Simply-supported Beam Research Articles

Simulation and detection of small crack-like surface flaws and slots in simply-supported beams using curvature analysis of analytical and numerical modal displacement data

Purpose The purpose of this paper is to employ analytical and numerical techniques to generate modal displacement data of damaged beams containing very small crack-like surface flaws or slots and to use the data in the development of damage detection methodology. The detection method involves the use of double differentiation of the modal data for identification of the flaw location and magnitude. Design/methodology/approach The modal displacements of damaged beams are simulated analytically using the Bernoulli-Euler theory and numerically using the finite element method. The principle used in the analytical approach is based on changes in the transverse displacement due to the localized reduction of the flexural rigidity of the beam. Curvature analysis is employed to identify and locate the structural flaws from the modal data. The curvature mode shapes are calculated using a central difference approximation. The effects of random noise on the detectability of the structural flaws are also computed. Findings The analytical approach is much more robust in simulating modal displacement data for beams with crack-like surface flaws or slots than the finite element analysis (FEA) approach especially for crack-like surface flaws or slots of very small depths. The structural flaws are detectable in the presence of random noise of up to 5 per cent. Originality/value Simulating the effects of small crack-like surface flaws is important because it is essential to develop techniques to detect cracks at an early stage of their development. The FEA approach can only simulate the effects of crack-like surface flaws or slots with depth ratio greater than 10 per cent. On the other hand, the analytical approach using the Bernoulli-Euler theory can simulate the effects of crack-like surface flaws or slots with depth ratio as small as 2 per cent.

Pedestrian-Induced Vibration of a Simply-Supported Beam

AbstractThis paper presents an analytical approach to analyze the vertical vibration of a simply supported beam subjected to pedestrian-induced loads. The loading time history of an individual footstep is simplified as a rectangular force pulse, and each identical footstep load acts at different locations along the beam depending on a step length. Although the loading model is very simple, it enables us to find analytical relations between the pacing parameters and the beam response. The results showed that the dependence of the pacing period, footstep contact duration, time delay of traveling between two pedestrians on the natural period of the beam as well as the step length can influence the dynamic response of the beam significantly.

Simply-supported multi-layered beams for energy harvesting

This paper develops an analytical model for predicting the performance of simply-supported multi-layered piezoelectric vibrating energy harvesters. The model includes the effects of material and geometric non-linearities, as well as axial pre-tension/compression, and is validated against experimental devices for a large range of base accelerations. Numerical and experimental investigations are performed to understand the benefits of using simply-supported devices compared to cantilevered devices. Comparisons are made in an unbiased manner by tuning the resonant frequency to the same value by modifying the geometry, and the results obtained indicate that simply-supported devices are capable of generating higher voltage levels than cantilever devices. The model is also used to investigate the benefits of using multi-layered devices to improve power density. Depending on harvester composition, power-per-unit-volume of piezoelectric material for a device is increased through the stacking of layers.

On the basic phenomenon of soil-structure interaction on the free vibration response of beams: Application to railway bridges

The dynamic transverse response of beam type bridges under railway traffic is addressed in this contribution. In particular, how soil-structure interaction may affect the critical or resonant velocities and the associated vibratory amplitudes is evaluated in detail. Resonance in beams, due to the circulation of equidistant loads, is highly influenced by the free vibration response that every single load leaves after traversing the structure. On this basis a numerical investigation is carried out analysing the effects of the wave propagation problem on the free vibration response of simply-supported beams in a wide range of travelling velocities. To this end a coupled three-dimensional boundary element-finite element model formulated in the time domain is used to reproduce the soil and structural behaviour, respectively. A catalogue of bridge deck typologies is defined, covering lengths, associated linear masses and fundamental frequencies that may experience high levels of transverse accelerations under resonant conditions, for nowadays existing trains and design velocities. Lengths ranging from 12.5 to 25m are evaluated, along with fundamental frequencies covering most common typologies. A homogeneous soil is considered with shear wave velocities in the interval 150 to 365m/s. From the single load free vibration parametric analysis conclusions are derived regarding the conditions of maximum free vibration and cancellation of the deck response. These conclusions are used afterwards to justify how resonant amplitudes of the bridge under the circulation of railway convoys may be affected by the soil properties, leading to substantially amplified responses or to almost imperceptible ones, and a numerical example is included to show the aforementioned situations.

Damage Detection of Axially Loaded Beam: A Frequency-Based Method

The present study utilizes an analytical method to formulate the three lowest modal frequencies of axially-loaded notched beam through both crack location and load level in a specific format that can be used in existing frequency-based crack-identification methods. The proposed formula provides a basis to shift into two states, one with axial loading and the other without any loading whatsoever. When any two natural frequencies in simply-supported beam with an open crack, subjected to axial load, are measured, crack position and extent can be determined, using a characteristic equation, which is a function of crack location, sectional flexibility, and eigenvalue (natural frequency). Theoretical results show high accuracy for service axial loads. In this range, errors for crack location and extent are less than 12% and 10%, respectively.

A stochastic model updating strategy-based improved response surface model and advanced Monte Carlo simulation

To improve the accuracy and efficiency of computation model for complex structures, the stochastic model updating (SMU) strategy was proposed by combining the improved response surface model (IRSM) and the advanced Monte Carlo (MC) method based on experimental static test, prior information and uncertainties. Firstly, the IRSM and its mathematical model were developed with the emphasis on moving least-square method, and the advanced MC simulation method is studied based on Latin hypercube sampling method as well. And then the SMU procedure was presented with experimental static test for complex structure. The SMUs of simply-supported beam and aeroengine stator system (casings) were implemented to validate the proposed IRSM and advanced MC simulation method. The results show that (1) the SMU strategy hold high computational precision and efficiency for the SMUs of complex structural system; (2) the IRSM is demonstrated to be an effective model due to its SMU time is far less than that of traditional response surface method, which is promising to improve the computational speed and accuracy of SMU; (3) the advanced MC method observably decrease the samples from finite element simulations and the elapsed time of SMU. The efforts of this paper provide a promising SMU strategy for complex structure and enrich the theory of model updating.

TWO-DIMENSIONAL COUPLED SOLUTION FOR THERMOELASTIC BEAMS VIA GENERALIZED DUAL-PHASE-LAGS MODEL

The generalized thermoelastic problem of a thick-walled simply-supported beam subjected to different applied heat source and mechanical loads at its surfaces is studied. The thermoelastic coupling differential equations of motion of the beam are established. The generalized thermoelasticity based on the dual-phase-lags (DPLs) theory is considered to treat this problem. An exact 2-D coupled solution is presented to deduce analytical expressions for the temperature, displacements and stresses. The time-harmonic motion behavior as well as the thermal and mechanical conditions at the bounded faces of the beam is used for this purpose. The effect of the DPLs on the field quantities against the axial and normal directions of the beam under thermomechanical loads is discussed. Final investigations to various thermoelastic models are made.

Data fusion of acceleration and angular velocity for improved model updating

In the traditional finite element (FE) model updating, translational responses, such as acceleration, have generally been employed to identify the structural properties. However, the boundary conditions of a structure are associated with both translational and rotational DOFs. Thus, the combinational measurement of translational and rotational responses (e.g., angular velocity) would increase accuracy of FE model updating of structures, especially in identifying their boundary conditions. This paper proposes data fusion of translational and rotational responses for improved system identification using FE model updating technique. In the proposed method, the accelerometers and gyroscopes are installed in between and near the supports of a structure, respectively, and FE model updating is carried out using the natural frequencies, the translational mode shapes obtained from accelerations, and the rotational mode shapes obtained from angular velocities. Numerical and experimental verifications are carried out on simply-supported beam structures. The verifications show that the proposed FE model updating strategy based on the data fusion results in more accurate assessment of both structural properties and boundary conditions than the traditional FE model updating using translational responses only.

구조물 모니터링을 위한 무선 스마트 센서 네트워크의 칼만 필터 기반 데이터 복구

사회기반시설물의 안전성을 효과적으로 평가하고 모니터링하기 위해 무선 스마트 센서가 개발되어 전 세계적으로 연구가 진행되고 있다. 무선 스마트 센서는 통상 계측 및 임베디드 데이터 연산, 무선 통신이 가능한 공통점을 갖고 있어 기존의 유선 기반 센서가 가진 단점을 극복할 수 있을 것으로 기대되고 있다. 그러나 구조물의 장기 모니터링의 경우 내구성이 충분하지 못해 발생하는 센서 고장이나, 환경적 이유로 인한 무선 통신이 불안정할 경우 계측 데이터를 가져올 수 없는 문제가 발생할 수 있다. 본 연구에서는 무선 스마트 센서 기반의 네트워크에서 이와 같은 문제로 센서 노드에 무선 통신으로 접근할 수 없는 경우를 대처하기 위해, 칼만 필터 기반의 데이터 복구를 수행하여 무선 스마트 센서 네트워크의 신뢰성을 향상시키는 기법을 제안한다. 본 논문에서는 무선 스마트 센서의 연산 기능을 활용하여 네트워크 내에서 계측된 가속도 데이터를 바탕으로 유실된 센서의 가속도 계측 데이터를 추정한다. 개발된 무선 스마트 센서 네트워크 시스템의 성능을 확인하기 위해 단순보 구조에서 실험을 수행하여 추정된 가속도 응답과 계측 값을 비교하였다. Extensive research effort has been made during the last decade to utilize wireless smart sensors for evaluating and monitoring structural integrity of civil engineering structures. The wireless smart sensor commonly has sensing and embedded computation capabilities as well as wireless communication that provide strong potential to overcome shortcomings of traditional wired sensor systems such as high equipment and installation cost. However, sensor malfunctioning particularly in case of long-term monitoring and unreliable wireless communication in harsh environment are the critical issues that should be properly tackled for a wider adoption of wireless smart sensors in practice. This study presents a wireless smart sensor network(WSSN) that can estimate unmeasured responses for the purpose of data recovery at unresponsive sensor nodes. A software program that runs on WSSN is developed to estimate the unmeasured responses from the measured using the Kalman filter. The performance of the developed network software is experimentally verified by estimating unmeasured acceleration responses using a simply-supported beam.

DEFLECTION PREDICTION OF TWO-SPAN CONTINUOUS CONCRETE BEAMS REINFORCED WITH GFRP BARS

Continuous beams are main structural members in many reinforced concrete (RC) applications such as parking garages and bridges, which are vulnerable to harsh environments. The use of glass fiber-reinforced polymers (GFRP) bars in such structures has proven to be a good solution to overcome the steel corrosion problem. In GFRP-RC beams, deflection is a concern and may govern the design; thus, attempts have been made to propose models to predict the effective moment of inertia and consequently the deflection. The current prediction models were verified against deflection of simply-supported beams. In this paper, available predictions models are compared to the experimental deflections of two-span continuous beams. Eight rectangular beams (200×300 mm) continuous over two equal spans of 2,800 mm each were constructed and tested to failure. All beams were reinforced with GFRP bars and stirrups. The test variables included concrete strength, longitudinal reinforcement ratio and transverse reinforcement ratio. The comparison revealed that the investigated models reasonably predicted the deflection at service load level while they underestimated the deflection at higher load levels. These models require to be modified in order to yield better predictions of the deflection at lower and higher load levels than service load.

A new analytical cone model for laterally-loaded vertical piles in cohesionless soil

ABSTRACTLaterally-loaded vertical piles are commonly designed by p-y curves or two-dimensional (2D) analytical methods. The former are vaguely correlated while the latter are oversimplified and conservative. Three-dimensional (3D) analysis can be used, but it is time consuming for practical applications. An improved analytical cone method is thus developed using spreadsheets. Subject to the lateral load under greenfield condition, the soil in front of a vertical pile is under constant volume plastic compression, heaving at the free boundary on top. Stress release occurs in the soil behind the pile. These mobilised soil masses are idealised as inclined right circular cones. The pile below the cone is subjected to the direct compression at the back and shear at the sides. Thus, the pile behaves as a simply-supported beam with one movable support at the loading point. This enables the rational prediction of the pile length required to satisfy the vertical tangent and the “zero-toe-kick” criteria. By plane of...

On the static stability of nonlocal nanobeams using higher-order beam theories

This paper investigates the effects of thermal load and shear force on the buckling of nanobeams. Higher-order shear deformation beam theories are implemented and their predictions of the critical buckling load and post-buckled configurations are compared to those of Euler-Bernoulli and Timoshenko beam theories. The nonlocal Eringen elasticity model is adopted to account a size-dependence at the nano-scale. Analytical closed form solutions for critical buckling loads and post-buckling configurations are derived for proposed beam theories. This would be helpful for those who work in the mechanical analysis of nanobeams especially experimentalists working in the field. Results show that thermal load has a more significant impact on the buckling behavior of simply-supported beams (S-S) than it has on clamped-clamped (C-C) beams. However, the nonlocal effect has more impact on C-C beams that it does on S-S beams. Moreover, it was found that the predictions obtained from Timoshenko beam theory are identical to those obtained using all higher-order shear deformation theories, suggesting that Timoshenko beam theory is sufficient to analyze buckling in nanobeams.

Buckling of Functionally-Graded Beams with Partially Delaminated Piezoelectric Layers

The critical buckling load of a functionally-graded simply-supported beam with partially delaminated piezoelectric layers is discussed. The governing equations of motion are derived using two different, i.e. Euler–Bernoulli and Timoshenko beam theories, and the buckling load was evaluated from the exact solution to the corresponding eigenvalue equation. The equations were simplified to some extent by shifting the coordinate origin such that there is zero bending-extension coupling. Effects induced by the delaminated length, asymmetry, piezoelectric thickness, voltage, and the functionally graded materials (FGMs) volume fraction are evaluated. The validity of results and the invoked assumptions were successfully verified with existing results and finite element calculations. There is some difference between the analytical buckling load and that calculated with the finite element method (FEM), proven small in amount but predictable in terms of the piezoelectric thickness. Further, a general formula is derived to evaluate the dimensionless critical buckling load with the parameters obtained from regression analysis of the solution that can be utilized for all cases where the materials are not far different in their mechanical properties. The results can be utilized as benchmark design tool.

Description of the performances of a thermo-mechanical energy harvester using bimetallic beams

Many recent researches have been focused on the development of thermal energy harvesters using thermo-mechanical or thermo-electrical coupling phenomena associated to a first-order thermodynamic transition. In the case of the bimetallic strip heat engine, the exploitation of the thermo-mechanical instability of bimetallic membranes placed in a thermal gradient enables to convert heat into kinetic energy. This paper is a contribution to the modeling and the comprehension of these heat engines. By restraining the study to the simply-supported bimetallic beams and using a Ritz approximation of the beam shape, this paper aims to give an analytical solution to the first mode of the composite beams and then to evaluate the efficiency of the harvesters exploiting these kinds of instability.

Harvesting heat with thermo-mechanically bistable beams: working principle and theoretical performances

The interest in miniaturizing heat engines to harvest low-grade heat has grown up with the development of wireless sensors requiring little energy to work. The bimetallic strip heat engines exploit the thermo-mechanical instability of composite membranes to convert heat into mechanical energy and have been proposed as an alternative to Seebeck thermoelectric generators based on the properties of semiconductors like bismuth tellurides. This article aims to describe the theoretical operation of these heat engines by explaining the nature and the conditions of occurrence of the thermo-mechanical instability of simply-supported beams. The thermodynamic cycle of the heat engine is explained and the performances of nine types of bimetallic beams are evaluated.

鉄筋を内蔵する集成材の繰り返し曲げ加力実験

A number of Japanese cedar, which ought to be used to buildings or other structures, would increase rapidly in a few years in Japan. However the stiffness and strength of the Japanese cedar’s timbers are much less than other timbers in North-America or Europe. In this paper, the construction approach to improve the weakness of the beam’s bending behavior are presented. Four 1/2-scale simply-supported beams, were tested to clarify the bending behavior and the measure to predict them. The construction approach means to use steel-reinforced lamellas made of the Japanese cedar’s timber, in tension zone and compression zone.

불균일 단면을 갖는 단순지지 보의 모달해석 및 실험

Beam-type structures with non-uniform cross sections are widely used in mechanical, architectural, and civil engineering fields. This paper deals with dynamic characteristics and vibration problems. Governing equations are first derived by using local coordinates. Their solutions are then assumed by using Galerkin's mode summation method. Bisection method is also applied in solving the determinant of the matrix which can provide natural frequencies. Whereas finite element methods adopt admissible functions satisfying only geometric boundary condition, in this study we apply Galerkin's mode summation method which uses eigen-functions satisfying both governing equations and boundary conditions. Modal analysis and experimental tests are finally performed using simply-supported beams with four different non-uniform cross-sections. Our analytical results then show good agreement with experimental ones.

Available rotation capacity of composite beams with high-strength materials under sagging moment

Using high-strength (HS) materials in steel–concrete composite beams is an efficient approach to reduce the self-weight and to improve the sustainability benefits of such members. Due to a lack of precise knowledge of the ductility of composite beams beyond their first yield as a result of the non-linear response of their constituent materials, a comprehensive investigation is reported herein to quantify the available rotation capacity of composite beams using HS materials when subjected to sagging bending. An advanced three-dimensional finite element model is utilised to calculate the end rotation of single-point-loaded simply-supported composite beams. The model has demonstrated to be adequate and reliable in terms of predicting the flexural strength and load–deflection response by comparisons with numerous experimental results reported elsewhere. As many as 1380 beams with various strengths of steel and concrete as well as a wide range of the degree of shear connection and geometries are modelled, and both solid slabs and composite slabs using profiled steel decking are incorporated. Analyses of this body of data indicate that the depth of the neutral axis is the most essential parameter for determining the available sagging rotation capacity, of which an increased value leads to a lower rotation capacity. The yield strength of the steel also has noticeable effects with higher values resulting in poorer rotation performance. The available rotation capacity is also sensitive to the span-to-depth ratio of the beams. In addition, the effects of varying cross-sectional geometries, different patterns of shear connector distributions as well as of geometric imperfections and residual stresses are clarified, which are shown to be slight and can therefore be ignored. Finally, a non-linear empirical equation is developed to predict the available rotation capacity of composite beams using HS materials, which covers the range of numerical data with satisfactory consistency. The outcomes provide an important basis for evaluating the ductility of composite beams by comparing with the required rotation capacity.

Shear strength of glass fiber reinforced polymer – reinforced concrete continuous beams without transverse reinforcement

The shear behaviour of continuous concrete beams reinforced with glass fiber reinforced polymer (GFRP) bars without shear reinforcement has not been studied yet. However, the current codes and guidelines for FRP-reinforced concrete (RC) structures provide shear provisions that are applicable to both simple and continuous members. These provisions are based on test results of simple beams that do not account for the continuity of indeterminate structures. In this paper, the test results of large-scale GFRP-RC continuous beams without transverse reinforcement are presented. The test variables are the type and ratio of longitudinal reinforcement, and concrete strength aiming to evaluate the shear strength of such beams. The test results showed that moment and shear redistribution occurred. Also, the shear capacity of GFRP-RC continuous beams increased as both the longitudinal reinforcement and the concrete strength increased. Moreover, it was found that the continuity affected the concrete contribution to the shear strength where the ratio of experimental-to-predicted shear strength by different codes and guidelines was less than that of simply-supported beams.

FRP Strengthening of Discontinuous Members

Deep beams, corbels, stepped beams, beams with openings and the like are structural members comprising discontinuous regions for which conventional flexural theory does not apply. To strengthen such members using fibre-reinforced polymer (FRP) systems, a rational approach based on strut-and-tie models (STMs) is desirable. This involves the strengthening of struts, ties and nodal zones where required. This study was carried out to illustrate the use of strut-and-tie models in FRP strengthening of reinforced concrete T-beams with a large opening created through the web. Five 3-meter long simply-supported beams were fabricated to simulate existing beams with webs punctured by openings of lengths varying from 180 to 720 mm and located at about 0.5 to 1.5 m from the near support. The depth of the opening was kept constant at 180 mm. Based on strut-and-tie modelling, the beams were strengthened using glass FRP sheets to carry the same ultimate load as a solid beam which had the same dimensions and internal steel reinforcement but without opening, that is, 200 kN. The beams were each subjected to a single load at one-third span from the other support. Test results indicated that all beams with strengthened opening performed as well as the solid beam without opening in terms of cracking and deflection characteristics, in addition to ultimate strength and ductility.