Rankine Formula Research Articles

Design Philosophy and Seismic Analysis of a Multi-Storey Building (B+G+10) with Flat Slabs in Seismic Zone II Using ETABS and SAFE Software

This journal paper presents the design philosophy and seismic analysis of a multi-storey building (B+G+10) with flat slabs located in Seismic Zone II. With the growing population and limited resources, vertical expansion has become a necessity. The economy's development has attracted foreign companies and innovative entrepreneurs, requiring the construction of high- end, safe workplaces. High-rise structures face challenges from wind and seismic loads, making proper design and analysis crucial for safety. The design philosophy emphasizes the use of reinforced concrete due to its versatility, durability, and fire resistance. Reinforced concrete, a composite material of concrete and steel reinforcement, ensures strain compatibility and load sharing between the materials. The paper discusses various design loads, including dead loads, live loads, wind loads, and earthquake (seismic) loads. Dead loads encompass the self-weight and permanent attachments to the structure, while live loads consider dynamic factors due to occupancy. Wind loads are assessed based on the local wind conditions, and earthquake loads are determined using seismic codes. Design principles for foundations, columns, and footings are addressed, with a focus on ensuring structural stability, safety, and uniform settlement. The depth of the foundation is calculated using Rankine's formula, considering soil properties. Column design is crucial, with short and long column classifications, and minimum eccentricity requirements are detailed. The seismic analysis of the building involves the use of ETABS and SAFE software, which enable engineers to model the structure and assess its response to seismic forces. The combination of a sound design philosophy and comprehensive seismic analysis tools ensures the building's ability to withstand various loads, thereby contributing to safety and sustainability.

Thermo-Mechanical Modeling of High-Strength Concrete Column Subjected to Moderate Case Heating Scenario in a Fire

This paper presents a numerically developed computer model to simulatethe thermal behavior and evaluate the mechanical performance of a fixedend loaded loaded High Strength Concrete Column (HSCC), subjectedto Moderate Case Heating Scenario (MCHS), in a hydrocarbon fire. Thetemperature distribution within the mid-height cross-sectional area of thecolumn was obtained to determine the thermal and mechanical responsesas a function of temperature. The governing two-dimensional transient heattransfer partial differential equation (PDE), was converted into a set of ordinary algebraic equations, subsequently, integrated numerically by usingthe explicit finite difference method, (FDM). A computer program, VisualBasic for Applications (VBA), was then developed to solve the set of ordinary algebraic equations by implementing the boundary as well as initialconditions. The predictions of the model were validated against experimental data from previous studies. The general behavior of the model as wellas the effect of the key model parameters were investigated at length in thereview. Finally, the reduction in the column’s compression strength and themodulus of elasticity was estimated using correlations from existing literature. And the HSCC failure load under fire conditions was predicted usingthe Rankine formula. The results showed that the model predictions of thetemperature distribution within the concrete column are in good agreementwith the experimental data. Furthermore, the increase in temperature ofthe reinforced concrete column, (RCC), due to fire resulted in a significantreduction in the column compression strength and considerably acceleratesthe column fire failure load.

"Reliability Analysis of Axially Compressed Steel Columns "

"In the past, the evaluation of the strength and safety of compressed steel columns has primarily relied on deterministic concepts. However, these concepts fail to account for the significant variability observed in buckling strength test results. This study aims to address this limitation by adopting a probabilistic approach to assess the buckling strength of steel columns; following the Indian standard specification IS 800-2007. In this investigation, the statistically variable nature of the material properties is considered by treating them as random variables and assuming a normal distribution. By incorporating this variability, a more comprehensive evaluation of the buckling strength can be achieved. The buckling strength of the steel column is computed using Merchant Rankine's formula, which takes into account deductive uncertainty. This formula considers various factors such as the slenderness ratio, live load to dead load ratio, and other relevant parameters to estimate the buckling strength of the column. Furthermore, the reliability of axially loaded columns is assessed using the First Order Second Moment (FOSM) technique. The effects of the slenderness ratio and the live load-to-dead load ratio on the reliability of the columns are specifically investigated and discussed. Overall, this study contributes to a more comprehensive understanding of the buckling strength of steel columns, incorporating statistical variability and providing insights into the reliability assessment of such structural elements."

SIMPLIFIED METHOD FOR CALCULATING THE PASSIVE EARTH PRESSURE OF COHESIVE BACKFILL AGAINST RIGID RETAINING WALLS

The horizontal slice element method for calculating the passive earth pressure against rigid retaining walls is extended to cohesive backfill. The formula of the nonlinear distribution of passive earth pressure is derived to satisfy the reciprocal theorem of shear stress. According to the Mohr-Coulomb theory, the lateral passive earth pressure coefficient can be determined, and a simplified expression of the total passive force is obtained. The influence of the shear stress increment between soil strips on the passive earth pressure is discussed by two calculation models. The results are compared with the existing classical theory. The comparison shows that the formula in this paper degenerates to the Rankine formula when there is no uniformly distributed load. If the backfill is cohesionless soil, the results are consistent with those in the literature that considered the soil arching effect. The calculated results are also consistent with the existing experimental results.

Simplified method for analysis of circular steel tubed-reinforced-concrete columns under eccentric compression

This paper develops a simple and general approach to analyse circular steel tubed- reinforced-concrete (TRC) columns including calculations of peak loads and plots of ascending branch of load-deflection curves. The proposed model relates cross-sectional capacity to actual column capacity by introducing a stability parameter so that column N-M curves can be determined directly. This stability parameter is calculated by establishing a rational mid-height deflection limit, which can be obtained through strain-compatibility and geometric relationship. It also provides a theoretical background to the classical Rankine formula. Moreover, a smooth transition of load-carrying capacity from short to slender columns is obtained. To generate the load-deflection curves, mid-height deflection is increased from 0 to the deflection limit while cross-sectional capacity is kept the same. The proposed method is validated against three series of test data including a total number of 40 TRC columns. It shows that the predictions compare well with the test results in three aspects, namely, mid-height deflection at peak load, peak load and ascending branch of the load-deflection curve. Moreover, the proposed method has a wide range of applications for TRC columns filled with both normal and high strength concretes with column slenderness ratio ranging from low to high. Most importantly, this model can be implemented using spreadsheet, which provides a simple and robust numerical tool for engineers to determine strength of TRC columns.

The effect of a shear bond in the Rankine method for the fire resistance of RC columns

The Rankine formula has been adopted by many natural codes for designing columns and frames at ambient temperature. The method had recently been adapted for structural fire resistance prediction by Tan and co-workers [Tang CY, Tan KH. Basis and application of a simple interaction formula for steel frames under fire conditions. J Struct Engrg ASCE 2001;127(10):1206–13; Tang CY, Tan KH, Ting SK. Basis and application of a simple interaction formula for steel columns under fire conditions. J Struct Engrg ASCE 2001;127(10): 1214–20; Tan KH, Tang CY. Interaction formula for reinforced concrete columns in fire conditions. ACI Struct J 2004;101(1):19–28; Tan KH, Tang CY. Interaction model for unprotected concrete filled steel columns under standard fire conditions. J Struct Eng 2004;130:1405–13]. The conventional Rankine method considers a linear interaction of two failure modes: plastic squashing and linear elastic buckling, ignoring the coupling interaction between the two modes. Consequently, the conventional method gives lower bounds to actual failure loads. This paper discusses the physical meanings of the Rankine approach and the effect of coupling interaction (termed “shear bond” for consistency with our earlier publications) between the collapse modes on the ultimate failure load. The purpose is to demonstrate that the Rankine method is a conservative, robust, and rational method for the structural design of columns under either ambient or elevated temperatures. The approach can be further adapted to provide more accurate failure load predictions by incorporating the shear bond effect into the Rankine formula. Material deterioration at elevated temperature and the resulting effects on the strength and stability of columns are also quantified. A computer code (SAFIR), developed at the University of Liege, was used to calibrate the modified Rankine method.

Structural response of restrained steel columns at elevated temperatures. Part 2: FE simulation with focus on experimental secondary effects

This paper presents finite element simulations of a series of elevated temperature tests on axially-restrained steel columns conducted at Nanyang Technological University. The test results have been presented in the companion paper (Part 1), which also includes analytical predictions of failure times based on the Rankine formula extended to the elevated temperature case. The columns with different axial restraint ratios were subjected to a constant axial load ratio. This paper focuses on accurate simulation of secondary effects arising from the experiments, namely, initial crookedness and boundary friction. An approach to correctly ascertain the axial restraint stiffness and rotational restraint stiffness due to inevitable friction effects is discussed in detail. Good agreement between FE predictions and experimental results on column internal force and lateral displacement validates the proposed numerical approach. In addition, the effects of load eccentricity, boundary friction and steel material models on column failure time are also examined. These effects are shown to have significantly influenced the structural response of heated columns.

Steel columns under fire—a neural network based strength model

This paper presents a strength model of steel columns under elevated temperatures using the artificial neural network. The many influencing parameters make it difficult to build an analytical steel strength model. Being a flexible model building method, the artificial neural network is an ideal tool to construct the complex relationship between the input and the output parameters accurately. A hybrid neural network, which combines the sigmoid neurons and the radial basis function neurons at the hidden layer, is proposed to better map the input–output relationship both locally and globally. The use of the genetic algorithm approach in searching the best-hidden neurons makes the hybrid neural network less likely to be trapped in local minima than the traditional gradient-based search algorithms. The genetic algorithm based hybrid neural network is applied to model the strength of steel columns under fire. The neural network results are compared with the modified Rankine formula.

Analytical Fire Resistance of Axially Restrained Steel Columns

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.

Rankine approach for steel columns in fire: numerical studies

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.

Basis and Application of Simple Interaction Formula for Steel Frames under Fire Conditions

For a long time, the Rankine method has been applied successfully to steel columns and frames subjected to increasing loads but maintained at constant ambient temperature. This paper extends the Rankine formula to frames under fire conditions. An original theoretical derivation of the formula, which is general and based on the virtual work principle, is provided in this paper. A simple expression has also been derived for the buckling coefficient of frames under fire conditions, which takes the deterioration of steel properties at elevated temperature into consideration. Comparison of test results shows that the Rankine formula yields very good predictions of critical temperatures of frames, with a mean value of 1.01 and a coefficient of variation of 9.2% for a total of 18 frames. Most importantly, it is shown that the fire resistance of steel frames can be obtained from the conventional frame analysis performed at ambient temperature.

Basis and Application of a Simple Interaction Formula for Steel Columns under Fire Conditions

For a long time, the Rankine method has been applied successfully to steel columns and frames subjected to increasing loads but maintained at constant ambient temperature. This paper extends the Rankine formula to frames under fire conditions. An original theoretical derivation of the formula, which is general and based on the virtual work principle, is provided in this paper. A simple expression has also been derived for the buckling coefficient of frames under fire conditions, which takes the deterioration of steel properties at elevated temperature into consideration. Comparison of test results shows that the Rankine formula yields very good predictions of critical temperatures of frames, with a mean value of 1.01 and a coefficient of variation of 9.2% for a total of 18 frames. Most importantly, it is shown that the fire resistance of steel frames can be obtained from the conventional frame analysis performed at ambient temperature.

사각튜브의 국부좌굴과 전체좌굴에 관한 연구 I

This paper presents the results of analytical invesstigation of the buckling behavior of thin-walled box-section column. Throughout this investigation, the single curve for finding the buckling stress at each effective slenderness ratio is derived by modification of the Rankine's formula. The applicable formula in the small slederness region is derived by considering the inelastic behavior of material. Additionally, the bifurcation criterion(slenderness ratio) which can distinguish between the local and overall buckling mode shapes is suggested by equating the local and overall buckling stresses. The overall buckling formula is closely concurrent with the experiments for the rectangular tubes.

Experiments on the buckling under internal pressure of thin torispherical ends of cylindrical pressure vessels

Ten cylindrical pressure vessels with torispherical ends, all of the same constant nominal thickness, were tested under internal pressure to observe the buckling behaviour of the toroidal knuckle. All specimens were of constant internal cylinder diameter to thickness ratio of 531·5. The sphere radius was equal to the cylinder diameter and only the torus radius was varied. Buckling was detected by rotating probes at the sphere/torus junction and at the mid-point of the torus. The buckling pressure increased with increasing torus radius and the two specimens with the largest torus radii did not buckle. For all specimens, the change of meridional shape with increase in pressure was measured. For one specimen, strain gauges were used to study the variation of circumferential strain on the inside and outside surface at the sphere/torus junction due to variation in thickness round the circumference. A simple theoretical expression for the buckling pressure, similar to the Rankine formula for a strut, is suggested.

STUDIES ON THE PROPERTIES OF STARCHED RAYON YARN BUNDLE

We made the wheat starched yarn bundles of the filaments of different denier, and examined the various physical properties.(1) Applying Euler's formula concerning the buckling of long pillar to calculate the flexual rigidity and Young's modulus, they were found increased in proportion to the number of filaments.(2) Golden & Rankine's formula were applied to obtain more detailed properties.(3) By means of cantilever, Young's modulus was again calculated.(4) Like in the previously reported, case the constricting elasticity was measured, and the similar tendeney was obtained.(5) We measured the quantity of starch of each sample, and found increased in proportion to the number of filaments.(6) In consideration of the above, we found that the effective degree in sizing increased as the number of filaments is increased.

The mechanical force on bodies of small susceptibility due to induced magnetization

The mechanical force on a body of any shape and of small susceptibility, when placed in a magnetic field, is expressed as an integral taken over the surface of the body. This principle is applied to the experimental arrangement adopted in Rankine's magnetic torsion balance, where a magnetic pole is attracted by the magnetization induced in a spherical body. The results obtained are (1) a direct proof of Rankine's formula for a point pole; (2) the correction for the earth's field; and (3) the correction for the finite spread of the poles in an actual magnet.

On Rankine's Formula for Earth Pressure

In a short course of lectures on “Railway Practice,” the author was recently called upon to deal with the mechanical principles involved in the design of retaining walls. What has come to be known as Rankine's method had to be explained, at all events, in its practical application. But the time at the author's disposal did not admit of the general consideration of the theory of stress by which Rankine in characteristic fashion leads up (in his Applied Mechanics) to the particular problem under discussion. The author had therefore to choose between omitting a demonstration or making a short cut, which at the same time should be of a rigorous nature.

3. Intimation of an Improvement in Rankine's Formula for Retaining Walls

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