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The notion of horizontal energy minimizers between C-C spaces is introduced. We prove the existence of such energy minimizers when the domain is a C2, noncharacteristic bounded open set in a C-C space and the target is a C-C space of Carnot type.

Traditionally, the Rankine equation has been applied to investigate the load bearing capacities of steel columns or frames at room temperature. Recently, the method has been extended to predict the fire resistance of a simply supported steel column by taking the temperature effects on steel material into consideration. This paper further extends the Rankine equation to predict the fire resistance of an isolated steel column from the surrounding structure. The complicated boundary restraints contributed by the adjoining structure are represented by a linear spring and a pair of rotational springs attached to the column ends. Both the boundary restraints and creep strain are incorporated into the Rankine equation. The predictions of Rankine approach are verified experimentally and numerically. Generally, it is shown that under different external load utilization factors and boundary restraint levels, Rankine approach yields accurate and slightly conservative predictions.

Using the standard fire resistance test to predict the fire resistance of a steel column is expensive and time consuming as well. On the other hand, numerical approximation using the finite element method assumes the availability of a nonlinear elastoplastic analysis software. This approach is not necessarily accurate as it does not explicitly take creep into account. As an alternative, theoretical analysis that can be performed manually is a convenient tool for engineers to quickly assess the column fire resistance. This study extends the traditional Rankine formula to the critical temperature prediction of an axially restrained steel column. A linear spring attached to the column top end accounts for the axial restraint of upper-story structure on the isolated heated column. The proposed Rankine approach incorporates both the axial restraint and creep strain. Comparison of test and numerical results using self-developed viscoelastoplastic analysis shows that the proposed Rankine formula yields very good column critical temperatures. Most importantly, it shows that axial restraint can significantly reduce the fire resistance of a column based on the adopted failure criterion.

Until now, the determination of fire resistance of reinforced concrete (RC) columns has essentially been based on tabulated data. Clearly, a more scientific approach based on an understanding of the fundamental behavior of columns in fire is timely and will be useful to structural engineers. This paper develops a simple and rational method to predict the fire resistance of RC columns subjected to four-face heating. The effects of elevated temperature on material deterioration with regard to the strength and stability of the columns are quantified. Both uniaxial and biaxial bending of columns is considered. The computer code \ISAFIR\N, developed at the University of Liege, was used to analyze reported experimental results and to simulate the deformation response. The approach comprises three steps. The first step is to determine the respective strength reduction factors of concrete and steel and the modulus of elasticity at elevated temperature on the basis of numerical simulations using \ISAFIR\N. The second step is to calculate the balanced failure point of the column, so as to determine whether the applied load eccentricity is small or large. The third step is to determine the failure load by a trial and error process. This method can be regarded as an extension of the existing American Concrete Institute method for ultimate strength predictions of RC columns at ambient temperature.

The popularity of high strength concrete in the construction industry propels the need for a revision in the current shear design equations for deep beams in EC2. This paper proposes shear design equations that are applicable for both normal and high strength concrete beams. The comparison study involves a total of 151 beams, out of which 24 were the ‘ideal’ beams. The idea of simulating ‘ideal’ beams using finite element modelling (FEM) was to provide more data points in a controlled manner to benchmark the proposed shear design equations. Additionally, parametric variations are introduced in the FEM simulation. It is noteworthy that the proposed shear design equations adequately predict the ultimate shear strengths of deep beams of various shear span-to-depth ratios (0·27–2·70), main reinforcement contents (1·23–5·80%), web reinforcement contents and concrete grades (covering both normal and high strength concrete from 25–100 MPa).

An elastoplastic analysis of semirigid steel frames subjected to elevated temperature is presented. Within the discretized structural framework, an updated Lagrangian approach is used to incorporate the nonlinear geometric effects on the equilibrium and constitutive conditions. Modeling of elastoplastic behavior is based on the generalized plastic hinge concept for beam-column elements that obey piecewise linearized yield criteria. Under the proportional temperature variation, the analysis for finite temperature increment becomes the standard form of a nonlinear complementarity problem in mathematical programming (MP) formalism. An iterative predictor–corrector scheme coupled with an MP-based method is used in the analytical procedures. Numerical studies are carried out on rigid and semirigid frames. It is found that the benefits of using a full-strength rigid connection over the semirigid one are not unlimited, particularly on the frames with fire-unprotected columns.

A finite element model has been extensively applied to verify the accuracy of the proposed Rankine formula for determining the fire resistance of bare steel columns. The parameters involved include relative slenderness ratio, load eccentricity, steel grade, and initial imperfections of initial crookedness and residual stresses. Through these parametric studies, it is possible to establish the safety limits on the application of the proposed Rankine formula. It is realised that incorporating a deflection control criterion in the evaluation of column elastic-critical loads helps to improve the overall performance of the formula. The Rankine formula is then compared to a series of 68 eccentrically-loaded column tests from the literature. The comparison study shows good agreement with both test results and analytical predictions.

This paper takes account of the effect of web reinforcement, be it in vertical, horizontal, inclined or orthogonal configurations. This is because test results have indicated that the contribution of web reinforcement to shear strength is indeed significant and should not be ignored. An original and rational strut-and-tie model for the tensile contribution of both reinforcement and concrete is proposed, which takes account of their influence on the principal tensile stress. The proposed model is verified against three case studies of a total of 116 beams. Generally, the predictions are not only accurate and consistent in each case study, but also conservative.

A linear programming (LP) formulation is proposed for the evaluation of the plastic limit temperature of flexibly connected steel frames exposed to fire. Within a framework of discrete models and piecewise linearized yield surfaces, the formulation is derived based on the lower-bound theorem in plastic theory, which leads to a compact matrix form of an LP problem. The plastic limit temperature is determined when the equilibrium and yield conditions are satisfied. The plastic mechanism can be checked from the dual solutions in the final simplex tableau of the primal LP solutions. Three examples are presented to investigate the effects of the partial-strength beam-to-column joints. Eigenvalue analysis of the assembled structural stiffness matrix at the predicted limit temperature is performed to check for structural instability. The advantage of the proposed method is that it is simple, computationally efficient, and its solutions provide the necessary information at the limit temperature. The method can be used as an efficient tool to a more refined but computationally expensive step-by-step historical deformation analysis.

The behavior of concrete exposed to fire is highly influenced by stress-temperature path. Any stress-temperature path consists of two basic paths: the path of constant temperature but subjected to increasing stress (TS path), and the path of constant stress but subjected to elevated temperature (ST path). The concrete mechanical strain for the TS path can be obtained experimentally, but it is difficult to obtain the mechanical strain for the ST path due to nonuniform temperature distribution within the concrete specimen. In this paper, through analyzing the stress versus mechanical strain curves of concrete from test results for the TS path, a new concept, known as the “temperature-flow mapping” method, is proposed. Based on this concept, temperature versus mechanical strain relationships of concrete for the ST path can be obtained from those curves for the TS path. Mathematical expressions for the concept are also given in the paper. It is found that the predictions based on this concept are in good agre...

Building structures subjected to fire will generally experience complex force-temperature paths. In this paper, these paths are expressed simply by two basic paths: the path of constant forces but subjected to elevated temperature (FT path), and the path of constant temperature but subjected to applying forces (TF path). A total of 13 beam specimens subjected to two such basic paths were tested. The results show that the fire resistance for the FT path is different from that for the TF path; the former is always greater than the latter, and the two are similar only when the temperature exceeds 525°C or thereabout. However, there are always great differences between their deformations. Therefore, the behaviors of reinforced concrete flexural members subjected to fire are closely dependent on the force-temperature paths they experience. Thus, in fire-resistant design, especially for statically indeterminate structures, to more accurately ascertain their fire resistances the force-temperature path factor can...

A finite element formulation for the analysis of two-dimensional steel frames either at ambient or elevated temperature is presented. The formulation includes both geometric and material nonlinearities, and a creep model. A new feature is the manner in which strain reversal is incorporated into the program. The program can analyze steel frames subjected either to (1) increasing external loads at ambient temperature or, (2) constant external loads at elevated temperatures. The program then gives predictions of (1) a collapse load or (2) critical temperature, whichever the case may be. This program has been applied to a number of experimental and analytical benchmark tests. Excellent agreement has been obtained in all these cases. Following on these benchmark tests, creep effects on the buckling of heated columns have also been studied. It shows that creep generally starts to be dominant beyond 400°C. In connection to this, the current simplified method for calculating the critical temperature in Eurocode 3...

Previous research work has found that catenary action can significantly increase structural resistance in addition to flexural capacity under column removal scenarios. However, whether reinforcements in beams can effectively function as ties to develop catenary action against progressive collapse is a big concern in current engineering practice because of the limited rotational capacity of RC beam-column connections. Therefore, four RC frame specimens were designed and tested to investigate their structural behavior under a column removal scenario. In addition to a specimen designed with conventional detailing in accordance with ACI 318-05, the other three specimens were designed with special detailing techniques at little or no additional cost, endeavoring to improve catenary action capacity at large deformations without reducing the structural resistance at small deformations. The special detailing included placing an additional reinforcement layer at the midheight of beam sections, partially de...

A series of experiments has been conducted to investigate the failure modes and ductility of composite beam-column joints in a middle-column-removal scenario. Composite web cleat and flush end plate connections were studied. The research included two types of tests: namely, middle joints under a sagging moment and side joints under a hogging moment. Five composite beam-column joints with reentrant steel profile decking were tested. The contributions from two types of mechanisms (namely, flexural action and catenary action) have been identified. The main objective of the experimental program is to study the behavior of composite joints in a middle-column-removal scenario. The test results demonstrate the ductility and load resistances of these five specimens in catenary action mode, and indicate that at the initial loading stage, the applied load was resisted by flexural action, while at the large deformation stage, the load was resisted by catenary action. It is also found that composite slabs could increase the load-carrying capacities of beam-column joints at both the flexural action and catenary action stages. The strengthened web cleat connection has a much higher load-carrying capacity than the normal web cleat connection because the former could sustain greater deformation.

The nature of progressive collapse is a dynamic event caused by accidental or intentional extraordinary loading. Most published experimental programs are conducted statically, without any consideration of the accidental loading and treating progressive collapse as threat independent. This paper demonstrates the more realistic process of progressive collapse in an experimental program on reinforced concrete subassemblages collapsed by a combination of dead weight loading and contact detonation. The dynamic results are represented systematically at different aspects and compared with previous published quasi-static experiments in terms of structural mechanisms, crack patterns and local failure modes. Moreover, the dynamic increase factor (DIF) of reinforcing bars and the dynamic load amplification factor (DLAF) are investigated and discussed. Following the above comparisons and the findings in the dynamic tests, previous quasi-static test results can be linked to actual progressive collapse behavior more convincingly. Finally, the dynamic tests also highlight the effect of contact detonation on structures, which are often not considered in quasi-static tests and design guidelines. The test results indicate that contact detonation causes uplift and out-of-plane actions to the subassemblage before their downward movement under gravity load, in which the strain rate of reinforcement is between 10−2 and 10−1/s. Moreover, the structural mechanisms are similar in both quasi-static and dynamic tests.

This paper presents the experimental observations and results of three one-quarter scale composite slab-beam systems, 3.15 m by 3.15 m in plan, and tested in fire conditions. The tests aimed to examine the effects of unprotected interior secondary beams and edge rotational restraint on the behaviour of floor assemblies. The test results show that continuity of reinforcement in the slab over the supporting beams, and the presence of interior beams, can reduce the slab deflection and enhance its load-bearing capacity. Interior beams can be left unprotected without leading to a structural failure. The interior beams play a major role in helping the slab to move from biaxial bending stage to membrane behaviour, enabling the slab to mobilize higher tensile membrane forces. Rotational restraint along the protected edge beams induces intense stress concentration above these beams, resulting in more severe concrete crushing at the four corners and wide cracks over the edge beams. In addition to the experimental study, a numerical model using ABAQUS has been developed to simulate the tests. The numerical predictions agree well with the experimental results, showing that the proposed model is reliable. A shortcoming of the study is that the fire resistance performance of the specimens cannot be compared with those in practical design because a real furnace fire and small-scale fire tests were used due to limits of the furnace. However, the experimental results do provide basic information on the membrane behaviour in fire and also allow analytical methods and numerical models to be validated. ASTAR (Agency for Sci., Tech. and Research, S’pore) Accepted version

The behaviour of structures subjected to progressive collapse is typically investigated by introducing column-removing scenarios. Previous experimental results show that large-deformation performances of reinforced concrete (RC) assemblages under a middle column removal scenario (MCRS) involve discontinuity due to bar slip and fracture near the joint interfaces. To consider the effects of the discontinuity on structural behaviour, a component-based joint model is introduced into macromodel-based finite-element analysis (macro-FEA), in which beams are modelled as fibre elements. The joint model consists of a series of non-linear springs, each of which represents a load transfer path from adjoining members to a joint. The calibration procedures of spring properties are illustrated systematically. In particular, a macro-bar stress–slip model is developed to consider the effects of large post-yield tensile strains and finite embedment lengths on the bar stress–slip relationship. Comparisons of simulated and observed responses for a series of RC assemblages indicate that macro-FEA incorporating the joint model is a practical approach to simulate the essential structural behaviour of RC assemblages under a MCRS, including catenary action. Finally, the macro numerical model is used to investigate the effects of boundary conditions, bar curtailment and beam depth on the structural behaviour of RC assemblages. The results suggest that beam depth affects the fixed-end rotation contributed by bar slip, and further significantly influences the development of catenary action.

The strut-and-tie method is a promising analytical tool for the design of disturbed regions where the conventional plane bending theory does not hold. These include pile caps, corbels, frame joints, deep beams, and so forth. This paper focuses on the application of a strut-and-tie model to prestressed deep beams. A simple and direct model is proposed, taking the prestressing effect into consideration. The model can be used for both pre- and posttensioned deep beams. Different configurations of web reinforcement can also be properly modeled in the proposed approach. A comparison shows that the predictions are in good agreement with experimental results of four case studies of 39 beams.

This paper presents a second-order elastic-plastic hinge method and a finite-element model for the analysis of plane frames in fire. Both methods are derived from the same two-node corotational beam element and are based on the plastic theorems. Hence, this paper first presents the extension of the three classical plastic theorems, viz, the Lower-Bound Theorem, Upper-Bound Theorem, and Uniqueness Theorem, to structural analysis incorporating thermal effects. The three plastic theorems are given new definitions, followed by original mathematical proofs. The basic formulations of the two analytical methods are then described. The proposed finite-element model has been validated against a series of published test and analytical results on different structure types. For comparison purposes, numerical examples are presented to examine the relative performance of the three methods investigated in this study. Discussions on the validity of the methods and their limitations are then addressed.

Microstructure development of eutectic solder alloy (63Sn/37Pb) after thermal cycling aging and its impact on the shear and fatigue failure of the solder joint has been investigated. The solder microstructure changes with the reflow process and subsequent thermal cycling environments in solder joint reliability tests from −40 to 125°C. In service, solder joints are subjected to thermal cycling aging, corresponding to power on–off cycling of the electronic equipment or cyclic environmental temperature loading, leading to thermal fatigue failures. Thus, it is important to study the effect of the microstructure changes, mechanical strength and fatigue resistance of solder before and after thermal cycling aging. A new specimen design has been developed to closely resemble the actual electronic packaging assembly condition. The joint is made simply by soldering a solder ball between two FR-4 substrates with copper pads using the reflow process. The study shows that the solder microstructure coarsened and intermetallic compound layers grew after 500, 1000 and 2000 thermal cycles. The shear and fatigue strength of the solder joint decreased with increased exposure to thermal cycling aging effects.

A simple analytical approach based on the Rankine principle has been developed to determine the ultimate resistance of steel frames in fire. The proposed Rankine approach gives an approximation of the frames' fire resistance through a simple interaction between two idealized structural behaviors—strength and stability. Here, the strength and stability of the structures are evaluated using the rigid-plastic and the elastic buckling analyses, both incorporating the thermal effects. The proposed approach is first verified using a finite-element model. The verification studies include the effects of column and frame slenderness ratios, beam-column stiffness ratio, steel grades, initial sway imperfections, and initial residual stresses. These studies indicate that frame slenderness ratio is an important parameter governing the behavior of simple frames in fire, and the performance of the proposed approach is related to it. The Rankine approach is then applied to a series of 18 test frames from the literature. ...

A simple and direct analytical formula has been derived to determine the compressive resistance of steel columns subjected to fire conditions. The approach is based on the Rankine principle, which allows the strength and the stability of a steel column at a particular temperature to be separately determined. Here, the evaluation of column strength is based on the rigid-plastic theory, while the stability consideration is based on the elastic-buckling theory. By considering the two factors in this manner, the proposed approach allows for their interaction. One important feature is that the same analytical formula can be used for both axially and eccentrically loaded columns. In addition, the proposed formula does not require any calibrations against test data. The proposed formula has been verified against a set of published column test results and other analytical predictions. The comparison study shows consistently good agreement with both the test and the analytical results.

In this paper, the shear stiffness in the weaker direction of Z-core sandwich panels is studied. The recurrence condition is assumed, and the shear stiffness is derived from the first principle. The contact interaction between the facing plates and the flanges of the Z-sections is considered. Compatibility equations and the shear strain of a typical segment are obtained from the Principle of Virtual Work. Formulas for the shear stiffness in the weaker direction are derived explicitly. It is shown that there are two shear stiffness values associated with the Z-core sandwich panels, depending on the direction of the shear force. The stronger value of a typical panel can be 60% higher than the weaker value. To verify this phenomenon experimentally, two steel sandwich panels with different stiffener spacing were fabricated and tested. Good agreement was obtained between the theoretical predictions and the experimental results.

An expression for the bit-error rate (BER) of 16 STAR-quadrature amplitude modulation (QAM) with differential encoding and detection in a Rician fading channel with diversity reception is obtained. Two types of intermediate frequency (IF) filters are considered in the analysis: the intersymbol interference (ISI)-free matched and nonISI-free Gaussian filters. BER curves for various ratios of the line-of-sight (LOS) power to the multipath power, Doppler spread frequencies, and orders of diversity are presented. It is shown that 16 STAR-QAM outperforms 16 DPSK under the same power-limited condition. For the Gaussian receive filter, a filter bandwidth of about 1.2 times the symbol rate is found to lead to a minimum error probability prior to the appearance of error-rate floors.

A component-based model is developed to predict the dynamic response of bolted-angle connections, such as web cleat connections and top and seat with web angle (TSWA) connections subjected to sudden column removal scenario. This model considers the behaviour of bolted-angles under large tension forces. A failure criterion determined from the test results is introduced into the model for the bolted-angle component to predict the connection resistance. The hysterical behaviour of each component under cyclic loads is also included for ensuring dynamic analysis. The proposed component-based connection models with detailed springs as well as their constitutive laws are implemented within a self-developed finite element programme FEMDYA to validate the model against both static and dynamic test results. A comparison study showed the capabilities of the component-based model in predicting the connection performance. Based on the failure criterion of the connection component, an accurate simulation of the fracture of the connections is conducted. Subsequently, the component model is incorporated directly into two types of 4-storey steel frames to simulate catenary action developed due to the loss of support to a middle column. The analysis results indicate that a value of up to 2.9 should be used as the dynamic increase factor for structures with web cleat connections when incorporating the dynamic effects into the nonlinear static load resistances. It is also shown that the ultimate load capacity of unbraced structure is much smaller than the one in the braced frame due to horizontal movements of adjacent columns under catenary action.

In this paper, a new generalized Drucker–Prager flow rule for concrete under compression with infinitesimal deformation is proposed, where the plastic volumetric strain is taken as the hardening parameter. The flow rule is proposed based on two fundamental phenomenological deformation properties of concrete, and on the fact that the deviatoric part of the plastic strain has no contribution to plastic volumetric strain. There is only one material parameter needing calibration for each grade of concrete in the flow rule. This parameter has a clear physical meaning to reflect the brittleness of concrete, and is defined as the brittleness index of concrete. The flow rule has a concise and simple form similar to the traditional Drucker–Prager flow rule, and for the case of uniaxial compression, the proposed flow rule is identical to the traditional Drucker–Prager flow rule. The effect of confinement stress is well considered by multiplying the hydrostatic part of the flow rule with a simple piecewise function, which differentiates the proposed flow rule from the traditional Drucker–Prager flow rule. An iterative method is proposed to determine the incremental stress–strain relationship when determining the flow rule. The correctness and reliability of the suggested flow rule are is validated using uniaxial, biaxial, and triaxial experimental results.

A C-core sandwich panel consists of two facing plates connected top and bottom to a core of C-shaped stiffeners. Sandwich panels of this type have high bending and shear-stiffness characteristic in the direction of the core. In the transverse direction, however, both stiffness properties are relatively weak, in particular the shear stiffness. In this paper, the shear stiffness in the weaker direction for C-core sandwich panels is derived and studied. Due to the geometry and discontinuity of the core configuration, the shear stiffness in the weaker direction must be derived from the first principle. By assuming the recurrence condition, the writers previously derived an approximate expression for the shear stiffness in the weaker direction. The work described here presents a more accurate formulation, taking into account the contact interaction between the facing plates and the flanges of the C-core stiffeners. The compatibility equations are obtained from Castigliano's second theorem. The shear strain of ...

Fire resistance of steel joints is one of the major concerns when designing structures against extreme hazards. However, little information is available for Circular Hollow Section (CHS) joints in fire condition, especially forT-joints. In order to gain greater insight into the static behaviour ofCHSconnections, this paper presents an experimental study of CHS T-joints subjected to in-plane bending with variation of brace-to-chord diameter ratios (β) under elevated temperatures. The tests were conducted with five specimens in isothermal heating condition where the joints were heated to the desired temperatures and then subjected to static load until failure occurred. The failure modes and ultimate strengths of such joints under various isothermal temperatures were investigated and compared with the corresponding joints at the ambient condition. Within the range of investigated parameters, at 700◦C, the joint strength was only 8.40 kNm while at ambient temperature, it was 43.25 kNm. Furthermore, at high temperature, there was a change in the failure modes of the joints. Cracks formed around the weld toes before the joints reach excessive deformation. These cracks affected post-yield strength in moment-rotation curves of the joints. In order to understand the development of cracks around the welded region, material tests were performed to determine the fracture strains of chord material beneath the center weld at high temperatures. To the authors’ best knowledge, these structural tests were among the first reported experimental investigations on the ultimate strength and failure mechanisms of tubular joints at elevated temperature.

Traditional design of connections is limited to moment-rotation characteristics and the information about the deformation capacities of connection components is quite limited. Thus, it is necessary to develop a new mechanical model, which could predict the deformation capacity of each connection component and the development of catenary action. In this paper, an experimental investigation of bolted-angle connections under tension is presented. Based on the experimental results, a new mechanical model of bolted-angle connections is developed. In the proposed model, the following issues are addressed: (i) the interaction between bolts and angles, (ii) failure criteria to determine the deformation capacities of connection components, and (iii) load limits due to bolt fracture. Finally, the proposed mechanical model is validated by the experimental results and the predictions of the model agree well with the test observations.

It is known from previous experiments that the load-carrying capacity for RC slabs at large deflection is governed and enhanced by the tensile membrane action. However, due to the intrinsic complexity of concrete properties, it is hard to capture the history of the load-deflection relationship for RC slabs. After reviewing several analytical reinforced concrete models, the author develops a simplified load-displacement semi-analytical model for laterally unrestrained simply supported RC slabs. Three stages have been identified for this model, which are elastic stage, transition stage and pure tensile membrane action stage. Compared with the existing RC slab tests, the author's model shows good agreement with the previous test results.

The computational effort to analyze a sandwich panel can be greatly reduced if an equivalent 2D thick-plate bending model is used instead of a detailed 3D finite-element model. The success of the transformation from a sandwich panel to a 2D thick-plate bending model depends on the equivalent elastic constants used. In this note, these elastic constants for Z-core sandwich panels are summarized. The elastic constant for the shear stiffness in the weaker direction for Z-core sandwich panels is derived. To ascertain the validity of the elastic constants, the results obtained from the 2D thick-plate bending model with different boundary conditions are compared with a detailed 3D finite-element model. From the results, it is shown that excellent agreement is obtained between the two models.

This paper is concerned with the ultimate load prediction of a sandwich panel manufactured from a steel plate assembly. The panel essentially consists of two facing plates, spot welded onto a corrugated steel core. All the plates of a panel have identical thickness. Three groups of panels with varying plate thicknesses were examined. The typical panel considered in this study had plan dimensions of 2·1 m × 1·0 m with a core consisting of top-hat stiffeners of depth 60 mm, placed side by side. The panel was simply supported across its y-direction boundaries and subjected to uniform lateral loading over its entire surface. The plastic hinge theory is used to evaluate the ultimate collapse load of the panel. In this respect, two approaches are compared: one based on fullsection properties and the other based on the effective width concept. It is shown that the effective section gives more reliable collapse-load predictions than the full section. Theoretical collapse-load predictions agree very well with the experimental collapse load for a series of 13 panels. To accentuate the salient features of the technique, a numerical example is presented.

Publisher Summary The chapter presents a numerical study of the rotational capacity of steel beams in fire. The study identifies the key factors affecting the available rotational capacity of steel beams in fire and also studies the interaction of these key factors. A series of UBs, UCs, and fabricated sections having different spans subjected to midspan point load are studied in the chapter. The numerical analysis is conducted using the commercially available finite element program MSC.MARC Mentat. It is found that the rotational capacity reduces at elevated temperatures because the highly non-linear stress-strain curve with reducing tangent modulus results in beams achieving its plastic moment capacity at a higher strain, thus reducing the inelastic rotation thereafter. Besides, three key factors that affect the rotation capacity are identified, namely, the local buckling of flange, the local buckling of web, and the lateral torsional buckling. Lateral torsional buckling often develops rapidly after the onset of flange buckling if there is no sufficient lateral restraint.

Publisher Summary The chapter discusses an alternative approach for lateral torsional buckling of unrestrained beams in fire. A beam subjected to flexure of a stronger axis may fail below its plastic moment capacity unless it is a short-span beam or it has sufficient lateral restraints. This limit state of structural usefulness where the deformation of a beam changes from predominantly in-plane deflection to a combination of lateral deflection and twisting, while the load capacity remains constant, before dropping off due to large deflections and yielding is called lateral torsional buckling. A laterally unrestrained long-span beam will fail by elastic lateral torsional buckling at a load much lower than its full plastic moment capacity. For a beam of intermediate span, failure may occur by inelastic lateral torsional buckling. The chapter proposes an alternative approach based on the existing BS5950 method at ambient temperature. The equation incorporates the material degradation factor at elevated temperature and stress-strain non-linearity factor. The non-linearity factor is introduced inside the Perry imperfection factor to consider the highly non-linear stress-strain relationship at elevated temperatures. This alternative approach matches finite element analysis (FEA) trend well and gives quite accurate predictions of test results by Vila Real et al. Applying this new approach, the over-conservatism of the EC3 approach for temperature below 500°C can be reduced. There is also continuity with the BS5950 equation at the ambient temperature, thus providing continuity in design between the ambient and the elevated temperatures.

This paper presents a simple and rational interaction formula -- the Rankine method -- for predicting the fire resistance of reinforced concrete (RC) columns under fire conditions. Conventionally, the Rankine method has been applied to RC columns and frames subjected to increasing loads but maintained at ambient temperature. Based on a theoretical approach, the Rankine method is extended to RC columns subjected to fire conditions by taking the temperature effects on steel reinforcement and concrete material into consideration. The accuracy of the method is verified through four sets of experimental test results comprising 76 columns. The method yields reasonably accurate predictions for the fire resistance of RC columns. In addition, the paper includes two worked examples to illustrate the flexible use of such a formula for either predicting the load-bearing capacity or the fire exposure time of an RC column subjected to the standard ISO 834 fire curve.

The paper studies the general behavior of a thermal strained steel beams under a fire condition. The governing factors under consideration comprise the load utilization factor, slenderness ratio and axial restraint ratio. Numerical analyses demonstrate that instead of the load utilization factor, it is beam axial restraint that substantially influences the response of beam towards the end of heating. This is due to the catenary action that is experienced by all beams under study. Most importantly, numerical study shows that the deflections which develop under an increasing temperature, are not controlled by the steel material degradation, but by member axial restraint. At very high temperature, the plastic and creep strain in together with post-buckling large displacement, retard the failure of a restrained beam.

This paper describes an experimental program on the size effect in reinforced concrete deep beams. A total of 12 large- and medium-sized specimens with overall height ranging from 500 mm to 1,750 mm and effective span from 1,500 mm to 4,520 mm were tested to failure under two-point symmetric top loading. The beams had compressive cylinder strengths of approximately 40 MPa and main steel ratio of 2.60%. Test results reveal that the ultimate shear stress is size-dependent and that Bazant's law can best describe this size effect. On the other hand, the diagonal cracking stress is hardly size-dependent. Besides the shear span-to-height ratio, the size effect also has a significant influence on the failure mode; larger deep beams are more brittle in comparison with smaller ones. Some plausible explanations are given to the source of size effect in deep beams. It may arise from different rates of release of fracture energies associated with crack propagation in beams of different sizes. This is demonstrated from the crack patterns of geometrically similar beams at the same nominal shear stress, where it is obvious that crack development was more extensive in larger specimens. The 12 test results are then compared with predictions from the current American Concrete Institute (ACI) Code, the U.K. CIRLA Guide-2, and the Canadian CSA Code. Comparison study shows that while the ACI Code predictions do not have uniform safety margin and estimations from CIRLA are generally unsafe for large deep beams, the strut-and-tie-model predictions in the Canadian Code yield uniform safety margin.

Twenty-two reinforced concrete deep beams with cylinder compressive strengths f 1 C generally exceeding 55 MPa (8000 psi) were tested under two-point symmetric top loading. Based on the main steel ratio ρ w the beams were organized into four groups with ρ W = 2.00, 2.58, 4.08, and 5.80 percent. Web reinforcement comprising 10 mm (0.4 m.) diameter plain mild steel stirrups at 300 mm (11.7 in.) centers was provided for all specimens, giving a vertical web steel ratio ρ ν of 0.48 percent. The beams were tested for different shear span-to-overall-height ratios a/h, ranging from 0.25 to 2.50 (equivalent to a/d from 0.28 to 3.14). The comparisons among the series serve to highlight the influence of ρ w and a/h ratio on the shear behavior of high strength concrete deep and short beams. It is shown that the a/h ratio (or equivalent a/d) dominates the failure modes while the beneficial effect of ρ w is more significant at the low end of a/d, say for a/d 1.50, the influence of the main steel ratio declines, except for the particularly high value of 5.80 percent, where the relative increase in shear strength due to main steel remains constantly high, regardless of a/d. The test results are then compared with predictions based on the current ACI Code, the Canadian Code, and the UK CIRIA Guide-2. It is shown that the ACT predictions are generally conservative, with the smallest standard deviation, though with a/d 1.50, a few cases are overestimated. The predictions from the Canadian Code are comparitively good, but the UK CIRIA Guide-2 estimations are generally unconservative, with the greatest scatter. The study shows that the CIRIA Guide-2 predictions may be unconservative for specimens with f' C ≥ 55 MPa (8000 psi) and with ρ W ≥2.58 percent.