End Constraints Research Articles

Steel Columns under Compression with Different Sizes of Square Hollow Cross-Sections, Lengths, and End Constraints

Steel Columns under Compression with Different Sizes of Square Hollow Cross-Sections, Lengths, and End Constraints

Elasticity solutions for functionally graded beams with arbitrary distributed loads

This paper derives the exact general elasticity solution for functionally graded rectangular beams subjected to arbitrary normal and tangential loads and with arbitrary end constraints. The general solution consists of bending moments and their integrals and derivatives, along with load-independent function sequences of the longitudinal coordinate. The method for determining function sequences has been established based on the stress function method. General solution formulas for stresses, strains and displacements have been derived and used to solve explicit special solutions for six cases involving concentrated forces, uniformly loads, and quadratically distributed loads with different displacement constraints scenarios. The results obtained are compared with existing exact solutions and those of Euler–Bernoulli and Timoshenko beams, and the errors of the latter two are analysed.

Higher‐order effect of axial force on free vibration and buckling of functionally graded sandwich beams

AbstractThis paper presents a novel higher‐order shear deformation beam theory for analyzing the stability and free vibration of functionally graded (FG) sandwich beams with emphasis on the effect of higher‐order moment (HOM) and cross‐sectional warping. The governing equation of axially loaded FG sandwich beams is derived from three‐dimensional equations of the theory of elastic waves in bodies with homogeneous initial stresses. The characteristic equations for typical end conditions are obtained exactly. The numerical results of the natural frequencies and critical loads are calculated and verified for special cases by comparing them with the existing solutions. The effects of gradient index, core thickness, HOM, and warping shapes on the natural frequencies and critical buckling loads are elucidated for different slenderness ratios and end constraints.

Simplified analytical model for prediction of collapse resistance of restrained steel beam-column substructure exposed to fire

This paper provides a simplified analytical model for the restrained steel beam-column substructure under a fire-induced progressive collapse scenario, which is applied to predict the collapse resistance of substructures under fire. The simplified model is formulated for axially and rotationally restrained two-span steel beam-column substructures under column-loss scenarios and involves rigid and semi-rigid connections. The response characteristics of restrained substructures exposed to fire under column-loss scenarios in each response stage are elaborately analyzed. A “negative” catenary stage, in consideration of the unique negative axial force effects induced by the temperature thermal expansion, was first introduced in the five-stage simplified model for restrained substructures exposed to fire against progressive collapse. Subsequently, the explicit calculation formulas for determining the vertical loads and deflection of both rigid and semi-rigid joint substructures exposed to fire were derived in a quantified way, based on rigid-plastic mechanisms and restraint coefficient methods. In addition, the corresponding numerical simulation analysis on a restrained two-span steel beam-column substructure with different end constraints and fire conditions was carried out to verify the applicability and reliability of established theoretical formulas. Comparisons of vertical load-deflection relationships between the calculation formulas and corresponding numerical simulation shows good accuracy. For more accurate prediction results, the error analysis was further conducted, and several theoretical formulas were modified, which were verified to be accurate and reliable after correction. With derived resistance functions, the resistance of steel beam-column substructures in fire-induced progressive collapse scenarios can be predicted.

Constrained Origami Artificial Muscle-Driven Robotic Manipulator Capable of Coordinating Twisting and Grasping Motions for Object Manipulation.

Grasping and twisting motions are vital when manipulating objects due to their fundamental role in enabling precision, adaptability, and effective interaction. However, few studies in soft robotics exploiting artificial muscles have achieved object manipulation in situ through the coordination of twisting and grasping motions akin to our forearm and hand's capabilities. Especially, when using the same artificial muscle module to achieve these two motions will greatly simplify the manufacturing and control complexity. Here, we introduce identical origami artificial muscle modules (OAMMs) subjected to distinct end constraints into the design of the robotic manipulator, allowing it to achieve independent grasping and twisting motions to achieve effective, precise object manipulation. Applying different end constraints to the identical OAMMs yields distinct motions at their ends, where utilizing a fixed end and a sliding end realizes pure translation, while opting for a fixed end and a rotating end enables pure rotation. The differentially constrained OAMMs then serve as soft actuators for the manipulator's torsional mechanism and grasping mechanism to accomplish independent, controllable twisting and grasping motions. The coordination of twisting and grasping motions finally enables the manipulator to complete various tasks, including installing a light bubble, pouring the water from a lidded bottle into a cup, and sorting and stacking puzzle blocks. Our study pioneers the utilization of OAMMs for precise and versatile object manipulation through the coordination of independent twisting and grasping motions.

Semi-analytical dynamic modeling and fluid-structure interaction analysis of L-shaped pipeline

The traditional semi-analytic modeling is mainly aimed at the straight pipeline with simple boundary conditions. In contrast, the actual aero-engine pipeline needs to consider the effects of complex configuration, boundary and multiple clamps, so proposing an improved semi-analytic method to model an arbitrary fluid-conveying L-shaped pipeline is challenging. Based on the Hamiltonian principle, the differential equations of the L-shaped pipeline are derived. Then, the modal shape functions are determined under the conditions that the elastic support points and the joints of the straight pipeline and the curved pipeline meet the deformation coordination requirements. The proposed semi-analytical model is verified by comparing the natural frequencies and mode shapes obtained from the finite element method using ANSYS and experiment; the error of each order natural frequency is less than 5 %. In addition, the model is verified under various working conditions: different center angle and boundary conditions (clamp position and pipeline end constraints). The effects of the fluid velocity and pressure on the natural characteristics of the L-shaped pipeline are also analyzed. The results show that under the critical pressure of 60.51 MPa and the critical velocity of 247 m/s, the natural frequency decreases with the increase of fluid velocity and pressure. The proposed method can provide theoretical support for efficient and accurate modeling of complex spatial pipeline systems.

Effect of out-of-plane dimension on simplified geometric models in stress analysis

Without exception, every physical object is three-dimensional. However, in many stress analysis situations the analyst is justified in using simplified two-dimensional idealizations of plane stress and plane strain, reducing the complexity of the problem. By obviating the need to mesh in the third dimension, this advantage also extends to numerical studies, helping economize significantly on time and computational power requirements. In plane stress idealization the out-of-plane stresses are zero, whereas in plane strain the out-of-plane strains are zero. These idealizations have variously been linked with the out-of-plane dimension, as well as the end conditions of the object under consideration. However, the exact correlation of the out-of-plane dimension with these idealizations remains ambiguous. Unlike the case of plane stress, there is much disagreement found in literature regarding the necessary conditions for the realization of plane strain, which needs to be addressed. In this article, finite element analysis was employed to study the effect of various out-of-plane dimensions and end constraints. The results show that there is no correlation between the out-of-plane dimension and plane strain, which depends only on the end constraints. However, the out-of-plane dimension does result in the transition from plane stress to generalized plane strain.

Bending performance of an inflation-powered bistable folding beam

Temporary structures necessitate lightweight, easy transportation, and rapid deployment. However, conventional metal temporary bridges are often constrained by weight and scale, while inflatable structures rely on the maintenance of internal over-pressure for stability. To address this issue, a bistable folding beam that can be unfolded through pneumatic actuation is created by integrating origami with inflatable structures. Utilizing planar laser cutting techniques, a novel folding beam is manufactured, followed by three-point bending tests to analyze its bending performance. The evaluation includes considering factors such as internal air pressure, support conditions, folding/unfolding cycles, and geometric parameters. The findings reveal that the beam’s foldable nature compromises its structural strength. Nevertheless, enhanced bending performance can be achieved through measures such as increased support end constraints, the addition of upper section plates, and adjustments to the angles of the basic units.

Shear and flexural deformations in flextegrity segmental beams inspired by Leonardo’s triangular masonry construction

In Tabula XCI verso of Codex Atlanticus, Leonardo da Vinci presents an ingenious masonry structure composed of segments in the shape of inverted triangles. These are assembled by contact in a chain to obtain a lintel or jack arch, where they are pressed together by the thrust of the end constraints. Drawing inspiration from Leonardo’s sketches, we show that, by connecting the segments in pair through elastic tendons, this system represents a new type of flextegrity beam. In a classical flextegrity, the contact surfaces of the segments are curved conjugate profiles, imposing a pure rolling motion along properly designed pitch lines: the consequent elongation of the tendon dictates the constitutive response as a function of the relative rotation of the segments. Here, the contact is through plane surfaces, so that the kinematics, besides the relative rotation, is characterized by segmental shearing. This system is theoretically analysed and a continuum model is derived as the length of the segments becomes small. Comparisons with experiments on three-dimensional-printed prototypes confirm the theoretical findings and highlight the possible competition between rotational and sliding types of deformation. Apart from the historical value, this type of construction can be used in innovative structures or metamaterials.

Study on the Strength and Deformation Characteristics of Coarse-Grained Soil under End-Restraint and End-Free (Microfriction) Conditions

In this study, for triaxial compression testing in geotechnical engineering, large-scale triaxial shear tests of coarse-grained soil were performed under end-restraint and end-free (microfriction) conditions utilizing a large-scale triaxial shear apparatus and a self-developed microfriction load-transfer plate. Under end-restraint and end-free (microfriction) circumstances, the strength and deformation characteristics of the coarse-grained soil were investigated. Further, the effect of the end restraint on the stress–strain characteristics of the coarse-grained soil was analyzed. The findings showed that the coarse-grained soil’s stress–strain curves under the end-restraint and end-free (microfriction) conditions displayed both softening under low confining pressures and hardening under high confining pressures. However, the stress–strain curves’ tendency to soften under low confining pressures was more pronounced under the end-free (microfriction) condition. Due to the influence of the end constraint, the peak strength of the coarse-grained soil increased under the same confining pressure, although the increase was less than 10% when the specimen had the typical height-to-diameter ratio of 2 in the drained triaxial test. Meanwhile, the coarse-grained soil first displayed volumetric contraction under the low confining pressures, followed by volumetric dilation, and it displayed volumetric dilation under the high confining pressures. The volumetric contraction tendency of the coarse-grained soil was more pronounced in the first stages of shearing when the low confining pressures were present. As the shear increased, it became more obvious that the coarse-grained soil had a tendency to dilate under the end-free (microfriction) conditions. Throughout the shearing operation, the coarse-grained soil exhibited a high propensity toward shear contraction. The coarse-grained soil’s strength values were not greatly impacted by the end limitation, but the deformation parameters were. Under the end-restraint conditions, the deformation parameter increased significantly, while the strength parameter only changed slightly.

Trajectory Optimization of Long-Range Guided Rocket-Propelled Vehicle Based on Pseudo-spectral Sequential Convex Programming

Aiming at the trajectory optimization problem of long-range guided rocket-propelled vehicles with multiple constraints, an improved pseudo-spectral sequence convex programming, which takes into account the efficiency and accuracy of optimization, is proposed. First, taking the change rate of the angle of attack and the bank angle as the control variables, the vehicle’s dynamic equation is discretized by using the flipped Radau pseudo-spectral method, and the trajectory optimization problem including start, end, and path constraints is transformed into a convex optimization problem by using the continuous linearization method, and then the solution is obtained by using the original dual interior point method. To make the optimized trajectory near the no-fly zone or the severe weather area more accurate, the concept of the predicted separation point is introduced to subdivide the trajectory. The simulation results show that the improved pseudo-spectral sequence convex programming has high discretization accuracy and fast solution speed, and can be applied to the online use of long-range guided rocket-propelled vehicles.

Development and Preliminary Application of Temperature Stress Test Machine for Cast-in-Place Inner Shaft Lining.

Over the past 20 years, as the depth and diameter of shaft lines increased in China, the cracking and water leakage of the inner walls of frozen shafts have become increasingly severe, resulting in significant safety threats and economic losses. Understanding the stress variation patterns of cast-in-place inner walls under the combined effects of temperature and constraint during construction is a prerequisite for evaluating the crack resistance performance of inner walls and preventing water leakage in frozen shafts. The temperature stress testing machine is an important instrument for studying the early-age crack resistance performance of concrete materials under the combined effects of temperature and constraint. However, existing testing machines have shortcomings in terms of applicable specimen cross-sectional shapes, temperature control methods for concrete structures, and axial loading capacity. In this paper, a novel temperature stress testing machine suitable for the inner wall structure shape, capable of simulating the hydration heat of the inner walls, was developed. Then, a reduced-scale model of the inner wall according to similarity criteria was manufactured indoors. Finally, preliminary investigations of the temperature, strain, and stress variations of the inner wall under 100% end constraint conditions were conducted by simulating the actual hydration heating and cooling process of the inner walls. Results show that the hydration heating and cooling process of the inner wall can be accurately simulated. After approximately 69 h of concrete casting, the accumulated relative displacement and strain of the end-constrained inner wall model were -244.2 mm and 187.8 με, respectively. The end constraint force of the model increased to a maximum value of 1.7 MPa and then rapidly unloaded, causing the model concrete to crack in tension. The temperature stress testing method presented in this paper provides a reference for scientifically formulating technical approaches to prevent cracking in cast-in-place concrete inner walls.

The Folded Pneumatic Artificial Muscle (foldPAM): Towards Programmability and Control via End Geometry

Soft pneumatic actuators have seen applications in many soft robotic systems, and their pressure-driven nature presents unique challenges and opportunities for controlling their motion. In this work, we present a new concept: designing and controlling pneumatic actuators via end geometry. We demonstrate a novel actuator class, named the folded Pneumatic Artificial Muscle (foldPAM), which features a thin-filmed air pouch that is symmetrically folded on each side. Varying the folded portion of the actuator changes the end constraints and, hence, the force-strain relationships. We investigated this change experimentally by measuring the force-strain relationship of individual foldPAM units with various lengths and amounts of folding. In addition to static-geometry units, an actuated foldPAM device was designed to produce continuous, on-demand adjustment of the end geometry, enabling closed-loop position control while maintaining constant pressure. Experiments with the device indicate that geometry control allows access to different areas on the force-strain plane and that closed-loop geometry control can achieve errors within 0.5% of the actuation range.

Analytical Solution and Exact Effective Thickness of Multilayered Laminated Glass Cantilevered Balustrades

Abstract Cantilevered laminated glass balustrades present design challenges. The cross-sectional warping allowed by the end constraint induces such strong asymmetrical deformations that traditional methods, defining the effective thickness of a monolith with equivalent bending properties, cannot be accurate. By using a recently proposed refined zig-zag theory for laminates, we analytically solve the representative problem of a short simply supported laminated beam with a long cantilevered overhang. The variables are the beam displacement and the mean sectional shear angle, defining the zig-zag warping of the cross section. Geometric and natural boundary conditions, as well as the matching condition at the intermediate roller constraint, necessary to solve the governing differential problem, are found variationally. The analytical solutions under concentrated and distributed loads exactly determine the effective thickness of the laminate. Comparisons are made with other approaches that that, however, apply only to three-layered packages. The expressions proposed here can be directly used in the design practice.

Instability Regions of a laminated Sandwich Plate with MRE Core Under Parametric Excitation Using Finite Element Method

This paper is concerned with the instability regions of a laminated sandwich plate with magneto-rheological elastomer (MRE) core under periodic axial loads. Structures operating under this unstable region cause large response even if the excitation force is small in magnitude. The governing equations of motion of the system are obtained by using finite element method [FEM].The parametric stable and unstable regions of the MRE cored composite sandwich plate for different system parameters and end constraints are found using modified Hsu method for simple parametric resonance condition. This analysis will help to design passive attenuation by using sandwich plates of varied dimensions, material parameters of both skin and core. Further active vibration reduction is possible by applying suitable magnetic field and operating the system outside of the instability region.

Stability of equal-leg angle section braces with torsional restraints

Cross bracing using the single leg angle sections are connected on both sides at the joints, affecting the stability capacity of the cross bracing because of the torque. The study on the stability capacity of equal-leg angle sections with the effects of torsional restraints at joints was insufficient. Additionally, there is no clear guidance from relevant design codes. In this study, an end constraint was designed to simulate the effect of torsional restraints for the angle sections. A total of 105 specimens of angle sections, divided into three kinds of pieces (A, B, C), were tested under three boundary conditions. Additionally, Finite Element Method modeling to study the effects of end constraint was parallelly conducted. Class B specimens had the worst stability. When the slenderness ratio is <120, the stability capacity of Class C specimens is larger. When the slenderness ratio is larger than 120, Class A specimens have better stability. The stability capacity of the equal-leg angle sections is positively related to the stiffness of the end constraint, when the stiffness is less than or equal to 200 kN•m/θ, the change of the stability capacity appears more dramatic, when the stiffness exceeds 200 kN•m/θ, the increase becomes slower. The effect of the slenderness ratio and the ratio of the torsional stiffness at nodes to the bending stiffness of columns were quantified through the stability capacities obtained from the experiment and FEMs to propose a modified method for effective length coefficients, which could optimize transmission tower design.

Experimental and numerical study on the stable bearing capacity of steel tubular cross bracing of a transmission tower

Taking the 1000 kV ultra-high voltage (UHV) steel tube transmission tower as the engineering research subject, the ultimate bearing capacity of the X-cross bracing was experimentally studied. The main tube was manufactured together to add the end constraint on the cross bracing used in the design, with six types of X-cross bracing specimens under two horizontal loading cases to study its ultimate bearing capacity and failure mode in the steel tube transmission tower. Under two loading cases as described, the two inclined tubes of the cross bracing are simultaneously compressed. One is in compression, the other is in tension. The magnitude of the force applied to the tubes remain the same. And then, the results show that the bearing capacities of the common U-joint and C-joint cross bracing specimens are low, especially the bearing capacity of the C-joint with diaphragm is insufficient as the inclined tubes of cross bracing is compressed simultaneously, while the bearing capacities of the intersecting joint and the cross-insert joint are excellent under both cases. Moreover, the corresponding ABAQUS finite element model with imperfection was verified using the test results. As a result, the stress distribution, failure mode and ultimate bearing capacity of the numerical simulation are in good agreement with the experimental results, which shows that results obtained from the subsequent numerical analysis are reasonable. The simplified finite element model is used to analyze the bearing capacity of the intersecting joint cross bracing with the different cross positions and the slenderness ratios under two loading cases. For the calculation according to the Technical Specification for Design of Steel Tube Towers for Overhead Transmission Lines, combined with the calculation results from 60 finite element models, the calculation formula of the slenderness ratio correction factor under two loading cases is elaborated. Compared with the existing formula, the improved formula is closer to the finite element calculation results, which helps reduce material waste during the stage of design.

Coupled effect of multi-factor on the vibration of submerged floating tunnel under impact load

The vibration of SFT under impact load is affected by various factors, i.e. load parameters, buoyancy-weight ratio, hydro-resistance, end constraint, on-way anchors as well as external axial force on both ends. Most of the current studies focused on the single effect of a few factors on a pinned SFT. Few of them addressed the coupled effect of the factors under different end constraints.In this study, the vibration characteristics of SFT under impact load was investigated extensively through the end constraint coupled to a series of normalized factors, i.e. non-dimension SFT length and bending stiffness, structural damping ratio, hydro-resistance coefficient and non-dimension axial force etc. The beam on elastic foundation (BOEF) model was adopted in this study and had been validated by the tube-cable coupled vibration (TCCV) model. The results showed that the natural frequency, deflection and acceleration of SFT were subjected to the joint impacts of end constraint, on-way anchors, SFT length and bending stiffness, hydro-resistance, structural damping and axial force. The end constraint had limited impact on natural frequency when the non-dimension length L˜ ≥10. Also, the effect of end constraint on deflection can be ignored when the non-dimension length L˜ ≥100. The best fitting curves showed that the maximum deflection decreased negative exponentially with the bending stiffness and linearly with the structural damping ratio and hydro-resistance coefficient. For all the four end constraints, the acceleration of SFT decreased negative exponentially with the SFT length. The effect of external axial force on natural frequency and deflection became significant only when the non-dimension axial force N˜ ≥10.

Effect of loading modes on torsional behavior of an extruded Mg alloy bar

The torsional deformation with shear stress is crucial for improving mechanical performance of wrought Mg alloys. Consequently, the difference of mechanical behavior between fixed-end and free-end torsion was investigated. It is found that strain hardening behavior is affected by end constraint. The decreased hardening rate is exhibited which is dominated by prismatic slip during fixed-end torsion. For free-end loading, two stages of strain hardening are presented during torsion, i.e., constant hardening rate and slowly decreased hardening rate. The basal slip dominates the onset of plastic deformation. The prismatic slip becomes dominated mechanism at large shear strain. However, the axial effect is sensitive to forward and reversal torsion. The difference of axial stress between forward and reversal torsion under fixed-end is attributed to twin growth. The difference of axial strain is small during forward and reversal torsion under free-end, which is attributed to the activation of tensile twinning.

Possible mechanism of spallation in rock samples under uniaxial compression

In uniaxial compression, rocks are often observed to have spallation failure at the lateral surface of the sample developing parallel to the direction of loading. While easily explained by a 2D concept of axial crack growth in uniaxial compression, the real 3D mechanism cannot be explained this way, as 3D crack growth in uniaxial compression is restricted to the size of the initial defect due to the phenomenon of wing wrapping. The wing wrapping can only be overcome in biaxial compression with the magnitude of the intermediate principal stress exceeding 5–8.5% of the axial load (depending on the type of crack-producing defect). In uniaxial compression, the role of the intermediate principal stress is played by the circumferential compressive stress induced due to the end constraint owning to the presence of end friction. The results of Finite Element modelling show that zones of biaxial compression are formed at the lateral surface near the sample ends. The sizes of the zones and the maximum magnitude of the circumferential stress depend upon the friction coefficient and the Poisson’s ratio of the rock. The modelling also revealed the presence of zones of tensile circumference stress which can potentially induce splitting failure.