Two-span Continuous Beams Research Articles

Strut-tie model for two-span continuous RC deep beams

In this study, a simple indeterminate strut-tie model which reflects complicated characteristics of the ultimate structural behavior of continuous reinforced concrete deep beams was proposed. In addition, the load distribution ratio, defined as the fraction of applied load transferred by a vertical tie of truss load transfer mechanism, was proposed to help structural designers perform the analysis and design of continuous reinforced concrete deep beams by using the strut-tie model approaches of current design codes. In the determination of the load distribution ratio, a concept of balanced shear reinforcement ratio requiring a simultaneous failure of inclined concrete strut and vertical steel tie was introduced to ensure the ductile shear failure of reinforced concrete deep beams, and the primary design variables including the shear span-to-effective depth ratio, flexural reinforcement ratio, and compressive strength of concrete were reflected upon. To verify the appropriateness of the present study, the ultimate strength of 58 continuous reinforced concrete deep beams tested to shear failure was evaluated by the ACI 318M-11\'s strut-tie model approach associated with the presented indeterminate strut-tie model and load distribution ratio. The ultimate strength of the continuous deep beams was also estimated by the experimental shear equations, conventional design codes that were based on experimental and theoretical shear strength models, and current strut-tie model design codes. The validity of the proposed strut-tie model and load distribution ratio was examined through the comparison of the strength analysis results classified according to the primary design variables. The present study associated with the indeterminate strut-tie model and load distribution ratio evaluated the ultimate strength of the continuous deep beams fairly well compared with those by other approaches. In addition, the present approach reflected the effects of the primary design variables on the ultimate strength of the continuous deep beams consistently and reasonably. The present study may provide an opportunity to help structural designers conduct the rational and practical strut-tie model design of continuous deep beams.

Stiffness and strength of composite truss beam to R.C. column connection in MRFs

The results of experimental and numerical investigations on the cyclic behavior of composite truss beam to reinforced concrete column connection of moment resisting frames are reported. The beams are constituted by a steel truss and end zone added rebar embedded within a concrete block cast in place, so that, after curing, the steel truss and the added rebar work together with the concrete matrix as reinforcement of the composite beam.Particularly, three specimens of two-span continuous beams were tested under monotonic and cyclic actions as three point bending test. With the goal to investigate about the behavior of the end zone of the beam integrated into a moment resisting frame before concrete casting strain gauges were placed on the steel reinforcements close to the joint, where plastic behavior was expected, in order to identify the role of steel truss elements in providing stiffness and strength in the plastic hinge region. Experimental results are reproduced by a fiber model. The results prove that in the joint region the bottom steel plate and the top truss chord are not effective in tension, while they are partially effective in compression. The large amount of unbounded reinforcing elements causes a concentration of plastic deformations in a small region at the beam edge; however, the beam exhibits large rotational ductility. In lacking of specific confining reinforcement at the beam end, when large rotations are experienced, the damage of unconfined concrete reduces section strength and ductility for cyclic action, causing a sudden loss of bonding between reinforcing rebars and damaged concrete.

Ultimate Unbonded Tendon Stress in CFRP Strengthened Post-Tensioned Indeterminate I-Beams Cast with HSCs

Based on the bending experiment for two-span continuous post-tension beams with unbounded tendons and externally applied CFRP sheets, the analysis of the stress increment of unbonded tendons is monitored in the loading process. Since self-compacting concrete (SCC) is a suitable innovation,, understanding the implementation of this type of concrete on the ultimate unbonded tendon stress is critical. For these aims, results of four continuous un-bonded post-tensioned I-beams in two groups were cast and monitored by electrical strain gauges andare presented here. In the first group, the beams (UPN1-12, SUPN1-12) consisted of high strength normal concrete (HSNC), while in the second group (UPS1-12, SUPS1-12) high strength self-compacting concrete (HSSCC) were tested. The beams are made which are compared with the theory proposed by different codes, and a preliminary modification is given for each code equation. The results of standard error of estimate Sy/x , indicates that for two types of HSCs (strengthened and non-strengthened beams), the ACI 318-2011 provides better estimates than AASHTO-2010 model, whereas this model provides better estimates as compared toBS 8110-97.Comparison of increase in experimental ultimate tendon stress of beams indicates that the increase in tendon stress at ultimate state in strengthened beams is lower than that in non-strengthened beams cast with HSCs.

Continuous Beams of Aluminum Alloy Tubular Cross Sections. I: Tests and FE Model Validation

The aims of this study are to generate experimental data and develop numerical models for aluminum alloy continuous beams, and to utilize the results to underpin the development of revised design methods for indeterminate structures. This paper presents an experimental program and finite-element (FE) analyses for two-span continuous beams (i.e., five-point bending) of square and rectangular hollow sections (SHSs and RHSs). The experimental program comprised 27 five-point bending tests with three different positioning of loads. The testing procedures and key results are reported. The test specimens were manufactured by extrusion, with 18 of grade 6061-T6 and 9 of grade 6063-T5 heat-treated aluminum alloys. The test specimens were nonslender sections, and mostly of Class 1 proportions. Generally, the specimens failed by the formation of a collapse mechanism comprising three plastic hinges. The distances between the supports and the loading points were varied in order to form the first plastic hinge in different locations, to achieve different load levels between the first hinge and collapse, and to change the rotation demands on the first hinge that formed. The FE models were developed and failure was defined as either when a plastic collapse mechanism was formed or the material fracture strain was reached on the tension flange, whichever occurred first. The numerical models were first validated against the experimentally obtained load-deflection responses, as well as the failure modes. The experimental and FE ultimate loads were both found to be beyond the theoretical loads corresponding to the formation of the first hinge as well as the calculated plastic collapse loads. A key characteristic of aluminum alloy, strain hardening, is shown to be particularly significant in both the experimental program and the numerical investigation. The validated FE models are used to generate numerical results through parametric studies in the companion paper. The development of design rules for indeterminate aluminum alloy structural systems is then described.

Factors affecting moment redistribution at ultimate in continuous beams prestressed with external CFRP tendons

A numerical investigation of redistribution of moments in continuous concrete beams prestressed with external carbon fiber reinforced polymer (CFRP) tendons at failure loads is described. A finite element analysis (FEA) model is introduced, and an extensive parametric study is carried out on two-span continuous beams. The factors examined in this study include the content of non-prestressed steel, tendon eccentricities, tendon area, effective prestress, span-to-height ratio, concrete strength, CFRP modulus of elasticity and load type. The results obtained from FEA are compared with those obtained from various codes. The study shows that the importance of some factors is not reflected in the codes. When used to calculate the degree of moment redistribution in these beams, the parameter εt (net strain in extreme tension steel) seems to be more reasonable than the parameter c/d (ratio of neutral axis depth to section effective depth). A simplified equation for calculating the degree of moment redistribution at ultimate is proposed.

Evaluation of Moment Redistribution in Normal-Strength and High-Strength Reinforced Concrete Beams

This article presents an investigation of redistribution of moments in normal-strength concrete (NSC) and high-strength concrete (HSC) continuous beams. A nonlinear computer model is introduced and validated by experimental results of HSC continuous specimens. A numerical test is performed on two-span continuous beams where the concrete strength varies between 30 and 90 MPa so as to cover both NSC and HSC beams. Various tensile steel ratios over the center support ρs2 and midspan ρs1 are used. Some rules related to moment redistribution are examined. The study confirms that the moment redistribution is dependent on the stiffness of the critical sections. The analysis also indicates that full redistribution of moments is more likely to be developed in HSC beams than in NSC beams. The results of a parametric study, where the ratio ρs2/ρs1 is fixed at 0.8, show that at a given steel ratio, a higher concrete strength registers higher redistribution of moments except when ρs2<1.55 precent. The effect of concrete strength on the moment redistribution is generally reflected in various codes.

Effect of longitudinal reinforcement and prestressing on stiffness of composite beams under hogging moments

To investigate the stiffness of steel–concrete composite beams subjected to hogging bending moments and their enhancement through prestressing using external tendons, laboratory experiments were conducted on five steel–concrete composite beams. Three of the beams were two-span continuous composite beams with box steel girders and the others were simply supported. The parameters investigated included the amount of longitudinal reinforcement and the arrangement of prestressing. From the experimental results, it was found that prestressing significantly improved the cracking moment of composite beams, and crack spacing was dominated by the transverse reinforcement spacing. The stiffness enhancement and crack control suggest the advantages to the mechanical performance of composite beams prestressed by external tendons. A theoretical model is proposed to estimate the effective stiffness of a composite beam subjected to hogging bending moments, taking into account the tension stiffening effect of concrete cracking.

The Anti-Bending Performance Study of Loaded Continuous Beam that Strengthened with Prestressed Laterally Bonded Carbon Fiber Sheets

We studied the strengthening performance of prestressed laterally bonded carbon fiber sheets (CFS) to two-span continuous beams in this paper, which included one unstrengthened beam, two non-prestressed laterally bonded CFS strengthening beams, and four prestressed laterally bonded CFS strengthening beams, and all of them were sticked with CFS on different styles on tension part of lateral beam. We discussed the stress and deformation of laterally bonded CFS strengthened continuous beams under different initial load cases and different prestressed impact, and studied the influence of initial load and prestressing to reinforcement effect. The results showed that the size of initial load and prestressing would affect the reinforcement effect of laterally bonded CFS to continuous beams; The smaller initial load, the better reinforcement effect; The reinforcement effect to continuous beams of prestressed laterally bonded CFS was better than the non-prestressed laterally bonded CFS. Corresponding to latter, the former also owned better deformation performance, less cracks number and width.

Ultimate response of stainless steel continuous beams

An experimental study of stainless steel continuous beams not susceptible to lateral torsional buckling is reported in this paper and the applicability of plastic design methods to such structures is considered. A total of 18 two-span continuous beams were tested. Three cross-section types – cold-formed square hollow sections (SHS), cold-formed rectangular hollow sections (RHS) and welded I-sections, and two material grades – austenitic EN 1.4301/1.4307 and lean duplex EN 1.4162, were considered. The geometric and material properties of the continuous beam test specimens were carefully recorded and supplemented by tests on simply supported specimens of the same cross-sections. The test specimens covered a wide range of cross-section slendernesses and two different loading positions were adopted. The experimental results were used to assess the degree of moment redistribution in indeterminate stainless steel structures and the applicability of both conventional and novel plastic design methods, including an extension of the continuous strength method (CSM). Comparisons indicated that conventional plastic design is applicable to stainless steel structures, while greater efficiency can be achieved by considering strain-hardening through the CSM.

Ultimate Tendon Stress in CFRP Strengthened Unbounded HSC Post-Tensioned Continuous I-Beams

The use of unbounded tendons is common in prestressed concrete structures and evaluation of the stress increase in unbonded tendons at ultimate flexural strength of such structure has posed a great challenge over the years. Based on the bending experiment for two-span continuous post-tension beams with unbounded tendons and externally applied CFRP sheets, the monitoring of the stress increment of unbounded tendons is made in the loading process. For these aims, in this paper there are presented results of two continuous un-bonded post-tensioned I-beams were cast with high strength concrete (HSC) and monitored by electrical strain gauges. The beams are made of which are compared with the theory proposed by different codes. The results indicate that the ACI 318-2011 provides better estimates than AASHTO-2010 model whereas this model provides better estimates than BS 8110-97. Comparison of experimental ultimate tendon stress increase of strengthened and non-strengthened beams casted with HSC indicates that increase in tendon stress at an ultimate state in strengthened unbounded post-tensioned beam is lower than non-strengthened unbounded post-tension beam casted with HSC.

Calculation Methods of CFRP Tendons Stress in Two-Span Prestressed Continuous Beams

Crack width, deflection and load bearing capacity of two-span concrete continuous beams partially prestressed with unbonded carbon fiber reinforced polymer (CFRP) tendons can be calculated accurately, the first premise is to calculate reasonably the stress increment of CFRP tendons. However, the law of stress increment of unbonded tendons is unclear. Moment-curvature analysis method and deformation compatibility condition were used to compile nonlinear full-range analysis programs of this kind of continuous beam. The comparison between simulated results and tested results of stress in CFRP tendons indicates that the simulation analysis programs are reliable.

Damage identification in beam type structures based on statistical moment using a two step method

Abstract This paper defines a novel damage index-strain statistical moment, and formulates the fourth strain statistical moment (FSSM) of beam-type structures under white noise excitation. Based on this newly defined strain statistical moment index and the least square optimization algorithm, a two-step damage identification method is proposed. This two-step method is operated like this: first use the difference curves of FSSMs before and after damage to locate damage elements; then for those identified damage elements, employ the model updating method based on the least square algorithm to assess their damage severity. Numerical studies on a simply supported beam and a two-span continuous beam are performed and the study results show that the newly defined index is effective to locate damages, even when the noise intensity is as high as 15 percent. Integrating with the least square-based model updating technique, the damage severities of beam-type structures can also be determined quantitatively. In this way, the proposed two-step method is verified and found to be capable of identifying damage positions and severities of beam-type structures and insensitive to measurement noise.

Shear strengthening of reinforced concrete continuous beams

This paper presents the results of an experimental investigation to evaluate the contribution of carbon-fibre-reinforced polymer sheets in enhancing the shear strength of continuous reinforced concrete beams. A total of five, two-span concrete continuous beams with rectangular cross-section were tested. One beam without strengthening was used as the control and the other four beams were strengthened with different arrangements of polymer sheets. The variables selected were various wrapping schemes and anchorage length of the polymer sheet. The aim was to develop a better understanding of the shear contribution of polymer and to investigate the potential for cost savings by minimising the area of externally bonded polymer sheets. Test results were compared with four existing shear prediction models available in the literature. The results indicate that the polymer sheet significantly enhanced the shear strength of the beams, and that the area of polymer sheet can be minimised with marginal compromise on the shear carrying capacity of strengthened concrete beams.

INTERACTION RESPONSE OF TRAIN LOADS MOVING OVER A TWO-SPAN CONTINUOUS BEAM

The objective of this study is to investigate the resonance and sub-resonance acceleration response of a two-span continuous railway bridge under the passage of moving train loadings. The continuous bridge is modeled as a Bernoulli–Euler beam with uniform span length and the moving train is simulated as a series of equidistant two degrees-of-freedom (2-DOF) mass–spring–damper units. The modal superposition method is adopted to compute the interaction dynamics of the train–bridge system. The numerical analyses indicate that (1) the train-induced resonance of the two-span continuous beam may result in significant amplification of the dynamic response of the train/bridge system; (2) for a two-span continuous beam, the first two resonant speeds may fall in the range of operating speeds of high-speed trains, which can lead to highly amplified vehicle responses; (3) due to the presence of sub-resonant peaks, the maximum acceleration of the two-span continuous beam need not occur at the midpoint of the beam; (4) inclusion of damping of a beam is helpful for reducing the train-induced resonant response on the beam, but the first two resonant peaks of the coupling system remain unchanged.

Mechanical State Model of Multispan Continuous Deep Beam under Concentrated Load

In the present study, the mechanical model of the multispan continuous deep beam under concentrated load was established on the state space theory, and with the model, the two-span continuous deep beam was calculated. And the calculation results are consistent with ANSYS results, which show that the model without the plane section assumption is accurate and applicable. The changes of the deflection and the section strain along the beam height were analyzed with the beam height variation. The results showed that the normal stress and the shear stress coexisted in the cross-section of multi-span continuous deep beam, which means that the shear stress can cause the cross section warping, while the normal stress can result in the section in parallel with the neutral layer extrusion. In addition, we incidentally found that, with the increase of the beam height, the impact of the buckling deformation and the extrusion deformation impact on the beam bending increase gradually. A new design method, which has been provided by the established mechanical model of the multi-span continuous deep beam, has extended the space of the calculation method on traditional continuous deep beam.

Application of the Influence Line on the Bridge Testing

In order to discuss the sensibility of influence line on the damage, a two-span continuous beam model is set up using the finite element method in the paper. The two damage positions at the middle span and 1/4 span are considered. And different damage degrees are simulated by the different crack depths. The shearing force and moment at four key positions of the beam are calculated under the moving unit load separately. Results show that the moment influence line at the middle support of the beam is very sensitive to the beam damage, while the moment influence line at other positions and the shearing force influence line at each position are insensitive. In the moment influence line of the beam middle support position, the obvious different values appear at the middle span when damage in the middle span. As damage in the 1/4 span, the obvious different also appear at the damage position, and the difference is increased unconspicuous up to the middle span, and then decreased soon. The conclusion is beneficial to determine the position of damage by means of comparing the moment influence line with that of undamaged bridge. Moreover, the difference value of moment influence line can also be used to estimate the damage degree.

Analytical solution of linear elastic beams cracked in flexure and strengthened with side plates

A new mathematical model and its analytical solution for the analysis of the stress–strain state of a linear elastic beam cracked in flexure and strengthened with plates on its lateral sides is presented. Both the longitudinal and the transversal interactions at the side plate/beam interface are considered. Linear behaviour of the contact connection is assumed. The method is based upon the linearised planar beam theory of Reissner. The weakening of the beam induced by the flexural crack is modelled conventionally as a rotational spring. The suitability of the theory is demonstrated in a case presentation involving the comparison between analytical results of the present beam (one-dimensional) model, the experiments and the numerical results of a full three-dimensional solid model created in the LUSAS finite element analysis software. An excellent agreement between the results is observed and the proposed formulation is found to be accurate and reliable. Finally, the solution is employed in an engineering analysis, discussing the effects of the material and the geometric properties of selected characteristic cases of the observed beams on the static and kinematic quantities, including the boundary conditions of the side plates, the longitudinal and the transversal stiffness of the connection, the size of the cracks, the span of the beam, and the length and the stiffness of the side plates. For the cracked cantilever beam, a substantial effect of any of these parameters is found. In contrast, for the cracked two-span continuous beam, only the effect of the stiffness of the side plates and the effect of the length of the beam spans are noticeable.

Flexural experimental study on continuous reinforced concrete beams strengthened with basalt fiber reinforced polymer/plastic

This paper discusses a new fibrous composite known as continuous basalt fiber reinforced polymer/plastic (BFRP). Compared with other fiber reinforced polymer/plastic, BFRP has many advantages, such as ductility, high thermal resistance, corrosion resistance and economic cost. To test mechanical properties and failure modes of flexural members strengthened with BFRP, flexural experiment is conducted on four two-span T-section continuous beams strengthened with BFRP and one un-strengthened comparative beam. The experimental result shows that the strengthened beams perform remarkably in terms of yield strength, ultimate strength and ductility. BFRP has good prospects in retrofitting and strengthening of concrete structures which require good ductility and corrosion resistance.

Long-term experiments of composite beams and connections

Composite steel–concrete solutions are commonly used in Australia for floor building applications. In these situations secondary beams are usually connected to primary beams or columns by means of composite web side plate connections, also known as fin or single plate joints, with the slab continuous at the connection. The rigidity of this arrangement can lead to extensive cracking at the joint and to significant compressive forces in the web side plate connection, which can compromise one of the design limit states. In this context, this paper focuses on serviceability limit state and presents the results of two sets of experiments aimed at investigating how the time-dependent behaviour of the concrete affects the response of such joints. The first set included six simply supported specimens subjected to a sustained negative moment with the connection placed at their mid-span. These were used to study the behaviour of the negative moment region of a continuous floor beam in isolation. The second set consisted of a full-scale two-span continuous composite beam with the joint specified at the internal support. The long-term response of these composite members was monitored measuring the changes in deflection, end rotation, strain and slip occurring over time. Design considerations are provided at the end of the paper and it is envisaged that these new experimental data will be useful for the validation and benchmarking of analytical and numerical models.

Some Qualitative Properties of Modes for Discrete Modes of Two-Spans Beam with Two Free-Ends

In this paper we give the conjugated system of the discrete system of the two-span beam with two free-ends, which is a discrete system of two-span continuous beam with two clamped-ends and an internal pin. The sign-oscillating property of the stiffness matrix of the conjugated system is proved. And the basic oscillating properties of the frequencies and “displacement” modes of the conjugated system are determined. Furthermore, some qualitative properties owned by displacement modes, rotation modes, moment modes and shearing force modes of the discrete system of the two-span beam with two free-ends are obtained.