End Restraint Effects Research Articles

The effective length factor of columns in steel framed buildings with end restraint effects

The present article presents a practical method to determine the effective length factor of multi-storey frames based on a rigorous analytical approach applied to a basic model judiciously parameterized in terms of geometry and load arrangement. The adopted novel model considers the different interactions and effects of varying the boundary conditions at the ends of the columns and beams that are connected to the column under consideration. Subsequently, parametric studies are presented to evaluate the influence of the boundary conditions at the ends of the columns located above and below the column under consideration on the effective length factor value K. It is worth specifying that these variations are considered for the braced and unbraced frame cases. A comparative study revealed substantial differences between the values of the effective length factors thus obtained and those calculated using the method recommended by Eurocode 3 and CCM97. The proposed charts and relating to the effective length factor value K associated with the double cross model adopted by Eurocode 3 and CCM97, are of practical significance as they are very useful in the conception and design of such frames.

Multifrequency-based tension intelligent identification for cables with unknown end-restraints using a metaheuristic algorithm

Vibration frequency methods are widely used for cable tension identification; however, calibrated a priori knowledge of the flexural stiffness, axial stiffness and end-restraints of cables cannot always be obtained in advance, leading to the inabilities of these methods in the identification from theoretical or empirical formulations. In this study, a multifrequency-based intelligent multiparameter identification method is proposed for the tension identification of a cable equipped with a single sensor. The framework of the method establishes an optimization objective function with the theoretical and on-site frequencies based on a cable dynamic model considering the effects of inclination, sagging, flexural stiffness and end-restraints. The identification domain is obtained from empirical formulas and dimensionless characteristic parameters, and the model is solved using a finite difference method. An efficient metaheuristic algorithm is introduced as an engine for the simultaneous identification of the cable tension and the parameters supposed to be calibrated first. The accuracy and efficiency of the method are demonstrated through comparative studies consisting of algorithm tests, tension identification tests of laboratory rigid short cables and some stay-cables of an on-site bridge.

Influence of moment modification factor in lateral torsional buckling of doubly and monosymmetric sections

Simply supported doubly symmetric prismatic beam subjected to uniform moment along the unsupported length is the simplest and basic case in the buckling analysis. The closed form solution for the elastic critical moment was derived based on the assumptions of perfect beam with standard restraint conditions at both ends of the beam. A perfect beam will be having linear-elastic behaviour without any type of geometrical or material imperfections. The standard conditions of restraint at each end of the beam in basic case are restrained against lateral movement, restrained against rotation about the longitudinal axis and free to rotate on plan. But in reality, loading conditions and conditions of restraint will vary widely from the basic case. An equivalent uniform moment factor is typically applied to elastic critical moment to account for the effects of variable moment and end restraints along an unbraced girder length. The values are derived numerically and varies with specifications. In this article equivalent uniform moment factor for IS 800-2007 and Eurocode 3 provisions are compared. The study is also extended to monosymmetric I sections.

Structural behaviour and design of end-restrained square tubed-reinforced-concrete columns exposed to fire

Tubed-reinforced-concrete columns are gaining increasing usage in engineering practices, whereas research on their fire behaviour is still limited and mainly concentrated on unrestrained columns. Since end-restrained columns might behave differently in fire as compared to the unrestrained columns, the fire behaviour of square tubed-reinforced-concrete columns with end restraints was investigated via numerical modelling in this study. The axial and rotational restraints were modelled using linear springs and then validated against fire test results. The effects of end restraints on the failure mode, axial force and buckling length at failure and fire resistance, and on the evolution of deformations, internal force and bending moment during heating, of end-restrained columns were systematically analysed through parametric studies. The axially-restrained columns mostly undergo axial contraction only in fire; their typical failure mode is uncontrolled overall axial deformation, except for those columns of relatively high axial restraints. An increase in axial restraint leads to a nearly linear decrease of axial force at failure and a linear increase of fire resistance. For rotationally-restrained columns, the typical failure mode is also uncontrolled overall axial deformation. As the rotational restraint increases, the bucking length at failure decreases and the fire resistance increases, which is particularly significant when the rotational restraint is relatively low. A practical design method is proposed for the fire resistance design of square tubed-reinforced-concrete columns with either axial, or rotational, or combined axial and rotational restraints. Both constant and temperature-dependent end restraints have been adopted and the outcome is that they provide very similar results.

Numerical assessment of the effects of end-restraints and a pre-existing fissure on the interpretation of triaxial tests on stiff clays

Conventional laboratory triaxial tests apply axisymmetric boundary conditions to a cylindrical sample which has an axisymmetric geometry. For a homogeneous sample this implies that the deformed shape of the sample should maintain an axisymmetric geometry during the test. Consequently, the sample should deform in a barrelling mode and if slip planes develop they should define a cup and cone-like failure surface. However, in many triaxial tests such behaviour is not observed, especially as failure is approached when a planar slip surface develops. Such a deformation mode is not axisymmetric. One reason for this behaviour is that a fissure pre-exists in the sample. Employing hydro-mechanically coupled three-dimensional finite-element analyses, this paper investigates the influence of a single fissure in a triaxial sample of stiff clay on its behaviour throughout the test, focusing on the fissure position, orientation, strength and stiffness, in conjunction with the sample's end-restraints (rough or smooth). The effects are quantified in terms of the sample's overall stiffness and strength, indicating that the presence of a fissure can affect the very small strain stiffness, and that it has a significant effect on the strength of the sample, demonstrating that the conventional methods used to interpret laboratory tests may give unconservative results. The results also show a significant effect of the conditions at the top and bottom surfaces of the sample, where in particular the lateral restraint and rough ends introduce ‘bending’ in the sample.

DEM modeling of large-scale triaxial test of rock clasts considering realistic particle shapes and flexible membrane boundary

This paper presents a novel framework for the discrete modeling of the large-scale triaxial test of rock clasts, considering both the realistic particle shapes and veritable flexible boundary condition. First, real-shaped particle models for the tested rock clasts are precisely reconstructed using the close-range photogrammetry technique. The rubber membrane was modeled by a series of bonded particles. Then, the laboratory procedures of the triaxial test, i.e., sample preparation, isotropic compression, and shearing, are reproduced in the DEM simulations with consideration of the veritable confining boundary. To ensure more reliable numerical results, a systematic DEM calibration framework is established to determine the modeling parameters based on a series of calibration experiments, including tensile test, suspension test, clast-membrane sliding test, and large-scale triaxial test. Finally, the proposed method is applied to investigate the effects of confining pressure on the macro- and micro-mechanical behaviors of rock clasts. The presented works lay a foundation for further studies on revealing the mechanisms of the conventional triaxial test, e.g., the effect of end restraint and rubber membrane. Moreover, the proposed systematic framework for calibration of modeling parameters can be applied to precisely capture the real mechanical properties of various types of granular rock-like materials in DEM simulations.

Experimental determination of the shear strength of peat from standard undrained triaxial tests: correcting for the effects of end restraint

Conventional triaxial tests on peats are strongly criticised due to the very high shear strength parameters obtained from standard data elaboration, leading to unrealistic factors of safety when used in geotechnical design and assessment. Various operational approaches have been proposed in the literature to overcome this difficulty; however, they seem to lack consistent mechanical background. Some of the issues related to the shear strength evaluation of peats from triaxial tests come from the non-uniform stress and strain states developing in the samples well before failure is attained, due to end restraint effects. Undrained triaxial compression tests were performed on reconstituted peat to examine the influence of end restraint on the deviatoric stress, excess pore pressure and deviatoric strain response. Samples were tested with standard rough end platens and with modified platens to reduce the friction between the sample and bottom and top caps. Four different initial height-to-diameter ratios were examined, to reduce the consequences of rough end platens on the sample response. The results indicate that end restraint contributes dramatically to overestimating the shear strength of peat, due to the increase in both the calculated deviatoric stress and the measured excess pore pressure at the bottom of the sample. Suggestions are given to quantify the influence of end restraint in the interpretation of standard data, in an attempt to suggest viable procedures to determine more reliable effective and undrained shear strength parameters from standard triaxial tests.

Implication of end restraint in triaxial tests on the derivation of stress–dilatancy rule for soils having high compressibility

Constitutive models for soils are developed and validated against laboratory tests assuming these give representative information on the true material behaviour. However, data from standard laboratory tests reflect the sample response rather than the true material behaviour, due to nonuniformities in stresses and strains generated over the experimental test. The work examines the implications of end restraint on the definition of the stress–dilatancy rule of highly compressible soils with a finite element numerical approach. The numerical model replicates a reconstituted peat, typically characterized by a combination of high compressibility and high friction angle, which increases the severity of end restraint effects. Simulated results show that the global measurements from standard triaxial tests with rough end platens would not give the proper stress–dilatancy rule, if they were interpreted as the response of a single soil element at the constitutive level. Both overestimation and underestimation of the true dilatancy compared to the material response can be observed, depending on the deformation mode. To support the validity of the numerical results, experimental findings from drained triaxial tests on reconstituted peat are presented. Practical indications are given on how the standard interpretation of drained triaxial tests data on peats can be improved.

Torsional vibration analysis of nanorods with elastic torsional restraints using non‐local elasticity theory

In this work, torsional vibration of nanorods with torsional elastic boundary conditions is presented via non-local elasticity theory. The present model developed based on non-local elasticity theory gives the opportunity to interpret size effect. Two torsional elastic springs are attached to a nanorod at both ends. A mathematical transformation known as ‘Stoke transformation’ is utilised to work out the Fourier series for the nanorods with torsional restraints. A coefficient matrix including torsional coefficients is determined by using non-local boundary conditions. A comparison is performed to validate numerical simulations with those given in the literature and the results agree with each other exactly. The non-local effects of torsional end restraints on the free torsional vibration response are investigated for both deformable and rigid boundary conditions.

Assessment of I-Section Member LTB Resistances Considering Experimental Test Data and Practical Inelastic Buckling Design Calculations

The current AASHTO and AISC Specification equations characterizing the lateral-torsional buckling (LTB) resistance of steel I-section members are the same, with minor exceptions, and are based in large part on unified provisions calibrated to experimental data. This paper takes a fresh look at the correlation of the flexural strength predictions from these equations with a large experimental data set compiled from research worldwide. To account fully for the moment gradient and end restraint effects present in the physical tests, the study employs practical buckling calculations using inelastic stiffness reduction factors (SRFs) based on the design resistance equations. The study focuses on uniform bending tests as well as moment gradient tests in which the transverse loads are applied at braced locations. Reliability indices are estimated in the context of building design. It is shown that a proposed modified form of the current resistance equations provides a more uniform level of reliability, as a function of the LTB slenderness, consistent with the target intended in the AISC Specification. The paper also calls attention to the limited experimental data pertaining to the inelastic LTB resistances in certain cases. The paper concludes by providing additional recommendations for LTB strength calculations in routine design, including illustrative plots conveying the impact of the proposed changes.

Determination of microscopic parameters of quartz sand through tri-axial test using the discrete element method

A calibration procedure, in which coupled effects of microscopic parameters are considered, is proposed to determine the values of the microscopic parameters in the Discrete Element Method (DEM) for Fujian quartz sand. Laboratory tri-axial tests are conducted to be compared with the DEM simulations and the effects of end restraint in the laboratory tests are eliminated through a digital image measurement system. Sensitivities of the macroscopic behaviour of the specimen to the microscopic parameters are analyzed through DEM simulations. Four coupled effects of the microscopic parameters on the macroscopic behaviour are investigated through a graphic method and then considered in the calibration procedure.

Design according to Eurocode 6 in practice

AbstractFor the structural design of masonry according to Eurocode 6 with the associated German national annex, the simplified method and the further simplified calculation method in Annex A are available. These procedures provide tools that can be used in practice to design standard cases quickly and easily. One feature of the verification of masonry walls under compressive loading is that no bending moments in the walls have to be determined as part of the determination of section forces and moments since the verification of the load‐bearing capacity of the wall is based solely on the acting vertical force. The effects of floor end restraint and buckling are dealt with by simple equations. One new feature of verification according to Eurocode 6 is that the effect of partially supported floors on the load‐bearing capacity of the wall can be included directly. The code is compact and simple to use and the further simplified calculation method is predestined for verification by manual calculation.

Slip and stress of tensile armors in unbonded flexible pipes close to end fitting considering an exponentially decaying curvature distribution

An analytical model is given to investigate the end-fitting effect on slip and stress of tensile armors in unbonded flexible pipes under tension, torsion and varying bending in the absence of friction. An exponentially decaying curvature distribution is assumed to represent controlled curvature over the end region. The deviation from the initial helical angle is taken to describe the armor wire path as the pipe is stretched, twisted and bent, which is determined by minimization of the strain energy functional using the Euler equation. The obtained simultaneous differential equations are numerically solved by transforming them into a boundary value problem. An analytical solution is found by neglecting the twisting rotation of the wire cross-section. The developed model is validated with a finite element simulation and geodesic based analytical expressions for constant curvature and with a current numerical model for varying curvature. Reasonable correlations are observed between the model predictions and the results from other methods. The validated model is then applied to typical flexible pipe designs to find the level and location of the greatest increases in stress. The results show that the end restraint could cause a significant stress increase in the armor wire at the end fitting vicinity and the critical location is close to the extreme fiber position on the compressive side of the pipe for typical cases. The effect of end restraint on layer bending stiffness is also evaluated.

Centrifuge Modeling of End-Restraint Effects in Energy Foundations

AbstractThis study presents the results from physical modeling experiments on centrifuge-scale energy foundations in dry sand and unsaturated silt layers. These experiments were performed to characterize end restraint effects on soil-structure interaction for energy foundations in different soils and include tests on foundations with semifloating and end-bearing toe boundary conditions and free-expansion and restrained-expansion head boundary conditions. Two scale-model energy foundations having different lengths were constructed from reinforced concrete to simulate end-bearing and semifloating conditions in soil layers having the same thickness. The foundations include embedded thermocouples and strain gauges, which were calibrated under applied mechanical loads and nonisothermal conditions before testing. The variables measured during the experiments include axial strain and temperature distributions in the foundation, temperature, and volumetric water content measurements in the soil, vertical displace...

Modeling of different bracing configurations in multi-storey concentrically braced frames using a fiber-beam based approach

This study presents a modeling approach for concentrically braced frames to be used in multi-storey buildings. The model for an inelastic beam-column brace consists of two inelastic force-based beam-column elements, each of which having five integration points and a discretized fiber section. The hysteretic response of such elements can be derived by integration of uniaxial stress-strain relations.To capture the effects of gusset end restraint, in addition to the two inelastic beam-column elements, this study uses an additional inelastic force-based beam-column element of length 2t (where t is the thickness of the gusset plate) at each end of the brace. This study presents the correlation of the axial force–axial displacement and the axial force–lateral displacement responses obtained from the brace model with the available experimental results.Based on the comparison of numerical hysteretic responses with the experimental results, it can be concluded that the brace model which includes two additional force-based beam-column elements at the ends of the brace can capture the hysteretic responses of axial force–axial displacement and axial force–lateral displacement more accurately. Finally, this study sets the limits of slenderness and the width-to-thickness ratio in which inelastic beam-column brace model can predict the hysteretic responses of a brace member with adequately accuracy.

Blast response and failure analysis of pile foundations subjected to surface explosion

This paper presents the response of pile foundations to ground shocks induced by surface explosion using fully coupled and non-linear dynamic computer simulation techniques together with different material models for the explosive, air, soil and pile. It uses the Arbitrary Lagrange Euler coupling formulation with proper state material parameters and equations. Blast wave propagation in soil, horizontal pile deformation and pile damage are presented to facilitate failure evaluation of piles. Effects of end restraint of pile head and the number and spacing of piles within a group on their blast response and potential failure are investigated. The techniques developed and applied in this paper and its findings provide valuable information on the blast response and failure evaluation of piles and will provide guidance in their future analysis and design.

Comments and discussion: “Elastic buckling of columns with end restraint effects”

Comments and discussion: “Elastic buckling of columns with end restraint effects”

Axially Restrained Beam-Column with Initial Imperfections and Nonlinear End Connections Subject to High Temperatures

A nonlinear formulation capable of accurately and efficiently predicting the nonlinear response of an axially restrained prismatic beam-column with initial geometric imperfections (i.e., initial curvature and connection eccentricities) and nonlinear temperature-dependent end connections under elevated temperatures is presented. The proposed model includes the effects of (1) an arbitrary thermal gradient along both the span and cross section of the member and (2) the nonlinear behavior and any stiffness and strength degradations of the beam-column material. However, high-temperature creep and shear deflection effects are not taken into consideration. The proposed model is generic allowing the implementation of any thermal regime and the use of any stress-strain-temperature curves of the beam-column material. The stress-strain-temperature curves of the material can be either theoretical or experimental providing a more realistic approach to the structural response of beam-columns subjected to different fire conditions. Two comprehensive examples are presented and discussed in detail showing the effectiveness and accuracy of the proposed iterative method on the nonlinear large-deflection behavior of slender prismatic beam-columns under high temperatures, including the combined effects of initial imperfections, end restraints, and nonlinear end connections.

Response to discussion on “Elastic buckling of columns with end restraint effects”

Response Paper to S.E. Djellab's Comments and discussion concerning: R. Adman,M. Saidani, “Elastic buckling of columnswith end restraint effects”, Journal of Constructional Steel Research 87 (2013) 1–5. In what follows is a response to the points of substance made in the discussion paper. The discusser raised issues relating to the rigour and accuracy of the results reported in the original paper. The response below has been presented following the same format adopted by the discusser.

Elastic buckling of columns with end restraint effects

It is a well established fact that the behaviour of columns as part of a structure is affected by the end restraints. The main aim of the current study is to develop a criterion of stability capable of predicting an impending failure by elastic buckling of a column of a structure. The rigidities at the ends of a column element are modelled using rotational and translational springs, which have been considered by taking into account their coupling effects. The role of the springs is to model the nodal restraints of any column of a given structure. This formulation offers significant practical advantages in the elastic buckling analysis of such structures. This approach is performed through a relationship to several parameters, such as the non-dimensional rotational and translational restraint indices and the effective length factor K. The approach was applied in analysing the elastic buckling of a number of structures and good results were obtained, thus justifying its reliability. In determining the effective length factor K, a marked difference was noted between the results obtained using the Eurocode approach and that proposed by the current study, particularly in the case of non-braced structures.