Load Contour Research Articles

Biaxial bending of RC rectangular column sections: Improved ‘load contour’ formulation

AbstractIn design practice, the safety of any given rectangular reinforced concrete column section is checked by ensuring that all possible combinations of axial compression (Pu) and biaxial moments (Mux, Muy) at the ultimate limit state lie within its three‐dimensional load–moment interaction failure surface. Although such a check is facilitated by commercial software packages, it is traditionally carried out with the help of an approximate formulation of the ‘load contour’ at any given axial load level, first proposed by Bresler. The present study proposes an efficient algorithm, using a ‘modified thick layer integration’ approach and a ‘nested bisection’ method, to generate the ‘exact’ load contour (and thereby, the complete interaction surface) for any symmetrically reinforced column section, complying with the strength design requirements of ACI 318 (or EC2 or IS 456). The algorithm also generates a measure of the safety margin, called ‘demand‐to‐capacity ratio’ (DCR), defined as the ratio of the resultant design moment (Mu) to the moment capacity (Muθ) at a given axial load level (Pu). For an optimal code‐compliant design, the peak value of DCR generated from various factored load combinations (obtained from frame analyses) must be close to 1.0. The extent to which the peak DCR is less (or greater) than 1.0 provides a measure of over‐design (or under‐design) in a given context. After validating the algorithm, a parametric study was carried out on a spectrum of 36 different cross‐sections having different aspect ratios, bar arrangements and reinforcement percentages. The study reveals that the exponent in the approximate load contour formulation depends not only on the axial load (as currently assumed in ACI: SP17, EC2 and IS 456), but also on the percentage of longitudinal reinforcement. Accordingly, modifications have been suggested to improve the accuracy in these code‐prescribed approximate formulations of the load contour, for more economical and safe design.

Derivation of a normalised failure surface for square CECFST columns embedded with circular steel tube: A unified approach

Concrete-Encased Concrete-Filled Steel Tube (CECFST) columns are a new type of composite columns that possess great strength, stiffness, ductility, corrosion and fire resistance. To fully realise their benefits, an accurate design approach of calculating the strength of these members is necessary. In this paper, the plastic stress distribution (PSD) and load contour (LC) methods in the design code of predicting the cross-section strength of CECFST columns is first evaluated through numerical and published experimental data. Considering the shortcomings of conventional design methods, a unified design approach is then developed. Compared to the conventional PSD method, the developed method can yield conservative resistance predictions for uniaxial eccentric loaded (UEL) CECFST columns. In addition, instead of using linear moment interaction curves to predict the strength of biaxial eccentric loaded (BEL) CECFST columns, elliptical moment interaction curves are adopted in the proposed design approach. Most importantly, a normalised 3D axial-load–biaxial-moment (n–my–mz) interaction surface can be generated through the proposed method, which only requires determining 6 critical points on the interaction surface. Unlike the conventional design approach, in which the UEL and BEL composite columns need to be respectively designed through PSD and LC methods, a unified design procedure can be achieved through the proposed approach. Besides, the confinement effect from both the steel tube and the stirrups can also be considered in the proposed approach and much higher accuracy and consistency can be achieved in predicting the cross-section strength of CECFST columns.

ON OPTIMAL DESIGN OF REINFORCED CONCRETE COLUMNS SUBJECTED TO COMBINED BIAXIAL BENDING AND AXIAL LOAD

This paper presents a new procedure for designing rectangular columns subjected to combined biaxial bending and axial load. The new procedure is realized within a new comprehensive structural analysis software for building analysis and design. Compared to existing engineering packages, the new program does not require pre-setting the distribution of bars before designing and gives more detailed results: not only the required reinforcement amount, but also a distribution of bars of a certain diameter along the sides of the column section. The program starts by inserting four reinforcement bars into the corners of the column, and checks if this reinforcement bars provide required resistance to the applied loads. If the reinforced concrete column fails to withstand the applied loads, then the program increases the reinforcement by adding new bars and checks required resistance again. This process continues until the reinforced concrete column is able to withstand the applied loads. The proposed procedure is realized within following methods: (I) calculation of the so-called column capacity interaction volume using a large number of interaction curves (usually 24 curves); (II) the elliptical approximation of the column capacity interaction volume (the load contour equation). Within the second method, the program calculates two interaction curves only: the first curve is for an eccentric case in X direction and the second curve is for an eccentric case in Y direction. In both methods, the interaction curves are calculated according to American concrete institute standards. The use of the new procedure within the second method gives results very close to the first method (the most accurate), but it requires much less time for calculation. This is very important when designing large structures with multiple columns.

Evaluation of Biaxial Strength Deterioration for Coastal Bridge Piers under Non-uniform Corrosion

Coastal bridge piers in the corrosive marine environment inevitably deteriorate over time, leading to a reduction in their strength. In the present study, a biaxial strength prediction method for non-uniform corroded bridge piers under axial force and biaxial bending moments is proposed by extending the applicable range of the load contour method. First, a method to establish the three-dimensional interaction surface is developed on the basis of the fiber-section model, which considers the two-dimensional chloride diffusion, the degradation of reinforcement and concrete. The proposed methodology is verified by a biaxial quasi-static test of the locally corroded bridge column specimens. Then, through further case study, the time-dependent characteristics of the interaction surface for bridge piers in the splash and tidal region and the atmospheric region are investigated. Finally, the expression of surface exponent is obtained from the interaction surface, which can be used in Bresler's load contour method to evaluate the deterioration in the biaxial strength of coastal bridge piers. The analytical results conclude that the expression of surface exponent can be divided into three stages including the constant stage, decrease stage and increase stage during the pier's service life. Axial compression ratio and corrosion degree are two important factors which influence the biaxial strength deterioration for different corrosion regions of the coastal bridge pier.

Influence of ambient conditions on evaporation and transport of respiratory droplets in indoor environment

Respiratory droplets are playing a significant role in the transmission of any flu type disease as well as SARS-Cov-2 virus. The presence of pathogens affects the evaporation of the liquid droplets along with ambient temperature and relative humidity (rh). Complete evaporation of droplets leads to the formation of aerosol or droplet nuclei which remain suspended in the air for a longer period of time and get spread over larger distances increasing the risk of disease transmission. In present work, a droplet evaporation model has been formulated considering the droplet as a salt solution and the formation of crystals has been taken into account which will be analogous to the aerosol formation. After the establishment of the evaporation model, the trajectories of the droplets are investigated considering a turbulent round jet model during exhalation. Aerosols are found to be spreading over distances of 8 to 9 m which is quite alarming. Large droplets get converted to smaller ones but the viral loading of the large droplets is much higher than the smaller as viral loading is proportional to initial size. This is highlighted by the viral load contour and the mean diameter line contour for a half-height window. Different weather conditions are investigated to observe the evaporation of droplets and the formation of aerosols in order to qualitatively analyse the risks associated with each city in specific weather conditions. Hot and dry conditions are most favourable to aerosol formation.

Strong-column–weak-beam criterion for reinforced concrete frames subjected to biaxial seismic excitation

In several earthquakes, numerous reinforced concrete (RC) frames subjected to seismic excitation exhibit a collapse pattern characterised by column hinges, although these frames are designed according to the strong-column–weak-beam (S-C–W-B) criterion. The effect of biaxial seismic excitation on the disparity between the design and the actual performance of RC frames is investigated in this study. First, a modified load contour method was proposed to derive a closed-form equation of the biaxial bending moment strength, which was verified based on numerical and experimental results. Subsequently, a group of time-history analyses of a simple frame modelled using fibre beam-column elements subjected to biaxial seismic excitation were conducted to verify that the current S-C–W-B criterion is inadequate for preventing the occurrence of column hinges. A biaxial overstrength factor was developed based on the proposed equation, and the reinforcement of columns was appropriately amplified using this factor to prevent the occurrence of column hinges under biaxial excitation; this approach was proved to be effective through another group of time-history analyses. Thus, the findings of this study can serve as a basis for improving the seismic design of RC frames, which will enable the construction of safer engineering structures.

A New Microhardness Testing Method on Grating Films

The manufacturing process of large-area, high-precision gratings is a very complicated and time-consuming process. The hardness testing of grating films is an important step in the entire process. In order to simplify the manufacturing process of gratings, we have proposed a new method for testing microhardness based on tool edge indentation. Also, it unified tool adjustment and microhardness testing steps in the grating manufacturing process. First, a mathematical model of the relationship between tool load and indentation contour length is established. The model parameters were then modified using tool indentation experiments with different loads. When measured with a nanoindenter, the average hardness of the grating film was 447 MPa. The hardness value of the grating film obtained by our proposed method is almost the same as that measured by the nanoindenter, and the maximum deviation is about 2.2% of the average hardness value. The experimental results show that our proposed method can replace the microhardness test method of using a nanoindenter. Therefore, the disadvantages of using a nanoindenter to test the hardness of a grating film are avoided, such as the limited sample size, the sensitivity of the indenter to the roughness of the film and the depth of the indentation, and the accuracy of film testing, and the efficiency of grating ruling can be improved.

Comparison between resistant load contours generated considering the parabolic-rectangular (DPR) and the rectangular (DR) stress-strain diagrams for rectangular sections under combined axial compression and biaxial bending

Abstract The aim of this study is to compare the load contour diagrams generated for rectangular RC cross-sections under combined axial compression and biaxial bending obtained by the two forms of analysis allowed by NBR 6118:2014 [1]: the first using the parabolic-rectangular stress-strain diagram (DPR) and the second using the rectangular (constant stress) diagram (DR). In order to compare the load contours generated, a reference cross-section was adopted for which the concrete strength class (from C20 to C90) and the deformation domains (4, 4a and 5) were varied for the study. It was studied whether the use of the different diagrams (DPR or DR) would provide greater (or smaller) resistant efforts for the same section. The results show that the use of the DR is only acceptable when the section is working up to the 4th domain. Above this domain, it was observed that the use of this diagram shows resistant efforts inferior to those calculated by the DPR. In addition, it was found that, for concretes with resistance class above C50, in oblique loading directions, the use of the DR presents higher resistant efforts than those calculated using the DPR.

Biaxial Interaction and Load Contour Method of Reinforced Concrete T-Shaped Structural Walls

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Reliability assessment of reinforced concrete rectangular columns subjected to biaxial bending using the load contour method

This paper presents the reliability assessment of reinforced concrete (RC) columns subjected to biaxial bending using the advanced first-order second-moment reliability method. The strength of an RC column under the biaxial bending condition is estimated by the load contour, which is utilized as the limit state function in the reliability analysis. The random variables include the strength parameters of the RC columns, the load parameters and the eccentricities of axial force. The Hasofer-Lind-Rackwitz-Fiessler algorithm with the gradient projection method is employed to calculate the most probable failure point and the reliability index. The surface exponents of the load contours are determined by the values presented by Bresler. The cubic spline interpolation is adopted to construct P-M interaction diagrams for uniaxial bending about the principal axes that are required in the load contour method. The sensitivities of the limit state function are calculated through the direct differentiation method. The reliability indices are calculated for a rectangular section and a square section. The effect of the surface exponent on the results of the reliability analysis is investigated for the rectangular section, and the uncertainty of the eccentricity of axial force is considered in the example for the square section.

Load contour method for biaxial design of reinforced concrete L-shaped structural walls

Abstract Biaxial behavior of L-shaped walls differs from that of rectangular or circular columns with symmetrical solid sections, due to the unsymmetrical section geometry. In this study, a straightforward design method for biaxially loaded L-shaped walls is developed by modifying the existing load contour method. Using the strain-compatibility section analysis method, a parametric study on the prototype L-shaped walls is performed to investigate the interaction of biaxial moment strengths at constant axial load. The results show that, due to the unsymmetrical section geometry of L-shaped walls, the contour of biaxial moment strengths is highly unsymmetrical in shape and significantly affected by the moment directions and axial loads. Based on the results, elliptical contour equations for the biaxial moment strengths of L-shaped walls are proposed. In addition, a design example illustrating the biaxial design procedure using the proposed method is given.

C형 및 H형 철근콘크리트 구조벽체의 2축 상호작용과 등하중법

C형 및 H형 단면의 구조벽체는 고층 건물에서 횡력저항시스템으로 널리 이용된다. 이러한 이형벽체는 축력과 함께 x축 및 y축에 대한 2방향 휨모멘트를 동시에 받으므로, 안전한 벽체설계를 위해서는 휨-압축 상호작용을 정확히 고려해야 한다. 이 연구에서는, 대칭단면을 갖는 기둥을 위하여 개발된 기존 등하중법을 수정하여, 2방향으로 재하된 C형 및 H형 벽체를 위한 근사설계방법을 제안하였다. 다양한 단면형상을 갖는 이형벽체에 대하여 2방향 모멘트강도를 계산할 수 있는 단면해석 프로그램을 개발하고, 실험결과와 비교를 통하여 프로그램의 정확성을 검증하였다. 또한 개발한 프로그램을 사용한 변수연구를 통하여, C형 및 H형 이형벽체 단면에 대한 2축 상호작용 특성을 분석하였다. 분석 결과, C형 및 H형 이형벽체의 2축 상호작용은 모멘트 방향과 압축력 크기에 의하여 크게 영향을 받는 것으로 나타났다. 이러한 변수연구를 통하여 일정한 압축력에서 2축 모멘트강도의 상관관계를 나타내는 정규화된 컨투어 설계식을 제안하였다. 또한 실무에서 쉽게 활용할 수 있도록, 제안된 컨투어 설계식을 사용한 이형벽체 설계절차와 설계예제를 제시하였다.

Deformation capacity of non-conforming r.c. columns under compressive axial load and biaxial bending

In a performance based approach, the knowledge of ultimate capacity of reinforced concrete (r.c.) columns subjected to simultaneous compressive axial load and biaxial bending becomes essential to correctly assess the seismic capacity of existing structures. In the present work, the reduction of ultimate deformation (i.e. rotational) capacity of r.c. members due to the two components of bending moment is analysed by using a specific simplified numerical model. The influence of axial load, geometry, amount of longitudinal reinforcement and mechanical properties on the ultimate chord rotation of r.c. columns is investigated on a dataset of 780 numerical tests on non-conforming columns with plain bars governed by flexural mode. Then, a practice-oriented formulation for the ultimate rotation domain, based on the Load Contour method extended to the chord rotations, is proposed and discussed.

Optimum Cost Design of Reinforced Concrete Columns Using Genetic Algorithms

The aim of this study is finding the optimum cost design of reinforced concrete columns with all loading conditions (axially, uniaxially and biaxially loaded) using the Genetic Algorithms GAs. Many design constraints were used to cover all the reliable design results, such as limiting the cross sectional dimensions, limiting the reinforcement ratio and even the behavior of the optimally designed sections. Each of the designed columns was handled by the GAs solver according to its loading condition specifications. The load contour method was used to design the biaxial sections with the adjustment of the plastic centroid. A long column constraint was introduced to limit the design procedure with the short columns only. The optimum results were compared with other published works, and a reduction in design cost of the biaxially loaded columns of about 26 % was achieved using the GAs design method while a small percent in the cost reduction ( 1 – 3 % ) was achieved for the uniaxially designed sections, while 50% was the cost savings in the axially loaded columns. It was proved that the genetic algorithm is capable for designing optimum columns sections despite the complex constraints that control the designing procedure

An improved design formula for a biaxially loaded slender RC column

Nonlinear analyses are conducted to evaluate the ultimate resisting capacity of slender reinforced concrete (RC) columns subjected to an axial load with biaxial bending moments. Consideration is given to the geometric nonlinearities caused by the P – Δ effect and the long-term behavior of concrete and to the material nonlinearities caused by the cracking of concrete and the yielding of steel. In addition, the biaxial stress state in an RC section is simulated on the basis of a fiber model. Because of the complexity of Bresler’s load contour method, which was introduced in the ACI 318 code, this paper introduces a new design approach to the construction of the failure surface of a slender RC column subjected to biaxial bending. Through a parametric study of slender RC columns, where consideration is given to the P – Δ effect and the time-dependent deformation of concrete, two regression formulas are proposed on the basis of the slenderness ratio and the creep deformation of concrete. Furthermore, the direct multiplication of the proposed formulas on the P – M interaction diagram for a short RC column subjected to axial force and a uniaxial bending moment enables a P – M interaction diagram to be generated for a slender RC column subjected to long-term axial force and biaxial bending moments. Correlation studies between analytical and experimental results are conducted with the objective of establishing the validity of the introduced numerical model. In addition, the ultimate resisting capacities calculated from the regression formula are compared with those obtained from rigorous nonlinear analyses and from the ACI formula, with the objective of establishing the relative efficiency of the proposed regression formula.

Design equations for the assessment and FRP-strengthening of reinforced rectangular concrete columns under combined biaxial bending and axial loads

A simple procedure is proposed for the assessment of reinforced rectangular concrete columns under combined biaxial bending and axial loads and for the design of a correct amount of FRP-strengthening for underdesigned concrete sections. Approximate closed-form equations are developed based on the load contour method originally proposed by Bresler for reinforced concrete sections. The 3D failure surface is approximated along its contours, at a constant axial load, by means of equations given as the sum of the acting/resisting moment ratio in the directions of principal axes of the sections, raised to a power depending on the axial load, the steel reinforcement ratio, and the section shape. The method is extended to FRP-strengthened sections. Moreover, to make it possible to apply the load contour method in a more practical way, simple closed-form equations are developed for rectangular reinforced concrete sections with a two-way steel reinforcement and FRP strengthenings on each side. A comparison between the approach proposed and the fiber method (which is considered exact) shows that the simplified equations correctly represent the section interaction diagram.

Effect of concrete strength on biaxial bending moment

This paper evaluates the ultimate strength of high-strength concrete (HSC) columns subjected to both axial load and biaxial bending. Fifteen square section (200 × 200 mm) columns with concrete compressive strength of 34, 57 and 64 MPa, respectively, were axially loaded with bi-directional eccentricities. The loading point was at an angle of 0°, 22·5° and 45° and the distances of eccentricity were determined to induce tension, balanced and compression failures, respectively. Test results were compared with the results of nonlinear analysis on P–M interaction curves and load contours. It was observed that, as concrete strength increases from 34 to 57 MPa, the concrete strength has a minor effect on the biaxial moment strength of reinforced columns, but as concrete strength increases up to 64 MPa, the concrete strength has a significant influence on biaxial flexural strength. This observed result is complementary when compared with the load contour method of ACI, in which biaxial bending design constants of columns are limited to concrete compressive strength of 41 MPa and estimate the effect of concrete strength to be minor.

A novel, accurate load-pull setup allowing the characterization of highly mismatched power transistors

The measurement of highly mismatched power transistors has always been a difficult problem. A novel, active load-pull technique providing an attractive solution is proposed in this paper. It consists of using suitable mismatched sources to drive the device under test. By using the proposed measurement setup, an electronic simulation of highly reflective loads very close to the edge of the Smith chart can be achieved (reflection coefficients larger than 0.9). Furthermore, the magnitude and phase of the reflected power waves at the output of the transistor under test are accurately controlled so as not to damage the component. Some examples of load contour mappings are given. They demonstrate the promising capabilities offered by this improved large signal measurement tool. >

Structural Load Estimates from Geographically Sparse Data

Structural engineers often must estimate wind or snow loads at a building site using data from weather stations some distance from the site. This paper describes how design loads at a site can be estimated using moving area-averaging rules that take into account inherent randomness in the data and the sparseness of the reporting stations. These rules provide load estimates or contour maps which are specified exactly by the given data.

Influence of Axle Load, Track Gage, and Wheel Profile on Rail-Vehicle Hunting

Analyses have been conducted on the influence of axle load, track gage, and wheel contour on the hunting behavior of simplified models of wheelsets for typical freight- and passenger-car suspensions. The capability of the wheel flange to limit hunting oscillations is found to increase with wheel axle load. The use of worn wheel contours or excessively tight gage is found to increase the susceptibility of the wheelset to excessive and unstable hunting oscillations. For freight-car applications, coulomb friction in the suspension (e.g., constant-contact side bearings) may act to increase the range of speeds over which hunting will not occur and may permit operation at higher speeds for extremely straight track. However, if track irregularities are sufficient to cause a breakout of the friction, drastic hunting oscillations leading to derailment can occur. Regions of stable limit-cycle hunting and unstable operating conditions are defined. Computational algorithms and computer programs for predicting the boundaries of stable, unstable, and limit-cycle behavior for the wheelset and more complex rail-car analytic models, using the describing-function type of analysis, are presented and reviewed.