Single-span Beam Research Articles

Free and forced small flexural vibrations of slightly curved slender composite beams with interlayer slip

This paper presents a beam theory for analyzing the dynamic bending response of slender slightly curved composite beams whose layers are flexibly connected and therefore subject to interlayer slip. The equations of motion and boundary conditions are derived using Hamilton’s principle, assuming separately for each layer the applicability of Euler–Bernoulli theory and a linear elastic relationship between the interlayer slip and the shear traction. For the problem of a three-layer slightly curved single-span beam with symmetric layer arrangement and soft-hinged bearings, analytical expressions for the natural frequencies and the eigenfunctions are derived. For the arbitrarily supported two-layer beam, on the other hand, a numerical solution scheme of the combined initial boundary value problem is presented. Several examples show how important it is to consider even very small deviations from the straight beam axis in the prediction of the dynamic response for slender beams with interlayer slip, in particular when all supports are immovable. The comparison of the beam solutions with the results of much more expensive FE analyses based on plane stress elasticity proves the accuracy of the presented theory.

NONLINEAR STATIONARY OSCILLATIONS OF BEAM GIRDER STRUCTURES FROM THE ACTION OF PERIODIC PULSES

Purpose. The aim of this work is to construct an approximate solution of stationary oscillations of a nonlinear equation with a cubic characteristic. The design scheme under consideration best corresponds to the operating conditions of the girder structure in the event of defects of the supporting parts. In this case, longitudinal forces may appear in the girder of the girder structure. Methods. The paper investigates nonlinear oscillations of girder structures of small bridges. The undamped oscillations of the beam resting on hinged fixed support parts on both sides from the action of periodic unilateral impulse loading are considered. The design scheme differs from the statically defined beam scheme, which takes into account defective support parts. In the article, the possible transition to the beam system on moving support parts is considered. To solve the nonlinear equation of the second order of a single-span beam taking into account the longitudinal force, a method of variable time scale used, developed by Academician of the Academy of Sciences of the USSR, professor Bondar M. G. The method of variable scale gives fairly approximate solutions to this class of problems. Results. An approximate analytical solution is performed by the method of variable time scale of stationary nonlinear oscillations of the girder structure taking into account the longitudinal force caused by one-sided pulses that are periodically repeated. According to the obtained formulas of the solution of the nonlinear equation with cubic characteristic it is easy to construct amplitude-frequency characteristics and forms of oscillations. Originality. For the first time the solution of the equation of oscillations of a beam taking into account longitudinal force is executed. The criteria of impact resonance for the system without friction, as well as taking into account the viscous friction. An expression for the amplitude-frequency characteristic of the system under consideration is obtained. Practical meaning. Studies of beam structure oscillations taking into account the longitudinal force gave practical value in the possibility of analytical calculation of oscillation forms, shock resonance criteria and amplitude-frequency characteristics of the system from the action of periodic unilateral impulse loads. The system of girder beams takes into account defective supporting parts (jamming of movable supporting parts) of girder structures of small and medium railway bridgesd.

Diagonal Tension Cracking Strength and Risk of RC Deep Beams

This study focuses on the shear serviceability of simple and continuous reinforced concrete deep beams. The test results of 81 simple deep beams (i.e., simply supported single-span deep beams) and 29 continuous deep beams, for which their diagonal tension cracking loads were reported, were collected from existing studies. On this basis, the diagonal tension cracking mechanism is discussed, and four existing models for diagonal tension cracking are evaluated. The evaluation results show the existing models fail to accurately reflect the influences of the main design parameters (including the shear span-to-effective depth ratio and main tensile reinforcement ratio) on the diagonal tension cracking strength. Therefore, a new equation for the diagonal tension cracking strength for simple and continuous deep beams is proposed. The proposed equation is verified to be superior to the existing models, showing an average value and a coefficient of variation for tested-to-predicted diagonal tension cracking strength ratios of 1.02 and 0.21, respectively. On the other hand, a probabilistic analysis is conducted to evaluate the diagonal tension cracking risk under service load, showing that 35% of deep beams exhibited a diagonal tension cracking load that is less than the service load, which indicated that diagonal cracks easily occur in RC deep beams under service loads.

METHODOLOGY AND RESULTS OF THE SERIES P-2 GLASS SLABS DURABILITY TEST

The paper presents the results of studies of glass plates of series P-2 for durability, which were made of float glass sheets, the methodology of experimental studies of glass plates for durability, as well as the design of the test facility. The analysis of literature sources on the subject made it possible to study the main factors of application of glass slabs as load-bearing structures. Glass is characterized by high strength, but glass structures are not widely used because of the possibility of their sudden brittle fracture and the lack of reliable calculation methods. The strength and reliability of glass structures depend on the type and strength of glass, manufacturing technology, the magnitude of external load and the duration of its action. Glass structures are characterized by the phenomenon of rheology and sudden brittle fracture, so the study of their durability, at different levels of external static load will ensure their reliable operation for the required time. Durability of a structure is the time from the beginning of static load action, which does not exceed the destructive load, with its further endurance until the destruction of the structure. For experimental studies of the durability of glass slabs, there was a need to create a research technique to ensure the stability of the external static load during the entire experiment with the possibility to record the growth of the deflection of the glass slab, the time and the nature of the failure up to the very moment of their sudden brittle failure. The paper describes the methodology and presents the initial results of the experimental study of glass beam slabs for durability. The slabs were hinged to two supports and operated as a single-span beam under a steady static load. Based on the experimental data obtained on the fracture of glass slabs during the action of a steady static load, we can try to predict their durability.

Общая устойчивость балок двутаврового постоянного сечения. Развитие расчетных положений

The paper presents the derivation of formulas for the method of calculating bent elements for general stability, given in SP 16.13330, which is based on the theory of thin-walled elastic rods and the system of differential equations of V. Z. Vlasov. Formulas are given for a single-span beam with a hinged mounting of supports. A general equation has been compiled for calculating the bending stability coefficient φ_1, which allows adapting the requirements of SP16.13330 standards to various types of load action in the beam span. The results are presented for twenty-five different types of external load action. The formulas take into account the arbitrary location of the load along the height of the beam section, from the lower edge of the lower belt to the upper edge of the upper belt. The values of the critical moment M_cr, at which a new form of equilibrium state arises in the beam, are determined, according to the conclusions of the work [6]. The critical moment M_cr is calculated in the form of a stability problem of the first kind – the loss of a stable position of an element of a rectilinear shape (the load acts along the line of the bending center, the neutral axis of the beam is rectilinear, the material is elastic). The formulas of the presented table can be used to perform verification calculations.

The Effect of Steel Beam Elastic Restraint on the Critical Moment of Lateral Torsional Buckling.

This paper reports the results of the next stage of the authors’ investigations into the effect of the elastic action of support nodes on the lateral torsional buckling of steel beams with a bisymmetric I-section. The analysis took into account beam elastic restraint against warping and against rotation in the bending plane. Such beams are found in building frames or frame structures. Taking into account the support conditions mentioned above allows for more effective design of such elements, compared with the boundary conditions of fork support, commonly adopted by designers. The entire range of variation in node rigidity was considered in the study, namely from complete freedom of warping to complete restraint, and from complete freedom of rotation relative to the stronger axis of the cross section (free support) to complete blockage (full fixity). The beams were conservatively assumed to be freely supported against lateral rotation, i.e., rotation in the lateral torsional buckling plane. Calculations were performed for various values of the indexes of fixity against warping and against rotation in the beam bending plane. In the study, formulas for the critical moment of bilaterally fixed beams were developed. Also, approximate formulas were devised for elastic restraint in the support nodes. The formulas concerned the most frequent loading variants applied to single-span beams. The critical moments determined in the study were compared, with values obtained using LTBeamN software (FEM). Good compliance of results was observed. The derived formulas are useful for the engineering design of this type of structures. The designs are based on a more accurate calculation model, which, at the same time offers simplicity of calculation.

Research on the Approximate Calculation Method of the Fundamental Frequency and Its Characteristics on a Tensioned String Bridge

As a new type of composite bridge, the dynamic structural characteristics of a tensioned string bridge need to be deeply studied. In this paper, based on the structural characteristics of a tensioned string bridge, the Rayleigh method is used to derive formulas for calculating the frequencies of vertical, antisymmetric and lateral bending vibrations. The characteristics of the vertical and lateral bending vibration frequencies are summarized. The fundamental frequencies of the antisymmetric vertical bending and lateral bending of the tensioned string bridge are the same as that of the single-span beam under the corresponding constraint conditions. The shape and physical characteristics of the main cable have no effect on the frequency. The vertical bending symmetrical vibration frequency of the tensioned string bridge is greater than the corresponding symmetrical vibration frequency of the simply supported beam. The shape and physical characteristics of the main cable have a greater impact on the vertical bending symmetrical vibration frequency than the lateral bending frequency, and the vertical bending symmetrical vibration frequency increases with an increasing rise-to-span ratio. The tension force of the main cable has no influence on the frequency of tensioned string bridges. The first-order frequency of the tensioned string bridge is generally the vertical bending symmetrical vibration frequency. By adopting a tensioned string bridge structure, the fundamental frequency of a structure can be greatly increased, thereby increasing the overall rigidity of the structure. Finally, an engineering example is applied with the finite element parameter analysis method to study the vibration frequency characteristics of the tensioned string bridge, which verifies the correctness of the formula derived in this paper. The finite element analysis results show that the errors between the derived formula in this paper and the finite element calculation results are less than 2%, indicating that the formula derived in this paper has high calculation accuracy and can meet the calculation accuracy requirements of engineering applications.

Flexural Strength and Behavioral Study of High-Performance Concrete Beams using Stress-Block Parameters

Most of the existing codes are using stress block parameters which were derived for normal strength concrete. Rectangular stress-block parameters used for normal strength concrete cannot be used safely for higher grade concrete like HPC, hence new stress-block parameters are established from the experimental investigations. Theses parameters can be made very much useful in the design of HPC members. Present research aims at behaviour study of HPC using stress block parameters. High performance concrete single span beams were tested under monotonic four-point bending. Considering the experimental stress-strain curves of HPC for grade 60, 80 and 100 MPa, an idealized stress block curve is established and the stress block parameters are derived. Based on the idealized stress block curve, the equations for ultimate moment of resistance, depth of neutral axis, limiting moment of resistance and maximum depth of neutral axis are proposed. Based on the observation of experimental load deformation curves, an ideal load deformation curve is proposed, which follows four significant events identified as, first cracking, yielding of reinforced steel, crushing of concrete with spalling of cover and ultimate failure. The predicted values compare well with the experimental values. The average location of the first crack observed was at 0.535 times the span of the beam from the left support of the observer in the tension zone.

Implementation and Testing of Short Span Footbridge System Made of UHPC

This paper presents the results of testing and the first application of a newly developed innovative system of the bridge superstructures made of UHPC material. Being designed for footbridges it is called short span footbridge system - SSFS. This system was developed for a wide range of uses mainly as footbridges for pedestrians and cyclists and without waterproofing or surfacing. Prestressed prefabricated elements work as single-span beams with a span of up to 20 meters. Thanks to the optimization of the geometry design and the advantageous use of the mechanical properties of the UHPC material in combination with the prestressing, it was possible to achieve a very small construction height. These subtle and light load-bearing components tend to move away from traditional concrete structures, their span-to-depth ratios are closer to steel beams, but the needs for corrosion protection are eliminated using UHPC, which is inherently very durable and therefore almost maintenance-free. SSFS elements can be manufactured with a variable width of up to approx. 3.2 m, but they can be connected transversely and therefore the width of the structure is not limited. The first part of the paper is devoted to the development of the system and real applications of the first few footbridges made of UHPC material and other possibilities of this technology for future applications. The second part of the paper mentions the design and experimental verification of anchoring devices that are used to manipulate small-span footbridges made of UHPC.

DEFORMABILITY OF REINFORCED CONCRETE BEAMS UNDER THE ACTION OF CYCLIC LOADING

The aim of the article is an experimental research of the influence of low-cycle sign-constant loading, as well as the most significant design factors on the deformability of reinforced concrete beam elements. In this regard, for experimental research, the authors developed a four-factor three-level Boxing plan B4. The experimental factors of the plan were varied according to the literature review, which showed that the most significant factors are the following: the value of the relative shear span a/h0, the concrete class C, the value (amount) of transverse reinforcement on the beams support sections ρsw, the level of sign-constant loading η. The samples were tested according to the scheme of a single-span beam, alternately loaded with two centre-point forces. The number of cycles of sign-constant loading was accepted as 10. According to the results of the experiment, using the COMPEX program, adequate mathematical models of the basic parameters of reinforced concrete specimens-beams deformability under the action of low-cycle sign-constant loading were derived, that reflect the influence of these factors both individually and in interaction with each other. Analyzing these models, the features of the development of tensile reinforcement and compressed concrete deformations, as well as beams deflections in the specified conditions, were established. In particular, the factors that have the greatest influence on deformations and deflections are the relative shear span and the level of low-cycle loading. Thus, with their increase, the relative deformations of tensile reinforcement increase by 51% and 52%, the relative deformations of compressed concrete by 40% and 37%, accordingly, by series. The increase of deflections is 43% and 40% with an increase of relative shear span and 38% and 12% with an increase of loading level, accordingly, by series.

Transmissibility performance assessment for drive-by bridge inspection

This paper presents results of numerical and experimental investigations for the feasibility assessment of drive-by bridge inspection, where the health status of a bridge is solely identified from the dynamic response of an inspection vehicle, passing over a target bridge. To this aim, a novel metric based on the transmission performance between the vehicle and the bridge is proposed, and it is demonstrated that the vehicle having a higher sensitivity index has superior performance, in moving state, to identify bridge frequencies, and consequently separate different states of a bridge from one another. In addition to numerical validation, experimental validation from a laboratory-based experiment is presented, where a two-axle vehicle model is adopted as an inspection vehicle and a single-span steel beam is employed as a target bridge. Vibration data are collected using three highly sensitive accelerometer sensors installed on the vehicle, one at the middle of each axle, i.e., the front axle, and the rear axle, and the third one at the center of gravity of the vehicle, to measure the pitching and bouncing modes of the vehicle. The contributions and the key findings of the work are as follows: First, this paper presents one of the early attempts of experimentally validating the feasibility of drive-by bridge inspection to identify an induced change in the structure of a bridge which has caused less than 2% variation in the fundamental frequency. Second, a novel index for quantifying the performance of transmission between the bridge and the vehicle is proposed. It is verified that a vehicle with a higher sensitivity index captures more bridge-related information, hence making the bridge modal frequencies more identifiable. Third, through experiments, successful identification of the first three vibration modes of the bridge, obtained from the moving vehicle tests, is verified under the condition of constant and slow speed, when there is ongoing excitation on the bridge. Fourth, the indices and criteria presented herein would be helpful in designing a real drive-by inspection system which can be potentially used for drive-by bridge inspection of a network of bridges with similar structural design in practice.

Experimental study of SSS of a steel concrete beam taking into account the rigidity of the connecting seam

The article presents the test results of a scale model of a single-span steel-reinforced concrete beam, which is as close as possible to the real structures of bridge spans with the combination of a reinforced concrete slab with steel beams using flexible pin stops. The distribution of the experimental values of shear deformations and normal stresses in the beam flanges is in good agreement with the calculated data, taking into account the shear stiffness of the joint, and differ in a number of values from those calculated by the classical approach.

Lateral-torsional deformations of single-span and two-span thin-walled beams with continuous bracing

C-section and Z-section cold-formed steel beams subjected to a uniformly distributed load on the top flange are analyzed. Single-span beams with each end pinned or fixed, and two-span beams with equal spans, are considered. Both gravity loading and wind uplift are treated. Continuous elastic restraints resist bending and twisting. The lateral restraint acts on the top flange. It is assumed that the beams are linearly elastic and that the cross section maintains its shape during deformation. Lateral-torsional deformation for the C-section beam involves twist and bending in the weak direction. For the Z-section beam, the deformation involves coupling between bending in both the weak and strong directions, along with the twist. The governing equilibrium equations are solved numerically. The problem is motivated largely by roof purlins in metal buildings. Numerical results demonstrate the effects of load magnitude, bending and twisting restraint stiffnesses, and roof slope (with top flange facing either upslope or downslope).

Mechanism of Transfer Story Beam-column in High-rise Buildings

The structural layout of the transfer floor is easy to cause the discontinuity of vertical components, Sudden change of floor lateral stiffness and floor shear capacity, they were unfavorable to the calculation of the overall index of the structure. In order to study the working mechanism of single span transfer beam on the first floor of high-rise building structure with or without interaction with wall, the mechanism of the beam and column was analyzed in the transfer layer of high-rise buildings, and the calculation methods of sectional moment for inertia beam and wall by using the calculation method of bending stiffness was put forward. A simple method for calculating the according to the different arrangement of the shear wall on the single-span transfer beam was deduced, the combined action of the transfer beam-shear wall and the non-combined action of the transfer beam-shear wall has been analyzed and calculated. The results show that, when the full span of the shear wall is arranged on the transfer beam and the deformation coordination is considered, the following conclusions are verified: the flexural rigidity of the transfer beam is suitable for the joint action of the beam and the shear wall of the upper layer ( 5 times amplification); Under the "arching effect", the transfer beam and the upper shear wall transform the vertical force into the horizontal support thrust, which increases the shear force of the frame pillar. It should be considered in the design. It can be used as a reference for engineering construction and design colleagues.

Dynamic analysis of multiple cracked Timoshenko beam under moving load–analytical method

When cracks start to surface in the surrounding areas of the structure, they create a local softness zone and influences on the dynamic response of the structure. The beams are more susceptible to shear and flexural cracks because of being subjected to shear and bending stress. In this study, the dynamic response of the single-span and multi-span damped beam under moving load with multi-crack and elastic boundary condition is studied based on Timoshenko’s theory. The Green’s function method is used to calculate the dynamic response of the cracked beam. In addition, the Green’s function method provides a solution for the differential equations. Moreover, the effects of the crack on the essential characteristics of the multi-span beams, especially the natural frequencies, are investigated. In this study, crack by itself is modeled in different situations and its effect on the behavior of the beam is analyzed. Also, the elastically restrained beam is modeled and its effect on the behavior of the beam is assessed. Because of the fact that the Euler–Bernoulli theory is also used in most beams, in this study, the results of the numerical examples are compared with the Euler–Bernoulli theory. Several examples are analyzed for a better understanding of the Timoshenko cracked beam.

Structural behaviour of steel cold-formed sigma beams strengthened with bonded steel tapes

In the paper the effectiveness of strengthening of thin-walled, cold-formed sigma beams using glued steel tapes is investigated. Special attention is paid to the evaluation of the possibility of increase the load capacity with simultaneous limitation of beams displacements by appropriate location of steel tapes on the sigma cross-section. The study is based on laboratory tests and numerical analysis results. The laboratory test were conducted on eight simply supported, single-span beams made of Σ200x70x2 profile by “Blachy Pruszyński”. Two beams were reinforced with steel tape placed on the inside surface of the upper flange, two with steel tape glued on the inside surface of the web, two beams with reinforcement located on the inside surface of the bottom flange, and two beams were tested as reference beams without reinforcement. Steel tape with the width of 50 mm and thickness of 1.5 mm, made of steel S235 was used as the reinforcement and was glued to the beams with SikaDur®-30 adhesive. Precise strain measurement was made using electrofusion strain gauges and displacements measurement was performed using two lenses of Aramis software with combination with the Tritop machine. Numerical models of the considered beams were developed in the program Abaqus. Work completes the comparative analysis and conclusions.

A novel and efficient method for damage detection in beam-like structures solely based on damaged structure data and using mode shape curvature estimation

Dynamic-based methods for detecting the location of damage in structures are generally based on changes caused by damage in the dynamic properties of structures such as modal frequencies and mode shapes. Mode shape curvature damage detection method is one of these techniques in which the difference in mode shape curvatures between undamaged and damaged structures is used to identify the location of damage. In this paper, a new damage detection method for beam-like structures is presented in which obtaining data from the analytical model of the undamaged structure is not required. The proposed method introduces the difference between mode shape curvature and mode shape curvature estimation of damaged structure as an indicator of identifying the damage location. Mode shape curvature estimation is solely done by using the displacement mode shape of the damaged structure. Hence, in the proposed method there is no need for the undamaged structure data. The curvature estimation can be easily and quickly done using simple mathematical relations. The efficacy of the proposed method in detecting the location of damage is evaluated using the finite element model of various damaged beams with different boundary conditions as well as experimental validation. Moreover, the classical relations of mode shape curvature estimation are presented for the first mode of single-span uniform beams with minor damage. The results indicate that the introduced index in magnifying the location of damage is as capable and efficient as the classical mode shape curvature method.

Laboratory and Numerical Analysis of Steel Cold-Formed Sigma Beams Retrofitted by Bonded CFRP Tapes.

In this paper, the retrofitting method of thin-walled, cold-formed sigma beams using bonded carbon fibre reinforced polymer (CFRP) tapes is proposed. The effectiveness of the presented strengthening method is investigated by the means of laboratory tests and numerical analysis conducted on simply supported, single-span beams made of 200 × 70 × 2 profile by “Blachy Pruszyński” subjected to a four-point bending scheme. Special attention is paid to the evaluation of possibility to increase the load capacity with simultaneous limitation of beams displacements by appropriate location of CFRP tapes. For this purpose, three beams were reinforced with CFRP tape placed on the inner surface of the upper flange, three with CFRP tape on the inner surface of the web, three beams with reinforcement located on the inner surface of the bottom flange, and two beams were tested as reference beams without reinforcement. CFRP tape with a width of 50 mm and a thickness of 1.2 mm was used as the reinforcement and was bonded to the beams by SikaDur®-30 adhesive. Precise strain measurement was made using electrofusion strain gauges, and displacement measurement was performed using two Aramis coupled devices in combination with the Tritop machine. Numerical models of the considered beams were developed in the Finite Element Method (FEM) program Abaqus®. Experimental and numerical analysis made it possible to obtain a very high agreement of results. Based on the conducted research, it was proved how important is the impact of the applied reinforcement (CFRP tapes) in thin-walled steel structures, with respect to the classic methods of strengthening steel building structures.

Correlation-Based Damage Identification and Quantification Using Modal Kinetic Energy Change

Many existing damage identification or quantification methods can be employed only if the internal and external mass changes are negligible when tested at two different states of a structure. This paper presents a new Modal Kinetic Energy (MKE)-based method to detect and quantify damage using modal properties of structures, which can be employed even in situations when mass change is not more than a certain extent. A new damage sensitivity parameter has been developed using measured modal characteristics of baseline structure. The MKE change (MKEC) concept is then employed to locate damage and to estimate relative perturbation at each element. The relative damage extent vector is estimated by searching the best correlation between the analytical and experimental MKEC vectors with the help of genetic algorithm optimization tool. The extent of damage is calculated after computing damage scaling coefficient using measured eigenvalue change vector. A numerical study is carried out on a simply supported single span beam to confirm its performance under various test conditions. The robustness of the proposed MKE method and the significance of mass variation in the damage detection approach are evaluated by comparing the damage quantification results with a traditional approach. Finally, the proposed damage detection method is applied on a two-span simply supported beam for single and multiple damage scenarios by extracting the modal properties experimentally. The results revealed that the proposed approach is capable of detecting and estimating single and multiple damages with reasonable accuracy even in moderate noise contaminated and mass change environments.

Simulation of crack growth in reinforced concrete beams using extended finite element method

Fracture mechanics and the crack propagation in reinforced concrete members is one of the most important approaches to structural safety analysis. Thanks to powerful software and increasingly fast computers, numerical simulation is well suited to analyse even more complex problems, such as crack growth, quickly and economically. In order to validate these analyses in their application, however, it is still necessary to confirm and compare numerous models of load-bearing and material behaviour by experimental investigations. This paper presents nonlinear numerical simulations of bending and shear crack propagation in reinforced concrete beams without transverse reinforcement. The smeared and discrete crack growth of haunched and non-haunched beams is analysed. Compared are the load deformation behaviour and curves which show the crack width in relation with the applied load without building a notch in the structure. The obtained results are presented with different mesh densities. The tests and the numerical simulations will provide a broad interpretation of crack growth and redistribution of inner forces in concrete members. As numerical method to map discrete cracks the extended finite element method (XFEM) is applied. For validation, the crack patterns of real experiments are compared with the results of the different finite element models up to two crack initiations. The evaluation is based on single span beams under bending and shear. The numerical simulations make it possible to predict the fracture behaviour of concrete members and shows good agreement with the experimental results.