Supported End Conditions Research Articles

Crack identification in elastically restrained vibrating rods

In this paper we consider the problem of identifying an open crack in a longitudinally vibrating rod with smooth variable profile by minimal eigenfrequency data. The crack is assumed to be open during vibration and it is modelled by an elastic spring acting along the rod axis. Most, if not all, the results available in the literature for this inverse problem refer to ideal end conditions, that is the rod is either under free or supported end conditions. As an example of almost optimal result, it is known that the knowledge of the fundamental (positive) natural frequency of the rod under free-free and cantilever end conditions allows for the unique determination of the crack, without any restriction on the damage severity. In this paper we show that the analysis of the analogous crack identification problem for rods under elastically restrained end conditions leads to different results and that, in general, the knowledge of the fundamental frequency belonging to two spectra associated to different end conditions is not sufficient for the uniqueness of the solution. The method we used to solve the inverse problem is of constructive type and it is based on general properties of the eigenfrequencies as functions of the position and severity of the crack. The identification procedure has been tested numerically on rods under various damage scenarios. Numerical results agree well with the theory, even in presence of noisy input data.

Fire response of restrained composite columns made with concrete filled hollow sections under different end-support conditions

The fire behaviour of composite columns made with concrete filled hollow sections under different structural boundary conditions was experimentally investigated and their results are presented and discussed in this paper. The main objectives of this research were therefore to investigate the influence of the section geometry, slenderness, section factor, boundary conditions, and stiffness of the surrounding structure to the thermal elongation of the columns on the structural performance of such columns exposed to fire. The critical time (fire resistance), the failure temperature distribution and the respective failure modes of the columns were then assessed. These experimental results were still compared with predictions from the currently European design rules (EN 1994-1-2:2005) in order to observe how unsafe they might be. Finally, results of this research study showed that circular composite tubular columns presented an enhanced fire performance, comparing to other sections (square and rectangular sections for instance). As the difference between the principal moments of inertia for the cross-section of a column increases, the effect of the boundary conditions in their fire resistance increases.

Load configuration, geometry and web reinforcement effects on failure modes of sandwich deep beams

In this article, an experimental investigation is carried out to understand the behaviour of sandwich deep beams. The shear strength, load transfer mechanism, integral action and failure modes are reported. Primary variables of this study include distribution of web reinforcement, beam geometry and type of loading. Twelve full-scale beams were tested with simply supported end conditions classifying them into two groups. Each series consists of six beams with rectangular and I-shaped cross-sectional geometry. Two types of loading, i.e. concentrated and uniformly distributed loads, were adopted to understand the influence of distribution of horizontal reinforcement along the depth of beam in the web. Using both types of cross-sections, two beams were tested under uniformly distributed loads, and all the remaining beams were subjected to concentrated load with two different shear span-to-depth ratios (a/d). Observations of the study with respect to strength and serviceability include initial cracking load, ultimate load, crack width, transverse deflection and out-of-plane deformations. The test results indicated that confined or unconfined I-beams in the web failed due to punching shear under concentrated or distributed loads. The modes of failure of rectangular beams with web confinement were diagonal splitting and shear-compression, whereas the unconfined beams failed in flexure. Shear strength expression for conventional deep beams is refined for sandwich beams, and the predicted shear strength agrees well with the experimental results.

Low-velocity impact response of functionally graded doubly curved panels with Winkler–Pasternak elastic foundation: An analytical approach

In this paper a nonlinear investigation is presented for impact response of functionally graded material (FGM) doubly curved panels which are resting on Winkler–Pasternak elastic foundation with simply supported end condition. The governing equations of the doubly curved panels are based on a Reddy’s higher order shear deformation theory with the von Kármán-type of kinematic nonlinearity. Moreover, the effect of heat conduction is considered and material properties of panels are assumed to be temperature dependent and vary along the thickness according to Mori–Tanaka micromechanical model. In addition, by using a modified Hertz’s contact law, the influence of material properties of the substrate layers on impact response is taken into account. The equations of motion are solved via an analytical procedure and a parametric polynomial solution is extracted. The influences of materials property gradient, initial velocity of the impactor, curvatures ratio, aspect ratio and the effect of elastic foundations on impact response of panels are discussed.

Member capacity of bolted steel angles in compression – Influence of realistic end supports

For compression members with primarily axial forces, steel angles with bolted connections are often used. Due to the eccentric connection on only one leg additional bending moments are acting on the member, leading to a complex load carrying behaviour with flexural and/or lateral torsional buckling phenomena. The paper mainly focuses on the presentation of the influence of realistic end support conditions at the gusset plates on the member capacity in compression.In the first part of the paper, about 300 existing test results for steel angles in compression with bolted connections are prepared in such a way that they can be compared with different design rules. This was achieved through the common representation of all test results as a function of the normalized slenderness λ¯v about the weak axis.In the second part, systematic numerical calculations are conducted for limit cases of the stiffness of the boundary condition of the load-introducing gusset plates (rigid/hinged). The results are also presented in dependency of the relative slenderness λ¯v. The consolidation of all tests and numerical results shows (besides the well-known difference between one-bolt and two-bolt connections) a very high influence of the end support of the gusset plate.

Nonlinear low-velocity impact response of functionally graded plate with nonlinear three-parameter elastic foundation in thermal field

Abstract In this paper a nonlinear investigation is presented for impact response of functionally graded material plates which are resting on a nonlinear three-parameter elastic foundation with simply supported end condition. In foundation, the influence of linear Winkler and Pasternak parameter is taken into account. The governing equations of the plate are derived based on a Reddy's higher order shear deformation theory with the von Karman-type of kinematic nonlinearity. The effect of thermal field is considered and material properties of the plates are assumed to be temperature dependent, as well and vary along the thickness according to Voigt model. In addition, by using a modified Hertz's contact law, the influence of material properties of the substrate layers on impact response is considered. The equations of motion are solved via an analytical procedure. The effects of deformation theories as well as thermal field, foundation parameters and impactor's parameters are discussed in results and discussion section.

Buckling resistance of axially loaded cold-formed steel columns

This paper reports an experimental campaign on the buckling behaviour of compressed single and built-up cold-formed steel columns. Four types of cross-sections were tested, namely, one single, one open built-up and two closed built-up, considering two end-support conditions, pin-ended and fix-ended. The obtained results were compared with the design predictions of EN1993-1-3:2004 and AISI S100-07. For pin-ended lipped channel columns the design predictions are in good agreement with the experimental results, however for the fix-ended columns the predictions may be conservative. For built-up columns was found that increasing the number of profiles may lead to unsafe design predictions.

Comparative Study of Linear and Geometric Nonlinear Load-Deflection Behavior of Flexural Steel Members

Background/Objectives: The proper analysis of beam is important for understanding the actual behavior and economical use of sections. The prime objective of this research paper is the comparative study of linear and geometric nonlinear load-deflection behavior of beams under vertical load. Methods/Statistical Analysis: In geometrically linear analysis, the equations of equilibrium are formulated before the deformation state and are not updated with the deformation, while in geometrical nonlinear analysis updated stiffness matrix is used at each load increment. In this study a three noded steel beam is formulated for linear and nonlinear analysis. The incremental central point load is applied to the beam. Linear and geometrically nonlinear deflection is computed for the beams. Three beams of the same length were taken for analysis, but having different thickness&support conditions. The linear and geometrical nonlinear load-deflection behavior is studied using STAAD PRO. The linear deflection is also computed by developing a finite element based code using MATLAB. Findings: The results obtained after the analysis of beams deformations are quite interesting. The percentage variation of linear and geometrical nonlinear deflection is very high for beam with lesser thickness. It was found that the support conditions also affect variation of deflection for the linear and nonlinear cases. The variation between linear and geometrical nonlinear deflection of beam is negligible when the ends are fixed. But for the same beam with simply supported end conditions the defections were found having variation up to 37 percentages. Geometrical nonlinearity is more when the load is very high and section is thin. At the initial stages of loading behavior of beam is linear only and it behaves nonlinear when we go for higher loads. Application/Improvements: Linear analysis is only an approximation, so for understanding the actual behavior of the structure and for the economical/optimized usage of sections it is suggested to go for nonlinear analysis.

Dynamic Buckling of Thermo-Electro-Mechanically Loaded FG-CNTRC Beams

Functionally graded carbon nanotube reinforced nanocomposites have drawn great attention in both research and engineering communities. The weak interfacial bonding between carbon nanotubes and the matrix, which traditionally hinders the application of carbon nanotube reinforced nanocomposites, can be remarkably improved through the graded distribution of carbon nanotubes in the matrix. Within the framework of classical beam theory, this paper investigates the dynamic buckling behavior of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes and integrated with two surface bonded piezoelectric layers. The governing equations of the beam subjected to an applied voltage, a uniform temperature and an axial periodic force are derived by applying Hamilton's principle. Numerical results are presented for beams with different distribution patterns and volume fractions of carbon nanotubes and end support conditions. The influences of the beam geometry, temperature change, applied voltage, static axial force component, boundary condition, carbon nanotube volume fraction and its distribution on the unstable regions of FG-CNTRC piezoelectric beams are discussed in detail.

EXPANDED POLYSTYRENE FIBRED LIGHTWEIGHT CONCRETE (EPSF-LWC) AS A LOAD BEARING WALL PANEL

This study was conducted to determine the optimum mix proportion of lightweight concrete (LWC) containing expanded polystyrene (EPS) and steel fiber which is designated as Expanded Polystyrene Fibred Lightweight Concrete (EPSF-LWC) for load bearing wall application. In order to produce LWC, EPS beads were chosen as lightweight aggregate because it gives advantages in term of energy absorbing capacity which suitable for structure that would be exposed to impact like shear wall. However, EPS beads possess zero strength. Therefore, steel fibre was added to improve LWC strength and also to reduce occurrence of micro and macro crack. In the mix design method, the percentage of EPS beads adding to the mix are differ while the percentage of steel fibre is same. The result showed optimum mix design was the one that contained 30% of EPS and 0.5 % of steel fibre and is designated as M8. The compressive strength EPSF-LWC of mix proportion designated as M8 is 19.51 MPa with density 1939 kg/m3. It is greater than 17 MPa as the requirement for structure component application that stated in the BS8110. Hence, reinforced and unreinforced EPS-LWC wall panels were constructed to determine the maximum loading that wall can sustain and deflection profile EPSF-LWC wall panel for the loaded to failure. The wall was set up under pinned-fixed end support condition. The sample was modelled using finite element analysis (FEA) for validation with experimental programme. The maximum loading capacity was found to be 908.20 kN and 853.40 kN for each reinforced (WR5) and unreinforced (WUR5) of EPSF-LWC wall panel. These loading were 31% to 35% less than finite element analysis. However, WR5 and WUR5 EPSF-LWC wall panel was deformed in single curvature profile for both experimental and FEA. Maximum deflection for WR5and WUR5 of EPSF-LWC recorded is 10.27 mm and 12.95 mm occurred at 0.7 heights (H) of wall panel. According to Euler buckling load theory, the location of maximum lateral displacement of wall panel sample is influenced by the type of fixity at end support of the sample.

Nonlinear analysis of functionally graded fiber reinforced composite laminated beams in hygrothermal environments, Part II: Numerical results

In this part, the extensive parametric studies performed are reported and numerical results are presented for the nonlinear vibration, nonlinear bending and thermal postbuckling of uniformly distributed and functionally graded fiber reinforced cross-ply and angle-ply laminated beams resting on Pasternak elastic foundations under different sets of hygrothermal environmental conditions. The numerical results reveal that a functionally graded reinforcement has a significant effect on the nonlinear vibration characteristics, nonlinear bending behaviors, and thermal postbuckling behaviors of fiber reinforced composite (FRC) laminated beams. The results show that the temperature/moisture variation has a moderately effect on the natural frequencies of the FRC laminated beam, but only has a small effect on the nonlinear to linear frequency ratios of the same beam. In contrast, it has a significant effect on the nonlinear bending load–deflection curves of the FRC laminated beam. The results confirm that the thermal postbuckling equilibrium path of mid-plane unsymmetric FG-FRC laminated beams with immovable simply supported end conditions is no longer the bifurcation type.

Size-dependent resonance and buckling behavior of nanoplates with high-order surface stress effects

This work presents a theoretical study of the resonance frequency and buckling load of nanoplates with high-order surface stress model. A classical thin plate theory based on Kirchhoff–Love assumption is implemented with surface effects. Circular and rectangular nanoplates with simply supported end conditions are exemplified. The size-dependent solutions are compared with the simplified solutions based on simple surface stress model, and also on the classical theory of elasticity. We aim to explore the scope of applicability so that the modified continuum mechanics model could serve as a refined approach in the prediction of mechanical behavior of nanoplates.

Cold-formed steel columns made with open cross-sections subjected to fire

An experimental study on the fire behaviour of cold-formed steel lipped channel (C) and built-up I (2C) slender columns with restrained thermal elongation is presented. The studied parameters were the stiffness of the surrounding structure, type of cross-section, end support conditions and initial applied load level on the columns. The results showed that increasing the stiffness of the surrounding structure and initial applied load level for the semi-rigid support conditions and both cross-sections, lead to a significant reduction of the critical temperature whereas for the pin-ended support conditions the reduction is supposed to be smaller.

Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories

This paper investigates the linear free vibration of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs). Two types of CNT reinforced beams, namely uniformly distributed CNT reinforced (UD-CNT) beams and functionally graded CNT reinforced (FG-CNT) beams, are considered. It is assumed that the SWCNTs are aligned along the beam axial direction and the distribution of the SWCNTs may vary through the thickness of the beam. The virtual strain and kinetic energies of the FG-CNT composite beam are obtained using the classic variational method of Hamilton’s principle and then solved by the p-Ritz method. Vibration frequency parameters for the FG-CNT beams based on the first order and third order beam theories are presented and the effects of CNT filler volume fraction, distribution, beam span to depth ratio and end support conditions on the free vibration characteristics of the beams are discussed. Comparison studies for UD-CNT and FG-CNT beams based on the first order and the third order beam theories are also performed and the differences in vibration frequencies between these two theories are highlighted.

Distortional–global interaction buckling of stainless steel C-beams: Part I — Experimental investigation

This paper presents experimental data for cold-formed stainless steel lipped channel beams featuring distortional–global interaction buckling. Three stainless steel alloys were employed in the tests, namely austenitic S30401, ferritic S44330 and lean duplex S32101. Coupon tests were carried out to determine the mechanical properties of the virgin sheet material as well as the work-hardened corner material of the press-braked sections. Geometric imperfections were measured for each specimen prior to testing. A total of 32 specimens were tested under four-point bending, six of which were tested with lateral bracing in order to determine the section capacity. The remaining 26 specimens, with spans ranging from 1.8m to 4.0m, were tested in a specially devised test set-up. The set-up featured a dual-actuator gravity simulator, flexible load-applying frames and idealized end support conditions, all of which contributed to minimizing uncertainties in the tests, thereby affording benchmark tests for analytical and numerical studies. Interaction buckling was observed in all tests of unbraced members.

NUMERICAL ANALYSIS ON NONLINEAR FREE VIBRATION OF CARBON NANOTUBE REINFORCED COMPOSITE BEAMS

This paper investigates the free vibration of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs). The distribution of the SWCNTs may vary through the thickness of a beam and are aligned along the beam axial direction. The virtual strain and kinetic energies of the carbon nanotube (CNT) composite beam are obtained using the classical variational method of Hamilton's principle, and the geometric nonlinearity of von Kármán sense is also included. The eigenvalue equation for free vibration of the beam is derived by the p-Ritz method. Vibration frequency parameters for the uniformly distributed (UD) and functionally graded (FG) CNT beams based on the first-order and third-order beam theories are presented and the effects of CNT filler volume fraction, distribution, beam length to depth ratio and end support conditions on the nonlinear free vibration characteristics of the beams are discussed. Comparison studies for UD-CNT and FG-CNT beams based on the first-order and the third-order beam theories are also performed and the differences in vibration frequencies and the nonlinear to linear frequency ratios between these two theories are highlighted.

Modeling and Analysis of Single-Plate Shear Connections under Column Loss

AbstractThis paper presents a computational assessment of the behavior of single-plate shear (shear tab) connections in gravity frames under column-loss scenarios. Two-span beam assemblies without floor slabs are considered under push-down loading of the unsupported center column. Both detailed and reduced modeling approaches are used in the computational assessment, and comparisons with experimental data are presented to establish confidence in the models. The models are used to investigate the influence of factors such as end support conditions, span length, connection strength, and postultimate connection behavior on the collapse resistance of gravity framing systems. Rotational capacities of single-plate shear connections under column-loss scenarios are, in some cases, less than half the values based on seismic test data, owing to the axial extension imposed on the connections in addition to rotation.

On the Direct Strength Method (DSM) design of cold-formed steel columns against distortional failure

This work reports the results of a numerical investigation on the influence of the cross-section geometry and end support conditions on the post-buckling behaviour and Direct Strength Method (DSM) design of cold-formed steel columns buckling and failing in distortional modes. The columns analysed exhibit (i) four end support conditions, (ii) different cross-section shapes, dimensions and lengths, and (iii) several yield stresses. These features were carefully selected in order to ensure, as much as possible, that all columns (i) buckle and fail in “pure” distortional modes and (ii) cover a wide (distortional) slenderness range. The post-buckling equilibrium paths and ultimate loads presented and discussed were obtained through Ansys elastic and elastic-plastic shell finite element analyses. Moreover, the ultimate strength data acquired are used to show that, regardless of the column geometry, the current DSM distortional design curve is not able to predict adequately (safely and accurately) the ultimate loads of columns with other than fixed end supports. The paper also includes a preliminary assessment/proposal of the modifications that must be included into this DSM design curve in order to overcome the above limitation.

Exact frequency equations of free vibration of exponentially functionally graded beams

Free vibration of axially inhomogeneous beams is analyzed. For exponentially graded beams with various end conditions, characteristic equations are derived in closed form. These characteristic or frequency equations can analytically reduce to the classical forms of Euler–Bernoulli beams if the gradient index disappears. The gradient has a strong influence on the frequency spectrum, and the natural frequencies noticeably depend on the variation of the gradient parameter and end support conditions. For certain beams with exponential gradients, there exists a critical frequency depending on the gradient parameter. Vibration can be only excited by propagating waves with frequencies in excess of the critical frequency, and otherwise vibration is prohibited for pseudo-frequencies lower than the critical frequency. For some gradient index with small change, the natural frequencies have an abrupt jump when across its critical frequencies. Obtained results can serve as a benchmark for other numerical procedures for analyzing transverse vibration of axially functionally graded beams. The minimal natural frequency can be sought for certain gradient index, and this helps engineers to optimally design vibrating nonhomogeneous beam structures. Obtained results also apply to free vibration of nonuniform beams with constant thickness and exponentially decaying width.

Experimental investigation on flexural behaviour of HSS stud connected steel-concrete composite girders

In this paper, experimental investigations on high strength steel (HSS) stud connected steel-concrete composite (SCC) girders to understand the effect of shear connector density on their flexural behaviour is presented. SCC girder specimens were designed for three different shear capacities (100%, 85%, and 70%), by varying the number of stud connectors in the shear span. Three SCC girder specimens were tested under monotonic/quasi-static loading, while three similar girder specimens were subjected to non-reversal cyclic loading under simply supported end conditions. Details of casting the specimens, experimental set-up, and method of testing, instrumentation for the measurement of deflection, interface-slip and strain are discussed. It is found that SCC girder specimen designed for full shear capacity exhibits interface slip for loads beyond 25% of the ultimate load capacity. Specimens with lesser degree of shear connection show lower values of load at initiation of slip. Very good ductility is exhibited by all the HSS stud connected SCC girder specimens. It is observed that the ultimate moment of resistance as well as ductility gets reduced for HSS stud connected SCC girder with reduction in stud shear connector density. Efficiency factor indicating the effectiveness of high strength stud connectors in resisting interface forces is estimated to be 0.8 from the analysis. Failure mode is primarily flexure with fracturing of stud connectors and characterised by flexural cracking and crushing of concrete at top in the pure bending region. Local buckling in the top flange of steel beam was also observed at the loads near to failure, which is influenced by spacing of studs and top flange thickness of rolled steel section. One of the recommendations is that the ultimate load capacity can be limited to 1.5 times the plastic moment capacity of the section such that the post peak load reduction is kept within limits. Load-deflection behaviour for monotonic tests compared well with the envelope of load-deflection curves for cyclic tests. It is concluded from the experimental investigations that use of HSS studs will reduce their numbers for given loading, which is advantageous in case of long spans. Buckling of top flange of rolled section is observed at failure stage. Provision of lips in the top flange is suggested to avoid this buckling. This is possible in case of longer spans, where normally built-up sections are used.