The Effects of the Longitudinal Axis of Loading upon Bending, Shear and Torsion of a Thin-Walled Cantilever Channel Beam

Three aluminium channel sections of US standard extruded dimension are mounted as cantilevers with x-axis symmetry. The flexural bending and shear that arise with applied axial torsion are each considered theoretically and numerically in terms of two longitudinal axes of loading not coincident with the shear centre. In particular, the warping displacements, stiffness and stress distributions are calculated for torsion applied to longitudinal axes passing through the section’s centroid and its web centre. The stress conversions derived from each action are superimposed to reveal a net sectional stress distribution. Therein, the influence of the axis position upon the net axial and shear stress distributions is established compared to previous results for each beam when loading is referred to a flexural axis through the shear centre. Within the net stress analysis is, it is shown how the constraint to free warping presented by the end fixing modifies the axial stress. The latter can be identified with the action of a ‘bimoment’ upon each thin-walled section.

General Torsional Stiffness Matching of Off-road Vehicle

Increasing frame torsional stiffness of off-road vehicle will lead to the decrease of body torsional deformation, but the increase of torsional loads of frame and suspension system and the decrease of wheel adhesive weight. In severe case, a certain wheel will be out of contact with road surface. Appropriate matching of body, frame and suspension torsional stiffnesses is a difficult problem for off-road vehicle design. In this paper, these theoretically analytic models of the entire vehicle, body, frame and suspension torsional stiffness are constructed based on the geometry and mechanism of a light off-road vehicle＇s body, frame and suspension. The body and frame torsional stiffnesses can be calculated by applying body CAE method, meanwhile the suspension＇s rolling angle stiffness can be obtained by the bench test of the suspension＇s elastic elements. Through fixing the entire vehicle, using sole timber to raise wheels to simulate the road impact on a certain wheel, the entire vehicle torsional stiffness can be calculated on the geometric relation and loads of testing. Finally some appropriate matching principles of the body, frame and suspension torsional stiffness are summarized according to the test and analysis results. The conclusion can reveal the significance of the suspension torsional stiffness on off-road vehicle＇s torsion-absorbing capability. The results could serve as a reference for the design of other off-road vehicles.

Torsional stiffness modeling and optimization of the rubber torsion bushing for a light tracked vehicle

Taking the rubber torsion bushing of a certain type of all-terrain tracked vehicle as the research object,a theoretical model of torsional stiffness was proposed according to the non-linear characteristics of rubber components and structural feature of the suspension. Simulations were carried out under different working conditions to obtain root mean square of vertical weighted acceleration as the evaluation index for ride performance of the all-terrain tracked vehicle,with a dynamics model of the whole vehicle based on the theoretical model of the torsional stiffness and standard road roughness as excitation input. Response surface method was used to establish the parametric optimization model of the torsional stiffness. The evaluation index showed that ride performance of the vehicle with optimized torsional stiffness model of suspension was improved compared with previous model fromexperiment. The torsional stiffness model of rubber bushing provided a theoretical basis for the design of the rubber torsion bushing in light tracked vehicles.

Nonlinear characteristics of torsional stiffness of a coilable mast with triangular section

With numerous applications coilable masts in high-precision astronomical observations,such as X-ray source observations,it is important to investigate mast stiffness.To date,there have been many studies on the bending stiffness of coilable masts,but few studies on their torsional stiffness,especially regarding the nonlinear characteristics of torsional stiffness of coilable masts under large torsional deformation.In this paper,a nonlinear analysis method is presented to examine the torsional stiffness of coilable masts with triangular sections.Based on the second-order bending buckling hypothesis of battens under large torsion deformation,the nonlinear relationship between torsional torque and torsional angle is obtained by analyzing torsional deformation and force of coilable masts.This method is used to analyze the torsional stiffness nonlinearity of a certain type of coilable mast which will be used in a practical application in the future and the results are verified by simulation and testing.The comparison results show that the error is within the acceptable range,which proves the effectiveness of the proposed method.

用于抵抗扭转采取边缘弹性约束下的单轴压缩下板材极限强度:一种新的综合方法

The ultimate strength of platings under compression is one of the most important factors to be addressed in the ship design.Current Rules for ship structural design generally provide explicit strength check criteria against buckling for simply supported and clamped platings.Nevertheless,ship platings generally exhibit an intermediate behaviour between the simple support and the clamped conditions,which implies that the torsional stiffness of supporting members should be duly considered.Hence,the main aim of this study is the development of new design formulas for the ultimate strength of platings under uniaxial compression,with short and/or long edges elastically restrained against torsion.In this respect,two benchmark studies are performed.The former is devoted to the development of new equations for the elastic buckling coefficients of platings with edges elastically restrained against torsion,based on the results of the eigenvalue buckling analysis,performed by Ansys Mechanical APDL.The latter investigates the ultimate strength of platings with elastically restrained edges,by systematically varying the plate slenderness ratio and the torsional stiffness of supporting members.Finally,the effectiveness of the new formulation is checked against a wide number of finite element(FE)simulations,to cover the entire design space of ship platings.

Effects of bottom bracings on torsional dynamic characteristics of horizontally curved twin I-girder bridges with different curvatures

Curved twin I-girder bridges (CTIGBs) have low torsional stiffness that makes them vulnerable to dynamic loads. This study investigates the effects of bottom bracings on the torsional dynamic characteristics of CTIGBs. Five types of bottom bracings are designed to investigate their effects on the dynamic characteristics of CTIGBs with different curvatures under free and forced vibrations. To perform numerical investigations, three-dimensional (3-D) finite element (FE) bridge and vehicle models are established using commercial ANSYS code, and then a vehicle-bridge interaction analysis approach is proposed. Road roughness profiles generated from power spectral density and cross spectral functions are also taken into account in the analyses. The numerical results show that torsional frequencies increase significantly after providing bottom bracings, and the increasing rate depends on the type of bottom bracings and their locations of installation. Bottom bracings can act as load transmitting members from one main girder to the others. Large negative bearing forces that have occurred in bridges with small radii of curvatures can be remarkably reduced by providing bottom bracing systems. It is found that the performances of several bottom bracing systems are effective in improving the torsional dynamic characteristics of the bridges in this study.

A Finite Element Cable Model and Its Applications Based on the Cubic Spline Curve

For accurate prediction of the deformation of cable in the towed system, a new finite element model is presented that provides a representation of both the bending and torsional effects. In this paper, the cubic spline interpolation function is applied as the trial solution. By using a weighted residual approach, the discretized motion equations for the new finite element model are developed. The model is calculated with the computation program complier by Matlab. Several numerical examples are presented to illustrate the numerical schemes. The results of numerical simulation are stable and valid, and consistent with the mechanical properties of the cable. The model can be applied to kinematics analysis and the design of ocean cable, such as mooring lines, towing, and ROV umbilical cables.

Dynamic load sharing characteristics and sun gear radial orbits of double-row planetary gear train

A new non-linear bending-torsional coupled model for double-row planetary gear set was proposed, and planet's eccentricity error, static transmission error, and time-varying meshing stiffness were taken into consideration. The solution of differential governing equation of motion is determined by applying the Fourier series method. The behaviors of dynamic load sharing characteristics affected by the system parameters including gear eccentricities error, ring gear's supporting stiffness, planet's bearing stiffness, torsional stiffness of first stage carrier and input rotation rate were investigated qualitatively and systematically, and sun gear radial orbits at first and second stage were explored as well. Some theoretical results are summarized as guidelines for further research and design of double-row planetary gear train at last.

Optimization of Torsional Stiffness for Heavy Commercial Vehicle Chassis Frame

The chassis frame is the backbone of a heavy commercial vehicle(HCV).Its main purpose is to securely carry the determined load under all designed operative situations.Thus,it should be rigid enough to withstand the various forces acting on it.The objective of this study was to develop a stiffness model to select an optimum cross section with determined torsional stiffness.Johnson’s method of optimization was adapted to develop a stiffness equation and select a section with a determined torsional stiffness and a requiredmass constraint.The stiffness obtained from the developed stiffnessmodel and finite element analysis(FEA)is a close match,which proves the validity of the proposed model.The section with the maximum torsional stiffness was used for frame-level optimization to improve the torsional and lateral stiffness of the overall chassis frame.The strain energy absorption capacity of the cross member improved by changing the section of the cross member.By using the optimized section,the torsional stiffness of the frame improved by 44%and a lateral stiffness of 10%was obtained.The Creo software was used for modeling and FEA was performed with the Hypermesh software.

Elastically restrained Bernoulli-Euler beams applied to rotary machinery modelling

Facing the lateral vibration problem of a machine rotor as a beam on elastic supports in bending,the authors deal with the free vibration of elastically restrained Bernoulli-Euler beams carrying a finite number of concentrated elements along their length.Based on Rayleigh’s quotient,an iterative strategy is developed to find the approximated torsional stiffness coefficients,which allows the reconciliation between the theoretical model results and the experimental ones,obtained through impact tests.The mentioned algorithm treats the vibration of continuous beams under a determined set of boundary and continuity conditions, including different torsional stiffness coefficients and the effect of attached concentrated masses and rotational inertias, not only in the energetic terms of the Rayleigh’s quotient but also on the mode shapes,considering the shape functions defined in branches.Several loading cases are examined and examples are given to illustrate the validity of the model and accuracy of the obtained natural frequencies.

Optimal design of automotive body B-pillar using simplified finite element model of body-in-prime combined with an optimization procedure

Optimization of an automotive body structure faces the difficulty of having too many design variables and a too large design search space. A simplified model of body-in-prime(BIP) can solve this difficulty by reducing the number of design variables. In this study, to achieve lighter weight and higher stiffness, the simplified model of BIP was developed and combined with an optimization procedure;consequently, optimal designs of automotive body B-pillar were produced. B-pillar was divided into four quarters and each quarter was modelled by one simplified beam. In the optimization procedure, depth, width, and thickness of the simplified beams were considered as the design variables.Weight, bending and torsional stiffness were also considered as objective functions. The optimization procedure is composed of six stages: designing the experiments, calculating grey relational grade, calculating signal-to noise ratio,finding an optimum design using Taguchi grey relational analysis, performing sensitivity analysis using analysis of variance(ANOVA) and performing non-dominated sorting and multi-criteria decision making. The results show that the width of lower B-pillar has the highest effect(about 55%) and the obtained optimum design point could reduce the weight of B-pillar by about 40% without reducing the BIP stiffness by more than 1.47%.

Improved Twin I-Girder Curved Bridge-Vehicle Interaction Analysis and Human Sensitivity to Vibration

The perceptible vibration of curved twin I-girder bridges under traffic loads is an important design consideration, because this bridge have rather low torsional stiffness that produce excessive vibrations. The objective of this investigation was to study the vibration of curved twin I-girder bridges due to moving vehicles and the effect of vibrations on bridge users. To this end, a comprehensive three-dimensional finite element models for bridge and vehicle are developed by using ANSYS code for studying bridge-vehicle interaction and the resultant sensitivity to vibration. Truck parameters include the body, the suspension and the tires. Gap and actuator elements are incorporated into the tire models to simulate the separation between the tires and road surface, and road surface roughness, respectively. Road roughness profiles are generated from power spectral density and cross spectral functions. To couple the motion of the bridge and vehicle, Lagrange multipliers and constraint equations are utilized through the augmented Lagrangian method. A parametric study is performed to identify the effect of various parameters on the vibration of the bridge. The results have been expressed in the form of human perceptibility curves. This study finds that the bridge response is significantly influenced by the road roughness, bump height at expansion joint and vehicle speeds. The results show that the inclusion of features such as increasing the torsional stiffness by providing additional stiffened bracing has major effects on the reduction of perceptible vibration.

Analytical and numerical analysis on the torsional performance of elliptically deformed tunnels in the longitudinal direction

Segmental lining structures constructed by shields/tunnel boring machines are often subjected to uneven longitudinal cross-section torsion as a result of eccentric external loads,which is extremely adverse to tunnel safety but has not received sufficient attention for a long time.To figure out the torsional performance of segmental tunnels,it is essential to assess the longitudinal torsional stiffness and active-torsion-rejection capability of a segmented tunnel.The aim of the paper is to derive an analytical solution to the longitudinal torsional stiffness of a segmental tunnel with existing elliptical deformation.The longitudinal torsional stiffness under different internal force combinations is deduced considering the longitudinal axial force and bending moment based on the equivalent continuous model and force balance equation.The validity of the analytical solution is verified by comparing it with finite element method results.Then,a parametric analysis,using the new analytical solution,is included to investigate the effect of the key parameters on torsional behaviors,including the segment size,the bolt size and the transverse bending stiffness,etc.It is found that:(1)the longitudinal torsional stiffness efficiency(LTSE)of the segmental tunnel decreases with the rise of segment thickness to diameter ratio but increases with the ring width to diameter ratio;(2)the LTSE reduces with the increase of the effective shear length but rises with the diameter of bolts;(3)the LTSE increases rapidly with the ratio of compression-torsion or bending-torsion.Furthermore,the envelope curve of the critical load(N0,M0)for a tunnel to actively resist a certain internal torque is given.The proposed solution can be easily utilized to determine the longitudinal torsional stiffness of segmental tunnels and is an effective tool for tunnel design and maintenance.

A novel structural modification to eliminate the early coupling between bending and torsional mode shapes in a cable stayed bridge

In this paper, a novel structural modification approach has been adopted to eliminate the early coupling between the bending and torsional mode shapes of vibrations for a cable stayed bridge model generated using ABAQUS software. Two lateral steel beams are added to the middle span of the structure. Frequency analysis is dedicated to obtain the natural frequencies of the first eight mode shapes of vibrations before and after the structural modification approach. Numerical simulations of wind excitations are conducted for the 3D model of the cable stayed bridge with duration of 30 s supporting on real data of a strong wind from the literature. Both vertical and torsional displacements are calculated at the mid span of the deck to analyze both the bending and the torsional stiffness of the system before and after the structural modification. The results of the frequency analysis after applying lateral steel beams declared a safer structure against vertical and torsional vibrations and rarely expected flutter wind speed. Furthermore, the coupling between the vertical and torsional mode shapes has been removed to larger natural frequencies magnitudes with a high factor of safety. The novel structural approach manifested great efficiency in increasing vertical and torsional stiffness of the structure.

Out-of-plane elastic buckling load and strength design of space truss arch with a rectangular section

The out-of-plane stability of the two-hinged space truss circular arch with a rectangular section is theoretically and numerically investigated in this paper.Firstly,the flexural stiffness and torsional stiffness of space truss arches are deduced.The calculation formula of out-of-plane elastic buckling loads of the space truss arch is derived based on the classical solution of out-of-plane flexural-torsional buckling loads of the solid web arch.However,since the classical solution cannot be used for the calculation of the arch with a small rise-span ratio,the formula for out-of-plane elastic buckling loads of space truss arches subjected to end bending moments is modified.Numerical research of the out-of-plane stability of space truss arches under different load cases shows that the theoretical formula proposed in this paper has good accuracy.Secondly,the design formulas to predict the out-of-plane elastoplastic stability strength of space truss arches subjected to the end bending moment and radial uniform load are presented through introducing a normalized slenderness ratio.By assuming that all components of space truss circular arches bear only axial force,the design formulas to prevent the local buckling of chord and transverse tubes are deduced.Finally,the bearing capacity design equations of space truss arches are proposed under vertical uniform load.