Effective Length Method Research Articles

The Chart Method of Multi-Columns Effective Length with Difference Axial Force

In practical engineering, multi-columns stability calculation is very common and very important. Determining the multi-columns effective length is the most important in stability calculation. So there are some nomo-graphs and formulas in Chinese Code for design of steel structures .The structures are very varied and calculation is too difficult. The nomo-graph and formulas in chinese standards are not fully applicable to a variety of situations. For example, in the multi-frame, the axial force is large difference between stories. If we determine the effective length with the nomo-graph and formulas in chinese standards, it will be unsafe in sometimes. This paper gives some nomo-graphs and formulas to determined the effective length. According to the method, we can determined the effective length quickly, and visually see the trend in the curves of effective length when the parameters change. The nomo-graph and formulas in this can be said to be a supplement to the chinese standards.

Effects of shape functions on flexural–torsional buckling of fixed circular arches

Because fixed arches have much higher flexural–torsional buckling resistance than pinended arches, they are used for engineering structures in many cases. However, studies on their flexural–torsional buckling behaviour have rarely been reported in the open literature hitherto. This paper investigates the elastic flexural–torsional buckling of fixed circular arches subjected to uniform compression and uniform bending because they play important roles in the design of steel arches against their flexural–torsional failure. One of the major difficulties in solving the flexural–torsional buckling problem of a fixed arch is to determine its accurate buckling shapes. The flexural–torsional buckling shapes are studied using a finite element (FE) method in association with eigenvalue analyses. It is found that the flexural–torsional buckling shape of a fixed arch becomes more complicated than the case of a straight beam-column or a shallow arch when the rise-to-span ratio increases, and so the theoretical analysis requires more terms of Fourier trigonometric series to describe the buckling shapes. Based on this, analytical solutions for flexural–torsional buckling loads of fixed arches are derived both by the Rayleigh–Ritz method and by solving differential equations for buckling deformations. Comparisons with the FE results show that the analytical solutions by the Rayleigh–Ritz method are reasonably accurate and that the analytical solutions by solving the equations for buckling deformations are exactly the same as the FE results. Simple approximate formulas for buckling loads of fixed arches with box-sections are proposed based on the extensive FE results for structural designers to use. The validity of the effective length method for the fixed arches is also discussed. It is found that in the case of circular arches the effective length method should not be used because the rise-to-span ratios and boundary conditions have complicated and significant influence on the buckling load.

Determining Method for Effective Length of Soft Ground in Tianjin Binhai

Tianjin Binhai has the properties of poor geological conditions and much weak soil. Cement mixing pile is used to the bridges and the roadbed of channel and culverts in Tianjin Avenue projection. But there are some problems about the effective length. The cement mixing pile is considered to be pure friction pile when the length is shorter than the effective length. According to the displacement coordination condition of the top surface of the pile-soil under rigid foundation, the computing method of effective length for composite foundation is derived from the settlement.

The Overall Stability Calculation Method for Sway Frame

There are three levels in stability calculating for sway frame. The first level is the traditional effective length method. The second level is the effective length method considering interaction of columns in one story and the third level is the effective length method considering inter-story interaction. The traditional effective length method may lead to unsafe design. Using the concept of equivalent negative stiffness, story stiffness to negative stiffness ratio factor and story support factor were proposed. Through the story stiffness to negative stiffness ratio factor, the weak-story and the inter-story support relationship can be found. Then, a formula for calculating the elastic stability capacity of sway frame is proposed, by which the inter-column interaction and inter-story interaction can be considered, and the finite element buckling analysis for stability capacity can be avoided. Through the stability capacity, the column effective length can be calculated. The results show that this simple calculation method has good precision and accuracy, can be used for engineering design and theoretical calculations.

Second-Order Analysis and Experiments of Semi-Rigid and Imperfect Domes

Since the recent introduction of second-order analysis in codes, engineers begin to realize the advantages of the second-order analysis in enhancing the safety and economy of the design output. The method is especially useful for design of complex structures like the shallow dome where failure is commonly due to snap-through buckling with the collapse load difficult to be assessed by the effective length method. In this paper, the second-order analysis is used to study the behavior and to determine the snap-through buckling load of the shallow domes. A comparison on the failure loads of a shallow dome between the laboratory test and second-order analysis is carried out to demonstrate the application of the second-order analysis. Further, the importance for inclusion of global imperfection in the analysis is demonstrated because the snap-through buckling load can be greatly reduced if the practically unavoidable and unfavorable global imperfection is included.

Ray-Trace is Inferior to Monte Carlo Dose Calculation Algorithm for Intracranial Robotic Radiosurgical Treatment: A Comparative Dosimetry Study

Ray trace (effective path length method; EPL) is inferior to the Monte Carlo (MC) dose calculation algorithm when treating tumors at tissue interfaces of varying densities, such as lung tumors. We hypothesized that the anatomic complexity of the skull base would introduce similar uncertainties, due to the multiple tissue interfaces in this location (e.g. bone, brain, and air). We set out to determine whether differences between the EPL and the MC dose calculations were more pronounced for lesions of the skull base when compared to those within the substance of the brain and what dosimetric and anatomic characteristics influence these differences.

Advanced design for trusses of steel and concrete-filled tubular sections

This paper presents an experimental and analytical investigation of buckling behavior of bare steel and concrete-filled steel (CFS) tubes used as columns and as members of trusses. The member resistances of the columns and trusses consisting of steel and CFS tubular members are compared to demonstrate the beneficial effects of the in-filled concrete, with their resistances predicted using the conventional effective length and second-order analysis methods of design in various international standards such as Eurocode 3 (EC3), Eurocode 4 (EC4), CoPHK, AISC-LRFD and AS5100. Test results are further used to validate the proposed second-order analysis, which skips the assumption of effective length, for accurate and reliable design of composite members. The present holistic approach of considering composite members as constituting elements in a truss represents a piece of original work on testing and design of structures as a system, rather than designing members in isolation in the traditional member-based design.

Structural Design in the Post-Effective Length Era

Before the publication of several new codes for structural design of steel structures in recent years, design of structures is based on the linear analysis and the effective length method, which relies on a nonexistent or erroneous assumption of buckling at undeformed geometry. After the publication of the several codes including the Eurocode-3 (2005), LRFD (2005) and the Hong Kong Steel Code (2005), interestingly all in 2005, a new method based on the equilibrium at deformed configuration emerges as a replacement of the old effective length method. In theory, the second-order and advanced analysis can be used for any structural forms but calibration for imperfections against coded and experimental results is essential for safety. Its application to various structural forms and materials has been successful with advantages of this new method manifested. This paper summaries the new design method on two structural forms as transmission line towers and steel, concrete and composite portal frames.

Second-order analysis and design of imperfect composite beam–columns

In current practice, the design of composite columns such as concrete-filled and concrete-encased steel columns is a member-based procedure which is based on linear analysis in conjunction with the effective length method, rather than a system-based approach utilizing the concept of advanced analysis. In principle, the advanced analysis can be used for any structural forms when imperfections are considered and proper plastic functions for cross-sections are used. The new method considering these issues needs to be calibrated with design codes and experimental tests. Surprisingly, the application of advanced analysis to composite structures is relatively rare in the literature and the advantages have not been exploited fully. In this paper, an “advanced analysis” design method for composite columns is proposed, and the results predicted by the proposed method are compared with the results calculated by Eurocode 4 and from laboratory tests. With the verification and calibration carried out in this study, without loss of generality, the proposed method can be applied to the design of composite structures beyond the scope of the conventional design method and without the assumption of an uncertain effective length factor, which implies an improved accuracy and convenience, as frame classification for the use of the effective length method or alignment charts is no longer needed.

PUSHOVER ANALYSIS BY ONE ELEMENT PER MEMBER FOR PERFORMANCE-BASED SEISMIC DESIGN

Nonlinear static (pushover) analysis is an effective and simple tool for evaluating the seismic response of structures and offers an attractive choice for the performance-based design. As such, it has generally been used in modern design due to its practicality. However, the nonlinear plastic design method consumes extensive computational effort for practical structures under numerous load cases. Thus, an efficient element capturing the nonlinear behavior of a beam-column will be useful. In this paper, the authors propose a practical pushover analysis procedure using a single element per member for seismic design. As an improvement to previous research works, both P – Δ and P – δ effects as well as initial imperfections in global and member levels are considered. Therefore, the section capacity check without the assumption of effective length is adequate for present design and the conventional individual element design is avoided. The uncertainty of the buckling effects and effective length method can be eliminated and so a more economical design can be achieved. Two benchmark steel frames of three-storey and nine-storey in FEMA 440 were analyzed to illustrate the validity of the proposed method.

Comparison of Planned Dose Distributions Calculated by Monte Carlo and Ray-Trace Algorithms for the Treatment of Lung Tumors With CyberKnife: A Preliminary Study in 33 Patients

To compare dose distributions calculated using the Monte Carlo algorithm (MC) and Ray-Trace algorithm (effective path length method, EPL) for CyberKnife treatments of lung tumors. An acceptable treatment plan is created using Multiplan 2.1 and MC dose calculation. Dose is prescribed to the isodose line encompassing 95% of the planning target volume (PTV) and this is the plan clinically delivered. For comparison, the Ray-Trace algorithm with heterogeneity correction (EPL) is used to recalculate the dose distribution for this plan using the same beams, beam directions, and monitor units (MUs). The maximum doses calculated by the EPL to target PTV are uniformly larger than the MC plans by up to a factor of 1.63. Up to a factor of four larger maximum dose differences are observed for the critical structures in the chest. More beams traversing larger distances through low density lung are associated with larger differences, consistent with the fact that the EPL overestimates doses in low-density structures and this effect is more pronounced as collimator size decreases. We establish that changing the treatment plan calculation algorithm from EPL to MC can produce large differences in target and critical organs' dose coverage. The observed discrepancies are larger for plans using smaller collimator sizes and have strong dependency on the anatomical relationship of target-critical structures.

QSPR for HLB Values of Nonionic Surfactants Using Two Simple Descriptors

By using the quantitative structure-property relationship (QSPR) method, the relationship between the hydrophilic-lipophilic balance (HLB) value and the chemical structure of surfactants were studied for nonionic surfactants of the ethoxylated alcohols and ethoxylated alkylphenols, for which the number of EO groups is less than 20. The obtained best QSPR model contains only two simple descriptors. One is zero order of Kier-Hall index (KH0); the other is the number of oxygen atoms in the hydrophilic group (N o). The correlation coefficient (R 2) is as high as 0.984. The resulted QSPR model shows a significantly higher accuracy with respect to the effective chain length method.

A more accurate reconstruction system matrix for quantitative proton computed tomography

An accurate system matrix is required for quantitative proton CT (pCT) image reconstruction with iterative projection algorithms. The system matrix is composed of chord lengths of individual proton path intersections with reconstruction pixels. In previous work, reconstructions were performed assuming constant intersection chord lengths, which led to systematic errors of the reconstructed proton stopping powers. The purpose of the present work was to introduce a computationally efficient variable intersection chord length in order to improve the accuracy of the system matrix. An analytical expression that takes into account the discrete stepping nature of the pCT most likely path (MLP) reconstruction procedure was created to describe an angle-dependent effective mean chord length function. A pCT dataset was simulated with GEANT4 using a parallel beam of 200 MeV protons intersecting a computerized head phantom consisting of tissue-equivalent materials with known relative stopping power. The phantom stopping powers were reconstructed with the constant chord length, exact chord length, and effective mean chord length approaches, in combination with the algebraic reconstruction technique. Relative stopping power errors were calculated for each anatomical phantom region and compared for the various methods. It was found that the error of approximately 10% in the mean reconstructed stopping power value for a given anatomical region, resulting from a system matrix with a constant chord length, could be reduced to less than 0.5% with either the effective mean chord length or exact chord length approaches. Reconstructions with the effective mean chord length were found to be approximately 20% faster than reconstructions with an exact chord length. The effective mean chord length method provides the possibility for more accurate, computationally efficient quantitative pCT reconstructions.

SECOND-ORDER ANALYSIS FOR DESIGN OF GLASS-SUPPORTING AND PRE-TENSIONED TRUSSES

Glass wall supporting systems, comprising of pre-tensioned high strength trusses made from alloy steel bars, behave nonlinearly under wind and other loads. For a proper analysis and design, second-order and large deflection analyses are mandatory as the structural response like stress and deflections are highly dependent on the small movements and temperature effects. Also, careful consideration must be given to various effects such as support settlement due to live loads and material creep, temperature change, pre-tension force and wind pressure. It is not uncommon to see many glass wall systems fail in a simulated wind load test under the simulated design wind speed. This paper presents a rigorous analysis and design of this type of structural systems used in two projects in Hong Kong. The key message is the need to use "section strength check" in place of the member design adopted in the conventional effective length method for design of practical structures behaving nonlinearly since the prescriptive buckling check formulae become inaccurate or even invalid. The use of stability function allowing for initial imperfections in place of cubic Hermite element is proposed here because of the high degree of nonlinearity.

A Model Specification for Stability Design by Direct Analysis

This paper presents a model specification for stability design by direct analysis. It is based on the stability provisions of the 2005 AISC Specification, rewritten around the Direct Analysis Method alone. The material is presented in the language and format of the AISC Specification, including "User Notes" and the italicizing of terms listed in the glossary where they first appear in a section. The focus on a single method has offered the opportunity to expand some of the provisions beyond what is in the current AISC Specification, both to improve clarity and to address issues that have arisen from use of the document. Where this involved substantive changes, they are explained in an appendix to this paper (Appendix A). A second appendix (Appendix B) outlines the purpose or physical significance of each of the important steps in the Direct Analysis Method by showing the correlation of these steps to the basic requirements for design of structures for stability. The "traditional" Effective Length Method is included in the correlation to show how that method differs from the Direct Analysis Method. A third appendix (Appendix C) provides guidance to the user on the modeling of structures for the application of the Direct Analysis Method.

Accurate Extraction of Effective Channel Length and Source/Drain Series Resistance in Ultrashort-Channel MOSFETs by Iteration Method

This paper presents a new extraction method for effective channel length (L eff) and source/drain series resistance (R SD) in ultrashort-channel MOSFETs using an iterative process, which is a modified channel resistance method (CRM). Although conventional methods for extracting L eff and R SD, such as the channel resistance and shift-and-ratio methods, are considered to be the most consistent techniques, they are not valid for shorter channel transistors, such as the total resistance (R tot = V DS/I DS) of MOSFETs, and do not scale proportionately with poly-gate length (L poly). This error results from the fact that these methods assume the effective mobility f(mueff) of long- and short-channel transistors to be the same. This assumption inevitably lowers the accuracy of the extracted channel length in ultrashort-channel MOSFETs. Therefore, an improved CRM using an iterative procedure has been proposed. This iteration method takes into account the fact that mobility is degraded in shorter channel devices. By compensating for the mobility in long-channel devices, more accurate approximations for L eff and R SD are extracted compared to conventional methods.

Inelastic storey-buckling factor of steel frames

This paper first briefly reviews the modified effective length method in the literature for design of steel columns considering the inter-column interaction with the limitations outlined. To overcome these limitations, a new storey-buckling approach for steel frames is presented, where each storey is treated as a whole at the sway buckling, and its resistance is evaluated using the presented storey-buckling factor, which can be obtained from the defined storey slenderness using the relationships illustrated by the column curves in current codes. In total, 30 frame examples, considering the effects of storey number, the span number and the configuration of axial forces in columns, are adopted to examine the proposed method. Comparisons between FE results show that the presented method has good performances in predicting the failure load of considered frames. In FE analysis, the geometric and material nonlinearities as well as two types of geometric imperfections and longitudinal residual stresses are considered. Predictions from the notional load approach are also compared with FE results for some cases, which shows that these predictions may underestimate or overestimate the failure load of certain frames. The interaction between the sway buckling of the whole storey and the non-sway buckling of the weakest column may reduce the failure load of steel frames. In this study, a reduction factor is presented to consider this mode interaction, and comparisons with FE results show that this method can reasonably assess the adverse effects of this mode interaction.

A Comparative Study of Alternative Approaches for Stability Design of Steel Frames

Approaches for stability design of steel frames include the traditional and the modified effective length methods and the notional load approach. This paper presents a storey-based stability factor approach in which storey slenderness and a stability coefficient for a storey are defined, the stability of a storey is checked as a whole. Also presented are detailed comparison of the modified effective length approach, the notional load approach and the proposed storey-based method with FEM. Seven frames, with different bases or beam-to-column connections, one or two spans, one or two stories, with or without leaning columns, uniformly or unevenly loaded, are analysed. Various imperfections are considered. All frames are assumed to be axially loaded to serve as benchmark problems for checking the priority of available methods. It is found that the notional load approach is ideal for frames that are uniformly loaded, and if the modified effective length factors of columns are within 1∼2. The modified effective length is not accurate enough in considering the inter-column interaction of a storey failing in an elastic-plastic state. The proposed storey-based stability factor approach shows good agreement with FEM for all the frames analysed.

Notional load method for industrial steel storage racks

The discussion in this paper is on the stability analysis and design of industrial steel storage racks. Studies were carried out to evaluate the current Effective Length Method, and to examine the Notional Load Method as an alternative design procedure. Presently, the design of industrial steel storage racks in the United States is carried out according to the Rack Manufacturers Institute (RMI) specification using the effective length method. The notional load method is a new method for stability analysis and design of structural steel systems introduced in the AISC Specification for Structural Steel Buildings (ANSI/AISC 360-05). The difference between these two methods is that in the effective length method the beam-column member within the frame is designed based on effective member length while in the notional load method it is designed based on actual member length and the analysis is based on a reduced stiffness in the structure. The finite element method, which considers both geometric and material nonlinearities, was used as the basis for evaluating the accuracy of these two design methods. The study was carried out for numerous cold-formed steel industrial storage rack configurations. The application to cold-formed steel structures had to include the frequently encountered torsional–flexural buckling and semi-rigid joints. Results showed that the effective length method is more conservative than the notional load method and the notional load method agrees better with the finite element results than the effective length method does. It is, therefore, recommended that the notional load method be considered as an alternative means for industrial steel storage rack design.

Monte Carlo dose calculations for a 6-MV photon beam in a thorax phantom

In this study we evaluated the accuracy of the Monte Carlo (MC) and effective path length (EPL) methods for dose calculations in the inhomogeneous thorax phantom. The Philips SL 75/5 linear accelerator head was modeled using the MCNP4C Monte Carlo code. An anatomic inhomogeneous thorax phantom was irradiated with a 6-MV photon beam, and the doses along points of the central axis of the beam were measured by a small ionization chamber. The central axis relative dose was calculated by the MCNP4C code and the EPL method in a conventional treatment planning system. The results of calculations and measurements were compared. For all measured points on the thorax phantom the results of the MC method were in agreement with the actual measurement (local difference was less than 2%). For the EPL method, the amount of error was dependent on the field size and the point location in the phantom. The maximum error was +19.5 and +26.8 for field sizes of 10 x 10 and 5 x 5 cm2 for lateral irradiation. Our study showed large, unacceptable errors for EPL calculations in the lung for both field sizes. The accuracy of the MC method was better than the recommended value of 3%. Thus, application of this method is strongly recommended for lung dose calculations, especially for small field sizes.