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We consider the problem of minimizing or maximizing the quantity λ (Ω) T q (Ω) on the class of open sets of prescribed Lebesgue measure. Here q > 0 is fixed, λ (Ω) denotes the first eigenvalue of the Dirichlet Laplacian on H 0 1 (Ω) , while T (Ω) is the torsional rigidity of Ω. The optimization problem above is considered in the class of all domains Ω , in the class of convex domains Ω , and in the class of thin domains. The full Blaschke–Santaló diagram for λ (Ω) and T (Ω) is obtained in dimension one, while for higher dimensions we provide some bounds. [ABSTRACT FROM AUTHOR]

We consider shape functionals of the form F q (Ω) = P (Ω) T q (Ω) on the class of open sets of prescribed Lebesgue measure. Here q > 0 is fixed, P (Ω) denotes the perimeter of Ω and T (Ω) is the torsional rigidity of Ω . The minimization and maximization of F q (Ω) is considered on various classes of admissible domains Ω : in the class A all of all domains, in the class A convex of convex domains, and in the class A thin of thin domains. [ABSTRACT FROM AUTHOR]

We establish some generalizations of the classical inequalities by Polya–Szego and Makai about torsional rigidity of convex domains. The main idea of the proof is in using an exact isoperimetric inequality for Euclidean moments of domains. This inequality has a wide class of extremal regions and is of independent interest. [ABSTRACT FROM AUTHOR]

Suspension bridges with three cable planes provide an excellent solution to the suspension bridges' downwarp problem with ultrawide decks, which implies their enormous demand and popularization prospect in the engineering scenarios. In this paper, a method for uniform allocation of dead load in the transverse direction to three cables of the suspension bridge with three cable planes (SB-3CP), which transforms the spatial stress mode into a plane model to simplify the calculation, is proposed. By altering the cross-sectional area of each hanger, the axial forces of the three hangers in the same cross section of the SB-3CP are equal. Therefore, the cross-sectional area and shape of the three cables are equal, improving the suspension bridge outlook in both the cross-sectional and facade views of the suspension bridge. Meanwhile, under the uniform allocation of the dead load, the side main cable bears a higher share of the dead load, which is conducive to improving the entire bridge's torsional rigidity. In this study, conditions for compatibility of deformation and energy conservation are utilized to derive the relationship between the axial rigidity of the three hangers in the same cross sections. The effects of axial rigidity of hangers, flexural rigidity of the deck in the transverse direction, and length between the hanging points on the difference in the axial rigidities of the three hangers are analyzed and discussed in detail. Finally, an SB-3CP with a main span of 2,320 m and a width of 75 m was taken as an example. The cross-sectional areas of each hanger of the bridge were calculated using the proposed method. Then, its accuracy was validated through the finite-element analysis. This method's design effect was verified by comparing the differences in the main cable diameter and torsional rigidity between the SB-3CP with uniform and nonuniform dead load distributions. [ABSTRACT FROM AUTHOR]

We discuss some of the geometric properties, such as the foliated Schwarz symmetry, the monotonicity along the axial and the affine-radial directions, of the first eigenfunctions of a Zaremba problem for the Laplace operator on annular domains. Together with the shape calculus, these fine geometric properties help us to prove that the first eigenvalue is strictly decreasing as the inner ball moves towards the boundary of the outer ball. [ABSTRACT FROM AUTHOR]

This paper deals with the Saint-Venant torsion of elastic, cylindrically orthotropic bar whose cross section is a sector of a circular ring shaped bar. The cylindrically orthotropic homogeneous elastic wedge-shaped bar strengthened by on its curved boundary surfaces by thin isotropic elastic shells. An analytical method is presented to obtain the Prandtl's stress function, torsion function, torsional rigidity and shearing stresses. A numerical example illustrates the application of the developed analytical method. [ABSTRACT FROM AUTHOR]

Torsional properties of single fibers have considerable effects on the performance of fabrics. A new torsion tester with high resolution is developed based on the torsion-balance method and the autocollimation principle for measuring the torsional properties of single fibers. The tester can measure the torque accurately using the torsion wire of known torsional rigidity. The angular displacement of the torsion wire is measured by a non-contact electronic autocollimator. The calibration results show that the maximum torque capability is around 6 × 10−6 N m with a theoretical resolution of about 8 × 10−11 N m. The instrument's validity is verified by comparing the measurement result with that obtained by a torsion pendulum. Torsion tests on four industrially microfibers, including glass fiber, polyimide fiber, carbon fiber, and gold wire, are performed with the tester, which gives shear moduli 71.0 ± 0.4, 5.6 ± 0.1, 22.6 ± 3.1, and 34.3 ± 0.5 GPa, respectively. Young's moduli of these fibers are also provided. It is demonstrated that these fibers have a high anisotropy in mechanical properties. [ABSTRACT FROM AUTHOR]

Since RV (rotation vector) reducer has a high reduction ratio with compact size, high efficiency and stiffness, the reducer is extensively used in applications requiring high precision in the restricted space like industrial robotic joints. Since the cycloid disk of the RV reducer makes contacts with many pin-rollers simultaneously, tolerance and friction between the cycloid disk and the pin-rollers significantly affect the contact force of the cycloid disks as well as the performance of the RV reducer. This paper presents an impact analysis of tolerance and friction on the RV reducer using FE method. A FE model for a two-disk RV reducer and the eccentric shaft is built considering tolerance and friction between the cycloid disk and the pin-rollers. Then, the FE analysis of an ideal rigid cycloid reducer was performed and verified with the theoretical calculation. The effects of tolerance and friction on multi-contact and the output torque of the RV reducer were investigated. Since the contact forces vary according to both tolerance and friction, the resulting torsional rigidity and output torque are affected considerably. Precisely, the tolerance mainly decreases the torsional rigidity while the friction primarily increases the output torque fluctuation. [ABSTRACT FROM AUTHOR]

A dislocation solution to evaluate the mode III stress intensity factors (SIFs) of the crack tips, torsional rigidity and the hoop stress around the cavities in an infinite rectangular cross-section bar coated by an FG layer under torsion is developed based on the Saint–Venant theory. The solution of the rectangular cross section with the FG coating weakened by a screw dislocation under torsion was first obtained in terms of the dislocation density function. Then, the problem is transformed into a group of Cauchy integral equations in the rectangular cross-section bar with the FG coating layer via the distributed dislocation approach. The singular integral equations are reduced to a set of algebraic equations to evaluate them numerically. After solving the integral equations, the SIFs of the crack tips, the hoop stresses around the cavities and the torsional rigidity of the entire domain are found. The obtained solution is validated by performing comparison studies in multiple cases. It is shown that the interaction between the cracks, the cracks length, variation of the shear module of the FG coating layer and distance of the crack tips from stress-free surfaces can significantly affect the SIFs of crack tips and torsional rigidity of the rectangular cross-section bar coated by an FG layer. [ABSTRACT FROM AUTHOR]

Using the membrane analogy, in 1934 Timoshenko derived the torsional rigidity of a rectangular prism of isotropic material as a function of its material shear modulus, width and thickness. However, he did not consider the energy conservation criterion, as it could be either unnecessary or replaced by other criteria in Timoshenko’s process. To confirm the correctness of Timoshenko’s solution, this work re-derives the torsional rigidity by considering all the equilibrium conditions, boundary conditions, symmetric and anti-symmetric conditions of displacement and stress, the energy conservation criterion, and even the energy minimization criterion. Using the TSAI technique, exact solutions for the displacements, strains, stresses and the torsional rigidity are derived perfectly. The derived torsional rigidity is in a completely different form from that derived by Timoshenko and is numerically identical. Interestingly, the solutions derived in this work verify that, when the values of the width and thickness of the rectangular prism are swapped, the value of the torsional rigidity remains the same, which makes perfect sense physically but is not discussed in Timoshenko’s process or any other research. This work presents a procedure considering all the mathematical details and the results remain correct when the width and thickness of the prism swap. This fact makes perfect sense physically, though has never been expounded before in Timoshenko’s or other researchers’ solutions, either for torsional rigidity or for the induced shear stresses and displacements.

This paper concerns an investigation on six large-scale Steel Fiber Reinforced Concrete (SFRC) beams tested in pure torsion. All beams had longitudinal rebars to facilitate the well-known space truss resisting mechanism. However, in order to promote economic use of the material, the transverse reinforcement (i.e. stirrups/links) was varied in the six large scale beams. The latter contained either no stirrups, or the minimum amount of transverse reinforcement (according to Eurocode 2), or hooked-end steel fibers (25 or 50 kg/m3). Material characterization were also carried out to determine the performance parameters of SFRC. The results of this study show that SFRC with a post-cracking performance class greater than 2c (according to Model Code 2010) is able to completely substitute the minimum reinforcement required for resisting torsion. In fact, the addition of steel fibers contributes to significantly increase the maximum resisting torque and maximum twist when compared to the same specimen without fibers. Moreover, SFRC provides a rather high post-cracking stiffness and a steadier development of the cracking process as compared to classical RC elements. This phenomenon improves beam behavior at serviceability limit state. The experimental results are critically discussed and compared to available analytical models as well as with other tests available into the literature. [ABSTRACT FROM AUTHOR]

In this paper we consider the so-called procedure of Continuous Steiner Symmetrization, introduced by Brock in [F. Brock, Math. Nachr.172 (1995) 25–48 and F. Brock, Proc. Indian Acad. Sci.110 (2000) 157–204]. It transforms every open set Ω ⊂⊂ ℝd into the ball keeping the volume fixed and letting the first eigenvalue and the torsional rigidity respectively decrease and increase. While this does not provide, in general, a γ-continuous map t ↦ Ωt, it can be slightly modified so to obtain the γ-continuity for a γ-dense class of domains Ω, namely, the class of polyhedral sets in ℝd. This allows to obtain a sharp characterization of the Blaschke-Santaló diagram of torsion and eigenvalue. [ABSTRACT FROM AUTHOR]

We prove that, in the class of convex polygons with a given number of sides, the regular n-gon is optimal for some shape optimization problems involving the torsional rigidity, the principal frequency of the Laplacian, or the logarithmic capacity. [ABSTRACT FROM AUTHOR]

Torsional problems commonly arise in frame structural members subjected to unsym­metrical loading. Saint-Venant proposed a semi inverse method to develop the exact theory of torsional bars of general cross sections. However, the solution to the problem using an analytical method for a complicated cross section is cumbersome. This paper presents the adoption of the Saint-Venant theory to develop a simple finite element program based on the displacement and stress function approaches using the standard linear and quadratic triangular elements. The displacement based approach is capable of evaluating torsional rigidity and shear stress distribution of homogeneous and nonhomogeneous; isotropic, orthotropic, and anisotropic materials; in singly and multiply-connected sections. On the other hand, applications of the stress function approach are limited to the case of singly-connected isotropic sections only, due to the complexity on the boundary conditions. The results show that both approaches converge to exact solutions with high degree of accuracy.

Let Ω be an open convex set in Rm with finite width, and let vΩ be the torsion function for Ω, i.e. the solution of -Δv=1,v∈H10(Ω). An upper bound is obtained for the product of ∥vΩ∥L∞(Ω)λ(Ω), where λ(Ω) is the bottom of the spectrum of the Dirichlet Laplacian acting in L2(Ω). The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area 1, it is shown that ∥vΩ∥L1(Ω)λ(Ω)≥π224, and that this bound is sharp [ABSTRACT FROM AUTHOR]

In this paper, we prove a variational formula of torsional rigidity under the Orlicz sum, and then introduce the Orlicz mixed torsional rigidity which yields the Orlicz-Brunn-Minkowski inequality and Orlicz-Minkowski inequality for torsional rigidity. As an application, we prove the existence of solution to the Orlicz-Minkowski problem for discrete measure and general measure associated with torsional rigidity. [ABSTRACT FROM AUTHOR]

Premise: Because of their own weight and additional wind forces, plants are exposed to various bending and torsional loads that sometimes require contradictory structural characteristics and mechanical properties. The resulting trade‐off between flexural and torsional rigidity can be quantified and compared using the dimensionless twist‐to‐bend ratio. Methods: The flexural rigidity of the stems of Carex pendula was determined by 2‐point bending tests. Additionally, 4‐point bending tests and torsional tests were carried out on segments of two internodes directly below the inflorescences to measure flexural and torsional rigidity. Anatomical investigations were performed to quantify the cross‐sectional distribution of their tissues. Results: The flexural rigidity of the stems, segments of the apical internode 1, and the more basal internode 2 differed significantly from each other, whereas the bending elastic moduli were not significantly different. The torsional rigidity of segments of internode 2 was a factor of 3.3 higher than that of internode 1, whereas the torsional moduli did not differ significantly. The twist‐to‐bend ratios of segments of internode 1 and 2 reached values between 85 and 403. Light microscopic investigations revealed a triangular stem possessing individual sclerenchyma strands, with internode 2 having significantly more strands than internode 1. Conclusions: In the case of Carex pendula, flexural and torsional rigidity are adapted to the given mechanical constraints by significant changes in morphometric variables (axial and polar second moment of area, number of sclerenchyma strands), whereas the material properties (bending and torsional modulus) do not change markedly along the stem. [ABSTRACT FROM AUTHOR]

Let λ 1 (R ~ 2 n + 1) be the first Dirichlet eigenvalues of the Laplacian associated to regular Reuleaux polygons R ~ 2 n + 1 of width w ~ 2 n + 1 = 1 , n = 1 , 2 , 3 , ... Let T (R ~ 2 n + 1) be their torsional rigidities. It is established that λ 1 (R ~ 2 n + 1) is a decreasing sequence, and T (R ~ 2 n + 1) an increasing sequence. Similar (but reversed) inequalities hold in the class of regular Reuleaux polygons circumscribed to the unit circle. [ABSTRACT FROM AUTHOR]

We provide a sharp double-sided estimate for Poincaré–Sobolev constants on a convex set, in terms of its inradius and N - dimensional measure. Our results extend and unify previous works by Hersch and Protter (for the first eigenvalue) and of Makai, Pólya and Szegő (for the torsional rigidity), by means of a single proof. [ABSTRACT FROM AUTHOR]

We study, in dimension n ≥ 2, the eigenvalue problem and the torsional rigidity for the p-Laplacian on convex sets with holes, with external Robin boundary conditions and internal Neumann boundary conditions. We prove that the annulus maximizes the first eigenvalue and minimizes the torsional rigidity when the measure and the external perimeter are fixed. [ABSTRACT FROM AUTHOR]

Substituting composite structures for conventional metallic structures has many advantages because of strength to weight ratio. Laminated composites, with their advantage of higher specific stiffness, gained substantiality in the field of torque carrying structures through many applications. Composite drive shafts has the ability of being the lighter and longer life drive train with higher critical speed. This work aim stop provide an alternative of conventional steel drive shaft for an automotive application and investigates for different combinations of composite materials and different stacking sequence opposite torque directional so determine the natural frequency of composite drive shaft with the help of ANSYS 18.1. [ABSTRACT FROM AUTHOR]

The distinguished names in the title have to do with different proofs of the celebrated Soap Bubble Theorem and of radial symmetry in certain overdetermined boundary value problems. We shall give an overeview of those results and indicate some of their ramifications. We will also show how more recent proofs uncover the path to some stability results for the relevant problems.

Lately, numerous studies have been performed to design composite monocoque's with high strength and low weight for various student-level racing contests. The objective of this paper is to develop an insightful methodology to design and developed a lightweight composite monocoque. The monocoque is designed to pass the mandatory static load tests laid down by the International Automobile Federation (FIA) Formula 3. These Formula 3 tests are considered the baseline of the desired structural integrity of the composite monocoque. The presented design technique emphasizes on a monocoque developed for Sports Car Club of America races. The three standard load tests performed on the monocoque are survival cell side test, fuel tank test and side intrusion test. A sandwich layup of bidirectional-woven carbon/epoxy prepreg and aluminum honeycomb is optimized for minimum weight while predicting the unknown properties of layup, and ensuring the monocoque does not experience failure. The approach intends to achieve minimum weight with high torsional rigidity and is capable of being used for the design and analysis of any kind of formula-type composite monocoque. [ABSTRACT FROM AUTHOR]

Given , we discuss the embedding of in. In particular, for we deduce its compactness on all open sets on which it is continuous. We then relate, for all q up the fractional Sobolev conjugate exponent, the continuity of the embedding to the summability of the function solving the fractional torsion problem in in a suitable weak sense, for every open set. The proofs make use of a non-local Hardy-type inequality in , involving the fractional torsion function as a weight. [ABSTRACT FROM AUTHOR]

• Saint-Venant torsion of cylindrical orthotropic solid elliptical cross section. • Appropriate analytical solution for torsion and Prandtl stress functions. • Upper and lower bounds for the torsional rigidity. This paper deals with the Saint-Venant torsion of elastic cylindrical orthotropic solid elliptical cross section. The origin of the cylindrical anisotropy coincides with the centroid of the elliptical cross section. By the use of principle of minimum of potential energy and principle of minimum of complementary energy approximate analytical solutions are derived for the torsion function and for the Prandtl's stress function. Obtained upper and lower bounds for the torsional rigidity show the accuracy of the derived approximate analytic solutions. [ABSTRACT FROM AUTHOR]

A unique fiber-reinforced polymer (FRP)–aluminum spatial truss structure with upper I-type, transverse beam braces was developed for deployable bridging, yielding the operational advantages of bestraddled erection bridges. Experimental testing and numerical simulation were performed to evaluate the torsional rigidity of a fabricated cantilever, full-scale experimental structure. The predictions obtained based on a computational finite element model were strongly consistent with the experimental results. Moreover, a numerical decomposition and reconstruction procedure was employed to understand the load-bearing mechanism of the structure. The results demonstrated that the improved transverse braces possessed adequate capacity for providing sufficient rigidity and lateral stability to the complete twin-treadway structure under torsion. The torsional center of the improved structure was located at the axis of symmetry of the twin-treadway bridge deck. The representative torsional rigidity of the twin-treadway module was approximately 87.5 kN·m2/degree. Compared to the original construction, the improved structure exhibited only minor discrepancies regarding the torsional rigidity, and consistent characteristics in terms of the load-bearing mechanism. The torsional rigidity of the improved twin-treadway structure was primarily generated by the vertical bending rigidities of its two parallel single treadways through the rigid transverse braces. This significant finding specifically pertains to the unique twin-treadway hybrid bridge. The results presented in this work are expected to provide valuable insights, which could, in turn, lead to further the development of similar lightweight structural systems. [ABSTRACT FROM AUTHOR]

We investigate interior and exterior overdetermined boundary value problems on triangles, which corresponds to studying stationary triangles for variational functionals under volume or perimeter constraint. We prove that in certain cases the only triangle supporting solution is the equilateral one. In some other cases, we obtain that all triangles support solutions, thus extending (through a simpler proof) what was recently shown by Hans Christianson. [ABSTRACT FROM AUTHOR]

We prove that the optimal cluster problem for the sum/the max of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity. In the specific case of the max of the first Robin eigenvalue, we are able to remove the convexity assumption on the cells. [ABSTRACT FROM AUTHOR]

A change in torsional stiffness of pneumatic tuners can be achieved by changing the pressure of the gaseous medium in the course of the mechanical systems' operation or while idle. Based on this statement, we tuned mechanical systems with torsional vibration when they are either idle or when they reach stable condition during operation. The principle of tuning mechanical systems with torsional vibration during operation when their condition is stable lies in the appropriate adaptation of basic dynamic properties (dynamic torsional stiffness and damping factor) of the pneumatic tuner to the dynamics of the system. This was achieved with a control system that helps create a closed control feedback loop. In this way, we can continuously change, that is, adapt the dynamic properties of the pneumatic tuner to the dynamics of the mechanical system so that no dangerous torsional vibrations occur while the system is in operation mode. [ABSTRACT FROM AUTHOR]

Abstract Quantitative assessment of bone fracture healing remains a significant challenge in orthopaedic trauma research. Accordingly, we developed a new technique for assessing bone healing using virtual mechano-structural analysis of computed tomography (CT) scans. CT scans from 19 fractured human tibiae at 12 weeks after surgery were segmented and prepared for finite element analysis (FEA). Boundary conditions were applied to the models to simulate a torsion test that is commonly used to access the structural integrity of long bones in animal models of fracture healing. The output of each model was the virtual torsional rigidity (VTR) of the healing zone, normalized to the torsional rigidity of each patient's virtually reconstructed tibia. This provided a structural measure to track the percentage of healing each patient had undergone. Callus morphometric measurements were also collected from the CT scans. Results showed that at 12 weeks post-op, more than 75% of patients achieved a normalized VTR (torsional rigidity relative to uninjured bone) of 85% or above. The predicted intact torsional rigidities compared well with published cadaveric data. Across all patients, callus volume and density were weakly and non-significantly correlated with normalized VTR and time to clinical union. Conversely, normalized VTR was significantly correlated with time to union (R2 = 0.383, p = 0.005). This suggests that fracture scoring methods based on the visual appearance of callus may not accurately predict mechanical integrity. The image-based structural analysis presented here may be a useful technique for assessment of bone healing in orthopaedic trauma research. [ABSTRACT FROM AUTHOR]

Wrapping technology is one of the effective ways of strengthening concrete elements. Several researchers reported the effectiveness of Glass fiber reinforced polymers and carbon fiber reinforced polymers for improving the strength of the concrete elements. Wrapping on three sides is one of the effective methods for strengthening the beams supporting slabs. Available literatures on the strength enhancement of “U” wrapped concrete elements subjected to torsional loads do not give a detailed insight, whereas this investigation is an attempt to address the issues with ferrocement “U” wrap. Ferrocement “U” wraps are preferred over FRPs because of its cost factor. The “U” wraps are found to provide better torque carrying capacity under all states of torsion while under reinforced members provide better toughness over other states of torsion. Similarly completely over reinforced beams provide more torque resisting capacity than the others. Increase of torsional capacity is more prominent in states of torsion while improvement in torsional strength with number of mesh layers in ferrocement “U” wrap is minimal.