• International journal of steel structures
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• v.18 no.5
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• pp.1497-1507
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• 2018

Most of previous studies on fire resistance of restrained steel beams neglected creep effect due to lack of suitable creep model. This paper presents a finite element model (FEM) for accessing the fire resistance of restrained high strength Q460 steel beams by taking high temperature Norton creep model of steel into consideration. The validation of the established model is verified by comparing the axial force and deflection of restrained beams obtained by finite element analysis with test results. In order to explore the creep effect on fire response of restrained Q460 steel beams, the thermal axial force and deflection of the beams are also analyzed excluding creep effect. Results from comparison infer that creep plays a crucial role in fire response of restrained steel beam and neglecting the effect of creep may lead to unsafe design. A set of parametric studies are accomplished by using the calibrated FEM to evaluate the governed factors influencing fire response of restrained Q460 steel beams. The parametric studies indicate that load level, rotational restraint stiffness, span-depth ratio, heating rate and temperature distribution pattern are key factors in determining fire resistance of restrained Q460 steel beam. A simplified design approach to determine the moment capacity of restrained Q460 steel beams is proposed based on the parametric studies by considering creep effect.

• Steel and Composite Structures
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• v.9 no.3
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• pp.191-208
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• 2009

Observations from experiments and real fire indicate that restrained steel beams have better fire-resistant capability than isolated beams. Due to the effects of restraints, a steel beam in fire condition can undergo very large deflections and the run away damage may be avoided. However disgusting damages may occur in the beam-to-column connections, which is considered to be mainly caused by the enormous axial tensile forces in steel beams resulted from temperature decreasing after fire dies out. Over the past ten years, the behaviour of restrained steel beams subjected to fire during heating has been experimentally and theoretically investigated in detail, and some simplified analytical approaches have been proposed. While the performance of restrained steel beams during cooling has not been so deeply studied. For the safety evaluation and repair of steel structures against fire, more detailed investigation on the behaviour of restrained steel beams subjected to fire during cooling is necessary. When the temperature decreases, the elastic modulus and yield strength of steel recover, and the contraction force in restrained steel beams will be produced. In this paper, an incremental method is proposed for analyzing the behaviour of restrained steel beams subjected to cooling. In each temperature decrement, the development of deformation and internal forces of a restrained beam is divided into four steps, in order to consider the effect of the recovery of the elastic modulus and strength of steel and the contraction force generated by temperature decrease in the beam respectively. At last, the proposed approach is validated by FE method.

• Steel and Composite Structures
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• v.8 no.1
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• pp.1-18
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• 2008

Observations from experiments and real fire indicate that restrained steel beams have better fire-resistant capability than isolated beams. Due to the effects of restraints, a steel beam in fire condition can undergo very large deflections and the run away damage may be avoided. In addition, axial forces will be induced with temperature increasing and play an important role on the behaviour of the restrained beam. The factors influencing the behavior of a restrained beam subjected to fire include the stiffness of axial and rotational restraints, the load type on the beam and the distribution of temperature in the cross-section of the beam, etc. In this paper, a simplified model is proposed to analyze the performance of restrained steel beams in fire condition. Based on an assumption of the deflection curve of the beam, the axial force, together with the strain and stress distributions in the beam, can be determined. By integrating the stress, the combined moment and force in the cross-section of the beam can be obtained. Then, through substituting the moment and axial force into the equilibrium equation, the behavior of the restrained beam in fire condition can be worked out. Furthermore, for the safety evaluation and repair after a fire, the behaviour of restrained beams during cooling should be understood. For a restrained beam experiencing very high temperatures, the strength of the steel will recover when temperature decreases, but the contraction force, which is produced by thermal contraction, will aggravate the tensile stresses in the beam. In this paper, the behaviour of the restrained beam in cooling phase is analyzed, and the effect of the contraction force is discussed.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2000.06a
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• pp.689-694
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• 2000

This paper presents a modeling method for the vibration analysis of cantilever beams with an elastically restrained root. Mass and stiffness matrices are derived explicitly by considering the elastically restrained root coupling effect between stretching and bending motion. Numerical results show that the two effects influence the vibration characteristics of rotating beams significantly. The results also present the magnitude of the elastic stiffness of the root to avoid the dynamic buckling. The method presented in this paper can be used to provide accurate predictions of the variations of natural frequencies of rotating beams with an elastically restrained root.

• Computers and Concrete
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• v.16 no.1
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• pp.67-97
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• 2015

This article presents a computing procedure developed to predict the torsional strength of axially restrained reinforced concrete beams. This computing procedure is based on a modification of the Variable Angle Truss Model to account for the influence of the longitudinal compressive stress state due to the axial restraint conditions provided by the connections of the beams to other structural elements. Theoretical predictions from the proposed model are compared with some experimental results available in the literature and also with some numerical results from a three-dimensional nonlinear finite element analysis. It is shown that the proposed computing procedure gives reliable predictions for the ultimate behaviour, namely the torsional strength, of axially restrained reinforced concrete beams under torsion.

• Steel and Composite Structures
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• v.1 no.4
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• pp.427-440
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• 2001

The growing use of unprotected or partially protected steelwork in buildings has caused a lively debate regarding the safety of this form of construction. A good deal of recent research has indicated that steel members have a substantial inherent ability to resist fire so that additional fire protection can be either reduced or eliminated completely. A performance based philosophy also extends the study into the effect of structural continuity and the performance of the whole structural totality. As part of the structural system, thermal expansion during the heating phase or contraction during the cooling phase in most beams is likely to be restrained by adjacent parts of the whole system or sub-frame assembly due to compartmentation. This has not been properly addressed before. This paper describes an experimental programme in which unprotected steel beams were tested under load while it is restrained between two columns and additional horizontal restraints with particular concern on the effect of catenary action in the beams when subjected to large deflection at very high temperature. This paper also presents a three-dimensional mathematical modelling, based on the finite element method, of the series of fire tests on the part-frame. The complete analysis starts with an evaluation of temperature distribution in the structure at various time levels. It is followed by a detail 3-D finite element analysis on its structural response as a result of the changing temperature distribution. The principal part of the analysis makes use of an existing finite element package FEAST. The effect of columns being fire-protected and the beam being axially restrained has been modelled adequately in terms of their thermal and structural responses. The consequence of the beam being restrained is that the axial force in the restrained beam starts as a compression, which increases gradually up to a point when the material has deteriorated to such a level that the beam deflects excessively. The axial compression force drops rapidly and changes into a tension force leading to a catenary action, which slows down the beam deflection from running away. Design engineers will be benefited with the consideration of the catenary action.

• Steel and Composite Structures
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• v.5 no.4
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• pp.305-326
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• 2005

The inelastic buckling behaviour of continuously restrained two and three-span continuous beams subjected to concentrated loads and uniformly distributed loads are studied in this paper. The restraint type considered in this paper is fully restrained against translation and elastic twist applied at the top flange. These types of restraints are most likely experienced in industrial structures, for example steel-concrete composite beams and half through girders. The buckling analysis of continuous beam consists of two parts, firstly the moment and shear distribution along the member are determined by employing force method and the information is then used for an out-of-plane buckling analysis. The finite element method is incorporated with so-called simplified and the polynomial pattern of residual stress. Owing to the inelastic response of the steel, both the in-plane and out-of-plane analysis, which is treated as being uncoupled, extend into the nonlinear range. This paper presents the results of inelastic lateral-torsional and lateral-distortional buckling load and finally conclusions are drawn regarding the web distortion.

• Structural Engineering and Mechanics
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• v.55 no.1
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• pp.1-27
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• 2015

This article presents a theoretical parametric analysis on the ultimate torsional behaviour of axially restrained reinforced concrete (RC) beams. This analysis is performed by using a computing procedure based on a modification of the Variable Angle Truss Model. This computing procedure was previously developed to account for the influence of the longitudinal compressive stress state due to the axial restraint conditions provided by the connections of the beams to other structural members. The presented parametric study aims to check the influence of some important variable studies, namely: torsional reinforcement ratio, compressive concrete strength and axial restraint level. From the results of this parametric study, nonlinear regression analyses are performed and some design charts are proposed. Such charts allow to correct the resistance torque of RC beams (rectangular sections with small height to width ratios) to account for the favorable influence of the axial restraint.

• Structural Engineering and Mechanics
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• v.4 no.1
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• pp.37-47
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• 1996

The paper considers the elastic distortional buckling of overhanging beams, which consist of an internal segment with a cantilevered segment continuous over an internal support. The beams were considered loaded by a concentrated load at the cantilever tip, and the beams were either partially restrained or laterally restrained over the internal support. An efficient line-type finite element developed previously by the author was modified to incorporate loading remote from the shear centre, as well as to allow for lateral buckling without distortion. Buckling loads were obtained for a range of geometry when the load was placed on the top flange, at the shear centre or on the bottom flange. Buckling mode shapes were also obtained, and conclusions drawn regarding the influence of distortion on the overall buckling load.

• Structural Engineering and Mechanics
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• v.16 no.2
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• pp.195-217
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• 2003

This paper, which is an extension of a previous work by Viola et al. (2002), deals with the dynamic stability of beams under a triangularly distributed sub-tangential forces when the effect of an elastically restrained end is taken into account. The sub-tangential forces can be realised by a combination of axial and tangential follower forces, that are conservative and non-conservative forces, respectively. The studied beams become unstable in the form of either flutter or divergence, depending on the degree of non-conservativeness of the distributed sub-tangential forces and the stiffness of the elastically restrained end. A non-conservative parameter ${\alpha}$ is introduced to provide all possible combinations of these forces. Problems of this kind are usually, at least in the first approximation, reduced to the analysis of beams according to the Bernoulli-Euler theory if shear deformability and rotational inertia are negligible. The equation governing the system may be derived from the extended form of Hamilton's principle. The stability maps will be obtained from the eigenvalue analysis in order to define the divergence and flutter domain. The passage from divergence to flutter is associated with a noticeable lowering of the critical load. A number of particular cases can be immediately recovered.