal moment without spatial buckling. The cross section that delivers the maximum or the minimum for the critical eigenvalue must be determined among all convex, simply connected domains. We demonstrate at the beginning the validity of static Euler’s approach for simply supported rod (hinged), twisted by the conservative moment. The applied method for integration of the optimization criteria delivers different length and volumes of the optimal twisted rods. Instead of the seeking for the twisted rods of the fixed length and volume, we directly compare the twisted rods with the different lengths and cross-sections using the invariant factors. The solution of optimization problem for twisted rod is stated in closed form in terms of the higher transcendental functions. In the torsion stability problem, the optimal shape of cross-section is the equilateral triangle.

scholarly journals An Integral Equation Related to the Exterior Riemann- Hilbert Problem on Region with Corners

2014 ◽  
Vol 4 (2) ◽  
Author(s):  
Zamzana Zamzamir ◽  
Munira Ismail ◽  
Ali H. M. Murid
Keyword(s):  
Integral Equation ◽  
Hilbert Problem ◽  
Boundary Integral ◽  
Connected Region ◽  
Fredholm Alternative ◽  
Alternative Theorem ◽  
Smooth Region ◽  
Simply Connected Region ◽  
Simply Connected ◽  
Riemann Hilbert Problem

Nasser in 2005 gives the first full method for solving the Riemann-Hilbert problem (briefly the RH problem) for smooth arbitrary simply connected region for general indices via boundary integral equation. However, his treatment of RH problem does not include regions with corners. Later, Ismail in 2007 provides a numerical solution of the interior RH problem on region with corners via Nasser’s method together with Swarztrauber’s approach, but Ismail does not develop any integral equation related to exterior RH problem on region with corners. In this paper, we introduce a new integral equation related to the exterior RH problem in a simply connected region bounded by curves having a finite number of corners in the complex plane. We obtain a new integral equation that adopts Ismail’s method which does not involve conformal mapping. This result is a generalization of the integral equation developed by Nasser for the exterior RH problem on smooth region. The solvability of the integral equation in accordance with the Fredholm alternative theorem is presented. The proof of the equivalence of our integral equation to the RH problem is also provided.

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