Rotational End Restraints Research Articles

Practical Formulae for Estimating Cable Tension with Unknown Rotational Restraints by the Frequency Method

To calculate the tension in cables with different boundary conditions, the relationship between cables with fixed–fixed and hinged–hinged boundary conditions in terms of the frequency was determined according to frequency characteristic equations of cables with the two boundary conditions. In this way, a simple calculation formula for tension with fixed–fixed boundary conditions was deduced. Similarly, a calculation formula for the tension in cables with a fixed–hinged boundary condition was proposed using the method. Results show that the proposed formulae, with high computational accuracy and wide ranges of application, can be used to calculate the cable tension under a dimensionless parameter (ξ) not lower than 6.9, so it is convenient to apply the formulae to calculate tension in practice. Meanwhile, changes in the frequency ratios of cables with different boundary conditions than those with a hinged–hinged boundary condition were analyzed. Results show that when ξ is not lower than 25, the frequency ratios of cables of various orders tend to be the same. The boundary coefficient(λ) was introduced. Given the cable stiffness, the tension and boundary coefficient(λ) can be calculated through linear regression. The method considers influences of unknown rotational end-restraints of cables and accurately calculates the cable tension. By using simulation examples and engineering examples, the method was verified to be accurate in calculating the cable tension, thus providing a novel, practical method for estimating tension in cables, booms, and anchor-span strands of suspension bridges.

Fire design of steel columns through second-order inelastic analysis with strain limits

A structural steel fire design approach through second-order inelastic analysis with strain limits is proposed and applied to the fire design of steel columns as a first step in the establishment of a novel fire design framework for steel structures in this paper. The proposed method is carried out using beam finite elements, utilising their computational efficiency. In the proposed design approach, the strength and stiffness deterioration of steel in fire, the spread of plasticity, global instability effects, indirect fire actions and thermal expansion are fully taken into account through second-order inelastic analysis, while strain limits are employed to consider the deleterious influence of local buckling on the ultimate resistance. Ultimate capacity of a steel member or system is determined by (i) the load or temperature level at which the predefined strain limit is attained or (ii) the peak load or critical temperature observed during the analysis, whichever occurs first. A systematic numerical parametric study is carried out through nonlinear shell finite element modelling, taking into account a high number of I-section and hollow section steel columns whose response is considered (i) using isothermal and anisothermal analysis techniques and (ii) with and without axial and rotational end-restraints. It is demonstrated that the proposed fire design approach consistently furnishes significantly more accurate capacity and limit temperature predictions for steel columns in fire relative to EN 1993-1-2 [1] design provisions.

Structural fire design of square tubed-reinforced-concrete columns with connection to RC beams in composite frames

The tubed-reinforced-concrete (TRC) column is an innovative type of steel-concrete composite column that is gaining increasing engineering usage, and structural fire performance should be an important consideration of its design. As affected by the adjacent members and beam-to-column joints, the fire behaviour of TRC columns integrated into a frame may largely differ from that of isolated columns with idealised end conditions. However, the fire performance of TRC columns in composite frames has not been studied, except for some work done on the fire performance of isolated TRC columns with certain end restraints. This paper presents finite element analysis on single-storey and single-span non-sway frames with square TRC columns, employing the software package ABAQUS. The thermal and deformation behaviour, development of axial stress, axial force and bending moment during heating, working mechanism and failure mode of the frame exposed to fire are numerically investigated. The difference between the fire performance of TRC columns in a frame and those of equivalent isolated columns is quantified. The fire resistance of the framed column is found to lie between those of the columns with rotational end restraints and columns with idealised pinned ends. The complex effects of the connected RC beam on the fire behaviour of TRC columns, e.g. the beneficial restraining effect and the negative effect due to restrained differential thermal expansion, are clarified. Finally, a practical fire resistance design approach is proposed for TRC columns considering the continuity and interactions between structural elements within a composite frame.

Flexural-torsional buckling of shear deformable steel circular arches under a central concentrated load in a thermal environment

When subjected to a constant temperature gradient, a steel circular arch will experiences non-uniform thermal expansion in its axial direction. This expansion will produces complex internal forces in the arch, which in turn will affect its flexural–torsional buckling behavior under a central concentrated radial load. Hitherto, research studies of the flexural–torsional buckling of such arches are scarce. The position of the effective centroid and shear center do not coincide with the geometric centroid of the section in a temperature gradient field, and this influences the flexural–torsional buckling response. In this paper, the flexural–torsional buckling of shear deformable circular arches with in-plane elastic rotational end restraints subjected to a central concentrated radial load at a constant temperature gradient field is studied. The theoretical solutions for the buckling load of the arch including the effect of the temperature gradient field are obtained and validated by ANSYS simulations. It is found that the buckling load decreases with an increase of the temperature differential when the included angle is small and increases with an increase of the temperature differential in case the included angle is larger than a certain value. The buckling loads including shear deformations are higher than those ignoring shear deformations at a bigger temperature differential. The influences of shear deformations on the buckling load are more significant at a bigger temperature differential.

Geometrically nonlinear buckling of graphene platelets reinforced dielectric composite (GPLRDC) arches with rotational end restraints

This paper presents an analytical investigation on the nonlinear buckling behavior of graphene platelets reinforced dielectric composite (GPLRDC) arches with rotational end restraints under applied electric and uniform radial load. The effective materials properties of GPLRDC, including the elastic modulus and dielectric permittivity, are estimated by employing effective medium theory (EMT). The analytical solutions for nonlinear equilibrium, critical load of limit point buckling and bifurcation buckling of the GPLRDC arch are derived according to the principle of virtual work. The critical geometric parameters and electric voltage governing the buckling mode switching behavior are also identified and discussed. The effects of graphene platelets (GPLs) weight fraction, size and geometry, applied DC voltage, AC frequency, as well as geometry of the arch on the buckling behavior are examined comprehensively. It is found that the dielectric property of the GPLRDC has significant effects on the buckling behavior of the GPLRDC when the GPLs concentration exceeds the percolation threshold. The nonlinear buckling behavior of the GPLRDC is quite sensitive to the AC frequency within a certain range. Furthermore, the change of the applied voltage can switch the buckling mode and even the number of limit points of the GPLRDC arch.

Lateral deformation behavior of eccentrically loaded slender RC columns with different levels of rotational end restraint at elevated temperatures

Purpose In literature, previous studies have focused on analyzing rienforced concrete (RC) columns with idealized end conditions when subjected to fire. In nature, full fixity or free rotation at column ends is not attained. Such ends may be considered partially restrained in rotation. This paper aims to shed a new light on the effect of different degrees of rotational restraint on the lateral deformation behavior of slender heated RC columns subjected to non-linear strain distributions produced by a time-dependent temperature history. Design/methodology/approach To find the strain distribution on the cross section, an iterative technique is adopted using Newton–Raphson method. By introducing a reliable calculation procedure, the lateral deformational behavior is expressed using numerical and searching techniques. A methodology is presented to calculate the effective length factor for RC columns at elevated temperature. Findings The results of the proposed model showed good agreement with available experimental test results. It was also found that the variation of rotational end restraint level has a considerable effect on the lateral deformation behavior of heated slender RC columns. In addition, the effectiveness and the validity of an analytical model should be verified by simultaneously validating the axial and lateral deformations. Moreover, the effective length factor for heated column is higher than that for the corresponding column at ambient temperature. Originality/value This paper shows the impact of different boundary conditions on the behavior of heated slender RC columns. It suggests powerful techniques to determine the lateral deflection and the effective length factor at high temperatures.

Closed-Form Solutions for Initial Thermal Postbuckling Response of Axially Restrained Columns

AbstractClosed-form equations for the initial thermal postbuckling response of axially restrained elastic columns with six well-known sets of rotational and lateral end restraints are presented. Th...

Structural fire resistance of partially restrained, partially composite floor beams, II: Modeling

This paper is Part 2 of a two-part investigation of the structural fire resistance of a partially composite, one-way spanning steel floor beam assembly with partial translational and rotational end restraint provided by shear tab connections to a supporting frame. A preceding paper (Part 1) described a pair of experimental tests on structurally identical composite floor specimens (one protected and the other unprotected) that are subjected to fire via a large modular furnace. In this paper, those experimental results are used to validate numerical models that conservatively capture the thermo-mechanical response of the specimen assemblies to fire. Models are developed to maximize simplicity toward implementing performance-based design of these assemblies in practice. Thermal analysis of the steel is performed using a multiple lumped mass approach, which can be implemented via spreadsheet or a simple, non-iterative programmed solution. Thermal analysis of the slab via a simple one-dimensional heat flow model is compared against a more detailed computational solution. Two types of structural finite element analyses were performed: one composed of shell elements (more complex) and another composed of fiber-beam elements (simpler). The results of all models show conservative agreement with the experimental data. These models are used to analyze the tested assemblies for variations in applied load and fire protection thickness to demonstrate the utility of the ASTM E119 standard fire test as a validation baseline for performance-based structural fire engineering evaluation.

Structural fire resistance of partially restrained, partially composite floor beams, I: Experiments

This paper is Part 1 of a two-part investigation of the structural fire resistance of a partially composite, one-way spanning steel floor beam assembly with partial translational and rotational end restraint provided by shear tab connections to a supporting frame. The goal of this study is to provide experimental data and validated numerical models that can aid the implementation of performance-based design and evaluation approaches for structural fire resistance of steel-framed composite floor systems that are typical of North American building practice. This paper describes a pair of experimental tests on structurally identical composite floor specimens that are subjected to fire via a large modular furnace. A subsequent companion paper (Part 2) presents the numerical studies. One specimen was designed to conform to a 2-h fire rating as prescribed by UL Design No. D902 – the other was tested with no passive fire protection (i.e. as bare steel). Both specimens were flexurally loaded and tested to structural failure under exposure to the ASTM E119 standard fire, and the experimental results are evaluated using the ASTM E119 thermal and structural failure criteria. The test results show similar correlations of midspan deflection to beam temperature for both specimens. The shear tab connections experienced permanent deformation but maintained adequate structural integrity to allow the specimens to reach flexural failure. Neither shear slip nor localized cracking could be visibly detected at the composite stud locations during post-test inspection, indicating that composite action was maintained up to flexural failure.

Analysis of rotational end restraint for cross beams of railway through truss bridges

Cross-beams of modern through truss bridges are connected to truss chord at its nodes and between them. It results in variable rotational end restraint for cross-beams, thus variable bending moment distribution. This feature is captured in three-dimensional modelling of through truss bridge structure. However, for preliminary design or rapid assessment of service load effects such technique of analysis may not be available. So an analytical method of assessment of rotational end restraint for cross-beam of through truss bridges was worked out. Two cases – nodal cross-beam and inter-nodal cross-beam – were analyzed. Flexural and torsional stiffness of truss members, flexural stiffness of deck members and axial stiffness of wind bracing members in the vicinity of the analyzed cross-beam were taken into account. The provision for reduced stiffness of the X-type wind bracing was made. Finally, general formula for assessment of rotational end restraint was given. Rotational end restraints for cross-beams of three railway through truss bridges were assessed basing on the analytical method and the finite element method (three-dimensional beam-element modelling). Results of both methods show good agreement. The analytical method is able to reflect effects of some structural irregularities. On the basis of the obtained results the general values of rotational end restraint for nodal and inter-nodal cross-beams of railway through truss bridges were suggested.

Stability of the neutral equilibrium of columns with a rotational end restraint by the generalized Fourier series

In this paper, stability of the neutral equilibrium and initial post-buckling of a column with a rotational end restraint is analyzed based on Koiter initial post-buckling theory. The potential energy functional is written in terms of the angle. By the generalized Fourier series of the disturbance angle, it is proved that the second-order variation of the potential energy is semi-positive definite at the neutral equilibrium. The stability of the neutral equilibrium is determined by the sign of the fourth-order variation for the buckling mode. For all values of the stiffness of the rotational end restraint, the neutral equilibrium is stable and the bifurcation equilibrium is upward in the initial post-buckling.

Adaption of active boundary conditions in structural fire testing

Purpose This paper aims to investigate the need for active boundary conditions during fire testing of structural elements, review existing studies on hybrid fire testing (HFT), a technique that would ensure updating of boundary conditions during a fire test, and propose a compensation scheme to mitigate instabilities in the hybrid testing procedure. Design/methodology/approach The paper focuses on structural steel columns and starts with a detailed literature review of steel column fire tests in the past few decades with varying axial and rotational end restraints. The review is followed with new results from comparative numerical analyses of structural steel columns with various end constraints. HFT is then discussed as a potential solution to be adapted for fire testing of structural elements. Challenges in contemporary HFT procedures are discussed, and application of stiffness updating approaches is demonstrated. Findings The reviewed studies indicate that axial and rotational restraints at the boundaries considerably influence the fire response of steel columns. Equivalent static spring technique for simulating effect of surrounding frame on an isolated column behavior does not depict accurate buckling and post-buckling response. Additionally, numerical models that simulate fire performance of a column situated in a full-frame do follow the trends observed in actual test results up until failure occurs, but these simulations do not necessarily capture post-failure performance accurately. HFT can be used to capture proper boundary conditions during testing of isolated elements, as well as correct failure modes. However, existing studies showed cases with instabilities during HFT. This paper demonstrates that a different stiffness updates calculated from the force-displacement response history of test specimen at elevated temperature can be used to resolve stability issues. Originality/value The paper has two contributions: it suggests that the provision of active boundary conditions is needed in structural fire testing, as equivalent static spring does not necessarily capture the effect of surrounding frame on an isolated element during a fire test, and it shows that force-displacement response history of test specimen during HFT can be used in the form of a stiffness update to ensure test stability.

Dynamic identification of axial force and boundary restraints in tie rods and cables with uncertainty quantification using Set Inversion Via Interval Analysis

A methodology is developed for the estimation of internal axial force and boundary restraints within in-service, prismatic axial force members of structural systems using interval arithmetic and contractor programming. The determination of the internal axial force and end restraints in tie rods and cables using vibration-based methods has been a long standing problem in the area of structural health monitoring and performance assessment. However, for structural members with low slenderness where the dynamics are significantly affected by the boundary conditions, few existing approaches allow for simultaneous identification of internal axial force and end restraints and none permit for quantifying the uncertainties in the parameter estimates due to measurement uncertainties. This paper proposes a new technique for approaching this challenging inverse problem that leverages the Set Inversion Via Interval Analysis algorithm to solve for the unknown axial forces and end restraints using natural frequency measurements. The framework developed offers the ability to completely enclose the feasible solutions to the parameter identification problem, given specified measurement uncertainties for the natural frequencies. This ability to propagate measurement uncertainty into the parameter space is critical towards quantifying the confidence in the individual parameter estimates to inform decision-making within structural health diagnosis and prognostication applications. The methodology is first verified with simulated data for a case with unknown rotational end restraints and then extended to a case with unknown translational and rotational end restraints. A laboratory experiment is then presented to demonstrate the application of the methodology to an axially loaded rod with progressively increased end restraint at one end.

Experimental investigation into the in-plane buckling and ultimate resistance of circular steel arches with elastic horizontal and rotational end restraints

This paper presents the experimental and simulation study of circular steel arches attached on the steel columns through the rigid connection, which are loaded to failure under static load. Local buckling of the top flange at the mid-span arch or the bottom flange at end of the arch occurs under the ultimate load. The deformation shape is approximately antisymmetrical with a 0.2 rise-to-span ratio and is symmetrical with a 0.1 rise-to-span ratio. A simplified model is proposed as an arch with elastic horizontal and rotational end restraints. The corresponding stiffness is determined as equivalent horizontal and rotational stiffness at each arch end under the ultimate load. Then a large amount of parameter analysis is conducted to investigate the influence of the elastic supported stiffness, load type, rise-to-span ratio, and slenderness on the ultimate resistance. The finding is that the reduction of the ultimate resistance for the arch with the horizontal and rotational elastic end restraints is maximal under the uniform full-span vertical load, minimal under the concentrated load. Finally, a design method of the steel arch with elastic horizontal and rotational end restraints is proposed taking both ultimate resistance and maximum horizontal displacement of the supports into account for engineering reference.

A simple formula for estimating the column ultimate load with effect of semi-rigid connections

Calculating the ultimate load of a framed column in sway structures involves, in the currently used design method, the calculation of the column effective length and utilizing the interaction formulae or tables. Therefore no allowance is made for the presence of semi-rigid connections in the currently used design method. In this paper, a new formula to calculate the column ultimate load is proposed using simple model to simulate the behavior of practical steel columns in sway structures. This formula takes into account the effect of semi-rigid connections, the column rotational end restraints and the ratio of the applied horizontal to vertical loads. Moreover, the proposed method has the advantage of allowing for the influence of the column end moments resulted from the presence of the externally applied loads. In order to verify the accuracy of the proposed formula, several examples are presented in which the column ultimate load have been calculated using the proposed formula and the well known computer program STAAD.Pro. Comparing the obtained results of the proposed method with those obtained from STAAD.Pro has shown good correlation and therefore the accuracy of the proposed method to predict the behavior of practical columns has been verified.

Multiple unstable equilibrium branches and non-linear dynamic buckling of shallow arches

An arch under a suddenly-applied load will oscillate about its equilibrium position. If the suddenly-applied load is sufficiently large, the oscillation may reach a position on the unstable equilibrium branch of the arch, triggering its dynamic buckling. In many cases, arches are supported by other structural members or by elastic foundations which provide elastic types of rotational restraints to the ends of the arch. When the rotational end restraints of an arch are not equal to each other, the in-plane non-linear equilibrium path of the arch may have multiple unstable branches, which will influence the dynamic buckling of the arch significantly. This paper investigates effects of multiple unstable equilibrium branches on the non-linear in-plane dynamic buckling of a shallow circular arch under a suddenly-applied central concentrated load. Two methods based on the energy approach are used to derive the dynamic buckling loads. It is found that the number and magnitude of dynamic buckling loads are influenced significantly by the number of unstable equilibrium branches, by the stiffness of the unequal rotational end restraints, and by the included angle and slenderness ratio of the arch.

In-plane stability of preloaded shallow arches against dynamic snap-through accounting for rotational end restraints

Because the non-linear equilibrium path of a shallow arch has unstable branches, when the arch is subjected to a suddenly-applied load, it may lose its stability by dynamic snap-through owing to excessive oscillations reaching its unstable branch. In practice, arches are usually subjected to a certain level of static loading before the sudden load is applied. The static load may have a profound influence on the dynamic snap-though instability of the arch. In addition, the rotational end restraints of an arch provided by adjacent structural members or/and elastic foundations have significant influences on the geometric non-linearity of the arch, and consequently they influence its stability against dynamic snap-through under a suddenly-applied load. An energy approach is used to derive the dynamic snap-through buckling loads of statically preloaded shallow circular arches having rotational end restraints under a suddenly-applied uniform radial load. It is found that the statically preloaded arches can be susceptible to dynamic snap-through buckling. It is also found that the dynamic buckling loads of the arches are influenced significantly by the magnitude and difference of the relative flexibility of the rotational end restraints. Arches having equal rotational end restraints may have an upper and a lower dynamic buckling loads, whereas arches having unequal rotational end restraints have only a unique dynamic buckling load.

Lateral–torsional elastic buckling of rotationally restrained arches with a thin-walled section under a central concentrated load

An arch often has elastic end restraints provided by the connected structures or elastic foundations. When the arch is subjected to an in-plane central concentrated load, the load produces combined non-uniform axial compressive and bending actions with complicated distributions along the arch length and these actions are significantly influenced by the stiffness of the end restraints. These combined axial compressive and bending actions increase with an increase of external loads and may reach the values, at which the arch suddenly deflects laterally and twists out of the plane of loading, and fails in a lateral–torsional buckling mode. Little research of the lateral–torsional buckling of such arches has been reported in the open literature. This paper derives the analytical solution for the elastic lateral–torsional buckling load of pin-ended circular arches with a thin-walled section and having in-plane elastic rotational end restraints under a central concentrated load, using the principle of stationary potential energy in conjunction with the Rayleigh–Ritz method. The analytical solution agrees with independent finite element results very well, which indicates that the analytical solution can provide accurate predictions for the lateral–torsional buckling loads of arches having in-plane rotational end restraints. The effects of the stiffness of rotational end restraints on the lateral–torsional buckling load are investigated. It is found that the change of the stiffness of rotational end restraints has significant effects on the lateral–torsional buckling resistance of arches.

Lateral-Torsional Buckling Analysis of Arches Having In-Plane Rotational End Restraints under Uniform Radial Loading

AbstractIn many cases, an arch may have in-plane elastic end restraints provided by the connected structures or elastic foundations, which may influence the elastic lateral-torsional buckling of the arch. However, little research of the lateral-torsional buckling of arches with elastic end restraints has been reported in the open literature. This paper analytically investigates the lateral-torsional buckling of pin-ended circular arches having in-plane elastic rotational end restraints under a uniform radial load. The analytical solutions for the prebuckling behavior of such an arch show that the uniform radial load produces combined axial compressive and bending actions in the arch and that the axial compressive force produced by the uniform radial load is approximately uniform along the arch axis. It is found that the stiffness of elastic rotational end restraints has significant effects on the magnitude and distribution of the axial compressive forces and bending moments. The axial compressive force de...

Stability behavior and free vibration of tapered columns with elastic end restraints using the DQM method

The stability behavior and free vibration of axially loaded tapered columns with rotational and/or translational end restraints are studied using the differential quadrature method (DQM). The governing differential equations are derived and transformed into a homogeneous system of algebraic equations using the DQM technique. The boundary conditions are discretized and substituted into the governing differential equations, then the problem is transformed into a two parameter eigenvalue problem, namely the critical load and the natural frequency. The solution of the eigenvalue problem yields the critical load for the static case (ω=0) and yields the natural frequencies for the dynamic case with a prescribed value of axial load (Po<Pcr). The obtained solutions were verified against those obtained from FEM and found in close agreement.