Double-span Beam Research Articles

Analytical model of compressive arch action for one-way RC beams and two-way framed substructures under column removal scenarios

This paper proposes a direct analytical model to evaluate compressive arch action (CAA) of reinforced concrete (RC) beam-column sub-assemblages under middle column removal scenario (MCRS), incorporating bending curvature of the double-span beam, and different horizontal and rotational restraint conditions at the beam ends. Based on the proposed approach, the structural engineer can easily obtain the CAA peak capacity and the corresponding CAA displacement by hand calculations. Unlike the previous models in which the accuracy highly relies on pre-assumed CAA displacement, this prediction approach directly derives the CAA displacement and CAA capacity through force equilibrium, geometric relationship, compatibility and boundary conditions, incorporating various horizontal restraint conditions, span-to-depth ratio and section profile of the bridging beam. Comparison with existing test results in available literature indicates that this model not only gives good predictions of peak capacity, displacement and maximum axial beam force in CAA, but also incorporates the effect of horizontal restraint stiffness and working stress of compression reinforcement. Besides, a further simplified explicit model is proposed to enhance usability in practical design, and sensitive studies are conducted to study how the simplification affects the prediction accuracy of vertical CAA capacity and displacement. Thereafter, the proposed model is extended to substructures with orthogonal beams in X- and Y- directions to predict CAA capacity under different column removal scenarios such as interior, penultimate, edge and corner column removal cases.

Alternate load path behaviour in structures with non-symmetrical design under column missing scenario

In the design and assessment of reinforced concrete (RC) structure against progressive collapse via single column removal scenario, also known as “alternate load path method”, development of secondary mechanisms, i.e. compressive arch action (CAA) and catenary action (CA) in double-span beams has been investigated and identified in many previous researches. The majority of the experimental tests are conducted on symmetrical double-span beams (identical span length and reinforcement detailing at both beam span). However, variations in details of beams adjacent to one another are not uncommon in RC building designs and constructions. Hence, a research study (as presented in this paper) with total of 5 RC frame tests are conducted to investigate the development of secondary mechanisms (CAA and CA) in a double-span beam with unsymmetrical 1) span length and 2) beam detailing. For each unsymmetrical case (span length and beam detailing), there will be 1 unsymmetrical double-span beam, and 2 symmetrical double-span beams (each representing one side of the unsymmetrical double-span beam) to serve as reference for quantifying the development of secondary mechanism in unsymmetrical double-span beams. The behaviour and development of secondary mechanisms are thoroughly analysed based on test observation, measured forces and displacements, etc.

Experimental investigation on alternate load path mechanisms in double-span beams with unsymmetrical span lengths and reinforcement ratios

In the design and assessment of reinforced concrete (RC) structures against progressive collapse via single column removal scenarios, also known as the “alternate load path (ALP) method”, the development of secondary mechanisms in double-span beams, i.e. compressive arch action (CAA) and catenary action (CA), has been thoroughly investigated and identified in many previous studies. The majority of the experimental tests were conducted on symmetrical double-span beams (identical span length and steel reinforcement ratio for both beams). However, variations in span length and reinforcement details of beams adjacent to one another are common in RC buildings due to architectural requirements. Hence, an experimental study with a total of 6 tests was conducted to investigate the development of ALP mechanisms (CAA and CA) in double-span RC frames with unsymmetrical 1) span length and/or 2) longitudinal reinforcement ratio. The 6 tests included 3 symmetrically-designed specimens that served as the baseline to facilitate comparison and quantification of the ALP development for the other 3 unsymmetrical continuous beams. The behaviour and development of ALP mechanisms in all the 6 specimens were analysed at structural level including general observations and measurements of the external forces, displacements, and strains. Last but not least, a simplified method that was initially developed based on symmetrical test data was employed to validate unsymmetrical test results, with the aim to provide a robust and simple tool for preliminary design considerations of the ALP mechanisms.

Experimental and numerical investigation on collapse behavior of precast reinforced concrete beam-column sub-assemblages with cast-in-place ECC joints

In order to investigate the effects of engineering cementitious composite (ECC) used as post-cast materials and anchorage length of beam longitudinal rebars in the ECC cast-in-place joints on collapse behavior of precast reinforced concrete (PRC) structures, static pushdown collapse tests on scaled PRC beam-column (BC) sub-assemblages were carried out. Three BC sub-assemblage specimens having a double-span beam and a middle column joint were designed and fabricated with the same geometric dimensions, including one monolithic specimen BC0 and two PRC specimens BC1 and BC2. The specimen BC0 was cast in site with normal concrete, while the double-span beams of specimens BC1 and BC2 were first cast with normal concrete and then the middle column joint was post cast with ECC. The anchorage length of beam longitudinal rebars embedded into the middle joint was 29d, 20d, and 10d (d is the diameter of beam rebar) respectively for specimens BC0, BC1, and BC2, which could produce different splice length in the middle joint. On the basis of tests, refined finite element (FE) models were established and reasonably verified by a detailed comparison between numerical and experimental results. Based on the validated FE models, a parametric analysis was performed to study the effects of horizontal constraint stiffness, anchorage length, and bond strength between beam longitudinal rebars and ECC on collapse resistance of PRC sub-assemblages. The results showed that the application of ECC as the post cast material in the joint of PRC sub-assemblages could achieve similar collapse behavior to the monolithic counterpart and reduce the anchorage length of beam longitudinal rebars with a suggested lower limit of 20d. The bond strength between rebar and ECC in the joint had obvious influence on the collapse resistance and deformation capacity at large deformation stage.

Bridge Structure Damage Identification Based on Dynamic Characteristics

With the increasing traffic volume and years of usage during the operation process, a bridge structure will experience aging and damage to different degrees, leading to the decline in bridge reliability and seriously affecting its operation safety. In this study, the bridge was abstracted into a beam structure for damage identification. Next, the influence of damage on the bridge structure was explored from the angles of its inherent frequency and displacement mode, respectively. Our results showed that whether the structure was damaged could be accurately judged by its inherent frequency, but the specific damage could not be further judged. Through the structural displacement curve, the rough range of structural damage could be judged; however, the damage could not be accurately positioned. The damage position could be accurately identified to some extent by taking the derivatives from the difference value of the structural displacement curve. The above conclusions were verified based on a double-span beam. We found that the above conclusions still held true for the double-span beam, thus proving their universality.

Blast-induced dynamic responses of reinforced concrete structures under progressive collapse

To study the response of reinforced concrete structures under progressive collapse induced by an actual blast event, an experimental programme on reinforced concrete frames subjected to contact detonation was conducted. The tested structure included a double-span beam with a middle joint, side columns and beam extensions connected to external restraints. The experimental programme employed similar geometry, reinforcing arrangement, material properties and boundary restraints from previous quasi-static as well as free-fall dynamic test series, which focused on the effects of horizontal restraint conditions on the mobilisation of catenary action. Compared to the corresponding static and free-fall tests conducted previously, the blast-induced tests simulated the condition of a progressive collapse event triggered by an explosive charge which is close to actual accidental/terrorist situations. As a result, initial effects/damages created by the initial blast effects – that is uplift of the double-span beam and blast pressure on beam and column – were witnessed and well captured. Damage patterns and failure modes from the blast tests were compared with those from related quasi-static and free-fall tests to emphasise the actual behaviour of structures under realistic explosive attacks. Finally, the blast tests also indicated the limitations of some simplified dynamic assessment methods when applying the single column removal assumption.

The Effect of Steel and Basalt Fibers on the Shear Behavior of Double-Span Fiber Reinforced Concrete Beams.

This study investigates the effects of adding different types of fibers to concrete mixes on the shear behavior of double-span fiber-reinforced concrete beams with or without shear reinforcement. As a part of the experimental study, a total of twenty-seven natural-scale double-span beams were tested. The beams, made of concrete with steel or basalt fiber, with fiber dosages of 78.5 and 5 kg/m3, were tested under shear force. The three tested series consisted of three beams with dimensions of 120 × 300 × 4150 mm, with various numbers of stirrups and contents of fiber reinforcement. During the tests, the shear capacity of the elements was determined. The values of support reactions, deflection in the middle of the span of both beam spans, deformations on the surface of the concrete member in the middle of the span in the compressive and tensile zone, and cracking (crack development and crack width) were also measured. The beams were tested using a digital image correlation (DIC) technique. Test results show that shear capacity increases in beams made of concrete with steel (1.87) or basalt fibers (1.23). Moreover, the failure mode changes from shear (brittle) to flexure-shear (less brittle). The experimental shear capacity of beams was compared with the theoretical values predicted by different design codes, i.e., fib Model Code 2010 and RILEM TC 162-TDF 2003. The results show that all the design codes underestimate the contribution of fiber-reinforced concrete beams to shear resistance and greatly overestimate the contribution of shear reinforcement.

Progressive collapse resistance of exterior reinforced concrete frames and simplified method for catenary action

This paper investigated the progressive collapse resistance of exterior reinforced concrete frames under middle column removal scenarios. Five half-scale exterior frames comprising of a double-span beam, a middle beam-column joint and two side columns were tested under a concentrated point load on the middle joint. Different joint links and reinforcement ratios in beams and column sizes were used in frames. The resistances, deformation capacities and failure modes of frames were experimentally obtained. Test results showed that catenary action in bridging beams resulted in flexural failure of side columns, and thus the load capacity of frames depended highly on the moment capacity of column sections. A simplified method was also proposed based on the moment capacity of column sections and the axial force-bending moment interaction diagram of beams. The axial tension force in beams and the associated catenary action capacity were calculated by using the proposed method. Comparisons with test data suggest that the simplified method could predict the load capacity and tension force in beams under catenary action with reasonably good accuracy.

Flexural Behaviour and Internal Forces Redistribution in LWAC Double-Span Beams.

This research study aimed to investigate the effect of the lightweight aggregate concrete and steel reinforcement interaction on the behaviour of continuous beams compared to the normal concrete of the same strength. This paper presents six full-scale, double-span beams with a rectangular cross-section made of both lightweight and normal concrete. The study confirmed that beams made of lightweight aggregate concrete achieve comparable flexural capacities to those made of NWC but their deformability and ductility are lower. Although the redistribution of internal forces depends mainly on the longitudinal reinforcement ratio, the influence of ultimate compressive strains of concrete is also noticeable. The ultimate compressive strains in LWAC are generally lower than in NWC. The lower rotational capacity of LWAC results in smaller degrees of moment redistribution in beams made of this concrete compared to normal concrete beams.

Experimental study on a novel method to improve progressive collapse resistance of RC frames using locally debonded rebars

The alternate path (AP) method is widely used to evaluate the capability of reinforced concrete (RC) frame structures to resist progressive collapse. Previous studies have found that the double-span beam, caused by an internal column removal, fails to sufficiently develop catenary action due to strain concentration and premature fractures of longitudinal reinforcements near beam ends. To enhance catenary action, a novel method is proposed in this study by embedding locally debonded rebars at the half-height of beams near beam ends. For verification, five RC beam-column substructures were quasi-statically loaded representing an internal column removal scenario. Two test parameters were considered, i.e., the amount of additional rebars and whether they were debonded from the surrounding concrete. Test results found that the vertical resistance of the RC substructures using the proposed method increased by 66%–159% compared to that of conventionally designed substructure. The resistance of RC substructures using locally debonded rebars increased by 13%–21% compared to the substructures using an identical amount of bonded rebars. The reason for this resistance improvement is that the locally debonded rebars can improve both the ultimate rotation capacities of beam ends and axial tension forces in beams, resulting in sufficiently developing catenary action. In addition, the resistance contribution at the catenary action stage was identified and the lesser impact on the seismic performance of RC frame structures was discussed when using the proposed rebars. Finally, an analytical model was proposed to predict the ultimate rotation of beam ends and the ultimate resistance of RC substructures with locally debonded and bonded rebars.

Parametric effects on composite floor systems under column removal scenario

This paper presents parametric studies on three-dimensional steel-frame-composite-floor systems (3D composite floor systems) subjected to column loss using macro-based finite element (FE) models and a verified analytical method. The FE modelling method is verified by four actual experimental tests with three important variables, viz. slab aspect ratio, boundary condition and degree of composite action between composite slabs and steel beams. To overcome the shortage of data acquisition in the actual composite floor system tests, the FE models can be used to investigate the effects of these variables on load-resisting mechanisms, such as flexure and catenary action in the double-span girder and the double-span beam over the missing column, and flexure and tensile membrane action in composite slabs. In addition, the parametric studies are extended to include slab thickness. In a similar manner, the analytical model is used to study the effects of slab aspect ratio and joint type on robustness of 3D composite floor systems. After evaluating the robustness of eight sub-structures with different combinations of extended-end-plate, flush-end-plate, web-cleat and fin-plate joints, a few combinations are recommended. Lastly, consistency between FE simulations and the analytical predictions is confirmed through a comparison of energy stored in different structural members.

Static and Dynamic Responses of Reinforced Concrete Structures under Sudden Column Removal Scenario Subjected to Distributed Loading

In this paper, static and dynamic experiments on reinforced concrete beam-column frames under single-column-removal scenario applying a multipoint loading method were conducted. One of the objectives was to investigate structural behavior compared with the single-point loading method, which has been popularly used in previous studies. By equally applying point loads at four locations of the double-span beam structure, the test setup successfully simulated the uniformly distributed loading condition, which is generally applicable for gravity loads in practice. Compared with the previously conducted static tests based on concentrated loading method, structural response from the static tests under multipoint loading condition differed not only on load-bearing capacity but also on failure sequence and displacement profile. On the basis of the test results, the analytical relationship between behaviors of the two loading methods was developed and verified. Compared with the static tests, the dynamic tests highlighted dynamic effects created by the column loss event and confirmed the structural behavior and failure modes observed in the static environment. The dynamic tests also verified the correctness and conservatism of the dynamic assessment framework using a previously proposed energy-based approach. Most important, both the static and the dynamic tests of structures under distributed loads showed less development of catenary action against progressive collapse compared with concentrated loading tests.

Nanobeams and AFM Subject to Piezoelectric and Surface Scale Effects

Vibration dynamics of elastic beams that are used in nanotechnology, such as atomic force microscope modeling and carbon nanotubes, are considered in terms of a fundamental response within a matrix framework. The modeling equations with piezoelectric and surface scale effects are written as a matrix differential equation subject to tip-sample general boundary conditions and to compatibility conditions for the case of multispan beams. We considered a quadratic and a cubic eigenvalue problem related to the inclusion of smart materials and surface effects. Simulations were performed for a two stepped beam with a piezoelectric patch subject to pulse forcing terms. Results with Timoshenko models that include surface effects are presented for micro- and nanoscale. It was observed that the effects are significant just in nanoscale. We also simulate the frequency effects of a double-span beam in which one segment includes rotatory inertia and shear deformation and the other one neglects both phenomena. The proposed analytical methodology can be useful in the design of micro- and nanoresonator structures that involve deformable flexural models for detecting and imaging of physical and biochemical quantities.

Three-Dimensional Composite Floor Systems under Column-Removal Scenarios

In this study, four 1/3-scaled enhanced steel frame–composite floor specimens were tested quasi-statically under a column-removal scenario. One specimen was the control test and the other three were different in terms of aspect ratio, degree of composite action, and boundary condition. The objective of this paper is to study the effects of these three parameters through comparing experimental results of the four specimens. The comparisons revealed the effects on load-carrying capacity, ductility, failure mode, deformation characteristics, and load redistribution among the remaining columns and reaction forces from surrounding structures. First, the load-carrying capacity of the specimen with slab panel dimensions of 2×2 m was around twice as great as those of specimens with slab panel dimensions of 2×3 m, while the deformation capacity of the former was much smaller than those of the latter. Second, weaker composite action decreased flexural resistance but improved ductility significantly, which was beneficial for development of tensile membrane action. Third, more flexible boundary conditions could not increase applied load beyond the stage dominated by flexure. For composite floor systems, flexural rigidity of steel columns was sufficient to support development of compressive arch action (CAA) so that additional horizontal restraints from surrounding structural members were not so beneficial at the CAA stage. However, the stiffness of additional restraints significantly affected the floor system at the catenary action stage. Finally, the typical yield pattern of composite slabs comprised negative yield lines that were formed around the perimeters and positive yield lines along the double-span girder, the double-span beam, and the paths connecting corner columns to the nearest loading points.

A simplified model for alternate load path assessment in RC structures

To assess the robustness of reinforced concrete structures under progressive collapses, alternate load path method by introducing a single column removal has been widely adopted by structural engineers. Numerical analyses of structures under several possibilities of single column removal may be time consuming, especially when non-linear finite element analyses are employed which require high computational costs and modelling skills. Towards this end, a simplified analytical model is proposed in this paper to facilitate engineers in predicting resistance of the affected substructure (frames or frame-slabs above the removed column), and allow them to perform a quick check on the adequacy of progress collapse resistance of the structure to stop or prevent the damage propagation to the remaining structure. The proposed model considers development of different bridging mechanisms in RC structures under a concentrated loading above the removed column, including compressive arch action, catenary action, and tensile membrane action. In addition to validation against test results, applications of the proposed analytical model to predict the bridging capacities of a 2-D two-storey frame and unsymmetrical double-span beams are also presented.

Analysis of dynamic coefficients for damage to the middle support of two-span and three-span continuous beams

The paper deals with the operation of continuous two- and three-span beams with damage to the middle support. For a two-span continuous beam, the dynamic coefficients for the action of concentrated forces over the middle support and span of the beam are theoretically substantiated. The results of numerical studies of double-span beams at different loads and time intervals of damage to the middle support are presented. The method of dynamic calculation, which can be used to calculate the survivability of steel structures, has been worked out. A good correspondence of the dynamic coefficients obtained numerically with theoretical values is established. Depending on the order of the formation of the design, the dynamic coefficients vary from 1 to 3. For survivability of the design, the pre-bending of the beams above the middle support is most dangerous. With the use of a well-grounded numerical technique, the dynamic coefficients for a continuous three-beam beam are investigated. For continuous beams, if the middle supports are damaged, it is recommended to use a dynamic coefficient of 2 for the load in the spans adjacent to the damaged support

Research in Redistribution of Bending Moments in the Beams of Reinforced Concrete Early Loaded

This paper presents results of a study of a reinforced concrete beam, simulating a double-span beam, early loaded. The beam was designed with limited redistribution of bending moments from the support to the span. The tested element was loaded at the operational load level, after the concrete had reached 80% of its predicted compressive strength. In a two-month test it was observed that the distribution of bending moments changed in the direction opposite to what was expected, i.e. from the span to the support. Moment redistribution in the tested beam leads to a reduction of security level in the middle support section of the beam.

Advanced numerical investigation into structural behaviour of high-strength cold-formed steel lapped Z-sections with different overlapping lengths

In order to improve buildability of cold-formed steel structures, a series of research and development projects have been undertaken by the authors to examine structural behaviour of bolted moment connections between cold-formed steel sections. In this article, a systematic numerical investigation with advanced finite element modelling technique into the structural behaviour of high-strength cold-formed steel lapped Z-sections under gravity loads is presented, and details of the modelling techniques are presented thoroughly. It aims to examine deformation characteristics of these lapped Z-sections with different overlapping lengths. After careful calibration of advanced finite element models of lapped Z-sections against test data, it is demonstrated that the predicted moment rotation curves of these models follow closely the measured data not only up to the maximum applied moments but also at large deformations. In general, all of these lapped Z-sections are unable to resist sustained loadings after section failure under combined bending and shear, and they exhibit sudden unloadings once the maximum applied loads are attained. Hence, the proposed finite element models are able to simulate highly non-ductile deformation characteristics of these Z-sections. While long overlapping lengths over internal supports in multi-span cold-formed steel purlin systems are often advantageous in terms of both ‘stiffness and strength’, more steel materials are used at the same time. Hence, it is very desirable to establish an efficient use of the lapped Z-sections with optimal overlapping lengths. A total of six models with different overlapping lengths are then extended to simulate the structural behaviour of lapped double-span beams, and extensive material and geometrical non-linear analyses have been carried out. It is found that lapped double-span beams with practical overlapping lengths tend to behave superior to continuous double-span beams in terms of both load resistances and deformations. Depending on the overlapping lengths of the lapped Z-sections, different system failure mechanisms have been clearly identified after significant moment redistribution within the beams, and their structural behaviour has been compared in a rational manner. Consequently, these models will be readily employed to investigate the structural behaviour of high-strength cold-formed steel lapped Z-sections under a wide range of practical loading and boundary loading conditions for possible development of effective design rules.

A Ductility-Centred Analytical Model for Axially Restrained Double-span Steel Beam Systems Subjected to Sudden Columns Loss

This paper presents a ductility-centred analytical model for robustness assessment of axially restrained double-span steel beam systems further to sudden columns loss. The main benefit of the model is its ability of linking the response of semi-rigid beam-to-column joints to the load carrying capacity of the system, and in addition, the model supports the use of codified methods for joint response. The analytical model is first validated through comparisons against detailed double-span FE models, where it is found that the load-joint rotation responses of the axially restrained systems can be accurately captured by the analytical model, especially at catenary stages. Comparisons are also made against the FE results excluding the axial restraints. The axial restraints are shown to play a crucial role in the development of catenary action. Following the verification study, the analytical predictions of the double-span beam systems with varying parameters were further presented and discussed, and an illustrative design example was finally given to clearly demonstrate the practical use of the analytical model for robustness assessment of building frames. The analytical model is design effective and hand-calculable, and thus it brings convenience to robustness assessment of structures in practice. The proposed method may be considered as a feasible alternative method to detailed nonlinear dynamic analysis.

Progressive Collapse of Exterior Reinforced Concrete Beam–Column Sub-assemblages: Considering the Effects of a Transverse Frame

Many experimental studies have evaluated the in-plane behavior of reinforced concrete frames in order to understand mechanisms that resist progressive collapse. The effects of transverse beams, frames and slabs often are neglected due to their probable complexities. In the present study, an experimental and numerical assessment is performed to investigate the effects of transverse beams on the collapse behavior of reinforced concrete frames. Tests were undertaken on a 3/10-scale reinforced concrete sub-assemblage, consisting of a double-span beam and two end columns within the frame plane connected to a transverse frame at the middle joint. The specimen was placed under a monotonic vertical load to simulate the progressive collapse of the frame. Alternative load paths, mechanism of formation and development of cracks and major resistance mechanisms were compared with a two-dimensional scaled specimen without a transverse beam. The results demonstrate a general enhancement in resistance mechanisms with a considerable emphasis on the flexural capacity of the transverse beam. Additionally, the role of the transverse beam in restraining the rotation of the middle joint was evident, which in turn leads to more ductile behavior. A macro-model was also developed to further investigate progressive collapse in three dimensions. Along with the validated numerical model, a parametric study was undertaken to investigate the effects of the removed column location and beam section details on the progressive collapse behavior.