Rotational Restraint Stiffness Research Articles

Quantifying the effect of end support restraints on vibration serviceability of mass timber floor systems: Testing

Design of mass timber floor systems is commonly governed by vibration serviceability due to high stiffness-to-weight ratio and low inherent damping of timber. Research and design practice have shown that static deflection under a concentrated load and fundamental natural frequency can be effective and robust indicators for vibration performance of mass timber floors. These design parameters are normally calculated by assuming simply supported conditions in existing design methods. However, such an assumption deviates from actual floor supports, especially in platform-framed buildings, and in-situ end support restraints have been widely recognized as a significant factor affecting the vibration performance. The purpose of the study is to quantify the influence of end support restraints on vibration serviceability of mass timber floors in platform construction through a comprehensive experimental program and analytical treatment. This paper is the first part and focus specifically on the experimental work on cross-laminated timber (CLT) panels. In particular, extensive laboratory tests have been conducted on different CLT floor panels with various end support restraints induced by top loads, self-tapping screws and steel angle brackets. The fundamental natural frequency and mid-span deflection under a concentrated load were measured for each end support configuration. The rotational restraint stiffness was determined by comparing results of restrained supports to those of simple supports and represented as end fixity factors. The analysis of test results shows that the CLT floor-to-wall connection exhibited inherent non-linear behaviour and such characteristic was more significant for higher top loads. Compared with screws and brackets, the top loads dominated the partially restrained effect but such dominance gradually diminished for lower-level top loads. In addition, support wall thickness notably impacts the support restraint. It was then suggested that the clear span could be used to determine deflection and frequency in the design, but further investigation is needed.

Buckling and critical temperatures of restrained H-section steel columns in fire considering dynamic effect

In this paper, a parametric analysis on the buckling and critical temperatures of restrained H-section steel columns in fire, considering the dynamic effect, is presented. The developed finite element model of steel columns was verified by test results. Then, the buckling and critical temperatures of steel columns, obtained by static and explicit dynamic analyses, were compared. Moreover, the effects of rotational and axial restraint stiffness ratios, as well as load ratios, were studied. It was found that as the rotational restraint stiffness ratio increased, the buckling and critical temperatures both rose, and the rotational restraint stiffness ratio affected the critical temperature more than the buckling temperature. Besides, when the axial restraint stiffness ratio and load ratio rose, the buckling and critical temperatures dropped significantly. Finally, by combining the findings of the research and existing calculation formulas in literatures, new simplified formulas for buckling and critical temperatures were proposed, respectively. The prediction results of the proposed calculation formulas agreed well with the parametric analysis results. The proposed formulas were further verified by test data. According to the comparative results, the proposed formulas could well predict the buckling and critical temperatures of steel columns in fire.

Frequency Variation of Beam with an Internal Hinge of Rotational Restraint

The frequency variation problem of a classical beam with a rotationally restrained internal hinge arbitrarily located is studied intensively. The characteristic equations of the natural vibration are derived rigorously upon the corresponding boundary conditions and the compatibility conditions at the interconnecting point such that the effects of both the rotational restraint stiffness and the hinge position on the natural frequencies and mode shapes are described comprehensively. In the context, the critical locations of an internal hinge are investigated extensively to obtain the extreme values of a natural frequency of the beam system, and the associated requirements are explored in depth. It appears that the essential conditions for obtaining the critical points with regard to the maximum or minimum frequency come, respectively, from the same compatibility relation of the rotational flexibility in view of the opposite criteria. Numerical results are provided with two representative uniform beams to show the variations of their dynamic characteristics and to gain new insights into the transition conditions in the two distinct cases.

Numerical modeling and design approach on restrained Q690 steel beams in fire

This paper presents a comprehensive numerical analysis of high-strength Q690 steel beams in fire. A FE model is developed and validated against the full-scale experiments on restrained high-strength Q690 steel beams in fire presented in a previous study by the authors. The FE model adequately replicates the deflection, axial force, and actual failure modes identified experimentally. An extensive parametric investigation was performed to determine the variables affecting the behavior of restrained Q690 steel beams in fire. The analysis indicates that several factors, including load ratio, cross-section temperature distribution, rotational restraint stiffness, and span-to-depth ratio are critical for the fire resistance of restrained high-strength Q690 steel beams. Furthermore, additional numerical analysis for the ultimate bearing capacity of high-strength Q690 steel beams under bending was carried out to estimate the flexural-torsional buckling behavior of Q690 steel beams at elevated temperatures. Additionally, based on the results of the FE models, design curves for determining the flexural-torsional buckling strength of high-strength Q690 steel beams at high temperatures were proposed. The findings were compared to the design curves specified by several design standards.

Experiments and a reverse-curved compressive arch model for the progressive collapse resistance of reinforced concrete frames

The compressive arch action (CAA) is a critical force-transfer mechanism to mitigate the vulnerability of progressive collapse for reinforced concrete frames. To deeply analyze the mechanical behavior of concrete beam-column sub-assemblages under a middle-column-removal scenario, the progressive collapse tests and numerical simulations were accomplished, in which the stress distribution and the force-flow characteristic under CAA were revealed. To accurately evaluate the peak capacity of CAA, a mechanical model of a reverse-curved compressive arch (RCCA) is put forward. The theoretical formulas are derived on the basis of the model, considering both the bending deformation and the axial deformation of the arch. The theoretical calculation results (including the peak capacities and the corresponding displacements) agree well with the experimental results of 20 specimens. Moreover, a parametric analysis on CAA peak capacity is investigated considering the following factors: the axial and rotational restraint stiffness of end supports, the initial connection gap of test setups, the depth-to-span ratio of beams, the concrete strength and the reinforcement ratio of beams. The research results demonstrate that the relative axial (rotational) restraint stiffness should not be less than 1 and the initial connection gap should be less than 1.0 mm to ensure sufficient boundary constraints. According to the parametric analysis results, a simplified formula for CAA peak capacity is established, in which multiple influence coefficients are proposed for the formula. The proposed formula is practical and simple to apply because no iteration or derivation is required, and good calculation accuracy is also verified by 20 specimens. Based on the formulas, the CAA capacities of different column-spacing concrete frames designed with various depth-to-span ratios of beams are compared with the necessary gravity loads. The research results show that the larger the depth-to-span ratio, the greater the bearing capacity margin.

Influence of rotational restraints on response of H-section steel columns under fire

In this paper, a parametric investigation on the influence of rotational restraints on the response of H-section steel columns under fire was conducted through a validated numerical model. Nine rotational restraint stiffness ratios (βR) were considered, from 0.0001 to 10.0. Other considered parameters include the slenderness ratio, axial restraint stiffness ratio, and load ratio. The analysis results showed that there are three failure scenarios of the columns with βR increases from 0.0001 to 10.0. Specifically, in failure scenario I, the failures are all dynamic failures; in failure scenario II, the failures change from dynamic failure to quasi-static failure; and in failure scenario III, the failures are all quasi-static failures. The occurrence of the failure scenarios is related to the slenderness ratio, axial restraint stiffness ratio, and load ratio. The dynamic effects of the columns in failure scenario I decrease first and then increase with the increase of βR. The dynamic effects of the columns in failure scenario II decrease with the increase of βR until the failure type of the columns change to quasi-static failure. For the above failure scenarios, there is a critical value of βR (namely βR,cr), after reaching which the influence of βR on the response of the columns becomes negligible. The value of βR,cr is less than 2.0 in general, and depends on the slenderness ratio, axial restraint stiffness ratio, and load ratio. The failure scenarios and βR,cr values of the H-section columns are the same with box-section columns for most of the cases.

Buckling of partially-compressed laminated composite plates

The web buckling is one failure mode of web crippling for thin-walled fiber-reinforced plastic (FRP) composite beams subjected to transverse concentrated or distributed compression load from both the top and bottom bearing plates, and the web between the top and bottom flanges can be modeled as a plate with the rotationally-restrained (RR) and partially-compressed loaded edges and the simply-supported (SS) unloaded edges. Explicit solution and universal analytical charts/formulas for buckling analysis of such RRSS composite plates partially compressed along the transverse direction are obtained. The unique Heaviside function and trigonometric series expansion are used to express the partial edge load with effective load length in the governing equilibrium equation. The explicit solution is established for the orthotropic plates subjected to concentrated point compression load in the middle; while the generic buckling analytical curves are developed for infinitely long laminated composite plates transversely compressed partially at the middle of edges as well as near the corners. The accuracy of the present method is validated with available simple analytical solution, numerical finite element method and experimental data. The effect of rotational restraint stiffness, plate aspect ratio, the location and length of partial edge compression load on buckling behavior of laminated composite plates is investigated in the parametric study. A practical guideline for laminated composite plates with the elastic rotational restraints along the loaded edges and partially compressed along the transverse direction is proposed, and it can be further applied in discrete plate analysis to predict the web buckling strength (one failure mode in web crippling) of FRP composite structural shapes under transverse loading at the locations of load bearing or supports.

Analytical models for load capacities of variable thickness reinforced concrete slabs considering compressive membrane action and boundary effects

The lateral restraint eccentricity and rotational restraint stiffness significantly influence the compressive membrane action (CMA) in reinforced concrete slabs. However, these two factors have rarely been studied and no analytical model considering them has been established. In addition, most analytical methods for evaluation of CMA were derived for slabs with constant thickness, even though slabs with variable depth are quite common in practice. Therefore, this paper develops analytical methods to predict the load capacities of laterally restrained slabs considering the effects of lateral restraint eccentricity, rotational restraint stiffness, and slab depth variation. Based on the compatibility conditions and equilibrium of forces, the axial forces and bending moments can be derived by solving equations as long as the deflection at the load application point is provided. Then the corresponding load can be determined and the load capacity of the slab can be obtained by iterative calculations. The proposed methods are then validated by comparing to finite element analysis and four groups of test results, in which the effects of lateral restraint eccentricity, rotational restraint stiffness, and slab depth variation are explicitly reflected. Finally, comprehensive parametric analyses are conducted by using the proposed models. The interaction effects of the design parameters on the CMA are revealed for slabs with strong and weak rotational restraints, respectively.

Dynamic buckling of rotationally restrained FG porous arches reinforced with graphene nanoplatelets under a uniform step load

This paper presents a dynamic buckling analysis for a rotationally restrained functionally graded (FG) graphene nanoplatelets (GPLs) reinforced composite (FG-GPLRC) porous arch under a uniform step load where GPL nanofillers are uniformly dispersed while the porosity coefficient varies along the thickness direction of the arch. The effective material properties of the FG-GPLRC porous arch are determined by the volume fraction distribution of materials. Analytical solutions for the symmetric limit point dynamic buckling and anti-symmetric bifurcation dynamic buckling loads of rotationally restrained FG-GPLRC porous arches are derived by using an energy-based approach. Critical geometric parameters that determine the dynamic buckling mode switching behavior are also identified and discussed. Depending on the geometric parameters and the rotational restraint stiffness, the FG-GPLRC porous arch can buckle in either a symmetric limit point mode or an anti-symmetric bifurcation mode dynamically. It is also found that the dynamic buckling load of the arch can be considerably improved by adding a small amount of GPLs as reinforcing nanofillers. The influences of the porosity coefficients, GPL weight fractions, arch dimensions and geometries on the dynamic buckling behavior of rotationally restrained FG-GPLRC porous arches are comprehensively investigated through extensive parametric studies.

Compressive local buckling of integrated fluted-core sandwich composite panels

Fluted-core sandwich structures, consisting of fiber-reinforced plastics, are frequently utilized in the aerospace industry owing to their excellent strength-to-weight ratio. However, the thin-walled structures are prone to local buckling, which reduces their load-bearing capacity and eventually leads to material failure. The current study presents analytical models for predicting the local buckling of fluted-core sandwich panels subjected to uniform compressive load. The proposed method is based on a discrete plate analysis approach, which considers the face-sheets between two webs as rotationally-restrained plates, wherein the rotational restraint stiffness is determined based on the geometric configuration and material properties. Herein, two boundary conditions, i.e., simply-supported and clamped, along the loaded edges were studied. Several deflection functions were proposed based on the classical Rayleigh–Ritz method, satisfying the boundary conditions. The experimental tests and finite element simulations were carried out to analyze the local buckling behavior, rendering excellent consistency and demonstrating the reliability of the proposed method. The effects of geometric parameters on the local buckling behavior were systematically studied and the results revealed that the current analytical models with high computational efficiency are reliable, exhibiting promise of eventual success in the optimization of preliminary engineering designs.

Analytical Prediction for Nonlinear Buckling of Elastically Supported FG-GPLRC Arches under a Central Point Load.

In this paper, we present an analytical prediction for nonlinear buckling of elastically supported functionally graded graphene platelet reinforced composite (FG-GPLRC) arches with asymmetrically distributed graphene platelets (GPLs). The effective material properties of the FG-GPLRC arch are formulated by the modified Halpin–Tsai micromechanical model. By using the principle of virtual work, analytical solutions are derived for the limit point buckling and bifurcation buckling of the FG-GPLRC arch subjected to a central point load (CPL). Subsequently, the buckling mode switching phenomenon of the FG-GPLRC arch is presented and discussed. We found that the buckling modes of the FG-GPLRC arch are governed by the GPL distribution pattern, rotational restraint stiffness, and arch geometry. In addition, the number of limit points in the nonlinear equilibrium path of the FG-GPLRC arch under a CPL can be determined according to the bounds of successive inflexion points. The effects of GPL distribution patterns, weight fractions, and geometric configurations on the nonlinear buckling behavior of elastically supported FG-GPLRC arches are also comprehensively discussed.

Geometrically Nonlinear Dynamic Response of Perforated Plates by Modified Differential Quadrature Method

The paper presents a modified differential quadrature (MDQ) method to investigate the dynamic response of perforated plates with elastically restrained edges under uniaxial impact compressive load. The perforated plate is divided into several separate plate elements that can be connected by using penalty function method (PFM) to ensure continuity along the shared edges. The in-plane stress distribution of the plate under the mechanical edge loading is determined by the pre-buckling analysis. To analyze the effect of elastically restrained edges on the dynamic response of perforated plates, artificial springs imposed for the edges are considered in the governing equilibrium equations. Verification analysis is carried out to demonstrate the efficiency and accuracy of the proposed method by comparing the results obtained with those available in the literature. Finally, the various effects of initial imperfection, rotational restrained stiffness, hole size and location, and shear load, on the dynamic response of perforated plates are investigated. The results show that the dynamic buckling load of perforated plates is significantly influenced by the rotational restraint stiffness, hole size and shear load as well as the initial geometric imperfection, whereas the effect of hole location can be neglected in the analysis of dynamic buckling of plates. Additionally, the results predicted by the proposed method can correlate well with the available numerical results.

Effect of rotational restraint conditions on performance of steel columns in fire

This paper presents a parametric analysis about the effect of rotational restraint conditions on the dynamic performance of steel columns in fire. A fine finite element model of restrained steel columns, which adopts an explicit dynamic solver, was developed and validated by test data in literatures. Nine values of rotational restraint stiffness ratio (βR) from 0.0001 to 10.0 were considered. Other considered parameters include the axial restraint stiffness ratio, load ratio and column slenderness ratio. Based on the results, a quantitative criterion of identifying dynamic and quasi-static failures of steel columns in fire was proposed. It is found that βR has a significant effect on the performance of steel columns in fire. With the increase of βR, three failure scenarios about the failure types of steel columns in fire may appear, namely, all the failures are dynamic failures (Scenario I), failures turn from dynamic failure into quasi-static failure (Scenario II), and all the failures are quasi-static failures (Scenario III). In failure scenario I, there exists a βR with which the column shows the minimal dynamic effects. Besides, when βR reaches a critical value (namely βR,cr), the effect of further increasing the value of βR on the response of steel columns becomes slight for all the above scenarios. βR,cr is affected by the axial restraint stiffness ratio, load ratio and column slenderness ratio, and is generally less than 1.5. In fire-resistant design, the steel columns with a βR greater than βR,cr can be considered as rotationally fixed at both ends. Moreover, with the increase of βR, the buckling and critical temperatures of steel columns both increase, and the increase in critical temperature is significantly larger than that in buckling temperature.

A Practical Approach for Fire Resistance Design of Restrained High-Strength Q690 Steel Beam Considering Creep Effect

Most of the previous studies on restrained steel beam behaviour in fire conditions have neglected the creep effect due to the lack of applicable creep models. A finite element model (FEM) is determined in this study to investigate fire resistance and the behaviour of restrained high-strength (RHS) Q690 steel beams in fire considering the high-temperature Fields & Fields creep model. A comparison of the results obtained by the FEM with those from previous tests proved the validity of the FEM. A second FEM without a creep model is also established to study the influence of creep on the fire resistance of restrained steel beams. Results showed that creep has a serious effect on the behaviour of restrained steel beams in fire. Thus, ignoring creep will possibly lead to unsafe designs. Several parametric studies are carried out using the validated FEM with the aim to investigate the influencing factors on the fire response of RHS Q690 steel beams. Analysis shows that some of the investigated factors, such as heating rate, cross-section temperature distribution, rotational restraint stiffness and span-to-depth ratio, have been found crucial in the fire resistance of RHS Q690 steel beams. Furthermore, a simplified approach is presented for RHS Q690 steel beams based on the results of the FEM, including the creep effect, to calculate the moment capacity. This approach is also suitable for calculating the critical temperature of RHS Q690 steel beams.

Novel design method for reinforced concrete decks in composite girders considering compressive membrane action

The deck slab in composite box girder is laterally and rotationally restrained, and the compressive membrane action (CMA) improves the stiffness and capacity of the slab under the vertical load. However, a rational design method for the slab in composite box girder is still lacking. The purpose of this paper is to establish theoretical models to predict the behaviors of the laterally and rotationally restrained slab at ultimate and serviceability limit states. First, a nonlinear whole process mechanism analysis of a laterally and rotationally restrained deck slab is conducted. Compared with previous studies, this research considers additional factors such as the rotational restraint stiffness, the lateral restraint eccentricity, two lateral restraints and the thickness variation of the deck slab. Subsequently, key performance indicators for the ultimate and serviceability limit states are proposed and investigated by a comprehensive parametric study. The parametric analyses results indicate the appreciable impacts of the rotational restraint stiffness, lateral restraint eccentricity and thickness variation of the deck slab. Furthermore, simplified formulas of the key performance indicators are obtained by regression. Finally, based on the mechanism analysis and the parametric study, design methods for predicting the load capacity and the load corresponding to the control crack width of the deck slab are proposed and then validated by the finite element analysis and test results.

A Method to Estimate Dynamic Buckling Response of an Unstiffened Plate Elastically Restrained Along All Edges Under In-Plane Impact

Unstiffened plates in structures are usually welded or fastened to supporting members, providing rotational restraint stiffness to the plate. Previous studies have shown that neglect of rotational restraint stiffness at the edges of a plate in a structure can introduce deviations in the analysis of dynamic elastic buckling. In this study, the in-plane impact-induced dynamic elastic buckling responses of isotropic imperfect unstiffened plates with four elastically restrained edges are analytically investigated, based on the large-deflection theory of thin plate. The evolution of the peak deflection predicted by the proposed analytical method is found to be consistent with the responses available from the literature. Then the method is further used to estimate the deformation map of an unstiffened plate with four elastically restrained edges, and the effects of rotational restraint stiffness, initial geometric imperfection and shock duration on the dynamic buckling response of the plate are examined. The results show that the critical dynamic buckling load and the maximum deflection response of the plates are significantly influenced by the rotational restraint stiffness as well as the first-order initial geometric imperfection, and thus cannot be neglected in the analysis of dynamic buckling.

Elastic buckling behaviours of grid cylindrical-lattice shells under axial compression

Grid Cylindrical-Lattice Shells (GCLSs) have found their wide application in high-rise buildings and TV towers as the outer cylindrical frame of tube-in-tube structures subjected mainly to axial compression. This paper presents the elastic buckling behaviour of Grid Cylindrical-Lattice Shells (GCLSs) under axial compression. Because of a lack of lateral braces in the radial direction of the GCLS along its height, the GCLS tends to fail by two buckling modes under axial compression, namely global flexural buckling and circumferential multi-wave buckling. First, the fundamental three-story GCLS is investigated by using the energy method. The restraining action of the ring beam on the columns in the two buckling modes is explored separately, and this exploration leads to the radial rotational restraint stiffness, tangential rotational restraint stiffness and translational restraint stiffness of the ring beams, denoted by βr, βt and α, respectively; in addition, the corresponding elastic buckling loads of the three-story GCLS in the two buckling modes are separately derived. Then, based on the three key parameters, simple fitting formulas are developed to separately predict the elastic buckling loads of the multi-story GCLS in the two buckling modes. This outcome provides fundamentals for further establishing load-carrying capacity design of GCLSs under axial compression.

Dynamic progressive collapse of steel moment frames under different fire scenarios

Dynamic mechanisms for progressive collapse of steel moment-resisting frames are studied under different durations of a column removal due to different fire scenarios. Development process of natural fire and gas temperature are modeled and predicted by software fire dynamics simulator (FDS). The steel temperature is then predicted reliably using the method recommended by EN19993–1-2 incorporating the predicted gas temperature by FDS. Steel frames and columns exposed to fire are simulated by BEAM188 in ANSYS with APDL and then validated against experimental data obtained from other researchers. It can be found that the failure duration of a column is different under standard and natural fire conditions, similarly under different boundary conditions, namely, axial load, initial imperfection, axial restraint stiffness ratio and rotational restraint stiffness. Finally, a step-by-step analysis procedure is established using SAP2000 API with Excel VBA to determine structural response versus duration of a column failure. The decay trend of maximum displacement can be described quantitatively by a one-phase exponential decay function, which is drawn after extensive analysis of dynamic response for different structures with various spans, member sections, loads, levels of a building and total number of floors

A Simple Empirical Formula for Predicting the Ultimate Strength of Ship Plates with Elastically Restrained Edges in Axial Compression

An investigation is conducted on the static ultimate limit state assessment of ship hull plates with elastically restrained edges subjected to axial compression. Both material and geometric non-linearities were considered in finite element (FE) analysis. The initial geometric imperfection of the plate was considered, while the residual stress introduced by welding was not considered. The ultimate strength of simply supported ship hull plates compared well with the existing empirical formula to validate the correctness of the applied boundary conditions, initial imperfection and mesh size. The extensive FE calculations on the ultimate strength of ship hull plates with elastically restrained edges are presented. Then a new simple empirical formula for plate ultimate strength is developed, which includes the effect of the rotational restraint stiffness, rotational restraint stiffness, and aspect ratios. By applying the new formula and FE method to ship hull plates in real ships, a good coincidence of the results between these two methods is obtained, which indicates that the new formula can accurately predict the ultimate strength of ship hull plates with elastically restrained edges.

Structural performance of slabs in composite box girder considering compressive membrane action

In steel-concrete composite box girders, the concrete slabs are laterally and rotationally restrained by diaphragms, surrounding slabs, and the steel box girder through shear connectors, thus developing compressive membrane action under vertical loading. However, the compressive membrane action in composite box girder bridge decks has rarely been considered, despite its significant effect on the structural performance of the slab. In this study, transverse behavior of slabs in composite box girder considering compressive membrane action was investigated and a new calculation method for assessing the load capacity of slabs with different boundary conditions was proposed. Static loading tests on six composite box girder deck slabs were first carried out. Elaborate finite element models were established and further validated by test results. Furthermore, the load capacities of the slabs were compared between the test results and several traditional theoretical predictions. It is indicated that the compressive membrane action significantly enhanced the load capacities of deck slabs. Moreover, this study reveals that ignoring the effects of rotational restraint stiffness and lateral restraint location on the compressive membrane action may lead to significant error. Based on the test results and the numerical parametric study, a new calculation method for the load capacity of laterally and rotationally restrained slabs was proposed. Comparisons with both the numerical and experimental results demonstrated that the proposed method can provide accurate and reliable predictions.